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 ${ \rm Ann}_R(H_{ \frak a}^{ c}(M))={ \rm Ann}_R(M/H_{ \frak b}^{ 0}(M))$.
Categories: math.AC