Codisjunctors and singular epimorphisms in the category of commutative rings

Journal

of Pure and Applied

Algebra

53 (1988) 39-57

39

North-Holland

CODISJUNCTORS AND SINGULAR EPIMORPHISMS CATEGORY OF COMMUTATIVE RINGS

IN THE

Y. DIERS DPparremenf

de

Valencie~nes,

France

Communicated Received

by P.T.

IS November

The calculus expect

MathPmatiques,

But one cannot

universal

algebra

extensive

making

elements.

is an important

construction

to an important

see which

one a priori.

a set of elements This new universal

study of it in the category

we will see emerging

Sciences,

Universitk

de

Valenciennes,

59326

Johnstone

correspond

algebras.

the invertible

de

1986

of fractions

that it would

U. E.R.

algebra.

in the category

If it is true that an algebra

invertible,

One might

of commutative of fractions

one still has to know what corresponds

construction

of commutative

in commutative

construction

unitary

will be revealed rings. Alongside

is a to

and we will make an the rings of fractions

‘rings of quotients’.

0. Introduction The calculus of fractions is an important construction in conlmutative algebra. One might expect that it would correspond to an important construction in the category of commutative algebras. But one cannot see which a priori. If it is true that an aIgebra of fractions is a universal algebra making a set of elements invertible, one still has to know what corresponds to the invertible elements. This new universal construction will be revealed and we will make an extensive study on it in the category of commutative unitary rings. Alongside the rings of fractions we will see emerging ‘rings of quotients’. In any category, a pair of morphisms (g, h) : CZA is said to be codisjoint if any morphism u : A -+A’ which satisfies ug = uh, necessarily has, as its target, a final object. One says that a morphism f: A -+B codisjoints a pair (g, h) : CZA if the pair (fg,fh) is codisjoint. A codisjunctor of (g, h) : CZA will be a universal morphism which codisjoints (g, h), that is, a morphism f: A -+ B which cod&joints (g, h) and which factorises in a unique way any morphism which codisjoints (g, h). The pair (g, h) is said to be cod~sju~ctable if it has a codisjunctor. In the category CRng of unitary commutative rings, one extends the terminology to ideals of an object A by identifying an ideal Zof A with the pair of canonical projections (r,, rz) : R GA of the congruence R module I on A. By associating to the pair (g, h) : CZA, the ideal Z(g, h) of A generated by the elements g(-u) - h(x) for XE C, 0022-4049/8X/$3.50

?‘ 1988, Elrevier

Science

Publishers

B.V. (North-Holland)

40

Y. Diers

one leads the study of the codisjunctor of (g,h) back to that of Z(g,h). It is then enough to study the codisjunctable ideals and their codisjunctors. Any principal ideal (a) of A is codisjunctable and has, as a codisjunctor, the ring of fractions A +A[@- ‘1. Any projective ideal of finite type Z of A is codisjunctable and has, as a codisjunctor, A -+ 1% Horn, tlGN

the localisation

morphism

(Z”, A)

of A for the I-adic Gabriel topology. One extends this result to n-projective ideals of A: these are the ideals Z of A such that the A-linear injection mapping i: Z”+Z is projective, i.e. such that, for any A-linear surjective mapping k: E+F and any A-linear map a : I+ F, there exists an A-linear map p : In + E, satisfying c-wi = k/l. One shows that an ideal of finite type with zero annihilator, is codisjunctable if and only if it is n-projective for some integer n. One obtains from this a characterisation of codisjunctable ideals of finite type. Any ideal is codisjunctable if and only if it contains a codisjunctable ideal of finite type with the same radical. One shows that a codisjunctable ideal of A is exactly an ideal Z such that the open set D(Z)= {p E Spec(A): ZQp} of Spec(A) is affine, and its codisjunctor is the restriction morphism Q! : A +A(D(Z)) of the structural sheaf A of A on the open set D(Z). A singular epimorphism is a morphism which is the codisjunctor of a pair of morphisms or, what amounts to the same, the codisjunctor of an ideal. One shows that these are precisely the epimorphisms f: A -+B which make B into an A-algebra of finite presentation and a flat A-module. In terms of relations, they are characterised by the existence of a sequence of elements et, . . . , ek of A and of a sequence of elements 6t, . . . , bk of B such that (a) (b) (c) These The

b,f(e,)+...+bkf(ek)=l, VbEB, Bn~thl, ViE[l,k], bf(er>Ef(A), V’a E ker(f), 31n E N, V’i E [l, k], aer = 0. are ‘rings of quotients’. notion of simultaneous codisjunctor

of

a set of

pairs

of morphisms

(g, h) : C3A with a variable source C and fixed target A, is an immediate extension of the notion of codisjunctor of a pair. It is a universal morphism f: A + B which simultaneously codisjoints all the pairs of the set. One is led back, as previously, to the notion of simultaneous codisjunctor of a set of ideals of A. Any set {(s): s ES} of principal ideals of A is simultaneously codisjunctable and has, as a simultaneous codisjunctor, the ring of fractions A +A[S-‘I. A set 9 of ideals of A is said to be N-projective if, for any ZE 4, there exist It, . . . , Z, E 9 such that the A-linear inclusion I. I,. ... . Z,+Z is projective. One shows that any IN-projective set 9 of ideals of finite type of A is simultaneously codisjunctable and has, as a simultaneous codisjunctor, the morphism A-+

Ii@ Hom,(Zt (I,, .. . . I,,)E.Y”)

. ..Z.,A)

Codisjunctors

and singular

41

epimorphisms

that is the localisation of A for the Gabriel topology generated by 9. One shows that a set 9 of finite type ideals of A with zero annihilators is simultaneously codisjunctable if and only if it is N-projective. One deduces from this a characterisation of simultaneously codisjunctable sets of ideals of finite type. For any set 9 of ideals D(Z), the set of prime ideals of A which do not of A, one writes D(S)= n,,, contain any ideal of 4. One shows that 9 is simultaneously codisjunctable if and only if the set D(9) is an affine subset of Spec(A), and that the simultaneous codisjunctor of 4 is then the restriction morphism Q,~ :A -A@(s)) of the structural sheaf a of A to the subset D(S). A semi-singular epimorphism is a simultaneous codisjunctor morphism of a set of pairs of morphisms with the same target, or, what is equivalent, a simultaneous codisjunctor of a set of ideals. One shows that these are precisely the epimorphisms f: A +B which make B a flat A-module.

1. Codisjunctors Let A be an arbitrary

category.

