The modular homology of inclusion maps and group actions
type: Item
instance of: scholarly article
title: The modular homology of inclusion maps and group actions
zbMATH Open document ID: 0857.16010
author: Johannes Siemons
author: Valeriy B. Mnukhin
published in: Journal of Combinatorial Theory. Series A
publication date: 1996-07-09T00:00:00Z
full work available at URL: On the modular theory of inclusion maps and--1996.pdf
review text: Given a finite set \(\Omega\) with power set \(2^\Omega\) and \(R\) an associative ring with 1, \(R2^\Omega\) is the free unitary left \(R\)-module with basis \(2^\Omega\) which becomes an associative ring with identity \(1\emptyset\) if the product \(f\cup g=(\sum f_\Delta\Delta)\cup(\sum g_\Gamma\Gamma)=\sum f_\Delta g_\Gamma(\Delta\cup\Gamma)\), \(\Delta,\Gamma\in 2^\Omega\). Then, if \((\Gamma)=\Gamma_1-+\Gamma_k\) is the formal sum of the distinct \((k-1)\)-subsets of the \(k\)-set \(\Gamma\), it follows that \(\partial(f\cup g)=\partial(f)\cup g+f\cup\partial(g)\) and \(\partial:M_k\to M_{k-1}\), where \(M_k\) is the submodule of \(R2^\Omega\) generated by the \(k\)-sets of \(\Omega\). The inclusion (boundary) mapping is thus a derivation, which leads to special properties useful in determining details of homology, including the characteristic \(p>0\) case. Given that many results have been obtained elsewhere, explicitly or in different guises, as recognized by the authors, they have nevertheless been able to take advantage of several nice devices (e.g., corollary 2.3 and lemma 2.4) to prove exactness of \(0\leftarrow M_0\leftarrow M_1\leftarrow\cdots\leftarrow M_m\) for all \(m\leq|\Omega|\) if \(R\) has characteristic 2 and a related notion of \(p\)-exactness (\(M_k\leftarrow M_{k+i}\leftarrow M_{k+p}\) is exact for \(0\leq k\leq m-p\) and \(i\), \(1\leq i\prec p\)) yielding \(0\leftarrow M_0\leftarrow M_1\leftarrow\cdots\leftarrow M_k\) \(p\)-exact if \(R\) has characteristic (prime) \(p\succ 0\) and \(2k\leq|\Omega|\). Having done this group actions on \(\Omega\) become linear maps on \(R2^\Omega\) and the machinery derived can be employed efficiently in obtaining a variety of expressions and forms including an inclusion-exclusion principle and version of Betti-numbers. The techniques employed are successful enough to be used in several similar papers with somewhat varying emphasis and look to be adequate to the tackling of further questions, some mentione…
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 55N20
zbMATH DE Number: 883815
zbMATH Keywords: boundary maps
zbMATH Keywords: finite sets
zbMATH Keywords: power sets
zbMATH Keywords: free unitary left modules
zbMATH Keywords: derivations
zbMATH Keywords: homology
zbMATH Keywords: \(p\)-exactness
zbMATH Keywords: group actions
zbMATH Keywords: linear maps
zbMATH Keywords: inclusion-exclusion principle
zbMATH Keywords: Betti numbers
reviewed by: J. Neggers
MaRDI profile type: Publication
OpenAlex ID: W2060291218
DOI: 10.1006/JCTA.1996.0051
description: scientific article; zbMATH DE number 883815