Local Algebra (Springer Monographs in Mathematics)

and Whence projdimA(xj M < sup (proj dimA[xj M[X], projdimAIXj M[X] sup (proj dimA M, proj dimA M = proj dimA M + 1,

+ I}

+ I)

qed. Finally, let us show that glob dim A[X] ~ glob dim A + 1 : Indeed, if m is an ideal of A such that htA m = dim A = glob dim A, we have globdimA[X] = dimA[X] > htAIXj(m[X],X} ~ htA[xj mIX] ~

htA m + 1.

+1

82

IV. Homological Dimension and Depth

Corollary.

If k is

11

field, k[Xl"" ,Xnl is regular.

Since every affine algebra is a quotient of a polynomial ring, we thus recover the properties of chains of prime ideals in affine algebras.

Remark. Let A = k[X1 , •.• ,Xnl/a where a is an ideal of the polynomial ring k[XI, ... ,Xnj. Let X be the corresponding affine variety. One says that X is non-singular if A is regular in the sense defined above. When k is perfect, the following are equivalent (cf. e.g. [Bour], Chap. X, §7): - X is non-singular; - X is smooth over k; - k' 0k A is regular for every extension k' of k.

4. A criterion for normality

Theorem 11. Let A be a noetherian local ring. For A to be normal, it is necessary and sufficient that it satisfies the following two conditions: (i) For every prime ideal p of A, such that ht( p) :::; 1, the local· ring Ap is regular (i.e. a field or a discrete valuation ring, according to ht(p) = 0 or 1). (ii) If ht(p) ~ 2, we have depth(Ap) ~ 2. Suppose A is normal, and let p be a prime ideal of A. If ht(p) :::; 1, Ap is regular (cf. Chap. III, prop. 9). If ht(p) ~ 2, let x be a nonzero element of pAp; then (loc. cit.) every essential prime ideal of xAp in Ap is of height 1; thus none of them is equal to pAp, which shows that depth(Ap) ~ 2 . Conversely, suppose that A satisfies (i) and (ii). If we already know that A is a domain, prop. 9 of Chap. III shows that A is normal. In the general case, one first proves that A is reduced (i.e. without nonzero nilpotent elements), and then that it is equal to its integral closure in its total ring of fractions. For the details, see [EGA], Chap. IV, tho 5.8.6, or [Bourj, Chap. X, p. 21, Rem. 1.

\

IJ: Relular Rlnp

83

5. Regularity in ring extensions In this section, A is a noetherian local ring, with residue field k, and B is a noetherian local ring containing A and such that m(A) C m(B). (Note that this condition is satisfied when B is a finitely generated A -module, thanks to Nakayama's lemma.) If B is regular, and A -flat, then A is regular.

Theorem 12.

By prop. 18, we have B

Q9A

Tor~(k, k) = Torf(B Q9 k, B Q9 k)

for all i.

Since B is regular, these modules are 0 for i > dim B. This implies that the same is true for Tor~(k, k) , because of the following lemma: Lemma 2.

If M is a finitely generated A -module, then: BQ9 A

M=O

=}

M=O.

Indeed, if M were 1= 0 , it would have a quotient which is isomorphic to k = A/m(A) and we would have B Q9A k = 0, contradicting the fact that m(A) is contained in m(B). Hence Tor~(k, k) = 0 for large i, which shows that A is regular. Theorem 13. Assume B is finitely generated as an A -module. (a) If A is regular, then: B is A -free {::} B is Cohen-Macaulay. (b) If B is regular, then: B is A -free {::} A is regular. Assume A is regular. By prop. 12, B is a Cohen-Macaulay ring if and only if depthA B = dim A, i.e. if and only if B is a Cohen-Macaulay A -module. By part ii) of cor. 2 to tho 9, this means that B is A -free. This proves (a) and half of (b). The other half of (b) follows from tho 12.

84

IV. Homological Dimension and Depth

Appendix I: Minimal Resolutions In what follows, we let A denote a noetherian lo~al ring, with maximal ideal m, and residue field k. Every A -module is assumed to be finitely generated. If M is such a module, we write M for the k -vector space M/mM.

1. Definition of minimal resolutions Let L, M be two A -modules, L being free, and let u : L -+ M be a homomorphism. Then u is called minimal if it satisfies the following two .conditions: a) u is surjective. b) Ker(u) c mL. It amounts to the same (Nakayama's lemma) to say that u: I -+ M is bijective. If M is given, one constructs a minimal u : L -+ M by taking a basis (ei) of the k-vector space M = M/mM , and lifting it to (ei)' with ei EM. Now let ... -

L

d i -

d

d

L1 -

.•. -

e

Lo -

M -

0

be a free resolution L. of M. Set: Ni

=

Im(Li

-+

L i - 1)

=

Ker(L i _ 1 -+ L i -

2 ).

Then L. is called a minimal free resolution of M if Li minimal for every i ~ 1 , and e: Lo -+ M is minimal as well.

-+

Ni is

Proposition 1. (a) Every A -module M has a minimal free resolution. (b) For a free resolution L. of M to be minimal, it is necessary and sufficient that the maps d : Ii -+ I i - 1 are zero. (a): Choose a minimal homomorphism e : Lo -+ M. If N1 is its kernel, choose a minimal homomorphism L1 -+ N1 , etc. (b): Since L. is a resolution, the homomorphisms

\

d: Li

-+

Ni

and

e: Lo

-+

M

are surjective. For these to be minimal, it is necessary and sufficient that t.heir kl'rneis NH I (resp. N 1 ) aw cont.aineo in mL, (resp. in mLo). which IIWlLns thaI. !.IH' 1)()lIl1dary opt'ral,or 011 L. is )':(!ro.

a

1ft'1 Appendix I: Minima! ReaolutiolUl

85

Corollary. 1£ L. = (Li) Is a minimal free resolution of M, the rank of Li is equal to the dimension of the k -vector space Tor:(M, k) . Indeed, we have: Tor:(M, k) ~ Hi(L. ® k) = H/L.) ~

"Ii'

Remark. In particular, the rank of Li is independent of the chosen resolution L.. In fact, it is easy to prove more: any two minimal free resolutions of M are isomorphic (non-canonically in general). See e.g. [Eil], or [Eis], §19.1.

2. Application Let L. = (Li) be a minimal free resolution of M, and let K. = (Ki) be a free complex, given with an augmentation Ko ...... M. We make the following hypotheses: ( Co) K 0 -+ M is injective. ( C i ) the boundary operator di : Ki -+ K i - 1 maps Ki into mKi - 1 and the corresponding map di : K/mKi -+ mKi_dm2 K i - 1 is injective. Since L. is a free resolution of M , the identity map M -+ M can be extended to a homomorphism of complexes

I: K.

-+

L.

Proposition 2. The map f is injective, and identifies K. with a direct factor of L. (as A -modules). We need to show that the Ii : Ki -+ Li are left-invertible. But, we have the following lemma (whose proof is easy):

Lemma. Let Land L' be two free A -modules, and let 9 : L ...... L' be a homomorphism. For 9 to be lcft-invertibl(~ (resp. right-invertihle), it is necessary alld sufficient that g: L ...... ;s illjective (resp. surjective).

r'

WI' t.hus nM
7.

:1{.

-t

I.

.UfO

injf'Ct.IVfI. Wf!

86

IV. Homological Dimension and Depth

a) i = O. We use the commutative diagram

Ko-£o

1

The fact that Ko b) i

~

-+

.

1

M~M. M is injective implies that Ko

-+

£0 is injective.

1. We use the commutative diagram

1

1

mKi _dm 2 K i _ 1 mL i _dm 2 L i _ l • Since /i-I: Ki-l -+ Li-l is left-invertible, so is the map -. 2 /i-I. mKi-l/m K i -

2

mLi-dm L i- 1 ;

I -+

by the condition (Ci ), we conclude that the "diagonal" ofthe square above is an injective map, whence the injectivity of K i -+ Li . Corollary. The canonical map Hi(K. ® k) for every i ~ 0 .

-+

Tor~(M, k) is injective

-

A

Indeed, Hi(K. ® k) Hi(K.) = Ki and Tori (M, k) corollary just reformulates the injectivity of the 1i .

3. The case of the Koszul complex From now on, we let M = k , the residue field of A. Proposition 3. Let x = (Xl, ... ,xs ) be a minimal system of generators of m, and let K. = K(x, A) be the corresponding Koszul complex. The complex K. (given with the natural augmentation Ko -+ k) satisfies the conditions (Co) and (Ci ) of §2. We have Ko = A and the map A -+ k is bijective. The condition ( Co ) is thus satisfied. It remains to check ( C i ). \ Set L = AS; let (el,"" e s ) be the canonical basis of Land (ei, ... ,e:) be the dual basis. We can identify Ki with 1\' L; the boundary map

Appendix I: Minimal H.eIolutiona

is then expressed in the following ma.nner:

87

.'IIi"';

j=.

=

d(y)

L

Xj(Y

L

ej).

j=1

The symbol L denotes the right interior product (cf. Bourbaki, Algebre III, §1l). We need to write d explicitly; for that, we identify Ki with L and mKi_dm2 K i - 1 with m/m2 01\i-1 L. Then the formula giving d becomes:

N

j=8

d(y) =

L

Xj(Y

L

-ej),

j=1

with obvious notation, Since the Xj form a basis of m/m 2 , the equation d(y) = 0 is equivalent to Y L = 0 for every j , whence Y = 0 , qed.

e;

Theorem.

We have dimk Tort(k, k) 2::

G) ,

with s = dimk m/m2 .

Indeed, prop. 3, together with the corollary to prop. 2, shows that the canonical map from Hi(K. 0 k) = Ki to Tort(k, k) is injective, and we have dimk Ki =

( ~.)•

,

Complements. There are in fact much more precise results (cf. the papers of Assmus [As], Scheja [Sc], Tate [T] quoted in the bibliography): Tor:(k, k) has a product (the product rh of [CaE]) which makes it a (skew) commutative, associative k -algebra with a unit element; the squares of elements of odd degree are zero, The isomorphism m/m2 --+ Tort(k, k) extends to a homomorphism of algebras ¢ : 1\ m/m2 --+ Tor:(k, k) which is injective [T]; we thus recover the theorem above. The ring A is regular if and only if ¢ is bijective (it is even enough, according to Tate, that the component of ¢ of some degree 2:: 2 is bijective). Moreover, Tor:(k, k) has a co-product [As] which makes it a "Hopf algebra", One can thus apply the structure theorems of Hopf-Borel to it; this gives another proof of the injectivity of ¢. One also obtains information on the Poincare series PA(T) of Tor:(k, k): 00

PA(T) =

L

ai Ti ,

where

ai =

dimk Tort(k, k) .

