Vol. 86, No. 1, 1980

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Extensions of pro-affine algebraic groups. II

Brian Lee Peterson

Vol. 86 (1980), No. 1, 277–285

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 Hu Hr of its unipotent radical Hu and any maximal reductive subgroup Hr. This suggests, for considering extensions of a unipotent pro-affine group U over F by H, only Hu 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(Hu,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 = Hu × Hr, 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
Received: 1 February 1978
Published: 1 January 1980
Brian Lee Peterson