On local categories of finite groups.
type: Item
instance of: scholarly article
title: On local categories of finite groups.
zbMATH Open document ID: 1267.20007
author: Fei Xu
published in: Mathematische Zeitschrift
publication date: 2012-12-19T00:00:00Z
review text: Let \(\mathcal P\) be a partially ordered set and let \(G\) be a group. Then \(\mathcal P\) is a \(G\)-poset if there is a group homomorphism \(G\to\Aut(\mathcal P)\) giving an action of \(G\) on \(\mathcal P\). The transporter category \(G\propto\mathcal P\) has the same objects as \(\mathcal P\) and morphisms from \(x\) to \(y\) are couples \((g,gx\leq y)\) where \(g\in G\) and where \(\leq\) is the order relation of \(\mathcal P\). For a small category \(\mathcal C\) denote the category algebra \(k\mathcal C\) as the \(k\) vector space with basis given by the morphisms in \(\mathcal C\) and composition of morphisms induces the multiplication in \(k\mathcal C\). The composition of two basis elements is put to \(0\) if they do not compose in \(\mathcal C\). The author recalls the main properties and basic examples of these constructions. In particular, the author recalls that for a category algebra one can always construct a trivial module, such as for group algebras, and Ext-groups including a notion of a cup product. The author shows as main result that for two modules \(\mathcal M\) and \(\mathcal N\) over \(k(G\propto\mathcal P)\), then \(\mathrm{Ext}^*_{k(G\propto\mathcal P)}(\mathcal{M,N})\) is finitely generated as \(\mathrm{Ext}^*_{k(G\propto\mathcal P)}(k,k)\)-modules. The author gives applications to fusion systems.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 18B40
zbMATH DE Number: 6116864
zbMATH Keywords: transporter categories
zbMATH Keywords: category algebras
zbMATH Keywords: finite generation of Ext algebras
zbMATH Keywords: fusion systems
MaRDI profile type: Publication
full work available at URL: s00209-011-0971-y
OpenAlex ID: W1998678625
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DOI: 10.1007/S00209-011-0971-Y
description: scientific article; zbMATH DE number 6116864