For example (IT], [As]), A is a "complete intersection" if and only if PA(T) is of the form (1 + T)n /(1 - T2)fl ,with n, dEN; for oUlPr all!llogOlls results, s(~! [Se]. In all ~meh ('f\..,PN, P A ('I') tllrllN Ollt. t.o he a rat.iolu\1 fUllctioli of 'f. III the orill;illlLl vt'rHioli of t.llt' Notc!H, it. WI\H j\..,k(·d wlu·tll<'r this is alw!\ys true: "Oil iKllore si fJ A('I') .",1. tuUjllllfH 11Il(' fOllrtioll rl\-

18

IV. Homolog;c..J O;menoloo and o.pth

/

tionnelle". This question, which some people misread 88 a conjecture, ,;Was solved negatively in 1982 by D. J. Anick ([AnI,2]), together with theinalogous question for loop spaces of simply connected finite complexes.,'

.

/

1

I

Appendix II: Positivity of Higher Euler-Poincare Characteristics

\ I.

We oh"",e the framewo,k of abelian cat"!!o"'''. More precisely, let\ be an abelian category given with n morphisms Xl,." , Xn from the identity functor into itself. This means that every E E C is given with endomor':::'~ phisms Xl (E), ... ,xn(E) , and that, for every f E Homc(E, E') , we have xi(E') 0 f = f 0 xi(E). In particular, the xi(E) belong to the center of Endc(E) , and they commute with each other. If J c I = [1, n] , the subcategory of C which consists of objects E such that xi(E) = 0 for i E J is written as CJ' Wehave C0 = C; besides this case, we will have to consider J = I and J = [2, n]. If E is an object in C, the Koszul complex K(x, E) is defined in an obvious way; its homology groups Hi(x, E) are objects of C, annihilated by each Xi; in other words, they are elements of CI. We now consider the higher Euler-Poincare characteristics formed by means of the Hi(X, E). First recall how one attaches (according to Grothendieck) a group K(V) to every abelian category V. One first forms the free group L(V) generated by the elements of V; if o -+ E' -+ E -+ E" -+ 0 is an exact sequence in V, one associates to it the element E - E' - E" of L(V); the group K(V) is the quotient of L(V) by the subgroup generated by these elements (for every exact sequence). If E E V, its image in K(V) is written as [E]; the elements of K(V) so obtained are called positive; they generate K(V); the sum of two positive elements is a positive element. This applies to the categories CJ' In particular, let E E C; we have Hi (x, E) E CI , and the alternating sum: (i=O,I, ... ) makes sense in the group K(C1 ). We can thus ask whether this characteristic Xi is 2:: 0 (in the sense defined above). We shall see that it is indeed the case if C has the following property: (N) Every E E C is noetherian, i.e. satisfies the ascending chain \ condition for sub objects. In other words,

89

Appendix II: Positivit.y or liigher guier-Poincare Characterilt.lca

Theorem.

If C has property (N), we have Xi(X, E)

E E C and every i

~

0 for every

~ 0.

We prove this by induction on n.

a)

The case n = 1 For simplicity, we write x instead of Xl. We have

Ho(x,E)

-~

=

Cokerx(E)

Hl(X,E)

and

=

Kerx(E).

The positivity of Xi(X, E) is clear for i > o. When i = 0, we have to show that Xo(x,E) = [Cokerx(E)]- [Kerx(E)] is a p sitive element of K(CI). For m = 1,2, ... ,let xm be the moth po r of x(E) , and let N m be the kernel of xm; the N m increase with . According to (N), the N m stabilize; let N be their limit, and let F = E/N. We have an exact sequence

o -+ N

-+

E

-+

F

-+

O.

The additivity of xo implies Xo(x, E) = Xo(x, N) + Xo(x, F). Since Ker x(F) = 0, Xo(x, F) is equal to [Ho(x, F)] , hence is ~ o. On the other hand, x(N) is nilpotent; this shows that N has a composition series whose successive quotients Qo. are annihilated by x. We have Xo(x, Qo.) = 0 for every Q', hence xo(x, N) = L Xo(x, Qo.) = 0, and we get

Xo(x, E) = Xo(x, F) b)

~

o.

Passing from n - 1 to n According to prop. 1 of part A, we have an exact sequence:

0-+ HO(Xl' Hi(X', E»

-+

Hi (x, E)

-+

Hl(Xl, Hi-l (x', E))

-+

0,

writing x' for the sequence (X2, ..• , x n ) . Passing to K(CI) , we can thus write:

=

[Hi (x, E)] where H:

[HO(Xl' HD]

+ [H1(Xl,H:_ 1)],

= Hi(x', E) . Hence:

Xi(X, E) = [Hl(Xl' H:_ 1)]

+L

(-l)m([Ho(Xl' H;+m)] - [H1(Xl, H:+ m )])

m~O

= [Hl(XlrHI_d]

+ L (-l)mXO(Xl,H:+ m )· m~O

Let J = [2, nJ. The Hj belong to CJ. By the induction hypothesis, the element X~ of J(CJ) defined by

X: -

L m>fl

(_1)m\H: tm l

IV. Homological Dimension and Depth

is ~ 0, i.e. equal to [Gil for some object G i of CJ . Since we have XO(XbGi)

=

L (-l)mXO(Xl,H:+

xo

m)

m;:::O

Hence Xi(x,E) = [HI(XI,H:_I)]+XO(XbGi). Since XO(XI,Gi ) ~ 0 by a), we have Xi (x, E) ~ 0, qed.

Example. Let A be a noetherian local ring, let Xl, ... , Xn be a s stem of parameters of A, and let C be the category of finitely generate A modules (with the endomorph isms defined by the Xi). The categoryCI is the category of A -modules annihilated by the Xi; the length defi~ an isomorphism from K(CI) onto Z, compatible with the order relations.~ The above theorem thus gives: Corollary. integer

is

~

If E is a finitely generated A -module and

is

~

0 , the

o.

Remark.

In the case of the above example, one can prove that Xi(E) = 0

implies

Hj(x, E) = 0

for j ~ i ~ 1 .

However, the only proof of this fact that I know of is somewhat involved (it consists of reducing to the case where A is a ring of formal power series over a discrete valuation ring or over a field). I do not know if there exists an analogous statement in the framework of abelian categories.

Exercises. 1) Assume that C has property (N). Recall that an object of C is simple if it is i- 0 and it has no nonzero proper sub object. a) Show that every nonzero object of C has a simple quotient. b) (Nakayama's lemma) Show the equivalence of the followng three properties of C : (i) E is simple => xi(E) =0 for i=l, ... ,n. (ii) Coker(xi(E)) = 0 for some i => E = 0 . (iii) Ho(x, E) = 0 => E = 0 .

,2) Let Xl, . . . , Xn be elements of a commutative noetherian ring A, and let B be an A -algebra. Let C be the category of left B -modules E which are finitdy generated as A -Olodllips (tlw endolI1orphisIlls x,(E) bl'ill)!, thosl' )!,ivl'll by t.hl· .r,). Show that C IUL<; propl·)'ty (N), IUIII that it

Appendix III: Graded-polynomilll AlgebrRII

91

has properties (i),(ii),(iii) of exerc. 1 if the Xi be lung to the nt.Jical of A. (Note that this applies in particular when B is the group algebra AIC] of a group C.)

Appendix III: Graded-polynomial Algebras All the results proved for local rings have analogues for graded algebras over a field. These analogues can be proved directly, or can be deduced 7 e local statements. We fullow the second method. ~/1.

Notation

We consider finitely generated commutative graded algebras over a field k:

A = EBA n ,

with Ao = k,

n2':O

together with graded A-modules M = EBMn such that M_ n = 0 for all sufficiently large n. We put m = m(A) = EBn~l An; it is a maximal ideal of A, with Aim = k. The completion Am of the local ring Am can be identified with the algebra of formal infinite sums: ao

+ al + ... + an + ... ,

with an E An for every n.

One has the following analogue of Nakayama's lemma: Lemma 1.

If M

=

mM , then M = 0 .

Indeed, if M -:I 0, choose n minimal such that Mn that Mn is not contained in mM.

-:I

0, and note

Lemma 2. If f : M --+ M' is a homomorphism of graded modules, and if MlmM --+ M'/mM' is surjective, then f is surjective.

This follows from lemma 1, applied to M' If (M) .

A graded A -modulI' M is caUl'd grAoflo-free if it hR.'! a bEl.'lis made up of hOIl\O)l;PIWOIIS e\PIIU'lltS. If 0111' d"lIotl"s by A(tl) IL fnoc' A -1111)(11111" with hasis a hO/ll0ll.elw()\ls dt~IIWIlt. of d"ll.ftOC· tl, M II grndtl
92

i

.IV. Homological Dimension and Depth

/

Lemma 3. The following properties are equivalent: . (i) M is graded-free. (ij) M if A -fiat (as a non-graded module). (iii) Tort(M, k) = O. • . It is clear that (i) ~ (ii) ~ (iii). To prove (iii) ~ (i), cho se a Ie-basis (xa,) of homogeneous elements of MjmM. Put do = detXQ) and select a representative XQ of Xo< in M do . Let L be the direct s of the free modules A(d a ). The Xa 's define a homomorphism f : L -+ M, which is surjective by lemma 2. Put N = Ker(f). We have an e~ct aequence: Tort(M, k)

NjmN

-+

-+

LjmL

-+

MjmM

-+

o.

By construction, LjmL -+ M jmM is an isomorphism, and by (iii) we have Tort(M, k) = O. Hence N jmN = 0, which implies N = 0 by lemma 1. Hence M is isomorphic to the graded-free module N, qed.

