Smoothness of limit functors

Abstract

Let S be a scheme. Assume that we are given an action of the one dimensional split torus \(\mathbb {G}_{m,S}\) on a smooth affine S-scheme \(\mathfrak {X}\). We consider the limit (also called attractor) subfunctor \(\mathfrak {X}_{\lambda }\) consisting of points whose orbit under the given action ‘admits a limit at 0’. We show that \(\mathfrak {X}_{\lambda }\) is representable by a smooth closed subscheme of \(\mathfrak {X}\). This result generalizes a theorem of Conrad et al. (Pseudo-reductive groups (2010) Cambridge Univ. Press) where the case when \(\mathfrak {X}\) is an affine smooth group and \(\mathbb {G}_{m,S}\) acts as a group automorphisms of \(\mathfrak {X}\) is considered. It also occurs as a special case of a recent result by Drinfeld on the action of \(\mathbb {G}_{m,S}\) on algebraic spaces (Proposition 1.4.20 of Drinfeld V, On algebraic spaces with an action of \(\mathfrak {G}_{m}\), preprint 2013) in case S is of finite type over a field.