Total Betti numbers of modules of finite projective dimension

Abstract

The Buchsbaum-Eisenbud-Horrocks Conjecture predicts that the $i^{\rm th}$ 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 $\sum_i \beta_i(M) \geq 2^{d}$. 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.