2. Graded-polynomial algebras (cf. [LIE], Chap. V, §5, and

[Bour 1, Chap. VIII, §6)

We say that A is a graded-polynomial algebra (in French: "algebre graduee de polynomes") if there exist homogeneous elements P 1 ,... ,Pd of A, such that the natural map k[X1"" ,Xd ]

-+

A,

defined by the Pi'S, is an isomorphism. If this is the case, the monomials

make up a basis of An . The Poincare series of A, 00

¢A(t) =

L

dimk(An)tn

n=O

is equal to d

1

II --, 1 - ta,

with a.. = deg(A). •

i=l

This shows that the sequence (ai) is independent of the choice of the I{ 's (if numbered so that a1 ~ a2 ~ '" ~ ad). The ai are called the basic degrees of A. Note that A is isomorphic to the symmetric algebra Sym( b,) , where L is the gra.ded-free A -module L = €a A( a,) .

Appendix 111: Graded-polynomial Algebrllll

93

3. A characterization of graded-polynomial algebras Let A:;: ffin2:0 An be a graded algebra satisfying the conditions of §l. Theorem 1. The following properties are equivalent: (1) A is a graded-polynomial algebra. (2) A is isomorphic (as a non-graded algebra) to a polynomial algebra. (3) A is a regular ring. (4) The local ring Am is regular. I

~

t. clear that (1) ~ (2) ~ (3) ~ (4). Let us show (4) ~ (1). First ote that mJm2 is a graded k -vector space of finite dimension. Let (Pl- ... ,Pd) be a basis of mJm2 made up of homogeneous elements; let ~; = deg(Pi) and choose a representative Pi of Pi in Aa;. Let A' be the graded-polynomial algebra k[X 1 , ... , Xd] , with deg(X i ) = ai, and let ! : A' -+ A be the unique morphism such that !(Xi ) = Pi for every i. One sees easily (by induction on n) that !(A~) = An for every n, Le. that! is surjective. The local map j: A:n, -+ Am (where m' = m(A' )) is surjective, and, since Am is regular, it is an isomorphism (use tho 10 of part D §2, for instance (*) ). This implies that ! is injective. Hence ! is an isomorphism, qed. Remark. Another property equivalent to (1), ... ,(4) is: (5) dim(A) = dimk(mJm2) .

4. Ring extensions Let B = ffin>o Bn be a commutative graded k -algebra containing A, and such that -Bo = k . Theorem 2. Assume that B is a graded-polynomial algebra. (a) If B is a graded-free A -module, A is a graded-polynomial algebra. (b) Conversely, if A is a graded-polynomial algebra, and if B is a finitely generated A -module, then B is a graded-free A -module.

In case (a), the local ring Bm(B) is regular, and is a flat Am -module. By tho 12 of part D §5, this implie" tha.t Am is regular; hence A is a graded-polynomial algehra, cf. tho 1. (0)

il.y

Alternative MlI;ument: If

film

.lilli, (m

A", Im:l)

<

Kp.r(j)

Wflffl

~ 0,

WfI

would hay., th., Inflqllal-

dilll A~, . . hul. it iM dC'N tllI,t buth clhlll'llllilllUI

/Uti

~lu,,1 to

'94

IV. Homological Dimension and Depth

'- A similar argument shows that (b) follows from tho 13 of part D §5, combined with lemma 3.

. Remark. In case (b), the A -module B has ~ basis (b a ) made up of homogeneous elements. Put ea = deg( ba ). Since B = Eaa Aba, the Poincare series
d

=

i=l

with d

1

d


II I-tbi' i=l

= dim A = dimB, one has d

(L tea) II (1 - t

(4.2)

b.)

i=l

i=l

Dividing both sides by (1 - t)d , and putting t

=1

gives:

(4.3) where [B: A] is the rank of the free A -module B, i.e. the nu ber of elements of the basis (b a ).

Example. Take B = k[X I , ..• ,Xd] with the standard grading: deg(Xi ) = 1 for 1 $ i $ d. Choose for A the subalgebra of the symmetric polynomials. We have A = k[P!, ... , Pd] , where PI"." Pd are the elementary symmetric polynomials: This shows that A is a graded-polynomial algebra with basic degrees 1,2, ... ,d. By tho 2, B is a graded-free A-module. Indeed, one can check that the monomials

Xr'l ···X;F'd,

with 0 $

mi


,

make up an A -basis of B . The identity (4.2) above becomes: d

(l-t)dTI(I+t+ ... +ti i=l

1)

=

(l-t)(I-t 2 )···(I-td ).

Appendlx III: Graded-polynomial AIKtlbru

95

5. Application: the Shephard-Todd theorem We are going to apply tho 2 to the proof of a theorem of Shephard-Todd ([ST]) which is very useful in Lie group theory. Let V be a k -vector space of finite dimension d, and W a finite subgroup of GL(V) such that (a) the order IWI of W is prime to the characteristic of k, (b) W is generated by pseudo-reflections. (A pseudo-reflection s of V is an automorphism of V such that Ker(s - 1) is a hyperplane of V. Such an automorphism can be written as x 1-+ x + u(x)v, where v E V - to} , and u is a non-zero linear form on V.) Let B = Sym V be the symmetric algebra of V , with the standard \irading; it is a graded-polynomial algebra, isomorphic to k[Xl' .. ' ,Xd]' The group W acts on B. Let A = B W be the subalgebra of B made p of the elements fixed by W. Theorem 3 ([ST J, see also [Ch3]). polynomial algebra.

The algebra A is a graded-

It is easy to see that B is a finitely generated A -module (e.g. because it is integral over A, and generated as an A -algebra by a finite number of elements). By tho 13, it will be enough to prove that it is a graded-free A -module. By lemma 3, this amounts to proving that the k -vector space E = Tort(B, k) is O. Note that E is a B -module, and that W acts on E.

Lemma 4. (i) We have E W

=0. (ii) The group W acts trivially on E' = E/m(B)E.

Assume Lemma 4. Since IWI is prime to the characteristic of k, the surjective map E --> E' gives a surjection E W --> E'w. By (i) we have EW = O. Hence E'w = 0 and by (ii) we have E' = 0, hence E = 0 by Nakayama's lemma applied to the B -module E. Proof of part (i) of Lemma 4 Since the order of W is invertible ill A

I

w(~

have

.1

Tor~(M, k)w = Tor~(Mw, k)

for every A -lIIodui(! M with projN:tor ~ L .... w til ).

1\11

A -UIIf'IU' H(:tion of W (e.,.

UIM!

t.he

IV. Homological DimefUlion and Depth

96

By taking M

= B , and i = 1 , this gives

....

E W = Tort{B,k)W = Tort{BW,k) = Tort{A,k)

O.

Proof of part (ii) of Lemma .4 Let s: x .- x + u{x)v be a pseudo-reflection belonging to W. The action of s on B is trivial modulo the ideal vB. (Note that v is an element of Bl .) Hence we may factor s - 1 on B as: A

,.,.

B-B-B, where >.. is A -linear, and It is multiplication by v. This gives an analogous factorization of the endomorphism (s - l)E induced on E by s - , namely

E

AE)

E~E,

where ItE is multiplication by v. In particular, the image of {s -1)E is contained in vE, hence in m{B)E. This shows that the pseudo- eflections belonging to W act trivially on E' = E /m{B)E. Since W is generated by pseudo-reflections, this proves (ii).

Remark.

There is a converse to tho 4: if BW is a graded-pol omial algebra, then W is generated by pseudo-reflections (see [ST], an also Bourbaki, [LIE], Chap. V, §5, no. 5).

Exercises

(cf. [LIE], Chap. V). Let B, A be as in tho 3, and call ai (i = 1, ... ,d) the basic degrees .of A. Assume char{k) = O. Let C = B ®A k = B/m{A)B. It is a graded k-vector space, with a.n action of W. Let wE W; if n ~ 0, denote by Xn{w) the trace of w acting on the n -th component of C . 1) Show that C ®k A is isomorphic to B as a graded A [W]-module. Deduce that the linear representation of W on C is isomorphic to the regular representation of W .

IWI = [B

2)

Prove that

3)

Show that cPA{t)

L n2:0

1 1-

t'h

: A]

= dimk C = nf=l at·

Xn{w)t n =

1 det{l - tw)

for wE W, where

Appendix III: Graded-polynomial AIa"br..

1

' " X (w) _ {I n 0

4)

Show that

5)

Use 3) and 4) to prove:

IWi wEW L.,
1

L

IWI wE w

97

if n = 0, if n > O.

1

det(1- tw)' d

6)

Show that det(t-w) divides the polynomial PA(t) =

for every wE W, and that PA(t) = lcrn det(t - w) .

II (t

a; -

1)

i=l

_w

~

",

\

7) Let V' be a subspace of V and let W' be a finite subgroup of \;L(V') generated by pseudo-reflections. Assume that every element of is the restriction of some element of W. Let aj be the basic degrees of W'. Show, by using 6), that the polynomial Wi - 1) divides the

1f'

II

polynomial

II

j

(t a ; -

1) . In particular, every

aj divides some ai'

'.~.

Chapter V. Multiplicities

A: Multiplicity of a Module In this section, A is a commutative noetherian ring; all A -modules are assumed to be finitely generated.

1. The group of cycles of a ring An element of the free abelian group Z(A) generated by the elements of Spec(A) is called a cycle of A (or of Spec(A)). A cycle Z is positive if it is of the form

Z =

L

n(p)p

with n(p) ;::: 0 for every p E Spec(A) .

