On the Buchsbaum property of associated graded rings
Let \(I\) be an \({\mathfrak m}\)-primary ideal in a Buchsbaum local ring \((A,{\mathfrak m})\). In this article, the author studies the Buchsbaum property of the associated graded ring \(G(I)=\bigoplus I^n/I^{n+1}\) of \(I\) when the equality \(I^2={\mathfrak q}I\) holds for some minimal reduction \({\mathfrak q}\) of \(I\). Namely: Theorem 1.1. Let \((A,{\mathfrak m})\) be a \(d\)-dimensional Buchsbaum local ring with infinite residue field and \(I\) an \({\mathfrak m}\)-primary ideal such that \(I^2={\mathfrak q}I\) for some parameter ideal \({\mathfrak q}=(a_1,a_2, \dots,a_d)A\) contained in \(I\). Let \(a_i^*\in G(I)\) be the initial form of \(a_i\). Then the following conditions are equivalent: (1) \(G(I)_M\) is a Buchsbaum ring. (2) \(a^*_1,a_2^*, \dots,a^*_d\) form a \(d^+\)-sequence on \(G(I)\). (3) \((a^2_1,a_2^2, \dots, a_d^2)A\cap I^n= (a^2_1,a^2_2, \dots, a^2_d)I^{n-2}\) for all \(3\leq n\leq d+1\). Corollary 1.2. Let \((A,{\mathfrak m})\) be a \(d\)-dimensional Buchsbaum local ring with infinite residue field and \(I\) an \({\mathfrak m}\)-primary ideal such that \(I^2={\mathfrak q}I\) for some parameter ideal \({\mathfrak q}=(a_1,a_2,\dots,a_d)A\) contained in \(I\). Suppose that \(I\supseteq \sum(a_1, \dots, \check a_i, \dots, a_d):{\mathfrak m}\). Then \(G(I)_M\) is a Buchsbaum ring and \(H_M^i (G(I))\) is concentrated in degree \(1-i\).
scientific article; zbMATH DE number 1234252
scientific article