Some isomorphisms in derived functors and their applications (Q2861577)

Let \(R\) denote a commutative ring and let \(M\) be an \(R\)-module. For an ideal \(\mathfrak{a} \subset R\) the authors study the local cohomology modules \(H^i_{\mathfrak{a}}(M)\). In particular, they show : (i) If \(H^i_{\mathfrak{a}}(M) = 0\) for all \(i < c\) then \(\text{Hom}_R(H^c_{\mathfrak{a}}(M), H^c_{\mathfrak{a}}(M)) \cong \text{Ext}_R^c(H^c_{\mathfrak{a}}(M),M)\). (ii) If \(H^i_{\mathfrak{a}}(M) = 0\) for all \(i \not= c\) then \(H^i_{\mathfrak{b}}(H^c_{\mathfrak{a}}(M)) \cong H^{i+c}_{\mathfrak{b}}(M)\) and \(\text{Ext}^i_R(R/\mathfrak{b},H^c_{\mathfrak{a}}(M)) \cong \text{Ext}_R^{i+c}(R/\mathfrak{b},M)\) for all \(i \geq 0\) and any ideal \(\mathfrak{b} \supset \mathfrak{a}\). Furthermore there is a natural homomorphism \(\text{Hom}_R(H^i_{\mathfrak{a}}(M), H^i_{\mathfrak{a}}(M)) \to \text{Hom}_R(H^i_{\mathfrak{b}}(M), H^i_{\mathfrak{b}}(M))\). These results and their applications extent those of \textit{M. Hellus} and \textit{J. Stückrad} [Proc. Am. Math. Soc. 136, No. 7, 2333--2341 (2008; Zbl 1152.13011)], Hellus, the reviewer (see [J. Algebra 320, No. 10, 3733--3748 (2008; Zbl 1157.13012)]) and the reviewer (see [Proc. Am. Math. Soc. 137, No. 4, 1315--1322 (2009; Zbl 1163.13010)]) to the case of modules.