Chapter II Algebraic Groups

CHAFTER IT Au;EBRAIC GROUPS

Throughout this chapter k denotes a model. I f and d(g) for SJ

write respectively _SEA

the k-functor which assigns t o each RE% -

is e

and

the set

.

5 1

A

k

A€$

XcA$g , we dk(_X). W e write % and

for

{e} whose only member

GROUP ScHEzVlES

A mnoid is a set together with an associative l a w of c m p s i t i o n which has

a (necessarily unique) unit element which we denote by

case may be. Given two mnoids M and N hcmamrphisn element of

M

e as the

we say that a map f:M+N

is a

if it c m t e s with the l a w s of c a p s i t i o n and sends the u n i t onto the unit ekcent of N

fran the f i r s t i f a sukmnoid of

0, 1 or

( t h i s second condition follms

is a group). A subset N

N

of a mnoid M is said to be

M’: (resp. a sukgroup of M ) i f it is stable under the l a w of

canpsition and contains the u n i t of M (resp. i f N is a suhnonoid of M -1 and i f for each x E N x exists and belongs to N ) Each mnoid contains

.

a largest subgroup, namely, the set of invertible elements. For each (not necessarily c m t a t i v e ) ring A tive group of

A

we write A+

for the multiplicative mnoid of

A”

the group of invertible elements of

A

.

A

Section 1

Groupfunctors and group schemes: definitions

1.1

Let

_X

nx: x _ x x

-

, and

be a k-functor. A l a w of c a n p s i t i o n on

phisn of functors -+

x 173

for the addiA*

for

is a m r -

ALx3EBRAIC GROUPS

174

11,

I t m u n t s to the same thing to be given for each &,ER on X(R) , such t h a t the naps _X($) : _X(R)+X(S)

-

nX(R)

W e say that

is associative i f , f o r each ?,ER

,

5 1, no

1

a l a w of ccknposition

are hcmmrphisns. (R)

is associative,

rX is a m m i d l a w i f , i n addition, each X(R)

has a unit ele-

nx

that is to say; i f the following cordition holds:

TI

X -

(Ass)

i s conarmtative.

W e say that

ment

which depends functiorally on R

E(R)

a mrphisn of functors that

nX

E~:$+&

. The family of

E(R)'s

defines

which we call the u n i t section. It follows

is a m m i d law-iff i n addition t o (Ass) it s a t i s f i e s the following

condition: There is a morphisn

E~:C+*X_

such that the

following diagram is c-mutative: (Un)

x

TI

tion

i s called a group l a w i f each X(R)

x*x-l

of

_X(R)

is a group. "he synanetrizing operai n t o X_(R) then depends functorially on R and

. Hence x and the following condition

defines a mrphisn a .X+y (Ass) , (Un)

3'-

TI

is a group l a w i f f it s a t i s f i e s (Sym) :

11,

5

1, no 1

175

GROUP SCHEMES There is a mrphisn u

-X :&+X

such that

the following diagram is m u t a t i v e :

(Id u ) x_' X-,xrz -

X

1%

E

-

rX

%

A k-mnoid functor (resp. a k-group-functor)

(z,nx)

is a p a i r

where g is

. We f&uently c d an abuse of notation by abbreviating this pair s i r p l y to g . W e say that a k-functor and nx

a mnoid (resp. group) l a w on

X is cammutative i f

%

_X(R) is ccemrutative f o r each RE-

following a x i m holds pr2

X

I

t

t h a t is, i f the

(sx:~x 2 + X x _X denotes the mrphism with cchnponents

and The diagram

is ccmnutative Given a k-mnoid-functor

5

, we

t o r which assigns to each RE$

or Xow f o r the k-mnoid func-0PP the opposite mnoid of _X(R)

write X

and

Given two k-mnoid-functors

-X

i n t o Y_ each mrphism f of -3

hammrphism for each RE$

.

(x,vy)

-

, we

call a hcmxn0rphi.m of

into Y_ for which _f(R) is a mnoid

; that is,

f

s a t i s f i e s the following two con-

Etions (the second being a consequence of the f i r s t i f iunctor) ;

Y_ is a k-group-

FLGEBRAIC GROUPS

176 The diagram

is a n n u t a t i v e

The diagram

is camrutative

The k-mnoid-functors

k% . The k-groupis denoted by gk.

form a category which we denote by

functors form a f u l l subcategory of

, which

Kn

If the underlying k-functor of a k-mnoid functor is a k-schene, we also say

that g i s a k-mnoid-schme or a k-mnoid. The expressions k-group-scheme, k-group are defined similarly.

Given a k-functor

1.2

and a k-functor defined by

-f and g-

-

g equipped with a law of canposition TTX y , the set $Fi(x,z) naturally carries a l a w of compos%ion

(_f,g)++a

oh where _h:Y-+XxX is the mrphisn with ccsnponents

x_-

. The relation

TT

opr x_ =pr -1 -2

holds i n

bI+lF&x~,~)

. If

vx

is a

monoid (resp. group) law, then M $ ~ ( ~ , ~is ) a mnoid (resp. group) f o r each k-functor

x

; in

%g(e+,z) (resp.

particular o

x_

Similarly, i f _X and

E

X

i s the u n i t element of the mnoid

is the inGerse of

x

4r

I

i n the group

are b m k-mnoid-functors,

mrphism of k-functors, then _f

and i f

$g(z,z) _f:y-+gis

. a

is a homomorphism i f f t h e following two

177 conditions are satisfied: (Hca-ni)

I n the mnoid rr%~(yxy,K)

(Han;)

~

~

O

x

E

, we

have

f

o

-

is the unit element of the mnoid

.

= ~ (_fopxl) ~ ( f pz21

Fl%g(e+?)

-

.

The category of k-mnoid functors (resp. k-group-functors) ob-

1.3

viously admits inverse limits, and the functors w i t h inverse limits. L e t us give

Given two k-mnoid-functors

_X

s ~ n eexamples.

_X (R)

(resp. k-groupfunctors)

,

, carmrute

RE-%

and

, the functor

g x y is naturally equipped with a monoid l a w (resp. group law), namely, the p r d u c t k-mnoid-functor each RE% ,

(resp. k-groupfunctor)

the product mnoid (resp. group)

which assigns to

zx

. If we assign

Z(R)xX(R)

%

its unique l a w of ccarrposition, we obtain a groupfunctor, sanetimes denoted by

0 or

1 , which is a f i n a l object i n the categories

2 of a k-mnoid-functor

A subfunctor

(resp. sub-group-functor) (resp. subgroup) of

of

_X

3% a d Grk

.

is said to be a sub-monoid-functor

1! i f , for each

,

Re$

g(R) is a submnoid

. There is then a unique l a w of ccmposition on

_X (R)

such that the canonical inclusion morphisn is a harransrphism: it is a mnoid (resp. group) law. I f

is a k-mnoid-scheme (resp. k-group-scheme)

_X

, we

apply the term sulmnoid (resp. subqroup) to those subfunctors of

g which

are a t the same time sukmnoid-functors (resp. sub-group-functors)

and sub-

scms. Given a sub-group-functor

of the group-functor

m m l (resp. central) i n central) i n _X(R) If

.

g i f , for each

R

,

g(R)

g:g+g is a hanarorpkisn of k-mnoid-functors,

submnoid-functor

Kerf

of

_X

, we

say t h a t

is

is noml (resp.

the kernel of

f

is the

11 such that

( K e r f ) (R) = K e r f ( R ) = {xtX(R) : f ( R ) ( x ) = l )

for each RE$

. If

X

functor of 5 ; thus f

is a normal sub-group-

is a mmrphism i f f

7 + e

A k-mnoid-functor

1.4

for RE&

-i f _ X-

is a k-group functor K s f

,

_X*(R)

~f

.

g has a largest sub-group-functor

is the set of invertible elements of X(R)

- -X*

is a k-schane, so i s

. To prove U s ,

let

X1

_X*

. i’brwer

;

be the pullback of

Au;EBRAIc GROUPS

178 the diagram

and let

x

x_-

X , A

%

(resp. j:Xl+X

_i:X + X

-1 -

jection p:xl+xXx

0 1, no 1

E

TI

-X

11,

) be the c a n p s i t i o n of

the canonical pro-

with px,:_~xg+._~(resp. with E ~ : X X X + )~ ; we thus

obtain a Cartesian diagram

-1 -1 such that pu(x)=(x /x) and -py(x)=(x,x ) isobtainedfm

that X*

X

that

II. by a pullback construction. This also

is affine (resp. algebraic) so i s

1.5

k'

Let

and x€_X*(R) , so

for RE-?

be a model. L e t

g

x* .

be a f u l l subcategory of

shows

g@

be a functor which which is stable under f i n i t e products, and let F:$+M& cxmtnkes w i t h f i n i t e products. (Then $€ g and F(e+) = $ ; mreover if

z,g€s , then i(z,g) : F(_Xx imrphim)

and the canonical mrphism

xxx€C-

x)

. If

+

F(5)x F@)

and i f

&€$

, with

and F ( E ~ ) is an nX is a l a w of ccsnposition on g then the

(resp. a k-group-functor) a kl-group-functor).

and i f

-X , -Y E s , then

If

T

F

F (px,)

-

ocsnposite mrphism

i s a l a w of composition

canponents

on F(5)

.

If

( 5 , ~ ~is) a kmnoid-functor

then (F(2) ,vF - i s a k'-mnoid-functor (resp. c:X+Y_ is a hormmrphisn of k-monoid-functors,

F(f):F(_X)-+F(Y) is a hcmmrphisn of k-mnoid-functors.

F':G-+$g be a second functor which c m t e s w i t h f i n i t e products, and l e t h:F-+F' be a functor mrphisn. Given a kmnoid-functor gcs ,

Let

h(3):F (g)+F' (5) i s then a

hcmcmorpkisn of k-monoid-functors. W e n m consider

scme examples of the b v e construction. a)

Let

k'€$

. The base-change

functor

$z+s;Ecutmutes with f i n i t e

products; it follows that, for each k-mnoid-functor XBkk'

5 , the kl-functor

canonically carries the structure of a k ' m n o i d functor ( t h i s m y

11,

p

1, no 1

GlWUP SCHEMES

also be verified directly frcm the formula b)

In the above situation, the functor

kvkX

179 (,Xmkk')(R) = X(kR) ) ,

kvk:se+%g a l s o carmutes w i t h

f i n i t e products (I, 5 1, 6.6); hence, f o r each k'-mnoid-functor

c) Given a f i e l d k for F the functor

, take f o r X++ % (3@)

i s a k-group-scheme

Q d(G)

phism $G: g+ s_P o(G)

(I,

the category of algebraic k-schemes and

5

. For each algebraic k-group-scheme

d) Again suppose that

2, 3.3) ; by the above, the canonical mr-

is a f i e l d , take f o r

k

and f o r F the functor

algebraic k-schemes,

5 ,

(I, § 1, 4.3) is a hammrphism.

-

9

, the

has a natural k-mnoid-functor structure.

k-functor

(I,

_X

E

the category of l o c a l l y

of connected components

no

4 , 6.6). For each locally algebraic k-group-scheme

k-groupschane and the canonical mrphism %:G+no(G) pkism. (This example w i l l be treated mre f u i i y i n

5

,

no (_GI is a is a group hommor_G

2, Sect. 2 1 .

e) Of course, the above constructions are not confined to categories of the which form $E& For example, they may be applied to the functor E+M&

.

assigns t o each set E

%

=%(/_XI

rE)

f i r s t bijection: it is a group law whenever E

5 , the phisns

% -+_X

mrphisn

functor

i s a group. For each k-mnoid

onto the set of mnoid hamornorphisms E -+ X (k)

5 = G . If

c a l mrphisn

Let

.

second bijection induces a bijection of the set of k-rronoid hamansr-

G is constant i f there is a mnoid

W e say t h a t the k-mnoid

1.6

; recall that we have

and S3\(%rX) zg(E,Z(k)) for each k-scheme 5 E is a mnoid, the natural k-mnoid structure on % arises f r m the

S,chk(Xr%)

If

the constant k-scheme

Spec k

yG:G(kIk+G

.

E and an is-

is connected, t h i s is equivalent to the canOni-

(I, 5 1, 6.10) being an isamorpkism i.e. the k-

G_ being a constant scheme.

Affine mnoids and bialgebras

. Specifying a law of cornpsition on

%A

is equivalent t o

specifying a k-algebra hmamrphisn AA: A-tABA.

k

Accordingly, the axiams of 1.1 may be rephrased as follows:

ALCJBFaIC GROUPS

180 The d i a g r a m

IdA@ hA A@A-A@(A@A)

$.\,A

k

k

I

1

AA@IdA

A A (A@A)@A k k k

@J

is catnutative. There is a k-algebra hcmmrphisn

EA:A+k

such that each of the following

cangositions is the identity:

(corn)

A+ABkA AA A+ABkA AA

IdA@E A E

>ABkk'A

@IdA

------+kBkA%A A

There is a k-algebra k m m r p h i s m

crA:A-+A

such that the follawing

d i a g r a m is camrmtative: A-

i n which s ( a @ b ) = b @ a

,

is c m t a t i v e .

11,

5

1, 120 1

11,

9 1, no 1 - k-bialgebra A

Definition: A :A+ABkA A

GROUP SCHEMES

is a pair

(A,AA)

181

, where

A &a

k-mdel and

is an algebra hcmmorphisn, called the coproduct of A , F c h (%I. The unique hcm~~rphismE ~ : A - + ~

s a t i s f i e s the axians (Coass) and which makes the diagram counit)

of

A

.

(e) c m t a t i v e is called the augmentation

A lmnmrphism of the bialgebra

k-algebra hamomorphism f :A + B (f@f)oAA= A,of

and

EBof = E

(A,AA)

into the bialgebra

(B,AB)

(or is a

satisfying the tho conditions

A '

In view of the above arguments, we m i a t e l y obtain the following: Proposition:

The functor A - S A

anti-equivalence between the cate-

-1

gory of k-bialgebras and the cateyory of affine k m n o i d s . Under this anti-

equivalence the k-bialgebras satisfying (Cosym) (resp. ( C m ) , (Cosym)

s

(Cocom)) , are associated with the affine k-groups (resp. the c m t a t i v e

affine k-mnoids, the cmnutative affine k-groups): -~

1.7

Let A

be a k-bialgebra,

.

G_=SzA

the associated k-nwmid,

H a k-mnoid-functor and f q ( A ) By 1.2, a necessary and sufficient con# dition f o r the morphisn f :G+_H which is canonically associated with f to be a monoid lmnmrphism is that the following t m requirements be met. - -

-

( H a y ) : Consider the three maps A r i l , i2:_H (A) -+g(ABkA) induced by the

q r c d u c t of A and the injections i,:a-a@l ( H a 2 ) : Consider the map

-

~ ( f )is the unit element of

C:g(A)+H(k)

.

H(k)

,

i3:a-1@a

. Then,

induced by the augmentation.

in the

Then

W e M i a t e l y deduce the following

L e n n ~: Let C _ = g A be'an affine k-monoid. Let il,i2 be the maps of into ABkA defined by i (x)=x@l and i2(x)=1@x 1

.

, we

(i)

in the monoid G(A@ A) =,An(A,ANkA) - k

(ii)

the u n i t element of the mnoid g(k)=,A(A,k)

have A = i l . i 2

;

is the auqnentation

o_f A ; (iii)

if

G

is a group, the involution uA of

IdA i n the group G(A) =,%(A,A)

.

A

A

is the inverse of

182

ALGEBRAIC GROWS

9 1, no

2

G , we can describe the bialgebra i n the following way:

1.8

Given an affine k-mnoid

structure of

O(C) =E\(_G,$)

a)

11,

the coproduct

b) the augmentation :

E

is defined by c) i f

Ef

is defined by

Rt-b?

= f (e)

k

, where

e is the u n i t element of G(k)

;

G is a group, the involution u

Let

d(G) .+

:

d(g

-b

9(G)

(of)(x)=f(x-’)

, for

_H be a closed subscheme of

and xcG_(R)

fEd((G)

,

xE_G(R)

G_ defined by an ideal

, we then have

,

RE& I

.

of d(G) ; i f

the equivalence

xcIj(R)i=>{f(x)=O for a l l f e I }

W e imnediately infer that

; is

a sukrmnoid of

G

i f f the following t w

conditions are satisfied:

The bialgebra structure of

structure of

.

d(E)=&)

/I is then the quotient of the bialgebra

Section 2

Examples of group s c h s

2.1

Groups defined by a k-mdule. Let V be a k d u l e . Define tsm

anmutative k-groupfunctors as follows: for each Rem%

, set

5

I I r

1, no 2

GROUP SCHEMES

V (R) = V f R

a

183

.

V-Da(V) is a contravariant functor and V-V is a covariant a functor. They both transform f i n i t e direct sums of k-mdules-into products of

Of course,

k-group-functors. If

, we

k'€m%

have canonical iscanorphisns

(Vgk') = Vagkk' k a

, we

RE$

If

symwtric algebra of the k-module

V ; if V

is an affine k-scheme iscmrphic to SiS(V) StV)

S(V) is the

have Q,(V) ( R ) = S ( V , R ) - S ( S ( V ) , R ) where

is given by 1 -

i s s m a l l , t h i s shows that ga(V)

. The bialgebra structure of

-

1.7; the coprcduct

A : S (V) * S (V)@ S (V)

is iladuced by the diagonal map V +Vx V imrphism

S (V x V)

mrphism x E:

-x of

+

S(V) *k=S(O)

Let

v

, taking

;

&scciated with the mrphism V + O

be a k-module kmm~rphism, and let

f:V+V'

account of the canonical

the involution u is given by the auto, and the augmentation is the hcxmrmrphism

S (V)BkS(V)

.

L&(f):pa(V')+Da(V) be the

induced l-mmamorphismof k-group-schemes. Then the foilawing conditions are equivalent:

i s surjective,

f

embedding.

is a mnmrphism,

Qa(f) -

ga(f)

is a closed

If V is projective and f i n i t e l y generated, then we have a canonical isomrphisn

to %+%I

2.2

V

=pa(%) , so fhat

. If

a _ -

sp S(%)

V'

is an a f f i n e algebraic scheme isomorphic

Va

is a sukm&le which is a direct factor of

is surjective, and so V' + V

_as

+

by a (R)= R

RE g

the additive k-group. I f

for the Z-group-functor defined

. W e then have canonical iSQn0rpkisn.s

and the underJ.ying k-functor of

ak

T:a k +O-k

, then

is a closed embeaaing.

The additive group. Write a for

V

is the affine line

$

ak =

a (k)= ka-

. W e call

%

is the identity function, the bialgebra

184

ALGEBRAIC GROUPS

of the affine algebraic k-group

%

w e have A T = T @ l + l @ T , ET=O

,

11,

9

1, no 2

is the f r e e cmnutative k-algebra

.

kCT1 ;

G is an affine k-monoid with bialgebra &(G)=A , the hamnorphisms of G into ak are the primitive elements of A , i.e. the functions x€d(_G) such t h a t A A x = x @ l + l @ x oT=-T

If

.

Now suppose that k i s an algebra over the f i e l d LF

where p i s a prime. P ‘ ak by setting F’x =$ , f o r each

W e then define an e.ndamrphisn F of

arad each x€ a ( R ) = R

P€_Y,

then have, for

,

RE$

Pr

.

Write

rak P

r

F :ak +ak ; we

for the kernel of

r a (R) = E x E R : x p = O ]

rak is an affine algebraic k-group w i t h bialgebra P r , where we identify T rrod $ w i t h the inclusion mrphism t

The k-group-functor k[T] /(‘J?

r

.

into Ok W e have A t = t @ l + l @ t rak P an affine k-mnoid, the hcmanorphisms of G

Of

,

E t = O

,

into

G_

is

are then in one-one

P r correspondence with the primitive elments of zero p -th gebra

. If

at=-t

power i n the bial-

.

cl(~)

The multiplicative group of an algebra,

2.3

Let

A be a k-algebra

(associative, q i p P e a w i t h a unit element, but not necessarily c m t a t i v e )

.

W e define a k-mnoid-functor by assigning t o each (A@kR)X ; we write

uA

the monoid RE% for the largest sub-group-functor of t h i s mnoid-

e then have functor. W A

u (R) =

If

A

(AfR)* for

RE$

.

is a f i n i t e l y generated projective k - d u l e ,

then vA

alqebraic k-scheme. To prove this, define an element d

d(A,) &t Cise

by setting, for each RE&

= $%(Aal$)

of

and xEABkR

is an affine

,

d(x) = deter-

of t h e ’ - ~ - e ~ ~ ~ ~ ~ ~a-++ t -ax ~rp ofh A i ~@t ~ ~ R (fig. ccmn. 11, § 5 , exerA

9). Then xEp (R)

open subset

-

of

iff Aa

-

d(x)

is invertible, so that

defined by the function d

W e give scsne examples of this construction below.

pA

is the affine

(cf. also 1 . 4 ) .

11,

5

1, no 2

2.4

V be a k-mdule. For each

The linear group. L e t

G(V@kR)

185

GROUP SCHEMES

RE$

be the mnoid of a l l erdmrphisms of the R-mcdule V @ R k &(V) by setting

, let

. Define

a k-mnoid-functor

W e then get a canonical bijection

I f we carry over the mnoid l a w of following law: i f

f ,gt%

to a ( V , V @ R )

&(V) (R)

(V,V@ R)

, the

product

gf

we obtain the

is the ccmposite of

the diagram V-V@R f

where

+gQ?R V@R@R

m is the multiplication i n R

Suppose that V

'*>V@R

.

is f i n i t e l y generated and projective over k ; then we have

the canonical bijections Pb&(VrV@RR)

f:

s ( h @ V , R )

it follows from t h i s that L(V)

mrphism S(%@V)=d ( & ( V ) ) follming way: for

RE%

rR) ;

is an affine algebraic k-scheme. The iso-

obtained above m y be explicitly defined in the

, f c b ( v ) (R)

function w@v a t the p i n t

&(S(%'W

f

and

WEVE%@V

, the value

of the

is

Rerrarks: The preceding argument shows mre generally that, i f

V

is a f i n i t e -

l y generated projective k-mxlule a d W is a mall k-module, then the k-func-

tor W ( W , V )

such that

is an affine k-scheme which is isorrprpkic to SpS(%@W) the other hand, i f

V

is f i n i t e l y generated and projective, the canonical

bijections kok(V)@R =zR(VQkR) k-mnoid-functor

.

show t h a t &(V)

associated with the k-algebra

is isamorphic t o the

6(V)

(2.3)

.

11,

ALGEBRAIC GROUPS

186

Now let us return to the general case. The linear group of

, is

Q(V)

the largest Sub-cjroup-functor

directly: define an element f Re&

functor &(VIf -nk

RtI&

of

&(V)

-nk

of d ( & ( V )

by r

i s f i n i t e l y generated and

by setting f ( x ) = d e t ( x ) for

defined by the function f

we have accordingly GL

V

we see imnediately that S ( V )

and call

=GL(kn)

GL

;

, denoted

is an affine algebraic scheme. W e can a l s o prove t h i s

projective, then E ( V )

,

1, no 2

of L(V) ; we then have, for RE&

It follows inmediately f r m 1 . 4 and 2.3 that i f

xe_L(V)(R)

V

5

is the affine open sub-

. In particular, we set

the linear k-group of order n ; f o r each

(R) = C;L(n,R)

.

For each f i n i t e l y generated projective k-module, the determinant defines a group k-KxlKmrpMsm

and called the special linear k-group of

whose kernel is denoted by &(V)

. W e set

= SL(kn) and call nkr n ; we then have for each Re$

V

$L

%

the special linear k-group of order

(R) = SL (n,R)

0 -modules. An 0 -module is a k-functor 8 w i t h a l a w of -k kcanposition together with a mrphism of functors _O,x_M+_M such that, for

2.5

set M(R) , taken with i t s l a w of c a n p s i t i o n and the map i s an R-module. For example. given a k-mdule V , the k-group-

, the

each RE& R%bJ(R)+M(R)

,

functors V

and D (V)

an

a_ 0 -module.

-a

of 2 . 1 are naturally endowed w i t h the structure of

-k

,a

Given two Qk-mdules _M and

hamarnorphim of _M

phisn of functors M+_N which induces, for each RE& mrphisn of M(R) F&cJ

(M,@

3-

into N_(R)

. For example, i f

into rJ_ is a mor-

, an

R-module hmw-

. The set of these hatmorphihisns w i l l be written

V

and W are k-rrcdules, we have the evident

11,

5

1, no 2

GROUP SCHEMES

187

The above maps are both bijective.

Propsition:

Proof: Consider f i r s t a diagram of -

0 -modules -k

where ~ ( k )is k-linear and g ( R )

R-linear.

we have a camutative diagram

that is

,

.

u = ~ ( k ) ~The map v e v a

-

injective.

g:V + W

3

. For each

RE

-

SJS (W)

of 2.1 irduces

Qa(W fS(W))~$g+(Pa(wf,Fa(V))

-

-

which may be described explicitly i n the follawing way: to YEP&&

-

(V,S (W) )

such that is assigned the mrphisn of functors gE%E(D (W) ,ga(V) w-g U(X)=X,Y h e r e RE% , A ~ ~ ( W , R ) = D , ( W(R) ) , and A,E$(s(w) , R ) X,(W=X

R by

, it

RCT]

u(TX) (v) =

whence

Xnyn (v)=0

i n t o S(W)

, we

An=X,I

for

n

is

let yn(v) be the ccanponent

( t X ) = tnhn

follows that

for each t E R ; by re-

TnAnyn(v)

nE IN

nfl

. Taking

infer t h a t yn=O

the proof i s complete.

Sn(W) -and

. Clearly

y f v ) , ;vcV

of degree n of placing

. Let

1

is thus surjective; it is obviously

a bijection

defined by

$

W e thus have u_(R)=u_(k)TR,

NOW consider the second map: the iscmrphism QW (, :)

S(V,S(W)) =

5

X t o be the inclusion map of

for n # l

, hence that

u=Qa(y,)

W

, and

Au;EBRAIC GRDUPS

188

11,

8 i, no

2

we infer f r m the proposition on canonibl i m r p h i s n s (2.4, remarks) that

Analogously, i f we write L(M)=m(M,M)

, we

ule M

and C&(M)=_L(M)*for each Ok&-

get canonical isomorphisns ZlCV,)

-

L(V)

,

GL(Va) li- @(V)

-

and

The definitions custamrily employed in the theory of modules exterad to gk"d-

ules. For instance, an gk-algebra is an O + d u l e pkism

M_Xu-+PJ

R d u l e M(R) CIk-"d"le on V

...

Va

-

which Mutes, for each

. I n virtue of the proposition,

are in one-one cori!espondence

,M together w i t h a mran algebra structure on the

the algebra structure on the

w i t h the k-algebra structures

Autcrmrphisn groups of algebraic structures.

2.6

2.4 may be gener-

alized t o the case i n which one is interested in the endcarorphisns of a k-mdule

V which carries additional algebraic structure (e.g., the struc-

ture of a not necessarily a s m i a t i v e algebra, guadratic form, involution, etc.)

. Consider, for example,

the case of an algebra. Thus l e t A be a (not

necessarily associative) k-algebra. Define the sub-mnoid-functor L(A)

by assigning to R e &

A @R ; also define &t (A)

the mnoid of a l l R-algebra-endmrphisns

by A&(A) k generated projective k-module, =(A)

=

End (A) nC& (A)

L(A)-+a(A@A,A)

A

is a closed subscheme of

hence an affine k-schem. To see this, observe t h a t image of the zero section of

. If

w(A@A,A)

the m p

of

is a finitely &(A)

,

(A) is the inverse

utader the mrphism

which assigns t o f t L ( A ) (R)

of

(A)

11,

B

1, no 2

189

GROW SCHEMES

A @ A @ R r; (A@R)@(A@R)

k k

into A8JkR

k R

k

.

Similar arguments apply in the case of a unitdl algebra. For example, the k-functor of autamrphisns of the u n i t a l k-algebra group-schene which we denote by order n .

Mn(k)

is an a f f i n e k-

m& and call the projective k-group

The endmrphism group of a scheme. Given a k-functor

2.7

f o r the mmid of endamrphisns of

g

of

, write

g and Au\(Z)

f o r the group of invertible e l m t s of this mnoid. Define the k-mnoid-functor End(X) and the k-group-functor Aut(X) by End (XI (R) =

%(X

f R)

AAt(X) (R) = Au&(Z@R)

k

If

_ X = S x A with A% t,

ALt(X)=&t(A)

.

, ve

.

, Re%

then have End(X)=E&d(A)

OPP I n particular, it follows f r m 2.6 that

OPP are a f f i n e algebraic s c h m s whenever

Aut(X)

k-schm. Observe that the underlying k-functor of (I, 52, Sect. 9 ) . If

9.3 we see anew that

X

X

and E sla (X )

and

is a finite locally f r e e

m(_X)is precisely

is a f i n i t e locally free k-scheme, by

m(x) i s an a f f i n e k-scheme.

~~(_X,g) using I, 5 2,

Pbreover, it follows

from 1.4 that Aut(X) is a schgoe (resp. an a f f i n e scheme) i f

N ( X )

is a

scheme (resp. an a f f i n e scheme).

the plyncanial P(T) corresponds t o the endcsrrorphism of

to RE&

and xE R

the element

P(x) of

R

.

0 which assigns -k These remarks rmin valid

i n the mre general situation i n which k is replaced by an a r b i t r a r y k⪙

accordingly w e g e t an isom>rphisn of k-functors

190

ALGEBRAIC GR3UPS

11,

9 1, no

2

G(Qk) (k[Tba

-

(of amrse, this i m r p h i s n does not preserve the mnoid structure!)

ck

Each aukamrphism of the k-scheme is of the form 2 n x-a + a x + a 2 x +...+a x , where ao,al,a 2,...,a Ek,alEk* , 0 1 n n for i 2 ard sufficiently lame r Proposition:

r

ai=O

.

Proof: I f -

P(T) = a + a T + ...+an? 0 1 show that al is invertible and a= :

defines an autcsoorphisn of 0

for

Keg

I t i s enough to show that for each f i e l d @ (al)

is invertible and @ (ai)=0

defines an autmorphism of is a field. If

Qk

for

, we

i22

Qk

, we

.

and sufficiently large r

i22

and each horncanorphism @:k+K ,

. Since

4 (ao)+.,.+ @ (a,)?

need only consider the case i n which k

Q (TI is a plynCanCa1 such that

P (Q(T) 1= T

, by

examining the

terms of P and Q of highest degree we see imnediately that P and Q are of degree 1 Conversely, i f ao, an satisfy the conditions of the p r o p sition, then, since the map x + + - aa -I+ ailx is bijective, we my replace

.

P

-1

by -aoal +al P

a =0 0

-1

...,

and. al= 1

0 1

.

2 +bjT3 Q(T)=T+b2T

. We m y thus confine ow attention to the case i n which

In t h i s case, we know that there is a formal p w e r series

+...

...,a,) 1

b.=P.(a2,a3, 1 1

.

such that P ( Q ( T ) ) = Q ( P ( T ) ) = T W e also have

where Pi

is a p o l y n d a l with integral coefficients.

...$

, it is easy r2+ 2 r 3 +...+ (i-l)ri is the weight of a mnanial : a t o show t h a t Pi is isobaric and has weight i-1 It follows t h a t bi=O If

.

for sufficiently large i

. This ccmpletes the proof.

Corollary 1: For a reduced k-scheme

, the mrphisns

the following form: there is an a € d ( x ) * =?a Re&

,& If -

Corollary 2:

xCX -( R )

a bc$(g) such that,

and each yER

, we

Diagonalizable groups. L e t

have f ( x ) ( y ) = a ( x ) y +b(x)

for aEk*

r

.

we have:

Q(b) (R)

g,

he a catmutative mnoid. Define a

Z-monoid-functor by Q(r) (R) = pxn(I',R ) ; for a group group-functor and we have For each RE&

for

k is a reduced r 3 , each autcmorphisn of the k-group

is hamthetic, i.e. of the form x h a x 2.8

are of

_f:s+Aut($)

= g(T,R*)

.

I?

, _D(r)

is a Z-

.

11, § 1, no 2

191

GRorJp SCHEMES

,

D ( r ) (R) = Mon(r,R ) = An(Z[I'l,R) where

Zk1

is mall, the Z-functor spzCrl

r

is the algebra of the mnoid

.

with coefficients i n

Z

. If

r

is thus an affine Z-scheme isawrphic t o

D(T)

LemM 1.7 enables us t o determine imndiately the bialgebra structure of

oLr1.. The maps A : zCrl+zCrl@zCrl , ~:zCrI+a and (when r is a group) ~:zCrl+aCrl are defined by n ( y ) = y @ y , ~ ( y ) = l u ( y ) = y-1 f o r y E r . Evidently g ( r x r ' ) =_D(r)Xg(r') and. we write & = I I ~ J ) , u =Q(Z) , nu =Q (Z/nZ) ; we then have, by definition I

~ ( R ) = R,*

-o ~ ( R ) = R , Q* =@Z[T]

,

p E%ZLT, T-~ ]

,

nu (R)=ha:xn=l) =~Z[T]/('?-l)

W e call LI the standard multiplicative group and

nu

the group of

th n

roots of unity. With the notation of 2.3, we have pk= pk

,

and, i f A€$

p(r), into % is the

Notice that the only hamwu3rphisn of

zero hcmxmr-

phisn. For each l-ammrphism corresponds to an elenent x =la y Y such that A x = x @ l + l @ x (2.21, hence such that

of

kCT

1

.

which imnediately inplies a = 0

, whence

2.9

be a k-mnoid-functor. An e l m t i f f (el=1 and and RE? I n other words, a character of

Y

f

Characters.

to(s)=M&G,

ek)

f ( x y ) = f ( x ) f ( y ) for

G

is called a character of x,yEG(R)

is a l-atrxmrphisn G-+C):

kQG,gk)

Let

*I'.

.

; the set of characters of

G_

is the mnoid

. It is customary to write the l a w of composition of for the value

additively; accordingly we write xf f

x =0

a t the elgnent xcG(R)

,

RE.&

I

t h i s mnoid

f (x) of the character

so that we have the f o m l a s

192

ALGEBRAIC GROUPS

,

f o r x,y€G(R)

s

acter of

factors through pk

.

group i r n r p h i c to Grk(s,pk) If S

. If

a d f,gEE?(g,$)

RE%

, and the

11,

_G

xE d(G) which s a t i s f y Ax = x @x ,

set g=s,

d&)= kCT1

. The characters of

d a l s P(T) which satisfy the formulas P ( O ) = l k

hand,

is reduced, it follows t h a t P = 1

For a f i e l d

k

linearly indepadent over k

that

.

,

EX

=1

.

For example,

ak are then the p l y P(?YT')=P(T)P(T')

.

In the general case, on the other

ak does have non-trivial characters (cf. § 2, 2.6 below).

Proposition:

Proof:

G is a

G is the multiplicative

m m i d consisting of

If

2

is a group, each char-

m m i d of characters of

is affine, the mnoid of characters of

, so t h a t

5, 1, no

Let

fl,

fo,fl,

...,f n f

...,f n

, distinct characters of

.

a k-monoid-functor are

be characters of the klru3noid-functor G

. Suppose

are linearly independent, and let 0

= a f +...+a f

n n

11

, aiEk ,

...+anfn(x'y')

f o ( x ) @ f o ( y= ) fo(xly') = a f ( x ' y ' ) t 11

+. ..+ fn (XI@ anfnfy) ,

= f l (x)@alfl ( y )

where x',y'

are the images of

fran a l l the f i

, it

x and y i n G_(R@S)

follows that there exist

zero, such t h a t for each RtM+

S€I&

and each xcG(R)

. If

f o is d i s t i n c t

and bicS

, not

all

we have

...+fn(x)@bn= 0 .

fl(x)@bl+ If

u:S+k

is a linear form on S such that u(bi)

follows t h a t there is a non-trivial linear relation,

.

u (bl) fl+. .+ u (b,) f n

=

0

,

are not a l l zero, it

11,

5

1, no 2

GROUP SCHEMES

193

a contradiction. C a r t i e r duality for f i n i t e locally f r e e c m t a t i v e groups

2.10

Define the biduality hcxmmrphism

a * G G'

+

g(Q(q)

as follows: for RE$

is the character of f onto f(gS)=gSES For each k'E,

and g t G ( K )

which sends fEQ(G) (S)

S€&

I

a,(g)

.

.

have g(G@kk')=_D(G)Bkk' and aG@k l = 0 1 ~ @ ~ k ' - k

-

b)

Let

be an affine k-mnoid and set A = d ( G )

_G

_D(GIR

%

. Assign the k - d u l e

the structure of an associative k-algebra by means of the fort ( f a g l a ) = (f@g,AAa) I where f,gE A and a E A ; the augmentation

tA= Mot&(A,k)

mulas E

A A .

of

A

.

is the u n i t elernent of

This algebra is related to

%:

t

defined as follows:

t~ we call

J ( p(G) )

t~ the cartier algebra of

via the lxrmmrphim

9(G) J ( D ( G ) ) +

if

RE_%

,

by 2.9

g(g) (R)

is the set of

such that AABRx =

X@X

R

and cAgR(x) = CEA(xi)ri = 1 i if

yct8(C)

I

REP&

and xcQ(s) (R)

by definition we set

ALGEBRAIC GROWS

194

11,

9 1, no

2

with the above notation. The fcsrrulas

This h x x ~ ~ r p h i sim s related to the biduality hrmrmorphism defined i n a) : given gcG(k) , l e t g:d(G)-+k be the hcarrarprphisn f-f(g) ; for each

and each xc_D(G)(R)

Rc%

, we

have

aG(g)(XI' X(g,)=(g,x)=a,($) -

whence aG(g)= BG(@

-

c)

-

;

(XI

.

is ccmmrutative ard affine, the cartier algebra tO(_C) of G is

If

comnutative. Accordingly, given a h o m m r p h i s m : ,B

-

the Canonically associated mrphism YG:

-+

-

&9(G)

which is defined explicitly as follaws: i f -

_G

of the form y +(y,x)

, where

b) which show that

-

8,

imqhisn. d)

If

E is

(I, 5 1, 4 . 3 ) .

.

X C A @ ~ R $ ~ ( G _ @ ~ R )Such a linear map i s a

Ax = x a x and

is a hcmamrphisn). This mans t h a t yG is an

-

; write

tm

for the canposite map

.

t ~tA :+k denotes the map y b y (1) mreover, since g (G) 13

and

.

