Cofiniteness of torsion functors of cofinite modules (Q2879358)

Let \(R\) be a Noetherian commutative ring with identity and \(I\) an ideal of \(R\). Recall that an \(R\)-module \(L\) is said to be \(I\)-cofinite if \(\mathrm{Supp}_RL\subseteq V(I)\) and \(\text{Ext}_R^i(R/I,L)\) is finitely generated for all \(i\geq 0\). Also, \(L\) is said to be \(I\)-weakly cofinite if \(\mathrm{Supp}_RL\subseteq V(I)\) and \(\text{Ext}_R^i(R/I,L)/X\) has finitely many associated prime ideals for all \(i\geq 0\) all submodules \(X\) of \(\text{Ext}_R^i(R/I,L)\).NEWLINENEWLINELet \(M\) be an \(I\)-cofinite \(R\)-module and \(N\) a finitely generated \(R\)-module. The authors prove that the \(R\)-modules \(\text{Tor}^R_i(N,M)\) are \(I\)-cofinite for all \(i\geq 0\), whenever \(\dim (\mathrm{Supp}_RN)\leq 2\) or \(\dim (\mathrm{Supp}_RM)\leq 1\). It is known that if \(\dim R/I=1\) and \(W\) is a finitely generated \(R\)-module, then \(H_I^j(W)\) is \(I\)-cofinite for all \(j\geq 0\). Thus in this case, \(\text{Tor}^R_i(N,H_I^j(W))\) is \(I\)-cofinite for all \(i\geq 0\) and all \(j\geq 0\).NEWLINENEWLINEAlso in the case \(R\) is local, the authors show that \(\text{Tor}^R_i(N,M)\) is \(I\)-weakly cofinite for all \(i\geq 0\), whenever \(\dim (\mathrm{Supp}_RN)\leq 3\) or \(\dim (\mathrm{Supp}_RM)\leq 2\).