Let us assume now that A is local, of dimension n. If p ;::: 0, let Zp(A) be the subgroup of Z(A) generated by the prime ideals p with dim Alp = p. The group Z(A) is the direct sum of its subgroups Zp(A) , for 0::; p ::; n . The cycles are related to A -modules in the following way: Let Kp(A) be the abelian category of A -modules M such that dimA M ::; p, K(A) be the category of all A -modules. It is clear that if O-M-+N-P-+O

is an exact sequence of K(A) and if M and P belong to Kp(A) , then N E Kp(A). Under these conditions, let M E K,,(A) , and let q be a prime ideal of A with dim Alq = p. Then the modulI' Mq over Aq is of finite length f(M q ) and this len!l;th obviously satisfies thl~ fol\owilll!; property: if 0 = Mo C ... C M, C ••• C M. = M III 1\ cnrnposition HNlell of M whoNf~ qlJot.ilmtH M,/M,_I are of I.h" form A/t, wlll'r.· t h." prlnlf' 1(\1'/11 of A t.hen it hOll 6Xactly f(M q ) '1IIIIth'llt.14 nf t.Il1' form A/q. I

100

V. Multiplicities

Thus let

z" : K,,(A)

-+

Z,,(A) be the function defined by

zp(M) =

L

€(Mq )q.

dimA/q=p

It is clear that zp is an additive function defined on the category Kp(A) and has values in the ordered group Z,,(A). The function zp is zero on K,,-l (A) . Conversely, it is clear that every additive function on Kp(A) which is zero on Kp-1(A) factors through zp. If A is a domain, then Zn(A) ~ Z for n = dim A, and zn(M) is the rank of the A -module M.

2. Multiplicity of a module Assume A is local; let m be its maximal ideal, and let a be an m -primary ideal. Then, for every nonzero A -module M, the Samuel polynomial P" (M, X) defined in Chap. II is of degree equal to dimA M . Furthermore, its leading term is of the type exr /r! , where r = dimA M and where e is an integer > 0 . The integer e is, by definition, the multiplicity of M for the primary ideal a. One writes it as e,,(M); more generally, p being a positive integer and M a module such that dimA M ~ p , we set e,,(M) if dim A M = p, ell ( M,p ) = { o if dimA M < p. It follows from the properties proved in Chap. II that e,,(M, p) is an additive function on Kp(A) , which is zero on Kp-l(A); we thus have the additivity formula:

e,,(M,p) =

L

f(Mq)e,,(A/q,p)

where q runs through the prime ideals with dimA/q = p (or those with dim A/q ~ p: it amounts to the same). In particular, if A is a domain of dimension n, we have

e,,(M, n) = rank(M)e,,(A).

= m, e,,(M) = em(M) is called the multiplicity of M. In particular em(A) is the multiplicity of the local ring A. If A is regular, its multiplicity is equal to 1, according to Chap. IV. Conversely, if the multiplicity of A is equal to 1 and if A is a domain, one can show that A is regular (cf. [Nag3], tho 40.6, and also [Bour],

If a

Chap. VIII, p. 108, exerc. 4); an example from Nagata shows that it is not enough to assume that A itself is a domain. Finally, let x be an ideal of definition of A, which is generated by Xl, .•• I Xn , where n == dim A. According to theorem 1 of Chap. IV, the

B:

IntenM!(~t.lon

Multiplicity or Two Modulea

. 101

i -th homology group of the KOf!zul complex K(x, M) ill of finite length hi(x, M) for every A -module M and every i ~ 0, and we have the formula n

ex(M,n) = L(-l)ih;(x,M). ;=0

B: Intersection Multiplicity of Two Modules 1. Reduction to the diagonal Let k be an algebraically closed field, let U and V be two algebraic sets of the affine space An (k) ~ k n , and let ~ be the diagonal of the product space An(k) x An(k) ~ Azn(k). Then ~ is obviously isomorphic to An (k) and the isomorphism identifies (U x V) n ~ with U n V. This reduces the study of the intersection of U and V to the study of the intersection of an algebraic set with a linear variety. This viewpoint already occurred in prop. 17 of Chap. III (Dimensions of intersections in affine space). In particular, in lemma 7, it helps to view A ®k A as the coordinate ring of An(k) x An(k) 1 and All' ®k Alq and (A ®k A)/1l as the coordinate rings of U x V and ~ (U and V irreducible). The isomorphism of (U x V)n~ with UnV translates into: All' ®A Alq ~ (All' ®k Alq) ®(A®kA) A, (1) where we identify A with (A ®k A)/lI. This formula generalizes as follows: let A be a commutative algebra with a unit element over a commutative field k (not necessarily algebraically closed); let M and N be two A -modules, B the k -algebra A ®k A , and 1I the ideal of B generated by the a® 1-1 ®a, a EA. Then (A ®k A) III is a k -algebra isomorphic to A, and A is given a B -module structure via this isomorphism. We have the formula ([CaE], Chap. IX, 2.8):

(2) So, if

- Ln ~ ... - Lo --+ A -+ 0 is an (A ®k A) -projective resolution of A, the bifunctor

(M, N) ...... (M

c;¢/r

N)

C;¢B

L.

is "resolving", I.e. Tor~ (M, N) is natumlly isomorphic. to the homology modules of the r.ompll'x (M ~k N) Mil L.. In part.icular, if A is tlw polynomial algebra klX., ... ,Xnl in U vlulahll'l4 X, ov/'r k. tI\(' KOHZIII complex KlI«X. o¢ 1 - 10¢ X.), IJ) i14" frl'C' rl!tl"lutioll of A, IUld Wtl hnvll: Tbr~(M,N} Q! H"(K"((X,001-1~X.),MOO.N)).

(3)

V. Multiplicitiea

102

,

"I

We recover the fact that k[Xll'" IXnl is regular! In what follows, the reduction to the diagonal will be used via formula (2) suitably generalized to completed tensor products (see below).

2. Completed tensor products Let A and B be k -algebras, with k, A, B noetherian, and l~t m, n denote ideals of A and B respectively such that A/m and B /n are k -modules of finite length. Let M (resp. N) be a finitely generated Amodule (resp. B -module) given with an m -good filtration (M p) (resp. n-good filtration (Nq». Then, M/Mp and N/Nq are k -modules o/finite length for every pair (p, q) of natural numbers. Then, for every natural number i, the modules Tor~(M/Mp, N/Nq ) form a projective system, and the completed Tori are defined by the formula: -k . k Tori (M,N) = hm Tori (M/Mp,N/Nq). (4)

---

(p,q)

For i

= 0, we obtain the completed tensor product

(cf. [Sa3]):

M ®k N = lim (M/Mp®k N/Nq).

(5)

(p,q)

The abelian groups thus defined have the following properties: , -k

1} The modules Tori (M, N) do not depend on the chosen good filtration for M or N, but only on M and on N (and of course on the maximal ideals of A and B containing m and n). 2) As the diagonal of N x N is a cofinal subset, it suffices to take the projective limit over this diagonal: -k

Tori(M,N}

-----

~

k

lim Tori (M/Mp,N/Np).

(6)

p

Similarly we may take the above limit over p and then over q: -k

Tori (M, N) ~ limlimTor~(M/Mp,N/Nq}

--p

q

~ limlimTor~(M/Mp,N/Nq}.

3)

q

p

Ttw canonical maps from M

~k

N into M/Mp

~Ir

N/Nq induce the

B:

'ntenec:~'on Multiplicity of Two Modul.

103

maps II

P

if ®k IV

and

-+

M ®k N,

and it is clear that M ®k N is naturally isomorphic to the completion of M ®k N for the (m ®k B + A ®k n) -adic topology.

4)

The ring A ®k B is complete for the t -adic topology, where

= m ®k B + A ®k n

t

--k

and the Tori (M, N) are complete modules for the t-adic topology. Since (A ®k B)/t = (Aim) ®k (Bin)

and

(M ®k N)/t(M ®k N)

= (M ImM) ®k (N/nN),

corollary 3 to proposition 6 of Chap. II is applicable and so A ®k B is noetherian and M ®k N is a finitely generated (A ®k B) -module. Moreover the formula l~X = 1+x+x 2 + ... shows that t is contained in the radical of A ®k B and the maximal ideals of A ®k B correspond to those of (Aim) ®k (Bin) . 5)

If 0

--t

M'

--t

M

Mil

--t

-+

0 is an exact sequence of finitely generated

A -modules, the exact sequences ... -+

Tor~(MlmPM, N InqN)

-+

Tor~(M" ImP Mil, N/nqN)

--t

Tor~_l(M' 1M' n mP M, N/nqN)

-+ ...

give an exact sequence ... -+

--k

Torn(M,N)

-+

--k

/I

Torn(M ,N)

-+

--k

Torn _l(M',N)

-+ ....

This follows from the following general property: if (Pi) and 1/1: (Pi) -+ (Pi') are two morphisms of projective systems of k -modules over an inductively ordered set, if the PI are artinian, and if the sequences

4> : (PI)

-+

pll Pi' -•• P., tjI. - ,

-

are exact, then the limit sequence

...-

lim P: - 11m P, - lim pt' ~. is exact. I Proof. Let (P,) be 8Jl .,I.!UU'llt KI'f(lIm PI -. 11m_. P:'). Lot E, be the prt~iml1K" of I), in TIll' HC't f:, III lIoll-t'lIIl1ty, "".t IIIUI " Iml.llml structur~ of ILtfilll! k -module. Let a l I,., tt. .. ...,t uf "II "mil" IIuhlJluclull1tl

P:.

or

.

.

104

,~,

V. Multiplicities

,

"

.",

of E i . Using the fact that P: is artinian, one checks readily that the family (eli) has properties (i), ... ,(iv) of Bourbaki E.III.58, tho 1, and this implies (loc. cit.) that ~Ei::/: 0. Hence (Pi). belongs to the image of limP: in lim Pi . 1 +--

+--

6) Suppose now that k is a regular ring of dimension n and suppose that M , viewed as a k -module, has an M -sequence {al,'" ,ar }, i.e. that there exist r elements ai in the radical of k such that aHI is not a zero-divisor in M I(al, ... ,ai)M, 0:5 i :5 r - 1 . -k Then Tori (M, N) = 0 for i > n - r. Indeed, it suffices to show this when N is a k -module of finite length, since -k

-k

Tori (M, N) = lim Tori (M, N Iqn N). +--

The exact sequence

o --M ~M -- MlalM --0, ----k

together with the fact that Tori (M, N) sequence: -k

al

.

= 0 if

i

> n,

gives the exact

-k

0-- Torn(M, N) -- Torn(M, N). -k

But a power of al annihilates N, hence also Torn(M, N) . It follows that -k

Torn(M, N) = 0, which proves our assertion for r = 1. If r > 1, we have -k

Torn(M, N) = 0, whence the exact sequence -k

al

-k

0-- Torn_I(M,N) -- Torn_I(M,N),

etc ....

