Cohen-Macaulay filtered modules and attached primes of local cohomology

For an ideal $\frak a$ in a Noetherian ring $R$ contained in the Jacobson radical of $R$, it is shown that if $M$ is a finitely generated $\mathfrak a$-relative Cohen-Macaulay $R$-module, then $\operatorname{Ann}_R(H_{\frak a}^{{\rm cd} (\frak a, M)}(M))= \operatorname{Ann}_R(M)$. As an application of this result, we show that if $M$ is a finitely generated $\frak a$-relative Cohen-Macaulay filtered $R$-module with the cohomological dimension filtration $\mathcal{M} =\{M_i\}_{0\leq i \leq c}$, then for each $0\leq i \leq c$, $\operatorname{Ann}_R(H_{\frak a}^i(M))=\operatorname{Ann}_R (M_i/M_{i-1})$, where $c={\rm cd} (\frak a, M)$. These generalize the main results of [Theorem 3.3] and [Theorem 2.11]. Also, we shall provide some new characterizations of the attached primes of top local cohomology module $H_{\frak a}^{{\rm cd} (\frak a, M)}(M)$ and give a short proof of the main results of [Theorem 2.2] and Theorem 2.7]. Finally, it is shown that if $M$ and $N$ are arbitrary $R$-modules (not necessarily finitely generated) such that $\operatorname{Att}_R(M)\subseteq\operatorname{Att}_R(N)$, then ${\rm cd}(\frak a, R/\operatorname{Ann}_R(M))\leq{\rm cd}(\frak a, R/\operatorname{Ann}_R(N)).$