- Email: [email protected]
Chapter I Introduction to Algebraic Geometry
Section 1
GeQnetric spaces
1.1
A geametric space E Definition: -
=
(x,0X
topological space X together with a sheaf of rings
each XEX , the stalk flx,x
(or simply
Qx) ~f
consists of a
QX such that, for
4
at
x is a local
ring. By
. The unique
abuse of notation, we shall o€ten write X instead of E
ox will be denoted by mx the residue field oJmx . If s is a section of ox a neighbourhooa of x , the by canonical image of in ox will be denoted by sx and called the germ and
maximal ideal of
over
K(X)
s
of s at x ; mreover, the canonical hage of s in the value s ( x ) of s at x at x lies in mx
1.2
.
K(x) will be called
. This value is thus zero iff the germ of
E l e : Let X be a topological space, and let
sheaf of genns of continuous cmplex valued functions on X
is local and its ux tions which vanish at x . the stalk
1.3
s
flX be the
. For each
x€X
maximal ideal is the set of g e m of func-
4)
-1e: Let (x, be a gecmetric space, and let P be a subset of X , endowed with the iladuced toplogy. Let i : P -+ X be the inclusion mp; then the restriction of definition the inverse image i' (
ox) 1
to P (dict.)
.
, written
Qx/P
, is by
,
ALGfBRAIc
2
Accordingly, if x€P
, we
have
(P,
oxIx. we
( oxlP)x =
c3,I~) i s called an open subspace of
For example, consider a section
s(x) #
0
, there
.
ox
s of
call
(P,
. If P is open i n (x, ax) .
( X I fix)
geoanetric space induced on P by
I,
ClWMETRy
over X
. If
5
the
QX\,ip)
, then
X
and
xu(
ox over a neighbourhood of
is a section t of
1, no 1
x
such
t = 1 It follaws that s t = 1 for a l l points y lying i n sane x x Y Y neighbourhood of x , so that the set of xu( such that s ( x ) # 0 i s open.
that
s
Such an open set is called a special open set and is written
Definition:
1.4
5 morphism of
.
Xs
g a w t r i c spaces f : ( X I
dx)
-f
(Y,dy) consists of a continuous map f' : X -+ Y and a hcmmxphism of f sheaves of rings f - of 8, i n t o t h e d i r e c t h a g e f . (8,) of such
that, f o r each xot 12
, g-e hcaxmrphlsm
local, i.e. s a t i s f i e s f (m )c% X
Y
fe
. If
W e s h a l l often write
f
for
is an open subset of
Y
containing
for the ring hcmxmrphisn induced by
fx :
.
of(x)*
ox
induced by f -f
dx
is an open subset of
U
f(u) f -f
, we write
.
fz :
and V uy(v) * dX(u) X
Ckqmsition of m o r p h i m of g m t r i c spaces is defined i n the obvious way.
Gemetric spaces and m r p h i m s between than thus define a category, denoted by Esg. A mrphism of geometric spaces f: X Mu-. gnbedding i f
1.5
f
induces an iscsclorphism of
ax and
Example: If
dy
+ Y
w i l l be called an open
X onto an open subspace of
.
Y
are t h e sheaves of germs of complex
valued continuous functions over X and Y
, each
continuous map f : X
+
defines a morphism of gecmetric spaces: with t h e above notation, w e need V
only set f U (s) = s o f '
1.6
are i n
Propsition:
,V , each
,
where f ' : U
12
functor d:
2 +
+
V
denotes the map induced by
is a category such that Ezq
has a d i r e c t l i m i t .
Obz
F1
f
Y
.
It is sufficient t o show f i r s t t h a t any family of g m t r i c spaces
Proof:
has a d i r e c t sum, and secondly that any pair of mrphisms
4)
frg : ( X r
(yr
0,)
has a cokemel. Now the direct sum
has as i t s underlying space the topological sum of the
8s / X1. = a)
QX
i
. The cokernel
(2,
dz)
~
of
( f , g)
i s the cokernel of the continuous maps f
Z
Xi and we have is constructed as follows: and g
i n the category
of topological spaces, and is therefore obtained by identifying i n Y the pints b)
f (x) and g(x) for each xEX ; if
p: Y
-P
Z is the canonical projection, each open set W c Z
, and
determines two open sets V = p-'(W)
U = f-'(V) = g-'(V) ; then V V such that f u ( s ) = gu ( s ) The
o z ( W ) is the ring of a l l sE 0Y (V) restrictions dz(W)+ dz(W') are induced by those of
canonical projection sions
W
%
: &,(W)
+
(Y,oy) +
8Y (V)
ing t h a t the stalks of
oz
dZ)
(Z,
is defined by p and the inclu-
. The only tricky p i n t
i n the proof is i n show-
are local rings, and t h i s is done as follcws.
since the hcmcmorphisms f, : flf(,) (1.3) ; similarly q-'{Vv) = Uu ' SO where W'
.
0, , and the
is an open subset of
W
are local, we have f-l(Vv) = -1 f-'?Vv) = g-'(Vv) and Vv = p ( W ' )
+
. -1 If
zEW'
u
the inverse of the germ of
w a t z is thus the germ of (vIvV) ; on the other hand, i f Z ~ W ' -1 p ( z ) does not meet Vv and v vanishes a t each p i n t of p-'(z) Fran
.
this we infer the following facts: W
first, i f W
w , w l ~$(w)
have non-inver-
tible germs a t z , then % (w) and % (w') vanish a t each p i n t of -1 W p ( z ) ; hence pv(w+w') also vanishes and so w + w' is not invertible;
4,
is a local ring. And secondly, i f w vanishes a t z , then % (w) vanishes a t each p i n t y€p-1( z ) ; thus p : -t is a local Y hamcmorphism. therefore W
oz oy
1.7
Example:
flX
Suppse that
8,
and
are the sheaves of germs
and Y and that the mrphisms
of ccsnplex valued continuous functions over X
f,9 are defined by ccPnposition with the underlying continuous maps. Then 0, m y be identified w i t h the sheaf of germs of c q l e x valued continuous functions over
1.8
2
.
M k :
and F1 2 E
Ob T E v1
,T is a category such t h a t
It can be s h a m that, i f
, then
each functor
d :2
+
has an inverse
JSE
limit.
Section 2
The prime spectrum of a rinq
2.1
we write
d(X) = 6X (X)
0 : 5s-S
for each qeanetric space X
each mrphism f : X
+
Y
and
Y
d ( f ) = f x (1.4)
, for
of g m t r i c spaces.
Spectral Existence Theorem: Spec A
for the functor such that
For each ring
and a honmnorphisn $A : A
+
A
, there
fl(Spec A)
is a geanetric space
satisfying the condition
("1 below: (*)
f
X
-
is a q m t r i c space and
there is a unique mrphism
4
: A + O(X)
f : X -+ Spec A A
4
i s a ring hcanomorphism,
such t h a t
$ = @(f)$A :
&X)
(Spec A , $A) is evidently Unique, since it is the solution of a universal problem. This universal problem means that the map f Ho ( f )$A
Such a pair
is a bijection
zg(X,Spec A)
describe the pair
ft+O(f)$,
.
(Spec A
),0
%(A,
&X))
. Instead of
a proof, we merely
and give the inverse of the bijection
I, 5 1, no 2
THE LANGUAGE
Description of of A
(Spec A, $A)
:
5
The points of Spec A are the prime ideals
(Alg. camn. 11, 54, no. 3 ) . If f a and pESpec A , we call the
canonical image of f in the field of quotients of A/p at p ; if a is an ideal of A
, we
the value of f
denote by D(a) the set of pints of
Spec A where at least one element f of a does not assume the value 0
The subsets D(a) of Spec A are the open sets of Spec A
.
.
Let S(a) be the set of a l l SEA which do not assume the value 0 at any pint of the open subset D(a) of Spec A D(a)
=
F(D(a))
D(b) =
. Thus
. We obtain a presheaf of rings over
S(a) =
S(b) if
Spec A by setting
NS(a)-l] (Alg. c m . 11, 5 2 , no. 1) and defining the restriction
hananorphisms in the obvious way. If a is the ideal generated by the single element s , and if As denotes the ring of fractions of A defined by the . I , then it is easy to verify multiplicatively closed subset ~l,s,s~,s~,.
.
that the canonical m p As
-+
Z$S(a)-']
is bijective. In particular,
. "he structure sheaf of Spec A is now defined to be the sheaf associ.atedwith F . The -1 stalk of this sheaf at p is the local ring A = A[: (A-p) ]. Finally, we P
F (Spec A) m y be identified with A
let -$IA
(by setting s
= 1)
be the canonical map of F (Spec A) into the ring of sections of
the associated sheaf.
.
We must now describe the inverse $ H g of the map f H&f) -$IA Let -$I : A +. O(X) be a hammrphism and let XEX By definition, g(x) will
.
be the inverse image of mx under the ccanposition
"he m p g is obviously continuous: if a is an ideal of A
, g-1 (D(a))
is the set of pints of X at which at least one element of $(a) does not vanish; the cmpsition
-1 thus factors through A [S (a) ], which defines a mrphisn F
(of the presheaf F into the direct image of requiredmrphism
flSpec A
g.(@X)
*
0,
under
g)
-+
g . ( (3),
, and
thus the
6
Example:
2.2
x
Let
8($x)$A = IdA
prime ideal of f mrphism $i:
. In v i r t u e of
dspc
o(x)
Definition:
2.3
+
($,
such that
s
s(x) = 0
the
. The
.(gX) is constructed as i n 2.1.
)
.
"depends functorially" on A ; i f
Spec A
such
Spec $(XI
For each ring A t h e g m e t r i c space Spec A
called the prime spectrum of A Of Course,
+
assigns t o each xEX
2.1,
d ( X ) consisting of a l l e
: X
1, no 2
O(X)
IXa g m t r i c space. I f we set A =
and @ = IdA i n 2.1, we g e t a unique mrphism qX that
5
I,
ALGEBRAIC GEaMFTRy
hcrtlcsnorphism, we write Spec 4 : Spec B
.
+
is -
is a ring
@ :A + B
Spec A f o r the unique mrphism
This m q h i s m is defined explicity as folsatisfying QB$= O ( S p ~ m s :the map (spec 0)s underlying spec Q sends q onto @-l(q) ; i f a
is an ideal of
A
, we
have
and the c a p s i t i o n
+
--+(3spec
A+B
factors through
B(D(B@(a)))
. As
XS (a)
a varies , we thus obtain a mrphism
frcm which we derive the required mrphism
I n particular, i f
SEA
an i s m q h i s m of
Spec As
and 4 : A
-t
As
is the canonical map,
onto the open subspace
(Spec A)
Spec $I
= D(As)
is of
Spec A .
2.4
For each ideal
For each subset V(&&P)
P
a of
of Spec A
A
, the
, set
closure
V(a1 = (Spec A)
5
of
P
- D(a)
.
thus coincides w i t h
I,
If
5
1, no 2 @ :A
For, i f
-+
THE
7
LANGUA(;E
is a hamnorphism and b is an ideal of
B
B
, it follows t h a t :
(Alg. c m . 11, 52, no. 6) , we have
Ja denotes the r a d i c a l of a
(Alg. cam. 11, §4, no. 3, corr. 2 of prop. 11).
, we
I n t h e particular case b = 0 image of
. For
Spec B
Spec $
see t h a t
V($-'(O))
is t h e closure of the
t o be dcminant ( t h a t is, f o r the image of
to be dense) r it i s thus necessary and s u f f i c i e n t t h a t
(Spec @)'
@-'(O)
be a n i l i d e a l .
If
2.5
is the ring z[T]
A
hanarnorphism @ : _Z[T] t r a r i l y chosen i n f i e d with
8(X)
.
-+
8(X)
d(X)
X
Definition:
If
$ : X
+
over x
X
2.6
Propsition: -+
0 (Spec A)
, 8(X))
A.J(,Z[T]
can be arbi-
may be identi-
o(X)
m y be i d e n t i f i e d w i t h the set of
. This j u s t i f i e s
the follavinq
Spec z[T]
tions
$A : A
, which
is a q m e t r i c space, a morphism
is called a function
.
$ (T)
, each
Applying the adjunction formula
i n t o Spec$T] X
is determined by
. It follows that
established above, we see that mrphisns of
of p l y n a n i a l s i n a variable T
; the ri.nq
O(X)
For each rinq A
, the
of 2.1 is an i s m r p h i s m .
i s called the rinq of func-
hamnorphism
. We show more generally that the presheaf
Proof: Set X = Spec A
ospec
F of
2.1 assumes the sane values as the associated sheaf A over the special open sets X = D(M) ,faSince X %I = X for f,gm , it is f f 4 fg sufficient to show that whenever X is covered by Xf ,...,Xf , we have f 1 n an exact sequence
.
V
F(Xf) -+ nF(X " .nF(Xf i fi w i l l' i j
I
where u, v, w are defined by u(a) = (a.) , v((bi)) = (b..) and 1 13 w((bi))= (cij) I ai , bij and cij denoting, respectively, the restrictions of a, b and bj to f' 'f.f. and xf.f i Xfi = X fK = Xffi and F(Xf) = i Af ,i j 11 f fi
. Since
it is sufficient to show that the sequence
v~~~~ .
, B= is exact. TO see this, set c = flat over C (Alg. ccatnn. 11, 5 3, prop. 15 identified w i t h
&
men B is faithfully cor.), and n A f f my ij
v and w being identified w i t h the maps b w b @ 1 and b-1
@ b
. Exact-
ness follows frran kmna 2.7 by setting M = C = Af '
2.7 Imma: E t C be a rinq , M 5 C-module and B a faithfully flat C-algebra. Then the sequence of C-dules
- -
0
*
M
-+
.
M f B
a n ( m @bl@. .@bn) f i
=
n > 0 is exact.
-f
i=n
1
i=O
MtfjBfB
-+
M?
B@B@B
c
(-1)im @blQp.. .@bn-i@l@bn-i+l@.
c
..@bn
-+
,
... ,
Proof:
Since B
i s faithfully f l a t over
C
, it
is enough t o show that the
sequence
is exact. But, i f we set
.
sn(m @bo@. .@bn+,) = m @boa.. .@bn-l@bnbn+l (n L 0 )
, we
have
so ( do@B) = Id
2.8 Proof:
(dn@B)
Corollary:
The functor
Set X = Spec B An(A, B)
*uI
(dn+f3B) = Id
+
.
AHSpec A
is f u l l y faithful.
i n Theorem 2.1. The map
FSg(Spec B, Spec A)
is the c a p s i t i n n of g ( A l @B) : %(A,
B)
-+
%(A,
O(Spec B)
with the bijection
d (XI
E(A,
7 Eg(X,
Spec A)
of 2.1. It is therefore i t s e l f a bijection.
2.9
Definition:
the mrphism spectral space
: X + Spec o ( X )
if
When X = Spec A
so t h a t
X
A geametric space
X
,
X
is called a prime spectrum fi
of 2.2 is an ismrphism.
X
is called a
has an open covering by prime sy3ctra.
it follows f r m 2.1, 2.2 and 2.6 that
JIx
= (Spec $A)
is a prime spectrum. Since the special open subsets of
are prime spectra (2.3) and form an open base f o r Spec A
, we
-1
,
Spec A
see mre
generally that each spec’qal space has an open base consisting of prime spectra. It follows that each open subspace of a spectral space is a spectral space.
Recall that a t o p l o g i c a l space
2.10
X is said to be irreducible i f
it is non-einpty and each f i n i t e intersection of n o n - q t y open subsets of X
is n o n - q t y . For example, f o r each t o p l o g i c a l space X and each point
XEX
, the
If
Proposition:
of
X
Proof:
TZ
closure
of
in
x
x
is an irreducible closed subset of
is a spectral space, the map x
X
onto the set of irreducible cLclsd subsets of In the case where X
I+
x
(3is a bijection .
x
is a spectnnn, the p r o p s i t i o n follows frcnn
Alg. c m . 11, 54, no. 3 , cor. 2 of prop. 1 4 . This special case imnediately
implies the general case. If
an3 i f x is the unique pint , x is called the generic point of F
F is an irreducible closed subset of
such t h a t F =
(XI
Emnple: For each family
2.11
write E i
. .
for the direct sum
i€E
CI
each mrphisn
S
i
X
.
(Si)iEE
of copies of
Spec
. To each geanetric space
X
5
we
and
f :X + U S iEE
we assign a map g : X
+
i
E
such that g(x) = i i f
xEX and f(x)ESi
. The
is locally constant, that is to say, it is continuous i f E is -1 assigned the discrete topology. If Xi = g (i), the canonical i s m r p h i s n
map g
Esg(Xi f i : Xi
, Spec_Z) +
Si
A n (2
, d(Xi))
(2.1) shows that the induced mrphisn
is thus a bi-
is detamined by i and Xi ; the map f - g
.
jection Esg(X,E')z =(X,E) 2
. 1
A spectral space
X
ismiorphian X G E i
is said to be constant i f there is a set E and an
.
An
2.12
Example:
Let
k
be a f i e l d and
X
a Boolean space, that is a
topological space with a base of m p a c t open sets. Let of rings which assigns to each open subset U of
X
4,
be the sheaf
the ring of locally
.
. For each
constant functions over U with range
k
for each ccanpact open subset U
the mrphism $
of
X
2.2 i s invertible (Stone)*. It follows that
3=
, we
xEX :
U
(x,0,)
U
ox
have
= k; Spec o ( U ) of
+
is a s w t r a l
space. 2.13
The theorem and remarks of 2 . 1 signify that the functor
Remark:
Spec :
go+ E
2
8': Esg + An'.
is the right adjoint of
* * l u r w
It thus transforms
direct limits of rings into inverse limits of gecmetric spaces. I n particular, for each diagram of rings of the form B Spec B NAC +(Spec
B
2A
C , the canonical mrphism Clwith m p n e n t s Spec (in,) and
s&
Spec (in,)
is invertible.
Section 3
g-functors
3.1
Definition:
Az-functor is a functor fran the category of models
g . The category
M ,into the category of sets
4 . 4
of Z-functors is denoted by
ME.
YCI
3.2
X p
Notational conventions:
and x€X(R.)
w
urider the map
5
, we
write $(x),xs
X(@) : X_(R)
+
X(S)
Y_ is a mrphism of
If
f
of
( f ( R ) )(x) for the h a g e of
-Y'
is a subfunctor of
:
+
If
A
2
, we
(SJ? A) (R) = $(A,R)
9A
simply x
xEg(R)
, we
for the h a g e of
write ;(XI
x under the map f (R)
: X(R)
-+
denotes the inverse h a g e of
-X , satisfying
.
write
, or
PJP and
Y , g - l ( Y-' )
i.e. the subfunctor X' of for each R?
.
2 , if
@ : W S is an arrow of
If
x
instead _Y(R)
. If
Y_' i n
-
X,
s ' ( R ) = {xEE(R) : f(x)EY'(R))
for the functor represented by A :
for R g
affine scheme of the ring A
. If
A
is a model, we say t h a t % A
is the
. With t h i s terminology, an affine scheme is
* See, e.g. J . L . KELLFJ, General Topology, Chapter 5, exercise S, Van nostrand, 1955
thus simply a representable functor. If
Sp f
I
the map $ If
R 2
and p€Z(R) , we write p#
functors which sends the map
$E(Sp - R) ( S )
onto
B
Emnple:
:
3 R +X
onto ME(Sp - R, -X)
The functor
0
.
*c(
onto ub = ( o ( R ) ) (IdR)
-
ME(X,g) (
$
= ($ (R) 1 (XI
if R e
, an
element $ 6 g ( $ T ] ,
R)
set
. A mrphism functions on X
i s determined by
Sp - i[T]
enable us to identify
@ (T)
Example:
is the functor G
. The ring of
set
with
0
$(T) ; thus
. Accordingly,
is an affine scheme.
3.4 P
.
fl@)
the maps $-
0
and xEX(R)
, we
its under-
i s the ring of p l y n m i a l s w i t h integer coefficients i n a variable
Z[T]
Y
, and
+ ($(R) 1 (x) and ($(R))(x) for R+
w i l l be called a function on X
is denoted by
T
(XI.
(x) = ( $ ( R ) )
E ?-(_x,g)
If
5 , the
#A”.
($+$I ( R ) ) (XI
. W e knm t h a t . The inverse
which assigns to each Rf”,
carries a natural ring structure: i f $,JI E ME(X 0) -1-
(($.$) ( R ) )
R%
f o r the mrphism of
lying set is called the affine line. For each?-functor *+I
5,
is a mrphism of
(X($)) - ( 0 ) = ps E_x(S)
is a bijection of X ( R )
p -p#
map sends UeA(Sp R, _X)
3.3
+
.
into :(AIR)
g(B,R)
H $ofof
, 5%
f :A
is the functor homcHnorphim which assigns to each
: Sp B + % A
of rank
($1
-n,r by $
G
-n,r
kt n
n=l
is denoted by
3.5
,
gr
- . If SX
A
functor
R-E(A,R)
be two integers
Rn+r
the -ge
2 0 ; the G r a s m i a n
of
. If
4:R-G
s 64RP
in
is a n x r m of
g,
s ~ t +d e ~ r the map
induced
is called the projective space of dimension r
%,r
. The functor
Example:
R w Es~(SpecR,
r
which assigns t o each RSM, the set of d i r e c t factors
n of the R-module
assigns to P
. If
,
Let
X
El
is called the projective line.
be a geometric space. The 5-functor
X) i s calfed the functor defined by X and is written
is a ring, ?(Spec A)
may, by 2.8, be identified with the
. Accordingly, w e have a canonical ismrphism
and
&s
A
.
S(Spec A)
If I is an ideal of A
, we can interpret the functor
g(D(1))
in a simi-
is the open subspace of Spec A consisting of all
lar fashion, where D ( I )
points where at least one sEI does not vanish (2.1). For if $ E
(9A) (R)
= &(A,R)
follows that Spec @
, we have
(Spec @)-'(D(I)) = D ( R $ ( I ) ) by 2.3. It factors through D ( 1 ) if and only if R = R $ ( I ) We
.
see accordingly that S(D(1)) may be identified with the subfunctor (SJ A)I
9A
of
for each R 2
3.6
satisfying
. We call
(9A ) I
the subfunctor of Sp A defined by I
Definition: Let - - X- be a 2-functor and. let
.
g be a subfunctor
of - -X We say that 2 is open in X (or is an open subfunctor of Xof for each &el A and each f : 9 A + 2 , the subfunctor f-l(g) -
S p A can be defined by an ideal I A mrphism
:
V_
5 of E-
+-
of -
A
of
m(sp - A, g) , set
c4
9A
c1
=
is said to be an open -ding
f E X(A)
-b
is such that, for each R € E
for which a R € g ( R )
,
)
-if,
(3.5).
X mnmrphisn and the image-functor is open in -
If f E
.
(3.2)
if
.
. The subfunctor
z - l ( U ) (R)
& is a L-'(u)
is the set of $:A+R
. We can thus reformulate the above definition by saying
that g is open in
5
if, for each AEM,and each a€X(A) -
, there is an
ideal I of A satisfying the following condition: for each arrow $:A+R of M
*r
, we have
. Then
iff R $ ( I ) = R
.
Example: Let X be agecanetric space, Y anopen subspaceof
3.7 X
a R € I ( R ) c X_(R)
SY
element of
.
is an open subfunctor of S X For if c1 : Spec A X is an -1 (SX) (A) , c1 (Y) is an open subset of Spec A and is therefore
of the form D ( 1 )
for same ideal I of A
+-
. This ideal
I
satisfies the
conditions of 3.6, We infer fram this that, i f
A
open iff V_ is of the form
(SJ
is a &el, A)I
.
a subfunctor g Ef S e A
For since
is
14
XGFBR?UC GMx.1FTRY
(sA) I
,
S_(D(I))
5 1, no
I,
. The con-
is open i n &S A for each ideal I
(Sp. A I I
3
X = Sp f = 1% i n definition 3.6. verse is established by setting - A and -
Example:
3.8
then
(sAIM
q : JS Af =
,
A$$
fEA
and i f
q:A+Af
is the canonical map,
Sp A is an open emkdding whose image functor is
. For instance,
and %?[TIT line 0
+
If
and f = T , % z [ T ] may be respectively identified with the affine
i n the case where A = HCT]
2 _Z[T,T-']
(3.3) and the subfunctor p
0 which assigns t o each
of
its
REM I
set of invertible elements.
mre
generally, i f
f:z<
is a 3-functor and
_X
, we
is a function on _X
zf
for the inverse image _f-l(u) ; we shall say that X is the sub-€ functor of -~ X where f does not vanish. This subfunctor i s ofin (the inverse image of an open subfunctor is an open subfunctor).
write
3.9 f+r
-',
.
Example:
Q be a direct factor of rank r of the group
Let
For each REM_ we identify R@ZQ with its image under
R ~ + ~ R B
~
~and+we ~write
n+ronto R /(RBZQ)
. Let
gQ be the subfunctor of Gnrr
E ; assigns t o each- I
the set of
Rn+r
U
. W e claim that
-Q
+.
rR for the canonical projection of
(R-linear)
ment of
ccanplments of
R
~
+
RgZQ
in
.
= S
iff
S gRP =
(5
nrr
( $ ) ) (P)
is a cmple-
In order for S BRP t o be a ccsnplement of S 8 Q , it is necessary and H n+r sufficient t h a t the map vs : S HRP + S /S BZQ induced by ?rS be bijective; s i n c e the damin and the range of
vs
Ge
t i v e d u l e s of the same rank, t h i s holds i f f coker vs -3 S
@
R
(Coker v ) = 0 R
.
vs
f i n i t e l y generated projecis surjective, i.e. i f f
Since Coker vR is a f i n i t e l y generated R-mdule, this last condition i s equivalent t o S$ (I) = S (Alg. cam. 11,
5
~
which
is open i n Snrr , that is to say, for Zach EE& there is an ideal I of R such that, i f $:wS
and each P E G (R) , -n,r is an arrow of g , we have S $ ( I ) n+r S C ~ ~i n Q S
(3.4)
, where
I
is the annihilator of
4, no. 4 , props. 17 and 1 9 ) .
Coker vR i n R
I, 9 1, no 3
THE m G U m
..
Now consider a basis el,e2,. ,e n+r of
15
zn+rover
such that
ze
Q = $
i>n
-i
If R is a n-cdel and P a cmplement of R C3zQ in Rn+r , we have the identities l @ e i = pi +.I aij @e j l’n for i_
.
.
.
Finally, we assign to each subset I of {l,...,n+rj of cardinality r the direct factor Q, of Zn+l consisting of the sums lniei such that
.
ni = 0 for ieI For each field K e , the excharge theorem says that G (K) is the union of the sets _V (K) We express this fact by saying -n,r QI that the subfunctors U of GnIr cover -n,r G . More generally, we make -Q1 the
3.10
.
Definition: Let X be a:-functor. A family
functors of X is said to cover union of the sets X(K)
.
if, - for each field K%
For example, if Y is a g m t r i c space and of Y
, then
(ZiliEIof sub, X(K)
is the
(YiIicI is an open covering
(SYi)iEI is an open covering of SY i.e. a cavering consist-
ing of open subfunctors. In particular, if A is a ring and
(fi,xi)iEI
is a finite family of pairs of elements of A such that
1
=
1
iEI
Xifi
then (sD(Afi))iEI is an open covering of (f2xi)icI a partition of unity in A
.
3.11
(Spec A) = Si A
Let & be a pfunctor, R a model and
. We call
(fi,xi)iEI a partition
16
ALGEBRAIC
of unity i n R
into Rf prio u
i
=
. W e associate w i t h these the sequence of
(3.10)
defined as follaws:
if
(resp. of Rf
a)
31
into R f . f
i
11
=
(fi,xiIiEI
, we
=
- -
, the
.
-X ( a1. 3. ) - p r j
is called local i f , for each
in R
R
set
p r ~priIj-w -x ( 7~1( . . ) ~ and
A wZ-functor X -
partition of unity
sets
ai (resp. a . 1 denotes the canonical map of
X_(ai) , pritjov
Definition:
I, § 1, no 3
GECMEmx
and each
RGbJ
sequence (*) above is exact.
(zi)iEI
b) X is called a scheme i f 5 is local and has a covering of affine open subfunctors Xi indexed by a set I belonging t o the fixed universe
.
g
The f u l l subcategory of ,ME, formed by schemes w i l l be written
zh.
W e observe irrunediately that an open subfunctor _Y of a schane ,X is i t s e l f a scheme. For, with the notations used i n the definition, choose models Ai -1 and isom3rphisms : Sp Ai + Xi By 3.7, the open subfunctor (Xis) of
.
si
+ Ai
is of the form
(Sp - AilIi
where Ii
is an ideal of
zi Ai
. As
s
m s through Ii , the affine open subfunctors (3Ails of 9 Ai cover (%AilIi If Zis denotes the image-functor of (9Ails under gi , the the family f o m the required covering of Y by affine open sub-
.
(-xis)
functors.
of a scheme
W e shall henceforth c a l l an open subfunctor
X.
scheme of
Example:
3.12
-
m y be identified with %(Spec
Esg(D(Rfi) Spec Rfi
n
D(Rfj), T )
, given
(*)
R,T)
t h a t D(Rfi)
i s the scheme induced by
the exactness of
, the functor -ST (*) , X(R) , X(Rfi)
For each gecmetric spce T
For i f we replace X by _ST i n the sequence &(Rfifj)
an open sub-
Spec R
.' Zg(D(Rfi) ,TI
D(Rf.) 3
=
D(Rf.f.) and that 11
means simply t h a t a mrphisn m : Spec R
tions satisfy the usual matching conditions.
and
and
over the open set D(Rfi)
termined by i t s restrictions to the open sets D(Rfi)
i s local.
+
T
. ?"nus
is de-
and that these restric-
I, $ 1, i no 3
_ST is a scheme i f
I t follows f r m t h i s that
a l s o small (i.e. the set underlying T of
TI
T
,
is a spectral space and is
T
is small and, f o r each open subset
is small).
c!~~(,(TI)
Example:
3.13
17
THE LANGUAGE
snlr is a scheme. For by 3.9
The grassmannian
s u f f i c i e n t t o show t h a t sequence
is a local functor; i f
G
-n,r
P E snIr(R)
it is
, the
i n lemma with the obvious arrows, is exact (set M = P , C = R , B = F R i fi n+r n+r and Pfifj C Rfifj , this simply means that P is 2.7) Since Pfi C Rfi
.
TR:
, i n other , or that
7 p f i of the inverse image i n ~ n + r of the s-ule words, that P E GnIr(R) is determined by the P E &(Rfi)
the map u
in
fi
vsn,r(~ .
Suppose now that we are given the Ji E G n I r ( R . ) ( J ~ )E
snIr(R)
I E that
fl
w i t h the product module J =
,I
B = TRfi
f1
such that J
inY+'(J)
B
and we identify the family
Ti over the r i n g
Suppose that v ( J ) = w ( J ) : we have t o show that there is an =
u(1)
. Now the assumption
, which
denotes the
in;
i n t o B @RB mRB ( i = 1, 2, 3)
(B @RB BRB)n+r
v(J) = w(J)
, the
B gRB @RB-suhzdule L
i s generated by
(J)
, is
of an R-suhndule show t h a t
I
I
of
such that
Rn+r
is a d i r e c t factor of
of
independent of
i
.
denote the
3 this lemna establishes t h e existence
, then
and
of
ith canonical i n j e c t i o n
Using t h e notation of Lemna 3.14 below, i f we l e t ui and u! maps induced by dy+r
means
(J) generate the same B @RB-suhndule K
and
(B @RB)n+r; i.t foL&owst h a t , i f
of
.
(3.11) i s i n j e c t i v e when ,X = G -n,r
(*)
J =
R"+r
Ffi.
I t thus remains t o
of rank r ; and t h i s is the con-
t e n t of lama 3.15 below.
L e t us intrcduce some terminology which we s h a l l use i n the
3.14
following lemna: Let h : R N
B
k a ring hcatamrphism,
B-module. W e say t h a t a map f : M
a
f(m-ii-n') = f ( m ) if
+
f
+
+
N
M
and Rrnodule,
is adapted to h
i f we have
f ( m ' ) and f ( m ) = h ( r ) f ( m ) f o r a l l m,m' E M
induces an isgnorphim
B@#
N
.
, r E R , and
Leima: - Let
R
be a ring, and. let B be a f a i t h f u l l y f l a t R-algebra.
Suppose we are given modules J,K,L
over
B, B gRBr B BRBBRB respectively,
u'n
and maps
cr3 ul,L
J U%K
,
-T
u ' 1
adaptea, respectively, to the ring hanrrnorphisms
such that do(b) = b @ 1 , dl(b) = l @ b , d;)(bm)= b @ c@1, d i ( b & ) = b 81 @c
,
u $ ~ = uiuo
d;(b W)
, u&
=
the inclusion map of
=
1@b@c f o r b,c E B
u$uo I
and
into
uiul
=
u'u 2 1
. Assume a l s o t h a t
. Then ____ if
I
induces an isanorphism
J
Ker(uo,ul)
=
I BRB 3 J
.
Proof: Inspect the diagram
J@
R
.i J
t
u1
dO@J J-B@B@
J
where i is the inclusion map, v1
are induced by
i,uo,u'o
j
K
the canonical injection,
R
J ,
v r v o r v ' o and.
and ul , and w i s t h e ccmposition
B @ B Q D J Bwk3 @ K
.
IwI
v ' 2 ,L
,
being ind.uced by u; By lemna 2 . 1 , t h e horizontal sequence a t t h e b t t m i s exact. Since v1 and w are bijections and w(do@J) = uI1v1
v;
w(dlW) = u;vl
, the
second horizontal sequence is exact.
I,
5
1, no 4
THE IANGLJAGF,
and v 0' (u160B) = uivo
Finally, since v~(uo@B) = uivo
and. v1
induce an iscanorphism v
J = Ker(u' , u l )
1 2
I
.
Lsmna: -L e t
3.15
a s M u l e of
d i r e c t factor of Proof:
M
and B
BBRM
of
, the
bijections
E 3RB = Ker(uo@B,ul@B) onto
be a ring,
R
19
M
a f i n i t e l y presented
a f a i t h f u l l y f l a t R-algebra. I f
,ths
I
factor of
is a direct
M
.
vo
R-module,
BBRI is a
It i s s u f f i c i e n t to s h m t h a t the canonical map
4
: HT(M,I)
+
HT(I,I)
is s u r j e c t i v e or, equivalently, t h a t B g R $
is surjective. This folbws f r a n
inspection of the camnutative square B @ H m ( M , I ) >-B
B F 4
Ji $ H T ( B a , B @I)->HT(B R
R
@ H%(I,I)
"k
2 I i B $1)
in which a l l the arrows are the "obvious" ones. For since
factor of
B BRM
,$
3.16
Remarks:
is a d i r e c t
is a surjection; on the other hand, since B BRI is
f i n i t e l y presented Over B bijection, a s is i
B@ I R
, and
,I
i s the same over R ; accordingly j
is a
BBR$ is surjective.
One can avoid using l
m 3.14 by employing the descrip-
t i o n of quasicoherent modules over Spec R given i n
5
2 , no. 1. Anyway, we
shall find t h i s leima useful l a t e r . Consider, on the other hand, the subfunctor GI of G which assigns to n,r n,r each model R the set of f r e e direct factors of Rn+r of rank n This
.
functor is covered by the a f f i n e open subfunctors U
QI
local.
Section 4
The q m t r i c r e a l i z a t i o n of a Z-functor
4.1
P r o p o s i t i o E The functor
-S
:E A
-+
FE3
of 3.9, but it i s n ' t
has a l e f t adjoint.
20
I, § 1, no 4
ALGEBRAIC Q3:OMETRY
Proof:
W e sketch a proof of this particular case of a w e l l knm theorem of
Let g be a2-functor and let
be the category of g-models: an 5 object of this category is an -F - d e l , i.e. a pair (R,p) consisting of a Kan.
and a p s ( R )
; a mqhism
of
into a s e c o d _Fmodel
mdel
R
(S,G)
is defined by a h m r p h i s n @fM_(R,S) such t h a t
(R,p)
80%$ = o # ) . If % : (MJD spec R , we set ( ~ =1 lij% . It -
such that (R, p)
I-*
each XEEs a bijection
'E(Fr$X)
: E+J(IFI
rx)
which is functorial i n X
. Let
each RCE
, a map
composition
_g(r): F ( R )
+
Esg
_I
(SX)(R)
- + where i ( p l is the canonical mrphism of
(i.e.
denotes the functor
thus remains t o construct, for
I
. If
pC_F(R)
, (g(R)) - (0)
is the
r
into 1 9 %
%(R,p)
-
define a mrphism g:F+sX and. we s e i g = @ (-F I X )( f ) -
g(R)
. .
G
f:lEl+X be a morphism; we must define, for
f i (pl ~ RC c l h % + X
S
-+
@(p) =
-
.
. The maps
It remains to show that the map $(FIX) is bijective. To this end l e t
6F
: (&)'
+
_ME, denote the functor
G s y t o v e r i f y that the morphisms p"
F
lim 6
+
F
. Now
( E l X)
( R , p ) w Sp R :
A ~ ( R , P )+ _F
. It
i s w e l l known and
induce an i s m r p h i s n
has been d e f i n s i n such a way as to make the
squares
camutative, where Since tions
= lim +
c1
P
%
c1
*
P '
ard
(2x1 (R)
F
-
+
,E(Sj
lim 6F +
-
R,
2x1
, $(FIX)
is the canonical isomorphism. is obtained fran the bijec-
by passage t o the limit aver the objects (R,p) ; hence $(F_,X)
my be identified with the inverse limit of the bijections
fore a bijection. If
= S_E A
, an F-mcdel
iS
simply a &el
c1
P
r
and is there-
carrying an A-algebra structure.
In t h i s case
% -
%m
coincides with
and its i n i t i a l object is the pair
(A ,IdA) ; acc&dingly we have a canonical i s m r p h i s m
i (IdA) : Spec A
I%Al
-f:E+F --
If
4.2
unique morphisn of
conmutative a5 X H
runs through
, we
(R , f ( p ) )
9.W e
Zf
If
-+
:-functor
c a l l 1 ~ 1 (resp. /:I) functor I ? I :F - e+ !_FI If
F is
(resp.
; w e write simply
points of
explicitly as fol-
denotes the functor
% ; consequently, Ef.
)" =
is
if -f
of
!_Fl
_F
(resp.
denotes, depending on the context,
F , or the
underlying s p c e of t h i s
x of this underlying space w i l l be called a point of
.
x g for x€\_FI W e shall take care not to confuse the
. W e s h a l l also c a l l the
,F w i t h the elgnents of _F(R) for R@I-
t o p l o g i c a l space which mderlies the geometric realization of of points or spectrum of a set of p i n t s of
is a mrphism of
,F
. A subset
_F ; we write ME
denote the image of
, we f) . The
is a mrphism of M&)
g m e t r i c realization
either the geanetric realization of
-F
-f
%
is called the geametric realization-functor.
a 2-functor, the symbol
realization. A p i n t
+
% lp % , a d t h i s -
_F
If/
can obtain
: +P
obviously have %*(M
l$n
induces a mrphism Definition:
for the
lfl:lE1+1_Fl
z g which makes the squares
lows: with the notation of 4.1, i f (R,p)
EVE , we write
i s a mrphism of
P c
P
of
for P c
and Q is a subset of
_F
the space
-F w i l l then by definition be
G -
)?I . Moreover,
, we
g:_F-tG_
(resp. g-'(Q))
Q) under the map lglg induced by g Similarly, g(_F) denotes the h a g e of /l?I under Is_lc ; which is not t o be confused w i t h the this image is thus a subset of P
.
image-functor
of
(resp. the inverse image of
l e t g(P) -
if
L
C J
which is denoted by --:
to each R€M the image of the map I
g(R) - :_F(R)*-(R)
t h i s image-functor assigns
.
Au;EBRAIc GEamrRy
22
open, closed) injective, ~ _ i f the _continuous map
note it by _fc
F i n a l l y , we write
Z-functor _ F .
0
-F
for
8IFI
is surjective (resp.
the map underlying f
.
dF
and call
1, no 4
gw is surjective (resp.
Henceforth we shall say that a mrphism g:-F+G_ of injective, open, closed). W e also c a l l
5
I,
and de-
the structure sheaf of the
-
h
Corollary: For each 2-functor - -
4.3
canonical ismrphism
-
Proof:
E(_F,% A)
I
, g((F, % A )
When A=$Tj
fied w i t h
(3(F)
,d(F))
there
.
is a
It follows from 3.5 and 2 . 1 that there are canonical isanorphisms
ME(F - ,%?A) "p+(F ,S(Spec A ) ) -3
W
%(A
F and each ring A
d( I,FI )
8(I_FI 1
%(ill
,d(\El))
,Spec F.)
is j u s t J ( _ F )
,&lgl))
and &(A
. W e thus infer the existence of
.
may be identi-
a canonical i s m r p h i s m
and hence the required ismrphism. This l a t t e r m y be defined
explicitly as follows: to each $ E ME(E ,SJ A) we assign the hOmcm0rphis-n +:A+d(F) such t h a t the map ($(a) (R) : r ( R ) + R sends x€g(R) onto ( $ ( x ) )(a) for each aEA and REM c
4.4
.
Now consider a :-functor Y ( F ) : -F + SlFl -
_F
and a geametric space X
and @(XI : ISXl
for the images of the identity morphisns of tions
@ ( F ,/!I)
and @(SX,X)-l
of 4.1. Let
+
. W e write
X
1x1
and _SX under the bijec-
p-'
F such that of _ME_ consisting of the functors -
be the full subcategory
Y(F)
is invertible, and
let E s ' be the f u l l subcategory of
E s consisting of a l l X such that @(X) is invertible. It follows from the well-known relations between 4, and Y that I ? [ @ induce an equivalence between and E ~ I
.
I,
5
1, no 4
THE
mGum
23
W e m y thus mnipulate the objects of these categories either gecanetrically
-
or functorially (be regarding then
(be regarding thm as belonging t o Esg')
as belonging to
El) .
For example, i f
A
and R are d e l s , the map
Y ( S p A) (R)
: :(AIR)
-+
%(Spec
R,Spec A)
, which,
by 2.8, is a bijection. It follaws
is precisely the map f++Spec f
that Sp , so t h a t Spec A ISp - A1 - A belongs to W e now describe further objects of ME' and Ezg' :
belongs t o Esg' UL-
.
rn
canparison Theoren:
x
a)
be a geanetcic space.
is invertible whenever X s a t i s f i e s condition there exists an open covering
(*)
such t h a t b)
c -
be -~ a 5-functor.
L e t _F
for Y (_F) : F -+ .s IF]F be a schane. -
@(XI
12x1
:
-+
x
- X by prime spectra Xi of
(Xi)iEI
to s a t i s f y
In order for
i
.
h
(*)
e and
to be invertible, it is necessary and sufficient that
and 5
The functors
Then
below:
is ismrphic to a model for each
and U(Xi)
IEU
(*)
induce quasi-inverse equivalences between the cate-
gory of schemes and the category of geanetric spaces satisfying (*)
.
The proof of t h i s theorem is deferred u n t i l 4.16, when we have a second desa t our d i s p s a l
cription of the g&tric-realization-functor
.
In the sequel we shall often make implicit use of the canparison theorem by arguing as i f a scheme were a geanetric space. Thus, for example, write f:x-+x or 5 for
1x1 . %en
I~\:lgl+\~Ior
ule
may
confusion is possible,
we notify the reader by using a phrase l i k e : "Taking the geanietric viewpoint". 4.5
H:$+g , we
A-
KCf
(K)
an3 b = ( H ( f ) )(a)
tion, we write Hx Hx(K)
is the set of
functors
. Given a functor
is the quotient of the dis19H by the smallest equivalence relation containing a l l
(a,b) for which there i s a mrphism
MH(L)
g
recall t h a t the d i r e c t limit
joint sum pairs
L e t K be the f u l l subcategory of f i e l d s of -
Hx
. If
x
for the subfunctor of clExfH(K)
and each Hx
f:K-+L such t h a t
a€H(K)
,
is an equivalence class modulo t h i s rela-
. Then
H
H
such that, f o r each K€K-
,
is the d i s j o i n t sum of these sub-
is indec~npo sable, that is to say, it is not the
d i s j o i n t sum two non-empty subfunctors. I n this s i t u a t i o n we shall call the Hx
the indeccsnposable m p n e n t s of
, and we may
H
identify
lip H with
the set of irdeccsnposable ccanpnents. 4.6
psable
.
4.7
Example:
Each representable functor of
Example:
Let
be a geometric space
X
the r e s t r i c t i o n of the 2-functor have
to
_SX:p;
into
and let H
B . For
Esg(Spec K r X) = ~ A ~ ( K ( xK) ) ,
H(K)
2
with the set of 4.8
XEx such t h a t
Example:
(2A) IE
of
p h i m of
A
K(X)
Let
S p A to
into K
A
. If
5
(SX)15 be , we thus
=
,
K(X)
is iso-
is the d i r e c t sum of the irdecmpsable
morphic to a model. Hence, since H functors represented by the
is indecm-
each K E E
xwhere x runs through the p i n t s of X whose residue f i e l d * )
2
, we
K(X)
see that 1 9 H m y be i d e n t i f i e d is i s m r p h i c to a mel.
be a model and l e t H be the r e s t r i c t i o n KEK
I
, an
, is determined by
element x € H ( K )
, i.e.
a hcancmor-
its kernel p which is a prime ideal,
.
and by the induced homcpnorphisn x : Fract(A/p) -+ K The map x t-+(p,x ) P P is thus a bijection of H(K) onto t h e d i s j o i n t sum UK(Fract(A/p) , K) so Pthat the indeccxnpxable cQnponents of H are represented by the residue
lp H
f i e l d s Fract(A/p) ard i d e a l s of 4.9
A
.
Proposition:
g e m e t r i c realization Proof: I f we l e t have, by 4.1, 4.8
may be identified w i t h the. set of prime
For each ;-functor
1,FI
of
, the
underlying set of the
f is canonically i s a m r p h i c to
l$n(FIE)
Xs denote the set of p i n t s of a geanetric space X and the c m u t a t i v i t y of d i r e c t l i m i t s , t h e follawing
canonical bijections:
.
w
5
I,
1, no 4
g ,a
W e now consider a &functor
4.10
25
THE LANGUAGE
functor
zp of
pF(R)
such that
subset P of _F
defined by l e t t i n g , f o r each REM_
f o r each hammrphism @:WK i n t o a f i e l d K€& Clearly ?PI$
= f
-
-Q
(Fp)
and i f
, we
Q = ~-'(P)C~SI
v e r i f y that
(3.2).
Example:
4.11
~ ~ ( F ~ I J E ).
via the formula P =
g : p ~is an arrow of
G
( i n the notation of 4.5).
1
-P
-1
and t h e sub-
be the set of
';xp(_Fl:),
so we can recover P f r m F If
, Ep(R)
If
is a gecmetric space ard _F
X
identified by 4.7 with a set of p i n t s XEX
EP
i s m r p h i c t o models. Thus
= _SX
,
P
whose residue f i e l d s
SP , where
may be identified with
may be
are
K(X)
P
is
and the r e s t r i c t i o n d X [ P of dx to P is an open subset of X , -FP i s an open subfunc-
assigned the topology iraduced by X P
. I n particular, i f %A .
t o r of 4.12
Propsition:
----
Let -
F
be a Z-functor. Then t h e map
duces a bijection between the open subsets of
in-
P w F p
and the open subfunctors
Il?/
of _ -F , Proof:
Let
inverse image of P
FP
i s open i n
111
-
for all
iff
(R,p)
open subfunctor of
(3R)Q
(R,p)
an object of
under the map i ( p ) : spec R
P
is open i n
Q
is an open subfunctor of
Sp R
,
be a subset of
P
. Since
-
Sp R
,F i f f Q
- . To
F
of
(4.111, and since we
-F
prove t h i s observe that
( ~ 1 (4.17. By d e f i n i t i o n ,
.
(9R) Q is an have further o*-l(_FP) = iff gp is open i n _F
is open i n Spec R i f f
Since P = l?y (FplK) f o r each subset P of
g
i s of t h e form
gEg(K)
I-F1
gp
, it
, it
.
remains t o show that
f o r some subset P of
is a subset of
gg(K)which is
saturated with respect to the equivalence r e l a t i o n defining
we set P = lip (lJIK)
and Q t h e
spec R f o r all (R,P) c*% Also, p + l ( ~ ~ ) i s an open subfunctor of
(4.10), w e see that P is open i n
each open subfunctor
-f
&P
i s e a s i l y shown that _V=F
-P
*
l$n ):?I(
; if
ALGEBRAIC G F 0 “ R Y
26 4.13
Propsition:
mivalent:
The following conditions on a ;-functor - -
(i)
F i s local. -
(ii)
For each model A of sets o v s
of sets over
.
Proof:
, the
151
.
G
, the
presheaf
g ( (9R)Uir F)
is
,)
f1
U
I+
,V(C+j“
3.11, l e t
(iii)=> (i) Using the notation of
Then _F (R
2
F
g)
U - E ( ( S p- AIU,
presheaf
1, no 4
is a sheaf
Spec A .
For each :-functor --
(iii)
5
1,
is a sheaf
Ui = (Spec R) fi
so t h a t exactness of sequence
3.11means that one determines a section of the presheaf by specifying the sections over the open sets
Ui
(*)
UHE((%
, provided
.
of
R)Ur
F)
these sections
s a t i s f y the usual matching conditions.
.
(ii)=> (iii) I f
V = i (p)-’(U) ME(G -u r F )
Am*
(~,p)E M
”G
, arid. i f u
is open i n
set
( 4 . 1 ) . W e thus have i s m r p h i s m s
-
ME+ ( l h (Sp RIVr f)
_ma
(Rr
1 2
EA((Sp NVr
F)
r
(Rr P)
P)
is an inverse l i m i t of sheaves
which shows that the presheaf
UHE(G~,_F)
(namely, the d i r e c t images i n
151 of the sheaves defined by
_F
over the
spaces Spec R) ; it i s therefore i t s e l f a sheaf. (i)=> (ii): I f
of (1)
U
, w e must
E(sUr
_F)
U
is open i n Spec A arid
(Ui)icI
is an open covering
show that t h e sequence -+
T T _ M E _ ( 2~ u~ ~~, (~ s) u i n ~E)u j , i i,j’”- -
i s exact. W e can, mreover, restrict our a t t e n t i o n t o s u f f i c i e n t l y fine cov-
erings, so l e t us assume that Ui = (Spec A) f i
g gi=@(fi) ,
mrphism of If
such that B =
1 B$ (f1, ) , that i
m(Sp - Br _F)
fiEA
. Let
$:A+B
is, an element of
is t h e inverse image of gi my be identified with -%@Vir F) Vi = (Spsc B)
ard _F(B . ) 41 It follows then f r m (i)t h a t the sequence A”*
,
.
be a
(SU)(B)
.
Ui urider Spec $
+nE(rir v M E ( _ S V n iE -F)V j r j i** 1,j L m
is exact ( t h i s is clear when I
the f i n i t e subsets of I) w i t h the d i r e c t limits of
is f i n i t e ; if not, pass t o t h e limit over
. Since su
,
sUi
9 B , gi and
and
suingJj as
may be identified
(B,$)
runs through
I, §
4
J-r
, we
the objects of _Msu
see t h a t
IF1 -
a&
Let F be a z-functor. The s t r u c t u r e sheaf of
Proposition:
~
is canonically i s m r p h i c to the sheaf of rings U e&(_Fu) (3.3) .
Proof: The presheaf -
0
(1) may be identified with an inverse
so is itself exact.
limit of exact s&ences, 4.14
27
THE LANGUAGE
is a sheaf i n v i r t u e of t h e f a c t that -U is a l o c a l functor (3.12 and 4.13). I f A€: ,F = % A
$(Spec :[TI)
U ++ O(F )
U is a special open set of the form (Spec A ) f
J A cal ismrphisms S f
8spec A (u) % Af
z -u F
% AM(Z[TI, w l *
fEA
, we
have canoni-
and
Af)
E(Eu,g) = o(Eu)
.
These isanorphisms induce the required canonical i s m r p h i s m when _F = S J A Now suppose that
is arbitrary. L e t
the inverse image of
U
urader
U
and l e t V be
be open i n
i ( p ) : Spec R
for
+
(R,p) € &
By 4 . 1 a d 4.10, we have
I n view of the d e f i n i t i o n of phism:
4.15
corollary:
subspace of Prmf:
1x1 -
I-f _Y
-
d , we obtain f r m this t h e required 1 1 1
is an open subfunctor of
.
5
(1.3).
1x1
(4.1).
ischnor-
is an open
This follows inmediately fran the description of t h e gecanetric reali-
zation given abave (4.9, 4.12 and 4 . 1 4 ) . 4.16
Proof of the c a p a r i s o n theorem:
satisfying condition
(*)
is isomorphic to a &el,
For each gecmetric space X
of 4.4 and f o r each XEX
so t h a t
the underlying sets ( 4 . 7 ) . If
(Ui)
@(X)
:
lSXl
+
x
, the
residue f i e l d
K(X)
induces a b i j e c t i o n of
is an open covering of
X by prime
ALGEBRAIC GECMEI'RY
28
spectra then QWi) :
15x1
open subspace of
lsuil
-+
ui
I,
is an iscamrphism by 4.4 and
lSXl
by 4.15. The topologies of
is invertible and a )
@(XI
5
is an
lSJil
ad X
t h e i r structure sheaves may thus be locally identified v i a
9 1, no
, and
also
Q ( X ) ; thus
is proved.
By 3.12, it rmins to show that the condition i n part b) of the canparison
(yi)iEI
theorem is s u f f i c i e n t . L e t _F be a scheme, of -F such that
, ard (gijci)
I€;
an a f f i n e open covering
, so
yield an open covering of
the prime spectra
. Then
an a f f i n e open covering of &JinLJj that
of 4.4. W e show t h a t Y (F) l_FI is inver- : _F tible by displaying the inverse Y ' of Y(g) ; one can define a mrphism s a t i s f i e s condition
Y'
:
SlF\
-+
(*)
-+
Y l l (SIU-Jjnl 1 = Y!3 I (-SlUijcil for a l l i f j , only set Y;
. Since
ci
Y(yi)
:
-ui
into
g
*
Section 5
Fibred products of schemes
5.1
Let
fibre3 product functor
-5
and
,Y
I f -X -
satisfies
and Z
Y_
%z
zi
j -if?
(Y
Also i f
gin
Y. -Xis X 21-1
covering of
, let
open covering of
9 %(R
gx
R B,S)
z- . Y
z-Y) xi@ ,
%,zY_iB) 2-S
and
so 'that the
need
zi -xi
-xi
X, gz:
-
Xz(R)Y(R)
in- _X X
-zx if
.
5 x z y is local.
I! g Z g L
in X and- Y
xi . Then
Let
and
, and
let
(Xis)
singzixi0 and is therefore o w n i n _X x zy .
be a f f i n e open coverings of
(Xis X
, we
. Recall t h a t t h e
,XI
are schemes, so i s
be t h e inverse images of
coincides with
_ME_
(5 X z x ) IR) = Z ( R ) -zy is open
I t follows e a s i l y from t h e d e f i n i t i o n s t h a t
(gi) be an a f f i n e
xi
zx
be mrphisms of
.
X
is open i n
proposition:
_X X
g:Y+g -
is invertible (4.4)
Y ( ~ ~ ~ 1 with - l the inclusion
. I t follows G d i a t e l y t h a t
f o r each R 3
Proof:
zIuiI
.
f:$+z and
such that
+
1
equal to the composition of
mrphismof -Ui
-XI
: S/gi/
by specifying mrphisms 4';
_F
zi
ziax
and
S l T
then obviously -
zi!iia form an a f f i n e open I
I,
5 1, no
THE LANGUAGE
5
29
More generally, i f
(zj,fkj) is a f i n i t e diagram of schemes, the inverse limit functor can be constructed with the help of fibred products. This in-
verse l i m i t is therefore a schane; i n particular,
arrow of z h
, the
kernel functor Er(_u,y)
if
, which
the ___ set K - e_ r ( u ( R ) ,v_(R)) = (x€X(R) ly(x) = y(x)1
is a double
g,_v:$:y
assigns to each REM,
is a scheme.
With the assumptions of the foregoing proposition, it follows easily fram 2.13 and 4.15 that, for each pair of mrphisms d:T+/XI ard e:WIY_I E g
Ifld=lsle
such that
.
, there is
a unique h : T
(I_X
d=lgx)h and e=l_fy)h In other w r d s ,
1x1 Is’
1x1
%
z
-f
~
~
I_X X zY_I -
,
~
~
of
such that
, ~is , a ~ fibred ~ y ~
x
.
)
i n the-categ%y $ SJ Wre generalprod;ct of the diagram I ly, the restriction to Sch of the functor ( ? ): E + E a cmutes w i t h f i n i t e v””
inverse limits.
W i t h the assumptions of p r o p s i t i o n 5.1, we naw examine the
5.2
spectral space
Ig K zxl
i t s residue f i e l d ard-
i n mre d e t a i l . L e t x be a p i n t of
E(X)
:
9
carries the unique pint w of
-+
spec
K(X)
onto
, and I E ( X ) I w : Ox+ Ju
onto x K (x)
. clearly
I E (x) I
is a mno-
morphismof
9.
Corollary:
With the assumptions of proposition 5.1, 1 s x, y, z
-of - X,_ Y, -
S U C ~t h a t
Z
f(x)=z=q(y)
& ( X ) X E(Y) E (zl
: *K(X)
SP
K ( Y ) -r
K ( Z r
induces a bijection of the set of prime ideals of
set of p i n t s
t€g x
zY -
with Spec
K ( X ) @ K(Y) K (Z)
Mareover,
E(X)
K(X)
(2.13).
E(:)
1
and
K(X)
@K(z) K ( y )
x SpeC
K(Z)
.
Onto the
Spec ~ ( y )may be identified
x ~ ( y )is a fibred product of m n m r p h i s n s , thus i t s e l f
E (Z)
tive. Finally, i f E(t)
x
-
a m m r p h i s n ; the following l
Iyx
x
- Z
which are projected onto xEX and
F i r s t recall that Spec
Proof:
be points
. T%en the mrphism
X
K(X)
IE (x)1
5 the following mrphisn:
K(X)
dX
is the canonical projection of
)_XI ,
t E _X
X
m implies that the induced map is injec-
zy is projected onto x a d y , the c m p s i t i o n s
I_fy€(t) I factor through
factors throiqh
E(X) Y
I E(X) I
~ ( y )and t
and
I ~ ( yI ) . Consequently
belongs to the image of
LgcaM: - I f L:g*x
5.3
t i v e (4.2).
Proof:
.
x,uG
Let
P1 F2
s a t i s f y f(x)=_f(u)=y :
into
K(U)
~ ( y, ) we have
K(X)
E(X)
. Since
@ K(U) K (Y)
fpl=fp2
@
K(Y)
K(u))
r
and by the canonical maps of
K(x)
fe(x) and f e ( u ) factor through
wherrce p =p 1 2
r
t € speck(^)
E(U)
r
Let
+I!
(K(X) @ K ( U ) ) K (y)
be t h e morphim induced by
and
is a mnanorphim of schemes, f is injec:
. Thus i f
r
Corollary: With the assmptions of proposition 5.1, let (x Y)' be the underlying sets of 1x1 - ,J-Y I,121 and J-X zx -Y I - Z Then the map 5.4
e e e x-,Y-,z-
(5% Yls Z
am^
-+
X e x Ye
- -8-
which sends
t€g xz Y- onto (g (t),_f (t)) is -xY -
Proof: This follows fran 5.2 and frcm the fact that -
5.5 CJ:Y~Z_ Proof:
K(X) @ K(Z)
.
surjective.
K(Y)
#
0
.
With the assumptions of proposition 5.1, if is surjective ( 4 . 2 ) , a s 9 : X x Y * X_. CoroLlary:
x
- z- -
This follows insnediately frcm 5 . 4
5.6
W e have j u s t described the points of
the inclusion m r p h i m of
r e s t r i c t i o n map, write
Evidently
E~
Proposition: pints y
E
X
:
Z
Y_
. To describe t h e
.
x
-.
does not depad on U
Et 5
be a scheine,
x€ly) Spec dx onto
such that
isamorphism of
_X
open subset of the-scheme & l e t j be in 5 a d l e t x€g I f q:d(U)+< U is the ~g dx * f o r the camposition
be an a f f i n e
local rings, let
x
a p i n t of
Then the mrphim
X and Px E~
t h e g m e t r i c space
:
3 dX +
(Px
, l!lxIPx)
the set of induces an (1.3) ,
Proof:
Observe t h a t Px
consists of all points
by an a f f i n e open set, so we may assume t h a t
X
x€m. . W e may replace & . The proposition now
t € x f o r which
t belongs to each open set containing x
Such a p i n t
=
SJ
A
follows frcm Alg. c m . 11, 52, no. 5, prop. 11 and f r m the description of
local rings i n Sp A
(2.1).
Proposition:
5.7 set (4.2)
of
onto
and
x
Xx
& spZ_be
mrphisms of schemes,
.
X_,Y and Z such that f (x)=z=q_(y) Let Q be t h e subconsisting of points t whose projections
z points of
Xry
Let f:Z+?
Y
z-
-satisfy
- and
x€CtXI
y
~
-
tx_"'d4_
v m .e n
Muces an isanorphism of
,axxy\Q)(1.3) .
(Q Proof:
- Z--
write pX (resp. P ,P )
(resp. by
1x1, I Z _ j )
Y
Z
.
1s;
x ~ { s ) (resp. y ~ M , z ~ ( s ) ) Since
ad
1x1
I:/':
over
i n the category
Y
.
Let
(5.1)
g g
-X
~ ( y : ) (* ~
K(Y))X~X_ -
w i t h the image-functor of
fibre
a
of
131
(Q ,Jxxy]Q) may
The proposition-%ow
y
follcws
be a point of
~ ( y -+) Y, is a mnanorphism, t h e same holds f o r t h e
Since ~ ( y :)
call the
evidently
r
~ : ~ be " ya mrphism of schemes and l e t
canonical projection (SJ K (y)
s such that
is the fibred product of
be identified w i t h the fibred product P X p XPZ Y f r a n 5.6 and 2.13. 5.8
151
f o r the gecmetric space induced by
on the subspace 'consisting of a l l points
f -over y
+
X_
E (y)
. The set of
. W e may thus
identify , which we write f -1 (y) and
points of
~ - l ( y ) is a subset of
(4.2).
Propsition:
that of
--
The topology of t h e space of points of
1x1 . 'f
X€X_, f(x)=y
then t h e local ring of f-'(y)
3 .&
@x
f-l(y)
-
is the local r i n g of
is induced by
5
a t x is canonically isanorphic to
at
x
,
32
Au;EBRAIc
Proof:
-f
W e m y reduce everything to the case i n which
being induced by a mrphisrn
, 1.e.
Spec(lc(y1 MBA)
@ :B+A
of
8
I,
GEcb.IEI?IY
1, no 6
5 = Sp A , y=9 B ,
5 . Then I (SJ
is just xfractions of A/@(y)A
the prime spectrum of the ring of
K (y)) x X I
.
with respect to the multiplicatively closed subset @(Bhy) The assertion about the toplogy now follows frm U g . c m . 11,
5
4 , cor. to prop. 13.
The second assertion follows f r m the canonical ismrphisms
it can also be deduced f r m the description of the local rings of a fibred product derived i n 5.7.
Section 6
Relativization
6.1
Let
be as-functor and l e t
, the
-s be written
A S
functor B w i & ( ( A , B )
. Each representable S-functor,
functor of the-form If
be the category of
-
$sA
k is a model, and
5
Smodels
.
into g For instance, i f , wfiich is represented by A , w i l l
(4.1). An S-functor is a functor 5 of &l A=(R,p)EM
M -s
i.e. one i s w r p h i c to a
,will
be called an affine S-scheme.
= Ss K
,
M
-2
coincides w i t h the category -M
k-models. An 5-functor is i n t h i s case called a k-functor.
i.e. a k-algebra belonging t o the fixed universe
E , we
If
A
of
is a k-model,
kA for SJ~A , 4 speak of affine k-schemes instead of affine S-schemes. In parti-
cuiar, when A is the algebra k[T] cally i s m r p h i c t o the k-functor underlying set, For each k-functor
tion on 5
. The set of
write &S
of plynanials in T
gk
,
A k
i s canoni-
which assigns t o each k-model
X , a mrphism s:g+k
these functions is written
R
its
is called a f E -
dk(z) and carries
a
k-algebra structure: addition an3 multiplication are defined as i n 3.3; if AEk and
g€uk(X) ,
Af
satisfies
( ( A ? ) ( R ) ) (x) = X ( f ( R ) ) (x) for each
. We c a l l +O the affine k-l& k = z , we have PIPI? and the k-functors
RE&
ard each x€X(R)
In the case
coincide with the
Z-functors considered so far.
MI
6.2
If
3
is a:-functor,
the theory of 5-functors reduces immedi-
ately t o the theory of &-functors. For l e t %/S
be the category
Of
Z-functors
u
I,
5
6
1,
33
THE! L A N G u m
over p:z+s of p? with tar_ _ _ -S : an object of t h i s category is a mrphism ; a m r p h i m of
get
_p:T-+
q:x-+g -
into
Fh
is a camnutative triangle of
of the form
Ccmposition of these triangles is effected i n the obvious way. The c a t q o r y
-
is related t o the category is: ME/S- -+ -5M E which assigns to
@JP
-
where
p*
:
3R
peg (R) (3.2)
.
-t
S is
Ivlvlg
of ?-functors v i a the functor the g f u n c t o r
_p:x+s
a s usual the mrphism canonically associated with is: E/?-+!s$ -
proposition:
The functor
Proof:
W e merely give a functor
verse f o r
i
defined of
S
5
. Let T
is an equivalence of categories. j
be an S-functor
.Els.-+ ME/S
u -
. men
Am
-
-p T ( ~ :)
Z-functor of
1-
-S
=
T
If
3k ,
sets I&(A,R)
for -+
onto
is an s-functor, we shall c a l l
Xz , where the
maps T ( R , ~ )
( Z ~ ( ~ S(R) )
(which is contained i n the d i s j o i n t sum-*-(R,;)) 6.3
which is a quasi-in-
let pT:zps be the image to be
(,T) (R)
d e r j, ; we have
sum i s taken over a l l &(R)
*
S'
pES(R)
.
the d e r l y i n g
T
Z-
. I
T_ and ~~:~rp the t sstructural projection. For example, i f R€; and A Z g , then (SAA) (R) is the d i s j o i n t sum of the
, where
H
is assigned a l l k-algebra structures cmptible
R
with the given ring structure. This d i s j o i n t sum may be identified with
, where 8 denotes the underlying ring of . cal isanorphism ,(S&A) SpzA .
_M(ZA,R)
**
*
c
T_ by giving
f:x-tY- be a m r p h i m of schgnes,
Sp - K(Y)
. W e thus have a canoni-
%
W e frequently define an 5-functor
let
A
y
zy and
a pint-of
-pT
. For instance, -p:
f-'(y)
-+
the canonical projection (5.8). By abuse of language, w e call the
K(y)-schme
such that
ZT = f-'(y)
~(y)-schgnew i l l also be de"not& by
minology of 5.8 is p s s i b l e .
and
f-'(y)
E ~ T the
f i b r e over
y ; this
when no confusion with the ter-
Fu;EBRAIc cEcMEmY
34
I,
5.1, no
6
-
Similarly, i f 5 and. Y are ~-functors, we deduce frcm 6.2 that the follow-
are z ~ and l zEr2):
ing diagram is catmutative (where the canpnents of
.c
I n general, given an S-functor
i s local i f
T all
we shall carry wex irrrplicitly t o
those results and definitions which apply expl i c i t l y t o
say that
y.l
,z
zz
. Thus w e shall
.,,.Tetc.is a
is local, that _T is an s-&mne if
T
,I! is open i n Z-T wreover, we set 1x1 = I z ~ (, and c a l l I T \ theugecmetric rea1i;ation of
schene, that a subfunctor *!
T
. Finally,
i n sections %ere
is open i f
k is constant and
110
confusion is likely,
we shall employ an abuse of notation and write Sp A
.
or
flk(x)
If
k€g , we
write &cS
k-schgnes, i.e. the
called the base
we define
($'I -
(RIP)
=
-f ( 0 )=P
zT' z (s-T')
canonical isomorphisn
w
-
S'
L
It follows that i f
when S ' = % k '
kT'
formed by the
f:s'-+S there is associated restriction and simply denoted by s? , i f T' is an S'-functor, and i f
T_'
(RIG)
where the sum i s taken over a l l a€S' (R)
call
S&A
&functors
although it Ldepedis prhiicily on _f:
-
for
(SJI k ) - s c h s .
a functor PlSl~--+@ (R,P)€&~
o(X)
for the f u l l subcategory of &M
W i t h each m r p h i m of
6.4
of
,
T'
such that f ( o ) = p
. W e thus have a
which makes the following square cmmte:
c -
-
3s
i s a scheme, so is S-T' -
= S p k and
f = 2
the k-functor derived f r m
4
, we
. write
T'
k-
for
-
, and
by the restriction of scalars 4
.
For example, i f lying
A€$
k-delof
A
we have k ( S A , A )
.
f:z'+S
To each mrphism
6.5
S+(kA)
primarily on f :
if
T
Zsg +lfvlfE
we assign a functor
the base extension functor and simply denoted by
called
although it deperds
?
is an 5-functor and R€Z
is the under-
where kA
r
2'
, we
define
(Tsl) ( R , d = T ( R , f ( a ) ) -
.
and f ( u ) € S ( ~ ) W e thus have, by definition,
where
u€S' - (R)
where
u runs through S ' (R)
. If
T € T ( R , ~ ( U ) ) i n the d i s j o i n t sum (T I ) (R)
z -5
onto
L
(,TI M.
(R) Xs(R)S'
i n u (T) denotes the canonical image of
G T ( R , f ( u ) ) , we obtain a bijection of
(R)
by Sending
inu(T)
Onto
( i n f ( u )( T )
-
W e therefore obtain a canonical i s m r p h i s n
z (T -s
1 )
c
' (,IF)%S' ,.. -
t h i s and 5.1 it follows t h a t if T is a scheme,
-
--S"
SO
is
*
z (T -5
*
-
+
.
. Frm
Lf f:?'-+ i s a mrphism of 2-functors, the base extension funcis r i g h t adjoint to the base r e s t r i c t i o n functor s?
Proposition:
tor ? Proof:
0)
r
whose second m p n e n t is the s t r u c t u r a l projection pr
-
r
S' If
.
T'
is an S'-functor and
bijection
x!?'IT) which i s functorial i n
:
%%(ST'
T'
r?)
and -T
mst define a
' .
is a family of
x (z' r T ) assigns
I, 8'1, no 6
ALGEBWC GEOMETRY
36
Thus we say that 2,'
by extension of scalars. If RE*%,
is derived f r m
rk,
we have (R) = T(kR) where kR is the wderlying k-albebra of R particular, if A S R q , and T_ = 4 S A we have
. In
and we infer the existence of a canonical isanorphism
(BkAIkI SJ~~ (A
k')
.
In virtue of this fact we occasionally write 2 8 k' for k
canonical bijection
x ( T f k', T) : k E ( k ( T t k 1 I r T) jp%l,E(T_%
6.6
I
I
f__
adjoint functor. "/'
:
even when
. The mrphism associated with '5 gkk '
T is not of the form %A
-
is then denoted by -pk tion. __
rkl
k(T Bkk')
I ;_T
by the
k l r T f k')
and is called the structural projec-
of 6.5 also possesses a right The base extension functor ? S' called the WeTl restriction or direct image : -P FSg r IEis given by the formula
bijection
arad T I
which is functorial in the maps T(R,~)+
(s&$~) ( ~ , p )
In the case where 5
-
sp k
v
2'
:
with any -g:T,,?'
, c(T,T')
associates
assigning to TE?(R,~) the ccmpsites
= S z k'
and f = Sp @ we shply write instead of For A=(R,p) , ( S J ? ~ ( R , ~ ) ) ~is ~ then identified with s /s k /k Sp (A Bkk') so that we get --k
111
=
v.
(k ?TI) /k-
(A)
s' (Amk k ' )
.
I,
5
1, no 6
THE LANGUAGE
I n this case
c(T,x')
31
can also be defined as associating with
g the family -
.
where A€-JI
be a m r p h i s n of mdels and suppose t h a t the
Proposition:
Let
k-dule
is projective and f i n i t e l y generated. Then
k'
$:k+k'
a)
if T'
b)
i f T' _ - -is a. k'-schane and. i f ,
gkz'
i s an a f f i n e k'-scheme,
is an a f f i n e k-scheme;
f o r each f i n i t e subset P
i s an a f f i n e open subscheme V_'
of - -T '
is a k-scheme.
of
T' -
PCU' - ,-then
such that
,t
h s
UkT'
c _ -
Proof:
Suppose f i r s t that 'J"=SSkA
where A = jSkl (E Bkk')
algebra of a k'-rrcdule of t h e fonn EBkk'
. If
R€I&
, we
is t h e symnetric then have canoni-
cal isamorphisms (kl,kT') TT (R) = & , ( A
where
tk'
,RBkk')
sTI( E B k k ' , R f
is the k-module dual t o k'
k')
W ( E ,RBk') k
(Alg. 11, 4, no. 1, prop. 1 and
no. 2, prop. 2 ) . rt follows that
I n the case i n which T'=%k,A
, where
A
i s an a r b i t r a r y k'-model, l e t
be t h e kernel of the canonical hamomorphisn of
c1
Bkk'
into A
. Then
with t h e amalgamated sum of the diagram
A may be i d e n t i f i e d within &I
where
$(A
I
is the canonical m p and
B ( I gkk') = 0
. Since
is a r i g h t
a d j o i n t functor, it c m t e s with inverse limits, and. SO
V k S p k l A , the f i b r e prcduct of a f f i n e schemes, is i t s e l f an a f f i n e scheme.
Now f o r b) : c l e a r l y
is open in
gz'
, E g l
is local whenever
is open i n
"k7k TI
-
x'
is. Furthemore, i f
: f o r consider t h e m r p h i m
38
I, 5'1, no 6
G€XPElXY
AU;EBRAIC
and the mrphism f ' : Sp '(AQDk') + T' -k k
associated with f
g o(Spk@)
-u'
of
g'o(@kl(@QDkk'))
f a c t o r s through
t h i s latter condition is s a t i s f i e d i f f
I denotes the ideal of A gkk'
. Since
ShI (A Bkk' )
defining the open subscheme
(ABkk')/I i n A
g'-'($)
(A QD k ' ) / I i s a f i n i t e l y generated A - d u l e ,
k
is equivalent t o saying t h a t B@(J) = B
.
This enables us to construct f o r each of
defined above. Clearly
C(SJ~A,Z')-'
GK' i f f
f a c t o r s through
. By 3.7,
where
by t h e b i j e c t i o n
. For
G T ' such that xEU -
where K€M i s a f i e l d and r*
x
, where
xEk%'
J
this
is t h e annihilator of
, an a f f i n e
open subschaw
is t h e equivalence class of an element
(4.5 and. 4.9)
p€M(k,K)
. Since
Spec KBkk'
has only a f i n i t e n m k r of points, there is an a f f i n e open subscheme U' of
x'
Igl I contains t h e image of the mrphism
such that
151: Spec K@k' k
It now s u f f i c e s to set 6.7
k'=k x 1
g
X
kn
. If
...
pri:k'+ki
,
there are canonical mrphisms
p. (R) is the map
-1
T_(R g p r . ): T ( R f k')
for each
. .
&t kl, ,k be n copies of k a d set n we assign ki the k'-algebra s t r u c t u r e derived frm the
ith canonical projection
such t h a t
IT'/
= U'_;
Example:
...
+
R%
k
1
. I_f
T
+
T(R $ki)
a
is a local functor, it follows inmediately f r m
d e f i n i t i o n 3.11 that the morphism
T + TTki
whose
ithc m p n e n t i s
I,
5
Pi
1, no 6
39
THE LANGUAGE
is an isamrphism (apply d e f i n i t i o n 3.11a) to the p a r t i t i o n
(eti,e' 1
of unity i n RiBkk' i l,
such t h a t
.
T
I n t h i s example, we see t h a t B u t we a l s o note that the functor
Ti
for let
n T = kyk -i
i s a scheme whenever
is a scheme.
/:\
i f not. Then t h e
Ti
T , while
form an open covering of
W e now return to the diagram of functors:
M E f
iE.9
I4
C I
considered i n section 4. Given a g m e t x i c space T
, we
define the category
of gecmetric spaces over T : a geanetric space over T is a mrphism
of EAg w i t h t a r g e t T q:YJT
T
eiEk'
6 foreach i if n > l n
6.8
EsT
, where
does not preserve open coverings: k/k such t h a t Ti(A) = T(A) if Aei = A
be the subfunctor of
and Y ( A ) = rd
e' i = lBkei
. A mrphism of aT with d m i n
and t a r g e t
p:X*
is a c m t a t i v e triangle of the form
f
Ccmpsition of mrphisms is defined i n the obvious way. The category %T functors
M is connected to -ST -
l?lTI?T : cS& To each object
p:X+T
%%T
BTwe
of
tural projection _Sp:sX+ST
v i a a pair of mutudly adjoint
(6.3)
assign t h e _ST-functor
. Conversely,
if
s$
z+sTz ,
!ElT: IE/+T
i s the mrphism assign& to the structural projection -pF: gF+ST bijection
@(zlj',T)-l m
of
4.1. W i t h the above notation,
the
w i t h struc-
by the
bijection
40
I,
AiXEBFKtC GJXNTPRY
6
enables u s to associate the c m u t a t i v e triangle (2) with the c m -
$(z&X)
b:ive
'9 1, no
triangle (1)below, where f ' = $ (ZF,X) ( f ) : n*
f
f' -
41~(F,p) : sT(lFIT,p)
Thus w e obtain a bijection
ing to each triangle (1) the m r p h i m -g:_F-ts+
&&((F_
by assiqn-
S ,+)
such t h a t
. I t follows
zg=z'
t h a t the functors ( ? I T and 5, induce quasi-inverse equivalences between the categories of _ST-schanes and. c
directly fran the canparison theorem (4.4) the f u l l subcategory of fies condition
k€g and l ? l T , STand If
such that
consisting of the p:X-tT
E2T
of 4.4.
(*)
, we
T = Spec k
shall also write
aT ; a geanetric space
p: X
+
satis-
X
mk
for / ? I k , Sk a d Spec k over Spec k w i l l be
called ageanetric k-space; by the spectral existence theorem, t o specify such a space is equivalent to specifying X and a structure of sheaves of k-algebras over
0X
s e t €pgk(Spec R, X)
'
Hence, i f of
%,
(skp)(R) m y be identified with the
RE-
fEEJg(Spec R, X)
such that u(f):u(X)+R is a
k-algebra hamcmorphim. Example:
6.9
(2.12)
. The sheaf ux
k q be a f i e l d and l e t X be a Boolean spce
Let
defined in 2.12 carries the structure of a sheaf of
k-algebras i n a natural way, and t h i s defines a mrphism
W e shall say that %=;,(p) RS
, the map
%(R)
7 TO J:(SPeC R,
p: X I k
-+
Spec k
i s the Boolean k-functor associated w i t h
f w gg of
spec
R, x t k )
in
spec
X
. . If
is c l e a r l y
R, XI
bijective (see also 2.11). Thus we get a canonical i s m r p h i s m
If k
X)
.
is an arbitrary model, we define the Boolean k-functor associated with
X by the formula %(R)
fixed universe J
5=
(Xz)k w
, it
, the k-functor -~ 5
rc
X-Pl
. It
X
belongs t o the
is a scheme: i n f a c t , as we have
Xz i s a scheme. B u t Xz is U is open i n X, Uz is an open subfunctor of
is sufficient to show t h a t
obviously local; also, i f
Xz
, for
= To~(SpecR, X)
. I t is thus sufficient t o show that
is a compact ~ o o l e a nspace. In this case
A*
A.
Uz
is-an affine scheme whenever LJ
Ti
is the topological inverse
I, § 1, no 6
41
THE LANGUAGE
limit of its discrete quotients V
. For each
R€M-,
we therefore have iso-
mrphisns UZ(R) = %(Spec
l$n To~(SpecR, V) = 1$n Vz(R)
R, U)
*
A ..
is an affine schane and
By 2.11, Vz=-sV'
-
raps of
H
V - into
,Z
. Hence
d(V,)
is the ring
-
2'
of a l l
is an inverse-limit of affine schemes, and
Uz
is therefore i t s e l f affine (2."1).
Its ring of functions is the d i r e c t l i m i t
of the rings
the ring of locally constant maps of
into
2
.
6.10
,z" , i n other mrds, Example:
E
be a set w i t h the discrete toplcgy a d l e t
be the map which assigns to each xEE
+E:E+%(k)
value
Let
x of
Spec k
into E
U
. For each k-functor
the constant map with X
, we
define a canonical
mP i(E,X):
+
i:%+z the canposition
which assigns to each
(k)
E -%Fk
W e claim that i f
{elk
_X
E z(E,X(k))=X(k)
f(k)rX(k) -
is local,
s_Pkk so that
i({e),x)
i(E,_X)
%
{el, form a covering of
assertion
therefore follows frcm 4.13.
g is
a local functor, we write
i(X(k) - ,X) - sends onto
is a bijection. For i f
is invertible for each
case, the
If
.
Spec k
tible.
, we
have
. In the general
by d i s j o i n t open subfunctors. The
y,:z(k)k+z
Iq((k)EE(_X(k) ,_X(k))
stant (or simply contant)-if there is a set If
2
e€E
for the mrphism that
. W e shall say that E
g
and an i s m r p h i s m
is
k-E-
%&; .
is connected, t h i s i s equivalent to asserting that yX be i n v e r
5 2
Q U A S I - C O MODULES; ~ ~ APPLICATIONS
Section 1
sheaves of modules over a geanetric space
X be a geanetric spce, and let U& be a sheaf of mcdules (or simply over X) For each open subset U ova the sheaf of rings is then by definition a x d u l e over the ring ux(U) ; mreof X , &(U) 1.1
Let
.
4
over, the transition maps &(U) mrphisns d X ( U )
+. %(V)
+&(V)
. Hence, i f
a module over the local ring a l l sheaves of d u l e s over
<
Jx
are canpatible w i t h the ring hanoa t x is xEX , the stalk of
.Ik,
=&@
; we set /&XI
K
(x) . The category of
sx . L"x
w i l l be denoted by
Now let f:X+Y be a mrphisn of geanetric spaces. If dd ( r e s p . 4 ) i S a sheaf of abelian groups over X (resp. over Y) , we write f . b%) f o r the direct image of & (resp. f' tk3 for the inverse image of # ) Thus we have f.
(4)(v) = & ( f - l ( v ) ) (4
where
v
is open i n Y
,
f.
sheaf of rings
f.
x , A(f-'(v))
is a module over ux(f-'(v))
When& E-q
"functorial i n V" f . (&)
.
, by
(fix)
, and f ' M x
.
for
=
carries the structure of a sheaf of modules over the
. To see this, notice t h a t for each open subset
. Fran the hcananorphisn
V of
arid this m u l e structure is fg: Jy+f.
(fix) , we derive for
restriction of scalars, the structure of a sheaf of modules over
QY The sheaf of ncdules thus defined w i l l be written f*&) , and we call it the direct image of the sheaf of modules &
.
1.2
Proposition:
The d i r e c t image functor
f,: -q-+MC&
l e f t adjoint. Proof:
XEX
Let
ard vz/ be sheaves of ridules over
c"f,
and (by
shall
has a
. Clearly
.
f a(4naturally carries the structure of a sheaf of modules over f ' (dy) Considering the geametric space XI = (X,f' (dY) , w e have a canonical bijection.
$W&)
: %(J,f*(&
which assigns t o v:&f. @)
the mrphisn u:f
*
I
0 -+A such
ampsition X
that
ux
is the
.
f o r each XEX
. The proposition thus follows frcm t h e existence of
a canoni-
cal bijection
which sends
h:f'
(&-+A onto the mrphism
h' : such that
dx f : f l y ) f ' ~
h(s) = h'(l69s)
1.3
Definition:
-+
f o r each section s of
Lf
f:X-+Y
f*(4
is a mrphism of geanetric spaces and
& is a sheaf of ncdules over dy r t h e sheaf of mdules
over flX -
is called the inverse image of J ~ r d e rf
W e point out that, i f
XEX
, we
~ h u sthe inverse image f*(&
.
&pf.
f * (&
aniiiswritten
f*dl/j
have
of
J
a s a sheaf of modules over X
to be confused with the inverse image f *
(4
is not
of the underlying sheaf of
groups. Proposition:
1.4
A-module : M-(x) @M (*)
Lf
M
, there
Let
A be a r i n g and X = Spec A
is a sheaf of modules
satisfying c o d i t i o n
(*)
belaw:
- x -and an A-linear map
over
is any sheaf of d u l e s over dx
map, then there is a unique mrphism $:&&
By definition, &(X)
$:W&fX) is any A-linear such that
-$ =
$(X)o@,
:
carries an $(X)-module structure. !This induces an
A-module structure i n v i r t u e of the isanorphim @A: A The prcof of t h e proposition is similar t o that of N
selves here t o constructing M and @M presheaf
. For each
over Spec A by s e t t i n g
. As
in
5
5
-+
dx(X)
of
5
1, 2.1.
1, 2.1. we confine
OUT-
1, 2.1, w e f i r s t define a
44
I,
-
M( D ( a ) )
for each ideal a of
=
5
2 , no 1
~ s ( a ) - 2' ~ M@XS(a)-'] ,
A ; i n t h i s way
A
we assiqn t o each open subset D(a)
of
spec A a module Over the ring A[s(~)-'I ; for example, for a special open set X w i t h fEA , M(X ) may be identified with the module of fracf f tions M of M w i t h respect to the multiplicatively closed subset 2 f3 (l,f,f ,f ,...) ; i n particular M(Spec A) may be identified with M
M"
be the associated sheaf of the presheaf
M
p ( D ( a ) ) : ~ [ S ( a ) - l ]x M[S(a)-']
. The action
-+
. Let
M[S(a)-'J
induces, by passage t o the associated sheaves, a mrphisn 4 p
:OxxZ&
which defines the structure of a sheaf of d u l e s on
M(x) into
be the canonical map os M
W e p i n t out that the functor
M
H
c a n p s i t i o n of the exact functor M
%
%'(XI
.
% . We
define $M
to
is exact, i n as m c h as it i s the
w i t h the "associated sheaf" functor.
I-+
This result could also have been derived f r m the structure of the s t a l k s of rJ
M ; if M
P
p€X
, the
= M [(A-p)-l]
.
stalk
Corollary:
1.5
P
of
a t p is the module of fractions
W i t h the assumptions of prop. 1 . 4 ,
$M i s an iso~~
mrphism.
Prcof: To prove t h i s corollary, we refer back t o 5 1, 2.6. -
It follows f r m
the exactness of the sequence
associated w i t h each f E A and each covering of
Proposition:
1.6 A-module c~M)-
jr
M
Let
y:B->A
(resp. each B k d u l e N)
(spec y),(iij
Xf
by special open sets
he a ring hcmmrphism. For each
, there
is a canonical isamorphism
(resp. (A gBN ) ~ (spec y)*(Ejj) where B~
B-module derived fran M by restriction of scalars.
is the
I,
5
2, no 1
Proof:
QUASI-U"T
X = Spec A
Let
,
Y = Spec B
MODULES
, f=
Spec y
rc/
I ) : (BM)w + f, (MI such that ismrphism: i f sEB , w e have
mrphism
(BM)y
. By
1.4, there is a unique
. This mrphism is an
= @M
+(Y)o@BM
z MY ( s )
(Ys) 7 (BM)s
45
G(f-l(YS))
.
In order to obtain the ismrphism i : ( A @ N)B
1f * & j
we observe that the functor
N
, d
H
f*(N) is the c a n p s i t i o n of two functors
which are both l e f t adjoints. It is therefore a l e f t adjoint for the mpos i t i o n of the corresponding r i g h t adjoints, that is, for the functor
3'-'
B T i ~ ). The
i s the canonical ismrphism of -1
q =Y
.
(PI
1.7
M(')
N
(A B ~ N ) -
+-+
.
,
is given an explicit description, we see that, f o r each p E Spec A
When i i
same argument may be applied to the functor
Recall that, i f
M
onto A @ N where pBqq
(A@BN)p
is an object of a given category, we write
f o r the direct sum of a family of copies of
Definition: quasi-coherent
1x1,
_Let _ X
if,
%-I
.
0 (I) -+ O(J) V
-+
V
formed by the quasi-
_ _ _ _If_ M Z A category of small A-modules.
