The associated graded modules of Buchsbaum modules with respect to \({\mathfrak m}\)-primary ideals in the equi-\(\mathbb{I}\)-invariant case
Let \((A,m)\) be a Noetherian local ring with maximal ideal \(m\) of dimension \(d\), and suppose that the residue field \(A/m\) is infinite. Let \(E\) be a finitely generated \(A\)-module of positive dimension \(s\). One denotes by \(h^i(E)\) the length of the \(i\)-th local cohomology module \(H_{m}^i(E)\) of \(E\). The author defines an invariant on \(E\), written \(\mathbb{I}(E)\), as \[ \mathbb{I}(E)=\sum_{i=0}^{s-1}{{s-1}\choose{i}}h^i(E). \] \(E\) is Cohen-Macaulay if and only if \(\mathbb{I}(E)=0\). When \(E\) is a Buchsbaum \(A\)-module, then this invariant \(\mathbb{I}(E)\) coincides with the Buchsbaum invariant of \(E\), i. e. \(l_A(E/qE)-e_q(E)\), where \(q\) is a parameter ideal of \(E\), and \(l_A(*)\) and \(e_q(*)\) denote the length and the multiplicity with respect to \(q\) (of an \(A\)-module). Let \(a\) still be an ideal in \(A\) and assume that \(l_A(E/aE)<\infty\). For the associated graded module of \(E\) with respect to \(a\), written \(G_a(E)\), one gets \(\mathbb{I}(G_a(E))\geq \mathbb{I}(E)\). The author remarks that most of the associated graded rings/modules, which are realized to be Buchsbaum, satisfy the equation \(\mathbb{I}(G_a(E))=\mathbb{I}(E)\). Consequently, these observations led him to the investigation of the case that the equation \(\mathbb{I}(G_a(E))= \mathbb{I}(E)\) holds. Actually, the author proves the following (under the above conditions): Let \(E\) be a Buchsbaum \(A\)-module \(E\) such that \(\mathbb{I}(G_a(E))=\mathbb{I}(E)\). Then the associated graded module \(G_a(E)\) is a Buchsbaum module over \(R(a)\), the Rees algebra of \(a\), and hence over \(G(a)\), the associated graded ring of \(a\). As a consequence it deserves to mention the following corollaries: (1) Let \(E\) be a Buchsbaum \(A\)-module. Then \(G_{q^n}(E)\) is a Buchsbaum \(G(q^n)\)-module for every parameter ideal \(q\) of \(E\) and integer \(n\geq 2\); (2) If \(A\) is a Buchsbaum ring of maximal embedding dimension, then \(G(m^n)\) is a Buchsbaum ring for all \(n\geq 2\). At the end of the paper the author gives some interesting examples concerning the relation between the equality \(\mathbb{I}(G_a(E))=\mathbb{I}(E)\) and the Buchsbaumness of \(G_a(E)\) and \(E\).
scientific article; zbMATH DE number 1423601
scientific article