Characterizations of Buchsbaum complexes
type: Item
instance of: scholarly article
title: Characterizations of Buchsbaum complexes
zbMATH Open document ID: 0671.13014
DOI: 10.1007/BF01168875
author: Mitsuhiro Miyazaki
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155380
review text: Let \(K\) be a field, \(\Delta\) a simplicial complex with vertex set \(V\subset \{x_ 1,\dots,x_ n\}\), K[\(\Delta\) ] the associated Stanley-Reisner ring, \(A=k[x_ 1,\dots,x_ n]\) and \({\mathfrak m}_ j=(x^ j_ 1,\dots,x^ j_ n)\). The author computes the modules \(\mathrm{Ext}^ i_ A(A/{\mathfrak m}_ j,K[\Delta])\) in terms of the reduced simplicial cohomology of certain subcomplexes of \(\Delta\). As a corollary he gets Hochster's fundamental theorem which relates the local cohomology of \(K[\Delta]\) with respect to \({\mathfrak m}\) and the reduced simplicial cohomology. After having recalled some criteria of Schenzel and Stückrad-Vogel for \(K[\Delta]\) to be a Buchsbaum ring, the author proves his characterization of the Buchsbaum property of Stanley-Reisner rings: \(K[\Delta]\) is Buchsbaum if and only if for all \(i<d\) the modules \(\mathrm{Ext}^ i_ A(A/{\mathfrak m},K[\Delta])\) and \(\mathrm{Ext}^ i_ A(A/{\mathfrak m}_ 2,K[\Delta])\) have the same length.
reviewed by: Winfried Bruns
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 4097527
zbMATH Keywords: Stanley-Reisner ring
zbMATH Keywords: Ext
zbMATH Keywords: Buchsbaum ring
MaRDI profile type: Publication
cites work: On graded rings. II: \(\mathbb{Z}\)-graded rings
cites work: Q4120658
cites work: Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay
cites work: Topological results in combinatorics
cites work: Cohen-Macaulay quotients of polynomial rings
cites work: On the number of faces of simplicial complexes and the purity of Frobenius
cites work: The Upper Bound Conjecture and Cohen-Macaulay Rings
cites work: Combinatorics and commutative algebra
cites work: Toward a Theory of Buchsbaum Singularities
OpenAlex ID: W2087943123
Recommended article: Buchsbaum\(^*\) complexes
Recommended article: Buchsbaum and Eulerian complexes
Recommended article: Betti numbers of Buchsbaum complexes.
Recommended article: Buchstaber invariants of universal complexes
Recommended article: On 2-Buchsbaum complexes
Recommended article: A characterization of two-dimensional Buchsbaum matching complexes
Recommended article: Buchsbaum complexes with linear resolutions
Recommended article: On Buchsbaum algebras
Recommended article: Characterizing certain semidualizing complexes via their Betti and Bass numbers
Recommended article: Socles of Buchsbaum modules, complexes and posets
description: scientific article; zbMATH DE number 4097527