Lengths of modules over short Artin local rings

Abstract

Let $(A,\mathfrak{m})$ be a short Artin local ring (i.e., ${\mathfrak{m}}^3=0$ and ${\mathfrak{m}}^2\ne 0$). Assume $A$ is not a hypersurface ring. We show there exists $c_A\ge 2$ such that if $M$ is any finitely generated module with bounded betti-numbers then $c_A$ divides $\ell (M)$, the length of $M$. If $A$ is not a complete intersection then there exists $b_A\ge 2$ such that if $M$ is any module with $curv(M)<curv(k)$ then $b_A$ divides $\ell ({\Omega }_A^i(M))$ for all $i\ge 1$ (here ${\Omega }_A^i(M)$ denotes the $i^{th}$-syzygy of $M$).