Cohomological dimension filtration and annihilators of top local cohomology modules

Let $\frak a$ 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 $\mathscr{M} =\{M_i\}_{i=0}^c$, where $ c={\rm cd} ({\frak a},M)$ and $M_i$ denotes the largest submodule of $M$ such that ${\rm cd} ({\frak a}, M_i)\leq i.$ Some properties of this filtration are investigated. In particular, in the case that $(R, \frak m)$ is local and $c= \dim M$, we are able to determine the annihilator of the top local cohomology module $H_{\frak a}^c(M)$. In fact, it is shown that ${\rm Ann}_R(H_{\frak a}^c(M))= {\rm Ann}_R(M/M_{c-1}).$ As a consequence, it follows that there exists an ideal $\frak b$ of $R$ such that ${\rm Ann}_R(H_{\frak a}^{c}(M))={\rm Ann}_R(M/H_{\frak b}^{0}(M))$.