On the coassociated primes of generalized local cohomology modules
Let \(\mathfrak{a}\) be an ideal of a complete noetherian local ring \((R,\mathfrak{m})\) and \(M\) and \(N\) two finitely generated \(R\)-modules. Assume that \(\text{pd}_R(M)\), the projective dimension of \(M\), is finite and let \(t>\text{pd}_R(M)\) be a positive integer. The main result of the paper under review asserts that \(\text{Hom}_R(R/\mathfrak{a},\text{Hom}_R(H^t_{\mathfrak{a}}(M,N),E_R(R/\mathfrak{m})))\) is finitely generated whenever \(H^i_{\mathfrak{a}}(M,N)\) is Artinian for all \(i>t\). Here \(H^i_{\mathfrak{a}}(M,N)={\varinjlim}_n\text{Ext}_R^i(M/\mathfrak{a}^nM,N)\) denotes the \(i\)-th generalized local cohomology functor. The notion of generalized local cohomology has been introduced by J. Herzog in 1974. As a nice application of this result, the author deduces that the set \(\text{Coass}_RH^c_{\mathfrak{a}}(N)\cap V(\mathfrak{a})\), \(c=\text{cd}(\mathfrak{a},N)\) is finite. Recall that \(\text{cd}(\mathfrak{a},N):=\max\{i\in \mathbb{N}_{0}: H^i_{\mathfrak{a}}(N) \neq 0\}.\) Also, recall that for an \(R\)-module \(L\), \(\text{Coass}_RL\), the set of coassociated prime ideals of \(L\), consists of any prime ideal \(\mathfrak{p}\) for which there exists an Artinian quotient \(Q\) of \(L\) such that \(\mathfrak{p}=(0:_RQ)\). Reviewer's remark: The reader should be careful about misprints.
scientific article; zbMATH DE number 5151972