f

F ’’ G ’’ M ’’ 0

be a ¬nite free representation of M , and n = rk G. Then MP is a free AP -module for a

prime ideal P of A if and only if In’r (f ) ‚ P .

Proof: One observes that MP is free if and only if µ(MP ) ¤ r and applies (16.3). ”

Extending the presentation to a free resolution the reader may formulate the gener-

alization of (16.10) describing the prime ideals P such that pd MP ¤ k.

B. Grade and Acyclicity

Let A be a local ring, P its maximal ideal, and M a ¬nitely generated A-module.

The length of a maximal M -sequence inside P is usually called depth M . In the following

de¬nition we replace P by an arbitrary ideal in a noetherian ring.

Definition. Let A be a noetherian ring, I ‚ A an ideal, and M a ¬nitely generated

A-module such that IM = M . Then the grade of I with respect to M is the length of a

maximal M -sequence in I. It is denoted by grade(I, M ).

The reader may consult [Mt], Ch. 6 for the de¬nition of “M -sequence.” There the

notation depthI (M ) is used for grade(I, M ) (Attention: The ¬rst edition of [Mt] di¬ers

considerably from the second one in regard to Ch. 6 !) It is easy to see that the grade

just de¬ned is always ¬nite; in fact, it is bounded by ht I.

Very often we shall have M = A, and therefore we introduce the abbreviation

grade I = grade(I, A).

For systematic reasons it is convenient to cover the case in which IM = M , too; thus we

put grade(I, M ) = ∞ if IM = M .

It is very important that grade can be computed from homological invariants.

(16.11) Theorem. Let A be a noetherian ring, I ‚ A an ideal, N a ¬nitely gen-

erated A-module such that Supp N = {P ∈ Spec A : P ⊃ I}. Then

grade(I, M ) = min{j : Extj (N, M ) = 0}

A

for every ¬nitely generated A-module M .

The case in which grade(I, M ) < ∞, thus IM = M , is an immediate consequence of

[Mt], Theorem 28 (and stated on p. 102 of [Mt]). If grade(I, M ) = ∞, one has MP = 0

for all P ∈ Supp N , thus Supp(Extj (N, M )) = … for all j.

A

The case N = A/I of (16.11) suggests that grade(I, M ) is an invariant of A/I rather

than an invariant of I. It would even justify to call grade(I, M ) the grade of N with

respect to M . We restrict ourselves to the case M = A.

Definition. Let N be a ¬nitely generated module over the noetherian ring A. The

grade of N is the grade of Ann N with respect to A, abbreviated grade N .

In order to avoid the ambiguity thus introduced we insist on the ¬rst meaning of

grade I whenever I is considered an ideal. An immediate corollary of (16.11):

207

B. Grade and Acyclicity

(16.12) Corollary. grade N ¤ pd N .

As a consequence of (16.11) one has grade(I, M ) = grade(Rad I, M ). This follows

also from the following local description of grade for which we refer to [Mt], p. 105,

Proposition:

(16.13) Proposition. With the notations introduced above one has

depth MP : P ∈ Spec A, P ⊃ I .

grade(I, M ) = inf

Another fact implied by (16.11) is the behaviour of grade along exact sequences

(which can of course be derived directly from the de¬nition of grade):

(16.14) Proposition. Let A be a noetherian ring, M1 , M2 , and M3 ¬nitely gen-

erated A-modules connected by an exact sequence

0 ’’ M1 ’’ M2 ’’ M3 ’’ 0.

Then one has for every ideal I of A:

grade(I, M3 ) ≥ min(grade(I, M1 ), grade(I, M2 )) ’ 1,

(a)

grade(I, M2 ) ≥ min(grade(I, M1 ), grade(I, M3 )),

(b)

grade(I, M1 ) ≥ min(grade(I, M2 ), grade(I, M3 ) + 1).

