Total Betti numbers of modules of finite projective dimension. (English) Zbl 1386.13040

Walker, Mark E.

Ann. Math. (2) 186, No. 2, 641-646 (2017).

The long standing Buchsbaum-Eisenbud-Horrocks conjecture on the Betti numbers of non zero finitely generated modules \(M\) with finite projective dimension over a commutative Noetherian ring \(R\), [D. Eisenbud and D. Buchsbaum, Am. J. Math. 99, 447–485 (1997; Zbl 0373.13006)], implies that the total rank \(\sum_i \beta_i(M)\) must be at least \(2^c\) where \(c=\text{height}_R(\mathrm{ann}_R(M))\). In the paper, the author proves that the above inequality holds in a larger number of cases. He supposes that \(R\) is a local Noetherian commutative ring and \(M\) has a finite length. He proves
Theorem 2. Assume \((R, m, k)\) is a local (Noetherian, commutative) ring of Krull dimension \(d\) and \(M\) is a non-zero \(R\) module of finite length and finite projective dimension. If either
(1) \(R\) is the quotient of a regular local ring by a regular sequence of elements and \(2\) is invertible in \(R\), or
(2) \(R\) contains \(\mathbb Z /p\) as a subring for an odd prime \(p\),
then \(\sum \beta_i(M) \geq 2^d\). Moreover, if the assumptions in (1) hold and \(\sum \beta_i(M)=2^d\), then \(M\) is isomporphic to the quotient of \(R\) by a regular sequence of \(d\) elements.
As a consequence, the inequality holds when the ring \(R\) and the module \(M\) are as in the Buchsbaum-Eisenbud-Horrocks conjecture. He proves
Theorem 1. Let \(R\) be a commutative Noetherian ring such that spec \((R)\) is connected, and let \(M\) be a nonzero, finitely generated \(R\)-module of finite projective dimension. Let \(P\) be a finite projective resolution of \(M\), and suppose that
(1) \(R\) is locally a complete intersection and \(M\) is \(2\)-torsion free or
(2) \(R\) contains \(\mathbb Z/p\) as a subring for an odd prime \(p\)
then \(\sum_i \text{rank}_R(P_i)\) must be at least \(2^c\) where \(c=\text{height}_R(\mathrm{ann}_R(M))\).

Reviewer: Sabine El Khoury (Beirut)

MSC:

13D02

Syzygies, resolutions, complexes and commutative rings

Keywords:

Betti numbers; Buchsbaum-Eisenbud-Horrocks conjecture

Citations:

Zbl 0373.13006

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