On the free resolutions of locally cohomology modules with respect to an ideal generated by a u.s. \(d\)-sequence. (English) Zbl 1267.13027

Summary: Let \(a\) be an almost complete intersection ideal of a commutative Noetherian local ring \(R\) and \(r\) be the number of elements of a minimal generating set of \(a\). Suppose that the \(i\)-th local cohomology module \(H^i_a(R)\) is finitely generated for all \(i< r\). We show that there exists a sequence \(x=x_1,\dots,x_r\) of elements in \(a\) which is both an \(a\)-filter regular and u.s. \(d\)-sequence on \(R\) and \(\Omega_R^{r-1}(H^{r-1}_a(R))\cong \Omega_R^{r+1}(R/(x))\) where, for an \(R\)-module \(M\), \(\Omega_R^{i}(M)\) is the \(i\)-th syzygy of \(M\).


13D02 Syzygies, resolutions, complexes and commutative rings
13D45 Local cohomology and commutative rings