Definition 1.0. (1) A pair of morphisms (g, h) : CZA with the same source and target, is said to be codisjoint if any morphism u : A +X that satisfies ug = uh necessarily has, as its target X, a final object. (2) One says that a morphism f :A -*B codisjoints a pair of morphisms (g, h) : CZA if the pair (fg, fh) is codisjoint. If the category A has no final object, a pair (g, h) : CZA is codisjoint if and only if there exists no morphism u : A +X satisfying ug= uh. If A has a final object, denoted 1, the existence of a pair of codisjoint morphisms implies that 1 is a strict In that case, a pair final object, i.e. any morphism U: 1 +X is an isomorphism. (g, h) : CZA is codisjoint if and only if the morphism A + 1 is the coequalizer of (g,h). In a category with a nonstrict final object, exists no pair of codisjoint morphisms. Definition

1.1. A codisjunctor

for instance

of a pair of morphisms

(g, h) : CZA

f: A + B which codisjoints which codisjoints A codisjunctor morphism.

a null object,

is a morphism

the pair (g, h) in such a way that any morphism (g,h) is factorised in a unique way through f.

of (g, h) is an epimorphism

which is defined

there

u : A AX

up to a unique

iso-

Definition 1.2. (1) A pair of morphisms is said to be codisjunctable if it admits a codisjunctor. (2) An object A is said to be codisjunctabfe if the coproduct of A with itself exists and the pair of injections A ZA flA is codisjunctable.

42

Y. Diers

Proposition

1.3. In a finitely cocomplete category, an object A is codisjunctable

and only if any pair of morphisms (g, h) : A ZC Proof. Let A be a codisjunctable

object

if

with source A is codisjunctable.

and d: A u A -tD be the codisjunctor

of

the pair of injections (q, r) : A 3 A u A. For a pair of morphisms (g, h) : A 3 C, we shall denote by k: A u A + C the morphism defined by kq =g and kr = h. Let (f:C+B,e:D+B) be the pushout of (k:AUA+C,d:AUA-+D). The pair (dq, dr) is codisjoint, therefore the pair (fg, fh) = (fkq, fkr) = (edq, edr) is codisjoint. Therefore the morphism f codisjoints the pair (g, h). Moreover, any morphism u : C-X which codisjoints (g, h) determines a morphism uk : A UA +X which codisjoints (q, r) and factorises in a unique way as the composite of d and a morphism u : D + X. The relation ud = uk implies the existence of a unique morphism w : B + X satisfying wf = u. The morphism f is thus a codisjunctor of (g, h) and, consequently, the pair (g, h) is codisjunctable. Conversely, if any pair (g, h) : A 3 C is codisjunctable, then, in particular, the pair (q, r) : A ZA u A is codisjunctable and the object A is codisjunctable. 0 Corollary 1.4. In a finitely cocomplete category, any quotient object of a codisjunctable object is codisjunctable. Proof. Let A be a codisjunctable object and f: A- B an epimorphism. For any pair of morphisms (g, h) : BZ C, the codisjunctor d: C+D of (sf, hf) is also a codisjunc0 tor of (g, h). The object B is therefore codisjunctable. Given an object A, we will write (p,, p2) : A x A + A, for the product of A by itself. We recall that a relation on A is a subobject r : R-A x A of A x A which can be identified with the pair of morphisms (r,, r2) : RZA defined by r, =plr and r2 =p2r. Definitions 1.0-1.2 can be applied to relations. Recall also that a congruence on A, or effective equivalence relation on A, is a relation which is the kernel pair of a morphism f: A -+B [5, p. 161. If the category A is complete and wellpowered, then the families of relations on A admit intersections, the congruences on A are stable under intersections and, for any pair of morphisms (g, h) : CZA, the intersection of congruences R = (r,, r2) through which (g, h) factors is a congruence

on A, said to be generated by the pair (g, h).

Proposition 1.5. In a well-powered complete category, a pair of morphisms (g, h) : CZ A is codisjunctable if and only if the congruence R on A generated by (g, h) is codisjunctable. The pair (g, h) and the congruence R then have the same codi
k: C-R

Codisjunctors and singular epimorphisms

satisfies target

ufrl = ufr2 also satisfies X is a final

object.

ufg = ufrl k = ufr2k= ufh,

The pair (fr,,fr2)

is therefore

43

thus

is such

codisjoint,

i.e.

that

its

f codis-

joints (rr, r2). Conversely, let us suppose that f codisjoints (rl, rz). Let u : B-+X be a morphism satisfying ufs = ufh. The kernel pair (m, n) : KZA of ufis a congruence on A which factorises the pair (g, h). It therefore factorises the congruence (r,, r2) in the form (r,,r2) =(mu,no). The relations ufr, =ufmv=ufnu= ufr, imply that X is a final object. Therefore, the pair (fg, fh) is codisjoint, i.e. f codisjoints (g, h). It follows that f codisjoints (g, h) if and only if f codisjoints (r,, r2) and, consequently, that f is a codisj~lnctor of (g, h) if and only if f is a codisjunctor of (r,, r2). q Let us suppose that we are in the category CRng of unitary commutative rings and ring homomorphisms that preserve the unit. To any ideal I of a ring A, is associated the congruence modulo I on A, and one says that a morphism f : A --t B codisjoints the ideal I if it codisjoints this congruence. This property corresponds to the fact that the quotient ring B/f(Z)B of B by the ideal generated by f(1) is null, or, in other words, to the existence of elements et, . . ..e., of I and of elements b [,...,b, of B satisfying blf(e,)+ .a. + b,f(e,) = 1. The ideal I is said to be codisjunctabfe if the congruence modulo I is codisjunctable, and the codisjunctor of this congruence is called the codisjunctor of the ideal 1. For a pair of morphisms (g, h) : C3A, the congruence on A generated by (g, h) is the congruence modulo Z(g, h) where I(g, h) = {a(g(x) - h(x)): a E A and x E C). According to Proposition 1.5, a morphism f: A -+ B codisjoints the pair (g, h) if and only if it codisjoints the ideal I(g, h), and it is a codisjunctor of (g, h) if and only if it is a codisjunctor of I(g, h). Let us study the particular case of principal ideals. Let (a) be the principal ideal of A generated by the element a. A morphism f: A +B codisjoints the ideal (a) if and only if the element f(a) is invertible in B. A codisjunctor of (a) is therefore a universal morphism which makes the element a invertible. It is the canonical morphism A -+A[a-‘] from A to the ring of fractions of A relative to the multip~icative set generated by a. The principal ideals are therefore codisjunctable. Besides, for any pair of morphisms of the form (g, h): Z[X]GA, the ideal Z(g, h) = {a(g(P) - h(P)): a EA and PE Z[X]) is the principal ideal generated byg(X) - h(X). Any pair (g, h) : z[X] 314 is therefore codisjunctable and has, as a codisjunctor, the canonical morphism to the ring of fractions: A -+A[(g(X) -h(X))‘1. In other words, e[X] is a codisjunctable object of CRng.