In the examples we will use, the algebra A has an A -sequence of n -k

elements. It follows that Tori (M', N) = 0 if M' is A -free and i > 0 . In this case the functor M

-k t-+

Tori (M, N) is the i -th derived functor of

~

-k

the functor M t-+ M ®k N. This implies that Tori (M, N) is a finitely generated A ®k B -module. The machine just constructed will only be used in the following two special cases:

k is afield, A ~ B ~ k[[Xl"" ,Xnll: -k ~ In this case the Tori are zero for i > O. Moreover, A ®k B is isomorphic to the ring of formal power series C ~ kIlX 1 , ... ,Xn ; Y., ... ,Yn ]]. Let now q (resp. ql) be an ideal of definition of A (resp. B),

a)

I\nd l,·t.

8

hI'

1.I\f'

i
P;t'lIl'TIlt .. d

by q

IWcl

q'. )"'1.

Ill'! put Oil

M,

B: InterMCtion Multiplicity of Two Module&

106

Nt M ®k N the q -adle, q' -adlc and .e-adic topologies, and consider the corresponding graded modules gr(M), gr(N) , gr(M ®k N) . The natural map M ®k N - M ®k N induces a homomorphism gr(M) ®k gr(N) -

gr(M ®k N).

One checks easily (d. [Sa3 J) that this homomorphism is bijective. Hence: dim(M

®k

N) = dim(M)

+ dim(N)

and e.(M

® N,dimM + dimN)

= eq(M,dimM). eq,(N,dimN).

Finally if

... - Kn - ... - Ko - M - 0 and

... - Ln - . " - Lo - M - 0 are A - and B -free resolutions of M and N, (~:)Kp ®k Lq)p+q=n)n is a C -free resolution of M ®k N. In particular if we identify A with the C-module C/~, where ~ = (Xl - Y I , ... ,Xn - Yn ) we have the equality Kp ®A Lq = (Kp ®k Lq) ®c A, whence the formula of reduction to the diagonal: "" ~ ToriA (M, N) = ToriC (M ®k N, A).

k is a complete discrete valuation ring, A ~ B ~ k[[X 1, .. · ,Xnll: The letter 'Tr denotes a generator of the maximal ideal of k and k denotes the residue field k/'Trk. Put: C = A®kB = k[[Xl, ... ,xn;YlI ... ,Ynll, b)

A = A/l!'A = k[[X1 , ... ,Xn]J, C = k[[Xl, ... ,Xn;Y1, ... ,Ynll. We have (M ®k N)/l!'(M ®k N) = (M/l!'M) ®k (N/l!'N). It follows that, if 11' is not a zero-divisor in M and N, we have dimM ®k N = dimM

+ dimN-1.

Finally, resolving M and N as in a), and taking a C -projective resolution of A = C /"0 , we end up with a spectral sequence: C

-k

A

Torp (A, Torq(M, N)) =* Torp+q(M, N). This spectral sequence degenerates if N ; it gives an isomorphism

- N) Tor,c (A, M ~,.

;W

11'

is not a zercrdivisor in M or

Tor"A (M, N). .1

106

V. Multiplicities

3. Regular rings of equal characteristic Let us look at case a) of §2 above, i.e. A = k[[Xb . :. ,Xnll , where k is a field. Here, the Koszul complex KG «Xi - }j), 0) is a free resolution of A = C 1"0. If M and N are two finitely generated A -modules, the Tort (M, N) may thus be identified with Torf (A, M ®k N) , i.e. with the homology modules ofthe Koszul complex KG«X j - }j),M ®k N): ~ N )) . ToriA (M, N) '" = Hi(K G «Xj - }j), M 0k

Theorem 1 of Chap. IV applies to this Koszul complex, an d gives the following result: (*) If M 0 A N has finite length, then the Tort (M, N) have finite lengths, and the Euler-Poincare characteristic i=n

X(M,N) = L(-l)i£(Tort(M,N)) i=O

is equal to the multiplicity e,,(M the ideal tJ. Thus X(M,N) ~ 0,

®k

N, n) of the C -module M

dimA M + dimAN = dimcM ®k N $ n, X(M, N) = 0 if and only if dimA M + dimA N

®k

N for

< n.

This result is easily generalized to the regular rings from algebraic geometry. First of all it is clear that every regular ring A is a direct product of a finite number of regular domains (a noetherian ring A such that, for every prime ideal p, Ap is a domain, is a direct product of a finite number of domains). If A is a domain, then A is called of equal characteristic if, for every prime ideal p, Alp and A have the same characteristic. We shall say that a regular ring A is of equal characteristic if its "domain components" are of equal characteristic, which is to say if, for every prime ideal p, the local ring Ap is of equal characteristic. Theorem 1. If A is a regular ring of equal characteristic, M and N are two finitely generated A -modules and q a minimal prime ideal of Supp(M 0A N) , then: (1) Xq(M,N) = E!:~imA(-l)i£(Tort(M,N)q) is ~ OJ (2) dimA q Mq + dimA q Nq $ htA q; (3) dim Mq + dim N q < htA q if and only if Xq(M, N) = O. If we localize at q, and complete, we have A A A Tori (M,N)q = Toriq(Mq,Nq) = Tor,q(Mq,Nq). A

HelH:e we

!lilLy

A

1I.'isume that A is <:omplde, and t.hat. q is its maximal

B: InterwecUon Multiplicity of Two Modul.

107

ideal. By Cohen's theorem, A is then isomorphic to k[[Xb ... ,XnlJ , and 'We apply (*) above. Moreover: Complement. 1£ a and b denote the annihilators of M and N in A, m the maximal ideal qAq of Aq , c the ideal generated by a + b in Aq , and if Xq (M, N) > 0 , we have the inequalities:

e~q(Mq,dimMq)·e~q(Nq,dimNq) $ Xq(M,N) $ e~q(Mq,dimMq)' e~q(Nq,dimNq).

Indeed, if k denotes a Cohen subfield of Aq, we have seen that Xq(M, N) is equal to the multiplicity ef (M ®k N, htA q) , where C = Aq ®k Aq and where 1I is the ideal of C generated by the a ®k 1-1 ®k a, a E Aq . But the ideal aq ®k Aq + Aq ®k bq = f annihilates Mq ®k N q and the assertion follows from the inclusions:

m

®k

Aq

+ Aq ®k m

:J 1I + f :J

C

®k

Aq

+ Aq ®k c.

4. Conjectures It is natural to conjecture that theorem 1 extends to all regular rings. On this subject, one can make the following remarks: a) Theorem 1 remains true without any regularity hypothesis if M is of the form A/(Xl, ... ,xr ), the Xi being an A-sequence. Then, indeed, M is of finite homological dimension, the Tort(M, N) are the homology modules of the complex K A « Xi), N) , and in particular

Xq(M,N) = e~:(Nq,r), where x is the ideal generated by the Xi. b) One may assume that A is a complete regular local ring. Indeed, we can reduce to this case by localization and completion. c) One may assume that M and N are of the form M = Alp, N = A/q, I' and q being prime ideals of A. Indeed X(M, N) is "bi-additive" in M and N, and the general case follows by taking composition series of M and N whose quotients are of the form A/I', A/q. In the case of equal r.harnd,('riHtic, Wf' first 111'1'11 II), then a) hy rNlllction to the diajl;onal. Thet-l(~ r«~markH will "lIow lUI to 1(t!IU!flLli1.e ttll'or4'l1l 1:

108

V. Multiplicities

5. Regular rings of unequal characteristic (unramified case) /. " Theorem 2. Theorem 1 remains true if the hypothesis" A is a regular ring of equal characteristic" is replaced by the more general hypothesis: A is a regular ring, and for every prime ideal p of A, the local ring Ap is of equal characteristic, or is of unequal characteristic and is unramified. (As a matter of fact, it suffices that this property is satisfied when p is a maximal ideal. Indeed, if A is a regular local ring of unequal characteristic and is unramified, every local ring Ap is of the same type or is of equal characteristic. ) Recall that a local ring of unequal characteristic is unramified if p fj. m 2 , where p denotes the characteristic of the residue field and m the maximal ideal. Cohen has proved (see [Co], [Sal], and [Bour],

Chap. IX,§2) that an unramified regular complete local ring of unequal characteristic is of the form k[[X b ... ,Xnll where k denotes a complete discrete valuation ring which is unramified (notation from §2, case b). By localization and completion, the proof of theorem 2 is reduced to that of:

Lemma 1. If A = k[[X 1 , ... ,Xnll, where k is a complete discrete valuation ring and if M and N are two finitely generated A -modules such that M ®A N is of finite length, then we have: (1) X(M,N) = L~~ol(-I)ie(Tor~(M,N» is;:::: OJ (2) dim M + dim N :::; dim A = n + I j (3) Moreover X(M,N) I- 0 if and only if dimM+dimN = dimA. (Note that we do not assume that k is unramified. The lemma is thus, in a sense, more general than theorem 2.) In view of remark c) of §4 it suffices to give the proof when M and N are "coprime" (Le. every endomorphism by scalar multiplication is injective or zero). We thus consider the different cases (11' denotes a generator of the maximal ideal of k):

0)

is not a zero-divisor in M or in N: We know (cf. §2,b) that

11'

A

C

~

Tori (M,N) = Tori (A,M ®k N), where

and thllt

dimr(M ~Ir N) -

(lim,. M t dim,. N - 1.

B: Int.el"8eCUon Multiplicity of Two Modul.

109

Moreover the Koszul complex KC«Xi - 1'1), C) is a free resolution of the C -module A = C /-0. Remark a) of §4 applies to XC (A, M ~k N) , and proves what we want.

(3)

annihilates M and is not a zero-divisor in N: Then M = M/lI'M = M, and M is a module over the ring A = k[Xi"" ,Xn ]. We have a spectral sequence ([CaE], Chap. XVI, §4, 2a and 3a): 11'

Tor:(M, Tor:(A,N» =* Tor:+q(M,N). The exact sequence 0 -+ A ~ A -+ A homological dimension lover A and that

-+

0 shows that

A is of

A®AN = N/lI'N, and

A-

Tori (A, N) = ",N = AnnN(lI')

= set of elements of N annihilated by

11'.