(cf the formulas

EX = 1

obtained by transposition fran the mltiplication m:A@A+A $(II(C_))

is

carmutative and f i n i t e locally free, we have a canonical iso-

% d i ~(ABAA)

mrphisn

,

is finite locally free, each k-linear map td(G)+R

k-mnmrphisn of unital k-algebras i f f of

an3 x € p ( G ) (R)

RE$

y G ( x ) : ' ~ ( ~ ) -is + ~the hcmmrphism yw(y,x) I f , i n addition,

t d ( ~ ) + ~ ( D, (we~ )have

.SMlarly,

is affine by b) ,

is a bialgebra whose coproduct and augmentation we denote simply by E

. For

~ € ~ d ( c RE$ ) I

and Xry'C_D(_G) (R)

r

w e have

11,

9

1, m 2

195

GROUP SCHEMES

It follows that

tm

bra structure over

and tn

t&c)

are the coprduct and a u m t a t i o n of a bialge-

; with respect

to this structure B,:td(G)+_D(d(G))

is a bialgebra isamorphisn.

-

F m this we deduce the

Proposition:

If

_G

locally free, so is

i =

k-mnoid-scheme which is ccmmtative and f i n i t e

p(c) and the biduality hcitmorphisn

an iscmmhisn.

a,:G_+Q(Q(G_))

-

&

Proof: Since -

a,(k)

if

, it remains to shm that -a @ k' for each k'E$ a G_gkk'- G_ k by b) ; thus is bijective. N m i f gE s ( k ) , we have aG(k)(g)=R,(G)

is enough to shm that 8, induces a bijection beb& the hcrmmrphic t and the 6 € & ( i ( G ) ) such t h a t A 6 = 6 & 6 and & 6 = 1 N m y maps yE t is a hcaKmorpkisn provided (y,l>= q(y)= 1 and (y,x*y)=(y,x>(y,y> for a l l

.

x,yCO(g) ; this last equation is equivalent to

(trm,,x@Y> = (Y@Y,X@Y) t hence to my = y @ y , The assertion now follms from the f a c t that 8,

-

bialgebra isanorphi'&. Remark:

If

G_

is a

is a k-mnoid which is catmutative and. f i n i t e locally free,

we have the mnnutative square

which yields a relation between the biduality hcmamrphisn a, and the canonical ismrphisn ~ ( G ) G ~ ~ o ( TO ~ ) show that this square-mutes, we

.

observe that a l l the relevant maps are algebra hamcmorpkisms. By applying an extension of scalars, we reduce the problem to showing that for each

Au;EBRAIc (;wxTps

196 mrphisn @:d(l;)*k of

$ we

9 1, no

2

have

-1 t d(CLG)BDG = ,$(can 1 8,

- --

Assming the notation of

11,

-

.

=g with gcG(k) I this last assertion is a consequence of the f a c t that the following diagram comnutes: b) I i f we have

@J

where, for each k-rrcdule M and each m E M , m' demtes the canonical h g e of

. (To prove that

m in

B,(g)

I=

, observe

aG(g)B,

and a'= &BH

-3 for each affine k-group H &

2.11

Duality for diagonalizable groups. Let

-

c&h

that

E,(g) =sl,(g)

dEQ(_H) (k) C @($)

I'

r'

and

t a t i v e mnoids. We shall d e t e m h e the hcmm~rphisms g ( T ' ) k + Q ( T ) k

.

-

be c m -

. First

of all observe that the lnorpkisms of functors

g:_D(r')k-+g(I'lk correspond

to the lnonoid hcmmrphisns g:r +k[r'

f

necessary and sufficient t h a t

kk]

-+

kCr']

, i.e.

1" . For

to be a l%xtm~rphisn, it i s

g give rise to a bialgebra hanomorphisn

t h a t the follawing two conditions be satisfied:

These conditions determine a continuous map Speck + T '

value rIk(k)

rl

on the closed &.open

subset

(which takes the

, i.e.

an elemznt of

. I t follaws that we have canonical mnoid iscanorpkisms

E%(g(I")k,E(r)k)

g g ( r , r i ( k )1

N

gr&(rk,ri)

.

197

In particular, we get a canonical isamrphisn

and so the elements of

RE&

r

yEr

, +gr)

r

may be identified with characters of

(R)=wx-n(r,g)

r

we have gy=g(y)

.

D(r)k : for

If Speck is connected, we deduce the existence of an imrphisn (I, 5 1, 6.10):

m r p y ) k , g ( r ) k )= g ( r , r i )

nrvu

.

In particular, the momid of characters of D(rIk m y be naturally identified with r Calling a kmnoid which is i m r p h i c to m D(IYk a diagonaliz-

.

able k-mnoid, we infer the:

Proposition: Suppose that Speck is connected. Then the functors

r w m k

and GMW)

(k)

are quasi-inverse antiequivalences between the category of snall CQrmUtative mnoids a d the category of diagonalizable k-mnoids. These antiequivalences associate finitely generated carmutative mnoids with diaqonalizable alqebrak-mnoids, and s n a l l axmutative groups with diagonalizable k-groups.

accordingly there is a canonical isamDrphism p(rk ) = Q(r)k ' Frcan the abwe results we infer the

Propsition: The functor G-D(G)

is an antiequivalence between the cate-

gory of diagonalizable kmnoids and the categoq of constant k-mnoids; - - and

G I+

(G)

is a quasi-inverse functor.

Boolean groups. Let r be a small Boolean topological group, i.e. a small topological group with a base of canpact open sets. We know (I, 9 1, 6.9) that the k-functor Tk defined by rk(R)=s(SpecR,I') is a scheme. Fram this description it is clear that rk naturally carries the 2.12

198

AIx;EBRAIc Gw3ups

11,

5 1, no 3

structure of a k-group functor.

r

For instance, i f

is a profinite toplogical group, the inverse limit of

, we

f i n i t e discrete groups Ti

have

(R)= s ( S p e c R , I ' ) = I+=

(SpecR, ri)

=@(riIk(R)

so that the k-groupfunctor

(riik

rk

.

,

i s the inverse limit of the constant k-groups

Section 3

Action of a k-group on a k-scheme

3.1

Definition:

Given a kmnoid-functor

on

a ( l e f t ) operation of a mrphim of functors

_X

(or sinply a ( l e f t )

_G

arad a k-functor

Goperation

on

X,

2 ) i.

11 : G x X _ + X such that, for each RE$

for g,g'cG(R)

,

xcK(R)

,w

.

m u(g,u(g',x))=1!(gg',x)

U n d a these conditions we shall say that G_ acts on

for u(g,x)

.

g

. W e shall write

Each mrphism of functors u_:Gx?-+X camnically induces mrphism p : G + m ( X )

saying that

p

. To say that

@ u(e,x)=x

(I, 9 2, 9.1)

gx

a

u_ is a @peration is equivalent to

is a lxxmrorphisn of mnoid-functors. The G-operations on

are accordingly i n one-one correspondence w i t h the hcmxmrphisms

G_+w(x)

Notice also that i f G is a group-functor, any hormmrphim G + R d . ( X ) factors through Aut(X) , so that the G-operations on _X are i n one-one correspoI.adence w i t h the k-groupfunctor l-rxtmorphisms G+Aut(X)

.

I f we express these conditions i n diagrammtic form, we obtain the following pair of axians:

11,

5

GRouPscHElMEs

1, 110 3

199

The diaqram

is curunutative.

a) i n a similar fashion we define the riqht Copera-

Remarks:

3.2

. These are i n one-one correspondence with the l-m~mrphisms of the o p p s i t e mnoid of G_ into E n d ( X ) . tions

u:X x G -+X

b) Suppse that such that

on g

G is

a k-group-functor.

f(g,x) = ( g , q ) for

Let

f:GxX-t_Gxg

gEG_(R) , xCX_(R) , RE$

i s an isamrphism and we obtain a camrmtative diagram:

be the mrphism

. This mrphism

It follaws that the mmhism u : G _ x X + X i s isanorphic to the projection

cxs-+_X

. For example,

5 is a k-scheme, 3.3

a)

if

G_ is a f l a t (hence faithfully f l a t ) k-group, and

u_ is faithfully f l a t .

Examples :

Let

gc$g

and r € z n

. If

ycr

,

{yIkx _X

is open in

rkx. 5

.

200

11,

If

5 is l s , we k v e

(I,

5

1, no 3

g-+& corresponds to a family of is a rk-operation on 5 i f f y -g(y) is

1, 6.10). Thus each mrphisn g:Tkx

mrphisns f ( y )

:x+x . Then

a hcmxmrphism of

g

Y into the monoid Em$(x)

. The rk-operations

thus i n one-one correspondence with the operations of the mnoid b)

5

X

on

r

are

on 3

.

A t t h i s point we could reproduce the r m k s of 1.5 concerning functors

commuting with f i n i t e products. However, w e confine ourselves to only one example of this type: l e t ting continuously on

Y,

. Let

r

be a Boolean space and

v : r r y + y be the map

a Boolean group ac-

(y,y) +y.y

, and

g

the ocsnposite mrphisn

where Yk

. If

is the canonical isamrphisn. Clearly 5 is a rk-operation on Spec k

i s connected, each rk-ation

in t h i s case the functor X n X

k

of

I,

5

on

xk

i s of t h i s type (for

1, 6.9 is f u l l y f a i t h f u l ) .

nG:G%G_-+G i n the k-mnoid-functor _G s a t i s f i e s the axioms (opass) and (0pGn) (cf. axicans (ASS) and (un) of 1.1) ~ h u sit is an operation of G on i t s e l f , called the l e f t translation operation. It is associated with the hcarmnorphisn y:G+m(G_) such that y(g1x = gx If G is a k-group-functor, then y factors through A u t ( s ) ; i f Rek% ard gcG_(R) , the l e f t translation y(g) is thus an autamorphisn of the R-functor G_QkP For example, i f k is a f i e l d , G is a k-group-scheme and

c)

The multiplication

.

.

.

g€G_(k) onto g

, the

translation y(g)

. Frgn t h i s it follows,

is an a u t a m p h i s n of

G which

sends e

for example, that the lccal rings of

G_ a t

i t s rational p i n t s are a l l i m r p h i c . The r i g h t translation operation is defined similarly: this is the r i g h t opera-

tion associated with the hcmamrphim 6 :G d)

Given a k-group-functor (g,x) Hg t (g)x =gxg-’

G_

, define

;’ this is the

-0PP

(G)

such that 6 (g)x = xg

a l e f t operation of

G_ on itself

inner automrphisn operation. This

by operation preserves the group structure of hanmoorpkisn

+E&i

5 , and

accordingly induces a

.

11,

9 1, no

3

GIiLlup

sr3EMEs

201

is the k-group-functor which assigns to each RE% where &t&(_G) autamrphi& group of the R-group-functor Definition:

3.4

GBkR

.

, _G

k t & k a k-functor

the

2 k-mnoid-functor

acting on X _ _ and _ _ _l e t p:G+Ehd(X) be the associated hmamrphism. a)

Given t m subfunctors y, Y_'

of

_X

, the transporter

of

y' into y

is the subfunctor TranspG(_Y,y') of _G defined as follows: f o r each RE& T r r G @ ' , X ) (R) is the set of gEG_(R) such that the ccmposite mrphissn

factors thmuqh 'PR

b)

Given a k-group-functor

, NormG (-Y ) (R)

RE$

p (9) o f

If

_X @ R

k

is the set of gcG - (R)

induces an autamDrphism of

, the

Y, is a subfunctor of gcg(R)

d) RE$

W e write G , g-(R)

96 G - (S)

x-G

Y @R

-k

for the subfunctor of p (g)xs =

nonnalizer

xs

xCX(R)

.

.

centralizer of

.

is the set of

we have

, the

such that the autanorphism

X

is the subfunctor

, CentG(_)

Re$

such that the endmrphism p(g)

identity on Y_BkR

_X

(Y) f 5 defined as follows: for each

G-

g t G ( y ) of G defined as follows: f o r each set of

2 cf

G and a subfunctor

is the subfunctor

of

c)

,

of

XakR

(R)

is the

induces the

defined as follows: for each

such t h a t for each Sc&

and each

E is a group-functor, (Y) is the largest sub-group-functor of G Transp,(y,x) and C e n t (Y) is a n o m l sub-group-functor of (Y) . GGIf

3.5

we have

Proposition:

Assuming the h

e notation, l e t RE&

. Then -

,

ALGEB€ucm w s

202

11,

5

1,

110

3

Proof: We have a Cartesian square

i n which

c1

is the mrphisn formed by canposing p with the obvious m r -

phism w(X)=H--(_X,_X)

of TranspG(y,y')

for R E k

+

%(x',s)

. By I, 5 2,

. By 1, 5 2,

; for this is precisely the definition

9.1 and 9.2, we have canonical i m r p h i m s

L?(X',_X)

9.3,

(R) may then be identified with the

set of families of m p s y' (S)-t_X(R@S) which are functorial with respect to S

. We have a similar identification for

-Hc+(xlrY)

. W u l o these identi-

fications, the Cartesian square G(R)-H-T(X',Z)

t

t

(R)

then yields the first formula. If G_ is a group, we infer without difficulty the formula for Norm ( Y ) (R) GThe last two fomlas are proved by means of the Cartesian squares

diag.

.

5

11,

1, no 3

GROUP SCHEMES

Let G -

Theorfa:

3.6

gnr 5

, and y'

If

a)

If

k-mnoid-functor

g

be subfunctors of

.

which acts on a k-func-

is a locally f r e e k-scheme (I, 9 2, 9.5)

X subfunctor of b)

x'

l e t Y_

203

, then -

T E U I S ~ , ( ~ ' , ~ ) is a closed subfunctor of

-

is a qroup, & _Y

_G

a locally f r e e k - s c h m , e

f

a)

To prove

apply I,

f o l l m s from I,

, observe

c)

5

5

which is also

x.

2, 9.7, whereas

t h a t by definition

2, 9.5 to diagram (2).

g is

separated, t h e n

6x

b)

11

is separated,

follaws from the f a c t that

is a closed embedding and

follows similarly, using diagram ( 3 ) .

d)

Corollary: Suppose that k is a f i e l d . Let G

3.7

k-group-

scheme acting on i t s e l f by conjugation. I f 2 is a closed subscheme of @ Y_'

is a subscheme of

G_

, then

Transp (Y',y) , C=t,(Y') G-

(Y) are a l l closed subfunctors o f 5

GProof: that

5

.

Since all schenes over a f i e l d are locally free, it is enough to show

5 is

separated, and this follows from the

3.8

u n i t section.

Lama: Let G beak-group-functor andlet <; is separated i f f

E~

.

5.

is a locally f r e e k-scheme and the k-functor

is a closed subfunctor of

Proof: -

.

G-

G-

then

G

(Y) is a closed sub-qroup-functor of 5

n

c) I_f g is a locally f r e e k-schene and the k-functor

If G

X

is a closed subfunctor of

Cent (Y) is a closed sub-mnoid-functor of d)

is a closed

_Y

-

E~:~+-+_G

-

kits

is a closed embeaaing. Wreover,

204

if

ALGEBRAIC

mms

11,

9 1, no

3

k is a field and 5 is a S C h e , t h z G is separated.

Proof: If

is separated, then eG is a closed epnbedding (I, 5 2 , 7.6b) -

)

.

Conversely, we have a Cartesian diagram f

GxGz--sG 4-

where f ( x , y ) = x y-1

for x,yeG(R)r Re$

- If

EG

is a closed embedding,

so is 6 (I, 5 2, 6.4). Finally, if k is a fieid and _G a k-sch-r G is a rational, hence closed, pint of G E~(-%)

.

Let G be a k-group functor acting on itself by conjugation. G We set G-= C e n t (GI and call Cent (G) the centre of G_ A subfunctor H 3.9

.

.

of _G is central iff it is contained in Cent(G) If 1I_ is a sub-groupfunctor of _G , it is clear that H is central (resp. n o m l ) in G iff CZtG (HI= G) ; we say that is characteristic in 5 if - G (resp. N-G_o n n (H)= -

i.e. if for each Re$ normalizes HBkR

.

, each autQn0rPki.m

of the R-group-functor SgkR

If _H is a normal (resp. characteristic) sub-group-functor of is a characteristic sub-groupfunctor of acteristic) in

G_

.

g and if K

is normal (resp. char-

Sd-direct products. Let G and & be k-g-roup-functors, and

3.10

let u_ : G_xg + g be a -peration

of

, then K-

, i.e.

on g which preserves the qroup structure

such that u_(g,hh')=u(g,h)u_(g,h') for h,h'e€J(R), gcG(R),

associated with (or, in other cmrds, the hamomorphim p:G+A&(_H) (H)) -& The semidirect product of G by g with respect to the given operation is Re$

g maps G_ into Aut

.

the k-functor H K G with the following group structure: for Re& g,g'cH(R) r set

,

11,

5

1, no 3

proposition: t:G+g

205

GROUP SCHEMES

k t _E

and

k-group-functors and let q:_E+G_

_G

. For

be hammrphisns such that q o $ = & l G

p (9) for the autcmrphisn of

(x

gcG(R)

r

RE$

.

, w2e

Then there q)BkR induced by I n t ( t ( g )1 is an i m r p h i s n of k-gmup-functors u_: ( g g ) X I G s _ E which makes the followP-

ing diagram comrmte:

Proof:

S m l y set u_(h,g)=h.$(g)

Let _E beak-group-functor,

a sub-group-functor fined by

of E

_H

if

Ref&,

hEEq(R)

and gcG_(R)

a n o m 1 sub-group-functorof

. Suppse the mqhism of k-functors

(h,g)uhg for geG(R) , he€J(R)

sider the inner a u m r p h i s m operation of

Re& _G

_E

.

and. G

gr_G+_E

de-

is an i m r p h i s m . Con-

on _H ; i f we assign _ H x _ C

the

corresponding semidirect product group structure, the iscnnorphism above i s a

ALGEBRAIC Gl"S

206

11,

5

1, no 3

group i m r p h i s m . W e also call (imprecisely:) E the semidirect product of

G by 5 . Example: the triagonal group.

3.11

Dd

I

r,

of

and LlJ-

Define the sub-k-functors (a. .I

as follows. If

* & G

13

( a . .)cD (R)<=>a = O

for

i#j

(a. . ) C T (R)<=>a = o 13 -nk ij

for

i>]

ij

-nk

17

cgn(R) ,

, set

R$&

zl

These are closed sub-groupschgoes of

(LA: for example, is the closed subscheme of defined by annihilating the functions ( a . . ) H a ij ij W e call T& the (upper) triaqonal group, %k the s t r i c t (upper) i>j triagonal group, and Ill the diaqonal group.

.

If

.

all.. ,antR*

matrix

(a.6. . ) 1

13

,

RC$

. Clearly

diag

: (pk)

n

, we write +

diag(al,.

..,an1

for the diagonal

D -nk

is a group iscmrphisn.

The group Tnk % I

Let

is the semidirect p r d u c t of

r-

(i(r),j (r)) (i, j)

be a bijection of the interval

for which 15 i < j <- n

is non-decreasing. For V_

by i t s n o m l subgroup

and we have the formula

s e t of pairs (r)

Q

and

-9J-

of *!L

0 < r
by

, we

,

c1 ,V2n (n-1)

1

such t h a t the map r-

onto the j (r)-i (r)

define the closed subschemes

11,

5

mm

1, no 3

f o r each RE&

me g(r)

.

,r

are normal s w o u p s of

For r #&(n+l)

207

SCHEMES

,

, and we

is a subgroup of

g(r)

have

which is isomorphic to a

More precisely, the mrphisn fr:y(r)+gk such that f r ((a. .) ) = ai(r+ll 17

k ’

,j (r+l)

is a group hchocar\orphisn whose kernel is g (r+l), and it induces an isamor-

phisn of

onto ak

. It follows imnediately that .

rect product of &r+l by y(r+l) Finally, i f xt (r-1)(R) , we have the formula -

RE$

g(’)

,

is the sfmidi-

(aij) C k ( R )

-1 ( I n t ( a . .)x) = a f (XI ; fr-1 - 13 i (r),i (r)a j (r),j (r)-r-1 -1 -1 i n particular x y x y ~ u ( (R) ~ ’ ) for y t y , ( ~ )

, x e-~ ( ~ - ’(R) )

.

and

5 2

LINEAR REPRESENTATIONS

Section 1

Definitions

p be a k-monoid-functor and. V a k-nd.de. presentation of G in V is a hanarorphisn of monoid-functors 1.1

Let

P :

linear re-

G -r L(V)

, we

i.e., for each R e &

are given a representation of G(R)

which depends functorially on R

module VNkR

A

i n t h e R-

. W e also call

(V,p)

a

k-G-module and define the category of k-(;-modules i n t h e obvious way. If

G

is a k-groupfunctor, then p f a c t o r s through a ( V ) ; the linear

E in

representations of

are thus i n one-one correspodence w i t h the

V

.

group l - K m K m r p h i s n s G * E ( V )

G be a k-momid-functor and l e t V be a k d u l e . By

Let

5

1, 2.5, this is

equivalent to being given the three following structures: (i)

a representation of

(ii)

a l e f t *peration

gcG_(R) (iii) Of

on Va

, the endatarphism of

such that f o r each RE&

VGkR

on ga (V) (R)

induced by g i s R-linear;

preserves-the Rlnodule structure of

-

1.2

Examples.

a) Take f o r

, the

presentations of

Tk

representations of

r

the constant k-mnoid

in V

. Accordingly, k k sn-dules

is projective of rank 1 , VWkR

, and

1 f o r each Re$

character x:G-+(lk f o r g'G_(R)

,

rk

.

associated

X

L(V) = Qk

, so

the category of k-r -modules is k

.

is a projective R-rodule of rank

that the l i n e a r representations of

i n V are i n one-one correspondence with the characters of X

-

i n V are i n one-one correspondence w i t h the linm

isamrphic to the category of V

g

Da (V) (R)

action

. One v e r i f i e s l n n d i a t e l y that the linear re-

r

with the abstract mnoid

c)

and. each

a r i g h t @operation on Da(V) such that, f o r each RE$

G (R)

b) I f

in V ;

G_

is associated the representation

vEVWR

,

Re&

we have

Suppose that the k-mnoid-functor

g

G

p:G+&(V)

P(g) ( v ) = x ( g ) v

.

. With the such that,

acts on the r a t on the a f f i n e

G_

11,

5

2, no 1

I n this way we construct a mrphism geG_(R)

,

which is a W q M s n of

p:G_+&(Lp(x) )

G in the k-rcd.de d(_X)

hence a linear representation of

k-mmid-functors, For

209

LINEAR REPRESEXTATIONS

,

Re&

.

p(g) is an algebra e n d m w h i s n of

d(_X)BkR

G i n the k-rcd.de d(S) such and that p(g) is an endm.orphisn of the algebra d(X)BkR f o r each RE& each geG(R) , yields a r i g h t W p e r a t i o n on g Conversely, a linear representation

of

p

.

d)

c ) ) induces a linear re-

G i n d(G) called the (right) regular representation of

presentation of

, written

a f f i n e k-monoid-scheme; the r i g h t

(5 1, 3.3

G_ on i t s e l f

translation operation of G_

that G & an

Suppose in particular

g

+

6 (gl ; accordingly we have

(6 (9)f ) (x)= f (xg)

.

Returning to the situation i n c) , suppose f o r shnplicity that k

e)

field. L e t

such that TranspG(Y,Y)=E then J

is a s&-k-G-dule

, we

. Let of

d(X)@kR

f i e 3 with the ideal of

se&

is a

3 which is stable under G_ , i.e. be the i d e a l of d@) defining Y_ ;

be a closed subschene of J

J(g)

. For

if

Re

h% ,

defining YBkR ; i f

JBkR

may be identi-

f e J a k R and yeY(S) - ,

have ( p ( g ) f )(y) = f(yg) = 0

so that

p(g)fkJ@R

.

he sequence

+4:)+ O

is therefore an

o+J+~(x)

exact sequence of k-G-modules. 1.3

Let

p:G_+L_(V)

be a linear representation of the k-rnonoid-

functor G in the k-mdule\ V

,

i f , f o r each Re$

we.

is said t o be pure the canonical map W @ R+VBkR is injective, i.e. i f

may be identified w i t h a subfunctor of

ple, i f

V

. A sub-k-module

W

is a d i r e c t factor of

such that W ' C W

%l,w(R)

Clearly,

V

. Let

. Define the subfunctor

k

Va

-

W'

W of

. This is the case, f o r examand W be of

= {gtG(R): p(g)x-xtW'@R,

i s a sub-mnoid-functor

of

2

V

G

pure

s M u l e s of

by s e t t i n g

xkW@JR}

.

. I n particular,

we have

210

ALGEBRAIC GROUPS

,

For simplicity we write C s t G ( W )

Cent

,

(W )

G a

&.

Transp (W W )

G a's

--

TranspG(W,W) and Norm (W)

Norm (W I-.

was a group.)

LemM: suppose that G_ is a groUpr and that W' is a sub-groupfunctor of

G

Proof:

Let

G

.

T r Y (W',W')

autamorphism. In particular, i f

2 , no 1

for

is f i n i t e l y generated. & T

. I n particular,

NIG(W') =

-

and geG_(R) ; the e n d w r p h i s n

Re&

5

(Recall that when we defined

-+a_

G

the l a s t object we assumed t h a t

11,

g t %lrw(R)

p(g)

of

V@R

is an

P(g) maps W W R

into W'@R and i d u c e s an injection WIWR/mR+W'@R/mR for each maximal ideal m OE R By f i g . m. 11, 9 3, prop. 11, p(g) induces an autanorphisn of W ' W R and we have ~ ( ~ ) - ' ( W W R I = W W R mus, i f ~ E W B R, we have

.

.

-1 x-x = p (4) (x-p (9)x)E

-1

p (9)

-

so that g-lEc+l

, as

rW(~)

G

was to

Let

Propsition:

1.4

k-=d W'

.

mnoids of

st,w

. I_f

_G

(

~

1

R) 8 =WQ R

be shown. be a linear representation of the

p:G_+&(V)

V

and

CstG(W)

r

G_

.

. If

G

space of

orthogonal to W1

5

for a l l pairs

(i,j)

-%I ,w

I-

Rmk:

and that

, then

. It follows t h a t

V

W

and satis-

mtG(W) and Norm (W)

-

(ai) generates W and if

defined by the functions

. Let

WrnspG(W,W) are closed sub-

I n virtue of L a m a 1.3, it i s enough to prove that

subschaw of

V

which are both d i r e c t factors of

G is a qm&, then

are closed subgroups of Proof:

p (9)

i n the finitely qenerated projective k-mdule

be two sub-k-mdules of

f y W'CW

1.5

r

gE(+

gwb.(p(g)ai-ai) 7

rw

7

Jw

i s a closed

( b . ) generates the sub3 W(R) i f f b . ( p ( g ) a . - a . ) = O 1 1 1 is the closed subschercle of G_

,

W e could also have observed that WA

-

is a closed subfunctor of

is a transporter, and then applied I, 9 2 , 9.7.

r

A

stable under G

pure s u b - k - a u l e if

W of the k-Ginsdule V

Transp (W,W)=G_

G

.

is said to be

Wa

-

11,

5

2, no 1

LINEAR REPRESENTATIONS

Suppose that k

211

is a field. W e say t h a t the representation p:G_+&(V)

siniple o r irreducible i f

. The representation

which are stable under G_

or cmpletely reducible i f

is

and 0 and V are the only subspaces of

V=O

p

V

is said t o be _semisimple

is a d i r e c t sum of shple k<-rrodules.

(V,p)

A direct sum of senisimple representations is semishple. A subrepresentation,

a quotient representation of a semisimple representation is semisimple. If

k is a field, the category of

If

V

k-(;-modules is always an abelian cateqory. it contains Jordan-Holder series;

is a f i n i t e dimensional k+mcdule,

the quotients of a Jordan-Holder series of we c a l l s-le

factors of

V

.

are sinple ks-modules which

V

is said to be isotypical i f a l l its simple

V

factors are isomrphic.

G

In particular, i f we denote the unit character of = (V a 0

(v,)- , and we -

G

set

J;-=v,

W e inmediately get Vm= (V ) (k) (V )

-a!!

o€

,

0

, we

have

.

But (V ) and k m@p ma are not necessarily identical, i n other words, the subfunctor (V )

am

va

.

by

(V ) (R) = (V@ R)

a!!!

may not be defined by a suktrcdule of

cannot arise i f proposition:

V

. Hawever,

am

t h i s situation

k is a field.

Suppse that k

is a field, and l e t

representation of the k-mnoid-functor

p:G-+&(V)

be a linear

5 in the k-vector space

.

V

. Then,

for each character

m of 5 , we

Proof:

and x E(V@ R) ; we show that xEVmBkR L e t (ai) k "@p , wr ite x = l v .1@ a i w i 6 vi€V L e t

Iet RE&

have

(V )

ma

be a base for the k-vector space R

s€$

and q € ~ _ ( ; s ) we have -

P ( g )s @ RXs 8 R - ~ ( gs ) Rxs @ R

which i n V@S@R may be written

= (V

am_

.

.

ALGEmaIc GRDWS

212

But since (ai) is a base for R over k p (g)vi@ 1 = vi@m(g) as required.

1.7

for each i

11,

5

2, no 2

, the last f o m l a implies t h a t vi€Vm , hence x € V @ R

. It follows that

-

mk

Proposition: Suppose that k is a field, and let

p:G-tL_(V)

be a linear representation of the k-mnoid-functor 5 in the k-vector space V Then the sum of the Vm m_ ranges through s\(G,Clk) is direct.

.

Proof: Let

-

be distinct characters of G

q r . . . r %

i = l,...,n w i t h v+...+v 1 n= O follows that in V @ R

.

. If

0 = p ( g ) ‘Vl+. ,+v 1

nR

=

R€M+

, and

and g€G(R)

let vi€V , Pi , it imnediately

.

Vl@E1 (9)+. .+vr@gn (g)

.

If the vi are not all zero, there is u€% such that the u(vi) are not all zero. We then have u(vl)~(g)+ u(v,)~~(~)= 0 , which contradicts Proposition 5 1, 2.9.

...+

Now assume that

k

is a field. The representation p:G+&(V)

is said to be

diagonalizable if V is the sum of the Vm or, in other mrds, if we can select a basis vi of V such that each &bspace hi is stable under G (the endmrphims p (9) them being representd over t h i s basis by diagonal mtrices). A direct sum of diagonalizable representations is diagonalizable. A

subrepresentation, a quotient representation of a diagonalizable representa-

tion is diagonalizable. We define the tensor product of t m representations in the obvious way; the tensor prcduct of t w diagonalizable representations is diagonalizable. Rmark: For each character z:G+Qk

, let

k,

be the k<-rrcdule such that

.

The above prok (R)=R and g-x=g(g)x for R C S , xER -and g e G ( R ) m_ position also follows frgn the facts that k, is a simple k-(;-module for each m and that k, is not iscsnorphic to -k- if _ m # ~ ,

-

Section 2

Linear representation of affine groups

Throughout thir section G denotes an affine k-mnoid with bialgebra

rr, 9

2, no 2

. Its

d(G)=A

213

LINEAR REPRGSENTATIONS

coproduct is denoted by A and its augmentation by A

(9 1, 1.6).

Let V be a k-mdule and let p :G +&(V)

2.1

be a mrphism of k-

functors. We have canonical i m r p h i m %E(G,L_(V))

Y

1

The mrphisn p

= &(V) (A)

y

S(V,VUkA)

.

accordingly induces a k-linear map

L+,:

V

-f

.

V@A k

L+, m y be defined as follows: if gOeG_(A) corresponds to the identity map of A , we have %(v)

=

p(go)vAe V@A k

.

For p to be a mnoid hcSrrrra3rpkisml it is necessary and sufficient that the

.

and (Ham2) of 9 1, 1.7 be satisfied for f '47 Adop ting the notation employed there, the prcduct i1(A$*i2(L+,) is by definition (cf. 5 1, 2.4) the ccsnposition of the mrphim corresponding to the unbroken arrws of the diagram: conditions (Hanl)

V@ i2 47 VjV'8A ->V@A@A I

;

I

I

1%

@A@A

V@ilu9A@A \y V@A@i2 V@A@A- - - - - , v @ A Q ~ A ~ ~ A - v @ A ~ A @ A Q P A - v @ A @ A

where m(a@Jb@c@d)=ac@M

V @rn

. Since the canpsition of the arrows

second line is the identity map, condition ( H c y ) following condition:

in the

is equivalent to the

ALGEBRAIC GROUPS

214

11,

5

2, no 2

(Modass)

is c m t a t i v e Condition

(Hm2) is equivalent to:

v

= V@k k

is carmutative Definition:

Let

A

be a bialgebra with coprcduct

A (right) ccamd.de is a pair

% :V -+VBkA

,

(.,el

where V

AA

and augmentation

k-module and

is a linear map satisfying (Wdass) & (Modun)

of the ccmdule (V),' into the m u l e such that h 4 = (4 @A) %

.

(W,h)is a linear map

By the above remarks, a k - @ d u l e structure on a k-module

to an A - d u l e structure on V

. A mrphisn

V

$:V+W

i s equivalent

(i.e. the category of ks-modules and the

category of A-cmrdules are i m q h i c ) . Given

If

47 , we imndiately construct the hcmKmDrphiSn

g€G(R)

corresponds to the mrphisn f:A*R

hence p(g)v =

(I4fBf)(Ap.)

, we

have g=G_(f)go),

.

II,

9

2, 110 2

LINEAR REPRESENTATIONS

The endmrphisn of

VBkR

associated w i t h g

215

is thus the wnposition

Remark: Left A-camcdule structures are defined similarly; these correspond

to linear representations of the opposite monoid of

SsA

.

The axicans (Coass) and (Coun) of 1.6 show that t h e coprcduct

2.2

endows A with an A-carnodule structure. This structure is k associated with the regular representation of G i n A (1.2 d ) ) For each AA:A+A@A

k-Wule

W

, write

t r i v i a l operation.

0 4J:V+V @$A

.

Wo

f o r the k-(;-Wule obtained by assigning W

For each k-(;-module V

, the

the

axian (Wdass) s i g n i f i e s that

is a k e d u l e hamroorpkisn. By (Modun)

47

has a k-linear

retraction. If

V

is f i n i t e l y generated arid projective, there is a k-linear map V -+i kn

w i t h a retraction; hence

is a homamorphisn of k-c;-modules w i t h a k-linear retraction. I n particular,

if

k i s a f i e l d , each f i n i t e dimensional linear representation of

G

can

be embedded i n a pa~erof t h e regular representation. 2.3

Let

p:G+&(v) be a linear representation of & in a f i n i t e l y

generated projective k-module

V

. By 5 1, 2.4,

L(V)

is an a f f i n e k-scheme

which is i m r p h i c to SJS(%@~VI, and the k-algebra hanomorpkisn S(\akV) + A

induces t h e k;linear

map

AT.GF.RRAIC GROUPS

216

for the image of w@v p

under

associated with v & w

cBkR

, and we

. By 5 1,

2.4,

11,

5

2, 110 2

call this the coefficient of it is defined by setting

field. The coefficient space of p , denoted by C(P) r , in other words, the vector subspace of A the coefficients cWrv , w e 4 , ~ E VIt is a subspace of A

Suppose that k is a

is defined to be the image of c generated by

.

0

which is stable under G ; we have a mncmrphim of k w u l e s V-tV @ C ( p ) , (5)O@V+ C (p) In particular we infer and an epimorpkism of k-+dules

.

the

be a finite dimen, and let C ( p ) be its

Proposition: Suppose that k is a field. __ Let p:G+_L(V) sional linear representation of an affine k-mnoid _G

coefficient space. V & C (p) have the sane simple factors. Yore-, p is sesnishple, isotypical or diagonalizable iff C ( p ) has the same

property. Finally observe that p is a closed Embeddrn ' g iff S ( ~ B V ) + A is surjective, i.e. i f the coefficients of 2.4

If (V,%)

p

generate A as a k-algebra.

is an A-cmdule and mcA is a character of G

r

we have

vm =

CvtV : %(v)=v@ml

this, notice that if gotg(A) is associated with the identity map of A , by definition we have p(go)vA=%(v) , hence %(v)=v@m if vEVm. Conversely, if p(g O)v A =v@m , we have p(g)vR=m(g)vR for each gcG_(R)r Re$ (Cf. 2.1). To prove

.

Similarly, a pure sub-k-module W of V is stable iff A W ~ W @ ~ A

2, no 2

11,

217

Example 1: Linear representations of diagonalizable groups

2.5 Let

LINEAR REPRESENTATIONS

r

be a mll c m u t a t i v e mnoid; l e t G_=D(rlk

( 5 I, 2.8).

If

sends vc V p h i m of

onto V

that A = k [ r l

is an A-cmcdule, the map

(V,%)

% :V

, so

-+

V@k[r]

lye p Y W @y , where

such that, for each vEV

I

is a family of endamr(py)re (v) vanishes f o r almost a l l y pY

.

In t h i s situation a i m (Modass) becomes

i.e.

py pyI=O for

yfy'

and py py=py

.

These two conditions are accordingly equivalent to asserting that i3-e p the projections of a grading of type gc_o(r)k(R) and V C V @ ~ R,

where we have identified elements of

. By 2 . 1 we have for

r

on V

r

with characters of

plained i n § 1, 2.11: This formula also shows that p (V) V

Y

Y

introduced i n 1.6 and our remarks above inply that V

of the V

Y

. W e sumnarize a l l this i n the

Proposition:

Let

izable k-mnoid. - If

Re-%

Y

be

,

Q(rIk , as ex-

-

is the k-module

is the d i r e c t sum

p:_D(T)k+ &(V) be a linear representation of a diagonal-

we identify

r

with a set of characters of _D(I'lk

.

,

i s an for y t r The functor V*(Vy)yCr V is the direct sum of the V Y equivalence of the category of k-g(r)k-mcdules with that of graded k-modules of type

r .

In particular, i f of

_D(TIk

k

is the spectrum of a f i e l d , each l i n e a r representation

is diagonalizable.

218

AIx;EBRAIc GROUPS

If

Remark:

Q(rlk acts on the a f f i n e k-scheme

d(_X) induces a

of the k-algebra

X I

then the

o(X) . Conversely,

1 . 2 c ) ) form an algebra grading of

r

11,

9

2, no 2

c”(X),

(see

each grading of type

.

_ D ( r l k - o ~ a t i o on n

Linear representations of t h e a d d i t i v e group. The is A=k[T] with AAT=T@l+l@T arki E ~ T = O bialgebra of the group a k W I f (V,%) is an A-camdule, s : V + V @ k [ T l maps v onto li=op(v)@Ti Example 2:

2.6

where

.