2. n-projective

ideals

Following Zimmermann [S] and Leroux 161, if we have two morphisms f: A -+ B and g: C-+D in a category A, we say that f is g-projective if, for any morphism m : B-+ D, there exists a morphism n : A + C satisfying gn = mf. We say that a mor-

44

Y. Diem

phism f: A + B is projective if it is g-projective for all regular epimorphisms g of A. If we work in the category Mod(A) of modules on a unitary commutative ring A, we get Definition 2.0. A pair (I,J) of ideals of A is said to be projective if J is included in I, and if the A-linear inclusion mapping J+lis a projective morphism in Mod(A). 2.1. A pair (I, J) of ideals of finite type of A is projective if and only if there exists a sequence e,, . . . , ek Of &mentS Of I and a SeqUenCt? p,, . . . , pk Of A-linear mappings from J to A, such that Vx E J, x = pI (x)e, f ... + v)k(x)ek . Proposition

Proof. Let us suppose that the pair (1, J) is projective. Denote by f: J-Z the Alinear inclusion, and let us choose a sequence of elements e,, . . . , ek which generate the ideal I. We denote by (p, : Ak+A)iGIl,kl the product of the object A, k times by itself, and by g : Ak + I the A-linear surjective mapping defined by g(x,, . . . , xk) = C:=i xjej. There is, by hypothesis, an A-linear mapping v : J+Ak satisfying goy?=f. The A-linear maps pi=piq: J+ A are then such that, for any XE J, one has: x=f(x>=go~(x)=g(~l(x),

...9vk(X))=V)I(X)el

f

“’

+vk(dek.

Conversely, let us assume the existence of the sequences e,, . . . ,ek and pi, . . . , pk satisfying the above mentioned condition. Then J is included in I. Let us denote by f : J+ I the A-linear inclusion. Let g : E-t F be an A-linear surjection and m : I+ F an A-linear mapping. There exists a sequence uI, . . . ,uk of elements of E such that We shall define the A-linear map v, : J-E by g(ui)=m(e,), . . . . g(uk) =rn(e,). (D(x) = q,(x)u,

+ ... + V)k(x)uI, . For

g(u?(x)) =g((P,(x)u,

X E

+ ...

J,

We

have

+V)k(X)~k)=V?I(X)g(~l)+“‘+V)k(X)g(~k)

=

pl(x)m(el) + .e++~k(x)m(ek)=m(y,l(x)el+.“+V7k(X)ek)

=

m(x) = m(f(x)).

As a result we have gy, =

mf.

The pair (Z, J) is therefore

projective.

0

Proposition 2.2. If (I, J) and (K, L) are two projective pairs of finite type ideals of A, then the pair of ideals (I. K, J’ L) is projective. Proof. According to Proposition 2.1, there exist a sequence al, . . . , ak of elements of I and a sequence of A-linear mappings (a,, . . . , (ok: J+A which are such that VXE J, and ly,,...,vp:L+A x=y,,(x)a, + . ..+vk(x)ak. Similarly there exist b,,...,b,~K jE[l,p], XEJ, such that Vy E L, y = lyl (y)b, + ... + t,u,(y)b,. For any iE[l,k], ye L, we have: p;(xy) =Y~;(x)E L, therefore vj(p;(XY)) is defined and equals v/i(u?,(XY)) = I,u,(Y)~~(x). It follows that for any i E [l, k] and any jE [l, p] one can define an A-linear mapping Ivjv7; : J. L -+A by (vjpj)(z) = wj(~j(Z)), and its value for Z=X_Y with XEJ and YEL is (~,~i)(~~)=~j(Y)~i(X)=~j(~)~~(~). The family of elements of 1. K and the family of A-linear mappings (ai~~)(,j),[i,kix[i,P~ wjp, : J. L + A then satisfy:

Codisjunctors

VXEJ,

VYEL,

and singular

epimorphisms

xy= ci;,c C$.(x)a. ,)(i, I

45

Wi(Y)bj) =C;,~;$;,l, wj(Y)%(x)a;b,

(k, P)

c

=

wjP,(xYwJ.

(i,j)=(l,

By linear extension it follows (1. K, J. L) is thus projective.

that 0

I)

VzeJ.

L, z= Ei::)EC,,,, vj~~(z)a,bj.

The pair

Corollary

2.3. If (I, J) is a projective pair of finite type ideals of A, then for any 0 n E iN the pair of ideals (I”, J”) is projective. Definition

2.4. An ideal I is n-projective for n E N* if the pair (Z, I”) is projective.

Proposition

2.5. (1) An ideal is I-projective if and only if it is projective. (2) An n-projective ideal is m-projective for nlm. (3) An ideal of finite type is n-projective if and only if there exist a sequence el, . . . . ek of elements of I and a sequence pl, . . . ,vk of A-linear mappings S-A such that VX E In, X = f$l, (X)e, + --. + V)k (X)+ . (4) If I is an n-projective ideal of finite type, then, for any p E IN, the pair (I”, I”“) is projective. Proof. (1) and (2) are immediate. (3) Follows from Proposition 2.1. (4) Follows from Corollary 2.3. q Theorem 2.6. Any n-projective ideal I of finite type in a unitary commutative A is codisjunctable and has as a codisjunctor the localisation morphism

ring

/,:A-+ l&r Hom,(I”,A) PE[h

of A for the I-adic Gabriel topology. Proof. We shall denote by @ the set of ideals of A containing a power of the ideal I. The set % is a Gabriel topology on A ([7, Chapter VI, $51 and [7, Chapter VI, Proposition 6.101) called the I-adic topology [7, Chapter VI, §4, Example 21. Following Gabriel [7, Chapter IX, 011, the A-module A(,,-, is defined by the filtering colimit

A(,-, = 1% Horn, (J,A) = Il,m Horn, (Ip, A) J Ed

[IE N

calculated in the category Mod(A) of A-modules. The A-module A(.$) has in fact the structure of a unitary commutative ring, the product of the element represented by lo : Ip + A by the element represented by v/ : I4 + A being the element represented

46

Y. Diers

by the A-linear map v. (D: Ip+q -+A defined by (w 1p)(x) = y(p(x)) 17, Chapter IX, $ I]. The morphism of unitary rings II : A -AcBj associates to the element a the element represented by the dilatation corresponding to a i.e. the morphism given by multiplication by a. As the ideal I is n-projective, there exist e,, . . . , ek E I and ‘b’xE/“, x=~t(x)er + “’ + $?k(+?k (Proposition 2.5.). cP,,...,V)k:In +A satisfying The elements el, . . . , f?,+define dilatations h,,, . . . , h, representing the elements l&r), . . . , I,(Q) of A(S). The A-linear maps pI, . . . ,v)~ represent elements @I, . . . , pk of Ac,9). For every element x of I”, one has 1, (x) = (~,oh,, + es. + qkohrk)(X). This implies that the relation 1 = @til(el)i .*’ +qkIf(ek) is satisfied in Ac:PI. Consequently, the morphism 1, codisjoints the ideal I. Let us show that 1, is a codisjunctor of I. Let us consider first a morphism f: A -+ L which codisjoints I and whose target L is a local ring. We denote by M the maximal ideal of L. The idea1 of L generated by f(1) is not included in M, therefore there exists some a E I such that f(a) EL -M, which means that f(a) is invertible in L. For an A-linear map v, : I”+,4 we write g

(p)

=

P

fcP(aP)) f(aP) .