,
-7-1
is a model, @A
M
x
exact
pa"4r
A N
of
0
, where
.
is called
&lv -+
f o r the f u l l subcategory of
coherent sheaves of modules. Similarly, i f
eA of the form
over vx
2 , and an
belonging t o the chosen universe
&9v
r r d u l e s over S p A
&
x€& , there is an o p n neighburhod-V
'of the form
the restriction of W e write
be a schene. A sheaf of modules
for each
I, J
sequence of M & c
indexed by the set I
M
J6lV
denotes
demtes the
is a quasi-coherent sheaf of
. To see this, notice that there is an exact sequence fram
A
(1) (J)*o -+A
,with
I J E ~
. Since the functor
M n 6i'
c m t e s with direct limits, t h i s exact sequence is transformed into the exact sequence
where X = % A . Conversely, i f there i s an exact sequence
vf‘ (L,L
then -_
is of the form 1
-t
M” : ~
4 for ,L,LIc_~~o~, , the canonical map
@o+/zI2) is the canpsition of the bijection
mod (L,L’) into mod (L.,c(5)) -A
-
-A by setting W L ard
&=Kt
(1.5) with the bijection $
.
A scheme
X
,+
of
F+
$,,og
$
obtained
in prop. 1.4. This proves that the functor -
M L’ is fully faithful. In particular m is of the form u:A(’) + A(J) is A-linear. Thus we have K G (Coker p)-
1.8
g
G , &ere
is said to be pasicanpact if its space of points
is quasiccmpact. g is said to be quasi-separated if its space of points is quasi-separated, that is, if the intersection of two quasicanpact open subsets is quasic2cmpct. For this cordition to be satisfied, it is sufficient that there be an open owering of 3 by affine open subschenes
Xi
such that
. For gim-1 may then be covered by ; if sEJ(zi) and t€flgj) , a finite number of affine open subschenes zijl (xi)sn(gj)t is the union of the affine open sets (zijl)sta d is therefore quasiccmpact. Hence the (zi) form an open base whose pairwise intersections
X.m -1 -j
is quasicanpact for each
(i,j)
are quasiccenpact, and the assertion follows.
Propsition: Consider a Cartesian square of
Y-X
sews
f -
where AIEEZ, A is flat over B
,
; is quasicanpact and quasiseparated.
Then for each quasicoherent sheaf of modules $ over - - Y
is bijective. (Clearly we write here f *
(5) instead of If I * (4(
, the canonical map
1x1) , see 5 1, 4.2) .
(xi) (resp.(Yijl)) be a finite affine open covering of Y_ -1 (Yijl) . Then we have the dia(resp.Y.nY.) . Let gi=f-l(Yi) and zijl=f -7
Proof: Let A1
gram
I, § 2,
47
1
i n which the arrows are the obvious ones. (For example, t h e ccsnponents of
index
zijl A
c
( i , j , l ) of
Xi and Sijl
a and b are induced respectively by the inclusions
. The f i r s t l i n e is exact since
c X.) -7
*: (4
i s f l a t over B ; since
is a sheaf, t h e second l i n e i s also exact.
Now, by 1.7, we may assume that each be of t h e forin
gi
@I
N
~i
whence, since t h e set of indices
f* -
is s u f f i c i e n t l y small f o r
!Zi
. By 1.5 and 1.6,
with N i W 13(yi)
A
L/1/ is a sheaf and
(4(zi)
Ci)
to
;
is f i n i t e ,
thus v i s invertible; so a l s o is w ; arid so is u
.
Rgnarks: With the above notation, suppose that the B d u l e A
t i v e and t h a t
xlxi
it follows that
is projec-
is quasicanpact, but not necessarily quasiseparated. Then
the canonical map
is i n j e c t i v e and. v A
i s invertible. Hence u
is invertible. Similarly, i f
is a f i n i t e l y generated projective B-module, our proposition
without r e s t r i c t i o n on of the set of irdices 1.9
x(Q) x(Ff f+
Proof:
true
v and w are then i n v e r t i b l e irrespective
and I i j l l
.
Corollary: L e t 5 be a scheme and l e t & be a sheaf of Then & is quasi-coherent i f f , f o r any a f f i n e open sub@ 5 and any f€B(g) , z(_V) is s m a l l arid t h e canonical map
mdules over 5 scheme
!il
. For
rfmains
me
.
is bijective. last c o d i t i o n on Jf simply tells us that the r e s t r i c t i o n o f $
to _V is identified with the sheaf of mcdules y(LJwderived from the d(g)-module d(LJ This implies, by 1.7, that J is quasi-coherent.
.
Corollary: (Structurethwrgn for quasi-coherent sheaves)
1.10
, the functor
M ++ % of 1.4 is an equivalence of the cate9ory of small B-modules onto the category of quasi-coherent sheaves of modules For any model B
over - -Sp B
.
Proof: By 1.7 we have already shown that is quasi-coherent if M m , and that the functor M is fully faithful. Conversely, by corollary 1.9 applied to U = & B , we see that any quasi-coherent sheaf of rrdules & is I--,
of the form
ii , with
M = d (s p B)
If 5 . If
Corollary: Let
1.11
sheaves of madules over a d Coker f
1
.
.
be a scheme and f : d + X a mrphism of and d ' are quasi-coherent, so are Ker f
Proof: Let V_ be affine and open in X ?., an arrow g for some arrow g of IK&(~)
1 L .
Ker f jg
Ker
<
7
2
flu is "isomorphic" to
. We thus have iscanorphisms
(Ker s ) ,~ Coker f ( U
Corollary: Let
1.12
. By 1.10, Coker
F z (Coker gi"
.
be a scheme a d O + ~ ~ & + c & " + O
.
an exact
sequence of quasi-coherent sheaves of modules over X Then, if g is affine and open in x , the sequence o+K'(LJ+/~~'~~)+~''oJ+o is exact.
.
Proof: We need only consider the case in which 5 = _V = S 3 A , A€g By 1.10 a d 1.4, the functor M !-+ with danain md and target the category of -A
quasi-coherent sheaves of modules over
has for its quasi-inverse the functor N++cN(y) This last functor is thus an equivalence; in particular, it
.
is exact. Section 2
Direct and inverse m g e s of quasi-coherent sheaves
2.1
Propsition: g t is quasi-coherent,
f:ZfY_ f * cu)
be a mrphisn of schemes. If
.
5
I,
2, no 2
Proof:
_v
(I) +
J(’)
v
+
g*
(XlV). Hence,
z(J) v d (J)
-
+xlV -+
induces an exact sequence
0
+
f*(& ITJ -
I, J€g , each exact sequence
if
-r
, if
herent over
-
-
, for
c*g), where
an inverse image
=II , the
have -f,
-
is i n general not quasi-co-
i s quasi-coherent over _X : take f o r
of sane copies of
sum :(I)
.
(observe that g*(Qv(’)) % flu(’))
0
With the notation of 2 . 1 , f * Gk6)
2.2
A
1x1 , l e t _V be its inverse image, and let iraduced by f . There is a canonical isatlorphism
be the mrphism
f *(4I _V
49
be open in
I!
Let
-g:g-+y
QUASI-COHERENTKIDU”
X
the “codiagonal” mrphism and for
f
is quasi-coherent over
V/
the d i r e c t
product being taken in
%I
such a product i s i n general not quasi-coherent i f
. We clearly
. It i s easily seen t h a t
I
is i n f i n i t e . Neverthe-
less, we shall see that the d i r e c t image of a quasi-coherent sheaf is quasicoherent under sane general conditions t o be specified below. Definition:
a)
f:s+x be a mrphisn of
Let
schfmes.
is said t o be qwsicchnpact i f the underlying continuous map of
f
f.
i s guasiccarrpact, that is i f , for each quasimpact open subset V o_f
1x1 ,
b)
If
Z-~(V)
is y u a s i m p c t .
f is said to be quasi-separated i f the diagonal morphia
_U
f-l(Y.)
has a c a r e r i ’ consisting ~ of open affine subschemes
is quasicmyact for each i
-1 (X. .) be a f i n i t e covering of
-17 affine open subscheme
z
of
, then
f-l(Xi)
xi ,
g-’(?)
f
xi
such that
is quasicchnpact. For l e t
by affine open subschemes; f o r each
is then the union of the affine -1 The open V such t h a t f (V)
zVij , hence quasiccanpact.
open subschms
is quasiccanpact thus form an open base; and the assertion follows. Applied to the diagonal morphism
this assertion says t h a t f. is 6_x/,i can be covered by affine open atd each
quasiseparated if by affine open
sij
(For observe that
such that
g.
.ail= $-& ( ~ i j y x i l ) ) . f
be covered by affine open
.
xi
:-‘(xi)
is quasiccanpact for each
(i,j,l)
13
Finally we can say that separated (1.8)
gij n X l
is quasiccanpact and quasi-separated i f f
xi
-1 such that f (Y. ) -1
_Y
can
is q u a s i c m p c t and quasi-
.
Au;EBRAIc c25omrF.Y
50
Proposition: Consider the diaqram of schemes
2.3
5
I,
f
2 , no 2
g
X+
+
y'
. If
-f is quasicanpact (resp. quasi-separated) then the canonical projection f Y l : 3 Y' * _Y' is quasiccanpact (resp. quasi-separated). 2(resp. 2)
Proof: Suppse for instance that f is quasiseparated; then _Y'
xi
can be covered by affine open (X.. I -17
. NOW let
-
(resp. Y.) such that g(y;)Cyi -1-1 be an affine open covering of _f ; set
(xi)
then we have
since
x. .n xil = 6-1~(Xijy. / x~xil1 -11 1 is quasicompact, as well as the mrphism
Hence
_f
1'
.
is quasiseparated (2.2)
Similarly, if
2.4
:;+xi
is quasicanpact, so is -fyl
-
Proposition:
If _g:Y+Z - -
(see 2 . 2 )
, so
.
is X! .@ij -13
.
is a quasicanpact ardi quasiseparated
mrphism of schemes, the direct image y * M of a quasicoherent sheaf of d u l e s L/ is quasicoherent. Moreaver, the functor induced by on quasicoherent sheaves preserves filtered direct lmts. CJ*
izqie,
Proof: We first show that, for each affine open subschane V of Z and each s€d(y) , the canonical map
(&(.I/,(WS
+
g*N) cys, , we
is bijective. By replacing _Y by q--('!) and set A = Bs We then have
.
(?* (4(V)
A
may assume that Z
=
=
Sp B -
@N(y) €3
and the first assertion follows fran 1.8 and 1.9. As for the secord assertion, it is e m q h to consider the case where Z is affine. It is then enough to show, that for quasicoherent sheaves on a quasi-
compact and quasiseparated scheme Y
, the functor
+
4x1
preserves
51 filtered direct limits. But, using the notation of 1.8, the functors
q+&xi)
OUT
and
x+“6Y.-111. 1
clearly preserve filtered direct limits. Thus assertion follows once aqain frcan the canonical exact sequence
We now suppose & to be quasiccanpact and quasiseparated. Then
2.5
the following coditions on a quasicoherent sheaf of modules & over 5 are equivalent: (i)
X can be covered by affine
such that &(g)
open subschemes
is a finitely generated IIx(~)-module; -
for any affine open subscheme _V of _X
(ii)
, A(!)is a finitely
dx(IJ-module; -
generated
if a quasicoherent sheaf 2 is the union of scme directed set (iii) of quasicoherent subsheaves Ni, then any mrphism ‘f : factors through SQne
A+
k.1 .
Proof:
(ii)+(i)
is clear, and
(i)=+ (iii) follows easily fran 1.10
(show that the restrictions f /I! factor through sane Mi1U) . In order to shm that (iii) implies (ii) , let g:y-tx be the inclusion mrphism and
let (Ni) be a directed set of s M u l e s covering sane We then have, by 1.10 and 2.4
Let
2.6 sheaf
dll
over
X
fl (U)-module N 5-
.
be quasiccmpact ard quasiseparated. A quasicoherent
satisfying the equivalent corditions (i) - (iii) of 2.5
will be called finitely generated. If g is any quasimpact open subscheme of _X am3 sheaf of d u l e s over _V
, it follows directly frm
any quasicoherent
2.4 that there is a quasi-
coherent sheaf
over
5
with
AlU
(take f o r instance
.*& , where
u:U+X - - - i s the inclusion rmrphisn). But there is a better r e s u l t , which says that
A may be chosen t o be f i n i t e l y generated
if
?'
is. This c l e a r l y fol-
lows from the
- - X be a quasimpact a d quasiseparated scheme and l e t Let
Proposition:
X over -
be a quasiccarrpact open subscheme. For any quasicoherent sheaf
U
Lkl of - 3; 1 , t h e r e is a & of '? such that c&=ul/Q .
axl any f i n i t e l y generated quasicoherent subsheaf
-~ f i n i t e l y generated quasicoherent subsheaf Proof: U
L
Let Ul,...,yn
, cover
be a f f i n e open subschgnes of _X
. Our proposition is t r i v i a l f o r
_X
.
open subscheme m e r e d by
n=O
. For
which, together with l e t -X '
n)o
be t h e
By i d u c t i o n on n we may suppose 'n-1 that there is a f i n i t e l y generated quasicoherent subsheaf Y' of ? ' ( X-I such
. It is c l e a r l y enough to e x t d
that A= & I ;1
generated quasicoherent subsheaf of
7 Isn
by
z'nEn , we
sane f i n i t e l y
which, matched together with
over X1nVn , w i l l supply us with the required
-U
d'l ~ ' n ~ tno
&. Replacing
reduce the proof to the case i n which
5
_X by
En
is affine.
- denote the inclusion mrphism. The invexse image I n t h i s case, l e t _u:v+X
Q of ,u* Wl under t h e camnical mrphism p-+ >* 61g) i s quasicoherent by 2.4 and s a t i s f i e s 9 [_V=&. By 1.10 Q is the union of the directed set of its f i n i t e l y generated quasicoherent subsheaves &' Thus is the union o f the r e s t r i c t i o n s k ' (-U and is equal t o sane by 2.5 (iii)
.
over
-
For any scheme
2.7
5 i s closed i n
Mod
Izl
5
t h e category
.
ml
of quasicoherent sheaves
under kernels and s m a l l direct limits. Hence
it is an abelian category with exact f i l t e r e d d i r e c t l i m i t s . I f ccknpact
2
is quasi-
and quasiseparated, it follows f r m 2.6 that the f i n i t e l y generated
objects of
s,
generator i n
-
generate this category. This implies t h e existence of a
, since
quasicoherent sheaves over
the i s m r p h i s m classes of f i n i t e l y generated
5
may c l e a r l y be indexed by sane small set.
In other w r d s , i f X is quasicanpact and quasiseparated, we can apply to the general r e s u l t s ahown for Grothendieck's AB5-categories with
%I
generators. For instance, i f
5
is a category with small mrphism sets
~ , ( x , y ), then any functor F : m , +
ICI
J1 preserving
d i r e c t limits has a r i g h t
1, § 2, mJ3
53
QUASI-COHERENT MODULES
-.
,K is t h e f u l l subcategory of Mod Izl such that y(U) is small f o r any open vCl_X[
adjoint. This holds i n particular i f
s”
formed by the sheaves
Taking f o r F the inclusion functor, we i n f e r that any
a quasicoherent sheaf
@“
x€g
may be assigned
following universal property: f o r each quasicoherent V% and each there is a unique ( J : & - + C & ~ ~
such t h a t q4(J=$
.
$:A+ &
W e may i n f a c t give a d i r e c t construction f o r Jqc : Assume f i r s t that
k=y,&)
of the form
subschene. Then we set (y*
(p) ,y* (q,) C
)
, where _v:v+z denotes pc= a ( y ) , and write
q,(y) $d (v) =
m r p h i s n such t h a t
zijl)
mrphism of
(1.4)
5
(resp. of
I
)
the inclusion of an a f f i n e open qZ:Pc+Y f o r the unique
. It is then easy to show that
consider a f i n i t e open covering (,Xi) I f _v i (resp. yiJ1) i s the inclusion
.
xi
i
= v*(dKi) (resp.
.
-1
viJ1 fl Xijl)
From
of
I d . (v)
is
L/
i s a solution of our universal problem.
In t h e general case, when &€$ (resp.
the
together with a m r p h i s n q4:p+&enjoying
-
our previous ranarks, we g e t solutions
of our universal problem relative to thus remains d i t i o n udq,=
uqc : qd)
Section 3
T < 1
and
v
W e m y therefore
Is)*Q by the conditions qv=vuyp and w d p . I t + y by the conto set dqC = Ker(v,w) and t o define quy:flC pu , where u :JqC + ? is the inclusion morphisn. The p a i r
define mrphisms v,w:
is the required solution of our universal problem.
Faithfully f l a t q u a s i m p a c t mrphisms
We give here another extremely useful property of quasicanpact morphisms. W e
f i r s t make a new definition.
Definition:
3.1
A mrphism of s c h m s
a point XEX- i f the map fx: Jg(x)+Ox
e
s
dX
_f:X+Y _ _ is said to be f l a t a -t a f l a t n\cdule over Jf
It is said to be f l a t i f it is f l a t a t each xcX_ ,
faithfully f l a t
.
ift
is f l a t and surjective.
94
In the case of affine schemes, flat) iff
-f
is f l a t (resp. f a i t h f u l l y
S2B
makes A a f l a t (resp. faithfully f l a t ) d u l e over B
$:B-+A
(Alg. c m . 11,
: S&A
5
3, prop. 15
and cor.). For a mrphism of schemes
f:&+x
to
be f l a t , it is thexefore necessary and sufficient t h a t the follming condition IXsatisfied: if L J , ~ are affine md open i n
is a f l a t module over
d(v)
Propsition:
3.2
.
f'
be a point of
is f l a t a t
x'
is f l a t (resp. faithfully f l a t ) .
'1 X'
f
-x
X_'
. -If -f -is f l a t
I
, g'
_ f ( v ) cI! , then
4,~)
Consider the c m t a t i v e square of schemes below,
which assume to be Cartesian. Let x' f. is f l a t a t x
_X,Y and
and
.If
-(XI)
(resp. faithfully f l a t ) ,
P
Y'
Proof:
5
BY
dxl
1, 5 . 7 ,
y'=f'(x')
and y=f(x)
f l a t wex
D
f'
by
5
Y' 1, 5 . 5 .
if
Ox
i s a ring of fractions of
. Hence
dxl
i s f l a t over
is f l a t over
J
Y
.
y @ J
, where 4,Oxwhich is itself
2 is
surjective, so is
d
9Y'
~ l s o if
Jy x
W e occasionally make use of a converse form of the foregoing propo-
Rgnark:
-f ' is f l a t a t x' and if q- i s f l a t a t y ' = f ' ( x ' ) then f is f l a t a t x=p(x') For is then f l a t over 8 , where y====(y')=f(x); Y by the p r o p s i t i o n dX, is f l a t over dx ; so Jx is f l a t over J (Mg. sition, namely, i f
m. I,
5
4'
.
3, no 4 , ran. 2)
.
Let k be a milel and
3.3
(resp. faithfully f l a t ) over k
pX -px: -
Y
-+
9k
Propsition:
X
a k-scheme. W e say that 3 is f l a t
i f the structural projection
i s f l a t (resp. faithfully f l a t ) . Let
rated k-schemes. I f -
k be a model, Z
- -ard- Y 2
two quasicanpact and quasisew-
is f l a t over k ard i f
d(Y)
is a f l a t k d u l e ,
I,
5
2,
55
QUASI-COHEBENT IWLXJLES
no 3
then the canonical map d(z)@k~(Y) -+ t,!)(pY) is bijective.
proof: Of course Z x Y- denotes the prduct in the category -ME . Let (gi) and (zijl) be finite affine open m e r i n g s of 5 and z,e , In the dia1 j gram
the two lines are exact. We see that v setting B=k ard A
=
(zi)
(resp. w) is an imrphism by
(zijl)) in propsition 1.8. It
(resp. A =
follows that u is invertible. 3.4
ffqc descent theorem: If f:X+Y _ _ _ is a faithfully flat quasi-
ocknpact mrphism of schemes and if Erl , p~~ are the canonical projections of the fibre product g p z onto its factors, then -
IPL-21 exact sequence of qeanetric spaces. Proof: By definition 3.1
If1
is surjective. By
fy f ( x ) = f ( x ' ) , there is a pint z€gx$
8 1, 5.4, if
such that x =
-
satis-
X,X'€~
p.xllzl ,
x' = pr2(z); since the converse is obvious, we may identify the underlying set of Y_ with the quotient of the underlying set of X obtained by identi-
.
fying pgl(z) with pg2(z) for z € g v It remains to show that the quotient toplogy WI. that the sq-ence
where g = fprl
=
fpg2
, is exact. We prove the
For each affine open subschgne show that the sequence
d(V)
-f
J ( y ) If d p
v-U)
carries
second assertion first:
of Y , if we set Q
=
f
-1
(v) , we must
is exact. If C=d(V)=M
,
is affine, this follows fran
B=d(U)
1, larma 2.7 by setting
. I n the general case, by 2.2,
(gi)
open covering
5
of
there is a f i n i t e affine
i s the d i s j o i n t sum of the
U-. ; i f
affine and the canonical morphisn of
induces a diagram
into
_V'
yi , _u' is
v
U
.
u'=iu , j w ' i , j w ' i The bottan sequence is exact since is affine; the top one i s likewise since i is injective.
such that
Finally,
1x1
carries the quotient toplcgy. For i f
of
5 , we
if
F is saturated, we therefore have
f(F)
W e now show t h a t
f
substituting
(9
A
i s closed.
-1
-
1 (f ( F ) ) , where F i s clos+ in (gl
(f ( F ) ) = f
for
and
and 2 = Sp 4
. Now
u'
Let
5 we may
for
sequence O+bB+A/a A@ b B
inl : A
+
of the image of
-
= V(b)
A
(9
= SS B
such t h a t
. By ,
F=V(a)
1, 2 . 4 ) . Frm the exact
we derive, by means of the f l a t extension
4:B+A
,
A$(b) = Ker inl
A @(A/a) s a t i s f i e s B
Spec(inl)
in,(x)
= x
is thus
V ( w ( b ) ) = f-'(V(b))
= f
-1
(f(F1) ~
It therefore remains t o show that the image of
, ant
assume t h a t
a be an ideal of
1, 2.4) a d b = @-'(a) ; we have f ( F )
where
is a closed subset
show below that
It follows that
3 = Sp
F
v'
01
. By 9 1, 2.4,
the closure
. Spec(inl)
coincides with
this follows fran 5 1, 5 . 4 applied t o the fibred product of f the diagram X--txt-Sp(A/a) : indeed, i f we set _F = _Sp(A/a) , the irrage
f - l ( f (F)
5
I,
2, no 3
57
QUASI-COHERENT M9DUIES
Corollary: Consider the c m t a t i v e diagram of schanes below. If
3.5
q- is faithfully f l a t and quasicanpact, and i f
is invertible, so i s
Proof:
If
f~ Y
- .z-
f
.
is invertible, so is
Thus, i n the diagram below, the first OrJO v e r t i c a l arrms are invertible, so therefore is the third provided the two horizontal sequences are exact. But
this follows frcan theoren 3.4 modulo the w e l l known identifications
3.6
L
Corollary:
Consider the c m u t a t i v e square of schemes below. I f
is a monmrphism and f. i s quasicanpact and faithfully f l a t , then there
is a unique mrphism j:_u-._Z
such that g=jf and
ii=y
.
58
I,
Prcof:
W i t h the notation of 3 . 4 , we have
fpr
1
= fpr -2
5
2, no 4
&pzl =
whence
yzprl = yfpg2 = &pgz ; hence gp.rl = gpg2 since A is a mncknorphisn. By 3.4 ard 5 1, 4.4, there is a unique mrphisn -j such that _u = j f ; hence
-i j-f-
= &u- = vf -_
, so that ;j
--
.
=
The functorial p i n t of view
Section 4
W e rum develop a purely "functorial" theory of quasicoherent modules on schemes
and show haw t h i s new notion overlaps the preceding "geanetrical" definition. 4.1
be a2-functor and let
Let
For any REM, a d any pE_S(R) Suppose each f i b r e M(R,p)
that, for any +%(R,S) which shall mean that
.
, call
M(R,p)
4
be an s-functor
the f i b r e of g
over
p
.
t o be given an R d u l e structure i n such a way
, the
induced map JI: M(R,p)
bJ(S,$(p))
+
is additive and s a t i s f i e s $(Am)
JI
(5 1, 6 . 1 ) .
=
is $-linear,
@(X)JI(m) for all
we then say that , w e t h e r with these module structures, is an S-module. If _M and 1 are two _ S d u l e s , a mrphism
m%(R,p)
and XER
f%s,-,(M,_N)
is called linear or an g-mdule mrphism i f
i s R-linear for a l l
f(R;p): _M(R,p)+ _N(R,p)
. These definitions ob-
(R,p)@I
-5
viously give rise t o a new category, which w e denote by
gs: the
abelian
category of S-mdules.
s-mdule g is called quasicoherent i f its fibres I j ( R , p )
are snall and i f , for any @@I(R,S), the induced map _M(R,p)MRS-+ @(S,Q(p)) i s invertible.
An
Ivbreover, i f
M(R,p)
n) for each
(R,p)
is a projective R-module of f i n i t e rank (resp. of rank
, we
say that
5
is a vector bundle (resp. a vector bursdle
. The f u l l subcategory of --s Mod S-rdules w i l l be denoted by 'Ws . of rank n) over
4.2
S
F i r s t examples:
structure for any
(~,p)
. This
a)
fonned by the quasicoherent
w i t h the usual R-ule
Take _M(R,p) = Rn
w i l l be written :Q
-
and called the
I,
5
2, no 4
QUASI-CDHEFENTMODULES
59
.
t r i v i a l vector bundle of rank n For the urderlyirg p-functor ( 5 1, 6.3) n we c l e a r l y g e t zM = sxg where Q i s the a f f i n e line (5 1, 3.3). Thus
i s a scheme i f
%-
is one.
_S
b) Let A be a mdel and l e t M be a small A-module. S e t g = *A Ma
, and
the functor M
Ws ; the functor S1-functor Ms
W e thus g e t a quasicoherent S-mdule
is clearly an m i v a l e n c e of a is quasi-inverse to it. H M_(A,Id ) M
H
be a mrphism of 2-functors. For any S-mcdule _M
, derived
fram
g
by base extension
vector bundle, i f
?
*
Mod
s" -2
-f
( 5 1,
, the
6.5) is c l e a r l y
-
g is so.
The base extension functor
@sl
defined i n c) has a r i g h t
adjoint. I n order to see this, we-only note her; that, f o r any sLmodule
and any
(R,p)€gsI (s&M'
-
onto
ySl is quasicoherent o r a
assigned an _ST-mule structure. This s'-module d)
GA(1.7)
A
7
c) Let f:S'-S
.
a l l p€E(A,R)
= M @ A ~for
Ma(R,p)
and
M'
the set ) (R, P)
=
zs IH( (Sps- (R, P ) 1-
g' 1
5 1, 6.6, carries a natural R-module structure. I n f a c t , any (R,p) is a function assigning sane x ( $ , p ' ) E g'(R',p') t o a
considered i n
xE
(sl/sM') I I
pair
(@,PI)
E g(R,R')xS' ( R ' )
such that S($) (0) = f ( R ' ) ( 0 ' )
addition and scalar multiplication by the formulas x1($,p1)
?sl - :
(1x1 ( $ , P I )
+ x2($,p'l': and
r e s t r i c t i o n functor
S'/S
:
=
(x,+x,)
$(M.x($,P')
gslws +
.
W e define =
($,PI)
. That the so defined Weil
is r i g h t adjoint to
zs Eslmay be prove3 as i n 5 1, 6.6. -+
W e shall see l a t e r that the Weil r e s t r i c t i o n of a quasicoherent m d u l e over
.
S' is i n general not quasicoherent over S_ Nevertheless, t h i s is obviously true i f both S and S' are a f f i n e schemes. 4.3
proposition:
h coherent _ _ S-module. _ - ThI' ~
fl
g t
5
b e a :-functor
is local
(5
1, 6.3)
i s a vector burdle, then - - M i s a scheme i f f Proof:
iff
_S
1 2 .
be a quasi-
e. Pbrmver, i f
3
Clearly the s t r u c t u r a l map _ P : ~ E $ ( 5 1, 6.3) has a section so t h a t
-S is a retract of
,p1
and thus is &a1 or a scheme provided the same is
zg . Conversely, f i r s t suppose of d e f i n i k n 9 I, 3.11 with 8 = zM . L e t true of
and l e t
I
n*
to be local; we use the notation ti€X(Rfi)
be elements such that
and set rli=p(Ci) E S ( R
v((Si)) = w((Si))
assigned the same image q. in lj nES(R) with imge n in S(Rfi)
S (Rf.
and M ( R f1 . f 1,nij)
are identified w i t h
5 1, 2.7 g ( R , n ) , there is a
)
. Since l j
i
. Since
and q j are for each i , there is a unique ,)
fl
qi
is quasicoherent, M(Rfirqi)
_M
and M ( R , ~ ) &R~f i c aRRf j ’
_M(R,n)cgRRfi
,
Hence by lemna
applied t o the case C=R
M =
unique 5 € bJ(R,?l) c 1((R) having
c X(R 1
B(Rfi,rli)
f
for each i
i
Suppose finally that _S
B =TTR i fi
.
and
6 i as image i n
is a scheme and t h a t _M is a vector buradle. Since
M i s kncwn to be local, it is enough t o show that, for any affine open subschane _V of 3 , the U-module derived fram g by the base extension +
S_
( 4 . 2 ~ ) ) is a scheme. The proof i s thus reduced to the case Fn which
5
and. 5 of the form
is of the form S E A
Ma
(4.2b))
. If
M
is free of
f i n i t e rank, we are through by 4.2a). In the general case M is projective, i.e. a retract of sane A”
. Thus
& I = M
zc a
is a scheme a s a retract of
(4.2a))
.
Let
and _N be vector bundles over
e_On=Z(An l a
m
r
r
M
4.4
bundle of
N
surrmand of
_M
, if
(R,p)€M
raturally i n the study of grassmanians get a subbundle
T
of rank n
be setting T ( R , ~ ) = %,r
( 5 1, 3.4,
for all R$
Let 2 be any 2-functor. the vector bundle T, over - -
Propsition:
by the base extension f
-
, we
3.9 a d 3.13). W e c l e a r l y
of the t r i v i a l bundle of rank n+r over
pcx””
the so-called tautoloqical bundle over G
f:$%n,r
71,r
p
.
c g n , r ( ~ ). %is
Tautology!
S
derived fran the tautological bundle
get a bijection frcm
A morphism
direct sumnand of rank n of
is
By assjgning to each morphism
E(S ,Gnfr)
of subbundles of rank n of the t r i v i a l vector bundle of rank Proof:
a &-
the f i b r e M(R,p) of M is a d i r e c t -s as an R-module. Vector bundles and. s u b b d l e s arise
for any
N(R,p)
; we c a l l _M
onto the s e t n+r
over
g
.
2 assigns by definition t o any a€S(R) a Rn+r
!
Notice that the preceding proposition is often given another equivalent formulation: i f
bJ
is an 5-mzx3.de,
j u s t c a l l a section of
- Pbg -gg-
the structural projection p each
ulf
[R,p)€M
...,at
”S
an element
generzte M_
, if
(§ 1, 6.3). This section
a(R,p) € _M(R,p) ul(R,p)
any section u of
u assigns to
. W e s h a l l say t h a t the section
,...,ot(R,p)
generate the R-mdule
I,
5
QUASI-COHERENT MODULES
2, no 4
-M(R,P)
.
61
for each (R,p) Equivalently, this means that the induced mrphism n Qs + g is an ephrphism of Mcd
.
-3
Now consider the vector bundle T' of rank r over G defined by -n,r n+r TI(R,P)= R /T(R,~) The imges of the natural basis elements of R ~ + ~ give us n+r sections E ~ , . . . , E of T' For any mrphism f : S-G n+r - -n,r of , we denote by clS, E j ? : (set the induced sections of n+rS
.
.
...,
-s .
We this assign to EiS(R,p)=Ei(R,fp) E T'(R,fp)~~(R,P)for any (R,p)€M ) any f a vector bundle of rank r over 2 together with n+r generating sections. Conversely, a vector bundle
and n+r
of rank r over
.
generating sections u l l . . , u n+r of _M determine a mrphism _s-tc -n,r : assign to any (R,p)€M the R-module of all relations between
-5
~~(RiP)i**.i u , ( R , P )
4.5
€
!
!(Rip)
We still have to relate the functorial to the geometrical point
of view: first consider a geametric space X together with a sheaf of modules
A over Ox . we get a module
g.li: over the associated ;-functor _SX ( 5 1, + X the mcdule of sections of p* #)
.
3.5) by assigning to any p: Spec R
In other words we set
(snfl (R,P)= P* (4)(Spec R)
.
When X is the geometric realization of a scheme _X is so over X
over Spec R if
, p*@)
is quasicoherent
. In this case, it follows directly fram
1.6 that Sd is a quasicoherent Sx-module.
.
be a module over some &-functor _Y Assign to any open of U_ the d u l e _N(!) of all sections of _"V- (the y-rndule
Conversely, let subfunctor derived fram
by the base extension
v+x) . This obviously provides us
with a presheaf of modules Over 1 Y I . The associated sheaf of modules will be denoted by
IN_I
. Notice for instance that
lMal
=
rc/
M if Y_
=
Sp A -
and
M € S A (1.4 and 4.2b)). This implies in the general case that coherent if is a schane and g is quasicoherent. Proposition: For any scheme 5
, the functors d + Sd
quasi-inverse equivalences between the category X and the category modules over -
modules over
1x1 .
Proof: By construction we have
uld
m
a
is quasi-
M * 1M[ provide
of quasicoherent
of quasicoherent sheaves of
=ll!l [
for any sheaf of mdules
.
I, 6.2, no 4
ALGEBRAIC GEQMETFS
62
Conversely, let _M be a quasicoherent K-mdule.
there is, by 4.2b)
, exactly
one i s m r p h i m
.
If
X = Sp -. A
A: 5
5 [FIT
each a f f i n e open subscheme _V
of _X an ismrphisn
As
4.3, these
g and sI_M[ are local by
provide us with a canonical i s m r p h i s n f :X+Y
Let
sheaf of mrdules over
-SCf*&
and
-4:
. For
0Y
any
$: Spec R
and
.
(?[MI)
-5 may be matched together and _M 2 _S .
be a morphism of geanetric spaces
(R,p) = (p*f*$) (Spec RI
5 (fw)
inducing the identi-
(S \MI) ( A , I ~ ~ )I n the general case, we may thus assign to
t y on M(A,IdA)
4.6
is a f f h e
.+
X
and
fl
be any
we have by d e f i n i t i o n
( ~ y c l f s x ( R , ~ l = ( ( p f ) * h (Spec R)
(sx4 are )canonically i d & t l f i e d . -
. Thus
For direct images t h e s i t u a t i o n is more involved. L e t ~-4% be a sheaf of d u l e s over
Ux and l e t u: Spec
be a morphism with
S +Y
we have S ( f * d ) (s,u) = ( u * f * d ) (spec S)
, whereas
S e
. By d e f i n i t i o n
(s@ys~) (s,a)
may be
described as t h e set of maps assigning to each ccmmtative square
a section of
, which of
p*(&)
Thus the canonical maps
course has to be f u n c t o r i a l i n
(u*f,A) (Spec S)
-+
( p * d ) (Spec R)
(R,P,T)
.
provide us with
a canonical morphism
j : sf,Lkd
But this
j
is not an i s m r p h i s m even i f
X,Y are geanetric r e a l i z a t i o n s
of schemes and d% is quasicoherent. Indeed, consider t h e case where
x=\x(,
Y=\YI
and f = \ g \
. men the ahveinention&
ccmmtative square is
mapped i n t o the p u l l back square
(Spec 9; x I
where
(Spec S); X
-
Pr2
,x
I
i s t h e qeanetric r e a l i z a t i o n of the scheme (2&
S ) 2 _X
.
I, § 2, no 5 I 1
Therefore sections of
63
QUASI-COHEREWPM3D~
(_SX/_SyS d)(S,u) pr;
may obviously be i d e n t i f i e d w i t h the set of
(A)over
(spec s):
x , or
) (Spec S) ; mreover, j (S,u) (prl*pr$ u*f, dtis * prl*pr5
A.
The question whether
equivalently with
is induced by the canonical m r p h i s n
is an iscmorphism plays sane role i n algebraic
j(S,u)
geametry. A simple example where t h i s is so is given i n 1.8. A simple counter-example is obtained as follaws: consider the p i n t w of &A-fmc-
el
tor
(the projective line) assigned to t h e subspace F (1,O) E _P (F ) of 9 1 -P is the prime f i e l d of characteristic p > 0 L e t X be t h e F2 where F ^P "P open subspace of ( P I ccanpkmentary t o w a d l e t f: x * spec -Z and
.
u: Spec F f,
-1 Spec2 be the canonical morphisms.It can then be sham t h a t
"P (4)(Spec z)
as pr;
-+
= $(X)
(Jx) ( (Spec F ) x -P
X)
(u*f,Jx) (Spec F Ep , where"P i s F LT] , the r i n g of polynanialS i n one variable. "P
fl-PL (P-1) 2 2 ; hence
When & is a scheme ard
4.7
shall sanetimes simply write
g is a
quasicoherent Ijmadule, we
, thus
instead of
identifying quasi-
coherent &-modules with quasicoherent sheaves of modules over
151
.
Notice a l s o that t h e definitions and statments of t h i s section may be e a s i l y
. W e entrust t h i s t a s k
exteladed to the r e l a t i v e case of k-schemes, w i t h kc:
to the reader.
Section 5
Affine mrphisms
5.1
Definition:
affine --
---
2
2-functors is said t o b"-
_ Sp_R i f , f o r each model R and each morphism g:
-+
Y_ , t h e fibred
Sp R p X_ is an a f f i n e scheme.
prduct If
fi mrphism z:K-+xof
is a 2-functor, we say t h a t a 5-functor _X
structural projection px:zps
is a f f i n e over _S
if t h e
is affine.
-,i.,
5.2
Example: I f
_Y
is an a f f i n e scheme,
f
is a f f i n e i f f
5
is
an a f f i n e scheme. W e see that the cordition is necessary by taking g - to be and 2 = Sp C , the fibred product =Sp B an i s m r p h i s m ; conversely i f -
-
Sp R
*
X
Y -
may be identified with
S p ( R p ) , which
is an a f f i n e schme.
proposition: If _f:x-ty is affine ad. if Y_
5.3
5 scheme,
Proof: We show first that
& is local, making use of the notation of def. 3.11 of 5 1. Let Ci€ X(Rfi) be elements such that v((C,))=w((E,)) ; set
.
Since ni and n j have the same image in y(Rf ) and is i- 1 ifj local, there is a unique nEY(R) whose image in Y(R ) is ni for all fi iEI If p.:R-.Rf is the canonical map, (pi,Ci) belongs to the fibred
q =f(C.)
.
product
1
(Sp - R;
i
n# : sp R
(Rf,) defined by 1
+
II_
. Since
Sp -R
55
is
representable, hence local, and the (pi'si) satisfy the usual ccmpatibility conditions, there is a unique for each i
. Obviously
image in Z(Rfi) is
FId
ti
R
5 ) (R) which maps onto (pi,<,) is the unique element of X(R) whose
-
( p r < ) € (Sp R Q
and 5
for all i
.
5 is local. We obtain an open covering of ,X by affine schemes by considering the fibred products JS R $ 5 attached to the open
This shows that
anbeddings -g: Sp - R
-f
-Y
for R€$
.
Let _S be a 2-functor. An S-alqebra is a pair
5.4
(&#a) consisting
of an S-module A_ a d a mrphism of 5-functors a:@fi-+& such that, for any R€$ aid any p€_S(R), a (R,p) is the multiplication of an R-algebra structure (associative,cmtative, with unit) canpatible with the R-module structure
.
of _A(R,p) The 3-algebra ($,a) is said to be quasi-coherent if A is quasi-coherent (4.11, If (€3,O) is a second 2-algebra, a mrphism of ( e r a ) into (B, B)
is defined to be a mrphism of S-functors f:&+g
f(R,p) is an R-algebra hcsocanorphisn for each ?-model shall simply write instead of (A+) -
.
As a first example we have the 5-algebra generally, if
_X
Qs
(R,p)
such that
. Henceforth we
defined by Qs(R,p) = R
is an 2-functor, the Weil-restriction
. More
I Q~ clearly bears x/s 2 is affine
an S-algebra structure. This ?-algebra is quasi-coherent 2.fover
5 .
Propsition:
2
2 2-functor, the functor
_X
-4b
Q~
is an anti-
equivalence between the category of S_-functors,which are affine over _S
-and the category of psi-coherent 2-algebras.
,
proof: Our propsition is a direct consequence of the definitions. We can associate with each quasi-coherent S-algebra
8
an S-functor JS
5 which we
I,
5
2, no 5
65
(9AIR
This means that
is a f f i n e over _S
s_p A -
. Hence
is represented by the RilFodel A ( R , p )
. Moreover,
i f we set _X = s_p -A
, (Io 1(~,p) x/s -5
, i.e.
is
.
identified w i t h t h e algebra of functions of
(Sp A)R
This means that t h e c a n p s i t i o n
is i s m r p h i c to the i d e n t i t y
(?@-: n o , ) Sp 0
functor. Finally, w e also have S_po(? s - 3
% Id
%i& A_(R,p)
because
( a Q (R,p) x ) is
- - -
TO-' is t h e R-algebra of functions of the fibred product of
Being a f f i n e , t h i s fibred product is identified with
S
When the 2-functor
5.5
sp'xg(lz) . (R,p)
-
i s a scheme, we may interprete the pre-
dx
(px)*(dx) of
the d i r e c t image
9
2.4. Moreover, t h e &rph.i&
+ (<)
s
-
under
t h e s t r u c t u r e of a sheaf of algebras ov&
bras
Jk over 5
Corollary:
pX
-5
(dx) -
S_ (by definitTon a sheaf of alge-
an associative and
r i n g haramrphisms and are cartpatible w i t h
If s
by
(p ) *
assigns
w i t h unit; moreover, the r e s t r i c t i o n maps
$-(U)
+
5-
(V) )
.
a scheme, t h e functor g k ( p x ) * ( < ) is an antiequiva-
lence between the category of S-schemes, ---
which are i f f i n e over
category of quasi-coherent sheaves of algebras
Proof:
by
asociates with each open subset Ucl
catmutative algebra &(u)
are d(U) -+ &I)
_ P ~ : ~ X is + ~quasi-coherent
* (gX)ir&&d
. Then
5
ceding p r o p s i t i o n i n the f o l l m i n g way. Suppose & is a f f i n e over
aver
.
W e only have to show that the morphism j : S(pX)*($1
-+
5
r
and t h e
2 g x de-
fined i n 4.6 is an isamrphisn. I n f a c t , as we knaw already-that 0 i s quasi-coherent, it is s u f f i c i e n t t o canpare S (p (R,p) and _x/_s -3 -5 Qx) (R,p) when p:' Sp - R 2 is an open anbeaahg. I n t h i s case both
(8
algebras are identified with
-f
J(Sp - R
-
5) I
AIi3I3RAIc m
66
m
Y
I,
9
5
2,
According to 5 . 4 w e get an antiequivalence, which is quasi-inverse to c-;'(p -5 * (d
by assigning as follcws a ;-functor
x
to any quasi-coherent sheaf of algebras is the set of pairs
of algebras
The maps
(p,X)
(xi)
+
Let
z-'(xi)
+
RI is an arrow of ,M u is the morphisn
, =en
Yi
.
for each i
: f l
Proof: With the notations of 5.1, -
S eRY X x
+
,
f
Qd+
.
induces an affine m r -
are affine over _Y
Sp R
.
it is sufficient t o show that the canoni-
is affine, assuming that the canonical pro-
are affine. W e m y thus suppose straight away that
= Sp R
(xi)
, hence
that
by a finer covering i f necessary, we m y assume -1 i s an affine open subschane of Then f is
is a scheme. Replacing
further that each
:
f is affine.
In particular, vector M l e s over a scheme
cal projection
p
, (spd) - ($1
2 mrphism of local 2-functors
f:&Y
an open COvWiny of
phisn ~ i :
and a m r p h i m of sheaves
p€_S(R)
define the "canonical projection''
p
Corollary:
5.6
; i f 4:R
R)
($(p),~(~)oX, ) where
++
over
consisting of a
(p,X)
X : A-p!(Js onto
(p,X)
mps
k
spd , c a l l 4 spectrum, _S : If RQ , (%& (R)
xi
(xi)
.
affine by 5.3. Being local and being covered by affine open subfunctors,
X
is a scheme.
The d i r e c t image
&= f , (8.)
each
since this is true for
the canonical isanorphim isamrphism _X
+
Sp + d
/xi . AS
g-l(yi)
+
Let
-
5 f
Z_
,
and ~ - p dare h ~ t hlocal functors, s_P d ( f - l ( y i ) ) = Sp - &Yi) induce an _X
5
(cf. the arquwnt of
Corollary:
5.7
is a quasi-coherent sheaf of algebras over Y
Y_
1, 4 . 1 6 ) .
-be-a diagram of
schemes such that
q- i s faithfully f l a t and quasicompact. I f the canonical projection f,:
-
y xz Y-
Proof:
+
is affine, so is f,
By restriction to an affine open subschane of
t o the case i n which
(xi)
.
5
z , w e reduce directly
i s affine. W e must now show that
is an affine open werirg of
, we may replace q-
5 is affine. I f by the c a n p s i t i o n
I,
5
2, no 6
67
-Y‘+Ycan where
y‘
g
c
z
yi
i s the d i s j o i n t sum of the
Suppose therefore that
. Since
and. C = &(?$:)
_Z,y and 3
;
. are a f f i n e and. set k=a(_Z) , B=@)
q- is then a f f i n e and surjective, the same holds f o r
sx: & xz Y- x
t h e canonical projection
-+
.
; therefore, since
so is _X Hence &ere is an a f f i n e open covering -1 (Xi) is a f f i n e -5
ccanpact,
5 5- Y_ (Xi)
is quasi-
of
5
;
since q
is a f f i n e , hence quasiccsrrpact. I t follows t h a t zinXj
is quasiccmpct and
lemna 1.8 implies that there is a canonical i s m r p h i s m
5
which allow us to identify qX ‘I with the mrphism - -
S(XrY)
z-
.
B u t this latter is invertible; so therefore is
Section 6
Closed embeddings
6.1
Definition: -
a)
Let
f:z+x
that f a closed & d i n g if _f each mrphism g: - s_P A _Y , t h e map -
-X
(3.5)
.
5 mrphism o f ~ - f u n c t o r s . We say
i s a f f i n e ard i f , f o r each AfM, @
+
’(:Sp
is s u r j e c t i v e b)
:f
.
-
A) : ~ (-S A) P
X_ @ 5 subfunctor
inclusion mrphism
&a
of
-+
Y -
J(S_PA p s)
, we
say that
closed Embedding.
5
is closed i n _Y
i f the
If
is a scheme ard
_Y
5.3. Accordingly w e call _X a closed subscheme of
, g-
Suppose that i n statement a)
_Y
.
is a scheme by
.
is of t h e form g=fo_h Then t h e mrphism _
L
X w i t h cunpnents Id and _h is a section of _f Sp A x SPA o ( s ) d ( Z )=Id , so that d(fQ A) is b i j e c t i v e , whence s=_f-l-'
s : Sp A
Hence
and
7 ,
is a closed subfunctor of
X_
-f
Sp -A
h = yxg$
SpA A
. This shows t h a t
that a closed gnbeddinq
_h is uniquely determined by
25 m m r p h i s n .
-
-g
'
S_pA
, so
.
Now suppose we are given a monanorphism of 2-functors _f:x+g By arguing as i n 5 1, 3 . 6 , we-see that f is a closed embedding i f f for each A€$ and each a % ( A ) there is an ideal I of A satisfying the f o l l m i n g condition: for each arrow $:A-+R of & , aREx(R) belongs t o the i m g e of X(R) in Y(R) iff -
$(I)
6.2
= 0
.
Fxample:
If, in
i s an a f f i n e scheme, then a necessary
6.1, _Y
and s u f f i c i e n t condition f o r f to be a closed mhdding is that X_ be an a f f i n e schene and d((f) be surjective. The necessity of the condition is v e r i f i e d by taking g: Sp - A + Y- t o be an i s m r p h i s m ; the converse is obvious If
.
I
is an ideal of
-
image y(1) of
Sp $
and $:A+A/I
A%
is the canonicalmp, the functor-
, we
is such that, for each R@J
I +(I)=OI . the map I wv(I) 2 2 b i j e c t i o n of closed subschgnes of Sp -A .
have
Y(I) (R) = I$%(A,R)
I t follaws that
onto t h e set of ----6.3 F (r,s) -n
simply _F
F (A) .!hen _F 2
in
system of
,
= G
= { (P,Q) %r,n-r
is closed i n _Y
. For
(A)xGs,n-s
I
(A)
FQ}
l e t a=(P,Q) and let Q'
be a cmplanent f o r
...,pn. be the projections onto Q' along Q of a ) .With generators of P . Suppose that pi = (ali,azi,...,a (n+r)i ; let
pl,
the notation of 6.1, the b n d i t i o n R @ P C f o r each
A
-r,n-rx %,n-s , r 5 s ( 5 1, 3 . 4 ) . write for the subfunctor of Y_ satisfying
Example: S e t
, or
the set of ideals of
(i,j)
A
. I t thus suffices t o set
R@Q A I =
1
i,j
is equivalent to
Aaji
(6.1).
$(y.)=O
I
9
I,
2 , no 6
69
QUASI-COHE~-~DWS
It is obvious that the canposition of
tm closed gnbedaings is f h a closed gnbedding. Similarly, i f in the diagram g--jut=-Y' of g f + y' is l i k e is a closed & d i n g , then the canonical projection -fyr: 6.4
ci:Xi+zi
w i s e . Finally, given a projective system of closed &dim$
,
the projective l i m i t
is a closed embedding. W e prove the l a s t assertion:
E 1 2 Y(A) , l e t aiEYi(A)
Q
6.1, there is an ideal Q
iR
be the projection of
of
Ii
1-1
1 8 X(R) i f f
for a l l
$(Ii)= 0
6.1 it thus suffices to set
6.5
Example:
(P1r
such that P1 c P2 c
'Gls,n-rs
Gr1,n-rt
C
Ps
. Hence
relation
aR belongs to the image of
. In order t o s a t i s f y the c r i t e r i o n of
...,rs) S
(R)i
w i t h coefficients i n R g is a
aXGr
s
,n-r
. The subfunctor
s
(R)
gn(rl,.. .,rS)
of
f o d by these flags is called the schene of f l a g s
. I t i s a closed subschew of
(n; rl,...rs)
of nationality
i
(n; rl,
E Grln-.rl
...
, the
. By
lsr,s.. .sr sn be an increasing sequence of inte-
Let
- - - rPs)
a wj.th irdex i
I = & . i
gers. A f l a s of nationalitv
sequence
and
such that, for each @:A+R
A
i s equivalent to $ (Ii) = 0
€_f. ( X . )
if A 2
zij
.
For l e t be the subfunctor of this l a s t scheme * * GrsJn-r, formed by sequences (Pl,...,Ps) such that Pic P ( i i j ) Then Eij is j the inverse h g e of Xn(ri,r.) under the canonical projection of
G q , n-r<
.
7
mrphism of
1x1
Proof:
$:K+L
of
f:z+x is a closed -ding,
Propsition:
6.6
Let
Y(K)
<(L)
iff
onto a closed subset of
1x1 .
be an extension of small f i e l d s and let
It follows easily fran 6.1 that pLEx(L) p
fs is a h m -
belongs to the image of
i s m r p h i s m of
XIK
nents of
(9
Z(K)
p
be an element
belongs t o the image of
. This implies that
glz
is an
onto the union of a collection of i r d w s a b l e ccmpo-
1, 4.5)
, so
that
f
is injective
(5
1, 4 . 9 ) . TO ccmplete
close3 in
2 , no 6
5
the proof, we may assume that
is the inclusion morphim.
5
I,
AtGEBRAIc GlxNmRy
70
i s a closed subschene of Y and tt-iat f W e show t h a t each closed subset P of is
1x1 . By the definition of
yl(P) is a closed subset of 1% phisn g: Sp g-l(P) - A -Y . Now -
(5 1, 4.11,
we must show that for each A% and each mr-
A1 = Spec A
1511 , which
is closed i n g-’(
-+
1x1
is easily
~-‘(XI (use the f a c t that X I ard ~ + -1 g (5)12 are both sums of collections of irdecmpsable ccanponents of YIE respectively). Since Z-l(x) was ass& to be of the form sp seen to be the set of points of
and
1x1)
, -g-’(
s_P(A/I)
= ~Q--~(_X)
I
, and
= Spec(A/I) i s closed i n Spec A
the
p r o p s i t i o n follaws.
~rollary:If - -X is aclosed subfunctorof Y_ tor of
X_
, there
y
is an open subfunctor
and
x
g
and
.
:1 1
of
P
:
-
c = &nv_,
y such that
cf
Prcof By the p r o p s i t i o n there is an open subset lLJl = P l i I x-I . If we set y=y (9 1, 4.12) , Vnx_ P p i n t s . Thus U = V- n X-
g is anopen subfuncsuch t h a t
have the same
I n general the map * 1x1 of the set of closed subfunctors of 2 into the set of closed subsets of is neither injective nor sur6.7
jective. For i f
I
I
and J are ha ideals of
have the same uxderlying space i f f being identical w i t h J
.
fi
=
A€&
,
and _V(J) (6.2)
!(I)
fi , which can ccw without I
This shows that _X
I+
is not necessaxily in-
j_X/
jective. On the other hand, l e t T be the gecanetric space whose underlying
set consists of two points 0, 1 , whose closed sets a r e @ , {Ol , { O , l l , arod such that the restriction d,({O,lI) -+ dT({lj) is the identity IMP of the prime-field with
p elements. I f
R%
, (ST) - (R)
is mpty i f
p
# 0 in
R and m y be identified w i t h the set of closed subsets of
Spec R i f p=O n R It can be sham that g@ and _Sr are the only closed subfunctors 3f ST ; it follows that the mp 3 is not necessarily surjective.
.
1x1
-f:X+Y --
be a closed (9 I, 4.1). AS a quotient-ring of A Let
6.8
gnbedding %
& let
4sp - A) , d’(9 A + 5)
is clearly quasi-coherent
-X
++
-ivy, 0
-
-
be a Y -d e l bears a natural
d ( 3 A $ 5)
. W e say that 0-x/y_ is_X/au quasi-coheremt it follows inmediately f m the definitions that t5e map
A-algebra structure. Wmover, the induced ~ - d u l e 0 Y-alqebra. In fact.
(A,p)
(Alp)
H
(4:l)
is a bijection between the closed subfunctors
X of Y - ard the
quasi-coherent quotient-x-algebras
. For s c h m s
0
of
be reformulated as follows i n tenns of
Y/Y sheaves of
Suppose that Y_ is a scheme and that
g:z-+X
is affine ard open in
this statement may
modules:
is a closed gnbedd%.
If
g:y-+y denotes the inclusionmrphism, defi-
and i f
nition 6.1 a ) ensures t h a t [g[g(y): 9 (V) 9 (f-l(Y)) is surjective. Hence yf I f / - : dy -+ _ f , ( f l ) is an epimorphism of sheaves. Set y = Kerlfl-f By 1.11
z-
-f
and 5.5,-
7
dy/ 3 , that
,F
.
is quasicoherent. By prop. 6.6, the support Supp(dy/g) of is the set of
of f e . Hence
&ge (F
x
,( ~ $ / f iIF) , where (dy/y)I_F
, and
= su&(JY/T)
-
Coy/?)Y # 0
yfY - such that
121
fe i s an isomorphism of
denotes the restriction of
f e - i s the map N e r l y i n g
.
f
7 be a quasicoherent ideal of oy-
Conversely, l e t
. For each
u",)
Y-suhcdule of
-
derived fran 0
+.
7 01!
-+
-+
, coincides
-g: s_P
fly/?+
A
-+
Y
with the
onto the geawtric space
, the
dy/j'
to
(i.e. a quasicoherent
sequence
0 is exact. Hence the canonical map
(% A) is surjective (1.12). Since - y (sp - A) into B = g*(dy/Y) s_P A G s(oy/J) may be identified with s_P B by lenna 6.9 belcw, it follows is a closed embedding. that the G n i c a l projection p : %(Jy/y) +.
of
A 7 g*(J
The image-functor of subscheme of
YY be
w-i l
Y defined-by
I!(?)
identify
-p&/7
[
with
(F
7 . With
written
and called the closed
the above notation, by 5.5, we may
,dY/7) IF) , where -
F = Supp(dy/7)
-
.
Fran these r e s u l t s , we deduce in particular the Proposition: Let - -Y
be a scheme.
Then
:
a)
The nap
7
I-+
v(T)
is a bi-
jection of the set of quasicoherent_Y-ideals onto the set of closed subschgnes
of 1 ; b)
A morphism of schemes
(4)
l _ f l f : Jy -+ _f* F c e of points of Proof:
_e_
g:z-+y is a closed
gnbedding, i f f
is a sheaf epimorphism and
-x
onto a closed subset of
'z
is a hcaneamrphim of the
1x1 .
. The condition is necessaryf by 6.6 and 6.8. the condition holds, the kernel of lsl- is a quasicohe-
It remains to prove b)
Conversely, i f
rent ideal (2.4 and 1.11). Setting F = Supp(
, it
131 onto the geQnetric spce
follows t h a t f (F ,(gy/Y)IF)
.
in-
NmBFAIc ciEmmRY
72
I,
5
2, no 6
f:gt+S is a mrphign of schanes, &3 p s i - c o h e r e n t sheaf of algebras over _S , we have a canonical isararpkim 6.9
-S' 3 Lf
b)
-
If
a)
Lama:
s&:
sp
g*c4
.
, we
,
Sp have a canonical isanorphism u -: -Sp B denotes-& sheaf over Spec R associated w i t h B (1.4)
RfM,
where
B%
, (8' xs Spd)(R)
a) For any RfM,
Proof:
, where
.
is the set of pairs
,-and where A phisn of sheaves of algebras over S If A' (p , ( f ( p ) , A ) )
p€S'(R)
.
: A +f , p f ( dSP R)
is a mr-
is associated with X
by the
bijection
of 1 . 2 , our canonical isomrphisn sends
E (9f * & ) (R)
(P,X')
Let
b)
i:R+B
@E(S_p B) (M)
.
onto
i
,
f
(3%) (M)
(+i,$ E
-
=
.
Sp i
-
Let
Corollary:
for each
,(f_(p) ,A)
onto
)
be the canonical isanorphism. For any MfM,
u satisfies p -Sp - B6.10
(p
f:px
,
serds
u(M)
( 1 . 6 ) . The i m r p h i s m u therefore
be a mrphisn of local Z--functors.
induces a closed gnbeddinq fi:
c-'(yi)
*
If,
!$ , then
3
f
a closed fmbedding.
Proof:
By 5.6 we need only consider the case i n which Y_ is a scheme. Then
i s a f f i n e by 5.6, and it r m i n s to s h o ~that 1-f: J * f, ( 3 . ~is an epimorphism of sheaves. But this follows from the f a c t that the &strictions
x
f
If[-f
of
to the open sets
6.11
. :1 1I
are epimorphisms
_Y
5 in
. If
~
, we
:
is~ a mrphisn + ~ of
Z_
Let
P
be a subset of
and
-
such that
Z K . Let
Y_
. With each f i e l d
# I p I maps Spec K into P
the inter-
, we
K%
z
is
and called the closure
J _f is the closed i m g e of also say t h a t I several examples of closed images: f
. By 6 . 4 ,
_Y
containing a given subfunctor
again closed. This intersection w i l l be written
of
(6.8).
W e return now to an a r b i t r a r y ;-functor
section of the closed subfunctors of
-
z
i s the image-functor f
. W e now consider
ard each element
associate a cow X(K,p)
pEx(K)
of
,
73
x = #X
(Kr 0)
and let _f:S-ty be the mrphism whose canponent of irdex p is p:'
g(~,p)
-+
. write pred
y
xred
-
/ ? Ired
for
.
(!?I
closed subfunctor _F of Y such that P we write simp1y
. W e say that
Pred
is reduced ard has the f o l l m i n g property: i f
i s reduced i f
. For each subset
Xred=
_t(T)C P
g:?:
6.12
is a ,punctor and P a subset of
If
l y contains the closure
-
P #
Ipredl
JY-
-~ ( 7=)prd 2 1111 Proof:
If
viously
P
Y(u)
1x1 . m:
of
hence
i
on P CI
u
. Similarly
. We now prove
J(Uf)
1x1 , IpredI
*
obvious-
it may h a p p n that
1x1
Y
t h e sheaf of
is quasicoherent ; b)
Fred
is the closure
P
~f
P
"$i"
c)
follows frcm a)
1x1
and each f€d(U)
is injective. Conversely, i f since g
x=g/fn E
1x1
nilpotent e l m n t s of
n uf
is annihilated on P
, the
canonical map
7 ( Uf ) ,
and f
gf
on
A necessary
i n 6.12.
B
Y
.
u - uf
if
. The i .
and s u f f i c i e n t c o d i t i o n f o r a scheme
fly'Y EY )
.
-I
be reduced (i.e. do not con-
It then s u f f i c e s to show that
NOW
;
is annihilated
implies that x belongs to the image of
Corollary:
Set P =
since we obviously have
a) : by 1.9, it is enough t o show that, f o r
Y_ to be reduced is that the local rings tain any n i l p t e n t elements apart fran 0 Proof:
'red
is bijective. Now we have y ( U ) fC u(U)f ard @ ( U ) 1 @(Uf)
equation g/fn = gf/fn+l 6.13
-
is a scheme:
a)
the space of p i n t s of
each affiGe open subset U of -+
L
,
is a quasicoherent sheaf of ideals contained i n 3 , then obSupp(dy/;J) , and so Fred (6.8). Assertion b) t h u s
P = Supp(dy/y) i: Y(U)
if
Y_
= I S E & ~ ( U ) J S ( ~ ) = O vXcunpi ,
2
follows f r a n a)
-
. By 6.7,
Iy[
of
P
such t h a t
; c)
C
factors through
L_e t_ Y _ be a scheme, P a subset of
f o r each open subset U
-
in
P
. This cannot occur, however,
Proposition: ideals of
of
part
is a reducedz-func-
T
tor, each mophism
such that
; it is the &lest
qedis the reduced
and say that
of
-
2
f o r the closed image of
'I! is a f f i n e and open
7Y
consists of
contains
y
ALGEBRAIC (3ImEmY
74
and i f
is annihilated a t each point o f
f6!Q)
every prime ideal of
of f a t y then (f IV)
fjv
,
Yl
fYE
, then
= 0
therefore nilpotent. I f
Y
t Y
Y
Y k-schemes. Then 5 x -
.
9
f
o
two reduced
_Y
zg is reduced.
& is called reduced i f
v
(zz)S x~ k (,Y) em
-z (-x x y )
c
5 I, 5.7, t h e local rings of , ( & x Y ) are the rings dxgkJy , X Q , y e . sGce dx and By
r i n g s of fractions of the
are reduced, they are
ox
contained i n products of f i e l d s (the local rings of
and
Y
prime i d e a l s ) . By Alg. VIII, 5 7 , no. 3, th. 1, it follows that
<%$
reduced, so that each ring of fractions of
is reduced.
at minimal
is
#
L?@
XkY
W e tlow turn our a t t e n t i o n to the clos& image o f a morphism of
schanes
f:z+x . Since the functor
of 1.4
M&
c m t e s with inductive
limits, the sum of a family of quasicoherent sheaves of i d e a l s of again p s i c o h e r e n t . Each sheaf of ideals of quasicoherent sheaf of ideals.
s'
$'
. Hence
y
1, 6.3, there is an isanorphism
6.15
no 6
is reduced.
I n accordance w i t h § 1, 6.3,
Proof:
2,
i s the germ Y is nilpotent. conversely, if =
implies that f
f
5
is contained in
f
Corollary: Le t k be a perfect f i e l d , _X
6.14
By
, and i s
U_
Y f o r a s u i t a b l e open neighburhccd V of
, so t h a t
EJ(v)
d(y)
I,
dy
therefore -
I t follows that Im f
=
dY
is
contains a l a r g e s t
v(%
(6.8)
, where
is the l a r g e s t quasicoherent ideal contained i n the kernel of the mrphisn
ox
l_flf:
+
induced by f
f*(Jx) -
.
This may be simplified when f_ is q u a s i c a n p c t and quasiseparated. For by 2.4 f*(~!$)
i s then quasicoherent,
f
Proof: g (v') -
that
f
r =K e ~ l f lf- .
be a diagram of schanE, where g is f l a t -1 i s quasicawact and cpasiseparated. Then w e have Im -fyl = s_ (Im $1
Proposition:
g@
y'+y.-X
SO
k t
-
'y
Let
I!
. Let
I!' he a f f i n e open subschemes of -Y and u' such that -U = -f -1 (v) and V_' = git(v') . By 1.8, the canonical map
ard
dcy)
Q0(")Q)
-
i s bijective. Ry varying
-f
J(U_',
-
I! and V_' , we derive a canonical i s m r p h i s n
.
I,
5
2, no 6
75
QUASI-COHERENT-IVK)DUL!ES
(see 1.6). If we set 4 =
Is[-f
-
and 4 '
=
allows us to identify
9*(4)
: f*(LJy)
-
with
Hence K e r 4 '
Ker g*($)
+
- -
1%
Y
I-,f
t h i s canonical i m o r p h i s n
g*(f*(dx))
-
-g*(Ker 4) , since the functor g- is exact. So
I n the following, we require a statement analogous t o prop. 6.15 in the case where f is not quasicanpact arid quasiseparatd. By way of canpensation assume provisionally t h a t we have U_ = Sp - B , B E g If 6.16
KerIfI
I & =
.
, it is clear that the largest quasicoherent sheaf of ideals 7 JV s a t i s f i e s J = &Y)" . Therefore = ~ ( 9=)~(~n/cy). be an affine open covering of X and l e t gcl:zcl+xbe the mrphisn f . If we set % = ~ e r l gf~ ,/ -then X= n % , YO = . c1
contained i n (X )
Let
-a i d u c d by
Now i f
is a ring hcanamorphisn which makes B '
p:B+B'
a projective Bincdule,
then
If
-V($
y
= sp - B'
B' UBJc1
and g =
(Y) )
s_p
-1 (anf) = ~ ( B w ~ / ( Y ) ) =
B , it follows t h a t -g
is the snallest closed subschane of
cyl -
But t h i s smallest closed subscheme is precisely Im
case of the following Proposition:
Et
u'
containing each
. This is a particular
be a quasicchnpact guasiseprated schme and let
_g:Y'+x -
be an affine morphism of schemes. If there is a f i n i t e a f f i n e open covering
(xi) of -
-1
(Y!))
such that (g is a projective J(Y.)incdule for all i -1 -1 then I m f y l = s_ ( I m f) for each morphism of schemes f:z+X ___
.
The proof of t h i s r e s u l t i n the general case i s sketched i n 6.17. W e shall use t h i s proposition in the follcwing form: L e t k - d e l which is projective over
k
k be a madel a d K
a
as a module. Then for each quasicanpact
,
I,
8 2 , no
6
, we
have
d' be the kernel of l e t Jqcbe the largest quasicoherent ideal contain& i n $ ard let
If- \
quasiseparated k-scheme (Imi?qK =
q.
f:_X-tY_
To sketch a proof of prop. 6.16 , let
6.17
qd
ard each m r p h i m of k-schenes
_Y
-+d be the inclusion wrphisn.
:JqC
Ebidently
(flc ,qW)
f
- ,
enjoys the
following universal property: for each quasicoherent sheaf of modules &6
wer Y- ard each mrphism @:l/td+Xthere is a unique $:k'+kqC such that qJ II, = @ For since CUC fly , Im @ is quasicoherent, hence is contained i n d'qc W e show f i r s t that if is quasiccmpact ard quasiseparated, this
.
.
universal problem has a solution whenever dzr is a Y-module satisfying the follming condition: clusion functor of satisfying
-1x1 (*)
i n Y , ck/(V) is mll (the ini n t o the category of sheaves of roodules over -
for each open V
nvsd
:
has a r i g h t adjoint).
(*)
Assume f i r s t that X i s of the form y*( 1, y:v+x k i n g the inclusion m r -
phisn of an affine open subschene a d q x : yqc
and write
9
a y-madule.
W e set
-+x for the unique mrphisn such that (1.4).
It is then easy to show that
(vx
qb(Y)$g(v) = Idde(v) is a solution of our universal problen.
is a x-module satisfying
In the general case, when
(xi)
(uijl))
x
1
= y:(Xlxi)
. = (resp. &111
1 71
a canonical exact sequence of Y d u l e s J7
(*)
,v* (q,) 1
above,
consider
of Y (resp. of YifiYj) Yi (resp. yijl) intn v-*i J 1 ( ~ l .~ ). In t h i s way we obtain
a finite affine open covering (resp. If yi (resp. yijl) is the inclusion mrphism of
2 , set
p= 8VI4
.
u
" t T1T 1u vw3 T i j lXi j l &
km
OUT
previous remarks,
and
are solutions of our universal problem relative to
m y therefore define mrphisms v,w : !?
and qw
=
w$p
. It thus remains
to set
Q
74
and
by the coriditions
NqC = Ker(v,w)
z"fj1.
W e
qv = v4p
and to define
.
I,
9
7
2,
77
QUASI-COHEREIW-M3DT.LES
q& : dqC +$
by t h e condition
inclusion mrphisn. The pair
, where
= pu
udq4
u
:PC +F is the
is the required solution of our
(dqC,qd)
universal problem. Returning n m t o the proof of 6.16 in the general case, it only remains to show t h a t
. To prove t h i s
&a is a family of quasicoherent sheaves of ideals of we need merely verify that the construction of e when
( ~ 4 ) " e l'cmtes"
-
w i t h the change of base functor -g:Y'+Y -
Section 7
Ehlbeddings
& anbedding i s a canposition of arrows gg - -f , where _ _ _ f- is a closed embedding an3 9 i s an open anbedding. Definition:
7.1
If
is aL-functor and _X a subfunctor of i f the inclusion mrphisn of
closed i n
X_
Y
, we'say
that
5
is locally
is an &ding.
into
If
-Y is a scheme, a locally closed subfunctor of Y_ is called a subscherne. 6.1 and
5
1, 3.11, each subscheme f o r a schane is i t s e l f a scheme.
Consider, for example, a scheme
(i.e. of
By
1x1) . Set
(5 1, 3.11
U =
1x1 -
and a locally closed subset P of (P-P)
. Then xu
is an open subscheme of
and 4.12)': When no confusion i s possible, we write Pred
intersection of the family of closed subfunctors of contains P
. W e may characterize
Y whose space of p i n t s is
P
.
Pred
xu
_U
f o r the
whose space of points
as the unique reduced subscheme of
The following assertions are imnediate consequences of the properties of open
and. closed embeddings: an anbedding is a m n m r p h i s n ;
the composition of t!m
embeddings is an embeading (cor. 6 . 6 ) ; f o r each diagram _X f , Y
3 x' , where
-f i s an embdding, the canonical projection f,' -
: XXY'
Y-
-+
y'
i s a l s o an embedding. 7.2
Proposition:
Let
the inclusion mrphism f of
be a scheme and & a subscheme such that
X
into Y is quasicanpact. Then
X
is open
78
i n t h e closure
2 & g (6.11).
f
f:g+y is a mnanorphiSn, separated
ard s u p p s e that
be open i n Y -
Proof: L e t
-*
(2.2). By 6.15,
i.e. w i t h x ; but
Xm-
g
5
is closed i n U-
. Since
X *Y X-
is invertible; hence f is quasicoincides w i t h t h e closure of _xnV_ i n , +
is open i n 5
Proposition:
7.3
:
%/Y
7
I, § 2,
ALGEBRAIC GECMETRY
. This m p l e t e s the proof.
f Consider the diagram of schemes & =+
Y_'
g - -is f a i t h f u l l y f l a t a d quasicanpact. I f the canonical projection
Y'
where
flr
:
is an open (resp. closed, resp. quasicanpact) gobedding, then -- f is-an open (resp. closed, resp. q u a s i c m p c t ) anbedding.
-X xY Y-'
Proof:
+.
If
-fyl
is an open embedding, it follows f r a n the equality
g -- l ( g ~ ~ )=) ~ G ~ ( X X Y '(8) 1, 5.4)
xthat f ( X )
-
subset of
subscheme of induced by f
such that
, then
Now suppose that
is an open subset of
IX_'
I
=f
x-
If
(3.4).
is the open
_XI
:x+z' is the mrphism hence so is f ' (3.5) .
(5) , ard i f f'
is invertible;
x
f'r
-fu,
the f a c t that Z ~ , ( X * Y Y 'is) an open
is a quasicanpact (resp. closed) embedaing. Then f
i s q u a s i c m p c t &-quasiseparated
Imf
since f y l has these properties ( c f . the
argument of 5 . 7 ) . I f 5'' = , and i f f'l:X+:''is induced by f , then -1 -1 2 (5") = Im gyl (6.15). It follows that -f";t,,~(5") is an open Embedding -
(resp. an iscsnorphisn). H e n c e
Definition: Amrphism
7.4
i f the diagonal mrphism 6x/y: X
f" is an open-embedding (resp. an i s m r p h i s n )
5
+
fi:x+x of is said t o be separated xpx is a closed enbeaaing.A2-functor
is said to be separated i f - t h e unique mrphism px:
-
X 9 2 +
is separated.
-
Referring back to 6.1, and recalling that a mrphisn g=(u,v) - - : Sp A + 5 Y* -X is determined by g ard v - , we see that f is separated i f f f o r each A%
and each pair of arrows (y,y) of Sp - A i n _X such t h a t f;=fy , Ker(s,_v) is closed i n SJ A Frm this and the definition we h e d i a t e l y infer the
.
following assertions: each monmrphism is separated; i f , i n the diagram
-x
1~ 9 y' of
gyl: - -X *Y Y'-
-+
,f
Y_' ; i f
is separated, then so i s the canonical projection
-g o -f
-
is separated, so is f ; a product of separated
mrphisms is separated. 7.5
Proposition: An a f f i n e mrphism
f:z+x
i s separated.
.
.
Let g,y: Sp A _X be a double arrow satisfying &=By definition, (Sy, A) k g is affine, so that there is a cartesian square of the
Prcwf: form
and a double arrow a,B: B Z A such that y'(3a )
=
_u
, y'(sp B)
= y
d
A ' Since we have
a@= @@ = Id
Ker(g,y)
is represented by
,
the elements a(b)-B(b)
A/I,
bEB
It f o l l m i n particular that
By setting
= cp
2 , we
is the ideal of
where I
. Hence
a)
f:x+x
Let
and and
a double
be mrphisms of f
a z ~ o wof
y: ?+
be the subfunctor of
-1
Then K E X ( ~ , ~ = ) w_
(A)
X_* X
1-
z$X
I$J
such that
A (R) =
For the f i r s t assertion, observe t h a t
:
y+ g;y
X,_Y,_Z are s c h e s
c (x,x)IxE X (R)1
ad. l e t for a l l R$
is locally closed i n
S
.
is the c a n p s i t i o n
x- (gf)Y .
and f Now & is derived fran by the change of base g;Id : ; since 6
where _h hascanpnents
6Y/z_
If
x
(by the lema below) ; hence Ker(g,y)
-z -Y
.
gf is a closed anbeddins
. If g is separated, A is closed i n X x-x , so that S . If X_ and are schemes, A is locally closed
in
h
.
a
such that gg = -fy
be the mrphism with ccanponents g,y
is closed in
3-x.
2 are affine.
is an gnbeddinq.
Ker(u,v_)
b)
& 5
. If
is a closed embeaaing.
is an enbeddinq, then f a) L e t
Prcwf: A
f:z-+x & g:x+z
q_ is separated, gf
.
be a separated mrphism (resp.
Then Ker(g,y) is closed (resp. locally closed)
g t
generated by
is closed i n Sp A
Ker(u,y)
f:g+x is separated i f g
mrphism of schemes) @ g,y:&
b)
A
infer that an affine scheme is separated.
Propmition:
7.6
%(Coker(a,&)) ,
a, 9 0)
Ker(g,y) = K e r @
I%
a closec3 embedding, it follows that
,Xix+Xz_Y
Y/_z
is
f I s the canposition of the two closed
.
80
I,
Fu;EBRAIc GEmEmY
embeddings Lennna:
h
ard
(sf), . The proof -
For each mrphisn of schemes
U_
Let
g a d
V
, the diagonal
?:?+:
irduced by
5
6x/y:
_ _
vary through the affine open subschemes of
f&J)'v . Then
satisfying
2 , no 8
of the second assertion is similar.
is an embedding.
Proof:
9
6i;y(i]$U)
=
I! and the morphism
U_
X x X
-t
- -Y -
X_
and
+
V$V_ -
coincides with-the nkphim induced by the map aQDb +a.b
.
into o ( U ) Since this map is surjective, (6.10) implies of d ( U ) @d(viG'(U) that 6 induces a closed esnbedding of X into the open subscheme of
$3
X 2 x which is covered by the Corollary:
7.7
_Vcg . -
The canpsition of two separated mrphisms is s e p -
rated. Proof:
Suppose
f:z+x
are separated. Let A be aRKdel an3
and g:Y+
X_ a double arrw such that gfu - _ - = gfv ___
9A
. Then
Ker(fg,_fir) is closed i n Sp - A since g- is separated. Since f is separated, it follows that Ker(y,v) is closed i n Ker(_fu,fy) Hence Ker(y,v) is closed i n (g,y) :
.
Example:
7.8
The projective s p c e
the flag schfsne FJrll..
.'rS)
En , the
q r a s m i a n Gnrr
&
are separated schanes. For the "diagonal" of
coincides with _F (r,r) , hence is closed by 6.3. Since a n product of separated functors is separated, G GrS,n-rs is s e p - r i ,n-rl rated. By 7 . 7 and 7 . 4 , a subfunctor of a separated functor is separated. The Gr n-rx Gr ,n-r
'..*
assertion therefore follows by 6.5.
Section 8
An affineness criterion for schemes
8.1
Affinity theoran:
Let - X
quasicoherent sheaf of ideals of defined by Proof:
4,
i s affine, &en so is
F i r s t recall that
structure sheaf i s
be a scheme and l e t
. If 5
.
7
be a nilpotent
the closed subscheme v ( 2
of
X
v(7) has the same space of p i n t s as ,X and its
Jx/g . Since
is enough t o show by -auction on
van)= _X n that
for sufficiently large n
v('Yn)
is affine. N c r ~ _ V ( f )
, it
qT(!fn/r"fl)
coincides with the closed subscheme
has vanishing square, we m y assume straight-away and suppose provisionally t h a t the map phism -f:Y+X - -
_V(yn+') that 7 2=O
of
. Since Yn/,f+' . Set
y=v(!f) ,
J(f) irduced by t h e inclusion mr-
is surjective. Since K e r f l ( f ) =
q ( X ) has vanishing
square,
Spec o ( f ) induces an i s m r p h i m of the underlying t o p l o g i c a l spaces of
spec
J c ~ )&
If1
+lYl
spec
3 ( ~.)NOW
and Spec &-f )
i n the carmutative square
are a l l hmecmorphisms,
(5 1,
2.2)
therefore so is
$J
121
.
is an i s m r p h i s m , it remains to prove that 1x1 and 1x1 have "the same" structure sheaf, t h a t is to say, that the canoniSpec &(XI - . This follows f r a n is b i j e c t i v e f o r each s€ d(X) cal map &X_)s + &Xs)
+
To show that
the diaqran
1 is
where since
,
d u l e s over
a f f i n e and CI
7
may be regarded a s a quasicoherent sheaf of
and y are bijections. Finally
d(_fs) and
'o(f) are
surjective. W e prove the latter contention, the proof of t h e former being similar. For each
s€&)
, write
zs
f o r the open subscheme of _X which has the
7(ys) , regarding 7 as a quasicoherent sheaf of mdules over Y . If a d ( g ) , there is a partition 1 = ~ ~ = l x i s iof unity i n d(y) and elements aitdx(zsi) same space of p i n t s as
whose images i n ailgsisj family equation
3'
8y (Y-si )
- a 3 ( 5Sisj I
ys
;t h a
g(gs)
may be identified with
are the r e s t r i c t i o n s a/Ysi
i s a section of
7
over
Also, a
~ s i n ~ s =j zsisj
ij Since the
.
is obviously a 1-cocycle of 7 f o r the mering (ysi) , the 1 H ( ( IY . I ) ,y ) = 0 , established below, implies that there are S1 It f o l l m that t h e restricsuch that aij = bi I Xs. - b j I
( a . .) I '
biE (5 , ) S1 tions of ai-bi
and aj-bj
to
zs.s1
l j
zsisj .
are the same, so that there is
a'E4(X) such -
Lama: - Let
8.2
. Hence
a'lXsi = ai-bi
that
be a ring,
A
d(z)( a ' )
= a
A-module,
M
.
. n 1 = liz1xisi
5
p a r t i t i o n of unity in A , u t _Y = Spec A Then the w h m l c q y qroups M w i t h respect to the covering (Y s a t i s f y H*((Y ,MI M __ and i si si H ((Y ),M= 0 i f i>O
o-f
.
si
It suffices to show that the sequence
Proof: 0
.+
G(Y)
+
Ijrjz(ysiny
.
57
.+
?-?-$Y
l r 7 ,
associated with M and the m e r i n g
(Y
s1
the same as the one obtained by s e t t i n g C=A
Section 9 k
.ny .m s1 53 Sl
...
i s exact. But t h i s sequence is
,
B
=psiin 5 1,
2.7.
Transporters
denotes a model throughout t h i s section.
.
ard each xcX_(R) This enables us to associate w i t h
f o r each RE$ arrow
g: -
X_xy -+
_Z
such t h a t
and yEY(R)
.
9.2
Corollary:
to the k-functor
Tf
R -+&E$SpkR)-
~
( g ( R ) ) (x,y) = ( f x ( R ) ) (y)
-
l
~
~
x Y , 2)
, mH?@,_Z) & ~
.
f o r R€w%
,
an xEX(R)
is canonically isanorphic
I,
5
2, no 9
Fxanple:
9.3
5
t h i s and
i s canonically iscmorphic
,
H?(%A,
in view of the canonical isamrphisms
1, 6.6 we deduce the existence of a canonical i s m r p h i s n
1, we arrive a t the following criterion: is a scheme i f the following three conditions hold:
H-%(SSA,_Z)
Z is a
is a f i n i t e l y generated projective k-mzdule, and f o r each f i n i t e
A
subset P PClLJI
z)
If AC?m
5
By Prop. 6.6 of
schene,
83
R * (R BkA)
to the k-functor
man
z
QUASI-COHEXENT-IWDULES
z
of
there is an affine open subscheme g of
2 such that
*
Definition:
9.4
a mnmrphisn,
Let p: X g Y _
and E': _X
-+
H 5-
Cy,_Z)
+
_Z
-%% ,
&:J'-+z
the mrphisn canonically associated
with - p by Prop. 9.1.
The t r a n s w r t e r of
written
, is
TranspD(Y,Z')
be a morphisn of @
2' relative t o 2
,
the pull-back of the diagram
(Transp ( Y , Z ' ) ) (R) m y thus be identified w i t h the set P- ~ ( Modulo ~ , of arrows g : SJ+ R -+ X such that p'g factors through ~
For each RE
$
.
1
prop. 9.1, the existence of such a factoring means that the canposition
factors through
Propsition:
9.5
and
z' . Let -
i:Z'-+z --
b e a c l o s e d erkdding gf k-functors
Y_ a locally f r e e k-schaue, that is to say, a scheme having a covering whose algebras of functions are f r e e k d u l e s . Then
by affine open
Hca(Y,i)
xi
: H?(Y,_Z')
-+
s a H&(:,_iZ ) closed
epnbeaaiq.
~
)
Proof:
Suppose first t h a t _Y = 5 % B where B is a free kmodule. By 9.3,
we must s h w that the canonical mrphism
mZ1
B/kB
-+
7 7 2
Bk-B
i.e. t h a t the following condition is satisfied (6.1): each a$m$B(A) = g ( B @ A ) , there is an ideal I of each hQtKmrpkism iff
Z'(B8.R)
,Mk , the element aR of Z(B@R) g'+z is a closed gnbedding, so i f
of
$:A*R
. Now
$(I)=O
are as above, there i s an ideal J of abare,
aR belongs to z ' ( B @ R )
is a sMlleSt ideal
and J c B @ K e r 4
of
I
is a closed embedding, for each A€-% and A such that, for
iff
and a
A
B @ A such that, €or each $
3
J
as
is free, there
(B@$) ( J ) = 0 ; since B
such that BE31
A
belongs to
. The conditions
I c Ker $
are accordingly equivalent, which proves the
= Ker(B@$)
first assertion.
In the general case, by 6.1 it is enough to show that, for each p: S&MxY where ME&
I
T_ = Transp (Y 2 ' )
P
-I-
is represented by a quotient of
M
. If
-+z
-pi denotes the restrictioi of p to the affine open subschgne SpkMxYi of S a M x Y , Ti = Wansp (xi,g') is represented by a quotient M/mi of M , by M/cimi and :Ti the morphisn q: nT.x - i-1
Ei
. It is nuw sufficient t o show that
z
T=
QT. , i.e. t h a t
.
1 -l
irduced by p factors through 3' But this follows frcm the fact that q-'(Z_') is a closed subscheme of ? z i r Y con-
taining each
9.6
(nT.)x
i -1
+
xi
Corollary:
1
If k
is a field,
closed embedding of k-functors, L?(y,&) Corollary:
9.7
With the notation of def. 9.4,
-p
Corollary: If
free k-scheme, H-(_Y,x)
Proof: By
5
1, 6 . 3 ,
is a closed embedding.
Lf:
_Y
a closed embeddinq, Transp (Y Z ' )
free k-scheme and 9.8
y is a scheme and _i:g'-+z 9
X
-I
-
is a locally is closed in _x
is a separated k-functor and
.
is a locally
I s separated.
X
is separated i f
,g is separated. By 7.4, 7.5 and * X -a k X-g-
7.7, this is the same as saying that the s%xctural projection p
is separated. Naw apply prop. 9.5 to the diagonal mrphism of
-
X -into
&" 3
.
5 3
AIx;EBmc S C r n E S
Section 1
F i n i t e l y presented mrphisms
1.1
Definition:
A-algebra
B
is s a i d to be f i n i t e l y presented
i f it is isamorphic t o the quotient of an algebra of plynanials
.
by a f i n i t e l y generated ideal I
over
, each
-Z
ideal of
.,Xn ]
AIXl,..
.,X n ]
is a f i n i t e l y generated algebra
is Noetherian, i n p a r t i c u l a r when A
When A
AIX1,..
is f i n i t e l y generated, so that an
A-algebra is f i n i t e l y presented i f f it i s f i n i t e l y generated.
Laam: -Let
1.2
. If
limit A
B
c1
a
is a f i n i t e l y presented A-algebra,
Ad f i n i t e l y presented A @A B,
be a directed system of rings, w i t h d i r e c t
(A )
.
A -algebra
S u p p s e that the ideal
Prcof:
B
is iscanorphic t o
def. 1.1 is generated by the plyncanials
of
I
.
such t h a t
B,
c1
there is an index a
P1,...,P r Choose a so that the image of Aa i n A contains a l l the coefficients of the plynanials Pi I f Q1,. ,Qr EA,[Xl,. ,Xn) are
.
mapped onto P1r...rP
r and generate the i d e a l I,
For example, w e may take A
. I t follows that
B
i f f there is a f i n i t e l y generated subring A.
f
generated Ao-algebra
of
A
Bo such that B
AmA Bo 0
1.3
Lama:
a)
If
f i n i t e l y presented wer B b)
If
$:BK
(b)
A
and a f i n i t e l y
is f i n i t e l y presented wer A and i f
,t
h s C
is f i n i t e l y presented wer A
C
.
of
B
i s f i n i t e l y generated.
These assertions follow e a s i l y fram lemna 1.2. For example, w e p r w e
. Let
A.
ad. Co a f i n i t e l y W i t h the ahme notation, l e t
be a f i n i t e l y generated subring of
p be t h e canonical projection of A,
.
is f i n i t e l y presented
B
generated Ao-algebra such that If
= A,[X1,...,Xn]/I,
is a surjective hcnmnorphism of f i n i t e l y presented A-algebras,
then t h e i d e a l K e r 4 Proof:
, set B,
..
to be the system of a l l f i n i t e l y generated
(A,)
subrings (i.e.A-subalgebras)
over
..
C
A@
4 3
AIXl,..
.
.,Xn]
is s u f f i c i e n t l y large, there are ci €Co
A
onto B = AIXl,.. such that
.,Xn]/I .
1@
I, 5 3, no 1
AzlGEBRAIc GEDMEma
86
A0
si = @(p(Xi)) ; if
generate Co
Tllr...rIls
over .A
, we have p l y n d a l
relations
oj
lA@
with coefficients
vj
'
.A
A
; thus,
= Q.(1 @ 5,,...,1
0
8
.A
5
)
is sufficiently large, we have
if .A
.
that the ti generate Co Also, if Plr...,Pr generate the ideal I , we have Pj(El,.. ,En) = 0 for sufficiently large .A Under these conditions, the hcnmnorphim of Ao[Xlr.. ,X 3 onto Co n which sends Xi onto factors through a hcnmrorphisn @o of Bo = = Qj(51r...rEn)
r
SO
.