(c)

With the notations and hypotheses as in (16.11) let

f1

F : · · · ’’ Fn ’’ · · · ’’ F1 ’’ F0

be a free resolution of N = Coker f1 . Put m = grade(I, M ) and consider the truncation

— — — — —

Fm : 0 ’’ F0 ’’ F1 ’’ · · · ’’ Fm’1 ’’ Fm

—

of the dual HomA (F, A). The inequality “¤” in (16.11) implies that Fm — M is acyclic.

This fact admits a far-reaching generalization:

(16.15) Theorem. Let A be a noetherian ring, and

fn f1

F : 0 ’’ Fn ’’ Fn’1 ’’ · · · ’’ F1 ’’ F0

n

a complex of ¬nitely generated free A-modules. Let rk = i=k (’1)i’k rk Fi and Jk =

Irk (fk ). Furthermore let M be a ¬nitely generated A-module. Then the following state-

ments are equivalent:

(a) F — M is acyclic.

(b) grade(Jk , M ) ≥ k for k = 1, . . . , n.

Proof: First we prove the implication (b) ’ (a) by induction on the length n of

F. One may suppose that F — M is acyclic, F given as the truncation

f2

F : 0 ’’ Fn ’’ Fn’1 ’’ · · · ’’ F2 ’’ F1

208 16. Appendix

of F. It “resolves” C — M , C being the cokernel of f2 . We have to show that the induced

homomorphism f 1 : C — M ’’ F0 — M is injective. By virtue of (16.4) the localizations

FP , P ∈ Ass M , are split-exact, and so is (F — M )P . Hence (Ker f 1 )P = 0 for all prime

ideals P ∈ Ass M . In order to derive a contradiction let us assume that Ker f 1 = 0.

Hence there exists an element b ∈ A, b neither a unit nor a zero-divisor of M , such that

b(Ker f 1 ) = 0. For a prime ideal Q minimal in Supp(Ker f 1 ) one then has depth MQ ≥ 1,

but depth(Ker f 1 )Q = 0.

Let m = depth MQ . Suppose ¬rst m ≥ n. Then an iterated application of (16.14),(a)

to the “M -resolution” F — M of C — M yields depth(C — M )Q ≥ 1. Let next 0 < m <

n. Applying (16.4) again, we see that F = (Coker fm+1 )Q is a free AQ -module, and

(F — M )Q decomposes into a split-exact tail

0 ’’ (Fn — M )Q ’’ · · · ’’ (Fm — M )Q ’’ F — MQ ’’ 0

and a shorter MQ -resolution of (C — M )Q

0 ’’ F — MQ ’’ (Fm’1 — M )Q ’’ · · · ’’ (F1 — M )Q .

By an iterated application of (16.14),(a) again: depth(C — M )Q ≥ 1. Since Ker f 1 ‚

C — M , depth(Ker f 1 )Q ≥ 1 as well, the desired contradiction.

In proving the implication (a) ’ (b) we may inductively suppose that grade(Jk , M )

≥ k ’ 1 for k = 1, . . . , n. Assume that grade(Jk , M ) = k ’ 1 for some k, and let P ⊃ Jk

be a prime ideal, depth MP = k ’ 1. In order to derive a contradiction we replace A by

AP , and, as above, split o¬ the tail of F. Then we substitute the right part of F for

F itself, and k for n, and conclude that it is enough to prove that depth M ≥ n. This

follows from the case n = 1, which we postpone, by induction: Let F be as above. If

depth M ≥ 1, there is a non-unit a ∈ A which is not a zero-divisor of M . Elementary

arguments imply the exactness of

(F — M ) — (A/Aa) = F — (M/aM ).

The inductive hypothesis yields depth M/aM ≥ n ’ 1, whence depth M ≥ n.

