Green functors and \(G\)-sets
type: Item
instance of: scholarly article
title: Green functors and \(G\)-sets
zbMATH Open document ID: 0882.19001
author: Serge Bouc
published in: Lecture Notes in Mathematics
publication date: 1997-10-26T00:00:00Z
review text: Let \(G\) be a finite group and \(R\) a commutative ring. This book deals with Mackey and Green functors for \(G\) over \(R\), and with (left) modules over a Green functor \(A\). These are the objects of categories denoted by \({\mathcal M}ack(G)\), \({\mathcal G}reen(G)\) and \(A\)-\({\mathcal M}od\), respectively. Given \(A\), there is a useful category \({\mathcal C}_A\) whose objects are the finite \(G\)-sets and whose morphisms are given by \(\Hom_{{\mathcal C}_A}(X,Y)= A(Y\times X)\). An important property of \({\mathcal C}_A\) is the fact that the category \(A\)-\({\mathcal M}od\) is equivalent to the category \({\mathcal F}unct_R({\mathcal C}_A)\) of all representations of \({\mathcal C}_A\) (whose objects are the \(R\)-linear functors \({\mathcal C}_A\to R\)-\({\mathcal M}od\)). There is a third category which is equivalent to \(A\)-\({\mathcal M}od\) and \({\mathcal F}unct_R({\mathcal C}_A)\). This is the category \({\mathcal E}nd_{{\mathcal C}_A}(\Omega)\)-\({\mathcal M}od\) \((=A(\Omega^2)\)-\({\mathcal M}od)\) where \(\Omega\) is the finite \(G\)-set which is the disjoint union of all \(G\)-sets \(G/H\) where \(H\) ranges over all subgroups of \(G\). The \(R\)-algebra \(A(\Omega^2)\) can be described by generators and relations. It generalizes the Mackey algebra \(\mu_R(G)\) introduced by Thévenaz and Webb (where \(A\) is the Burnside ring Green functor \(b\)) and an algebra defined by Yoshida (where \(A\) is the fixed point functor \(FP_R\) and \(A\)-\({\mathcal M}od\) is the category of all cohomological Mackey functors). The category \(A\)-\({\mathcal M}od\) can also be investigated via functors \({\mathcal H}_A(-,-): A\)-\({\mathcal M}od\times A\)-\({\mathcal M}od\to {\mathcal M}ack(G)\) and \(-\otimes_A-:{\mathcal M}od\)-\(A\times A\)-\({\mathcal M}od\to {\mathcal M}ack(G)\) \((=b\)-\({\mathcal M}od)\). The functors \({\mathcal H}_A(-,-)\) and \(-\otimes_A-\) share the usual properties of hom and tensor functors. They can be used to develop a Morit…
Mathematics Subject Classification ID: 19-02
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 1077996
zbMATH Keywords: Yoshida algebra
zbMATH Keywords: Burnside ring
zbMATH Keywords: Mackey algebra
zbMATH Keywords: Mackey functors
zbMATH Keywords: Morita theory
zbMATH Keywords: Green functors
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0095821
OpenAlex ID: W1566506739
DOI: 10.1007/BFB0095821
description: scientific article; zbMATH DE number 1077996