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.