The vatue g, (lo) does not depend on the choice of the element that f(b) is invertible, the relations

imply

a, since for b E I such

the relation f(M”)) ------=_

f@‘)

f(9(aP))

.ftaP) ’

One then defines an A-linear mapping gP: Horn, (ZYA)-+L by associating to cp the element g,(p). For p = 0, the A-module Horn, (I’, A) = HomA (A, A) can be identified with the A-module A, and the A-linear mapping go can be identified with f. Let us suppose that p I q E N and write q =p + m. The A-linear inclusion Zq -+ Zp induces, by composition, the A-linear mapping a4,, : Horn, (Ip, A) -+ Horn, (Zq,A). The relations g (a 4

4P

fp))=f(v(aq)) -= f(aq)

_f(v(a“‘“N =f(a’“dapN =f’(a’Vf(cpW)

f(aP+“>

f(amaP>

f(a’“If(aP>

= g,. The family of morphisms (gP : Horn, (14 A) --t ~5)~~M imply the equality g,a, therefore defines an inductive cone and an A-linear map g : AC.+)-+ L. The mapping g is a morphism of unitary rings according to the relations g2 P

(wco>

=f(v(ul(azP)N =f~v(apMaPN f(a2”) f(ap)f(ap)

Codisjunctors

and singular

=f(W(aP))f(~(~P))=g f@">

f(aP)

epimorphisms

(w).

g

(u?)

/-’ ’

p

and the relation

Moreover, the morphism g satisfies the relation gl,=g, =f. The morphism f thus factorises through the morphism II. Let us show that this factorisation is unique. Let m : A(,-, -+ L be a morphism which satisfies mZ1= f. Let p E IN and p E HomA (I’, A). of A which respectively multiply by ap We denote by h,,,, hrpCufljthe dilatations and by ~(a”). For XEA, we have y(hoij(x)) = v(aPx) =xp(d’) = hvp(a,Jl(x).It follows by q, we that poh,il = hrpCojJj.If we denote by @EA~.~) the element represented have u,. [,(a”) = I,(cp(a”)). Consequently, r7r(@)m(Zl(aP)) = m(Z,(~(a~))), therefore m(p). f(a”) =f(cp(cP)) and m(@) =f(p(a”))/f(aP). The morphism rn is therefore equal to the morphism g. Next, let us consider an arbitrary ring B. For every prime ideal q of B, we denote by /q: B+ B, the localisation of B at q. We know that the morphism (Z,) : B+ in CRng. It follows that if f,g: A,,,,zB n qtSpecCBjB, is a regular monomorphism are two morphisms satisfying fl,=gl,, then, for every qE Spec(B), we have I,fll = I,gl,, thus lqf = I,g and as a consequence, f = g. The morphism I1 is therefore epimorphic. Finally, let us consider a morphism f: A + B which codisjoints I. For every q E Spec(B), the morphism f,f : A + B, codisjoints I and has as target a local ring. It can thus be factorised through I, in a morphism g, : A(,‘*-)+ B,. The morB, then satisfies (g,)/[= (/,)f. Since i, is epimorphic phism (g,) : A(S)* IITYESpec~B~ and (/,) is regular monomorphic, there exists a morphism g:A(,*,-)-t B satisfying g/,=f. This completes the proof that the morphism 1, is a codisjunctor of I. morphism Now, we are left to prove that I, : A +A(,,) is precisely the localisation of A for W. Recall that the torsion of A defined by r(A)=

U

radical

{XEA:

Ann(Z”)=

of A associated

2pdN,

to ,% is the ideal f(A)

Zpx=O},

PEN

and the localised A-module

module

corresponding

A.,- = (A/t(A))C,+-, = I$ JE.7

Horn,

to A, for the Gabriel

(J,A/f(A))

= I&I Horn,

topology

@, is the

(Zp, A/r(A))

j7EIPd

[7, Chapter IX, Q11. The canonical morphism q : A -A/t(A) naturally defines a morphism q(,+) : ACsj --f (A/f(A))(,,-, This morphism is a monomorphism since Z(A) is the kernel of the morphism Zc,F, [7, Chapter IX, Lemma 1.2.1. Let us show that it is also surjective. Consider an element of (A/t(A))(.,-, represented by v/: P-, A/t(A). As the ideal Z is n-projective of finite type, the pair of ideals (ZP,Z”p) is

48

Y. Diem

projective (Proposition 2.5.) and, consequently, if we linear inclusion, there exists an A-linear mapping v,: qv = V/U.Then v, represents an element of A(.@), whose ment represented by w. It follows that q(.i*jinduces an and A,+, and consequently, localised ring of A for 9.

3. Codisjunctable

that the morphism 0

write cz : Znp-+ZP for the AZ”“+A which is such that image under q(.$, is the eleisomorphism between AGr)

I, : A *AC,“*‘) is indeed

the Gabriel

ideals

Theorem 3.0. For an ideal I of a unitary commutative tions are equivalent : (1) The ideal I is codisjunctable.

ring A, the following asser-

(2) The radical of I is a codisjunctable ideal. (3) The ideal I contains a codisjunctable ideal of finite type with the same radical as I. (4) The set D(Z) = {p E Spec(A): ZQp] is an affine open subset of Spec(A). The codisjunctor of I is then the restriction morphism et : A-A(D(Z)) of the structural sheaf of A to the open set D(I). Proof. (1) H (2). We denote by Z?(Z) the radical of I. Let f: A +B be a morphism of CRng. If f codisjoints I, the inclusion ZCR(Z) implies that f codisjoints R(Z). Conversely, let us suppose that f codisjoints R(Z). There exist e,, . . . ,e, ER(Z) and bl, . . . , b, E B satisfying b, f (e,) + ... + b,f(e,) = 1. There exists p E N such that ef, . . . , e,” E I. The relation (blf(e,) + ... + b,f(e,))“r = 1 is of the form clf(ef)+ ... + c, f (et) = 1, which implies that f codisjoints I. It follows that a morphism f : A -B is a codisjunctor of Z if and only if it is a codisjunctor of Z?(Z). (2) =) (3). Let f: A + B be a codisjunctor of I. There are et, . . . , e, EZ and b ,,...,b,EBsuch that blf(el)+ *.*+ b,f(e,) = 1. Let us denote by Z, the ideal of A generated by et, . . . , e, . The morphism f codisjoints IO and any morphism g : A+ C which codisjoints I,, codisjoints I, therefore it can be factorised in a unique way through f. The morphism f is therefore a codisjunctor of I,. The inclusion Z,CZ implies that R(Z,)CR(Z). Let p be a prime ideal of A which does not contain the ideal I. The localisation morphism I,,: A +A, of A at p, codisjoints the ideal I, consequently it can be factorised through f and therefore codisjoints the ideal Z,. It follows that IO is not included in p. Any prime ideal containing IO contains therefore I. As a result, we have the inclusion R(Z)cR(Z,J and hence the equality R(Z,) =R(Z). (3) = (2). Follows from the equivalence (1) ti (2). (1) * (4). Let f: A + B be the codisjunctor of I. Let q E Spec(B) and p =f - ‘(4). We denote by l,, : A -+A, the localisation morphism of A at p, and lq : B-B, that of B at q. We denote by & : A,+ B, the localisation morphism off at q defined by .&J = l,f.

