Free resolutions of simplicial posets
type: Item
instance of: scholarly article
title: Free resolutions of simplicial posets
zbMATH Open document ID: 0882.06004
author: Art M. Duval
published in: Journal of Algebra
publication date: 1998-02-03T00:00:00Z
review text: A simplicial poset is a finite poset with minimal element in which every interval is isomorphic to a Boolean lattice. This notion generalizes the face poset of a simplicial complex. Indeed, any simplicial poset may be realized as the face poset of a regular cell complex in which every cell is isomorphic to a simplex. R. Stanley defined a ring \(A_P\) associated with a simplicial poset \(P\) which generalizes the face ring (or Stanley-Reisner ring) of a simplicial complex. In the paper under review, the author extends several important calculations from face rings of simplicial complexes to face rings of simplicial posets. The \textit{Betti polynomial} of module \(M\) is the generating function for the ranks of the terms in a minimal free resolution of \(M\). The first main result of the paper calculates the Betti polynomial of \(A_P\) as a module over \(k[V]\), the polynomial ring on the set \(V\) of vertices of \(P\). The result is expressed in terms of ordinary (topological) Betti numbers of subcomplexes of the realization of \(P\). The proof follows the general outline of the proof in the simplicial complex case [\textit{M. Hochster}, ``Cohen-Macauley rings, combinatorics, and simplicial complexes'', in: Ring Theory II, Proc. Second Oklahoma Conference 1975, 171-223 (1977; Zbl 0351.13009)], but uses a refinement of the usual \({\mathbb{Z}}^n\)-grading of the Koszul complex (\(n=|V|\)). A similar, but more complicated analysis leads to the second main result, a calculation of the cohomology \(H^*(A_P):=H^*({\mathcal K}({\mathbf x}^\infty,A_P))\) of a certain chain complex \({\mathcal K}({\mathbf x}^\infty,A_P)\) determined by combinatorially defined local cohomology can be used to compute depth of modules. Thus the author is able to prove that (i) the depth of \(A_P\) is a topological invariant, that is, depends only on the realization of \(P\) as a topological space, and (ii) the depth of \(A_P\) is equal to \(1+m\), where \(m\) is maximal such that the \(m\)-ske…
Mathematics Subject Classification ID: 06A11
zbMATH DE Number: 1013928
zbMATH Keywords: simplicial poset
zbMATH Keywords: face ring
zbMATH Keywords: free resolution
zbMATH Keywords: Betti polynomial
zbMATH Keywords: local cohomology module
zbMATH Keywords: depth
zbMATH Keywords: Hilbert series
reviewed by: Michael Falk
MaRDI profile type: Publication
full work available at URL: jabr.1996.6855
OpenAlex ID: W2062250542
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DOI: 10.1006/JABR.1996.6855
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description: scientific article; zbMATH DE number 1013928