The spectral sequence reduces to the exact sequence: ... -+

Tort.dM, ",N)

-+

Tor:{M, N)

-+

Tort2(M,,,,N)

-+ ... ,

-+

Tort(M, N/lI'N)

whence XA(M, N) = XA(M, N /lI'N) - XA(M, ",N) . But we assumed that wN = 0 ; whence

XA(M,N) = XA(M,N/lI'N) ~ 0, dimAM + dimAN /lI'N :::; n and the second inequality is strict if and only if XA (M, N /11' N) = O. Since = dimA N /lI'N = dimA N - 1, we get what we want.

dimA M = dimAM and dimAN /lI'N

"y)

annihilates M and N: We again consider M as an A -module, and N as an A -module; the spectral sequence now gives: 11'

XA(M, N) = XA(M, N /lI'N) - XA(M, wN). But in this case N/lI'N = ",N = N, hence XA(M, N) = 0; it remains

to check that dimA M + dimA N = dimA M + dimA N < n. But since M ® A N = M ®A N is an A -module of I1l1it.l~ length, and the lemma has been proved for A, we have dimA M + dim A N S n, q(~d.

110

V. Multiplicities

6. Arbitrary regular rings It is not known yet (.) how to extend properties (1) and (3) of tho 1 to such rings. On the other hand, one can prove the inequality (2) (the "dimension formula" of algebraic geometry, cf. Chap. III, prop. 17). More precisely:

Theorem 3. Let A be a regular ring, p and q be two prime ideals of A, t a prime ideal of A minimal among those containing p + q. Then: htA p + htA q

~

htA t.

By localizing at t, one may assume that A is local with maximal ideal t. In that case, the theorem may be reformulated as follows: (*) If M and N are finitely generated A -modules, then

e(M ®A N) <

00

==> dimM + dimN

~

dimA.

To prove (*), one may assume that A is complete. By a theorem of Cohen [Co], A can be written as At/aA t where At is a ring of formal power series over a complete discrete valuation ring, and a is a nonzero element of At. If we view M and N as At -modules, one shows as in the case 'Y) of §5 that )(Al (M, N) = 0, and lemma 1, applied to At , shows that dim M + dim N < dim Al , and dimM

+ dimN

~

dimA

=

(dim At) -1,

qed. Let us also mention the following result:

Theorem 4. Let A be a regular local ring of dimension n, let M and N be two nonzero finitely generated A -modules such that M ®A N is of finite length, and let i be the largest integer such that Tort(M, N) 1=- O. We have: i = proj dim(M) + proj dim(N) - n. (.) In 1996, 0. Gabber has proved property (1), i.e. the fact that )((M, N) is O. See [Be] and [R4].

~

Moreover, half of (3) had already been proved in 1985 by P. Roberts (see

IRl],(R3]) and II. Gillet-C. Soule [GiSI; namely

dim M

+ dim N <

dim A

~

X(M, N)

It iH HUll unkllowlI wllI't1wr 1.111' ('OIlVI'rHC, iml'lic'lltillll iH trull.

= O.

B: Intflr1M!Ction

Multipli(~ity

of Two Modul.

J 11

Proof (Grothendieck): Let k be the residue field of A. We are going to determine in two different ways the largest integer r such that the "triple Tor" Tor~(M, N, k) is ~ 0: a) The spectral sequence Tor: (Tor:(M, N), k) ===> Tor~q(M, N, k) shows that Tor1(M,N, k) = 0 if j > i + n, and that Tor~+n(M, N, k) = Tor~(Tor~(M, N), k) ~ 0,

since Tor~(M,N) is a nonzero A-moduleoffinitelength. Hence r =n+i . . b) The spectral sequence Tor:(M, Tor:(N, k» ===> Tor:+q(M, N, k) shows that r = proj dim(M) + proj dim(N) (use the "maximum cycle principle"). Whence n + i = proj dim(M) + proj dim(N) , qed.

Corollary.

The hypotheses being those of theorem 4, in order that Tor~(M,N) = 0

for i > 0,

it is necessary and sufficient that M and N are Cohen-Macaulay modules and that dimM + dimN = n.

We can write the integer i from theorem 4 in the following form: i

= (proj dim(M) + dim M - n) + (proj dim(N) + dim N - n)

+ (n -

dimM - dimN) = (dimM - depthM) + (dimN - depthN) + (n - dimM - dimN). But each term between parentheses is ~ 0 (for the first two, according to Chap. IVj for the third, according to tho 3). Thus i = 0 if and only if each of these terms is zero, whence the result we want. Remark.

When the hypotheses of the corollary are satisfied, we have X(M, N)

=

£(M ®A N)j

it is probable that the converse is true, and that one has X(M, N)

< £(M ®A N)

if either M or N is not Cohen-Macaulay. More generally, one may conjecture that each of the "higher Euler-Poincare characteristics" X,,(M, N) =

L (_l)i £(Tor~+,.(M, N»,

r = 1, ... , n,

,~o

is ~ 0, and that x,- = 0 if and ollly if eR.r.h of tllf! Tor~+ r (M. N) i8 Zf!rO, cf. ChIll'. IV. Apl'l'lldlX 11. This iK 8t Ic!ut trlJt! In tho f'{JIIIU dlRfllC:tNIMtlc

112

V. Multiplicities

case, according to Auslander-Buchsbaum. That explains why the Gr6bner definition of multiplicities (in terms of f(M ®A N) ) gives the right result only when the varieties are locally Cohen-Maca.,ulay (see the examples constructed by Grabner himself, [Grob D.

C: Connection with Algebraic Geometry 1. Tor-formula Let X be an algebraic variety, defined over a field k. For simplicity, we suppose that k is algebraically closed, and X is irreducible. Let U, V, W be three irreducible subvarieties of X , W being an irreducible component of U n V. Suppose that W meets the open set of smooth points of X , i.e. that the local ring A of X at W is regular (the equivalence "smooth" = "regular" follows from the fact that the ground field k is perfect). Then (cf. part B,§3): dim U

+ dim V

~

dim X

+ dim W.

(1)

When there is equality in this formula, the intersection is called proper at W (and one says that U and V intersect properly at W). Let Pu and Pv be the prime ideals of the local ring A which correspond to the subvarieties U and V. By hypothesis, A is regular, and A/(pu + pv) has finite length. The Euler-Poincare characteristic . dimX

L

XA(A/Pu,A/pv) =

(-I)ifA(Tor~(A/pu,A/pv»

i=O

is defined; this is an integer

~

0 (cf. part B).

Theorem 1. (a) If U and

V do not intersect properly at W, we have XA(A/l'u, A/pv) = O. (b) If U and V intersect properly at W, XA(A/pu, A/l'v) is > 0 and coincides with the intersection multiplicity i(X, U . V, W) of U and V at W, in the sense of Weil, Cheva1ley, Samuel (cf. [W],[Ch2 ],[Sa3]).

Assertion (a) follows from theorem 1 of part B. We will prove (b) in §4, after having shown that the function I(X,U· V, W) = :\A(A/Pu,A/pv) satisfies the formal properties of an "intersection multiplicity".

c:

C()lmec~lon

with

All~hralc Georn.~r)'

2. Cycles on a non-singular affine variety

/

113

..

Let X be a non-singular affine variety, of dimension n, and with coordinate ring A. If a EN, and if M is an A -module of dimension ::; a, the cycle za(M) is defined (cf. part A); it is a positive cycle of dimension a, which is zero if and only if dim M < a . Proposition 1. Let a, b, c E N such that a + b = n be two A -modules such that: dimM ::; a,

dimN

~

b,

+ c.

dimM ®A N

~

Let M , N

c.

(2)

Then the cycles za(M) and zb(N) are defined, they intersect properly, and the intersection cycle za(M)· Zb(N) (defined by linearity from the function I of §1) coincides with the cycle

zc(TorA(M, N))

=

2) _1)' zc(Tor~(M, N)).

(3)

Let W be an irreducible subvariety of X of dimension c, corresponding to a prime ideal t of A; let B be the local ring At" (i.e. the local ring of X at W). By definition, the coefficient of W in the cycle zc(TorA(M, N)) is equal to ~

.

A

L.., (-1)' £B(Tori

B

(M, Nh) = X (Mr, Nt).

This coefficient is thus "biadditive" in M and N, and is zero if either dim M < a or dim N < b, cf. part B, §3. The same is obviously true for the coefficient of W in za(M) . zb(N). We are thus reduced to the case where M = Alp, N = Alq , the ideals p and q being prime and corresponding to irreducible subvarieties U and V of X of respective dimensions a and b. In this case, the coefficient of W in zc(TorA(M, N)) is equal to XB(BlpB,BlqB) = /(X,U· V, W), qed. Remarks. 1) Proposition 1 gives a very convenient method for computing the intersection product z· z' of two positive cycles Z and z' , of dimensions a and b, intersecting properly: choose modules M and N for the cycles z and z' such that dim M®N has the desired dimension (this is automatically the case if Supp(M) = SlIpp(Z) and Supp(N) = Supp(z')), and the cycle z·z' we want iH simply !.lit! "eyd(! of Tor(M, N) ", i.e. the alt.ernat.ing sum of t.he eyries of Tor,(M. N) . . 2) In the case of al!l;d)mk varietlt!H whkh Ilrt! not neceflHluily I\ffine, coht~fY'nt .~hl'tl1l~lJ rl'pllu'(' mO<\IIII'II. If M iN Much 11 Nlwllf. wit.h dimSllpp(M) 'S a (whidl Wl'lll/'l(1 wrltf! dlmM <:; (1). ollt' dl·IiI\l!1I ill all ohvioliN w.'y till" r"d,' z,,(M). I'Wl'llfIlUlln I rnmahlll valid, with

114

V. MultiplicitieK

the modules 1br~ (M, N) being replaced by the sheaves 'IOTi(M,N) the ':to'C being taken over the structural sheaf Ox of X .

I

3. Basic formulae We shall see that the product of cycles, defined by means of the function I from §l (Le. by taking the "Tor-formula" as definition), satisfies the fundamental properties of intersection theory; these properties being local, we may suppose that the varieties we consider are affine and non-singular. This will allow us to apply proposition 1 from the preceding section.

