On lengths of modules over certain Artinian complete intersections

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.