,

and f o r each v E V

piC%(V)

pi(v) = O

f o r almost a l l

i

. The co-

rnodule aim may be written:

or

pjopi= ( ( i l j ) ) p i + j

v or

po =I%

(Pi)i t N

I

where

= (VQ?E*)e(v, = p o w

, and

I

. A k L T k m c d u l e structure is thus defined by giving a sequence of end.mrphim of

each v t V

( ( i , j ) )= ( i + j ) : / i ! j :

, pi(v)

such t h a t

V

vanishes f o r almost a l l

we have

pjopi=

i

. For

( ( i , j ))pi+j

Re&

ard, f o r

and t c a ( R ) = R

W

W e now analyse these equations i n two particular cases: a)

Characteristic 0 : I n this case t h e r i n g

Setting

pl=X

, we

k

is a Q-algebra.

have pi =$/i! and, f o r each vcV

for sufficiently large

i

(“X is l o c a l l y nilpotent”)

,

d ( v ) vanishes

. we have

m

For

RE&

and

t c c1 (R) = R , we

have

linear representations of ak -i n

l o c a l l y nilpotent endmrphisms of

b)

v V

p (t) = exp t (XBkR)

are i n one-one correspondence w i t h the

.

Characteristic p # 0 : in this case the r i n g k

p prime.

. Accordingly

is an F -algebra w i t h P

The c a l c u l a t i m is elementary although a t r i f l e technical. W e g e t t h e

,

11,

8

2,

110

LINEAR REPRESENTATIONS

2

iem

following result: for

set

I

S. =p i ;

l P V which satisfy the following conditions:

f o r each vcV If

and sufficiently large i

n = n + n p+ 0

w e have

1

P

...+n 2r

n

=

,

...snr

"0 "1 so s1

.

r

the si are m ~ h i m of

.

, is

0 snisp-l

219

the p-adic expansion of

nc[N

,

.

no! "1!. .nr!

Setting

SP

exp(Si)0 = l + s . X + . . . + -

l

i

(p-11 !

1

xP-l

,

we get

and tca(R)=R , we have

For Re&

i

m

p ( t ) = v e x p t P (sifR)

i=O

Conversely, a family

(

s

~

of ) endmrphisms ~ ~ ~ of

V

satisfying the a h v e

conditions defines, via the above f o m l a s , a linear representation of

. For

in V

instance, taking V=k

,

able us t o determine the characters of Proposition:

L e t p :a k - + g ( V ) -

then ?k=o

.

Proof:

vEV

for

Let

j >i

. Then

, v#O

pn(pi(v) 1 = ( SO

ak

.

be such that pi(v)# 0

. For i f

n>O

1

(vf = 0

t h a t %(pi(v))= pi(v) and pi(v)

ak

preceding renarks en-

be a linear representation of

and l e t i t N

pi(V)cVak

, the

GL-(V)=pk

,

,

is invariant ( 2 . 4 ) .

ak

.I f

V #O

, pi(v)# 0

,

AIx;EBRAIc m w s

220

IIr

that k is an algebra over

.A

and set G _ = rak

[F

P s M l a r to the one above shows t h a t the

e shows that

calculation

P linear representations

G correspond to the cxnmting families (si) s i .,r-l V whose p* powers vanish, via the formula

The same argument as h

2, 110 2

pk . suppose

Example 3: L i n e a r representation of the group

2.7

5

V+O

p

of

of erldmrphisns of

implies V'+O

.

in V are in one-one Pk' V whose p* pawers vanish. More

I n particular, the linear representations of correspondence with the endmrphisms of particularly, t h e characters of

correspond to the elements of Pak pth powers vanish; hence we g e t a canonical isamorpkism

k whose

which yields a canonical isamrphism D ( a ) =

- p k By the h

p%'

e arquwnts, this iscmmrphism is given by t h e pairing

Escample 4: Linear representations of

2.8

&(U).

Suppose t h a t k

is a f i e l d and U

is a finite dimensional k-vector space. Identify Qk w i t h

a sub-k-mnoid of

&(U)

e.ndcxmrphism u -UX p :L (U)+ &(V)

,V

such that

example, i f

and each xc R

. For each linear representation

accordingly carries a natural k+k-module

, where

is a sub-k-&(U)-mdule of

of degree n

U akR

the

structure and

denotes the subspace of V formed by the n p(x) ( v @ l ) = v @ x n f o r all Re$ and all xcR Clearly Vn

we have V=@ntNVn

v

by assigning to each Re&

of

. If

v=&

V =V

n

and

V

,V p

V

. W e call

.

the homgeneous ccanponent of V n is said to be h a q e n e o u s of degree n For

satisfies

V

.

p ( 4 ) 'vl@.. .@vn' = g(vl)@. .Qg(vn)

.

,

11,

5

2, no 2

221

LINEAR REPEENTATIONS

then V is hrcgeneous of degree n

.

Identify the algebra of functions of

L(U) w i t h the symnetric algebra

5

S(%@U) as i n (i=l,.. .In) Cf

, we

1, 2.4.

W i t h the notation of 2.3,

have (9) = (f,@..

@...@fn,U1@...

1

.@

if

fie%

and uicU

.

fn, q ( u p . .@y(un) )

accordingly coincides with t h e canonical map of

i n t o s (5@u) of

S(%@U)

nent of

. The image of

of degree n

S(%@U)

.

t h i s rrap is the space of m e n m s polyncanials

This space is therefore t h e hcmgeneous acanpo-

of degree n

. Thus it follows f r a n proposition 2.3

that

t h e simple factors of a hamqeneous k-& (U) -module of degree n already appear

as simple factors of

.

hbre generally, 2.3” shms that i f

G, is a closed sulroomid of

L(U) , the

smle f a c t o r s of any k-G=module appear as sirrrple factors of the k s d u l e s 8%

for

ntN

.

a k e d u l e structure i n such a way that

AIxEmAIc Gwsups

222

..@ xn@Yl@. .

g

*@

Y),

= g (x1I@.

.

.@

11, § 2, no 3

.

g (xn)@g (yl)@. .@g (ym)

.

. Then the simple f a c t o g

Proposition:

g b e a closed subgroup of

of each k + d u l e

appear as simple factors of the k-(;-rrodules

Praof: Set

U =Vx%

&(V)

,

ard consider the l-mcmrphisn p :C&(V)+&(U)

p (9) (x,y) = (g(x), 6 ( y ) 1

.

n,mcN.

defined

We claim t h a t this is a closed embedding. For and the it can be s p l i t into the obvious closed &&ding &(V)X &(%I+ &(U)

by

mnamrphisn

p

1 : &(V)+L(V)x_L(%) L

(x,y) ranges through V*% 2.8 to the closed anbedaing E

as

&

such that p l ( g )

. In vFrtue of

+a(V)

= (g,&)

. Now

if

t h i s , it is enough to apply

8 &(U) , noting

that the k - G (V) -fiodules

.

are direct sums of d u l e s iscsnorphic to the QDPV q

Existence of linear representations (in the case of a base

Section 3

field) Throughout this section k is a field. Given a linear representation p :G_+&(V)

3.1

the intersection of any family of stable flxbspaces of space of

V

. In particular, f o r each subset of

V

of the k-mnoid V

G

,

is a stable sub-

there is a smallest stable

subspace of V which contains it: we call this the stable subspace generated by the subset.

Lemna: -L e t

p

p :

+& (V)

be a linear representation of

the a f f i n n k m n o i d

V

generates a f i n i t e

. Then each f i n i t e dimensional vector subspaceof

dimensional stable subspace. Proof:

Let

A

dule l a w of

V

be the bialgebra of

G , and l e t

:V+V@ A k

. It is sufficient to show that each element

be the c m -

x of

V

5

11,

2, m 3

LINEAR RE!?FE!SE"IONS

223

belongs to a f i n i t e dimasional s t a b l e subspace. L e t

. Set

the k-vector space A =

1x.@a 1 i ~

J.

be a base f o r

(ai) iE I

'

Axim (Fbdass) yields

i n which we have set

.

(Pkxlun) gives x = lixi€

xi

contains x

,

. Axim

vi=lj x.@b ji

A a . =C.b. . @ a j ; this gives A 1 711 (ai) The vector subspace W of

and s a t i s f i e s %WCW@kA

generated by the

A

, and accordingly meets

the re-

quirements.

Larma: rmid G - - Let H be a closed suhmrmid of the a f f i n e k m ~Let - I be the ideal of A = i l ( G ) defining H Then H_=TranspG(I,I)i n the regular representation of G ( 1 . 2 d) ) 3.2

.

.

Pmf:

Let R t $

fcI@R

, we

have

. W e show that

claim that

.

& ( h f f c m R For each-S€&

( 6 ( h ) f )(x)=f(xh)=O , hence

Conversely, l e t gE G(R) each

K(R)=TranspG(I,I)(R)

satisfy

a f f i n e algebraic k-rrormid and l e t

of

Proof: Let -

such that

V

p

g

G_ to act on A

= 0

, whence

(U,V)

Let G _G

for

be an

. Then there

and a vector sub-

-

G_ and I

the ideal defining

.

. Allow

s of the regular representation. By 3.1 there is a

by m

A

such t h a t V

a k-algebra and I n V generates t h e ideal I A

.

gEH(R)

be a closed suhmnoid of

f i n i t e dimensional stable subspace V of k-algebra

, we

i s a closed a b d d i n g and _H=TranspG(U,U)

be the bialgebra of

A

. Then

6 (g)f c I@ R f o r each f€ I@ R

p:g+_L(V)

space U

and

arad each x t g ( S )

Existence theorem f o r l i n e a r representations.

3.3

hcIl(R)

& ( h ) f e I @ R as claimed.

we havq, f ( g ) = ( S ( g ) f )(el

fEI

. If

is f i n i t e l y generated)

. Set

generates A

as

(since G is algebraic, the

U =I n V

. hQ

claim that the pair

s a t i s f i e s the conditions of the tharem.

a) W e have T r I Y G ( U , U ) = g

the action of

G_

OK

A

. Since

U

generates the i d e a l

preserves the algebra structure of A

I

, and since , we have

.

Au,;EBRAIc m w s

224

Finally, by the lam, we have Transp

G

b)

p :

onto the coefficient

tion of

~ ~ : A - t to k V

so that

c

3.4

2,

no 3

.

, we

c

and grG_(R),

have for RE&

c

WIV

is surjective.

S (%@V)-+ A

sends

(2.3). I f we take for w the restsic-

WIV

. It follows that the coefficients

=v

WIV

accordingly

5

is a closed embeading. For the hchnroorphFsn S (%@V)+A

-+L(V)

w@v'&@V

(I,I)=H

11,

Corollary: An algebraic k-mnoid

mrphic to a closed sulnmnoid of an L(kn)

.

G

generate A

I

and

is affine i f f it is iso-

Proof: The coM3ition is obviously sufficient; theorem 3.3 necessary (take g = .

shows that it is

+)

3.5

Corollary:

Let

.

G be an

a f f i n e algebraic k-group and l e t

be a closed submnoid of

G Then: -~

a)

H_ is a subgroup of

5

b)

there is a f i n i t e dimensional linear representation G_+C&(V')

;

li n e (i.e. a 1-dimensional vector subspace) Proof: a) of

3.3. Suppose U

.,xnE V@ R

I

has dimension n

. Set

a 1.3. For b) I take a linear

we have

= I V E V B R : vA(D@R)=O)

k

V'= A%

such that, for

W e have, for each RE$

U@R k

V'

and a such that H=Nom (D) Y

and a subspace U of V satisfying the conditions

sider the representation G_+GJ,(V')

xl,..

D of

follows m i a t e l y fran 3.3 and l

representation G+&(V)

H

k

I

D = h"v

RE$

and con, g€G_(R) and

.

5

11,

2, no 3

LINEAR REPRESEXTTATIONS

so that g (u)E U@ S 3.6

-

Let _G be an a f f i n e algebraic k m m i d . Then the

l a r g e s t subgroup functor

, where

Proof:

By 3.4,

.

and gE Transp, (U,U) (R) =_H (R)

Corollary:

form Gf

fE

225

13(~)

g* of

G

is a character of

we may assume that

G_

.

is a closed suhmnoid of

5I-I G_L,k

is a group by 3.5, and is accordingly equal to G*

take f o r

f

&(kn)

. Then

. Hence we may

the r e s t r i c t i o n to 5 of the determinant function. Proposition:

3.7

G -of the -

is an a f f i n e open subgroup of

G -

Let - G_ be an affine k-mnoid Iresp. k-qroup).

Then there is an inverse system (resp. k-groups)

I€2

,

(ci)iEI of

, and a

a f f i n e alqebraic k m m i d s

coherent system of lxmxmrphism

G+Gi

such that

a)

the maps

d(Gi)+ d ( G ) and - d(Gi)+J(G.1

are a l l injective;

7 b) d(g) is the union of the images of the Proof:

Let

spaces of A

(3.1)

(Vi) E i I

A=@)

be a directed family of f i n i t e dimensional vector sub-

which are stable under

. As we have seen i n the proof

S(%i€9Vi)+A

&si) .

contains, V

i

G_

of theorem 3.3, the image

. It follows that

f i n i t e l y generated k-bialgebras

A

i

and such that UiVi A

generates Ai

of

is t h e directed union of

. This proves the proposition i n the case

G is a k-group, let di be the image in A of the determinant function d e t E d ( & ( V i ) ) = S(hi@Vi) Then A is the diand S J ( A ~ ) ~is ~ a closed s-noid of g ( V i ) , rected union of the (Ai)d

where

is a k-mnoid;

if

.

hence a k-group (3.5). Remark:

Consider the case of groups. It can be shown that the mrphim

and "+% are f a i t h f u l l y f l a t and are epimrpkisms in the category of a f f i n e k-groups. I t follows t h a t each a f f i n e k-group is the inverse limit,

E+Gi

i n this category, of a "strict" inverse system of a f f i n e algebraic k-groups.

Throughout M s paragraph

G

a t m u t a t i v e k-groupfunctor (g,m) _tgm

, which preserves

g(rrthn')=gm+gm' f o r Re&

denotes a kmnoid-functor. A _Gllu3dule is a

g cxpippcd with a @operation written t h e group structure of

,

,

_M

.

, i.e.

satisfies

of Grrodules is a hcmxmrphisn of k-group functors which cmtnutes with the m p e r a geE(R)

rn,m'c&(R)

A harnsanorphia

tions. Section 1

The Hcchschild ccanplex and the exact cohcrology sequence

1.1

Let

g be a @module. For each n 2 0 assign t h e set

c" (G,M)

=

jq$

the c m t a t i v e group structure defined by t h e group l a w of ~n element

of

c n ( ~ , g )is

called an n-ccchain of

accordingly a systen of n-ccchains -R f

e Cn (G_(R),M(R) 1

G i n _M

(5 1, 1.2) . ; this is

11,

6

3 , IIO 1

HXHSCHILD COHWJLXX

a0 , we see which '%=%

writing out rnEbJ(k)

for

, so that

for all R e &

, that

G into

and that the l-cob&ies

w h e r e rnEg(k1 1.3

0 0 H ~ ( G , M ) =z ~ , g ) is the set of

a0 and 3' , we see that the l-cocycles are t h e crossed ham-

By writing out

mrphisns of

inmediately that

227

(cf.

5

is, the mrphims f:G_-+M satisfying

are the t r i v i a l crossed ha-ramrphisns of the f o m

1, 3.10).

Given a errcdule hchnamorphism

_F:M-+N_ , we define

in the obvious

way a kmamrphisn of ccnnplexes C'(G,f)

: C'(G,N)

-+

,

C'(G,X)

whence group hmsoorphisns

Suppse that G_ is an a f f i n e k-mnoid with bialgebra A be n a t U a l l y identified with

product of

n copies of o-+I$

A

M(A%)

. If

where An'

-+

?(_G,_M)

-+M_+PJ"+O

M' (R)

+

M(R)

-f

E"(R)

-t

0

may

is the k-algebra tensor

is an exact sequence of e m d u l e s (i.e. i f f o r each REw& 0

. Then

t h e sequence

is exact) I for each n

Proposition:

we then have an exact sequence

I

If G is

an affine k-momid, the functor

B-Hi(G19

I

~

w x h & ranqes throuqh the catwow of G-mdules, is the derived functor of G the functor M++M-(k) Proof:

It is enough to prove that

d u l e _M there is a e d u l e %(G,~(M))=o

for n > O

Take _E (I$ = L%(gl@

Assign

I

.

H;)(G,?)

i s effaceable, i.e. for each G-

_E(g) and a m o m r p h i s m &*g(&) , such

that

where

E(M) the group-functor structure induced by that of

together w i t h

such t h a t f' (h)= f (hg) : this tums E_ (M_) into a ( + d . u l e . The mrpkism I3-+E(Pj) which assigns to mcFl(R) the mrphism : +% i s a mnmrphisn (since such that & ( g ) ='% for gtG_(S) I SCI& ,

the -=ation

& sR

cm(e)=m) , and is ccsnpatible w i t h the action of g(f,)

(h) = fJhg)

= ( h g )= ~ h(gm)

LemM: For each catmutative k-group-functor F p G ,H_ChIX(G'D)= 0

for n > O .

G (since =f

I

gm

(h)

we have

I

229

and so

(s )

n

is a hmtopy operator.

0 1 1.4 As usual we can extend the definition of H O G , @ and HO(G,M_) , as well as the initial stages of the cohamology exact sequence, to m r e general situations. We will only need to do this in the sinrplest case, which follows:

Let _Mr E ' , E" be k-group functors on which G, operates by group d m r phisms and let

v M_ Y M"

M_'

be hornanorpkisms of k-group-functors which are ccnnpatible with the action of

G and are further such that for each 1 + PJ' (R) is exact (i.e. g ( R ) y(R)

, with

that

HOG,!')

1

uR)

M(R)

'3) M"(R)

, the sequence +

1

induces an iscanorphim of

4'(R) onto the kernel of

surjective), S u p p s e further that M' is defined.

y(R)

is cmtative, so

G 1 Proposition: Under the above conditions there is a map a :M" (k) Ho (G,M_') such that in the sequence of maps -+

230

AIx;EBRAIc GRDUPS

0

0

& vo=Ho(G,v) , wehave Ker%=l

where u o = H 0 ( G , d -1

a (o)=mo

.

Proof: The f i r s t ism e q u a l i t i e s are imnediate. -

, choose

m"tM"'(k) -

11,

mcM(k)

we have y(gm)= gmt'=m'l=y(m)

,

5

=muo ,

Kerv -0

We now define

.

a

3 , no 2

. If

such that y ( m ) = m " For Re$ and gcG(R) -1 , hence gm ,m may be written i n the form

cR

g ( f R ( g ) ) , with f,(g) (R) ; one v e r i f i e s imnediately that the define I f mlEu(k) is a a mrphism _f:_G-+g'which is a 1-cocycle of G_ i n g' second e l m t such that

m =u(m')m 1 -

, hence

If

4.1.m-1l

= g ( g m 1 ) . g m . m -1*u_(m'

m"

.

,

Remark:

, we

1

-

mDdu10 B (G,M_')

a(m")

such that

is independent of the

. We now verify the last assertion.

can choose me @(k) and a (mu')=0 ; conversely, i f m'-'. f ( g ) =0 f o r each there is m' c_M'(k) such that 'rn' -R

m"€ Im(yo)

gcG(R)

f

m ; t h i s we denote by

a (m") = 0 ,

is m'cb-l' (k)

such that

It follows that the class of

choice of

, there

y(?)=m"

.

Re$

. Then

In general

Section 2

u(m')m is invariant under G and is projected onto

a is

not a group hcammrphism. However, it is one i f

&tensions a d cohmology of degree 2

Throughout this section we assure that G is a k-group-functor. 2.1

Definition:

Let

extension of

G & 8 is

a sequence of k-group-functor

M_4E_J+ satisfying t h e f o l l m i n g two conditions:

a)

for each Re$

, the

sequence

k-group-functor.

& H-

hcammrphisns

,

231

is exact, b)

The H-extension ( E , i,p)

and

H-extension @,A,p)

such that p s _ = I d

(El ,j=' ,g') are said to be equivalent if. there

such that f o i = & , 'p l 0 f = p

is a hamanorpkism f:E-+E_' The

I= .

thse is a mrphism of k-functors _s:_G-+_E

.

is said to be inessential if there is a k-group-

functor hcanarrorphism s:G-+_E such that 120. = IdG

-

.

One verifies without difficulty that equivalence of H-extensions is indeed an equivalence relation. In the language which has just been formulated, propH-extension of G @

sition 3.10 of § 1 beccanes:

g is inessential iff

it is equivalent to an extension defined by the semidirect product of G b_r with respect to a suitable operation of G_ e n H. Notice that condition b) since _s(R)

implies that p ( R )

is a section of p ( R )

is surjective for each R e w

%

. Conversely:

lknna: - - If G is an affine k - e p , condition b) is equivalent to: b')

for each R e +

the map p ( R )

is surjective;

andto b")

the map p>d(s)) is surjective.

Proof: Since b) =>b') =>b")

, it is enough to prove

b") *b)

. &t

we have

a m t a t i v e diagram

2.2

Given an H-extension 5

4 E_ 9 G

Goperation on g in the following way: Since

, we define a is normal in E , E acts

of G_ by M_

by inner automorphisms in _M ; since g is carmrutative,

acts trivially

E

and t h e action of

I n t (x)_i (m)

,

f o r xEE(R)

ture of

G

factors through

mE&(R)

. Acc0rairiq-l.y we have

= i - (p(x)m)

,

. This operation preserves the group struc-

RE$

and depends only on the G i v a l e n c e class of the given extension.

W e call this operation the -G-operation on _M defined by the qiven class of

extensions.

W e say that the H-extension E,

, i.e.

PJ

4 E_ g

if the lroperation on

Proposition:

2.3

of H-extensions

Let

G_

is central i f

i(MJ

is c e n t r a l i n

defined by t h i s extension is t r i v i a l .

M_ be a G d u l e . Then the set of classes

of G by defininq the given g - o p r a t i o n on & i s canoni2 cally identifiable with HO(G_,M)

.

If the abave extension defines the given Goperation on M_ m i a t e l y that fs€ z2 G,M) is

h:G

+bj

that

. If

, one v e r i f i e s

~ ' : G - + E is another section of p

, there

p ' ( g ) = &(h_(g) )_s(g) , and we obtain without d i f f i c u l t y 2 depends only on the so that the class of -fs in HO(G,bJ)

such

fsl= f,+ alh ,

extension i n question; m r w e r , it d e d s only on the class of t h i s extension. b) Given a 2-cocycle

a s follows:

f:cxC_+M_ of

on the product

E=MxG

G_

in

impose

, define

G

an H-extension of

the group l a w

(m,g)(m',g') = (m+gm'+ f ( g r g l )I gg')

,

,

. Set

a t e d w i t h _f

m,m'ckl(R)

R E 4

L(m)=(m,e)

.

and P(m,g)=g If f ' is a 2-cocycle which is c o h ~ l o g o u sto f , one shows e a s i l y that the H-extension associated with f,' is equivalent to the H-extension associfor g , g ' E G ( R )

.

c ) I t remains to verify t h a t t h e trm constructions above are mutually

inverse, and this is inmediate.

The proof h

e inmediately inplies the

Proposition:

Let

M k a carmutative k-groupfunctor, and suppose that

is canmtative. Then the set of classes of H-extensions 2

2.5

M_+E+_G such that G acts

Hs(G,E) , where

E is axmutative is canonically identifiable with

.

t r i v i a l l y on

5

Remarks :

1) As usual we can define directly the Baer sum of t w o H-extensions. This 2 corresponds to the addition given i n HO(_GrM)

.

2)

Here we have used a very restricted type of epimorphisn (those possessing

a section), and, accordingly, a very restricted type of extension. 3) The bijection of

Z

1

onto the set of sections of the semidirect pro-

duct Mw G d e s c r h i n § 1, 3.10 m y be generalized as follaus: l e t

iE - E :M-T tG_

(E)

be an H-extension of

autmorphisn of bijection of

Z

aocycle -f:G+M_

and x€_E(R)

by

. Let us define an

(~)-autcanorphisnto be an

E which induces the identity on _M and

(G,!)

Proposition:

. W e obtain a

onto the group of ( ~ ) - a u t m r p h i s n sby assigning to the

the autQnorphism u_

such that g ( x ) = s ( f _ -( p ( x ) ) ) xf o r RGBk

W e shall be concerned with the case in which

3.1

, where

G_

haml logy of a linear representation

Section 3

Va -

.

1

G_

V

8

is of

the form

is a k-module.

Suppose that G_ is an affine k-mnoid and let A=d((G) be its

234

ALx;EBRAIc GEMcrps

5

11,

3 , no 3

bialgebra. L e t p:G_-+L_(V) be a linear representation of G and let

% : v + v @kA

thecanp la:

be the corresponding d u l e l a w

c”(G,v) = V ~ A B A B .. .@A

an where

ani

=

: v@A@

n+l

1 (-iiia2 ~

k

(n factors A)

@ n+l

is defined. by

A

.@an)= % ( v al@.. ~ .Nan

a:(v@aiB..

2.1). L A C’(G,V)

,

i=O

n.+ v

(5 2,

I

ar(v@ai@,..@an)= v @ a@...@Aa@...@a 1 i n @...@a @l a:+l(v@al@...@a n = v @ a1 n

.

1
,

Then we have a canonical isamrphisn of oanplexes

Proof: By definition, we have a n (?(G,V

P

r:

Va(A

) = V@ABn

-

,

which gives canonical iscamrphisms An : P ( G , V )

a

Cn(G,V)

.

W e must rcw cunpare the boundary operators. For this p w p s e let gi , @n which correspond to the i = 1,. ,n , be the elements of G(A ) 2 A&(A,A@”)

..

n injections a + + 1 ~ . . . @ 1 @ a ~ l . . .(a ~ li n the ithplace). If

we have by definition f(gl,.. 4,) = A n (f) -

~ ~ P -G , v

.

1 transforms the operator a: of C * (G,v,) n into the operator 3; of C’(G,V) Take x = v B a1@ @anELp(G,V) ; we must n n s~ that i f f=~;;l(x) , we Mve ‘n+l a O-f But we have

L e t us s h , for example, that

n n An+laof = (a0Q (glI...,gn+l)

.

=sox .

= glf(g2,...fgn+l)

...

= gL ( v @ l @ a l @ ...@a n1

11,

9

3, no 3

(since g

235

HccHscHII;D coHapIy3ILx;y

g E G ( A @ ~ ) under the map assxii

is the image of

EG(A@(~+') i+l

ated w i t h the hammrphism

...

a@...@a w l @ a , @ @a 1 n n

of

A @ ~into

. The latter product is the image of v @ 1@al@. ..@ a n A@~+')

under the composite map

. Accordingly, n %(v)@alW...@an , i.e. a 0 x .

where u(a@b)=a.b

3.2

If

tor E

, we

is a linear representation of the k-mnoid-func-

p:G_+&(V)

H~(G,v) for the group

write

exampleI

0 H (G_,V) = V'(k)

=

a

If

this latter product is

s,.

~ ( G , v 0 -

@.

G_ is affine, proposition 3.1 supplies an

8(G,V)

^.

n u s we have, for

8(C'(G,V))

immism

;

G in particular, we x q a i n the description of Vgiven i n 5 2, 2.4.

Corollary: 0 -+V'+ V

5 is an

Suppose that

YVll -+

affine k-mnoid, and l e t

0 be a n exact sequence of k-(;-nodules such that p:V + V "

a_ k-linear section. Then we have an exact sequence of m t a t i v e groups @ -+

... Proof:

G

V'-+

-f

G V-

+

H"(G,V')

For each Re-$+

G

V"-+

, the

-+

1

H (E,V')

Hn(G,V)

-t

+

...

H"(G_,V")

-f

X-+l (G,V) -+

has

...

sequence

is exact; apply 1.3. 3.3

Proposition:

Suppose k

monoid. Then the functor V++H'(G,V)

i s a f i e l d and

, where

V

is an affine k-

ranges through the cateqory

236

ALGEBRAIC GROUPS

of k-G-dules, Proof:

i s the derived functor of the functor V-V-

By 3 . 2 , it is enough to show t h a t

each k-Cj-module

V

, where G

for n>O

= 0

.

G

5

3 , no 3

.

is effaceable, i.e. for

H'(G,?)

there is a k e m d u l e E(V)

such that #(G,E(V)) V@A

11,

and a mmrphisn V

Take E(V)

+

E(V)

to be t h e k-vector space

acts t r i v i a l l y on V and on A through its regular re-

presentation. W e know

(5 2 ,

2.2) that

mdules; all that remains is to v e r i f y that

G

Suppose that

3.4

Lerma:

hialgebra of

G _ _and _ _l e t

is a mnanorphism of k s -

%:V+E(V)

H"(GE(v)

=

o

n >o

.

is an a f f i n e k m n o i d ; l e t A be the

kinodule. Then #(G,VNkA)

V

for

= 0

for

n>O.

Proof:

Let

Now apply

3.6

Re$

; we have canonical bijections

l a m a 1.3.

Comllary:

with the assumption of proyosition 3.3, 1 s C-

be the category of A-cmdules

(A =

(9 ) . Then we

have canonical isorror-

phisns €?(C,V)

Proof: 3.6

we have t h e

&(k,V) Ohio&

. G

i m r p h i s m V-- C5-

(k,V)

Suppose that G is a f f i n e and l e t k'e,T

. W e have a

237

canonical isomorphisn

k

over

If k' is = t

, we accordingly obtain canonical isamorphisns

.

H"(GFk' ,V@k') = €?(G,V)@k' k k

3.7 Propsition: Suppse that k a f i e l d and that g is an affine k-group. Then the following conditions are equivalent: (i) for - n>O (ii)

.

For each linear representation G_+c;L(V) , we have H"(g,V)= 0

For each finite dimensional linear representation G_-+&(V) 1 we have H (G,V)= 0

.

,

(iii)

Each linear representation of G_ is semisimple.

(iv) simple.

Each finite dimensional linear representation of G_ is semi-

(V)

The regular representation of G is semisinple.

Proof: -

(i) => (ii):

Trivial.

(iv)=> (iii): By

5

2, 3.1.

(iii)=> (v): Trivial. (v) => (iii): By

5

2, 2.3.

(iii)=> (i): By 3.3. (ii)=> (iv): Given two k+rrcdules

U,V which are finite dimensional over

k , assign m ( U , V ) a k-(;-dule structure as follows: if RCI& fy m(U,V)QDkR with R ( U fR

, identi-

V fR)

-1 by means of the canonical bijection; then set (gf)( u ) = g f( g u) for gcG(R) , f C ~ ( U , V ) ~ and R uCUgkR N m let O-+V'-+V-+V"-+O be an exact se-

.

quence of k-@mdules of k+-mdules 0 -+ & ( V " , V ' )

of finite dimension over k -+

& x ( V I ' , V )

hence a c o ~ l c g y exact sequence

-+

) " V , ( & I

. We have an exact sequence -+

0

,

238

Au7EBRAIc GROUPS

R(V",V)-

G

-+

11,

~ ( V " , V " ) G + H 1 (G I %(V",V')

5

3, no 4

.

1

It follows that the identity map on V" l i f t s to be a k-linear map V"+V

which is %invariant, which means that the original sequence s p l i t s .

Section 4

Calculation of various cohcmlcqy groups

4.1

Propsition:

r

be a mnoid and let

k-qroup, on which the constant k m n o i d

be a camutative

acts i n a m a n n e r anpatible with

M _ . Then we have canonical iscrmrphisns

the group structure of

d(r$)

rk

g

2

Hi(r,H(k))

(where the second member is the ithcohmdogy group of the mnoid

.

r-module M(k) 1 Proof:

By

5

r

i n the

1, 1.5, we have

a d the stardard canplexes C'

(rk&

and C' (r,hJ(k) )

are canonically iso-

mrphic. Proposition:

4.2 _G +L(v) a

E t 5

linear representation of G

.

diagonalizable k m n o i d and Then we have Hn(GrV)=

o f~r

n>O. Proof:

Take G = D ( T I k ; by 3 . 4 , it is enough to show that %:V-V@khl

has a retraction r which is Ginvariant. Let p , y c r , be the projections Y associated with the grading of V (5 2, 2.5) ; set r (lvy@y)=lp (v ) Then

we have '.%=I% and + r = ( r @ A ) n ( V @ A A ) Revark:

3.7 and 4.3

When

5

G

is a group ard k

with A=kLrJ

.

Y Y

.

i s a f i e l d , it is s u f f i c i e n t to invoke

2, 2.5.

Corollary:

L e t- G -

be an affine k-mnoid. Suppose there is a

239 faithfully f l a t

k’C$

be a l i n e a r representation of _G

- Inanediate proof:

is diagonalizable.

such that GBkk’

. >m H~(G,v)=o

k-module

k

.

from 4 . 2 a d 3.6.

4~

W e m proceed to the cohomolcqy of

4.4

f ~ rn > o

Let G+_L(V)

. The a f f i n e algebra of

%

is

acting t r i v i a l l y on the

k k l , which

imnediately yields

the standard cmplex. W e have

Hence

0

H (%rk)

Proposition;

a)

If

k

k

Q

is isarrorphic to k ; correspndinq t o

x-1~

,

.

xCa(R)=R, RE$

At k

, then

, we

the ring Sk(%,ak)

have the homothetic map

, w i t h p primer the ring zk(akrakf 13 P ismorphic t o the non-comrmtative rinq of plyncanials k h l , where FA = A %

b)

rrf

for

k

A€ k

is an alqebra over i?

. Corresponding to

A€ k

we have the h t h e t i c map x W A X

to - F t h e Frobenius e n d m r p h i s n x ~2

By derivation, we obtain P’ (X+Y)=P’(X) P=ax+Q Q

, we

, where

have

Q =0

Q’=O

,

.

. Hence

and Q(X+Y)=Q(X)+Q(Y)

P = aX

P’

. If

is a constant k

,@

a and

is an algebra over

and the ring in question m y be identified w i t h

ALGEBRAIC WUPS

240

. If

11,

5

3 , no 4

k is an algebra over F , we have Q(X)=R(XP) and R(X+Y) = P B y induction on the degree, we may assume t h a t we have sham that R(X)+R(Y)

k

.

n-1

x +a2&'+. ..+ an$ 1

R = a

so that

P = aoX +a xP+.

1

By assigning a e k

;

..+ anXPn

to the polyncmial aX and F to the polynmial Xp

, we

obtain the required. i m r p h i s n .

Corollary: There is a canonical isanorpkisn of k-mnoids

4.5

End

-Gr

(a)

Q

cox

-Q

By definition,

4.6

d7.bk ,k)

m y be identified w i t h Sc&(<,%)

arid so naturally carries the structure of a l e f t module over If

E k ( % , .~ )

k is an algebra over F this structure behaves i n such a way that, i f P ' n P(X1,. .,Xn) 6 C (ak&) ,

.

then

(Em(X1

xn1

I . . . ,

= P(X1,

...,xn1P

The boundary operators are accordingly Bn(ak,k) and #(ak,k)

o < i < p , let WC AX,Y 1 for

anti

W(X,Y)

Theorem:

a)

if

k k L l i n e a r , so that

are a l l kk.krcdules. If

us agree to write

p

Zn(ak,k)

for the image of

for the integer

W(X,Y)

i n IF L x , ~ 3 P

/p

. we also write

.

is a field. Then

k is of characteristic 0

, we

2 have H (ak,k)= 0 ;

is of characteris-tic p > 0 , the plynanials r W(X,Y) XYP (r Z O )

-

,

is a prime nurmber and

the polyncanial

Suppose that k

b) if k

.

a

2 fmm a basis for a k b h t c d u l e canplment for B (ak,k)

2

Z (%,k)

11, § 3, no 4

Proof:

HCCHSCHILD COHOPDILGY

241

Since the boundary operator is hcmx~eneouswith respect to the total 2

degree, 2 (ak,k)

is a graded subspace of

to consider the kcanogeneous

ccknponents.

k[X,Y]

, so

that it is sufficient

Thus let

.

+. .+ an-lXln-l + an? ,

P = aoxn + a,?-%

n >o

be a polynomial such that P ( X , Y ) + P ( X + Y 12) = P(X,Y+Z)+P(Y,Z)

(*)

By derivation w i t h respct to X and setting X = O

I

we get

By derivation with respect to

I

vie get

P;(x+Y,o)

Z

and setting Z = O

= P;(X,Y)+P'(Y,o)

i.e. P; (x,Y)

I

Y

p-1

= a,[ (X+Y) n-1-

I f we m invoke Euler's formula, we see t h a t

(**I

n~ = ~ & + YYP = ' al[(x+y)n-?-ynJ

+ (anml- a,) CX (X + Y) n-l

-$I

.

W e 1l0w distinguish several cases: a) al#an-,

. It follows frpm (**)

that Q = X ( X + Y ) n - l - ?

is the diffe-

rence of a cOqCle and a coboundary, hence a cocycle. Replacing (*)

and X by -Y

yields

(y+z)n-l-y"-1-zn-1 If

p =0

If p + o modulo p

=

, this last formula implies , the formula implies ~t and (**)

*lies

that

P

0

P

by

Q

in

.

that

n =2

, which contradicts

n = l + p r ; hence n is cohcxmlcgous to

al# an-1

is invertible

(a, -a

n-1)Q/n

.

.

r r Q = X(X+Y)P -xp

But

=

xyp' .

.

and p=O or n F 0 m d p 5 (**I we then have P = al((X+Y)n-X"-Y") /n and P is a cobo~ndary. b) al=an-,#0

c) al = an-1# 0 , p # 0 and n = 0 mod p cient

(E:)

. I n t h i s case the bincmial coeffi-

...

(n-1) (n-2) (n-ptl) 1.2.. (pl)

=

.

is congruent to +l modp

. If

n#p

, it

follows that al((x+Y)

n-1-

p-1,

contains the t e r m *a ?-pYp-l , which is absurd since the f o m r expression 1 Up-') = a?; is precisely P; (X,Y) Hence n = p , so that P; =al ( (X+Y)

.

.

Similarly P i = a %' so that the p a r t i a l derivatives of Pal% vanish. 1 X ' Hence P =alW +aOXP+a Yp ; but t h i s can only be a cocycle i f a. =a = 0 P P I n t h i s case P;=P;=O so that P = P l ( X P,Y P1 d) al=an-l=O , p#O

.

where P1

is a cocycle of degree n/p < n

( i f P#O)

.

.

By induction on the degree of the polynomial, it follows from the preceding discussion that the polynanCals i n question do indeed generate, 2

the k[_F>&ule

Z (ak,k)

.

Moreover, the Fs(XYpr ) (r > 0 rent degrees.

Finally, Since the

cobundaries, and X@€

,s 20)

F%

k[X] = C1 (a ,k)

k

2

m d B (ak,k)

and the F%(s 20) are a l l of diffe-

XYP'

are m t Symnetric, they cannot be

is not a cobo~ndarysince the co!mxiiay of vanishes. T k i s canpletes the proof.

