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which N is (n ’ 1)-torsionless, as follows by induction or from the preceding proposition.
Dualizing one obtains an exact sequence

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

We split this into two exact sequences:
0 ’’ N — ’’ F — ’’ K ’’ 0,
0 ’’ K ’’ M — ’’ Ext1 (N, A) ’’ 0.
A

Dualizing again one gets exact sequences

Exti’1 (N — , A) ’’ Exti (K, A) ’’ 0 for i ≥ 1,
A
A
Exti (Ext1 (N, A), A) ’’ Exti (M — , A) ’’ Exti (K, A) for i ≥ 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 ’ 1, one has
grade Ext1 (N, A) ≥ n ’ 1,
A

and therefore Exti (Ext1 (N, A), A) = 0 for all i, 0 ¤ i ¤ n ’ 2 (cf. (16.11)). One readily
A A
concludes that
Exti (M — , A) = 0 for i = 2, . . . , n ’ 2.
A
Since n > 2, N is re¬‚exive, and therefore the linear map F —— ’ N —— is the original
epimorphism F ’ N , whence Ext1 (K, A) = Ext1 (M — , A) = 0, too. ”
A A

(16.35) Remark. The hypothesis
pd MP < ∞ for all prime ideals P with depth AP < n ’ 1
is only needed to ensure grade Ext1 (N, A) ≥ n’1. In any case depth NP = depth AP for
A
all prime ideals P with depth AP < n ’ 1, and arguing with an injective resolution of AP
one also concludes grade Ext1 (N, A) ≥ n’1 if the localizations AP with depth AP < n’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 classi¬cation theorems. One of the few
examples identi¬es the ideals I in a noetherian ring for which A/I has a free resolution
of length 2:
f g
0 ’’ Am ’’ Am+1 ’’ A.
(1)
Let f be given by the matrix U and put δi = (’1)i+1 [1, . . . , i, . . . , m + 1]. Choosing the
map h : Am+1 ’ A by sending the i-th element of a basis of Am+1 to δi , i = 1, . . . , m + 1,
one certainly obtains a complex
f h
0 ’’ Am ’’ Am+1 ’’ A.
(2)
The acyclicity criterion (16.15) applied to the exact sequence (1) yields that grade I ≥ 1,
grade Im (f ) ≥ 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 ∼ Coker f ∼ Im (f ).
= =
Since grade Im (f ) ≥ 2 and, thus, Ext1 (A/Im (f ), A) = 0, the natural homomorphism
A
A— ’’ (Im (f ))— is an isomorphism, whence every map Im (f ) ’’ A is a multiplication
by an element a ∈ A. So I = aIm (f ); because of grade I ≥ 1, a cannot divide zero:
(16.36) Theorem. Let A be a noetherian ring, and I ‚ A an ideal for which A/I
has a free resolution as (1) above. Then there exists an element a ∈ 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 ¬rst modern version by Burch
[Bh.1]. One should note that its hypotheses are ful¬lled if A is a regular local ring or a
polynomial ring over a ¬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 classi¬ed as “folklore”, even if
some of them should have been documented in the literature.
Proposition (16.1) is a theorem of Fitting [Fi], thus the notion “Fitting invariant”.
Our de¬nition of “rank” 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
“depth” and its extension “grade”. Another good source for the theory of grade is
Kaplansky™s book [Ka], pp. 89 “ 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 de¬ned below
(16.11) has been introduced in Rees™ fundamental paper [Re], the equality in (16.11)
serving as the de¬nition.
The utmost important acyclicity criterion (16.15) is (almost) identical with [BE.2],
Theorem. It is closely related to the lemme d™acyclicit´ of Peskine and Szpiro ([PS],
e
(1.8)). Our proof may be new (though perhaps not original).
The notion “perfect” goes back to Macaulay ([Ma], p. 87). Our de¬nition which is
copied from Rees [Re] is an abstract and generalized version of Gr¨bner™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 Rees™ 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, §§ 5, 6.
Subsection E is based on Auslander and Bridger™s monograph [ABd]. It is di¬cult
to say something about the notion “n-torsionfree” and its relatives not being contained
in [ABd] already. However, the treatment in [ABd] su¬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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