Some splitting theorems for extension and torsion functors of local cohomology modules (Q1643102)

In the paper under review, the authors firstly prove the following theorem. (see also, Theorem 1.1. of [\textit{Nguyen Tu Cuong} and \textit{Pham Hung Quy}, J. Algebra 331, No. 1, 512--522 (2011; Zbl 1227.13012)]. Theorem 1. Let \(M\) be a finitely generated module over a Noetherian ring \(R\) and \(\mathfrak{a}\) be an ideal of \(R\). Suppose that there exists a positive integer \(t\) such that \(\mathfrak{a}^nH^t_{\mathfrak a}(M)=\mathfrak{a}^nH^{t+1}_{\mathfrak a}(M)=0\) for some \(n\in\mathbb{N}\). Then, for every \(\mathfrak{a}\)-filter regular element \(x\in M\), there exists a positive integer \(m\) with \(m\geq n\) such that \(H^t_{\mathfrak{a}}(M/x^mM)\cong H^t_{\mathfrak a}(M)\oplus H^{t+1}_{\mathfrak a}(M)\). Thereafter the authors prove some similar splitting results for the extension and torsion functors of the local cohomology modules by assuming \(\mathfrak{a}^nH^t_{\mathfrak a}(M)=0\) and replacing the assumption \(\mathfrak{a}^nH^{t+1}_{\mathfrak a}(M)\) (in the statement of the aforementioned theorem) by other presumptions. More explicitly, splitting of \(\mathrm{Ext}^k_R\big(R/\mathfrak{a},H^t_{\mathfrak{a}}(M/x^mM)\big)\) as \(\mathrm{Ext}^k_R\big(R/\mathfrak{a},H^t_{\mathfrak{a}}(M)\big)\oplus \mathrm{Ext}^k_R\big(R/\mathfrak{a},H^{t+1}_{\mathfrak{a}}(M)\big)\) and similar splitting for \(\mathrm{Tor}^R_k\big(R/\mathfrak{a},H^t_{\mathfrak a}(M/x^mM)\big)\) is proved; and to this aim, e.g. for the case of extension functor, some conditions on the vanishing of some \(\mathrm{Ext}^i_R\big(R/\mathfrak{a},H^j_{\mathfrak a}(M)\big)\) and \(\mathrm{Ext}^i_R\big(R/\mathfrak{a},x^vH^j_{\mathfrak a}(M)\big)\) is assumed (in place of the condition \(\mathfrak{a}^nH^{t+1}_{\mathfrak a}(M)=0\) of the statement of Theorem 1).