4.7 Proposition: Suppose that k is a f i e l d , and l e t K ?eb i i closed subsroup of ak Then the canonical maps H (aktk)+H W,k) , i 2 0 2 2 and Hs (0:k,k)+Hs(_H,k) are surjective.

.

~

H=ak , assume +&at H#a, . L e t 5 , and let n be the degree of a

Proof: Since the assertion is t r i v i a l for I be the ideal of

polyracrmial

k[T]

P

such that I =(PI

generated by

and A W W U

k[Tl=A defining I,T,

.kt

UckCT]

...,?-' . The canonical map

. By what we have'already proved,

,i :A@ i+fU+l)

be the k-vector subspace of

. Similarly,

Z1(g,k)

is bijective is the kernel of

U-+A+A/I

Zi(\,k)

is the kernel of the induced map

11,

5

3, 110 4

Let Ec Zi(H,k)c (A/IIBi i

a

and

243

HC€XSCHILD COHCKJLCGY

x = o -I

. Replacing i

, so that a

x=o

, we

2 by xE UBi

and xtZi(aklk)

assertion. If i = 2 and T is symnetric, so is x

have

a ixE "@i+l

. ~ k i proves s the f i r s t

, which p m e s

the s m n d

assertion. 4.8

Then of -

2

.

H (

w.

Corollary: Suppose k is a f i e l d of characteristic p#O ak,k) i s a k-vector space of dimension 1 generated by the class

P

Proof: By 4.6 -

2 H ( cx ,k)

is generated by the class of Pk class cannot be zero because the coboundaries belonging to

and 4.7

k[X$(XP)@kkCX]/(XP) 4.9

W2

Resrark:

are a l l of degree < p

w . This

.

2 C l e a r l y W(X,Y) E Zs ( a l l )

. Henceforth we shall write

for the Z-group whose underlying Z-schene is Q2 and. is such that (x,y) + (x'ry') = ( x + x ' r y + y ' - W ( X , X ' ) )

for R E 5

and x,y,x',y'ER

. By

L

4.6 a ) , we have !Y2Q=aq

.

CALCULUS ON GRXJP SCHENES

5 4

D-IAL

Section 1

Infinitesimal points of a groupfunctor

1.1

Let

T

,write

R[Tl/(T2)

f o r the class of

E

,

be a ring; i f

R

RCTI

is the algebra of plyncanials i n and R ( E )

T mod T2

f o r the quotient algebra

which is called the algebra of dual numbers over

,

a deccanposition R ( E ) = R @ E R and hatnmrphisms i : R + R ( & ) defined by

i(1)=1, p ( l ) = l

,

p ( & ) = O , such that

p :R(&)+R

pi=I%

there is an endmrphism ua

Associated w i t h each a t R

.

of

. W e have

R

.

R(E)

,

such that

ua(l)=l , u ~ ( E ) = ~ ; E we have p a = p and u a i = i P/breover, t h e m p a HU is a hanmnorphisn of the mnoid R" i n t o the mnoid of endmrphisms a

of the R-algebra

1.2

.

R(E)

Let

G_

be a k-group-functor.

G ( i ) :G(R)+G(R(E))

h-rphisms

be the kernel of

G(p)

. Since

1 * L i e ( G ) (R)

+

and G(p) : G ( R ( E ) ) + G ( R )

x t L i e ( G ) (R)

,

G(R(E))

(R)

mrpkisn G_(ua) of

,

f G(R) -P

+

. Let

the

ke(G) (R)

1 and G(p)

. W e now de-

by setting, for each gtG_(R) and

.

the hcmmrphism ua: R ( E ) + R ( E )

G_(R(E)) which is ccanpatible with

endomorphism L i e ( G ) (u,)

, consider

have a split exact sequence

i

and p for G ( i )

-1 Ad(g)x = i ( g ) x i ( g )

Similarly, for each a t R

, we

p i = 1%

i n which we have shnply written fine a _G(R)-operationon

For each Re$

of the group ke(G) (R)

.

induces an endo-

-p and &

, hence an

W e abbreviate simply t o

u the hma'mrphism a - L i e (G) (ua) of R into the endmrphisn monoid of the p u p Lie@) (R) For xE Lie(G) (R) we s e t ax = u ( a ) (x)

.

The two operations defined b

.

e preserve the group structure of L i e ( G ) (R) and are ccanpatible with one another: i f gt_G(R) , x,x'ELie(G) (R) and

atR

,

we have Ad(g)' ( X X ' ) = (Fd(g).x)(Ad(g)*x')

a(xx') = (ax)(ax')

11,

8

DlE3RENTIAL CALCULUS

4, 110 1

245

e constructions are f u n c t o r i a l i n RE%

Since a l l of t h e h

fact defined a k - p u p f u n c t o r

Lie(G)

, as well as operations

, we

have in

G x Lie (G) + Lie (G) -

gkx lie(^)

+

lie(^)

which are o ~ n ptible a boffi with one another and with the group l a w of 1.3

Let

G_

-

i b l e w i t h the mrphisms

and p

r e l a t i v e t o G_ and H_

. I t follaws that

induces a l-cammrphism

L i e (g) (R)

(G) (R)

:

+ L i e (H)(R)

and a hcmomorphism of s p l i t exact sequences:

I n particular we obtain the f o m l a s :

1.4

If

k'ki&

Lie(G@k') by 1 . 2

and I,

9

,

we have

= L>(G)@k'

k

and. u

1, 6.5. W e then v e r i f y that t h e operations

r e l a t i v e to the group GBkk'

m y be obtained from those of t h e group G be

extension of scalars. 1.5

.

and g be k-group-functors and f:G+g a l-xmmmrphism. and f ( R ) : G(R)+H(R) are ccanpat-

The haromorpkismS f ( R ( € ) ) : ~ ( R ( € ) ) " H ( R ( € ) )

-f (R(E)

Lie(G)

Finally, we see imnediately t h a t the functor

products of k-group-functors

&$i

1+G_L-IqK_

Lie

pxdiicts of k-group-functors.

transforms p.loreover, i f

246

11, § 4, no 2

ALamVlIc GRDUPS

is an exact sequence of k-group-functors 1 + G_(R)

-+(R)

H(R)

, the sequence

(i.e. f o r each Re$

'lR)&(R)

is exact), then the s q a e 1

Lie(f)

-+

Lie(K)

is exact.

Section 2

Examples

2.1

Modules.

cgnsider the map eR : (R)

€"R(€1tL,e(M)

.

g be an Qk-rrdule (5 1, 2 . 5 ) .

Let

of M(R)

'-tE%(E)

As

For each Re&

. Clearly

i n t o M(R(E))

& , we obtain a mrphism of

R ranges through

k-

group-functors e :

-f

Lie(M)

($ on

which is cmpatible with the actions of phim g

is evidently an i m r p h i s m i f

M=V,

V .

The linear group.

2.2

the k-functors defined i n f-Id+Ef inverse of

R(E)

of

Id - Ef

I , ( @ R(E)

5

(R)

Let

fl and

Lie(Fl)

or M = D (V)

-

-a

, for

i n t o _L(bl) ( R ( E ) )

, let %

. Clearly

Id+Ef

and t h e latter belongs to the kernel of +

I n this way we define a mrphism of k-functors

is of the form Va or D (V) -0

r

L(E) (i): &(MI (R)

we deduce from

be t h e map R(E)

.

GL(M) (R)

g:&(E'I) + L i e ( G L (I$)

5

a k-module

a d g(g)

g be an Q k d u l e ,

1, 2.5. For each RE%

(p) : GL(M) ( R ( E ) 1

. This m r -

is t h e

. When

M

1, 2.5 that t h e hancmorphism

(R(E))

-f

induces an i m r p h i s m

&(_MI (R);R(E)

(~(€1)

.

This implies the last assertion of the following proposition; the other ass--

tions are t r i v i a l . Propsition:

xt

fi

gk&ule.

For Re%

,

x,x'f&(&J) (R)

,

11,

If

9

4 , no 2

DIFFERENTIAL CALCULUS

k-module and i f

.V

-E:_L (E)-+ L i e (GL (5)) 2.3

_M is i m r p h i c to Va

-

is an i m r p h i s n .

Autamorpkisms of an algebra.

. By t h e above discussion,

1, 2.6)

t h e subfunctor _F of

-a

&(A)

, the31

be a (not necessarily

Let A

&t

of

(A)

a (A)

m y be i d e n t i f i e d with

Lie(Aut(A))

such that

x€_F(A) (R) <=> E(x)E&t(A)

If

or t o D (V)

Consider the sub-group-functor

associative) k-algebra.

(9

247

a,b€A@R and x€I,(A) (R)

,

(R(E))

w e have i n A @ R ( E )

This implies t h e Proposition: Er(A)

Let

A

@ (not necessarily associative) k-alqebra. &&

be the subfunctor of &(A) such that

vations of the R-algebra

.

8 For each by

.

Autamsrphisns of a scheme. Let

2.4

L i e (Aut (3)1 (k)

k (€1 -model

is t h e set of d e r i -

D s ( A ) (R)

Then the i s a m r p h i s n k of 2.2 induces an isamrphisn E r (A) 1 Lie (Aut (A) ) A@ R

X

E :5(A)-+ Lie -(GL (A) )

be a k-functor and l e t

.

R we thus have a permutation of

) which reduces to t h e i d e n t i t y when

ER=O

_X (R)

(also denoted

. I f c:?+Gk

is a function,

and x t g ( S ) , then C ( + X ~ ( ~ )i )s of t h e form a + E b with a,bES Setting E = O , w e obtain a =_f(x) S e t t i n g b = (D'f) (x) , we have SEG

.

accordingly f(+xS(,)

-

=

f M + E(D+f) 8 (XI

+-

*

Since this f o m l a is f u n c t o r i a l with respect to S

, we

see that the maps

.

248

x

-+

11, § 4 , no 2

AT&EmAIc GROUPS

X

. Moreover, one v e r i f i e s

X

(D-f) (x) define a new function D-f :X+O+

6-

e a s i l y that the operator

X X D- :f W D - f

4 -

6-

W e now turn our attention to the action of /_X@kk(E)I

.

is a k-derivation of the algebra &_XI

6-

of the k(E)-functor Xgkk(E)

on the geanetric r e a l i z a t i o n

6

. Given a geanetric k-space

T

, let

be the g m t r i c k (€1 -space which has the same underlying space as T

T (E)

a d satisfies L!&(~)=

LIT(€)= dT@sflT . If

T = 1x1 , it is easy to see that

IzQDkk(E)[

there is a canonical isarrorpkisn

IX_l

(E)

clusion map A - + A ( E ) induces a hameanorphism i of (the prime ideals of

A(E)=A@EA

: if

, the

A€$

in-

S p e c A ( ~ ) onto SpecA

, where

are of the form P&EA

.

pc SpecA )

,

and an i m r p h i s m d s p e c A ( ~ ) l i * ( J ) Since the functDrs Spec A ( € ) carmrute w i t h d i r e c t limits, the argument of (E) X M IX_$k(E) [ and X w I, § 1, 4.1 shows that there is a unique i s o m r p h i s m of functors j : IX@kk(E) (E) such t h a t , f o r each A€$ , j ( S p e c A ) is the can-

1x1

(x)

1x1

-+

posite isQnorphism

[Spk(EIA(El I

[%kA@kk(E)[ (I,

9 1,

S p e c A ( ~1 ) (SpecA) (E)

6.5 and 4 . 1 ) .

X . Since,

Mre generally, let V_ be an open subfunctor of

formula imnediately ahme, ygkk(E) of points,

and U_ obviously have t h e same space

6 induces the i d e n t i t y on

Since, with the notation of

5

.

of

obtain a k-derivation D

($

U_Nkk(c) such that

.

Dg

I f 5 is a scheme (resp. an a f f i n e scheme) , the "ap

-

(resp. ~f Proof

Lie ( A u t g) (k)

.

4 I+

D

6

(resp.

onto the set of k-derivations of

1.

(sketch) : W e merely give the inverse of the map

i n which

Ugkk(€)

6 (V) of d(I_I) By varying of the structure sheaf dx of & . -

$(LJ)@k(E)k=Id This defines a k-derivation

is a bijection of

P of

the space of p i n t s

1, 4.10, we have U N k k ( E ) = (X@kk(E))p, we

see that 6 induces an a u t m r p h i s n

, we

by the displayed

is a scheme. To each k-derivation

D

of

4 dX

I-+

-

D

6

6I-+D'X 6 dx -

)

i n the case

we assign the

.

11,

5

DIETREBTIAL CALCULUS

4, no 2

autamorphisn ii, of

0X (€1 = < + € J X

V_ i s an open subscherne of @ of

such that $ ( a + ~ b =) a + ~ ( b + D a ) where

&- and.- a,bE(7$)

IX@kk(E)I 7 j _ X j (€1

and the a u b m r p h i s n $ of the s t r u c t u r e sheaf

Der(dX) be the k-module fo&

where 8

. The required a u t m r p h i s m

induces t h e i d e n t i t y on the underlying toplqical space of

XBkk(E)

Let

249

is a scheme. I f

RE&

by the k-derivations of t h e sheaf

, define

and dEDer(dx)

u

of the sheaf of R-algebras dxgR as follows: i f scheme of

fix(€) -

5 , then d(vgkR); S(LJBkR and %(LJgkR)

a derivation

Der (X) (R) = D e r (c$ C9R)

dx , -

%

is an a f f i n e open subis obtained from d ( g )

by an extension of scalars, W e my therefore define a functor D e r ( X ) that

.

such

. The proposition then inplies the existence of

a canonical isomorphism of k-functors

_ Lie(Aut(3)) _ =

Ds(X)

,

.

which is ccknpatible with t h e group l a w s and the action of 3 ‘; ~ r o u p sof invariants.

2.5

let f:g+Aut

-@

Let _G

and

H

be k-group-functors

(GI be a ha-mmrphisn. For each RE$ -

and

each hE_H(R)

,

is an automDrpNsm of the R-group GBkR , so that Lie(f (h) ) is an F r m this we derive a hanawrphisn autamorphism of Lie(GBkR) = Lie(G)BkR f (h)

g (R)

.

-+

A u S ( L i e (G)BkR) and by varying R we get a hcmcanorphism

; Aut(Lie(G)) +.

Lie(G)BkR

, this

Proposition:

. Since t h e actions of

Let

preserve the group s t r u c t u r e of (Lie(Q) Gr--

homsnorphisn f a c t o r s through &t _G

a hanawrphisn. T M Proof:

H(R)

8

.

k-group-functors

and l e t f:H+Aut - a (G) - _be is a sub-grourfunctor of G Le(&=ce(G)-.

It is enough to show that

trary. Nay by 9 1, 3.5 we have

Lie

(8)(k)=&e(_G)-H (k)

since k

is arbi-

250

11,

FLGEBRAIC GRou??s

Section 3

I n f i n i t e s i m l points of a group schene

3.1

Consider a k-schgne Y_ and a yeY(k)

of

associated with y

,

# i.e. the mrphisn y

section of the canonical projection

. Write

p:y+$ -

. Let

5 a,

110

3

& be the section

. This mqhisn is a

:$+Y_

arid is accordingly an embsdding

f o r the k-mcdule w . (%) formed by the sections Y A of t h e W u l e wi of the gnbeaaing i_ (I, 5 4 , 1 . 3 ) . I f A€% and (I,

5

2 , 7.6 b) 1

, there

w

-is a canonical isamorpkFsm

w = 1/12 , where

I is the kernel Y of y:A+k If k i s a f i e l d , y may be identified with a rational point 2 of Y_ (I, 5 3, 6.8) arrd w with m/my Y Returning now to the general case, l e t be the f i r s t neighbourhooa of &

g =Sp A

i n U_

.

xi-

arid let

%

-1

+ x -i

.

L2

+_y

.

be the can~nicalfactoring of (I, 5 4 , 1.1) Since il irduces an isomrphi& of + e onto the closed subschene of defined by an i d e a l of

xi

vanishing square, Xi

the

be identified w i t h

&(xi)

(I, 5 2, 8.1) .-By I, 5 4, 1.5, may algebra k@w i f we assign t h e k d u l e k e w the is a f f i n e

Y

Y l

multiplication such that =

( h , S ) (A',<')

(AX'

, A<'+ X ' O

.

f o r X,X'Ek , (l('Ewy The mrphism il:e++Xi and pi,: associated w i t h the hcaoan0rpNsms (A,S) * 1 of- k @ w in Y 1- (1,O) of k in k@w Y '

xi+%

are

k , and

is functorial w i t h respect to the "pointed schene" Y &,y) For let Y_' be a k-schene, y'EY' (k) , & ' = y ' # and g:Y_' - +Y_ a mrphisn such that g ( y ' ) = y B y I, 5 4, 1.3, g induces a mrphisn The construction of

w

.

.

into w =wi,(%) which we denote by w +w hence a m p of w =w ($1 _i g' Y & Y' w ASSt h i s notation, the ham-mrphism d ( ~ . ) - + d (associated ~~) with g -1 & mrphism y;+yi induced by g sends (h,C)Ek@w onto

.

-

(h,wg(€J)Ek@wyl -

.-

?Axthemre, it is clear, that the mps w

w

F 3

:w

Y'

+a

(YrY')

Y

:w +u

Y

(YIY')

induce an i m r p h i s m w @ w Y Y

(YlY')

and

, where

11,

5

DIFFERENTIALCALCULUS

4 , 110 3

251

With the notation of 3.1, l e t J be an i s 1 of

3.2

k of vanish-

in9 square. To each k-linear map d:w -+J we assiqn the hmnmrphisn

Y k&w + k which sends (A,<) onto A+d(c) , next, the corresponding mrphisn Y dl:-%+Yi of , and finally the element a (d) of _Y(k) associated Y w i t h the-canposition

Kt

,

an ideal of k of vanishing square and q:Y- (k)+Y_(k/J) the map induced by the canonical projection induces an isamrphism of Mc&(wy,J) k+k/J T B a :i$x&wy,J)+~(k) Y -1

Proposition:

.

0

s

q

(q(y))

be a k-scheme,

y€Y_(k)

J

.

The problem is a special case of the situation discussed in I, 5 4 , -1 Let 1.5. W e merely give the inverse map of q (q(y)1 into k&d+(wy,~) zE q (q(y)1 ; since % coincides with the f i r s t neighbourhccd of

Proof:

.

__.

%(k/J)+Qk

, z#:q-+y factors through xi , hence . Evidently z(wy)CJ , am3 so the required inverse map

in Qk=%

zcY, ( k ) = s ( k @ u,k) -1 Y assigns to z the induced map of If

3.3

R€$

, we may

w

Y

into J

.

apply the preceding proposition to the

yBkR(~) , the canonical image t = yR ( E ) of y i n Y _ ( R ( E ) ) arxl the ideal ER of R ( E ) Now wt m y be identified with w 8 R ( E ) Yk (apply I, 9 4 , 1 . 6 to the case i n which f_ is the canonical projection R ( E ) -scheme

.

.

Accordingly s&R(~)-+Spk) %(E)

w B R ( E ) ,ER) ( y k

, hence with

(wt,&R)

s(wy,R)

may be identified with

. Thus we obtain the

ALGEBRAIC GROWS

252

Corollary:

Let

Y be a k-scheme, ___

ated w i t h the hammrphim a+bE*a there is a bijection (Try)

,

of

R(E)

-1

q

-(wy,R)Z

-

RE&

11,

5

3

4,

q:Y(R(E))+Y_(R) the map associ-

(q(y))

R

& y€Y(k)

.Th3

which is functorial i n R

be a k-linear

Let us recall the definition of t h i s bijection: l e t d:w,+R . i

map and l e t d": k@w * R ( E ) be the hamomorpkism such that d"(X,c) = Y # a+d(S)E If DEq-'(q(y)) is the image of d under this bijection, D is

.

the cchnposition R(E)

If

R=k

,

92 d" >Yi------+g -

we call q-I(q(y))

& y

the tanqent space to

. When

k is

a field, this space may, by the corollary, be identified with the Zariski tangent space

(wy,k)

(I, § 4, 4 . 1 5 ) .

Now consider a k-group

3.4

G

. Let

e be the unit e l m t of

.

and l e t E=EG=e# be the unit section of G_ Set wG h = w e = w l a r l y , i f f:G+_H is a l-armcqhisn of k-groups, we write w

f/k

(3.1).

rf

G_ i s a k-,

'GJk

''tcn,/k)

E -

G(k)

. simi-

for w

f

pG:G+e+ is the canonical projection, k , we have canonical iso%/k i s the sheaf of differentials of E Proposition:

mrphisns

where E

G/k

and

'G h

p*G

-G

G/k

is the quasicoherent module over + e

1

I

associated with w G/k '

-

Since wQ'k = w ~ ' '_G/k may be identified with the module wE of the enbedding E The f i r s t famula then follows fran I, § 4, 2.2. The Proof:

.

second follows fran I,

5 4, 1.6 and from the Cartesian square

11,

9 4,

IIO 3

3.5 J

253

D I E T R E N T m CALCULUS

Kt G

Theorem on infinitesimdl points:

k a k-group and l e t

k of vanishing square. Then the map ae :h k x & ( ~ ~ ~ , J ) + G ( k )

be an i d e a l of

-

of 3.2 is a group hammrphisn ard the sequence a

0

-+

b'b&(~~/~,J)

-

G(k)

-f

G(k/J)

is exact. Proof: The second assertion follows fram 3.2. -

that, by 3.1, the functor

fer a s usual (cf.

5

@,y)++lc&(u

1, 1.5) t h a t , i f

G_

To e s t a b l i s h t h e f i r s t , notice

k) catmutes with products. W e inY' is a k-group, the k-linear map

imposes a group structure on bt.r&(~~,~,k)

Mo&(avG,k)

is Zn isamorpkism. The following lama shows t h a t %(w e : with the natural addition i n Mc&(uGIk,k) a

-

L e t M be Lam: -

be a

f o r a l l x,yCM

, so

m(x,y)=m(l,y)m(x,l)=yx

.

Corollary:

3.6 (R)

M

e

is m t a t i v e and

that m(x,y)=m(x,l)m(l,y)=xy ; similarly

Let

5 @ k-group. Then f o r each

eqU ipped w i t h the R-opration defined in 1.2,

t

The k-functor

LA@)

R

.

Re&

,

is an R-nodule

Clearly the i m r p h i s m of 3 . 3 , where y=G_ and y = e

with the actions of

, is ccnrrpatible

,pipped with

accordingly an Ok-rrcd,ule If

.

, then

we have x = m ( x , e ) = m ( ( l , e )( x , l ) ) = m ( l , e ) m ( x , l ) =I.m(x,l)=m(x,l) ;

similarly m(l,y)=y

Proof:

,k) coincides

hcmmrphism. If M contains an elanent

such that m(e,x)=m(x,e)=x f o r a l l xeM

Proof:

"G

a set quipped w i t h a l a w of camp0s i t i o n p s s e s s i n g a u n i t

element and l e t m : M x M + M

m(x,y)=xy

i n such a way that

(5 1, 2.5)

the Ox-operation defined i n 1.2 is -k canonically isamorphic to Qg(wG/k)

f:c+H i s a hammrphim of k-groups,

.

L i e ( f ) may be i d e n t i f i e d w i t h

-Da(uf/k) W e now i n t r d u c e a notion which w i l l f a c i l i t a t e our calculations

3.7

a great deal. Let us write the group l a w of

st&

ard

c1

is an element of

s

Lie(G)

additively. I f

Re&

of vanishing square, there is a unique

,

254

ALGEBRAIC GROUPS

hammrphisn of R-algebras

of

R (E)

-+

11,

which sends

S

onto a

E

4, no 4

imge of

urider the ccsrrposite lxxcmrphism

x€Lie(G)(R)

L i e @ ) (R)

+

G_(R(d)

G(S)

+

w i l l he w i t t e n

ecLx ( i n G ( R ( & ) )

we thus have i n

G(S)

, we

EX

have x = e

I). For x,yc&e(G)

(R)

,

,

ea(x+y) = eaxeaY

(1)

while the definition of the external law of

follows: . f o r xc=(Cj)

e

(2)

(R)

, we

and acR

L i e ( G ) (R) m y be written as

have i n G_(R(E))

If g:G_*H is a hchoorrPrphism of k-group-schemes, i f c1

ax a(&(f)x) f(e = e

Section 4

The L i e algebra of a group-schm

and i f

S

be a k-group-schenne. W e shall assign to Lie(G) the structure of an

'Qk-Lie algebra"

.

The adjoint representation.

4.1

morphism a ( g ) of

ture of

xcLie(G) (R)

of vanishing square, ws have i n _H(S)

(3)

E

,

(Ea)X - €(ax) - e

is an R - d e l w i t h an elanent

Let

. !?!he

5

L i e ((;I (R)

. If we write

GL(Lie (G))

functor of linear autcmrphisms of

: _G

+

a d gcG_(R)

. The auto-

defined i n 1 . 2 preserves the R 4 u l e struc-

Lie(G) (R)

morphic to the group-functor

RE&

Let

@(w

6(Lie(G))

Lie(G)

1

G/k OPP

(cf.

5

1, 2.5) for the k-group-

(which, by loc. cit., is iso-

, we

derive a I.lomanorphism

,

which we call the adjoint representation of _G ,

If xEL&(Cj)

(R)

and gt_G(R) , we thus have i n _G(R(E))

,

= e&i(g)x

(4)

geExg-l

4.2

The bracket.

By 2.2

and

5

1, 2.5,

Ke(C;L(L&((;))

may be

11,

5

4 , no 4

255

DlFFERENTIALmS

identified with L ( L i e (G)) ; hence we g e t a canonical gkk-rodule i m r p h i s n ad = Lie(Ad) : L i e ( G )

-f

L(Lie(G))

,

thus a "bilinear" mrphim LLe(G)xL>(G)+Lie(G) ( a d x ) y f o r x,yt&e(_G) (R)

and

which serids

. Set

Re4

(x,y)

onto

. I n v i r t u e of

( a d x ) y = Lx,y]

the identification we made i n 2.2, we g e t t h e f o m l a (5)

i.e.

Ad(eEX) = Id

+~ a d ( x )

M ( e E X ) y = y + E [ x , y ] , where x,yEL&e(G) (R)

and where ACJ(eEX) belongs

to the algebra L ( L i e G ) ( R ( E ) ) = L_(LieG) (R) & ' €L(LieG_)(R)

Proposition:

Let

two elenents of

x,yELie(G) (R)

RE%

and l e t a,B be

SE&

S of vanishinq square. Then i n G(S) we have

eaxeBye"xe-By

(6)

. kt

.

.

= eaBh,yl

Proof: It is enough to prove t h e contention when -

S =R(E,E')

generated by two elements of vanishing square. Noting that

is the R-algebra

, we

S " R ( E ) (E')

g e t successively eE~eE

ye-Ex = eE 1 s(eEx) y

( (4)

= eE ' (y+ELx,yl)

( (5) 1

-- eE1y,eE' ( E ~ I Y I )

( (1)1

- e ~ f ~ ~. e, Y~ I~ f = eE E Lx, y

I

1

y

( ( 2 )1

I

which by (1) gives the required r e s u l t .

, which

eE E ' L X ' y l

by (6) @lies that

k,y]=L-y,xl=-Ly,xl EE'U

it is enough to observe that the equality e For i f

4

: R(E)+R(E,E')

EX s ' y -Ex e e e

'E'Y

e

W e note in passing t h a t the above proofs also gives

i s the mrphisn of

EE~V

=e

&

. To prove this .

implies u = v

such that

$(E) =EE'

then Spec $

induces a h n e a m r p h i m of the underlying spaces of

Spec R ( E , E ' )

and Spec R ( E )

,

-

,

and a m m r p h i m of the s t r u c t u r e sheaf of

.

S p e c R ( ~ ) i n t o its d i r e c t image i n s p e c R ( ~ , ~ ' ) It follows that Spec$ is an epimorpkisn of

sk , so t h a t

other words, f o r each k-scheme

3

,

s_p$ _X($)

is an e p m r p h i s n of :X(R(E) + & ( R ( E , E ' ) )

&\

is an

. In

256

Azx;EBRAIc

injection

mws

11,

9

4 , no 4

. we endcw L i e ( G ) (R) with the R-algebra structure defined by

For each RE&

the Lie product. W e show later (4.5) that t h i s gives u s a L i e algebra. Notice

t h a t it follows from t h e d e f i n i t i o n of the Lie product (or frcm (6) and the preceding remark) that i f

f :G +€J is a hmmmphisn of k-group-schemes,

L A (f) (R) : L i e (G) (R) + L i e (H)(R)

4.3

Let

of the form Va

-

p:G+m(M)

is an R-algebra hamcanorphism f o r each RE$

be a l i n e a r representation of G

. By 2 . 2 we g e t a mrphism

or Da(V) -

then

, where

.

is

which implies Lie(p) i x , y l = (~>(p)x) (~*(p)y)

(8)

Iiie(p)x,~>(p)yl

=

, we

For each Re$

endow &()!

- ( ~ i e ( p ) y(L%(P)X) )

.

(R) with the L i e algebra structure which un-

d e r l i e s its R-algebra structure. From 2 . 2 and t h e preceding discussion we deduce the Proposition:

Let

p:G+C&(_M) -

G , where

5 D (V) m, f o r each R t & , L>(P) (R) -a L i e (G)(R) into the L i e algebra L (E) (R)

of the form Va

mrphism of

.

be a linear representation of

.

to be the a d j o i n t representation of

In particular, i f we take

p

tain

, which,

h , y 3 = Lad(x) ,ad(y) 1

1 2

is a ham-

G

, vie

Ob-

i n v i r t u e of the antisymnetry of t h e L i e

product, i s p r w i s e l y the Jacobi identity:

p

11,

4 , no 4

-

257

Now l e t X be 'a k-functor and let G X_ ' 5 be a E-operation

4.4

on

DIFFERENTIAIL cAI;cuLUS

X , which we

write in the form

w i t h the hcxmmrphim P :G +&t of the sheaf of k-algebras

ture

(b,El

dx

(SIX)

.

L i e ( A u t 5 ) (k)+Der(Jx) ; by composing : L i e ( G ) (k)

+

Der

equipped with

f o r DyEcDer(dx))

=D'E-EeD

p'

Let

p (g)x

.

, so

(dx)be

t h a t it is associated

t h e set of derivations

its natural k-Lie

I n 2.4 we defined a map

this with

Lie(p)

we g e t a map

Der(Jx)

-

By definition, we have accordingly f o r each o p n subfunctor

f€d(g)

, each

, each

Re&

and each xE&e(G_) (k)

meLJ(R)

. If

f (m)+~ ( p (' x ) f )(m)

L

such that

(10)

f(p(ecix)m)= f ( m ) + u ( p l( x ) f ) (m)

Proposition:

_L e_t _X _be a k-functor and l e t p ' ..: L i e (G) (k)+Der ( d ) defined

L i e (G) (k)

Proof: have

x

above is an anti-hamrsnorphim

-

(XI r p ' (y) ] = p ' ([y,x]

)

. Let

arld m c u ( R )

. By

. Thus

p:_G+Aut(X) be a hamaru>rphisn.

i n t o t h e k-Lie alqebra Der (dx)

variables of vanishing square. L e t RE$

following

. .

I n v i r t u e of 2.4, it is enough to show that i f [p'

, the

$(a+bc)=a+ba where a , b E R

we obtain

of

, each

g

we m y apply to this formula

satisfies a = 0

ciER

the hamomorphisrn I $ : R ( E ) + R

-

of

( w i t h the normal abuse of notation) : f ( p (eEX)m) =

relation i n R(E)

Then the map

algebra strut-

k(E,E')

x,yELie(G) (k)

r

we

be the r i n g generated by two

I! be an open

subfunctor of

_X

,

fE ~ L J

(6) ard (10) , we have

By means of a step-by-step calculation using (10) we see t h a t t h e right-hand side of this equation kcaxes EE'

(Cp'

( Y I P ' (XI-

P ' ( X ) P ' (y) I f ) (m)

which implies t h e required r e l a t i o n ,

_U,f,R having been chosen a r b i t r a r i l y .

r

2 58

Azx;EBRAIc GROUPS

4.5

11,

9 4,

no 4

The preceding discussion may be applied to the particular case

i n which G acts on i t s e l f by translations. Accordingly the harmnorphism y:G *Aut (G) (9 1, 3.3 c) ) gives rise to an algebra-antihcmmxphisn y I : Lie (G_) (k)+ Der (dG)

-

. Therefore we have by definition

.

f o r x,yEL&(_G) (k)

,

Since y:G+Aut(G)

is a mnanorphism,

a€ k

y ' : Lie(G) (k)+Der(dG) is injective.

It follaws t h a t L i e ( G ) (k) is a k-Lie algebra (it remained-to shcw that

cx,x] = 0

f o r xE Lie(G) (k)

, while

the abme m

the Jacobi i d e n t i t y ) . Replacing k by a variable Proposition:

For each RE&

, L*(G)

(R)

n

t gives a new proof of

RE_Y(

, we

g e t the

is an R-Lie algebra.

By d i r e c t manipulation of (11) and (12) we derive the usual formulas:

4.6

Let

dEDer(JG) and l e t $cLie(AutG) (k)

, define

( 2 . 4 ) . For each g€G(k)

F r m the definikion of

D

o

s a t i s f y d =Do

gd and dg by the fonnulas

we imnediately obtain the following equivalent

11,

5

259

DIFFERENTIAZl CAJXULUS

no 4

4,

v

is open i n G_ and fCd(g) , let be the functions which s a t i s f y (fg) (x) = -1 ( g f ) ( y ) = f ( y g ) ; thenwehave ( g d ) f = ( d ( f g ) ) g and ( d g ) f =

descriptions of

gd arid dg : i f

f g € d ( y ( g ) - k ) and gfEd(6(g)-1Lj) f(gx) and -1 g (d(gf))

.

Fram the definitions we inmediately obtain the fonriulas g6' (x) = 6 '

(15)

(XI

.

for gEG(k)

and x€L&e(g) (k)

A derivation

dE Der ( dG) is said to be left &anslation) invariant (resp.

r i g h t (translation) i n k r i a n t ) i f , f o r each RE& have g%=% Proposition:

(resp. %g=% 1

arid each g€G(R)

(2.4).

(k)+Der(JG) (resp. 6'

The map y ' :

-

induces an antiimmrphism (resp. an immrphim) algebra of

Der (& )

derivations. Proof:

G

:&e(g)

y :_G +&t(_G)

, we

of

= 6 (4)-'& (9)

L'I

(k)+ Der(JG) )

LLe(G) (k) onto the

Obviously it is enough to prove the assertion for y '

the autcatlorphism u'

u(x)

of:

sub-

formed by the right (resp. left) translation invariant

on &t(_G)as follows:

i.e. by

, we

if

RCL$

,

g€G(R)

Let

and uE&tR(GgkR)

-1 G @ R by gu(x)=u(xg)g

.

.

f o r xcG_(S)

G_ act

, define

,

Set&

,

W e quickly infer t h a t the homcarprpkism

irduces an immrphism of G onto jY&

(G)

. For i f

u(xg-') g =

have u ( e ) x = u ( x ) , so that u = y ( u ( e ) ) ; the converse is clear.

(s)

It follaws from 2.5 that Lie (y): k e (G) L i e (Aut ) induces an iscanorphim G Lie(G)+Lie(Aut(G_))--; taking the values on k of the two members, we get the

required assertion. Corollary:

4.7

(a) The following assertions are equivalent for

xE Lie(G) (k) :

is right-and-left

translation invariant.

(i)

y ' (x)

(i')

6' (x) is right- and - l e f t translation invariant.

(ii)

y ' ( x ) = 6' (x)

.

260

m B R A I C GFOUPS

The Lie algebra of right-and-left

0

is catmutative and y

&

5 4,

no 4

translation invariant derivations of G induces the bijection of (LAG_)-(k)

b)

G

11,

6

onto t h i s algebra. Proof:

a)

follows fram ( 1 4 ) , (15) and 2.5.

G , written

The L i e algebra of

4.8 L i e @ ) (k)

Lie(G@R)

. Thus for each and let

L e t Re$

. Then the canonical m p

Sc&

, is

Lie(G)

the k-algebra

we have an algebra isomorphism L i e ( G ) (R)

RE&

.

(a) and 4.6.

(b) follows f r m

L i e @ ) ( R ) + L i e ( G ) (S)

2

is

cmpatible with the algebra structure of the t m oomponents and with the ring hancanorphism R+S

. Frcan this we derive a canonical S-algebra

hoarmnorphisin

This hcmmrphism is not bijective in general. In particular, the canonical

( L i e ( Q ) -+LAe(GJ is not always an isa-mrphisn, the upshot of ? which is that the k-algebra L i e ( G ) is not in general sufficient to determine

hxmmrphism

the Qk-algebra

Lie(G)

. However,

there is an irr'prtdnt case in which it is

sufficient: Proposition: (i)

The following conditions are equivalent.

The canonical homcanorphism Lie(G)a+&e(G)

the map Lie(G_)BkR-+Lie(G) (R) (ii) (iii)

The

&c;n-cdule

The - _k-kdule _ _

QG,k

w G/k

-.

is an

isamOrphiSm,

is bijective for each RE%

.

i.e.

is f i n i t e l y qenerated and locally free.

is f i n i t e l y generated and projective.

Proof: (ii) <=> (iii)

: by p r o p s i t i o n 3.4.

(iii)<=>(i) : by proposition 3.6 and Iug. 11,

5

5, prop. 8.

I n particular, the above conditions hold when: (a) k

is a f i e l d and G is locally algebraic over

(b) when G_ i s m t h over 4.9

k

k

(by (iii)) ;

(by (ii)and (I, § 4 , 4.131)

.

Let us sum up these results in one particular case:

Suppose that k

is a f i e l d . To each locally algebraic k-group

functorially a f i n i t e dimensional k-Lie algebra

G we assign

Lie(_G)=x, a linear

11,

5

4, no 4

261

DLFFERENTIAL-S

and, for each Re& , a map x + eEX of g@R These assignments s a t i s f y (11, ( 2 ) , (51 and 16) i mreover,

representation & d : G + g ( c ~ ) into G ( R ( E ) )

.

for each RE_%

, the

sequence 1 + 9 @ R

m y be identified w i t h

k

a

G(R)

+

1 is exact.

.

p:ak+@(V) be a linear representation of

mrphi.-

of

pi

v

such that

Lie(GL(V))

if

k

2, 2.6 there are endo-

. . Identifying

t c ~ E % Applying

for

p (eEX) = I d + €xpl

get

L a Q-algdxa we obtain by (t)

p ( t ) = expL&(p)

k

5

by means of f o m l a ( 7 ) of 4.3 we g e t &(p)x=xpl

with &(V)

I n particular,

p(t)v=ltipi(v)

, we

this f o m l a w i t h t = eEX =EX

k

,

tcRE4

.

