On the \(k\)-regular sequences and the generalization of \(F\)-modules (Q2914428)

Let \(R\) be a commutative Noetherian ring with nonzero identity and \(M\) a finitely generated \(R\)-module. Let \(k\geq -1\) be an integer. \textit{N. Q. Chinh} and \textit{L. T. Nhan} [Algebra Colloq. 15, No. 4, 599--608 (2008; Zbl 1159.13008)] have introduced the notion of \(k\)-regular sequences as a generalization of the notion of filter regular sequences. A sequence \(x_1,x_2,\dots ,x_t\) of elements of \(R\) is said to be a \(k\)-regular \(M\)-sequence if \(x_i\) doesn't belong to any member of NEWLINE\[NEWLINE \{\mathfrak p\in Ass_R(\frac{M}{(x_1,x_2,\dots, x_{i-1})M})|\dim R/\mathfrak p>k\}NEWLINE\]NEWLINE for all \(i=1,2,\dots, t\) and \(\dim_R(\frac{M}{(x_1,x_2,\dots, x_t)M})>k\). The authors extend some properties of filter \(M\)-regular sequences to \(k\)-regular \(M\)-sequences. In particular, for a \(k\)-regular \(M\)-sequence \(x_1,x_2,\dots ,x_t\), they establish the following results:NEWLINENEWLINE1) Assume that \(I\) is an ideal of \(R\) containing \(x_1,x_2,\dots ,x_t\) such that \(\dim R/\mathfrak p>k\) for all NEWLINE\[NEWLINE\mathfrak p\in \mathrm{Supp}_R(\frac{M}{(x_1,x_2,\dots, x_t)M})-V(I).NEWLINE\]NEWLINE Then \(H^i_{I}(M)\cong H^i_{(x_1,x_2,\dots, x_t)}(M)\) for all \(i<t\). (Here for an ideal \(J\) of \(R\), \(H^i_{J}(M)\) denotes the \(i\)th local cohomology module of \(M\) with respect to \(J\).)NEWLINENEWLINE2) For each \(i>0\), the dimension of \(i\)th homology module of the Koszul complex of \(M\) with respect to \(x_1,x_2,\dots, x_t\) is at most \(k\).