Linear tori with the action of finite groups
type: Item
instance of: scholarly article
title: Linear tori with the action of finite groups
zbMATH Open document ID: 0846.20057
DOI: 10.1007/BF02304411
published in: Mathematical Notes
publication date: 1996-05-20T00:00:00Z
review text: The main object of the paper is a linear representation \(V\) of a finite group \(G\) with an additional structure, namely with a \(G\)-stable torus \(T\) embedded into \(V\) as an open subset which is called a linear torus. If the action of \(G\) on \(T\) is free, \(T\) is called a linear \(f\)-torus. One can introduce a natural partial order on the set of linear tori: \(T \geq T'\) if there is a \(G\)-equivariant map \(T \to T'\). It turns out that in the set of linear \(f\)-tori there is only one maximal equivalence class with respect to this relation, and it can be distinguished by the following condition: the image of \(K_0 (T/G)\) in the cohomology of \(T/G\) is trivial. The author uses this result to define partially stabilized cohomology \(H^*_{\text{par}} (G,F)\) with coefficients in a discrete module \(F\) as the image of \(H^* (G,F)\) in \(H^*(T/G, F)\) where \(T\) is a linear \(f\)-torus from the maximal equivalence class. This notion was introduced earlier [the author, Mat. Sb. 183, No. 5, 3-28 (1992; Zbl 0789.14022)] as an attempt to give a more computable analog of stabilized cohomology.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 857370
zbMATH Keywords: linear representation
zbMATH Keywords: finite group
zbMATH Keywords: linear tori
zbMATH Keywords: \(G\)-equivariant map
zbMATH Keywords: partially stabilized cohomology
author: Fedor Bogomolov
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
cites work: STABLE COHOMOLOGY OF GROUPS AND ALGEBRAIC VARIETIES
cites work: Chern classes of complex Galois representations
full work available at URL: https://doi.org/10.1007/bf02304411
OpenAlex ID: W2093555682
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description: scientific article; zbMATH DE number 857370