Filter Regular Sequences and Generalized Local Cohomology Modules

Abstract

Let $\mathfrak a, \mathfrak b$ be ideals of a commutative Noetherian ring $R$ and let $M, N$ be finite $R$-modules. The concept of an $\mathfrak a$-filter grade of $\mathfrak b$ on $M$ is introduced and several characterizations and properties of this notion are given. Then, using the above characterizations, we obtain some results on generalized local cohomology modules $\mathrm{H }^{i}_{\mathfrak a}(M,N)$. In particular, first we determine the least integer $i$ for which $\mathrm{H }^{i}_{\mathfrak a}(M,N)$ is not Artinian. Then we prove that $\mathrm{H }^{i}_{\mathfrak a}(M,N)$ is Artinian for all $i\in \mathbb N_0$ if and only if $\dim {R}/({\mathfrak a+\mathrm{Ann\, }\, M+\mathrm{Ann\, }\, N})=0$. Also, we establish the Nagel–Schenzel formula for generalized local cohomology modules. Finally, in a certain case, the set of attached primes of $\mathrm{H }^{i}_{\mathfrak a}(M,N)$ is determined and a comparison between this set and the set of attached primes of $\mathrm{H }^{i}_{\mathfrak a}(N)$ is given.