5

2, 2.6a)

,

i s an F -algebra, similar argun-tents apply to the group pgk

P algebra my be identified with

, we

have, for each t t R c %

4.11

. Let

i n a k-module V

c1

of 4.3. By

us determine the hamwrphism =(p)

$k

+

Take G=a By 2 . 1 L i e ( G ) k ' , where eEX=Ex f o r xERE,Yc ; the L i e algebra

ak is the m t a t i v e Lie algebra k

Let

If

G(R(E)

Fixample 1: the additive group.

4.10 of

+

Example 2:

k

, and

if

,

tp=O

diagonalizable groups.

is a small catmutative group

(5 1, 2.8)

R(E)*= {a+Eb:aER*

: its Lie

is a linear representation of

p

such t h a t

.

. For

p ( t ) = -Lie(@)

Take

(t)

.

G=D(rIk , where r

we have

RE$

, bER} ,

which M i a t e l y yields an exact sequence 0

+

G(r,R) 2 D(r) ( R ( E ) )

i n which u(b)=1+ Eb

and e"=l

+EX

L i e algebra of

4.12

+

. It follows that

= w G/k may be identified with k

Example 3: linear groups.

jective k-module; take G _ = g ( V )

+

. By 2.2

Let

r

is iSCarOrphic to G r ( r , R ) BZk ; i n particular, t h e

V be a f i n i t e l y generated pro-

and 4.3,

Lie(G) (R)

, with -1

joint representation being given by a ( g ) x = g o x o g

is a sub-group-scheme of

G_

,

,

.

r a l l y identified with the L i e algebra & (V) (R)

If

1

L i e (G) (R)

. Accordingly we have

uk

_D(r)(R)

Lie (H) (R)

.

my be natu-

eEX = I d + EX

, the ad-

m y be identified with the

11, 9 4, no 5

74IGFIBMC GROUPS

262

.

Lie subalgebra of L(V) (R) consisting of all x for which Id+~xeg(R(~))

, then

For instance, if E=E&(V) is equivalent to Tr(x)= 0

Section 5

.

xE Lie(H) (R) iff det(1d +EX)= 1

, which

Differential aperators

In t h i s section, for each k-scheme 2,

1x1

of k-rrcdules over

AJ;

.

denotes the category of sheaves

Let f:_X+g be a mrphisn of k-sc-s and let d :dy*$,(dx) Y be a mrphism of &Z For each open subscheme g of y and. each- $Ed($ 5.1

.

denote by (ad$)d the element of A#(Ju,f*(Jx) [ g ) such that ( (ad @)dl (x)=$d(x)- d($x) for xE o ( V ) -and (it being assumed that defined by the ham-f, (dX) is assiqned the structure of a module over Oy mrpksn [ f [ ' : O y + f + ( d x )induced. by f ) . L

-

-

Definition: _A k-deviation of order a pair

5n

(Era) consisting of a mrphisn

dEAJ$(dx,f,

'9._')

, with origin f:_X+Y of & Scl

Y and target X and a

such that

... (ad$n)d = 0

(ad@,) (ad$l) for each open subscheme

g of 2

and all sequences $o,...,$n

We call

of d(g) .

(f,d) a k-deviation, or simply a deviation, if there is a natural number n such that (f,d) is a k-deviation of o r d a 5 n We also say that d is a k-deviation of f

, and write

.

d for

(f,d) whenever there is no

possibility of confusion. Finally, for each mrphisn f of %k~ r go def (I, 5 1, 1.4) Like mrphisns, deviations notes the k-deviation (f, If\-) will be represented by arrows: ~( -f d) --% or~ (f,d): Y+X_

.

Let (frd):x+X and (g,e):

z-+

.

be deviations of order 5x1 and L p respectively. Write d e E$&(dz, ( g f ) (dx) ) for the mrphisn which assigns to each open subschene V_ of Z the &site map 2

+

11, 9 4, no 5

DIFFERFSPTIFL(XLCUWS

It is easily shown that d e

263

is a k-deviation of g f _

L

of order sn+p: use

the formula

here + E ~ ( _ u ) and where +g is the image of

+ in

J(g-l(v_)) -

. set

_the canposite deviation of (_g,e) and (<,a)(?,el = (gfrde); we call (gfrde) (f,d) Cchnposition of deviations is associative and its unit elanents are the deviations of the form Id0

.

.

0

Notice that (gf) _ _ =f

0 0

:

Example: Suppose that X=$=S&k

5.2

. Then a mrphisn

_f:q+y

of & cS is called a section of and a deviation d of f is called a distribution of carried by f If _f factors through an open subschme of and if + E d ( v ) we say that d(@)Ed(e+)=k is the value of the distribution d at 4 . Setting u = (frd) we also write

.

d(4) = Igdu

=

l+h4du(y)

.

is associated with

where f EY(k)

Notice in particular that $(g,)=($dfo

b -

f.

ExTle: Let X_ be a k-scheme. A differential operator on

5.3

X

of order s n is a k-deviation of order r n of the identity mrphism of 8 The set of differential operators on 5 is written Dif(g) ; the subset consisting of those of order L n is written Difn(X) By 5.1 r Dif (5) is a k-subalgebra of A$(& ,& ) : the algebra of differential operators on _xx_

.

.

.

we have D(fg)= fD(g) for each open in _X and all f,gE d(U) If g = 1 we thus see that D(f)= fD(1) Accordingly we may identify with Difo(X) by assiqing to + E d ( X-) the differential operator f H f $ . If DE Difo(X)

.

.

a(&)

If DE Dif (5)

+ Dif (X)

,

~ ( 1€ )

+ (resp. Difn(X))

4 ~is) called the constant term of

D

. we w r i t e

for the subset 06 Dif (3) (resp. Difn(g))

.

conwhere

+ Difl (X)= Der(dx)

sisting of all D for which D(1)= 0 We then have D e r ( ~ ? ~ )denotes the set of k-derivations of the sheaf 8, This isprove3 + and + E ~ ( u _ ) as foiims: if DE Difl(_X) g is an open subscheme of

X

.

264

Au;EBRAIC

GRDUPS

( a d 4 ) D C Difo(U_) ; hence there is a function a($)Ed(!)

then

. setting

~($9) = $ ~ ( g ) + a ( $ ) gf o r g € O ( g )

5.4

methcd. f o r calculating the deviations of f d-

for the element of x t g , d;f sends

f

x

_Xk

over an open set U

7,-

.

~ ( d z @ (dy) g ' , 8,) 7,

on

-

.

a($) =

-

1

, write

$c

(dY) generated by

9,

denotes the section of €'

(dy) -

-

U..

over

induces a bijection of t h e set of k-deviations

t$~&(d~@~c* (dy), 9,)

onto the set of elements of

Clearly the map d t+d-

n+l

d€ $ ( d y f g .

induced by the section I$ of

of order 5 n n+l

this bijection,

. If

f o r the sheaf of i d e a l s of !)xC3kf'

Proposition: The map d ++df

-

follows that

-

sections of the foxm -l@$-($t)@ 1 , where

Proof:

such that

( d @f'(8 ) , dx)defined as foll0ws:for each

%I-

yf

mrmer, we write

f

, it

4 , no 5

f : g + x be a morphisn of k - s c h s . W e ncw formulate a

kt

vanishing on

g=1

5

is a derivation. The converse is obvious.

D ( @ ) so t h a t D

of

11,

f

-

is a bijection of

-

Y &(dy ,f. (dx) )

. I t follows d i r e c t l y from the def&tions-that,

-

onto

under

correspond to morphisms which vanish

deviations of order s n

f o r the g-ule (dz@kf* (dY)/$+I . me order r n are thus i n one-one correspondence with the

In future we shall write k-deviations of elements of

f

of

~ - -c ,Jx) G. The X-module

5.5

7; m y

consider an a r b i t r a r y gnbeaaing scheme of

z

such that

also be constructed a s follows. F i r s t

i:_X+z of

k-schemes; let

be an open sub-

i_ is the m p o s i t i o n of a closed embedding j:X+v

and the inclusion mrphisn of

in

. If

7

is the kernel of t h e rmrphisn

-jf :Jv +j,(dx) induced by' j , it is clear that the closed subscheme -V(mnR) - (I,-§ 2, 6.8) of V depends only on & and not on t h e choice of c

V_

. W e denote this closed subschene by

2:

-

and call it the n*

neighbourhccd

II,

5

no 5

4,

DIFFERENTIAL CALCULUS

265

.

o_f i & 2 W e entrust to the reader the task of generalizing t o n* neighkourkcds the functorial properties of leading n e i g h b m h c d s descrifsed i n section 1

.

Let f:_X+y be a mrphism of k-schemes and let X be the 5 TM the qeametric r e a l i z a t i o n of the nth n e i g m u r -

Proposition:

space of p i n t s of

X

hood of

-f

.

with respect t o the anbedding y:X +X_ x Y_ w i t h canponents

is canonically i m r p h i c to

Proof:

Set

and

=_X

induces a hQneanorphisn of

=y

X

.

Id

and

( ~ -~ 7 % ) and apply the preceding remarks; c l e a r l y y

onto the space of p i n t s of

mains to oompare the structure sheaves. L e t

and

$

Zn

-Y

. It thus re-

be t h e presheaues of

k-algebras over g such that

. By ccanparing t h e s t a l k s of

vanishes on '$+l

the sheaves in question, vie see

that the induced m r p h i m

i s an isamrphisn.

Remarks:

Henceforth we shall identify

ixnwrphim. I f d'

:pi

-y

- -Y

dx

d

i s a deviation of

lZnl Y

with

($,pi) by means of this -

f:g+x of order g n and

if

is the mrphim associated with d (5.4) , we may reconstruct f d frckn dE by observing that d- is a deviation of t h e m r p h i m fn:X + Z n induced by f and d is the ccmposite deviation +.

11,

266

For each mrphisn of k-schaws - -

Corollary:

5.6

quasiooherent X-module

.

Proof: With the notation of ink1

zn

-Y Identifying have

q (XI= x

d

++

-s n

is a

by means of the canonical iscamrphisn, we

for xE 2" ,-and the Xrrodule Y

Zn

-Y

3

is the direct image under

. The proof zs ccpnpleted by invoking I, . men the map

Corollary: _ L e.t h:B+A be a mrphisn of d ( s B ) is a bijection of the set of k-deviations of onto the set of k-linear maps D:B + A for a l l bo,...,bntB

(adbn)D = 0

-

PZ1

2, 2.4.

5.7

,

xxy + L

q - of the sheaf of functions of

9

4 , no 5

5.5, l e t q - be the ccmpsite mrphisn

with

Zn

-Y

~

f:X+Y

9

such that

.

cpkh of order

(adbO)(adbl).

..

Of course we set

for b,x€B

. A map

D

satisfying the conditions of the corollary W i l l be

.

called a k-deviation of

h of order d n

Proof of the corolhy:

F i r s t assume the notation of 5.5 for the case in

which :=%A

. By

5.6 and I, § 2, 1.10, we have accordingly

%(pn,d ) = _rCna,(<(X) - Z Z

rvLu"

,A)

.

Moreover, by arguing as i n 5.4 we see that the k-linear maps D:B+A

satis-

fying the condition of the corollary are in one-one correspondence w i t h the

elements of gA(A@kB/J"+l

,A) ,, where

J

is the ideal of

by the elements l @ b-h(b)@1: t h i s follows by assigning to A :A€3kB/?+1+A

the map b;

fran the fact that

X (18b mod?+')

q(x) = A @ ~ B / J(5.5) " + ~. -

A@JkB generated

. The corollary now follows

11,

5

5.8

Given A€$ , set PnA/k = ( A g A ) where J is the ideal ABkA generated by the elgnents a @ l - l @ a, a € A Endow Pn with

4, no 5

267

DIFFERENTIAt-S

.

of

the A-algebra structure induced by the hammrphism a w a@1ncdJnp; set 6 (a)= 18 a mod f'l f o r a€ A By 5.6, there is a canonical bijection

.

A -kd (Pn A/k ,A)

Difn(S&A)

, the d i f f e r e n t i a l

A E B A ( P i f i ,A)

namely, i f sends a € A

onto

;

.

(X(G(a))E A

operator associated w i t h X

Pn is functorial i n A ; given a hatnmrphisn h:A-+B A/k n of p+ , we write Ph/k: P>k-+ PEIk for the map induced by hgkh :AgkA + BakB I f S is a multiplicatively closed subset of A and h is the canonical map of A into A[S-l] ,P" induces a bijection The construction of

.

-+

h/k

Similarly, i f Pn h/k

a is an ideal of

A and h:A+A/a is the canonical map, n n n PA/k/ (aAlk+ A 6 (a)1 onto PA/a,

induces a bijection of

5.9

If

Example:

A = k[T]

T E ~ l r n s d J " ' ~ and

bysending T t h mod hM1

, respectively.

aTi

such t h a t P(T+S) =

if

PE k[T]

, whence

-+

get an i s a m r p h i m

onto Tncdhntl

With the k[T]-linear

Xi(hlrnsdhn+l) = 6

a : k[T] -

, we

l@T1m3nvsd'~

Xi : k[T,h]/(hn+l) such that

.

ij

-+

map

k[T]

is associated the d i f f e r e n t i a l operator

k[T]

I(-.a

i aT1

and

PISi

P(T+S)Ek[T,S]

. We have accordingly

11, 3 4, no 5

ALGFBRAIC WUPS

268

aTr ( F ) ~ - i . =

aTi

It f o l l m that

a

3

aT1

aT'

a

-

so that i n particular,

is the f r e e k[T]-dule

and so Dif (C&)

a

a

Id=-,aTo

5.10

Example:

T.+hmdhn+l

,

k[T ,T1j/k hence

Tr+

(z)Tr-2h2+.

aT1

It follows that operators

Dif

,

3/aTi

5.11

Let

uijl

,

'.'

, there

..+ fn)T'-"hn modhn+l

(uk) iElN

is a k[T,T -1]-algebra iso-

, of

(n€Z)

, such

6 (TI

onto

.

is the f r e e k[T,T-']-dule

generated by the

that

be a mrphism of

f:X_+Y_

iEI

a , mi

onto k[T,T-l,h]/(hn+l) which sends n+l onto (T+hIrxrcdh -

a) a f f i n e open subschemes LJi schemes Vi -

#

By 5.8 and 5.9

n mrphism of P

T'-'h+

, *.-

generated by

&$

. Assume that we are given:

, iEI , covering 5 f(LJi)cyi ;

Y_ such that

.

zi:vi+Yi

;

b) a f f i n e open sub-

c) a f f i n e open subschemes

l€Iij covering FinU Let be the mrphisn induced by -j ,f and l e t h i = d (f 1 By 5.7, each deviation Di of hi of order s n k -i is associated w i t h a deviation di of f i of order s n In order f o r there

,

.

t o be a deviation d of

.

_f

of order 5 n

which induces di

it is necessary and sufficient that, f o r each

xq

j 1

M u c e d by

di

i,j,l

, the

f o r each

deviations of

and d . coincide. I n other words, the cmposite 7

i

,

11,

9

4, no 6

DIFFERENTIAL-S

269

maps below coincide:

z

Now suppose we are given a k-scheme

of

z , it is clear that the k-linear

. If

respect to the cavering of

13k (W)@ D - k i

Z

If

(_f,d)

and

each

the Di

s a t i s f y the above

s a t i s f y the "matching" conditions with

z x g by the a f f i n e open subschesnes _WxU

i and each _W

.

. If

in

Accordingly there is a deviation Z x d of

dk (w)@ D for - k i

is an a f f i n e open subscheme

map

is a k-deviation of dk(W)@khi or order "matching" conditions, the

W

zz(_f

of order

. This deviation , one

e is a deviation of a mrphism g:z+T -

-i ' which induces

sn

& x d depends only on

defines e x 5 in a similar

fashion. Set e x d = ( e x z )( T r d ) ; then we also have e x d = ( g x d ) ( e x Finally, i f order s n

R%

we write d @ R

I

k

which induces, for each

or i

%

f o r the deviation of

, the

k-linear m p

Section 6

Invariant d i f f e r e n t i a l operators on a group schane

6.1

Let

G_

be a k-mnoid-schane and l e t

section. A distribution of

5 carried by cG

bution a t the origin, o r s-ly (resp. DistnG_) order of

LI

5

n)

. If

. W e set

cG:

%+G

y)

c@kR

.

of

be its u n i t

(5.2) -will be called a d i s t r i -

a distribution on

G . We

write D i s t G

for the k-module of distribution (resp. distributions of

u

Dist'G

DistG

,

~ ( 1=) due k

= { U c D i s t G : u(1)= 0)

w i l l be called the constant term

270

ALGEBRAIC GRCUPS

and

+

D i s t G = (Dist+C_) n D i s t n G

n-

e be the unit element of

9 4, no

6

.

. If

is a f i e l d

m y be n+l identified w i t h the space of k-linear m p s de + k vanishing on me (apply

Let

5.4 t o the case f = s G )

G(k)

11,

k

r

DistnG

.

If p t D i s t m G and VE D i s t n G , define the Convolution product v*vc

Distmt,G_

to be the canposite deviation

!The convolution product is obviously associative

, the algebra

algebra structure on D i s t G Given a harrrrprphisn

write f(p)

Clearly

g:G_+g (u)

f(u)cDistng and

-.

of distributions on G

of kmnoid-schemes, and a pcDistmG_

(Dist 4

or

a d induces an associative

, we

for the capsite deviation

Distf

: DistG - +DistH

is a k-algebra hcmmr-

phisn.

Mrewer, if RE&

, the map

p”1.1~

(5.11) of

DistG

into D i s t ( G a k R )

is acmpatible with the convolution product and may accordingly be exteladed

to a harvmrphisn of algebras ( D i s t G)QkR + D i s t ( G @ R) k 6.2 J

Example:

.

Suppose that G_ is affine and has bialgebra A

be the kernel of the augmentation

.

sA By 5.4 and 5.6

, DistmG_

. Let

may be

.

such that p (pl)= 0 +1 If v:A+k i s a second k-linear map satisfying v (3 ) = 0 , u * v is the canposite k-linear map identified with the k m d u l e of k-linear maps p:A+k

For example, i f ei(T’)= A i j

G=ak

. Then

W e have E ~ * E = .(

G=vk

, let

k[T,T-ll/(T-l)i+l

Distc$

ci:k[T1-tk

be the k-linear IMP satisfying

i s , t h e free k d u l e generated by

i+j

..

vi

-1

.

E ~ , E ~ , E ~ , . . .

) E ~ + ~

7

If

, let

: kLT,T

1

-t

k

be the linear map which vanishes on

and s a t i s f i e s v , ( ( T - l ) j ) = A

. Then we have ij

.

wi(T’)=(:)

.

,

11,

5

4, no 6

where

denotes the coefficient of

(l+T)J ( j €Z

, iEIN)

. If

I n this case, D i s t p k by v1

271

DIFl?EIWiTIAL(XCLXUJS

k

Ti i n the series developnent of

is an algebra over Q

, then

we have

is therefore the free ccarmutative k-algebra generated

( t h i s does not hold i n the general case).

Consider a k-sc-

6.3

the k - m n o i d - s c h

associated w i t h g

_G

. If

on _X

g and a right operation g :5 x c +X_

. Let

ptDistmG

p :G

-0PP

be the hancanorpksm

+A&t&

we write

I

of

p ' (p)

for the

CaIpsite

devia-

tion (5.1) :

Clearly p ' (p) is a deviation of order s m

5

i.e. of the identity mrphisn of operator on Proposition:

X

of order

.

For each k-schene &

mrphism p : G +AUt X -oPJ? - mrphisn. Proof:

Lm

Since p '

, the map

of the canposite mrphisn

. Accordingly ,

p'(p)

k-mnoid-scheme

is a differential

and each

m-

p ' :D i s t G-t - Dif X_ i s a k-algebra hano-

i r obviously k-linear, it is enough to show that

p l ( p * v ) = p ' ( p ) . p 1 (v) for

p,vtDistG

.

T h i s follows fran the c m t a t i v e diagram below: i n this diagram, the

CQ~TU-

tativity of the t m base triangles follows frcm the definition of p ' ; also 0 the ccmpsite deviation of x x nG I 3 s G_ x v and 3 x p coincides w i t h

XX(U*V)

.

-

27 2

11,

6

The definition of the convolution product (6.1) m y be rephrased

6.4

as follows. Let _V be an OF

u- and

5 4, no

l e t $€

B&J)

subschane of

. Then we have

the function induced by $ on fom:

G such t h a t

E~

factors through

@d(p*v)= J($nG)d(px v ) - r where $wG

T,~@)

-

. W e w r i t e tkis last f o m l a i n the

is

I@db*v)= ( @ ( x y ) d ( u ( x ) x v ( y ) )

Also, assuming the notation of 6.3, let

,

.

be an open subschene of

5 ,

and x € y ( k ) By 5.2 ( p ' ( p ) $ ) (XI is the value a t # p ' (u)$ of the distribution x" carried by x :$+v j accordingly it is also the value a t Q of the canposite distribution

$ChJ

p€Dist(G)

which coincides (5.11) with

6.5

W e now apply the results of the preceding discussion t o the

case i n which G_

is a k-group-scheme,

X_=g

and g =IT

.

G

. In this case

is the hamanorpkisn 6 : G +&tG_ of 5 1, 3 . 3 ~ ) -0PP If G_ is affine, With bialgebra A , we may regard a distribution

p

p E D i s tG

11, fj 4, no 6

DIFFERENTIAL CALCULUS

273 (6.2), and 6' ( 1 ~ ) as the

as a linear fonn on A which vanishes on capsite map A-

aA

ABA-

A @ku

ABk-

A

k

(s.6.).

i S'(E~)=~/ ~T(6.2 and 5.9). If G=LI k' we have 6' (vi)= Tia/aTi (6.2 and 5.10) For example, if G=%

, we have

.

In general, if g EG(k) and D is a differential operator on G_

, we write

gD for the campsite deviation

i.e. the differential operator such that (gD)($) = (D($g))g-I for each section $ of dG over an apen p (cf. 4.6). We say that D is left invariant if, for each %EwI&

and each g EG(R)

we have gDR = DR

Invariance Theorem. For each k-group-scheme

.

s,

6' : Dist_G+DifG_ is an ismrphisn of the algebra of distributions of G onto the subalgebra of Difg

f o d by the left invariant differential operators.

Consider a differential operator D on G_ and the value D(e) of D at the origin, i.e. the cchnposite deviation 0 D E G-G G k e + Then by definition we have

$dD(e)= (D$)(e)

. Let

be an open subscheme

11, 5 4, no 6

zlLmBRAIc GROUPS

274

of _G

, @Ed(;)

,

.

xEF(k)

If

D

is l e f t invariant, we then have

Since U s calculation may be repeated after an arbitrary change of base, we D = 6 (D (e))

see that

Finally let 1 ~ .E D i s t G ( 6 ' (!.I)@)

we see that

if

is l e f t invariant.

D

. I f we set (x)

=

(6' (u) (el) ($1

i

x = e in the formula

@(xg)du(g)

=

i

+(es)du(9) =

so that 6 ' (u) (e)=1-1 and DI-.D(e) For example, i f g Eak(k) = k P(T)

,

G_=%,

u ($) ,

is the required inverse map of

6'

.

is the d i f f e r e n t i a l operator P(T) a/aT1 , and i then we have gD =p(T-g) a/aT ; this operator is invariant if D

is constant. Similarly, i f

i then we have gD = g P (T/g) a/aTi

G=uk ,

D = P ( T ) a/aTi

and g cpk(k) = k*

; this operator is invariant i f

i P (TI= AT

I

XEk. Ranark: G

The invariance theor.em may be generalized t o k-mnoid-schemes. I f

is a kmonoid-sew, let us say that a d i f f e r e n t i a l operator D on _G

is l e f t q u i v a r i a n t i f , f o r each open subscheme each RE$

and each g€G(R)

isanorpkism of

DistG

, we

have D,(@,q)

onto the subalgebra of

, each (PE d(U_), =(DR+,)g . Then 6 ' is an Dif G formed by the l e f t of

G

equivariant d i f f e r e n t i a l operators.

6.6

on a k-scheme

Let g : G x g + X

3 and

let

fine a map p : D i s t G_ +Dif

p :

be a l e f t operation of a k - m n o i d - s c b

+&tX

be the associated h a r a m p h i m . De-

5 by assigning t o u E D i s t 5 the carpsite devi-

ation

X L The map p '

G_

$ % X _

is then an ant-

u

GX3-x

0

-

rphisn of k-algebras:

,

11,

0

4, no 6

DIFWTENTIALCALCULUS

men ~ = and g

g=.rr

E

,

is an anti-morphisn of

y'

. If,

(5 1, 3.32)

is the ha-mmrphisn y of

p

G is a qroup,

moreover,

275

onto the alqe-

DistG_

bra of r i q h t invariant d i f f e r e n t i a l operators, provided one defines r i q h t invariance as follows: i f (Dg) (I$) = q-l(D(qI$))

then

D

DEDif G

, where

and qEG(k)

I$' is a section of

, define Dg dG over an

is said to be riqht invariant i f , f o r eac;

RE$

by the formula

open V_

(4.6) ;

and each q E G ( R )

,

we have D g=D R

6.7

R '

G

If

is a k-group-schgne,

p EDist

G

and g EG_(k)

then

(M g ) u is by definition the canposite deviation

-%W e then have

subscheme of

(Int9)0

u

G<

-

.

I$d(Adq)p)=jO(qtq-')du(t) G such that 1 EU(k)

.

for

I$ E &()!

, where

is an open

&guinq as i n 4.7, we infer the

Propsition:

Qgt _G

k-groupscheme: a)

f i

, the

p EDistG -

following

conditions are equivalent:

(u)

is l e f t invariant.

( i)

y'

(it)

6' (p)

(ii)

y'(u1 = S'(u)

(iii)

U E (DistG)'

is r i q h t invariant.

.

, i.e.

(Adq)uR=uR

for a l l RE&

and gE_G(R)

b) The algebra of left-and-riqht invariant d i f f e r e n t i a l operators on G caranutative and y '

6' induce the same isanorphim of

Dist(g)G

this algebra. Remark:

If

u,vEDistG

, we

have ~ ' ( v ) & ' ( L I )= 6 ' ( u ) y ' ( v )

. For

if

i s

.

AUEBRAIC GROUPS

276

It follows that ~ ' ( 1 1 ) = Sl(1.1)

Dist(G)'

, we

that p*v =v*p

11,

is contained in the centre of

Dist5

4, no 6

. For i f

have

for each v E D i s t G_

.

W e now relate the above r e s u l t s to those of section 4. Let G_

6.8

be a k-group-scheme, l e t GE be the f i r s t neighburhocd of

to

0

E~

-

: %+G

, and

with respect

let

€1

sk-G-g

e+

-E

€2

(I,5 4 , 1.1). By 5.4 and 5.5, the deviaG D i s t l G , are of the tions of E~ of order i 1 , i.e. the distributions 0 form V E ~, where v:gE++ is a deviation of E~ Since GE i s affine with algebra &gE)=k@ (3.11, the deviations v of E~ are associ%/k correspond ated with linear forms k@w Gfi+ k ( 5 . 7 ) Elements of

be the canonical factoring of

E

.

.

to linear forms which are zero on k

mrphism

t

DistlG

Write

v

G

%(~~/~,k),

+

: DistlG

LieG

Distg

. Accordingly we have a canonical iso-

11,

5

4, no 6

277

DIFFERENTIAL CRICULUS

for the c a n p s i t i o n of this canonical ismqhism w i t h the isamorphism -1 L i e g = tp&(wGfir k) of 3.6. The inverse isomorpkism vG m y be explicitly described a s follaws: let n: Spk(E)+e+ be the deviation of the canonical

embedding % + S-p k ( ~ ) such that ~ ( 1 ) = 0 and n ( ~ ) = l; i f -1 5 € L i e G c G _ ( k ( ~ ) ,) vG (6) is then the ccsnposite deviation L

applying a change of base, we see that tile equation ( 6 ' ( g ) $ ) (m) r a i n s true f o r a l l R€&

and m€y(R)

y ' (p) is proved similarly.

Proof: By (12) of 4.5, 6.5 and 6.8, we have

=

6 ' (u*v-v*u)

= 6 ' (vG (ll*v-v*ll) )

.

(6

4) (m) =

. The equation

=

11,

Section 7

Infinitesimdl groups

7.1

Definition: E~

*

4, no 7

A k-group-scheme G is said to be infinitesimal is f i n i t e locally free and its unit sec-

i f the canonical projection Cj+-%

Q n

8

: g , + ~ induces a ticmmxphisn of

f i r s t condition means that

G

, and

is a f i n i t e l y

is a f f i n e and that

generated projective kmodule. By I, presented algebra over k

.

[$(onto [ G [

5

5, 1.6,

the kernel

I

d(G) is thus a f i n i t e l y of J ( E:~ d(Q ) +k is a f i n i t e -

l y generated ideal (I, § 3, 1.3). The second condition-in the d e f i n i t i o n means G_(K)

that

reduces to the u n i t element for each f i e l d K

S

, or

that

I

is

a nilideal. Since I is f i n i t e l y generated, this l a s t condition is equivalent ’

to saying that I i s nilpotent. By 5.4 and 5.6, D i s t G f i e d with

may then be identi-

R ( d ( G ),k) =td(g) and the multiplication i n

D i s t G_

m y be ob-

tained by transposing the c o p d u c t of d ’ ( ~ _ ):

Also, it follows f r m a x i m DistG

(Coun)

is the augmentation

E

10 (GI

(5 1, 1.6)

:J(G_)+k of

that the unit element of

d(G) .

Througbut the rest of t h i s section, we write

A : D i s t (5)+ D i s t (5)C 3 D i s t (5)

k

for the map derived by transposition of the multiplication c

md(g

in

d(G) :

W e then have, by definition,

(Al.r,a@b) = (l.r,a.b) for

p€Dist(g)

and x,y€I

, we

,

a,Md(G)

get

Moreover we have

. Setting

a = c r + x and b = B + y

, where

a,Wk

279 ( p @ l + l @ p , a @ b ) = aCp,b) +B(p,a) = ( p ,2aB+ay+Bx)

.

Accordingly we have Ap = p @ l + l @ p

iff

2

, i.e.

! ~ ( 1 ) = 0 and ~ ( ) =1O

Finally, we write

7.2

E:

iff

+

pEDistlG

.

.

DistG_+k f o r the map pt-+u(1)

-

Let _ _G

Proposition:

a_ k-schane. Then the map

p

-right _

p'

be an infinitesirrdL k-group and let

1s.

of 6.3 induces a bijection of the set of

Goperations on 5 onto the set of algebra hanaru3rphisns v:DistG+DifX_ suchthat v ( p ) ( l ) = ~ ( p ) v b ) (fg)

,

where LIE D i s t G

open subschenes of Proof:

Let

Au

-

x-

-

=lipi@

( f ) ) (V(Wi) (9))

f,gEd(g)

wi

X.

Z X €

x_xe+-

i s the identity and into _V

1(V(Pi)

i

u_:_XrG-+X be a right G-operation on

mrphisn

uxG_

=

/_X

, hence

XE

G

1

G XxG-

u-

_V

5

ranges through the

. Since the campsite

x_

is a hmmrphism, u_ induces a mrphisn of

a -Goperation on

,

for each open subscheme V_ of

This f a c t allaws us to confine our attention to the case i n which

& is

affine. In this case, we show that the map p I-+ p '

establishes a one-one correspond-

ence between right Goperations ! on X_ and hamcanorpkisms v:Dist G-t Dif X_ such t h a t v (p) (1)=E ( p )

for f,gC8@)

. If

projective, and i f the composite map k

M,N,P

and

are k-modules, where P is f i n i t e l y generated and

A :M+N@ P

k

is a k-linear map, write

1' : M @ k % + N

for

.

ALGEBRAIC

280

We knm that A H A '

mws

11,

is a bijection of S ( M , N @ P )

5

4,

no 7

onto Mc&(M@tp,N)

Consider i n particular a linear map

Then

a'

:

d(X)@d(X)@Dist G+ &(_XI

sends f @ g @ p onto A'(fg@u)

13'

sends f @ g @ p onto

is compatible with the multiplication (i.e.

Accordingly A a' = 8'

, while

, i.e.

a =6 ) iff

iff X'(fg@IJ) = C(X"f@IJi)X'(g@vi)) i

for a l l p E D i s t G _

and g,fEd(g)

a) the equality X (1)= 1 is equivalent to

One shows similarly that

X' ( 1 @ p ) =

E(U)

for a l l 1.1 ; b)

is equivalent to asserting that

the n-ap

the c m t a t i v i t y of the diagrams

.

is iduced by a haamrphim p:G f@U-p'(p)

.

+#(X)

A ' : c!(g)@ D i s t (G) -

(unital) l e f t d u l e over D i s t (G) -

Naw if X

.

OPP

+Aut(X)

turns

d (X)

,

is precisely

I'

kt0 a

(f) Our assertion follows from this and the f a c t that

.

5

11,

4,

DIFFERENTIALCi4LCULUS

no 7

X is induced by a i-mmt~~rphisnp i f f X makes the diagrams c~nrmte,and is s u c h t h a t X ( 1 ) = 1

Proposition:

7.3

a)

Ef

If

A(x)=x@x

x,yEG(k) =&(A,k) CDistG

volution product x*y Proof:

.

cl=B

.

(*)

and

.

,

x is an algebra hcnwrcprphism i f f x.y=.ir (x,y) coincides with the con-

!?

is an algebra hammrphism provided that f o r f ,g&

x ED i s t G

(**)

_G=SpA be an infinitesimal k-group.

, then

x E D i s t (9 =Mc&A,k)

E(x)=~ a 4

b)

-Let .

and

281

we

have (Ax,f@g) = (x,f.g) = (x,f)(x,g) = ( x @ x , f @ g )

so that x.y Ranark:

Let

= x*y

X

.

beak-scheme,

g an i n f i n i t e s i m a l k q m u p a n d

:G +Aut(X) a hamnorphism. I f x E D i s t G s a t i s f i e s E ( X ) = 1 and -0PP Ax=x@x I it f o l l c k directly fram the definitions that the endcm~qhism p

1 p (x) 1-€

P'(X)

.

of

dx -

induced by

p (x)

coincides with the d i f f e r e n t i a l operator

5 5

IEALLY AuJEBR4IC CROUPS OVER A FIELD

5

Troughout

5 , we shall a s m t h a t k is a field. L e t

, and

closure of k which belongs t o h$ which are separable over k

. We denote by

be an algebraic

let ks be the set of elements of

7l the Galois group'of ks over

i.e. the profinite topological group of k-automrphisms of of &I

1; k,

. Those members

which are f i e l d s are called extensions of k .

Section 1

The neutral component, &ale groups

1.1

Neutral component theorem: Let - - G be a locally algebraic

Go the open subscheme of

e the u n i t e l m t of G(k)

k-group,

G car-

ried by the connected component of e . Then (a)

-Go

(b)

For each extension K

(C)

The connected components of

i s a characteristic subgroup of G _ .

of k , we 5

have Go Bk K = ( G C3k K)' =irreducible,

.

algebraic over k

and a l l have the same dimensions.

5

Proof : (b) Since G is locally algebraic,

5 4,

c m . 11,

prop.10). Accordingly

,

Go

is locally connected (Mug.

i s closed and open in G . For each

Go Bk K i s closed and open and connected by I ,5 4 ,6.9 and hence coincides with the connected component of e in GC3k K

extension K of k

(a) By I

.

-

, 5 4 , 6.11, Go x go G_o

x

so

G

,

is connected, hence the mrphisn IT

x G_

--L G_

factors through Go : similarly, the morphia 0

G 0 - - q L G factors through G . It follows that Go is a sub-groupscheme of G _ . W e show

that it is characteristic. If K i s an extension of k

, in virtue

of ( b ) ,

Go @k K is invariant under each autmrphism of the K- scheme G Bk K preserving e

. In particular,

G-@

k

K

is invariant for each autmrphism of

11,s 5 , no 1

the K-group

L0CALI;Y

GBkK

. Now let

RE-?

u (x) E Go akR into Spec R

. Let

, and

u_ be an a u t m r p h i s n of the

and let

G g k R : w e must show t h a t , i f

R-group

283

ALGEBRAIC GROUPS

x

G0BkR , then

is a p i n t of

I be the prime ideal arising as the projection of x

let K be the f i e l d of fractions of R / I ; then x is the

image of a point x' of

G0BkK

, and

u(x) -

is the imaqe of

by the preceding discussion belorigs t o G o @ k K

, so that

which

%(XI)

we have

~ ( xE )G o @ R a s required.

k

(c) W e show f i r s t t h a t Go is irreducible. I t is enough to prove t h a t Go@k is irreducible, and by (b) we may confine ourselves t o the case in which k

is algebraically closed. If C," is notirreducible, then there are two closed points x,yEGo such that x belongs t o exactly one irreducible component

G , and

of

ring

d

y belongs t o a t l e a s t two d i s t i n c t irreducible components. The

then has exactly one minimal prine ideal and the rinq

0 at

least Y two; but this i s impossible since these rings are isomorphic ( 5 1,3.3 c) ) X

.

Let ,U be an open subscheme of Go which contains e and is algebraic over

. By lemma 1 . 2 below, the morphisn

k

Qx

g -+ go

induced by

tive; hence Go is quasicompact and so algebraic over k

(

T~

is surjec-

s i,3.8

and

I 153I2.1).

Finally l e t

be an arbitrary connected component of

G

and let x be a

closed pint of H . Let N be a f i n i t e quasi-Galois extension of k containing

K(X)

, and'consider

the projection morphism p :

This morphism is closed and open by I

-1

p (x) are rational over -

N

,5 5 , 1.6.

,(GB~N)-+G_ .

A l l the pints of t h e stalk

(for by I , § l , s e c t . 5 the residue f i e l d s of

these p i n t s are the residue f i e l d s of the local factors of

,

K ( x ) @ ~ N and

are accordingly isomorphic t o N ) . If y is such a point, and i f u i s the -Y translation ( l e f t translation, for example) of G_ BkN which sends e onto connected component of y is _u ( G o @ N) , and is therefore irreduY - k cible. But then p ( u ( G o @ N) i s irreducible, closed and o p n i n g , so - y - k that it coincides with 5 , which implies t h a t is irreducible. Finally,

y

, the

H@ N is the union of the g ( G o@ N) as y ranges through p-'(x) , hence - k Yk is algebraic over N , and its dinension is the same as that of Go@ N , k i.e. t h a t of so It follows t h a t jj i s algebraic over k and has the same

.

dinaension as Go ( I , 0 3

, 6.2 1.

