Betti numbers for modules of finite length

Abstract

Let R R be a Gorenstein local ring of dimension d > 5 d > 5 and let M M be a module of finite length and finite projective dimension. If M M is not isomorphic to R R modulo a regular sequence, then the Betti numbers of M M satisfy β i ( M ) > ( i d ) {\beta _i}(M) > (_i^d) for 0 > i > d 0 > i > d , and ∑ i = 0 d β i ( M ) ≥ 2 d + 2 d − 1 \sum \nolimits _{i = 0}^d {{\beta _i}(M) \geq {2^d} + {2^{d - 1}}} .