ńņš. 44 |

Dualizing one obtains an exact sequence

0 ā’ā’ N ā— ā’ā’ F ā— ā’ā’ M ā— ā’ā’ Ext1 (N, A) ā’ā’ 0.

A

We split this into two exact sequences:

0 ā’ā’ N ā— ā’ā’ F ā— ā’ā’ K ā’ā’ 0,

0 ā’ā’ K ā’ā’ M ā— ā’ā’ Ext1 (N, A) ā’ā’ 0.

A

Dualizing again one gets exact sequences

Extiā’1 (N ā— , A) ā’ā’ Exti (K, A) ā’ā’ 0 for i ā„ 1,

A

A

Exti (Ext1 (N, A), A) ā’ā’ Exti (M ā— , A) ā’ā’ Exti (K, A) for i ā„ 0.

A A A A

217

F. The Theorem of Hilbert-Burch

Since NP is a free AP -module for all prime ideals P such that depth AP < n ā’ 1, one has

grade Ext1 (N, A) ā„ n ā’ 1,

A

and therefore Exti (Ext1 (N, A), A) = 0 for all i, 0 ā¤ i ā¤ n ā’ 2 (cf. (16.11)). One readily

A A

concludes that

Exti (M ā— , A) = 0 for i = 2, . . . , n ā’ 2.

A

Since n > 2, N is reļ¬‚exive, and therefore the linear map F ā—ā— ā’ N ā—ā— is the original

epimorphism F ā’ N , whence Ext1 (K, A) = Ext1 (M ā— , A) = 0, too. ā”

A A

(16.35) Remark. The hypothesis

pd MP < ā for all prime ideals P with depth AP < n ā’ 1

is only needed to ensure grade Ext1 (N, A) ā„ nā’1. In any case depth NP = depth AP for

A

all prime ideals P with depth AP < n ā’ 1, and arguing with an injective resolution of AP

one also concludes grade Ext1 (N, A) ā„ nā’1 if the localizations AP with depth AP < nā’1

A

are Gorenstein rings. Similarly one can replace the condition on M in (16.33) by the

hypothesis: AP is Gorenstein for all prime ideals P such that depth AP < n. (Observe

that Ext1 (N, A) = 0 for the module N constructed in the proof of (16.33).)

A

F. The Theorem of Hilbert-Burch

Commutative algebra is not very rich in classiļ¬cation theorems. One of the few

examples identiļ¬es the ideals I in a noetherian ring for which A/I has a free resolution

of length 2:

f g

0 ā’ā’ Am ā’ā’ Am+1 ā’ā’ A.

(1)

Let f be given by the matrix U and put Ī“i = (ā’1)i+1 [1, . . . , i, . . . , m + 1]. Choosing the

map h : Am+1 ā’ A by sending the i-th element of a basis of Am+1 to Ī“i , i = 1, . . . , m + 1,

one certainly obtains a complex

f h

0 ā’ā’ Am ā’ā’ Am+1 ā’ā’ A.

(2)

The acyclicity criterion (16.15) applied to the exact sequence (1) yields that grade I ā„ 1,

grade Im (f ) ā„ 2. On the other hand, I1 (h) = Im h = Im (f ), so it forces the complex (2)

to be exact, too, and we have an isomorphism

I ā¼ Coker f ā¼ Im (f ).

= =

Since grade Im (f ) ā„ 2 and, thus, Ext1 (A/Im (f ), A) = 0, the natural homomorphism

A

Aā— ā’ā’ (Im (f ))ā— is an isomorphism, whence every map Im (f ) ā’ā’ A is a multiplication

by an element a ā A. So I = aIm (f ); because of grade I ā„ 1, a cannot divide zero:

(16.36) Theorem. Let A be a noetherian ring, and I ā‚ A an ideal for which A/I

has a free resolution as (1) above. Then there exists an element a ā A which is not a

zero divisor, such that I = aIm (f ).

This theorem is often called the Hilbert-Burch theorem since it has appeared in a

special form in [Hi], pp. 239, 240 and has been given its ļ¬rst modern version by Burch

[Bh.1]. One should note that its hypotheses are fulļ¬lled if A is a regular local ring or a

polynomial ring over a ļ¬eld, grade I = 2, and A/I is a Cohen-Macaulay ring.

218 16. Appendix

G. Comments and References

Many of the auxiliary results in this section may be classiļ¬ed as āfolkloreā, even if

some of them should have been documented in the literature.

Proposition (16.1) is a theorem of Fitting [Fi], thus the notion āFitting invariantā.

Our deļ¬nition of ārankā and its treatment are borrowed from Scheja and Storch ([SS],

section 6). It may have appeared elsewhere.

In Subsection B we have given references to Matsumura [Mt] for the basic notion

ādepthā and its extension āgradeā. Another good source for the theory of grade is

Kaplanskyā™s book [Ka], pp. 89 ā“ 103. Our notation grade(I, M ) is his G(I, M ); (16.14),

for example, is an exercise on p.103 of [Ka]. The grade of a module as deļ¬ned below

(16.11) has been introduced in Reesā™ fundamental paper [Re], the equality in (16.11)

serving as the deļ¬nition.

The utmost important acyclicity criterion (16.15) is (almost) identical with [BE.2],

Theorem. It is closely related to the lemme dā™acyclicitĀ“ of Peskine and Szpiro ([PS],

e

(1.8)). Our proof may be new (though perhaps not original).

The notion āperfectā goes back to Macaulay ([Ma], p. 87). Our deļ¬nition which is

copied from Rees [Re] is an abstract and generalized version of GrĀØbnerā™s ([Gb], p. 197).

o

The description of the relationship between the properties of being perfect and being

Cohen-Macaulay as given above, is just a technical elaboration of Reesā™ results ([Re], p.

41).

Our treatment of the process of dehomogenization has been inspired by unpublished

lecture notes of Storch, it is certainly the standard one nowadays. A detailed discussion

is to be found in [ZS] , Ch. VII, Ā§Ā§ 5, 6.

Subsection E is based on Auslander and Bridgerā™s monograph [ABd]. It is diļ¬cult

to say something about the notion ān-torsionfreeā and its relatives not being contained

in [ABd] already. However, the treatment in [ABd] suļ¬ers from a rather heavy technical

apparatus, and the inclusion of subsection E should be regarded as an attempt to make

the results of [ABd] directly accessible.

The version of the Hilbert-Burch theorem given above has been drawn from [BE.3],

Theorem 0. It can be greatly extended, cf. [BE.3], Theorem 3.1.

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ńņš. 44 |