September 2017 Total Betti numbers of modules of finite projective dimension
Mark E. Walker
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Ann. of Math. (2) 186(2): 641-646 (September 2017). DOI: 10.4007/annals.2017.186.2.6

Abstract

The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that the $i^{\mathrm{th}}$ Betti number $\beta_i(M)$ of a nonzero module $M$ of finite length and finite projective dimension over a local ring $R$ of dimension $d$ should be at least ${d \choose i}$. It would follow from the validity of this conjecture that $\sum_i \beta_i(M) \ge 2^d$. We prove the latter inequality holds in a large number of cases and that, when $R$ is a complete intersection in which $2$ is invertible, equality holds if and only if $M$ is isomorphic to the quotient of $R$ by a regular sequence of elements.

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Mark E. Walker. "Total Betti numbers of modules of finite projective dimension." Ann. of Math. (2) 186 (2) 641 - 646, September 2017. https://doi.org/10.4007/annals.2017.186.2.6

Information

Published: September 2017
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2017.186.2.6

Subjects:
Primary: 13D02

Keywords: Betti numbers , Buchsbaum-Eisenbud-Horrocks Conjecture

Rights: Copyright © 2017 Department of Mathematics, Princeton University

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Vol.186 • No. 2 • September 2017
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