Codisjunctors

(a) Let us show that the morphism codisjoints

I, the morphism

and singular

epitnorphisms

& is an isomorphism.

l,f =&,I,, codisjoints

49

Since the morphism

I. As the morphism

f

& is local, it

preserves proper ideals. Therefore, the morphism Ip codisjoints I and can be factorised through the morphism f in a morphism m. The morphism f being epimorphic, it follows that l,=yqm. As the morphism & is local, there is a morphism n : B,-tA,, such that nl,= m. It is easy to prove that y1 and & are inverse to each other and, as a result, & is an isomorphism. (b) By the localisation property of flatness [l, Chapter Ii, $3, Proposition 151, it follows from (a) that the morphism f: A -+B makes B into a flat A-module. (c) Let us denote by Spec(f) : Spec(B)-+Spec(A) the mapping defined by Spec(f)(q) =f-‘(4). One knows that Spec(f) is a continuous mapping. We are going to show that it induces a homeomorphism between Spec(B) and its image in Spec(A). Let J be an ideal of B. We write K=f ‘(J) and we denote by L the ideal of B generated by f(K). Then L C J. We denote by qK : A + A/K, qJ: B+ B/J, qL : B-, B/L, h : B/L + B/J the canonical morphisms and g : A/K-B/L, m : A/K-, B/J the morphisms which satisfy gq,= qLf and mqK= qJf. Then (g, q,.) is the pushout of (qK, f). Consequently, g is an epimorphism. It follows that (l,,,, h) is the pushout of (m,g). Since the morphism f makes B a flat A-module, m being a monomorphism implies that h is a monomorphism and therefore an isomorphism. It follows that L = J. The ideal J is thus generated by f(K) and the pair (m, qJ) is the pushout of (qK, f). As a first consequence of this, we note that the mapping Spec(f) is injective since, if q, TE Spec(B) are such that Spec(f)(q) = Spec(f)(r), then f(f-‘(4)) and f(f-‘(r)) g enerate equal ideals in B, therefore q = r. Next, we find that for any QgSpec(B) and P=f-‘(Q), if we denote the canonical morphisms by qr : A 4 A/P and B-B/Q, we have QED(J) * qQ does not factorise through qp does not factorise through qK es PE D(K) @ Spec(f)(Q) ED(K) G Q E Spec(f)- ‘(D(K)). It follows that D(J) = Spec(f)-‘(D(K)), which proves that Spec(f) induces a homeomorphism between Spec(B) and its image. (d) Let us show that the image of Spec(f) is the open set D(Z). Let q E Spec(B) and p =f ‘(4). By (a), the localisation morphism I,, : A +A, codisjoints I. Therefore ZCp, i.e. PED(Z). Conversely, consider an element PED(Z). Then ZCp. The morphism f,,: A-A,] codisjoints Z and, therefore, it can be factorised through f in a morphism m : B+A,. If we denote by Mp the maximal ideal of A, and q=m-‘(M,), then Spec(f)(q)=f-‘(q)=f~‘(m-‘(M,))=l,-’(M,~)=p. Therefore qQ

qJ

:

@

p E WSpec(f )). (e) From what was proved previously, it follows that the morphism of affine schemes (Specdf), p) : (Spec(B), B) --f (Spec(A),a) [4, Chapter I, 1.6.5.1 induces an isomorphism of schemes between (Spec(B), @ and the scheme (D(Z), a/D(Z)), the restriction of the scheme (Spec(A),A) to the open set D(Z). This proves that the open set D(Z) is affine [4, Chapter I, 2.1.1.1. The restriction morphism e, : A -A@(Z)) is then isomorphic to f. (4) * (1). Suppose that the open set D(1) is affine, that is, the scheme (D(Z),A/D(Z)) is affine, or equivalently, that there exists a ring B such that we have an isomorphism

Y. Diem

50

of schemes (D(I), A/D(I))= &XC(B), E). The restriction morphism on the open set D(I) defines a morphism f : A -+ B such that Spec(f) : Spec(B) -+ Spec(n) induces a homeomorphism between Spec(B) and D(I) and such that, for any q E Spec(B), the We will show that ,f: A + B is a fibre morphism & : A/ I(~)+B, is an isomorphism. codisjunctor of I. Let A4 be a maximal ideal of B. The ideal p =f- ‘(Mj belongs to D(1), therefore Icfp and f(l) KM. The image f(l) is therefore not contained in any maximal ideal of B, consequently f codisjoints the ideal I. Let us show that f is an epimorphism. Let (g, h) : B3 L be a pair of morphisms of CRng satisfying &f==hf with target a Local ring L. We denote by Mthe maximal ideal of L. The prime ideals g.-‘(M) and h-‘(M) have the same inverse image p under f. They are therefore equal and we write q =g- ‘(M) = h ‘(M). The localisation morphism /4 : B-t B, factorises the two morphisms g and fi in the form g = g, /q and h = h, lq. The relation lc8f =fq14, implies the relations h,&f,t,= h,Z,f =g,t,f =g,&l,. Since /,, is an epimorphism and f4 is an isomorphism, it follows that hr =gl and h =g. This proves that f is an epimorphism since any object of CRng is a sub-object of a product of local objects. Let g : A -+C be a morphism which codisjoints I. Let rE Spec(C). We denote by f, : C-+ C,. the morphism localising C at r and M, the maximal ideal of C,. The composite morphism I,.g codisjoints the ideal f. Therefore (t,.g)(I)stM,, and g-‘(r) ED(I). The image of the so IQ(/,.g)-‘(M,)=g-‘(Z~~‘(M,.))=g~’(r), mapping Specfg) is therefore included in D(1). The morphism of schemes (Spec(g), & : (Spec(C), c) + ((Spec(A), A) then induces a morphism of schemes (Spec(C), a -+ (D(Z),a/D(r)) and consequently a morphism of rings h : B-t C which satisfies hf =g. U

4. Codisjunctable

ideals of finite type

Recall that the annihilator of an ideal I of A is the ideal Ann(Z) = {x E A : xl= {0} } and that an ideal I is said to be dense if its annihilator is nul1. Xt is immediate that if I is a dense ideal, then, for any n E R\I,the idea1 I” is dense. Proposition 4.0. A codisju~cta~le ideal of finite type is dense if and only if its codisjunctor is a monomorphism. Proof. Let 1 be a codisjunctabie ideal of finite type of A, generated by e,, . . . , ek, and let f: A -+B be its codisjunctor. For every i E (1, k], the canonical morphism le, : A +A [e? ‘1 codisjoints I, therefore can be factorised through f. Let us suppose that I is dense and let XE ker(f). For every iE [1,/c], we have /&)==O, therefore there exists p;~ N satisfying xep = 0. If we write p = sup(p;: itz [l, k]), then we have xef = 0 for any i E [ 1, k]. Since the ideal I @ is generated by the elements ep, . . . ,e$, it follows that xIkP= (01. Since the ideal Ikp is dense, the element x is null. The codisjunctor f is therefore a monomorphism. Conversely, let us suppose that f is a monomorphism. Let XEA be an ele-

Codisjunctors and singular epimorphisms

ment bJ(et)

such that xl=

(0).

