On minimax modules and generalized local cohomology with respect to a pair of ideals
Let \(R\) be a commutative Noetherian ring with identity and \(I\) and \(J\) be two ideals of \(R\). There are many generalizations of the notion of local cohomology in the literature. One of these generalizations was introduced by \textit{J. Herzog} [Komplexe, Auflösungen und Dualität in der localen Algebra. Universität Regensburg (Habilitationsschrift) (1970)]. Let \(M\) and \(N\) be two \(R\)-modules. The generalized local cohomology modules \(\text{H}_ {I}^i(M,N)\) are defined by \[\text{H}_{I}^i(M,N):=\underset{n}{\varinjlim}\ \text{Ext}^i_R(M/I^n M,N),\ i\in \mathbb{N}_0.\] Another generalization of local cohomology was introduced by \textit{R. Takahashi} et al. [J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)]. For each \(i\in \mathbb{N}_0\), \(\text{H}_{I,J}^i(\sim)\), the \(i\)-th local cohomology functor with respect to the pair \((I,J)\), is defined to be the \(i\)-th right derived functor of \(\Gamma_{I,J}(\sim)\). Here, \(\Gamma_{I,J}(\sim)\) is a subfunctor of the identity functor that for every \(R\)-module \(X\) is determined by \[\Gamma_{I,J}(X):=\{x\in X\mid I^nx\subseteq Jx \ \text{for some} \ n\in \mathbb{N}\}.\] There is a unify generalization of the above generalizations. The \(i\)-th generalized local cohomology functor, \(\text{H}_{I,J}^i(M,\sim)\), with respect to the pair \((I,J)\) is the \(i\)-th right derived functor of the functor \(\Gamma_{I,J}(\Hom_R(M,\sim))\). In the paper under review, the authors examine some finiteness properties of these local cohomology modules. Assume that \(R\) is local with the unique maximal ideal \(\mathfrak{m}\) and \(M\) and \(N\) are finitely generated. Also, suppose that \(p:=\text{pd}_RM< \infty\) and \(d:= \dim_RN\). Among other things, the authors show that \(\text{H}_{I,J}^{p+d}(M,N)\) is Artinian and \[\text{Att}_R(\text{H}_{I,J}^{p+d}(M,N))=\text{Att}_R(\text{H}_{I}^ {p+d}(M,N/JN))\\ =\{{\mathfrak p}\in \text{Ass}_R(N/JN)\mid \text{H}_{I}^{p+d}(M,R/{\mathfrak p})\neq 0 \}.\] (Recall that for an Artinian \(R\)-module \(A\), the set of attached prime ideals of \(A\) is defined by \[\text{Att}_RA:= \{{\mathfrak p}\in \text{Spec}\ R \mid {\mathfrak p}=\text{Ann}_RB \ \ \text{for some quotient module}\ B \ \text{of}\ A \}.)\]
scientific article; zbMATH DE number 7384625
scientific article