After all, we have reduced the implication (a) ’ (b) to the following statement: Let

A be a local ring, f : F1 ’ F0 a homomorphism of ¬nitely generated free A-modules,

M a ¬nitely generated A-module for which f — M is injective; if depth M = 0, then f

embeds F1 as a free direct summand of F0 . Suppose, not. Then there is an element

e ∈ F1 which belongs to a basis of F1 such that f (e) ∈ P F0 , P being the maximal ideal

of A. On the other hand there is an x ∈ M , x = 0, such that P x = 0. Now e — x = 0,

but f (e — x) ∈ P F1 — x = F1 — P x = 0, a ¬nal contradiction. ”

In view of (16.5),(d) and (16.7), the reader may prove that Irk +1 (fk )M = 0 for

k = 1, . . . , n if F — M is exact.

Undoubtedly the most important case of the theorem is the one in which M = A;

and very often the following weaker version of (b) ’ (a) is all one needs:

(16.16) Corollary. If F — AP is split-exact for all prime ideals P such that

depth AP < n, then F is exact.

209

C. Perfection and the Cohen-Macaulay Property

C. Perfection and the Cohen-Macaulay Property

In (16.12) it is stated that the projective dimension of a module always bounds its

grade: grade M ¤ pd M . The modules for which equality is attained, are of particular

importance and merit a special attribute:

Definition. Let A be a noetherian ring. A ¬nitely generated A-module M such

that grade M = pd M , is called perfect. By the usual abuse of language, an ideal I is

called perfect if A/I is a perfect A-module.

Perfect modules are distinguished by having a perfect resolution: They have a pro-

jective resolution P of ¬nite length whose dual P — is acyclic, too, cf. (16.11). P — resolves

Extg (M, A), g = grade M , and Extg (Extg (M, A), A) = M .

A A A

Perfect modules are “grade unmixed”:

(16.17) Proposition. With the notations of the de¬nition, a prime ideal P of

A is associated to M if and only if MP = 0 and depth AP = grade M . Furthermore

grade P = grade M for all P ∈ Ass M .

Proof: Because of Ass M ‚ Supp M , we may suppose MP = 0. If depth AP =

grade M , then depth MP = 0 since always

grade M ¤ grade MP ¤ pd MP ¤ pd M,

and

depth MP = depth AP ’ pd MP

by the equation of Auslander and Buchsbaum. Conversely, if depth MP = 0, necessarily

depth AP = grade M . It remains to prove that grade P = depth AP . Suppose grade P <

depth AP . Then there is a prime ideal Q ⊃ P such that grade P = grade Q = depth AQ ≥

pd MQ ≥ pd MP . Contradiction. ”

The preceding proof shows that pd MP = grade MP = grade M for all prime ideals

P in the support of a perfect module M .

The main objects of our interest will be certain perfect ideals I in a polynomial ring

A = B[X1 , . . . , Xn ]. In the investigation of A/I it is often important to know that an

ideal in A/I has grade ≥ 1. In this connection the following proposition will be very

useful.

(16.18) Proposition. Let A be a noetherian ring, I a perfect ideal in A, and J ⊃ I

another ideal. Then

grade J/I ≥ grade J ’ grade I,

where J/I is considered an ideal of A/I, of course. If J is perfect, too, one has equality.

Proof: Let Q ⊃ J/I be a prime ideal, and P the preimage of Q in A. Then

depth(A/I)Q = depth(A/I)P = depth AP ’ pd(A/I)P ≥ grade J ’ grade I,

and this is enough by (16.13). If J is perfect, too, we obtain equality by ¬rst choosing P

as an associated prime of J, and Q as its image in A/I. ”

We say that a noetherian ring A is a Cohen-Macaulay ring if each of its localizations

AP is Cohen-Macaulay. For modules we adopt the analogous convention. The theory of

Cohen-Macaulay rings and modules is developped in [Mt], Sect. 16; cf. also [Ka], Chap. 3.

Perfection and the Cohen-Macaulay property are closely related:

210 16. Appendix

(16.19) Proposition. Let A be a Cohen-Macaulay ring, M a ¬nitely generated

A-module.

(a) If M is perfect, M is a Cohen-Macaulay module.