Since f codisjoints

I, there

51

exist bi, . . . . bkcB

+ ... + bkf (ek) = 1. For any i E [ 1, k], we have xe; = 0. It follows

satisfying that

f(x)=f(x)(b,f(e,)+...+bkf(ek))=blf(xe,)+...+b,f(xek)=0. Therefore

x= 0. The ideal Z is therefore

4.1. A dense ideal of finite n-projective for some integer n.

Theorem

dense.

0

type is codisjunctable

if and only if it is

Proof. By Theorem 2.6, any n-projective ideal of finite type is codisjunctable. Conversely, we look at a codisjunctable dense ideal Z generated by e,, . . . , ek. By Theorem 3.0, the codisjunctor of I is the restriction morphism Q, : A *~(D(I)). For every p E IN, we define the A-linear mapping op : Horn, (14 A) -A@(Z)) by

For p =0 and Z’=A, the A-module Horn, (A,A) can be identified with the Amodule A, and the morphism a0 : Horn, (Z’, A) -a(D(Z)) can be identified with the morphism @I : A +A(D(Z)). The family of morphisms (ap)l-‘E N is in fact the family of arrows of an inductive cone (ap : HomA (Zq A) +a(D(Z))),. ,,,,which defines an A-linear mapping a : 1% HomA (Zp, A) -A@(Z)). PfN

Let us show that (Y is a monomorphism. For every iE [ 1, k], the element (&eP))/e”

Let v,: Z”+A be such that is null in A[e, ‘1, therefore

or,(v,)=O. there are

q,E N such that eFy,(ef)=O,

so v(e,?+Y’)=O. On writing q=sup{qj: iE [l,k]}, we for every iE[l,k]. It follows that, for any XEZ’~+~“(, we have have p(e, ptq)=O v(x)=O, i.e. the restriction of ~7 to Z(P+q)k is null. This proves that the morphism a is a monomorphism. Now we show that a is surjective. Consider an element s=(a,/ef, . . . . ak/e[) of A@(Z)). If we choose p big enough, the relations a,eg= tzjey are satisfied for every (i, j) E [ 1, k] x [ 1, k]. Then for every i E [l, k], we have

e,?s= It follows that VXEZ~~, XSEQJA). The morphism ec being injective (Proposition 4.0), there exists, for each XEZ~~, a unique element p(x) of A such that xs= @{(p(x)). In this way, we define an A-linear mapping p : Zkp+ A. For i E [I, k], the relations el(v(e,kp)) = eys = e,(e,ka;) imply q(e,kp) = e,kai. The equality okp(p) = s follows. The morphism cr is therefore surjective. Since the morphism Q[ codisjoints the ideal I, there exist s,, . . . . sk E A@(Z)) satisfying st @t(et) + ‘.. + s,@,(e,) = I. The elements s , ,..., s, are of the form s,=cx,(~,) ,..., sk=a,(pk) with V, ,..., qkE Horn, (I”, A). We denote by h+, . . . , h,, : Z‘*Z” the restrictions of the dilatations

52

Y. Diers

with respective multipliers e i, . . . ,ek. The relation a,,(f~+)@~(ei)+ ... +~&(rp,)&(e~)=l can now be written a,(plOhel + ... + q)kohek) = 1. There exists then an integer m E N such that Vx E I”‘, pl (x)e, + -1. + pk(x)ek =x. The ideal Z is therefore m-projective. 0 Proposition valence

4.2. If Z is an ideal of finite type of A, the set I= {x: x E I} of equielements of I in the quotient ring AlUnG, Ann(Y), is a dense

classes of

ideal called the dense ideal associated to I. Proof.

It is immediate

that 1 is an ideal of A/U,~,,

Ann(Z”).

Let RE Ann(Z)

and

I. For each i E [l, k], we have X4= 0, el, ..-, ek a sequence of elements generating therefore we have njE N such that xe; E Ann(Z”‘). On taking n = sup{n;: iE [l, k]}, we have, for every ie [l, k], xejE Ann(Z”), so xeiZn = 0. It follows that xZn+’ = 0, therefore x E Ann(Z”+ ’ ), and x=8. The annihilator of I is therefore null and I is dense. 0

if and only if the dense ideal I associated to I is n-projective for some integer n E N *. The codisjunctor of Z can then be factorised through the quotient morphism q : A + A/U,,~ N Ann(I”) in a monomorphism which is a codisjunctor of 1. Theorem 4.3. An ideal of finite type I of A is codisjunctable

Proof. Let e,, . . . , ek be elements which generate I. Let f : A -+B be a morphism which codisjoints I and x an annihilating element of a power I” of I. There are elements b ,,...,bkeB satisfying b,f(e,)+... + bkf(ek) = 1. The relation (b,f(e,) + ... + bkf(ek))k” = 1 is of the form c,f(e;) + ... + ckf (ei) = 1. Since x6?,’= 0 for every iE[l,k], We have f(X)=f(X)(C,f(e~)+‘..+ckf(e~))=C~f(Xe;)+...+ckf(Xe~)=0. The morphism f thus can be factorised through the quotient morphism q :A + I. Ann(Z”) in a morphism g: A/U no N Ann(Y) -+ B which codisjoints I is such Conversely, any morphism g : A/U, EN Ann(Z”) --f B which codisjoints that the composite gq codisjoints I. The morphisms f : A + B which codisjoint Z are precisely those which factorise through q in a morphism g : A/UnEN Ann( B which codisjoints I. It follows that a morphism f : A + B is a codisjunctor of Z if and only if it can be factorised through q in a morphism g which is a codisjunctor of I. The ideal I is therefore codisjunctable if and only if the ideal I is codisjunctable. However, the ideal Zis dense and of finite type. Therefore the ideal fis codisjunctable if and only if it is n-projective (Theorem 4.1.). Moreover, by Proposition 4.0 0 the codisjunctor of 7 is monomorphic.

A’lLN

5. Singular epimorphisms Definition 5.0. A singular epimorphism pair of morphisms.

is a morphism

which is codisjunctor

of a

Codisjunctors and singular epimorphisms

In a well-powered tor of an equivalence

complete relation

category

any singular

(Proposition

epimorphism

53

is the codisjunc-

1.5.).

