On affine algebraic homogeneous spaces
type: Item
instance of: scholarly article
title: On affine algebraic homogeneous spaces
zbMATH Open document ID: 0554.14016
DOI: 10.1007/BF01194008
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let G be a connected affine algebraic group over an algebraically closed groundfield k, X a homogeneous space of G and H an isotropy group of G on X. It is proved that if X is an affine variety (resp. isomorphic to affine space) then \(\dim R_ uG\geq \dim R_ uH\) (resp. \(R_ uG\) is transitive on X), where \(R_ u\) denotes the unipotent radical. The proof makes use of étale cohomology with proper supports. If Y is a smooth k- variety, let \(H_ c^ j(Y)\) be the j-th space of étale cohomology of Y with proper supports, coefficients in \({\mathbb{Z}}/\ell\), \(\ell\) is prime, prime to the characteristic of k, and let m(Y) be the smallest j for which this group is \(\neq 0\). Then it is shown that \(m(X)=\dim X+\dim R_ uG-\dim R_ uH.\) The theorem then follows because \(m(X)\geq \dim X\) (resp. \(m(X)=2\dim X)\) if X is affine (resp. isomorphic to affine space). Relations with known results of E. Cline - B. Parshall - L. Scott, R. W. Richardson and M. Raynaud are discussed.
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14L35
zbMATH DE Number: 3882578
zbMATH Keywords: connected affine algebraic group
zbMATH Keywords: unipotent radical
zbMATH Keywords: étale cohomology with proper supports
MaRDI profile type: Publication
cites work: Q5580322
cites work: Arithmetic subgroups of algebraic groups
cites work: Induced modules and affine quotients
cites work: Q3871006
cites work: Modules projectifs universels
cites work: Affine Coset Spaces of Reductive Algebraic Groups
cites work: Questions of rationality for solvable algebraic groups over non-perfect fields
author: Armand Borel
description: scientific article; zbMATH DE number 3882578