Functors on categories of finite group representations
type: Item
instance of: scholarly article
title: Functors on categories of finite group representations
zbMATH Open document ID: 0567.20001
DOI: 10.1016/0022-4049(85)90102-1
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: In this interesting paper the author studies induction and restriction on the functor category from finitely generate \(\bar kG\)-modules to \(\bar k- \)spaces, \(\bar k\) a field and G a finite group. This is of great importance from the representation theory of G because of the work of Auslander, Reiten e.a. The functors ''induction'', ''restriction'' and ''conjugation'' are defined for functors on the level of modules. These functors enjoy nice properties, e.g. Mackey's formula holds. In connection with relative projectivity of finitely presented functors the existence of vertices and sources is shown - also here the vertex is a p-group, \(p=char \bar k\). Many of the results for modules hold for functors; however, a projective functor is not necessarily \(\{\) \(1\}\)- projective. Using this and a relative trace map, the author obtains an elegant proof of Carlson's strong correspondence theorem. To mention just one of the many interesting results in this paper: Theorem: Let V be an indecomposable \(\bar kG\)-module with vertex P and source S. Let H be a subgroup of G which contains the stabilizer of S in G. Let W be an indecomposable summand of \(V_ H\) which has vertex P and source S. If \(0\to W_ 2\to W_ 1\to W\to 0\) is an almost split sequence of \(\bar kH\)-modules, then \(0\to W_ 2\uparrow^ G\to W_ 1\uparrow^ G\to W\uparrow^ G\to 0\) is the direct sum of the almost split sequence \(0\to V_ 2\to V_ 1\to V\to 0\) and a split exact sequence.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 3904811
zbMATH Keywords: induction
zbMATH Keywords: restriction
zbMATH Keywords: functor category
zbMATH Keywords: finitely generate \=kG-modules
zbMATH Keywords: relative projectivity
zbMATH Keywords: finitely presented functors
zbMATH Keywords: vertices
zbMATH Keywords: sources
zbMATH Keywords: Carlson's strong correspondence theorem
zbMATH Keywords: almost split sequence
zbMATH Keywords: split exact sequence
author: James A. Green
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90102-1
OpenAlex ID: W2008788210
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description: scientific article; zbMATH DE number 3904811