284

11,s 5,

ALGEBRAIC GNU'PS

no 1

It remains t o prove the

1.2 Lemma: -

L e t U be dense and open i n the algebraic k-qroup G . Then the com--

-

I -

p s i t e mrphism

gxv_

71

G ,

c_xc_

G,

is faithfully f l a t .

Proof : This mrphism is f l a t by i s surjective, i.e. that

5 1 ,3.2

b)

, so

LJ(c) . L e t

G(c) =

it remains to show t h a t it

q E G ( c ) and let y(g) be

the l e f t translation which it defines. Since LJE and y ( g ) aG(g$ are dense and open in $ , they have non-empty intersection and there are u,vEU(c) with

-1

u=gv

, hence

If

1.3

G

g = u v , which is what we are required t o prove.

i s a locally algebraic group, Go w i l l be called the

neutral component of G ; it is an algebraic and irreducible open subqroup

of

5 . The theorem

t h a t the diwnsion of

-lies

d i m s = dimGo = Kdimde Finally, since Go i s an open subgroup of S

G is f i n i t e , and

. , we

have L i e G = L i e Go

notice that [LieG_ : kl = [ m / m 2 : k l

e

Proposition:

1.4

Let

e

that

2 Kdimd

e

=

dimg

. Also

.

_G be a locally algebraic k - w .

'Innen the

following conditions are equivalent: i s &tale.

(i)

G

(ii)

_GBk ks

(iii)

Go = + e

.

(iv)

de

= k

.

(V)

Lie

G

is constant.

= 0

.

Proof : (i)<=> (ii) : cf. I , 5 4 (i) = > (iv) : we have

K(e) = k , hence

(iv) <=> (v) : L i e G_ = 0 (iv) => (iii): apply

, 6.2

. de =

k

if

5

is equivalent t o rne / meL = 0

1 ,5 3

, 4.2

is etale.

, hence

t o the canonical projection

to m = 0 e 4

+ e

.

.

II,§ 5 , no 1

285

LLXAT.LY ATX;EBRAIC GROUPS

(iii)=> (ii): i f

=

GO

, then

+ e

( G B~ E l o

k z . This inplies that each connected component of

, so

i s i s o m r p h i c to

that

G C3k i;

is constant.

W e know that G i s an a f f i n e algebraic k-group

( $ 1 ,2.6) ; we also know t h a t

. It follows that

Lie E is the space of k-derivations of K ( § 4 ,2 . 3 )

i s &ale i f f t h e extension n = [K:k] 2 3

,

Klk

is not constant, f o r

&tK

Example: By § 4 , 4 . 1 1 ,

(&tK)(k)

has a t most n elements;

En ,

(&t K ) ( C ) is isomr-

Lie( u ) {xEk:nx= 0 ) .It n k n l k # 0 . Suppose t h e latter holds;

we have

follows from 1 . 4 t h a t n ~ ki s &ale i f f

=ndE)

i s constant i f f #k)

&t K

is separable. Observe moreover t h a t i f

on t h e o t h e r hand, s i n c e K @ k i s isomorphic t o k phic to t h e syrrunetric group on n letters.

then

E @k

Example: Let K be a f i n i t e extension of k and l e t _ G = &tK.

1.5

1.6

by t r a n s l a t i o n w e in-

GB i; t h e connected component of g

f e r that f o r each closed point g of

is isormrphic to

;

=

, i.e.

i f k contains t h e nth roots of

unity. I n t h i s case

p(k) is a c y c l i c group of order n and each p r i m i t i v e n nth root of unity accordingly defines an iscanorphism of ( Z / n q onto 1.7

If

k=k

S

, each

acts on t h e group G(ks)

.

.

&ale k-group G is constant. I n general, IT

Proposition : The functor _Gt-+G(ks) is an equivalence between t h e category of &tale k-groups and the category of small d i s c r e t e groups on which II acts

continuously (TI - groups").

Proof: Let C be t h e category of &ale k-schemes and C' then the functor X (I

, § 4 ,6.4) , hence

++

X(k ) S

induces an equivalence between

C

and C '

also an equivalence between t h e category of groups in C

onto that of groups in C'

1.8

that of IT-sets;

Proposition:

t h e r e is an &ale k-group

.

Let

_G be a l o c a l l y algebraic k-group.

and a homom3rphism

Then

% :G -+

TJG) that, f o r each &ale k-group H and each hommrphism -2 : 5-2 there is a unique homomorphism

71 (G) 0-

g : T0(G) - -+ g such t h a t

f_=

C J

s, . Moreover gcis

f a i t h f u l l y f l a t and f i n i t e l y presented; its fibres are the i r r e d u c i b l e components of G_ and i t s kernel i s G_"

.

286

11,s 5, no 1

ALGEBRAIC GROUPS

TIJG)

Proof : Let

be the scheme of connected components of G, (I ,5 4 ,6.6) %:G is faithfully f l a t and

W e know that the canonical morphia

.

-+

G,

f i n i t e l y presented and t h a t its fibres-are the connected components of

hence the irreducible components of G ( 1 . 1 ~) .) By 5 1 , 5.1 d) , there is on IT

a unique k-group structure such that

(G)

0-

%

is a group hommrphism. L e t

H_ be an &ale k-group and l e t _f :_G -+ be a hamomorphism. By I, 5 4 , 6.5 there is a unique mrphism g : IT (G) -+ I! such that f = 2% : by the lemma

below,

-

Lemma:-Let R be a mdel, G _ , _ K and H -

1.9

-p : G

0-

g - is a horcmrorphism.

-+

R-group-schemes,

a faithfully f l a t quasicompact homomorphism

K

mrphism of schemes. E n 9 is a hcm-awrphism j.ff Proof: h e way round is obvious. Conversely, i f have

2 g:K+H -

gp

2

is a hmmrphism.

g - -p i s a hmmrphism, we

' H ( g x g ) (gxE) = ITg(9Pxgp)= gpITG - -= -

~

T

I

~

(

'~

x

~

)

Since pxp is faithfully f l a t and quasicompact, it is an epimrphism of schems, so that TI ( g x g ) = g n and 2 is a homomorphism. H - - I ( A s the solution of a universal problem, the pair

1.10

is evidently unique. W e call

If K is an extension of k

IT

0-

, by

.

I, 5 4 , 6 . 7 , we have a canonical ismrphism

If _H is another Locally algebraic k-group,

,

(TAG),

(G) the group of connected camponents o f _G

we have a canonical isomorphism

( 1 § 4 r 6.11)

Since the connected components of G_ are algebraic over k (1.1), G is

2-

yebraic over k i f f the set of connected components i s f i n i t e , i.e. i f the k-group

TI

0

(G_)

is finite.

11,s 5, no 2

287

LOZALLY ALGEBRAIC GROUPS

Section 2

Smooth groups

2.1

Smoothness theorem f o r groups over a f i e l d :

Let G be a

local-

l y algebraic k-group. Then the following conditions are equivalent: (i)

G_ is smooth.

(ii)

go

(iii)

G

(iv)

The completion ring

series (v) duced (vi)

.

is smooth. is smooth a t e

k [ [X1

,...,Xn1 I

.

. A

0e

There i s a perfect f i e l d

[Lie(_G) : K ] = dimG

is isomorphic t o an algebra of formal parer such that the ring

KE$

de @k

K

is re-

.

( v i i ) For each R E $ such that [ R : k ] < + m and each ideal I 2 that I = 0 , the homomorphism G ( R ) 4 G(R/I) is surjective.

of

R

proof : (i)=> (ii)=> (iii): t r i v i a l .

(iii) => (iv) : cf. 1 ,§4 , 4 . 2 . (iv) => (v) : t r i v i a l .

-

(v) => (i): assume (v), and l e t Since @k

is separable, the ring

Since it is enough t o show that

be an algebraic closure of K

K €I&

de 8-k

S. %

K

.

= ( l5’ @ K ) BK I? is reduced. e k is smooth (I ,§4 , 4.1) , we my as-

sume that k i s algebraically closed and &e reduced. By translation, G_ is

then reduced a t a l l of i t s closed p i n t s By I

, § 4 ,4.12,

there i s a dense open

v

( 5 1, 3

in

. 3 ~ ) and ) ~ so is reduced.

G which is smooth over

G is the union of the smooth open subschemes

y(g)_U for

g E G(k)

k

. But

, and

therefore smooth over k . (iii)<=> (vi) : this follows from the equalities dimG = Kdim

[Lie(G_): k l = [ m e / m e

L

:kl

,

from the definition in I ,8 4 ,4 . 1 and from I I (i)<=> ( v i i ) : cf. 1 , 5 4 , 5 . 1 1 .

4 ,2.2

.

de and

is

288

11,§ 5, no 2

ALGEBRAIC GROUPS

f:G-+g is a f l a t hommrphism of l o c a l l y

Corollary: f I

2.2

algebraic k-groups, and i f G is m t h , then so is H. Proof : Since f @I But

d H f e -9+ Gre

(by (v? of 2.1)

i s f l a t f we m y assume that k is algebraically closed.

k

is injective, so that

.

Corollaq : Let

2.3 perfect, then

d-H r e

is reduced and H_ is m t h

be a locally algebraic k-group.

sred is a m t h subqroup of G .

proof: Since k is perfect,

the mrphism %ed

Ired

-

GredxGred

-

G x G

If. k is

is reduced ( I ,§2 ,6.14)

so that

71

G + G _

factors through G similarly, the composite mrphism -red ’ U-

G

sed

G -

. It follows that

factors through GTd

Gred

is a subgroup of

G ; condition

(v) of the theorem implies that it is smooth. 2.4 sion 0 .

Let G i s perfect, 5 is

Corollary:

If

k

Proof : Since

%

‘ied by 1.4

be a locally algebraic k-group of dimen-

the semidirect product of

gred

b i Go

.

is swdzh, connected, and 0-dimensional, it

. The camposite mrphism

-+G_--+T (G) is G red 0 accordingly a mnmrphism. But, by 1.8, it is faithfully f l a t and quasicom-

is identical with

pact: hence it is an isomorphism, which conrpletes the proof (§1,3.10)

.

0-dimnsional connected groups, i.e. those which are spectra of local k-algebras of f i n i t e rank, are called infinitesimal (54,6.1)

Each 0-dimensional

locally algebraic group is accordin-the

.

semidirect

product of an &ale group by an infinitesimal group, provided the base f i e l d

is perfect. 2.5

Remark: W e show in 5 6

s e c t . 1 that i f k is of characteris-

t i c 0 , each locally algebraic k-group is mth.

11,s 5 , no 2

289

L K A L L Y ALG%BRAIC GROUPS

Examples: The group a is mth:it is entertaining t o verik fy this by means of condition ( v i i ) . If V is a f i n i t e dimensional k-vector 2.6

space, the group g ( V ) is smooth; t h i s m y also be verified directly from (v ii). If

r

i s a f i n i t e l y generated commutative group, the diagonalizable

g ( r )k i s

algebraic group

r

smooth i f f the torsion of

and the characteristic

of k a r e relatively prime (a vacuous condition i f k is of characteristic 0 ) . To see t h i s , apply condition ( v i ) : the dimension of

p('r)k is isomorphic t o

Lr ( r , k ) .

over Q

, while uk

In particular,

"'rk

is the Kmll di-

the L i e algebra of

is m t h .

W e now give a p a r t i a l generalization of theorem 2.1. Let A E B ,

2.7

and consider the A-group

proposition: P1,..

r

k [ r l , i.e. the rank of

mension of

Let G

.,Pr E A[X.1 7. I

GL -nA

which we identify w i t h a n o p subset of

be a closed sub-group-schm

be such that, for each

G(R) = { ( x i j ) E % ( R )

: P

REgA

( x . .) = 1 11

of &lG

, we

i

1 s

have

... = Pr(x.1 3. ) = 0 1 .

Suppose that for each s E QA , the L i e algebra of the K ( s )-qroup 2 has rank n - r K(s) Th% G_ i s smooth over A Proof : In virtue

.

.

over

0%

I , 5 4 , 4.2 it is enough t o show that for each pint

nAArX,

of G I the dP.(x) a r e linearly independent elements of 1

For t h i s purpose we m y assum that and that x

is rational over A . Since R

( I, 5 4

has rank

13 i s a f i e l d (replace A- by

A

the A-vector space with basis the dX dPi(x)

,

n2 - r

G @A K(S)

ij

G/A

x

@A K(X) K(S)

.

)

(x) is the quotient space of

by the subspace generated by the

def. 2.10 and 2.6 b) ) , it i s enough t o show that R G , A ( ~ )

. By translation

to verify that R

G/A

(

x being rational over A ) , it i s enough

2 (e) has rank n - r

. But

RG/A(e)

vector space Lie(G) which w a s assumed t o have rank

n

2

is the dual of the

-r

.

This proposition enables us t o show t h a t the classical groups are m t h over tion

a . Take f o r example

G=cLna

defined by the single equa-

d e t ( X . . )= l ; since for each f i e l d k the L i e algebra of

mension n2

( .§ 4

, 4.12 ) , an.is

s m x ~ t hover 72

.

zL*

has di-

Au=EBRAIc m u P s

290

Smoothness of centralizers theorem:

2.8

algebraic k-group, ___

_H _a

G-

Kt

k-group and f:_H-AutGrG

is a natural linear representation H

11,s 5 , no 2

p : E-+

( ~ iGe)

G be a smoth locally

a hormmorphism; there

. ~fH1 (

, L i e G_) =

o ,

swath locally algebraic k-group.

Proof : By 0 1 , 3.6 and 3.8,

G-H is a c b s e d sukgroup of G , and hence a

locally algebraic k-group. To establish its smoothness, by 2.1 it i s enough t o show that i f

has f i n i t e rank over k and i f I is an ideal of A

A€,%

of vanishing square, then the h o m r p h i s m Define the k-group-functors

By 5 4 , 3 . 5 ,

G and -1

G2

g(A)

-+

H G-( A / I )

i s surjective.

by

there is an exact sequence

where

-

has a description similar t o t h a t of the isomrphism R)

M & ~ c~ ./(-

of 5 4 , 3 . 3 .

,

induces autmrphisms of G (R) and g(hLBR -1 G2(R) and f ( h ) induces an automrphism of Lie(G) @ R , hence an automrIf

hEg(R)

L i e ( G ) (R)

.

phisn o f L i e ( _ G ) @ I @ R I t follows irnnediately from the definitions that the hommrphisms u ( R ) and y(R) of the sequence (*) are compatible with the actions of

H(R)

tors acted upon by

Now apply 0 3

sets :

,1 . 4

. Accordingly we get an exact sequence of k-group-func-

g:

t o this s i i h t i o n ; we get an “exact sequence” of pointed

11,

5

5 , no 3

IccALLY

0 H H0 (H,_G 1) = G - ( A )

What are these various terms? By 5 1 , 3 . 5 , 0 H HO(H,G2) is the set of elements of G-(A/I) of

. But,

G(A)

so that

HiO ( g , ( L i e ( G ) @ I ) a ) 2 H:(_H,Lie(Gla)

-

1-

0 H (g,Lie(G))81

1

So i f

I over k , we have

- &A)

H (E,Lie(G)) = 0 ,

81

-

Hence we get the "exact sequence":

H

H

~-(A/I)

-

H

G-(A)

while

is surjective (2.1),

. Moreover, we have

[ f o r i f n is the rank of

,

which are images of e l m n t s

since _G is Smooth, G(A)---*G_(A/I)

0 H ~ ( _ H , G =~ )G'(A/I)

291

ALGEBRAIC GROUPS

ap

( L i e ( G ) @ I ) =- ( L i e ( 5 )

a

1.

.

1

H (g,Lie(G))@I

is surjective, and this is what

+ _G-(A/I)

we had t o prove. Remark: The above proof still works when H_ is a k-mnoidH

G-

functor, provided one already knows that

i s a locally algebraic

k-she.

Section 3

Orbits

3.1

Proposition : g t

k-groupfunctor,

algebraic k-schemes acted upon by which is ccanpatible with the (a) I_f

g is non-empty,

then f

is faithfully f l a t .

(b)

If G(k)

subset of

u.

3

f ( X ) r e d 4 _U

g(c)

(a) Since

, where

g(E) #

g -+

, then

be a mrphi&

x(E) ,

f(X) is a locally closed

c(E) acts transitively on

2% , t h z

f factors

the f i r s t mrphim is faithfully f l a t and the

second i s an embeddinq. F u r t h e m r e Proof:

f:

_U is reduced and Cj(c) acts transitively on

a c t s transitively on X($

(c) r f _X is reduced and _X

5,and l e t actions of g.

k t 5 and g

f(X)red

i s stable under

G.

0 ( I , § 3 , 6 . 9 ) and G(k) acts transitively on

is surjective and so therefore is f ( I , 5 3 , 6.11). By I , 5 3

I

x(c),

3.6,

11, § 5, no 3

AiXBRAIC GROUPS

292

induces a f l a t mr-

there is a non-empty open subschm _V of Y_ such that phisn of f by

-

X Bk k

.

-1(g) into _Y To show that f is f l a t , we my replace

-

-

, Y_ Bk k ,...,u mk k , and hence assume that

k i s algebraically

closed. Then G(k) acts transitively on _Y(k) and G(k) # @

X ,_Y ,..,_V

a;the

translates

for g EG_(k) accordingly cover Y_. It follows that f is f l a t over

-

(b) I n virtue of I ,5 3 ,3.11, the projection morphism k(y g k k ) 3 U_ is open and surjective. To show that f(g i s locally closed, it is enough t o

lu

Bk I , i n other mrds ( f BkE) (X gkE) , i s locally closed. Accordingly we m y assme k to be algebraically closed.

prove that its inverse imge i n

,

By I, 5 3 3.9,

f(g) i s

; hence there is a subset U of _f@) hi&is open and dense i n f(5) . Thus C-’(U) is open i n 1x1 . Since the translates 4 Z - b = f-1(gu) , gEG(k) cover 1x1 ( U being

a constructible subset of

r

non-empty because 3 is non-empty),

g(XJ

is the union of the gg

, hence

open in i t s closure. (c) Let

x=

~

f be the closed image of f and l e t & =

subscheme of T carried by

f(x) . By

zR of

, the

P (R) =

fR(gR) be the set of p i n t s of

yR

be the open

(a) it i s enough t o show that, for each

RE$

subscheme

f(g),,

is stable under G ( R )

and 1,52,6.11, w e then have

. For t h i s ,

zR . Using the notation of

let I ,5 1 ,4.10

( I ,5 2 ,6.15) and it is clear that the l a s t expression i s stable under G(R) Proposition: Let G be a snooth locally algebraic k-group

3.2

actinq on a locally algebraic k-scheme (a)

Let

5.

be a reduced closed subschene of

i f f - Y ( E ) is stable under -

3 . Then

Y is stable under

5

g(c) ,

I f -Y i s a stable subscheme of (b) stable under G _ .

5, then

(c) Each non-empty stable subscheme of

-

(1x1- iy[)red

5 of minimum dimension is closed.

.

11, § 5 , no 4 Proof:

UXALLY

(a) By 1 , 5 4 , 6 . 3 applied t o the projection

p : C_x g duced. A mrphism through

x(c) .

(b) Identifying

-

I

-

x(E)

(c)

X thus

z(E) w i t h the set of

factors through

5 Bk

fixed and induces a continuous autmrphism of

have dim(I_YI-I_Y/)red< dimz 3.3

Let

( I ,53

x(c) i n &(c). Since each

is a non-empty stable subscheme of

, whence

by (b)

Xof

is re-

? i f f -p(E) factors

closed points of

closure of

-

, cxU_

GxY-

fixed; (a) now applies. Similar reasoning works

red

(c) If

-+

lglred(E) i s the

and 6.81, leaves

293

AI;GEBRAIc GROUPS

, 6.6

gE_G(E)

_X(E), it leaves for (1 I - 1 I)red .

minhum dimension, we

1x1 -/XI=@

and _Y is closed.

G be a reduced alq&raic k-group and let X be an alge-

G.

braic scheme acted upon by f : G + g defined by

, consider the mrphism , RE$ . W e m y now apply

Given xE&(k)

f ( g ) = gx for

gEG_(R)

3 . l ( c ) t o t h i s mrphism: this explains why we shall call the subscheme f(G)&

of

X

the & t

of

X. G

Proposition: Suppose t h a t k i s algebraically closed, and l e t algebraic k-group acting on a non-empty algebraic k - s c h m

a pint

be a s m o ~ t h

X . Then there is

x E s ( k ) with a closed orbit. be a non-empty stable subscheme of

Proof : Let

of

g ; accordingly

of minimum dimension.

xEY_(k) , the o r b i t of x is

By 3.2 t h i s subschm is closed. Moreover, i f

a stable subsch&

X

it coincides with

and is therefore closed.

(3.2 (b)) ,

Y

-red

Section 4

The group of rational points over an algebraically closed f i e l d

4.1

Suppose that k is algebraically closed. For each locally al-

gebraic k-scheme

5 3,6.6

X

identify X(k) w i t h the set of closed points of

and 6.8). W e know ( I ,§ 3,6.9) that

A

++

AnX(k)

5

(I,

is a bijection of

the family of closed (resp. open, locally closed, irreducible, constructible) subsets of & onto the corresponding family of subsets of Z(k) k-qroup-functor acting on X and

is a reduced subscheme of

. If

X, we

G_

is a

imme-

diately obtain

Norm (Y) (k) G-

= Norm

-G(k)-

(Y(k)) -

,

C e n t ( Y ) (k) = Cent

G-

---G(k)

(Y(k)) -

.

--

In particular, i f

Nom (Y) (resp. -

is closed (resp. i f

wtG(!J) ) is a closed -

X

is separated), we know t h a t

subfunctor of G . I f , i n addition, G

is a reduced k-group, it follows that it normalizes Y_ (resp. centralizes Y_) (resp. centralizes g(k) ) .

i f f G(k) n o m l i z e s Y(k)

Now i f k i s arbitrary closed subgroup of

G , we

5

i s a smooth k-group, and

g a smooth

m y apply the above results to the reduced

G 8 j; and H Bk I? , thus obtaining Norm (H)(;) = Norm - (€j (I?) ) k G*-(k) and Cent (H)(k) = Cent - (H(E)) In particular _H i s normal (resp. central)

k-groups

G-

.

__ G(k) -

i f f G_ i s normal (resp. central) in iff

G(E)

G(E) . For

example, G_ is comnutative

is comtative. If 8 and U_ are locally algebraic k-schemes, the product top-

4.2

logy of X(k) x x ( k ) is not in general the topology of

(XxY)(k)

(for in-

stance, i f G i s a locally algebraic k-group, G(k) i s not i n general a t o p logical group). However, i f A (resp. B ) is a subset of X(k)

-

AXE = AxB

-X x _ U

: To prove this, notice

that, i f

wfiich sends y onto (a,y) for yEY(R)

m p of Y(k) into

(g x X ) ( k )

- -

. I t follows that

aEA

,

have

, the mrphism of

R E 4 ax

, we

C

-

hto

induces a continuous

a x B . Hence

a x B c A x B , s o t h a t A x B ~ A x B ~ A x Thereverseinclusion B . a is obvious. Accordingly, for each mrphism f : X x y -+ , we have

A x B C U

g A x B ) cf(Ax1

z

.

Proposition: Suppose t h a t k is alqebraically closed and let

4.3

G be a locally algebraic k-group.

(a) The map

5 cf g(k) is a

bijection of the set of open (resp. closed re-

duced) subqroups of G onto the set of open (resp. closed) subqroups of G(k) (b) Lf A E d B Kre constructible (resp. irreducible, resp. dense construct i b l e ) subsets of g ( k )

=el

resp.

, then

A.B

is constructible (resp. A.B

i s irredu-

A.B = G(k) ) .

(c) The closure of a subgroup of G(k) i s a sulqroup of G(k)

. Moreover,

each constructible subgroup of g ( k ) is closed. Proof:

(a) Clearly, i f

H i s an open (resp. closed reduced) subgroup of

.

G_,

then H_(k) is an open (resp. closed) subgroup of G_(k) Conversely, i f L is

an open (resp. closed) subgroup of G(k)

, let g

be the open (resp. closed

.

11, 9 5, no 4

I E A L L Y Azx;EBRAIC GRarps

295

reduced) subscheme of g whose space of pints is the open (resp. closed)

-

-

subset L ' of _G such that L' n g k ) = L

Hxg

It is irrunediate that the mrphim

GxCj

factors through € asI does , the mrphism

Hand so

€J is

"G

G

~

indeed a subgroup of G_.

(b) Let A and B be constructible subsets of g(k) and let A' and B ' be the corresponding constructible subsets of G _ . The subset C ' of G x G de-1 (A') " 2 r 1

-1

(B')

is constructible, hence also the subset T ( C ' ) of G But obviously we have 71 (C') n G(k) = A.B The irreducibility G G assertion is proved similarly (4.2 and I , 5 4 ,4.11) Finally, if A and B fined by

C' =

.

Er

.

.

are constructible and dense, A' and B ' are constructible and dense in IC_[ and hence contain dense open subsets V and W of U = VOW

, then

U(k) U(k) = G(k) (1.2)

;'

x

0

.

is a schem-automorphisn of G_ ,

=

.

--I-= ; (H) - H Since i.fi = Finally, if H is a constructible sukqroup G_

is an autmrphim of G(k) and so

.

(1,53,3.2). But if

that A.B = G(k)

SO

(c) Let H be a subgroup of G(k) ; since

lC_l

(4.2)

is a subgroup of G(k)

of G(k)

let H' be the closed reduced sukqroup of G_ such that g'(k) =

(

H' exists by (a)) . Applying (b) to g' , we get H'(k) = fi

=

i?

H.H = H , so

t h a t H is closed.

4.4

Lemma : Let R E d

p :R

S be d e l s and let

+

S be a m r -

phisn which makes S a faithfully flat R-module. Let G be an P.-group-schem and let

be a subscheme of _G. Then H is a subgroup of G iff H m R S

a subsroup of G mR S

.

-

g

Proof: Consider the cowsite mrphim f:ExE

Then f factors through

71

G_xc

+

G.

g iff the canonical embedding

imrphism, and by I,5 2 ,3.5 this holds iff E-'(g)

(l(_H)

gRS

3

-+

EXH_

is an

(g x 5)gRS is

an isomorphism. A similar argument applies to the composite mrphism

296

FLGEBRAIC GROUPS

4.5

cormnutative, smooth, s d rated by x

.

Proof: Let

f: 1 k

image of

LakE

EnG(E)

H(c)

_G be

5, no 4

.

:(I) = x . Let

be

EJBkG is the closed

( 1 , § 2 , 6 . 1 1 ) : by 1 , § 2 , 6 . 1 6 ,

and it is carried by the closure E of the image of lg@k K l -

f(E) : 'Z

is the closure of the image of

B'(g) , where K' is

is then of the form

G_BkE. Hence _HBkK =

g'

an extension of k

I!

. It follows t h a t

g

is a subgroup of G_ ( 4 . 4 ) ;

E(E) is c m t a t i v e .

the closed subgroup of _G generated by x . I f K is

the subgroup of

I

G(G) , hence is the

a closed reduced subgroup of

by 2 . 1 it i s m t h and by 4 . 1 conmutative since

We c a l l

4

generated by x . By 4.3 (a) and (b), E n s ( ; )

closure of the subgroup of G(j;)

of x in

e-

is the closure of the subgroup of G(E)

G_ be the hommrphism such that

-t

the closed image of f But

5

an algebraic k-group and l e t x c G ( k ) . be the smallest closed subgroup of G for which x E g ( k ) Then _H Proposition :

Let fl

Kt

11,

( G B k K ) ( K ) = G(K)

G @ k K generated by the canonical image

is H B k K .

Example : If x E G_(k) has order n < + m

,g

is isomorphic t o

the constant group (z/nz) k ' Proposition: Suppose that k i s algebraically closed and l e t

4.6

G_ be an algebraic k-group. Let

be a family of constructible and

(Ai).€ I

irreducible subsets of G(k) containing the unit element. s t H be the subgroup of G(k) generated by the A i .

H is closed and irreducible.

Wreover, there is a sequence B1,

nS2dim G , of subsets of G(k)

chosen f r m the A

i-

and the Ai

-1

...'B n '

I

,

such that

H = R B

... Bn .

Proof : Clearly w e may assume that each A-' is a member of the family (A?. i Consider the collection of subsets of G(k) of the form AilA i2... AiP I as ranges through the s e t of f i n i t e sequences of elements of I . P Each subset of this fonn i s an irreducible closed subset of G(k) (4.3 (b) 1. (i1,...'i)

If

A jl A j 2 . . .

Ajs

i s maximal ( q 5 n )

, then

the inclusion

A.B c A.B

(4.2)

Aj, = A jl...

Ah

implies Aj,

for a l l

il:

... A j q . A i l...

...,iP .

A$

=

Ail

... A i p . A j

11,

5

5 , no 4

i s a subgroup of G(k) containing H . Thus we

In particular, have

297

LCCALLY ALGEBRAIC GROUPS

, so that,

C A

J,**-

group of G defined by

i= This inplies that

H =

i

by 4.3 (b) applied t o the closed reduced sub-

(4.3 (a) ) , we get

Ajl.

... Ajs

i , and

also yields the f i n a l assertion.

Corollary: Suppose that k i s algebraically closed and l e t

4.7

be a locally algebraic k-group.

Let

A

B be closed subgroups of G(k)

where B is irreducible. Then the group of cornrmtators ( A , B )

irreducible. Moreover,

if

n = dim G

, each

element of

( A , B)

5 ,

is closed and is the pro-

duct of a t most 2n c o m t a t o r s .

l e t B be the image of B under the map a b c, a b ab Since B i s the set of rational p i n t s of the image of a a mrphism B -+ 5 , where g i s the closed reduced subgroup of G such that Proof : For each aEA

-1 -1

g(k) =

,

B

Ba

.

i s a constructible and irreducible subset of

Go(k) which

contains the ,unit element. The proof i s completed by applying 4.6 family

to the

(Ba)aEA * Existence theorem for the derived group: L e t G be a smo~th

4.8

Let B(G) be the derived group of _G , i.e. the subfuncdefined as follows: f o r each R E & , B(G)(R) is the set of

alqebraic k-qroup. tor _ _ of G g E G(R)

for which there is S €&

, faithfully

f l a t and f i n i t e l y presented

over R , such t h a t gs belongs t o the group of commutators of G_(S)

(a) $(G)

is a smooth closed subgroup of G

, which

i s connected i f

. Then

5 is

connected; (b) f o r each algebraically closed extension K of cornrmtators of G(k)

.

of

k , a ( G ) ( K ) is the yroup

Proof: F i r s t consider the group g(k) and its group of commutators H : w e

/ (G(E) ,Go&) ) is f i n i t e . Let K = G ( c )/ ( G ( c ),Go(E) 1 : then since the image of Go(E) in X is central, the centre of K i s of f i n i t e index show that

H

in K . By a classical result in group theory*, it follows t h a t the group of

*

See f o r example B. HUPPERT, Endliche Gruppen I, chap. IV I § 2 Springer-Verlag, 1967.

, Satz

2.3,

ALGEBRAIC GROWS

298

IIr

5

5, no 4

COnUtutators of K is f i n i t e , But this group of commutators is precisely

G(E) ,Go(E))

H /(

. By 4.7,

(

Cj(E) ,Go(E) )

G(k) ; moreover, i f each element of

and hence H are closed subgroups of

g(E) ,Go(E) ) i s the product of a t

H /(

most q c m t a t o r s of K , and i f n i s the dimension of _G

, then

each ele-

ment of H i s the product of a t most N = ( 2n+ q) c o m t a t o r s . 2N Now we consider the mrphism g:_G -+ _G -1 -1 have g ( x ty I ...,%,yN) = x1 1 1 y1 x1y1

such that f o r each RErn% we 5-1yN-1%yN r and let D be its closed image ( 1 , § 2 , 6 . 1 1 ) . Ey I , § 2 , 6 . 1 S r U @ is the closed bmge

...

of

; it

_u Bk

E n G (g) = H

D is

Bk El

. Since

is the closed reduced sub-

by the preceding remarks,

G @k

group of 2.1,

k i s carried by the closure E of the image of

U cBk whose set of rational pints is H (4.3 (a))

therefore a smooth sulqroup of G

, which

. By 4.4

and

is connected i f G is

connected. W e now show that f

:G2N

4

Q

D = U (G)

. To do t h i s , notice f i r s t that the mrphisn

induced by g is dominant. Hence, by I , § 3 , 3 . 6 ,

dense open subfunctor of G2N such that D

. Accordingly,

fly

is f l a t and

the composite mrphism

"

-

flu

x

flu

~

'

71

EX!?

is flat ( § I r3 . 2 ) and surjective (1.2). If

there i s a

f(g) is

dense in

* D

g€g(R)

, consider

the induced

C a r t e s i a n square

If S

(vi)

i s a f i n i t e affine open covering of _V

, and

if

S = Il

, then

is faithfully f l a t and f i n i t e l y presented over R and we have a c o m t a -

t i v e diaqram

I SO

!

g#

.

t h a t gs belongs t o the group of c o m t a t o r s of G ( S ) and gED(G)(R)

11,

9

5, no 5

X C A I L Y ALGEBRAIC GROUPS

299

Conversely, since the mrphism of G x G into G h i c h sends (x,y) onto -1 -1 x y xy factors through r it is plain that the group of commutators of G(R)

is contained in Q(R)

. If

gEG_(R) and i f gs belongs to the group of

commutators of g(S) , hence to Q ( S ) 9 E _D(R)

,

S being faithfully f l a t over R , then

by I,5 2 ,3.6 applied to the c m t a t i v e diagrm

This proves ( a ) . To prove (b) , remark that, f o r each

S E& -i

rated over K , by the Nullstellensatz there is cpE_i&(S,K) tion of the theorem, of 4.9

One proves similarly the

Proposition:

Let

subgroups o_f _G

be an algebraic k-group, - H and

r h -e-r e H

,

% , faithfully

gs E (g (S) ,g(S ) )

,..is a

X

tw smooth closed

is connected. Then the subfunctor

such that f o r each RE$ there is S E

; with the nota-

g = cp(gs) then belongs to the group of c m t a t o r s

.

rp(g(S))

f i n i t e l y gene-

(_H,_K)(R) is the s e t of

(_H,K_)

of G

g€G(R) f o r which

f l a t and f i n i t e l y presented over R , such that

smooth, connected and closed sukqroup of G

Section 5

Hamomorphisms of algebraic groups

5.1

Proposition:

Let

f:G_

4

.

be a hammrphisn of algebraic

k-groups.

(a) The h q e dim f(_G)

f(G) of -2

is a closed subset of _H

, and we have

= dimG-dimgrf:.

(b) L f _f is a mnomrphism (i.e.

if

K sf =

) , then f

is a closed em-

bedding. Ho _ is (c) If g is reduced and if f i s surjective (resp. and i f f o : Go-+ surjective) then f is faithfully f l a t (resp. __ flat).

(d)

If G

is reduced (resp. smooth), then

f ( Q e d is a closed reduced

g , and f

(resp. smooth) subqroup of

-_ the f i r s t m r p h i m _ i s faithfully

Proof :

factors into G 4 f(G)red-+_H where

f l a t and the second is a closed embeddinq.

(a) L e t _G a c t on _G and H by translations ( gEG(R) sends g ' E G ( R )

onto g q ' Since

11, § 5, I10 5

ALmE3wc GROUPS

300

f(G(G)) is

dingly

f(G)

and h E H ( R ) onto _f(g)h_ 1. By 3.1 ( b ) ,

f(c)

, it

a subgroup of H(E)

is locally closed.

i s closed (4.3 ( c ) ) , and accor-

e defer the proof of the second part i s closed by I ,§ 3,6.11. W

of (a) u n t i l a f t e r we have proved

(a).

(b) W e may assume that k i s algebraically closed ( 1 , § 2 , 7 . 3 ) . By I , § 3 , 4 . 7 ,

there is dense open subscheme V_ of _H such that

gl-'(Y)

is an embedding.

By translation f is an embedding and by (a) it i s closed. (c) If f i s surjective, it is faithfully f l a t by 3.1 (a)

. If

f" : Go-+ €lo

i s surjective, it is f l a t by the preceding remark, hence, by translation, f is f l a t (by I , § 2 , 3 . 2 rem. we my assume that k is algebraically closed).

-f(G)red

(d) Apply 3.1 (c) ; we simply show that the f o m r is obviously stable under

G

G

, it

is a subqroup of G . Since

is sufficient t o show that it

-

i s stable under the product. N o w w e have the c m t a t i v e diagram

f (GI red x f (GIred

i x

&

-

where _pxp - i s faithfully f l a t and where 2,3.6,

1%

H H_

& is an embeaaing. Applying I ,

we obtain the required conclusion. If G is smooth, so i s

_f (GIred

by 2 . 2 . Finally, w e take up the last assertion of (a). W e may replace k by

( I,

§3,6.2) and hence assume that k is algebraically closed. Then G and -red are subgroups of G_ and _H (2.3) and by (d) we have a faithfully f l a t 'red mrphism Gred -C g(GIred h o s e kernel is ( g r $ 1 By I , 5 3,6.3,

mred .

wehave

5.2

d i m f _ ( Q = d i m G -dim(&rf)nGred=dimG_ - d i m K e r f . -red Corollary:

Let 5

conditions are equivalent: (i) G

is affine;

be an algebraic k-group. Then the following

11,

5

5, no 5

301

JAXXLY AU;EBRAIC GROUPS p :g -+

(ii) there is a faithful linear representation

GL(V)

f

G

f i n i t e dimensional k-vector space. (Recall t h a t p is f a i t h f u l i f it is a mnmrphism. ) Proof: By 5.1 (b) and 5 2 , 3:4. Proposition : L e t _f : G -+

5.3

k-groups. Suppose that

-

are equivalent, and they imply that (i) L&(_f) (ii) K e r _E

: -(G)

g be a

G i s smooth over is

sllooth :

i s surjective;

Lie(@

i s smooth and

g

homomorphism of algebraic

k . Then the following conditions

f(G)red

H;

is open in

is snooth.

(iii) f

Proof: I& K=@r

f , g = Lie(G_) , ,h= L i e @ )

.

A .,

, &=

Lie@)

'p=

Lie(?)

.

By

5.1, we have dimG =dimg + dimf_(G) Forewer, evidently [&:k]+[cp(c$ :k] =

-

[g:kl ; finally, since _G is m t h , we have dims = [g:kl I

. It follows inme-

diately t h a t

I f (i)holds, the left-hand side of the above equation is zero, hence so also a r e the three expressions on the right-hand side. This implies that and H_ are smooth'and t h a t

.