.
ci
...,
.
.
Ao[X1, Xn]/(Plr...,Pr) onto Co Therefore @ = Af$o and Ker @ 0 image of Af ( K e r @o) in B , and. is hence finitely generated.
is the
0
LamGl: - Let A' be a faithfully flat A-algebra. Then an A-alqebra
1.4 B
is finitely presented iff A'@AB
is finitely presented over A'
.
proof: The latter condition is obviously necessary without restriction on A' ; we shaw that it is sufficient. Let B'
run through a l l the finitely
generated subalgebras of B ; then we have 1 9 A'@ B' =A'@ lim B' A A +
A'@ B A
,
so that, since A'@ AB is finitely generated over A' , A'BAB' =A'@ AB for scme subalgebra B' . The assumption that A' is faithfully flat over A then implies that B' = B , so that B is the quotient of an algebra AIX1, ,Xn] by an ideal I Let I' run through all finitely generated ideals contained in I . Then we have
...
.
A'f B = A' [Xlr. m. rXn]/ (A'g I) and A'aAI is finitely generated (1.3)
,
. Since
A'@ I = 1 9 A'@ I' A A we have A'BAI
=
A'@ I' for sane I' A
, whence
I = I'
.
1.5 Lerma: - let B be an A-algebra and let 1 = lxifi be a partition of Unity Of B If Bfi is finitely presented over A for each i , then - B is finitely presented over A
.
.
Proof: Let Bo run through the finitely generated A-subalgebras of B containing
the fi and the xi. The equalities
87
lhB
Ofi
+
= B hply fi
Bofi
for sane Bo ; hence
= B
fi
that B of
.
TTB This implies is faithfully f l a t over B~ , we b v e B=B, i o f :I is of the form A [ X ~ , . . ' . , X , J / I If Qi , Pi are representatives
and, since
.
xi, f i
i n AIXl,...,Xn]
, then
1 - IiQiPi
and i f
..,Xn ]/I ) P i
(AIXl,.
is the ideal generated by
I'
is f i n i t e l y presented over A
I
(W')
. By
of the map ( A [ x ~ , . . . , x ~ ~ / I ' ) B is Pi Pi f i n i t e l y generated f o r each i , hence so are I/I' (Alg. cam. I I r f $5
h m a 1.3, the kernel
-+
no. 1) and I . 1.6
Definition:
f i n i t e l y presented
y of X @a
V_
llf,
of
A mrphism of schemes
for each p i n t such that
_Y
XCLJ
a f i n i t e l y presented algebra over f
x@
,
.
f:z-+x is said t o be locally
, there
are affine open subschemes
f(x)tV_ , f(g)CV
O(L$
(!I(!)
i s said to be f i n i t e l y presented
if
f
is quasicarrpact, quasiseparated
and l o c a l l y f i n i t e l y presented. 1.7
Proposition:
If
@:WA
is a mrphism of models, the following
assertions are equivalent. (i)
A
is a f i n i t e l y presented B-algebra.
(ii)
S J
@
Sp 4
(iii)
Proof:
By
5
is locally f i n i t e l y presented. is f i n i t e l y presented.
2, 2.2 it is clear that
(ii)<=> (iii)
trivial. W e prove (ii)=> (i); set 3 = If
Xr
,V and
9A ,
. Also
= a B
(i)=> (ii) is
and f = @-@
are as in 1 . 6 , there is a t E B such that f(x)€YtC V
.
.
Thus hence
8 ( g N t ) ) is f i n i t e l y presented over o(y),
Now substitute g
4(t)
; w e may then assume that
for
of 1.6. I n this case, there is an
d ( z , ) = B ( t ~ ) ~ = A, ~ and
As
= Bt
zs ,
dlso over
B
.
v=Y_ in the notation
SEA such that xQs'=u ; whence
is f i n i t e l y presented over B
with f i n i t e l y many of these
, so
. By covering
(i) follavs fran lami 1.5.
2
88
z4LGEBRAIC GExmFTW
5
3 , no 1
L e t f:X+Y be a locally f i n i t e l y presented mrphisn, affine open subschews of X_ and _Y such that f(U')cV -' Corollary:
1.8
and V'
U'
men - @(u') Proof:
-f
I,
By 1.7, it is enough t o show that the morphisn
t@(Y)
there is a
u' ,
~, f_(x)
Corollary:
1.9
iff
CpV'
S E & ~ ) such that
) ~i s f i n~i t e l y presented over I1(gs) yt
and
c
f_(gs)c yt
A mrphisn of schemes
that:
, there
a)
for fixed i -1 f (Xi) ;
b)
for each t r i p l e ___.
c)
for each pair
Proof:
. Then
x e chosen as i n 1.6,
.
are only f i n i t e l y many
( i , j , k ) , X nx -ij - i R fi,j)
, d(Xij)
LWI) , and we
$:z-+x is f i n i t e l y presented xi
be covered by affine open subsch&s
ard X
flyi)
v,V_
induced by
us c g-l(yt)nv_nv_' .
O(us) = 9
have xE gs c
and i f
f':g'+y'
. Moreover, there is an
d(yt)
i s f i n i t e l y presented over
E
xtu'
such that f(x) E yt
~ ( g - ' ( y ~ ) n g1)
Since
J(v') .
is a f i n i t e l y presented algebra over
is locally f i n i t e l y presented. If
.
zij
I
zij
and they cover
i s quasicanpact;
i s a f i n i t e l y presented algebra over
I
This follows imnediately fran 1.8 and
5
2, 2.2.
.
o(yi)
is a noetherian ring, so is O ( X . .) by c) In t h i s case the -11 underlying topolq.icdl spce of X j is noetherian. Each open subscheme of
If
zij
C?(Yi) 1.10
is therefore quasiccmpact, so that c o d i t i o n b)
is noetherian for each i Proposition:
.
is implied by c)
if
a) The canposition of two locally f i n i t e l y pre-
sented (resp. f i n i t e l y presented) mrphim is lccallv f i n i t e l y presented (resp. f i n i t e l y presented). b)
I n the diagram of schemes g
5 _Y
y' , if f is locally f i n i t e l y Pre-
sented (resp. f i n i t e l y presented) then the canonical projection
I, 5 3, no 1
gpx'
f,,
:
c)
If
-
-
ALGEBRAIC SCHENEs
+.
_Y' is locally finitely presented (resp. finitely presentd).
and
9.f
89
g are locally finitely presented (resp. if
is finite-
go$
ly presented and if g is quasiseparated and locally finitely presented), thLn
f is locally finitely presented (resp. finitely presented).
-
Proof: b) follms hediately fran 1.6 and
5
2, 2 . 3 . Assertion a) follows
fran 1.8 a d the fact that the camposition of t m quasiseparated mrphisns
f:&+x
g:x+z
is quasiseparated. For 6x/z: +. X g X is the canpsition -_ (which is quasiccanpact) and the inclusion morphism :
3 X ~+ _ X~X X xX x x + g 65 , which is derived fran
of ~
x-
-
-
$3 -fzx--f * 3 so is
1~ -gy
-+
1; by the "change of base" This inclusion Ghhism i s therefore quasicanpact, and
.
6_x/E '
It remains to prove c) g - € E h (y,z)
,
f
6y/z:
_Y
-+
. We prove the "resp." part.
If f € E h (8,Y)and
is the canpsition
and . Notice that, since gf '4i (g<)y. By a) , it is enough to show that h
where & has canpnents presented, so is
L
x/z
presented. Now h is derived fran 6
*
Y_
+
is finitely is finitely
y%Y_ by the "change of base"
fxy : 3 % ~ Y ~ .Y since 6 -(it's a mmrphism!) -zx/z_ is quasisepited is locally finitely presented. and quasicanpact, it r m i n s to shm that %/_z For t h i s purpose we may assume that Y_ and z are affine. In this case, we must show that the kernel of the canonical map of o(y)@ 0(x) onto 8 em -+
is finitely generated; and this is clear, since it is generated by the
-
gi@l
l@gi
r
where
(gi)l
U(Z) * Proposition: Consider the diagram of schemes _X
1.11
9 Y' ,
- where g is faithfully flat and quasicmpct. If the canonical projection fyl: ZgE' + y ' is locally finitely presented (resp. finitely presented),
then f
is locally finitely presented (resp. finitely presented).
Proof: One easily shows that if so is f
if g,,
-
g,,
i s quasicanpact and quasiseparated,
(cf. the arq-ment of 5 2,-5.7). It therefore remains to show that is locally finitely presented, so is f To see this, let V_ and
.
& and Y_ such that f(tl)c_V , l e t (V_li) -1 be a f i n i t e family of a f f i n e open subschemes of y' covering g (41) By 1.8, o(Vix $1 is f i n i t e l y presented over fl(yi) f o r each i , hence
V
be a f f i n e open subschanes of
.
-
TU(!ll)
is f i n i t e l y presented over
cl(v_)
over
.
1.12
A mrphism of schemes
generated i f , f o r any point
g and 2 of y , such f.
u(v)
is f i n i t e l y presented
is said to be locally f i n i t e l y
g:x+Y
, there are a f f i n e open subschemes x€y! , f(x)Ey , f ( 1 ) C V and (I(!) is
of
xEg
that
f i n i t e l y generated algebra wer if
and, by 1.4,
#(v) . We say that
a
f is f i n i t e l y generated
is quasiccanpact am3 locally f i n i t e l y generated.
I n statements 1.7, 1.8, 1.10a), b) and. 1.11, " f i n i t e l y presented" may be replaces by " f i n i t e l y generated", whereas by statements a ) a)
and. b) below.
?:&+!
A mrphism of schanes
ings of Y d i t i o n s all
and
X
hold:
f o r any i
a2) f o r any
, (xij)
(i,j)
algebra over (3(yi) b)
If
~gaf
.
and
zij
such that con-
f -1
is a f i n i t e m e r i n g of
(l(xij)
(xi)
,
is a f i n i t e l y generated
. If
gag is f i n i t e l y
is quasiseparated, then f is f i n i t e l y generated.
Algebraic schemes
Throughout the rest of 2.1
,
xi
is locally f i n i t e l y generated, so is f_
generated and g -
Section 2
is f i n i t e l y generated i f f there are w e r -
by a f f i n e open subschemes
and a,) al)
and 1 . 1 0 ~ ) should be replaced
1.9
5
Definition:
3,
k denotes a &el.
A_ k-scheme
5 is said t o be locally k-algebraic
(resp. k-algebraic) i f the structural morphism jX: f i n i t e l y presented (resp. f i n i t e l y presented).
-
zz -+ SJ
m
k
is locally
5
I,
3, no 2
f:&+x
Any mrphism of k-algebraic k-schemes
-g:Y'+Y - - is a
If
91
Au;EBRAIc SCHEavIES
is f i n i t e l y presented by 1 . 1 0 ~ ) .
second mrphism of k-algebraic k-schemes, the pull back
i s a k-algebraic k-scheme by 1.10a) and b ) , since the structural pro_X$y' jection
is the c a p x i t i o n
px yl -v-
-
Hence a f i n i t e inverse limit of k-algebraic k-schemes is k-algebraic. The same r e s u l t holds f o r locally k-algebraic k-schemes.
is k-algebraic i f f
By 1.9, a k-scheme
ing
7
has a f i n i t e affine open cover-
are quasicanpact and (!?(X.)
such that the 2 . f l X
(X.) -7
presented over k
5
. If
-Q.
f o r each j
k
-7
is f i n i t e l y
is a ncetherian ring, the k-scheme
3 i s locally k-algebraic (resp. is k-algebraic) i f f the structural mrphism
-px_ : ,$
+
Sl k
is locally f i n i t e l y generated (resp. is f i n i t e l y generated) ,
that is i f f 2 has an affine open (resp. open and f i n i t e ) covering
(Xi)
is a f i n i t e l y generated k-algebra; i n that case,
Spec d(gi) is a noetherian topological spce; hence each open subset, and i n particular
such that
xifl g
-
3
is autcmatically quasiccmpct.
3 is locally
Each open subscheme Y_ of a locally k-algebraic k-scheme k-algebraic. I f , i n addition,
3 is k-algebraic, k-algebricity of
is
equivalent t o quasicanpctness. By abuse of language, we shall sanetimes confuse "algebraic" w i t h
ic"
.
Proposition:
A k-scheme
system of k-models j ective
Proof: B
.
(A,)
5 is locally k-alqebraic i f f f o r any directed
, the
canonical map 1 9 X(Aa) + X(l$n A,)
map
l+ I&(B,A,) + I&(B,l$n A ) Q f a c t is entrusted t o the reader. Suppose now f i r s t that the maps A
0,'
(Aol)
the
is injective. The proof of this w e l l known
@ :
1 9 X(Aa) +
jective. L e t _V be an affine open subscheme of
l$n
is bi-
When X - is affine, the proposition reduces t o the f a c t that a k d e l
is a f i n i t e l y presented k-algebra i f f for any directed system
A =
"k-algebra-
and denote by -p, : Sp -A
+
Sp - Aa
z
(19Aa)
, set
and paa
are always bi-
B = d@) : cp AB
+
, Sp A,
the
Au;EBRAIc GEOMETRY
92
1,
5
3, no 2
t r a n s i t i o n mDrphisms. By hypothesis any f:B+A is iraduced by some -1 -1 g : @ Aa * 5 r and t h e r e l a t i o n -A implies the existence pa (2 (UJ) = Sp of a p a r t i t i o n of unity 1 = Cxiyi i n A , h e r e the yi are the m g e s of -1 s a w elements i n Aa vanishing outside g (LJ Such a r e l a t i o n must exist -1 -1 already i n A for sane B > a Hence we obtain -pclB(g(V_)) = S&A B B ' which means that f : W A is induced by scme f':B+Aa , or i n other wrds that
.
.
the map lpn_%(B,Aa)
+
i s surjective. As &(B,Aa)
I&(B,A)
are identified w i t h subsets of Hence
B
am3 Z(A)
_X(Aa)
, this
and
&IM(B,A)
map is even b i j e c t i v e .
5 i s locally k-algebraic.
is f i n i t e l y presented and
X is locally k-algebraic. W e f i r s t prove that $ grs_ : 9 A, 5 be t w o mrphisms such that --a fp = .
Conversely, suppose that is injective. L e t
.
W e w a n t t o p r o v e that --up f p = %a@ for sane B 2 a By taking sane p a r t i t i o n of unity 1 = Cxiyi in A it is easy to reduce the proof t o the case, i n a f and g are factored through a f f i n e open subschemes g and V of which -1 -1 X Then we can prove as above that p (g ( U ) ) = 9 AB f o r same 8 2 a -
.
-aB
. AS
we may therefore suppose that g =
k-algebra, the maps u ( f ) , d ( g ) : d(U) For such a
B we obtain zpaB= gpaB
Consider f i n a l l y a m r p h i s n @ :
1 = Cxiyi
of
A
9A
d ( v ) is a f i n i t e l y presented are equalized by sane Aa + A B .
Aa
-+_X
-hi
Axi
:
+
.
as A l$(Aalxi r xi the f i r s t part of t h e proof,
Over'
hi
Let
limit k . I f r f o r m e index
-xi
of
hai and h
(ka)
5
such t h a t
zi . For s u f f i c i e n t l y large still denote by xi
i s induced by s m
large a an3 thus define a mrphisn Corollary:
There e x i s t a p a r t i t i o n of unity
, which we
i s t h e image of sane elanents i n Aa
.
-
and a f f i n e open s u b s c h m s
is factored through scme
2.2
.
-
kai
coincide on
& : @ Aa * X
Ax
,
a
xi
. mre-
zi . By
: S i A 7
ax1
%(Aa)xixj
such that
Fapa
for
= h -
.
be a directed system of mdels w i t h d i r e c t
a i s a locally ka-algebraic scheme and a quasiccmpact, quasiseparated ka-scheme, t h e canonical nap X
is bijective.
i
x (resp. of Zinx. 1 -7
. Set
= Q(x)
and
=
O(gijk)
. me assertion
follaws from proposition 2 . 1 by taking d i r e c t limits i n the exact sequence
2.3
let
W i t h t h e assumptions of 2.2,
Corollary:
of algebraic ka-schemes. ~- If
h@
k
kol
_h:X-ty be a mrphism
i s i n v e r t i b l e (resp. a m n m r p h i m , an
B > a such t h a t &
open embedding, a closed abddinq), then there is
y B.
is i n v e r t i b l e (resp. a mmrphism, an open embedding, a closed embedding) Proof:
kor h '
@ :f
:
f o r s u f f i c i e n t l y large y If
h @k kcl
mrphisn
B is s u f f i c i e n t l y large, by 2.2 there
hC3 k is invertible and
If
is a mrphism
+
X C3 k
h ' f k = (h f k1-I
such t h a t
ka
. Since
.
is a mmrphism, it i s enough to apply the abave to t h e diagonal
&z/y
:X + -
5xX y_-
(which is i n v e r t i b l e i f f
& is a monomorphim)
.
Now suppose that h L&ko
k is an open embedding. By t h e above argument, % is a mnmrphiSm f o r s u f f i c i e n t l y large D ; we may thus assume
straightaway that X_ is a subfunctor of , h being t h e inclusion mrphism. &t (Y.) be a f i n i t e a f f i n e open covering of Y_ If " g k is open i n -1 for each i there are functions fimR, ...,firi E fl(,Yi)% k such that
.
is the union of the open subschemes the
fij
(yi
Ck)Eij
of
yi
subschme of
C3 k
kolB
covered by the
. This -lies
When _X f&k
that
is closed in ' g k
kernel of t h e canonical map
. For s u f f i c i e n t l y l a r g e
d ( x . ) @k lkcl
. Let
-1
,
gij
B
g f&ky i s open in '&kY
, we
E
5 be the open
. Since z C3kgk = _X fLk , 5 f' k y = 3 &5 f o r s u f f i c i e n t l y
(Y. @ k )
by t h e f i r s t part of the proof we have large y
&k
are the images of functions gij E
choose
fia.,...,f*i
-
to generate t h e
Let
-Ti
be the closed subscheme of
the quotient
(4XJ
f ka)/(gill a
xi%k &Iir
whose algebra of functions is
I
being define3 as b e . W e have
the gij
for sufficiently large y
. Hence
all i
is closed i n
, so t h a t
_X
&p
Proposition:
2.4
direct l i m i t k
Let
Let
(&nu 1
k is closed in Xi 4 Y 6-10). yi Eky ( 5
T -a
k
for
21
(ka) be a directed system of models with
. For each algebraic k-schane
an algebraic ka-schm Proof:
)
i
B
T
there is an index a and
is i s m r p h i c to
such that
(Ti) be a f i n i t e affine open covering of E d(Ti) be functions such t h a t _TinT
T
5
k a
.
; let
fij is covered by the r(iljli j open family (T 1 ij , 1 5 R 5 r ( i , j ) By 1.2, for sufficiently large p fR .. there are algebraic k-schemes z i and functions g? such that ‘1 fl
Let
I...,
, .
Ti7
.
be the open subschm of
_Ti
a
By 2.2 and 2.3, for sufficiently large
such that
+ijf k a
01
there are i s m r p h i s n s
is the identity of
If a. is sufficiently large,
for which
covered by the open family
+iJ
induces an i s m r p h i m
(Ti) f3
ij
911
.
I, § 3,
3
for a l l
(i,j,k)
where IT;
z;
ALCZGBRAIC SCHEMES
95
. I t is then sufficient to take fox
Ta
the k - s c h m
is the spectral space obtained by matching together the
$ta1
along the open subspaces
2.5
Corollary:
generated subring ko of i s i s m r p h i c to
X@
Qko
If _h &k
such that the
so is
h%kB
a finitely
01
.
f3
5 &k
i s an inclusion mrphism (2.3) ; since
(5
g k ) n xi
(xi)
of affine open s u b s c h m s
are closed i n
. The open subscheme z of
Z'
.
Xo such that &
f o r sufficiently large
xi
for each i
xi
cuvered by the
&k
algebraic k-scheme. For sufficiently large
B
I
via the ismrphisms
Le t h:X-+x be a mrphism of algebraic k -schemes. --
i s q u a s i m p c t , there is a f i n i t e family
k -scheme
I
.
Proof: W e m y assume t h a t
%k
01
k ard an alqebraic ko-scheme
is an a k d d i n q ,
in cover 3 &k
k
A k-scheme 5 is algebraic i f f there is -
k
Corollary:
2.6
-6 B,
,
ST;
I
I
and
is an
there is an algebraic
such that
PBk
;
furthemore, for suffiiciently large /3 there are m r p h i m s
such that
h2hl
gnbedding,
b2
(2.2); for sufficiently large
= h_ &kB
B
,
is a closed
an open fmbedding by 2 . 3 .
Section 3
Constructible subsets of an algebraic schane. F l a t mxphisns.
3.1
Let
a subset
U
of
X be a topological space. W e shall say provisionally that
X has the property C i f the intersection of
quasimpact open subset of
x
is constructible i f
where U
P
X
U with each
is quasiccmpact. W e say that a subset P of
is a f i n i t e union of sets of the form
and V are open subsets of
X with the property
constructible set clearly has the property
C
, we
C
un
CV
,
. Since any
see that an open subset U
ALGEBRAIC GM=METwI
96 of
x
is constructible i f f
U
1,
has the property C
5
3, no 3
.
it follcws that 1
1'P h
so t h a t
i s a f i n i t e union of sets of the form
v . n...nvj n(:(u. u...uui ' 1
Hence i f
S
l1
is constructible, so i s CP
P
structible subsets of
r
)
.
. It fo1h.m
that the family of con-
X is closed under f i n i t e union, f i n i t e intersection
and canplanentation. X i s quasiccanpact and quasiseparated, the constructible
If
3.2
open subsets of
coincide with the quasicanpact open subsets. Lf X
X
&
a noetherian space, t h a t is, i f each family of open subsets has a maximal mEmber (under inclusion), then every open subset of
follows t h a t the constructible subsets of
is constructible. It
X
X a r e t h e n p r e c i s e l y the f i n i t e
unions of locally closed subsets. Proposition: X
The following conditions on a subset
P
of a noetherian space
are equivalent:
(i)
P is constructible.
(ii)
For each irreducible closed subset F
i s dense i n F Proof:
such that PI7 F
X
contains a non-empty open subset of
F
.
(i) => (ii): Suppse that P =
where Pi PinF
,P
of
u
P
lssn i
is locally closed in X
f o r each i
is dense in F f o r a t least one
l o c a l l y closed in F
closed i n F subset U of
. Since F
.
, hence of the form Ti = F , we have K
. If
is dense i n F
P nF
i ; for such an i UnK = F
where U
and pi
,
PinF
is
is open and K
contains the open
(ii)=> (i): Since each decreasing sequence of closed subsets of
,
X
is
is
(ultimately) stationary, we may argue by noetherian induction by assuming that the implication (ii)=> (if holds w i t h i n any closed set s t r i c t l y con-
tained i n X P
.
If
X = AUB
is reducible, w i t h A and B closed, and i f
s a t i s f i e s (ii), then P ~ A and P ~ Ba l s o s a t i s f y (ii); i n this case
PnA
and P n B are unions of sets which are locally closed i n A and B
, and so the
hence a l s o i n X
same applies to P
. If
P
i s not dense, apply the induction hypothesis to
P
is dense,
P
contains an open set U
. "hen
P-U
X
,
is irreducible and
. If,
on t h e other hand,
s a t i s f i e s (ii) , and. is
therefore constructible by the induction hypothesis. I t follows that P = U
U
(P-U)
is constructible. W e now apply the r e s u l t s of 3 . 1 ardi 3.2 t o t h e gecmetric real-
3.3
lgl of an algebraic k-scheme _X
ization
. Since 11
is quasiccanpact and
quasiseparatd, the constructible open subsets of
X_
(i.e. of
c i s e l y the quasicanpact open subsets. Furthermore,
if
k
rmdel, then
/XI
are pre-
is a noetherian
is a noetherian t o p l o g i c a l space. To prove t h i s , observe
t h a t f o r each a f f i n e open subscheme g of
&
, d(LJ
ated k-algebra, hence is noetherian. It follows that noetherian; since
1x1)
1x1
is a f i n i t e l y generILJl
is covered by f i n i t e l y m y such
= Spec d(V_) is
l_Vl , 151
is
a l s o noetherian. Propsition:
subset of
-f:Y+X --
_ Let_ -X
be an algebraic k-scheme and let P be a constructible
X . Then t h e r e
such that
P=z(x)
is an a f f i n e algebraic k-scheme
.
Suppose f i r s t of a l l t h a t
proof:
arid a m r p h i m
P
is the union of two constructible sub-
. Suppose a l s o that we have constructed two m r p h i m s such that P1= f1(Y -1) and P2=f2(!12) . Then P is f :Y +X and f2:Y2'+11 -1 -1 the image of the map underlying (f,,f,): _Y1U_Y2 X . Accordingly, we m y sets P1
and P2
-+
confine o u a t t e n t i o n to the case i n which P is of the form IYIn CI ; covering U by a f i n i t e open family (Fa) ar61 replacing P by the iga,lnP , we reduce to the case i n which g is affine. W e can then cover
uny
.-
by the special a f f i n e o p subschemes
fl,...,fn
E
O(V_) . I t
and to choose f o r
gfl,...,g
is now s u f f i c i e n t to set
g- the m r p h i s n
f
n
with
ilzduced by t h e canonical projection of
I,
Corollary:
3.4
Let
x
5
3, no 3
be a p i n t of an algebraic k-schane
let
P
be a constructible subset of
iff
P
contains each point y
X
.ms
such that x
X
P is a neighbourhccd of
is i n the closure of
(y}
.
x
Proof: The condition is obviously necessary. Conversely, suppose the cond i t i o n holds; we may then assume that % is affine; furthermore, since
is constructible, there is an a f f i n e algebraic k-scheme f:x+g such t h a t
P =
and a mrphism
. If
f(Y)
P
:9Jx+ x
EX
5 1, 5.6,
is the mrphism defined i n
the space of pints of
(s+, dX)xz~ is
anpty, i.e.
or 1 8 1 = 0 Q(X)
. Since
not v d s h i n g a t x
dx = l$n O ( _ X ),~as
, we
runs .throughthe functions
s
l@-l(x)l = 0 i n a t least one of the rings d(&)s@b(x)d'(x)Hence the undeslyi& space of -s X x Y is empty, which im37
.
/zs/
p l i e s that
have
(5
is contained i n P
1, 5.4).
Lemna: - Let A be a f i n i t e l y generated algebra over an integral d m i n B If..M is a f i n i t e l y generated B-module, M is free over 3.5
B 9
.
.
for sane 0 # g E B
Proof:
g
The following e l m n t a r y proof is due to D m e r . Clearly w e may
suppose t h a t A = BIT1r..
.,Tn]
generated by a single element m
is an algebra of p l y n a n i a l s and that M is (replace M i f
necessary by the cyclic
quotients of sane ccanposition series and take f o r g t h e product of the elanents of
v = (vl,
A
associated with these d i f f e r e n t quotients). For any
...,vn c,." X:,."-t$
.
/ u /= vl+. .+un
set
mii T' = T ; '
..T.:
. Further-
X ( u ) < X(u) i f +1 in t h e lexicographic ordering on $ ( / v / r V l , rvnf < ( \ L I / , ? . I ~ , . . . , V ~ ) is such that T E L T i , we then clearly have that A(u)
mre, l e t
...
be the bijection such that
.
implies A(utai) < X ( ~ + E ~ ) Furthemore, for each r=A(p)€l Mr =
1
BTUm and l e t
X (u)ir
I
u
be the ideal of
B
,
set
annihilating Mr+l/M r
.
( t h i s B-module is generated by the residue class of T’m) = T . 1 BT!Jm C T M = TEiM u 111 i h(u) A (U) MA (11+Ei) *
. W e have clearly
1 BT’+€im
Hence I
C I
.
P+Ei
, and mre
generally I
I
C
U
P
if
p< p
in the product
this order has the property that, for any subset S of , the subset so of minimal e l m t s of s is f i n i t e (otherwise take any
order of
#
u
! I ? But
i n f i n i t e sequence of d i s t i n c t elments i n So
and construct successively
i n f i n i t e subsequences such that the f i r s t canpnents increase, then also the second ones
O#g E
.... contradiction!). Take now
GIu.
Then g
Generic flatness theorem:
3.6
over a noetherian model and l e t
_f(z)
such that the imaqe
Say,
Proof:
f-l(V)
(Of course,
v
zi
M
g
is free.
be alqebraic schemes
_f:x+x be a dcaninant mrphism, that is to . m, i f is reduced,
onto V_
.
of
such that f
induces a faithfully
is said to be dense in Y_ i f the space of p i n t s of
is dense i n
subschemes
,
Let -
IXI.) Let Y1,...,Y r be (Alg. m. 11, 9 4, no. 3 , prop. 10). By
V_
. Thereforeg
is dense i n
there is a dense open subscheme f l a t mrphism of
such that
is mt free. Hence the B d u l e
is cyclic free or zero for a l l r
(Mr+,)g/(Mr)g
and
annihilates a l l the quotients Mr+l/Mr
i.e. such that Mr+l/Mr
A-’(r)fS
S = { p d : I,#O}
such that
the irreducible c m p n e n t s of replacing
-x
1x1
separately by open
and _X by the inverse images of these open subschenaes, we m y assume that 1x1 is irreducible. If we then replace _Y by a (dense) affine open subscheme, we may assume that = SAB ,
Igil
= Yi-UjziYj
where B is an integral damin. Let _Xl,...,X of
X_
; if
scme of the
zi
, w e may
assume that each
there exist mn-arrpty open V . C _Y into
vi
-S
be an affine open covering
we replace _Y by a smaller affine open subschem and suppress
-1
-xi
is d d n a n t over
such that the mrphisn frcan
. Thus i f
f-’(yi)tXi
induced by f are faithfully f l a t , then we may set y = niyi
are therefore l e f t with the case i n which g
. We
and f is defined by a m m r p h i s m $:B+A , which is necessarily injective (5 I, 2.4). To ocanplete the proof, it m w suffices to set &A
= SJ~A
in l a m a 3.5 abwe and take f o r _V
the special open subscheme Y thus taking the open subscheme X 7 ’ -4 (g)
-f - 5 y )
.
for
Ef
Corollary:
3.7
k
F:z+x is a daninant
is noetherian and i f
mrphism of algebraic k-schemes, then the h a g e g(5) contains a dense open subset of
1x1 .
Proof:
Apply 3.6
to the mrphism
) y \ and. lyredl
the t o p l o g i c a l spaces
lzrdl
, f(5)
contains the open subset
-xred"-Yred
constructed i n
and
3.6.
tobenoetherian.
Lf
mrollary:
c:x+x
constructible subset of Proof:
_ I
is a mrphism of alqebraic k-schemes,
2 , then
W e f i r s t assume, t h a t
of locally closed subsets Pi
k
the image f- (P)
of
X'
i s dense i n Y' , then f is the inverse image of
& a
is constructible.
i s ncetherian; then P
is a f i n i t e union
and we my assume straightaway that _X coin-
an irreducible closed subset of f@)nY'
P
P
cides with the reduced subschane carried by one of the Pi
If
1x1
are the same, as are
Ivl
. Since
induced by _f
In the three follawing corollaries, we do not assume the d e l
3.8 k
-fr&
1x1
. N w let
Y' be
; it is enough t o show that, i f
(5) contains an open subset of Y'
Y' i n
I _ x ~ , we
have
5' and g' are the reduced closed subschanes carried i s thus sufficient to apply Cor. 3 . 7 to the mrphism
(3.2).
f ( x ' ) = f(g)nYi . I f by X' ard Y' , it
f':_X'-+x'
induced by
In the general case, we m y assume that
(X,(= P by Prop. 3.3. By Props. 2.2 and 2 . 4 , there i s a f i n i t e l y generated subring ko of k and a mrphism of algebraic ko-schemes such t h a t f m y be identified w i t h f 63 k
-0kO
.
fo:$o+xo
If
is the canonical projection, we thus have
-f (6) = q-l(fo(zo) ) by 5 1, 5 . 4 . Since q --'($(%)) by def. 3.1. 3.9
fo(zo) i s constructible, so i s
Corollary: For each morphism of locally alqebraic k-schemes
f:Z+g , the follawing assertions are equivalent: (i)
-f isopen.
(ii)
For a l l y,y'EY
, such that
y
is i n the closure of
and each x€f_-l(y) there is x€f-'(y') the closure of { X I )
.
such that
x
{y']
is i n
1,
5
3, no 3
AIGEmAIc SCHEMES
For each xu(
(iii)
,
101
_fl induces a surjection of
Jx
Spec
df (XI Proof:
(ii)<=> (iii): This follows imnediately frcm
5
1, 5.6.
is an open quasiccmpact subset of
&
,
(ii)=> (i): If
V
. The assertion then follows f r m 3.4. (iii):If not, s e t y=f(x) . Let y ' be a pint of
is con-
f(V)
structible by 3.8 (i)=>
does not belong to the imge of the unit of
K(Y')@
affine, we have
where s ( x )
some s
I
JY
dX
Spec
dx . Then
+ o . I t follows that the u n i t of
i.e. Ifl-l(Y')nl_Xsl
so that y'ft f (gs)
,a
spec(^(^')@
is zero. Assuming, as we may,
=
dy which dx)= a , and
Spec
thaty OYand 5
are
~ ( y ' ) m J ( , ) J ( z ) ~is zero for
-
a
contradiction.
Corollary: A f l a t mrphisn of alqebraic k-schemes is o m .
3.10
Proof: With the notation of cor. 3.9, we must show t h a t the maps
dx Spec Jz(x) induced , hence faithfully f l a t dffollows f r m ALg. cam. 11, 5
Spec
-+
(Alg. conan. I,
5
dX
is f l a t over
3, prop. 9 ) . The assertion now
2, no. 5, cor. 4 of prop. 11.
Corollary: If k
3.11
are surjective. Now
by f
is a f i e l d and
canonical projection Erl: X_xy
-+
5,g
sk-schemes,
g i s open.
Proof: C l e a r l y w e may reduce the problem to the case i n which X=S&A X=S%B -
, A,BE$
. If
B
i s f i n i t e l y generated over
k
,
,
mkB is f i n i t e l y
presented over A and 3.11 follows frmi 3.10. In the general case it is enough to show that i f
sWkB
, then ~ ~ ( X x is y )open ~
has f i n i t e l y generated subalgebra Bo the i m g e of
(_Xrxo)s i n
1x1
r a i n s to show that the imge of
such that
in
1x1 . Now
s W k B 0 ; setting
B
Y =QB -0
0 '
i s open by our previous remarks. It thus (_X r
x)
and
(_X r
yo)
same. 'Ib see this, write p: ( Z X ~-+ )5 ~ and _po: ( ~ mrphisns induced by the canonical projections. For each
in s
-+
1x1
X~
are the
for ~ the )
~ € ,5we
~
then have
102
ALGEBRAIC c333ETm
I,
5
3,
3
110
since the canonical rmp (K (x)BkB0) + (K (x)akB) is i n j e c t i v e , the mrphisn -1 -1 p (x) + po (x) is daninant f o r each x (5 1, 2.4) This canpletes the proof. -
.
3.12
Corollary: For each model k a d each mrphism of locally
algebraic k-schemes
f:@x
, the
m i n t s xEx_
Proof: Evidently we m y assume t h a t
. Let
_X
, Ap
A such t h a t A is f l a t over k ; P' is f l a t wer A , hence also wer k
By 3.4 it is therefore enough t o show that vious remafks, U
{p)
PEU
. Hence,
if
, then u
Naw i f
U
.
P'
is constructible. By our pre-
contains a p i n t p E Spec A
if
U
-
, by
(k/q)s
meets t h e closure
is noetherian, it is s u f f i c i e n t (3.2) to show that if
k
contains a non-enpty open subset of {pl
q=c(p)
f l a t over
are a f f i n e , with algebras A
and
p' be a prime ideal of
for each prime ideal p c p '
-
is f l a t , form an
f
X.
ogen subset U fo
and k
, where
.
3.5 (or 3.6) there is an sEk-q
.
such that
is
Since we are assuming that
and since Torl(A,k/q) k is a f i n i t e l y generated Arnodule (see 3.13 below) there is a
If
p'
q'=Z (p' )
gEA-p
such that
is a prime ideal of
, we
A
such that
and s g f p ' , and if
p'+
have accordingly
gs' .
By lug. ccann. 111, 5 5 , th. 1 and and A p l / ~ l A p is l f l a t over kq'/ prop. 2, Apt is f l a t over k , which proves t h e corollary when k is q' noetherian. Now assume that k is not noetherian; choose p
over k a
.
is f l a t P W i t h the notation of 3.14 below, this r e s u l t implies that there is
t EAo-po
such that Aot
is f l a t over ko
Spec A
. If
t'
so that A
is t h e image of
t
I,
8
3,
in A
3
, then
xGEE3RAIc SCHEMES
p E (Spec A) t , and
proof i n the general case.
m: g t
3.13
k-alqebra,
k
Tori(M,N)
At,
103
, which
is f l a t over k
k be a noetherian ring,
A
completes t h e
a f i n i t e l y generated
a f i n i t e l y qenerated Amodule (resp. kmodule)
M (resp. N)
is a f i n i t e l y qenerated Amodule f o r each i
.
Proof: Since A is a quotient of an algebra of plyncmials
kCX1r..
.m a
.,XnI ,
3 , assume A t o be f l a t over k n k using an acyclic To prove the l m , we need only calculate Tori(M,N)
we m y , by replacing A by k k ll...,X resolution of
M
by f r e e f i n i t e l y generated A-modules.
Lemna:Let -
3.14
.
A be a f i n i t e l y presented algebra over a rinq k
.
such t h a t A is f l a t over k P there is a f i n i t e l y qenerated subrinq ko of k , a f i n i t e l y qenerated
ard let p be a prime ideal of ko-alqebra .A
A
and a rinq hammrphism $o : A.
for
md a E A o
+
A
such that
a)
@,(Sa) = <@,(a)
b)
the map A @ k + A induced bv Q0 is bijective ; O ko -1 if po=@o (p) , A is f l a t over ko OPO
c)
;
.
Proof: By generated
1.2, there is a f i n i t e l y generated subring kl
kl-algebra
A1
, and
of
k
,a
finitely
a hamamorphism $I~:A~+A satisfying con-
d i t i o n s a ) and b) above mutatis mutardis. Let ko be a f i n i t e l y generated subring containing irduced by of
@1
pI pol p1
kl
. Set
, set
.A = Al@Jklk0 ard l e t $o be the hamamorphism
p1 = +;l(p)
i n kl kor kl :
and l e t q,
so,
q1 be the inverse images
ALGEBRAIC
104
Since we have
R 0 = Tor (B ,R/q R ) = 1% 1 1
and s i n c e Tor;l(B1
1,
GEoMETIiy
5
3 , no 3
Tory (Bo,Ro/qlRo)
is a f i n i t e l y generated B l d u l e we m y , by 1 1 1 taking ko s u f f i c i e n t l y large, assume f+atthe canonical image T of R T o r F ( B 1 ,R1/qlR1) i n Torp(Bo , R 0/q lR o) is zero. I n this case., ko satis,R / q ' R 1
f i e s t h e condition of t h e l m :
is a presentation of t h e
R1-"u>dule
U@Ro
V@Ro
g2-B
-C
i s a presentation of t h e
B1 ' then
1R 1
@ R 0-
1 R1
0
Ro-ndule
B1Q9R1 R 0
0
. Hence
is a quotient of
is a f i e l d ! ) . Since Torl (B1 RorRo/qlRo) (observe that R1/qlRl R1 Torl ( B ,~ t /q c are quotients of ~ e r ( u R /q R l and 1 1 1 R1O 1 0 ~ e r ( ~lP,l/qlRl) u , it follows that the image of Torl''
Since B
0
I?
i n Torlo
( B Q~ ~
0 '10 /ql
is a r i n g of f r a c t i o n s of
B @ R
(B1, tl/qlR1)
1 R1
IIo) 0
r
generates t h i s last R 0-module.
we have
hence t h e canonical image T generates t h e B o 4 u l e Tor1 'Bo ,Ro/qlRo)
which is t h e r e f o r e zero. Since Bo/qlBo
and
is a r h q of f r a c t i o n s of
it is f l a t over Bo
, SO
Ro/qlEo,
is f l a t over
.
!Lo
Section 4
IWnmrphisns of algebraic schemes
4.1
Propsition: Let ; we assume
X
a)
I_f
grg:z+
the induced maps
b)
If
f =@ -X
.
5, th. 1 and prop. 2 ,
y€x be points of W k-schemes
be locally alqebraic over k
PJg
such that
.
f(x)=q(x)=y, and if
coincide, then there is an open subscheme _V
fxrgx:Q@x
.
EU_ & fly
=gIU_
i k a k-alqebra hammrphim, then there is an open sub-
$: dy+ GX
scheme V_ cf
~ € 5
are two mrphisms of
such that x
o_f _X
1 to
5
that, by Alg. cam. 11,
of &E
g and a m r p h i m f:U-ty
such that
f(x)=y @
a ) : W e m y assume without loss of generality that _X and y
Proof:
are affine. Thus l e t
rated by blr...,b n Since A = X
for each
i
Z=kpkA and X=S&B
. Then we have
l$ns(bps , and , there is an
the same image in As
$(bi)
, where
for each
and. $(bi) i
. Now set
b) : W e may again assume that be such that
s ( x ) #0
. Let
p:As+dx
and suppose that the k-algebra relations
Pl(blr..
.,bn)
=
B
As
that
...
is gene-
@,$ E&(B,A)
@(bi), and. $(bi)
Zs
=
S&As
and q:B+8
Y
.
= 0
have
. Let
sf A
be the canonical maps,
is defined by generators
, there
B
have the same image i n Ax
Z=SjkA and y=S&B
... = Pt(blr.. .,bn)
l i m i t of the rings of fractions
, with
g=Spk$ and. g=S%$
, s p x , such
stA
the k-algebra
. Since
blr...,b
dx
n
and
is the d i r e c t
is an s and elements
...
...
€A ,an ) = = Pt(al, r a n ) = 0 and n s such t h a t Pl(al, p(all=$(q(bl)), p(an)=@(q(bn)) Hence there is a hcmmorphisn jr:B+As
alr...,a
such that
...,
+(bi)=ai
, and we
.
need only set g=zs and
= Ss
+
t o canplete
the proof. 4.2
Corollary:
Let
f:z+x
be a mrphisn of locally algebraic
.
106
I, 9 3 ,
ArGEBRAIc GlzcMmm
Ex: df(x)+ox
k-schemes and let x EX. I f t h e map open subschemes Q
V_
M u c e s an i m r p h i s n V_ 3 y
Proof: If -
, by
yg(x)
Set
v'
.
4.1 b) there is an open subscheme
mrphism CJ:!'+~ such that take
is b i j e c t i v e , there are
2 such that x€V_ , f(x)EY
o f 3
so s ~ lthat l
g'=f--'(y')
ZCJ
y€V'
, g(y)=x
and
CJ
4
=f-l
v'
of
. By 4 . 1
-Y
and
f
and a
a) , we m y
Y -x coincides with the inclusion mrphism
.
yl+x .
and l e t f':g'-+y' be the mrphism induced by f By 4 . 1 a) there is an open subscheme of _V' such t h a t x € Q and g- ( f ' / g ) coincides with the inclusion mrphisn of Q i n _X Now set I! = CJ-1 (g)
.
x€x .-W scheme
f
1 of &
f:z+x
Let
Definition:
4.3
.
be a morphia of schemes arad let
t o be a local anbedding a t x i f there is an open subsuch that
~ € 2 &
i s an embedding.
_flV_
be a local embedding i f it is a local embedding a t each x€& For instance, l e t A = Z[U,V)/(W)
,
2 B = ZkJ,V]/( (V-U +U)V)
and Y_ = Sp -B
. For each model
and
Y(M) = {(xry)EM2 : (y-x 2+x)y;o)
The mrphisn
g:&+x
(x-y,y2+y)E Y(M) sends
g(Z)
such that
M
we then have
.
said to
, and X = Sp -A
(x,y)E&(M) onto
mps
f(M)
2 is
i s a local embedding, but not an embedding. For the map
(l,O)€Y(Z)
and
(O,l)€_X(Zl
onto
(l,O)€Y(@
, SO
that f ,
is not a mnmrphism. On the other hand, l e t a,b (resp. u,v) be the images Of
(resp. in B ) . Then we have f_ = SJ 4 , where 4 : W A is such 4 (u) = a-b and 6 (v) = b2+b Now one v e r i f i e s e a s i l y t h a t 4 in-
UrV
that
in A
duces surjections of
.
B
U
onto Aa ;. Z[a,a-l]
and
% :Nb,b-']
.
ALGEBRAIC SCHEMES
Pbreover, i n the ring of fractions A
107
we have
1-a+b
+
a = @(ul
and
which shms that
subschemes
induces a surjection of
@
za
. Since the
onto A1-a+b
B
1-u $ and &l-a+b cwer 3 , it follaws that f
I
is a
local embedding.
z:x-?j
If
4.4
5
with if
f
f is a local embedding a t xf &
1, 6.3, that :
$ -+ z-Y
Proposition:
is a mrphism of k-schemes, we say, in accordance
is a local embedding a t x
Let
suriective.
.
affine. If all.. ,a
-X
is surjective; we may assume that _X and
ti(x)#O @
. Hence
. If we set
(bi) /$ (si) = ai/l
algebra of $,(&)
flk(X) , then we $=ok(Z) , si,biEok(Y)
n generate the k-algebra
$(bi)/4(si) = ai/l f ~ k ( ~ ,) where x 1 -
ti (ai$ (si)-$(bi) ) = 0
have
and
ok(c_X) for which
and t = tl...tn$(s) the equations n Since t/l belongs to the subalready hold i n flk(c_X)
. Hence
which implies that
are
f o r same ti€
s = sl...s
.
generated by the ai/l
where vE ok(y)
iff
is a local atkdding a t a p i n t xE 11
woof: Suppose that f
s , (f (x))#0
f:x-+x a mrphisn
be a locally alqebraic k-schgne ard
gf k-schemes. Then f
gx: d z ( x ) + ~ xis
(resp. a local embedding)
(resp. a local embedding).
$
,
induces a surjection of
is a local embedding a t x
f
is of the form @ (v)/@ ( s ) ~
t/l
ok(x)sv
dk@) ,
onto
. The converse is obvious.
Lama: - L e t 5 be an irreducible scheme (i.e. such t h a t the t o p l o g i c d l space is irreducible) g:x+x an injective
4.5
1x1
f. i s an anbedding.
gnbedding. Then
proof: that
yx
Let
x , let Cx
x€gx , g(yx)c17x arsd
. Since
g(yx)
xf
U I 1-x
IU -x
yx
be open subschemes of
5 and
f
induces a closed -ding
of
that
is dense i n
i s closed i n
jective, we have
and
\X_l
, f(vX )
[yxl , we have g(ux)
I
=
g-'(Yx)
5 into the open subscheme of
, so
that
is dense in
=
f
g(3)nlVxl
f(~)nly~l. Since
f
such
into ;
since
is in-
induces a closed embedding of
covered by the
yx (5
2, 6.10).
108
ALGEBWC czEDmrRY
I,
8
31 no 4
The irreducibility assumption is essential for the truth of the above 1em-m.
5'
For let
be the schene satisfying
X I (MI = { ( x , y ) ~: y~ - ~ 0
. The mrphisn
for each madel M
jective local gnbedding
of example 4.3 then induces an in-
f:g+x
E':X'+y'
Proposition:
4.6
and l+y is invertible] f_ is not an gnbedding.
but
Suppose that k
is noetherian. f I
f:z+x
mncplmrphism of alqebraic k-schemes, then there is a dense open subschane
-X such that flu is an injective local embeddw.
U - of -
F i r s t of a l l consider a mrphism of schemes g -: s c
Proof:
m m r p h i s n i f f the diagonalmrphism K
is a f i e l d ,
T
=
3K
and
. Then
9 is a
is an isomorphism. If
*sx$
5 = Sp A , this last assertion holds
i f f the
or A=K k It follows that a mnomorphism of a nonmpty scheme i n t o the spectrum of a
map a @3b +a.b
of
A @ A into A is bijective, so t h a t A={O)
.
f i e l d i s an ismrphism.
In the general case, i f If
x€g -1
2': 2
that of
, set
.
y=q(x) and y'= &S
(5 1,
dx is the local
r i q of
3 at x ,
a t x is ~ x / ~ x m y, so it follows from the above that g
g-'(y)
induces an ismrphism
~ ( y ) 9 /9m
x XY
.
dx i s Artin, t h i s implies that
If
o
is surjective. For i f J ' , and m' are the imges of _9x: JY+ Jx 1 Y Y m i n Jx , then ~ ( y ) dx/~xmy is equivalent to I;' = 9'+ m l J Y x Y Y X i s to 8, = d; by Nakayama's l e t m a (m' being n i l p t e n t )
.
Y
axd. x i s the generic p i n t
This results applies when g= f
an irreducible m p n e n t of one p i n t , so that
dx
Corollary:
4.7
5.3).
~ ( y ) The canonical projection
is a mnmrphism. If
(y)+f'
g- is injective
g- is a motlomDrphisn,
dense open subscheme
-
2-
1x1 . For by 5
is Artin.
me
1, 5.6
,,
Spec
(5
axd. '
that
1, 2.10) of
dx then has
just
assertion now follows frcm 4 . 4 .
W i t h the assumptions of prop. 4.6, there is a
of g
such that
L-l@)
is non-empty arid flg-l(y)
is an embeddinq.
...,xn
Proof:
Let
of
and l e t yi
yj
1x1
. Since
xl, f
be the generic pints of the irreducible m p n e n t s
= f(xi)
. Write
yi S
is injective, we m y assume y1
if
yi i s i n the closure of to be maximal with respect to
I,
5
3,
4
this ordering. Then there is an open subschene ylc[yll
and y i g l ~ ~ 1if i > 1 the irreducible ccsrrponents of
If
an embedding; for
of
such that
are the generic p i n t s of (4.6) , then yi is d i s t i n c t f r m
ul,...,um
I_Vl
yl is small enough,
-1
(YJ) I
is open in
yl is
This proves the corollary when
such that
of
struction, using the mrphisn Example:
4.8
schemes over a f i e l d that
v1
[ylI does not meet the image of 4.5 then implies that the restriction of g t o f-'(pl) is
1x1- IUI.
v'
. If
1x1 -
...,f(u,) . If
f_(ul),
scheme
109
Azx;EBRAIc SCHEMES
IV_'
I
f':
and is therefore irreducible.
{x,)
dense. If not, consider the open sub-
1 . Then apply the preceding con1 -1 f (v')+x' induced by 5 ... = IY_I-{y
.
Observe that a mmrphiism of locally algebraic
k is not necessarily a lccal anbedding. For suppose
is of characteristic #2 ; take for 31 am3 _Y the algebraic schemes
k
such that X(M) = { t € M
: 1-t
is invertible]
3 2 2 Y(M) = I(u,v)EM2 : u-=u -v 1 The mrphisn
~
:
, "% 2
2
such ~ t+h a t ~ f ( M ) (t)= (1-t , t(1-t 1) f o r each M€&
a mnoroorphism, but it i s not a local embedding a t the pht of with the elanent -1 of that
--
0
X(k) (To prove t h a t f. is a mmrphism, show (cf. 5 4, belaw) ; by the theoren of 5 4, 3.1 belaw, X is then an open embedding, and a l l that rmins is t o show
z/x : -X & x Ythat &z/x is bijective, 6
+
.
is associated
i.e. that f ( K )
is injective f o r each f i e l d K S )
"* I
I \
I
Y
Section 5
The Krull dimension of a m t h e r i a n ring
5.1
Definition:
Let
is the supreitnnn of the lenqths n irreducible closed subsets of T
T
.
G & n
, the
xET
local dimension of
W e write
dim T
X
, the
For each ring A
the Krull dimension of A
A
If
Let
... f: g,
B
, then we
... $ Fn
of
(resp. the local
. By 5 1, 2.10 , Kdim A po% ... $ pn of prime
A
. ThTh
5
be a subring of
Kdim A = Kdim B
is a chain of prime ideals of
.
A
-. T
B be a r i n g and let A
by Alg. c m . V, qi+ln A # q.nA 1 Kdim A Kdim B Conversely, i f of
dim T =
n of a l l chains
.
qo
$
is the infimum of t h e dimen-
and is written Kdim A
such that B i s an inteqral extension of Proof:
.
F1
dimension of the topological space Spec A is called
Proposition:
5.2
T Kt x
P@ , we set
i s the s u p r m of the lengths
ideals of
chains Fo
(resp. dim T) f o r t h e dimension of
T at x ). If
dimension of
ofall
x & T
sions of open n e i g b u r h d s of
of
be a topological space. T&dimension
T
B
.
, then
2, cor. 1 of prop. 1 so that
po$
... g pn
is a chain of prime ideals
can construct inductively a sequence of ideals qo,ql...
such that q . c qi+l 1
a d Anqi
.
B
( A l g . c m . V,
= pi
5
of
2 , cor. 2 of th. 1).
Hence K d h B
2 KdhA
5.3
Corollary: Let k be a f i e l d and let A be a f i n i t e l y gene-
rated k-alqebra without zero divisors. fractions of
A
Let
zation l
, then
poZ
... g p r
m (Alg. comn. V,
be a chain of prime ideals of
5
al,
...,ah(.)
A
3, no. l), there are elfments
algebraically independent over k B = kLa l,...,an]
Fract(A) denotes t h e f i e l d of
t r k d q Fract(A) t h e deqree of transcendence of
k
Fract(A)
Proof:
,
If
, such
. By al
the normali-
,...an
of
A
that A is an i n t e g r a l extension of
and pinB is the ideal of B generated by ao=O, (where h ( i ) i s an increasing function of i) Since
.
,
fi
I,
3, no 5
ALGEBRAIC
s m s
111
+
< n , and hence Kdim A = PinB P i + p (loc. cit.), it follows t h a t r of B K d i m B < n . On the other hand, since the prime ideals (all...,a form a chain of
p
of
have K d h A = Kdim B
w e have
A
Kdim A = Kdim A P Prcof:
, we
+ trkdeg FYact(A/p)
Using the notation of 5.3, set r=l
.
n
fi
(Alg. cam. V,
ththearen"
p r i m ideals q h ( l ) ~ q h ~ l ) - l ~ . . . ~ qofl
.
, po={Ol
Kdim A 2 Kdim A + Kdh(A/p) , since the second -s P of the lengths of prime ideal chains containing p
the "going dawn
2n
With the assumption of 5.3, for each prime ideal
Corollary:
5.4
n
B of length
and pl=p
. Evidently
is j u s t the supremum
. On the other hand, by
2, th. 3 ) , there i s a chain of A
qh(l)=p
such that
and q . n B = (al,...,as) for each i ; hence K d i m A > h(1) Ybreover, since 1 1 PA/p is an integral extension of B' = k [ ~ ( l ) + l l . . . r a 1 , we have n
.
t r k d q Fract(A/p) = Kdim(A/p) = Kdim B' = n-h(1) ;
so that n = Kdim A = h ( l ) + ( n - h ( 1 ) )5 Kdim A P
+ Kdim(A/p)
W e now turn to the problem of calculating the Krull dimension
5.5
of certain rings which are not covered by corollary 5 . 3 above. For this pur-
p s e we shall use another formulation of the m u l l dimension: Let
of
E
be a p a r t i a l l y ordered set. If
x E E such that a < x < b
called the deviation of
symbols
, +m
-m
partially ordered sets E
, set
decreasing sequence from E
2
dev[ai+l,ai]
n
5
dev E
, we
determine by i d u c t i o n on n
for which d w E < n : i f dev E =
-m
; if
E
the
is discrete (i.e.
t d n a t e s ) set dev E < O ; now suppose that we such that dev F 5 n-1 ; then
...
from E
such t h a t
for each i is f i n i t e . Finally, set d w E = +oo i f ,
for each n EN it is not the case t h a t dev E For example,
,
is Artin (i.e. each s t r i c t l y
E
i f each decreasing sequence alla2, n-1
for the set
a quantity dev E
W e shall assign to E
have determined the partially ordered sets F
set dev E
[a,b]
E : t h i s w i l l be a natural n m h r or one of the
. To define
a < b implies a = b )
.
,write
a,bEE
dev E = 0 means t h a t E
5
n
.
i s Artin and not discrete. Accordingly
I, 5 3, no 5
ALGEBRAIC cT33lmm
112 wehave d e v N = O
but dev Y = 1 and d e v Q = + o o (where iB
are assigned their natural orderings)
.
Z’
and Q
5.6
We rn list sane elementary properties of the deviation function:
a)
If f:E+F is a strictly hcreasinq map between partially
-.
.
ordered sets (i.e. a < b implies f (a) < f (b)) , then dev E 5 dev F For the truth of the assertion is obvious when dev F = ; suppse it holds when dev F < n ; we prove it for the case dev F = n Let alra2 r . . . be a decreasing sequence fran E such that de~[a~+~,a~] 2 n ; by the induction hypothesis, we then have dedf (ai+l),f(ai)1 2 n SO that the sequence f (a,) ,f (a,)
,...
..
is finite. Hence alra2,.
is finite.
If E,F are mn-enpty partially ordered sets, dev(ExF) = sup(dev E,dev F) We show that dev(ExF) 5 sup(dev E,dev F) by b)
.
induction on the pair (dev E,dev F) (the reverse inequality follms directly fran a)). Tb this end suppose that d w E 5 dev F = n ; the assertion is be a decreasing sequence fran obvious when n = -m If not, let xl,x2...
.
Ex F such that dev[xi+lrxi] 2 n for each i
.
. Let
xi= (airbi); if the
sequence x1,x2.. were infinite, then we muld have dev[ai+lrai] 5 n-1 and de~[b~+~,b~] 5 n-1 for sufficiently large i By the induction h p -
.
thesis, this muld imply
Let E be a partially ordered set and let Sc(E) be the set
C)
of infinite sequences el,e2,
.
...
fran E such that en is constant for
sufficiently large n If (ei),(fi) ESc(E) for all i Then we have
.
dev S c ( E ) = 1 + dev E
-,
, set
(ei) 5 (fi) if e i <- fi
.
. The assertion is obvious e=(ei)ESc(E), set e, = lhneen .
To prove t h i s we argue by induction on n = dev E
.
so suppose that n,O If Let a1-(ali , a2=(azi) be a decreasing sequence fran Sc (E) such that if n
=
...
d ~ [ a ~ + ~ ,2a n+l ~ J for each i dev[a(itl),rai,] aij = aim
. Were the sequence infinite, we would ha6e
< n for sufficiently large i ; choose j 0 so that
a(it-1)j = a(i+l)cc for j > j,
. Let
I, 5 3, no 5
ALGEBRAIC s m s
.
(ei) * (el,...,ejo,(ejo+l,. .) 1
be the map
113
. By
(a) , (b) arid the irduction
hypothesis we have
This shows that Sc(E) 5 1 n
=
dev E = 0
+ dev E
, so suppose that
. The reverse inequality is clear if
n is finite ard
...
infinite decreasing sequence bl,b2,
> 0
. Then there is an
frm E such that
dev[bi+l,bi] 2 n-1 for each i ; if aiESc(E) is such that aij=bi for each i , then dev[ai+l,ai] 2 n by the induction hypothesis, so that dev Sc(E)
2 n+l
.
Let Cr(E) be the subset of Sc(E) consisting of increasing
d)
sequences. Then dev Cr(E) = 1 + dev E dev Cr(E) 5 dev Sc(E) 5 1
+ dev E
. For by a) we have
; the reverse inequality is proved as in
c). If E is noetherian, each increasing sequence fran E belongs to Sc(E) arid
SO
to Cr(E)
.
Given a ring A and a module M
5.7
, let
-
dev M be the deviation iff of the set of sulmodules of M , ordered by inclusion. Then dev M = &{O}
. If
sA is the underlying A-module of A
, we write
dev A instead
of devSA (althoughwe are only considering canrmtative rings here, do not be misled into thinking that the notion of deviation is useless in the general case! 1
.
If N is a suhncdule of M , then dev
a)
M = sup(dev N, dev M/N)
.
For the map PI-+ (PnN,P/P nN) is strictly increasing, so d w M 5 sup(dev N, dev M/N) by 5.6 a). The reverse inequality is obvious.
If A is noetherian and M is a finitely generated A-module,
b)
= {O} such that, we know that there is ccanposition series M = M 2 M '...3Mn 0 1 is ismrphic to A/pi for some prime ideal pi of for each i , M./Mi+l 1 A (Alg. comn. IV, 5 1, th. 1). By loc. cit., th. 2 , we have dev M =
sup dev(A/p)
.
, where
p rhqes over the minimal prime ideals containing
114
ALGEEwac cEmEmY
5.8
I,
5
3, no 5
kt A L a (camutativd ncetherian rinq.
Propsition:
KdimA=devA.
Proof: To shaw that Kdim A <- dev A
, we m y
then enough to shm that dev A/p > d w A/q
assume that d w A <
whenever
,
s#O
p $ q . For t h i s purpse we may assume that . Then we have an i n f i n i t e sequence A 3 A s z ) A s 2
2
dw(Asi/qsi)
i t y is obvious when Kdim A
. Let
0
, that
such that
that
Kdim A
. If
Kdim A = Kdim(A/p)
s - ~ = o, then we have
is a f i n i t e l y generated A d u l e such
M
5 sup Kdim(A/q)
< Kdim A
(apply 5.7 b) , the induction hypothesis ard the f a c t that q2Ann M
, hence
if
q meets
sequence of ideals such that
Corollary:
dev(Ir/Ir+l)
metherian rinq A Proof:
Let -(A)
t i o n of
A
,
If x
Kdim A
algebra where F'
111,
I
5
- + Kdim(A/Ax)
of
.
assign the graded ideal
A
nAxn)/(I
nmn+l)
. Since
, where
T
@(A)
, clearly
IwGr(1) i s s t r i c t l y
, where
F
i s the set of
is obviously a quotient of the graded
is an ideterminate, we have d w F c dev F '
is the set of graded ideals of A/-
2
belonqs tD the radical of a (camutative)
3 , th. 2 and prop. 6, the map
&(A)
(A/&) [ T I
of ideals of
1
2, prop. 91, the sequence is f i n i t e .
increasirq. Accordingly we have Kdim A < dev F graded ideals of
, then
be the graded rirq associated with the (Ax)-adic f i l t r a -
. To each
CCPTITL.
5
2
Kdh A < 1
G ~ ( I )= By Alg.
Kdim A/q < Kdim A
Consequently, if I 31 I... is a
S).
Since S - ~ A is m t i n (mg. m. IV, 5.9
is f i n i t e and
s € A which do not belong t o any prime ideal p
dev M = sup dev(A/q) q3AnnM
if
. The inequal-
dev A < - Kdim A
; suppose then that
= SCO
S be the set of
such that
and the assertion follows.
= dev(A/q)
W e show finally, by induction on Kdim A
->
~ ( 0 ); l e t
3...
dev(Asi/ASi+l)
it is
are prime
p and q
ideals such that
s €q
+db ;
F'
Cr(E)
(A/Ax)[T]
. But i f
E
is the set
. The corollary now follows f r m
,
I,
5
3, no 5
115
Azx;EBRAIc SCHEMES
5.6 d) ard 5.8. 5.10 i
Let
Corollary: w
, and l e t n
m
A
be a noetherian local rinq with maximal
be a natural number. Then the following assertions are
equivalent. (i)
KdimAZn
(ii)
There is a sequence al,
A/C
the rinq Proof:
~a~
...,an
of elenents of
rn such that
i s Artin.
(ii)=> (i): This follows inmediately f r a n 5.9 by induction on n
(i)=> (ii): This is obvious when
n=O ; i f
n >0
,
m is not a minimal
prime ideal ( f i g . cam. IV, 5 2, prop. 9 ) . Hence there is an element
m which belongs to no minimal prime ideal.
ad. 5.9, we have, s e t t i n g Z = A / h l duction on n
S2,.
..,?I n
, w e may
.
(ii)
such t h a t
= Kdim A
.%/L.%i
-
1
of
Kdim A
. Arguing by in-
..,an are the al,a2, ...,a satisfies n
is Artin. If
Z2,...,Z n i n m , the sequence
Corollary:
5.11
Kdim
By the d e f i n i t i o n of
al
assume that we have proved the existence of a sequence
fran m/Fal
representatives of
,
.
a2,.
With the assumptions of 5.10, w e have
Kdim A < h/m2:A/m] proof:
If al,
then m = 5.12 ACB B
r
. If
lAai
...,an
are the representatives i n m of a basis f o r m/m
2
,
(by Nakayama's l m ) .
Corollary:
Let
the maximal ideal m
and of A
A
B be noetherh 1
4 rinqs such that
is contained i n the mima1 i d e a l of
m
KdhBZKdhA+KdhB/Eln. Esuality occurs i f
B
Proof: By induction -
is f l a t over A on Kdim A
of n i l p t e n t elements of
.
. For each ring
C ; we then have
C
, l e t rC
rA = rB n A
denote the set
. Hence we may replace
and B by B/BrA , which thus enables us to reduce the problan to the case in which r = O I f Kdim A = 0 , the assertion is then t r i v i a l A
A
by A/rA
.
. If
since m = { O }
(Alg. c m . IVr and Alg.
c(Hrm.
Kdim B/Bx
5
5
Kdim A > 0
, let x Em
be a non-zero divisor of
2 , prop. 10) , W e then have
IV, 5 2, prop. 10)
Kaim(A/Ax)
Kdim A/&
, Kdh B/BX 2
+ Kdim(B/Fm)
Kdim B
= Kdim A
-1
A
-1
(5.9
d
by the inductive hypothesis; the re-
quired inequality follows. If B is f l a t over A
,
x does not divide 0
B and a l l the inequalities above may be replaced by equalities.
in
Corollary: I f A is a m e t h e r i m local ring w i t h maximal ideal
5.13 m r -
K d h A = K d h A ,
i n the m-adic topoloqy.
is the ccanpletion of A
where
Corollary: If k is a f i e l d and T1,...,T are indeterminates, n
5.14
then
Kdim kCCT1r
Proof:
k[ ITl,.
..,Tn]]
...,Tn 33 = n .
is the cunpletion of the localization of
.,Tn) . The r e s u l t now follows from 5 . 3 ,
a t the ideal
(Tl,..
Section 6
Algebraic schemes over a f i e l d Throughout t h i s section k
k[T1,.
..,Tn 3
5.4 and 5.13.
denotes a f i e l d
and an R-scheme X , we write dim X o r dim X f o r di.rlX/ and dimxIgI , and speak of the dimension of 11 o r the xGiven a &el
6.1
R
.
local dimension of
X
Dimension theoren:
Let x be a
ard l e t
be the irreducible m p o n e n t s of
x
. Then
X1,.
-i s proof:
..,Xr
at x
point of a locally algebraic k-scheme
Clearly dim X = sup.dim X
. Let
IX_l
p s s i n q throuqh
be the prime ideal of
x i Pi which carried onto the generic pint of Xi by the map (5 1, 5.6 and 5 1, 2.10). If we identify Xi Spec dX + 1
X-
1x1
closed reduced subscheme of of
5
2, 6.11)
, dx/pi
2
carried by
is the local ring of
Xi Xi
X
ox
w i t h the
(i.e. t h e subschme Ri red at x
.
By
5
1, 5.6 we see
I,
9
3,
NGEBRAIC SCHE?ES
6
fiX
Kdim(0x/ p1, ) , so
= supi
0X
are the minimal prime i d e a l s of
that pl, ...,pr mim
117
that it i s enough to show that
_X
dimxXi =
.
dim Xi = Kdim(ox/pi) + trkdeg K(x) W e may thus assume t h a t
. It follows that & is a f f i n e , the
is irreducible and reduced. I f
theoren follows f r a n 5.3 and 5.4. I f not, l e t w be t h e generic p i n t of
1x1 . For each non-empty a f f i n e open subscheme
dim _V = trkdeg
, whence
K(W)
dim
I! =
V_ of
, we
_X
sup dim _V = tr deg k
then have
, which
K(W)
re-
duces the general case to t h e a f f i n e case. Corollary:
6.2
L e t Klk
be an extension of f i e l d s and l e t
a locally algebraic k-scheme. Then a )
a x € z BkK , if dimx(X-@kK) Proof:
-p:
Since dim
a(X@kK)
+
zl,...,X-n
Let
y
.
X
dim X = dim XBkK ; b) x &
is the projection of
g , we
&
3 carried by the irreducible
be t h e reduced subschgnes of
is the generic p i n t of and x E {xi}
passing through
3
passing through
xi
ziBkK
x@ kK containing . Mreover, i f yi
. The irreducible ccmpnents of
zi mkK
gi of
(6.1)
dim X . = dim X!
-1
-1
K(w')
Xi
f o r each
A
,
,
is a r i n g of frack and hence is an i n t e g r a l d a m i n .
K
A@ K
is irreducible and by 5 . 3
is an algebraic extension of
p i n t s of
XBkK
. For this purpose we my obvious-
dim X.63 K = tr deg Fract(A@ K) = tr deg FYact(A) = dim -1 k K k k K
not
is a f f i n e w i t h algebra A ; several cases then arise:
tions of an algebra of p l p d a l s over
If
XgkK
thus contain x and are irreducible c m p n e n t s of
is a pure transcendental extension of
Accordingly Z i g k K
such that
("forget" the irreducible CQTlPonents of
. I t is therefore enough to show that
ly assume that Xi K
then there i s a p i n t zi€XBkK
X.
-1
and apply 3.11)
irreducible m p n e n t
If
have dim X = Y-
X and t h e canonical projection x xi s surjective, we see that b) iinplies a ) . Now t o p r m e b ) .
x are then irreducible ccklrponents of the schemes ZiBkK
i
f o r each
= sup dim
C m p n e n t s containing y ; the irreducible ccenponents of
p(xi) = y
& be
and Xi
is a quotient of
, then
K(W) @
and we have dim -1 X! = tr deg k
K
p(w') = w K k
K(w')
, =
an3 i f
w'
because p
Xi
.
and w are the generic is open (3.11). Since
~ ( w ' ) is an integral extension of
tr deg k
K(W)
= dim
-xi .
K(W)
Finally, i n the general case, there is a pure transcendental subextension K'
of K such that KIK' is algebraic.
Then dim X! = dim X. @ K' = dim -1 -1 k
Xi
.
Corollary: Let f:g+x be a mrphism of locally alqebraic
6.3
k-schemes, let x
and l e t y=f(x)
be a point of
dimX < dim Y+dimxf -1 (y) x-y-
.
. Then we have
.
Equality occurs when f is f l a t a t x
at (Ix an3 0 are the local r h g s of g and Y x and y I and i f m is t h e maximal i d e a l of 0 dx/oXmy is the local -1 Y Y' ring of f (y) a t x ~y 5.12, MimOx 5 ~ a i m d + ~ a i m ( r /l O m ) , whence Y x X Y Proof:
By
5
1, 5.8, i f
.
dimxX
- trkdeg
K(X)
< dim Y Y-
-
- trkdeg ~ ( y +) dimxg-'(y) - trK(y)deg K ( X )
(6.1) and the required inequality follows. When f
is f l a t a t x
, the in-
e q u a l i t i e s may be replaced by equalities (5.12).
corollary: If f:X+x is a m r p h i s n of l o c a l l y algebraic -1 k-schemes, the function x-dim f ( f ( x ) ) is upper seni-continuous. 6.4
X-
, let
Proof: For any e ED\3 Take any x
i n the closure
Xe
xe
x E X w i t h dimxfl(f(x) ) > e
be the set of
of
and define &'
Xe
as the reduced sub-
3 carried by the Frreducible canponents X1,...,X9 of ye passhg through x For the mrphism f ' : 5' f(g')red induced by f we schane of
.
+
have by 6.3 dimxZ-'L(x)
2
I n order to prove t h a t x € X e -dim f '
dim
5'
(x) ?-dim
dimxf'-'f'
, it
f(X')red + dim 5' .
is therefore enough to show that
such that (sl)rea+ ding1 2 e . For t h i s purpose take Xi dim Xi . By 3.6 there i s a t€Xe"Xi such that g' i s f l a t a t
=
Hence 1
~n
the other M,
dim Z 2 e l i e i n
i1(f- (t) w i t h l_f(t) = dim f-lf (t)2 e . t- -
a l l the irreducible m p n e n t s
, so
that dimtf'
-
z
of
t
.
.
k
. Thena) __
Let & and
Corollary:
6.5
be locally algebraic schemes over
Y
dim_Xx_Y=dim_X+dimY_ ; b)
and
projections x
y
_Y
_X
I
1Lf
is a p i n t of
z
th&h dimzzr_Y = dimxz
_Xxu
+ dim Y Y-
.
Proof:
with
.
Since dimgXY_=s u p d i m X x Y , weneedonlyprave b) Wemy r assume that and are affine, and by 6.2, that k is algebraically
X
closed. By 6.1, w e m y replace
assume that
z is closed (i.e. associated with a maximal ideal of
f
-1
. If
3, prop. 1 (iii) , it follows that
5
By Alg. comn. V, k = K(Y)
,f: _X
XY
s (zxy)Xy(% + dimx& .
(y)
dim Y Y-
of X
extension of Proof:
Let
k
. Hence by 6.3
~ ( y ) 3)
K(Z)
If
dim&xY = dim Y Y-
of x
K(X)
.
and affine and contain x
, let
k
x is closed i n Since
,
A/p
( A / p c ~ ( x ) ), so p
is a f i e l d
for any _V
is algebraic over k
. conversely, ,
6.7
corollary:
6.8
If
2
K(X)
is maximal and
x is closed, p is maximal
5
3, prop. 1 (iii).
x of a k-scheme
X
is identical with k
K(X)
C l e a r l y one obtains a bijection of
g(k)
is said t o be
.
onto the set of rational pints of
Spec k # 15 \ : Spec k -.\XI Accordingly, we simply write Xfk)
by assigning t o each i;G(k) the image of the unique p i n t of
urder the mrphism
.
for the set of rational points of 6.9
. If
is a l o c a l l y alqebraic k-scheme, each locally
A pint
rational i f i t s residue f i e l d
-X
if
is a f i n i t e
& is closed.
Definition:
Propsition:
If
=
is a f i n i t e l y generated k-algebra. The
A
proof is ccanpleted by applying Alg. cam. V,
closed p i n t of
+ dimzf-1 (y)
be its algebra
A
of functions and let p be the prime ideal CorrespndiXq to x
is f i n i t e over
, whence
.
is a locally alqebraic k-schem, then a
X
is closed i f f the residue f i e l d
V_ be open
k =
o(_X 1))
i s the canonical projection, we then have
-t
Propsition:
6.6
point x
. W e may accordingly
by a p i n t of
z
k
g
.
is algebraically closed, and i f
5 &a
locally alqebraic k-scheme, the map P H P nx(k) is an isanorpkism of the
120
I, § 3, no 6
ALGEBRAIC G2umn-a
lattice of closed sets (resp. open sets, constructible sets) ~f
onto
the l a t t i c e of closed sets (resp. open sets, constructible sets) of the sub-
of 5
space &(k)
F i r s t consider the case in which
Proof:
of
runs through the closed subset
P
. We construct an inverse map by assigning t o each closed subset F X(k) its closure i? i n 1x1 . For F =?@(k) , so it is enough t o prove
__.
of
.
I&/
that P = PQXk)
if
1x1
is closed i n
P
/;I
affine open g i n 3 such that
8Q)).By
If
P
# PTD((k) then there is an
meets P but not
contains a point x which is closed i n mima1 ideal of
,
g
. Then
P ns(k)
P
(that is, associated with a
6.6 ardt 6 . 7 , it follows that x E_X(k)
,a
contra-
diction. The assertion about the l a t t i c e of open sets follows fran the above by passage
to ccmplanents. Finally, it is clear that each constructible subset of X(k)
, where
i s of the form P n x ( k )
Accordingly, we have to shm that i f = QnK(k)
then P n x ( k )
implies
1x1 .
i s a constructible subset of
P
P
P = Q
and Q are constructible in &
. By setting
U=PQ
, or
,
U=Q-P
we reduce the problem t o proving that U ng (k) = @ implies U =@ for con-
. But this holds when restriction on U . structible U
is locally closed, and hence without
U
Remark: Now that we know that under the assumptions of prop.
6.10
izl
the l a t t i c e s of open sets of
6.9,
that the theories of sheaves over
over
151
ng(k)) =
T(U)
each sheaf that
T'(U
1x1
and s ( k ) are iscmrphic, we see and x ( k ) are equivalent: explicitly,
is associated w i t h the sheaf
where U
i s open i n
131
.
T i over
X(k)
such
Since any mrphisn
of locally algebraic k-schemes sends rational points onto rational p i n t s , it
follows that, i f
k
is algebraically closed, the functor
3 + (X_(k),f l t X )
which is defined on the category of locally algebraic k-schemes and take; values in
akis fully faithful. where X_
(X(k),8 l X ) sets of Serre. If
k
X(k)
its
The gecpnetric spaces of the form
is a separated algebraic k-scheme are the algebraic
i s not algebraically c;osed, we obtain analogous results by replacing by the set of closed points of
true when
5
131
. Finally, proposition
6.9 renains
is an arbitrary k-scheme, provided one replaces X(k) by the
I, S 3 ,
M
6
&ALGEBRAICSCHEMES
set of locally closed points of
1x1
121
. Unfortunately, a mrphism of k-schemes
does not necessarily send locally closed pints onto locally closed pints: Let k again be arbitrary, and let
6.11 closure of
k
Corollary: f(E) :
proof:
xcr<)
-+
.
A mrphism of alqebraic k-schemes f:X_+Y
Since the canonical projection of
.
is surjective iff
~ _ ( k )is surjective.
it is enough to s h m that g @ and 6.9
be the algebraic
k
2
(x@&
onto zy
is surjective,
is surjective. But this follows f r m 3.8
Section 1
The d u l e of an embedding
1.1
Let
1
such that
subscheme of
j:z+y
i:&+Y_ be an embedding of schmes arid l e t
i s the composition of a closed enbedding
and the inclusion m r p h i s n of f
mrphisn
8v+j, - (Jx) induced
j-:
schene
of
be an open
by
i n Y_
-j , it
. If
9
i s t h e kernel of the
i s clear t h a t the closed sub-
d e p e n d s only on _i and not on
xi
. Accordingly,
we
denote t h i s closed subscheme by xi and call the f i r s t n e i q h b o u r h d of & (or of A) & Moreover, i2:yi +y denotes t h e inclusion mrphism and i.L:&+Xi t h e mrphism such that _i =&,il W e then have V and
.
L1=ll
xi i s obviously functorial i n 11 . For each c m t a t i v e
and & ' are embeddings, there i s a unique mrphism _h:y;
such that such that ; if
--j
*
The construction of
g
xi=-
.
-
= _i,f
and 3%; = i-2-h
. W e say t h a t
h
i s i d u c e d by
g - (resp. f ) i s the i d e n t i t y mrphisn, we say t h a t
g (resp. g)
.
W e make analogous definitions f o r -dings
Lama: - Let
1.2
k ,bea &el,
+yi &
h is induced
by
of k-schemes.
&:&+y
j:Z_-+T gnbeklinqs of
k-schgnes. Then the mrphism
i d u c e d by Proof:
L2 x -&: Y-i x
T -j
+
Y_xT is invertible.
W e imnediately rduGe t h e proof to the case in which Y_ = S k A
T = S&B , where
& and
j
,
are d e f i n d by the canonical projections of
A
5
I,
s
4 , no 1
m mwHIsMs
and B onto the quotients A/I
123
. Then
and B/J
u is induced by the canoni-
c a l isamrphisn of
(A@ B>/(I*B B k k
+
I8J +A@ J ~ ) k k
W e now reinstate the notation of 1.1. The quasicoherent &-mdule
1.3
-j*(!f/r ) 2
& and is denoted by is called the module of the &ding Clearly the canonical ismrphism Y/r2 +j*(wi) is invertible. Also, each point v € y has an affine open neiqhbourhooa V_ such that f i n i t e l y generated ideal, then the equation w . = O
open embedding. For i f
(by Nakayama's l m )
=0
U
of
v
. The mcdule
which the mkdding Now let
that
x€_X and v = j ( x )
-i'
v'
w
i
is an
implies that
i n a certain sense thus measmes the extent t o
be an open subscheme of 1' (1.1) contained i n q-'(V)
1':z'
is the c a n p s i t i o n of a closed anbedding in
j": dyl +ji(dxl) and i f
(Y)
0
+oi, -
y'
. Then
cx:T+%
factors through
W e shall say tkat the mrphism 2 ) = wi, -j'*(~'/~' -
g induces
then so is the square
. If
7'
such
with the
a mrphism of
V_'
into
is t h e kernel of
i s the-inclusion mrphism, then
3' and
j'*(r) -
is induced by f
~emna:
is a
for a suitable open neighbourhood
which for simplicity we also denote by g
1.4
(f//.rk=
.
fails to be open.
inclusion mrphism of
? * ( a ) :_s*
2
then
7IU =0
and
'y(LJ)
implies that
1
wi
if
i d u c e s a c m t a t i v e diagram
of
and g
i'*g*(g/!f2) = f*(w.) into L
.
If the c m t a t i v e square
(*)
-above is
~&&iy, -4
124
I, 5 4 ,
1'
Y! -L
L> y_'
'1
i2
xi---
Y
the induced m r p h i m
woof: if
(*)
f* (wi)
-
is Cartesian i f f
+
y'
wil
-
W i t h the assumptions of l
is the image of
m 1.4,
_f*(wi) +w;
q is a m m r p h i s n . The l a t t e r holds hy&~ conditions of l m 1.6 below.
.
4
in
g*(Y)
-
dvl -
.
But
is an isamorphim whenever is f l a t , and a l s o under the
Consider the diagram of schemes
1.5
where p&=Iq(
, go
of vanishing &mre,
is a closed subschene of
2.
z
defined by a z-ideal
a d -j is the inclusion mrphisn.
an embedding; mreover,
with
&s an epimrphim.
is an epimorphism, so are p and r
q
1
Z . = g a d the module w
By
2
52, 7.6b), _i is
of
-j m y be identified Defining so =gi-, we examine the mrphhns f:Z +Y_ such t h a t -3
j
the square
pf-g
cmnutes.
+?
LemM: The mp which assigns to f we mrphisn gi;(wi) induced by % ard f is a bijection of the set of $:Z+_Y such that &=q @ &%=f
onto. &z Proof:
,p -
(gi;(y
-0 For sirnplicity identify the space of points of
by means of kernel wi I
.
5 w i t h that of
xi
i-l:g+yi Accordingly $: dy; dz is an epimrphism, w i t h f (pi2)': +c9 is a sec€ion of &i we may then X X-i
and
.
-
identify
8
with
Yi
dx@wi -
in such a way that
/
respectively-by the canonical projection of
ox
i n dx@wi
Ll
Ox@wi
. This done,
-
and. PA2 are defined onto dX and the in-
since we have Z.=Z , f 3 necessarily factors G o u g h th; f i r s t neighbourhcd ; we again-write f
clusion mrphisn of
xi . The underlying map of
4 in
f o r the induced m r p h i m of
cides with that of go and the h&rphisn f f f f= 9- (since pf=c$ and _f- (wi) c
/JX
foll&s that
f induced by f-
f
xi
this f
:'2 dx@oi + go* (dz) s a t i s f i e s - go,Cf) (since &go=fj) . I t
(r)
i s uniquely defined by the mrphism 4 E ~ ( w i l g o *
. The mrphkm
+ I of the l
$(wL)
coin-
)
m is obvi&sly assign&
t o $I by the canonical bijection
(5 2,
1.21. Since $I
kma:
1.6
XI---=+x
is arbitrary, the l
m follows.
Consider t h e diagram of sch-s i
,f X =_
-
+
_p
. Let 1' be the f i b r e product u x2-I , q:Y_' -tZ anm the canonical projections, and :_XI +XI the mrphism with cmp-
such that p i =I%
p':XI
if
> n e n
1%'
-
. Then each of
the squares i n the diagram
i s Cartesian and the mrphism f*(wi) ible
-
.
Proof:
Observe that, by
5
2 , 7.6b)
,
+
wil
-
induced by f
is an -ding:
9 is invert-
the r i g h t hand square
is Cartesian by construction, and so is the middle one by 1.4. E5y reduction
to the category of sets we see that the m a r e with sides f , g, A 2-1 ' i ' = -i' and s2i,i=A is Cartesian. Therefore so is the left-hand one. Identifying
xi-
with
Sp(dX@ui) - -
as explained i n 1,5, w e deduce that; ! l
-
is
I, 5 4, no 2
ALGEBRAIC GEmErRY
126
Section 2
The module of differentials
2.1
Definition: Let g:x+g be a mrphism of schemes; for 1
take the fibre product
%/_s
:
R
2%
.
g x ,& , far 1 the diaqoml Trclllphi~n
8 xsg The quasi&herent d x d u l e wi is then denoted by and calied the d u l e of differentials ,f X over S -+
When k
.
is a k-scheme, we also write R
is a d e l and
T/k
for the
module of differentials associated with the structural projection pT:
$ -+
S_p k
. We then say that
Q
_xfi
k.
&er
Proposition: Let
2.2
e_:?-+X
a section of 9
e*("/s)
we -
--
.
is the d u l e of differentials of T
e:E-+sbe a mrphim
of schemes and let
. wen there is a canonical iscrorphism
Proof: Apply lemma 1.6 to the diagram
%/s
E
and to the Cartesian squares
For exarrrple, if 5 is the spectrum of a field k and x is the image of Spec k under g
, then
2
E * ( R ~ / ~ ) may be identified w i t h mx/mx
--
, where
I, 9 4, no 2
m
X
SMOOTH MLIRPHISMS
is the fiaxirnal ideal of
127
dx .
Consider the conmutative square of schms
2.3
Proposition: If the square
Y*(Q~~~I
+
(**)
abare is Cartesian, the mrphism
Qxl/s, is invertible.
Proof: Apply l
m 1.6 to the diagram
-
We now apply the results of 1.5 to the diagram
2.4
i b Z !3+Y -
-zO
%/s
- _x
XSs
-+
@l We m y assume that go is a closed subschm of
z defined by a Z-id-1 2
. Then with each $ €mod (g*(R ) ,8) *_Zo -0 2 5 4 there is associated a mrphism of schemes _f: z 5 xs& such that prlf = g-
of vanishing square; set go
-+
-
The first condition means that has g as its first and F J = 6 q g o ccanponent. Denoting the second ccanponent of & g + J, , the condition
.
-_ fj = 6x/sgo means that gj -- = Q+J,)j
Module 1.5, g+J,= cr f m y be explicitly constructed in the following way. 2Set 6 = 6 ; the space of pints of the first neighbourhood (5 XS& may
XIS
then be identified with that of
with
8X @QxIs , so that -
_ _
canonical projection of
dl
3 and the structure sheaf of (5
and ~~6~ onto
&
correspond respectively to the
dx
-
and to the inclusion mrpfisn
of
I, 5
AIXEBRAIC GFEMERY
128
ox-
in p e n
XIS
-
4,
no 2
.
The mrphism g2d2: X + X is then associated with a morphim dx + d..@Rz/s -6 of the form xtz(x,dx) , where d:dX+fR is called the universal deriva&t
z/s s . under these- conditions, g - +$
o$kelative to
f
has the same under-
lying map as -g , and \ g + $ l - : dx+ 2*(oz) is themrphism f x-lgl-(x) +$(ax) , where O E p & ( R X / s , & ( f ) ) is assigned to JI by the
-
canonical bijections
- -
We deduce from 1.5 the Proposition: g t
-
z9
2
9S
zo
mrphisms of schemes and let
2
be the
of vanishinq square. ThIh the map $ g (9" (RXiS) ) onto the set of - + $ is a bijection of GE -mrphisms -g ' : _Z + Xsuch that _qg' _ =qg - - and - _ g'IZ-0 =glZ --0 ' closed subscheme of
defined by a Z-ideal
( g + $ ) t $ ~ ' = g + ( J , + $ ' ) and
We leave it to the reader to verify that 2.5
When 5 and S axe affine, we m y reformulate 2.4
following way: suppose that
s = SJ
k and g = Sp -A
, where
g_+O=g
.
in the
k,AEg
. We
write R for the module of differentials of the k-algebra A , i.e. the A/k d u l e of sections of R over _X By definition, we then have _XIS RAjk= 1/12 , where I is the kernel of the k-algebra hcxmprphisn
.
a@b -a.b
of ABkA into A
. I@reover,
the universal derivation of
dx
into R _X/s (2.4) induces a IMP of A into RA/k which we denote by d-. This map d sends aEA onto the residue class rod 'I of l @ a - a @ l E I Proposition: The map d:A+R
i k a k-derivation. For each k-derivation D A/k of A into an A-module M , there is a unique A-linear map f: QAk+M such that D =f d .L)
.
