Extensions of pro-affine algebraic groups. II

Brian Lee Peterson

Vol. 86 (1980), No. 1, 277–285

DOI: 10.2140/pjm.1980.86.277

Abstract

Introduction. We fix an algebraically closed field F of characteristic zero throughout. It is known that any pro-affine algebraic group H over F is the semidirect product H_u ⋅ H_r of its unipotent radical H_u and any maximal reductive subgroup H_r. This suggests, for considering extensions of a unipotent pro-affine group U over F by H, only H_u is relevant. More precisely, one is led to ask whether, given a homomorphism H → O(U) = Aut(U)∕Inn(U) for which Ext(H,U) is nonempty, the restriction map Ext(H,U) → Ext(H_u,U)^H is bijective. The author has shown that this is the case if U is affine. We will show that for unipotent pro-affine U, the above restriction map is injective and that it is surjective in the case where H = H_u × H_r, provided that Ext(H,U) is nonempty. We will also obtain necessary and sufficient conditions that Ext(H,U) be nonempty in case both H and U are affine, U unipotent.

Mathematical Subject Classification

Primary: 14L25, 14L25

Milestones

Received: 1 February 1978

Published: 1 January 1980