Lengths of modules over short Artin local rings

TL;DR

This paper investigates the lengths of modules over short Artin local rings, establishing divisibility properties related to Betti numbers and syzygies, especially distinguishing cases where the ring is not a hypersurface or a complete intersection.

Contribution

It introduces divisibility conditions for module lengths over short Artin local rings, linking algebraic properties to Betti numbers and syzygy modules, expanding understanding of module structure.

Findings

Existence of a constant dividing module lengths for modules with bounded Betti numbers.

Divisibility of lengths of syzygies for modules with curvature less than that of the residue field.

Results apply specifically to non-hypersurface and non-complete intersection 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 $M$).