Proof: Uniqueness follows f r m the observation that since the elanents 1@a - a w 1 obviously generate I
, the
R as an A d u l e . A/k As for existence, write E:AmkA + M for the map a@b +a.Db ; this map vanishes on I2 and induces the required map f on 1/12 da generate
.
This last assertion ties up with prop. 2.4 in the follawing way: Let P:AX be a k-algebra Ixmxmrphisn, J an ideal of C of vanishing square and
.
I,
5
4, no 2
129
SMOOTH PKIWHISMS
p:C->C/J the canonical projection. The k-algebra hcnamrphisms
are then of the form p '
that pp'pp' of
i n t o the A-nmlule
A
J
.
= p+D
, where
The upshot of Proposition 2.5 i s that
2.6
of a universal problem. Thus we may identify
Let
a)
(Xi)itI
i s a k-derivation
D
(RA2A/k, d) is a solution
d) with every solution
(R
of the same universal problem. A few examples:
such
p':A-tc
Ah'
be a family of indeterminates; set A = k[ (Xi)
Let M be the f r e e A-module generated by a family
3
and let D:AM
be the derivation such that
Since QAIk
is obviously a solution of our universal problem, we see that
(MID)
is the f r e e A-module generated by the dxi
=
ti
.
(a.) be a family of eleI %J A generated by these elements. +IsAda.) and D: A/a-+M thermp
Again l e t A 13e arbitrary; l e t
b) m n t s of
A
and l e t a be the ideal of
Let M be the
(A(a)-mdule
RUk/(aQm
, ,
derived from d by passage t o the quotient. Then
the universal problem with respect to A/a
d k
is a solution of
(M,D)
so that
Consider a multiplicatively closed subset S of
C)
A
. Let
M
7 S-lA@ARA/k and D: S-lA S-lQ the m p A/k A/k (s(da) - a ( d s ) ) / s 2 Then (M,D) i s a solution of the universal problem a/s -1 w i t h respect to S A and k , whence
be the
S-lA-module
S-lR
-+
.
F r m this it follows i n particular t h a t for each k-scheme
x of
I! , the
stalk (RX/k)x
of
R
X/k indeed, i f we assume X -to be affine,
algebra d)
O(_X) .
5
and each p i n t
my be identified w i t h R
c",
-.
Wk m y be obtained by localizing the
Consider the ccinnutative square of rings
.
130
Au;EBRAIc
The reap a ~ d J(a) i
GECMETRY
is a k-derivation of
a k-linear map R$/+: RAIk are the canonical maps,
+
and by 2'. 5 induces and $ , j induces an isarorphism Q+: QAkgkt' RB,e
RBB/e
fi$,$
i n t o fi
A
. I n the case whereB/ pB = A g k e
( 2 . 3 ) . Moreover, one v e r i f i e s d i r e c t l y that the B-module
fiAjkgke and the
a g k x +da@kx form a solution of the universal problem with
derivation
.
t
respect to B and
X
k t k he a d e l and l e t
Proposition:
2.7
be a locally
algebraic k-scheme. Then f o r each a f f i n e open subscheme V_ E f
4(v)-mziiule
- (v)
we have fi
Proof:
(v)
.
.
. Since
R d,(y) /k
Z/k ,Xnl/(all.. ,ar 1
k[X1,..
2 ,
is f i n i t e l y presented.
(5 3 ,
1.7)
, we
Ok(y) is of the form
need only apply 2.6a) and b) to ob-
t a i n the desired conclusion. In Section 4 we shall need a corollary to Prop. 2.7 which we
2.8
prove; namely, l e t C be a k-model,
an ideal of
J
~ 3 w
C
.
of vanishing
the canonical projection, and 0 E_X(C/J) Suppose that we n are given a p a r t i t i o n 1 = 1i=l x.f of unity i n c and sCane 5 , E X ( C ) 1 i p:C+C/J
square,
such t h a t p
f
i
(Si) EX(C JJ
fi f l
Then there i s a t; Eg(C)
coincides with the image of
0
.
such that p(5) = n
gij be the image of ti i n g ( C f , , ) l j ard LJij denote (% C ) and ( S p C I f ,
To prove this assertion, let
R =q'*
(R
X/k
ui
and let
)
of which are open i n SJ C X-(Cf,f / J f . f 1 i j i j
c!!
= t;fj + Q i j
T:~ H % ( R / u , J I ~ ) 1
H
--
((IJJ),T)
$i E t;! . = ij
=
. Since cij
, by 2.4 there are
and
'Jij E
. Clearly the family
tji
fi
17
(yi)
it w i l l f o l l a J that
i
I j
, both
,) such that
11
is a 1-cocycle of the sheaf of
spc
. If we can show that
IJJ~/IJJ~ -ajlyij
J ' =~ ~
, 'here
= 5,# t + i , then we have ( ~ j y ~ . ~If ~t;il s~a t ~i s f)i e s -i t;' & is a local functor, there is a j i i n the above notation. Since
t; EX(C)
w i t h hmge
c;
i n X_(C ) fi
for a l l i
. This
.
; set
have the same irrage i n
(RIUij,J(v qij
($. ,)
f o r the covering
o , then
f
f o r each i
5 is the required
solution. It therefore remains to Show that H1((ui)rT) it is enough to show that
=
0
. By 5 2 ,
1.10 and 8.2,
3“ is quasicoherent. To do this we choose a cover-
ing of Sp - C form4 by affine open subschemes _V such that the images are contained in affine open subschanes of ,X . By 2.7 ,
q#
(c)
is then a finitely presented O(v)-moduIe. It now follows f r m Alg. cam. 11,
5 2, prop. 19(i) that the sheaf Fly
: _w
JIg
+-%(n(g--JE,
-
is quasicoherent. 2.9
f:Z-+Y
Proposition:
g:_Y+z are mrphigns of schemes,
then the smence
of E ~ X &
- is exact.
-
Proof: First we reduce the problem to the case in which X = S p A , Y = SpB -Z = Sp C , f = Sp - 4 , 2 = Sp $ It is then sufficient to show that, for each A-module M 0
+
-
,“the sequence
HQnA(QA/BrM)
-
.
Hy(QA/CrM)
-
---+
HQ”A(QB/C
A, M)
is exact. But this is a consequence of the fact that the sequence 0 --+ DerB(A,M)
Derc (A,M)
D”rc (B,M)
is exact, where Derk(K,N) is the module of derivations of the k-algebra K with values in the K-ndule N
x a pint of 5 with for the vector space (Q ) @ K ( X ) Z / k x L”,
Let k be a model, g a k-schae,
2.10
residue field Over
.
K(X)
.
Proposition:
K(X)
If
is a point of 5
_X
. Write
Qm(x)
is a locally algebraic schane over a field k
, ths
[QZ/k(x):K(x)
3 ?di”,X_.
x
,
ALGEBRAIC ( 2 x m T R Y
132 Proof:
of
X
R- - (XI X/k
, we may
that
=
(5
W e a l s o know that di",rs = d i m 2 by
. It follows f r m 2.3
B w~ i t h projection ~ x
4 4 , no 3
k and l e t E be a p i n t
be an algebraic closure of
Let =
I,
3, 6.2)
. Replacing
3 by -2 and k
assume k to be algebraically closed. I f t h e required in-
equality is f a l s e ,
Qz/k
w i l l be generated by less than n = dimxz sections
over sane open neighbwrhood
U of
. Choose a closed pint
x
which belongs to a l l the irreducible ccsnponents of
in U
y
5 passing through x
.
.
But this contraThen we have dim X 5 dim X = Kdim d and Rx2x/k(y) m/m2 X' Y Y d i c t s the inequality [m : K ( y )3 2 Kdhl dy- ( 5 3, 5.11) . J
rm2 Y
Section 3
Clean mrphisms
3.1
Cleanness theorem. s t
x be a point of
mrphisn,
x
f:g+X
be a l o c a l l y f i n i t e l y presented
and let y = g ( x )
. The follawinq assertions
are then equivalent:
_X/x(X)
fl
= 0
.
(i)
fi
(ii)
There i s an open subschew _V
=
,x/x @sx
the diagonal mrphisn 6u/y:
--
K(X)
g+
x-
x U
the following property: f o r each scheme _Z r
the equations gIZred =
( iv)
For each local ring
containing x
such that
is an open embedding.
There is an open neiqhbourhooa _T! o f
(iii)
g,h:Z!
of 2
x
in
_X
which has
and each pair of mrphisms
-hl-Zred and (flrl)g=(f/Y)_h C , each ideal I pf C of
2=h .
vanishing
square and each catmutative square
9 )I are local hamanorpkisms, there is x:dx*C such t h a t $=xcx _and c a n o x = $ .
i n which @ phism
a t mst one hommor-
I,
5
4,
SMOOTH PDWHISMS
3
l-0
(V)
ox/my (Ix
133
is a sewable f i n i t e dqebraic extension of
Proof: Wemy assume that -
K
(y)
.
.
and y = Sp - k Since RA/k is a f i n i t e l y generated A - d u l e , the equation R (x) = 0 is equivalent t o x = s A
$4
z/x ) x = 0 , or again, to the existence of such that R z/x =;I Q-U/Y_ = 0 .
an open neighbourhood _V of
(R
Replacing
(i)+(ii):
that RAA
. Our
= 0
2
by the open subschane
1
b e , we m y assume
statenent then follows by 1.3.
_s and -h
(ii&=+(iii): Let _m:_Z - + g x U be the mrphism having ccnrponents Y-
u/z . Since
Then r ~ 1 - Zfactors ~ ~ through 6
-Zr&
has the same underlying space as _Z
.
whence g=h (iii)+(iv):
x,x'
If
x
,
6
94
.
is an open anbedding and
_m factors through
6uv/y
- -
,
are two hancenorphisms satisfying assertion (iv) ,
apply (iii) to the p a i r of ccarrposite mrphisn~
(iv)=+(i):
c be the local ring a t x of the first neighbourhood
Let
x
+ X of in hence C / I =
X xL.X
ox
. Then we have
(2.4) ; l e t
C = ~x6(Rx/y)x
and we may set $I =Id
,
+ =gx , L
-
where g:X - -6 - + -Y
x,x': Clx=tc
canonical mrphisn. The local hcmamrphim
I = (f?g x ' x ;
associated with
prl0 62,pr2062 : x 6 z g x then s a t i s f y assertion (iv). Thus we have Since the dx-mcd.ule
is generated by the elements dg
(Rxx/y)x
6 cdx (2.5) , it follc&-that
BBkB
= (3
X/Y ) x
= 0
.
=XI
x=x' . (5) -x (5) ,
By 2.6 we have
(v)====+(i):
here B
(Q
is the
/m 4
X
Y
X
. If
B
is a separable f i n i t e alqebra over
~ ( y ),
is semisiqle and the kernel J of the c a n o n i d map B@lkB-+B satis2
fies J =J (i)==3(v) :
. Hence
R
2
B/k
S e t Y_'=
=J/J = O
sp
.
and (Rx/,)x=O
~ ( y )and
=f
-1
(y)
. Then we have
134
Au;EBRAIc
with the above notation. Replacing = Szk
assum t h a t
X
I, 5 ' 4 , no 3
c.zmErRY
by 5'
for saw f i e l d k
, we
and Y_ by U_'
. By 2.10
, we
may then
then have dimx& = 0
and dx are f i n i t e algebras over k ( 5 3, 6.1 and 6.6) . If -kso tish aant algebraic closure of k , it remains to show that dX"k'; is a product of copies of k . Now, f o r each pint
,
K(X)
with projection x
Since that
x
o
=
I
we have
-X , we
is a closed point of 2
R- -(x) = m-/mX/k x x
, whence
Definition:
3.2 be a point of
. Then
x
Let
5
.
-
(2) = k
m - = ~ and X
is said to be clean
presented and sl,/x(x) = 0 By 3.1 and
K
k =J-X
. By 2 2, .
f:s-+x be a mrphisn of
there is an open neighbourhDod 1
to be etale a t x
have
Olf x
. Lf, mreove.rI
schemes and let x
(ornon-ramified)
such that
3, 3.13, the set of p i n t s of
fig
I!
is etale (resp. clean) i f
. If
,f:
k
Cg , we
st x
if
is locally f i n i t e l y
f is f l a t a t x
,
is said
f
x a t which f is etale (resp.
clean) is open. W e say that f is etale fresp. clean) i f clean) a t each pint of
it follows
f is etale (resp. of k-schanes
say t h a t a mrphisn f
is etale (resp. clean). we aim say that a
-X is e t a l e (resp. clean) i f the structural projection
k-schm
gx: & + 9 k
is an etale
is etale (resp. clean). Finally, i f
(resp. clean) k-algebra i f the k-scheme
A
e& ,
!~J~A
we say that A
is etale (resp.
clean). Proposition:
3.3 Proof:
i s the inverse image of the diagonal of
Ks(f,q)
mrphism
Given a diaqram of schemes
52' Xzy with ccanponents f and 4,
-
f
-z
h -+
z
where
U_ xzy under the
. Now apply 3.17ii) .
.
Corollary: Let p:Y+z be a clean mrphism and _s a section of p Then - _ _ s is an open gnbedding. -
I, § 4 ,
-Prcof: -
4
SMCXt'H P.IDRPHISMS
s induces an ixmorphisn of
. I-f
a p i n t of _X
onto K s ( % , I f $ )
_u is etale a t x
, then
the map
(nu/) x:
u* ("/z)
Proof:
W e may assume without loss of generality that
+.
--
.
-
k t _x-!&ysz be a diagram of schemes and x
Propsition:
3.4
&
135
("/z)
1s bijective.
--
Q
is etale. Now con-
sider the diagram
Since t h e right-hand square i s Cartesian and
(u E l ) * ( Q y /-z )g prl
embedding,
6$/:/y(~incl)
4.1
-
34
is an open
mi be identified w i t h R_x/z ; the assertion
Definition:
X , we
-
WhCl
follaws.
3Icoth mrphisms
Section 4
x of
-
and 11 xzg is invertible (1.4). Since 6
induced by
-
+
g xZg is f l a t , the mrphism
say that f
Given a mrphism of schemes
L:?
+_Y
and a p i n t
is smooth a t x when the following conditions
are s a t i s f i e d : a)
of
there is an open neighburhood _V
x such that fig i s f i n i t e l y
presented; is f l a t a t x ;
b)
f
c)
tsl_x/_y(x):
f
K
(x)1 2 dimxi' (f (x))
.
is said to be m t h i f it is smooth a t each p i n t of
X
. If
zllxm3mc c2xMETRY
136
is a Cartesian square of schemes and x'
4' (x')
(2.3).
=
x
is a pint of 5'
5 4, no 4
such that
, then we have
Hence
and similarly
-1 -1 dim f (_f(x))= dimxI-f' ( f '
( X I ) )
X-
(5
I,
is snooth at x' ; and conversely, if ct f ' ( x ' ) Given k point x
f is smooth at x , t h s g' g' is smoth at x' and g is flat
It follows fran 5 2, 3.2 that f i
3, 6.2).
, then
f
is snooth at x
.
€5 , we say
that a k-scheme & is k - m t h (resp. k-smoth at the if the structural projection -px: & + SJ k is s11l0ot.h (resp.
-
no risk of confusion, we shall also say "smooth" instead of "k-smooth". Suppose in particular that - k is a field and k' is
mth at x ) . When there is
an extension of k
. The residue fields of points
~ ' € 5k k8' projected
onto x are them the residue fields of pints of S i K(x)Nkk' and 5.7).
It follms that, if k' contains
K(X)
.
(5 1,
, then there is a
5.2
rational
point x'CgBkk' which is projected onto x By what we have already said, it follows that, to verify t h a t g k - m m t h at x we need only verify
that 4.2
XBkk'
is k ' - m t h at the rational point x'
...,Tr]
Let B be a model, A = BITl,
n d a l s in r indeterminates and
g=
.
the algebra of p l y -
. We knm that
R is a free A/B (2.6) ; accordingly for each x € 5 , the A-module with base dT1, ,dT, r a g e s dTi(x) of the dTi. form a base for QxjB(x) = RA/B8A K (x) If PEA, write @(x) for the h g e of dp in R (x) Then we have X/B *(XI = 1~ aP~ ( x ) d T ~ ( x ) i
...
Sz
A
.
.
137 where
-€ A aTi
Smoothness theorem: x a pint
schemes,
ap and ,,i(x)EK(x)
L e t f:X+!j of X g&
.
be a locally f i n i t e l y presented mrphism of
y=f(x)
.
(i)
f is smx~tha t
(ii)
There is an open neighbourhood
and a mrphism q:U_+Y_x
on
x
. Consider the following assertions: x
,a
natural nLmJ3er n
which is etale a t x and s a t i s f i e s
For each local rinq C€&
(iii)
of
, each
ideal I
of
C
=
K,OCJ
.
of vanishing
square and each conmutative square
where @
x: ox +C
and
are local hammrphisns, there is a local hmmmrphism
)I
such that
Ji=x-fx
&
~ a n o x = @
.
There are affine open subschemes _V,_V
(iv)
..
..
of _X,y
such that
f ( 1 - 1 ) ~ y , plyncmials pl,. ,pS€ 8 ~C T )~ , ., T ~ I and an open anbeddins g * y(pl,. ..,pS) such that the matrix (aPi/aT.7 (-h ( x ) ) i s of rank s -f IF = p r o 5 (Ebeing the canonical projection of the subscheme V_(P1,. ..,P S ) of yxgr defined by P1,...,Ps onto Y .
h: -
ox
(V)
and
-
...,TnII .
p b r a of f o m l p e r series d “T1,
Y
Then we have holds when
oy
(i)<=> ( i i ) < = > , ( i i i ) < = > ( i v ) (v) <=
i s metherian and
K(X) =K(y)
ox A
(Iy are ncetherian and
i m r p h i c to an al-
. The implication
.
The proof of the STTDOthness theorem is deferred u n t i l Section 5
(i) =>(v)
. I n order to
reduce it to manageable proportions we have l e f t out m y details: we advise the reader to approach it only when he is feeling particularly industrious! 4.3
schemes
Corollary:
f:X-+y is
The set of points of 5 a t which a mrphism of
stcoth is open.
Proof:
By the equivalence
(i) <=> (ii) of 4.2.
and Z Y-+ Z-
Corollary: Given mrphisns of schemes f : X + Y
4.4
if
,
& g is m t h a t f(x) , then gf_ is smooth a t
f is mth a t x
x .
Proof:
Apply criterion (ii)of 4.2 and the obvious f a c t that the composition
of two etale mrphisms is etale.
Corollary: Given a locally f i n i t e l y presented morphim of
4.5
schemes
f:_X+Y , the
follminq assertions are equivalent:
is smooth.
f
(i)
+
(ii)
For each model
each
w EY(C) and each
uEX(C)
Proof:
, each ideal vE&(C/I) , L f f(v) C
and f ( u ) = w
such that
.
I =
of w
C/I
of vanishinq square, E Y_(C/I) , then there is C
Assertion (ii)msans that, given a CQrmUtative diagram
#
%(C/I )
i SEC
W
#
2
.
X such that u# = v and _fu#=w# If C is a #local ring and u ,w send the closed points of @(C/I) and SJ C respectively onto x t & and y , then assertion (ii)is equivalent to assertion
9 C there is a u:%
can
-+
if
(iii)of the smoothness theorem. According to this theorem, then, _f
srooth i f f (ii)holds for each local ring C Conversely, i f
f
5 0 %
: %(C/ I )
enough to show that:
if
SP
* (Sp C L
placing X by the C-schm
c
C)X$+Sp C such that the ~csnpo# Rehas cm&nents can and v
: (SJ
3 T with structural projection
2 is a smooth C-schem and
I
of vanishing square, then the canonical map T(C) +?(C/I) To prove this last assertion, l e t
C
r
1 2
(ii =>(i) ,
is srooth, the existence of u# i s equivalent t o the
existence of a section 5 of f sition
. I n particular
l e t 11 be the image of P
-sp f
is
r
it is
ideal of
C
is surjective.
. For each prime ideal p of T ( C /I ) . By (iii)of the smmthness P P
q ET(C/I)
11 i n
.
I,
9 4, no 4
SMOOTH MOWHISMS
139
.
theorem, there is a 5 E ;(C which is projected onto ?-I By 5 3 , 4.1, P P P there is an f EC , f fp and a SfEx(Cf) which is projected onto the image of rl in x(Cf/If)
qf
. Fran these
such that the n which is projected onto
(9C)
fl,...,f
q
- .
cover Sp C The existence of a 5 E;(C) fi ET(C/I) now follows frm 2.8.
Corollary: Given a model k and a locally alqebraic k-schm
4.6 X
f's one can choose a sequence
, the following assertions are equivalent:
21 isk-smooth. (ii) For each Cc 4 and each ideal I the canonical map &(C) -+ x(C/I) is surjective. (i)
of
C of vanishing square,
Corollary: Let f:g-+x be a locally finitely presented m r -
4.7
~
phism of schmes. The following assertions are equivalent:
.
(i)
f is etale
(ii) (iii)
f is smooth and dimx_f-'(f_(x)) = 0 for each xE2
-f
.
satisfies assertion (ii) of 4.5 and the element u of that
assertion is unique.
z
For each schane 2 , each subschme 2' of defined by a nilpotent ideal, and each pair (a,%') consistins of mrphjsns g:_z-+u _g ' : Z- ' + X- such that _ f s _ ' = g ( c, there is a unique _h:Z-t_X such that -g = -f h(iv)
and
h(F' =g' .
Proof:
(i) => (ii): T h i s follows imnediately fran the definitions. -1
(ii) => (i): If dhXf
(_f(x))= 0 , then n = O
in assertion (ii) of the
snoothness theorem. (i) => (iii): The existence of u follows fran 4.5; uniqueness follows fran assertion (iii) of the cleanness theorem (3.1): for if u and u' satisfy the conditions of 4.5(ii), then u#=u'# by 3.1, whence u=u' (iii) => (iv): If
2 is the ideal defining
cmtative square
. p q
2'r
x
T
.
,Z' , we must show that, for each
1 40
ALGEBRAIC GFcMETRY
such that q; = ~ ' ~ + ~ l 2 W e may assume without loss of generality that
there is a unique
.
fg;+l=gr+l
-
y(#r+l)* _X
case, condition (iii)inplies that, _Z
%I
I,
, there
. H e n c e there is a % This h is the required mrphisn.
-_
have fix,,y=O
,
t h e uniqueness f o r each V_
. By 3.1
is smooth (4.2)
f
,
.
also
(i) follows.
and
Corollary: Let k be a d e l l k '
4.8
and
-
(iv) = > ( i )(iv) : implies (iii), so
V ( f and )
2 = O - 11Inthis of
_hv: _V+X such that -%=g[u , then _hv=-$ly by v i r t u e of unique _h:g '5- s&h that &=&lV_
Of
4 , no 4
f o r each a f f i n e open subscheme
i s a unique mrphisn gnz' - = g_' I U_n z -l I f VCrJ
.
9
a k-model which is f i n i t e l y
generated and projective a s a k-nloclule, and X a l o c a l l y algebraic k'scherne If X is each f i n i t e subset of which is contained in an a f f i n e open subset. mmth over
proof:
7 7 - X
k'/kBy the construction of
. W e must show t h a t ,
I _
k
, then
k'
is m t h over k
9
( v k X ) (C) + ( v k X ) (C/I)
But by the definition of the functor X(CBkk')
+
X((C/I)@kk')
is l o c a l l y algebraic over
I X k'/k
and each i d e a l I of C of vani-
for each C €-IY
shing square, the map
,
1, 6.6
.
, which, by
/k
i s s u r j e c t i v e (4.6).
t h i s map may be i d e n t i f i e d with
4.6, i s surjective.
Corollary: Let x be a point of a s c h m
4.9
l o c a l l y alqebraic over a f i e l d the local r i n q
Proof:
dx
k
,
If X
which is
is m t h over k & x
,then
is an i n t e q r a l domain.
6y 4.1, there is a r a t i o n a l pint x E X @ k ~ ( x )which is projected
onto x
. Since
8% is
f a i t h f u l l y f l a t over
is injective, and similarly f o r O2 is an integral d m i n , so is @ x ' Remark:
A
+
dE
A similar arcjument s h m s t h a t ,
dx , the
. Since 8,
if
4
hcxmmrphism K
[CT,,
(XI
is stmoth a t x
ox Jz -+
....,Tn 11
, then
the
ox
i s regular, i.e. is of f i n i t e hawlogical dimension. For since ring is f l a t over dX , we have
8%
f o r a l l dx-modules
M,N
. If
w e know that
K
(x)"T1,
...,Tn 11
has hamological
5
I,
4, no 4
141
, we
dimension n
for
i>n
have
, whence OX
for
i>n
Tori (M,N) = 0
.
Given a reduced locally algebraic scheme over a
Corollary:
4.10
perfect f i e l d k
, the
set
1
of points x E.X
open and dense i n
X.
Proof:
is open. To show that
By 4 . 3 , _V
assume that
g
s b that _V
X_
n
K
where A
. For
such a p i n t
is of dimension
Iv&lK(<) ( C ? K ( 5 ) , k l ~ ( < ) )
(5)/k
n over
K(<)
is reduced, we have
K(S)
is of dimension
f a c t that
x
Section 2. Hence
5 is
smooth a t
obviously
5
.
. Since
K(<)
is separable over
n over
K(S)
, so
(Alg. V, =
5
0 , so 5
that
9, th. 2 ) . I n view of the
that
R
-
K([)/k - "z/k)< by
The product of two irreducible schemes over an
Corollary:
4.11
11, we m y
5 , the transcendence degree of t h e
~ ( 5 )over k is n = d h X 5
residue f i e l d
,
is smooth is
is dense, that is, contains the generic pints of the irreducible
canponents of k
is dense in
X
is a f i n i t e l y generated k-algebra. W e now
-
= SpkA
a t which
algebraically closed f i e l d is irreducible.
Assume w e have proved the assertion f o r the product of two a f f i n e
Proof:
irreducible schemes.
Let _X and
we have to show that i f
W_ and
Wl
be two a r b i t r a r y irreducible k-schemes;
are open and non-eqty i n g x y
, then
their intersection is also non-empty. Now obviously there are non-empty that
w
meets
Since _Vry is
, u' ,
y , V_' , _V" of y such v,xy , y1 meets ylx yI and g"c u n g l and y"c ynXt irreducible, _W n (usIT) , and hence W , meets y'' ;
a f f i n e open subschemes V_
similarly w_' meets
w n (KJ'x y ) r g ~ n
( ! 1 I
yl'
x
x
,V"
,'I
. Since
of
X_ and
y"x v_l'
is irreducible, it follows that
ytl) i s non-empty, and s i m i l d y f o r _w
W e now turn to the a f f i n e case. Since we m y assume that
duced, it is enough to show that, i f
A
5
and
.
are re-
and B are two k-algebras which
are b t h integral d m i n s , then t h e product
.
ABkB
is an i n t e g r a l d m i n .
1 42
ALGEBRAIC GEKNEIRY
For this purpose w e may obviously assume that
A
A
A
If
i :A + A
m
injection of B[[T l,...,T
n
is the canonical map (which is injective:)
11
f :X
...,Tn ]]@,,B , which,
,
(@i) akI$
is an
as a subring of
is an integral dcsoain. Hence A @ B is an integral damin. k
+Y be a dominant mrphisn of
k-schemes which are algebraic, irreducible
and reduced. Then the set of points of 3 a t which f anddensein
Proof: If
and an ismrphism
Corollary: L e t k be a f i e l d of characteristic 0 and l e t
4.12 c -
A
.
k[[T1,...,Tn]]
A@JkB into k"T1,
4 , no 4
is f i n i t e l y generated. By
4.2(v) and 4.10 there i s then a maximal ideal m of
@ : Am
5
I,
5 . is said to be dcaninant i f
Recall that f
x and y are the generic points of
5 and
residue f i e l d s
Since
X
are reduced, the local rings of
that
K
is smooth is open
K(X)
and
K(Y)
sl,/p
=
--
_f(g) is dense in
and ,U
, then
f(x)
,
f
. Since
x and y coincide with the
. It follows frcm the r e s u l t s of Section 2 =
K
(X)/K (y) '
(x) is a separable f i n i t e l y generated extension of
Since -€ is f l a t a t x
= y
1x1 .
is m t ha t x
K
(y)
, we
have
. The r e s u l t now follms frcan
4.3.
4.13
Corollary:
snooth a t x tx -1 dhxf (f(x))
.
, then
*
Given a mrphisn of schmes
_ _
i s a free
0X-module
f:X+Y , If f i s
of rank
I, § 4 1 no 4
SXOTH MIRPHISMS
143
Proof: With the notation of 4.2 (ii) , i f we set 2 = g 0n , then i s a free 0 9 -module of rank n (2.6a)) . Since -g*(Rz/u) x
(XI
by 3 . 4 , we see that
is free of rank
(R,&x
n
1x -
. Setting -y =g (x) ,
corollary 9 3, 6.3 applied to the mrphism g ’ : f-’(y) implies that dimf
(fi5/x) g(x)
+On -K(Y)
induced by 4
I
-1
(y) =dim On = n , g’ k 1 - K (y) -
X-
and the assertion follcws.
Corollary:
4.14 k-schemes
f:g+Y_ , a
x
k - m t h at y
point x o f _X
, then
m t h at
(i)
f is -
(ii)
The map ?/k(’)@K
jective.
, a mrphism of algebraic y = f ( x ) , if X g k - s m o o t h
Given a rrcdel k
x
,
.
(y)
K(X)
R
-+
_X/k
(x) induced by
Proof: (i) => (ii): By 2.9 we have an exact sequence “Y/k’y -
the functor ?
@
52
ox
K(X)
X/k
%y x‘
at
the following assertions are equivalent:
is in-
’
* (‘_X/k)x +. ‘ R ~ / $ x-+
transforms t h i s sequence into the sequence
,:
( y ) @ K(X) K (y)
-+
52g/kw
+
x/ll fx)
R
-+
0
.
, the three
terms of -1 this equation are identical with dimx( ( & @ k ~ ( z ) ) dimxg (y) and But if z
denotes the canonical &rage of
y
in Sp k
,
dim fY@ K ( z ~ ) respectively. The desired q u a l i t y now follows f r m Y- k
9
3 , 6.3.
144
I, § 4 , no 5
ATXEBRAIC GECBETRY
f i n i t e rank (4.131, assertion (ii)means that an isomorphisn of
(9 ) @ -Y/k y 6y 3, prop. 6 ) . Since
comn. 11, 8 each k-derivation of vation of
dX
Y
)
@
Y/k y 8 onto a direct factor o?
8x
.
. If
D:
dx-+ I
: dx +C
X I
canox' = 4
such that
is a k-derivation extending )I -x'fx
Remark: The vector space over
t
G x ~ k ( X ) = k'kd -K
(XI (n_X/k(X)
I
K
x iff
K
space a t y
(v) = > ( iwe ) :know that
of
Z-'(y)
( f i g . m. 111,
at x
and ifix
8 5 , prop.
dx
is f l a t over
4). If
ax A
3
K(Y)
-
dx= Jx/Jxmy
is its maximal ideal, we have
( 2 . 2 ) . These iscmrphisms a d the isanorphisn
*lY
f
. W e may rephrase is smooth a t
.
5.1
Y
into
t h e map x = x ' + D
g at x
(x)= K (y) by saying that
Proof of the snoothness theoren
0
.
I
Y
K(X))
Section 5
over
(using t h e notation
induces a surjection of the tangent space a t x onto the tangent
f
A
there is a
(x)
is sometimes called the (Zariski) tangent space of
cor. 4.14 i n the case i n which
I
is then a k-derivation of
X
is a hammrphisn satisfying J, = x f x and canox = 4
4.15
I
( 2 . 5 ) . This enables us t o verify assertion l i i i ) of
of 4.2 (iii))The k-linear map $ -x'f I
(A1-s-
(Qyklx
the stmthness theorem ( 4 . 2 ) : for since g is k-smooth a t x k-algebra hommorphisn
is
('XJk'x
-+
Q&/k (2.6) (Qy/k)y "13y/k and into an -ox--module M m y be extended t o a k-deri-
0
into M
ax
(R
[[T1,.
..,Tn]]
B
Y
if
A
dX
is f l a t
is the local ring
1,
5
145
4, no 5
Likewise,
-
dim f - l ( y ) = Kdim 0 = Kdim X
X-
(5
c?
dX
= n
3, 5.14).
(ii)= > ( iSince ): g is f l a t a t x -
5.2 g(x)
, f.
is f l a t a t x
we may c l e a r l y replace
-f - l ( y )
+
Sp
K
(y)
. To show that
g
by f-l(y)
. The proof
is thus reduced t o the case i n which 2 = Sp -
we deduce f r m the canonical isanorphism
that
dimx& = dim
5
3 , 6.3 we have X'
g (X) -
By 2.10, we have
is f l a t a t
and f by the induced mrphisn
Setting
( 3 . 4 ) . On the other hand, by
and prl
+ dimxY-l ( -g (x))
.
K
(y)
.
I, 5 4, no 5
Au;EBRAIC GFXXEIRY
146 whence dimx%
= dim
X’ = n
9 (XI-
( 5 3,
6.1) , and the implication follows.
We ncw lead up to the implication (i) => (ii) by proving the
5.3
following result: given a field k and a pint x of an affine algebraic scheme X_
, if
hXk(x)
:K
(x)3 s. dhxz
k-schanes g-: g +$which is etale at x particular case in which = s_P k
, then there is a mrphim
.
of (i) => (ii) in the
. This proves
...,fn€dk (2) such that the canonical images
Consider functions fl,
(X) form a base for R
of the differentials dfiER
Xk;
Z/k
(x) over
dfi (x)
. We
K(X)
claim that the mrphisn g:g+Qk with cmpnents fl,...,f n satisfies the n required condition. For let Ti be the ith canonical projection of Gk
onto Qk and let g* - (dTi) be the canonical image of dTiE )EQ( ,Q g*(R ) (g) The canonical mrphim g*(R
.
- $/k g*(dTi) -
-
, so
onto dfi
-
the map g*(R
$p)x
$/2 -+
“_./k
R
-x/k x
in
sends
is surjective. Replacing
-X
by a smaller affine open subscheme, we m y assume that g* (R - -k @/k’ + “2_X/k = O and it remains to shcrw is an epimorphism. By 2.9, we then have
that g is flat at x
“-./$
.
To prove t h i s last assertion, consider
set g = X W k K
, q-= g-@kK
an extension K of the field k and
. For sufficiently large extension, K , there is a
rational pint 2 E X which is projected onto x EX
(5 1, 5.2 and
5 . 7 ) . If
is flat at X , then g is flat at x , because 9, is flat over dx Since [Q- (2): K ( Z ) ] = [Oxk(x) : K ( X ) I by 2.3, and and $g(x) X/K dim-z = dim X (5 3, 6.2) , we may assume that x is rational. Setting XXn 4 z =g(x) , we then have dx/mz (3x = k (3.1) It follows that d , -+ dx is -
.
,-
.
ax
the Gz-adic filtrations and apply Alg. surjective (assign 0Z and m. 111, 5 2, 110 8, wr. 2 of th. 1). Now if z Ekn is of the form i (bl,--.,bn)
8
...,T -b )
(T1-bl,
is the ccanpletion of the local ring of k[T l,...,Tn]
, and
at
is hence i m r p h i c to a ring of f o w l p e r Series. A 4 In particular, dZ is an integral damin. If Jz dx were not bijective, “4
-f
then we would have
dimxz = Kdimd X
=
A
A
Kdimgx < Kdimdz = n
,
5
I,
4 , no 5
SMCDrH MOwHIm
contradicting the hypothesis n
dx
that
6,
is f l a t over
.
dhxX_
5
147
. This shows that
.
set ~ = r l ( ~, ) B = D ( Y )
A
5 and
(i)=>(ii) :W e may c l e a r l y assume that
5.4
"
d, dx
_Y are a f f i n e .
F i r s t of all suppose t h a t B is noetherian. A s in 5.3, choose i n such a way that t h e inages dfi(x) form a base f o r
,x/z (x)
R
and hence
fl,...,fnE
of t h e d i f f e r e n t i a l s dfiE R
Z/X
A
(X)
. we claim that the mrphisn -g : z +- ~~9 - with is etale a t x . The equation Q n(x) = 0 _x/lcx_o -1
is ccanponents flfll...,f n established a s i n 5.3. Also, by 5.3 applied to the K (y)-scheme f (y) , the g is f l a t a t x (y) I n other mrphism &) o;(y) induced by -
. By M g .
-+
z = g- ( x )
wxds, i f
5
, 0x/mY 3X
is f l a t over d /m
5, no 4, prop. 3, dX i s f l a t over
Lf
dz
B U o t noetherian, we apply the
and x = p
. Using the notation of
.
an open
yo
go:
(Sp ko)x Qn
Uo
-+
Z
cmn. 111,
lemna of 5 3, 3.14, setting B = k
go:
, then
SJ A.
-f
S l ko
we have
-1
in gp .A
such that po
€zo and an etale mrphisn
such that JolUo = prlogo
. W e f i n a l l y set
(ii)=> (iii) : Clearly we may assume that & and
5.5
is the
is (yo) ( 5 3 , 6 . 2 ) . It follows that $'(yo) PoBy the regnarks above, -fo i s smooth a t po ; hence there is
and dim f - l ( y ) = dim f
.
13
Y
that len'ma, i f
structural projection and i f we set yo=,fo(po) -1 -1 f (Y) f o ( Y ~ ) @ ~ ( ~ ~ ) KI ( whence Y)
pm t h a t po
=
.
.
a f f i n e and that g = ~I f
g - is etale a t x
,
g
g are
is etale i n a neighbur-
hood of x , so that we may assume that we are given an etale mrphism s:X -+y x on such that f =pr1og With this s u p p s i t i o n i n force, we prove the following assertion which obviously implies (iii),namely: for each C €2 , each i d e a l I f C of vanishing square, each w E Y -( C ) and each vCX_(C/I)
.
such that f (u)= w -
v =
, there is m e u E&(C) such that uC / I C/I (to see that this implies (iii), set w = ( E (3+))b f(v) =w
kX(% Wb and.
u =
(Ex@
X))b
)
Y
*
=V I
@
ALGEBRAIC GEDMETRY
148
5
I,
For the proof of t h i s assertion, suppose that g(v) = (f_(v),E1,...,cn)
. Let
with Ei€c/l
cl,...,c
n
-
El, ...,c
be representatives of
4 , no 5
-
c
in
,
.
n Replacing f_ by g - and w by ( W r C l r . . . r ~ n) , we see that it is enough t o prwe the assertion in the case in which f is etale. In this case, set Y ' = SxC
,
-
Y" = Sp(C/I)
duced diagram
, 2'
"6 -
= Y_'
II
3'' = y"
X
9 and consider the in-
#
W"
gyl,
The mrphisn
v#
. By 3.3
such that
has a section _ s ' ' : ~ " + ~ " x y X - with cQnponents I$,,
and-§ 2, 7.6b), there is an open &d closed subscheme
_f,,
Z''
induces an isomorphism of
fy the underiying topological spaces of
&(P) s J (-X 1 ) / I ~ ( g1.' ) So l e t L
onto
&'' and
X'
Y'' . Now we
and
-
A''
5"
of
can identi-
(since we have
z' be the open and closed subscheme of -X '
3'' . Then f,, induces a ~ ' € 5 'and y'= f Y l (x') r dy,/IJYl* I&/IL?~, is bijective. By N a k a y m ' s lema, +dxl , it f o l l m that dyl surjective. Since Jx, is f l a t over J Y' that gyl , induces an isamorpkism of Z,' onto y ' . Hence _fyl tion E~ and we need only set u = ( If~ ~ 2 ' ) ~
which has the same underlying space as
hameamor-
phisn of
the map
z' onto
y'
. Wreover,
if
-
.
is then
JxI , SO has a sec-
Before proving the equivalence of (iii)with (iv) , which we
5.6
leave u n t i l 5.7, we make scane prefatory rararks. Suppse that X_ and _Y are affine and set A = cl(g) over
B
, we
,
.
B =J(_Y) Since A
m y assume t h a t & is the closed subschane of
by a f i n i t e l y generated ideal P Let
R
, and that at x
be the local ring of Y_X$
Setting Q =Px
r,
i s a f i n i t e l y presented algebra
, we
have R =R/Q
p h i s ~ ~ ~and IJ turn )I = x o f x implies that x
ox and
f
is induced by prl:y
, and K'
that of
. With the notation of C
Y_%$
x
X
or + y .
at x
into B-algebras, and the equality T1,...,T
.
(iii) , the hornanor-
is a B-algebra ham~~rphim. Thus l e t
be the images of the indetenninates
defined
tl,...,tr
r urader the m p s i t i o n
-
I, 8 4, no 5
SMOOTH MORPHISMS
-
149
-
if tl,...,t r are representatives of tl,...,t r in C , we evidently obtain a ccmnutative square of -4 :
A
C-R
such that I(Ti)=ti
,
.
i = lr...rr
Given this A , the other mrphisms A':R+C of $ such that canoA' are of the form A'=X-D , where D is B-linear and satisfies
, we
Denoting the equivalence classes rrod Q of x a d y by 2 and A(x)D(y) + A(y)D(x) = $(x)D(y) + $(y)D(x)
We can then assign I the
can
.
D (xy) = A (x)D(y) + A (y)D(x) have
=
.
(R/Q)-module structure derived from $ and the
canonical(C/I)-module structure. The anditions imposed on D then mean simply that D is a B-derivation of R into the (R/Q)-module I see that the existence of a mrphim x:R/Q-+C of
&
. We now
such that canox = $
is equivalent to the existence of 'aderivation D such that D IQ
=
A IQ
. We
now reformulate this condition in mre erudite terms: Let 6: R
I
-f
RRIB(BR(R/€?) be the derivation x + d x @ l
that each B-derivation D of R into an (R/&ncdule expressed in the form D = t 6 , where 2 linear map. Wreover, since 6 (Q ) = 0 2 1 j : Q/Q -,RRIB@.,(R/Q)
e:
I
. It follows frm 2.5 M may be uniquely
1
Q~/~@~(R/Q) + M is an (R/Q)6 induces an (R/Qflinear map
.
2 If h:Q/Q2 +I denotes the (R/Q)-linear map induced by X (A (Q = O!) I the existence of x is then equivalent to the existence of an (~/~)-lin= map A
such that
Aj
=x . We deduce fram this that assertion (iii) of the smoth-
ness theorem means that j is an i m r p h i m of Q/Q2 onto a direct factor WQ) of the (R/Q)-module oA/B~R
.
To prove t h i s assertion, notice that, if j is such an iscanorphim, there
.
is obviously an extension A of A The necessity of the condition is proved 2 2 by setting C=R/Q , I=Q/Q I $=Id and taking for A the canonical map of R onto R/Q2
. Under these conditions,
A
is in fact the identity
,
150 map of Q/Q
(iii) <=> (iv) : W e may assme t h a t
1
the notation of 5.6,
.,6Tr
K
are a f f i n e . w i t h
Y
is a f r e e ( R / Q ) - d u l e w i t h base
nRjBBR(R/Q)
jcp) =
5
has image
i n Q/Q
2
I
c giaTi .
is t h e residue f i e l d of
(x)
& and
( 2 . 6 ) . Pbreover, if pEPCBIT1,...,Trl
we have If
4 , no 5
.
2
5.7 AT1,..
5
I,
U E B R A I C GEKBETRY
R
, assertion
( i v ) of t h e mxkhness
-
... -
sends t h e generators p1 , ,P of S R/Q 2 1 onto the elements of a base f o r (QR/,QRR/Q) @ R / Q ~ ( ~ , ) By Alg. c m . Q/Q 11, § 3, prop. 6 (it is unnecessary to assume that M is f r e e i n t h i s pro-
theorem simply means that $3
p o s i t i o n ) , this implies that
(XI
K
is an i s m r p h i s m onto a d k e c t f a c t o r , and
j
a s s e r t i o n (iii)follaws by 5.6.
j is such an iSdQrphism, it is emugh t o choose P1,...Ps€P
Conversely, i f
i n such a way that
Fl,.. .,PS
form a minimal system of generators f o r
.
Q/Q
2
Then t h e matrix ((8Pi/aT.) (x)) has a rank s mreover, P1, ...'P form a 3 S minirral system of generators f o r Q , so that v(P,, ,PSI and V(P) Coinc i d e on a neighburhood of
this neighbourhood and
.
V(PlI.. .IPS)
4
x
(5 3,
4.2).
where B E M
, and that 3
nomials P1,..
s
rank
.
.,P
S
.. .
To obtain ( i v ) he take
to be the inclusion m r p h i m of
(iii)=> (i) : Since (iii) <=> ( i v )
5.8
, we
.,Tnl
t o be
V_ i n
= Sp -B
m y assume that
is t h e closed subscheme of
C BLT1,..
y
or
K
such that the matrix
defined by ply-
( (aPi/aTj)
Under these conditions, l e t Bo be the subring of
(x)) has
B generated
by t h e c o e f f i c i e n t s of t h e Pi ; l e t
xo = 9 B~ , zo= S and l e t x
0
B l r~ . .~. ~T ~
x i n -c X
. Then
rl/(~l~...~~s) ((aPi/aTj) (xo)) has
s and by 4 . 1 it s u f f i c e s t o show that the m r p h i s n
rank at x
If
be the projection of
~
. We may accordinqly assume t h a t
( (aPi/aTj) (x))
has rank
s
, some
.
zo+yo
is m t h
B is noetherian. s is in-
square s h t r i x of order
.
W of x Let X ' -1 be a closed p i n t belonging t o t h e closure of x i n yng (y) Since the set of p i n t s of 3 a t which f is smooth is open (by (i) <=> (ii)and 3 . 2 ) ,
v e r t i b l e . Therefore this holds throughout a neighbourhood
.
,
9
I,
4, no 5
MIRPHISMS
SXXTH
it is emugh to show that
is closed in f-'(y)
.
U n d e r these conditions,
f
. Thus we my assume that
is smooth a t x'
is a f i n i t e algebraic extension of
K(X)
lennna 5.10 below, there is then an
0Y-algebra
local and such that the residue f i e l d B'/n'
9
151
-
B'
K(Y)
x
. By
which is noetherian, f l a t
coincides w i t h
K(X)
. Applying
1, 5.2 and 5.7 t o f. and the c a r p s i t i o n E
- (3Y
Sp + B'-Sp
we see that there is an x'
n'
'Sp - B'
Y f which is projected onto x and
X_ xu(s_P B ' )
and satisfies K ( x ' ) =B'/n' is smooth a t x'
Y
=K(x)
.
. This we reduce
By 4 . 1 it is enough to show that
t o the case in which
K
(x)= K ( y )
.
.
5.9 below, we thus have (iii)=>(v) = > ( i )
we now prove that (iii)=>(v) w h e n Jy
5.9
is noetherian
2 tl,...,t be a base f o r m /(mx+m ) and l e t ti be a n 5 Y X representative of ti i n m . Setting S = 0 LLT1,.. .,Tn]] , we claim that K(x)=K(Y)
. Let
X
A
, is
i = l , ..., n
Y
bijective. For i f
Y
%
4 :S -+ such that (p (Ti) = ti , s is the maximal i d e a l of S , we have
t h e continuous hcanarrr3rphim of d -algebras
...&
S / ( s L t m S) % K ( Y ) C ~ K ( Y ) T ~ KK(y)Tn ~
Y
and
ax/(m:+m hence there is an $,(ti)
=Ti
.
By
8) Y X
-
K ( y ) ~ K ( yl )~t . . . ~ K ( y ) t n; 2
# -algebra hcmxmrphism q0: (lx+ S/ ( s +m S ) such Y Y (iii) , there is a factoring of q0 of t h e form
that
'.$+ S / s 3 . Continuing
similarly, $1 f a c t o r s through a mrphism q 2 : way we build a catmutative diagram of 8 -algebras Y
--
-, ;- ;4su
s/s3;;,
s/s
2
2
s/ ( s iinYS'
i n this
BY passing to the inverse l i m i t we see that the $n induce an hamanorphisn $: 2 s / ( s + m S)
;
Y
. By construction
A
dX+ S
since $$
i s an
m r p h i s n of
3
.
2
S
the s-adic f i l t r a t i o n ,
mrphism of t h e graded algebra associated w i t h cor. 3 of th. 1, $I$ is an autcanorphisn of 2
2
2
+
13Y-algebra
Y
2
Accordingly, i f we assign
4, no 5
$4 induces t h e i d e n t i t y map on -algebra hamrorphim, $I$ induces an auto-
my/" e s / ( s + m s) Y Y
s/s2
$
I,
ALI;EBRAIc GlxMmRY
152
mx/mx * mx/ (mx+m
S
dJ
o
-f
induces an auto-
cam. 111,
5
. Using the exact sequence
S
Y X
+$
. By Alg.
2,
,
we v e r i f y s i m i l a r l y t h a t $+ is an a u t m r p h i s n of
dx . The claim follaws. A
Lama: Le t A be a local r i n g with residue f i e l d K be a f i e l d extension of K . Then t h e r e is a f l a t local A-algebra
5.10
and l e t
L
B
residue f i e l d is i m r p h i c to L algebraic extension of
, we
K
. If
may take
A
B
is noetherian and L
L
over
our a t t e n t i o n t o the case i n which the extension L of a s i n g l e element. I f
, set
L
is a f i n i t e
to be noetherian.
By well-ordering a set of generators O f
Proof:
~ ! d - ~
K K
, we may
confine
is generated by
is t h e f i e l d K(T) of r a t i o n a l f r a c t i o n s in one
w h e r e p is t h e prime i d e a l of ALTl conP ' sisting of a l l polynomials whose c o e f f i c i e n t s belong t o t h e maximal i d e a l m
variable T of
t
A
. It therefore remains
i s algebraic over
P = -
al,
B=ALTI
+ T+ n
B = ACT]/PA[T]
.
.
.
Hence B
5.11
, where
all.. ,a are representatives of n , it is s u f f i c i e n t to set P = a 1+ a2~+...+a~T"-'+pand For B is obviously a free A-nusdule; mreover, mB is
contained i n t h e r a d i c a l of
B/mB = L
t o consider t h e case i n which L = K L t ]
and has a minimal p l y n a n i a l of t h e form
... +anT"-l+ T" . I f
1 - 2 in A ,a
...
k
B (Alg. m.V,
5
2, cor. 3 cf th. 1) and
is local and the lemna is provd.
Remark:
The proof given i n 5.9 shows that i n a s s e r t i o n (iii)
of the m t h n e s s theorem we m y i m p s e f u r t h e r conditions on t h e r i n g C For example,
if
k
is a f i e l d and
X
.
i s a l o c a l l y algebraic k-scheme, the
abwe arguments imply t h e following result:
& isk-smooth i f f , for each
I,
5
4, no 6
153
SMCUEI IvK)FU?HISE
local k-algebra
C
such that
k:k] < + m
and for each ideal I
of
C
of
.
is surjective (cf. 4.6)
vanishing square, the canonical map X_(C) + z ( C / I )
W e leave the proof of the following r e s u l t as an exercise f o r the reader: i f
i s an i n f i n i t e f i e l d and 5 is a locally algebraic k-scheme, then & is
k
k - m t h a t each of its rational p i n t s i f f f o r each integer
canonical map
g(k"J?1/(?+') 1
-f
, the
nrl
_X(k"r]/(T")1
i s surjective.
E t a l e schemes over a f i e l d
Section 6
Throughout this section,
a separable closure of
denotes a f i e l d belonging t o
k
such that
k
Galois group of the extension ks/k Proposition:
6.1
5
of
and
X
Ak-schm
.
k
denotes
ks
II denotes the t o p l o g i c a l
is etale i f f the space of points
X
is discrete and the local rinqs of
tensions of Proof:
.
ksc;
,
are a l l separable f i n i t e ex-
Clearly any schesne satisfying the latter conditions is etale over
g
Conversely, i f
is e t a l e over k
, each
point x
€2
k
.
i s closed (3.1 and.
so that each irreducible ccarrponent of & reduces to a single point. Since each affine open subset contains only a f i n i t e number of irreducible
5
3, 6.61,
CCklp?Onents, it follows t h a t the underlying space of over, the local rings
this, set m = O Y
ox
are separable f i n i t e extensions of
k
More-
(to see
i n 3.1 ( v ) ) .
Corollary:
6.2
5 i s discrete.
A k-scheme ~-
X
i s etale i f f
XBkks
is a constant
ks-sche. Proof:
Clearly X is etale over
k -scheme S
6.3
T
i s etale i f f Corollary:
k
iff
XBkks
i s etale over k
, and
a
is constant. If
_f:g+y is a smmth mrphism of schemes and
154
ALGEBRAIC (zKm3rRY
y is reduced, then so is g
.
I, s.4, no 6
,
By 4 . 2 ( i i ) we m y assume that & = S l B
Proof:
-
and f is etale.
Y_ = Sp A
There is a product of f i e l d s A ' C S and an injective hammrphisn A + A ' since Sp(BNAA') 3 %€3 Sp A' -
Y-
1s f l a t over A
B
(6.1). As
,
-
such that, for each x EE
an open subgroup of acts on X(k ) S
JI via
, the
Sp k
s'
centralizer
{y EII 1 yx =XI
, let
of
%\
of
x i n ll is
be the image under x# of the unique
w
K(O)
+ks is associated w i t h x#
which enables us to identify the residue f i e l d
Corollary:
on which Il acts
+
then a hcammrphisn
ated subextension of
E
. For instance, i f 5 i s a locally algebraic k-scheme, k . I f xEg(ks) and i f x# : S x ks X i s the S
mrphism associated w i t h x p i n t of
;
is reduced.
BgAA'
is injective and B is reduced.
B+BQAA'
A I l - - is by definition a a l l set
6.4
TI
,
is etale over Sp A'
ks
. This shows that
The functor zt+z(ks)
K(W)
with a f i n i t e l y gener-
i s a II-set.
X(ks)
is an equivalence of the f u l l subcategory
formed by the etale k-schms with the cateqory of Il-m.
Proof: W u l o the characterization of etale schemes f o m l a t e d i n prop. 6.1, this corollary is nothing mre than a variant of Galois theory. Sirrpsly observe that, if K is a f i n i t e subextension of ks , (Sp K) (ks) is II/II' , where II' is the Galois group of ks over K Since each II-set E is the d i r e c t sum of Il-sets of the form II/II' and since the functor x y X _ ( kS ) preserves direct sums, we see that E is of the form X(ks) , where & is etale over k .W e leave the rest of the proof to the reader.
.
Proposition:
6.5
f
gx:&
m k +
. Then there
no CX_)
be a locally algebraic scheme over a
T ~ ( X ) and a mmhism
is an etale k-scheme
w i t h the following universal property: f o r each mrphisn
-f:X_-+E of X
i n t o an etale k-scheme
such t h a t f =gqx -
.
E_
, there
is a unique g:n0(&) + g
Moreover , of
X
is faithfully f l a t and its fibres are the connected. ocarqeonents qX ( i . e . t h e open subschemes of _X whose spaces of points are the connect-
edccanpo nents of
Proof:
1x0
.
F i r s t consider the case i n which
3
= Sp -A
, where
A
is a f i n i t e l y
generated k-algebra. If we can show that A contains a ntutimal separable k-subalgebra
As
of f i n i t e rank, then the mrphism S%A+S%As A
prove the existence of
into A has the required universal property. To
S
As
, consider
the connected mn-pnents
X
A "A1%
the underlying noetherian space of
. Clearly we have
dx(Xi) , and the algebras
where Ai=
Accordingly-the unit of
= Als
Now suppose that
x
..."Ans .
i
is a
.
=\
formed
of _X and inclusion mrphisms
set n0(g) = ~ ~ , J associated w i t h the inclusion m p of restriction
V
NGW
o(LJ
+
b(v)
serid.s
jc:no (EI) +no (V)
8(v)
O;Jl)e ~t q-v : ~ - + n ~ ( u ) J(~=J) i n t o J ( ~ J; i f onto
V
o(v)
-jc% =';[v'; f r m the construction of direct limits in-Es
mrphisn
such that V
( 1 no (g) I , 1 2 1 )~ etale k-scheme. By the caparison theorem
limit of the diagram
(no@)
.
m
between them. Clearly ,X m y be identified w i t h the d i r e c t limit of
diagram.
&s
,
bi/mi:k] This shows that has a largest element Ais
5 i s arbitrary and consider the diagram of
by the affine open subschemes _V
sV: v + u
is a f i e l d . I f
Ai 5
the u p a r d directed system formed by these K S
...%An
of
n
i s i t s sole non-zero idempotent, so that each
Ai
maximal ideal of Ai ' it follows that [K:k] A
Xl,...,X
cannot be further decamposed.
Ai
f i n i t e l y generated separable subalgebra K of
NaJ set
induced by
c
the inclusion map of
,$
V
the mrphism
,
V_CV_ the
and induces the unique
. I t follows imnediately
(6 1, no.
1) that the d i r e c t
i s the geometric realization of an
(5 1, no.
4 and 6.8)
, the diagram
; evidently the mrphism
then has a direct l i m i t ~ ~ ( gi n) &cS
gX:x+n0?g) derived from the mrphisms qu by passage t o t$e d i r e c t l i m i t has the required universal property ("the i e f t adjoint functor
no ccmnutes
w i t h d i r e c t limits"). To prove the f i n a l assertion of the proposition, we observe that the functor
n
0
c m t e s with d i r e c t sums, which reduces the problem to the case i n which
5 is
connected. Under these conditions the image of
therefore contains only a single point w of scheme of
no(X)
induced by qx -
-
whose only point is w
, then
i n question, so that
clearly
= n (X)
a field!).
6.6
Definition:
0-
(y,c$
no(x)
and i f
.
qx is connected and is the open subIf
-
-g:X+Y_
denotes the mrphism
is a solution of the universal problem
. Hence
q
-x_
is surjective and f l a t
(
With the a s s q t i o n s of 6.5, we call no(g)
is
"m
ALGEBmIc (ZXNmRY
156
k-scheme of ~nne~ted. c m pnents of g
I, § 4, no 6
-
qx the canonical projection.
11 and
6.7 Proposition: Given a locally algebraic k-scheme field extension K/k mv .i % , E K , then the unique mrphim
a
such that qJ3kK = $ q is invertible. x-ekK Proof: Just as before we reduce the problem to the case in which g is affine. We must then prove the following assertion: if A is a finitely pre-
__I_
sented k-algebra and if :A is the largest etale k-subalgebra of A , then k AsmkK = (A@kKIK s To prove this, consider the set % of field extensions k LE& such that AsWkL = (AWkL): for each finitely presented k-algebra A
.
We show that a)
If
L
a Galois extension of k
,
. For if r
denotes the K &lois group, then acts on A @ ~ K and normalizes ( A @ ~ K ) ~ BY a g . VIII, 5 4, prop. 7 , it follcrws that (AQ~K): = V@~K here A CVCA j.s
r
Since V is etale over k iff V@ K k b)
ks
rf
ks
Lt5
is etale over K
is a separable closure of k
, then k t %
S
. For in order to prove that
we may assume that Spec A
k
S
.
, we have V=As
.
an algebraic closure of
is connected. If p is the characteristic ex-
ponent of k , then each a€ABkk has scane p e r ap S that the projection
is a haneanorphisn, so that Spec A@ k kS kS As =k inplies (ABk kIs = k,. S
.
. It follows
is connected. In other wrds
S
c) If T is an indeterminate,
in A
k(T)6 %
argument of (a) to the group of autcanorphims
. To prove this, apply the
f
of x ( T ) of the form
.
s
I, 5 4, no 6
m MOwmsMs
157
aT+b
-x d) If
e)
If
&
KC%
LEEK
, then
and LE%
k'KcL
.
and a d - b c f O
with a,b,c,dCk
. This is clear.
LC%
,then
KC%
. This is also clear.
- K is the Union of an upnrard directed system of extensions KiC% f) If then KC % Again clear.
.
,
It now follows fran a),b),c),d) and e) that each finitely generated extension belongs to
5 . S o by f ) ,
every extension has t h i s property.
Corollary: The following assertions are equivalent for a
6.8
l o c a l l y algebraic k-scheme
g
:
(i) 5 is geametrically connected (that is to say, K@kK is connected for each extension K of k) ; (ii) if ks is a separable closure of k , g@kks is connected; (iii) IT^(^) is i m r p h i c to SJ k k
.
Corollary: I_f
6.9
X
is a connected locally algebraic k-scheme
which contains a rational point, then Proof: If
5 is connected,
.
T,,(&)
2
is geametrically connected.
is of the form S x K for some separable
finite extension K of k If, in addition, & contains a rational pint, , so that k = K k-tlj, hence a mrphisn Sp k+S%K there is a mrphisn Sp -k k
.
Corollary: Let - - -X _and . Y be locally algebraic k-schemes. Then the canonical morphim I T , ( ~ x -+~ )~ ~ (x 5 r 0)( x ) is invertible. 6.10
Proof: In virtue of 6.5 and 6.7 we may confine our attention to the case in which k is algebraically closed and & and Y_ are connected. We must show that _x x Y_ is connected. Since each open subscheme of _X x X contains a rational point (x,y) and
(5
3, no 6.), it suffices to show that any trm rational points
(x',y') belong to the same connected component. Now this is
certainly true for (x,y) and (x,y') (which both belong to the connected subset (9I C ( X ) ) X Y_ ) , it is also true for (Xry') and (x'ry')
(which belong to
X K (9
IC
( y ' 11 G
&
)
, and the corollary
follaws.
Corollary:. With the assmptions of 6.10, if y is connected and 2 is qeanetrically connected, then XxY_ is connectd. 6.11
Proof:
By 6.10 and 6.8, we have
§ 5
PROPER WRPHISMS
Section 1
Integral mrphisns
1.1
Definition:
f
,Let f:X+Y_
be an a f f i n e mrphism of schemes.
is said to be i n t e g r a l (resp. f i n i t e , f i n i t e locally f r e e , of rank n) if,
I! of 2 , O(_f-l@)) is
f o r each a f f i n e open subschane
an integral algebra
(resp. a f i n i t e algebra, a f i n i t e l y generated projective module, a projective module of rank
.
over
n)
@(y)
is a f r e e 0 -module X Y Y (Mg. m. 11, 5 . th. 1). The rank n(y) of this module is l o c a l l y constant is f i n i t e l o c a l l y f r e e and i f
If f
yEy
, g*(&)
by Alg. ccinn. 11, 5, cor. 2 of prop. 2. Accordingly Y_ can be covered by closed and open subschews
-+xn
f :f-'(Xn) -n When k E g
xn ,
nEN,
such that the mrphign
is of rank n f o r each nED
induced by _f
and -g is a mrphism of \AS
(resp. f i n i t e , f i n i t e locally f r e e , of rank
, we n)
-
say that g provided
.
is i n t e g r a l
zg has t h e same
property *
a f f i n e mrphisn of schemes
f:X
+.Y_ :
.
(i) f is i n t e g r a l (resp. f i n i t e , f i n i t e locally f r e e , of rank n) (ii) Each point
~ ( Z - ~ CisV an )
has an a f f i n e open neiqhbourhood V_ such that
yE Y_
i n t e g r a l algebra (resp. a f i n i t e algebra, a f i n i t e l y generated
projective module, a projective module of rank
Proof: Clearly a IJ If
-1
(v))
l&n
I
_V
we have
yEy
o(v) .
is, f o r instance, an i n t e g r a l algebra over
, we
then have
is also an i n t e g r a l algebra over
,
a f f i n e and open i n I
over J(y) .
.
is a f f i n e and open i n
so that d(f-'(y', ) fiE d(U)
n)
(i)=>(ii)Conversely, suppose that f o r each
such that (3(f
vt
an
The following assertions are equivalent f o r
Proposition:
1.2
f?(v') . I f
V_
may then be covered by open subschemes U
such t h a t the algebra
is
*i
I
.
o(u,
for each i If A = d(y) and ) &?) i fi x~ d(p-l(~) , it follms that AL.J~ is a finitely generated A -mdule fi i for each i By. Alg. cmn. 11, 5 5 , cor. to prop. 3, A L x j is finitely is integral over
.
generated over
A
, hence D(_f-l(V))
is integral over 8(u)
Let us say that a mrphim of schemes
1.3
closed if, for each mrphism of schemes g:Y'+Y_ f,': Z X Y' -+ y' is closed.
-f:Z*x
.
is universally
, the canonical projection
Y-
-
proposition: An affine morphisn of schemes is universally closed iff it is integral. Proof: This imnediately reduces to the case in which the schgnes are affine. The contention then follows fram:
LemM: For each hcmxmrphism of rodels
$:A*B
are equivalent: (i)
, the
follming assertions
.
B is an integral algebra over A Ss $ : J S B * @ A is universally closed.
(ii) (iii) For any indeterminate T is closed..
, the m p
Spec $ L T l : Spec B L T l - + Spec ALT]
Proof: (i) =>(iil. If B is integral over A , BWAA' is integral over A' It is then enough to observe that Spec $I is a closed map whenever B
.
is integral over A
(Alg. m.V,
5
2 , no 1, remark 2 ) .
.
(ii) => (iii) This is clear.
.
(iii) => (i)
Let btB
and consider the cmtative square
where B' is the localization of B at b , A' is the subring of B' generated by l/b and the image of A , $ ' is the inclusion map, and
B map T onto l/b a Closed map,
SO
. Since
is Spec $ '
is dense in Spec A'
c1
. Since
(5 1, 2 . 4 ) ,
c1
and P are surjective and Spec @[TI
I
is is injective, the h g e of Spec B
4' so that Spec $ '
is surjective. Since no
.
prime ideal of B' contains l / b , the sane holds in A' It follows that l/b is invertible in A' , that is, b / l E A ' Thus we have the equation
.
b
$(a,)
1
1
- = -
+-
$(a,) b
+...+-
4 (an) bn
whence S
b
= $
(a,) bs-'+.
for sufficiently large s
..+ $ (an)bs-n-1
.
Proposition: If f : X-+ Y- is an integral and surjective m r -
1.4
phisn of schemes, then dim & = dim
.
Proof: We have dim as
=
SUP dim I! and dim 5
v
=
SUP dim -1
r
v
. We m y then assume B =&) , 41 =d(_f). Factoring
luns through the affine open subschemes of
and.
that
are affine and set A = @@)
,
A and B by their nilradicals, we m y also assum that A and B have no non-zero nilpotent elements. Under these conditions $ is injective 2.4).
We may therefore a s s m that B is an integral extension of A
apply
5
(5 1,
. Naw
3, 5.2 to 6htplete the proof.
1.5
Proposition: If a m m r p h i m of schemes
z:z-+x is a finite
mrphisn, it is a closed gnbedding.
xi
Proof: By covering by affine open subschemes , and replacing _f by -1 the induced mrphisms f (xi) , we reduce t h i s to the case in which X = SPA and = Sp B Since the diagonal mrphism :g+zX$ is
-
-
.
-+xi
%/x
an iscBnorphism, the canonical map A@ BA+A is invertible. The s& then holds for the canonical map (A/I-A)@~/~(A/~A) + A/nA for each 1~ximi1 ideal n of B
. We then have
h/nA:B/nI2=LA/nA:B/n]
and so the map B/n+A/nA
, so that
is surjective. By Alg. c m . 11,
b/nA:B/n]=O,l
5
3 , prop. 11,
B+A is also surjective. 1.6
Proposition: Any finite locally free mrphim is finitely
162
5, no 2
I,
presented. Proof:
Since such a morphism is affine, the problem reduces to proving the
following assertion: a B-algebra is f i n i t e l y presented whenever the underlying B-module of
A
i s projective and f i n i t e l y generated. By § 3, 1.4 and
Alg. ccmn. 11, § 5 , th. 1, this reduces to the case i n which A
...an . Suppse then that we have
B-module w i t h base al,
. Clearly the kernel of
b!.EB 17
that
the hamchnorpNsm 4: BLTl,...,TnI
reader w i l l verify that, mre generally, a B-algebra whenever the underlying B-module of Corollary: Prcof:
If
A
This follavs frcn 1.3, 1.6, and
kc&
,with
+ A
such
. The
A
is f i n i t e l y presented
is f i n i t e l y presented.
A f i n i t e locally f r e e mqhism is closed
Section 2
R
i s the ideal generated by the elements TiTj- leb:jTa
(Ti)= ai
c$
is a free
aiaj= lebijaa
5
and open.
3, 3.11.
The valuation criterion for properness r
a 1k-
V
i s said to k discretely valued i f its underlying
ring is a discrete valuation ring, that is to say, a ring which is principal, l0CaI-r
and not identical with its f i e l d of fractions. I f
valued,
V
is discretely
SppV then has exactly tm p i n t s , one open, the other closed.
Definition:
2.1
A morphism of schemes
f:x+Y_
is said t o be
proper i f it is separated, f i n i t e l y generated and universally closed. If
k e g , a mqhism
A k-scheme
- of g
2%is
is proper.
said to be proper i f g,
,X is called ccanplete i f the structural projection
is proper. Notice that any closed. embedding i s proper. 2.2
If
g:X+_Y
is a proper mrphism of schemes,
.
f,:
FX:$
-
X$
+
S ek
-,Z_
Conversely, i f Y_ can be cave& by is proper for each mrphism g:Z_-ty -1 such that the induced mrphisns _fi:f_ are
open subschemes Y
-i i ,
(xi) -+xi
then f is proper. This follows easily fram 5 3 , 1.9 and the f a c t that the mrphism f, abwe is closed i f the fiz are closed. proper for each
Moreover:
-
-
I,
5
5, no 2
PRDPER FKIRPHISMS
163
Proposition: _ Let_ f : X-+ Y- and q:_Y+z be t m mrphisns of schemes.
.
If
Then:
f and 9 are proper, so is c ~ o f (b) fI gof is proper and g is separated, f is proper. (c) I_f gof- is proper, f i s surjective, and g is separated and f i n i t e l y (a)
a
generated, then 5 is proper.
(a) follows frm the correspnding properties of separated and.
Proof:
f i n i t e l y generated mrphisns. Assertion (b) m y be proved i n the same way
5
(c) of
as
3, 1.10. Finally, (c) becames clear when one observes that the
a s s w t i o n s r m a i n t r u e a f t e r a "change of base"
X
.
Let - k be an algebraically closed. f i e l d
Corollary:
2.3
h:T+Z
and let
be a ccanplete, connected and reduced k-schgne. Then, f o r each k-model
my be identified with A
dA(XWkA)
Proof:
By the l
m of
5
.
2, 1.8, we have flA(X$A);
enough to show that we have k
$(x) . L e t
f
A
,
. It is then
{(X)QkA
_h:X +gk be a function on
5
-
and l e t -g:zQk +. &S k be the structural mrphism; then g is separated and -g*& is proper. Were ,h surjective, then g- w u l d be proper (prop. 2 . 2 ( c ) ) , thich is f a l s e by 1.3. Since
. Since
h ( 3 ) is a p i n t of gk gkk
, which s h s that
is closed (prop. 2 . 2 ( b ) ) am3 connected,
5 is reduced, h, then factors through
the m p k +d(_X) i s surjective.
Lama: -Let
2.4
b&)
A
be
a noetherian local integral d m i n of
dimension 2 - 1 , m its maximal ideal, a f i n i t e l y generated extension of of -
L
such that v(x) 3 - 0 -i f
xEA
K
K
its f i e l d of fractions,&
L
. Then there is a discrete valuation if
annv(x)>O
xEm
.
v
.
be a set of generators of m Since K d h A 2 - 1 n n+l is not of f i n i t e length. Accordingly the graded. ring gr(a1 = an m /m for instance, is not nilthe residue class mod m2 of one of the x i f xo Hence no relation holds of the form potent i n g r ( A ) Proof:
xo,xl,...,x
Let
.
r-1 xo = P(xo,xl,...,x
) , where P is a homageneous p l y n a n i a l of degree n r 21 with coefficients i n A I f C is the subring of K generated by A
and xl/xo,.
..,xn/x o
, we
.
therefore have W = xoC f c
p i s a minimal p r i m ideal of and p2 n A P
=
m
. If
D
C
containing
xo
. It follows that, i f
, we
is the integral closure of
have K d h C = 1 P i n K , and n P
C
,
I, 5 5, no 2
ALGEBRAIC GEaMETRY
164
a maxim11 ideal of D , then Dn is a discrete valuation ring of K with maximal ideal nDn , such that nDnnA =m (Alg. c m . VIIl 5 2, prop. 5 ) . The valuation w associated w i t h Dn is positive on A Thus one may take for v any extension of w to L (Alg. c m . VII, 5 8, prop. 6 and 5 10, prop. 2)
.
.
k be a noetherian rodel, f:X+u a morphisn of algebraic k-schemes, x a point of 5 , y=f(x) If y I E m is a disL a m: Let -
2.5
tinct fran y
I
.
then there is a discretely valued k-21
V with field of
fractions L and moqhisms g:spkV+_Y h:S%L+_X such that fh=gjSpkL and &(L+L)={XI , and that -g maps the closed pint of s v onto y '
% -
I1 be an affine open neighbourhood of
-
.
in the reduced subY_ carried by Replacing f by the irduced morphisn -1 , we m y assume that Y_=Y-1 . Now set A=%' ; since Y_ is f (u,) assumed to be irreducible and reduced, and y is its generic point, the field Setting L = K ( x ) consider of fractions K of A is precisely 8 = K ( Y ) Y the valuation v of 2.4 and the k d e l V consisting of ttL for which . It is then sufficient to set h=E(x) (5 1, 5.2) and to take for v(t),O - the carpsition of cy,S:& g + y (5 1, 5.7) with the mrphisn induced by the inclusion m p of 13 into V Y'
Proof: Let
.
+xl
y'
.
.
I
Properness theorem: For each noethwian ring k and each
2.6
morphim f:_X+_Y
of algebraic k-schemes, the following assertions are equi-
valent: (i) f_ is universally closed. the (ii) For each discretely valued k - d e l V with field of fractions L , __ 9 X(V1 yX(L) _X(L) with ccanponents f(V) E d X(inc1) is surjective. +
Proof: In virtue of the canonical isanorpkisms m%~(~k~,z):
%*E(S&V,_Y)?
z(L)
and
Y_(V) , assertion (ii) means that, for each cmtative square h
(*I
can
165 there is an R : g k V + . X
h = & o .~
such that -g=fak - - and
(i)=> (ii) : S e t _Z = SJ V k
,
_Z I = Sp L -k
and consider the diagram
and -fz are the canonical projections and the canpnents of m_ are can a d h The required mrphisms are of the form yxo2 , where s_ is a section of f such that m = y % W e n m shcw that, since f, is
where g
-8
.
.
-Z
Gists.
closed, such an
TO prove t h i s l a s t assertion, l e t
.
y be the unique p i n t of
x=_m(y) Since -2 f is closed, there is an yl is the & i q e closed p i n t of where -
dx
dx1
dxl
=O
, we
r
X,
and s e t
that fz(xl)=
y’ ,
therefore have
ml#Ker(ay)
L
,
. If
Ker(ay)
i s the maximal ideal of
B-’(m1)
V
ay factors through a retraction
6:
m’
V ; since
. Accordingly the
. Since
i n L contains V and is distinct fran L
proper subring of
y, y’:
XI,
dxl a t the prime ideal
is the local ring of
is the maximal ideal of B-’(Ker(cry))
x’€IX) such
z1
z . we then have the following
m t a t i v e diagram for the local rings of
where
-
image of
is a maximal
dxl+V
of
B
.
The composition
yields the required section s
.
e must show that, for each mrphism (ii)=> (i): W jection
Ez-
: zxy
-
5
is algebraic over k
+.
z
f:z -+Y_
, the
canonical pro-
is closed. To achieve t h i s we assume f i r s t that
. By observing that
-fz
-
also satisfies (ii)mtatis
z
mutadis, we reduce the problem to s b i n g t h a t a mrphisn f:g+Y_ i s closed whenever it s a t i s f i e s (ii) NOW i f ~ € ,3 y =L(x) and y ' e m , we may
.
e i n such a way that _h(S&L)
choose the square (*) h sends the closed point of
x' denotes the hage of f(x')=x
(5 I,
={XI
and that g-
V onto y' (2.5). With the above notation, 4 the closed point of S a V under 1 , we have S
if
. This showsthat the image of an irreducible closed subset i s closed
2.10). Since each closed subset is a f i n i t e union of irreducible closed
subsets, the assertion is proved.
2 be arbitrary. W e must show that, for each closed subscheme _F of Z_ x$ , f Z ( F ) is a closed subset of g . Ey replacing 1 by the members of -1 an i f f i n e open covering (xi) , and _X by the open subschemes _f (xi) ,
Now l e t
we f i r s t reduce the problem to the case in which
. If
is affine w i t h algebra
z by affine open subschmes, we further reduce the problem to the case i n which z is affine with algebra C . Thus let Co be B
we n m replace
a f i n i t e l y generated B-subalgebra of
,
,
the k-scheme S&Co the mrphisn induced by the inclusion map of
zx,$-
C
Zo
Co into * go%$ C , and Fo the closed h a g e of po IFo If V_ is affine and open i n _X + _Z x U is the mrphisn induced by po , the closed and. i f -poU : 0 _yU U ~ ) 2, prop. 4.14) . b g e F- of P O ~ _nF(_zx u) is precisely Fo I ( _ Z ~ X ~ (9 -0 Since we obviously have
po
:
.
x-
"0'
for each _V
, we
see that
Now w e have
is iladuced by the inclusion map of
a closed subset of
But t h i s follows fram the f a c t that _f,'(z)
-
z
Co
into C
and it i s enough to show
is a noetherian space f o r each
.
167
, so
zez
that
for sufficiently large subalgebras
2.7
Corollary:
over a noetherian &el
of
p:X+S is a
If
, the
k
Co
C
.
morphisn of algebraic schemes
following assertions are equivalent:
p is a separated mrphism.
(i)
(ii) If & (V) g (V) -+
V
syL)z(L) -
are chosen as i n theorem 1 , the map
L
with
ccsnponents ~ ( v )
x/s:X
Since the diagonal mrphisn
Proof:
_x/s is proper.
a closed fmbedding i f f 2.8
6
6
X(inc1) +
X
s-X
i s injective.
is an &ding,
6
Now apply theorem 2.6 to
6
x/s
z//s
is
*
With the assmptions of theorem 2.6, the following
Corollary:
assertions are equivalent: (i) f (ii)
is proper.
Ef. V and
is bijective
.
2.9
L
are chosen as i n theorem 2.6, the map X(V) *_Y(V)&#(L)
Corollary:
ring k
5
(i)
, the
If
& is
an alqebraic scheme over a noetherian
foll&ing assertions are equivalent:
is a ccsnplete k-scheme.
(ii) For each discretely valued k-model
x(inc1) :X(V) +_X(L)
L
, the
I t i s enough to apply cor. 2.8 to the structural mrphism
Proof:
each A t & 2.10
Proof:
with f i e l d of fractions
is bijective.
-&: $ + S b k , observing that
EW
V
a
.
.
Corollary:
is reduced t o a single p i n t for
The Grassmnn functor
Apply corollary 2.9;
is a d i r e c t factor of
(Sbk) (A)
v""
if
P
sn,r
is a ccgnplete scheme
is a direct factor of
Ln+l
,
Pn?+l
(Alg. V I I , 4 , cor. theoran 1) I
It follows f r m cor. 2.10 that G C3 k -n,r Z
is a ccsnplete k-scheme f o r each k q
.
168
Au;EBRAIc GEx3ME;TRy
Algebraic curves
Section 3
Throughout t h i s section, Definition:
3.1
.
denotes a f i e l d belonging t o M -
k
& algebraic curve
over
k
k - s c h which
*i
.
is algebraic, irreducible, separated ard of dimension 1 An algebraic curve Over k is said t o be regular i f the local rings a t closed m i n t s are discrete valuation rings. Proposition:
3.2
Each
slaooth
The converse holds i f the f i e l d k Pmf:
If an algebraic curve _X
each closed p i n t 1
(5 4,
13,
4.9).
xez
algebraic curve over k
is regular.
is perfect.
is snooth over k
, the
local ring dx a t
is an integral damin and has hamlogical dimension
It is therefore a discrete valuation rixg (for the ideals of
dx
are projective mdules, hence free of rank 1). ca-wsrsely, i f
discrete valuation ring, l e t 1 be a rational p i n t of &@
K(X)
k
.
is a
which i s
and let t be a uniformizing element of @ men X is the local ring of d x @ k ~ a( t~a) maximal ideal m ard we have
projected onto x
8%/tJES. ( 8X @k I C ( X ) ) / t ( d X @k K ( X ) ) m T If
k
is perfect,
K(x)~K(x)
It follaws that
x @ k ~ ( is ~ )snooth a t
X
Ranark:
k
X
and
-
(K(X)@K(X)),,,
i s sgnisimple so that Q-/t$
Hence m z = t 0-
X and g is
Jz
is a f i e l d .
SmDOth a t
x
.
Ushg the "same mthcd" one can show that an algebraic s c k o v a
a perfect f i e l d is mth i f f i t s local rings are "regular". 3.3
, we
Given an algebraic curve 5 Over k
generic p i n t and ~ ( 5 )for the residue f i e l d of
write
.
LO(&)If
dcminant mrphism of algebraic curves, we have f (w (_XI ) = w (g) K ( f ) :K(y)'K(X)
for the hcmmrphism induced by f
set of daminant mrphisns of
X_
into
.
w(g)
for its
-f:-X+y is a
. W e then write
and z ( X , x ) for the
I,
5
5, no 3
Proposition:
PROPER mRPHIsMs
Let
-
. If
5 is reqular )i s a bijection of %(?,?) onto
be algebraic curves over
is canplete (2.1), the map f + ~ ( f &(K(X)),K(Z))
169
k
% k ~ ( ~ ( gis ) ) contained in w ( f , g- ) , which is closed in X (5 2, 5.6). Since _X is regular, hence reduced, we have Ks(_f,g) =X , so that g = g .
(5
1, 5.2). Hence
(x)
Now suppose we are given a hQoomorphisn v: K K (5): we construct a g such that ~ ( g ) = v For each closed p i n t xc_X , dJx is a discrete valu-+
.
K(X)
ation ring whose f i e l d of fractions i s
dx+y
there is a mrphisn $(:Spk
gx
of
is c q l e t e , by 2.9
_Y
such t h a t
X
has an extension g - :U + _ Y t o an a f f i n e open n e i g m u r -
his mrphisn
h00d
. Since
(if
x
t o th€ hclmmrphisn dy closed p i n t s of
-X
9
-+
sends the closed p i n t onto y ~ y , apply induced by $1
Llx
. If
x
, x'
5
3, 4 . 1
are d i s t i n c t
g , we have
i n view of the uniqueness property proved m e . W e then obtain the required
mrphisn g- by "mtching tcqether" the mrphims -g" Remark:
m n - q t y closed s u b s c h Sc$(_X,y)
-+
Sc$(y,x)
, then
group 3.5
Of
.
a cconplete scheme over k
_V
of
& , the canonical map
be
f o r any
I f 5 is a regular canplete algebraic curve over
the autamrphisn qroup of _X K(5)
. Then,
X
i s bijective.
corollary:
3.4 k
1, 4.13)
By the same methods, we can prove the following result: Let
a regular algebraic curve and
-
(5
.
Corollary:
If
is a n t i i m r p h i c to the k-autmmrphisn
X ans 3 are regular c q l e t e algebraic
170
curves over k
, X
3.6
Corollary:
k
I, § 5, mJ 3
ALmBRAIC GFmEcRY
, X
is i m q h i c to y
Lf
iff K(X)
is a regular c q l e t e algebraic curve over
iff K(X)
is i m r p h i c to the projective l i n e P1azk
.
transcendental extension of
k
Proof:
5 1, 3.9,
Setting n = r = l i n
.
is i m q h i c t o K(_U)
is a pure
we see t h a t the open subschmes
LJcl1
r;r I21
defined there are i m r p h i c to @ .%[TI ; accordingly _P C 3 k con1 2 iscmorphic t o S%k[T] ; it follows that C 3 k tains an open s u b s c h m U
and
-{11 z ~ ( 3 )is the f i e l d of fractions k ( T ) of
3.7
k[T]
.
Theorem on t h e c l a s s i f i c a t i o n of curves:
The functor _X
+K@)
formed by regular is an anti-equivalence of t h e f u l l subcateg-ory of &cS ccsnplete alqebraic curves and d d n a n t morphisns w i t h the f u l l subcategory of & I
formed by f i n i t e l y generated f i e l d extensions of k of transcendence
degree 1 Proof:
.
Since t h e functor
~
W
(g) is f u l l y f a i t h f u l (3.3) , it is s u f f i c i e n t
K
to construct, f o r each f i n i t e l y generated f i e l d extension Klk of transcendence degree 1, a regular ccknplete algebraic curve g such that K(&) = K
.
To t h i s end, l e t
and k(V)=K
.
T
denote t h e set of valuation rings V
The m g n b e r s of this set
T
such that k c S K
then consist of the f i e l d K
together with sane d i s c r e t e valuation rings (Alg. c m . V I , th. 1).W e endow
T
9
10, cor. 1 to
with a toplogy by c a l l i n g a subset open i f it is e i t h e r
empty o r it contains K and its ccanplernent i n T f i e s these conditions, we set
o T ( U ) = vT!V
is f i n i t e . If
U
satis-
; by taking inclusion maps
r e s t r i c t i o n s , we thus define a sheaf of k-algebras
dT . W e show t h a t
as (T,OT)
i s t h e geometric realization of a curve satisfying t h e required conditions. Let
t be a uniformizing element of a ring VCT
c m . VI,
8 1,
over k ( t ) VI,
8
different fran K
th. 3, 1,k is transcendental over k
. Let
A
be the i n t e g r a l closure of
1, cor. 2 to prop. 3, the rings V ' C T
. Thus
k[t]
such that
in K t$V'
K
. By ALg.
is algebraic
. By ALg.
m.
dcminate the
t-lk[t-'] . By Alg. m. V I , 5 8, prop. 2 (b), there are only f i n i t e l y * m y such V' . I n other words, the rings V " t T such that t C V " form an open subset U of T . By Alga CCBITII. VIIr 5 2, cor. 2 of prop. 5 and th. 1, U is the set of local r i n g s of A . Since f o r local ring of
k[t-l] a t t h e ideal
I,
5
5, no 3
each S E A
, we
ideals of
A
= nA , where p runs through s Ps P s , we see on the one hand that (LJ,
have As = n(A )
not m t a i n i n g
canonically isamsqhic to Spec A
a t a point V"
U
171
PROPER PDFPHISMS
, and
is a spectral space. Since the k-algebra (Alg. c m . V,
generated k[t]-mdule
9
V
, we
see that
(T,
dT)
defined above is a f i n i t e l y
A
(T, (3 1 is the gecanetric T W e claim that _X_ is the required
3, th. 21,
.
5
realization of an algebraic k-scheme
dTIU) is
on the other that the local r i n g of
. By varying
is precisely V"
the prime
curve.
To prove this, observe that since T is irreducible and of dimension 1 , _X has the same properties; we have already seen that K coincides w i t h the
X
local r i n g of (ii) of 2.9.
a t its generic p i n t . It then remains to v e r i f y assertion
W i t h the notation of 2.9, i f the image of a mrphisn
, -g,( dX)
is a closed point
x
a subextension of
L of f i n i t e d q e e over k
5
yx
1, th. 3,
spkv
. on the other hand, i f
g -
to SJkV
3.8 over
.
, the
(5
K
3, 6. 5). By Alg. m.V I ,
g has a (unique) extension to
that
x i s the generic point, set v'=~-'(v) -X
X
a): i f
is a regular and complete algebraic curve
description of the g m t r i c realization of
b) : Let
keg
. Let
K
.
P(S,T)=$-SP+l+aS-b=O
p >0
be a f i e l d extension of
d q r e e one generated by tw~elements
S and
. The kernel of
T
k
of transcendence
such that
the mrphisn +:k[S,T]-+k' of
$ ( S ) = s and $(TI= t is then the ideal m of k[S,T] geneP S -a Accordingly the local ring V=k[S,Tk is a d i s c r e t e
such that
rated by
3
and let k'= k ( s , t ) be an k of degree p2 generated by elements s and t such that be a f i e l d of characteristic
sp=a € k and tP= bE k
$
.%
proof of the c l a s s i f i c a t i o n theorem yields, with the help of
LCH) ,a
extension of
s_:SkL+z
(x) , and is therefore
induced by -gx defines t h e (unique) required extension
Remarks: k
, so
factors through V
hcmxmqhism V ' + V of
i s isamsrphic to
.
valuation ring and its residue f i e l d is k!
On the other hand, by
4.2 ( i v ), _SEkk[S,T) is s n x ~ t ha t each p i n t
d ~ ( x )= -(SP(x)-a)dS(x) i.e. a t each point other than m braic curve _X
such that
+o
. Thus we
K(~)=K
x t O2 -k
5
4,
such that
, see that a regular ccarrplete alge-
cannot be m t h . I f we set
172
AKW?AIC(;EOMETRy
V1=VWkk'
, we
-X@Jkkt
9
5 , no 3
is local, so is k is not generated by a single element,
also have V'/rnVT qkf@kk' ; since k*@ k'
V' ; since the mximal ideal of k'Wkk'
v' is
1,
not a discrete valuation rinq. Accordinqly the algebraic curve over
k' i s not regular.
