Cohomological dimension filtration and annihilators of top local cohomology modules
Volume 139 / 2015
Abstract
Let $\mathfrak a$ denote an ideal in a Noetherian ring $R$, and $M$ a finitely generated $R$-module. We introduce the concept of the cohomological dimension filtration $\mathscr {M} =\{M_i\}_{i=0}^c$, where $ c=\mathop {\rm cd}\nolimits (\mathfrak a,M)$ and $M_i$ denotes the largest submodule of $M$ such that $\mathop {\rm cd}\nolimits (\mathfrak a,M_i)\leq i.$ Some properties of this filtration are investigated. In particular, if $(R, \mathfrak m)$ is local and $c= \dim M$, we are able to determine the annihilator of the top local cohomology module $H_{\mathfrak a}^c(M)$, namely ${\rm Ann}_R(H_{\mathfrak a}^c(M))= {\rm Ann}_R(M/M_{c-1}).$ As a consequence, there exists an ideal $\mathfrak b$ of $R$ such that ${\rm Ann}_R(H_{\mathfrak a}^{c}(M))={\rm Ann}_R(M/H_{\mathfrak b}^{0}(M))$. This generalizes the main results of Bahmanpour et al. (2012) and Lynch (2012).
