On extensions of algebraic groups with finite quotient

Michel Brion

Vol. 279 (2015), No. 1-2, 135–153

DOI: 10.2140/pjm.2015.279.135

Abstract

Consider an exact sequence of group schemes of finite type over a field $k$,

$$1\to N\to G\to Q\to 1,$$

where $Q$ is finite. We show that $Q$ lifts to a finite subgroup scheme $F$ of $G$; if $Q$ is étale and $k$ is perfect, then $F$ may be chosen étale as well. As applications, we obtain generalizations of classical results of Arima, Chevalley, and Rosenlicht to possibly nonconnected algebraic groups. We also show that every homogeneous space under such a group has a projective equivariant compactification.

Keywords

algebraic groups, finite quotients, extensions, equivariant compactifications

Mathematical Subject Classification 2010

Primary: 14L15

Secondary: 14L30, 20G15

Milestones

Received: 27 May 2015

Revised: 12 July 2015

Accepted: 12 July 2015

Published: 21 December 2015