Local cohomology and Serre subcategories (Q949602)

Let \(R\) denote a commutative noetherian ring. A Serre subcategory \(\mathcal S\) of the category of \(R\)-modules satisfies condition \((C_{\mathfrak a})\) provided (i) \(M = \Gamma_{\mathfrak a}(M)\) and (ii) \(0:_M \mathfrak a \in {\mathcal S}\) implies \(M \in {\mathcal S}.\) Let \(\mathcal S\) be a Serre category. Let \(M\) denote an \(R\)-module. The authors define the notion of \(\mathcal S\)-regular sequences as a sequence \(x_1,\dots,x_n\) of \(R\) such that \((x_1,\dots,x_{j-1})M:_M x_j/(x_1,\dots,x_{j-1})M\) belongs to \(\mathcal S.\) Now let \(\mathcal S\) denote a Serre category satifying \((C_{\mathfrak a})\) for an ideal \(\mathfrak a\) of \(R.\) Then there is a characterization of the property when \(H^i_{\mathfrak a}(M) \in \mathcal S\) for all \(i < n\) in terms of Ext-modules, Koszul homology, generalized local cohomology and when \(M\) is finite in terms of \(\mathcal S\)-sequences. These are far reaching generalizations of the case when \(H^i_{\mathfrak a}(M)\) is an Artinian \(R\)-module for all \(i < n.\) There are various applications, e.g. to associated primes resp. to the support of local cohomology modules. In the case of \(M/\mathfrak aM \not\in {\mathcal S}\) the authors define the \({\mathcal S}-\text{depth }_{\mathfrak a} M.\) Finally they descrcibe when \(H^i_{\mathfrak a}(M) \in {\mathcal S}\) for all \(i > n\) for instance in terms of the corresponding question for minimal associated primes of \(M.\)