The \(k\)-Buchsbaum property for some polynomial ideals (Q1890382)

Let \(R\) be a standard graded ring over a field \(K\), and let \(\mathfrak m\) be the maximal homogeneous ideal. A graded module \(M\) over \(R\) is said to be \(k\)-Buchsbaum if its local cohomology modules \(H^i_{\mathfrak m} (M)\), \(0 \leq i \leq d\), are annihilated by \({\mathfrak m}^k\), where \(d+1\) is the Krull dimension of \(M\). In this article, the authors focus on the case where \(R = R_{r+1} := K[x_0,\dots,x_r]\) and \(M = R/\mathfrak a\), where \(\mathfrak a\) is a homogeneous ideal and \(K\) is infinite. An equivalent (and, according to the authors, more workable) definition for \(k\)-Buchsbaum ideals, and some technical reformulations, is given in terms of systems of parameters. Much earlier, \textit{H. Bresinsky} and \textit{W. Vogel} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 39, 143--159 (1993; Zbl 0837.13008)] had given an algorithm to test if such an ideal is Cohen-Macaulay or Buchsbaum. (The former corresponds to \(k=0\); the latter means that \(R_{r+1}/{\mathfrak a}\) and \(R_{r+1}/({\mathfrak a},F_0,\dots,F_i )\), \(0 \leq i \leq d\), are 1-Buchsbaum for any system of parameters.) The central problem addressed in this paper is to find an algorithm to determine, or at least bound from above, the \(k\)-Buchsbaumness of an ideal, for larger \(k\), without explicit computation of Ext-modules or local cohomology modules. The authors are successful in the case of certain binomial ideals, using Gröbner bases computations, and in fact they find the smallest such \(k\).