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 $\mathfrak{m}^2 \neq 0$). Assume $A$ is not a hypersurface ring. We show there exists $c_A \geq 2$ 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 $b_A \geq 2$ such that if $M$ is any module with $curv(M) < \ curv(k)$ then $b_A$ divides $\ell(\Omega^i_A(M))$ for all $i \geq 1$ (here $\Omega^i_A(M)$ denotes the $i^{th}$-syzygy of $M$).