Bockstein cohomology of associated graded rings

Abstract

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ \wrt \ $I$ and let $\mathbb{R}=A[It,t^{-1}]$ be the extended Rees ring of $A$ with respect to $I$. Notice $t^{-1}$ is a non-zero divisor on $\mathbb{R}$ and $\mathbb{R}/t^{-1}\mathbb{R}=G$. So we have \textit{Bockstein operators} $β^i \colon H^i_{G_+}(G)(-1) \rt H^{i+1}_{G_+}(G)$ for $i\ge 0$. Since ${\beta }^{i+1}(+1)\circ {\beta }^i=0$ we have \textit{Bockstein cohomology} modules $B{H}^i(G)$ for $i=0,\dots ,d$. In this paper we show that certain natural conditions on $I$ implies vanishing of some Bockstein cohomology modules.