Large lower bounds for the betti numbers of graded modules with low regularity

Abstract

Suppose that $M$ is a finitely-generated graded module of codimension $c\ge 3$ over a polynomial ring and that the regularity of $M$ is at most $2a-2$ where $a\ge 2$ is the minimal degree of a first syzygy of $M$. Then we show that the sum of the betti numbers of $M$ is at least ${\beta }_0(M)(2^c+2^{c-1})$. In addition, if $c\ge 9$ then for each $1\le i\le \lceil c/2\rceil$, we show ${\beta }_i(M)\ge 2{\beta }_0(M)\left(\genfrac{}{}{0ex}{}{c}{i}\right)$.

14 pages, 2 figures