5.1. In the category CRng, any singular epimorphism is the codisjunctor of a radical ideal of finite type.

Proposition

of an Proof. Any singular epimorphism f: A +B of CRng, being a codisjunctor equivalence relation on A, is a codisjunctor of an ideal of A (notation of Section 0 1) and even of a radical ideal of finite type by Theorem 3.0. Theorem 5.2. For a morphism f : A + B of CRng, the following valent :

assertions are equi-

(1) f is u singular epimorphism.

(2) f is an epimorphism of rings which makes B a finitely presentable A-algebra and a flat A-module. (3) There exist e,, . . . , ek E A and bl, . . . , bk E B such that (4 b,f(e,)+...+bkf(ek)= 1, (b) VbEB, &ElN, ViE[l,k], bf(er)Ef(A), (c) V’aEker(f), &E/N, ViE[l,k], aey=O. Proof. (1) * (2). Let Z= (et, . . . , ek) be an ideal of A with codisjunctor f. The Aalgebra B is a filtered colimit of A-algebras of finite presentation. Therefore, there exists in the category A/CRng, a filtered colimit (aj : (Bj,fj)‘(B, f))j,, such that the A-algebras Bj are of finite presentation. Then (~j : Bj-t B)j~J is a filtered colimit in CRng. Since the morphism f codisjoints the ideal Z, there exist b,, . . . , b,,E B such that b, f (el) + ee. + b,f(e,) = 1. It is easy to prove that there exists an object j of J and elements b,j, . . . , 6, E Bj, satisfying bljfJ(e,) + ... + bkjJ;(ek) = 1. Therefore the morphism h codisjoints the ideal Z and, hence, it can be factorised through f in a morphism g : B + Bj. The relations ojgf= Oljfj =f imply ajg = 1,. The A-algebra B is then of finite presentation as a split quotient of the A-algebra of finite presentation Bj. It has been proved elsewhere, in the proof of Theorem 3.0 (‘(1) * (4)‘, part (b)), that the A-module B is flat. (2) =) (1). We shall use the notation of the proof of Theorem 3.0 ‘(1) * (4)‘. Let q E Spec(B) and p =f l(q). The relation l,f =yqlP, together with the fact that l4 and f are epimorphisms, implies that fq is an epimorphism. The localisation property of the flatness implies that the morphismfq is flat. It follows that & is local, epimorphic and flat. Therefore it is an isomorphism. Then we show, as in the proof of Theorem 3.0, that the mapping Spec(f) : Spec(B)+Spec(A) induces a homeomorphism between Spec(B) and its image Xin Spec(A). It follows that Xis an affine subset of Spec(A), that we have an isomorphism B=A(X), and that the morphism f is isomorphic to the restriction morphism ex : A -A(X), but A(X)=

1% A@(Z)) Xc D(I)

and

ex=

I&r XcD(I)

e1

54

Y. Diem

where @[:A -+A(D(Z)) is the morphism restricting to D(Z). Since the morphism f: A + B is an object of finite presentation in the category A/CRng, there exists an ideal Z satisfying XC D(Z) and a morphism g : A(X) +a(D(Z)) satisfying g-f= er. If p E D(Z), the morphism lP : A -+A, can be factorised through eI, therefore through f, and consequently, p EX. It follows that X=D(Z). Therefore X is an open affine set. According to Theorem 3.0, the ideal Zis a codisjunctable and has as a codisjunctor the morphism ex : A -A(X), isomorphic to the morphism f. The morphism f is therefore a singular epimorphism. (l)=,(3). Let Z=(ei,..., ek) be an ideal of A with f as codisjunctor. By Theorem 3.0, the morphism f is isomorphic to the restriction morphism e[:A+A(D(Z)). Since f codisjoints I, there exist bi, . . . , bk E B such that b&e,) + ... + b&e,) = 1. An element s of A(D(Z)) can be written as .S= (at/e:, . . . , ok/e;) with a;ej” = ajcr for every (i,j)~ [l,k] x [l,k]. Then

It follows that for any element b E B, there exists n E N such that bf(el) ef(A) for every i E [l, k]. Moreover, if a E ker(el), then (a/l, . . . , a/l) = 0 in A@(Z)), therefore there exists n E N such that e,‘a= 0 for every in [l, k]. So the morphism f satisfies conditions (a)-(c). (3) * (1). Let us show that the morphism f: A -+ B is a codisjunctor of the ideal Z generated by e,, . . . , ek. Condition (a) implies that f codisjoints the ideal I. Let g : A *L be a morphism which codisjoints I, with target a local ring L. If A4 is the maximal ideal of L, then g(Z) c M. Thus there exists a E Z such that g(a) is invertible. Let ye B. By condition (b) there exists n E N such that V’i E [l, k], yf(el) Ef(A). Since the ideal Ink is generated by e;, . . . , el, one has the inclusion yf (Ink) cf(A). We write m = nk. Then yf(a’“) Ef(A). So there exists an element x of A satisfying f(x) =yf(am). We write h(y) =g(x)/g(a”). Let us show that the element h(y) does not depend on the choice of m nor x. Let us suppose that f (x’) =yf (a”“). The relationsf(a”x’)=f(a’“)f(x’)=f(aM)yf(am’)=f(x)f(a’n’)=f(am;C>implya’i’x’-a”~~ ker(f). By condition (c), there exists p E N such that Vi E [l, k], (amx’- &x)ef = 0. Since the ideal Zkp is generated by ef, . . . , e,P, it follows that (amx’- a’n’x)Zkp = 0. Thus (amx’- ~“‘x)a@ = 0, so G”+~~x’= o”‘+~“x. Hence g(a”‘kp)g(x’) =g(a”“kp)g(x) and g(x)/g(a”)=g(x’)/g(a”‘). In this way, we define a mapping h: B-tL which is a morphism in CRng. For XEA, the relation f(xu”)=f(x)f(a”) implies h(f(x))= g(xa”‘)/g(a”) =g(x). So hf =g. The morphism g can thus be factorised through f. This factorisation is unique, since any morphism h which satisfies hf = g is such that g(x)=hCf(x))=h(yf(am))=h(y)h(f(am))=hCv)g(am) and thus h(y)=g(x)/g(a”). Since any object C of CRng is a regular subobject of a product of local rings, it follows first, that the morphism f is an epimorphism, and second, that any morphism g : A -+ C which codisjoints Z can be factorised in a unique way through f. The morphism f is therefore a codisjunctor of Z and, consequently, it is a singular epimorphism. 0

