Cofiniteness properties of generalized local cohomology modules
Let \(I\) be an ideal of a commutative Noetherian ring \(R\) and \(X,Y\) two \(R\)-modules. This paper studies some finiteness properties of the generalized local cohomology modules \[\text{H}_I^i(X,Y):=\underset{n}{\varinjlim} \ \text{Ext}^i_R(X/I^n X,Y);\ \ i\geq 0.\] Let \(n\) be an integer. Recall that an \(R\)-module \(M\) is said to be \(\text{FD}_{\leq n}\) if there exists a finitely generated submodule \(N\) of \(M\) such that \(\dim_R M/N\leq n\). An \(R\)-module \(M\) is called minimax if it has a finitely generated submodule \(N\) such that the \(R\)-module \(M/N\) is Artinian. Also, an \(R\)-module \(M\) is called weakly Laskerian if it has a finitely generated submodule \(N\) such that the \(R\)-module \(\text{Ass}_RM/N\) is finite. An \(R\)-module \(M\) with \(\text{Supp}_RM\subseteq \text{V}(I)\) is called \(I\)-cofinite (resp. \(I\)-cominimax, \(I\)-weakly cofinite) if \(\text{Ext}_R^i(R/I,M)\) is finitely generated (resp. minimax, weakly Laskerian) for all \(i\geq 0\). Let \(t\) be a non-negative integer and \(M\) a finitely generated \(R\)-module. Assume that \(\text{Ext}_R^i(R/I,X)\) is finitely generated (resp. minimax, weakly Laskerian) for all \(i\leq t+1\) and \(\text{H}_I^i(M,X)\) is \(\text{FD}_{\leq 1}\) for all \(i<t\). Then, the main result of this paper asserts that \(\text{H}_I^i(M,X)\) is \(I\)-cofinite (resp. \(I\)-cominimax, \(I\)-weakly cofinite) for all \(i<t\) and \(\text{Ass}_R(\text{H}_I^t(M,X))\) is finite.
scientific article; zbMATH DE number 7324959
scientific article