2 (Cj)red is open in

Now assume (ii); we derive (iii) Since

g is

(5.1),

K_

f(G) red is open i n H I and (ii)follows. H_ and smxth

smooth (2.1) and f i s f l a t (5.1). To show t h a t f i s smooth,

by extension of the base f i e l d w e may confine ourselves t o the case in which k

is algebraically closed ( I

, 4 ,4 . 1 ) .

Since the set of p i n t s of _G a t

f i s s m o o t h i s o p e n ( I , § 4 , 4 . 3 ) , i t i s e n o u g h t o s h o w t h a t f is which -

smooth a t each rational point gEG_(k) -1

fy that

. For this purpose we need only veri-

( f ( 9 ) ) i s smooth, and t h i s i s the case because the l e f t trans-1 Kerf onto f ( f ( g ))

f_

lation y(q) :C, --+ G_ induces an ismrphism of

Finally, (iii)implies (i)by I , 5 4 r 4 . 1 4 or 4.15. 5.4

m* -

Example: Take for = G ( M(k) )

gEGAR)

,

n

RE$

(

5 1,2.6

, the

)

5 the

,and

group

g* , f o r g

the group

for f the m r p h i m which assigns t o

automorphim m

-+i

qrq-l of Mn(R)

. The L i e algebra

of

may be identified with Mn(k)

,2 . 2 )

( §4

Der ( M ( k ) ) of derivations o f n the m p L i e ( f ) If xE Lie@) , w e have the algebra

.

e

EX

--Ex

me

it follows that Lie(fJ

m

xm-nor

=

. Let us compute

5 4, 2.3)

:

sends x E &In(k) onto the inner derivation (cf. Cartan and Eilenberg, Homological

Lie(f) is then surjective. Applying 5.3,

Algebra, chap. I X , 5.1 and 7.8),

m&

(

( l + E x ) r n ( l - ~ x )= m + E ( x m - m x )

. By a classical result

we infer that

Bd w i t h

and that of

Mn(k)

is srrmth. Moreover, we know ( Alg. VIII, § 1 0 , no. 1 ,

are of the form r nb gmg-l. Kerf_ is a smooth subqroup of GL by 5.3 : since the centre of Mn(C) consists of scalars --nk only, ( g r consists of h m t h e t i c maps. It follows that K e r f is the skgroup D_ of CLA , i s m r p h i c t o pk , such t h a t , €or each R E +M , D ( R ) i s the set of homthetic maps.

cor. t o th. 1) t h a t the I;-autmrphisms of Mn&)

I t follows f r m 5.1 that f is faithfully f l a t . Finally,

5.5

Corollary : Let ~

k-groups. SuEpose that (a) L i e ( ? )

5

be a hcmmrphism of algebraic

is smooth over k

is bijective i f f

is an open embedding

(b) f

g

f : C,

g f iff

f

. Then

f(G)red

I s B t a l e and

is open in H_:

is a mnmrphism and L i e @ ) is bi-

jective.

Proof :

(a) By 5.3, L i e @ ) is bijective i f f

is open and Ker f

is

msoth and has zero L i e algebra, i.e. i s &ale ( 1 . 4 ) . (b) One way round is obvious. Conversely, any m n m r p h i a f. i s a closed

embedding (5.1 (b)) : i f of

E.

-

Lie(f)

The induced morphia G_

is bijective,

f(G)red

f(GIred

is an open subgroup

is a mnmrphism which is also

faithfully f l a t (5.1 ( d ) ) , hence a strict epimrphism (1,52,3.4). It follows that G_

g(G)red

is an ismrphism and

5.

i s an open embedding.

Corollary: Let 2 , b e a smooth subqroup of an algebraic

5.6 k-group

-+

5.

Then - - H is an open subgroup of

g

iff

Lie(H_) = Lie

Proof : This i s j u s t 5.5 (b) applied t o the embedding _H

---t

G_

.

(g).

11,

5

5, no 5

LD3I.J-Y XGEBRAIC GIEOUPS

LemM : L e t G be a locally alqebraic k-group.

5.7

(a) &t

G . L e t H act on L i e ( 5 ) via the adjoint reG. Let E = m (H) , c = C s t (H) (cf. 9 1 , 3.7). Then we

be a su?qroup of

presentation of have (b)

H

rf

p :G

Lie(!)/Lie(g)

G

= (Lie(G)/Lie(E))-

.

is a f i n i t e dimensional linear representation of

a(V)

-+

I.-

G-

,

L i e ( c ) = Lie(G)-

and i f

G,

are two vector subspaces of V , then the L i e algebra of the

W'CW

of

subgroup

G ( 52,1.3) is

consisting of all x E L i e ( G )

the sub-lie algebra Lie(G&,,w of

such that

Lie(p)(x)E L(V)

proof : (a) "he assertion about C follows from

mrphism

-H Let

303

-

G

. "hen

x €Lie@)

Int

x €Lie@

F

---=Aut

(G)

5 4 ,2 . 5

maps W

Lie(G)

.

W'

applied to the ho-

. and each h € !(MI

i f f for each MEm%

I

we have

by 5 1 , 3.5. But t h i s l a t t e r condition m y be written in the form

hence i n the f o q ,

i.e.

%-

@(h)%€ Lie@)@M

. Then

(b) Let

xELie(G_)

W@k(E)

into W ' @ k ( E )

5.8 (a)

, which

implies the required result.

x belongs t o Lie((&,

i.e. i f f

- -

,w )

L i e ( p ) ( x ) raps

iff

e E x - I d maps

into W'

W

.

Corollary: Let H be a smooth subgroup of an algebraic k-group.

If

H

(Lie(G)/Lie(g))-= 0

Proof : L e t

(H)

2

H is an open sukqroup of

is self-normalizing i n L i e @ )

(b) I_f L i e @ )

qroup of

, then

G-

and

(H)

G-

then

is an open subgroup of

and 2' be the L i e algebras of

By 5 . 7 (a) and ( b ) , we have

I

Norm

(H)

G-

NormG(H) . -

is an open subN g G ( L i e ( H ))

.

and N s G ( L i e ( H _ )I -

.

11,

ALGEBRAIC GRDUPS

304

5, 110 5

Q

Now apply 5.6 t o complete the proof. 5.9

Let 2

Corollary:

be an algebraic k-group, Q a conmutative

s w r o u p of G and H a stxmth subgroup of G_ containing C e n t ( Q )

1

H (Q ,L i e @ ) ) = 0

, then

_H i s an open subgroup of

- - -

-

Proof : W e have the exact sequence ( §3 1 . 4 or § 3 , 3 . 2

o

Q

Lie(@

Since _H 3

Cent

(Q)

G-

It follows that

5.10

,

L i e( G) -

Q

Let

-:

C e n t (Q)

G.

-

-G_-

open subgroup of its successor. Proof: W e already know that

H ' ( Q

, hence

1

(g,Lie(g))

L i e ($

. Q

= Lie (G-k

.

.

_G be a smoth algebraic k-group and Q a If

1

H (Q,L i e ( G ) 1 = 0

-G - G -

Norm (Q) and Norm (Cent ( Q ) )

_G-

is affine) :

if H

. Ef

Q = O ; also QCE (Lie(_G)/Lie(H)): now apply 5.8 (a)

closed m t a t i v e subgroup of them

Q

(Lie(G)/Lie(g) 1-

L i e (g) 2 L i e (G)- (5.7 (a))

.

NLT(K) -

G-

-

-

= 0

,

is 91100th (2.8). Moreover, w e have

CSt,(g)

, Lie(cztG(q))=

and G r k ( Q r a J

are mmth, and each is an

H1(B,Lie(G);)0

Q we have accordingly by 5.7 ( a ) . Since Q acts t r i v i a l l y on Lie(G_)-, H1( 8 ,L i e ( C s t (Q))

G-

* H1(Q,k)@Lie(G)Q sz g ( & , % ) @ L i e ( G ) '

Applying corollary 5.9 t o the pair

is an open subgroup of

.

t (Q)) we infer t h a t C e n t ( Q ) GGBut Norm (Q) is contahed be-

tween these two, and this completes the proof. 5.11

Remark : By 5 3 , 4 . 2 and 4.3

*-

, the

above corollary applies i n

particular when Q- is diagonalizable (or, mre generally, when diagonalizable)

.

.

(9,s

Norm ( CentG(Q))

---G----

= 0

9 @k

is

THECHARACTERISTIC 0 CASE

5 6

5

Throughout

6

, we

assume that k i s a f i e l d of characteristic 0 .

Section 1

The enveloping algebra and invariant differential owrators

1.1

Let

phism Lie

G

'L

G be a locally algebraic k-group. The canonical isomor-

Dist'G

I

phism

of 5 4

1

c : U ( L i c G ) -+ where U ( L i e Lie, I

,52

G)

, 6.8

Dist

may be extended t o an algebra homomor-

G

denotes the enveloping algebra of

LieG

.

( G r . et. alg. de

,prop. 1). This homomorphism i s corrpatible with the f i l t r a t i o n s ,

i . e . we have 5 2 , no.6).

c ( U ( L i e G ) c D i s t G_ n n

Cartier's theoren:

Kt 5

for each

nE%J (Gr. e t . alg. de Lie, I

,

he a locally algebraic k-group. Then

a)

G issmth,

S)

the canonical isomorphism c : W(Lie

GI

-

Dist

5

i s bijective,

Proof : Let e be the unit element of _G(k), regarded as a p i n t of

5 4 , 5.4 , we

G

. By

may identify D i s t G w i t h the space of k-linear forms on 8 ne n+l which vanish , hence also as the space of forms on 5 which vanish on m e2 e L e t I be the ideal of the symmetric algebra S (me/me ) generated on fien+' by me/m: , 2(me/m e2 ) the completion of S ( m /m 2 ) in the I-adic topology, 2 e Dn the space of k-linear forms on 2 (me/me ) which vanish on f n + l and

.

Each section

s : m /m

eA

---t

tinuous homomorphism S (me/m:) Dn

, and

m of the canonical projection extends t o a cone -

hence induces mps

C

de

; the transpose map sends

h n : D i s t G_ n

-

below that the composition

U( L i e _G )

-+

h D i s t G_ --.+

---+

D

n

and h : D i s t s

D i s t G into nshow

-+0 . W e

D

2

i s bijective. Since h is injective (for the homomorphism g ( m /m ) -+ be e_e induces a surjection of the graded algebras associated with the I-adic and

fh-adic filtrations; by Alq.com. I11 , $ 2 ,cor. 2 to th.1 it is therefore surjective); it follows that h is bijective, and so also are c and A 2 A (for if this last map is not injective, by Alg. corn. I11 , S(m,/m,) + Oe A n

5 2 , cor. to prop.5, its kernel is not contained in I

for sufficiently . 2 large n , which contradicts the surjectivity of h). Since s(m /m ) is isoe e mrphic to the algebra of formal power series in [me/m: : k ] variables, a) and b) follow. To show that h oc is bijective, we prove that, for each n € P J

ced map :

Un /un-l

-

, the indu-

Dn /Dn-l

(where Un= lir (LieG ) ) is bijective. To see this, choose a basis W~,...,W n d for m /m and a dual basis El,. ,<, for Disti Ncdk ( me/m2 ,k) e e By the Poincare- Birkhoff - Witt theorem ( Gr. et. alq. de Lie I , 5 2 ,no. 7 ) ,

*

..

"n"n-1

.

then has a basis consisting of the residue classes mod Un-l

the pantities 5

1

... sp , where

CI

1

...

+

... +

a=

CI

n

Of

. Similarly, the residue

classes of w 81 d ' with 6 + + Bd = n constitute a basis for 1 '''wd 1 In/In+l . The bijectivity of h c now follows from the canonical isomr0

phism Dn/Dn-l --+Pld,(In/In+l,k) and from 1.2 helow. Lemma: With the notation of 1.1, we have

1.2

provided ui #

Bi for at least one i ,

... 5,ad ) )I, :u ... wdd) c1

( h (~(6:~

=

ell!

ci2!

... old! .

Proof: It is enough to show that, m r e generally, we have

a1, ..., an E mehe'

where xl,...,xn €Lie G ,

and bi = s (ailE me

.

Now we have, by definition (h(c(xl...xn)),al...an) = / bl...bnd(xl*...*x). If

IT^

: GX

...

n

X G -+G

satisfies

IT

n

(sll...,CJn1 =

gl---

gn

for REP&

11, 8 6 , no I

THE CHARACTERISTIC

and giEG_(R) , then by § 4 %

e

e -k + -k

xl* ...* x

, 6.1,

x ...x

n

%i

is the corrpsite deviation

x1 x . . . x x n

Moreover, i f G

i s the leading n e i g h b o u r h d of

section

-+

x

i

- ck E '

E

G'

G

and

-j':sE-+ G i s

307

CASE

0

:TI

G X ...XG_C--G_.

$

with respect t o the u n i t

t h e inclusion mrphism, then each

is a composition of the form

=

/ ( ( bl...b

n

)TI

( j x...x n -

1))d(ylX ... xyn) .

Now, by 1.3 below, we have (bl.. .bn)

... x j- )

vn(j x

= 1

(ai@.

.. @ 1+ ... + 1@ ...@ ai + ci)

sn ( a o ( l ) @ . . . @u (an ) )

= oEC

where

I

c . E M 2 and M is the maximal i d e a l of 1

de/me B

... @

de/me

.

This and t h e equality

Lema:

1.3

for b E m e , s

-

W i t h the notation of 1.1 g@ 1 . 2 , the hommrphism

c=bmdm

2

e

.

308

oe

FIx;EBRAIc

GRMJPS

-...

de/m;

Oehe

%Wk

2

Proof : Let

fin:

by

... x j ) : 5 x G-E x ...x _G

rn(G_x j x

6,(b)

3

:b @ l@ . .@ . 1+1@b@ ...@I+

i d e a l of + b"'

1@ b" C,XG

GX

-E

de@de/me2

, where

'

Gx

G

the composition of

@

...@

-

b"'E me@ me/me2

G

-E

mrphism, we s i m i l a r l y

E

be t h e hommrphism induced

. It

G_

6, m 1

i s enough t o show that where N is t h e maximal

mod N2

. Now, when n = 2 , we have 6 2 (b) = b ' @ 1+ . Since the composition _G G x %-+

Je/me2

'IT2

...

$.C

5

11,

is the i d e n t i t y , we see t h a t b = b'

% x GE - -+ G x G-E w i t h o b t a i n b" = b . The general

. Noting

that

( G X j ) i s the inclusion 2 - case may be i n f e r r e d f r m 'IT

this by induction, using t h e formula 2

6n = ( ( ~ de/me ~ 8@

...@

which follows from t h e f a c t that

5 4,

n

=

2

71

)

0

(

0

n-1

Recall t h a t wjth each element

1.4 (

71

%/me

v2 x

8-l).

x E L i e G_ we have a s s o c i a t e d

4.5) a d e r i v a t i o n 6 ' ( x ) ( r e s p . y ' ( x ) ) on

-

,

6n-1

G which i s l e f t ( r e s p . r i q h t )

t r a n s l a t i o n i n v a r i a n t . This map extends t o a homomorphism (resp. antihomomrphism) of algebras 6' : O ( L i e G ) where

Dif G

Dif

G

( r e s p . y' :U ( L i e

is t h e algebra of d i f f e r e n t i a l operators on Dif G

Corollary : The map 6 ' : U ( 1 , i e C )

G)

--).

5

Dif

I. ( 5 4 ,5.3)

( r e s p . y' : U ( L i e C,)

3

.

I.

Dif

)

)

induces an isomorphism ( r e s p . an anti-isEmrphism) of t h e alqebra U ( L i e G ) onto t h e algebra of d i f f e r e n t i a l operators on

G

riqht)

which are l e f t ( r e s p .

translation invariant. proof : Immediate from 1.1 and 5 4

Since t h e isomrphism c of 1.1 i s compatible with t h e a d j o i n t

1.5

representation of G_ S u p p s e now t h a t U(Lie

G)

G

, it

induces an isomorphism

+( D i s t

G )G- .

G

aT ( L i e

L i e (3 generates t h e algebra Pi(Lie G_)

Corollary : SuFp ' o-t

C,)G-

g ) , a l l of which are f i n i t e n Lie G But ) , we g e t U ( L i e C , I G = U ( L i e 5)

(which is t h e union of t h e

c i s e l y t h e c e n t e r of

U(I,ie

is connected. Applying 2 . 1 ~ ), which is proved below, to

dimensional and stable under since

,6.5.

U(Lie

5)

. Applying

, the

5 4 ,6.7

is connected: a )

following conditions a r e equivalent :

.

right-hand s i d e i s pre-

, we

o b t a i n the

Lef x E U ( L i e

. Then

the

11,

5

6, no 2

309

THECHARACTERISTIC 0 CASE

(i)

y'(x) i s riqht-and-left

(i')

S'(x) i s r i q h t r g d - l e f t translation invarian:.

(ii) y'ix) =

(iii) x

S'(x)

translation invariant.

.

is i n the centre-of- U(Lie 5)

.

6' induce the same isomrphism of the centre of n ( L i e G ) ____

b) The maps y' a@

onto the alqebra of riqht-and-left-translation invariant differential operators on

G.

Section 2

Relationships between groups and L i e algebras

2.1

Proposition:

Let G

be a locally alqebraic k-qroup.

(a) Let _H and X be subqroups of G ; &f _H i s connected, then Lie

H

C

. Lf

Lie g

are connected,

_H

g = K .iff

Lie

g =L i e X

(b) I_f fl and -f2 are hommrphisms of G i n t o a k-qroup G' connected, th31

f =f

(c) I_f p : G -+

G(V)

-1

-2

iff -

Lie f

-1

= Lie

_HCg

, and

.

iff

5.

(d) L e t

g

iff

Liep

a.

be a connected subqroup of G . Then we have*

I f , i n addition,

iff

G,

i s a f i n i t e dimensional linear representation of

and i f G_ i s connected, then a vector subspace of V is stable under G G LieG it is stable under L i e G . W e have V- = V ; mreover, p is s-le (resp. semisimple) iff

&

if G

G

is connected, then H i s normal (resp. central)

L i e fI is an ideal of L i e G (resp. is i n the centre of

L i e G_ )

.

G_

(e) If -G is connected, thh G_ i s c o m t a t i v e i f f L i e G_ is c o m t a t i v e , G is f i n i t e i f f the centre of L i e G is zero. and the c e n t r e of -

- _ - - - - - - - - - - - - - - - - - - - * If g is a Lie algebra and h i s a subalgebra of g , we set

- - _ _

Cent h = ( x E g : [ x , h ] = O ~, Norm h = { x E g : [ x , h l c h ) g

g

.

ALGFJ3WC GRiUPs

310 Prmf : (a)

gcg

11,

&ng=g. Since

i s equivalent t o

t h i s l a s t condition i s equivalent by S 5 ,5.6 t o

. The

i.e. t o L i e _HcLie 5

Lie

=

Ix

, hence

c2 .

(c) Let W be a vector subspace of V

. Now

(Lie G)w,LJ = L i e

By the same argument,

(Lies)

0 ,w

G

W is stable under

G , i.e.

= L i e G_

,

is s m t h by 1.1,

( L i e _H)n(Lieg ) = L i e

&(R) = f g€G(R): f l f q ) = f ( a ) ) -2 -

c ~ i e : ( L i e g$(x) = ( L i e f 2 ) ( x ) }

and f = f i s equivalent t o _K = G -1 -2 finally t o L i e f = L i e -1

by 8 5 , 5 . 7

2

,

second assertion follows immediately.

G such that (b) L e t K be a subgroup of -

Then

5. 6,

,

t o L i e _K = L i e G_ by (a) , hence

. Then we have iff

.

Norm

G

W =

L i e ( m W) = ( L i e G_)

G

G

i f W i s stable under L i e G_

, hence by

.

w ,w

(a) i f f

G

i s equivalent t o Cent W = G , hence t o G Lie G and so t o W c V The second assertion follows W C V-

.

from the f i r s t . (d) Writing _C = C e n t G

-

Lie

N

Lie

NormH , we G-

_N =

c = ( L i e G ) -H

By (c), t h i s gives

argument,

,

,

have by 5 5 , 5.7

(Lie g)/(Lie g) = ( ( L i e G ) / ( L i e

Lie G = (Lie G ) L i e H = Cent -Lie G_ ( L i e = N o G(Lier j )

.

yie

u)

H

E))-

.

; by the same

The last assertion thus follows from ( a ) .

(e) This follows inmediately from

(a).

L e t G_ be a connected al-gebraic k-group. Then the Corollary : -

2.2

following conditions are equivalent:

(i)

The L i e algebra

(ii)

Each normal connected cormnutative subgroup of

(iii)

G

L i e G_

is semisimple.

5

is zero.

has f i n i t e centre, and a l l f i n i t e dimensional linear representa-

tions of G are semisimple.

(iv)

_G

has f i n i t e centre, and the adjoint representation of G in L i e 5

i s semisbple.

(v)

C, has f i n i t e centre, and

G

has a f i n i t e dimensional semisimple li-

near representation whose kernel i s f i n i t e .

5

11,

Proof: then Lie G

THE n

no 2

6,

5

(i)=> (ii): If

Lieg

S

T

I

C 0 W E

3ll

is a normal connected commutative sukgroup of G _ ,

i s a commutative ideal of

Lie G_

, and

i s therefore 0 i f

is semisimple.

(ii) =>

(i): Let

h

be a commutative ideal of

Lie G_

. Then X = (C=t_h)O

i s a connected subgroup of G whose L i e algebra i s Cent . this l a t t e r i s an ideal of

Lies

,

so

5

Lie

G (h)

(

5 5 , 5.7) ;

i s a normal connected commutative

subgroup of G_ whose L i e algebra contains h . I f (ii)holds, we thus have

h=O. (el.

(i) => (iii): By 2 . 1 (c) and

(iii)=> (iv) : Trivial.

(iv) => of

G

(v) : The Lie algebra of the kernel of the adjoint representation

is the kernel of the adjoint representation of

Lie G_

. If g

has f i -

n i t e centre, the adjoint representation of G is therefore f i n i t e , and, i f (iv) holds, the adjoint representation i s s e m i s m l e . (v)

=>

(i):

nel, then

If V is the space of the representation and K_ i s i t s ker-

L i e K = 0 i s the kernel of the associated representation of L i e G

in V . By 2 . 1 (c) and G r . e t alq. de L i e , I , 5 6 , prop. 5 ,

tive. Since G has f i n i t e centre, the centre of Lie G

Proposition: &t

i t s derived group (

if g

and each g € G ( R )

is connected,. we have Lie g ( c ) By the proof of

an open subscheme _U of G_2N

for

smallest vector subspace d

,

Ad(g) - I d

of

=f)(s) be L i e G_

maps ( L i e G ) @ R

d@R;

Proof : a)

9

G be an algebraic k-qroup and l e t

Q ( G ) is the

such t h a t , for each REM%

b)

i s zero. Hence

5 5 , 4.8). Then

the Lie algebra of

into

i s reduc-

is semisimple.

2.3

a)

LieG

Lie C,

5 5 ,4.8 we

= [Lie

G,Lie

a .

my choose a natural number N and

i n such a way t h a t the mrphism f of

g

into

satisfying

(gl,...,l-Q€g(R)

,

R€LI

, is

faithfully f l a t . Since _U and g(G) are

smooth (1.1), t!?e set of points a t which f i s smth i s dense and open i n U_

312

11, 8.6, no 2

ALGEBRAIC cZaJPs

( I ,5 4

,4.12).

W e may assume that k is algebraically closed, and choose a

u = (g,, ...,%,I

rational p i n t

of V_ a t which f is smooth, hence a t which

the tangent map t o f_ is,sur]ective

(I

, 5 4 , 4.15) . By

gent space t o !a t this point may be identified with

the tangent space t o g(G) a t f ( u ) into L i e g(G)c L i e G

x . ,yiE Lie 1

may be identified w i t h

Lie

g(G)

G

. If

d is the subspace of

and xELieG_

gEG(k)

. The

(Lie G )

2N

such t h a t

of the theorem, we see immediately that e Also, i f

, similarly,

(Lie G_)2N

a t the pint u thus corresponds t o a map t of

tangent map t o

for

translation the tan-

, we

Ed

defined i n the statement

Lie G

i s a normal subqroup of G ( ~ ( E ) ) .

obtain directly

EX -1 EAd(9)X geExX=ge g g = e

. I t follows a t once t h a t L i e g(G) C d . Noreover, in GJR(E))

so that- g and eEx commute modulo eEd

gEG_(R), xELie

for b)

, so

into d

(Lie G)2N

that

@R

W i t h the notation of

d of L i e s

, which

5 2 , 1.3,

proves that Lie

L i e S(G)3 d

t maps

,

.

9 ( c ) is the smallest vector subspace

such that, i n the adjoint representation of G _ , we have

.

G = G Applying 2 . l ( a ) and using 6 5 , 5.7 b) , we infer that i f + . , L i e G_ connected, d i s the snallest vector subspace of Lie G_ such that [Lie

G , Lie G ]

C

d

, which

Definition:

2.4

bra of

Lie

group of

5

establishes b)

Let

is

.

be a locally alqebraic k-group.

A subalge-

is said t o be algebraic i f it i s the L i e alqebra of a sub-

5.

By 2 . l ( a )

groups of

,

RE-?

is a bijection of , the map g o L i e 5 onto the s e t of algebraic subalgebras

the s e t of connected subof

Lie

s.

Clearly the intersection of algebraic subalgebras is algebraic. In particu-

l a r , for each subalgebra h of gebra A(h) of

Lie

G, , there i s a smallest algebraic subal-

Lie G_ containing h ; t h i s we c a l l the algebraic hull of h .

11,

9

6,

THE cWWiCTERISTIC

no 2

CASE

0

313

Lemma:-L e t 5 be a locally algebraic kgebra of L i e 5 & W,W' two vector subspces of L i e

h a L i e subal-

2.5

and W ' [h,W] C W ' -

C

W . Then

[A(h), W ]

W'

C

.

5

such t h a t

0 0 Proof : Consider the adjoint representation of G and the subgroup -%,,w of Go ( 5 2 , 1.3) . Its L i e algebra i s the set of a l l x E L i e such that [ x , ~C] W ' ( 5 5 , 5.7) . Since it contains h , it also contains A(h) .

m: Let

2.6

(a)

Let

h be a subalgebra of

A(h) : w e have the algebra

G be a locally alqebraic k-group. Lie G_

. Then

[h,h] = [A(h),A(h)1

each ideal of h is an ideal of

A(h)/h is commutative. F u r t h e m r e

is algebraic.

[h,h]

(b) The derived ideal, the radical, the nilpotent radical, and the Cartan subalqebras of

5

Lie

are a l l alqebraic.

Proof : (a) Let k be an ideal of the subalgebra h ; then we have

so that [A(h),hl

[A(h),k] C

plies that

by 2.5 and k i s ax ideal of

k

C

[h,hl ; applying 2.5 again, we get A(h)/li is commutative and

A(h)

a@)i s

[A(h),A(h)] = [h,h]

[A(h),A(h)] = [h,hl and so

by 2.5. Hence A ( r )

Lie

. We have

is an ideal of

--

have A ( r ) = r

G

[Lie Lie

. The nilpotent radical of

[h,h]

. By

K = ( N o r m h)O. By 5 5 , 5.7,

2.7

subalgebra of

let

2.3 b), the Lie

is algebraic.

Lie

; by (a) , it

Lie

G

is solvable, and we

i s [ L i e g,L,ie G]

T~~ G(h)= h

L i e _H = No

nr

Lie

5

; it

and

*

Lie

5

which coincides w i t h i t s derived algebra is algebraic. Lie

i s alqebraic.

Corol.lary : Each f i n i t e dimensional k-Lie alqebra which coin-

cides with its derived algebra is the L i e alqebra of an affine alqebraic group.

is

Coro1.la.q : Let G be a locally alqebraic k-qroup. Then each

I n particular, each semisimple subalgebra of 2.8

k

which im-

. Finally,

therefore algebraic. Finally l e t h be a Cartan subalgebra of let

C

G i s algebraic. Let r _G,r]C r , so that [ L i e _G , A ( r ) 3 C r

(b) W e already know t h a t the derived ideal of be the radical of

. Similarly

[A(h),A(h)l C [h,hl

be the connected subgroup of G_ w i t h L i e algebra A(h) algebra of

[h.k]

m B R A I C GROUPS

314

11,

0

6, no 2

,th.1) , there i s q ( V ) , hence into

Proof: By Ado's theorem ( G r . e t alg.de L i e , I , 5 7 , n o . 3 a mnomrphism of the given L i e algebra into an a.lgebra the Lie algebra of a group c;L(V)

. Now apply 2 . 7 .

I n particular, each semisimple k-Lie algebra is the L i e algebra of a n affine algebraic k-group, and 2.2 applies. Proposition:

2.9

(a)

f : 5 -+

Let

g

f i n i t e kernel. i f f (b)

Let

i s bijective.

cp : Lie G_ + Lie

-s

G and H be connected algebraic k-groups.

be a homomorphism. Then f i s faithfully f l a t and has

Lie f

be a hommrphism of k-Lie alqebras, and suppo-

. Then there

G_ = 9 ( G )

se t h a t

Let

,and

is a faithfully f l a t homomorphism with f i n i t e

a hommrphism f : 5'-+I such that L i e

kernel

p :G'

Proof:

(a) Imediate from 1.1 and 5 5 , 5 . 1 and 5 . 5 .

k c ( L i e _G) x (Lie El)

(b) Let

(Lie G)x (Lie _H) Lie _G' = k

which is isomrphic t o

. By

Lie

(a), the projection

hommrphism whose kernel is f i n i t e . If we have Lie f = c p p o ( L i e p) 2.10

-

Corollary:

.

( L i e 2)

be the graph of cp ; t h i s is a subalgebra of

G

and hence identical with i t s

derived algebra. By 2.7 there i s a connected subgroup G' that

= cp o

and G

1-

f

.

G

p : 5' -+G

:GI

of

Gxg

such

is a faithfully f l a t

i s the second projection,

+ ;

be connected alqebraic k-qroups,

2

both identical w i t h t h e i r respective derived groups and l e t cp : L i e s

1

Lie G

-2

be an isomrphism. Then there i s a connected algebraic

k-group G_ arid fiiithfully f l a t hommrphisms w i t h f i n i t e kernels

and

f 2 :G

-+g2

such that

L i e f 2 = cp

(Lie 2,)

.

fl:g+G_l

Corollary: -L e_t G be a connected algebraic k-group which coin-

2.11

cides with i t s derived group and s a t i s f i e s the following condition: (SC)

5'+ G_ ,

Each faithfully f l a t homomorphism with f i n i t e kernel where G' is connected, i s an isomorphism.

Then for each locally algebraic k - B

_H

,

f

-

+ Lie f is a bijection of

H) onto the s e t of k-algebra hommrphisms cG rk (G - I -

Lie

G

Lie

g

.

Proof: "he map i n question is injective by 2 . l ( b ) and surjective by 2 . 9 .

11,

5

6, no 3

315

THE CHARACTERISTIC 0 CASE

Section 3

The exponential map

3.1

Let

G be a k-group-functor and l e t R E % . W e denote the elemts of G(R[ [TI ] ) by function symbols such as f (T) Given an R-algebra

.

which i s linearly topologized and complete, and a topologically nilpo-

Sc&

tent element t of S

, we

write f ( t ) for the element of G ( S ) which is the

bage of f (T) under the continuous morphism of R[ [TI 1 i n t o S which sends T onto t

. Thus we w i l l have,

the element

f o r instance, the elenaent f

f(T+T') of G ( R [ [ T , T ' ] ] )

there is a unique element exp (Tx) E X

(b) exp(T+T')x =

+

G_(R(E))

of

Let

E~

eXp(m)q ( T ' X )

,..., n E

,

Then for each xELie(G@R) such that

G ( R [ [TI ] )

in G ( R " T r T ' I I ) [XryI

= 0

*

, we

have

be n variables of vanishing square and l e t

R = R(E~,...,E ) = Rn-l(~n) n n

xn

of G ( R ( E ) )

.

Moreover, ~- if x,yELie(G_@R), and i f

Proof:

.

and l e t G be a k - B .

Proposition: Let RE-%

(a) e x p k x ) = e

, etc.

(E)

. Consider the elerent

.

= e E l x . . eEnx

Xn

of G ( R ) defined by n

,

. By 5 4 ,4.2, the element Xn is invariant under permutations of the variables E~ . Now consider the R-hommrphism an: R[T]/?+'4Rn + ... + E~ . A straightforward argument shows that, when such that a (T) = n 1

h e r e x E Lie (G @R)

E

k

,

a is a bijection of R[T]/T"+' onto the subn f o m d by the invariants under the group Sn of permutations of

has characteristic

0

ring of R n the E ~ It . follows that there is a unique element En of that

G(RIT]/T"+l)

such

a (E ) = X n : t o see t h i s , let _V be an affine open subscherw of G con-

n n taining the origin whose r b q i s A. Since $ Rn and space of p i n t s and the composition

factors through

E

G

, we have

.

R

have the s m

XnE_U(Rn)LM+(A,Rn) Since we have

AIx;EBRAIc GRWPS

316

belongs to

g(Im

En

E

an)

, and

11,

is therefore of the form a (E )

LJ(R[T]/?+l)

E C_(R[T]/T"+l)

n n

.

0

6, no 3

, where

Now consider the conmutative diagram

a

R[T] / ?+l

R[T]

Rn

I"

a

/T"

Rn-l

where pn is the canonical m p and

nihilates

E

.

W e have %(Xn)

sends

E

, so that

i

onto

E

i

for i # n and an-

p (E ) = Enql n n such that E n = E ( T mod

= Xn-l

n is a unique element E(T) E G ( R [ [ T ] ] )

. Hence there T"")

for

each n . To prove this, take U_ and A as above; each En corresponds t o a

hommrphisn A

--+

. Hence these form a n inverse l i m i t system,

R[Tl/T"-'

which i n t u r n yields a hamomorphisn

A+

ment E(T) of _U(R[[T]]) C G_(R[[TI]) G(R[[T]])

such that

( 8 1 , lemma 3.81,

. Let

, associated

w i t h an ele-

of

E'(T) be another el-t

for each n

E n = E'(T mod ?+')

# # W ( E ( T ) ,E'(T) )

R[[T]]

. Since G

is closed i n a R " T 1 1

is separated

(1,§2,7.6)

that 1 = 0 and

. By hypothesis we have E(T) = E'(T) .

E(T) E G(R[ [TI ] )

meets the conditions (a) and

and i s accordingly d e f i n e d by an ideal I of R[[T]I

+1

I&?

R[ [TI] € o r each n

, so

W e now show that the element

.

(b) %is i s immediate in the case of (a), for

we have for each n a c m t a t i v e diagram

a

2n

1

1

R2n

( E . ) = ~ ~ 6 3for 1 l S i S n and u ( E . ) = 1 @ n i n i w h i l e v (T) = 1@ T + Tc3 1 S i n c e b y construction w e have n where i and i2 are the injections of R into R @ R 1 n n n '

where u

.

, where

.

E ( E ) = X = eElX A s for (b),

~€ o ~ r n-+ l~
u (X

= i (X i (X n 2 n 1 J 2 r l it f o l l o w s that

are the injections of R[T]/'l? 2 into its tensor square. Accordingly E(T+T') and E(T)E(T') have the same

vn(EZn) = j1(En) j2(En)

j,

image in G(R[ [T,T']]/(Tn+l,T'n+l))

and j

+1

so that E(T+T') = E(T)E(T') b y the sam

11,

9

6, no 3

317

THE CHARACTERSTIC 0 CASE

aqumnt as above. L e t us show f i n a l l y that E (T) is t h e unique element of

meets the conditions in question. I f lows inmediately by induction that

s a t i s f i e s (b) I it fol-

F(T) = exp(Tx) F ( ?Ti)= 1

1

G(R[ [TI ] ) which

i n S(R[[T1,

F(Ti)

Now i f F(T) s a t i s f i e s ( a ) , w& i n f e r t h a t in G_(Rn) we have

which -lies

x,y

we have

... e EnX

(e But

and i f

E L i e (@3 R)

[x,y1 = 0

, then

exp(T(x+y)) =

. I n v i r t u e of our arguments above, it is enough to show that

exp(nC) exp(Ty)

in G(Rn)

and f i n a l l y F(T) = E(T) . L a s t l y

F ( T md '?+I) = E ( T mod Tn+')

w e show that i f

...,Tn1 1 ) .

( e €1'

... e

= eE1(X+Y)

e 'iXe 'iY= eEi(x+y)and t h e e E i

En (X+Y 1

c o m t e by § 4 ,4.2

eEi

and

... e

,

and the contention fol1-,1s.

Remark : Under the conditions of 3.1, i f

3.2

if

f (T+T') = f (T) f(T')

such that

in G(R)

f (T) = exp(Tx)

, so t h a t

, there

in G_(R[ [TIT']])

f(T)EG(R[[T]]) and

is a unique

. To see this, note that we have

f ( 0 ) = 1 ; it follows that

xELie(E@R) f (0) = f ( 0 ) f (0)

f(E)EG(R(E)) is projected on-

to 1 and is accordingly of t h e form e E Xfor a uniquely determined xELie(_G@R).Thus we have 3.3

f ( T ) = exp(Tx)

Example: Take G = &(V)

k-vector space and x E

it0

. i! e

EX

... ( I d +

n = (Id+E~X)

X

Id +

.

( E +~ * *

= Id + t x +

where t = a n ( T mod

. Then

Ti xi

C

To v e r i f y this, notice t h a t w e have

formula.

where V is a f i n i t e dimensional

d;, (V) 8 R 2 y R ( V @ R)

~ ( T X =)

=

.

2"+1) . By

E

by § 4

, 4.2,

...

n

x)

n

+ E ~ x) +

... +

= Id+EX

... +

E~

so that

E ~ x)

(tn/n!)xn

passing t o t h e l i m i t we obtain the required

3.4 (a)

Corollary:

If

(b) Lf

(d)

11, § 6, no 3

ALGEBRAIC GEiDupS

318

aER

, then

qEG(R)

If G

Let

G , R E d x be as in 3 . 1 .

exp(aT)x = e x p T ( a x )

, then

.

gexp(Tx)f'= exp(TAd(g)x)

.

is locally algebraic, then i n % ( L i e (_GI ( R [ [TI I

we have

Proof : ( a ) ,(b) and (c) follow immediately from the uniqueness assertion of 3.1;

(d) follows from (c) applied t o the hommrphism

Ad: G 3 = ( L i e ( G ) )

and f r m 3.3.