. a)

Commutativity Obvious, because of the commutativity of each Tori'

Associativity We consider three positive cycles z, z', z" of respective dimensions a, a', a". We assume that the products z· z', (z· z') . z", z'· z" and z· (z' . z") are defined, and we have to prove that (z . z') . z" = z· (z' . Zll).

b)

Let A be the coordinate ring of the given ambient variety X , and let n be its dimension. Choose an A -module M with support equal to that of z, and such that za(M) = Z; let M' and M" be such modules for z' and z". The desired formula is proved from the "associativity" of Tor. According to [CaE], p. 347, this associativity is expressed by the existence of the triple Tor Tor~(M, M', Mil) and the two spectral sequences: Tor:(M, Tor:(M', Mil)) ==='> Tor:+q(M, M', Mil)

(4)

Tor:(Tor:(M, M'), M") ==='> Tor:+q(M, M', Mil). (5) Set c = a + a' + a" - 2n, and b = a' + a" - n. Since the intersections considered are proper, we have and dim M ® M' ® Mil < c. dim M' ® Mil ::; b One can thus define the cycles Yq Xp,q

zb(Tor:(M', M")), = zc(Tor:(M, Tor:(M',M"))),

= zc(Tor~(M, M', Mil)). The invariance of Euler-Poincare characteristics through a spectral sequence, applied to (4), gives: Xi

L(-l}i Xi = L(-l)p+qxp,q. ,..'1

c: Connection

with Algebraic Geometry

115

But proposition 1 shows that

~) -l)Pxp,q = z· Yq

L:( -l)qYq

and

Thus

=

ZI.

z".

q

p

L

(-l)ixi

=

z· (Zl . z").

i

The same argument, applied to (5), gives

L (-l)ixi

= (z· Zl) . z",

whence the associativity formula we want.

c)

Product formula Consider two non-singular varieties X and X', and two cycles Zl, Z2 (resp. z~, z~) on X (resp. X'). We suppose that Zl' Z2 and z~ . z~ are defined. Then the product cycles Zl x zi and Z2 x z~ (on X x X') ,intersect properly, and we have: (Zl x zD . (Z2 x z~) = (Zl'

Z2)

x (z~ . z~).

(6)

Indeed, we may assume that these cycles are ~ 0, and that X and X' are affine, with coordinate rings A and A'. If M 1 , M 2 , M{, M2 are modules corresponding to Zl, Z2, z~, z~, one checks that the cycle associated to Ml ®k M{ (viewed as a module over the ring B = A ®k A' of X x X' ) is equal to Zl x z~ [this fact could even be taken as the definition of the direct product of cycles]. The formula to be proved then follows from the "Kiinneth formula":

Tor~(Ml ® MLM2 ® M 2) =

EB

Tort(M1, M2) ®k Tort' (MLM2)·

i+j=h

d)

Reduction to the diagonal Let ~ be the diagonal of X x X . We have to show the formula Zl . Z2 = (Zl x

Z2) . ~,

(7)

valid when the cycles Zl and Z2 intersect properly. Let A be the coordinate ring of X , and let B = A ®k A be that .of X x X. If Ml and M 2 are modules corresponding to Zl and Z2 respectively, we have Tort(M1 , M2) = Torf(M I ®k M 2, A), cr. part B, §l. Formula (7) then followtl by taking the alternating sum of the eyclml on both lIidml.

U6

V. Multiplicities

4. Proof of theorem 1 We want to show that the functions I and i coincide. We begin by treating the case where U is a complete intersection in W; this means that the ideal Pu of §l is generated by h elements Xl, ..• ,Xh, with h = dim X - dim U = dim V - dim W . We have Tort(A/pu,A/pv) = Hi(x,A/pv) , cf. Chap. IV, cor. 2 to prop. 3. Hence XA(A/Pu,A/pv) =

2::C- 1)if(Hi (x,A/pv)),

and by theorem 1 of Chap. IV, this gives XA(A/pu, A/pv) = ex(A/pv),

where x denotes the ideal of A/pv generated by the images of the Xi. But according to [Sa4), p. 83, the multiplicity ex (A/pv) is equal to i(X, U . V, W) , which proves the equality I = i in this case. The general case reduces to the previous one, by using reduction to the diagonal, which is valid for both I and i. Since ~ is non-singular, it is locally a complete intersection, and the hypotheses of the preceding case are satisfied, qed.

5. Rationality of intersections For simplicity we restrict ourselves to the case where X is an affine variety with coordinate ring A over k. Let ko be a subfield of k. We say (in Wei! style) that X is defined over ko if one has chosen a ko -subalgebra Ao of A such that A = Ao ®ko k . Let Mo be an Ao -module (finitely generated, as usual), with dim Mo :::; a. We can view Mo ®ko k as an A-module, and we have . dim(Mo ® k) :::; a, which allows us to define the cycle za(Mo ® k). A cycle z of dimension a on X is called rational over ko if it is the difference of two cycles za(Mo ® k) and za(Mo ® k) obtained in this way. The abelian group of cycles rational over ko has a basis consisting of the "prime cycles" za(Ao/Po ® k) , where Po ranges over the set of prime ideals of Ao such that dim(Ao/po) = a. This definition of the rationality of cycles is equivalent to that given by Wei! in [W]; this non-trivial fact can be proved by interpreting the "order of inseparability" which appears in Weil's book in terms of tensor products of fields (cf. [ZS I, Chap. III, p. 118, tho 38). Theorem 2 (Weil). ko ,

IllId ,'illdl

Let z and z' be two rycles on X . rational over 1.' is ru/.ill/lltl IIW'" ku .

t //11/ ;:. z' i.'i 1/"tilll·d. '1'1""1 ;:.

C: C'.onnectlon with Algebraic Geometry

117

We can assume that z and z' are positive, thus corresponding to Ao -modules Mo and M~. The theorem then follows from the formula: Tor~(Mo ® k,M~ ® k) = TortO(Mo, M~) ® k.

6. Direct images Let f : X -+ Y be a morphism of algebraic varieties (over an algebraically closed field k, to fix ideas), and let Z be a cycle on X of dimension a. The direct image f.(z) of z is defined by linearity, starting from the case where z = W , an irreducible subvariety of X. In this case, set: 0 if the closure W' of f(W) has dimension < a, { f.(W) = dW' if dim W' = a, where d = [k(W) : k(W')] is the degree of the map f : W -+ W'. This operation preserves dimension. It is mainly interesting when f is proper (do not confuse the properness of a morphism with that of an intersection!), because of the following result: Proposition 2. Let f : X -+ Y be a proper morphism, let z be a cycle on X of dimension a, and let M be a coherent sheaf on X such that za(M) = z. Let Rq f(M) be the q -th direct image of M , which is a coherent sheaf on Y ({EGA], Chap. III, tho 4.1.5). (a) We have dimRo f(M) ~ a and dimRqf(M) < a for q ~ 1. (b) We have

f.(z)

=

za(Ro f(M»

=

L (-l) Qza(Rq f(M». Q

The proof is done by reducing to the case where the restriction of f to the support of Z is a finite morphism, in which case the Rq f(M) are zero for q ~ 1.

7. Pull-backs They can be defined in diverse sit.llations. We consider only the following: Let f : X -+ Y be a morphislJI, wit.h Y heinf.!; non-singular, and let :t and y be cycles on X and Y respmtively. Set Ixl = Supp(x) and 1111 = Supp(y) . Then: dim Ixl n /-1 (lyl) ~ dim lxl - codlm 1!l1. The "proper" eMf! is that wh,·rt· dlllr" IH tMllIl\lity. In t.lmt CftlI(l, Oll(l d"f1l1flH all intorfKldlnn eye-In .r./ 11 wit.h Mippuft mllt.nlne.. 1 III Ixl n f - 1(1",1) hy eit.lll'l lilli' tIt' I ht, 1'"lIlIwillJ(, III1·t/UH\fI

118

V. Multiplicities

a) Reduction problem being variety V, for X into V x Y cycle ')'( x) on such that:

to a standard intersection: assume X to be affine (the local), which allows it to be embedded in a non-singular example an affine space. The map Z f-+ (z,/(z» embeds , and thus allows us to identify any eycle x on X with a V x Y . Then one defines x· f y as the unique cycle on X

')'(x 'f y) = ')'(x). (V x y), (8) the intersection product of the right hand side being computed on the nonsingular variety V x Y. One checks that the result obtained is independent of the chosen embedding X -+ V . b) Choose coherent sheaves M and N over X and Y of respective cycles x and y, and define x· f y as the alternating sum of the cycles of the sheaves 'roti(M, f*(N)) , the 'rOti being taken over Ox (and being sheaves on X); since Y is non-singular, the 'rOti are zero for i> dim Y , and the alternating sum is finite.

Special case: Take x = X . The cycle X· f Y is then written f* (y) and is called the pull-back of y. Recall the conditions under which it is defined: i) Y is non-singular, ii) codim 1-1 (Iyi) = codim Iyl . No hypothesis on X is necessary.

Remarks. 1) When X is non-singular, we have (9) x'fY = x·/*(y), provided that both sides are defined. 2) The special case when Y is a line is the starting point of the theory of linear equivalence of cycles. Projection lormula It is the formula: valid when

I

I.(x·f y) = I.(x), y, is proper and both sides are defined.

(10)

The proof can be done by introducing sheaves M and N corresponding to the cycles x and y, and using two spectral sequences with the same ending and with the E2 terms being respectively Rq 1('rotp(M,N))

and

'roti(Rj I(M),N),

the 'rOt being taken over Oy (cf. [EGA J, Chap, III, prop. 6.9.8). When X is non-Hin~1I1ur. thiH forllllliu takeH tht! Ht.ul\dard form:

J.(.'"

r(,,)) -

J.(.r)·"

(II )

C: ConnfK:tion wit.h Algebraic Geomet.ry

,IlY

Exercises (see also [Fl, Chap. 8). 1) Let Z !!.. y 1-. X, and let x, y ,Z be positive cycles on X, Y ,Z. Suppose that X and Yare non-singular. Prove the following formula (valid whenever the products which appear are defined):

=

=

(12) Recover (for I = 9 = 1 ) the associativity and commutativity of the intersection product. For X = Y, I = 1 , deduce the formula: z'g(Y'Jx)

{z'gY)'Jgx

(z·Jgx)·gY·

(13)

g*(x·y) = g*(x)'gY,

whence g*(x. y) = g*(x)· g*(y) when Z is non-singular.

