Cofiniteness of extension functors of cofinite modules (Q555588)

Let \(R\) be a commutative noetherian ring with identity. Let \(I\) be an ideal of \(R\) and \(M,N\) two \(R\)-modules. The paper under review investigates the \(I\)-cofiniteness of the Ext-modules \(\mathrm{Ext}_R^i(N,M)\). Recall that an \(R\)-module \(X\) is said to be \(I\)-cofinite if \(\mathrm{Supp}_RX\subseteq V(I)\) and \(\mathrm{Ext}_R^i(R/I,X)\) is finitely generated for all \(i\geq 0\). Also, an \(R\)-module \(X\) is said to be \(I\)-weakly cofinite if \(\mathrm{Supp}_RX\subseteq V(I)\) and \(\mathrm{Ass}_R(\mathrm{Ext}_R^i(R/I,X)/Y)\) is finite for all \(i\geq 0\) and all submodules \(Y\) of \(\mathrm{Ext}_R^i(R/I,X)\). Assume that \(M\) is \(I\)-cofinite and \(N\) is finitely generated. The authors prove that if either \(\dim M\leq 1\) or \(\dim N\leq 2\), then \(\mathrm{Ext}_R^i(N,M)\) is \(I\)-cofinite for all \(i\geq 0\). In particular, this immediately implies that \(\mathrm{Ext}_R^j(N,H_J^i(L))\) is \(J\)-cofinite for all one-dimensional ideals \(J\) of \(R\), all finitely generated \(R\)-modules \(L\) and all \(i,j\geq 0\). This is because for any ideal \(J\) of \(R\) with \(\dim R/J\leq 1\) and any finitely generated \(R\)-module \(L\), it is known that \(H_J^i(L)\) is \(J\)-cofinite for all \(i\geq 0\); see [\textit{K. Bahmanpour} and \textit{R. Naghipour}, J. Algebra 321, No. 7, 1997--2011 (2009; Zbl 1168.13016)] Also, the authors show that if \(R\) is local and either \(\dim M=2\) or \(\dim N=3\), then \(\mathrm{Ext}_R^i(N,M)\) is \(I\)-weakly cofinite for all \(i\geq 0\).