Non-Artinian local cohomology with respect to a pair of ideals (Q2862500)

Let \(R\) be a commutative Noetherian ring, \(I,J\) ideals of \(R\) and \(M\) an \(R\)-module. Let NEWLINE\[NEWLINEW(I,J)=\{\mathfrak{p}\in\mathrm{Spec}(R)\mid I^n\subseteq\mathfrak{p}+J {\text{ for an integer n}}\gg1\}NEWLINE\]NEWLINE be a subset of \(\mathrm{Spec}(R)\). The \((I,J)\)-torsion submodule \(\Gamma_{I,J}(M)\) of \(M\) is defined as the set which consists of all element \(x\) of \(M\) with \(\mathrm{Supp}(Rx)\subseteq W(I,J)\). For an integer \(i\), \textit{R. Takahashi, Y. Yoshino} and \textit{T. Yoshizawa} [J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)] defined the local cohomology functor \(\mathrm{H}_{I,J}^i(-)\) with respect to \((I,J)\) to be the \(i\)-th right derived functor of \(\Gamma_{I,J}(-)\). Note that if \(J=0\) or \(J\) be a nilpotent ideal, then \(\mathrm{H}_{I,J}^i(-)\) coincides with the ordinary local cohomology functor \(\mathrm{H}_{I}^i(-)\) with the support in the closed subset \(V(I)\). On the other hand, if \(J\) contains \(I\), then \(\Gamma_{I,J}\) is the identity functor and \(\mathrm{H}_{I,J}^i(-)=0\) for \(i>0\).NEWLINENEWLINEIn the paper under review the authors give a generalization of some known result of classical local cohomology for the local cohomology with respect to a pair of ideals.NEWLINENEWLINEOne of the main result in this paper is as follows:NEWLINENEWLINEFor a non-negative integer \(t\) the following hold: {\parindent=6mm \begin{itemize}\item[(a)] If \(\mathrm{Ext}_R^{t-j}(R/I,\mathrm{H}_{I,J}^j(M))\) for all \(j<t\) and \(\mathrm{Hom}_R(R/I,\mathrm{H}_{I,J}^t(M))\) are finite (resp. Artinian), then \(\mathrm{Ext}_R^t(R/I,M) \) is finite (resp. Artinian). \item[(b)] If \(\mathrm{Ext}_R^{t+1-j}(R/I,\mathrm{H}_{I,J}^j(M))\) for all \(j<t\) and \(\mathrm{Ext}_R^t(R/I,M)\) are finite (resp. Artinian), then \(\mathrm{Hom}_R(R/I,\mathrm{H}_{I,J}^t(M))\) is finite (resp. Artinian). NEWLINENEWLINE\end{itemize}} In addition, the authors show that: if \((R,\mathfrak{m})\) is a local ring, \(J\) a non-nilpotent ideal and \(M\) a finite \(R\)-module, then \(\mathrm{H}_{\mathfrak{m},J}^i(M)\) is not Artinian for some non-negative integer \(i\).