Bockstein cohomology of Maximal Cohen-Macaulay modules over Gorenstein isolated singularities

Abstract

Let $(A,\mathfrak{m})$ be an excellent equi-charateristic Gorenstein isolated singularity of dimension $d \geq 2$. Assume the residue field of $A$ is perfect. Let $I$ be any $\mathfrak{m}$-primary ideal. Let $G_I(A) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$ and let $\mathcal{R}_I(A) = \bigoplus_{n \in \mathbb{Z}}I^n$ be the extended Rees algebra of $A$ with respect to $I$. Let $M$ be a finitely generated $A$-module. Let $G_I(M) = \bigoplus_{n \geq 0}I^nM/I^{n+1}M$ be the associated graded ring of $M$ with respect to $I$ (considered as a $G_I(A)$-module). Let $BH^i(G_I(M))$ be the $i^{th}$-Bockstein cohomology of $G_I(M)$ with respect to $\mathcal{R}_I(A)_+$-torsion functor. We show there exists $a \geq 1$ depending only on $A$ such that if $I$ is any $\mathfrak{m}$-primary ideal with $I \subseteq \mathfrak{m}^a$ and $G_I(A) $ generalized Cohen-Macaulay then the Bockstein cohomology $BH^i(G_I(M))$ has finite length for $i = 0, \ldots, d-1$ for any maximal Cohen-Macaulay $A$-module $M$.