Total Betti numbers of modules of finite projective dimension

TL;DR

This paper investigates the Buchsbaum-Eisenbud-Horrocks Conjecture, providing proofs for the lower bounds of the sum of Betti numbers of modules with finite projective dimension over local rings.

Contribution

It proves the conjecture's claim that the sum of Betti numbers is at least 2^d in many cases, advancing understanding of module invariants.

Findings

Sum of Betti numbers at least 2^d in many cases

Supports the conjecture relating Betti numbers and ring dimension

Provides partial proofs for the Buchsbaum-Eisenbud-Horrocks Conjecture

Abstract

The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that if M is a non-zero module of finite length and finite projective dimension over a local ring R of dimension d, then the i-th Betti number of M is at least d choose i. This conjecture implies that the sum of all the Betti numbers of such a module must be at least 2^d. We prove the latter holds in a large number of cases.