Artinianness of local cohomology modules defined by a pair of ideals (Q454565)

Let \(I,J\) be two ideals of a commutative Noetherian ring \(R\) and \(M\) a finitely generated \(R\)-module. The paper under review examines the local cohomology modules of \(M\) with respect to the pair \((I,J)\). The notion of local cohomology with respect to a pair of ideals is a generalization of the usual notion of local cohomology which was introduced in [\textit{R. Takahashi, Y. Yoshino} and \textit{T. Yoshizawa}, J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)]. For each integer \(i\geq 0\), the \(i\)th local cohomology module of \(M\) with respect to the pair \((I,J)\) is denoted by \(H_{I,J}^i(M)\). The authors proved several results concerning the Artinianness of the modules \(H_{I,J}^i(M)\). For instance, they showed that if \(\dim R/I+J=0\), then \(H_{I,J}^i(M)/JH_{I,J}^i(M)\) is a \(I\)-cofinite Artinian \(R\)-module for all \(i\geq 0\).