Some results on local cohomology modules defined by a pair of ideals
Let \((R,\mathfrak m)\) be a commutative noetherian local ring. Let \(I,J\) be two ideals of \(R\) and set \(W(I,J)=\{\mathfrak p\in \text{Spec} R|I^n\subseteq \mathfrak p+J \;\;\text{ for some positive integer} \;\;n\}\). The functor \(\Gamma_{I,J}\) is a subfunctor of the identity functor that for any \(R\)-module \(N\) determined by \[ \Gamma_{I,J}(N)=\{x\in N|\text{Supp}_R(Rx) \subseteq W(I,J)\}. \] For each non-negative integer \(i\), the \(i\)-th right derived functor of \(\Gamma_{I,J}\) is denoted by \(H_{I,J}^i\). Theses functors were introduced and studied by \textit{R. Takahashi, Y. Yoshino} and \textit{T. Yoshizawa} [J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)]. Note that the notion \(H_{I,J}^i\) generalizes the notion of usual local cohomology functor \(H^i_I\) which corresponds to the case \(J = 0\). In the present paper, the authors extend two known nice properties of usual local cohomology functors to the functors \(H^i_{I,J }\). Let \(M\) be a \(d\)-dimensional finitely generated \(R\)-module. Their first result asserts that \(H^d_{I,J }(M)\) is Artinian. Because the set \(W(I,J)\) is stable under specialization, the Artinianness of the module \(H^d_{I,J }(M)\) is already known [see e.g. Theorem 2.6 and Lemma 3.2 in \textit{K. Divaani-Aazar, R. Naghipour} and \textit{M. Tousi}, Commun. Algebra 30, No. 8, 3687--3702 (2002; Zbl 1054.14006)]. As their second result, they show that \(\inf \{i\in \mathbb{N}_0| H^i_{I,J }(M)\;\;\text{is not Artinian}\}=\inf \{\text{depth} M_{\mathfrak p}|\mathfrak p\in W(I,J)-\{\mathfrak m\} \}\).
scientific article; zbMATH DE number 5601045