3.5

Corollary :

and let x € Lie(GJ@ R (i) p :E

Let G

be an affine algebraic k-group, l e t

. Then the followinq conditions a r e equivalent.

RE&

There i s a faithful f i n i t e dimensional linear representation 3

&(V)

such t h a t L i e ( p ) (x) is nilpotent.

(ii) For each f i n i t e dimensional linear representation p of G ,

Lie(p)(x)

i s nilpotent. (iii)exp(Tx) E G(R[T])

.

(iv) There is a hormmrphism f : c1 R Proof : (ii)=> (i): By § 2

(i) => (iii): By 3.3,

-

ST R

such that

, 3.4.

exp(TLie(p)x)EGL(V)(RITI)

L i e (f) (1)=

. By 3.4

x

,

(c) , we have

the c o m t a t i v e square

P1

here p1 and p2 correspond .to exp(T Lie(p)x) and exp(Tx) is surjective

(

. Since

J(p)

9 5 I 5.1) and can i s injective, p2 factors through R[T]

(iii)=> (iv) : L e t

S € h ; €or each

.

t E S = a ( S ) , consider the hommrphism

11,

5

6,

110

THECHARACTERISTIC 0 CASE

3

R[T] +S which sends T onto t

, and

319

t h e image f (t) of exp(Tx1 under this

homxnsrphism. W e thus obtain a mrphism f : aR-tG; it i s immediately seen t o be a hommrphism ( ( b ) of 3.1). W e then have

L i e ( f ) (1)= x i n v i r t u e of

(a) of 3.1. (iv) =

(ii) : By 5 2 , 2 . 6 .

Suppose R = k .The homomorphism f whose existence is asserted

3.6

by ( i v ) is uniquely determined (2.1 (b)) x # 0

. To see t h i s ,

sion 0

, hence

, and

notice that its kernel is a subgroup of

%

of dimen-

&tale (1.1), while a(K) has no non-zero subgroups.

When t h e conditions of 3.5 are m e t , we say t h a t x i s n i l p

3.7

t e n t , and we write exp(x) f o r t h e e l e n t of exp(Tx1 if

it is a mnomrphism when

R = k

under the hommrphisn

, we

have

R[Tl -R

G ( R ) which is the image of

.

which sends T t o 1 Accordingly,

f ( t )= exp(tx) f o r each

tES

, S€M+

.

I f x is nilpotent, we may replace T by 1 i n corollary 3 . 4 ; i n p a r t i c u l a r ,

we obtain the formulas

Similarly, i f x and y are two nilpotent e l e m n t s of [x,yl = 0 3.8

, we

have

exp(x+y)= exp(x1 exp(y) by 3.1

.

Lie(G)@R

, and

if

Remark : It follows from 3.5 t h a t the subalgebra of Lie(G)

generated by a nilpotent element is algebraic. 3.9

Let

k [ [TI 1-

be the subring of

k [ [TI ] consisting of formal

power series a r i s i n g a s solutions of linear d i f f e r e n t i a l equations with cons t a n t coefficients. I f k = @ , these are l i n e a r combinations of formal p e r series of the form P(T)exp(aT) where PEk[T] and a E k

. If

k = R , they

are l i n e a r combinations of formal p e r series of the form HT) exp(aT)sin/bT) , P(T)exp(aT)cos(bT), where PEk[Tl and a , b E k

.

Azx;EBRAIC GWWPS

320

_ _G be an affine k-group. T k k , P r o p o s i t i o n : Let

11,

for

x E Lie(G)

belonqs to G(k[[T]lexp)

.

Prmf : Let -

k [ [T,T'] ] be the h o m r p h i s n

3.1 ( b ) ,

6 : k [ [TI 1

-

have _G(G)exp(Tx) = exp(T+T')X E G(k"T11

,

5

6,

no 3

exp(Tx)

f ( T ) H f (T+T')

. By

E ~ ~ k " T ' 1 1 ) i whence

e x p ( ~ x )E c _ ( 6 j 1 ( ~ ( k [ [ T 1 18 k ~ ~ T ' 3 1 ~ ~ = G ~ 6 1 €3 ~ kk"T'11)) ~[Tll k k

since

is affine. It i s therefore enough t o prove the following fEk[[T]]

Lemna:Let -

3.10

€(T+T') Ek"Tll$c Proof:

k"T'l1

.

fEk[[Tll

exp-

iff

*

If f (T+T' 1 =

C ai(T) b i ( T ' ) 1

r

by applying a derivation w i t h respect t o T

n times and setting TI = 0 we

obtain

which shows that the

hence that a

1'

f E k[ [TI 1

...,ar E k [ [ T ] ]

generate a finite dimensional vector space,

f(")(T)

.

Conversely, i f f E k [ [TI 1 exp exp such that for each n we have

T a y l o r ' s f o m l a now applies, t o give

where bi(T')

1

= C 7 b n n. i , n TIn

.

, there

exist

THE CHARACTERISTIC p

5 7

5 7,

In

if

rp:k+A?

,

REEL

is a hommrphism of models and i f

rpii denotes

obtained from R by r e s t r i c t i o n of s c a l a r s . The external l a w of

t h e k-&el rpR i s then

,

(X,x)++rp(A)x

XEk

sgkL : an element of

S%I=

# 0 CASE

which we denote by

smqA

,

. Similarly,

xER

if

S€Ek

, we

set

is then a linear combination of elements

S@,a

( s E S,X €

sv@$,X = s@,rp(u) X

satisfying

A?)

€or

uEk.

Throuqhout

5

p

7,

denotes a fixed prime number and k

Section 1

The Frobenius mrphism

1.1

L e t f be the endomorphism of k such t h a t

If

we write

R €$

for x E R from

X

. For (fR)

g(’) : X“)

‘Y

REgk

R

-+

f

f o r t h e mrphism of & i

R

g , we write

for

(‘I

. Finally,

. Similarly, i f

REL$

is t h e mrphism of

g : we

f

denote it by

F

-x_

. Accordingly

, we

f R ( x )=

AEk

2

.

. Then we have

,

g(‘)(R)

= u _ ( ~ R ) for

R€k$

i s c a l l e d the Frobenius mrphism w i t h

e

k

for

which assigns t o

If E -k Tl and EM& , w e hzvc (&@,1) Thus, if k = F , we have f = 14, , SO t h a t P ZB 1 ( i n general, of course, F # 1% ) . In the general case, i f n 2 0

such t h a t

such that

o r s b p l y _F

-5

f ( A ) = Ap

g:X+y is a mrphism of @b

3 into

the mrphism of

t h e mp X(fR) : x’(R) -+X( R)=X(’)(R) domain

%-E

P-

f o r t h e functor derived

)’(,

( I ,5 1 , 6 . 5 )

by the extension of s c a l a r s f

X(D) ( R ) =

-

f :R

each k-functor

JF -model.

-

define

.

X-(p)@kA?

and and

X

x (8)by

.

(353) Ex

= F @ 1.

-3

k

=dP)BkL =

Z(pn) (R) = z ( f n R ) f o r

<:

we have g (9’’) = (>((p)) Similarly, we define n F“ X-+ x _ (pn) by the formula _FX(R) = g ( f i ) i f RE$ and i f R+ fnR -3 sends x onto xp Then , h i c h we abbreviate t o _F“ , is t h e composition RE$

--

1.2 T(‘)

.

Exarrple:

$-

Let

T

be a geometric k-space

(I,5 1 , 6 . 8 ) and l e t

be t h e g e o m t r i c k-space which has the s m und.erlying topological

mws

ALQBRAIC

322 space as T

0T@f k FT : T

, and

11,

whose structure sheaf i s the sheaf of k-algebras

be the nvsrphism of

R E M , consider the k-functor $(T(')) wke f (u-,u-) : Spec R -+ ) ' ( T of

68 1 f

and a

sk . Writing

mrphism

sheaf of k-algebras of

Spec R

, l e t u'

u z ( d ) by

u(T)

O

_Sk(FT)=

.

E

SkT

proposition:

1.3

which are functorial i n X_

and are such t h a t Set T = 1121,

Proof:

i s invertible when _X

for the structure

dT

into the sheaf of f : k-k

. As

which define an

0

:

IF^^^

[X(P)Ik+ = _F

klk i s a scheme.

(zlk (P)

f o r each

0 : _X+skIglk i s the canonical mrphim

which arises from the adjointness of S

v(2) to

u(E)

There e x i s t mrphisms

i n 1 . 2 . If

8

i s m r p h i s n satisfies

~ E _ \ E , satisfyinq v ( 5 )

v(&)

U E S ~ ( T ( ~,) i.e. ) a

(u',u')

(ue,uf-)

2; _Sk(T)(') . This

p (T) :Sk(T('I)

-+

the restriction of scalars

R varies, we accordingly obtain maps

isomrphism

cp A .

Of

be the composition

Clearly u ' is a mrphism of the sheaf of k-algebras of k-algebras derived from

. Let

asociated w i t h the i d k t i t y map

Esq k T and the mrphism dTBf k -4JT induced by the maps Given

@T(p) =

U ++dT(U)Brk )

(i.e. the associated sheaf of the presheaf -+

7, m 1

5

-k

to

] ? I k , it

is enough t o take

be the mrphism assigned t o the composition

by the bijection

A s an application of t h i s proposition consider the case in which

246%

. For each

R€I&

whence an isomorphism

, we

then have a canonical bijection

z=SE k A

,

11,

5

If

'p

THE CHARACTERISTIC P f O

7 , no 1 : Agfk

.

Fx = S 4 cp

X(A)

-

Spec

-F" : -G

-+

. This

-

i s a homomorphism. We w r i t e

G

W e say that

has heiqht

2 n

obviously has height

5 n

Observe t h a t

F-

Fllg

if

L i e (G) = L i e (FGJ

L i e G I ( k ) = L i e ( G ) (k)

, or that

= G_

, and

E

shows and

for the kernel of

G

,

it is enough t o v e r i f y t h a t

L i e ( G ) (k) GFG(k(E))

so factors through k

, w e have a

R"

. For each k-group-functor

. . To see this, -

irrunediately from the f a c t t h a t t h e h m m r p h h hilates

G

P-

and 11 is an autoomorphism of

RE-? commutative diagram

.

. But

-

this follows

fk(&)k : (E)

anni-

$(E)

Examples :

1.5

If _G

Hence b) If

=

% , then

$=

-

G

~ ( x=)xp

.

pn%

= Q(T)k

, whence

be i d e n t i f i e d with

c)

= G(fR)

latter i s a c h a r a c t e r i s t i c subqroup of G _ . To see this,

observe that, i f

a)

.

: Spec A --c Spec A @f k

(p

dp) is n a t u r a l l y endowed w i t h the s t r u c t u r e of a k-group-functor

t h a t gG : G

G

X -

G is a k-group-functor t h e fonrmla G(I')(R)

If

that

F"-

I

Since l_F

the same i s t r u e of 1.4

a @ A ++ apA , one shows e a s i l y t h a t f is b i j e c t i v e i n v i r t u e of the proposition,

i s t h e homomorphism

A

-+

323

CASE

%

,(P) =

, RE$

, arid

(1.1) m c :

xE%(r,R*)

g(x) =

, we

xP(y) = x ( y P = x(py) for y E r n

. For example,

g(r/p\rlk

If G i s constant, then )'(,

=

G

xP

have

for ~ E R E M +

c(')

5

and

. Consequently, ps

,pPk = pnpk

and E = I d

=

.

. If G

my

*

i s etale

, then g

i s an isomorphism. 1.6

Proposition:

tural number

2 0

i)

G

Let 5

. Then t h e

=k-group-scheme

and l e t n be a na-

followinq conditions are equivalent:

has height 2 n

.

324

ALGEBRAIC CJuxrPs

-G

ii) O(G)

, then

if

is affine ;

I (G)

7, no 2

is the kernel of the augmentation of

the pth p e r of each element of I (g) vanishes.

(ii) => (i): If G_ = = A

Proof:

5

11,

, and

, then g(x)

if xE&Pl(A,R) = _G(R)

is the compsition fR

X

A - R - R ,

_F" (x)(a)=

so that

, which

x ( a$)

implies that

En(x)

factors through the

augmentation of A . (i) => (ii): "he Cartesian square 'L

and the fact that

191 '

* G _

#

is bijective together inply that

jective, hence that the canonical projection p : G _ hence that the unit section

E~ : $-

G_

-

-G

-

---f

gk is in-

is bijective, and

-+

(gi)

is a closed embedding. If

is an affine open cover of , E - ~ ( U _ . ) is affine for each i. Since -1 -1 gi = -pc ( E ~(gi)), it follows from I , S Z , 5.6 and 5.2 that 5 is affine. The remainder of the argument is inmediate.

he p*-pwer

Section 2

operation i n Lie(G)

Throughout &?is section G denotes a k-group-scheme. th

We now define a map of Lie(G) into itself called the p -p-

2.1

wer operation and written x t-+ k(E1,

..., P) E

zero. Set

0

'4 . Let

x E Lie (5): consider the algebra

obtained by adjoining to k variables =

E

+...+

E

and

= E

-.- .

all of square €1 * * €P Then we have ' S I = 0 , lr2 = 0

1 P 1 EP and it is easily shown that the subalgebra of k(E1,

and i '

T

..., P) E

generated by cs

is t h e algebra of elements invariant under all permutations of the

. Consider the element

eEIX eE2x

... e " 3

of Ker ( G ( k ( E 1 , . .

This element is invariant under all permutations of the

Arguing

a.s

in 5 6

E

i

(

.

,E

P

))

+G(k)

0 4 , 4.2 (6) ) .

, 3.1 , we infer that it belongs to Ker (_G(k(a,lr)-t G(k))

.

.

11,

9

7, m 2

THE CHARACTERISTIC &O

I f we apply t o this element t h e homomorphism of

annihilates u

, we

o b t a i n an element of

k ( u , n ) onto k ( n )

K e r ( G ( k ( n ) ) --+ _G(k))

, where y € L i e @ ) . S e t ( G ( k ( o , n ) )+ G ( k ( n ) ) ) , we then have

thus of the form eV

Ker

eElx

2.2

... eEP"

e

(El..

325

CASE

y = xrpl

which

, which

. Modulo

is

[PI

.EP)X

Examples :

s=%

1) S e t

and i d e n t i f y k with L i e @ ) v i a t h e ma&> x

( 54,4.11).

+EX

Then we have

eElx SO

2)

that

x[']

Take

, and

= 0

G =

... eEfl =

E )x = P

,

ux

the pth-power operation i n L i e ( G ) i s zero.

Q(r), , where r

i s a small c o m t a t i v e group, and i d e n t i f y

with L i e ( G ) v i a the map

E(r,k)

... +

( El+

x

+--+

1+Ex (54

,4.11)

. men

we have

so t h a t e7ix 3)

[PI

1+d and x[']

=

, where

Take G_ = G&(V)

2

=

,

V i s a f i n i t e l y generated p r o j e c t i v e k-module.

* Id+EX

I d e n t i f y L ( V ) w i t h Lie(G) v i a t h e m p

x

computation as the one a v e then gives

x[Pl = xp

2.3

a) if

Proposition. : L e t

v

G;

+

isomorphism ( 5 4

, 6.8 )

Ed

terms i d e n t i c a l w i t h x)

1

-

(x) [PI

=

i k a k-scheme,

Der (X)C Dif

c)

.

L

C

VG

I_f

'

. The

D i s t (G) and we have

belongs t o

b)

x E L i e (G)

- D i s t l (G) Lie (G) is t h e canonical x E D i s t (G) 1- + - D i s t ( G ) , then x * ... * x ( p If

+

.

( 5 4 r 4.12)

v (x*x*

-G

g :G

--+

...*x ) .

A&(?)

a hommrphism, E d

u ' : L i e ( G ) -+

(5) the c o r r e s p n d i n g antihommrphism, we have u'(x)'

I n the algebra

Dif(_G),we have

y'(x[P])

=

y'(x)P

,

6'(x[P]) =

S l ( X 1P

.

= u'(x'")

.

The notation i s taken from

__ Proof :

54,

lows from b) applied t o the hommrphisms ( 5 1 , 3 . 3 ) . By § 4 , 6 . 6 ,

we have y ' ( x *

tive, a) follows from c ) by 5 4 , 6 . 8 . Let

11, § 7, no 3

ALGEBRAIC GR(3upS

326

be open in

sections 4 and 6. Assertion c) foly :G 4 &t(G)

...* x) = y ' ( x ) '

-0Pp

-+&t(G)

y' i s injec-

It i s therefore enough t o prove b ) .

, f E @(u)

& and l e t R€$

6 :G ; since

and m E g ( R )

.By

definition

we have f ( u ( e E X ) m )= f ( m ) + c ( u ' ( x ) f ) ( m );

Setting

ci =

0

, we

get

Section 3

Lie p-algebras

3.1

Definition : L e t I > eb

For O < r < p

set

s (x ,x ) =

r 0 1

k-Lie alqebra and l e t

--r1 C ad xu ( l ) ad xu(2) ... ad

x0' x 1E R

.

x u(p-1) (xl)

[l,p-11 4 {O,l) which assume r times the

where u ranqes throuqh the maps value 0 . For instance,

s (x ,x ) 1 0 1

.

coincides w i t h

t ~ ~ r t ~ for ~ r p~ =~3 l ] 3.2

Proposition:

Let

[x x ] or 1

A be a k-algebra

cessarily commutative). Given a,b € A Then we have the Jacobson formulas

, set

for

p = 2 and with

(associative, but not ne-

(ad(a)b ) = [arb]= ab

- ba .

11, fj 7 , no 3

THE -STIC

Proof : Setting L (b) = % ( a ) = ab a (ad(ap))(b) = (LE which gives a )

.

W E

$0

-

ALSO, if alI...'a

w e have

I

-

(b) = (La

<)

P

327

RaIP (b) = ad(a)' (b)

€ A , we have

To see t h i s , notice t h a t the right hand side of ( *) is

where

(i1,...'ir) ranges through the s t r i c t l y increasing sequences of na-

t u r a l numbers in the interval

[l,p-11

and

denotes the

(jlr...,-j-l-r)

s t r i c t l y increasinq sequence whose members are the integers in [l,p-11

. This

t i n c t from i l I . . . ' i r

But the formula

k[x] if

,

(x-1)

implies that

-

dis-

sum may be written

xp-l = x-1

p-1-r

$-'+

$-2

+

... + 1

which holds in

p-1

(pl-r) = 1 , vihich gives (*)

(-1)

.

Now for b) :

x x € A we have 0' 1

where F ( r ) is the set of maps of

{0,1) which assume r times

[ l , p ] into

the value 0 . If we assign t o each s E S the Ira? w € F ( r ) such t h a t S -P -1 -1 ( r ) } , we obtain a surjective map of S onto F ( r ) (0)= {s-'(l),...'s

w

-P

S

,

such t h a t the inverse h q e of each w E F ( r ) has r!(p-r)! elements. Setting a =...=a = x 1 r and

, ar+l=...=a

P

=x I

, we

have x

Similarly we obtain s (x ,x ) = r 0 1

--1 r

1

r! (p-1-r):

Using ( * ) ' we obtain the required formula.

-

-

ws (1)' '"WS (p) = as (I)' * as (p)

ts

- p-1

... ad at(p-1) (ap ) .

ad at(l)

328

ALGEBRAIC GFKxJPs

3.3 on

& k-Lie

Definition : IEt l?

R is a map

X-+X[~]

of

11,

th

alqebra.

p

5

7 , no 3

-)-aver operation

L into i t s e l f satisfying the following condi-

tions :

,

Ip-ATA 1)

(Ax)["

=

Ap x["

(P-AL 2 )

ad(x[")

=

(ad(,:))"

(p-X 3)

(x+y)[PI

=

Afk , x E l ;

, xE& ;

,[PI + ,[PI +

c sr(x,y) O
,

x,yEL

.

e p-alqebra over k is a pair consistinq of a k-Lie alqebra & and a th p -power operation on R

u

.

The upshot of pro-msition 3.2 i s that the L i e product

fhe pth-power operation xLpl=

2

[x,yl = xy

- yx

and

endow each associative k-algebra A with

the structure of a L i e p-algebra over k . In particular, each L i e subalgebra

of A which is stable under the pth - power operation is a L i e p-algebra. Given two L i e p-algebras

,a

L,L'

hommrphism c p : L +R'

and u, (,Ip1

a k-linear map satisfying c p ( [x,yl) = [cp(x),cp(y)I for a l l x,yEL

. The category of

i s by definition ) =

u, (x)[PI

Lie p-alc,ebras is denoted by LLpk.

From 2.3 w e immediately infer the

th Proposition: For each k-group-scheme C,, t h h p -power oper-

3.4

ation definied i n 2 . 1 endows L i e ( G ) w i t h the structure of a L i e p-algebra

. If

k-scheme and g : _G +Aut(g) a hommrphism, OPP u ' : L i e ((2)--t Der (3) i s a homomorphism of L i e p-algebras.

over k

&

-1

then

W e have, naturally, assigned Der(g) the pth-power operation induced by the

p* 3.5

- power

operation in the associative algebra Dif (_X) Theorem: -

.

Let 1 be a L i e p-algebra, which i s also a f i n i t e l y

generated projective k-module. Then there i s a k-group-scheme isomorphism of Lie p-algebras

at : L * L i e @ ( & ) )

E(L)

and an

such that the followinq

condition holds: (*)

for each k-group-schem cp : L-+Lie(G) cp = Lie(?)

O a L

, there

g

and each hommrphism of L i e p-algebras

i s a unique hommrphism _f : g ( R ) - j G _

such t h a t

*

Remark : Condition (*) mans t h a t the map :++Lie(:)

0

aR

is a bijection

11,

9

Gr (

E(&),G

*k

7 , no 3

the pair

)

THE CHARACTERISTIC $0

'L +

. As

L A k ( L,Lie(G) )

329

CASE

the solution of a universal problem,

( g ( L ),aL) is "unique".

I n proving this theorem we w i l l make use of the results t o be proved i n 3.6

-

3.10. F i r s t of a l l consider two arbitrary L i e p-algebras

3.6

over k and a L i e algebra homomorphism cp : L - - + l ' th

r i l y preserve the p

,

XE k

x,yEf

. By

(which does not necessa-

c ( x ) = cp(x)1'

-power operation). Set

R and L'

- cp(x[pl) ; l e t

1) , we have c(Ax) = 'X c ( x ) : by (p-AL 3 ) , we have I)

(PAL

c(x+y) = c ( x ) + c ( y ) ; by (p-AL 21, we have

[ c ( x ),cp(y)I = 0

. If we apply

these r e s u l t s t o the case i n which cp is the canonical map of L into its enveloping algebra U ( L ) for each x E 1

. If

, we

see that c ( x ) belongs t o the center of U ( L )

P , the

( x . ) i s a system of generators of the k - d u l e 1

l e f t ideal of U ( 1 ) generated by the e l m n t s c ( x ) i s two-sided and coincides w i t h the ideal generated by the c(xi)

. Let

U["(L)

be the quotient alye-

bra of V ( L ) by t h i s ideal arid l e t j be the composite map L -+U(L)+U[P3(L) Clearly j i s a homomorphism of L i e p-algebras ( w i t h respect t o the L i e p-algebra structure on

(1) derived from i t s associative algebra structure).

Let

Proposition: a )

A be an associative unital (not necessarily c o m t a -

tlve) alqebra equipped w i t h the structure of a L i e p-algebra defined i n 3 . 3 . Then f o r each hongmrphism of Lie p-algebras homomorphism of unital alqebras b)

If the k-module

!f.

g : U["

7'

IIS j ( x i S

( L ) -+A

:&

-+

(&)

.

cp = g

, where

j

.

1 is totally

has a basis consisting of 1 and the

... j ( x ir )nr

nl 1

= j(xi )

A , there i s a unique

such that

i s freely generated by

ordered, then the k - d u l e products

'p

such t h a t

i
... < ir

g@

O < n . < p for each j E [ l , r ] 7 Proof: a ) follows immediately from the universal property of the enveloping

. A s for b) , identify L with its image i n U ( L ) and set c . = c(xi) = < - x p l . Then the c . belong t o the centre of U(L) and gene(&) . Let Ur be the r k e the kernel J of the canonica: ma
algebra U ( L ) of 1

1

d

U

PI

de Lie I ,

it follows from the Poincar&- Birkhoff -Witt theorem ( G r . e t alg.

5

2 ,no. 7, t h . 1 ) that the mnomials

.

330

Au;EBRAIc GROUPS

II x ni II c m i , Osn. < p i

1

,

11,

5

7, no 3

,

in. 2 0 1

form a basis for the k-mule U(1); and the result follows instantly. Corollary: If the k-module l? is finitely generated and pro-

3.7

jective, then U[p’(l?)

is a finitely qenerated projective k-module and the

canonical m p j : .P+u[P](P)

-_ Proof :

is injective.

-1

For each multiplicatively closed subset S of k , we may endow S C

with the structure of a Lie p-algebra by setting

for x E P and s E S ty in k such that

. In particular, consider a partition

1

=

ef i

Z t i. fi of uni-

is a free module over kf for each i. Clearly the i pair (U[p’(L)fi,jfi) is a solution of the universal problem of 3.6 relative

to k

fi

and

efi.

It follows that U[pl(P)fi is isomorphic to tJ[”(l,,)

is therefore free, and that jf

UIP1(e)

and

is injective for each i. Accordingly

i is finitely generated and projective and j is injective.

3.8

m e construction of

u [”(P)

is functorial in .P

( UIP1(P)

and the universal property of the pair

,j

)

. This fact

yields the followinq

results: a) There is a unique homomorphism of unital algebras that cOj=O

.

E

:U

[PI( e ) - + k such

b) There is a unique homomorphism of u n i t a l algebras A : U[pl(P)+U[pl(f)@ U [PI(l?) such that A ( j ( x ) )= I@j(x)+j(x)@l

c) There is a unique antihmmrphism of unital algebras such that n(j (x)) = - j(x) for xtc P . If I J ~ U [ ~ ~ ( Pand ,)

u

= C

u.@v. 1

c

1

, we have

Ui 8 Vi =

-

C u, 8 Av.

c

G(Ui)

C v i

= C

€4

hi8 vi

vi = u

c ‘7(Ui)vi

=

ui

E(U)

.

rl : U ‘ ” ( 1 )

for X U . +U[”](l)

11,

5

7, no 3

THE CHARACTERISTIC $0

CASE

331

To prove the above formulas it is enouqh t o verify them when

u =j(x)

, xER , and

RE-%

we simply denote the maps A@R

E:

or

t h i s i s inmediate. Henceforthr if U = U [ p l ( & )

and

U9R--tU$UfR-~(U€4R)@(IJ@R) k k R k by A and

u=l

U@R-

F@R

and

’L

k@R-+R k

k

respxtively.

Let L be

proposition:

3.9

a L i e p-alqebra over k which i s a l s o

let g(!) (R) be a f i n i t e l y qenerated projective k-module. For each F? E k the submnoid of ( U [ p l ( l ’ ) q R ) x f o m d byQ.x such that A x = x @ x , EX=^

% -

.

myl a)

E ( L ) is a f i n i t e locally free k-groupscheme of height 2 1

.

b) W e have a commutative square Bl

u [PI(.p )

where the riqht hand vertical arrow is induced by the canon-gcal isomrphism ?J

Lie( l3(1))-+

D i s t ; ( g ( Y ) ) of

bras and BL is (

Bg TB,)

Proof :

o

A

54

6.8,

clR

i s an isoomorphism of Tie p-a&-

alqebra isomorphism such t h a t

, with

the notation of § 4

7.1

Observe first that since x E g ( f ) (R)

that x i s invertible,

.

implies

is a group. I&

E ( R ) (R)

E = F

0

By

AoB,

=

(x) = 1 , so EJ‘”](R) ,k 1 ; then,

ri (x)x= E

A = %(

equipped with the multiplication t(U$U)

A $ A L

where

u

element

= u [ P 1 ( ~,) A E:

. Moreover,

t

-%

U ‘

=

i s an associative commutative k-algebra w i t h unit

since U‘P](R) i s a projective k-module of f i n i t e rank

( 3 . 7 ) , w e have a bicluality isomorphism

As in 5 4 , 7 . 3

i : u [ P ~ ( ~ ) $ ’LR - + ~ ( A , R )

.

, we

@ I ~ R, i ( x )

see that, for

XEU[~](!)

i s a homomorphism

NXEBRAIC GROUPS

332

-

of unital k-algebras i f f

g(1)

%

S2

A

,

. Accordingly

x E E ( L ) (R)

so that E(C)

f i e s that the coprcduct A A : A

11,

9

7 , no 3

i induces an isomorphism

i s a f i n i t e k c a l l y free k-scheme. One veri---f

A@A

associated with the group structure

E ( l ) is derived by transposition of the multiplication U@U -+ U (ape now prove b) . By definition, L i e ( E ( C ) ) may be identiply 5 1 , 1.8 a ) ) . W fied with the s e t of elements of E ( l ) ( k ( & ) ) of the form 1+ E X . I f of

R'

=

{x

€UJ['](l?)

:

A(x) = x @ l + l @ x } ,

is a bijection of

the map a : x w l + E x

onto L i e ( E ( & ) ). This map i s

e'

in fact an isomorphism of L i e p-algebras, for i f X € k and x,yEL'

whence

I a ( x ) , a ( y ) l = a ( [ x , y l ) i and finally

m ~ dE l +

...+

E

P The canonical map

we claim that

, so

j( f ) = l '

is free w i t h basis

is injective (3.7) and maps L into A!'

U1pl(L)

. To prove t h i s we m y assume that

(xi)i E1

(cf. 3 . 7 ) . Then U['](L)

sis consisting of the

where n = (n ) iiEI

the k - d u l e

and 0 2 n .

p

1

is free and has a ba-

for each i

. It

is easily shown that

i

; the claim

follows immediately.

W e have t h u s constructed an isomorphism a k : L -+ L i e ( E _ ( L ) ) U["(L)

U'p'(g)

--+

. I n virtue of

and 2.3, this extends to an algebra hom-

mrphism

b --

1.

r+s=n c UI: @ us

(r+s)i= ri+ s

the universal property of

.

ni

A(un) = where we have set

have

a(xlP = a ( x [PI )

that

j : L-+

, we

D i s t ( E,(L))

.

5

11,

THE CHARACTERISTIC $0

7 , no 3

If J i s the kernel of the augmentation

may be identified with the s e t of

1.1 : A

-

of A

F

A

--+

CASE

333

, we

know that D i s t ( E ( L ))

which vanish on a power of J .

k

Accordingly we get a canonical injection

y : Dist(g(1))

M L"% v (A,k)

(1);

= UJ']'

in view of the definition of the convolution product and the f a c t t h a t m u l t i -

U[pl(L)

plication in

y i s a homomorphism of unital algebras. One verifies t h a t

transpsition,

y BQj = j

and the coproduct of A correspnd t o one another by

by writing out the definitions of y and

i s the identity and t h a t $

1

. It

pL

follows that y B

k

is bijective.

E(l) is 5 1 . If fEA , we have S'(x)(fp) = 0 for each xELie( E ( L ) ) , because 6 ' ( x ) is a derivation. Since L i e ( E ( 1 ) ) generates D i s t ( E ( t ) ) , we therefore have 6 '(u) f'" = 0 f o r each u E D i s t ' ( E-( 1 ) ) . By duality, this gives f p = 0 i f f ( 1 ) = 0 , and E ( 1 ) has Finally we show that the height of

height 5 1

.

3.10

P r o p s i t i o n : L e t X be a k-scheme. Then for each homomorphism

-

of Lie p-algebras p :

E(L)opp

such t h a t

J, : k

&t(g)

J,= p'

Proof : By 3.9

0

clL

.

-+

Der (9 )

-X

, there

-

meets the conditions of prop. p :

g([)opp -+

5

4 ,7.2,

Aut

Dif _X

v(1.1)(1) = ~ ( 1 . 1 1

5

4

(X) . Assuming

the image of l?

this, given

is a unique

v Be j = J,

. If

v

the notation of this p r o p -

1.1 € D i s t ( E ( 1 ) )

such t h a t

and

7.2 ) is a subalgebra of D i s t ( g ( k ))

3.11

such t h a t

it i s associated w i t h a (unique)

sition, one shows easily that the s e t of

(

, there

b) and the universal property of U[pl(l?)

algebra homomorphism v : D i s t ( E_(k')) homomorphism

is a homomorphism

(thus correspnding t o a right E_(L)-operation on _X )

, it

. Since this

coincides with D i s t ( g ( L ) )

subalgebra contains

and the assertion follows.

Theorem 3.5 i s now an inrmediate consequence of 3 . 1 0 . 'p

: 1 -+ Lie(G_)

l e t I$ be the cornpsition

To see

Au;EBRAIc GROWS

334

9 7,

110 4

the corresponding hommrphism. Since 6'p (x) is

p : _E(P)opp-+ & t

and

11,

,

l e f t translation invariant €or each xEk? a E E ( l ) (k)

P (a) (9g') = 9 p (a) (9') for

,

so i s p

g,g' E C,

served by changes of base. It follows that

;

hence we have

, and

the property i s pre-

f : a + - +p ( a ) ( l ) i s the required

unique hommrphism. Corollary:

3.12

Let E & & ' be two

L i e p-alqebras which are

also f i n i t e l y qenerated projective k-modules. Then the map which assigns t o cp E L i

(1,l') the unicpe hommrphism

&

Lie( E(cp) 1

o

at = aL, O rp

, is

_E((P)

a bijection

: _E(R) -+

L Ak ([,el)

_E(&') % 3

such t h a t

s k ( g ( l ? ) , g ( l ' ).)

Proof : Immediate from theorem 3 . 5 .

Section 4

Groups of height $ 1over a f i e l d

W e assume that k i s a f i e l d (of characteristic p )

.

proposition : The functor &t+g(C) is an equivalence between

4.1

the cateqory of f i n i t e dimensional Lie p-alqebras and the cateqory of alqe-

b

e k-groups of height 5 1

.

Proof : In virtue of 3.9 and 3 . 1 2 it is sufficient t o show that each algebraic k-group of height 5 1 is isomrphic t o a group E ( E )

.'

In f a c t mre ge-

nerally we have the Structure theorem for groups of height 5 1 : J z t G be a

4.2

k-group-scheme. Then the following conditions are equivalent : (i)

-

C, i s alqcbraic, G(k) = e , and the canonical homomorphism U

( L i e (G))-+D i s t (G)

i s algebraic,

(ii)

is bijective.

c(E) =

is generated by Lie(G) (iii) G

(iv)

e

.

, and

the unital alqebra

is algebraic and of height $ 1

.

a/

Dist(G_)

There exists a f i n i t e dimensional L i e p-alqebra such that is isomrphic t o

E(l)

.

11, § 7 , no 4

THE CH?J?ACIEFUSTIC p#O

For each k-qroup _H

(v)

, the

335

Grk(G,g)-+LAk(Lie(G_) , L i e @ ) )

canonical map

is bijective.

(3(G) is isomorphic t o the quotient of an alqebra

i s affine,

(vi)

of polynomials k [X1,.

.., X n ]

by the ideal generated by the

X p

.

Proof : (i)=> (ii) : Trivial. (ii)=> (iii): This is proved in the same way as the f i n a l portion of 3 . 9 .

(vi) => (iii) : By 1 . 6 . (i) => (v)

:

c f . the proofs of 3.10 and 3 . 1 1 .

(v)

:

Set

=> (iv)

R = Lie(G)

; (v) then applies t o give a h o m m ~ h i s m

. Aplyinq 3.5,

-f : G 4 g(R)

such t h a t

Lie(2) = a[

9 : G ( L ) -+

such t h a t

Lie (9) - 0 a[ = Id

-g o f

= Id

G_

,

(iv) => (vi)

f - " =~ Id :

.

4G)

, and

basis for the k-vector space m mod m2 ~p

such t h a t cp (X.)

G = E ( L ) , we

1

(v) again and 3.5, we get

F i r s t l e t G_ be an algebraic k-group of height 2 1

be the augmentation ideal of (1.6)

. Applying

we get a hommrphism

: k

=

have

[xl,- - - rXnl / (XiP )

m

i'

[

+

let

n = [m/rn2 : k]

, there

. If

, let m

(m.) i s a

i s a surjective hom&rphism

J(G)

I f L is a f i n i t e dimensional L i e p-algebra and i f

L : kl

[8(G): k l

=

[m/m2 : k]

. Hence, by

= [nist(C,):k] =

3.9 and 3.6, we have

[UJ["](L) : k ]

= p"

and cp i s bijective. (iii)=> (i) :

such that

Set

l? = L c i e ( G ) a2d consider the hommrphism f

L i e ( _ f j 0al? = Idl?

. Assume the notation of

and set mi= &f)(mi)E @ ( E ( f ) ) ; the homomorphism k [X1

,..., X n ] / (X ip )

into

d(E ( L ) )

sends X

i

: E ( L ) --t G_

the preceding paragraph d(f)Ocp

of

onto mi and is therefore bi-

jective by what we have already seen. Since cp i s surjective, cp and are bijective, and

4.3 a)

d(f)

i s an isomorphism.

Corollary:

Let G

be a locally alqebraic k-qroup.

The map F t++Lie(H) induces a bijection of the set of subgroups of

of height 5 1 onto the set of sub-Lie-p-algebras

of

Lie@)

. If

H_ a d

Ji

ALc;EBRAIc Gwxrps

336

11,

are two subqroupsAf G and i f H has heiqht 2 1

.

Equivalent t o L i e (H) c L i e (K) b)

Irf g1

, the

inclusion _HHJ

and -f2 are h o m r p h i s m s of G into a k-qreup G’

heiqht 2 1

, then

c) I f p :_G-+&(V)

f = f, 1

iff

Lie(fl) = L i e ( _ f 2 )

9 7 , no 4

.

&

and i f G has

5 stable under 5

is a f i n i t e dimensional linear representakion of

, then a

and if _G has heiqht 2 1

i f f it is stable under Lie(_G)

vector subspace of V is

. W e have

V- = &e(’);

moreover p i s simple

orsemisimple i f f L i e ( p ) has the same property. d)

If -

i s a subgroup of _G of heiqht 2 1

Lie(Cent (HI 1 GLie

e)

gf

=

(mG (El I = -

_G has height 5 1

I

I

we have

CentLie(G) (Lie(;)) Nonyie

( L i e @) I

the map & + L i e @ )

I

.

is a bijection of the set Of

normal sdqroups of G onto the set of p i d e a l s of Lie(G) (_a p-ideal behq ~ [ P ’ E I1. ____. a subspace 1 such that xELie(g) and y e 1 mly [x,y] EI The proof of this result is similar t o t h a t of

5 6 , 2.1

.