GeQnetric spaces
1.1
A geametric space E Definition: -
=
(x,0X
topological space X together with a sheaf of rings
each XEX , the stalk flx,x
(or simply
Qx) ~f
consists of a
QX such that, for
4
at
x is a local
ring. By
. The unique
abuse of notation, we shall o€ten write X instead of E
ox will be denoted by mx the residue field oJmx . If s is a section of ox a neighbourhooa of x , the by canonical image of in ox will be denoted by sx and called the germ and
maximal ideal of
over
K(X)
s
of s at x ; mreover, the canonical hage of s in the value s ( x ) of s at x at x lies in mx
1.2
.
K(x) will be called
. This value is thus zero iff the germ of
E l e : Let X be a topological space, and let
sheaf of genns of continuous cmplex valued functions on X
is local and its ux tions which vanish at x . the stalk
1.3
s
flX be the
. For each
x€X
maximal ideal is the set of g e m of func-
4)
-1e: Let (x, be a gecmetric space, and let P be a subset of X , endowed with the iladuced toplogy. Let i : P -+ X be the inclusion mp; then the restriction of definition the inverse image i' (
ox) 1
to P (dict.)
.
, written
Qx/P
, is by
,
ALGfBRAIc
2
Accordingly, if x€P
, we
have
(P,
oxIx. we
( oxlP)x =
c3,I~) i s called an open subspace of
For example, consider a section
s(x) #
0
, there
.
ox
s of
call
(P,
. If P is open i n (x, ax) .
( X I fix)
geoanetric space induced on P by
I,
ClWMETRy
over X
. If
5
the
QX\,ip)
, then
X
and
xu(
ox over a neighbourhood of
is a section t of
1, no 1
x
such
t = 1 It follaws that s t = 1 for a l l points y lying i n sane x x Y Y neighbourhood of x , so that the set of xu( such that s ( x ) # 0 i s open.
that
s
Such an open set is called a special open set and is written
Definition:
1.4
5 morphism of
.
Xs
g a w t r i c spaces f : ( X I
dx)
-f
(Y,dy) consists of a continuous map f' : X -+ Y and a hcmmxphism of f sheaves of rings f - of 8, i n t o t h e d i r e c t h a g e f . (8,) of such
that, f o r each xot 12
, g-e hcaxmrphlsm
local, i.e. s a t i s f i e s f (m )c% X
Y
fe
. If
W e s h a l l often write
f
for
is an open subset of
Y
containing
for the ring hcmxmrphisn induced by
fx :
.
of(x)*
ox
induced by f -f
dx
is an open subset of
U
f(u) f -f
, we write
.
fz :
and V uy(v) * dX(u) X
Ckqmsition of m o r p h i m of g m t r i c spaces is defined i n the obvious way.
Gemetric spaces and m r p h i m s between than thus define a category, denoted by Esg. A mrphism of geometric spaces f: X Mu-. gnbedding i f
1.5
f
induces an iscsclorphism of
ax and
Example: If
dy
+ Y
w i l l be called an open
X onto an open subspace of
.
Y
are t h e sheaves of germs of complex
valued continuous functions over X and Y
, each
continuous map f : X
+
defines a morphism of gecmetric spaces: with t h e above notation, w e need V
only set f U (s) = s o f '
1.6
are i n
Propsition:
,V , each
,
where f ' : U
12
functor d:
2 +
+
V
denotes the map induced by
is a category such that Ezq
has a d i r e c t l i m i t .
Obz
F1
f
Y
.
It is sufficient t o show f i r s t t h a t any family of g m t r i c spaces
Proof:
has a d i r e c t sum, and secondly that any pair of mrphisms
4)
frg : ( X r
(yr
0,)
has a cokemel. Now the direct sum
has as i t s underlying space the topological sum of the
8s / X1. = a)
QX
i
. The cokernel
(2,
dz)
~
of
( f , g)
i s the cokernel of the continuous maps f
Z
Xi and we have is constructed as follows: and g
i n the category
of topological spaces, and is therefore obtained by identifying i n Y the pints b)
f (x) and g(x) for each xEX ; if
p: Y
-P
Z is the canonical projection, each open set W c Z
, and
determines two open sets V = p-'(W)
U = f-'(V) = g-'(V) ; then V V such that f u ( s ) = gu ( s ) The
o z ( W ) is the ring of a l l sE 0Y (V) restrictions dz(W)+ dz(W') are induced by those of
canonical projection sions
W
%
: &,(W)
+
(Y,oy) +
8Y (V)
ing t h a t the stalks of
oz
dZ)
(Z,
is defined by p and the inclu-
. The only tricky p i n t
i n the proof is i n show-
are local rings, and t h i s is done as follcws.
since the hcmcmorphisms f, : flf(,) (1.3) ; similarly q-'{Vv) = Uu ' SO where W'
.
0, , and the
is an open subset of
W
are local, we have f-l(Vv) = -1 f-'?Vv) = g-'(Vv) and Vv = p ( W ' )
+
. -1 If
zEW'
u
the inverse of the germ of
w a t z is thus the germ of (vIvV) ; on the other hand, i f Z ~ W ' -1 p ( z ) does not meet Vv and v vanishes a t each p i n t of p-'(z) Fran
.
this we infer the following facts: W
first, i f W
w , w l ~$(w)
have non-inver-
tible germs a t z , then % (w) and % (w') vanish a t each p i n t of -1 W p ( z ) ; hence pv(w+w') also vanishes and so w + w' is not invertible;
4,
is a local ring. And secondly, i f w vanishes a t z , then % (w) vanishes a t each p i n t y€p-1( z ) ; thus p : -t is a local Y hamcmorphism. therefore W
oz oy
1.7
Example:
flX
Suppse that
8,
and
are the sheaves of germs
and Y and that the mrphisms
of ccsnplex valued continuous functions over X
f,9 are defined by ccPnposition with the underlying continuous maps. Then 0, m y be identified w i t h the sheaf of germs of c q l e x valued continuous functions over
1.8
2
.
M k :
and F1 2 E
Ob T E v1
,T is a category such t h a t
It can be s h a m that, i f
, then
each functor
d :2
+
has an inverse
JSE
limit.
Section 2
The prime spectrum of a rinq
2.1
we write
d(X) = 6X (X)
0 : 5s-S
for each qeanetric space X
each mrphism f : X
+
Y
and
Y
d ( f ) = f x (1.4)
, for
of g m t r i c spaces.
Spectral Existence Theorem: Spec A
for the functor such that
For each ring
and a honmnorphisn $A : A
+
A
, there
fl(Spec A)
is a geanetric space
satisfying the condition
("1 below: (*)
f
X
-
is a q m t r i c space and
there is a unique mrphism
4
: A + O(X)
f : X -+ Spec A A
4
i s a ring hcanomorphism,
such t h a t
$ = @(f)$A :
&X)
(Spec A , $A) is evidently Unique, since it is the solution of a universal problem. This universal problem means that the map f Ho ( f )$A
Such a pair
is a bijection
zg(X,Spec A)
describe the pair
ft+O(f)$,
.
(Spec A
),0
%(A,
&X))
. Instead of
a proof, we merely
and give the inverse of the bijection
I, 5 1, no 2
THE LANGUAGE
Description of of A
(Spec A, $A)
:
5
The points of Spec A are the prime ideals
(Alg. camn. 11, 54, no. 3 ) . If f a and pESpec A , we call the
canonical image of f in the field of quotients of A/p at p ; if a is an ideal of A
, we
the value of f
denote by D(a) the set of pints of
Spec A where at least one element f of a does not assume the value 0
The subsets D(a) of Spec A are the open sets of Spec A
.
.
Let S(a) be the set of a l l SEA which do not assume the value 0 at any pint of the open subset D(a) of Spec A D(a)
=
F(D(a))
D(b) =
. Thus
. We obtain a presheaf of rings over
S(a) =
S(b) if
Spec A by setting
NS(a)-l] (Alg. c m . 11, 5 2 , no. 1) and defining the restriction
hananorphisms in the obvious way. If a is the ideal generated by the single element s , and if As denotes the ring of fractions of A defined by the . I , then it is easy to verify multiplicatively closed subset ~l,s,s~,s~,.
.
that the canonical m p As
-+
Z$S(a)-']
is bijective. In particular,
. "he structure sheaf of Spec A is now defined to be the sheaf associ.atedwith F . The -1 stalk of this sheaf at p is the local ring A = A[: (A-p) ]. Finally, we P
F (Spec A) m y be identified with A
let -$IA
(by setting s
= 1)
be the canonical map of F (Spec A) into the ring of sections of
the associated sheaf.
.
We must now describe the inverse $ H g of the map f H&f) -$IA Let -$I : A +. O(X) be a hammrphism and let XEX By definition, g(x) will
.
be the inverse image of mx under the ccanposition
"he m p g is obviously continuous: if a is an ideal of A
, g-1 (D(a))
is the set of pints of X at which at least one element of $(a) does not vanish; the cmpsition
-1 thus factors through A [S (a) ], which defines a mrphisn F
(of the presheaf F into the direct image of requiredmrphism
flSpec A
g.(@X)
*
0,
under
g)
-+
g . ( (3),
, and
thus the
6
Example:
2.2
x
Let
8($x)$A = IdA
prime ideal of f mrphism $i:
. In v i r t u e of
dspc
o(x)
Definition:
2.3
+
($,
such that
s
s(x) = 0
the
. The
.(gX) is constructed as i n 2.1.
)
.
"depends functorially" on A ; i f
Spec A
such
Spec $(XI
For each ring A t h e g m e t r i c space Spec A
called the prime spectrum of A Of Course,
+
assigns t o each xEX
2.1,
d ( X ) consisting of a l l e
: X
1, no 2
O(X)
IXa g m t r i c space. I f we set A =
and @ = IdA i n 2.1, we g e t a unique mrphism qX that
5
I,
ALGEBRAIC GEaMFTRy
hcrtlcsnorphism, we write Spec 4 : Spec B
.
+
is -
is a ring
@ :A + B
Spec A f o r the unique mrphism
This m q h i s m is defined explicity as folsatisfying QB$= O ( S p ~ m s :the map (spec 0)s underlying spec Q sends q onto @-l(q) ; i f a
is an ideal of
A
, we
have
and the c a p s i t i o n
+
--+(3spec
A+B
factors through
B(D(B@(a)))
. As
XS (a)
a varies , we thus obtain a mrphism
frcm which we derive the required mrphism
I n particular, i f
SEA
an i s m q h i s m of
Spec As
and 4 : A
-t
As
is the canonical map,
onto the open subspace
(Spec A)
Spec $I
= D(As)
is of
Spec A .
2.4
For each ideal
For each subset V(&&P)
P
a of
of Spec A
A
, the
, set
closure
V(a1 = (Spec A)
5
of
P
- D(a)
.
thus coincides w i t h
I,
If
5
1, no 2 @ :A
For, i f
-+
THE
7
LANGUA(;E
is a hamnorphism and b is an ideal of
B
B
, it follows t h a t :
(Alg. c m . 11, 52, no. 6) , we have
Ja denotes the r a d i c a l of a
(Alg. cam. 11, §4, no. 3, corr. 2 of prop. 11).
, we
I n t h e particular case b = 0 image of
. For
Spec B
Spec $
see t h a t
V($-'(O))
is t h e closure of the
t o be dcminant ( t h a t is, f o r the image of
to be dense) r it i s thus necessary and s u f f i c i e n t t h a t
(Spec @)'
@-'(O)
be a n i l i d e a l .
If
2.5
is the ring z[T]
A
hanarnorphism @ : _Z[T] t r a r i l y chosen i n f i e d with
8(X)
.
-+
8(X)
d(X)
X
Definition:
If
$ : X
+
over x
X
2.6
Propsition: -+
0 (Spec A)
, 8(X))
A.J(,Z[T]
can be arbi-
may be identi-
o(X)
m y be i d e n t i f i e d w i t h the set of
. This j u s t i f i e s
the follavinq
Spec z[T]
tions
$A : A
, which
is a q m e t r i c space, a morphism
is called a function
.
$ (T)
, each
Applying the adjunction formula
i n t o Spec$T] X
is determined by
. It follows that
established above, we see that mrphisns of
of p l y n a n i a l s i n a variable T
; the ri.nq
O(X)
For each rinq A
, the
of 2.1 is an i s m r p h i s m .
i s called the rinq of func-
hamnorphism
. We show more generally that the presheaf
Proof: Set X = Spec A
ospec
F of
2.1 assumes the sane values as the associated sheaf A over the special open sets X = D(M) ,faSince X %I = X for f,gm , it is f f 4 fg sufficient to show that whenever X is covered by Xf ,...,Xf , we have f 1 n an exact sequence
.
V
F(Xf) -+ nF(X " .nF(Xf i fi w i l l' i j
I
where u, v, w are defined by u(a) = (a.) , v((bi)) = (b..) and 1 13 w((bi))= (cij) I ai , bij and cij denoting, respectively, the restrictions of a, b and bj to f' 'f.f. and xf.f i Xfi = X fK = Xffi and F(Xf) = i Af ,i j 11 f fi
. Since
it is sufficient to show that the sequence
v~~~~ .
, B= is exact. TO see this, set c = flat over C (Alg. ccatnn. 11, 5 3, prop. 15 identified w i t h
&
men B is faithfully cor.), and n A f f my ij
v and w being identified w i t h the maps b w b @ 1 and b-1
@ b
. Exact-
ness follows frran kmna 2.7 by setting M = C = Af '
2.7 Imma: E t C be a rinq , M 5 C-module and B a faithfully flat C-algebra. Then the sequence of C-dules
- -
0
*
M
-+
.
M f B
a n ( m @bl@. .@bn) f i
=
n > 0 is exact.
-f
i=n
1
i=O
MtfjBfB
-+
M?
B@B@B
c
(-1)im @blQp.. .@bn-i@l@bn-i+l@.
c
..@bn
-+
,
... ,
Proof:
Since B
i s faithfully f l a t over
C
, it
is enough t o show that the
sequence
is exact. But, i f we set
.
sn(m @bo@. .@bn+,) = m @boa.. .@bn-l@bnbn+l (n L 0 )
, we
have
so ( do@B) = Id
2.8 Proof:
(dn@B)
Corollary:
The functor
Set X = Spec B An(A, B)
*uI
(dn+f3B) = Id
+
.
AHSpec A
is f u l l y faithful.
i n Theorem 2.1. The map
FSg(Spec B, Spec A)
is the c a p s i t i n n of g ( A l @B) : %(A,
B)
-+
%(A,
O(Spec B)
with the bijection
d (XI
E(A,
7 Eg(X,
Spec A)
of 2.1. It is therefore i t s e l f a bijection.
2.9
Definition:
the mrphism spectral space
: X + Spec o ( X )
if
When X = Spec A
so t h a t
X
A geametric space
X
,
X
is called a prime spectrum fi
of 2.2 is an ismrphism.
X
is called a
has an open covering by prime sy3ctra.
it follows f r m 2.1, 2.2 and 2.6 that
JIx
= (Spec $A)
is a prime spectrum. Since the special open subsets of
are prime spectra (2.3) and form an open base f o r Spec A
, we
-1
,
Spec A
see mre
generally that each spec’qal space has an open base consisting of prime spectra. It follows that each open subspace of a spectral space is a spectral space.
Recall that a t o p l o g i c a l space
2.10
X is said to be irreducible i f
it is non-einpty and each f i n i t e intersection of n o n - q t y open subsets of X
is n o n - q t y . For example, f o r each t o p l o g i c a l space X and each point
XEX
, the
If
Proposition:
of
X
Proof:
TZ
closure
of
in
x
x
is an irreducible closed subset of
is a spectral space, the map x
X
onto the set of irreducible cLclsd subsets of In the case where X
I+
x
(3is a bijection .
x
is a spectnnn, the p r o p s i t i o n follows frcnn
Alg. c m . 11, 54, no. 3 , cor. 2 of prop. 1 4 . This special case imnediately
implies the general case. If
an3 i f x is the unique pint , x is called the generic point of F
F is an irreducible closed subset of
such t h a t F =
(XI
Emnple: For each family
2.11
write E i
. .
for the direct sum
i€E
CI
each mrphisn
S
i
X
.
(Si)iEE
of copies of
Spec
. To each geanetric space
X
5
we
and
f :X + U S iEE
we assign a map g : X
+
i
E
such that g(x) = i i f
xEX and f(x)ESi
. The
is locally constant, that is to say, it is continuous i f E is -1 assigned the discrete topology. If Xi = g (i), the canonical i s m r p h i s n
map g
Esg(Xi f i : Xi
, Spec_Z) +
Si
A n (2
, d(Xi))
(2.1) shows that the induced mrphisn
is thus a bi-
is detamined by i and Xi ; the map f - g
.
jection Esg(X,E')z =(X,E) 2
. 1
A spectral space
X
ismiorphian X G E i
is said to be constant i f there is a set E and an
.
An
2.12
Example:
Let
k
be a f i e l d and
X
a Boolean space, that is a
topological space with a base of m p a c t open sets. Let of rings which assigns to each open subset U of
X
4,
be the sheaf
the ring of locally
.
. For each
constant functions over U with range
k
for each ccanpact open subset U
the mrphism $
of
X
2.2 i s invertible (Stone)*. It follows that
3=
, we
xEX :
U
(x,0,)
U
ox
have
= k; Spec o ( U ) of
+
is a s w t r a l
space. 2.13
The theorem and remarks of 2 . 1 signify that the functor
Remark:
Spec :
go+ E
2
8': Esg + An'.
is the right adjoint of
* * l u r w
It thus transforms
direct limits of rings into inverse limits of gecmetric spaces. I n particular, for each diagram of rings of the form B Spec B NAC +(Spec
B
2A
C , the canonical mrphism Clwith m p n e n t s Spec (in,) and
s&
Spec (in,)
is invertible.
Section 3
g-functors
3.1
Definition:
Az-functor is a functor fran the category of models
g . The category
M ,into the category of sets
4 . 4
of Z-functors is denoted by
ME.
YCI
3.2
X p
Notational conventions:
and x€X(R.)
w
urider the map
5
, we
write $(x),xs
X(@) : X_(R)
+
X(S)
Y_ is a mrphism of
If
f
of
( f ( R ) )(x) for the h a g e of
-Y'
is a subfunctor of
:
+
If
A
2
, we
(SJ? A) (R) = $(A,R)
9A
simply x
xEg(R)
, we
for the h a g e of
write ;(XI
x under the map f (R)
: X(R)
-+
denotes the inverse h a g e of
-X , satisfying
.
write
, or
PJP and
Y , g - l ( Y-' )
i.e. the subfunctor X' of for each R?
.
2 , if
@ : W S is an arrow of
If
x
instead _Y(R)
. If
Y_' i n
-
X,
s ' ( R ) = {xEE(R) : f(x)EY'(R))
for the functor represented by A :
for R g
affine scheme of the ring A
. If
A
is a model, we say t h a t % A
is the
. With t h i s terminology, an affine scheme is
* See, e.g. J . L . KELLFJ, General Topology, Chapter 5, exercise S, Van nostrand, 1955
thus simply a representable functor. If
Sp f
I
the map $ If
R 2
and p€Z(R) , we write p#
functors which sends the map
$E(Sp - R) ( S )
onto
B
Emnple:
:
3 R +X
onto ME(Sp - R, -X)
The functor
0
.
*c(
onto ub = ( o ( R ) ) (IdR)
-
ME(X,g) (
$
= ($ (R) 1 (XI
if R e
, an
element $ 6 g ( $ T ] ,
R)
set
. A mrphism functions on X
i s determined by
Sp - i[T]
enable us to identify
@ (T)
Example:
is the functor G
. The ring of
set
with
0
$(T) ; thus
. Accordingly,
is an affine scheme.
3.4 P
.
fl@)
the maps $-
0
and xEX(R)
, we
its under-
i s the ring of p l y n m i a l s w i t h integer coefficients i n a variable
Z[T]
Y
, and
+ ($(R) 1 (x) and ($(R))(x) for R+
w i l l be called a function on X
is denoted by
T
(XI.
(x) = ( $ ( R ) )
E ?-(_x,g)
If
5 , the
#A”.
($+$I ( R ) ) (XI
. W e knm t h a t . The inverse
which assigns to each Rf”,
carries a natural ring structure: i f $,JI E ME(X 0) -1-
(($.$) ( R ) )
R%
f o r the mrphism of
lying set is called the affine line. For each?-functor *+I
5,
is a mrphism of
(X($)) - ( 0 ) = ps E_x(S)
is a bijection of X ( R )
p -p#
map sends UeA(Sp R, _X)
3.3
+
.
into :(AIR)
g(B,R)
H $ofof
, 5%
f :A
is the functor homcHnorphim which assigns to each
: Sp B + % A
of rank
($1
-n,r by $
G
-n,r
kt n
n=l
is denoted by
3.5
,
gr
- . If SX
A
functor
R-E(A,R)
be two integers
Rn+r
the -ge
2 0 ; the G r a s m i a n
of
. If
4:R-G
s 64RP
in
is a n x r m of
g,
s ~ t +d e ~ r the map
induced
is called the projective space of dimension r
%,r
. The functor
Example:
R w Es~(SpecR,
r
which assigns t o each RSM, the set of d i r e c t factors
n of the R-module
assigns to P
. If
,
Let
X
El
is called the projective line.
be a geometric space. The 5-functor
X) i s calfed the functor defined by X and is written
is a ring, ?(Spec A)
may, by 2.8, be identified with the
. Accordingly, w e have a canonical ismrphism
and
&s
A
.
S(Spec A)
If I is an ideal of A
, we can interpret the functor
g(D(1))
in a simi-
is the open subspace of Spec A consisting of all
lar fashion, where D ( I )
points where at least one sEI does not vanish (2.1). For if $ E
(9A) (R)
= &(A,R)
follows that Spec @
, we have
(Spec @)-'(D(I)) = D ( R $ ( I ) ) by 2.3. It factors through D ( 1 ) if and only if R = R $ ( I ) We
.
see accordingly that S(D(1)) may be identified with the subfunctor (SJ A)I
9A
of
for each R 2
3.6
satisfying
. We call
(9A ) I
the subfunctor of Sp A defined by I
Definition: Let - - X- be a 2-functor and. let
.
g be a subfunctor
of - -X We say that 2 is open in X (or is an open subfunctor of Xof for each &el A and each f : 9 A + 2 , the subfunctor f-l(g) -
S p A can be defined by an ideal I A mrphism
:
V_
5 of E-
+-
of -
A
of
m(sp - A, g) , set
c4
9A
c1
=
is said to be an open -ding
f E X(A)
-b
is such that, for each R € E
for which a R € g ( R )
,
)
-if,
(3.5).
X mnmrphisn and the image-functor is open in -
If f E
.
(3.2)
if
.
. The subfunctor
z - l ( U ) (R)
& is a L-'(u)
is the set of $:A+R
. We can thus reformulate the above definition by saying
that g is open in
5
if, for each AEM,and each a€X(A) -
, there is an
ideal I of A satisfying the following condition: for each arrow $:A+R of M
*r
, we have
. Then
iff R $ ( I ) = R
.
Example: Let X be agecanetric space, Y anopen subspaceof
3.7 X
a R € I ( R ) c X_(R)
SY
element of
.
is an open subfunctor of S X For if c1 : Spec A X is an -1 (SX) (A) , c1 (Y) is an open subset of Spec A and is therefore
of the form D ( 1 )
for same ideal I of A
+-
. This ideal
I
satisfies the
conditions of 3.6, We infer fram this that, i f
A
open iff V_ is of the form
(SJ
is a &el, A)I
.
a subfunctor g Ef S e A
For since
is
14
XGFBR?UC GMx.1FTRY
(sA) I
,
S_(D(I))
5 1, no
I,
. The con-
is open i n &S A for each ideal I
(Sp. A I I
3
X = Sp f = 1% i n definition 3.6. verse is established by setting - A and -
Example:
3.8
then
(sAIM
q : JS Af =
,
A$$
fEA
and i f
q:A+Af
is the canonical map,
Sp A is an open emkdding whose image functor is
. For instance,
and %?[TIT line 0
+
If
and f = T , % z [ T ] may be respectively identified with the affine
i n the case where A = HCT]
2 _Z[T,T-']
(3.3) and the subfunctor p
0 which assigns t o each
of
its
REM I
set of invertible elements.
mre
generally, i f
f:z<
is a 3-functor and
_X
, we
is a function on _X
zf
for the inverse image _f-l(u) ; we shall say that X is the sub-€ functor of -~ X where f does not vanish. This subfunctor i s ofin (the inverse image of an open subfunctor is an open subfunctor).
write
3.9 f+r
-',
.
Example:
Q be a direct factor of rank r of the group
Let
For each REM_ we identify R@ZQ with its image under
R ~ + ~ R B
~
~and+we ~write
n+ronto R /(RBZQ)
. Let
gQ be the subfunctor of Gnrr
E ; assigns t o each- I
the set of
Rn+r
U
. W e claim that
-Q
+.
rR for the canonical projection of
(R-linear)
ment of
ccanplments of
R
~
+
RgZQ
in
.
= S
iff
S gRP =
(5
nrr
( $ ) ) (P)
is a cmple-
In order for S BRP t o be a ccsnplement of S 8 Q , it is necessary and H n+r sufficient t h a t the map vs : S HRP + S /S BZQ induced by ?rS be bijective; s i n c e the damin and the range of
vs
Ge
t i v e d u l e s of the same rank, t h i s holds i f f coker vs -3 S
@
R
(Coker v ) = 0 R
.
vs
f i n i t e l y generated projecis surjective, i.e. i f f
Since Coker vR is a f i n i t e l y generated R-mdule, this last condition i s equivalent t o S$ (I) = S (Alg. cam. 11,
5
~
which
is open i n Snrr , that is to say, for Zach EE& there is an ideal I of R such that, i f $:wS
and each P E G (R) , -n,r is an arrow of g , we have S $ ( I ) n+r S C ~ ~i n Q S
(3.4)
, where
I
is the annihilator of
4, no. 4 , props. 17 and 1 9 ) .
Coker vR i n R
I, 9 1, no 3
THE m G U m
..
Now consider a basis el,e2,. ,e n+r of
15
zn+rover
such that
ze
Q = $
i>n
-i
If R is a n-cdel and P a cmplement of R C3zQ in Rn+r , we have the identities l @ e i = pi +.I aij @e j l’n for i_
.
.
.
Finally, we assign to each subset I of {l,...,n+rj of cardinality r the direct factor Q, of Zn+l consisting of the sums lniei such that
.
ni = 0 for ieI For each field K e , the excharge theorem says that G (K) is the union of the sets _V (K) We express this fact by saying -n,r QI that the subfunctors U of GnIr cover -n,r G . More generally, we make -Q1 the
3.10
.
Definition: Let X be a:-functor. A family
functors of X is said to cover union of the sets X(K)
.
if, - for each field K%
For example, if Y is a g m t r i c space and of Y
, then
(ZiliEIof sub, X(K)
is the
(YiIicI is an open covering
(SYi)iEI is an open covering of SY i.e. a cavering consist-
ing of open subfunctors. In particular, if A is a ring and
(fi,xi)iEI
is a finite family of pairs of elements of A such that
1
=
1
iEI
Xifi
then (sD(Afi))iEI is an open covering of (f2xi)icI a partition of unity in A
.
3.11
(Spec A) = Si A
Let & be a pfunctor, R a model and
. We call
(fi,xi)iEI a partition
16
ALGEBRAIC
of unity i n R
into Rf prio u
i
=
. W e associate w i t h these the sequence of
(3.10)
defined as follaws:
if
(resp. of Rf
a)
31
into R f . f
i
11
=
(fi,xiIiEI
, we
=
- -
, the
.
-X ( a1. 3. ) - p r j
is called local i f , for each
in R
R
set
p r ~priIj-w -x ( 7~1( . . ) ~ and
A wZ-functor X -
partition of unity
sets
ai (resp. a . 1 denotes the canonical map of
X_(ai) , pritjov
Definition:
I, § 1, no 3
GECMEmx
and each
RGbJ
sequence (*) above is exact.
(zi)iEI
b) X is called a scheme i f 5 is local and has a covering of affine open subfunctors Xi indexed by a set I belonging t o the fixed universe
.
g
The f u l l subcategory of ,ME, formed by schemes w i l l be written
zh.
W e observe irrunediately that an open subfunctor _Y of a schane ,X is i t s e l f a scheme. For, with the notations used i n the definition, choose models Ai -1 and isom3rphisms : Sp Ai + Xi By 3.7, the open subfunctor (Xis) of
.
si
+ Ai
is of the form
(Sp - AilIi
where Ii
is an ideal of
zi Ai
. As
s
m s through Ii , the affine open subfunctors (3Ails of 9 Ai cover (%AilIi If Zis denotes the image-functor of (9Ails under gi , the the family f o m the required covering of Y by affine open sub-
.
(-xis)
functors.
of a scheme
W e shall henceforth c a l l an open subfunctor
X.
scheme of
Example:
3.12
-
m y be identified with %(Spec
Esg(D(Rfi) Spec Rfi
n
D(Rfj), T )
, given
(*)
R,T)
t h a t D(Rfi)
i s the scheme induced by
the exactness of
, the functor -ST (*) , X(R) , X(Rfi)
For each gecmetric spce T
For i f we replace X by _ST i n the sequence &(Rfifj)
an open sub-
Spec R
.' Zg(D(Rfi) ,TI
D(Rf.) 3
=
D(Rf.f.) and that 11
means simply t h a t a mrphisn m : Spec R
tions satisfy the usual matching conditions.
and
and
over the open set D(Rfi)
termined by i t s restrictions to the open sets D(Rfi)
i s local.
+
T
. ?"nus
is de-
and that these restric-
I, $ 1, i no 3
_ST is a scheme i f
I t follows f r m t h i s that
a l s o small (i.e. the set underlying T of
TI
T
,
is a spectral space and is
T
is small and, f o r each open subset
is small).
c!~~(,(TI)
Example:
3.13
17
THE LANGUAGE
snlr is a scheme. For by 3.9
The grassmannian
s u f f i c i e n t t o show t h a t sequence
is a local functor; i f
G
-n,r
P E snIr(R)
it is
, the
i n lemma with the obvious arrows, is exact (set M = P , C = R , B = F R i fi n+r n+r and Pfifj C Rfifj , this simply means that P is 2.7) Since Pfi C Rfi
.
TR:
, i n other , or that
7 p f i of the inverse image i n ~ n + r of the s-ule words, that P E GnIr(R) is determined by the P E &(Rfi)
the map u
in
fi
vsn,r(~ .
Suppose now that we are given the Ji E G n I r ( R . ) ( J ~ )E
snIr(R)
I E that
fl
w i t h the product module J =
,I
B = TRfi
f1
such that J
inY+'(J)
B
and we identify the family
Ti over the r i n g
Suppose that v ( J ) = w ( J ) : we have t o show that there is an =
u(1)
. Now the assumption
, which
denotes the
in;
i n t o B @RB mRB ( i = 1, 2, 3)
(B @RB BRB)n+r
v(J) = w(J)
, the
B gRB @RB-suhzdule L
i s generated by
(J)
, is
of an R-suhndule show t h a t
I
I
of
such that
Rn+r
is a d i r e c t factor of
of
independent of
i
.
denote the
3 this lemna establishes t h e existence
, then
and
of
ith canonical i n j e c t i o n
Using t h e notation of Lemna 3.14 below, i f we l e t ui and u! maps induced by dy+r
means
(J) generate the same B @RB-suhndule K
and
(B @RB)n+r; i.t foL&owst h a t , i f
of
.
(3.11) i s i n j e c t i v e when ,X = G -n,r
(*)
J =
R"+r
Ffi.
I t thus remains t o
of rank r ; and t h i s is the con-
t e n t of lama 3.15 below.
L e t us intrcduce some terminology which we s h a l l use i n the
3.14
following lemna: Let h : R N
B
k a ring hcatamrphism,
B-module. W e say t h a t a map f : M
a
f(m-ii-n') = f ( m ) if
+
f
+
+
N
M
and Rrnodule,
is adapted to h
i f we have
f ( m ' ) and f ( m ) = h ( r ) f ( m ) f o r a l l m,m' E M
induces an isgnorphim
B@#
N
.
, r E R , and
Leima: - Let
R
be a ring, and. let B be a f a i t h f u l l y f l a t R-algebra.
Suppose we are given modules J,K,L
over
B, B gRBr B BRBBRB respectively,
u'n
and maps
cr3 ul,L
J U%K
,
-T
u ' 1
adaptea, respectively, to the ring hanrrnorphisms
such that do(b) = b @ 1 , dl(b) = l @ b , d;)(bm)= b @ c@1, d i ( b & ) = b 81 @c
,
u $ ~ = uiuo
d;(b W)
, u&
=
the inclusion map of
=
1@b@c f o r b,c E B
u$uo I
and
into
uiul
=
u'u 2 1
. Assume a l s o t h a t
. Then ____ if
I
induces an isanorphism
J
Ker(uo,ul)
=
I BRB 3 J
.
Proof: Inspect the diagram
J@
R
.i J
t
u1
dO@J J-B@B@
J
where i is the inclusion map, v1
are induced by
i,uo,u'o
j
K
the canonical injection,
R
J ,
v r v o r v ' o and.
and ul , and w i s t h e ccmposition
B @ B Q D J Bwk3 @ K
.
IwI
v ' 2 ,L
,
being ind.uced by u; By lemna 2 . 1 , t h e horizontal sequence a t t h e b t t m i s exact. Since v1 and w are bijections and w(do@J) = uI1v1
v;
w(dlW) = u;vl
, the
second horizontal sequence is exact.
I,
5
1, no 4
THE IANGLJAGF,
and v 0' (u160B) = uivo
Finally, since v~(uo@B) = uivo
and. v1
induce an iscanorphism v
J = Ker(u' , u l )
1 2
I
.
Lsmna: -L e t
3.15
a s M u l e of
d i r e c t factor of Proof:
M
and B
BBRM
of
, the
bijections
E 3RB = Ker(uo@B,ul@B) onto
be a ring,
R
19
M
a f i n i t e l y presented
a f a i t h f u l l y f l a t R-algebra. I f
,ths
I
factor of
is a direct
M
.
vo
R-module,
BBRI is a
It i s s u f f i c i e n t to s h m t h a t the canonical map
4
: HT(M,I)
+
HT(I,I)
is s u r j e c t i v e or, equivalently, t h a t B g R $
is surjective. This folbws f r a n
inspection of the camnutative square B @ H m ( M , I ) >-B
B F 4
Ji $ H T ( B a , B @I)->HT(B R
R
@ H%(I,I)
"k
2 I i B $1)
in which a l l the arrows are the "obvious" ones. For since
factor of
B BRM
,$
3.16
Remarks:
is a d i r e c t
is a surjection; on the other hand, since B BRI is
f i n i t e l y presented Over B bijection, a s is i
B@ I R
, and
,I
i s the same over R ; accordingly j
is a
BBR$ is surjective.
One can avoid using l
m 3.14 by employing the descrip-
t i o n of quasicoherent modules over Spec R given i n
5
2 , no. 1. Anyway, we
shall find t h i s leima useful l a t e r . Consider, on the other hand, the subfunctor GI of G which assigns to n,r n,r each model R the set of f r e e direct factors of Rn+r of rank n This
.
functor is covered by the a f f i n e open subfunctors U
QI
local.
Section 4
The q m t r i c r e a l i z a t i o n of a Z-functor
4.1
P r o p o s i t i o E The functor
-S
:E A
-+
FE3
of 3.9, but it i s n ' t
has a l e f t adjoint.
20
I, § 1, no 4
ALGEBRAIC Q3:OMETRY
Proof:
W e sketch a proof of this particular case of a w e l l knm theorem of
Let g be a2-functor and let
be the category of g-models: an 5 object of this category is an -F - d e l , i.e. a pair (R,p) consisting of a Kan.
and a p s ( R )
; a mqhism
of
into a s e c o d _Fmodel
mdel
R
(S,G)
is defined by a h m r p h i s n @fM_(R,S) such t h a t
(R,p)
80%$ = o # ) . If % : (MJD spec R , we set ( ~ =1 lij% . It -
such that (R, p)
I-*
each XEEs a bijection
'E(Fr$X)
: E+J(IFI
rx)
which is functorial i n X
. Let
each RCE
, a map
composition
_g(r): F ( R )
+
Esg
_I
(SX)(R)
- + where i ( p l is the canonical mrphism of
(i.e.
denotes the functor
thus remains t o construct, for
I
. If
pC_F(R)
, (g(R)) - (0)
is the
r
into 1 9 %
%(R,p)
-
define a mrphism g:F+sX and. we s e i g = @ (-F I X )( f ) -
g(R)
. .
G
f:lEl+X be a morphism; we must define, for
f i (pl ~ RC c l h % + X
S
-+
@(p) =
-
.
. The maps
It remains to show that the map $(FIX) is bijective. To this end l e t
6F
: (&)'
+
_ME, denote the functor
G s y t o v e r i f y that the morphisms p"
F
lim 6
+
F
. Now
( E l X)
( R , p ) w Sp R :
A ~ ( R , P )+ _F
. It
i s w e l l known and
induce an i s m r p h i s n
has been d e f i n s i n such a way as to make the
squares
camutative, where Since tions
= lim +
c1
P
%
c1
*
P '
ard
(2x1 (R)
F
-
+
,E(Sj
lim 6F +
-
R,
2x1
, $(FIX)
is the canonical isomorphism. is obtained fran the bijec-
by passage t o the limit aver the objects (R,p) ; hence $(F_,X)
my be identified with the inverse limit of the bijections
fore a bijection. If
= S_E A
, an F-mcdel
iS
simply a &el
c1
P
r
and is there-
carrying an A-algebra structure.
In t h i s case
% -
%m
coincides with
and its i n i t i a l object is the pair
(A ,IdA) ; acc&dingly we have a canonical i s m r p h i s m
i (IdA) : Spec A
I%Al
-f:E+F --
If
4.2
unique morphisn of
conmutative a5 X H
runs through
, we
(R , f ( p ) )
9.W e
Zf
If
-+
:-functor
c a l l 1 ~ 1 (resp. /:I) functor I ? I :F - e+ !_FI If
F is
(resp.
; w e write simply
points of
explicitly as fol-
denotes the functor
% ; consequently, Ef.
)" =
is
if -f
of
!_Fl
_F
(resp.
denotes, depending on the context,
F , or the
underlying s p c e of t h i s
x of this underlying space w i l l be called a point of
.
x g for x€\_FI W e shall take care not to confuse the
. W e s h a l l also c a l l the
,F w i t h the elgnents of _F(R) for R@I-
t o p l o g i c a l space which mderlies the geometric realization of of points or spectrum of a set of p i n t s of
is a mrphism of
,F
. A subset
_F ; we write ME
denote the image of
, we f) . The
is a mrphism of M&)
g m e t r i c realization
either the geanetric realization of
-F
-f
%
is called the geametric realization-functor.
a 2-functor, the symbol
realization. A p i n t
+
% lp % , a d t h i s -
_F
If/
can obtain
: +P
obviously have %*(M
l$n
induces a mrphism Definition:
for the
lfl:lE1+1_Fl
z g which makes the squares
lows: with the notation of 4.1, i f (R,p)
EVE , we write
i s a mrphism of
P c
P
of
for P c
and Q is a subset of
_F
the space
-F w i l l then by definition be
G -
)?I . Moreover,
, we
g:_F-tG_
(resp. g-'(Q))
Q) under the map lglg induced by g Similarly, g(_F) denotes the h a g e of /l?I under Is_lc ; which is not t o be confused w i t h the this image is thus a subset of P
.
image-functor
of
(resp. the inverse image of
l e t g(P) -
if
L
C J
which is denoted by --:
to each R€M the image of the map I
g(R) - :_F(R)*-(R)
t h i s image-functor assigns
.
Au;EBRAIc GEamrRy
22
open, closed) injective, ~ _ i f the _continuous map
note it by _fc
F i n a l l y , we write
Z-functor _ F .
0
-F
for
8IFI
is surjective (resp.
the map underlying f
.
dF
and call
1, no 4
gw is surjective (resp.
Henceforth we shall say that a mrphism g:-F+G_ of injective, open, closed). W e also c a l l
5
I,
and de-
the structure sheaf of the
-
h
Corollary: For each 2-functor - -
4.3
canonical ismrphism
-
Proof:
E(_F,% A)
I
, g((F, % A )
When A=$Tj
fied w i t h
(3(F)
,d(F))
there
.
is a
It follows from 3.5 and 2 . 1 that there are canonical isanorphisms
ME(F - ,%?A) "p+(F ,S(Spec A ) ) -3
W
%(A
F and each ring A
d( I,FI )
8(I_FI 1
%(ill
,d(\El))
,Spec F.)
is j u s t J ( _ F )
,&lgl))
and &(A
. W e thus infer the existence of
.
may be identi-
a canonical i s m r p h i s m
and hence the required ismrphism. This l a t t e r m y be defined
explicitly as follows: to each $ E ME(E ,SJ A) we assign the hOmcm0rphis-n +:A+d(F) such t h a t the map ($(a) (R) : r ( R ) + R sends x€g(R) onto ( $ ( x ) )(a) for each aEA and REM c
4.4
.
Now consider a :-functor Y ( F ) : -F + SlFl -
_F
and a geametric space X
and @(XI : ISXl
for the images of the identity morphisns of tions
@ ( F ,/!I)
and @(SX,X)-l
of 4.1. Let
+
. W e write
X
1x1
and _SX under the bijec-
p-'
F such that of _ME_ consisting of the functors -
be the full subcategory
Y(F)
is invertible, and
let E s ' be the f u l l subcategory of
E s consisting of a l l X such that @(X) is invertible. It follows from the well-known relations between 4, and Y that I ? [ @ induce an equivalence between and E ~ I
.
I,
5
1, no 4
THE
mGum
23
W e m y thus mnipulate the objects of these categories either gecanetrically
-
or functorially (be regarding then
(be regarding thm as belonging t o Esg')
as belonging to
El) .
For example, i f
A
and R are d e l s , the map
Y ( S p A) (R)
: :(AIR)
-+
%(Spec
R,Spec A)
, which,
by 2.8, is a bijection. It follaws
is precisely the map f++Spec f
that Sp , so t h a t Spec A ISp - A1 - A belongs to W e now describe further objects of ME' and Ezg' :
belongs t o Esg' UL-
.
rn
canparison Theoren:
x
a)
be a geanetcic space.
is invertible whenever X s a t i s f i e s condition there exists an open covering
(*)
such t h a t b)
c -
be -~ a 5-functor.
L e t _F
for Y (_F) : F -+ .s IF]F be a schane. -
@(XI
12x1
:
-+
x
- X by prime spectra Xi of
(Xi)iEI
to s a t i s f y
In order for
i
.
h
(*)
e and
to be invertible, it is necessary and sufficient that
and 5
The functors
Then
below:
is ismrphic to a model for each
and U(Xi)
IEU
(*)
induce quasi-inverse equivalences between the cate-
gory of schemes and the category of geanetric spaces satisfying (*)
.
The proof of t h i s theorem is deferred u n t i l 4.16, when we have a second desa t our d i s p s a l
cription of the g&tric-realization-functor
.
In the sequel we shall often make implicit use of the canparison theorem by arguing as i f a scheme were a geanetric space. Thus, for example, write f:x-+x or 5 for
1x1 . %en
I~\:lgl+\~Ior
ule
may
confusion is possible,
we notify the reader by using a phrase l i k e : "Taking the geanietric viewpoint". 4.5
H:$+g , we
A-
KCf
(K)
an3 b = ( H ( f ) )(a)
tion, we write Hx Hx(K)
is the set of
functors
. Given a functor
is the quotient of the dis19H by the smallest equivalence relation containing a l l
(a,b) for which there i s a mrphism
MH(L)
g
recall t h a t the d i r e c t limit
joint sum pairs
L e t K be the f u l l subcategory of f i e l d s of -
Hx
. If
x
for the subfunctor of clExfH(K)
and each Hx
f:K-+L such t h a t
a€H(K)
,
is an equivalence class modulo t h i s rela-
. Then
H
H
such that, f o r each K€K-
,
is the d i s j o i n t sum of these sub-
is indec~npo sable, that is to say, it is not the
d i s j o i n t sum two non-empty subfunctors. I n this s i t u a t i o n we shall call the Hx
the indeccsnposable m p n e n t s of
, and we may
H
identify
lip H with
the set of irdeccsnposable ccanpnents. 4.6
psable
.
4.7
Example:
Each representable functor of
Example:
Let
be a geometric space
X
the r e s t r i c t i o n of the 2-functor have
to
_SX:p;
into
and let H
B . For
Esg(Spec K r X) = ~ A ~ ( K ( xK) ) ,
H(K)
2
with the set of 4.8
XEx such t h a t
Example:
(2A) IE
of
p h i m of
A
K(X)
Let
S p A to
into K
A
. If
5
(SX)15 be , we thus
=
,
K(X)
is iso-
is the d i r e c t sum of the irdecmpsable
morphic to a model. Hence, since H functors represented by the
is indecm-
each K E E
xwhere x runs through the p i n t s of X whose residue f i e l d * )
2
, we
K(X)
see that 1 9 H m y be i d e n t i f i e d is i s m r p h i c to a mel.
be a model and l e t H be the r e s t r i c t i o n KEK
I
, an
, is determined by
element x € H ( K )
, i.e.
a hcancmor-
its kernel p which is a prime ideal,
.
and by the induced homcpnorphisn x : Fract(A/p) -+ K The map x t-+(p,x ) P P is thus a bijection of H(K) onto t h e d i s j o i n t sum UK(Fract(A/p) , K) so Pthat the indeccxnpxable cQnponents of H are represented by the residue
lp H
f i e l d s Fract(A/p) ard i d e a l s of 4.9
A
.
Proposition:
g e m e t r i c realization Proof: I f we l e t have, by 4.1, 4.8
may be identified w i t h the. set of prime
For each ;-functor
1,FI
of
, the
underlying set of the
f is canonically i s a m r p h i c to
l$n(FIE)
Xs denote the set of p i n t s of a geanetric space X and the c m u t a t i v i t y of d i r e c t l i m i t s , t h e follawing
canonical bijections:
.
w
5
I,
1, no 4
g ,a
W e now consider a &functor
4.10
25
THE LANGUAGE
functor
zp of
pF(R)
such that
subset P of _F
defined by l e t t i n g , f o r each REM_
f o r each hammrphism @:WK i n t o a f i e l d K€& Clearly ?PI$
= f
-
-Q
(Fp)
and i f
, we
Q = ~-'(P)C~SI
v e r i f y that
(3.2).
Example:
4.11
~ ~ ( F ~ I J E ).
via the formula P =
g : p ~is an arrow of
G
( i n the notation of 4.5).
1
-P
-1
and t h e sub-
be the set of
';xp(_Fl:),
so we can recover P f r m F If
, Ep(R)
If
is a gecmetric space ard _F
X
identified by 4.7 with a set of p i n t s XEX
EP
i s m r p h i c t o models. Thus
= _SX
,
P
whose residue f i e l d s
SP , where
may be identified with
may be
are
K(X)
P
is
and the r e s t r i c t i o n d X [ P of dx to P is an open subset of X , -FP i s an open subfunc-
assigned the topology iraduced by X P
. I n particular, i f %A .
t o r of 4.12
Propsition:
----
Let -
F
be a Z-functor. Then t h e map
duces a bijection between the open subsets of
in-
P w F p
and the open subfunctors
Il?/
of _ -F , Proof:
Let
inverse image of P
FP
i s open i n
111
-
for all
iff
(R,p)
open subfunctor of
(3R)Q
(R,p)
an object of
under the map i ( p ) : spec R
P
is open i n
Q
is an open subfunctor of
Sp R
,
be a subset of
P
. Since
-
Sp R
,F i f f Q
- . To
F
of
(4.111, and since we
-F
prove t h i s observe that
( ~ 1 (4.17. By d e f i n i t i o n ,
.
(9R) Q is an have further o*-l(_FP) = iff gp is open i n _F
is open i n Spec R i f f
Since P = l?y (FplK) f o r each subset P of
g
i s of t h e form
gEg(K)
I-F1
gp
, it
, it
.
remains t o show that
f o r some subset P of
is a subset of
gg(K)which is
saturated with respect to the equivalence r e l a t i o n defining
we set P = lip (lJIK)
and Q t h e
spec R f o r all (R,P) c*% Also, p + l ( ~ ~ ) i s an open subfunctor of
(4.10), w e see that P is open i n
each open subfunctor
-f
&P
i s e a s i l y shown that _V=F
-P
*
l$n ):?I(
; if
ALGEBRAIC G F 0 “ R Y
26 4.13
Propsition:
mivalent:
The following conditions on a ;-functor - -
(i)
F i s local. -
(ii)
For each model A of sets o v s
of sets over
.
Proof:
, the
151
.
G
, the
presheaf
g ( (9R)Uir F)
is
,)
f1
U
I+
,V(C+j“
3.11, l e t
(iii)=> (i) Using the notation of
Then _F (R
2
F
g)
U - E ( ( S p- AIU,
presheaf
1, no 4
is a sheaf
Spec A .
For each :-functor --
(iii)
5
1,
is a sheaf
Ui = (Spec R) fi
so t h a t exactness of sequence
3.11means that one determines a section of the presheaf by specifying the sections over the open sets
Ui
(*)
UHE((%
, provided
.
of
R)Ur
F)
these sections
s a t i s f y the usual matching conditions.
.
(ii)=> (iii) I f
V = i (p)-’(U) ME(G -u r F )
Am*
(~,p)E M
”G
, arid. i f u
is open i n
set
( 4 . 1 ) . W e thus have i s m r p h i s m s
-
ME+ ( l h (Sp RIVr f)
_ma
(Rr
1 2
EA((Sp NVr
F)
r
(Rr P)
P)
is an inverse l i m i t of sheaves
which shows that the presheaf
UHE(G~,_F)
(namely, the d i r e c t images i n
151 of the sheaves defined by
_F
over the
spaces Spec R) ; it i s therefore i t s e l f a sheaf. (i)=> (ii): I f
of (1)
U
, w e must
E(sUr
_F)
U
is open i n Spec A arid
(Ui)icI
is an open covering
show that t h e sequence -+
T T _ M E _ ( 2~ u~ ~~, (~ s) u i n ~E)u j , i i,j’”- -
i s exact. W e can, mreover, restrict our a t t e n t i o n t o s u f f i c i e n t l y fine cov-
erings, so l e t us assume that Ui = (Spec A) f i
g gi=@(fi) ,
mrphism of If
such that B =
1 B$ (f1, ) , that i
m(Sp - Br _F)
fiEA
. Let
$:A+B
is, an element of
is t h e inverse image of gi my be identified with -%@Vir F) Vi = (Spsc B)
ard _F(B . ) 41 It follows then f r m (i)t h a t the sequence A”*
,
.
be a
(SU)(B)
.
Ui urider Spec $
+nE(rir v M E ( _ S V n iE -F)V j r j i** 1,j L m
is exact ( t h i s is clear when I
the f i n i t e subsets of I) w i t h the d i r e c t limits of
is f i n i t e ; if not, pass t o t h e limit over
. Since su
,
sUi
9 B , gi and
and
suingJj as
may be identified
(B,$)
runs through
I, §
4
J-r
, we
the objects of _Msu
see t h a t
IF1 -
a&
Let F be a z-functor. The s t r u c t u r e sheaf of
Proposition:
~
is canonically i s m r p h i c to the sheaf of rings U e&(_Fu) (3.3) .
Proof: The presheaf -
0
(1) may be identified with an inverse
so is itself exact.
limit of exact s&ences, 4.14
27
THE LANGUAGE
is a sheaf i n v i r t u e of t h e f a c t that -U is a l o c a l functor (3.12 and 4.13). I f A€: ,F = % A
$(Spec :[TI)
U ++ O(F )
U is a special open set of the form (Spec A ) f
J A cal ismrphisms S f
8spec A (u) % Af
z -u F
% AM(Z[TI, w l *
fEA
, we
have canoni-
and
Af)
E(Eu,g) = o(Eu)
.
These isanorphisms induce the required canonical i s m r p h i s m when _F = S J A Now suppose that
is arbitrary. L e t
the inverse image of
U
urader
U
and l e t V be
be open i n
i ( p ) : Spec R
for
+
(R,p) € &
By 4 . 1 a d 4.10, we have
I n view of the d e f i n i t i o n of phism:
4.15
corollary:
subspace of Prmf:
1x1 -
I-f _Y
-
d , we obtain f r m this t h e required 1 1 1
is an open subfunctor of
.
5
(1.3).
1x1
(4.1).
ischnor-
is an open
This follows inmediately fran the description of t h e gecanetric reali-
zation given abave (4.9, 4.12 and 4 . 1 4 ) . 4.16
Proof of the c a p a r i s o n theorem:
satisfying condition
(*)
is isomorphic to a &el,
For each gecmetric space X
of 4.4 and f o r each XEX
so t h a t
the underlying sets ( 4 . 7 ) . If
(Ui)
@(X)
:
lSXl
+
x
, the
residue f i e l d
K(X)
induces a b i j e c t i o n of
is an open covering of
X by prime
ALGEBRAIC GECMEI'RY
28
spectra then QWi) :
15x1
open subspace of
lsuil
-+
ui
I,
is an iscamrphism by 4.4 and
lSXl
by 4.15. The topologies of
is invertible and a )
@(XI
5
is an
lSJil
ad X
t h e i r structure sheaves may thus be locally identified v i a
9 1, no
, and
also
Q ( X ) ; thus
is proved.
By 3.12, it rmins to show that the condition i n part b) of the canparison
(yi)iEI
theorem is s u f f i c i e n t . L e t _F be a scheme, of -F such that
, ard (gijci)
I€;
an a f f i n e open covering
, so
yield an open covering of
the prime spectra
. Then
an a f f i n e open covering of &JinLJj that
of 4.4. W e show t h a t Y (F) l_FI is inver- : _F tible by displaying the inverse Y ' of Y(g) ; one can define a mrphism s a t i s f i e s condition
Y'
:
SlF\
-+
(*)
-+
Y l l (SIU-Jjnl 1 = Y!3 I (-SlUijcil for a l l i f j , only set Y;
. Since
ci
Y(yi)
:
-ui
into
g
*
Section 5
Fibred products of schemes
5.1
Let
fibre3 product functor
-5
and
,Y
I f -X -
satisfies
and Z
Y_
%z
zi
j -if?
(Y
Also i f
gin
Y. -Xis X 21-1
covering of
, let
open covering of
9 %(R
gx
R B,S)
z- . Y
z-Y) xi@ ,
%,zY_iB) 2-S
and
so 'that the
need
zi -xi
-xi
X, gz:
-
Xz(R)Y(R)
in- _X X
-zx if
.
5 x z y is local.
I! g Z g L
in X and- Y
xi . Then
Let
and
, and
let
(Xis)
singzixi0 and is therefore o w n i n _X x zy .
be a f f i n e open coverings of
(Xis X
, we
. Recall t h a t t h e
,XI
are schemes, so i s
be t h e inverse images of
coincides with
_ME_
(5 X z x ) IR) = Z ( R ) -zy is open
I t follows e a s i l y from t h e d e f i n i t i o n s t h a t
(gi) be an a f f i n e
xi
zx
be mrphisms of
.
X
is open i n
proposition:
_X X
g:Y+g -
is invertible (4.4)
Y ( ~ ~ ~ 1 with - l the inclusion
. I t follows G d i a t e l y t h a t
f o r each R 3
Proof:
zIuiI
.
f:$+z and
such that
+
1
equal to the composition of
mrphismof -Ui
-XI
: S/gi/
by specifying mrphisms 4';
_F
zi
ziax
and
S l T
then obviously -
zi!iia form an a f f i n e open I
I,
5 1, no
THE LANGUAGE
5
29
More generally, i f
(zj,fkj) is a f i n i t e diagram of schemes, the inverse limit functor can be constructed with the help of fibred products. This in-
verse l i m i t is therefore a schane; i n particular,
arrow of z h
, the
kernel functor Er(_u,y)
if
, which
the ___ set K - e_ r ( u ( R ) ,v_(R)) = (x€X(R) ly(x) = y(x)1
is a double
g,_v:$:y
assigns to each REM,
is a scheme.
With the assumptions of the foregoing proposition, it follows easily fram 2.13 and 4.15 that, for each pair of mrphisms d:T+/XI ard e:WIY_I E g
Ifld=lsle
such that
.
, there is
a unique h : T
(I_X
d=lgx)h and e=l_fy)h In other w r d s ,
1x1 Is’
1x1
%
z
-f
~
~
I_X X zY_I -
,
~
~
of
such that
, ~is , a ~ fibred ~ y ~
x
.
)
i n the-categ%y $ SJ Wre generalprod;ct of the diagram I ly, the restriction to Sch of the functor ( ? ): E + E a cmutes w i t h f i n i t e v””
inverse limits.
W i t h the assumptions of p r o p s i t i o n 5.1, we naw examine the
5.2
spectral space
Ig K zxl
i t s residue f i e l d ard-
i n mre d e t a i l . L e t x be a p i n t of
E(X)
:
9
carries the unique pint w of
-+
spec
K(X)
onto
, and I E ( X ) I w : Ox+ Ju
onto x K (x)
. clearly
I E (x) I
is a mno-
morphismof
9.
Corollary:
With the assumptions of proposition 5.1, 1 s x, y, z
-of - X,_ Y, -
S U C ~t h a t
Z
f(x)=z=q(y)
& ( X ) X E(Y) E (zl
: *K(X)
SP
K ( Y ) -r
K ( Z r
induces a bijection of the set of prime ideals of
set of p i n t s
t€g x
zY -
with Spec
K ( X ) @ K(Y) K (Z)
Mareover,
E(X)
K(X)
(2.13).
E(:)
1
and
K(X)
@K(z) K ( y )
x SpeC
K(Z)
.
Onto the
Spec ~ ( y )may be identified
x ~ ( y )is a fibred product of m n m r p h i s n s , thus i t s e l f
E (Z)
tive. Finally, i f E(t)
x
-
a m m r p h i s n ; the following l
Iyx
x
- Z
which are projected onto xEX and
F i r s t recall that Spec
Proof:
be points
. T%en the mrphism
X
K(X)
IE (x)1
5 the following mrphisn:
K(X)
dX
is the canonical projection of
)_XI ,
t E _X
X
m implies that the induced map is injec-
zy is projected onto x a d y , the c m p s i t i o n s
I_fy€(t) I factor through
factors throiqh
E(X) Y
I E(X) I
~ ( y )and t
and
I ~ ( yI ) . Consequently
belongs to the image of
LgcaM: - I f L:g*x
5.3
t i v e (4.2).
Proof:
.
x,uG
Let
P1 F2
s a t i s f y f(x)=_f(u)=y :
into
K(U)
~ ( y, ) we have
K(X)
E(X)
. Since
@ K(U) K (Y)
fpl=fp2
@
K(Y)
K(u))
r
and by the canonical maps of
K(x)
fe(x) and f e ( u ) factor through
wherrce p =p 1 2
r
t € speck(^)
E(U)
r
Let
+I!
(K(X) @ K ( U ) ) K (y)
be t h e morphim induced by
and
is a mnanorphim of schemes, f is injec:
. Thus i f
r
Corollary: With the assmptions of proposition 5.1, let (x Y)' be the underlying sets of 1x1 - ,J-Y I,121 and J-X zx -Y I - Z Then the map 5.4
e e e x-,Y-,z-
(5% Yls Z
am^
-+
X e x Ye
- -8-
which sends
t€g xz Y- onto (g (t),_f (t)) is -xY -
Proof: This follows fran 5.2 and frcm the fact that -
5.5 CJ:Y~Z_ Proof:
K(X) @ K(Z)
.
surjective.
K(Y)
#
0
.
With the assumptions of proposition 5.1, if is surjective ( 4 . 2 ) , a s 9 : X x Y * X_. CoroLlary:
x
- z- -
This follows insnediately frcm 5 . 4
5.6
W e have j u s t described the points of
the inclusion m r p h i m of
r e s t r i c t i o n map, write
Evidently
E~
Proposition: pints y
E
X
:
Z
Y_
. To describe t h e
.
x
-.
does not depad on U
Et 5
be a scheine,
x€ly) Spec dx onto
such that
isamorphism of
_X
open subset of the-scheme & l e t j be in 5 a d l e t x€g I f q:d(U)+< U is the ~g dx * f o r the camposition
be an a f f i n e
local rings, let
x
a p i n t of
Then the mrphim
X and Px E~
t h e g m e t r i c space
:
3 dX +
(Px
, l!lxIPx)
the set of induces an (1.3) ,
Proof:
Observe t h a t Px
consists of all points
by an a f f i n e open set, so we may assume t h a t
X
x€m. . W e may replace & . The proposition now
t € x f o r which
t belongs to each open set containing x
Such a p i n t
=
SJ
A
follows frcm Alg. c m . 11, 52, no. 5, prop. 11 and f r m the description of
local rings i n Sp A
(2.1).
Proposition:
5.7 set (4.2)
of
onto
and
x
Xx
& spZ_be
mrphisms of schemes,
.
X_,Y and Z such that f (x)=z=q_(y) Let Q be t h e subconsisting of points t whose projections
z points of
Xry
Let f:Z+?
Y
z-
-satisfy
- and
x€CtXI
y
~
-
tx_"'d4_
v m .e n
Muces an isanorphism of
,axxy\Q)(1.3) .
(Q Proof:
- Z--
write pX (resp. P ,P )
(resp. by
1x1, I Z _ j )
Y
Z
.
1s;
x ~ { s ) (resp. y ~ M , z ~ ( s ) ) Since
ad
1x1
I:/':
over
i n the category
Y
.
Let
(5.1)
g g
-X
~ ( y : ) (* ~
K(Y))X~X_ -
w i t h the image-functor of
fibre
a
of
131
(Q ,Jxxy]Q) may
The proposition-%ow
y
follcws
be a point of
~ ( y -+) Y, is a mnanorphism, t h e same holds f o r t h e
Since ~ ( y :)
call the
evidently
r
~ : ~ be " ya mrphism of schemes and l e t
canonical projection (SJ K (y)
s such that
is the fibred product of
be identified w i t h the fibred product P X p XPZ Y f r a n 5.6 and 2.13. 5.8
151
f o r the gecmetric space induced by
on the subspace 'consisting of a l l points
f -over y
+
X_
E (y)
. The set of
. W e may thus
identify , which we write f -1 (y) and
points of
~ - l ( y ) is a subset of
(4.2).
Propsition:
that of
--
The topology of t h e space of points of
1x1 . 'f
X€X_, f(x)=y
then t h e local ring of f-'(y)
3 .&
@x
f-l(y)
-
is the local r i n g of
is induced by
5
a t x is canonically isanorphic to
at
x
,
32
Au;EBRAIc
Proof:
-f
W e m y reduce everything to the case i n which
being induced by a mrphisrn
, 1.e.
Spec(lc(y1 MBA)
@ :B+A
of
8
I,
GEcb.IEI?IY
1, no 6
5 = Sp A , y=9 B ,
5 . Then I (SJ
is just xfractions of A/@(y)A
the prime spectrum of the ring of
K (y)) x X I
.
with respect to the multiplicatively closed subset @(Bhy) The assertion about the toplogy now follows frm U g . c m . 11,
5
4 , cor. to prop. 13.
The second assertion follows f r m the canonical ismrphisms
it can also be deduced f r m the description of the local rings of a fibred product derived i n 5.7.
Section 6
Relativization
6.1
Let
be as-functor and l e t
, the
-s be written
A S
functor B w i & ( ( A , B )
. Each representable S-functor,
functor of the-form If
be the category of
-
$sA
k is a model, and
5
Smodels
.
into g For instance, i f , wfiich is represented by A , w i l l
(4.1). An S-functor is a functor 5 of &l A=(R,p)EM
M -s
i.e. one i s w r p h i c to a
,will
be called an affine S-scheme.
= Ss K
,
M
-2
coincides w i t h the category -M
k-models. An 5-functor is i n t h i s case called a k-functor.
i.e. a k-algebra belonging t o the fixed universe
E , we
If
A
of
is a k-model,
kA for SJ~A , 4 speak of affine k-schemes instead of affine S-schemes. In parti-
cuiar, when A is the algebra k[T] cally i s m r p h i c t o the k-functor underlying set, For each k-functor
tion on 5
. The set of
write &S
of plynanials in T
gk
,
A k
i s canoni-
which assigns t o each k-model
X , a mrphism s:g+k
these functions is written
R
its
is called a f E -
dk(z) and carries
a
k-algebra structure: addition an3 multiplication are defined as i n 3.3; if AEk and
g€uk(X) ,
Af
satisfies
( ( A ? ) ( R ) ) (x) = X ( f ( R ) ) (x) for each
. We c a l l +O the affine k-l& k = z , we have PIPI? and the k-functors
RE&
ard each x€X(R)
In the case
coincide with the
Z-functors considered so far.
MI
6.2
If
3
is a:-functor,
the theory of 5-functors reduces immedi-
ately t o the theory of &-functors. For l e t %/S
be the category
Of
Z-functors
u
I,
5
6
1,
33
THE! L A N G u m
over p:z+s of p? with tar_ _ _ -S : an object of t h i s category is a mrphism ; a m r p h i m of
get
_p:T-+
q:x-+g -
into
Fh
is a camnutative triangle of
of the form
Ccmposition of these triangles is effected i n the obvious way. The c a t q o r y
-
is related t o the category is: ME/S- -+ -5M E which assigns to
@JP
-
where
p*
:
3R
peg (R) (3.2)
.
-t
S is
Ivlvlg
of ?-functors v i a the functor the g f u n c t o r
_p:x+s
a s usual the mrphism canonically associated with is: E/?-+!s$ -
proposition:
The functor
Proof:
W e merely give a functor
verse f o r
i
defined of
S
5
. Let T
is an equivalence of categories. j
be an S-functor
.Els.-+ ME/S
u -
. men
Am
-
-p T ( ~ :)
Z-functor of
1-
-S
=
T
If
3k ,
sets I&(A,R)
for -+
onto
is an s-functor, we shall c a l l
Xz , where the
maps T ( R , ~ )
( Z ~ ( ~ S(R) )
(which is contained i n the d i s j o i n t sum-*-(R,;)) 6.3
which is a quasi-in-
let pT:zps be the image to be
(,T) (R)
d e r j, ; we have
sum i s taken over a l l &(R)
*
S'
pES(R)
.
the d e r l y i n g
T
Z-
. I
T_ and ~~:~rp the t sstructural projection. For example, i f R€; and A Z g , then (SAA) (R) is the d i s j o i n t sum of the
, where
H
is assigned a l l k-algebra structures cmptible
R
with the given ring structure. This d i s j o i n t sum may be identified with
, where 8 denotes the underlying ring of . cal isanorphism ,(S&A) SpzA .
_M(ZA,R)
**
*
c
T_ by giving
f:x-tY- be a m r p h i m of schgnes,
Sp - K(Y)
. W e thus have a canoni-
%
W e frequently define an 5-functor
let
A
y
zy and
a pint-of
-pT
. For instance, -p:
f-'(y)
-+
the canonical projection (5.8). By abuse of language, w e call the
K(y)-schme
such that
ZT = f-'(y)
~(y)-schgnew i l l also be de"not& by
minology of 5.8 is p s s i b l e .
and
f-'(y)
E ~ T the
f i b r e over
y ; this
when no confusion with the ter-
Fu;EBRAIc cEcMEmY
34
I,
5.1, no
6
-
Similarly, i f 5 and. Y are ~-functors, we deduce frcm 6.2 that the follow-
are z ~ and l zEr2):
ing diagram is catmutative (where the canpnents of
.c
I n general, given an S-functor
i s local i f
T all
we shall carry wex irrrplicitly t o
those results and definitions which apply expl i c i t l y t o
say that
y.l
,z
zz
. Thus w e shall
.,,.Tetc.is a
is local, that _T is an s-&mne if
T
,I! is open i n Z-T wreover, we set 1x1 = I z ~ (, and c a l l I T \ theugecmetric rea1i;ation of
schene, that a subfunctor *!
T
. Finally,
i n sections %ere
is open i f
k is constant and
110
confusion is likely,
we shall employ an abuse of notation and write Sp A
.
or
flk(x)
If
k€g , we
write &cS
k-schgnes, i.e. the
called the base
we define
($'I -
(RIP)
=
-f ( 0 )=P
zT' z (s-T')
canonical isomorphisn
w
-
S'
L
It follows that i f
when S ' = % k '
kT'
formed by the
f:s'-+S there is associated restriction and simply denoted by s? , i f T' is an S'-functor, and i f
T_'
(RIG)
where the sum i s taken over a l l a€S' (R)
call
S&A
&functors
although it Ldepedis prhiicily on _f:
-
for
(SJI k ) - s c h s .
a functor PlSl~--+@ (R,P)€&~
o(X)
for the f u l l subcategory of &M
W i t h each m r p h i m of
6.4
of
,
T'
such that f ( o ) = p
. W e thus have a
which makes the following square cmmte:
c -
-
3s
i s a scheme, so is S-T' -
= S p k and
f = 2
the k-functor derived f r m
4
, we
. write
T'
k-
for
-
, and
by the restriction of scalars 4
.
For example, i f lying
A€$
k-delof
A
we have k ( S A , A )
.
f:z'+S
To each mrphism
6.5
S+(kA)
primarily on f :
if
T
Zsg +lfvlfE
we assign a functor
the base extension functor and simply denoted by
called
although it deperds
?
is an 5-functor and R€Z
is the under-
where kA
r
2'
, we
define
(Tsl) ( R , d = T ( R , f ( a ) ) -
.
and f ( u ) € S ( ~ ) W e thus have, by definition,
where
u€S' - (R)
where
u runs through S ' (R)
. If
T € T ( R , ~ ( U ) ) i n the d i s j o i n t sum (T I ) (R)
z -5
onto
L
(,TI M.
(R) Xs(R)S'
i n u (T) denotes the canonical image of
G T ( R , f ( u ) ) , we obtain a bijection of
(R)
by Sending
inu(T)
Onto
( i n f ( u )( T )
-
W e therefore obtain a canonical i s m r p h i s n
z (T -s
1 )
c
' (,IF)%S' ,.. -
t h i s and 5.1 it follows t h a t if T is a scheme,
-
--S"
SO
is
*
z (T -5
*
-
+
.
. Frm
Lf f:?'-+ i s a mrphism of 2-functors, the base extension funcis r i g h t adjoint to the base r e s t r i c t i o n functor s?
Proposition:
tor ? Proof:
0)
r
whose second m p n e n t is the s t r u c t u r a l projection pr
-
r
S' If
.
T'
is an S'-functor and
bijection
x!?'IT) which i s functorial i n
:
%%(ST'
T'
r?)
and -T
mst define a
' .
is a family of
x (z' r T ) assigns
I, 8'1, no 6
ALGEBWC GEOMETRY
36
Thus we say that 2,'
by extension of scalars. If RE*%,
is derived f r m
rk,
we have (R) = T(kR) where kR is the wderlying k-albebra of R particular, if A S R q , and T_ = 4 S A we have
. In
and we infer the existence of a canonical isanorphism
(BkAIkI SJ~~ (A
k')
.
In virtue of this fact we occasionally write 2 8 k' for k
canonical bijection
x ( T f k', T) : k E ( k ( T t k 1 I r T) jp%l,E(T_%
6.6
I
I
f__
adjoint functor. "/'
:
even when
. The mrphism associated with '5 gkk '
T is not of the form %A
-
is then denoted by -pk tion. __
rkl
k(T Bkk')
I ;_T
by the
k l r T f k')
and is called the structural projec-
of 6.5 also possesses a right The base extension functor ? S' called the WeTl restriction or direct image : -P FSg r IEis given by the formula
bijection
arad T I
which is functorial in the maps T(R,~)+
(s&$~) ( ~ , p )
In the case where 5
-
sp k
v
2'
:
with any -g:T,,?'
, c(T,T')
associates
assigning to TE?(R,~) the ccmpsites
= S z k'
and f = Sp @ we shply write instead of For A=(R,p) , ( S J ? ~ ( R , ~ ) ) ~is ~ then identified with s /s k /k Sp (A Bkk') so that we get --k
111
=
v.
(k ?TI) /k-
(A)
s' (Amk k ' )
.
I,
5
1, no 6
THE LANGUAGE
I n this case
c(T,x')
31
can also be defined as associating with
g the family -
.
where A€-JI
be a m r p h i s n of mdels and suppose t h a t the
Proposition:
Let
k-dule
is projective and f i n i t e l y generated. Then
k'
$:k+k'
a)
if T'
b)
i f T' _ - -is a. k'-schane and. i f ,
gkz'
i s an a f f i n e k'-scheme,
is an a f f i n e k-scheme;
f o r each f i n i t e subset P
i s an a f f i n e open subscheme V_'
of - -T '
is a k-scheme.
of
T' -
PCU' - ,-then
such that
,t
h s
UkT'
c _ -
Proof:
Suppose f i r s t that 'J"=SSkA
where A = jSkl (E Bkk')
algebra of a k'-rrcdule of t h e fonn EBkk'
. If
R€I&
, we
is t h e symnetric then have canoni-
cal isamorphisms (kl,kT') TT (R) = & , ( A
where
tk'
,RBkk')
sTI( E B k k ' , R f
is the k-module dual t o k'
k')
W ( E ,RBk') k
(Alg. 11, 4, no. 1, prop. 1 and
no. 2, prop. 2 ) . rt follows that
I n the case i n which T'=%k,A
, where
A
i s an a r b i t r a r y k'-model, l e t
be t h e kernel of the canonical hamomorphisn of
c1
Bkk'
into A
. Then
with t h e amalgamated sum of the diagram
A may be i d e n t i f i e d within &I
where
$(A
I
is the canonical m p and
B ( I gkk') = 0
. Since
is a r i g h t
a d j o i n t functor, it c m t e s with inverse limits, and. SO
V k S p k l A , the f i b r e prcduct of a f f i n e schemes, is i t s e l f an a f f i n e scheme.
Now f o r b) : c l e a r l y
is open in
gz'
, E g l
is local whenever
is open i n
"k7k TI
-
x'
is. Furthemore, i f
: f o r consider t h e m r p h i m
38
I, 5'1, no 6
G€XPElXY
AU;EBRAIC
and the mrphism f ' : Sp '(AQDk') + T' -k k
associated with f
g o(Spk@)
-u'
of
g'o(@kl(@QDkk'))
f a c t o r s through
t h i s latter condition is s a t i s f i e d i f f
I denotes the ideal of A gkk'
. Since
ShI (A Bkk' )
defining the open subscheme
(ABkk')/I i n A
g'-'($)
(A QD k ' ) / I i s a f i n i t e l y generated A - d u l e ,
k
is equivalent t o saying t h a t B@(J) = B
.
This enables us to construct f o r each of
defined above. Clearly
C(SJ~A,Z')-'
GK' i f f
f a c t o r s through
. By 3.7,
where
by t h e b i j e c t i o n
. For
G T ' such that xEU -
where K€M i s a f i e l d and r*
x
, where
xEk%'
J
this
is t h e annihilator of
, an a f f i n e
open subschaw
is t h e equivalence class of an element
(4.5 and. 4.9)
p€M(k,K)
. Since
Spec KBkk'
has only a f i n i t e n m k r of points, there is an a f f i n e open subscheme U' of
x'
Igl I contains t h e image of the mrphism
such that
151: Spec K@k' k
It now s u f f i c e s to set 6.7
k'=k x 1
g
X
kn
. If
...
pri:k'+ki
,
there are canonical mrphisms
p. (R) is the map
-1
T_(R g p r . ): T ( R f k')
for each
. .
&t kl, ,k be n copies of k a d set n we assign ki the k'-algebra s t r u c t u r e derived frm the
ith canonical projection
such t h a t
IT'/
= U'_;
Example:
...
+
R%
k
1
. I_f
T
+
T(R $ki)
a
is a local functor, it follows inmediately f r m
d e f i n i t i o n 3.11 that the morphism
T + TTki
whose
ithc m p n e n t i s
I,
5
Pi
1, no 6
39
THE LANGUAGE
is an isamrphism (apply d e f i n i t i o n 3.11a) to the p a r t i t i o n
(eti,e' 1
of unity i n RiBkk' i l,
such t h a t
.
T
I n t h i s example, we see t h a t B u t we a l s o note that the functor
Ti
for let
n T = kyk -i
i s a scheme whenever
is a scheme.
/:\
i f not. Then t h e
Ti
T , while
form an open covering of
W e now return to the diagram of functors:
M E f
iE.9
I4
C I
considered i n section 4. Given a g m e t x i c space T
, we
define the category
of gecmetric spaces over T : a geanetric space over T is a mrphism
of EAg w i t h t a r g e t T q:YJT
T
eiEk'
6 foreach i if n > l n
6.8
EsT
, where
does not preserve open coverings: k/k such t h a t Ti(A) = T(A) if Aei = A
be the subfunctor of
and Y ( A ) = rd
e' i = lBkei
. A mrphism of aT with d m i n
and t a r g e t
p:X*
is a c m t a t i v e triangle of the form
f
Ccmpsition of mrphisms is defined i n the obvious way. The category %T functors
M is connected to -ST -
l?lTI?T : cS& To each object
p:X+T
%%T
BTwe
of
tural projection _Sp:sX+ST
v i a a pair of mutudly adjoint
(6.3)
assign t h e _ST-functor
. Conversely,
if
s$
z+sTz ,
!ElT: IE/+T
i s the mrphism assign& to the structural projection -pF: gF+ST bijection
@(zlj',T)-l m
of
4.1. W i t h the above notation,
the
w i t h struc-
by the
bijection
40
I,
AiXEBFKtC GJXNTPRY
6
enables u s to associate the c m u t a t i v e triangle (2) with the c m -
$(z&X)
b:ive
'9 1, no
triangle (1)below, where f ' = $ (ZF,X) ( f ) : n*
f
f' -
41~(F,p) : sT(lFIT,p)
Thus w e obtain a bijection
ing to each triangle (1) the m r p h i m -g:_F-ts+
&&((F_
by assiqn-
S ,+)
such t h a t
. I t follows
zg=z'
t h a t the functors ( ? I T and 5, induce quasi-inverse equivalences between the categories of _ST-schanes and. c
directly fran the canparison theorem (4.4) the f u l l subcategory of fies condition
k€g and l ? l T , STand If
such that
consisting of the p:X-tT
E2T
of 4.4.
(*)
, we
T = Spec k
shall also write
aT ; a geanetric space
p: X
+
satis-
X
mk
for / ? I k , Sk a d Spec k over Spec k w i l l be
called ageanetric k-space; by the spectral existence theorem, t o specify such a space is equivalent to specifying X and a structure of sheaves of k-algebras over
0X
s e t €pgk(Spec R, X)
'
Hence, i f of
%,
(skp)(R) m y be identified with the
RE-
fEEJg(Spec R, X)
such that u(f):u(X)+R is a
k-algebra hamcmorphim. Example:
6.9
(2.12)
. The sheaf ux
k q be a f i e l d and l e t X be a Boolean spce
Let
defined in 2.12 carries the structure of a sheaf of
k-algebras i n a natural way, and t h i s defines a mrphism
W e shall say that %=;,(p) RS
, the map
%(R)
7 TO J:(SPeC R,
p: X I k
-+
Spec k
i s the Boolean k-functor associated w i t h
f w gg of
spec
R, x t k )
in
spec
X
. . If
is c l e a r l y
R, XI
bijective (see also 2.11). Thus we get a canonical i s m r p h i s m
If k
X)
.
is an arbitrary model, we define the Boolean k-functor associated with
X by the formula %(R)
fixed universe J
5=
(Xz)k w
, it
, the k-functor -~ 5
rc
X-Pl
. It
X
belongs t o the
is a scheme: i n f a c t , as we have
Xz i s a scheme. B u t Xz is U is open i n X, Uz is an open subfunctor of
is sufficient to show t h a t
obviously local; also, i f
Xz
, for
= To~(SpecR, X)
. I t is thus sufficient t o show that
is a compact ~ o o l e a nspace. In this case
A*
A.
Uz
is-an affine scheme whenever LJ
Ti
is the topological inverse
I, § 1, no 6
41
THE LANGUAGE
limit of its discrete quotients V
. For each
R€M-,
we therefore have iso-
mrphisns UZ(R) = %(Spec
l$n To~(SpecR, V) = 1$n Vz(R)
R, U)
*
A ..
is an affine schane and
By 2.11, Vz=-sV'
-
raps of
H
V - into
,Z
. Hence
d(V,)
is the ring
-
2'
of a l l
is an inverse-limit of affine schemes, and
Uz
is therefore i t s e l f affine (2."1).
Its ring of functions is the d i r e c t l i m i t
of the rings
the ring of locally constant maps of
into
2
.
6.10
,z" , i n other mrds, Example:
E
be a set w i t h the discrete toplcgy a d l e t
be the map which assigns to each xEE
+E:E+%(k)
value
Let
x of
Spec k
into E
U
. For each k-functor
the constant map with X
, we
define a canonical
mP i(E,X):
+
i:%+z the canposition
which assigns to each
(k)
E -%Fk
W e claim that i f
{elk
_X
E z(E,X(k))=X(k)
f(k)rX(k) -
is local,
s_Pkk so that
i({e),x)
i(E,_X)
%
{el, form a covering of
assertion
therefore follows frcm 4.13.
g is
a local functor, we write
i(X(k) - ,X) - sends onto
is a bijection. For i f
is invertible for each
case, the
If
.
Spec k
tible.
, we
have
. In the general
by d i s j o i n t open subfunctors. The
y,:z(k)k+z
Iq((k)EE(_X(k) ,_X(k))
stant (or simply contant)-if there is a set If
2
e€E
for the mrphism that
. W e shall say that E
g
and an i s m r p h i s m
is
k-E-
%&; .
is connected, t h i s i s equivalent to asserting that yX be i n v e r
5 2
Q U A S I - C O MODULES; ~ ~ APPLICATIONS
Section 1
sheaves of modules over a geanetric space
X be a geanetric spce, and let U& be a sheaf of mcdules (or simply over X) For each open subset U ova the sheaf of rings is then by definition a x d u l e over the ring ux(U) ; mreof X , &(U) 1.1
Let
.
4
over, the transition maps &(U) mrphisns d X ( U )
+. %(V)
+&(V)
. Hence, i f
a module over the local ring a l l sheaves of d u l e s over
<
Jx
are canpatible w i t h the ring hanoa t x is xEX , the stalk of
.Ik,
=&@
; we set /&XI
K
(x) . The category of
sx . L"x
w i l l be denoted by
Now let f:X+Y be a mrphisn of geanetric spaces. If dd ( r e s p . 4 ) i S a sheaf of abelian groups over X (resp. over Y) , we write f . b%) f o r the direct image of & (resp. f' tk3 for the inverse image of # ) Thus we have f.
(4)(v) = & ( f - l ( v ) ) (4
where
v
is open i n Y
,
f.
sheaf of rings
f.
x , A(f-'(v))
is a module over ux(f-'(v))
When& E-q
"functorial i n V" f . (&)
.
, by
(fix)
, and f ' M x
.
for
=
carries the structure of a sheaf of modules over the
. To see this, notice t h a t for each open subset
. Fran the hcananorphisn
V of
arid this m u l e structure is fg: Jy+f.
(fix) , we derive for
restriction of scalars, the structure of a sheaf of modules over
QY The sheaf of ncdules thus defined w i l l be written f*&) , and we call it the direct image of the sheaf of modules &
.
1.2
Proposition:
The d i r e c t image functor
f,: -q-+MC&
l e f t adjoint. Proof:
XEX
Let
ard vz/ be sheaves of ridules over
c"f,
and (by
shall
has a
. Clearly
.
f a(4naturally carries the structure of a sheaf of modules over f ' (dy) Considering the geametric space XI = (X,f' (dY) , w e have a canonical bijection.
$W&)
: %(J,f*(&
which assigns t o v:&f. @)
the mrphisn u:f
*
I
0 -+A such
ampsition X
that
ux
is the
.
f o r each XEX
. The proposition thus follows frcm t h e existence of
a canoni-
cal bijection
which sends
h:f'
(&-+A onto the mrphism
h' : such that
dx f : f l y ) f ' ~
h(s) = h'(l69s)
1.3
Definition:
-+
f o r each section s of
Lf
f:X-+Y
f*(4
is a mrphism of geanetric spaces and
& is a sheaf of ncdules over dy r t h e sheaf of mdules
over flX -
is called the inverse image of J ~ r d e rf
W e point out that, i f
XEX
, we
~ h u sthe inverse image f*(&
.
&pf.
f * (&
aniiiswritten
f*dl/j
have
of
J
a s a sheaf of modules over X
to be confused with the inverse image f *
(4
is not
of the underlying sheaf of
groups. Proposition:
1.4
A-module : M-(x) @M (*)
Lf
M
, there
Let
A be a r i n g and X = Spec A
is a sheaf of modules
satisfying c o d i t i o n
(*)
belaw:
- x -and an A-linear map
over
is any sheaf of d u l e s over dx
map, then there is a unique mrphism $:&&
By definition, &(X)
$:W&fX) is any A-linear such that
-$ =
$(X)o@,
:
carries an $(X)-module structure. !This induces an
A-module structure i n v i r t u e of the isanorphim @A: A The prcof of t h e proposition is similar t o that of N
selves here t o constructing M and @M presheaf
. For each
over Spec A by s e t t i n g
. As
in
5
5
-+
dx(X)
of
5
1, 2.1.
1, 2.1. we confine
OUT-
1, 2.1, w e f i r s t define a
44
I,
-
M( D ( a ) )
for each ideal a of
=
5
2 , no 1
~ s ( a ) - 2' ~ M@XS(a)-'] ,
A ; i n t h i s way
A
we assiqn t o each open subset D(a)
of
spec A a module Over the ring A[s(~)-'I ; for example, for a special open set X w i t h fEA , M(X ) may be identified with the module of fracf f tions M of M w i t h respect to the multiplicatively closed subset 2 f3 (l,f,f ,f ,...) ; i n particular M(Spec A) may be identified with M
M"
be the associated sheaf of the presheaf
M
p ( D ( a ) ) : ~ [ S ( a ) - l ]x M[S(a)-']
. The action
-+
. Let
M[S(a)-'J
induces, by passage t o the associated sheaves, a mrphisn 4 p
:OxxZ&
which defines the structure of a sheaf of d u l e s on
M(x) into
be the canonical map os M
W e p i n t out that the functor
M
H
c a n p s i t i o n of the exact functor M
%
%'(XI
.
% . We
define $M
to
is exact, i n as m c h as it i s the
w i t h the "associated sheaf" functor.
I-+
This result could also have been derived f r m the structure of the s t a l k s of rJ
M ; if M
P
p€X
, the
= M [(A-p)-l]
.
stalk
Corollary:
1.5
P
of
a t p is the module of fractions
W i t h the assumptions of prop. 1 . 4 ,
$M i s an iso~~
mrphism.
Prcof: To prove t h i s corollary, we refer back t o 5 1, 2.6. -
It follows f r m
the exactness of the sequence
associated w i t h each f E A and each covering of
Proposition:
1.6 A-module c~M)-
jr
M
Let
y:B->A
(resp. each B k d u l e N)
(spec y),(iij
Xf
by special open sets
he a ring hcmmrphism. For each
, there
is a canonical isamorphism
(resp. (A gBN ) ~ (spec y)*(Ejj) where B~
B-module derived fran M by restriction of scalars.
is the
I,
5
2, no 1
Proof:
QUASI-U"T
X = Spec A
Let
,
Y = Spec B
MODULES
, f=
Spec y
rc/
I ) : (BM)w + f, (MI such that ismrphism: i f sEB , w e have
mrphism
(BM)y
. By
1.4, there is a unique
. This mrphism is an
= @M
+(Y)o@BM
z MY ( s )
(Ys) 7 (BM)s
45
G(f-l(YS))
.
In order to obtain the ismrphism i : ( A @ N)B
1f * & j
we observe that the functor
N
, d
H
f*(N) is the c a n p s i t i o n of two functors
which are both l e f t adjoints. It is therefore a l e f t adjoint for the mpos i t i o n of the corresponding r i g h t adjoints, that is, for the functor
3'-'
B T i ~ ). The
i s the canonical ismrphism of -1
q =Y
.
(PI
1.7
M(')
N
(A B ~ N ) -
+-+
.
,
is given an explicit description, we see that, f o r each p E Spec A
When i i
same argument may be applied to the functor
Recall that, i f
M
onto A @ N where pBqq
(A@BN)p
is an object of a given category, we write
f o r the direct sum of a family of copies of
Definition: quasi-coherent
1x1,
_Let _ X
if,
%-I
.
0 (I) -+ O(J) V
-+
V
formed by the quasi-
_ _ _ _If_ M Z A category of small A-modules.
