On the generalized local cohomology of minimax modules (Q2816916)

Let \((R,\mathfrak{m})\) be a Noetherian commutative ring with identity and \(I\) an ideal of \(R\). An \(R\)-module \(X\) is called \textit{minimax} if it has a finitely generated submodule \(Y\) such that \(X/Y\) is an Artinian \(R\)-module. Let \(M\) be a finitely generated \(R\)-module and \(N,L\) two minimax \(R\)-modules such that \(\mathrm{Supp}_RL\subseteq V(I)\). The paper under review investigates the finiteness properties of the generalized local cohomology modules NEWLINE\[NEWLINE\text{H}_{I}^i(M,N):=\underset{n}{\varinjlim} \text{Ext}^i_R(M/{I}^nM,N);\;\;i\in \mathbb{N}_0.NEWLINE\]NEWLINE The main result of this paper asserts that \(\text{Ext}^j_R(L,\text{H}_{I}^i(M,N))\) is minimax for all \(i,j\geq 0\) if any of the following conditions is satisfied: {\parindent=0.7cm\begin{itemize}\item[(i)] \(\dim R/I\leq 1\); \item[(ii)] \(\sup\{i | \text{H}_{I}^i(R)\neq 0\}=1\); or \item[(iii)] \(\dim R\leq 2\). NEWLINENEWLINE\end{itemize}} In particular, the Bass and Betti numbers of \(\text{H}_{I}^i(M,N)\) are finite for all \(i\geq 0\) if any one of these conditions holds.