On the \( h \)-triangles of sequentially \((S _r)\) simplicial complexes via algebraic shifting
type: Item
instance of: scholarly article
title: On the \( h \)-triangles of sequentially \((S _r)\) simplicial complexes via algebraic shifting
zbMATH Open document ID: 1263.13022
author: M. R. Pournaki
author: Siamak Yassemi
author: S. A. Seyed Fakhari
published in: Arkiv för Matematik
publication date: 2013-03-27T00:00:00Z
review text: As the title suggests, the paper under review studies the \(h\)-triangles of sequentially \((S_r)\) simplicial complexes via algebraic shifting. The notion of a sequentially (\( S_r\)) simplicial complex, introduced by \textit{H. Haghighi} et al. [Proc. Am. Math. Soc. 139, No. 6, 1993--2005 (2011; Zbl 1220.13014)], gives a generalization of two properties for simplicial complexes: being sequentially Cohen-Macaulay and satisfying Serre's condition (\( S_r\)). Let \(\Delta\) be a \((d - 1)\)-dimensional \((S_r)\) simplicial complex with \(\Gamma\) (\(\Delta\)) as its algebraic shifting. When \(i\leq r-1\), the \(i\)th pure skeleton \(\Delta^{[i]}\) of \(\Delta\) is Cohen-Macaulay. This fact enables the study of the \(f\)-vectors of the sequential layers of \(\Delta\) and \(\Gamma(\Delta)\), which leads to the main result (Theorem 3.2), namely, algebraic shifting preserves the \(h\)-triangles: \( h_{i,j}(\Delta)= h_{i,j}(\Gamma(\Delta))\) for every \( i \) and \( j \) with \(0 \leq j \leq i \leq r - 1\). Several corollaries, including the study of iterated Betti numbers, are deduced from this main result.
reviewed by: Yi-Huang Shen
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 52B05
zbMATH DE Number: 6149023
zbMATH Keywords: \(h\)-triangles
zbMATH Keywords: sequentially \((S_r)\) simplicial complex
zbMATH Keywords: algebraic shifting
MaRDI profile type: Publication
full work available at URL: s11512-011-0160-6
OpenAlex ID: W2004305128
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DOI: 10.1007/S11512-011-0160-6
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description: scientific article; zbMATH DE number 6149023