,
-7-1
is a model, @A
M
x
exact
pa"4r
A N
of
0
, where
.
is called
&lv -+
f o r the f u l l subcategory of
coherent sheaves of modules. Similarly, i f
eA of the form
over vx
2 , and an
belonging t o the chosen universe
&9v
r r d u l e s over S p A
&
x€& , there is an o p n neighburhod-V
'of the form
the restriction of W e write
be a schene. A sheaf of modules
for each
I, J
sequence of M & c
indexed by the set I
M
J6lV
denotes
demtes the
is a quasi-coherent sheaf of
. To see this, notice that there is an exact sequence fram
A
(1) (J)*o -+A
,with
I J E ~
. Since the functor
M n 6i'
c m t e s with direct limits, t h i s exact sequence is transformed into the exact sequence
where X = % A . Conversely, i f there i s an exact sequence
vf‘ (L,L
then -_
is of the form 1
-t
M” : ~
4 for ,L,LIc_~~o~, , the canonical map
@o+/zI2) is the canpsition of the bijection
mod (L,L’) into mod (L.,c(5)) -A
-
-A by setting W L ard
&=Kt
(1.5) with the bijection $
.
A scheme
X
,+
of
F+
$,,og
$
obtained
in prop. 1.4. This proves that the functor -
M L’ is fully faithful. In particular m is of the form u:A(’) + A(J) is A-linear. Thus we have K G (Coker p)-
1.8
g
G , &ere
is said to be pasicanpact if its space of points
is quasiccmpact. g is said to be quasi-separated if its space of points is quasi-separated, that is, if the intersection of two quasicanpact open subsets is quasic2cmpct. For this cordition to be satisfied, it is sufficient that there be an open owering of 3 by affine open subschenes
Xi
such that
. For gim-1 may then be covered by ; if sEJ(zi) and t€flgj) , a finite number of affine open subschenes zijl (xi)sn(gj)t is the union of the affine open sets (zijl)sta d is therefore quasiccmpact. Hence the (zi) form an open base whose pairwise intersections
X.m -1 -j
is quasicanpact for each
(i,j)
are quasiccenpact, and the assertion follows.
Propsition: Consider a Cartesian square of
Y-X
sews
f -
where AIEEZ, A is flat over B
,
; is quasicanpact and quasiseparated.
Then for each quasicoherent sheaf of modules $ over - - Y
is bijective. (Clearly we write here f *
(5) instead of If I * (4(
, the canonical map
1x1) , see 5 1, 4.2) .
(xi) (resp.(Yijl)) be a finite affine open covering of Y_ -1 (Yijl) . Then we have the dia(resp.Y.nY.) . Let gi=f-l(Yi) and zijl=f -7
Proof: Let A1
gram
I, § 2,
47
1
i n which the arrows are the obvious ones. (For example, t h e ccsnponents of
index
zijl A
c
( i , j , l ) of
Xi and Sijl
a and b are induced respectively by the inclusions
. The f i r s t l i n e is exact since
c X.) -7
*: (4
i s f l a t over B ; since
is a sheaf, t h e second l i n e i s also exact.
Now, by 1.7, we may assume that each be of t h e forin
gi
@I
N
~i
whence, since t h e set of indices
f* -
is s u f f i c i e n t l y small f o r
!Zi
. By 1.5 and 1.6,
with N i W 13(yi)
A
L/1/ is a sheaf and
(4(zi)
Ci)
to
;
is f i n i t e ,
thus v i s invertible; so a l s o is w ; arid so is u
.
Rgnarks: With the above notation, suppose that the B d u l e A
t i v e and t h a t
xlxi
it follows that
is projec-
is quasicanpact, but not necessarily quasiseparated. Then
the canonical map
is i n j e c t i v e and. v A
i s invertible. Hence u
is invertible. Similarly, i f
is a f i n i t e l y generated projective B-module, our proposition
without r e s t r i c t i o n on of the set of irdices 1.9
x(Q) x(Ff f+
Proof:
true
v and w are then i n v e r t i b l e irrespective
and I i j l l
.
Corollary: L e t 5 be a scheme and l e t & be a sheaf of Then & is quasi-coherent i f f , f o r any a f f i n e open sub@ 5 and any f€B(g) , z(_V) is s m a l l arid t h e canonical map
mdules over 5 scheme
!il
. For
rfmains
me
.
is bijective. last c o d i t i o n on Jf simply tells us that the r e s t r i c t i o n o f $
to _V is identified with the sheaf of mcdules y(LJwderived from the d(g)-module d(LJ This implies, by 1.7, that J is quasi-coherent.
.
Corollary: (Structurethwrgn for quasi-coherent sheaves)
1.10
, the functor
M ++ % of 1.4 is an equivalence of the cate9ory of small B-modules onto the category of quasi-coherent sheaves of modules For any model B
over - -Sp B
.
Proof: By 1.7 we have already shown that is quasi-coherent if M m , and that the functor M is fully faithful. Conversely, by corollary 1.9 applied to U = & B , we see that any quasi-coherent sheaf of rrdules & is I--,
of the form
ii , with
M = d (s p B)
If 5 . If
Corollary: Let
1.11
sheaves of madules over a d Coker f
1
.
.
be a scheme and f : d + X a mrphism of and d ' are quasi-coherent, so are Ker f
Proof: Let V_ be affine and open in X ?., an arrow g for some arrow g of IK&(~)
1 L .
Ker f jg
Ker
<
7
2
flu is "isomorphic" to
. We thus have iscanorphisms
(Ker s ) ,~ Coker f ( U
Corollary: Let
1.12
. By 1.10, Coker
F z (Coker gi"
.
be a scheme a d O + ~ ~ & + c & " + O
.
an exact
sequence of quasi-coherent sheaves of modules over X Then, if g is affine and open in x , the sequence o+K'(LJ+/~~'~~)+~''oJ+o is exact.
.
Proof: We need only consider the case in which 5 = _V = S 3 A , A€g By 1.10 a d 1.4, the functor M !-+ with danain md and target the category of -A
quasi-coherent sheaves of modules over
has for its quasi-inverse the functor N++cN(y) This last functor is thus an equivalence; in particular, it
.
is exact. Section 2
Direct and inverse m g e s of quasi-coherent sheaves
2.1
Propsition: g t is quasi-coherent,
f:ZfY_ f * cu)
be a mrphisn of schemes. If
.
5
I,
2, no 2
Proof:
_v
(I) +
J(’)
v
+
g*
(XlV). Hence,
z(J) v d (J)
-
+xlV -+
induces an exact sequence
0
+
f*(& ITJ -
I, J€g , each exact sequence
if
-r
, if
herent over
-
-
, for
c*g), where
an inverse image
=II , the
have -f,
-
is i n general not quasi-co-
i s quasi-coherent over _X : take f o r
of sane copies of
sum :(I)
.
(observe that g*(Qv(’)) % flu(’))
0
With the notation of 2 . 1 , f * Gk6)
2.2
A
1x1 , l e t _V be its inverse image, and let iraduced by f . There is a canonical isatlorphism
be the mrphism
f *(4I _V
49
be open in
I!
Let
-g:g-+y
QUASI-COHERENTKIDU”
X
the “codiagonal” mrphism and for
f
is quasi-coherent over
V/
the d i r e c t
product being taken in
%I
such a product i s i n general not quasi-coherent i f
. We clearly
. It i s easily seen t h a t
I
is i n f i n i t e . Neverthe-
less, we shall see that the d i r e c t image of a quasi-coherent sheaf is quasicoherent under sane general conditions t o be specified below. Definition:
a)
f:s+x be a mrphisn of
Let
schfmes.
is said t o be qwsicchnpact i f the underlying continuous map of
f
f.
i s guasiccarrpact, that is i f , for each quasimpact open subset V o_f
1x1 ,
b)
If
Z-~(V)
is y u a s i m p c t .
f is said to be quasi-separated i f the diagonal morphia
_U
f-l(Y.)
has a c a r e r i ’ consisting ~ of open affine subschemes
is quasicmyact for each i
-1 (X. .) be a f i n i t e covering of
-17 affine open subscheme
z
of
, then
f-l(Xi)
xi ,
g-’(?)
f
xi
such that
is quasicchnpact. For l e t
by affine open subschemes; f o r each
is then the union of the affine -1 The open V such t h a t f (V)
zVij , hence quasiccanpact.
open subschms
is quasiccanpact thus form an open base; and the assertion follows. Applied to the diagonal morphism
this assertion says t h a t f. is 6_x/,i can be covered by affine open atd each
quasiseparated if by affine open
sij
(For observe that
such that
g.
.ail= $-& ( ~ i j y x i l ) ) . f
be covered by affine open
.
xi
:-‘(xi)
is quasiccanpact for each
(i,j,l)
13
Finally we can say that separated (1.8)
gij n X l
is quasiccanpact and quasi-separated i f f
xi
-1 such that f (Y. ) -1
_Y
can
is q u a s i c m p c t and quasi-
.
Au;EBRAIc c25omrF.Y
50
Proposition: Consider the diaqram of schemes
2.3
5
I,
f
2 , no 2
g
X+
+
y'
. If
-f is quasicanpact (resp. quasi-separated) then the canonical projection f Y l : 3 Y' * _Y' is quasiccanpact (resp. quasi-separated). 2(resp. 2)
Proof: Suppse for instance that f is quasiseparated; then _Y'
xi
can be covered by affine open (X.. I -17
. NOW let
-
(resp. Y.) such that g(y;)Cyi -1-1 be an affine open covering of _f ; set
(xi)
then we have
since
x. .n xil = 6-1~(Xijy. / x~xil1 -11 1 is quasicompact, as well as the mrphism
Hence
_f
1'
.
is quasiseparated (2.2)
Similarly, if
2.4
:;+xi
is quasicanpact, so is -fyl
-
Proposition:
If _g:Y+Z - -
(see 2 . 2 )
, so
.
is X! .@ij -13
.
is a quasicanpact ardi quasiseparated
mrphism of schemes, the direct image y * M of a quasicoherent sheaf of d u l e s L/ is quasicoherent. Moreaver, the functor induced by on quasicoherent sheaves preserves filtered direct lmts. CJ*
izqie,
Proof: We first show that, for each affine open subschane V of Z and each s€d(y) , the canonical map
(&(.I/,(WS
+
g*N) cys, , we
is bijective. By replacing _Y by q--('!) and set A = Bs We then have
.
(?* (4(V)
A
may assume that Z
=
=
Sp B -
@N(y) €3
and the first assertion follows fran 1.8 and 1.9. As for the secord assertion, it is e m q h to consider the case where Z is affine. It is then enough to show, that for quasicoherent sheaves on a quasi-
compact and quasiseparated scheme Y
, the functor
+
4x1
preserves
51 filtered direct limits. But, using the notation of 1.8, the functors
q+&xi)
OUT
and
x+“6Y.-111. 1
clearly preserve filtered direct limits. Thus assertion follows once aqain frcan the canonical exact sequence
We now suppose & to be quasiccanpact and quasiseparated. Then
2.5
the following coditions on a quasicoherent sheaf of modules & over 5 are equivalent: (i)
X can be covered by affine
such that &(g)
open subschemes
is a finitely generated IIx(~)-module; -
for any affine open subscheme _V of _X
(ii)
, A(!)is a finitely
dx(IJ-module; -
generated
if a quasicoherent sheaf 2 is the union of scme directed set (iii) of quasicoherent subsheaves Ni, then any mrphism ‘f : factors through SQne
A+
k.1 .
Proof:
(ii)+(i)
is clear, and
(i)=+ (iii) follows easily fran 1.10
(show that the restrictions f /I! factor through sane Mi1U) . In order to shm that (iii) implies (ii) , let g:y-tx be the inclusion mrphism and
let (Ni) be a directed set of s M u l e s covering sane We then have, by 1.10 and 2.4
Let
2.6 sheaf
dll
over
X
fl (U)-module N 5-
.
be quasiccmpact ard quasiseparated. A quasicoherent
satisfying the equivalent corditions (i) - (iii) of 2.5
will be called finitely generated. If g is any quasimpact open subscheme of _X am3 sheaf of d u l e s over _V
, it follows directly frm
any quasicoherent
2.4 that there is a quasi-
coherent sheaf
over
5
with
AlU
(take f o r instance
.*& , where
u:U+X - - - i s the inclusion rmrphisn). But there is a better r e s u l t , which says that
A may be chosen t o be f i n i t e l y generated
if
?'
is. This c l e a r l y fol-
lows from the
- - X be a quasimpact a d quasiseparated scheme and l e t Let
Proposition:
X over -
be a quasiccarrpact open subscheme. For any quasicoherent sheaf
U
Lkl of - 3; 1 , t h e r e is a & of '? such that c&=ul/Q .
axl any f i n i t e l y generated quasicoherent subsheaf
-~ f i n i t e l y generated quasicoherent subsheaf Proof: U
L
Let Ul,...,yn
, cover
be a f f i n e open subschgnes of _X
. Our proposition is t r i v i a l f o r
_X
.
open subscheme m e r e d by
n=O
. For
which, together with l e t -X '
n)o
be t h e
By i d u c t i o n on n we may suppose 'n-1 that there is a f i n i t e l y generated quasicoherent subsheaf Y' of ? ' ( X-I such
. It is c l e a r l y enough to e x t d
that A= & I ;1
generated quasicoherent subsheaf of
7 Isn
by
z'nEn , we
sane f i n i t e l y
which, matched together with
over X1nVn , w i l l supply us with the required
-U
d'l ~ ' n ~ tno
&. Replacing
reduce the proof to the case i n which
5
_X by
En
is affine.
- denote the inclusion mrphism. The invexse image I n t h i s case, l e t _u:v+X
Q of ,u* Wl under t h e camnical mrphism p-+ >* 61g) i s quasicoherent by 2.4 and s a t i s f i e s 9 [_V=&. By 1.10 Q is the union of the directed set of its f i n i t e l y generated quasicoherent subsheaves &' Thus is the union o f the r e s t r i c t i o n s k ' (-U and is equal t o sane by 2.5 (iii)
.
over
-
For any scheme
2.7
5 i s closed i n
Mod
Izl
5
t h e category
.
ml
of quasicoherent sheaves
under kernels and s m a l l direct limits. Hence
it is an abelian category with exact f i l t e r e d d i r e c t l i m i t s . I f ccknpact
2
is quasi-
and quasiseparated, it follows f r m 2.6 that the f i n i t e l y generated
objects of
s,
generator i n
-
generate this category. This implies t h e existence of a
, since
quasicoherent sheaves over
the i s m r p h i s m classes of f i n i t e l y generated
5
may c l e a r l y be indexed by sane small set.
In other w r d s , i f X is quasicanpact and quasiseparated, we can apply to the general r e s u l t s ahown for Grothendieck's AB5-categories with
%I
generators. For instance, i f
5
is a category with small mrphism sets
~ , ( x , y ), then any functor F : m , +
ICI
J1 preserving
d i r e c t limits has a r i g h t
1, § 2, mJ3
53
QUASI-COHERENT MODULES
-.
,K is t h e f u l l subcategory of Mod Izl such that y(U) is small f o r any open vCl_X[
adjoint. This holds i n particular i f
s”
formed by the sheaves
Taking f o r F the inclusion functor, we i n f e r that any
a quasicoherent sheaf
@“
x€g
may be assigned
following universal property: f o r each quasicoherent V% and each there is a unique ( J : & - + C & ~ ~
such t h a t q4(J=$
.
$:A+ &
W e may i n f a c t give a d i r e c t construction f o r Jqc : Assume f i r s t that
k=y,&)
of the form
subschene. Then we set (y*
(p) ,y* (q,) C
)
, where _v:v+z denotes pc= a ( y ) , and write
q,(y) $d (v) =
m r p h i s n such t h a t
zijl)
mrphism of
(1.4)
5
(resp. of
I
)
the inclusion of an a f f i n e open qZ:Pc+Y f o r the unique
. It is then easy to show that
consider a f i n i t e open covering (,Xi) I f _v i (resp. yiJ1) i s the inclusion
.
xi
i
= v*(dKi) (resp.
.
-1
viJ1 fl Xijl)
From
of
I d . (v)
is
L/
i s a solution of our universal problem.
In t h e general case, when &€$ (resp.
the
together with a m r p h i s n q4:p+&enjoying
-
our previous ranarks, we g e t solutions
of our universal problem relative to thus remains d i t i o n udq,=
uqc : qd)
Section 3
T < 1
and
v
W e m y therefore
Is)*Q by the conditions qv=vuyp and w d p . I t + y by the conto set dqC = Ker(v,w) and t o define quy:flC pu , where u :JqC + ? is the inclusion morphisn. The p a i r
define mrphisms v,w:
is the required solution of our universal problem.
Faithfully f l a t q u a s i m p a c t mrphisms
We give here another extremely useful property of quasicanpact morphisms. W e
f i r s t make a new definition.
Definition:
3.1
A mrphism of s c h m s
a point XEX- i f the map fx: Jg(x)+Ox
e
s
dX
_f:X+Y _ _ is said to be f l a t a -t a f l a t n\cdule over Jf
It is said to be f l a t i f it is f l a t a t each xcX_ ,
faithfully f l a t
.
ift
is f l a t and surjective.
94
In the case of affine schemes, flat) iff
-f
is f l a t (resp. f a i t h f u l l y
S2B
makes A a f l a t (resp. faithfully f l a t ) d u l e over B
$:B-+A
(Alg. c m . 11,
: S&A
5
3, prop. 15
and cor.). For a mrphism of schemes
f:&+x
to
be f l a t , it is thexefore necessary and sufficient t h a t the follming condition IXsatisfied: if L J , ~ are affine md open i n
is a f l a t module over
d(v)
Propsition:
3.2
.
f'
be a point of
is f l a t a t
x'
is f l a t (resp. faithfully f l a t ) .
'1 X'
f
-x
X_'
. -If -f -is f l a t
I
, g'
_ f ( v ) cI! , then
4,~)
Consider the c m t a t i v e square of schemes below,
which assume to be Cartesian. Let x' f. is f l a t a t x
_X,Y and
and
.If
-(XI)
(resp. faithfully f l a t ) ,
P
Y'
Proof:
5
BY
dxl
1, 5 . 7 ,
y'=f'(x')
and y=f(x)
f l a t wex
D
f'
by
5
Y' 1, 5 . 5 .
if
Ox
i s a ring of fractions of
. Hence
dxl
i s f l a t over
is f l a t over
J
Y
.
y @ J
, where 4,Oxwhich is itself
2 is
surjective, so is
d
9Y'
~ l s o if
Jy x
W e occasionally make use of a converse form of the foregoing propo-
Rgnark:
-f ' is f l a t a t x' and if q- i s f l a t a t y ' = f ' ( x ' ) then f is f l a t a t x=p(x') For is then f l a t over 8 , where y====(y')=f(x); Y by the p r o p s i t i o n dX, is f l a t over dx ; so Jx is f l a t over J (Mg. sition, namely, i f
m. I,
5
4'
.
3, no 4 , ran. 2)
.
Let k be a milel and
3.3
(resp. faithfully f l a t ) over k
pX -px: -
Y
-+
9k
Propsition:
X
a k-scheme. W e say that 3 is f l a t
i f the structural projection
i s f l a t (resp. faithfully f l a t ) . Let
rated k-schemes. I f -
k be a model, Z
- -ard- Y 2
two quasicanpact and quasisew-
is f l a t over k ard i f
d(Y)
is a f l a t k d u l e ,
I,
5
2,
55
QUASI-COHEBENT IWLXJLES
no 3
then the canonical map d(z)@k~(Y) -+ t,!)(pY) is bijective.
proof: Of course Z x Y- denotes the prduct in the category -ME . Let (gi) and (zijl) be finite affine open m e r i n g s of 5 and z,e , In the dia1 j gram
the two lines are exact. We see that v setting B=k ard A
=
(zi)
(resp. w) is an imrphism by
(zijl)) in propsition 1.8. It
(resp. A =
follows that u is invertible. 3.4
ffqc descent theorem: If f:X+Y _ _ _ is a faithfully flat quasi-
ocknpact mrphism of schemes and if Erl , p~~ are the canonical projections of the fibre product g p z onto its factors, then -
IPL-21 exact sequence of qeanetric spaces. Proof: By definition 3.1
If1
is surjective. By
fy f ( x ) = f ( x ' ) , there is a pint z€gx$
8 1, 5.4, if
such that x =
-
satis-
X,X'€~
p.xllzl ,
x' = pr2(z); since the converse is obvious, we may identify the underlying set of Y_ with the quotient of the underlying set of X obtained by identi-
.
fying pgl(z) with pg2(z) for z € g v It remains to show that the quotient toplogy WI. that the sq-ence
where g = fprl
=
fpg2
, is exact. We prove the
For each affine open subschgne show that the sequence
d(V)
-f
J ( y ) If d p
v-U)
carries
second assertion first:
of Y , if we set Q
=
f
-1
(v) , we must
is exact. If C=d(V)=M
,
is affine, this follows fran
B=d(U)
1, larma 2.7 by setting
. I n the general case, by 2.2,
(gi)
open covering
5
of
there is a f i n i t e affine
i s the d i s j o i n t sum of the
U-. ; i f
affine and the canonical morphisn of
induces a diagram
into
_V'
yi , _u' is
v
U
.
u'=iu , j w ' i , j w ' i The bottan sequence is exact since is affine; the top one i s likewise since i is injective.
such that
Finally,
1x1
carries the quotient toplcgy. For i f
of
5 , we
if
F is saturated, we therefore have
f(F)
W e now show t h a t
f
substituting
(9
A
i s closed.
-1
-
1 (f ( F ) ) , where F i s clos+ in (gl
(f ( F ) ) = f
for
and
and 2 = Sp 4
. Now
u'
Let
5 we may
for
sequence O+bB+A/a A@ b B
inl : A
+
of the image of
-
= V(b)
A
(9
= SS B
such t h a t
. By ,
F=V(a)
1, 2 . 4 ) . Frm the exact
we derive, by means of the f l a t extension
4:B+A
,
A$(b) = Ker inl
A @(A/a) s a t i s f i e s B
Spec(inl)
in,(x)
= x
is thus
V ( w ( b ) ) = f-'(V(b))
= f
-1
(f(F1) ~
It therefore remains t o show that the image of
, ant
assume t h a t
a be an ideal of
1, 2.4) a d b = @-'(a) ; we have f ( F )
where
is a closed subset
show below that
It follows that
3 = Sp
F
v'
01
. By 9 1, 2.4,
the closure
. Spec(inl)
coincides with
this follows fran 5 1, 5 . 4 applied t o the fibred product of f the diagram X--txt-Sp(A/a) : indeed, i f we set _F = _Sp(A/a) , the irrage
f - l ( f (F)
5
I,
2, no 3
57
QUASI-COHERENT M9DUIES
Corollary: Consider the c m t a t i v e diagram of schanes below. If
3.5
q- is faithfully f l a t and quasicanpact, and i f
is invertible, so i s
Proof:
If
f~ Y
- .z-
f
.
is invertible, so is
Thus, i n the diagram below, the first OrJO v e r t i c a l arrms are invertible, so therefore is the third provided the two horizontal sequences are exact. But
this follows frcan theoren 3.4 modulo the w e l l known identifications
3.6
L
Corollary:
Consider the c m u t a t i v e square of schemes below. I f
is a monmrphism and f. i s quasicanpact and faithfully f l a t , then there
is a unique mrphism j:_u-._Z
such that g=jf and
ii=y
.
58
I,
Prcof:
W i t h the notation of 3 . 4 , we have
fpr
1
= fpr -2
5
2, no 4
&pzl =
whence
yzprl = yfpg2 = &pgz ; hence gp.rl = gpg2 since A is a mncknorphisn. By 3.4 ard 5 1, 4.4, there is a unique mrphisn -j such that _u = j f ; hence
-i j-f-
= &u- = vf -_
, so that ;j
--
.
=
The functorial p i n t of view
Section 4
W e rum develop a purely "functorial" theory of quasicoherent modules on schemes
and show haw t h i s new notion overlaps the preceding "geanetrical" definition. 4.1
be a2-functor and let
Let
For any REM, a d any pE_S(R) Suppose each f i b r e M(R,p)
that, for any +%(R,S) which shall mean that
.
, call
M(R,p)
4
be an s-functor
the f i b r e of g
over
p
.
t o be given an R d u l e structure i n such a way
, the
induced map JI: M(R,p)
bJ(S,$(p))
+
is additive and s a t i s f i e s $(Am)
JI
(5 1, 6 . 1 ) .
=
is $-linear,
@(X)JI(m) for all
we then say that , w e t h e r with these module structures, is an S-module. If _M and 1 are two _ S d u l e s , a mrphism
m%(R,p)
and XER
f%s,-,(M,_N)
is called linear or an g-mdule mrphism i f
i s R-linear for a l l
f(R;p): _M(R,p)+ _N(R,p)
. These definitions ob-
(R,p)@I
-5
viously give rise t o a new category, which w e denote by
gs: the
abelian
category of S-mdules.
s-mdule g is called quasicoherent i f its fibres I j ( R , p )
are snall and i f , for any @@I(R,S), the induced map _M(R,p)MRS-+ @(S,Q(p)) i s invertible.
An
Ivbreover, i f
M(R,p)
n) for each
(R,p)
is a projective R-module of f i n i t e rank (resp. of rank
, we
say that
5
is a vector bundle (resp. a vector bursdle
. The f u l l subcategory of --s Mod S-rdules w i l l be denoted by 'Ws . of rank n) over
4.2
S
F i r s t examples:
structure for any
(~,p)
. This
a)
fonned by the quasicoherent
w i t h the usual R-ule
Take _M(R,p) = Rn
w i l l be written :Q
-
and called the
I,
5
2, no 4
QUASI-CDHEFENTMODULES
59
.
t r i v i a l vector bundle of rank n For the urderlyirg p-functor ( 5 1, 6.3) n we c l e a r l y g e t zM = sxg where Q i s the a f f i n e line (5 1, 3.3). Thus
i s a scheme i f
%-
is one.
_S
b) Let A be a mdel and l e t M be a small A-module. S e t g = *A Ma
, and
the functor M
Ws ; the functor S1-functor Ms
W e thus g e t a quasicoherent S-mdule
is clearly an m i v a l e n c e of a is quasi-inverse to it. H M_(A,Id ) M
H
be a mrphism of 2-functors. For any S-mcdule _M
, derived
fram
g
by base extension
vector bundle, i f
?
*
Mod
s" -2
-f
( 5 1,
, the
6.5) is c l e a r l y
-
g is so.
The base extension functor
@sl
defined i n c) has a r i g h t
adjoint. I n order to see this, we-only note her; that, f o r any sLmodule
and any
(R,p)€gsI (s&M'
-
onto
ySl is quasicoherent o r a
assigned an _ST-mule structure. This s'-module d)
GA(1.7)
A
7
c) Let f:S'-S
.
a l l p€E(A,R)
= M @ A ~for
Ma(R,p)
and
M'
the set ) (R, P)
=
zs IH( (Sps- (R, P ) 1-
g' 1
5 1, 6.6, carries a natural R-module structure. I n f a c t , any (R,p) is a function assigning sane x ( $ , p ' ) E g'(R',p') t o a
considered i n
xE
(sl/sM') I I
pair
(@,PI)
E g(R,R')xS' ( R ' )
such that S($) (0) = f ( R ' ) ( 0 ' )
addition and scalar multiplication by the formulas x1($,p1)
?sl - :
(1x1 ( $ , P I )
+ x2($,p'l': and
r e s t r i c t i o n functor
S'/S
:
=
(x,+x,)
$(M.x($,P')
gslws +
.
W e define =
($,PI)
. That the so defined Weil
is r i g h t adjoint to
zs Eslmay be prove3 as i n 5 1, 6.6. -+
W e shall see l a t e r that the Weil r e s t r i c t i o n of a quasicoherent m d u l e over
.
S' is i n general not quasicoherent over S_ Nevertheless, t h i s is obviously true i f both S and S' are a f f i n e schemes. 4.3
proposition:
h coherent _ _ S-module. _ - ThI' ~
fl
g t
5
b e a :-functor
is local
(5
1, 6.3)
i s a vector burdle, then - - M i s a scheme i f f Proof:
iff
_S
1 2 .
be a quasi-
e. Pbrmver, i f
3
Clearly the s t r u c t u r a l map _ P : ~ E $ ( 5 1, 6.3) has a section so t h a t
-S is a retract of
,p1
and thus is &a1 or a scheme provided the same is
zg . Conversely, f i r s t suppose of d e f i n i k n 9 I, 3.11 with 8 = zM . L e t true of
and l e t
I
n*
to be local; we use the notation ti€X(Rfi)
be elements such that
and set rli=p(Ci) E S ( R
v((Si)) = w((Si))
assigned the same image q. in lj nES(R) with imge n in S(Rfi)
S (Rf.
and M ( R f1 . f 1,nij)
are identified w i t h
5 1, 2.7 g ( R , n ) , there is a
)
. Since l j
i
. Since
and q j are for each i , there is a unique ,)
fl
qi
is quasicoherent, M(Rfirqi)
_M
and M ( R , ~ ) &R~f i c aRRf j ’
_M(R,n)cgRRfi
,
Hence by lemna
applied t o the case C=R
M =
unique 5 € bJ(R,?l) c 1((R) having
c X(R 1
B(Rfi,rli)
f
for each i
i
Suppose finally that _S
B =TTR i fi
.
and
6 i as image i n
is a scheme and t h a t _M is a vector buradle. Since
M i s kncwn to be local, it is enough t o show that, for any affine open subschane _V of 3 , the U-module derived fram g by the base extension +
S_
( 4 . 2 ~ ) ) is a scheme. The proof i s thus reduced to the case Fn which
5
and. 5 of the form
is of the form S E A
Ma
(4.2b))
. If
M
is free of
f i n i t e rank, we are through by 4.2a). In the general case M is projective, i.e. a retract of sane A”
. Thus
& I = M
zc a
is a scheme a s a retract of
(4.2a))
.
Let
and _N be vector bundles over
e_On=Z(An l a
m
r
r
M
4.4
bundle of
N
surrmand of
_M
, if
(R,p)€M
raturally i n the study of grassmanians get a subbundle
T
of rank n
be setting T ( R , ~ ) = %,r
( 5 1, 3.4,
for all R$
Let 2 be any 2-functor. the vector bundle T, over - -
Propsition:
by the base extension f
-
, we
3.9 a d 3.13). W e c l e a r l y
of the t r i v i a l bundle of rank n+r over
pcx””
the so-called tautoloqical bundle over G
f:$%n,r
71,r
p
.
c g n , r ( ~ ). %is
Tautology!
S
derived fran the tautological bundle
get a bijection frcm
A morphism
direct sumnand of rank n of
is
By assjgning to each morphism
E(S ,Gnfr)
of subbundles of rank n of the t r i v i a l vector bundle of rank Proof:
a &-
the f i b r e M(R,p) of M is a d i r e c t -s as an R-module. Vector bundles and. s u b b d l e s arise
for any
N(R,p)
; we c a l l _M
onto the s e t n+r
over
g
.
2 assigns by definition t o any a€S(R) a Rn+r
!
Notice that the preceding proposition is often given another equivalent formulation: i f
bJ
is an 5-mzx3.de,
j u s t c a l l a section of
- Pbg -gg-
the structural projection p each
ulf
[R,p)€M
...,at
”S
an element
generzte M_
, if
(§ 1, 6.3). This section
a(R,p) € _M(R,p) ul(R,p)
any section u of
u assigns to
. W e s h a l l say t h a t the section
,...,ot(R,p)
generate the R-mdule
I,
5
QUASI-COHERENT MODULES
2, no 4
-M(R,P)
.
61
for each (R,p) Equivalently, this means that the induced mrphism n Qs + g is an ephrphism of Mcd
.
-3
Now consider the vector bundle T' of rank r over G defined by -n,r n+r TI(R,P)= R /T(R,~) The imges of the natural basis elements of R ~ + ~ give us n+r sections E ~ , . . . , E of T' For any mrphism f : S-G n+r - -n,r of , we denote by clS, E j ? : (set the induced sections of n+rS
.
.
...,
-s .
We this assign to EiS(R,p)=Ei(R,fp) E T'(R,fp)~~(R,P)for any (R,p)€M ) any f a vector bundle of rank r over 2 together with n+r generating sections. Conversely, a vector bundle
and n+r
of rank r over
.
generating sections u l l . . , u n+r of _M determine a mrphism _s-tc -n,r : assign to any (R,p)€M the R-module of all relations between
-5
~~(RiP)i**.i u , ( R , P )
4.5
€
!
!(Rip)
We still have to relate the functorial to the geometrical point
of view: first consider a geametric space X together with a sheaf of modules
A over Ox . we get a module
g.li: over the associated ;-functor _SX ( 5 1, + X the mcdule of sections of p* #)
.
3.5) by assigning to any p: Spec R
In other words we set
(snfl (R,P)= P* (4)(Spec R)
.
When X is the geometric realization of a scheme _X is so over X
over Spec R if
, p*@)
is quasicoherent
. In this case, it follows directly fram
1.6 that Sd is a quasicoherent Sx-module.
.
be a module over some &-functor _Y Assign to any open of U_ the d u l e _N(!) of all sections of _"V- (the y-rndule
Conversely, let subfunctor derived fram
by the base extension
v+x) . This obviously provides us
with a presheaf of modules Over 1 Y I . The associated sheaf of modules will be denoted by
IN_I
. Notice for instance that
lMal
=
rc/
M if Y_
=
Sp A -
and
M € S A (1.4 and 4.2b)). This implies in the general case that coherent if is a schane and g is quasicoherent. Proposition: For any scheme 5
, the functors d + Sd
quasi-inverse equivalences between the category X and the category modules over -
modules over
1x1 .
Proof: By construction we have
uld
m
a
is quasi-
M * 1M[ provide
of quasicoherent
of quasicoherent sheaves of
=ll!l [
for any sheaf of mdules
.
I, 6.2, no 4
ALGEBRAIC GEQMETFS
62
Conversely, let _M be a quasicoherent K-mdule.
there is, by 4.2b)
, exactly
one i s m r p h i m
.
If
X = Sp -. A
A: 5
5 [FIT
each a f f i n e open subscheme _V
of _X an ismrphisn
As
4.3, these
g and sI_M[ are local by
provide us with a canonical i s m r p h i s n f :X+Y
Let
sheaf of mrdules over
-SCf*&
and
-4:
. For
0Y
any
$: Spec R
and
.
(?[MI)
-5 may be matched together and _M 2 _S .
be a morphism of geanetric spaces
(R,p) = (p*f*$) (Spec RI
5 (fw)
inducing the identi-
(S \MI) ( A , I ~ ~ )I n the general case, we may thus assign to
t y on M(A,IdA)
4.6
is a f f h e
.+
X
and
fl
be any
we have by d e f i n i t i o n
( ~ y c l f s x ( R , ~ l = ( ( p f ) * h (Spec R)
(sx4 are )canonically i d & t l f i e d . -
. Thus
For direct images t h e s i t u a t i o n is more involved. L e t ~-4% be a sheaf of d u l e s over
Ux and l e t u: Spec
be a morphism with
S +Y
we have S ( f * d ) (s,u) = ( u * f * d ) (spec S)
, whereas
S e
. By d e f i n i t i o n
(s@ys~) (s,a)
may be
described as t h e set of maps assigning to each ccmmtative square
a section of
, which of
p*(&)
Thus the canonical maps
course has to be f u n c t o r i a l i n
(u*f,A) (Spec S)
-+
( p * d ) (Spec R)
(R,P,T)
.
provide us with
a canonical morphism
j : sf,Lkd
But this
j
is not an i s m r p h i s m even i f
X,Y are geanetric r e a l i z a t i o n s
of schemes and d% is quasicoherent. Indeed, consider t h e case where
x=\x(,
Y=\YI
and f = \ g \
. men the ahveinention&
ccmmtative square is
mapped i n t o the p u l l back square
(Spec 9; x I
where
(Spec S); X
-
Pr2
,x
I
i s t h e qeanetric r e a l i z a t i o n of the scheme (2&
S ) 2 _X
.
I, § 2, no 5 I 1
Therefore sections of
63
QUASI-COHEREWPM3D~
(_SX/_SyS d)(S,u) pr;
may obviously be i d e n t i f i e d w i t h the set of
(A)over
(spec s):
x , or
) (Spec S) ; mreover, j (S,u) (prl*pr$ u*f, dtis * prl*pr5
A.
The question whether
equivalently with
is induced by the canonical m r p h i s n
is an iscmorphism plays sane role i n algebraic
j(S,u)
geametry. A simple example where t h i s is so is given i n 1.8. A simple counter-example is obtained as follaws: consider the p i n t w of &A-fmc-
el
tor
(the projective line) assigned to t h e subspace F (1,O) E _P (F ) of 9 1 -P is the prime f i e l d of characteristic p > 0 L e t X be t h e F2 where F ^P "P open subspace of ( P I ccanpkmentary t o w a d l e t f: x * spec -Z and
.
u: Spec F f,
-1 Spec2 be the canonical morphisms.It can then be sham t h a t
"P (4)(Spec z)
as pr;
-+
= $(X)
(Jx) ( (Spec F ) x -P
X)
(u*f,Jx) (Spec F Ep , where"P i s F LT] , the r i n g of polynanialS i n one variable. "P
fl-PL (P-1) 2 2 ; hence
When & is a scheme ard
4.7
shall sanetimes simply write
g is a
quasicoherent Ijmadule, we
, thus
instead of
identifying quasi-
coherent &-modules with quasicoherent sheaves of modules over
151
.
Notice a l s o that t h e definitions and statments of t h i s section may be e a s i l y
. W e entrust t h i s t a s k
exteladed to the r e l a t i v e case of k-schemes, w i t h kc:
to the reader.
Section 5
Affine mrphisms
5.1
Definition:
affine --
---
2
2-functors is said t o b"-
_ Sp_R i f , f o r each model R and each morphism g:
-+
Y_ , t h e fibred
Sp R p X_ is an a f f i n e scheme.
prduct If
fi mrphism z:K-+xof
is a 2-functor, we say t h a t a 5-functor _X
structural projection px:zps
is a f f i n e over _S
if t h e
is affine.
-,i.,
5.2
Example: I f
_Y
is an a f f i n e scheme,
f
is a f f i n e i f f
5
is
an a f f i n e scheme. W e see that the cordition is necessary by taking g - to be and 2 = Sp C , the fibred product =Sp B an i s m r p h i s m ; conversely i f -
-
Sp R
*
X
Y -
may be identified with
S p ( R p ) , which
is an a f f i n e schme.
proposition: If _f:x-ty is affine ad. if Y_
5.3
5 scheme,
Proof: We show first that
& is local, making use of the notation of def. 3.11 of 5 1. Let Ci€ X(Rfi) be elements such that v((C,))=w((E,)) ; set
.
Since ni and n j have the same image in y(Rf ) and is i- 1 ifj local, there is a unique nEY(R) whose image in Y(R ) is ni for all fi iEI If p.:R-.Rf is the canonical map, (pi,Ci) belongs to the fibred
q =f(C.)
.
product
1
(Sp - R;
i
n# : sp R
(Rf,) defined by 1
+
II_
. Since
Sp -R
55
is
representable, hence local, and the (pi'si) satisfy the usual ccmpatibility conditions, there is a unique for each i
. Obviously
image in Z(Rfi) is
FId
ti
R
5 ) (R) which maps onto (pi,<,) is the unique element of X(R) whose
-
( p r < ) € (Sp R Q
and 5
for all i
.
5 is local. We obtain an open covering of ,X by affine schemes by considering the fibred products JS R $ 5 attached to the open
This shows that
anbeddings -g: Sp - R
-f
-Y
for R€$
.
Let _S be a 2-functor. An S-alqebra is a pair
5.4
(&#a) consisting
of an S-module A_ a d a mrphism of 5-functors a:@fi-+& such that, for any R€$ aid any p€_S(R), a (R,p) is the multiplication of an R-algebra structure (associative,cmtative, with unit) canpatible with the R-module structure
.
of _A(R,p) The 3-algebra ($,a) is said to be quasi-coherent if A is quasi-coherent (4.11, If (€3,O) is a second 2-algebra, a mrphism of ( e r a ) into (B, B)
is defined to be a mrphism of S-functors f:&+g
f(R,p) is an R-algebra hcsocanorphisn for each ?-model shall simply write instead of (A+) -
.
As a first example we have the 5-algebra generally, if
_X
Qs
(R,p)
such that
. Henceforth we
defined by Qs(R,p) = R
is an 2-functor, the Weil-restriction
. More
I Q~ clearly bears x/s 2 is affine
an S-algebra structure. This ?-algebra is quasi-coherent 2.fover
5 .
Propsition:
2
2 2-functor, the functor
_X
-4b
Q~
is an anti-
equivalence between the category of S_-functors,which are affine over _S
-and the category of psi-coherent 2-algebras.
,
proof: Our propsition is a direct consequence of the definitions. We can associate with each quasi-coherent S-algebra
8
an S-functor JS
5 which we
I,
5
2, no 5
65
(9AIR
This means that
is a f f i n e over _S
s_p A -
. Hence
is represented by the RilFodel A ( R , p )
. Moreover,
i f we set _X = s_p -A
, (Io 1(~,p) x/s -5
, i.e.
is
.
identified w i t h t h e algebra of functions of
(Sp A)R
This means that t h e c a n p s i t i o n
is i s m r p h i c to the i d e n t i t y
(?@-: n o , ) Sp 0
functor. Finally, w e also have S_po(? s - 3
% Id
%i& A_(R,p)
because
( a Q (R,p) x ) is
- - -
TO-' is t h e R-algebra of functions of the fibred product of
Being a f f i n e , t h i s fibred product is identified with
S
When the 2-functor
5.5
sp'xg(lz) . (R,p)
-
i s a scheme, we may interprete the pre-
dx
(px)*(dx) of
the d i r e c t image
9
2.4. Moreover, t h e &rph.i&
+ (<)
s
-
under
t h e s t r u c t u r e of a sheaf of algebras ov&
bras
Jk over 5
Corollary:
pX
-5
(dx) -
S_ (by definitTon a sheaf of alge-
an associative and
r i n g haramrphisms and are cartpatible w i t h
If s
by
(p ) *
assigns
w i t h unit; moreover, the r e s t r i c t i o n maps
$-(U)
+
5-
(V) )
.
a scheme, t h e functor g k ( p x ) * ( < ) is an antiequiva-
lence between the category of S-schemes, ---
which are i f f i n e over
category of quasi-coherent sheaves of algebras
Proof:
by
asociates with each open subset Ucl
catmutative algebra &(u)
are d(U) -+ &I)
_ P ~ : ~ X is + ~quasi-coherent
* (gX)ir&&d
. Then
5
ceding p r o p s i t i o n i n the f o l l m i n g way. Suppose & is a f f i n e over
aver
.
W e only have to show that the morphism j : S(pX)*($1
-+
5
r
and t h e
2 g x de-
fined i n 4.6 is an isamrphisn. I n f a c t , as we knaw already-that 0 i s quasi-coherent, it is s u f f i c i e n t t o canpare S (p (R,p) and _x/_s -3 -5 Qx) (R,p) when p:' Sp - R 2 is an open anbeaahg. I n t h i s case both
(8
algebras are identified with
-f
J(Sp - R
-
5) I
AIi3I3RAIc m
66
m
Y
I,
9
5
2,
According to 5 . 4 w e get an antiequivalence, which is quasi-inverse to c-;'(p -5 * (d
by assigning as follcws a ;-functor
x
to any quasi-coherent sheaf of algebras is the set of pairs
of algebras
The maps
(p,X)
(xi)
+
Let
z-'(xi)
+
RI is an arrow of ,M u is the morphisn
, =en
Yi
.
for each i
: f l
Proof: With the notations of 5.1, -
S eRY X x
+
,
f
Qd+
.
induces an affine m r -
are affine over _Y
Sp R
.
it is sufficient t o show that the canoni-
is affine, assuming that the canonical pro-
are affine. W e m y thus suppose straight away that
= Sp R
(xi)
, hence
that
by a finer covering i f necessary, we m y assume -1 i s an affine open subschane of Then f is
is a scheme. Replacing
further that each
:
f is affine.
In particular, vector M l e s over a scheme
cal projection
p
, (spd) - ($1
2 mrphism of local 2-functors
f:&Y
an open COvWiny of
phisn ~ i :
and a m r p h i m of sheaves
p€_S(R)
define the "canonical projection''
p
Corollary:
5.6
; i f 4:R
R)
($(p),~(~)oX, ) where
++
over
consisting of a
(p,X)
X : A-p!(Js onto
(p,X)
mps
k
spd , c a l l 4 spectrum, _S : If RQ , (%& (R)
xi
(xi)
.
affine by 5.3. Being local and being covered by affine open subfunctors,
X
is a scheme.
The d i r e c t image
&= f , (8.)
each
since this is true for
the canonical isanorphim isamrphism _X
+
Sp + d
/xi . AS
g-l(yi)
+
Let
-
5 f
Z_
,
and ~ - p dare h ~ t hlocal functors, s_P d ( f - l ( y i ) ) = Sp - &Yi) induce an _X
5
(cf. the arquwnt of
Corollary:
5.7
is a quasi-coherent sheaf of algebras over Y
Y_
1, 4 . 1 6 ) .
-be-a diagram of
schemes such that
q- i s faithfully f l a t and quasicompact. I f the canonical projection f,:
-
y xz Y-
Proof:
+
is affine, so is f,
By restriction to an affine open subschane of
t o the case i n which
(xi)
.
5
z , w e reduce directly
i s affine. W e must now show that
is an affine open werirg of
, we may replace q-
5 is affine. I f by the c a n p s i t i o n
I,
5
2, no 6
67
-Y‘+Ycan where
y‘
g
c
z
yi
i s the d i s j o i n t sum of the
Suppose therefore that
. Since
and. C = &(?$:)
_Z,y and 3
;
. are a f f i n e and. set k=a(_Z) , B=@)
q- is then a f f i n e and surjective, the same holds f o r
sx: & xz Y- x
t h e canonical projection
-+
.
; therefore, since
so is _X Hence &ere is an a f f i n e open covering -1 (Xi) is a f f i n e -5
ccanpact,
5 5- Y_ (Xi)
is quasi-
of
5
;
since q
is a f f i n e , hence quasiccsrrpact. I t follows t h a t zinXj
is quasiccmpct and
lemna 1.8 implies that there is a canonical i s m r p h i s m
5
which allow us to identify qX ‘I with the mrphism - -
S(XrY)
z-
.
B u t this latter is invertible; so therefore is
Section 6
Closed embeddings
6.1
Definition: -
a)
Let
f:z+x
that f a closed & d i n g if _f each mrphism g: - s_P A _Y , t h e map -
-X
(3.5)
.
5 mrphism o f ~ - f u n c t o r s . We say
i s a f f i n e ard i f , f o r each AfM, @
+
’(:Sp
is s u r j e c t i v e b)
:f
.
-
A) : ~ (-S A) P
X_ @ 5 subfunctor
inclusion mrphism
&a
of
-+
Y -
J(S_PA p s)
, we
say that
closed Embedding.
5
is closed i n _Y
i f the
If
is a scheme ard
_Y
5.3. Accordingly w e call _X a closed subscheme of
, g-
Suppose that i n statement a)
_Y
.
is a scheme by
.
is of t h e form g=fo_h Then t h e mrphism _
L
X w i t h cunpnents Id and _h is a section of _f Sp A x SPA o ( s ) d ( Z )=Id , so that d(fQ A) is b i j e c t i v e , whence s=_f-l-'
s : Sp A
Hence
and
7 ,
is a closed subfunctor of
X_
-f
Sp -A
h = yxg$
SpA A
. This shows t h a t
that a closed gnbeddinq
_h is uniquely determined by
25 m m r p h i s n .
-
-g
'
S_pA
, so
.
Now suppose we are given a monanorphism of 2-functors _f:x+g By arguing as i n 5 1, 3 . 6 , we-see that f is a closed embedding i f f for each A€$ and each a % ( A ) there is an ideal I of A satisfying the f o l l m i n g condition: for each arrow $:A-+R of & , aREx(R) belongs t o the i m g e of X(R) in Y(R) iff -
$(I)
6.2
= 0
.
Fxample:
If, in
i s an a f f i n e scheme, then a necessary
6.1, _Y
and s u f f i c i e n t condition f o r f to be a closed mhdding is that X_ be an a f f i n e schene and d((f) be surjective. The necessity of the condition is v e r i f i e d by taking g: Sp - A + Y- t o be an i s m r p h i s m ; the converse is obvious If
.
I
is an ideal of
-
image y(1) of
Sp $
and $:A+A/I
A%
is the canonicalmp, the functor-
, we
is such that, for each R@J
I +(I)=OI . the map I wv(I) 2 2 b i j e c t i o n of closed subschgnes of Sp -A .
have
Y(I) (R) = I$%(A,R)
I t follaws that
onto t h e set of ----6.3 F (r,s) -n
simply _F
F (A) .!hen _F 2
in
system of
,
= G
= { (P,Q) %r,n-r
is closed i n _Y
. For
(A)xGs,n-s
I
(A)
FQ}
l e t a=(P,Q) and let Q'
be a cmplanent f o r
...,pn. be the projections onto Q' along Q of a ) .With generators of P . Suppose that pi = (ali,azi,...,a (n+r)i ; let
pl,
the notation of 6.1, the b n d i t i o n R @ P C f o r each
A
-r,n-rx %,n-s , r 5 s ( 5 1, 3 . 4 ) . write for the subfunctor of Y_ satisfying
Example: S e t
, or
the set of ideals of
(i,j)
A
. I t thus suffices t o set
R@Q A I =
1
i,j
is equivalent to
Aaji
(6.1).
$(y.)=O
I
9
I,
2 , no 6
69
QUASI-COHE~-~DWS
It is obvious that the canposition of
tm closed gnbedaings is f h a closed gnbedding. Similarly, i f in the diagram g--jut=-Y' of g f + y' is l i k e is a closed & d i n g , then the canonical projection -fyr: 6.4
ci:Xi+zi
w i s e . Finally, given a projective system of closed &dim$
,
the projective l i m i t
is a closed embedding. W e prove the l a s t assertion:
E 1 2 Y(A) , l e t aiEYi(A)
Q
6.1, there is an ideal Q
iR
be the projection of
of
Ii
1-1
1 8 X(R) i f f
for a l l
$(Ii)= 0
6.1 it thus suffices to set
6.5
Example:
(P1r
such that P1 c P2 c
'Gls,n-rs
Gr1,n-rt
C
Ps
. Hence
relation
aR belongs to the image of
. In order t o s a t i s f y the c r i t e r i o n of
...,rs) S
(R)i
w i t h coefficients i n R g is a
aXGr
s
,n-r
. The subfunctor
s
(R)
gn(rl,.. .,rS)
of
f o d by these flags is called the schene of f l a g s
. I t i s a closed subschew of
(n; rl,...rs)
of nationality
i
(n; rl,
E Grln-.rl
...
, the
. By
lsr,s.. .sr sn be an increasing sequence of inte-
Let
- - - rPs)
a wj.th irdex i
I = & . i
gers. A f l a s of nationalitv
sequence
and
such that, for each @:A+R
A
i s equivalent to $ (Ii) = 0
€_f. ( X . )
if A 2
zij
.
For l e t be the subfunctor of this l a s t scheme * * GrsJn-r, formed by sequences (Pl,...,Ps) such that Pic P ( i i j ) Then Eij is j the inverse h g e of Xn(ri,r.) under the canonical projection of
G q , n-r<
.
7
mrphism of
1x1
Proof:
$:K+L
of
f:z+x is a closed -ding,
Propsition:
6.6
Let
Y(K)
<(L)
iff
onto a closed subset of
1x1 .
be an extension of small f i e l d s and let
It follows easily fran 6.1 that pLEx(L) p
fs is a h m -
belongs to the image of
i s m r p h i s m of
XIK
nents of
(9
Z(K)
p
be an element
belongs t o the image of
. This implies that
glz
is an
onto the union of a collection of i r d w s a b l e ccmpo-
1, 4.5)
, so
that
f
is injective
(5
1, 4 . 9 ) . TO ccmplete
close3 in
2 , no 6
5
the proof, we may assume that
is the inclusion morphim.
5
I,
AtGEBRAIc GlxNmRy
70
i s a closed subschene of Y and tt-iat f W e show t h a t each closed subset P of is
1x1 . By the definition of
yl(P) is a closed subset of 1% phisn g: Sp g-l(P) - A -Y . Now -
(5 1, 4.11,
we must show that for each A% and each mr-
A1 = Spec A
1511 , which
is closed i n g-’(
-+
1x1
is easily
~-‘(XI (use the f a c t that X I ard ~ + -1 g (5)12 are both sums of collections of irdecmpsable ccanponents of YIE respectively). Since Z-l(x) was ass& to be of the form sp seen to be the set of points of
and
1x1)
, -g-’(
s_P(A/I)
= ~Q--~(_X)
I
, and
= Spec(A/I) i s closed i n Spec A
the
p r o p s i t i o n follaws.
~rollary:If - -X is aclosed subfunctorof Y_ tor of
X_
, there
y
is an open subfunctor
and
x
g
and
.
:1 1
of
P
:
-
c = &nv_,
y such that
cf
Prcof By the p r o p s i t i o n there is an open subset lLJl = P l i I x-I . If we set y=y (9 1, 4.12) , Vnx_ P p i n t s . Thus U = V- n X-
g is anopen subfuncsuch t h a t
have the same
I n general the map * 1x1 of the set of closed subfunctors of 2 into the set of closed subsets of is neither injective nor sur6.7
jective. For i f
I
I
and J are ha ideals of
have the same uxderlying space i f f being identical w i t h J
.
fi
=
A€&
,
and _V(J) (6.2)
!(I)
fi , which can ccw without I
This shows that _X
I+
is not necessaxily in-
j_X/
jective. On the other hand, l e t T be the gecanetric space whose underlying
set consists of two points 0, 1 , whose closed sets a r e @ , {Ol , { O , l l , arod such that the restriction d,({O,lI) -+ dT({lj) is the identity IMP of the prime-field with
p elements. I f
R%
, (ST) - (R)
is mpty i f
p
# 0 in
R and m y be identified w i t h the set of closed subsets of
Spec R i f p=O n R It can be sham that g@ and _Sr are the only closed subfunctors 3f ST ; it follows that the mp 3 is not necessarily surjective.
.
1x1
-f:X+Y --
be a closed (9 I, 4.1). AS a quotient-ring of A Let
6.8
gnbedding %
& let
4sp - A) , d’(9 A + 5)
is clearly quasi-coherent
-X
++
-ivy, 0
-
-
be a Y -d e l bears a natural
d ( 3 A $ 5)
. W e say that 0-x/y_ is_X/au quasi-coheremt it follows inmediately f m the definitions that t5e map
A-algebra structure. Wmover, the induced ~ - d u l e 0 Y-alqebra. In fact.
(A,p)
(Alp)
H
(4:l)
is a bijection between the closed subfunctors
X of Y - ard the
quasi-coherent quotient-x-algebras
. For s c h m s
0
of
be reformulated as follows i n tenns of
Y/Y sheaves of
Suppose that Y_ is a scheme and that
g:z-+X
is affine ard open in
this statement may
modules:
is a closed gnbedd%.
If
g:y-+y denotes the inclusionmrphism, defi-
and i f
nition 6.1 a ) ensures t h a t [g[g(y): 9 (V) 9 (f-l(Y)) is surjective. Hence yf I f / - : dy -+ _ f , ( f l ) is an epimorphism of sheaves. Set y = Kerlfl-f By 1.11
z-
-f
and 5.5,-
7
dy/ 3 , that
,F
.
is quasicoherent. By prop. 6.6, the support Supp(dy/g) of is the set of
of f e . Hence
&ge (F
x
,( ~ $ / f iIF) , where (dy/y)I_F
, and
= su&(JY/T)
-
Coy/?)Y # 0
yfY - such that
121
fe i s an isomorphism of
denotes the restriction of
f e - i s the map N e r l y i n g
.
f
7 be a quasicoherent ideal of oy-
Conversely, l e t
. For each
u",)
Y-suhcdule of
-
derived fran 0
+.
7 01!
-+
-+
, coincides
-g: s_P
fly/?+
A
-+
Y
with the
onto the geawtric space
, the
dy/j'
to
(i.e. a quasicoherent
sequence
0 is exact. Hence the canonical map
(% A) is surjective (1.12). Since - y (sp - A) into B = g*(dy/Y) s_P A G s(oy/J) may be identified with s_P B by lenna 6.9 belcw, it follows is a closed embedding. that the G n i c a l projection p : %(Jy/y) +.
of
A 7 g*(J
The image-functor of subscheme of
YY be
w-i l
Y defined-by
I!(?)
identify
-p&/7
[
with
(F
7 . With
written
and called the closed
the above notation, by 5.5, we may
,dY/7) IF) , where -
F = Supp(dy/7)
-
.
Fran these r e s u l t s , we deduce in particular the Proposition: Let - -Y
be a scheme.
Then
:
a)
The nap
7
I-+
v(T)
is a bi-
jection of the set of quasicoherent_Y-ideals onto the set of closed subschgnes
of 1 ; b)
A morphism of schemes
(4)
l _ f l f : Jy -+ _f* F c e of points of Proof:
_e_
g:z-+y is a closed
gnbedding, i f f
is a sheaf epimorphism and
-x
onto a closed subset of
'z
is a hcaneamrphim of the
1x1 .
. The condition is necessaryf by 6.6 and 6.8. the condition holds, the kernel of lsl- is a quasicohe-
It remains to prove b)
Conversely, i f
rent ideal (2.4 and 1.11). Setting F = Supp(
, it
131 onto the geQnetric spce
follows t h a t f (F ,(gy/Y)IF)
.
in-
NmBFAIc ciEmmRY
72
I,
5
2, no 6
f:gt+S is a mrphign of schanes, &3 p s i - c o h e r e n t sheaf of algebras over _S , we have a canonical isararpkim 6.9
-S' 3 Lf
b)
-
If
a)
Lama:
s&:
sp
g*c4
.
, we
,
Sp have a canonical isanorphism u -: -Sp B denotes-& sheaf over Spec R associated w i t h B (1.4)
RfM,
where
B%
, (8' xs Spd)(R)
a) For any RfM,
Proof:
, where
.
is the set of pairs
,-and where A phisn of sheaves of algebras over S If A' (p , ( f ( p ) , A ) )
p€S'(R)
.
: A +f , p f ( dSP R)
is a mr-
is associated with X
by the
bijection
of 1 . 2 , our canonical isomrphisn sends
E (9f * & ) (R)
(P,X')
Let
b)
i:R+B
@E(S_p B) (M)
.
onto
i
,
f
(3%) (M)
(+i,$ E
-
=
.
Sp i
-
Let
Corollary:
for each
,(f_(p) ,A)
onto
)
be the canonical isanorphism. For any MfM,
u satisfies p -Sp - B6.10
(p
f:px
,
serds
u(M)
( 1 . 6 ) . The i m r p h i s m u therefore
be a mrphisn of local Z--functors.
induces a closed gnbeddinq fi:
c-'(yi)
*
If,
!$ , then
3
f
a closed fmbedding.
Proof:
By 5.6 we need only consider the case i n which Y_ is a scheme. Then
i s a f f i n e by 5.6, and it r m i n s to s h o ~that 1-f: J * f, ( 3 . ~is an epimorphism of sheaves. But this follows from the f a c t that the &strictions
x
f
If[-f
of
to the open sets
6.11
. :1 1I
are epimorphisms
_Y
5 in
. If
~
, we
:
is~ a mrphisn + ~ of
Z_
Let
P
be a subset of
and
-
such that
Z K . Let
Y_
. With each f i e l d
# I p I maps Spec K into P
the inter-
, we
K%
z
is
and called the closure
J _f is the closed i m g e of also say t h a t I several examples of closed images: f
. By 6 . 4 ,
_Y
containing a given subfunctor
again closed. This intersection w i l l be written
of
(6.8).
W e return now to an a r b i t r a r y ;-functor
section of the closed subfunctors of
-
z
i s the image-functor f
. W e now consider
ard each element
associate a cow X(K,p)
pEx(K)
of
,
73
x = #X
(Kr 0)
and let _f:S-ty be the mrphism whose canponent of irdex p is p:'
g(~,p)
-+
. write pred
y
xred
-
/ ? Ired
for
.
(!?I
closed subfunctor _F of Y such that P we write simp1y
. W e say that
Pred
is reduced ard has the f o l l m i n g property: i f
i s reduced i f
. For each subset
Xred=
_t(T)C P
g:?:
6.12
is a ,punctor and P a subset of
If
l y contains the closure
-
P #
Ipredl
JY-
-~ ( 7=)prd 2 1111 Proof:
If
viously
P
Y(u)
1x1 . m:
of
hence
i
on P CI
u
. Similarly
. We now prove
J(Uf)
1x1 , IpredI
*
obvious-
it may h a p p n that
1x1
Y
t h e sheaf of
is quasicoherent ; b)
Fred
is the closure
P
~f
P
"$i"
c)
follows frcm a)
1x1
and each f€d(U)
is injective. Conversely, i f since g
x=g/fn E
1x1
nilpotent e l m n t s of
n uf
is annihilated on P
, the
canonical map
7 ( Uf ) ,
and f
gf
on
A necessary
i n 6.12.
B
Y
.
u - uf
if
. The i .
and s u f f i c i e n t c o d i t i o n f o r a scheme
fly'Y EY )
.
-I
be reduced (i.e. do not con-
It then s u f f i c e s to show that
NOW
;
is annihilated
implies that x belongs to the image of
Corollary:
Set P =
since we obviously have
a) : by 1.9, it is enough t o show that, f o r
Y_ to be reduced is that the local rings tain any n i l p t e n t elements apart fran 0 Proof:
'red
is bijective. Now we have y ( U ) fC u(U)f ard @ ( U ) 1 @(Uf)
equation g/fn = gf/fn+l 6.13
-
is a scheme:
a)
the space of p i n t s of
each affiGe open subset U of -+
L
,
is a quasicoherent sheaf of ideals contained i n 3 , then obSupp(dy/;J) , and so Fred (6.8). Assertion b) t h u s
P = Supp(dy/y) i: Y(U)
if
Y_
= I S E & ~ ( U ) J S ( ~ ) = O vXcunpi ,
2
follows f r a n a)
-
. By 6.7,
Iy[
of
P
such t h a t
; c)
C
factors through
L_e t_ Y _ be a scheme, P a subset of
f o r each open subset U
-
in
P
. This cannot occur, however,
Proposition: ideals of
of
part
is a reducedz-func-
T
tor, each mophism
such that
; it is the &lest
qedis the reduced
and say that
of
-
2
f o r the closed image of
'I! is a f f i n e and open
7Y
consists of
contains
y
ALGEBRAIC (3ImEmY
74
and i f
is annihilated a t each point o f
f6!Q)
every prime ideal of
of f a t y then (f IV)
fjv
,
Yl
fYE
, then
= 0
therefore nilpotent. I f
Y
t Y
Y
Y k-schemes. Then 5 x -
.
9
f
o
two reduced
_Y
zg is reduced.
& is called reduced i f
v
(zz)S x~ k (,Y) em
-z (-x x y )
c
5 I, 5.7, t h e local rings of , ( & x Y ) are the rings dxgkJy , X Q , y e . sGce dx and By
r i n g s of fractions of the
are reduced, they are
ox
contained i n products of f i e l d s (the local rings of
and
Y
prime i d e a l s ) . By Alg. VIII, 5 7 , no. 3, th. 1, it follows that
<%$
reduced, so that each ring of fractions of
is reduced.
at minimal
is
#
L?@
XkY
W e tlow turn our a t t e n t i o n to the clos& image o f a morphism of
schanes
f:z+x . Since the functor
of 1.4
M&
c m t e s with inductive
limits, the sum of a family of quasicoherent sheaves of i d e a l s of again p s i c o h e r e n t . Each sheaf of ideals of quasicoherent sheaf of ideals.
s'
$'
. Hence
y
1, 6.3, there is an isanorphism
6.15
no 6
is reduced.
I n accordance w i t h § 1, 6.3,
Proof:
2,
i s the germ Y is nilpotent. conversely, if =
implies that f
f
5
is contained in
f
Corollary: Le t k be a perfect f i e l d , _X
6.14
By
, and i s
U_
Y f o r a s u i t a b l e open neighburhccd V of
, so t h a t
EJ(v)
d(y)
I,
dy
therefore -
I t follows that Im f
=
dY
is
contains a l a r g e s t
v(%
(6.8)
, where
is the l a r g e s t quasicoherent ideal contained i n the kernel of the mrphisn
ox
l_flf:
+
induced by f
f*(Jx) -
.
This may be simplified when f_ is q u a s i c a n p c t and quasiseparated. For by 2.4 f*(~!$)
i s then quasicoherent,
f
Proof: g (v') -
that
f
r =K e ~ l f lf- .
be a diagram of schanE, where g is f l a t -1 i s quasicawact and cpasiseparated. Then w e have Im -fyl = s_ (Im $1
Proposition:
g@
y'+y.-X
SO
k t
-
'y
Let
I!
. Let
I!' he a f f i n e open subschemes of -Y and u' such that -U = -f -1 (v) and V_' = git(v') . By 1.8, the canonical map
ard
dcy)
Q0(")Q)
-
i s bijective. Ry varying
-f
J(U_',
-
I! and V_' , we derive a canonical i s m r p h i s n
.
I,
5
2, no 6
75
QUASI-COHERENT-IVK)DUL!ES
(see 1.6). If we set 4 =
Is[-f
-
and 4 '
=
allows us to identify
9*(4)
: f*(LJy)
-
with
Hence K e r 4 '
Ker g*($)
+
- -
1%
Y
I-,f
t h i s canonical i m o r p h i s n
g*(f*(dx))
-
-g*(Ker 4) , since the functor g- is exact. So
I n the following, we require a statement analogous t o prop. 6.15 in the case where f is not quasicanpact arid quasiseparatd. By way of canpensation assume provisionally t h a t we have U_ = Sp - B , B E g If 6.16
KerIfI
I & =
.
, it is clear that the largest quasicoherent sheaf of ideals 7 JV s a t i s f i e s J = &Y)" . Therefore = ~ ( 9=)~(~n/cy). be an affine open covering of X and l e t gcl:zcl+xbe the mrphisn f . If we set % = ~ e r l gf~ ,/ -then X= n % , YO = . c1
contained i n (X )
Let
-a i d u c d by
Now i f
is a ring hcanamorphisn which makes B '
p:B+B'
a projective Bincdule,
then
If
-V($
y
= sp - B'
B' UBJc1
and g =
(Y) )
s_p
-1 (anf) = ~ ( B w ~ / ( Y ) ) =
B , it follows t h a t -g
is the snallest closed subschane of
cyl -
But t h i s smallest closed subscheme is precisely Im
case of the following Proposition:
Et
u'
containing each
. This is a particular
be a quasicchnpact guasiseprated schme and let
_g:Y'+x -
be an affine morphism of schemes. If there is a f i n i t e a f f i n e open covering
(xi) of -
-1
(Y!))
such that (g is a projective J(Y.)incdule for all i -1 -1 then I m f y l = s_ ( I m f) for each morphism of schemes f:z+X ___
.
The proof of t h i s r e s u l t i n the general case i s sketched i n 6.17. W e shall use t h i s proposition in the follcwing form: L e t k - d e l which is projective over
k
k be a madel a d K
a
as a module. Then for each quasicanpact
,
I,
8 2 , no
6
, we
have
d' be the kernel of l e t Jqcbe the largest quasicoherent ideal contain& i n $ ard let
If- \
quasiseparated k-scheme (Imi?qK =
q.
f:_X-tY_
To sketch a proof of prop. 6.16 , let
6.17
qd
ard each m r p h i m of k-schenes
_Y
-+d be the inclusion wrphisn.
:JqC
Ebidently
(flc ,qW)
f
- ,
enjoys the
following universal property: for each quasicoherent sheaf of modules &6
wer Y- ard each mrphism @:l/td+Xthere is a unique $:k'+kqC such that qJ II, = @ For since CUC fly , Im @ is quasicoherent, hence is contained i n d'qc W e show f i r s t that if is quasiccmpact ard quasiseparated, this
.
.
universal problem has a solution whenever dzr is a Y-module satisfying the follming condition: clusion functor of satisfying
-1x1 (*)
i n Y , ck/(V) is mll (the ini n t o the category of sheaves of roodules over -
for each open V
nvsd
:
has a r i g h t adjoint).
(*)
Assume f i r s t that X i s of the form y*( 1, y:v+x k i n g the inclusion m r -
phisn of an affine open subschene a d q x : yqc
and write
9
a y-madule.
W e set
-+x for the unique mrphisn such that (1.4).
It is then easy to show that
(vx
qb(Y)$g(v) = Idde(v) is a solution of our universal problen.
is a x-module satisfying
In the general case, when
(xi)
(uijl))
x
1
= y:(Xlxi)
. = (resp. &111
1 71
a canonical exact sequence of Y d u l e s J7
(*)
,v* (q,) 1
above,
consider
of Y (resp. of YifiYj) Yi (resp. yijl) intn v-*i J 1 ( ~ l .~ ). In t h i s way we obtain
a finite affine open covering (resp. If yi (resp. yijl) is the inclusion mrphism of
2 , set
p= 8VI4
.
u
" t T1T 1u vw3 T i j lXi j l &
km
OUT
previous remarks,
and
are solutions of our universal problem relative to
m y therefore define mrphisms v,w : !?
and qw
=
w$p
. It thus remains
to set
Q
74
and
by the coriditions
NqC = Ker(v,w)
z"fj1.
W e
qv = v4p
and to define
.
I,
9
7
2,
77
QUASI-COHEREIW-M3DT.LES
q& : dqC +$
by t h e condition
inclusion mrphisn. The pair
, where
= pu
udq4
u
:PC +F is the
is the required solution of our
(dqC,qd)
universal problem. Returning n m t o the proof of 6.16 in the general case, it only remains to show t h a t
. To prove t h i s
&a is a family of quasicoherent sheaves of ideals of we need merely verify that the construction of e when
( ~ 4 ) " e l'cmtes"
-
w i t h the change of base functor -g:Y'+Y -
Section 7
Ehlbeddings
& anbedding i s a canposition of arrows gg - -f , where _ _ _ f- is a closed embedding an3 9 i s an open anbedding. Definition:
7.1
If
is aL-functor and _X a subfunctor of i f the inclusion mrphisn of
closed i n
X_
Y
, we'say
that
5
is locally
is an &ding.
into
If
-Y is a scheme, a locally closed subfunctor of Y_ is called a subscherne. 6.1 and
5
1, 3.11, each subscheme f o r a schane is i t s e l f a scheme.
Consider, for example, a scheme
(i.e. of
By
1x1) . Set
(5 1, 3.11
U =
1x1 -
and a locally closed subset P of (P-P)
. Then xu
is an open subscheme of
and 4.12)': When no confusion i s possible, we write Pred
intersection of the family of closed subfunctors of contains P
. W e may characterize
Y whose space of p i n t s is
P
.
Pred
xu
_U
f o r the
whose space of points
as the unique reduced subscheme of
The following assertions are imnediate consequences of the properties of open
and. closed embeddings: an anbedding is a m n m r p h i s n ;
the composition of t!m
embeddings is an embeading (cor. 6 . 6 ) ; f o r each diagram _X f , Y
3 x' , where
-f i s an embdding, the canonical projection f,' -
: XXY'
Y-
-+
y'
i s a l s o an embedding. 7.2
Proposition:
Let
the inclusion mrphism f of
be a scheme and & a subscheme such that
X
into Y is quasicanpact. Then
X
is open
78
i n t h e closure
2 & g (6.11).
f
f:g+y is a mnanorphiSn, separated
ard s u p p s e that
be open i n Y -
Proof: L e t
-*
(2.2). By 6.15,
i.e. w i t h x ; but
Xm-
g
5
is closed i n U-
. Since
X *Y X-
is invertible; hence f is quasicoincides w i t h t h e closure of _xnV_ i n , +
is open i n 5
Proposition:
7.3
:
%/Y
7
I, § 2,
ALGEBRAIC GECMETRY
. This m p l e t e s the proof.
f Consider the diagram of schemes & =+
Y_'
g - -is f a i t h f u l l y f l a t a d quasicanpact. I f the canonical projection
Y'
where
flr
:
is an open (resp. closed, resp. quasicanpact) gobedding, then -- f is-an open (resp. closed, resp. q u a s i c m p c t ) anbedding.
-X xY Y-'
Proof:
+.
If
-fyl
is an open embedding, it follows f r a n the equality
g -- l ( g ~ ~ )=) ~ G ~ ( X X Y '(8) 1, 5.4)
xthat f ( X )
-
subset of
subscheme of induced by f
such that
, then
Now suppose that
is an open subset of
IX_'
I
=f
x-
If
(3.4).
is the open
_XI
:x+z' is the mrphism hence so is f ' (3.5) .
(5) , ard i f f'
is invertible;
x
f'r
-fu,
the f a c t that Z ~ , ( X * Y Y 'is) an open
is a quasicanpact (resp. closed) embedaing. Then f
i s q u a s i c m p c t &-quasiseparated
Imf
since f y l has these properties ( c f . the
argument of 5 . 7 ) . I f 5'' = , and i f f'l:X+:''is induced by f , then -1 -1 2 (5") = Im gyl (6.15). It follows that -f";t,,~(5") is an open Embedding -
(resp. an iscsnorphisn). H e n c e
Definition: Amrphism
7.4
i f the diagonal mrphism 6x/y: X
f" is an open-embedding (resp. an i s m r p h i s n )
5
+
fi:x+x of is said t o be separated xpx is a closed enbeaaing.A2-functor
is said to be separated i f - t h e unique mrphism px:
-
X 9 2 +
is separated.
-
Referring back to 6.1, and recalling that a mrphisn g=(u,v) - - : Sp A + 5 Y* -X is determined by g ard v - , we see that f is separated i f f f o r each A%
and each pair of arrows (y,y) of Sp - A i n _X such t h a t f;=fy , Ker(s,_v) is closed i n SJ A Frm this and the definition we h e d i a t e l y infer the
.
following assertions: each monmrphism is separated; i f , i n the diagram
-x
1~ 9 y' of
gyl: - -X *Y Y'-
-+
,f
Y_' ; i f
is separated, then so i s the canonical projection
-g o -f
-
is separated, so is f ; a product of separated
mrphisms is separated. 7.5
Proposition: An a f f i n e mrphism
f:z+x
i s separated.
.
.
Let g,y: Sp A _X be a double arrow satisfying &=By definition, (Sy, A) k g is affine, so that there is a cartesian square of the
Prcwf: form
and a double arrow a,B: B Z A such that y'(3a )
=
_u
, y'(sp B)
= y
d
A ' Since we have
a@= @@ = Id
Ker(g,y)
is represented by
,
the elements a(b)-B(b)
A/I,
bEB
It f o l l m i n particular that
By setting
= cp
2 , we
is the ideal of
where I
. Hence
a)
f:x+x
Let
and and
a double
be mrphisms of f
a z ~ o wof
y: ?+
be the subfunctor of
-1
Then K E X ( ~ , ~ = ) w_
(A)
X_* X
1-
z$X
I$J
such that
A (R) =
For the f i r s t assertion, observe t h a t
:
y+ g;y
X,_Y,_Z are s c h e s
c (x,x)IxE X (R)1
ad. l e t for a l l R$
is locally closed i n
S
.
is the c a n p s i t i o n
x- (gf)Y .
and f Now & is derived fran by the change of base g;Id : ; since 6
where _h hascanpnents
6Y/z_
If
x
(by the lema below) ; hence Ker(g,y)
-z -Y
.
gf is a closed anbeddins
. If g is separated, A is closed i n X x-x , so that S . If X_ and are schemes, A is locally closed
in
h
.
a
such that gg = -fy
be the mrphism with ccanponents g,y
is closed in
3-x.
2 are affine.
is an gnbeddinq.
Ker(u,v_)
b)
& 5
. If
is a closed embeaaing.
is an enbeddinq, then f a) L e t
Prcwf: A
f:z-+x & g:x+z
q_ is separated, gf
.
be a separated mrphism (resp.
Then Ker(g,y) is closed (resp. locally closed)
g t
generated by
is closed i n Sp A
Ker(u,y)
f:g+x is separated i f g
mrphism of schemes) @ g,y:&
b)
A
infer that an affine scheme is separated.
Propmition:
7.6
%(Coker(a,&)) ,
a, 9 0)
Ker(g,y) = K e r @
I%
a closec3 embedding, it follows that
,Xix+Xz_Y
Y/_z
is
f I s the canposition of the two closed
.
80
I,
Fu;EBRAIc GEmEmY
embeddings Lennna:
h
ard
(sf), . The proof -
For each mrphisn of schemes
U_
Let
g a d
V
, the diagonal
?:?+:
irduced by
5
6x/y:
_ _
vary through the affine open subschemes of
f&J)'v . Then
satisfying
2 , no 8
of the second assertion is similar.
is an embedding.
Proof:
9
6i;y(i]$U)
=
I! and the morphism
U_
X x X
-t
- -Y -
X_
and
+
V$V_ -
coincides with-the nkphim induced by the map aQDb +a.b
.
into o ( U ) Since this map is surjective, (6.10) implies of d ( U ) @d(viG'(U) that 6 induces a closed esnbedding of X into the open subscheme of
$3
X 2 x which is covered by the Corollary:
7.7
_Vcg . -
The canpsition of two separated mrphisms is s e p -
rated. Proof:
Suppose
f:z+x
are separated. Let A be aRKdel an3
and g:Y+
X_ a double arrw such that gfu - _ - = gfv ___
9A
. Then
Ker(fg,_fir) is closed i n Sp - A since g- is separated. Since f is separated, it follows that Ker(y,v) is closed i n Ker(_fu,fy) Hence Ker(y,v) is closed i n (g,y) :
.
Example:
7.8
The projective s p c e
the flag schfsne FJrll..
.'rS)
En , the
q r a s m i a n Gnrr
&
are separated schanes. For the "diagonal" of
coincides with _F (r,r) , hence is closed by 6.3. Since a n product of separated functors is separated, G GrS,n-rs is s e p - r i ,n-rl rated. By 7 . 7 and 7 . 4 , a subfunctor of a separated functor is separated. The Gr n-rx Gr ,n-r
'..*
assertion therefore follows by 6.5.
Section 8
An affineness criterion for schemes
8.1
Affinity theoran:
Let - X
quasicoherent sheaf of ideals of defined by Proof:
4,
i s affine, &en so is
F i r s t recall that
structure sheaf i s
be a scheme and l e t
. If 5
.
7
be a nilpotent
the closed subscheme v ( 2
of
X
v(7) has the same space of p i n t s as ,X and its
Jx/g . Since
is enough t o show by -auction on
van)= _X n that
for sufficiently large n
v('Yn)
is affine. N c r ~ _ V ( f )
, it
qT(!fn/r"fl)
coincides with the closed subscheme
has vanishing square, we m y assume straight-away and suppose provisionally t h a t the map phism -f:Y+X - -
_V(yn+') that 7 2=O
of
. Since Yn/,f+' . Set
y=v(!f) ,
J(f) irduced by t h e inclusion mr-
is surjective. Since K e r f l ( f ) =
q ( X ) has vanishing
square,
Spec o ( f ) induces an i s m r p h i m of the underlying t o p l o g i c a l spaces of
spec
J c ~ )&
If1
+lYl
spec
3 ( ~.)NOW
and Spec &-f )
i n the carmutative square
are a l l hmecmorphisms,
(5 1,
2.2)
therefore so is
$J
121
.
is an i s m r p h i s m , it remains to prove that 1x1 and 1x1 have "the same" structure sheaf, t h a t is to say, that the canoniSpec &(XI - . This follows f r a n is b i j e c t i v e f o r each s€ d(X) cal map &X_)s + &Xs)
+
To show that
the diaqran
1 is
where since
,
d u l e s over
a f f i n e and CI
7
may be regarded a s a quasicoherent sheaf of
and y are bijections. Finally
d(_fs) and
'o(f) are
surjective. W e prove the latter contention, the proof of t h e former being similar. For each
s€&)
, write
zs
f o r the open subscheme of _X which has the
7(ys) , regarding 7 as a quasicoherent sheaf of mdules over Y . If a d ( g ) , there is a partition 1 = ~ ~ = l x i s iof unity i n d(y) and elements aitdx(zsi) same space of p i n t s as
whose images i n ailgsisj family equation
3'
8y (Y-si )
- a 3 ( 5Sisj I
ys
;t h a
g(gs)
may be identified with
are the r e s t r i c t i o n s a/Ysi
i s a section of
7
over
Also, a
~ s i n ~ s =j zsisj
ij Since the
.
is obviously a 1-cocycle of 7 f o r the mering (ysi) , the 1 H ( ( IY . I ) ,y ) = 0 , established below, implies that there are S1 It f o l l m that t h e restricsuch that aij = bi I Xs. - b j I
( a . .) I '
biE (5 , ) S1 tions of ai-bi
and aj-bj
to
zs.s1
l j
zsisj .
are the same, so that there is
a'E4(X) such -
Lama: - Let
8.2
. Hence
a'lXsi = ai-bi
that
be a ring,
A
d(z)( a ' )
= a
A-module,
M
.
. n 1 = liz1xisi
5
p a r t i t i o n of unity in A , u t _Y = Spec A Then the w h m l c q y qroups M w i t h respect to the covering (Y s a t i s f y H*((Y ,MI M __ and i si si H ((Y ),M= 0 i f i>O
o-f
.
si
It suffices to show that the sequence
Proof: 0
.+
G(Y)
+
Ijrjz(ysiny
.
57
.+
?-?-$Y
l r 7 ,
associated with M and the m e r i n g
(Y
s1
the same as the one obtained by s e t t i n g C=A
Section 9 k
.ny .m s1 53 Sl
...
i s exact. But t h i s sequence is
,
B
=psiin 5 1,
2.7.
Transporters
denotes a model throughout t h i s section.
.
ard each xcX_(R) This enables us to associate w i t h
f o r each RE$ arrow
g: -
X_xy -+
_Z
such t h a t
and yEY(R)
.
9.2
Corollary:
to the k-functor
Tf
R -+&E$SpkR)-
~
( g ( R ) ) (x,y) = ( f x ( R ) ) (y)
-
l
~
~
x Y , 2)
, mH?@,_Z) & ~
.
f o r R€w%
,
an xEX(R)
is canonically isanorphic
I,
5
2, no 9
Fxanple:
9.3
5
t h i s and
i s canonically iscmorphic
,
H?(%A,
in view of the canonical isamrphisms
1, 6.6 we deduce the existence of a canonical i s m r p h i s n
1, we arrive a t the following criterion: is a scheme i f the following three conditions hold:
H-%(SSA,_Z)
Z is a
is a f i n i t e l y generated projective k-mzdule, and f o r each f i n i t e
A
subset P PClLJI
z)
If AC?m
5
By Prop. 6.6 of
schene,
83
R * (R BkA)
to the k-functor
man
z
QUASI-COHEXENT-IWDULES
z
of
there is an affine open subscheme g of
2 such that
*
Definition:
9.4
a mnmrphisn,
Let p: X g Y _
and E': _X
-+
H 5-
Cy,_Z)
+
_Z
-%% ,
&:J'-+z
the mrphisn canonically associated
with - p by Prop. 9.1.
The t r a n s w r t e r of
written
, is
TranspD(Y,Z')
be a morphisn of @
2' relative t o 2
,
the pull-back of the diagram
(Transp ( Y , Z ' ) ) (R) m y thus be identified w i t h the set P- ~ ( Modulo ~ , of arrows g : SJ+ R -+ X such that p'g factors through ~
For each RE
$
.
1
prop. 9.1, the existence of such a factoring means that the canposition
factors through
Propsition:
9.5
and
z' . Let -
i:Z'-+z --
b e a c l o s e d erkdding gf k-functors
Y_ a locally f r e e k-schaue, that is to say, a scheme having a covering whose algebras of functions are f r e e k d u l e s . Then
by affine open
Hca(Y,i)
xi
: H?(Y,_Z')
-+
s a H&(:,_iZ ) closed
epnbeaaiq.
~
)
Proof:
Suppose first t h a t _Y = 5 % B where B is a free kmodule. By 9.3,
we must s h w that the canonical mrphism
mZ1
B/kB
-+
7 7 2
Bk-B
i.e. t h a t the following condition is satisfied (6.1): each a$m$B(A) = g ( B @ A ) , there is an ideal I of each hQtKmrpkism iff
Z'(B8.R)
,Mk , the element aR of Z(B@R) g'+z is a closed gnbedding, so i f
of
$:A*R
. Now
$(I)=O
are as above, there i s an ideal J of abare,
aR belongs to z ' ( B @ R )
is a sMlleSt ideal
and J c B @ K e r 4
of
I
is a closed embedding, for each A€-% and A such that, for
iff
and a
A
B @ A such that, €or each $
3
J
as
is free, there
(B@$) ( J ) = 0 ; since B
such that BE31
A
belongs to
. The conditions
I c Ker $
are accordingly equivalent, which proves the
= Ker(B@$)
first assertion.
In the general case, by 6.1 it is enough to show that, for each p: S&MxY where ME&
I
T_ = Transp (Y 2 ' )
P
-I-
is represented by a quotient of
M
. If
-+z
-pi denotes the restrictioi of p to the affine open subschgne SpkMxYi of S a M x Y , Ti = Wansp (xi,g') is represented by a quotient M/mi of M , by M/cimi and :Ti the morphisn q: nT.x - i-1
Ei
. It is nuw sufficient t o show that
z
T=
QT. , i.e. t h a t
.
1 -l
irduced by p factors through 3' But this follows frcm the fact that q-'(Z_') is a closed subscheme of ? z i r Y con-
taining each
9.6
(nT.)x
i -1
+
xi
Corollary:
1
If k
is a field,
closed embedding of k-functors, L?(y,&) Corollary:
9.7
With the notation of def. 9.4,
-p
Corollary: If
free k-scheme, H-(_Y,x)
Proof: By
5
1, 6 . 3 ,
is a closed embedding.
Lf:
_Y
a closed embeddinq, Transp (Y Z ' )
free k-scheme and 9.8
y is a scheme and _i:g'-+z 9
X
-I
-
is a locally is closed in _x
is a separated k-functor and
.
is a locally
I s separated.
X
is separated i f
,g is separated. By 7.4, 7.5 and * X -a k X-g-
7.7, this is the same as saying that the s%xctural projection p
is separated. Naw apply prop. 9.5 to the diagonal mrphism of
-
X -into
&" 3
.
5 3
AIx;EBmc S C r n E S
Section 1
F i n i t e l y presented mrphisms
1.1
Definition:
A-algebra
B
is s a i d to be f i n i t e l y presented
i f it is isamorphic t o the quotient of an algebra of plynanials
.
by a f i n i t e l y generated ideal I
over
, each
-Z
ideal of
.,Xn ]
AIXl,..
.,X n ]
is a f i n i t e l y generated algebra
is Noetherian, i n p a r t i c u l a r when A
When A
AIX1,..
is f i n i t e l y generated, so that an
A-algebra is f i n i t e l y presented i f f it i s f i n i t e l y generated.
Laam: -Let
1.2
. If
limit A
B
c1
a
is a f i n i t e l y presented A-algebra,
Ad f i n i t e l y presented A @A B,
be a directed system of rings, w i t h d i r e c t
(A )
.
A -algebra
S u p p s e that the ideal
Prcof:
B
is iscanorphic t o
def. 1.1 is generated by the plyncanials
of
I
.
such t h a t
B,
c1
there is an index a
P1,...,P r Choose a so that the image of Aa i n A contains a l l the coefficients of the plynanials Pi I f Q1,. ,Qr EA,[Xl,. ,Xn) are
.
mapped onto P1r...rP
r and generate the i d e a l I,
For example, w e may take A
. I t follows that
B
i f f there is a f i n i t e l y generated subring A.
f
generated Ao-algebra
of
A
Bo such that B
AmA Bo 0
1.3
Lama:
a)
If
f i n i t e l y presented wer B b)
If
$:BK
(b)
A
and a f i n i t e l y
is f i n i t e l y presented wer A and i f
,t
h s C
is f i n i t e l y presented wer A
C
.
of
B
i s f i n i t e l y generated.
These assertions follow e a s i l y fram lemna 1.2. For example, w e p r w e
. Let
A.
ad. Co a f i n i t e l y W i t h the ahme notation, l e t
be a f i n i t e l y generated subring of
p be t h e canonical projection of A,
.
is f i n i t e l y presented
B
generated Ao-algebra such that If
= A,[X1,...,Xn]/I,
is a surjective hcnmnorphism of f i n i t e l y presented A-algebras,
then t h e i d e a l K e r 4 Proof:
, set B,
..
to be the system of a l l f i n i t e l y generated
(A,)
subrings (i.e.A-subalgebras)
over
..
C
A@
4 3
AIXl,..
.
.,Xn]
is s u f f i c i e n t l y large, there are ci €Co
A
onto B = AIXl,.. such that
.,Xn]/I .
1@
I, 5 3, no 1
AzlGEBRAIc GEDMEma
86
A0
si = @(p(Xi)) ; if
generate Co
Tllr...rIls
over .A
, we have p l y n d a l
relations
oj
lA@
with coefficients
vj
'
.A
A
; thus,
= Q.(1 @ 5,,...,1
0
8
.A
5
)
is sufficiently large, we have
if .A
.
that the ti generate Co Also, if Plr...,Pr generate the ideal I , we have Pj(El,.. ,En) = 0 for sufficiently large .A Under these conditions, the hcnmnorphim of Ao[Xlr.. ,X 3 onto Co n which sends Xi onto factors through a hcnmrorphisn @o of Bo = = Qj(51r...rEn)
r
SO
.
