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of rank r. Furthermore let
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

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