On affine algebraic homogeneous spaces (Q759812)

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.