.
ci
...,
.
.
Ao[X1, Xn]/(Plr...,Pr) onto Co Therefore @ = Af$o and Ker @ 0 image of Af ( K e r @o) in B , and. is hence finitely generated.
is the
0
LamGl: - Let A' be a faithfully flat A-algebra. Then an A-alqebra
1.4 B
is finitely presented iff A'@AB
is finitely presented over A'
.
proof: The latter condition is obviously necessary without restriction on A' ; we shaw that it is sufficient. Let B'
run through a l l the finitely
generated subalgebras of B ; then we have 1 9 A'@ B' =A'@ lim B' A A +
A'@ B A
,
so that, since A'@ AB is finitely generated over A' , A'BAB' =A'@ AB for scme subalgebra B' . The assumption that A' is faithfully flat over A then implies that B' = B , so that B is the quotient of an algebra AIX1, ,Xn] by an ideal I Let I' run through all finitely generated ideals contained in I . Then we have
...
.
A'f B = A' [Xlr. m. rXn]/ (A'g I) and A'aAI is finitely generated (1.3)
,
. Since
A'@ I = 1 9 A'@ I' A A we have A'BAI
=
A'@ I' for sane I' A
, whence
I = I'
.
1.5 Lerma: - let B be an A-algebra and let 1 = lxifi be a partition of Unity Of B If Bfi is finitely presented over A for each i , then - B is finitely presented over A
.
.
Proof: Let Bo run through the finitely generated A-subalgebras of B containing
the fi and the xi. The equalities
87
lhB
Ofi
+
= B hply fi
Bofi
for sane Bo ; hence
= B
fi
that B of
.
TTB This implies is faithfully f l a t over B~ , we b v e B=B, i o f :I is of the form A [ X ~ , . . ' . , X , J / I If Qi , Pi are representatives
and, since
.
xi, f i
i n AIXl,...,Xn]
, then
1 - IiQiPi
and i f
..,Xn ]/I ) P i
(AIXl,.
is the ideal generated by
I'
is f i n i t e l y presented over A
I
(W')
. By
of the map ( A [ x ~ , . . . , x ~ ~ / I ' ) B is Pi Pi f i n i t e l y generated f o r each i , hence so are I/I' (Alg. cam. I I r f $5
h m a 1.3, the kernel
-+
no. 1) and I . 1.6
Definition:
f i n i t e l y presented
y of X @a
V_
llf,
of
A mrphism of schemes
for each p i n t such that
_Y
XCLJ
a f i n i t e l y presented algebra over f
x@
,
.
f:z-+x is said t o be locally
, there
are affine open subschemes
f(x)tV_ , f(g)CV
O(L$
(!I(!)
i s said to be f i n i t e l y presented
if
f
is quasicarrpact, quasiseparated
and l o c a l l y f i n i t e l y presented. 1.7
Proposition:
If
@:WA
is a mrphism of models, the following
assertions are equivalent. (i)
A
is a f i n i t e l y presented B-algebra.
(ii)
S J
@
Sp 4
(iii)
Proof:
By
5
is locally f i n i t e l y presented. is f i n i t e l y presented.
2, 2.2 it is clear that
(ii)<=> (iii)
trivial. W e prove (ii)=> (i); set 3 = If
Xr
,V and
9A ,
. Also
= a B
(i)=> (ii) is
and f = @-@
are as in 1 . 6 , there is a t E B such that f(x)€YtC V
.
.
Thus hence
8 ( g N t ) ) is f i n i t e l y presented over o(y),
Now substitute g
4(t)
; w e may then assume that
for
of 1.6. I n this case, there is an
d ( z , ) = B ( t ~ ) ~ = A, ~ and
As
= Bt
zs ,
dlso over
B
.
v=Y_ in the notation
SEA such that xQs'=u ; whence
is f i n i t e l y presented over B
with f i n i t e l y many of these
, so
. By covering
(i) follavs fran lami 1.5.
2
88
z4LGEBRAIC GExmFTW
5
3 , no 1
L e t f:X+Y be a locally f i n i t e l y presented mrphisn, affine open subschews of X_ and _Y such that f(U')cV -' Corollary:
1.8
and V'
U'
men - @(u') Proof:
-f
I,
By 1.7, it is enough t o show that the morphisn
t@(Y)
there is a
u' ,
~, f_(x)
Corollary:
1.9
iff
CpV'
S E & ~ ) such that
) ~i s f i n~i t e l y presented over I1(gs) yt
and
c
f_(gs)c yt
A mrphisn of schemes
that:
, there
a)
for fixed i -1 f (Xi) ;
b)
for each t r i p l e ___.
c)
for each pair
Proof:
. Then
x e chosen as i n 1.6,
.
are only f i n i t e l y many
( i , j , k ) , X nx -ij - i R fi,j)
, d(Xij)
LWI) , and we
$:z-+x is f i n i t e l y presented xi
be covered by affine open subsch&s
ard X
flyi)
v,V_
induced by
us c g-l(yt)nv_nv_' .
O(us) = 9
have xE gs c
and i f
f':g'+y'
. Moreover, there is an
d(yt)
i s f i n i t e l y presented over
E
xtu'
such that f(x) E yt
~ ( g - ' ( y ~ ) n g1)
Since
J(v') .
is a f i n i t e l y presented algebra over
is locally f i n i t e l y presented. If
.
zij
I
zij
and they cover
i s quasicanpact;
i s a f i n i t e l y presented algebra over
I
This follows imnediately fran 1.8 and
5
2, 2.2.
.
o(yi)
is a noetherian ring, so is O ( X . .) by c) In t h i s case the -11 underlying topolq.icdl spce of X j is noetherian. Each open subscheme of
If
zij
C?(Yi) 1.10
is therefore quasiccmpact, so that c o d i t i o n b)
is noetherian for each i Proposition:
.
is implied by c)
if
a) The canposition of two locally f i n i t e l y pre-
sented (resp. f i n i t e l y presented) mrphim is lccallv f i n i t e l y presented (resp. f i n i t e l y presented). b)
I n the diagram of schemes g
5 _Y
y' , if f is locally f i n i t e l y Pre-
sented (resp. f i n i t e l y presented) then the canonical projection
I, 5 3, no 1
gpx'
f,,
:
c)
If
-
-
ALGEBRAIC SCHENEs
+.
_Y' is locally finitely presented (resp. finitely presentd).
and
9.f
89
g are locally finitely presented (resp. if
is finite-
go$
ly presented and if g is quasiseparated and locally finitely presented), thLn
f is locally finitely presented (resp. finitely presented).
-
Proof: b) follms hediately fran 1.6 and
5
2, 2 . 3 . Assertion a) follows
fran 1.8 a d the fact that the camposition of t m quasiseparated mrphisns
f:&+x
g:x+z
is quasiseparated. For 6x/z: +. X g X is the canpsition -_ (which is quasiccanpact) and the inclusion morphism :
3 X ~+ _ X~X X xX x x + g 65 , which is derived fran
of ~
x-
-
-
$3 -fzx--f * 3 so is
1~ -gy
-+
1; by the "change of base" This inclusion Ghhism i s therefore quasicanpact, and
.
6_x/E '
It remains to prove c) g - € E h (y,z)
,
f
6y/z:
_Y
-+
. We prove the "resp." part.
If f € E h (8,Y)and
is the canpsition
and . Notice that, since gf '4i (g<)y. By a) , it is enough to show that h
where & has canpnents presented, so is
L
x/z
presented. Now h is derived fran 6
*
Y_
+
is finitely is finitely
y%Y_ by the "change of base"
fxy : 3 % ~ Y ~ .Y since 6 -(it's a mmrphism!) -zx/z_ is quasisepited is locally finitely presented. and quasicanpact, it r m i n s to shm that %/_z For t h i s purpose we may assume that Y_ and z are affine. In this case, we must show that the kernel of the canonical map of o(y)@ 0(x) onto 8 em -+
is finitely generated; and this is clear, since it is generated by the
-
gi@l
l@gi
r
where
(gi)l
U(Z) * Proposition: Consider the diagram of schemes _X
1.11
9 Y' ,
- where g is faithfully flat and quasicmpct. If the canonical projection fyl: ZgE' + y ' is locally finitely presented (resp. finitely presented),
then f
is locally finitely presented (resp. finitely presented).
Proof: One easily shows that if so is f
if g,,
-
g,,
i s quasicanpact and quasiseparated,
(cf. the arq-ment of 5 2,-5.7). It therefore remains to show that is locally finitely presented, so is f To see this, let V_ and
.
& and Y_ such that f(tl)c_V , l e t (V_li) -1 be a f i n i t e family of a f f i n e open subschemes of y' covering g (41) By 1.8, o(Vix $1 is f i n i t e l y presented over fl(yi) f o r each i , hence
V
be a f f i n e open subschanes of
.
-
TU(!ll)
is f i n i t e l y presented over
cl(v_)
over
.
1.12
A mrphism of schemes
generated i f , f o r any point
g and 2 of y , such f.
u(v)
is f i n i t e l y presented
is said to be locally f i n i t e l y
g:x+Y
, there are a f f i n e open subschemes x€y! , f(x)Ey , f ( 1 ) C V and (I(!) is
of
xEg
that
f i n i t e l y generated algebra wer if
and, by 1.4,
#(v) . We say that
a
f is f i n i t e l y generated
is quasiccanpact am3 locally f i n i t e l y generated.
I n statements 1.7, 1.8, 1.10a), b) and. 1.11, " f i n i t e l y presented" may be replaces by " f i n i t e l y generated", whereas by statements a ) a)
and. b) below.
?:&+!
A mrphism of schanes
ings of Y d i t i o n s all
and
X
hold:
f o r any i
a2) f o r any
, (xij)
(i,j)
algebra over (3(yi) b)
If
~gaf
.
and
zij
such that con-
f -1
is a f i n i t e m e r i n g of
(l(xij)
(xi)
,
is a f i n i t e l y generated
. If
gag is f i n i t e l y
is quasiseparated, then f is f i n i t e l y generated.
Algebraic schemes
Throughout the rest of 2.1
,
xi
is locally f i n i t e l y generated, so is f_
generated and g -
Section 2
is f i n i t e l y generated i f f there are w e r -
by a f f i n e open subschemes
and a,) al)
and 1 . 1 0 ~ ) should be replaced
1.9
5
Definition:
3,
k denotes a &el.
A_ k-scheme
5 is said t o be locally k-algebraic
(resp. k-algebraic) i f the structural morphism jX: f i n i t e l y presented (resp. f i n i t e l y presented).
-
zz -+ SJ
m
k
is locally
5
I,
3, no 2
f:&+x
Any mrphism of k-algebraic k-schemes
-g:Y'+Y - - is a
If
91
Au;EBRAIc SCHEavIES
is f i n i t e l y presented by 1 . 1 0 ~ ) .
second mrphism of k-algebraic k-schemes, the pull back
i s a k-algebraic k-scheme by 1.10a) and b ) , since the structural pro_X$y' jection
is the c a p x i t i o n
px yl -v-
-
Hence a f i n i t e inverse limit of k-algebraic k-schemes is k-algebraic. The same r e s u l t holds f o r locally k-algebraic k-schemes.
is k-algebraic i f f
By 1.9, a k-scheme
ing
7
has a f i n i t e affine open cover-
are quasicanpact and (!?(X.)
such that the 2 . f l X
(X.) -7
presented over k
5
. If
-Q.
f o r each j
k
-7
is f i n i t e l y
is a ncetherian ring, the k-scheme
3 i s locally k-algebraic (resp. is k-algebraic) i f f the structural mrphism
-px_ : ,$
+
Sl k
is locally f i n i t e l y generated (resp. is f i n i t e l y generated) ,
that is i f f 2 has an affine open (resp. open and f i n i t e ) covering
(Xi)
is a f i n i t e l y generated k-algebra; i n that case,
Spec d(gi) is a noetherian topological spce; hence each open subset, and i n particular
such that
xifl g
-
3
is autcmatically quasiccmpct.
3 is locally
Each open subscheme Y_ of a locally k-algebraic k-scheme k-algebraic. I f , i n addition,
3 is k-algebraic, k-algebricity of
is
equivalent t o quasicanpctness. By abuse of language, we shall sanetimes confuse "algebraic" w i t h
ic"
.
Proposition:
A k-scheme
system of k-models j ective
Proof: B
.
(A,)
5 is locally k-alqebraic i f f f o r any directed
, the
canonical map 1 9 X(Aa) + X(l$n A,)
map
l+ I&(B,A,) + I&(B,l$n A ) Q f a c t is entrusted t o the reader. Suppose now f i r s t that the maps A
0,'
(Aol)
the
is injective. The proof of this w e l l known
@ :
1 9 X(Aa) +
jective. L e t _V be an affine open subscheme of
l$n
is bi-
When X - is affine, the proposition reduces t o the f a c t that a k d e l
is a f i n i t e l y presented k-algebra i f f for any directed system
A =
"k-algebra-
and denote by -p, : Sp -A
+
Sp - Aa
z
(19Aa)
, set
and paa
are always bi-
B = d@) : cp AB
+
, Sp A,
the
Au;EBRAIc GEOMETRY
92
1,
5
3, no 2
t r a n s i t i o n mDrphisms. By hypothesis any f:B+A is iraduced by some -1 -1 g : @ Aa * 5 r and t h e r e l a t i o n -A implies the existence pa (2 (UJ) = Sp of a p a r t i t i o n of unity 1 = Cxiyi i n A , h e r e the yi are the m g e s of -1 s a w elements i n Aa vanishing outside g (LJ Such a r e l a t i o n must exist -1 -1 already i n A for sane B > a Hence we obtain -pclB(g(V_)) = S&A B B ' which means that f : W A is induced by scme f':B+Aa , or i n other wrds that
.
.
the map lpn_%(B,Aa)
+
i s surjective. As &(B,Aa)
I&(B,A)
are identified w i t h subsets of Hence
B
am3 Z(A)
_X(Aa)
, this
and
&IM(B,A)
map is even b i j e c t i v e .
5 i s locally k-algebraic.
is f i n i t e l y presented and
X is locally k-algebraic. W e f i r s t prove that $ grs_ : 9 A, 5 be t w o mrphisms such that --a fp = .
Conversely, suppose that is injective. L e t
.
W e w a n t t o p r o v e that --up f p = %a@ for sane B 2 a By taking sane p a r t i t i o n of unity 1 = Cxiyi in A it is easy to reduce the proof t o the case, i n a f and g are factored through a f f i n e open subschemes g and V of which -1 -1 X Then we can prove as above that p (g ( U ) ) = 9 AB f o r same 8 2 a -
.
-aB
. AS
we may therefore suppose that g =
k-algebra, the maps u ( f ) , d ( g ) : d(U) For such a
B we obtain zpaB= gpaB
Consider f i n a l l y a m r p h i s n @ :
1 = Cxiyi
of
A
9A
d ( v ) is a f i n i t e l y presented are equalized by sane Aa + A B .
Aa
-+_X
-hi
Axi
:
+
.
as A l$(Aalxi r xi the f i r s t part of t h e proof,
Over'
hi
Let
limit k . I f r f o r m e index
-xi
of
hai and h
(ka)
5
such t h a t
zi . For s u f f i c i e n t l y large still denote by xi
i s induced by s m
large a an3 thus define a mrphisn Corollary:
There e x i s t a p a r t i t i o n of unity
, which we
i s t h e image of sane elanents i n Aa
.
-
and a f f i n e open s u b s c h m s
is factored through scme
2.2
.
-
kai
coincide on
& : @ Aa * X
Ax
,
a
xi
. mre-
zi . By
: S i A 7
ax1
%(Aa)xixj
such that
Fapa
for
= h -
.
be a directed system of mdels w i t h d i r e c t
a i s a locally ka-algebraic scheme and a quasiccmpact, quasiseparated ka-scheme, t h e canonical nap X
is bijective.
i
x (resp. of Zinx. 1 -7
. Set
= Q(x)
and
=
O(gijk)
. me assertion
follaws from proposition 2 . 1 by taking d i r e c t limits i n the exact sequence
2.3
let
W i t h t h e assumptions of 2.2,
Corollary:
of algebraic ka-schemes. ~- If
h@
k
kol
_h:X-ty be a mrphism
i s i n v e r t i b l e (resp. a m n m r p h i m , an
B > a such t h a t &
open embedding, a closed abddinq), then there is
y B.
is i n v e r t i b l e (resp. a mmrphism, an open embedding, a closed embedding) Proof:
kor h '
@ :f
:
f o r s u f f i c i e n t l y large y If
h @k kcl
mrphisn
B is s u f f i c i e n t l y large, by 2.2 there
hC3 k is invertible and
If
is a mrphism
+
X C3 k
h ' f k = (h f k1-I
such t h a t
ka
. Since
.
is a mmrphism, it i s enough to apply the abave to t h e diagonal
&z/y
:X + -
5xX y_-
(which is i n v e r t i b l e i f f
& is a monomorphim)
.
Now suppose that h L&ko
k is an open embedding. By t h e above argument, % is a mnmrphiSm f o r s u f f i c i e n t l y large D ; we may thus assume
straightaway that X_ is a subfunctor of , h being t h e inclusion mrphism. &t (Y.) be a f i n i t e a f f i n e open covering of Y_ If " g k is open i n -1 for each i there are functions fimR, ...,firi E fl(,Yi)% k such that
.
is the union of the open subschemes the
fij
(yi
Ck)Eij
of
yi
subschme of
C3 k
kolB
covered by the
. This -lies
When _X f&k
that
is closed in ' g k
kernel of t h e canonical map
. For s u f f i c i e n t l y l a r g e
d ( x . ) @k lkcl
. Let
-1
,
gij
B
g f&ky i s open in '&kY
, we
E
5 be the open
. Since z C3kgk = _X fLk , 5 f' k y = 3 &5 f o r s u f f i c i e n t l y
(Y. @ k )
by t h e f i r s t part of the proof we have large y
&k
are the images of functions gij E
choose
fia.,...,f*i
-
to generate t h e
Let
-Ti
be the closed subscheme of
the quotient
(4XJ
f ka)/(gill a
xi%k &Iir
whose algebra of functions is
I
being define3 as b e . W e have
the gij
for sufficiently large y
. Hence
all i
is closed i n
, so t h a t
_X
&p
Proposition:
2.4
direct l i m i t k
Let
Let
(&nu 1
k is closed in Xi 4 Y 6-10). yi Eky ( 5
T -a
k
for
21
(ka) be a directed system of models with
. For each algebraic k-schane
an algebraic ka-schm Proof:
)
i
B
T
there is an index a and
is i s m r p h i c to
such that
(Ti) be a f i n i t e affine open covering of E d(Ti) be functions such t h a t _TinT
T
5
k a
.
; let
fij is covered by the r(iljli j open family (T 1 ij , 1 5 R 5 r ( i , j ) By 1.2, for sufficiently large p fR .. there are algebraic k-schemes z i and functions g? such that ‘1 fl
Let
I...,
, .
Ti7
.
be the open subschm of
_Ti
a
By 2.2 and 2.3, for sufficiently large
such that
+ijf k a
01
there are i s m r p h i s n s
is the identity of
If a. is sufficiently large,
for which
covered by the open family
+iJ
induces an i s m r p h i m
(Ti) f3
ij
911
.
I, § 3,
3
for a l l
(i,j,k)
where IT;
z;
ALCZGBRAIC SCHEMES
95
. I t is then sufficient to take fox
Ta
the k - s c h m
is the spectral space obtained by matching together the
$ta1
along the open subspaces
2.5
Corollary:
generated subring ko of i s i s m r p h i c to
X@
Qko
If _h &k
such that the
so is
h%kB
a finitely
01
.
f3
5 &k
i s an inclusion mrphism (2.3) ; since
(5
g k ) n xi
(xi)
of affine open s u b s c h m s
are closed i n
. The open subscheme z of
Z'
.
Xo such that &
f o r sufficiently large
xi
for each i
xi
cuvered by the
&k
algebraic k-scheme. For sufficiently large
B
I
via the ismrphisms
Le t h:X-+x be a mrphism of algebraic k -schemes. --
i s q u a s i m p c t , there is a f i n i t e family
k -scheme
I
.
Proof: W e m y assume t h a t
%k
01
k ard an alqebraic ko-scheme
is an a k d d i n q ,
in cover 3 &k
k
A k-scheme 5 is algebraic i f f there is -
k
Corollary:
2.6
-6 B,
,
ST;
I
I
and
is an
there is an algebraic
such that
PBk
;
furthemore, for suffiiciently large /3 there are m r p h i m s
such that
h2hl
gnbedding,
b2
(2.2); for sufficiently large
= h_ &kB
B
,
is a closed
an open fmbedding by 2 . 3 .
Section 3
Constructible subsets of an algebraic schane. F l a t mxphisns.
3.1
Let
a subset
U
of
X be a topological space. W e shall say provisionally that
X has the property C i f the intersection of
quasimpact open subset of
x
is constructible i f
where U
P
X
U with each
is quasiccmpact. W e say that a subset P of
is a f i n i t e union of sets of the form
and V are open subsets of
X with the property
constructible set clearly has the property
C
, we
C
un
CV
,
. Since any
see that an open subset U
ALGEBRAIC GM=METwI
96 of
x
is constructible i f f
U
1,
has the property C
5
3, no 3
.
it follcws that 1
1'P h
so t h a t
i s a f i n i t e union of sets of the form
v . n...nvj n(:(u. u...uui ' 1
Hence i f
S
l1
is constructible, so i s CP
P
structible subsets of
r
)
.
. It fo1h.m
that the family of con-
X is closed under f i n i t e union, f i n i t e intersection
and canplanentation. X i s quasiccanpact and quasiseparated, the constructible
If
3.2
open subsets of
coincide with the quasicanpact open subsets. Lf X
X
&
a noetherian space, t h a t is, i f each family of open subsets has a maximal mEmber (under inclusion), then every open subset of
follows t h a t the constructible subsets of
is constructible. It
X
X a r e t h e n p r e c i s e l y the f i n i t e
unions of locally closed subsets. Proposition: X
The following conditions on a subset
P
of a noetherian space
are equivalent:
(i)
P is constructible.
(ii)
For each irreducible closed subset F
i s dense i n F Proof:
such that PI7 F
X
contains a non-empty open subset of
F
.
(i) => (ii): Suppse that P =
where Pi PinF
,P
of
u
P
lssn i
is locally closed in X
f o r each i
is dense in F f o r a t least one
l o c a l l y closed in F
closed i n F subset U of
. Since F
.
, hence of the form Ti = F , we have K
. If
is dense i n F
P nF
i ; for such an i UnK = F
where U
and pi
,
PinF
is
is open and K
contains the open
(ii)=> (i): Since each decreasing sequence of closed subsets of
,
X
is
is
(ultimately) stationary, we may argue by noetherian induction by assuming that the implication (ii)=> (if holds w i t h i n any closed set s t r i c t l y con-
tained i n X P
.
If
X = AUB
is reducible, w i t h A and B closed, and i f
s a t i s f i e s (ii), then P ~ A and P ~ Ba l s o s a t i s f y (ii); i n this case
PnA
and P n B are unions of sets which are locally closed i n A and B
, and so the
hence a l s o i n X
same applies to P
. If
P
i s not dense, apply the induction hypothesis to
P
is dense,
P
contains an open set U
. "hen
P-U
X
,
is irreducible and
. If,
on t h e other hand,
s a t i s f i e s (ii) , and. is
therefore constructible by the induction hypothesis. I t follows that P = U
U
(P-U)
is constructible. W e now apply the r e s u l t s of 3 . 1 ardi 3.2 t o t h e gecmetric real-
3.3
lgl of an algebraic k-scheme _X
ization
. Since 11
is quasiccanpact and
quasiseparatd, the constructible open subsets of
X_
(i.e. of
c i s e l y the quasicanpact open subsets. Furthermore,
if
k
rmdel, then
/XI
are pre-
is a noetherian
is a noetherian t o p l o g i c a l space. To prove t h i s , observe
t h a t f o r each a f f i n e open subscheme g of
&
, d(LJ
ated k-algebra, hence is noetherian. It follows that noetherian; since
1x1)
1x1
is a f i n i t e l y generILJl
is covered by f i n i t e l y m y such
= Spec d(V_) is
l_Vl , 151
is
a l s o noetherian. Propsition:
subset of
-f:Y+X --
_ Let_ -X
be an algebraic k-scheme and let P be a constructible
X . Then t h e r e
such that
P=z(x)
is an a f f i n e algebraic k-scheme
.
Suppose f i r s t of a l l t h a t
proof:
arid a m r p h i m
P
is the union of two constructible sub-
. Suppose a l s o that we have constructed two m r p h i m s such that P1= f1(Y -1) and P2=f2(!12) . Then P is f :Y +X and f2:Y2'+11 -1 -1 the image of the map underlying (f,,f,): _Y1U_Y2 X . Accordingly, we m y sets P1
and P2
-+
confine o u a t t e n t i o n to the case i n which P is of the form IYIn CI ; covering U by a f i n i t e open family (Fa) ar61 replacing P by the iga,lnP , we reduce to the case i n which g is affine. W e can then cover
uny
.-
by the special a f f i n e o p subschemes
fl,...,fn
E
O(V_) . I t
and to choose f o r
gfl,...,g
is now s u f f i c i e n t to set
g- the m r p h i s n
f
n
with
ilzduced by t h e canonical projection of
I,
Corollary:
3.4
Let
x
5
3, no 3
be a p i n t of an algebraic k-schane
let
P
be a constructible subset of
iff
P
contains each point y
X
.ms
such that x
X
P is a neighbourhccd of
is i n the closure of
(y}
.
x
Proof: The condition is obviously necessary. Conversely, suppose the cond i t i o n holds; we may then assume that % is affine; furthermore, since
is constructible, there is an a f f i n e algebraic k-scheme f:x+g such t h a t
P =
and a mrphism
. If
f(Y)
P
:9Jx+ x
EX
5 1, 5.6,
is the mrphism defined i n
the space of pints of
(s+, dX)xz~ is
anpty, i.e.
or 1 8 1 = 0 Q(X)
. Since
not v d s h i n g a t x
dx = l$n O ( _ X ),~as
, we
runs .throughthe functions
s
l@-l(x)l = 0 i n a t least one of the rings d(&)s@b(x)d'(x)Hence the undeslyi& space of -s X x Y is empty, which im37
.
/zs/
p l i e s that
have
(5
is contained i n P
1, 5.4).
Lemna: - Let A be a f i n i t e l y generated algebra over an integral d m i n B If..M is a f i n i t e l y generated B-module, M is free over 3.5
B 9
.
.
for sane 0 # g E B
Proof:
g
The following e l m n t a r y proof is due to D m e r . Clearly w e may
suppose t h a t A = BIT1r..
.,Tn]
generated by a single element m
is an algebra of p l y n a n i a l s and that M is (replace M i f
necessary by the cyclic
quotients of sane ccanposition series and take f o r g t h e product of the elanents of
v = (vl,
A
associated with these d i f f e r e n t quotients). For any
...,vn c,." X:,."-t$
.
/ u /= vl+. .+un
set
mii T' = T ; '
..T.:
. Further-
X ( u ) < X(u) i f +1 in t h e lexicographic ordering on $ ( / v / r V l , rvnf < ( \ L I / , ? . I ~ , . . . , V ~ ) is such that T E L T i , we then clearly have that A(u)
mre, l e t
...
be the bijection such that
.
implies A(utai) < X ( ~ + E ~ ) Furthemore, for each r=A(p)€l Mr =
1
BTUm and l e t
X (u)ir
I
u
be the ideal of
B
,
set
annihilating Mr+l/M r
.
( t h i s B-module is generated by the residue class of T’m) = T . 1 BT!Jm C T M = TEiM u 111 i h(u) A (U) MA (11+Ei) *
. W e have clearly
1 BT’+€im
Hence I
C I
.
P+Ei
, and mre
generally I
I
C
U
P
if
p< p
in the product
this order has the property that, for any subset S of , the subset so of minimal e l m t s of s is f i n i t e (otherwise take any
order of
#
u
! I ? But
i n f i n i t e sequence of d i s t i n c t elments i n So
and construct successively
i n f i n i t e subsequences such that the f i r s t canpnents increase, then also the second ones
O#g E
.... contradiction!). Take now
GIu.
Then g
Generic flatness theorem:
3.6
over a noetherian model and l e t
_f(z)
such that the imaqe
Say,
Proof:
f-l(V)
(Of course,
v
zi
M
g
is free.
be alqebraic schemes
_f:x+x be a dcaninant mrphism, that is to . m, i f is reduced,
onto V_
.
of
such that f
induces a faithfully
is said to be dense in Y_ i f the space of p i n t s of
is dense i n
subschemes
,
Let -
IXI.) Let Y1,...,Y r be (Alg. m. 11, 9 4, no. 3 , prop. 10). By
V_
. Thereforeg
is dense i n
there is a dense open subscheme f l a t mrphism of
such that
is mt free. Hence the B d u l e
is cyclic free or zero for a l l r
(Mr+,)g/(Mr)g
and
annihilates a l l the quotients Mr+l/Mr
i.e. such that Mr+l/Mr
A-’(r)fS
S = { p d : I,#O}
such that
the irreducible c m p n e n t s of replacing
-x
1x1
separately by open
and _X by the inverse images of these open subschenaes, we m y assume that 1x1 is irreducible. If we then replace _Y by a (dense) affine open subscheme, we may assume that = SAB ,
Igil
= Yi-UjziYj
where B is an integral damin. Let _Xl,...,X of
X_
; if
scme of the
zi
, w e may
assume that each
there exist mn-arrpty open V . C _Y into
vi
-S
be an affine open covering
we replace _Y by a smaller affine open subschem and suppress
-1
-xi
is d d n a n t over
such that the mrphisn frcan
. Thus i f
f-’(yi)tXi
induced by f are faithfully f l a t , then we may set y = niyi
are therefore l e f t with the case i n which g
. We
and f is defined by a m m r p h i s m $:B+A , which is necessarily injective (5 I, 2.4). To ocanplete the proof, it m w suffices to set &A
= SJ~A
in l a m a 3.5 abwe and take f o r _V
the special open subscheme Y thus taking the open subscheme X 7 ’ -4 (g)
-f - 5 y )
.
for
Ef
Corollary:
3.7
k
F:z+x is a daninant
is noetherian and i f
mrphism of algebraic k-schemes, then the h a g e g(5) contains a dense open subset of
1x1 .
Proof:
Apply 3.6
to the mrphism
) y \ and. lyredl
the t o p l o g i c a l spaces
lzrdl
, f(5)
contains the open subset
-xred"-Yred
constructed i n
and
3.6.
tobenoetherian.
Lf
mrollary:
c:x+x
constructible subset of Proof:
_ I
is a mrphism of alqebraic k-schemes,
2 , then
W e f i r s t assume, t h a t
of locally closed subsets Pi
k
the image f- (P)
of
X'
i s dense i n Y' , then f is the inverse image of
& a
is constructible.
i s ncetherian; then P
is a f i n i t e union
and we my assume straightaway that _X coin-
an irreducible closed subset of f@)nY'
P
P
cides with the reduced subschane carried by one of the Pi
If
1x1
are the same, as are
Ivl
. Since
induced by _f
In the three follawing corollaries, we do not assume the d e l
3.8 k
-fr&
1x1
. N w let
Y' be
; it is enough t o show that, i f
(5) contains an open subset of Y'
Y' i n
I _ x ~ , we
have
5' and g' are the reduced closed subschanes carried i s thus sufficient to apply Cor. 3 . 7 to the mrphism
(3.2).
f ( x ' ) = f(g)nYi . I f by X' ard Y' , it
f':_X'-+x'
induced by
In the general case, we m y assume that
(X,(= P by Prop. 3.3. By Props. 2.2 and 2 . 4 , there i s a f i n i t e l y generated subring ko of k and a mrphism of algebraic ko-schemes such t h a t f m y be identified w i t h f 63 k
-0kO
.
fo:$o+xo
If
is the canonical projection, we thus have
-f (6) = q-l(fo(zo) ) by 5 1, 5 . 4 . Since q --'($(%)) by def. 3.1. 3.9
fo(zo) i s constructible, so i s
Corollary: For each morphism of locally alqebraic k-schemes
f:Z+g , the follawing assertions are equivalent: (i)
-f isopen.
(ii)
For a l l y,y'EY
, such that
y
is i n the closure of
and each x€f_-l(y) there is x€f-'(y') the closure of { X I )
.
such that
x
{y']
is i n
1,
5
3, no 3
AIGEmAIc SCHEMES
For each xu(
(iii)
,
101
_fl induces a surjection of
Jx
Spec
df (XI Proof:
(ii)<=> (iii): This follows imnediately frcm
5
1, 5.6.
is an open quasiccmpact subset of
&
,
(ii)=> (i): If
V
. The assertion then follows f r m 3.4. (iii):If not, s e t y=f(x) . Let y ' be a pint of
is con-
f(V)
structible by 3.8 (i)=>
does not belong to the imge of the unit of
K(Y')@
affine, we have
where s ( x )
some s
I
JY
dX
Spec
dx . Then
+ o . I t follows that the u n i t of
i.e. Ifl-l(Y')nl_Xsl
so that y'ft f (gs)
,a
spec(^(^')@
is zero. Assuming, as we may,
=
dy which dx)= a , and
Spec
thaty OYand 5
are
~ ( y ' ) m J ( , ) J ( z ) ~is zero for
-
a
contradiction.
Corollary: A f l a t mrphisn of alqebraic k-schemes is o m .
3.10
Proof: With the notation of cor. 3.9, we must show t h a t the maps
dx Spec Jz(x) induced , hence faithfully f l a t dffollows f r m ALg. cam. 11, 5
Spec
-+
(Alg. conan. I,
5
dX
is f l a t over
3, prop. 9 ) . The assertion now
2, no. 5, cor. 4 of prop. 11.
Corollary: If k
3.11
are surjective. Now
by f
is a f i e l d and
canonical projection Erl: X_xy
-+
5,g
sk-schemes,
g i s open.
Proof: C l e a r l y w e may reduce the problem to the case i n which X=S&A X=S%B -
, A,BE$
. If
B
i s f i n i t e l y generated over
k
,
,
mkB is f i n i t e l y
presented over A and 3.11 follows frmi 3.10. In the general case it is enough to show that i f
sWkB
, then ~ ~ ( X x is y )open ~
has f i n i t e l y generated subalgebra Bo the i m g e of
(_Xrxo)s i n
1x1
r a i n s to show that the imge of
such that
in
1x1 . Now
s W k B 0 ; setting
B
Y =QB -0
0 '
i s open by our previous remarks. It thus (_X r
x)
and
(_X r
yo)
same. 'Ib see this, write p: ( Z X ~-+ )5 ~ and _po: ( ~ mrphisns induced by the canonical projections. For each
in s
-+
1x1
X~
are the
for ~ the )
~ € ,5we
~
then have
102
ALGEBRAIC c333ETm
I,
5
3,
3
110
since the canonical rmp (K (x)BkB0) + (K (x)akB) is i n j e c t i v e , the mrphisn -1 -1 p (x) + po (x) is daninant f o r each x (5 1, 2.4) This canpletes the proof. -
.
3.12
Corollary: For each model k a d each mrphism of locally
algebraic k-schemes
f:@x
, the
m i n t s xEx_
Proof: Evidently we m y assume t h a t
. Let
_X
, Ap
A such t h a t A is f l a t over k ; P' is f l a t wer A , hence also wer k
By 3.4 it is therefore enough t o show that vious remafks, U
{p)
PEU
. Hence,
if
, then u
Naw i f
U
.
P'
is constructible. By our pre-
contains a p i n t p E Spec A
if
U
-
, by
(k/q)s
meets t h e closure
is noetherian, it is s u f f i c i e n t (3.2) to show that if
k
contains a non-enpty open subset of {pl
q=c(p)
f l a t over
are a f f i n e , with algebras A
and
p' be a prime ideal of
for each prime ideal p c p '
-
is f l a t , form an
f
X.
ogen subset U fo
and k
, where
.
3.5 (or 3.6) there is an sEk-q
.
such that
is
Since we are assuming that
and since Torl(A,k/q) k is a f i n i t e l y generated Arnodule (see 3.13 below) there is a
If
p'
q'=Z (p' )
gEA-p
such that
is a prime ideal of
, we
A
such that
and s g f p ' , and if
p'+
have accordingly
gs' .
By lug. ccann. 111, 5 5 , th. 1 and and A p l / ~ l A p is l f l a t over kq'/ prop. 2, Apt is f l a t over k , which proves t h e corollary when k is q' noetherian. Now assume that k is not noetherian; choose p
over k a
.
is f l a t P W i t h the notation of 3.14 below, this r e s u l t implies that there is
t EAo-po
such that Aot
is f l a t over ko
Spec A
. If
t'
so that A
is t h e image of
t
I,
8
3,
in A
3
, then
xGEE3RAIc SCHEMES
p E (Spec A) t , and
proof i n the general case.
m: g t
3.13
k-alqebra,
k
Tori(M,N)
At,
103
, which
is f l a t over k
k be a noetherian ring,
A
completes t h e
a f i n i t e l y generated
a f i n i t e l y qenerated Amodule (resp. kmodule)
M (resp. N)
is a f i n i t e l y qenerated Amodule f o r each i
.
Proof: Since A is a quotient of an algebra of plyncmials
kCX1r..
.m a
.,XnI ,
3 , assume A t o be f l a t over k n k using an acyclic To prove the l m , we need only calculate Tori(M,N)
we m y , by replacing A by k k ll...,X resolution of
M
by f r e e f i n i t e l y generated A-modules.
Lemna:Let -
3.14
.
A be a f i n i t e l y presented algebra over a rinq k
.
such t h a t A is f l a t over k P there is a f i n i t e l y qenerated subrinq ko of k , a f i n i t e l y qenerated
ard let p be a prime ideal of ko-alqebra .A
A
and a rinq hammrphism $o : A.
for
md a E A o
+
A
such that
a)
@,(Sa) = <@,(a)
b)
the map A @ k + A induced bv Q0 is bijective ; O ko -1 if po=@o (p) , A is f l a t over ko OPO
c)
;
.
Proof: By generated
1.2, there is a f i n i t e l y generated subring kl
kl-algebra
A1
, and
of
k
,a
finitely
a hamamorphism $I~:A~+A satisfying con-
d i t i o n s a ) and b) above mutatis mutardis. Let ko be a f i n i t e l y generated subring containing irduced by of
@1
pI pol p1
kl
. Set
, set
.A = Al@Jklk0 ard l e t $o be the hamamorphism
p1 = +;l(p)
i n kl kor kl :
and l e t q,
so,
q1 be the inverse images
ALGEBRAIC
104
Since we have
R 0 = Tor (B ,R/q R ) = 1% 1 1
and s i n c e Tor;l(B1
1,
GEoMETIiy
5
3 , no 3
Tory (Bo,Ro/qlRo)
is a f i n i t e l y generated B l d u l e we m y , by 1 1 1 taking ko s u f f i c i e n t l y large, assume f+atthe canonical image T of R T o r F ( B 1 ,R1/qlR1) i n Torp(Bo , R 0/q lR o) is zero. I n this case., ko satis,R / q ' R 1
f i e s t h e condition of t h e l m :
is a presentation of t h e
R1-"u>dule
U@Ro
V@Ro
g2-B
-C
i s a presentation of t h e
B1 ' then
1R 1
@ R 0-
1 R1
0
Ro-ndule
B1Q9R1 R 0
0
. Hence
is a quotient of
is a f i e l d ! ) . Since Torl (B1 RorRo/qlRo) (observe that R1/qlRl R1 Torl ( B ,~ t /q c are quotients of ~ e r ( u R /q R l and 1 1 1 R1O 1 0 ~ e r ( ~lP,l/qlRl) u , it follows that the image of Torl''
Since B
0
I?
i n Torlo
( B Q~ ~
0 '10 /ql
is a r i n g of f r a c t i o n s of
B @ R
(B1, tl/qlR1)
1 R1
IIo) 0
r
generates t h i s last R 0-module.
we have
hence t h e canonical image T generates t h e B o 4 u l e Tor1 'Bo ,Ro/qlRo)
which is t h e r e f o r e zero. Since Bo/qlBo
and
is a r h q of f r a c t i o n s of
it is f l a t over Bo
, SO
Ro/qlEo,
is f l a t over
.
!Lo
Section 4
IWnmrphisns of algebraic schemes
4.1
Propsition: Let ; we assume
X
a)
I_f
grg:z+
the induced maps
b)
If
f =@ -X
.
5, th. 1 and prop. 2 ,
y€x be points of W k-schemes
be locally alqebraic over k
PJg
such that
.
f(x)=q(x)=y, and if
coincide, then there is an open subscheme _V
fxrgx:Q@x
.
EU_ & fly
=gIU_
i k a k-alqebra hammrphim, then there is an open sub-
$: dy+ GX
scheme V_ cf
~ € 5
are two mrphisms of
such that x
o_f _X
1 to
5
that, by Alg. cam. 11,
of &E
g and a m r p h i m f:U-ty
such that
f(x)=y @
a ) : W e m y assume without loss of generality that _X and y
Proof:
are affine. Thus l e t
rated by blr...,b n Since A = X
for each
i
Z=kpkA and X=S&B
. Then we have
l$ns(bps , and , there is an
the same image in As
$(bi)
, where
for each
and. $(bi) i
. Now set
b) : W e may again assume that be such that
s ( x ) #0
. Let
p:As+dx
and suppose that the k-algebra relations
Pl(blr..
.,bn)
=
B
As
that
...
is gene-
@,$ E&(B,A)
@(bi), and. $(bi)
Zs
=
S&As
and q:B+8
Y
.
= 0
have
. Let
sf A
be the canonical maps,
is defined by generators
, there
B
have the same image i n Ax
Z=SjkA and y=S&B
... = Pt(blr.. .,bn)
l i m i t of the rings of fractions
, with
g=Spk$ and. g=S%$
, s p x , such
stA
the k-algebra
. Since
blr...,b
dx
n
and
is the d i r e c t
is an s and elements
...
...
€A ,an ) = = Pt(al, r a n ) = 0 and n s such t h a t Pl(al, p(all=$(q(bl)), p(an)=@(q(bn)) Hence there is a hcmmorphisn jr:B+As
alr...,a
such that
...,
+(bi)=ai
, and we
.
need only set g=zs and
= Ss
+
t o canplete
the proof. 4.2
Corollary:
Let
f:z+x
be a mrphisn of locally algebraic
.
106
I, 9 3 ,
ArGEBRAIc GlzcMmm
Ex: df(x)+ox
k-schemes and let x EX. I f t h e map open subschemes Q
V_
M u c e s an i m r p h i s n V_ 3 y
Proof: If -
, by
yg(x)
Set
v'
.
4.1 b) there is an open subscheme
mrphism CJ:!'+~ such that take
is b i j e c t i v e , there are
2 such that x€V_ , f(x)EY
o f 3
so s ~ lthat l
g'=f--'(y')
ZCJ
y€V'
, g(y)=x
and
CJ
4
=f-l
v'
of
. By 4 . 1
-Y
and
f
and a
a) , we m y
Y -x coincides with the inclusion mrphism
.
yl+x .
and l e t f':g'-+y' be the mrphism induced by f By 4 . 1 a) there is an open subscheme of _V' such t h a t x € Q and g- ( f ' / g ) coincides with the inclusion mrphisn of Q i n _X Now set I! = CJ-1 (g)
.
x€x .-W scheme
f
1 of &
f:z+x
Let
Definition:
4.3
.
be a morphia of schemes arad let
t o be a local anbedding a t x i f there is an open subsuch that
~ € 2 &
i s an embedding.
_flV_
be a local embedding i f it is a local embedding a t each x€& For instance, l e t A = Z[U,V)/(W)
,
2 B = ZkJ,V]/( (V-U +U)V)
and Y_ = Sp -B
. For each model
and
Y(M) = {(xry)EM2 : (y-x 2+x)y;o)
The mrphisn
g:&+x
(x-y,y2+y)E Y(M) sends
g(Z)
such that
M
we then have
.
said to
, and X = Sp -A
(x,y)E&(M) onto
mps
f(M)
2 is
i s a local embedding, but not an embedding. For the map
(l,O)€Y(Z)
and
(O,l)€_X(Zl
onto
(l,O)€Y(@
, SO
that f ,
is not a mnmrphism. On the other hand, l e t a,b (resp. u,v) be the images Of
(resp. in B ) . Then we have f_ = SJ 4 , where 4 : W A is such 4 (u) = a-b and 6 (v) = b2+b Now one v e r i f i e s e a s i l y t h a t 4 in-
UrV
that
in A
duces surjections of
.
B
U
onto Aa ;. Z[a,a-l]
and
% :Nb,b-']
.
ALGEBRAIC SCHEMES
Pbreover, i n the ring of fractions A
107
we have
1-a+b
+
a = @(ul
and
which shms that
subschemes
induces a surjection of
@
za
. Since the
onto A1-a+b
B
1-u $ and &l-a+b cwer 3 , it follaws that f
I
is a
local embedding.
z:x-?j
If
4.4
5
with if
f
f is a local embedding a t xf &
1, 6.3, that :
$ -+ z-Y
Proposition:
is a mrphism of k-schemes, we say, in accordance
is a local embedding a t x
Let
suriective.
.
affine. If all.. ,a
-X
is surjective; we may assume that _X and
ti(x)#O @
. Hence
. If we set
(bi) /$ (si) = ai/l
algebra of $,(&)
flk(X) , then we $=ok(Z) , si,biEok(Y)
n generate the k-algebra
$(bi)/4(si) = ai/l f ~ k ( ~ ,) where x 1 -
ti (ai$ (si)-$(bi) ) = 0
have
and
ok(c_X) for which
and t = tl...tn$(s) the equations n Since t/l belongs to the subalready hold i n flk(c_X)
. Hence
which implies that
are
f o r same ti€
s = sl...s
.
generated by the ai/l
where vE ok(y)
iff
is a local atkdding a t a p i n t xE 11
woof: Suppose that f
s , (f (x))#0
f:x-+x a mrphisn
be a locally alqebraic k-schgne ard
gf k-schemes. Then f
gx: d z ( x ) + ~ xis
(resp. a local embedding)
(resp. a local embedding).
$
,
induces a surjection of
is a local embedding a t x
f
is of the form @ (v)/@ ( s ) ~
t/l
ok(x)sv
dk@) ,
onto
. The converse is obvious.
Lama: - L e t 5 be an irreducible scheme (i.e. such t h a t the t o p l o g i c d l space is irreducible) g:x+x an injective
4.5
1x1
f. i s an anbedding.
gnbedding. Then
proof: that
yx
Let
x , let Cx
x€gx , g(yx)c17x arsd
. Since
g(yx)
xf
U I 1-x
IU -x
yx
be open subschemes of
5 and
f
induces a closed -ding
of
that
is dense i n
i s closed i n
jective, we have
and
\X_l
, f(vX )
[yxl , we have g(ux)
I
=
g-'(Yx)
5 into the open subscheme of
, so
that
is dense in
=
f
g(3)nlVxl
f(~)nly~l. Since
f
such
into ;
since
is in-
induces a closed embedding of
covered by the
yx (5
2, 6.10).
108
ALGEBWC czEDmrRY
I,
8
31 no 4
The irreducibility assumption is essential for the truth of the above 1em-m.
5'
For let
be the schene satisfying
X I (MI = { ( x , y ) ~: y~ - ~ 0
. The mrphisn
for each madel M
jective local gnbedding
of example 4.3 then induces an in-
f:g+x
E':X'+y'
Proposition:
4.6
and l+y is invertible] f_ is not an gnbedding.
but
Suppose that k
is noetherian. f I
f:z+x
mncplmrphism of alqebraic k-schemes, then there is a dense open subschane
-X such that flu is an injective local embeddw.
U - of -
F i r s t of a l l consider a mrphism of schemes g -: s c
Proof:
m m r p h i s n i f f the diagonalmrphism K
is a f i e l d ,
T
=
3K
and
. Then
9 is a
is an isomorphism. If
*sx$
5 = Sp A , this last assertion holds
i f f the
or A=K k It follows that a mnomorphism of a nonmpty scheme i n t o the spectrum of a
map a @3b +a.b
of
A @ A into A is bijective, so t h a t A={O)
.
f i e l d i s an ismrphism.
In the general case, i f If
x€g -1
2': 2
that of
, set
.
y=q(x) and y'= &S
(5 1,
dx is the local
r i q of
3 at x ,
a t x is ~ x / ~ x m y, so it follows from the above that g
g-'(y)
induces an ismrphism
~ ( y ) 9 /9m
x XY
.
dx i s Artin, t h i s implies that
If
o
is surjective. For i f J ' , and m' are the imges of _9x: JY+ Jx 1 Y Y m i n Jx , then ~ ( y ) dx/~xmy is equivalent to I;' = 9'+ m l J Y x Y Y X i s to 8, = d; by Nakayama's l e t m a (m' being n i l p t e n t )
.
Y
axd. x i s the generic p i n t
This results applies when g= f
an irreducible m p n e n t of one p i n t , so that
dx
Corollary:
4.7
5.3).
~ ( y ) The canonical projection
is a mnmrphism. If
(y)+f'
g- is injective
g- is a motlomDrphisn,
dense open subscheme
-
2-
1x1 . For by 5
is Artin.
me
1, 5.6
,,
Spec
(5
axd. '
that
1, 2.10) of
dx then has
just
assertion now follows frcm 4 . 4 .
W i t h the assumptions of prop. 4.6, there is a
of g
such that
L-l@)
is non-empty arid flg-l(y)
is an embeddinq.
...,xn
Proof:
Let
of
and l e t yi
yj
1x1
. Since
xl, f
be the generic pints of the irreducible m p n e n t s
= f(xi)
. Write
yi S
is injective, we m y assume y1
if
yi i s i n the closure of to be maximal with respect to
I,
5
3,
4
this ordering. Then there is an open subschene ylc[yll
and y i g l ~ ~ 1if i > 1 the irreducible ccsrrponents of
If
an embedding; for
of
such that
are the generic p i n t s of (4.6) , then yi is d i s t i n c t f r m
ul,...,um
I_Vl
yl is small enough,
-1
(YJ) I
is open in
yl is
This proves the corollary when
such that
of
struction, using the mrphisn Example:
4.8
schemes over a f i e l d that
v1
[ylI does not meet the image of 4.5 then implies that the restriction of g t o f-'(pl) is
1x1- IUI.
v'
. If
1x1 -
...,f(u,) . If
f_(ul),
scheme
109
Azx;EBRAIc SCHEMES
IV_'
I
f':
and is therefore irreducible.
{x,)
dense. If not, consider the open sub-
1 . Then apply the preceding con1 -1 f (v')+x' induced by 5 ... = IY_I-{y
.
Observe that a mmrphiism of locally algebraic
k is not necessarily a lccal anbedding. For suppose
is of characteristic #2 ; take for 31 am3 _Y the algebraic schemes
k
such that X(M) = { t € M
: 1-t
is invertible]
3 2 2 Y(M) = I(u,v)EM2 : u-=u -v 1 The mrphisn
~
:
, "% 2
2
such ~ t+h a t ~ f ( M ) (t)= (1-t , t(1-t 1) f o r each M€&
a mnoroorphism, but it i s not a local embedding a t the pht of with the elanent -1 of that
--
0
X(k) (To prove t h a t f. is a mmrphism, show (cf. 5 4, belaw) ; by the theoren of 5 4, 3.1 belaw, X is then an open embedding, and a l l that rmins is t o show
z/x : -X & x Ythat &z/x is bijective, 6
+
.
is associated
i.e. that f ( K )
is injective f o r each f i e l d K S )
"* I
I \
I
Y
Section 5
The Krull dimension of a m t h e r i a n ring
5.1
Definition:
Let
is the supreitnnn of the lenqths n irreducible closed subsets of T
T
.
G & n
, the
xET
local dimension of
W e write
dim T
X
, the
For each ring A
the Krull dimension of A
A
If
Let
... f: g,
B
, then we
... $ Fn
of
(resp. the local
. By 5 1, 2.10 , Kdim A po% ... $ pn of prime
A
. ThTh
5
be a subring of
Kdim A = Kdim B
is a chain of prime ideals of
.
A
-. T
B be a r i n g and let A
by Alg. c m . V, qi+ln A # q.nA 1 Kdim A Kdim B Conversely, i f of
dim T =
n of a l l chains
.
qo
$
is the infimum of t h e dimen-
and is written Kdim A
such that B i s an inteqral extension of Proof:
.
F1
dimension of the topological space Spec A is called
Proposition:
5.2
T Kt x
P@ , we set
i s the s u p r m of the lengths
ideals of
chains Fo
(resp. dim T) f o r t h e dimension of
T at x ). If
dimension of
ofall
x & T
sions of open n e i g b u r h d s of
of
be a topological space. T&dimension
T
B
.
, then
2, cor. 1 of prop. 1 so that
po$
... g pn
is a chain of prime ideals
can construct inductively a sequence of ideals qo,ql...
such that q . c qi+l 1
a d Anqi
.
B
( A l g . c m . V,
= pi
5
of
2 , cor. 2 of th. 1).
Hence K d h B
2 KdhA
5.3
Corollary: Let k be a f i e l d and let A be a f i n i t e l y gene-
rated k-alqebra without zero divisors. fractions of
A
Let
zation l
, then
poZ
... g p r
m (Alg. comn. V,
be a chain of prime ideals of
5
al,
...,ah(.)
A
3, no. l), there are elfments
algebraically independent over k B = kLa l,...,an]
Fract(A) denotes t h e f i e l d of
t r k d q Fract(A) t h e deqree of transcendence of
k
Fract(A)
Proof:
,
If
, such
. By al
the normali-
,...an
of
A
that A is an i n t e g r a l extension of
and pinB is the ideal of B generated by ao=O, (where h ( i ) i s an increasing function of i) Since
.
,
fi
I,
3, no 5
ALGEBRAIC
s m s
111
+
< n , and hence Kdim A = PinB P i + p (loc. cit.), it follows t h a t r of B K d i m B < n . On the other hand, since the prime ideals (all...,a form a chain of
p
of
have K d h A = Kdim B
w e have
A
Kdim A = Kdim A P Prcof:
, we
+ trkdeg FYact(A/p)
Using the notation of 5.3, set r=l
.
n
fi
(Alg. cam. V,
ththearen"
p r i m ideals q h ( l ) ~ q h ~ l ) - l ~ . . . ~ qofl
.
, po={Ol
Kdim A 2 Kdim A + Kdh(A/p) , since the second -s P of the lengths of prime ideal chains containing p
the "going dawn
2n
With the assumption of 5.3, for each prime ideal
Corollary:
5.4
n
B of length
and pl=p
. Evidently
is j u s t the supremum
. On the other hand, by
2, th. 3 ) , there i s a chain of A
qh(l)=p
such that
and q . n B = (al,...,as) for each i ; hence K d i m A > h(1) Ybreover, since 1 1 PA/p is an integral extension of B' = k [ ~ ( l ) + l l . . . r a 1 , we have n
.
t r k d q Fract(A/p) = Kdim(A/p) = Kdim B' = n-h(1) ;
so that n = Kdim A = h ( l ) + ( n - h ( 1 ) )5 Kdim A P
+ Kdim(A/p)
W e now turn to the problem of calculating the Krull dimension
5.5
of certain rings which are not covered by corollary 5 . 3 above. For this pur-
p s e we shall use another formulation of the m u l l dimension: Let
of
E
be a p a r t i a l l y ordered set. If
x E E such that a < x < b
called the deviation of
symbols
, +m
-m
partially ordered sets E
, set
decreasing sequence from E
2
dev[ai+l,ai]
n
5
dev E
, we
determine by i d u c t i o n on n
for which d w E < n : i f dev E =
-m
; if
E
the
is discrete (i.e.
t d n a t e s ) set dev E < O ; now suppose that we such that dev F 5 n-1 ; then
...
from E
such t h a t
for each i is f i n i t e . Finally, set d w E = +oo i f ,
for each n EN it is not the case t h a t dev E For example,
,
is Artin (i.e. each s t r i c t l y
E
i f each decreasing sequence alla2, n-1
for the set
a quantity dev E
W e shall assign to E
have determined the partially ordered sets F
set dev E
[a,b]
E : t h i s w i l l be a natural n m h r or one of the
. To define
a < b implies a = b )
.
,write
a,bEE
dev E = 0 means t h a t E
5
n
.
i s Artin and not discrete. Accordingly
I, 5 3, no 5
ALGEBRAIC cT33lmm
112 wehave d e v N = O
but dev Y = 1 and d e v Q = + o o (where iB
are assigned their natural orderings)
.
Z’
and Q
5.6
We rn list sane elementary properties of the deviation function:
a)
If f:E+F is a strictly hcreasinq map between partially
-.
.
ordered sets (i.e. a < b implies f (a) < f (b)) , then dev E 5 dev F For the truth of the assertion is obvious when dev F = ; suppse it holds when dev F < n ; we prove it for the case dev F = n Let alra2 r . . . be a decreasing sequence fran E such that de~[a~+~,a~] 2 n ; by the induction hypothesis, we then have dedf (ai+l),f(ai)1 2 n SO that the sequence f (a,) ,f (a,)
,...
..
is finite. Hence alra2,.
is finite.
If E,F are mn-enpty partially ordered sets, dev(ExF) = sup(dev E,dev F) We show that dev(ExF) 5 sup(dev E,dev F) by b)
.
induction on the pair (dev E,dev F) (the reverse inequality follms directly fran a)). Tb this end suppose that d w E 5 dev F = n ; the assertion is be a decreasing sequence fran obvious when n = -m If not, let xl,x2...
.
Ex F such that dev[xi+lrxi] 2 n for each i
.
. Let
xi= (airbi); if the
sequence x1,x2.. were infinite, then we muld have dev[ai+lrai] 5 n-1 and de~[b~+~,b~] 5 n-1 for sufficiently large i By the induction h p -
.
thesis, this muld imply
Let E be a partially ordered set and let Sc(E) be the set
C)
of infinite sequences el,e2,
.
...
fran E such that en is constant for
sufficiently large n If (ei),(fi) ESc(E) for all i Then we have
.
dev S c ( E ) = 1 + dev E
-,
, set
(ei) 5 (fi) if e i <- fi
.
. The assertion is obvious e=(ei)ESc(E), set e, = lhneen .
To prove t h i s we argue by induction on n = dev E
.
so suppose that n,O If Let a1-(ali , a2=(azi) be a decreasing sequence fran Sc (E) such that if n
=
...
d ~ [ a ~ + ~ ,2a n+l ~ J for each i dev[a(itl),rai,] aij = aim
. Were the sequence infinite, we would ha6e
< n for sufficiently large i ; choose j 0 so that
a(it-1)j = a(i+l)cc for j > j,
. Let
I, 5 3, no 5
ALGEBRAIC s m s
.
(ei) * (el,...,ejo,(ejo+l,. .) 1
be the map
113
. By
(a) , (b) arid the irduction
hypothesis we have
This shows that Sc(E) 5 1 n
=
dev E = 0
+ dev E
, so suppose that
. The reverse inequality is clear if
n is finite ard
...
infinite decreasing sequence bl,b2,
> 0
. Then there is an
frm E such that
dev[bi+l,bi] 2 n-1 for each i ; if aiESc(E) is such that aij=bi for each i , then dev[ai+l,ai] 2 n by the induction hypothesis, so that dev Sc(E)
2 n+l
.
Let Cr(E) be the subset of Sc(E) consisting of increasing
d)
sequences. Then dev Cr(E) = 1 + dev E dev Cr(E) 5 dev Sc(E) 5 1
+ dev E
. For by a) we have
; the reverse inequality is proved as in
c). If E is noetherian, each increasing sequence fran E belongs to Sc(E) arid
SO
to Cr(E)
.
Given a ring A and a module M
5.7
, let
-
dev M be the deviation iff of the set of sulmodules of M , ordered by inclusion. Then dev M = &{O}
. If
sA is the underlying A-module of A
, we write
dev A instead
of devSA (althoughwe are only considering canrmtative rings here, do not be misled into thinking that the notion of deviation is useless in the general case! 1
.
If N is a suhncdule of M , then dev
a)
M = sup(dev N, dev M/N)
.
For the map PI-+ (PnN,P/P nN) is strictly increasing, so d w M 5 sup(dev N, dev M/N) by 5.6 a). The reverse inequality is obvious.
If A is noetherian and M is a finitely generated A-module,
b)
= {O} such that, we know that there is ccanposition series M = M 2 M '...3Mn 0 1 is ismrphic to A/pi for some prime ideal pi of for each i , M./Mi+l 1 A (Alg. comn. IV, 5 1, th. 1). By loc. cit., th. 2 , we have dev M =
sup dev(A/p)
.
, where
p rhqes over the minimal prime ideals containing
114
ALGEEwac cEmEmY
5.8
I,
5
3, no 5
kt A L a (camutativd ncetherian rinq.
Propsition:
KdimA=devA.
Proof: To shaw that Kdim A <- dev A
, we m y
then enough to shm that dev A/p > d w A/q
assume that d w A <
whenever
,
s#O
p $ q . For t h i s purpse we may assume that . Then we have an i n f i n i t e sequence A 3 A s z ) A s 2
2
dw(Asi/qsi)
i t y is obvious when Kdim A
. Let
0
, that
such that
that
Kdim A
. If
Kdim A = Kdim(A/p)
s - ~ = o, then we have
is a f i n i t e l y generated A d u l e such
M
5 sup Kdim(A/q)
< Kdim A
(apply 5.7 b) , the induction hypothesis ard the f a c t that q2Ann M
, hence
if
q meets
sequence of ideals such that
Corollary:
dev(Ir/Ir+l)
metherian rinq A Proof:
Let -(A)
t i o n of
A
,
If x
Kdim A
algebra where F'
111,
I
5
- + Kdim(A/Ax)
of
.
assign the graded ideal
A
nAxn)/(I
nmn+l)
. Since
, where
T
@(A)
, clearly
IwGr(1) i s s t r i c t l y
, where
F
i s the set of
is obviously a quotient of the graded
is an ideterminate, we have d w F c dev F '
is the set of graded ideals of A/-
2
belonqs tD the radical of a (camutative)
3 , th. 2 and prop. 6, the map
&(A)
(A/&) [ T I
of ideals of
1
2, prop. 91, the sequence is f i n i t e .
increasirq. Accordingly we have Kdim A < dev F graded ideals of
, then
be the graded rirq associated with the (Ax)-adic f i l t r a -
. To each
CCPTITL.
5
2
Kdh A < 1
G ~ ( I )= By Alg.
Kdim A/q < Kdim A
Consequently, if I 31 I... is a
S).
Since S - ~ A is m t i n (mg. m. IV, 5.9
is f i n i t e and
s € A which do not belong t o any prime ideal p
dev M = sup dev(A/q) q3AnnM
if
. The inequal-
dev A < - Kdim A
; suppose then that
= SCO
S be the set of
such that
and the assertion follows.
= dev(A/q)
W e show finally, by induction on Kdim A
->
~ ( 0 ); l e t
3...
dev(Asi/ASi+l)
it is
are prime
p and q
ideals such that
s €q
+db ;
F'
Cr(E)
(A/Ax)[T]
. But i f
E
is the set
. The corollary now follows f r m
,
I,
5
3, no 5
115
Azx;EBRAIc SCHEMES
5.6 d) ard 5.8. 5.10 i
Let
Corollary: w
, and l e t n
m
A
be a noetherian local rinq with maximal
be a natural number. Then the following assertions are
equivalent. (i)
KdimAZn
(ii)
There is a sequence al,
A/C
the rinq Proof:
~a~
...,an
of elenents of
rn such that
i s Artin.
(ii)=> (i): This follows inmediately f r a n 5.9 by induction on n
(i)=> (ii): This is obvious when
n=O ; i f
n >0
,
m is not a minimal
prime ideal ( f i g . cam. IV, 5 2, prop. 9 ) . Hence there is an element
m which belongs to no minimal prime ideal.
ad. 5.9, we have, s e t t i n g Z = A / h l duction on n
S2,.
..,?I n
, w e may
.
(ii)
such t h a t
= Kdim A
.%/L.%i
-
1
of
Kdim A
. Arguing by in-
..,an are the al,a2, ...,a satisfies n
is Artin. If
Z2,...,Z n i n m , the sequence
Corollary:
5.11
Kdim
By the d e f i n i t i o n of
al
assume that we have proved the existence of a sequence
fran m/Fal
representatives of
,
.
a2,.
With the assumptions of 5.10, w e have
Kdim A < h/m2:A/m] proof:
If al,
then m = 5.12 ACB B
r
. If
lAai
...,an
are the representatives i n m of a basis f o r m/m
2
,
(by Nakayama's l m ) .
Corollary:
Let
the maximal ideal m
and of A
A
B be noetherh 1
4 rinqs such that
is contained i n the mima1 i d e a l of
m
KdhBZKdhA+KdhB/Eln. Esuality occurs i f
B
Proof: By induction -
is f l a t over A on Kdim A
of n i l p t e n t elements of
.
. For each ring
C ; we then have
C
, l e t rC
rA = rB n A
denote the set
. Hence we may replace
and B by B/BrA , which thus enables us to reduce the problan to the case in which r = O I f Kdim A = 0 , the assertion is then t r i v i a l A
A
by A/rA
.
. If
since m = { O }
(Alg. c m . IVr and Alg.
c(Hrm.
Kdim B/Bx
5
5
Kdim A > 0
, let x Em
be a non-zero divisor of
2 , prop. 10) , W e then have
IV, 5 2, prop. 10)
Kaim(A/Ax)
Kdim A/&
, Kdh B/BX 2
+ Kdim(B/Fm)
Kdim B
= Kdim A
-1
A
-1
(5.9
d
by the inductive hypothesis; the re-
quired inequality follows. If B is f l a t over A
,
x does not divide 0
B and a l l the inequalities above may be replaced by equalities.
in
Corollary: I f A is a m e t h e r i m local ring w i t h maximal ideal
5.13 m r -
K d h A = K d h A ,
i n the m-adic topoloqy.
is the ccanpletion of A
where
Corollary: If k is a f i e l d and T1,...,T are indeterminates, n
5.14
then
Kdim kCCT1r
Proof:
k[ ITl,.
..,Tn]]
...,Tn 33 = n .
is the cunpletion of the localization of
.,Tn) . The r e s u l t now follows from 5 . 3 ,
a t the ideal
(Tl,..
Section 6
Algebraic schemes over a f i e l d Throughout t h i s section k
k[T1,.
..,Tn 3
5.4 and 5.13.
denotes a f i e l d
and an R-scheme X , we write dim X o r dim X f o r di.rlX/ and dimxIgI , and speak of the dimension of 11 o r the xGiven a &el
6.1
R
.
local dimension of
X
Dimension theoren:
Let x be a
ard l e t
be the irreducible m p o n e n t s of
x
. Then
X1,.
-i s proof:
..,Xr
at x
point of a locally algebraic k-scheme
Clearly dim X = sup.dim X
. Let
IX_l
p s s i n q throuqh
be the prime ideal of
x i Pi which carried onto the generic pint of Xi by the map (5 1, 5.6 and 5 1, 2.10). If we identify Xi Spec dX + 1
X-
1x1
closed reduced subscheme of of
5
2, 6.11)
, dx/pi
2
carried by
is the local ring of
Xi Xi
X
ox
w i t h the
(i.e. t h e subschme Ri red at x
.
By
5
1, 5.6 we see
I,
9
3,
NGEBRAIC SCHE?ES
6
fiX
Kdim(0x/ p1, ) , so
= supi
0X
are the minimal prime i d e a l s of
that pl, ...,pr mim
117
that it i s enough to show that
_X
dimxXi =
.
dim Xi = Kdim(ox/pi) + trkdeg K(x) W e may thus assume t h a t
. It follows that & is a f f i n e , the
is irreducible and reduced. I f
theoren follows f r a n 5.3 and 5.4. I f not, l e t w be t h e generic p i n t of
1x1 . For each non-empty a f f i n e open subscheme
dim _V = trkdeg
, whence
K(W)
dim
I! =
V_ of
, we
_X
sup dim _V = tr deg k
then have
, which
K(W)
re-
duces the general case to t h e a f f i n e case. Corollary:
6.2
L e t Klk
be an extension of f i e l d s and l e t
a locally algebraic k-scheme. Then a )
a x € z BkK , if dimx(X-@kK) Proof:
-p:
Since dim
a(X@kK)
+
zl,...,X-n
Let
y
.
X
dim X = dim XBkK ; b) x &
is the projection of
g , we
&
3 carried by the irreducible
be t h e reduced subschgnes of
is the generic p i n t of and x E {xi}
passing through
3
passing through
xi
ziBkK
x@ kK containing . Mreover, i f yi
. The irreducible ccmpnents of
zi mkK
gi of
(6.1)
dim X . = dim X!
-1
-1
K(w')
Xi
f o r each
A
,
,
is a r i n g of frack and hence is an i n t e g r a l d a m i n .
K
A@ K
is irreducible and by 5 . 3
is an algebraic extension of
p i n t s of
XBkK
. For this purpose we my obvious-
dim X.63 K = tr deg Fract(A@ K) = tr deg FYact(A) = dim -1 k K k k K
not
is a f f i n e w i t h algebra A ; several cases then arise:
tions of an algebra of p l p d a l s over
If
XgkK
thus contain x and are irreducible c m p n e n t s of
is a pure transcendental extension of
Accordingly Z i g k K
such that
("forget" the irreducible CQTlPonents of
. I t is therefore enough to show that
ly assume that Xi K
then there i s a p i n t zi€XBkK
X.
-1
and apply 3.11)
irreducible m p n e n t
If
have dim X = Y-
X and t h e canonical projection x xi s surjective, we see that b) iinplies a ) . Now t o p r m e b ) .
x are then irreducible ccklrponents of the schemes ZiBkK
i
f o r each
= sup dim
C m p n e n t s containing y ; the irreducible ccenponents of
p(xi) = y
& be
and Xi
is a quotient of
, then
K(W) @
and we have dim -1 X! = tr deg k
K
p(w') = w K k
K(w')
, =
an3 i f
w'
because p
Xi
.
and w are the generic is open (3.11). Since
~ ( w ' ) is an integral extension of
tr deg k
K(W)
= dim
-xi .
K(W)
Finally, i n the general case, there is a pure transcendental subextension K'
of K such that KIK' is algebraic.
Then dim X! = dim X. @ K' = dim -1 -1 k
Xi
.
Corollary: Let f:g+x be a mrphism of locally alqebraic
6.3
k-schemes, let x
and l e t y=f(x)
be a point of
dimX < dim Y+dimxf -1 (y) x-y-
.
. Then we have
.
Equality occurs when f is f l a t a t x
at (Ix an3 0 are the local r h g s of g and Y x and y I and i f m is t h e maximal i d e a l of 0 dx/oXmy is the local -1 Y Y' ring of f (y) a t x ~y 5.12, MimOx 5 ~ a i m d + ~ a i m ( r /l O m ) , whence Y x X Y Proof:
By
5
1, 5.8, i f
.
dimxX
- trkdeg
K(X)
< dim Y Y-
-
- trkdeg ~ ( y +) dimxg-'(y) - trK(y)deg K ( X )
(6.1) and the required inequality follows. When f
is f l a t a t x
, the in-
e q u a l i t i e s may be replaced by equalities (5.12).
corollary: If f:X+x is a m r p h i s n of l o c a l l y algebraic -1 k-schemes, the function x-dim f ( f ( x ) ) is upper seni-continuous. 6.4
X-
, let
Proof: For any e ED\3 Take any x
i n the closure
Xe
xe
x E X w i t h dimxfl(f(x) ) > e
be the set of
of
and define &'
Xe
as the reduced sub-
3 carried by the Frreducible canponents X1,...,X9 of ye passhg through x For the mrphism f ' : 5' f(g')red induced by f we schane of
.
+
have by 6.3 dimxZ-'L(x)
2
I n order to prove t h a t x € X e -dim f '
dim
5'
(x) ?-dim
dimxf'-'f'
, it
f(X')red + dim 5' .
is therefore enough to show that
such that (sl)rea+ ding1 2 e . For t h i s purpose take Xi dim Xi . By 3.6 there i s a t€Xe"Xi such that g' i s f l a t a t
=
Hence 1
~n
the other M,
dim Z 2 e l i e i n
i1(f- (t) w i t h l_f(t) = dim f-lf (t)2 e . t- -
a l l the irreducible m p n e n t s
, so
that dimtf'
-
z
of
t
.
.
k
. Thena) __
Let & and
Corollary:
6.5
be locally algebraic schemes over
Y
dim_Xx_Y=dim_X+dimY_ ; b)
and
projections x
y
_Y
_X
I
1Lf
is a p i n t of
z
th&h dimzzr_Y = dimxz
_Xxu
+ dim Y Y-
.
Proof:
with
.
Since dimgXY_=s u p d i m X x Y , weneedonlyprave b) Wemy r assume that and are affine, and by 6.2, that k is algebraically
X
closed. By 6.1, w e m y replace
assume that
z is closed (i.e. associated with a maximal ideal of
f
-1
. If
3, prop. 1 (iii) , it follows that
5
By Alg. comn. V, k = K(Y)
,f: _X
XY
s (zxy)Xy(% + dimx& .
(y)
dim Y Y-
of X
extension of Proof:
Let
k
. Hence by 6.3
~ ( y ) 3)
K(Z)
If
dim&xY = dim Y Y-
of x
K(X)
.
and affine and contain x
, let
k
x is closed i n Since
,
A/p
( A / p c ~ ( x ) ), so p
is a f i e l d
for any _V
is algebraic over k
. conversely, ,
6.7
corollary:
6.8
If
2
K(X)
is maximal and
x is closed, p is maximal
5
3, prop. 1 (iii).
x of a k-scheme
X
is identical with k
K(X)
C l e a r l y one obtains a bijection of
g(k)
is said t o be
.
onto the set of rational pints of
Spec k # 15 \ : Spec k -.\XI Accordingly, we simply write Xfk)
by assigning t o each i;G(k) the image of the unique p i n t of
urder the mrphism
.
for the set of rational points of 6.9
. If
is a l o c a l l y alqebraic k-scheme, each locally
A pint
rational i f i t s residue f i e l d
-X
if
is a f i n i t e
& is closed.
Definition:
Propsition:
If
=
is a f i n i t e l y generated k-algebra. The
A
proof is ccanpleted by applying Alg. cam. V,
closed p i n t of
+ dimzf-1 (y)
be its algebra
A
of functions and let p be the prime ideal CorrespndiXq to x
is f i n i t e over
, whence
.
is a locally alqebraic k-schem, then a
X
is closed i f f the residue f i e l d
V_ be open
k =
o(_X 1))
i s the canonical projection, we then have
-t
Propsition:
6.6
point x
. W e may accordingly
by a p i n t of
z
k
g
.
is algebraically closed, and i f
5 &a
locally alqebraic k-scheme, the map P H P nx(k) is an isanorpkism of the
120
I, § 3, no 6
ALGEBRAIC G2umn-a
lattice of closed sets (resp. open sets, constructible sets) ~f
onto
the l a t t i c e of closed sets (resp. open sets, constructible sets) of the sub-
of 5
space &(k)
F i r s t consider the case in which
Proof:
of
runs through the closed subset
P
. We construct an inverse map by assigning t o each closed subset F X(k) its closure i? i n 1x1 . For F =?@(k) , so it is enough t o prove
__.
of
.
I&/
that P = PQXk)
if
1x1
is closed i n
P
/;I
affine open g i n 3 such that
8Q)).By
If
P
# PTD((k) then there is an
meets P but not
contains a point x which is closed i n mima1 ideal of
,
g
. Then
P ns(k)
P
(that is, associated with a
6.6 ardt 6 . 7 , it follows that x E_X(k)
,a
contra-
diction. The assertion about the l a t t i c e of open sets follows fran the above by passage
to ccmplanents. Finally, it is clear that each constructible subset of X(k)
, where
i s of the form P n x ( k )
Accordingly, we have to shm that i f = QnK(k)
then P n x ( k )
implies
1x1 .
i s a constructible subset of
P
P
P = Q
and Q are constructible in &
. By setting
U=PQ
, or
,
U=Q-P
we reduce the problem t o proving that U ng (k) = @ implies U =@ for con-
. But this holds when restriction on U . structible U
is locally closed, and hence without
U
Remark: Now that we know that under the assumptions of prop.
6.10
izl
the l a t t i c e s of open sets of
6.9,
that the theories of sheaves over
over
151
ng(k)) =
T(U)
each sheaf that
T'(U
1x1
and s ( k ) are iscmrphic, we see and x ( k ) are equivalent: explicitly,
is associated w i t h the sheaf
where U
i s open i n
131
.
T i over
X(k)
such
Since any mrphisn
of locally algebraic k-schemes sends rational points onto rational p i n t s , it
follows that, i f
k
is algebraically closed, the functor
3 + (X_(k),f l t X )
which is defined on the category of locally algebraic k-schemes and take; values in
akis fully faithful. where X_
(X(k),8 l X ) sets of Serre. If
k
X(k)
its
The gecpnetric spaces of the form
is a separated algebraic k-scheme are the algebraic
i s not algebraically c;osed, we obtain analogous results by replacing by the set of closed points of
true when
5
131
. Finally, proposition
6.9 renains
is an arbitrary k-scheme, provided one replaces X(k) by the
I, S 3 ,
M
6
&ALGEBRAICSCHEMES
set of locally closed points of
1x1
121
. Unfortunately, a mrphism of k-schemes
does not necessarily send locally closed pints onto locally closed pints: Let k again be arbitrary, and let
6.11 closure of
k
Corollary: f(E) :
proof:
xcr<)
-+
.
A mrphism of alqebraic k-schemes f:X_+Y
Since the canonical projection of
.
is surjective iff
~ _ ( k )is surjective.
it is enough to s h m that g @ and 6.9
be the algebraic
k
2
(x@&
onto zy
is surjective,
is surjective. But this follows f r m 3.8
Section 1
The d u l e of an embedding
1.1
Let
1
such that
subscheme of
j:z+y
i:&+Y_ be an embedding of schmes arid l e t
i s the composition of a closed enbedding
and the inclusion m r p h i s n of f
mrphisn
8v+j, - (Jx) induced
j-:
schene
of
be an open
by
i n Y_
-j , it
. If
9
i s t h e kernel of the
i s clear t h a t the closed sub-
d e p e n d s only on _i and not on
xi
. Accordingly,
we
denote t h i s closed subscheme by xi and call the f i r s t n e i q h b o u r h d of & (or of A) & Moreover, i2:yi +y denotes t h e inclusion mrphism and i.L:&+Xi t h e mrphism such that _i =&,il W e then have V and
.
L1=ll
xi i s obviously functorial i n 11 . For each c m t a t i v e
and & ' are embeddings, there i s a unique mrphism _h:y;
such that such that ; if
--j
*
The construction of
g
xi=-
.
-
= _i,f
and 3%; = i-2-h
. W e say t h a t
h
i s i d u c e d by
g - (resp. f ) i s the i d e n t i t y mrphisn, we say t h a t
g (resp. g)
.
W e make analogous definitions f o r -dings
Lama: - Let
1.2
k ,bea &el,
+yi &
h is induced
by
of k-schemes.
&:&+y
j:Z_-+T gnbeklinqs of
k-schgnes. Then the mrphism
i d u c e d by Proof:
L2 x -&: Y-i x
T -j
+
Y_xT is invertible.
W e imnediately rduGe t h e proof to the case in which Y_ = S k A
T = S&B , where
& and
j
,
are d e f i n d by the canonical projections of
A
5
I,
s
4 , no 1
m mwHIsMs
and B onto the quotients A/I
123
. Then
and B/J
u is induced by the canoni-
c a l isamrphisn of
(A@ B>/(I*B B k k
+
I8J +A@ J ~ ) k k
W e now reinstate the notation of 1.1. The quasicoherent &-mdule
1.3
-j*(!f/r ) 2
& and is denoted by is called the module of the &ding Clearly the canonical ismrphism Y/r2 +j*(wi) is invertible. Also, each point v € y has an affine open neiqhbourhooa V_ such that f i n i t e l y generated ideal, then the equation w . = O
open embedding. For i f
(by Nakayama's l m )
=0
U
of
v
. The mcdule
which the mkdding Now let
that
x€_X and v = j ( x )
-i'
v'
w
i
is an
implies that
i n a certain sense thus measmes the extent t o
be an open subscheme of 1' (1.1) contained i n q-'(V)
1':z'
is the c a n p s i t i o n of a closed anbedding in
j": dyl +ji(dxl) and i f
(Y)
0
+oi, -
y'
. Then
cx:T+%
factors through
W e shall say tkat the mrphism 2 ) = wi, -j'*(~'/~' -
g induces
then so is the square
. If
7'
such
with the
a mrphism of
V_'
into
is t h e kernel of
i s the-inclusion mrphism, then
3' and
j'*(r) -
is induced by f
~emna:
is a
for a suitable open neighbourhood
which for simplicity we also denote by g
1.4
(f//.rk=
.
fails to be open.
inclusion mrphism of
? * ( a ) :_s*
2
then
7IU =0
and
'y(LJ)
implies that
1
wi
if
i d u c e s a c m t a t i v e diagram
of
and g
i'*g*(g/!f2) = f*(w.) into L
.
If the c m t a t i v e square
(*)
-above is
~&&iy, -4
124
I, 5 4 ,
1'
Y! -L
L> y_'
'1
i2
xi---
Y
the induced m r p h i m
woof: if
(*)
f* (wi)
-
is Cartesian i f f
+
y'
wil
-
W i t h the assumptions of l
is the image of
m 1.4,
_f*(wi) +w;
q is a m m r p h i s n . The l a t t e r holds hy&~ conditions of l m 1.6 below.
.
4
in
g*(Y)
-
dvl -
.
But
is an isamorphim whenever is f l a t , and a l s o under the
Consider the diagram of schemes
1.5
where p&=Iq(
, go
of vanishing &mre,
is a closed subschene of
2.
z
defined by a z-ideal
a d -j is the inclusion mrphisn.
an embedding; mreover,
with
&s an epimrphim.
is an epimorphism, so are p and r
q
1
Z . = g a d the module w
By
2
52, 7.6b), _i is
of
-j m y be identified Defining so =gi-, we examine the mrphhns f:Z +Y_ such t h a t -3
j
the square
pf-g
cmnutes.
+?
LemM: The mp which assigns to f we mrphisn gi;(wi) induced by % ard f is a bijection of the set of $:Z+_Y such that &=q @ &%=f
onto. &z Proof:
,p -
(gi;(y
-0 For sirnplicity identify the space of points of
by means of kernel wi I
.
5 w i t h that of
xi
i-l:g+yi Accordingly $: dy; dz is an epimrphism, w i t h f (pi2)': +c9 is a sec€ion of &i we may then X X-i
and
.
-
identify
8
with
Yi
dx@wi -
in such a way that
/
respectively-by the canonical projection of
ox
i n dx@wi
Ll
Ox@wi
. This done,
-
and. PA2 are defined onto dX and the in-
since we have Z.=Z , f 3 necessarily factors G o u g h th; f i r s t neighbourhcd ; we again-write f
clusion mrphisn of
xi . The underlying map of
4 in
f o r the induced m r p h i m of
cides with that of go and the h&rphisn f f f f= 9- (since pf=c$ and _f- (wi) c
/JX
foll&s that
f induced by f-
f
xi
this f
:'2 dx@oi + go* (dz) s a t i s f i e s - go,Cf) (since &go=fj) . I t
(r)
i s uniquely defined by the mrphism 4 E ~ ( w i l g o *
. The mrphkm
+ I of the l
$(wL)
coin-
)
m is obvi&sly assign&
t o $I by the canonical bijection
(5 2,
1.21. Since $I
kma:
1.6
XI---=+x
is arbitrary, the l
m follows.
Consider t h e diagram of sch-s i
,f X =_
-
+
_p
. Let 1' be the f i b r e product u x2-I , q:Y_' -tZ anm the canonical projections, and :_XI +XI the mrphism with cmp-
such that p i =I%
p':XI
if
> n e n
1%'
-
. Then each of
the squares i n the diagram
i s Cartesian and the mrphism f*(wi) ible
-
.
Proof:
Observe that, by
5
2 , 7.6b)
,
+
wil
-
induced by f
is an -ding:
9 is invert-
the r i g h t hand square
is Cartesian by construction, and so is the middle one by 1.4. E5y reduction
to the category of sets we see that the m a r e with sides f , g, A 2-1 ' i ' = -i' and s2i,i=A is Cartesian. Therefore so is the left-hand one. Identifying
xi-
with
Sp(dX@ui) - -
as explained i n 1,5, w e deduce that; ! l
-
is
I, 5 4, no 2
ALGEBRAIC GEmErRY
126
Section 2
The module of differentials
2.1
Definition: Let g:x+g be a mrphism of schemes; for 1
take the fibre product
%/_s
:
R
2%
.
g x ,& , far 1 the diaqoml Trclllphi~n
8 xsg The quasi&herent d x d u l e wi is then denoted by and calied the d u l e of differentials ,f X over S -+
When k
.
is a k-scheme, we also write R
is a d e l and
T/k
for the
module of differentials associated with the structural projection pT:
$ -+
S_p k
. We then say that
Q
_xfi
k.
&er
Proposition: Let
2.2
e_:?-+X
a section of 9
e*("/s)
we -
--
.
is the d u l e of differentials of T
e:E-+sbe a mrphim
of schemes and let
. wen there is a canonical iscrorphism
Proof: Apply lemma 1.6 to the diagram
%/s
E
and to the Cartesian squares
For exarrrple, if 5 is the spectrum of a field k and x is the image of Spec k under g
, then
2
E * ( R ~ / ~ ) may be identified w i t h mx/mx
--
, where
I, 9 4, no 2
m
X
SMOOTH MLIRPHISMS
is the fiaxirnal ideal of
127
dx .
Consider the conmutative square of schms
2.3
Proposition: If the square
Y*(Q~~~I
+
(**)
abare is Cartesian, the mrphism
Qxl/s, is invertible.
Proof: Apply l
m 1.6 to the diagram
-
We now apply the results of 1.5 to the diagram
2.4
i b Z !3+Y -
-zO
%/s
- _x
XSs
-+
@l We m y assume that go is a closed subschm of
z defined by a Z-id-1 2
. Then with each $ €mod (g*(R ) ,8) *_Zo -0 2 5 4 there is associated a mrphism of schemes _f: z 5 xs& such that prlf = g-
of vanishing square; set go
-+
-
The first condition means that has g as its first and F J = 6 q g o ccanponent. Denoting the second ccanponent of & g + J, , the condition
.
-_ fj = 6x/sgo means that gj -- = Q+J,)j
Module 1.5, g+J,= cr f m y be explicitly constructed in the following way. 2Set 6 = 6 ; the space of pints of the first neighbourhood (5 XS& may
XIS
then be identified with that of
with
8X @QxIs , so that -
_ _
canonical projection of
dl
3 and the structure sheaf of (5
and ~~6~ onto
&
correspond respectively to the
dx
-
and to the inclusion mrpfisn
of
I, 5
AIXEBRAIC GFEMERY
128
ox-
in p e n
XIS
-
4,
no 2
.
The mrphism g2d2: X + X is then associated with a morphim dx + d..@Rz/s -6 of the form xtz(x,dx) , where d:dX+fR is called the universal deriva&t
z/s s . under these- conditions, g - +$
o$kelative to
f
has the same under-
lying map as -g , and \ g + $ l - : dx+ 2*(oz) is themrphism f x-lgl-(x) +$(ax) , where O E p & ( R X / s , & ( f ) ) is assigned to JI by the
-
canonical bijections
- -
We deduce from 1.5 the Proposition: g t
-
z9
2
9S
zo
mrphisms of schemes and let
2
be the
of vanishinq square. ThIh the map $ g (9" (RXiS) ) onto the set of - + $ is a bijection of GE -mrphisms -g ' : _Z + Xsuch that _qg' _ =qg - - and - _ g'IZ-0 =glZ --0 ' closed subscheme of
defined by a Z-ideal
( g + $ ) t $ ~ ' = g + ( J , + $ ' ) and
We leave it to the reader to verify that 2.5
When 5 and S axe affine, we m y reformulate 2.4
following way: suppose that
s = SJ
k and g = Sp -A
, where
g_+O=g
.
in the
k,AEg
. We
write R for the module of differentials of the k-algebra A , i.e. the A/k d u l e of sections of R over _X By definition, we then have _XIS RAjk= 1/12 , where I is the kernel of the k-algebra hcxmprphisn
.
a@b -a.b
of ABkA into A
. I@reover,
the universal derivation of
dx
into R _X/s (2.4) induces a IMP of A into RA/k which we denote by d-. This map d sends aEA onto the residue class rod 'I of l @ a - a @ l E I Proposition: The map d:A+R
i k a k-derivation. For each k-derivation D A/k of A into an A-module M , there is a unique A-linear map f: QAk+M such that D =f d .L)
.
