Cohomological dimension filtration and annihilators of top local cohomology modules

Abstract

Let $\mathfrak{a}$ denote an ideal in a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of the cohomological dimension filtration $\mathcal{M}=\{M_i{\}}_{i=0}^c$, where $c=\mathrm{cd}(\mathfrak{a},M)$ and $M_i$ denotes the largest submodule of $M$ such that $\mathrm{cd}(\mathfrak{a},M_i)\le i.$ Some properties of this filtration are investigated. In particular, in the case that $(R,\mathfrak{m})$ is local and $c=\dim M$, we are able to determine the annihilator of the top local cohomology module $H_{\mathfrak{a}}^c(M)$. In fact, it is shown that ${\mathrm{Ann}}_R(H_{\mathfrak{a}}^c(M))={\mathrm{Ann}}_R(M/M_{c-1}).$ As a consequence, it follows that there exists an ideal $\mathfrak{b}$ of $R$ such that ${\mathrm{Ann}}_R(H_{\mathfrak{a}}^c(M))={\mathrm{Ann}}_R(M/H_{\mathfrak{b}}^0(M))$.

arXiv comment10 pages. arXiv admin note: text overlap with arXiv:1312.1253