A fixed point criterion for linear reductivity

Abstract

Let G be a linear algebraic group over an algebraically closed field. If for all actions of G on smooth schemes, the fixed point scheme is smooth, then G is linearly reductive under either of the additional assumptions: (a) the ground field is characteristic zero, or (b) G is connected, reduced, and solvable. Let K be an algebraically closed field and G a linear algebraic group over K. Say G has the smooth fixed point property if for all actions of G on a smooth scheme, the fixed point scheme is also smooth. Fogarty [1I has shown that if G is linearly reductive, then G has the smooth fixed point property. One can ask the converse question: If G is not linearly reductive, is there an action of G on a smooth scheme with a nonsmooth fixed point scheme? In this note we show how to construct such an action for any G that is a split extension by a unipotent subgroup. This gives an affirmative answer to the question for any class of groups where G being not linearly reductive implies G is a split extension by a unipotent subgroup. In particular this includes all groups of characteristic zero and in arbitrary characteristic connected reduced solvable groups. Let G be a linear algebraic group over K which is a split extension by the unipotent subgroup U. We first show that we may assume that U is the direct sum of copies of the additive group. If G modulo a normal subgroup has an action with a nonsmooth fixed point scheme, then certainly G does. Using this we can replace G by G modulo the commutator subgroup of U, and hence we may assume G is commutative. In characteristic zero this already implies U is the direct sum of copies of the additive group. In characteristic p there are truncated Witt groups; however, if we take G modulo p * U we may assume G is the direct sum of additive groups by a theorem of Serre [2]. We now construct an action of G on affine space with a nonreduced fixed point scheme. Using our assumption that G is a split extension, we Received by the editors April 16, 1974. AMS (MOS) subject classifications (1970). Primary 20G15. Copyright (C 1975, American Mathematical Society