The finiteness of co-associated primes of local homology modules (Q2472550)

Let \(R\) be a commutative ring. A topological \(R\)-module \(M\) is called linearly topologized if its zero element has a base of neighborhoods consisting of open submodules. Also a Hausdorff linearly topologized \(R\)-module \(M\) is said to be semi-discrete if every submodule of \(M\) is closed. Assume that \(M\) is a such module. Let \(I\) be an ideal of \(R\) and \(i\) a non-negative integer. This paper provides some finiteness results for the set of coassociated prime ideals of the local homology module \(H_i ^{I}(M):={\varprojlim}_n \text{Tor}_i ^R (R/ I^n, M)\). More precisely \(H_i^{I}(M)\) has finitely coassociated prime ideals, if one of the following conditions hold: (i) the \(R\)-modules \(H_j^{I}(M)\) are finitely generated, for all \(j<i\), and (ii) \(I\subseteq \sqrt{\text{Ann}_R(H_j^{I}(M))}\) for all \(j<i\). In the case \((R,\mathfrak m)\) is a complete local ring, the author by using the Matlis duality, provides some finiteness results of associated prime ideals of the local cohomology module \(H^{i}_{I}(M):={\varinjlim}_n\text{Ext}^{i}_{R}(M/\mathfrak a^{n}M)\) in the following cases: (i) the \(R\)-modules \(H_j^{I}(M)\) are Artinian, for all \(j<i\), and (ii) \(I\subseteq \sqrt{\text{Ann}_R(H_j^{I}(M))}\) for all \(j<i\).