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 $\R = A[It,t^{-1}]$ be the extended Rees ring of $A$ with respect to $I$. Notice $t^{-1}$ is a non-zero divisor on $\R$ and $\R/t^{-1}\R = G$. So we have \textit{Bockstein operators} $\betaⁱ \colon H^{i_{G_+}(G)(-1)} \rt H^{{i+1}{G+}(G)$} for $i \geq 0$. Since $\beta^{{i+1}(+1)\circ} \betaⁱ = 0$ we have \textit{Bockstein cohomology} modules $BH^i(G)$ for $i = 0,\ldots,d$. In this paper we show that certain natural conditions on $I$ implies vanishing of some Bockstein cohomology modules.