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 $d\ge 2$. Assume the residue field of $A$ is perfect. Let $I$ be any $\mathfrak{m}$-primary ideal. Let $G_I(A)={⨁}_{n\ge 0}{I}^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$ and let ${\mathcal{R}}_I(A)={⨁}_{n\in \mathbb{Z}}{I}^n$ be the extended Rees algebra of $A$ with respect to $I$. Let $M$ be a finitely generated $A$-module. Let $G_I(M)={⨁}_{n\ge 0}{I}^{n}M/I^{n+1}M$ be the associated graded ring of $M$ with respect to $I$ (considered as a $G_I(A)$-module). Let $B{H}^i(G_I(M))$ be the $i^{th}$-Bockstein cohomology of $G_I(M)$ with respect to ${\mathcal{R}}_I(A{)}_{+}$-torsion functor. We show there exists $a\ge 1$ 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 $B{H}^i(G_I(M))$ has finite length for $i=0,\dots ,d-1$ for any maximal Cohen-Macaulay $A$-module $M$.