On the Buchsbaum associated graded modules with respect to \(\mathfrak m\)-primary ideals whose reduction numbers are at most one
Let \((A,m)\) be a Noetherian local ring with maximal ideal \(m\) and suppose that the residue field \(A/m\) is infinite. Let \(E\) be a finitely generated \(A\)-module of positive dimension \(s\). \textit{Y. Nakamura} [J. Algebra 209, 345-366 (1998; Zbl 0942.13002)] gave necessary and sufficient conditions for the associated graded rings of \(m\)-primary ideals, whose reduction numbers are at most one, to be Buchsbaum. After that, the second author proved [\textit{K. Yamagishi}, J. Algebra 225, No. 1, 1-27 (2000; see the preceding review Zbl 0982.13001)] statements similar to those given by Nakamura, but without any hypothesis on the reduction numbers of \(m\)-primary ideals. Let \(a\) still be an ideal in \(A\) and assume that \(l_A(E/aE)<\infty\). In the case where the reduction numbers of \(m\)-primary ideals are very small, especially at most one, it may be hoped that there are some ''nice'' relations between the local cohomology modules of \(G_a(E)\), the associated graded module of \(E\) with respect to \(a\), and the structure of the given ideal \(a\). One denotes by \(h^i(E)\) the length of the \(i\)-th local cohomology module \(H_{m}^i(E)\) of \(E\) and defines an invariant on \(E\), written \(\mathbb{I}(E)\), as \[ \mathbb{I}(E)=\sum_{i=0}^{s-1}{{s-1}\choose{i}}h^i(E). \] Let \(q=(a_1,\ldots,a_s)\) be a parameter ideal of \(E\). One defines an \(A\)-submodule of \(E\) as follows \[ \sum(q;E)=\sum_{i=1}^s[(a_1,\ldots,\widehat{a}_i,\ldots,a_s)E:a_i]+qE. \] The main results in this paper are the following: (1) Let \(E\) be a Buchsbaum \(A\)-module of dimension \(s\) and let \(a\) be an ideal such that \(l_A(E/aE)<\infty\). Then the following two statements are equivalent (i) \(G_a(E)\) is a Buchsbaum \(G(a)\)-module such that \([H_{\mathcal{M}}^i(G_a(E))]_n=0\) for \(n\neq 1-i\) \((0\leq i<s)\) and \(a(G_a(E))\leq 1-s\), where \(a(G_a(E))\) stands for the \(a\)-invariant of \(G_a(E)\). (ii) Some (resp. every) parameter ideal \(q\) of \(E\) such that \(q\subseteq a\) satisfies \(a^2E=qaE\) and \(\sum(q;E)\subseteq aE\). When this is the case, one also has \(\mathbb{I}(G_a(E))=\mathbb{I}(E)\) and \([H_{\mathcal{ M}}^s(G_a(E))]_{1-s}=aE/\sum(q;E)\). (2) Under the same conditions as above, the following two statements are equivalent (i) \(G_a(E)\) is a Buchsbaum \(G(a)\)-module such that \([H_{\mathcal{M}}^i(G_a(E))]_n=0\) for \(n\neq -i\) \((0\leq i<s)\) and \(a(G_a(E))\leq 1-s.\) (ii) Some (resp. every) parameter ideal \(q\) of \(E\) such that \(q\subseteq a\) satisfies \(a^2E=qaE\) and \(aE\cap\sum(q;E)=qE\). When this is the case, one also has \(\mathbb{I}(G_a(E))=\mathbb{I}(E)\) and \([H_{\mathcal{M}}^s(G_a(E))]_{1-s}=aE/qE\). At the end of the paper the authors give some examples concerning the above theorems.
scientific article; zbMATH DE number 1565857
scientific article