2) Same hypotheses as in 1), with the difference that Y may be singular, but that 9 is proper (it suffices that its restriction to Supp(z) is so). Prove the formula: g*(Z 'Jg x)

= g*(z)'J x,

valid when both sides are defined. (For

(14)

1= 1, one recovers (10).)

3) Give the conditions of validity for the formula: (YI ~ Y2) '!Ixh (Xl x X2) = (YI·ft Xl) x (Y2'h X2).

(15)

4) Let I : Y -+ X, f' : Y -+ X' , with X, X' non-singular, let 9 = (j, I') : Y -+ X X X'. Let x, x' , Y be cycles on X, X' , Y. Give conditions of validity for the formula: (Y'JX)'f'x' = (Y'f'x')'Jx

=

Y·g(xxx' ).

5) Let I : Y -+ X and 9 : Z -+ X, with X non-singular. Let be cycles on Y, Z. Define (under the usual properness conditions) a "fiber product" y'x z, which is a cycle on the fiber product Y Xx Z of Y and Z over X. What does this give when 9 = I? And when X is reduced to a point? y,

Z

8. Extensions of intersection theory It is clear that the "Tor-formula" allows us to define the intersection of two cycles in more general cases than those of classical algebraic geometry. For example: i) It applies to analytic (or formal) .~I,acr.8. There is no difficulty, since every local ring which is involwd ill of ('qu,,1 chlLmderiHtic. In the CI18e of COTnl,ier analytic H\llu:eH, 1.111' inll~rHI,,·t.iOIl prod III'l HI) ohl.aitll,,1 coilH:ideH with that dPlilU'd topolo~iCILlly hy Bord-lIlu'IIiJ(C'f. 1'( IBolI]; thili iii provtld by reduction to the "t,lellltmtary" CIUltl 4.10 of thulr paper.

120

V. Multiplicities

ii) It applies to every regular scheme X provided that the conjectures of part B have been checked for the local rings of such schemes; this is especially the case when these local rings are of equal characteristic. [Even when X is a scheme of finite type over a field k, this gives an intersection theory a little more general than the usual one; indeed, if k is not perfect, it may happen that X is regular without being smooth over k; but Weil's theory applies only to the smooth case. ] iii) More generally, intersection theory applies to every scheme X which is smooth over a discrete valuation ring C. One can indeed show that the local rings of X satisfy the conjectures from part B [ the proof is done by a process of reduction to the diagonal which is analogous to - and simpler than - the one used in part B, §5 ]. This case is important, because it gives the reduction of cycles of Shimura, [Sh]. We briefly indicate how: Let k (resp. K) be the residue field of C (resp. its field of fractions). The scheme X is a disjoint sum ofits closed subscheme Xk = X®ck and its open subscheme X K = X ®c K ; the scheme Xk is of finite type over k (it is an "algebraic variety" over k); similarly, XK is of finite type over K. One sometimes says, rather incorrectly, that Xk is the reduction of XK·

Every dimension; cycle z of dimension

cycle on X k defines, by injection, a cycle on X of the same every cycle z of dimension a on XK defines by closure a dimension a + 1 on X. The group Zn(X) of cycles on X of n thus decomposes into a direct sum: Zn(X)

=

Zn(Xk) E9 Zn-l(XK)'

The projection Zn(X) - Zn-l (X K ) is given by the restriction of cycles. From the point of view of sheaves, the cycles from Z(Xk) correspond to coherent sheaves M over X which are annihilated by the uniformizer 7r of C; those from Z(XK) correspond to coherent sheaves M which are flat over C (i.e. torsion-free); this decomposition into two types has already played a role in part B, §5. Now let z E Zn(XK), and let z be its closure. We can view Xk as a cycle of codimension 1 on X . The intersection product

z=

Xk'

z

(computed on X)

belongs to Zn(Xk) ; it is called the reduction of the cycle z. Moreover this operation can be defined without speaking of intersections (and without any hypothesis of smoothness or regularity); from the point of view of sheaves, it amounts to associating to every coherent sheaf M that is fiat over C the sheaf M / 7r M. The hypothesis of smoothness comes only for proving the formal properties of the operation of reduction: compatibility with products, direct images, intersection products; the proofs can be done, as in the preceding sections, by working with identities between sheaves, or, at worst, with spectral sequences. '1'111' mtl'7'Mdum thl'ory on X RiveH more than tlu' uWrP fY'durtinn of ("1idl~.~. Thllli if ;r 1L1lt! I' Ilfl' '"yd.'!' 011 X h , till' 1"0111110111'111 of J' . T

C: Connection wit.h Algf'hraic Geomet.ry

121

in Z(X",) gives an interestin~ invariaut of the pair x, x' (8B8uluing, of course, that the intersection of x and x' is proper); this invariant is related to the "local symbols" introduced by Neron, [Ne].

Bibliography

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mbliography

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Index

acyclic complex on a module, 52 additivity formula, 100 annihilator, 4 artinian (ring), 21 Artin-Rees theorem, 17 associated prime ideal, 6 associativity formula, 114 Auslander theorem, 71 Auslander-Buchsbaum theorem, 78 basic degrees, 92 binomial polynomials, 19 canonical topology, 19 chain of prime ideals, 29 chain of prime ideals, extremity, 29 chain of prime ideals, length of, 29 chain of prime ideals, maximal, 45 chain of prime ideals, origin, 29 chain of prime ideals, saturated, 45 codifferent, 40 Cohen's theorem, 80 'W Cohen-Macaulay lIlodule, 63 Cohen-Macaulay ring, 63 Cohen-SeidmllH'rg firtlt theorem, 31 ColIPIl-Sc-idc-llherg l'IflCond I.liC'ClrC-lIl, :12 compl4'ted tC'IlHClr product, 102 rom "Ic-tc'd '( hr ,III:.! c'ydl', ~,~,

cycle, direct image, 11 7 cycle, intersection, 117 cycle, positive, 99 cycle, pull-back of, 118 cycle, rational over a field, 116 cycles, reduction of, 120 Dedekind ring, 39 depth, 61 difference operator, 20 dimension of a module, 33 dimension of a ring, 29 direct image (of a cycle), 11 7 discrete valuation ring, 37 divisorial ideal, 39 domain, 1 embedded prime ideal, 8 equal characteristic, 106 essential prime ideal, 10 Euler-Poincare characteristic, 56 factorial ring, 39 filtered module, 11 filtl'red ring, 11 finit.e l«mgt.h, 6 Hilt mod ule, 2 KI(lIII~1

homologkal dimetlHiotl, 70 Kmcll'C 1- fre .. •• !H

Krlul,·,I· pClIVIIClIllII~1 nlJ(4'hm. 92 Iflul,·,IIII'IIIIII,·, 14 ... ",I'ICI rill". 14

12M

Ind~x.·

highp.r Euler-Poincare characteristics, 57, 88, 111 Hilbert polynomial, 22 Hilbert theorem, 22 homological dimension, 70 ideal of definition, 33 induced filtration, 11 injective dimension, 70 integer-valued polynomial, 20 integral, 30 integral closure, 32 integrally closed, 32 intersect properly, 112 intersection cycle, 117 irreducible, 4 irreducible component, 5 Jacobson radical, 1 Koszul complex, 51 Krull theorem, 37 lie over, 31 local ring, 1 m -adic filtration, 12 m -adic topology, 12 M -sequence, 59 maximal ideal, 1 minimal homomorphism, 84 minimal free resolution, 84 multiplicity, 100 Nakayama's lemma, 1 noetherian module, 4 noetherian ring, 4 non-singular variety, 82 normal ring, 37 normalization lemma, 42 Nullstellensatz, 44 Poincare series, 92 polynomial-like function, 21 primary (suhlllodule, ideal), 10 primary df'COIIlP{lI!itioll, 10

prime Ideal, product. forllllllll, 115 projection formulll, 118 projectivl'! dimension, 70 proper intersection, 112 pseudo-reflection, 95 pull-back (of a cycle), 118 q -good (filtration), 17 quotient filtration, 11

radical, 3 reduction to the diagonal, 49, 101, 115 reflexive module, 78 regular ring, 75 regular system of parameters, 78 right interior product, 87 Samuel polynomial, 24-25 Samuel theorem, 24 semilocal ring, 1 Shephard-Todd theorem, 95 shift, 52 spectrum, 4 strict morphism (of filtered modules), 12 support (of a module), 5 symbolic power (of a prime ideal), 38 system of parameters, 36 Tor-formula, 112 unramified (local ring), 108 Wei! theorem, 116 Zariski ring, 19 Zariski topology, 4 zero-divisor (in a module), 53

Index of Notation

p(n)

A p , Mp : 3

vp

rad(a)

:3

alg dimk A : 44

Spec(A) , V( a) : 4

K(x) , K(x,M)

Ann(M)

:4

Supp(M)

:5 :6

AnnM(x) = Ker(xM) : 52 K(x) ,K(x,M) : 53

Ass(M) FA : 11

:

38

S-lA, S-IM : 2

:

38

: 51

Hp(x,M) : 53 ex(E, n) : 57,65

1m, Coim : 11-12

depthAM : 61

Ker, Coker : 11-12

proj dimA M , inj dimA M globdimA : 70

.4, if : 13 gr(M) : 14 ~

: 20

X(M, n) : 22 Q(M), Q(M, n)

: 22

P«M,)) , P«(Mi),n) : 24-25 Pq(M) , Pq(M,n) :25 ht(a)

: 29--30

dimA : 29 dimM : 33

m(A) : 33

A(d) : 91 ¢>A(t) : 92 zp(M) : 100, 113 ea(M) , ea(M,p) : 100

-

T ori : 102 , i(X, U . V, W)

: 112, 116

I(X, U . V, W)

: 112, 116

/.(Z) , f.(W) .Z'fy:117

/"(y) : 118

.' .

: 117

: 70