Proof: Uniqueness follows f r m the observation that since the elanents 1@a - a w 1 obviously generate I
, the
R as an A d u l e . A/k As for existence, write E:AmkA + M for the map a@b +a.Db ; this map vanishes on I2 and induces the required map f on 1/12 da generate
.
This last assertion ties up with prop. 2.4 in the follawing way: Let P:AX be a k-algebra Ixmxmrphisn, J an ideal of C of vanishing square and
.
I,
5
4, no 2
129
SMOOTH PKIWHISMS
p:C->C/J the canonical projection. The k-algebra hcnamrphisms
are then of the form p '
that pp'pp' of
i n t o the A-nmlule
A
J
.
= p+D
, where
The upshot of Proposition 2.5 i s that
2.6
of a universal problem. Thus we may identify
Let
a)
(Xi)itI
i s a k-derivation
D
(RA2A/k, d) is a solution
d) with every solution
(R
of the same universal problem. A few examples:
such
p':A-tc
Ah'
be a family of indeterminates; set A = k[ (Xi)
Let M be the f r e e A-module generated by a family
3
and let D:AM
be the derivation such that
Since QAIk
is obviously a solution of our universal problem, we see that
(MID)
is the f r e e A-module generated by the dxi
=
ti
.
(a.) be a family of eleI %J A generated by these elements. +IsAda.) and D: A/a-+M thermp
Again l e t A 13e arbitrary; l e t
b) m n t s of
A
and l e t a be the ideal of
Let M be the
(A(a)-mdule
RUk/(aQm
, ,
derived from d by passage t o the quotient. Then
the universal problem with respect to A/a
d k
is a solution of
(M,D)
so that
Consider a multiplicatively closed subset S of
C)
A
. Let
M
7 S-lA@ARA/k and D: S-lA S-lQ the m p A/k A/k (s(da) - a ( d s ) ) / s 2 Then (M,D) i s a solution of the universal problem a/s -1 w i t h respect to S A and k , whence
be the
S-lA-module
S-lR
-+
.
F r m this it follows i n particular t h a t for each k-scheme
x of
I! , the
stalk (RX/k)x
of
R
X/k indeed, i f we assume X -to be affine,
algebra d)
O(_X) .
5
and each p i n t
my be identified w i t h R
c",
-.
Wk m y be obtained by localizing the
Consider the ccinnutative square of rings
.
130
Au;EBRAIc
The reap a ~ d J(a) i
GECMETRY
is a k-derivation of
a k-linear map R$/+: RAIk are the canonical maps,
+
and by 2'. 5 induces and $ , j induces an isarorphism Q+: QAkgkt' RB,e
RBB/e
fi$,$
i n t o fi
A
. I n the case whereB/ pB = A g k e
( 2 . 3 ) . Moreover, one v e r i f i e s d i r e c t l y that the B-module
fiAjkgke and the
a g k x +da@kx form a solution of the universal problem with
derivation
.
t
respect to B and
X
k t k he a d e l and l e t
Proposition:
2.7
be a locally
algebraic k-scheme. Then f o r each a f f i n e open subscheme V_ E f
4(v)-mziiule
- (v)
we have fi
Proof:
(v)
.
.
. Since
R d,(y) /k
Z/k ,Xnl/(all.. ,ar 1
k[X1,..
2 ,
is f i n i t e l y presented.
(5 3 ,
1.7)
, we
Ok(y) is of the form
need only apply 2.6a) and b) to ob-
t a i n the desired conclusion. In Section 4 we shall need a corollary to Prop. 2.7 which we
2.8
prove; namely, l e t C be a k-model,
an ideal of
J
~ 3 w
C
.
of vanishing
the canonical projection, and 0 E_X(C/J) Suppose that we n are given a p a r t i t i o n 1 = 1i=l x.f of unity i n c and sCane 5 , E X ( C ) 1 i p:C+C/J
square,
such t h a t p
f
i
(Si) EX(C JJ
fi f l
Then there i s a t; Eg(C)
coincides with the image of
0
.
such that p(5) = n
gij be the image of ti i n g ( C f , , ) l j ard LJij denote (% C ) and ( S p C I f ,
To prove this assertion, let
R =q'*
(R
X/k
ui
and let
)
of which are open i n SJ C X-(Cf,f / J f . f 1 i j i j
c!!
= t;fj + Q i j
T:~ H % ( R / u , J I ~ ) 1
H
--
((IJJ),T)
$i E t;! . = ij
=
. Since cij
, by 2.4 there are
and
'Jij E
. Clearly the family
tji
fi
17
(yi)
it w i l l f o l l a J that
i
I j
, both
,) such that
11
is a 1-cocycle of the sheaf of
spc
. If we can show that
IJJ~/IJJ~ -ajlyij
J ' =~ ~
, 'here
= 5,# t + i , then we have ( ~ j y ~ . ~If ~t;il s~a t ~i s f)i e s -i t;' & is a local functor, there is a j i i n the above notation. Since
t; EX(C)
w i t h hmge
c;
i n X_(C ) fi
for a l l i
. This
.
; set
have the same irrage i n
(RIUij,J(v qij
($. ,)
f o r the covering
o , then
f
f o r each i
5 is the required
solution. It therefore remains to Show that H1((ui)rT) it is enough to show that
=
0
. By 5 2 ,
1.10 and 8.2,
3“ is quasicoherent. To do this we choose a cover-
ing of Sp - C form4 by affine open subschemes _V such that the images are contained in affine open subschanes of ,X . By 2.7 ,
q#
(c)
is then a finitely presented O(v)-moduIe. It now follows f r m Alg. cam. 11,
5 2, prop. 19(i) that the sheaf Fly
: _w
JIg
+-%(n(g--JE,
-
is quasicoherent. 2.9
f:Z-+Y
Proposition:
g:_Y+z are mrphigns of schemes,
then the smence
of E ~ X &
- is exact.
-
Proof: First we reduce the problem to the case in which X = S p A , Y = SpB -Z = Sp C , f = Sp - 4 , 2 = Sp $ It is then sufficient to show that, for each A-module M 0
+
-
,“the sequence
HQnA(QA/BrM)
-
.
Hy(QA/CrM)
-
---+
HQ”A(QB/C
A, M)
is exact. But this is a consequence of the fact that the sequence 0 --+ DerB(A,M)
Derc (A,M)
D”rc (B,M)
is exact, where Derk(K,N) is the module of derivations of the k-algebra K with values in the K-ndule N
x a pint of 5 with for the vector space (Q ) @ K ( X ) Z / k x L”,
Let k be a model, g a k-schae,
2.10
residue field Over
.
K(X)
.
Proposition:
K(X)
If
is a point of 5
_X
. Write
Qm(x)
is a locally algebraic schane over a field k
, ths
[QZ/k(x):K(x)
3 ?di”,X_.
x
,
ALGEBRAIC ( 2 x m T R Y
132 Proof:
of
X
R- - (XI X/k
, we may
that
=
(5
W e a l s o know that di",rs = d i m 2 by
. It follows f r m 2.3
B w~ i t h projection ~ x
4 4 , no 3
k and l e t E be a p i n t
be an algebraic closure of
Let =
I,
3, 6.2)
. Replacing
3 by -2 and k
assume k to be algebraically closed. I f t h e required in-
equality is f a l s e ,
Qz/k
w i l l be generated by less than n = dimxz sections
over sane open neighbwrhood
U of
. Choose a closed pint
x
which belongs to a l l the irreducible ccsnponents of
in U
y
5 passing through x
.
.
But this contraThen we have dim X 5 dim X = Kdim d and Rx2x/k(y) m/m2 X' Y Y d i c t s the inequality [m : K ( y )3 2 Kdhl dy- ( 5 3, 5.11) . J
rm2 Y
Section 3
Clean mrphisms
3.1
Cleanness theorem. s t
x be a point of
mrphisn,
x
f:g+X
be a l o c a l l y f i n i t e l y presented
and let y = g ( x )
. The follawinq assertions
are then equivalent:
_X/x(X)
fl
= 0
.
(i)
fi
(ii)
There i s an open subschew _V
=
,x/x @sx
the diagonal mrphisn 6u/y:
--
K(X)
g+
x-
x U
the following property: f o r each scheme _Z r
the equations gIZred =
( iv)
For each local ring
containing x
such that
is an open embedding.
There is an open neiqhbourhooa _T! o f
(iii)
g,h:Z!
of 2
x
in
_X
which has
and each pair of mrphisms
-hl-Zred and (flrl)g=(f/Y)_h C , each ideal I pf C of
2=h .
vanishing
square and each catmutative square
9 )I are local hamanorpkisms, there is x:dx*C such t h a t $=xcx _and c a n o x = $ .
i n which @ phism
a t mst one hommor-
I,
5
4,
SMOOTH PDWHISMS
3
l-0
(V)
ox/my (Ix
133
is a sewable f i n i t e dqebraic extension of
Proof: Wemy assume that -
K
(y)
.
.
and y = Sp - k Since RA/k is a f i n i t e l y generated A - d u l e , the equation R (x) = 0 is equivalent t o x = s A
$4
z/x ) x = 0 , or again, to the existence of such that R z/x =;I Q-U/Y_ = 0 .
an open neighbourhood _V of
(R
Replacing
(i)+(ii):
that RAA
. Our
= 0
2
by the open subschane
1
b e , we m y assume
statenent then follows by 1.3.
_s and -h
(ii&=+(iii): Let _m:_Z - + g x U be the mrphism having ccnrponents Y-
u/z . Since
Then r ~ 1 - Zfactors ~ ~ through 6
-Zr&
has the same underlying space as _Z
.
whence g=h (iii)+(iv):
x,x'
If
x
,
6
94
.
is an open anbedding and
_m factors through
6uv/y
- -
,
are two hancenorphisms satisfying assertion (iv) ,
apply (iii) to the p a i r of ccarrposite mrphisn~
(iv)=+(i):
c be the local ring a t x of the first neighbourhood
Let
x
+ X of in hence C / I =
X xL.X
ox
. Then we have
(2.4) ; l e t
C = ~x6(Rx/y)x
and we may set $I =Id
,
+ =gx , L
-
where g:X - -6 - + -Y
x,x': Clx=tc
canonical mrphisn. The local hcmamrphim
I = (f?g x ' x ;
associated with
prl0 62,pr2062 : x 6 z g x then s a t i s f y assertion (iv). Thus we have Since the dx-mcd.ule
is generated by the elements dg
(Rxx/y)x
6 cdx (2.5) , it follc&-that
BBkB
= (3
X/Y ) x
= 0
.
=XI
x=x' . (5) -x (5) ,
By 2.6 we have
(v)====+(i):
here B
(Q
is the
/m 4
X
Y
X
. If
B
is a separable f i n i t e alqebra over
~ ( y ),
is semisiqle and the kernel J of the c a n o n i d map B@lkB-+B satis2
fies J =J (i)==3(v) :
. Hence
R
2
B/k
S e t Y_'=
=J/J = O
sp
.
and (Rx/,)x=O
~ ( y )and
=f
-1
(y)
. Then we have
134
Au;EBRAIc
with the above notation. Replacing = Szk
assum t h a t
X
I, 5 ' 4 , no 3
c.zmErRY
by 5'
for saw f i e l d k
, we
and Y_ by U_'
. By 2.10
, we
may then
then have dimx& = 0
and dx are f i n i t e algebras over k ( 5 3, 6.1 and 6.6) . If -kso tish aant algebraic closure of k , it remains to show that dX"k'; is a product of copies of k . Now, f o r each pint
,
K(X)
with projection x
Since that
x
o
=
I
we have
-X , we
is a closed point of 2
R- -(x) = m-/mX/k x x
, whence
Definition:
3.2 be a point of
. Then
x
Let
5
.
-
(2) = k
m - = ~ and X
is said to be clean
presented and sl,/x(x) = 0 By 3.1 and
K
k =J-X
. By 2 2, .
f:s-+x be a mrphisn of
there is an open neighbourhDod 1
to be etale a t x
have
Olf x
. Lf, mreove.rI
schemes and let x
(ornon-ramified)
such that
3, 3.13, the set of p i n t s of
fig
I!
is etale (resp. clean) i f
. If
,f:
k
Cg , we
st x
if
is locally f i n i t e l y
f is f l a t a t x
,
is said
f
x a t which f is etale (resp.
clean) is open. W e say that f is etale fresp. clean) i f clean) a t each pint of
it follows
f is etale (resp. of k-schanes
say t h a t a mrphisn f
is etale (resp. clean). we aim say that a
-X is e t a l e (resp. clean) i f the structural projection
k-schm
gx: & + 9 k
is an etale
is etale (resp. clean). Finally, i f
(resp. clean) k-algebra i f the k-scheme
A
e& ,
!~J~A
we say that A
is etale (resp.
clean). Proposition:
3.3 Proof:
i s the inverse image of the diagonal of
Ks(f,q)
mrphism
Given a diaqram of schemes
52' Xzy with ccanponents f and 4,
-
f
-z
h -+
z
where
U_ xzy under the
. Now apply 3.17ii) .
.
Corollary: Let p:Y+z be a clean mrphism and _s a section of p Then - _ _ s is an open gnbedding. -
I, § 4 ,
-Prcof: -
4
SMCXt'H P.IDRPHISMS
s induces an ixmorphisn of
. I-f
a p i n t of _X
onto K s ( % , I f $ )
_u is etale a t x
, then
the map
(nu/) x:
u* ("/z)
Proof:
W e may assume without loss of generality that
+.
--
.
-
k t _x-!&ysz be a diagram of schemes and x
Propsition:
3.4
&
135
("/z)
1s bijective.
--
Q
is etale. Now con-
sider the diagram
Since t h e right-hand square i s Cartesian and
(u E l ) * ( Q y /-z )g prl
embedding,
6$/:/y(~incl)
4.1
-
34
is an open
mi be identified w i t h R_x/z ; the assertion
Definition:
X , we
-
WhCl
follaws.
3Icoth mrphisms
Section 4
x of
-
and 11 xzg is invertible (1.4). Since 6
induced by
-
+
g xZg is f l a t , the mrphism
say that f
Given a mrphism of schemes
L:?
+_Y
and a p i n t
is smooth a t x when the following conditions
are s a t i s f i e d : a)
of
there is an open neighburhood _V
x such that fig i s f i n i t e l y
presented; is f l a t a t x ;
b)
f
c)
tsl_x/_y(x):
f
K
(x)1 2 dimxi' (f (x))
.
is said to be m t h i f it is smooth a t each p i n t of
X
. If
zllxm3mc c2xMETRY
136
is a Cartesian square of schemes and x'
4' (x')
(2.3).
=
x
is a pint of 5'
5 4, no 4
such that
, then we have
Hence
and similarly
-1 -1 dim f (_f(x))= dimxI-f' ( f '
( X I ) )
X-
(5
I,
is snooth at x' ; and conversely, if ct f ' ( x ' ) Given k point x
f is smooth at x , t h s g' g' is smoth at x' and g is flat
It follows fran 5 2, 3.2 that f i
3, 6.2).
, then
f
is snooth at x
.
€5 , we say
that a k-scheme & is k - m t h (resp. k-smoth at the if the structural projection -px: & + SJ k is s11l0ot.h (resp.
-
no risk of confusion, we shall also say "smooth" instead of "k-smooth". Suppose in particular that - k is a field and k' is
mth at x ) . When there is
an extension of k
. The residue fields of points
~ ' € 5k k8' projected
onto x are them the residue fields of pints of S i K(x)Nkk' and 5.7).
It follms that, if k' contains
K(X)
.
(5 1,
, then there is a
5.2
rational
point x'CgBkk' which is projected onto x By what we have already said, it follows that, to verify t h a t g k - m m t h at x we need only verify
that 4.2
XBkk'
is k ' - m t h at the rational point x'
...,Tr]
Let B be a model, A = BITl,
n d a l s in r indeterminates and
g=
.
the algebra of p l y -
. We knm that
R is a free A/B (2.6) ; accordingly for each x € 5 , the A-module with base dT1, ,dT, r a g e s dTi(x) of the dTi. form a base for QxjB(x) = RA/B8A K (x) If PEA, write @(x) for the h g e of dp in R (x) Then we have X/B *(XI = 1~ aP~ ( x ) d T ~ ( x ) i
...
Sz
A
.
.
137 where
-€ A aTi
Smoothness theorem: x a pint
schemes,
ap and ,,i(x)EK(x)
L e t f:X+!j of X g&
.
be a locally f i n i t e l y presented mrphism of
y=f(x)
.
(i)
f is smx~tha t
(ii)
There is an open neighbourhood
and a mrphism q:U_+Y_x
on
x
. Consider the following assertions: x
,a
natural nLmJ3er n
which is etale a t x and s a t i s f i e s
For each local rinq C€&
(iii)
of
, each
ideal I
of
C
=
K,OCJ
.
of vanishing
square and each conmutative square
where @
x: ox +C
and
are local hammrphisns, there is a local hmmmrphism
)I
such that
Ji=x-fx
&
~ a n o x = @
.
There are affine open subschemes _V,_V
(iv)
..
..
of _X,y
such that
f ( 1 - 1 ) ~ y , plyncmials pl,. ,pS€ 8 ~C T )~ , ., T ~ I and an open anbeddins g * y(pl,. ..,pS) such that the matrix (aPi/aT.7 (-h ( x ) ) i s of rank s -f IF = p r o 5 (Ebeing the canonical projection of the subscheme V_(P1,. ..,P S ) of yxgr defined by P1,...,Ps onto Y .
h: -
ox
(V)
and
-
...,TnII .
p b r a of f o m l p e r series d “T1,
Y
Then we have holds when
oy
(i)<=> ( i i ) < = > , ( i i i ) < = > ( i v ) (v) <=
i s metherian and
K(X) =K(y)
ox A
(Iy are ncetherian and
i m r p h i c to an al-
. The implication
.
The proof of the STTDOthness theorem is deferred u n t i l Section 5
(i) =>(v)
. I n order to
reduce it to manageable proportions we have l e f t out m y details: we advise the reader to approach it only when he is feeling particularly industrious! 4.3
schemes
Corollary:
f:X-+y is
The set of points of 5 a t which a mrphism of
stcoth is open.
Proof:
By the equivalence
(i) <=> (ii) of 4.2.
and Z Y-+ Z-
Corollary: Given mrphisns of schemes f : X + Y
4.4
if
,
& g is m t h a t f(x) , then gf_ is smooth a t
f is mth a t x
x .
Proof:
Apply criterion (ii)of 4.2 and the obvious f a c t that the composition
of two etale mrphisms is etale.
Corollary: Given a locally f i n i t e l y presented morphim of
4.5
schemes
f:_X+Y , the
follminq assertions are equivalent:
is smooth.
f
(i)
+
(ii)
For each model
each
w EY(C) and each
uEX(C)
Proof:
, each ideal vE&(C/I) , L f f(v) C
and f ( u ) = w
such that
.
I =
of w
C/I
of vanishinq square, E Y_(C/I) , then there is C
Assertion (ii)msans that, given a CQrmUtative diagram
#
%(C/I )
i SEC
W
#
2
.
X such that u# = v and _fu#=w# If C is a #local ring and u ,w send the closed points of @(C/I) and SJ C respectively onto x t & and y , then assertion (ii)is equivalent to assertion
9 C there is a u:%
can
-+
if
(iii)of the smoothness theorem. According to this theorem, then, _f
srooth i f f (ii)holds for each local ring C Conversely, i f
f
5 0 %
: %(C/ I )
enough to show that:
if
SP
* (Sp C L
placing X by the C-schm
c
C)X$+Sp C such that the ~csnpo# Rehas cm&nents can and v
: (SJ
3 T with structural projection
2 is a smooth C-schem and
I
of vanishing square, then the canonical map T(C) +?(C/I) To prove this last assertion, l e t
C
r
1 2
(ii =>(i) ,
is srooth, the existence of u# i s equivalent t o the
existence of a section 5 of f sition
. I n particular
l e t 11 be the image of P
-sp f
is
r
it is
ideal of
C
is surjective.
. For each prime ideal p of T ( C /I ) . By (iii)of the smmthness P P
q ET(C/I)
11 i n
.
I,
9 4, no 4
SMOOTH MOWHISMS
139
.
theorem, there is a 5 E ;(C which is projected onto ?-I By 5 3 , 4.1, P P P there is an f EC , f fp and a SfEx(Cf) which is projected onto the image of rl in x(Cf/If)
qf
. Fran these
such that the n which is projected onto
(9C)
fl,...,f
q
- .
cover Sp C The existence of a 5 E;(C) fi ET(C/I) now follows frm 2.8.
Corollary: Given a model k and a locally alqebraic k-schm
4.6 X
f's one can choose a sequence
, the following assertions are equivalent:
21 isk-smooth. (ii) For each Cc 4 and each ideal I the canonical map &(C) -+ x(C/I) is surjective. (i)
of
C of vanishing square,
Corollary: Let f:g-+x be a locally finitely presented m r -
4.7
~
phism of schmes. The following assertions are equivalent:
.
(i)
f is etale
(ii) (iii)
f is smooth and dimx_f-'(f_(x)) = 0 for each xE2
-f
.
satisfies assertion (ii) of 4.5 and the element u of that
assertion is unique.
z
For each schane 2 , each subschme 2' of defined by a nilpotent ideal, and each pair (a,%') consistins of mrphjsns g:_z-+u _g ' : Z- ' + X- such that _ f s _ ' = g ( c, there is a unique _h:Z-t_X such that -g = -f h(iv)
and
h(F' =g' .
Proof:
(i) => (ii): T h i s follows imnediately fran the definitions. -1
(ii) => (i): If dhXf
(_f(x))= 0 , then n = O
in assertion (ii) of the
snoothness theorem. (i) => (iii): The existence of u follows fran 4.5; uniqueness follows fran assertion (iii) of the cleanness theorem (3.1): for if u and u' satisfy the conditions of 4.5(ii), then u#=u'# by 3.1, whence u=u' (iii) => (iv): If
2 is the ideal defining
cmtative square
. p q
2'r
x
T
.
,Z' , we must show that, for each
1 40
ALGEBRAIC GFcMETRY
such that q; = ~ ' ~ + ~ l 2 W e may assume without loss of generality that
there is a unique
.
fg;+l=gr+l
-
y(#r+l)* _X
case, condition (iii)inplies that, _Z
%I
I,
, there
. H e n c e there is a % This h is the required mrphisn.
-_
have fix,,y=O
,
t h e uniqueness f o r each V_
. By 3.1
is smooth (4.2)
f
,
.
also
(i) follows.
and
Corollary: Let k be a d e l l k '
4.8
and
-
(iv) = > ( i )(iv) : implies (iii), so
V ( f and )
2 = O - 11Inthis of
_hv: _V+X such that -%=g[u , then _hv=-$ly by v i r t u e of unique _h:g '5- s&h that &=&lV_
Of
4 , no 4
f o r each a f f i n e open subscheme
i s a unique mrphisn gnz' - = g_' I U_n z -l I f VCrJ
.
9
a k-model which is f i n i t e l y
generated and projective a s a k-nloclule, and X a l o c a l l y algebraic k'scherne If X is each f i n i t e subset of which is contained in an a f f i n e open subset. mmth over
proof:
7 7 - X
k'/kBy the construction of
. W e must show t h a t ,
I _
k
, then
k'
is m t h over k
9
( v k X ) (C) + ( v k X ) (C/I)
But by the definition of the functor X(CBkk')
+
X((C/I)@kk')
is l o c a l l y algebraic over
I X k'/k
and each i d e a l I of C of vani-
for each C €-IY
shing square, the map
,
1, 6.6
.
, which, by
/k
i s s u r j e c t i v e (4.6).
t h i s map may be i d e n t i f i e d with
4.6, i s surjective.
Corollary: Let x be a point of a s c h m
4.9
l o c a l l y alqebraic over a f i e l d the local r i n q
Proof:
dx
k
,
If X
which is
is m t h over k & x
,then
is an i n t e q r a l domain.
6y 4.1, there is a r a t i o n a l pint x E X @ k ~ ( x )which is projected
onto x
. Since
8% is
f a i t h f u l l y f l a t over
is injective, and similarly f o r O2 is an integral d m i n , so is @ x ' Remark:
A
+
dE
A similar arcjument s h m s t h a t ,
dx , the
. Since 8,
if
4
hcxmmrphism K
[CT,,
(XI
is stmoth a t x
ox Jz -+
....,Tn 11
, then
the
ox
i s regular, i.e. is of f i n i t e hawlogical dimension. For since ring is f l a t over dX , we have
8%
f o r a l l dx-modules
M,N
. If
w e know that
K
(x)"T1,
...,Tn 11
has hamological
5
I,
4, no 4
141
, we
dimension n
for
i>n
have
, whence OX
for
i>n
Tori (M,N) = 0
.
Given a reduced locally algebraic scheme over a
Corollary:
4.10
perfect f i e l d k
, the
set
1
of points x E.X
open and dense i n
X.
Proof:
is open. To show that
By 4 . 3 , _V
assume that
g
s b that _V
X_
n
K
where A
. For
such a p i n t
is of dimension
Iv&lK(<) ( C ? K ( 5 ) , k l ~ ( < ) )
(5)/k
n over
K(<)
is reduced, we have
K(S)
is of dimension
f a c t that
x
Section 2. Hence
5 is
smooth a t
obviously
5
.
. Since
K(<)
is separable over
n over
K(S)
, so
(Alg. V, =
5
0 , so 5
that
9, th. 2 ) . I n view of the
that
R
-
K([)/k - "z/k)< by
The product of two irreducible schemes over an
Corollary:
4.11
11, we m y
5 , the transcendence degree of t h e
~ ( 5 )over k is n = d h X 5
residue f i e l d
,
is smooth is
is dense, that is, contains the generic pints of the irreducible
canponents of k
is dense in
X
is a f i n i t e l y generated k-algebra. W e now
-
= SpkA
a t which
algebraically closed f i e l d is irreducible.
Assume w e have proved the assertion f o r the product of two a f f i n e
Proof:
irreducible schemes.
Let _X and
we have to show that i f
W_ and
Wl
be two a r b i t r a r y irreducible k-schemes;
are open and non-eqty i n g x y
, then
their intersection is also non-empty. Now obviously there are non-empty that
w
meets
Since _Vry is
, u' ,
y , V_' , _V" of y such v,xy , y1 meets ylx yI and g"c u n g l and y"c ynXt irreducible, _W n (usIT) , and hence W , meets y'' ;
a f f i n e open subschemes V_
similarly w_' meets
w n (KJ'x y ) r g ~ n
( ! 1 I
yl'
x
x
,V"
,'I
. Since
of
X_ and
y"x v_l'
is irreducible, it follows that
ytl) i s non-empty, and s i m i l d y f o r _w
W e now turn to the a f f i n e case. Since we m y assume that
duced, it is enough to show that, i f
A
5
and
.
are re-
and B are two k-algebras which
are b t h integral d m i n s , then t h e product
.
ABkB
is an i n t e g r a l d m i n .
1 42
ALGEBRAIC GEKNEIRY
For this purpose w e may obviously assume that
A
A
A
If
i :A + A
m
injection of B[[T l,...,T
n
is the canonical map (which is injective:)
11
f :X
...,Tn ]]@,,B , which,
,
(@i) akI$
is an
as a subring of
is an integral dcsoain. Hence A @ B is an integral damin. k
+Y be a dominant mrphisn of
k-schemes which are algebraic, irreducible
and reduced. Then the set of points of 3 a t which f anddensein
Proof: If
and an ismrphism
Corollary: L e t k be a f i e l d of characteristic 0 and l e t
4.12 c -
A
.
k[[T1,...,Tn]]
A@JkB into k"T1,
4 , no 4
is f i n i t e l y generated. By
4.2(v) and 4.10 there i s then a maximal ideal m of
@ : Am
5
I,
5 . is said to be dcaninant i f
Recall that f
x and y are the generic points of
5 and
residue f i e l d s
Since
X
are reduced, the local rings of
that
K
is smooth is open
K(X)
and
K(Y)
sl,/p
=
--
_f(g) is dense in
and ,U
, then
f(x)
,
f
. Since
x and y coincide with the
. It follows frcm the r e s u l t s of Section 2 =
K
(X)/K (y) '
(x) is a separable f i n i t e l y generated extension of
Since -€ is f l a t a t x
= y
1x1 .
is m t ha t x
K
(y)
, we
have
. The r e s u l t now follms frcan
4.3.
4.13
Corollary:
snooth a t x tx -1 dhxf (f(x))
.
, then
*
Given a mrphisn of schmes
_ _
i s a free
0X-module
f:X+Y , If f i s
of rank
I, § 4 1 no 4
SXOTH MIRPHISMS
143
Proof: With the notation of 4.2 (ii) , i f we set 2 = g 0n , then i s a free 0 9 -module of rank n (2.6a)) . Since -g*(Rz/u) x
(XI
by 3 . 4 , we see that
is free of rank
(R,&x
n
1x -
. Setting -y =g (x) ,
corollary 9 3, 6.3 applied to the mrphism g ’ : f-’(y) implies that dimf
(fi5/x) g(x)
+On -K(Y)
induced by 4
I
-1
(y) =dim On = n , g’ k 1 - K (y) -
X-
and the assertion follcws.
Corollary:
4.14 k-schemes
f:g+Y_ , a
x
k - m t h at y
point x o f _X
, then
m t h at
(i)
f is -
(ii)
The map ?/k(’)@K
jective.
, a mrphism of algebraic y = f ( x ) , if X g k - s m o o t h
Given a rrcdel k
x
,
.
(y)
K(X)
R
-+
_X/k
(x) induced by
Proof: (i) => (ii): By 2.9 we have an exact sequence “Y/k’y -
the functor ?
@
52
ox
K(X)
X/k
%y x‘
at
the following assertions are equivalent:
is in-
’
* (‘_X/k)x +. ‘ R ~ / $ x-+
transforms t h i s sequence into the sequence
,:
( y ) @ K(X) K (y)
-+
52g/kw
+
x/ll fx)
R
-+
0
.
, the three
terms of -1 this equation are identical with dimx( ( & @ k ~ ( z ) ) dimxg (y) and But if z
denotes the canonical &rage of
y
in Sp k
,
dim fY@ K ( z ~ ) respectively. The desired q u a l i t y now follows f r m Y- k
9
3 , 6.3.
144
I, § 4 , no 5
ATXEBRAIC GECBETRY
f i n i t e rank (4.131, assertion (ii)means that an isomorphisn of
(9 ) @ -Y/k y 6y 3, prop. 6 ) . Since
comn. 11, 8 each k-derivation of vation of
dX
Y
)
@
Y/k y 8 onto a direct factor o?
8x
.
. If
D:
dx-+ I
: dx +C
X I
canox' = 4
such that
is a k-derivation extending )I -x'fx
Remark: The vector space over
t
G x ~ k ( X ) = k'kd -K
(XI (n_X/k(X)
I
K
x iff
K
space a t y
(v) = > ( iwe ) :know that
of
Z-'(y)
( f i g . m. 111,
at x
and ifix
8 5 , prop.
dx
is f l a t over
4). If
ax A
3
K(Y)
-
dx= Jx/Jxmy
is its maximal ideal, we have
( 2 . 2 ) . These iscmrphisms a d the isanorphisn
*lY
f
. W e may rephrase is smooth a t
.
5.1
Y
into
t h e map x = x ' + D
g at x
(x)= K (y) by saying that
Proof of the snoothness theoren
0
.
I
Y
K(X))
Section 5
over
(using t h e notation
induces a surjection of the tangent space a t x onto the tangent
f
A
there is a
(x)
is sometimes called the (Zariski) tangent space of
cor. 4.14 i n the case i n which
I
is then a k-derivation of
X
is a hammrphisn satisfying J, = x f x and canox = 4
4.15
I
( 2 . 5 ) . This enables us t o verify assertion l i i i ) of
of 4.2 (iii))The k-linear map $ -x'f I
(A1-s-
(Qyklx
the stmthness theorem ( 4 . 2 ) : for since g is k-smooth a t x k-algebra hommorphisn
is
('XJk'x
-+
Q&/k (2.6) (Qy/k)y "13y/k and into an -ox--module M m y be extended t o a k-deri-
0
into M
ax
(R
[[T1,.
..,Tn]]
B
Y
if
A
dX
is f l a t
is the local ring
1,
5
145
4, no 5
Likewise,
-
dim f - l ( y ) = Kdim 0 = Kdim X
X-
(5
c?
dX
= n
3, 5.14).
(ii)= > ( iSince ): g is f l a t a t x -
5.2 g(x)
, f.
is f l a t a t x
we may c l e a r l y replace
-f - l ( y )
+
Sp
K
(y)
. To show that
g
by f-l(y)
. The proof
is thus reduced t o the case i n which 2 = Sp -
we deduce f r m the canonical isanorphism
that
dimx& = dim
5
3 , 6.3 we have X'
g (X) -
By 2.10, we have
is f l a t a t
and f by the induced mrphisn
Setting
( 3 . 4 ) . On the other hand, by
and prl
+ dimxY-l ( -g (x))
.
K
(y)
.
I, 5 4, no 5
Au;EBRAIC GFXXEIRY
146 whence dimx%
= dim
X’ = n
9 (XI-
( 5 3,
6.1) , and the implication follows.
We ncw lead up to the implication (i) => (ii) by proving the
5.3
following result: given a field k and a pint x of an affine algebraic scheme X_
, if
hXk(x)
:K
(x)3 s. dhxz
k-schanes g-: g +$which is etale at x particular case in which = s_P k
, then there is a mrphim
.
of (i) => (ii) in the
. This proves
...,fn€dk (2) such that the canonical images
Consider functions fl,
(X) form a base for R
of the differentials dfiER
Xk;
Z/k
(x) over
dfi (x)
. We
K(X)
claim that the mrphisn g:g+Qk with cmpnents fl,...,f n satisfies the n required condition. For let Ti be the ith canonical projection of Gk
onto Qk and let g* - (dTi) be the canonical image of dTiE )EQ( ,Q g*(R ) (g) The canonical mrphim g*(R
.
- $/k g*(dTi) -
-
, so
onto dfi
-
the map g*(R
$p)x
$/2 -+
“_./k
R
-x/k x
in
sends
is surjective. Replacing
-X
by a smaller affine open subscheme, we m y assume that g* (R - -k @/k’ + “2_X/k = O and it remains to shcrw is an epimorphism. By 2.9, we then have
that g is flat at x
“-./$
.
To prove t h i s last assertion, consider
set g = X W k K
, q-= g-@kK
an extension K of the field k and
. For sufficiently large extension, K , there is a
rational pint 2 E X which is projected onto x EX
(5 1, 5.2 and
5 . 7 ) . If
is flat at X , then g is flat at x , because 9, is flat over dx Since [Q- (2): K ( Z ) ] = [Oxk(x) : K ( X ) I by 2.3, and and $g(x) X/K dim-z = dim X (5 3, 6.2) , we may assume that x is rational. Setting XXn 4 z =g(x) , we then have dx/mz (3x = k (3.1) It follows that d , -+ dx is -
.
,-
.
ax
the Gz-adic filtrations and apply Alg. surjective (assign 0Z and m. 111, 5 2, 110 8, wr. 2 of th. 1). Now if z Ekn is of the form i (bl,--.,bn)
8
...,T -b )
(T1-bl,
is the ccanpletion of the local ring of k[T l,...,Tn]
, and
at
is hence i m r p h i c to a ring of f o w l p e r Series. A 4 In particular, dZ is an integral damin. If Jz dx were not bijective, “4
-f
then we would have
dimxz = Kdimd X
=
A
A
Kdimgx < Kdimdz = n
,
5
I,
4 , no 5
SMCDrH MOwHIm
contradicting the hypothesis n
dx
that
6,
is f l a t over
.
dhxX_
5
147
. This shows that
.
set ~ = r l ( ~, ) B = D ( Y )
A
5 and
(i)=>(ii) :W e may c l e a r l y assume that
5.4
"
d, dx
_Y are a f f i n e .
F i r s t of all suppose t h a t B is noetherian. A s in 5.3, choose i n such a way that t h e inages dfi(x) form a base f o r
,x/z (x)
R
and hence
fl,...,fnE
of t h e d i f f e r e n t i a l s dfiE R
Z/X
A
(X)
. we claim that the mrphisn -g : z +- ~~9 - with is etale a t x . The equation Q n(x) = 0 _x/lcx_o -1
is ccanponents flfll...,f n established a s i n 5.3. Also, by 5.3 applied to the K (y)-scheme f (y) , the g is f l a t a t x (y) I n other mrphism &) o;(y) induced by -
. By M g .
-+
z = g- ( x )
wxds, i f
5
, 0x/mY 3X
is f l a t over d /m
5, no 4, prop. 3, dX i s f l a t over
Lf
dz
B U o t noetherian, we apply the
and x = p
. Using the notation of
.
an open
yo
go:
(Sp ko)x Qn
Uo
-+
Z
cmn. 111,
lemna of 5 3, 3.14, setting B = k
go:
, then
SJ A.
-f
S l ko
we have
-1
in gp .A
such that po
€zo and an etale mrphisn
such that JolUo = prlogo
. W e f i n a l l y set
(ii)=> (iii) : Clearly we may assume that & and
5.5
is the
is (yo) ( 5 3 , 6 . 2 ) . It follows that $'(yo) PoBy the regnarks above, -fo i s smooth a t po ; hence there is
and dim f - l ( y ) = dim f
.
13
Y
that len'ma, i f
structural projection and i f we set yo=,fo(po) -1 -1 f (Y) f o ( Y ~ ) @ ~ ( ~ ~ ) KI ( whence Y)
pm t h a t po
=
.
.
a f f i n e and that g = ~I f
g - is etale a t x
,
g
g are
is etale i n a neighbur-
hood of x , so that we may assume that we are given an etale mrphism s:X -+y x on such that f =pr1og With this s u p p s i t i o n i n force, we prove the following assertion which obviously implies (iii),namely: for each C €2 , each i d e a l I f C of vanishing square, each w E Y -( C ) and each vCX_(C/I)
.
such that f (u)= w -
v =
, there is m e u E&(C) such that uC / I C/I (to see that this implies (iii), set w = ( E (3+))b f(v) =w
kX(% Wb and.
u =
(Ex@
X))b
)
Y
*
=V I
@
ALGEBRAIC GEDMETRY
148
5
I,
For the proof of t h i s assertion, suppose that g(v) = (f_(v),E1,...,cn)
. Let
with Ei€c/l
cl,...,c
n
-
El, ...,c
be representatives of
4 , no 5
-
c
in
,
.
n Replacing f_ by g - and w by ( W r C l r . . . r ~ n) , we see that it is enough t o prwe the assertion in the case in which f is etale. In this case, set Y ' = SxC
,
-
Y" = Sp(C/I)
duced diagram
, 2'
"6 -
= Y_'
II
3'' = y"
X
9 and consider the in-
#
W"
gyl,
The mrphisn
v#
. By 3.3
such that
has a section _ s ' ' : ~ " + ~ " x y X - with cQnponents I$,,
and-§ 2, 7.6b), there is an open &d closed subscheme
_f,,
Z''
induces an isomorphism of
fy the underiying topological spaces of
&(P) s J (-X 1 ) / I ~ ( g1.' ) So l e t L
onto
&'' and
X'
Y'' . Now we
and
-
A''
5"
of
can identi-
(since we have
z' be the open and closed subscheme of -X '
3'' . Then f,, induces a ~ ' € 5 'and y'= f Y l (x') r dy,/IJYl* I&/IL?~, is bijective. By N a k a y m ' s lema, +dxl , it f o l l m that dyl surjective. Since Jx, is f l a t over J Y' that gyl , induces an isamorpkism of Z,' onto y ' . Hence _fyl tion E~ and we need only set u = ( If~ ~ 2 ' ) ~
which has the same underlying space as
hameamor-
phisn of
the map
z' onto
y'
. Wreover,
if
-
.
is then
JxI , SO has a sec-
Before proving the equivalence of (iii)with (iv) , which we
5.6
leave u n t i l 5.7, we make scane prefatory rararks. Suppse that X_ and _Y are affine and set A = cl(g) over
B
, we
,
.
B =J(_Y) Since A
m y assume t h a t & is the closed subschane of
by a f i n i t e l y generated ideal P Let
R
, and that at x
be the local ring of Y_X$
Setting Q =Px
r,
i s a f i n i t e l y presented algebra
, we
have R =R/Q
p h i s ~ ~ ~and IJ turn )I = x o f x implies that x
ox and
f
is induced by prl:y
, and K'
that of
. With the notation of C
Y_%$
x
X
or + y .
at x
into B-algebras, and the equality T1,...,T
.
(iii) , the hornanor-
is a B-algebra ham~~rphim. Thus l e t
be the images of the indetenninates
defined
tl,...,tr
r urader the m p s i t i o n
-
I, 8 4, no 5
SMOOTH MORPHISMS
-
149
-
if tl,...,t r are representatives of tl,...,t r in C , we evidently obtain a ccmnutative square of -4 :
A
C-R
such that I(Ti)=ti
,
.
i = lr...rr
Given this A , the other mrphisms A':R+C of $ such that canoA' are of the form A'=X-D , where D is B-linear and satisfies
, we
Denoting the equivalence classes rrod Q of x a d y by 2 and A(x)D(y) + A(y)D(x) = $(x)D(y) + $(y)D(x)
We can then assign I the
can
.
D (xy) = A (x)D(y) + A (y)D(x) have
=
.
(R/Q)-module structure derived from $ and the
canonical(C/I)-module structure. The anditions imposed on D then mean simply that D is a B-derivation of R into the (R/Q)-module I see that the existence of a mrphim x:R/Q-+C of
&
. We now
such that canox = $
is equivalent to the existence of 'aderivation D such that D IQ
=
A IQ
. We
now reformulate this condition in mre erudite terms: Let 6: R
I
-f
RRIB(BR(R/€?) be the derivation x + d x @ l
that each B-derivation D of R into an (R/&ncdule expressed in the form D = t 6 , where 2 linear map. Wreover, since 6 (Q ) = 0 2 1 j : Q/Q -,RRIB@.,(R/Q)
e:
I
. It follows frm 2.5 M may be uniquely
1
Q~/~@~(R/Q) + M is an (R/Q)6 induces an (R/Qflinear map
.
2 If h:Q/Q2 +I denotes the (R/Q)-linear map induced by X (A (Q = O!) I the existence of x is then equivalent to the existence of an (~/~)-lin= map A
such that
Aj
=x . We deduce fram this that assertion (iii) of the smoth-
ness theorem means that j is an i m r p h i m of Q/Q2 onto a direct factor WQ) of the (R/Q)-module oA/B~R
.
To prove t h i s assertion, notice that, if j is such an iscanorphim, there
.
is obviously an extension A of A The necessity of the condition is proved 2 2 by setting C=R/Q , I=Q/Q I $=Id and taking for A the canonical map of R onto R/Q2
. Under these conditions,
A
is in fact the identity
,
150 map of Q/Q
(iii) <=> (iv) : W e may assme t h a t
1
the notation of 5.6,
.,6Tr
K
are a f f i n e . w i t h
Y
is a f r e e ( R / Q ) - d u l e w i t h base
nRjBBR(R/Q)
jcp) =
5
has image
i n Q/Q
2
I
c giaTi .
is t h e residue f i e l d of
(x)
& and
( 2 . 6 ) . Pbreover, if pEPCBIT1,...,Trl
we have If
4 , no 5
.
2
5.7 AT1,..
5
I,
U E B R A I C GEKBETRY
R
, assertion
( i v ) of t h e mxkhness
-
... -
sends t h e generators p1 , ,P of S R/Q 2 1 onto the elements of a base f o r (QR/,QRR/Q) @ R / Q ~ ( ~ , ) By Alg. c m . Q/Q 11, § 3, prop. 6 (it is unnecessary to assume that M is f r e e i n t h i s pro-
theorem simply means that $3
p o s i t i o n ) , this implies that
(XI
K
is an i s m r p h i s m onto a d k e c t f a c t o r , and
j
a s s e r t i o n (iii)follaws by 5.6.
j is such an iSdQrphism, it is emugh t o choose P1,...Ps€P
Conversely, i f
i n such a way that
Fl,.. .,PS
form a minimal system of generators f o r
.
Q/Q
2
Then t h e matrix ((8Pi/aT.) (x)) has a rank s mreover, P1, ...'P form a 3 S minirral system of generators f o r Q , so that v(P,, ,PSI and V(P) Coinc i d e on a neighburhood of
this neighbourhood and
.
V(PlI.. .IPS)
4
x
(5 3,
4.2).
where B E M
, and that 3
nomials P1,..
s
rank
.
.,P
S
.. .
To obtain ( i v ) he take
to be the inclusion m r p h i m of
(iii)=> (i) : Since (iii) <=> ( i v )
5.8
, we
.,Tnl
t o be
V_ i n
= Sp -B
m y assume that
is t h e closed subscheme of
C BLT1,..
y
or
K
such that the matrix
defined by ply-
( (aPi/aTj)
Under these conditions, l e t Bo be the subring of
(x)) has
B generated
by t h e c o e f f i c i e n t s of t h e Pi ; l e t
xo = 9 B~ , zo= S and l e t x
0
B l r~ . .~. ~T ~
x i n -c X
. Then
rl/(~l~...~~s) ((aPi/aTj) (xo)) has
s and by 4 . 1 it s u f f i c e s t o show that the m r p h i s n
rank at x
If
be the projection of
~
. We may accordinqly assume t h a t
( (aPi/aTj) (x))
has rank
s
, some
.
zo+yo
is m t h
B is noetherian. s is in-
square s h t r i x of order
.
W of x Let X ' -1 be a closed p i n t belonging t o t h e closure of x i n yng (y) Since the set of p i n t s of 3 a t which f is smooth is open (by (i) <=> (ii)and 3 . 2 ) ,
v e r t i b l e . Therefore this holds throughout a neighbourhood
.
,
9
I,
4, no 5
MIRPHISMS
SXXTH
it is emugh to show that
is closed in f-'(y)
.
U n d e r these conditions,
f
. Thus we my assume that
is smooth a t x'
is a f i n i t e algebraic extension of
K(X)
lennna 5.10 below, there is then an
0Y-algebra
local and such that the residue f i e l d B'/n'
9
151
-
B'
K(Y)
x
. By
which is noetherian, f l a t
coincides w i t h
K(X)
. Applying
1, 5.2 and 5.7 t o f. and the c a r p s i t i o n E
- (3Y
Sp + B'-Sp
we see that there is an x'
n'
'Sp - B'
Y f which is projected onto x and
X_ xu(s_P B ' )
and satisfies K ( x ' ) =B'/n' is smooth a t x'
Y
=K(x)
.
. This we reduce
By 4 . 1 it is enough to show that
t o the case in which
K
(x)= K ( y )
.
.
5.9 below, we thus have (iii)=>(v) = > ( i )
we now prove that (iii)=>(v) w h e n Jy
5.9
is noetherian
2 tl,...,t be a base f o r m /(mx+m ) and l e t ti be a n 5 Y X representative of ti i n m . Setting S = 0 LLT1,.. .,Tn]] , we claim that K(x)=K(Y)
. Let
X
A
, is
i = l , ..., n
Y
bijective. For i f
Y
%
4 :S -+ such that (p (Ti) = ti , s is the maximal i d e a l of S , we have
t h e continuous hcanarrr3rphim of d -algebras
...&
S / ( s L t m S) % K ( Y ) C ~ K ( Y ) T ~ KK(y)Tn ~
Y
and
ax/(m:+m hence there is an $,(ti)
=Ti
.
By
8) Y X
-
K ( y ) ~ K ( yl )~t . . . ~ K ( y ) t n; 2
# -algebra hcmxmrphism q0: (lx+ S/ ( s +m S ) such Y Y (iii) , there is a factoring of q0 of t h e form
that
'.$+ S / s 3 . Continuing
similarly, $1 f a c t o r s through a mrphism q 2 : way we build a catmutative diagram of 8 -algebras Y
--
-, ;- ;4su
s/s3;;,
s/s
2
2
s/ ( s iinYS'
i n this
BY passing to the inverse l i m i t we see that the $n induce an hamanorphisn $: 2 s / ( s + m S)
;
Y
. By construction
A
dX+ S
since $$
i s an
m r p h i s n of
3
.
2
S
the s-adic f i l t r a t i o n ,
mrphism of t h e graded algebra associated w i t h cor. 3 of th. 1, $I$ is an autcanorphisn of 2
2
2
+
13Y-algebra
Y
2
Accordingly, i f we assign
4, no 5
$4 induces t h e i d e n t i t y map on -algebra hamrorphim, $I$ induces an auto-
my/" e s / ( s + m s) Y Y
s/s2
$
I,
ALI;EBRAIc GlxMmRY
152
mx/mx * mx/ (mx+m
S
dJ
o
-f
induces an auto-
cam. 111,
5
. Using the exact sequence
S
Y X
+$
. By Alg.
2,
,
we v e r i f y s i m i l a r l y t h a t $+ is an a u t m r p h i s n of
dx . The claim follaws. A
Lama: Le t A be a local r i n g with residue f i e l d K be a f i e l d extension of K . Then t h e r e is a f l a t local A-algebra
5.10
and l e t
L
B
residue f i e l d is i m r p h i c to L algebraic extension of
, we
K
. If
may take
A
B
is noetherian and L
L
over
our a t t e n t i o n t o the case i n which the extension L of a s i n g l e element. I f
, set
L
is a f i n i t e
to be noetherian.
By well-ordering a set of generators O f
Proof:
~ ! d - ~
K K
, we may
confine
is generated by
is t h e f i e l d K(T) of r a t i o n a l f r a c t i o n s in one
w h e r e p is t h e prime i d e a l of ALTl conP ' sisting of a l l polynomials whose c o e f f i c i e n t s belong t o t h e maximal i d e a l m
variable T of
t
A
. It therefore remains
i s algebraic over
P = -
al,
B=ALTI
+ T+ n
B = ACT]/PA[T]
.
.
.
Hence B
5.11
, where
all.. ,a are representatives of n , it is s u f f i c i e n t to set P = a 1+ a2~+...+a~T"-'+pand For B is obviously a free A-nusdule; mreover, mB is
contained i n t h e r a d i c a l of
B/mB = L
t o consider t h e case i n which L = K L t ]
and has a minimal p l y n a n i a l of t h e form
... +anT"-l+ T" . I f
1 - 2 in A ,a
...
k
B (Alg. m.V,
5
2, cor. 3 cf th. 1) and
is local and the lemna is provd.
Remark:
The proof given i n 5.9 shows that i n a s s e r t i o n (iii)
of the m t h n e s s theorem we m y i m p s e f u r t h e r conditions on t h e r i n g C For example,
if
k
is a f i e l d and
X
.
i s a l o c a l l y algebraic k-scheme, the
abwe arguments imply t h e following result:
& isk-smooth i f f , for each
I,
5
4, no 6
153
SMCUEI IvK)FU?HISE
local k-algebra
C
such that
k:k] < + m
and for each ideal I
of
C
of
.
is surjective (cf. 4.6)
vanishing square, the canonical map X_(C) + z ( C / I )
W e leave the proof of the following r e s u l t as an exercise f o r the reader: i f
i s an i n f i n i t e f i e l d and 5 is a locally algebraic k-scheme, then & is
k
k - m t h a t each of its rational p i n t s i f f f o r each integer
canonical map
g(k"J?1/(?+') 1
-f
, the
nrl
_X(k"r]/(T")1
i s surjective.
E t a l e schemes over a f i e l d
Section 6
Throughout this section,
a separable closure of
denotes a f i e l d belonging t o
k
such that
k
Galois group of the extension ks/k Proposition:
6.1
5
of
and
X
Ak-schm
.
k
denotes
ks
II denotes the t o p l o g i c a l
is etale i f f the space of points
X
is discrete and the local rinqs of
tensions of Proof:
.
ksc;
,
are a l l separable f i n i t e ex-
Clearly any schesne satisfying the latter conditions is etale over
g
Conversely, i f
is e t a l e over k
, each
point x
€2
k
.
i s closed (3.1 and.
so that each irreducible ccarrponent of & reduces to a single point. Since each affine open subset contains only a f i n i t e number of irreducible
5
3, 6.61,
CCklp?Onents, it follows t h a t the underlying space of over, the local rings
this, set m = O Y
ox
are separable f i n i t e extensions of
k
More-
(to see
i n 3.1 ( v ) ) .
Corollary:
6.2
5 i s discrete.
A k-scheme ~-
X
i s etale i f f
XBkks
is a constant
ks-sche. Proof:
Clearly X is etale over
k -scheme S
6.3
T
i s etale i f f Corollary:
k
iff
XBkks
i s etale over k
, and
a
is constant. If
_f:g+y is a smmth mrphism of schemes and
154
ALGEBRAIC (zKm3rRY
y is reduced, then so is g
.
I, s.4, no 6
,
By 4 . 2 ( i i ) we m y assume that & = S l B
Proof:
-
and f is etale.
Y_ = Sp A
There is a product of f i e l d s A ' C S and an injective hammrphisn A + A ' since Sp(BNAA') 3 %€3 Sp A' -
Y-
1s f l a t over A
B
(6.1). As
,
-
such that, for each x EE
an open subgroup of acts on X(k ) S
JI via
, the
Sp k
s'
centralizer
{y EII 1 yx =XI
, let
of
%\
of
x i n ll is
be the image under x# of the unique
w
K(O)
+ks is associated w i t h x#
which enables us to identify the residue f i e l d
Corollary:
on which Il acts
+
then a hcammrphisn
ated subextension of
E
. For instance, i f 5 i s a locally algebraic k-scheme, k . I f xEg(ks) and i f x# : S x ks X i s the S
mrphism associated w i t h x p i n t of
;
is reduced.
BgAA'
is injective and B is reduced.
B+BQAA'
A I l - - is by definition a a l l set
6.4
TI
,
is etale over Sp A'
ks
. This shows that
The functor zt+z(ks)
K(W)
with a f i n i t e l y gener-
i s a II-set.
X(ks)
is an equivalence of the f u l l subcategory
formed by the etale k-schms with the cateqory of Il-m.
Proof: W u l o the characterization of etale schemes f o m l a t e d i n prop. 6.1, this corollary is nothing mre than a variant of Galois theory. Sirrpsly observe that, if K is a f i n i t e subextension of ks , (Sp K) (ks) is II/II' , where II' is the Galois group of ks over K Since each II-set E is the d i r e c t sum of Il-sets of the form II/II' and since the functor x y X _ ( kS ) preserves direct sums, we see that E is of the form X(ks) , where & is etale over k .W e leave the rest of the proof to the reader.
.
Proposition:
6.5
f
gx:&
m k +
. Then there
no CX_)
be a locally algebraic scheme over a
T ~ ( X ) and a mmhism
is an etale k-scheme
w i t h the following universal property: f o r each mrphisn
-f:X_-+E of X
i n t o an etale k-scheme
such t h a t f =gqx -
.
E_
, there
is a unique g:n0(&) + g
Moreover , of
X
is faithfully f l a t and its fibres are the connected. ocarqeonents qX ( i . e . t h e open subschemes of _X whose spaces of points are the connect-
edccanpo nents of
Proof:
1x0
.
F i r s t consider the case i n which
3
= Sp -A
, where
A
is a f i n i t e l y
generated k-algebra. If we can show that A contains a ntutimal separable k-subalgebra
As
of f i n i t e rank, then the mrphism S%A+S%As A
prove the existence of
into A has the required universal property. To
S
As
, consider
the connected mn-pnents
X
A "A1%
the underlying noetherian space of
. Clearly we have
dx(Xi) , and the algebras
where Ai=
Accordingly-the unit of
= Als
Now suppose that
x
..."Ans .
i
is a
.
=\
formed
of _X and inclusion mrphisms
set n0(g) = ~ ~ , J associated w i t h the inclusion m p of restriction
V
NGW
o(LJ
+
b(v)
serid.s
jc:no (EI) +no (V)
8(v)
O;Jl)e ~t q-v : ~ - + n ~ ( u ) J(~=J) i n t o J ( ~ J; i f onto
V
o(v)
-jc% =';[v'; f r m the construction of direct limits in-Es
mrphisn
such that V
( 1 no (g) I , 1 2 1 )~ etale k-scheme. By the caparison theorem
limit of the diagram
(no@)
.
m
between them. Clearly ,X m y be identified w i t h the d i r e c t limit of
diagram.
&s
,
bi/mi:k] This shows that has a largest element Ais
5 i s arbitrary and consider the diagram of
by the affine open subschemes _V
sV: v + u
is a f i e l d . I f
Ai 5
the u p a r d directed system formed by these K S
...%An
of
n
i s i t s sole non-zero idempotent, so that each
Ai
maximal ideal of Ai ' it follows that [K:k] A
Xl,...,X
cannot be further decamposed.
Ai
f i n i t e l y generated separable subalgebra K of
NaJ set
induced by
c
the inclusion map of
,$
V
the mrphism
,
V_CV_ the
and induces the unique
. I t follows imnediately
(6 1, no.
1) that the d i r e c t
i s the geometric realization of an
(5 1, no.
4 and 6.8)
, the diagram
; evidently the mrphism
then has a direct l i m i t ~ ~ ( gi n) &cS
gX:x+n0?g) derived from the mrphisms qu by passage t o t$e d i r e c t l i m i t has the required universal property ("the i e f t adjoint functor
no ccmnutes
w i t h d i r e c t limits"). To prove the f i n a l assertion of the proposition, we observe that the functor
n
0
c m t e s with d i r e c t sums, which reduces the problem to the case i n which
5 is
connected. Under these conditions the image of
therefore contains only a single point w of scheme of
no(X)
induced by qx -
-
whose only point is w
, then
i n question, so that
clearly
= n (X)
a field!).
6.6
Definition:
0-
(y,c$
no(x)
and i f
.
qx is connected and is the open subIf
-
-g:X+Y_
denotes the mrphism
is a solution of the universal problem
. Hence
q
-x_
is surjective and f l a t
(
With the a s s q t i o n s of 6.5, we call no(g)
is
"m
ALGEBmIc (ZXNmRY
156
k-scheme of ~nne~ted. c m pnents of g
I, § 4, no 6
-
qx the canonical projection.
11 and
6.7 Proposition: Given a locally algebraic k-scheme field extension K/k mv .i % , E K , then the unique mrphim
a
such that qJ3kK = $ q is invertible. x-ekK Proof: Just as before we reduce the problem to the case in which g is affine. We must then prove the following assertion: if A is a finitely pre-
__I_
sented k-algebra and if :A is the largest etale k-subalgebra of A , then k AsmkK = (A@kKIK s To prove this, consider the set % of field extensions k LE& such that AsWkL = (AWkL): for each finitely presented k-algebra A
.
We show that a)
If
L
a Galois extension of k
,
. For if r
denotes the K &lois group, then acts on A @ ~ K and normalizes ( A @ ~ K ) ~ BY a g . VIII, 5 4, prop. 7 , it follcrws that (AQ~K): = V@~K here A CVCA j.s
r
Since V is etale over k iff V@ K k b)
ks
rf
ks
Lt5
is etale over K
is a separable closure of k
, then k t %
S
. For in order to prove that
we may assume that Spec A
k
S
.
, we have V=As
.
an algebraic closure of
is connected. If p is the characteristic ex-
ponent of k , then each a€ABkk has scane p e r ap S that the projection
is a haneanorphisn, so that Spec A@ k kS kS As =k inplies (ABk kIs = k,. S
.
. It follows
is connected. In other wrds
S
c) If T is an indeterminate,
in A
k(T)6 %
argument of (a) to the group of autcanorphims
. To prove this, apply the
f
of x ( T ) of the form
.
s
I, 5 4, no 6
m MOwmsMs
157
aT+b
-x d) If
e)
If
&
KC%
LEEK
, then
and LE%
k'KcL
.
and a d - b c f O
with a,b,c,dCk
. This is clear.
LC%
,then
KC%
. This is also clear.
- K is the Union of an upnrard directed system of extensions KiC% f) If then KC % Again clear.
.
,
It now follows fran a),b),c),d) and e) that each finitely generated extension belongs to
5 . S o by f ) ,
every extension has t h i s property.
Corollary: The following assertions are equivalent for a
6.8
l o c a l l y algebraic k-scheme
g
:
(i) 5 is geametrically connected (that is to say, K@kK is connected for each extension K of k) ; (ii) if ks is a separable closure of k , g@kks is connected; (iii) IT^(^) is i m r p h i c to SJ k k
.
Corollary: I_f
6.9
X
is a connected locally algebraic k-scheme
which contains a rational point, then Proof: If
5 is connected,
.
T,,(&)
2
is geametrically connected.
is of the form S x K for some separable
finite extension K of k If, in addition, & contains a rational pint, , so that k = K k-tlj, hence a mrphisn Sp k+S%K there is a mrphisn Sp -k k
.
Corollary: Let - - -X _and . Y be locally algebraic k-schemes. Then the canonical morphim I T , ( ~ x -+~ )~ ~ (x 5 r 0)( x ) is invertible. 6.10
Proof: In virtue of 6.5 and 6.7 we may confine our attention to the case in which k is algebraically closed and & and Y_ are connected. We must show that _x x Y_ is connected. Since each open subscheme of _X x X contains a rational point (x,y) and
(5
3, no 6.), it suffices to show that any trm rational points
(x',y') belong to the same connected component. Now this is
certainly true for (x,y) and (x,y') (which both belong to the connected subset (9I C ( X ) ) X Y_ ) , it is also true for (Xry') and (x'ry')
(which belong to
X K (9
IC
( y ' 11 G
&
)
, and the corollary
follaws.
Corollary:. With the assmptions of 6.10, if y is connected and 2 is qeanetrically connected, then XxY_ is connectd. 6.11
Proof:
By 6.10 and 6.8, we have
§ 5
PROPER WRPHISMS
Section 1
Integral mrphisns
1.1
Definition:
f
,Let f:X+Y_
be an a f f i n e mrphism of schemes.
is said to be i n t e g r a l (resp. f i n i t e , f i n i t e locally f r e e , of rank n) if,
I! of 2 , O(_f-l@)) is
f o r each a f f i n e open subschane
an integral algebra
(resp. a f i n i t e algebra, a f i n i t e l y generated projective module, a projective module of rank
.
over
n)
@(y)
is a f r e e 0 -module X Y Y (Mg. m. 11, 5 . th. 1). The rank n(y) of this module is l o c a l l y constant is f i n i t e l o c a l l y f r e e and i f
If f
yEy
, g*(&)
by Alg. ccinn. 11, 5, cor. 2 of prop. 2. Accordingly Y_ can be covered by closed and open subschews
-+xn
f :f-'(Xn) -n When k E g
xn ,
nEN,
such that the mrphign
is of rank n f o r each nED
induced by _f
and -g is a mrphism of \AS
(resp. f i n i t e , f i n i t e locally f r e e , of rank
, we n)
-
say that g provided
.
is i n t e g r a l
zg has t h e same
property *
a f f i n e mrphisn of schemes
f:X
+.Y_ :
.
(i) f is i n t e g r a l (resp. f i n i t e , f i n i t e locally f r e e , of rank n) (ii) Each point
~ ( Z - ~ CisV an )
has an a f f i n e open neiqhbourhood V_ such that
yE Y_
i n t e g r a l algebra (resp. a f i n i t e algebra, a f i n i t e l y generated
projective module, a projective module of rank
Proof: Clearly a IJ If
-1
(v))
l&n
I
_V
we have
yEy
o(v) .
is, f o r instance, an i n t e g r a l algebra over
, we
then have
is also an i n t e g r a l algebra over
,
a f f i n e and open i n I
over J(y) .
.
is a f f i n e and open i n
so that d(f-'(y', ) fiE d(U)
n)
(i)=>(ii)Conversely, suppose that f o r each
such that (3(f
vt
an
The following assertions are equivalent f o r
Proposition:
1.2
f?(v') . I f
V_
may then be covered by open subschemes U
such t h a t the algebra
is
*i
I
.
o(u,
for each i If A = d(y) and ) &?) i fi x~ d(p-l(~) , it follms that AL.J~ is a finitely generated A -mdule fi i for each i By. Alg. cmn. 11, 5 5 , cor. to prop. 3, A L x j is finitely is integral over
.
generated over
A
, hence D(_f-l(V))
is integral over 8(u)
Let us say that a mrphim of schemes
1.3
closed if, for each mrphism of schemes g:Y'+Y_ f,': Z X Y' -+ y' is closed.
-f:Z*x
.
is universally
, the canonical projection
Y-
-
proposition: An affine morphisn of schemes is universally closed iff it is integral. Proof: This imnediately reduces to the case in which the schgnes are affine. The contention then follows fram:
LemM: For each hcmxmrphism of rodels
$:A*B
are equivalent: (i)
, the
follming assertions
.
B is an integral algebra over A Ss $ : J S B * @ A is universally closed.
(ii) (iii) For any indeterminate T is closed..
, the m p
Spec $ L T l : Spec B L T l - + Spec ALT]
Proof: (i) =>(iil. If B is integral over A , BWAA' is integral over A' It is then enough to observe that Spec $I is a closed map whenever B
.
is integral over A
(Alg. m.V,
5
2 , no 1, remark 2 ) .
.
(ii) => (iii) This is clear.
.
(iii) => (i)
Let btB
and consider the cmtative square
where B' is the localization of B at b , A' is the subring of B' generated by l/b and the image of A , $ ' is the inclusion map, and
B map T onto l/b a Closed map,
SO
. Since
is Spec $ '
is dense in Spec A'
c1
. Since
(5 1, 2 . 4 ) ,
c1
and P are surjective and Spec @[TI
I
is is injective, the h g e of Spec B
4' so that Spec $ '
is surjective. Since no
.
prime ideal of B' contains l / b , the sane holds in A' It follows that l/b is invertible in A' , that is, b / l E A ' Thus we have the equation
.
b
$(a,)
1
1
- = -
+-
$(a,) b
+...+-
4 (an) bn
whence S
b
= $
(a,) bs-'+.
for sufficiently large s
..+ $ (an)bs-n-1
.
Proposition: If f : X-+ Y- is an integral and surjective m r -
1.4
phisn of schemes, then dim & = dim
.
Proof: We have dim as
=
SUP dim I! and dim 5
v
=
SUP dim -1
r
v
. We m y then assume B =&) , 41 =d(_f). Factoring
luns through the affine open subschemes of
and.
that
are affine and set A = @@)
,
A and B by their nilradicals, we m y also assum that A and B have no non-zero nilpotent elements. Under these conditions $ is injective 2.4).
We may therefore a s s m that B is an integral extension of A
apply
5
(5 1,
. Naw
3, 5.2 to 6htplete the proof.
1.5
Proposition: If a m m r p h i m of schemes
z:z-+x is a finite
mrphisn, it is a closed gnbedding.
xi
Proof: By covering by affine open subschemes , and replacing _f by -1 the induced mrphisms f (xi) , we reduce t h i s to the case in which X = SPA and = Sp B Since the diagonal mrphism :g+zX$ is
-
-
.
-+xi
%/x
an iscBnorphism, the canonical map A@ BA+A is invertible. The s& then holds for the canonical map (A/I-A)@~/~(A/~A) + A/nA for each 1~ximi1 ideal n of B
. We then have
h/nA:B/nI2=LA/nA:B/n]
and so the map B/n+A/nA
, so that
is surjective. By Alg. c m . 11,
b/nA:B/n]=O,l
5
3 , prop. 11,
B+A is also surjective. 1.6
Proposition: Any finite locally free mrphim is finitely
162
5, no 2
I,
presented. Proof:
Since such a morphism is affine, the problem reduces to proving the
following assertion: a B-algebra is f i n i t e l y presented whenever the underlying B-module of
A
i s projective and f i n i t e l y generated. By § 3, 1.4 and
Alg. ccmn. 11, § 5 , th. 1, this reduces to the case i n which A
...an . Suppse then that we have
B-module w i t h base al,
. Clearly the kernel of
b!.EB 17
that
the hamchnorpNsm 4: BLTl,...,TnI
reader w i l l verify that, mre generally, a B-algebra whenever the underlying B-module of Corollary: Prcof:
If
A
This follavs frcn 1.3, 1.6, and
kc&
,with
+ A
such
. The
A
is f i n i t e l y presented
is f i n i t e l y presented.
A f i n i t e locally f r e e mqhism is closed
Section 2
R
i s the ideal generated by the elements TiTj- leb:jTa
(Ti)= ai
c$
is a free
aiaj= lebijaa
5
and open.
3, 3.11.
The valuation criterion for properness r
a 1k-
V
i s said to k discretely valued i f its underlying
ring is a discrete valuation ring, that is to say, a ring which is principal, l0CaI-r
and not identical with its f i e l d of fractions. I f
valued,
V
is discretely
SppV then has exactly tm p i n t s , one open, the other closed.
Definition:
2.1
A morphism of schemes
f:x+Y_
is said t o be
proper i f it is separated, f i n i t e l y generated and universally closed. If
k e g , a mqhism
A k-scheme
- of g
2%is
is proper.
said to be proper i f g,
,X is called ccanplete i f the structural projection
is proper. Notice that any closed. embedding i s proper. 2.2
If
g:X+_Y
is a proper mrphism of schemes,
.
f,:
FX:$
-
X$
+
S ek
-,Z_
Conversely, i f Y_ can be cave& by is proper for each mrphism g:Z_-ty -1 such that the induced mrphisns _fi:f_ are
open subschemes Y
-i i ,
(xi) -+xi
then f is proper. This follows easily fram 5 3 , 1.9 and the f a c t that the mrphism f, abwe is closed i f the fiz are closed. proper for each
Moreover:
-
-
I,
5
5, no 2
PRDPER FKIRPHISMS
163
Proposition: _ Let_ f : X-+ Y- and q:_Y+z be t m mrphisns of schemes.
.
If
Then:
f and 9 are proper, so is c ~ o f (b) fI gof is proper and g is separated, f is proper. (c) I_f gof- is proper, f i s surjective, and g is separated and f i n i t e l y (a)
a
generated, then 5 is proper.
(a) follows frm the correspnding properties of separated and.
Proof:
f i n i t e l y generated mrphisns. Assertion (b) m y be proved i n the same way
5
(c) of
as
3, 1.10. Finally, (c) becames clear when one observes that the
a s s w t i o n s r m a i n t r u e a f t e r a "change of base"
X
.
Let - k be an algebraically closed. f i e l d
Corollary:
2.3
h:T+Z
and let
be a ccanplete, connected and reduced k-schgne. Then, f o r each k-model
my be identified with A
dA(XWkA)
Proof:
By the l
m of
5
.
2, 1.8, we have flA(X$A);
enough to show that we have k
$(x) . L e t
f
A
,
. It is then
{(X)QkA
_h:X +gk be a function on
5
-
and l e t -g:zQk +. &S k be the structural mrphism; then g is separated and -g*& is proper. Were ,h surjective, then g- w u l d be proper (prop. 2 . 2 ( c ) ) , thich is f a l s e by 1.3. Since
. Since
h ( 3 ) is a p i n t of gk gkk
, which s h s that
is closed (prop. 2 . 2 ( b ) ) am3 connected,
5 is reduced, h, then factors through
the m p k +d(_X) i s surjective.
Lama: -Let
2.4
b&)
A
be
a noetherian local integral d m i n of
dimension 2 - 1 , m its maximal ideal, a f i n i t e l y generated extension of of -
L
such that v(x) 3 - 0 -i f
xEA
K
K
its f i e l d of fractions,&
L
. Then there is a discrete valuation if
annv(x)>O
xEm
.
v
.
be a set of generators of m Since K d h A 2 - 1 n n+l is not of f i n i t e length. Accordingly the graded. ring gr(a1 = an m /m for instance, is not nilthe residue class mod m2 of one of the x i f xo Hence no relation holds of the form potent i n g r ( A ) Proof:
xo,xl,...,x
Let
.
r-1 xo = P(xo,xl,...,x
) , where P is a homageneous p l y n a n i a l of degree n r 21 with coefficients i n A I f C is the subring of K generated by A
and xl/xo,.
..,xn/x o
, we
.
therefore have W = xoC f c
p i s a minimal p r i m ideal of and p2 n A P
=
m
. If
D
C
containing
xo
. It follows that, i f
, we
is the integral closure of
have K d h C = 1 P i n K , and n P
C
,
I, 5 5, no 2
ALGEBRAIC GEaMETRY
164
a maxim11 ideal of D , then Dn is a discrete valuation ring of K with maximal ideal nDn , such that nDnnA =m (Alg. c m . VIIl 5 2, prop. 5 ) . The valuation w associated w i t h Dn is positive on A Thus one may take for v any extension of w to L (Alg. c m . VII, 5 8, prop. 6 and 5 10, prop. 2)
.
.
k be a noetherian rodel, f:X+u a morphisn of algebraic k-schemes, x a point of 5 , y=f(x) If y I E m is a disL a m: Let -
2.5
tinct fran y
I
.
then there is a discretely valued k-21
V with field of
fractions L and moqhisms g:spkV+_Y h:S%L+_X such that fh=gjSpkL and &(L+L)={XI , and that -g maps the closed pint of s v onto y '
% -
I1 be an affine open neighbourhood of
-
.
in the reduced subY_ carried by Replacing f by the irduced morphisn -1 , we m y assume that Y_=Y-1 . Now set A=%' ; since Y_ is f (u,) assumed to be irreducible and reduced, and y is its generic point, the field Setting L = K ( x ) consider of fractions K of A is precisely 8 = K ( Y ) Y the valuation v of 2.4 and the k d e l V consisting of ttL for which . It is then sufficient to set h=E(x) (5 1, 5.2) and to take for v(t),O - the carpsition of cy,S:& g + y (5 1, 5.7) with the mrphisn induced by the inclusion m p of 13 into V Y'
Proof: Let
.
+xl
y'
.
.
I
Properness theorem: For each noethwian ring k and each
2.6
morphim f:_X+_Y
of algebraic k-schemes, the following assertions are equi-
valent: (i) f_ is universally closed. the (ii) For each discretely valued k - d e l V with field of fractions L , __ 9 X(V1 yX(L) _X(L) with ccanponents f(V) E d X(inc1) is surjective. +
Proof: In virtue of the canonical isanorpkisms m%~(~k~,z):
%*E(S&V,_Y)?
z(L)
and
Y_(V) , assertion (ii) means that, for each cmtative square h
(*I
can
165 there is an R : g k V + . X
h = & o .~
such that -g=fak - - and
(i)=> (ii) : S e t _Z = SJ V k
,
_Z I = Sp L -k
and consider the diagram
and -fz are the canonical projections and the canpnents of m_ are can a d h The required mrphisms are of the form yxo2 , where s_ is a section of f such that m = y % W e n m shcw that, since f, is
where g
-8
.
.
-Z
Gists.
closed, such an
TO prove t h i s l a s t assertion, l e t
.
y be the unique p i n t of
x=_m(y) Since -2 f is closed, there is an yl is the & i q e closed p i n t of where -
dx
dx1
dxl
=O
, we
r
X,
and s e t
that fz(xl)=
y’ ,
therefore have
ml#Ker(ay)
L
,
. If
Ker(ay)
i s the maximal ideal of
B-’(m1)
V
ay factors through a retraction
6:
m’
V ; since
. Accordingly the
. Since
i n L contains V and is distinct fran L
proper subring of
y, y’:
XI,
dxl a t the prime ideal
is the local ring of
is the maximal ideal of B-’(Ker(cry))
x’€IX) such
z1
z . we then have the following
m t a t i v e diagram for the local rings of
where
-
image of
is a maximal
dxl+V
of
B
.
The composition
yields the required section s
.
e must show that, for each mrphism (ii)=> (i): W jection
Ez-
: zxy
-
5
is algebraic over k
+.
z
f:z -+Y_
, the
canonical pro-
is closed. To achieve t h i s we assume f i r s t that
. By observing that
-fz
-
also satisfies (ii)mtatis
z
mutadis, we reduce the problem to s b i n g t h a t a mrphisn f:g+Y_ i s closed whenever it s a t i s f i e s (ii) NOW i f ~ € ,3 y =L(x) and y ' e m , we may
.
e i n such a way that _h(S&L)
choose the square (*) h sends the closed point of
x' denotes the hage of f(x')=x
(5 I,
={XI
and that g-
V onto y' (2.5). With the above notation, 4 the closed point of S a V under 1 , we have S
if
. This showsthat the image of an irreducible closed subset i s closed
2.10). Since each closed subset is a f i n i t e union of irreducible closed
subsets, the assertion is proved.
2 be arbitrary. W e must show that, for each closed subscheme _F of Z_ x$ , f Z ( F ) is a closed subset of g . Ey replacing 1 by the members of -1 an i f f i n e open covering (xi) , and _X by the open subschemes _f (xi) ,
Now l e t
we f i r s t reduce the problem to the case in which
. If
is affine w i t h algebra
z by affine open subschmes, we further reduce the problem to the case i n which z is affine with algebra C . Thus let Co be B
we n m replace
a f i n i t e l y generated B-subalgebra of
,
,
the k-scheme S&Co the mrphisn induced by the inclusion map of
zx,$-
C
Zo
Co into * go%$ C , and Fo the closed h a g e of po IFo If V_ is affine and open i n _X + _Z x U is the mrphisn induced by po , the closed and. i f -poU : 0 _yU U ~ ) 2, prop. 4.14) . b g e F- of P O ~ _nF(_zx u) is precisely Fo I ( _ Z ~ X ~ (9 -0 Since we obviously have
po
:
.
x-
"0'
for each _V
, we
see that
Now w e have
is iladuced by the inclusion map of
a closed subset of
But t h i s follows fram the f a c t that _f,'(z)
-
z
Co
into C
and it i s enough to show
is a noetherian space f o r each
.
167
, so
zez
that
for sufficiently large subalgebras
2.7
Corollary:
over a noetherian &el
of
p:X+S is a
If
, the
k
Co
C
.
morphisn of algebraic schemes
following assertions are equivalent:
p is a separated mrphism.
(i)
(ii) If & (V) g (V) -+
V
syL)z(L) -
are chosen as i n theorem 1 , the map
L
with
ccsnponents ~ ( v )
x/s:X
Since the diagonal mrphisn
Proof:
_x/s is proper.
a closed fmbedding i f f 2.8
6
6
X(inc1) +
X
s-X
i s injective.
is an &ding,
6
Now apply theorem 2.6 to
6
x/s
z//s
is
*
With the assmptions of theorem 2.6, the following
Corollary:
assertions are equivalent: (i) f (ii)
is proper.
Ef. V and
is bijective
.
2.9
L
are chosen as i n theorem 2.6, the map X(V) *_Y(V)&#(L)
Corollary:
ring k
5
(i)
, the
If
& is
an alqebraic scheme over a noetherian
foll&ing assertions are equivalent:
is a ccsnplete k-scheme.
(ii) For each discretely valued k-model
x(inc1) :X(V) +_X(L)
L
, the
I t i s enough to apply cor. 2.8 to the structural mrphism
Proof:
each A t & 2.10
Proof:
with f i e l d of fractions
is bijective.
-&: $ + S b k , observing that
EW
V
a
.
.
Corollary:
is reduced t o a single p i n t for
The Grassmnn functor
Apply corollary 2.9;
is a d i r e c t factor of
(Sbk) (A)
v""
if
P
sn,r
is a ccgnplete scheme
is a direct factor of
Ln+l
,
Pn?+l
(Alg. V I I , 4 , cor. theoran 1) I
It follows f r m cor. 2.10 that G C3 k -n,r Z
is a ccsnplete k-scheme f o r each k q
.
168
Au;EBRAIc GEx3ME;TRy
Algebraic curves
Section 3
Throughout t h i s section, Definition:
3.1
.
denotes a f i e l d belonging t o M -
k
& algebraic curve
over
k
k - s c h which
*i
.
is algebraic, irreducible, separated ard of dimension 1 An algebraic curve Over k is said t o be regular i f the local rings a t closed m i n t s are discrete valuation rings. Proposition:
3.2
Each
slaooth
The converse holds i f the f i e l d k Pmf:
If an algebraic curve _X
each closed p i n t 1
(5 4,
13,
4.9).
xez
algebraic curve over k
is regular.
is perfect.
is snooth over k
, the
local ring dx a t
is an integral damin and has hamlogical dimension
It is therefore a discrete valuation rixg (for the ideals of
dx
are projective mdules, hence free of rank 1). ca-wsrsely, i f
discrete valuation ring, l e t 1 be a rational p i n t of &@
K(X)
k
.
is a
which i s
and let t be a uniformizing element of @ men X is the local ring of d x @ k ~ a( t~a) maximal ideal m ard we have
projected onto x
8%/tJES. ( 8X @k I C ( X ) ) / t ( d X @k K ( X ) ) m T If
k
is perfect,
K(x)~K(x)
It follaws that
x @ k ~ ( is ~ )snooth a t
X
Ranark:
k
X
and
-
(K(X)@K(X)),,,
i s sgnisimple so that Q-/t$
Hence m z = t 0-
X and g is
Jz
is a f i e l d .
SmDOth a t
x
.
Ushg the "same mthcd" one can show that an algebraic s c k o v a
a perfect f i e l d is mth i f f i t s local rings are "regular". 3.3
, we
Given an algebraic curve 5 Over k
generic p i n t and ~ ( 5 )for the residue f i e l d of
write
.
LO(&)If
dcminant mrphism of algebraic curves, we have f (w (_XI ) = w (g) K ( f ) :K(y)'K(X)
for the hcmmrphism induced by f
set of daminant mrphisns of
X_
into
.
w(g)
for its
-f:-X+y is a
. W e then write
and z ( X , x ) for the
I,
5
5, no 3
Proposition:
PROPER mRPHIsMs
Let
-
. If
5 is reqular )i s a bijection of %(?,?) onto
be algebraic curves over
is canplete (2.1), the map f + ~ ( f &(K(X)),K(Z))
169
k
% k ~ ( ~ ( gis ) ) contained in w ( f , g- ) , which is closed in X (5 2, 5.6). Since _X is regular, hence reduced, we have Ks(_f,g) =X , so that g = g .
(5
1, 5.2). Hence
(x)
Now suppose we are given a hQoomorphisn v: K K (5): we construct a g such that ~ ( g ) = v For each closed p i n t xc_X , dJx is a discrete valu-+
.
K(X)
ation ring whose f i e l d of fractions i s
dx+y
there is a mrphisn $(:Spk
gx
of
is c q l e t e , by 2.9
_Y
such t h a t
X
has an extension g - :U + _ Y t o an a f f i n e open n e i g m u r -
his mrphisn
h00d
. Since
(if
x
t o th€ hclmmrphisn dy closed p i n t s of
-X
9
-+
sends the closed p i n t onto y ~ y , apply induced by $1
Llx
. If
x
, x'
5
3, 4 . 1
are d i s t i n c t
g , we have
i n view of the uniqueness property proved m e . W e then obtain the required
mrphisn g- by "mtching tcqether" the mrphims -g" Remark:
m n - q t y closed s u b s c h Sc$(_X,y)
-+
Sc$(y,x)
, then
group 3.5
Of
.
a cconplete scheme over k
_V
of
& , the canonical map
be
f o r any
I f 5 is a regular canplete algebraic curve over
the autamrphisn qroup of _X K(5)
. Then,
X
i s bijective.
corollary:
3.4 k
1, 4.13)
By the same methods, we can prove the following result: Let
a regular algebraic curve and
-
(5
.
Corollary:
If
is a n t i i m r p h i c to the k-autmmrphisn
X ans 3 are regular c q l e t e algebraic
170
curves over k
, X
3.6
Corollary:
k
I, § 5, mJ 3
ALmBRAIC GFmEcRY
, X
is i m q h i c to y
Lf
iff K(X)
is a regular c q l e t e algebraic curve over
iff K(X)
is i m r p h i c to the projective l i n e P1azk
.
transcendental extension of
k
Proof:
5 1, 3.9,
Setting n = r = l i n
.
is i m q h i c t o K(_U)
is a pure
we see t h a t the open subschmes
LJcl1
r;r I21
defined there are i m r p h i c to @ .%[TI ; accordingly _P C 3 k con1 2 iscmorphic t o S%k[T] ; it follows that C 3 k tains an open s u b s c h m U
and
-{11 z ~ ( 3 )is the f i e l d of fractions k ( T ) of
3.7
k[T]
.
Theorem on t h e c l a s s i f i c a t i o n of curves:
The functor _X
+K@)
formed by regular is an anti-equivalence of t h e f u l l subcateg-ory of &cS ccsnplete alqebraic curves and d d n a n t morphisns w i t h the f u l l subcategory of & I
formed by f i n i t e l y generated f i e l d extensions of k of transcendence
degree 1 Proof:
.
Since t h e functor
~
W
(g) is f u l l y f a i t h f u l (3.3) , it is s u f f i c i e n t
K
to construct, f o r each f i n i t e l y generated f i e l d extension Klk of transcendence degree 1, a regular ccknplete algebraic curve g such that K(&) = K
.
To t h i s end, l e t
and k(V)=K
.
T
denote t h e set of valuation rings V
The m g n b e r s of this set
T
such that k c S K
then consist of the f i e l d K
together with sane d i s c r e t e valuation rings (Alg. c m . V I , th. 1).W e endow
T
9
10, cor. 1 to
with a toplogy by c a l l i n g a subset open i f it is e i t h e r
empty o r it contains K and its ccanplernent i n T f i e s these conditions, we set
o T ( U ) = vT!V
is f i n i t e . If
U
satis-
; by taking inclusion maps
r e s t r i c t i o n s , we thus define a sheaf of k-algebras
dT . W e show t h a t
as (T,OT)
i s t h e geometric realization of a curve satisfying t h e required conditions. Let
t be a uniformizing element of a ring VCT
c m . VI,
8 1,
over k ( t ) VI,
8
different fran K
th. 3, 1,k is transcendental over k
. Let
A
be the i n t e g r a l closure of
1, cor. 2 to prop. 3, the rings V ' C T
. Thus
k[t]
such that
in K t$V'
K
. By ALg.
is algebraic
. By ALg.
m.
dcminate the
t-lk[t-'] . By Alg. m. V I , 5 8, prop. 2 (b), there are only f i n i t e l y * m y such V' . I n other words, the rings V " t T such that t C V " form an open subset U of T . By Alga CCBITII. VIIr 5 2, cor. 2 of prop. 5 and th. 1, U is the set of local r i n g s of A . Since f o r local ring of
k[t-l] a t t h e ideal
I,
5
5, no 3
each S E A
, we
ideals of
A
= nA , where p runs through s Ps P s , we see on the one hand that (LJ,
have As = n(A )
not m t a i n i n g
canonically isamsqhic to Spec A
a t a point V"
U
171
PROPER PDFPHISMS
, and
is a spectral space. Since the k-algebra (Alg. c m . V,
generated k[t]-mdule
9
V
, we
see that
(T,
dT)
defined above is a f i n i t e l y
A
(T, (3 1 is the gecanetric T W e claim that _X_ is the required
3, th. 21,
.
5
realization of an algebraic k-scheme
dTIU) is
on the other that the local r i n g of
. By varying
is precisely V"
the prime
curve.
To prove this, observe that since T is irreducible and of dimension 1 , _X has the same properties; we have already seen that K coincides w i t h the
X
local r i n g of (ii) of 2.9.
a t its generic p i n t . It then remains to v e r i f y assertion
W i t h the notation of 2.9, i f the image of a mrphisn
, -g,( dX)
is a closed point
x
a subextension of
L of f i n i t e d q e e over k
5
yx
1, th. 3,
spkv
. on the other hand, i f
g -
to SJkV
3.8 over
.
, the
(5
K
3, 6. 5). By Alg. m.V I ,
g has a (unique) extension to
that
x i s the generic point, set v'=~-'(v) -X
X
a): i f
is a regular and complete algebraic curve
description of the g m t r i c realization of
b) : Let
keg
. Let
K
.
P(S,T)=$-SP+l+aS-b=O
p >0
be a f i e l d extension of
d q r e e one generated by tw~elements
S and
. The kernel of
T
k
of transcendence
such that
the mrphisn +:k[S,T]-+k' of
$ ( S ) = s and $(TI= t is then the ideal m of k[S,T] geneP S -a Accordingly the local ring V=k[S,Tk is a d i s c r e t e
such that
rated by
3
and let k'= k ( s , t ) be an k of degree p2 generated by elements s and t such that be a f i e l d of characteristic
sp=a € k and tP= bE k
$
.%
proof of the c l a s s i f i c a t i o n theorem yields, with the help of
LCH) ,a
extension of
s_:SkL+z
(x) , and is therefore
induced by -gx defines t h e (unique) required extension
Remarks: k
, so
factors through V
hcmxmqhism V ' + V of
i s isamsrphic to
.
valuation ring and its residue f i e l d is k!
On the other hand, by
4.2 ( i v ), _SEkk[S,T) is s n x ~ t ha t each p i n t
d ~ ( x )= -(SP(x)-a)dS(x) i.e. a t each point other than m braic curve _X
such that
+o
. Thus we
K(~)=K
x t O2 -k
5
4,
such that
, see that a regular ccarrplete alge-
cannot be m t h . I f we set
172
AKW?AIC(;EOMETRy
V1=VWkk'
, we
-X@Jkkt
9
5 , no 3
is local, so is k is not generated by a single element,
also have V'/rnVT qkf@kk' ; since k*@ k'
V' ; since the mximal ideal of k'Wkk'
v' is
1,
not a discrete valuation rinq. Accordinqly the algebraic curve over
k' i s not regular.