Inheritance of the Buchsbaum and Cohen-Macaulay properties in subcomplexes
type: Item
instance of: scholarly article
title: Inheritance of the Buchsbaum and Cohen-Macaulay properties in subcomplexes
zbMATH Open document ID: 0840.06006
author: Dean E. Smith
published in: Advances in Mathematics
publication date: 1996-07-01T00:00:00Z
review text: Let \(\Sigma\) be a finite simplicial complex with underlying topological space \(|\Sigma |\). Let \(k [\Sigma]\) be the Stanley-Reisner ring of \(\Sigma\) over a field \(k\). It has been conjectured by M. Hochster that the property of \(k [\Sigma]\) to be Cohen-Macaulay is independent of the simplicial structure of \(\Sigma\), depending only on the space \(|\Sigma |\). Proving this conjecture, \textit{J. R. Munkres} [Michigan Math. J. 31, 113-128 (1984; Zbl 0585.57014)] defined two topological invariants \(\alpha\) and \(\gamma\) of \(\Sigma \). Namely, \(\gamma (\Sigma)\) is the smallest integer \(j\) for which at least one of the relative cohomological groups \(\widetilde H^j (|\Sigma |, |\Sigma |- p)\) for \(p \in |\Sigma |\) is not trivial. Further, \(\alpha (\Sigma)\) is the least integer \(j\) so that one of the groups \(\widetilde H^j (|\Sigma |)\), \(\widetilde H^j (|\Sigma |, |\Sigma |- p)\), \(p \in |\Sigma |\), does not vanish. It is clear that \(\dim \Sigma \geq \gamma (\Sigma) \geq \alpha (\Sigma) \geq - 1\). In the case \(\alpha (\Sigma) = \dim \Sigma\), the ring \(k [\Sigma]\) is to be Cohen-Macaulay and it is to be Buchsbaum if \(\gamma (\Sigma) = \dim \Sigma\). This paper is devoted to further investigations of the invariants \(\alpha (\Sigma) \) and \(\gamma (\Sigma)\). Here are partial statements of its main results (Theorems 2.3 and 2.5): Let \(T \subseteq V (\Sigma)\) be the subset of the vertex set of a complex \(\Sigma \) such that, for \(x,x' \in T\), \(x \neq x'\), \(\{x,x' \} \notin \Sigma\). Let \(\Sigma'\) be the subcomplex \(\Sigma' = \Sigma - \{\sigma \in \Sigma \mid \sigma\) contains some \(x \in T\}\). Then \(\alpha (\Sigma') \geq \alpha (\Sigma) - 1\) and \(\gamma (\Sigma') \geq \gamma (\Sigma) - 1\). Note that associating with a finite poset \(P\) its order complex \(\Sigma (X)\) [\textit{R. P. Stanley}, Combinatorics and commutative algebra (1983; Zbl 0537.13009)], one can speak about invariants \(\alpha\) and \(\gamma\) of posets…
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 05E25
zbMATH DE Number: 814486
zbMATH Keywords: Cohen-Macaulay poset
zbMATH Keywords: Buchsbaum poset
zbMATH Keywords: Stanley-Reisner ring
zbMATH Keywords: order complex
reviewed by: Valeriy B. Mnukhin
MaRDI profile type: Publication
full work available at URL: aima.1995.1043
OpenAlex ID: W1975507091
DOI: 10.1006/AIMA.1995.1043
description: scientific article; zbMATH DE number 814486