On the associated graded module of an ideal generated by an unconditioned strong \(d\)-sequence (Q1570021)

Let \(A\) denote a commutative ring. Let \(M\) denote an \(A\)-module. Let \(x_1,\ldots,x_s\) denote a sequence of elements of \(A\) generating the ideal \(\mathfrak q\). Let \(G_{\mathfrak q}(A)\), respectively \(G_{\mathfrak q}(M)\), denote the form ring of \(A\) with respect to \(\mathfrak q\), respectively the form module of \(M\) with respect to \(\mathfrak q\). Let \(h_i\), \(i = 1,\dots,s\), denote the initial forms of \(x_i\) in the form ring \(G_{\mathfrak q}(A)\). Then the authors prove -- among others -- the following main result: If \(x_1,\dots,x_s\) is an unconditioned strong \(d\)-sequence, then it is an unconditioned \(\mathfrak q\)-filter regular sequence and the sequence \(h_1,\dots, h_s\) constitutes an unconditioned strong \(d\)-sequence on \(G_{\mathfrak q}(M)\). The converse is true provided \(A\) is Noetherian and \(M\) is finitely generated. The main technical tool for the authors' considerations is a natural homomorphism of the Koszul complex of a system of elements to a certain complex of modules of generalized fractions. As an application the authors deduce a result shown by \textit{S. Goto} and \textit{K. Yamagishi} [see ``The theory of unconditioned strong \(d\)-sequences and modules of finite local cohomology'' (Preprint)], characterizing the finiteness of the local cohomology of \(M\) and the local cohomology of the form module \(G_{\mathfrak q}(M)\).