A remark on the local cohomology modules of a union of disjoint matroids
type: Item
instance of: scholarly article
title: A remark on the local cohomology modules of a union of disjoint matroids
zbMATH Open document ID: 1354.13024
author: Nguyên Công Minh
published in: Vietnam Journal of Mathematics
publication date: 2016-10-27T00:00:00Z
review text: Let \(\Delta\) denote a simplicial complex on the vertex set \(\{1,\ldots,n\}\). Let \(S = K[x_1,\ldots,x_n]\) be the polynomial ring in \(n\) variables over the field \(K\). Let \(S/I\) denote the Stanley-Reisner ring, where \( I = I_{\Delta}\) denotes the Stanley-Reisner ideal of \(\Delta\). In recent research there is much attention about the Cohen-Macaulay-ness (Buchsbaum-ness, \(k\)-Buchsbaum-ness, generalized Cohen-Macaulay-ness etc.) of \(S/I^{(t)}\) for various \(t\geq 1\) in terms of combinatorial data of \(\Delta\). Here \(I^{(t)}, t \geq 1,\) denotes the \(t\)-symbolic power of \(I\). As a main result the author proves a generalization of results of the author's joint work with \textit{Y. Nakamura} [Nagoya Math. J. 213, 127--140 (2014; Zbl 1291.13041)] about the \(k\)-Buchsbaum-ness of \(\Delta\), a union of disjoint matroids. As an application it follows that \(\Delta\) is a matroid if and only if \(S/I^{(t)}\) is \(k\)-Buchsbaum for some \(t \geq \max\{3, k/2 +1\}\) (resp. equivalently for all \(t \geq 1\)).
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 05E45
zbMATH DE Number: 6644454
zbMATH Keywords: symbolic powers
zbMATH Keywords: \(k\)-Buchsbaum
zbMATH Keywords: generalized Cohen-Macaulay
zbMATH Keywords: local cohomology modules
MaRDI profile type: Publication
full work available at URL: s10013-015-0161-z
OpenAlex ID: W1124817088
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DOI: 10.1007/S10013-015-0161-Z
description: scientific article; zbMATH DE number 6644454