Bockstein cohomology of associated graded rings

TL;DR

This paper investigates Bockstein cohomology modules associated with the associated graded rings of Cohen-Macaulay local rings, establishing conditions under which these modules vanish, thus advancing understanding of their algebraic structure.

Contribution

It introduces new conditions on ideals that lead to the vanishing of specific Bockstein cohomology modules in associated graded rings.

Findings

Certain natural conditions on the ideal I imply vanishing of some Bockstein cohomology modules.

The work connects properties of the ideal I with the algebraic structure of Bockstein cohomology.

Results contribute to the understanding of the cohomological behavior 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} $\beta^i \colon H^i_{G_+}(G)(-1) \rt H^{i+1}_{G_+}(G)$ for $i \geq 0$. Since $\beta^{i+1}(+1)\circ \beta^i = 0$ 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.