Codisjunctors and singular epimorphisms

6. Simultaneous

55

codisjunctors

The notion of simultaneous codisjunctors is an immediate extension of the notion of codisjunctors. If we have an object A and a set 0 of pairs of morphisms of the form (m, n) : XZA with variable source X, we say that a morphism f: A + B simul-

taneously codisjoints g if it codisjoints each of the pairs of % and a simultaneous codisjunctor of g is a morphism f: A -B which simultaneously codisjoints FZ and which factorises in a unique way all the morphisms which simultaneously codisjoint g. The set % is said to be simultaneously codisjunctable if such a simultaneous codisjunctor exists. If each pair of @Zis codisjunctable, then the set ‘8 is simultaneously codisjunctable and has, as a simultaneous codisjunctor, the cointersection epimorphism of the set of codisjunctors of pairs in g. But it is possible that the set @Zis simultaneously codisjunctable without each of the pairs of g being codisjunctable. If we suppose the category to be complete and well-powered, then each pair of morphisms of % generates an equivalence relation on A. If we write .9?? for the set of these equivalence relations, then a morphism f: A + B simultaneously codisjoints g if and only if it simultaneously codisjoints 5? (proof of Proposition 1.5). This implies the following: Proposition 6.0. A set F? of pairs of morphisms with target A is simultaneously codisjunctable if and only if the set 32 of equivalence relations on A generated by the pairs of g is simultaneously codisjunctable. The two sets % and ~52have then the same simultaneous codisjunctor. 0 If we work in the category CRng, to a set 9 of ideals of a ring A, we associate the set 99 of equivalence relations module the ideals of 9. We say that a morphism f: A -+B simultaneously codisjoints 9 if it simultaneously codisjoints %? or, what comes to the same thing, if it codisjoints each ideal of 9. The simultaneous codisjunctor of Z is then called the simultaneous codisjunctor of 9. Let us study the special case where 9 is a set of principal ideals. Let 9= {(s):s~S}.Amorphismf:A -+ B simultaneously codisjoints 9 if the elements f(s) are invertible for each s E S. A simultaneous codisjunctor of 4 is therefore a universal morphism making the elements of S invertible. It is the canonical morphism A-tA[K ‘1 from A to the ring of fractions of A relative to the multiplicative subset generated by S. Sets of principal ideals are therefore simultaneously codisjunctable. Definition 6.1. A set 9 of ideals of A is said to be N-projective if, for every 1~ &, there exists a finite sequence I,, . . . , Z, of ideals of 9 such that the pair (1, Z. I, . . . I,,) is projective. According to Proposition 2.1, a set 9 of finite type ideals of A is N-projective if and only if for every IE g, there exist a sequence e,, . . . , ek of elements of I, a

Y.Diets

56

sequence I,, . . . , Z, of ideals of 4 and a sequence CJJ,,. . . , cpk of A-linear mappings II, . . . [,-+A such that Vx~lf, . ..I., x=~,(x)el+...+~k(x)ek. Theorem 6.2. Any N-projective set 9 of ideals of finite type of A is simultaneously codisjunctable and has, as a simultaneous codisjunctor, the localising morphism

l.B: A -+ Ii&n Hom,(I, (II,..., I,,)E,‘I(P!)

. ..I.,A)

of A for the Gabriel topology generated by 9. From now on, we leave it to the reader to adapt to simultaneous codisjunctors, the previous proofs for codisjunctors. Cl

Proof.

If 9 is a set of ideals of A, we write LB(@) for the set of radicals of ideals of g andD(g)=n IE,I D(I), the set of prime ideals on A containing no ideal belonging to 9. Then 0(%(4))=D(S). Theorem 6.3. For a set LJ of ideals of a unitary commutative statements are equivalent : (1) The set 9 is simultaneously codisjunctable.

ring A, the following

(2) The set B(4) of radicals of ideals of 9 is si~lultaneously CodisjLinctable. (3) There exists a set 9, 01~ideals of finite type of A such that 9, is simultaneously codisjun~table and D(9 ) = D(@,). (4) The set D(.g) is an affine subset of Spec(A). The simultaneous codisjunctor of 9 is then the restriction morphism Ds : A -+ A(D(#)) of the structural sheaf of A to the subset D(9). El Proposition 6.4. Let 4 be a simultaneous codisjunctable set of ideals offinite type of A. The ideals of 4 are dense if and only if the simultaneous codisjunctor of JJ is a monomorphism. U Theorem 6.5. A set 4 of dense ideals of finite type of A is simultaneously

table if and only if it is N-projective. Proposition

codisjunc-

0

6.6. If .a is a set of ideals of finite type of A, the set s= {E ZE @ > of

ideais f= (x: x E Z> made up of equivalence classes of elements of I, in the quotient ring A/U (I,,,,,, I,,)E.Y”%’ Ann(Z1 .**I,,), is a set of dense ideals said to be associated f0 9.

q

of ideals of finite type of A is simultaneously codisjunctable if and only if the associated sef d of dense ideals is N-projective. The simultaneous codisjunctor of 9 can be factorised through the quotient morphism A + A/U (I,,,.., I,i)E,g(D‘) Ann(Z1 ... Z,) in a monomorphism which is codisjunctor for 3. 0 Theorem 6.7. A set 4

51

Codisjunctors and singular epimorphisms

7. Semi-singular Definition

epimorphisms

7.0. A semi-singular epimorphism

is a simultaneous

codisjunctor

phism of a set of pairs of morphisms with the same target. Any cointersection of singular epimorphisms is a semi-singular the converse is not true.

epimorphism,

In the category CRng, any semi-singular epimorphism codisjunctor of a set of radical ideals of finite type. 0

Proposition

7.1.

simultaneous

morbut

is a

Theorem 7.2. For a morphism f : A + B of CRng, the following

assertions are eguivalent : (I) f is a semi-singular epimorphism. (2) f is an epimorphism of rings which give B a flat A-module structure. (3) f satisfies the two following properties: (a) V~EB, Be, ,..., e,EA, ?Ib,,..., b,EB,

@) VxEker(f),

vie [Lnl, yfk)Ef(A);

1 and

,cIf(e;)b;=

Be, ,..., e,EA,

,$,f (e,)b; = 1

and

Zb, ,..., b,EB,

ViE[l,n],

xe,=O.

0

7.3. The singular epimorphisms in CRng are precisely the semi-singular epimorphisms f : A j B which turn B into an A-algebra of finite presentation. 0 Proposition

References [l] N. Bourbaki,

Algebre

[2] Y. Diers, Morphismes Fast.

Commutative disjoints

Paris, Publ.

1961) Ch. 1 and II. I.R.M.A.

de I’Universite

de Lille I, Vol. VIII,

3, No. 4, 1985.

[3] R. Godement,

Topologie

Algebrique

[4] A. Grothendieck

and J.A.

[5] P.T.

Topos

Johnstone,

[6] P. Leroux,

Theory

Injective

Munchen,

1969.

et Theorie

des Faisceaux

(Hermann,

Paris,

Elements

de Geometric

Algebrique

(Springer,

(Academic

Press,

d’effacement,

Rings of Quotients

Zimmerman,

Universitat,

Dieudonne,

Sur les structures

[7] B. Stenstrom, [8] W.

(Hermann,

et disjoncteurs,

Math.

(Springer,

Strukturen

London,

und

Berlin,

1973). Berlin,

1971).

1977).

Z. 121 (1971) 329-340. 1975).

M-injektive

Objekte,

Thesis,

Ludwig-Maximilians-