(b) If M is a Cohen-Macaulay module, pd M < ∞, and Supp M is connected, then M

is perfect.

Proof: Suppose pd M < ∞, P ∈ Supp M . Then

depth MP = depth AP ’ pd MP

and

dim MP = dim AP ’ ht Ann MP = depth AP ’ grade Ann MP .

Therefore MP is perfect if and only if MP is a Cohen-Macaulay module.

It only remains to prove that M is perfect if its localizations MP , P ∈ Supp M , are

perfect and Supp M is connected. Evidently, the crucial point is that pd MP is constant

on Supp M . The local perfection of M implies that

{P : pd MP = k} = {P : pd MP ≥ k} © {P : grade MP ¤ k}

for all k, 0 ¤ k ¤ pd M . Both sets on the right side are given as intersections of ¬nitely

many closed sets, each of which is the locus of vanishing of a (co)homology module;

cf. (16.11) for the rightmost set. Therefore the set on the left side is closed, too. Since

Supp M is connected, it is nonempty if and only if k = pd M . ”

In particular, perfection and the Cohen-Macaulay property of M are equivalent if

A is a polynomial ring over a ¬eld or the integers, and M is a graded A-module. (Note

that Supp M is connected if and only if A/Ann M has no nontrivial idempotents.) For

such modules perfection can even be tested at a single localization.

(16.20) Proposition. Let A be a polynomial ring over a ¬eld, P its irrelevant

maximal ideal, and M a graded A-module. Then the following conditions are equivalent:

(a) M is a Cohen-Macaulay module.

(a ) M is perfect.

(b) MP is a Cohen-Macaulay AP -module.

(b ) MP is a perfect AP -module.

Only the implication (b) ’ (a) needs a proof, and it follows immediately from the

fact that a minimal graded resolution of M over A becomes a minimal resolution of M P

over AP upon localization.

A very important invariant of a local Cohen-Macaulay ring A is its canonical (or:

dualizing) module ωA , provided such a module exists for A. We refer to [HK] and [Gr]

for its theory. The canonical module is uniquely determined (up to isomorphism). A is

a Gorenstein ring if and only if it is Cohen-Macaulay and ωA = A. A regular local ring

is Gorenstein, and a Cohen-Macaulay residue class ring A = S/I of a local Gorenstein

ring S has a canonical module:

ωA = Extg (A, S), g = grade I.

S

Let now A be an arbitrary Cohen-Macaulay ring. An A-module is called a canonical

module of A if it is a canonical module locally. We denote a canonical module by ωA

211

D. Dehomogenization

though it is no longer unique in general: if M is a projective module of rank 1, ωA — M is

a canonical module, too. The characterization of Gorenstein rings remains valid. For the

existence theorem quoted we have to require that grade IP = grade I for all P ∈ Supp A

now. Since the ideals I of interest to us are perfect, this is not an essential restriction.

If S is Gorenstein, I ‚ S a perfect ideal, and A = S/I resolved by

P : 0 ’’ Gg ’’ · · · ’’ G0 , Gi projective,

one has a very direct description of ωA : it is the (A-)module resolved by P — .

D. Dehomogenization

The principal objects of our interest are two classes of rings. The rings in the ¬rst

class are graded, and every ring A in the second one arises from a ring R in the ¬rst

one by dehomogenization: A = R/R(x ’ 1) for a suitable element x ∈ R of degree 1.

The rings A and R are much closer related than a ring and its homomorphic images in

general, and very often it will be convenient to derive the properties of A from those of

R. (Geometrically, R is the homogeneous coordinate ring of a projective variety and A

the coordinate ring of the open a¬ne subvariety complementary to the hyperplane “at

in¬nity” de¬ned by the vanishing of x.)

Let R = i≥0 Ri be a graded ring, x ∈ R1 a non-nilpotent element. The natural

homomorphism π : R ’ A, A = R/R(x ’ 1), factors through S = R[x’1 ] in a canonical

way, so one has a commutative diagram