Filter depth and cofiniteness of local cohomology modules defined by a pair of ideals (Q2928443)

Let \(I,J\) denote two ideals of a local ring \((R,\mathfrak{m})\) and let \(M\) be a finitely generated \(R\)-module. In their paper [J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)], \textit{R. Takahashi} et al. introduced the local cohomology \(H_{(I,J)}^i(M), i \in \mathbb{N},\) with respect to the pair of ideals \((I,J)\). It specializes to the ordinary local cohomology \(H^i_I(M), i \in \mathbb{N},\) in the case of \(J = 0\). In the present paper the authors generalize results on filter depth and cofiniteness known for the ordinary local cohomology to the local cohomology with respect to a pair of ideals. Among their results they prove a formula for the least integer \(i\) such that the local cohomology module \(H^i_{(I,J)}(M)\) is not an Artinian \(R\)-module. Moreover they prove an estimate for an integer \(t\) such that \(H^i_{(I,J)}(M)\) is \((I,J)\)-cofinite for all \(i < t\), which is related to a result of \textit{L. Chu} and \textit{Q. Wang} [J. Math. Kyoto Univ. 49, No. 1, Article ID 10, 193--200 (2009; Zbl 1174.13024)]. Further results concern the cohomological dimension with respect to \((I,J)\) and Serre categories related to the generalized local cohomology.