The cohomology of the finite general linear group
type: Item
instance of: scholarly article
title: The cohomology of the finite general linear group
zbMATH Open document ID: 0655.20031
DOI: 10.1016/0022-4049(88)90024-2
author: Ove Kroll
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let F be a finite field of characteristic p, \(\ell\) a prime \(\neq p\), and G the F-points of the general linear group of n by n matrices. The author extends a result of Quillen's by computing the cohomology of G with coefficients in \({\mathbb{Z}}/\ell\) k\({\mathbb{Z}}\), when card(F) is congruent to 1 mod \(\ell\) k (resp. \(\ell^{k+1})\) if \(\ell\) is odd (resp. 2). The cohomology is the product of a free polynomial and a free exterior algebra over \({\mathbb{Z}}/\ell\) k, with generators in degrees 2,4,...,2n and 1,3,...,2n-1 resp. From here it is not too hard to deduce Quillen's original theorem for \({\mathbb{Z}}/\ell\) coefficients (with no restriction except \(\ell \neq p)\) and results where F is replaced by its algebraic closure. The methods involve Quillen's detection theorem to get upper bounds on the cohomology and transfer-restriction arguments from parabolic subgroups of G to construct the cohomology classes that are claimed to exist.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 4069189
zbMATH Keywords: finite field
zbMATH Keywords: general linear group
zbMATH Keywords: cohomology
zbMATH Keywords: free exterior algebra
zbMATH Keywords: Quillen's detection theorem
zbMATH Keywords: cohomology classes
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90024-2
OpenAlex ID: W1987645958
cites work: Q4143495
cites work: Q3240881
cites work: The Sylow 2-subgroups of the finite classical groups
cites work: Corrections and Additions to "On the Degrees and Rationality of Certain Characters of Finite Chevalley Groups"
cites work: Homology of classical groups over finite fields and their associated infinite loop spaces
cites work: Q5648522
cites work: Q4102036
cites work: The cohomology of Chevalley groups of exceptional Lie type
cites work: The Adams conjecture
cites work: On the cohomology and K-theory of the general linear groups over a finite field
cites work: Invariants of Finite Reflection Groups
cites work: Sylow p-Subgroups of the General Linear Group Over Finite Fields of Characteristic p
description: scientific article; zbMATH DE number 4069189