Monomial resolutions of trivial source modules
type: Item
instance of: scholarly article
title: Monomial resolutions of trivial source modules
zbMATH Open document ID: 1009.20011
author: Robert Boltje
author: Burkhard Külshammer
published in: Journal of Algebra
publication date: 2002-07-04T00:00:00Z
review text: Let \(G\) be a finite group and \(F\) a field of characteristic \(p>0\). For an \(FG\)-module \(V\), a subgroup \(H\) of \(G\) and a homomorphism \(\phi\colon H\to F^\times\) a generalized Brauer construction \(\overline V(H,\phi)\) was defined by the authors in a previous paper, and it turned out that it is related to the canonical induction formula for trivial source modules. In the present paper, for fixed \(V\) and varying \((H,\phi)\), conjugation, restriction, corestriction and transitivity maps between the spaces \(\overline V(H,\phi)\) are defined and properties of these maps are investigated. In Section 2 of the paper the notion of a `Brauer sheaf' is introduced. Its objects are families of \(F\)-vector spaces indexed by pairs \((H,\phi)\) together with conjugation, restriction, corestriction and transitivity maps satisfying certain natural compatibilities. The generalized Brauer construction gives rise to Brauer sheaves, and another source of examples comes from the category \(_{FG}\mathtt{mon}\) of finite \(G\)-equivariant line bundles. In Section 3 a monomial resolution of a trivial source module \(V\) is defined as a chain complex \(M_*\) in \(_{FG}\mathtt{mon}\) using the embeddings of \(V\) and \(M_*\) into the category of Brauer sheaves in a way compatible with the canonical induction formula for \(V\). A necessary and sufficient condition for the existence of a monomial resolution of \(V\) is the existence of a so called `Brauer filtration' of \(V\). If such a filtration exists, \(V\) can be regarded as an object in another category of sheaves \(\text{Sh}_F(G)\) previously introduced by the first author. This leads in Section 4 to the study of monomial resolutions of sheaves in \(\text{Sh}_F(G)\). The construction of the monomial resolution is not functorial. In the last section conditions are given under which \(FG\)-homomorphisms can be extended to chain maps between monomial resolutions, and a list of open questions is stated.
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 1763579
zbMATH Keywords: group algebras
zbMATH Keywords: generalized Brauer construction
zbMATH Keywords: canonical induction formula
zbMATH Keywords: Brauer sheaves
zbMATH Keywords: Brauer filtrations
zbMATH Keywords: finite groups
zbMATH Keywords: trivial source modules
zbMATH Keywords: monomial resolutions
MaRDI profile type: Publication
OpenAlex ID: W2084287395
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DOI: 10.1006/JABR.2001.8816
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description: scientific article; zbMATH DE number 1763579