On lengths of modules over certain Artinian complete intersections

TL;DR

This paper investigates the lengths of modules over certain Artinian complete intersections, establishing divisibility properties related to support varieties and applying results to modules over specific polynomial quotient rings.

Contribution

It introduces a divisibility result for module lengths over Artinian complete intersections defined by regular sequences with specific properties, linking support varieties and module lengths.

Findings

Existence of a projective space element $eta_A$ with divisibility conditions on module lengths.

Divisibility of module lengths by a fixed integer depending on the algebra.

Application to polynomial quotient rings showing prime divisibility of module lengths.

Abstract

Let $(Q,\mathfrak{n})$ be a regular local ring of dimension $c \geq 2$ with algebraically closed residue field $k = Q/\mathfrak{n}$. Let $f_1, f_2, \ldots f_{c-1}, g$ be a regular sequence in $Q$ such that $ f_i \in \mathfrak{n}^2$ for all $i$ and $g \in \mathfrak{n}$. Set $A = Q/(f_1,\ldots, f_{c-1}, g^r)$ with $r \geq 2$. Notice $A$ is an Artinian complete intersection of codimension $c$. We show that there exists $\alpha_A \in \mathbb{P}^{c-1}(k)$ such that there exists integer $m_A \geq 2$ (depending only on $A$) with $m_A$ dividing $\ell(M)$ for every finitely generated $A$-module $M$ with $\alpha_A \notin \mathcal{V}(M)$ (here $\ell(M)$ denotes the length of $M$ and $\mathcal{V}(M)$ denotes the support variety of $M$). As an application we prove that if $k$ be a field and $R = k[X_1, \ldots, X_c]/(X_1^{a_1}, \ldots, X_c^{a_c})$ with $a_i \geq 2$ and $c \geq 2$. Let $p$ be a prime number and assume $p$ divides two of the $a_i$. Then $p$ divides $\ell(E)$ for any $A$-module with bounded betti numbers.