Fixed point schemes of additive group actions

Abstract

THE Bore1 fixed-point theorem [2, Prop. 15.5., p. 641 states that the action of a connected solvable algebraic group on a complete variety has a fixed-point. An equivalent statement of the theorem is that restriction of ttale coverings to the fixed-point variety is a faithful functor, and the object of this paper is to show that the functor is an equivalence for solvable groups whose factors are isomorphic to the additive group of the ground field. More generally, let S be a base prescheme, let G be an S-group (that is, a group scheme over S), and let 0 : G x s X + X be an action of G on an S-scheme X. Then restriction of &ale coverings of X to the fixed point scheme Xc is an equivalence of categories when G is a solvable with factors isomorphic to the additive group scheme of S and X is proper and finitely presented over S.