Buchsbaum and Eulerian complexes
type: Item
instance of: scholarly article
title: Buchsbaum and Eulerian complexes
zbMATH Open document ID: 0839.13008
DOI: 10.1016/0022-4049(95)90011-X
published in: Journal of Pure and Applied Algebra
publication date: 1996-06-19T00:00:00Z
review text: Let \(\Delta\) denote a finite simplicial complex. For a fixed field \(K\) let \(K [\Delta]\) denote its Stanley-Reisner ring over \(K\). Then \(\Delta\) is called a Cohen-Macaulay (resp. Buchsbaum) complex over \(K\) if and only if \(K [\Delta]\) is a Cohen-Macaulay (resp. Buchsbaum) ring. This is characterized topologically by the vanishing of the simplicial cohomology of the links of \(\Delta\) with coefficients in \(K\) [see \textit{G. A. Reisner}, Adv. Math. 21, 30-49 (1976; Zbl 0345.13017); respectively the reviewer, Math. Z. 178, 125-142 (1981; Zbl 0472.13012)]. For a Buchsbaum complex \(\Delta\) let \(\beta (\Delta) : = (\beta_0, \dots, \beta_{d - 1})\) with \(\beta_i : = \dim_KH_i (\Delta,K)\) and \(0 \leq i \leq d - 1 : = \dim \Delta\). Then \(\chi (\Delta) = \sum^{d - 1}_{i = 0} (-1)^i \beta_i (\Delta)\) is the Euler characteristic of \(\Delta\). Moreover \(\Delta\) is called an Eulerian complex provided it is pure and for every face \(F \in \Delta\) one has \(\chi (\text{lk}_\Delta F) = 1 + (-1)^{d(F)}\), where \(d(F)\) denotes the dimension of the link \(\text{lk}_\Delta F\). Then the authors describe the possible \(\beta (\Delta)\)'s for Buchsbaum-Eulerian, Eulerian, and semi-Eulerian complexes. Furthermore, they describe the ring-theoretic Betti numbers of \(K [\Delta]\) in a free resolution, in particular the depth of \(K [\Delta]\), in the case of Buchsbaum-Eulerian complexes. It turns out that for an Eulerian complex any depth between one and the dimension is possible.
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 06A11
zbMATH DE Number: 737476
zbMATH Keywords: Stanley-Reisner ring
zbMATH Keywords: Buchsbaum complex
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Eulerian complex
zbMATH Keywords: Betti numbers
zbMATH Keywords: depth
author: Clara S. Chan
author: Douglas S. Jungreis
author: Richard A. Stong
reviewed by: Peter Schenzel
MaRDI profile type: Publication
cites work: Betti numbers of Buchsbaum complexes.
cites work: Q3963718
cites work: Quotient algebras of Stanley-Reisner rings and local cohomology
cites work: Q4154669
cites work: Characterizations of Buchsbaum complexes
cites work: Topological results in combinatorics
cites work: Q3827224
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: Q5522742
cites work: Combinatorics and commutative algebra.
cites work: Q3748279
cites work: Flag \(f\)-vectors and the \(cd\)-index
full work available at URL: 0022-4049(95)90011-x
OpenAlex ID: W2110659829
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description: scientific article; zbMATH DE number 737476