Existence of slices on a tame context

Abstract

We study the ramification theory for actions involving group schemes, focusing on the tame ramification. We consider the notions of tame quotient stack introduced in Abramovich et al. (Ann Inst Fourier (Grenoble) 58(4):1057–1091, 2008) and of tame action introduced in Chinburg et al. (Duke Math J 82(2):269–308, 1996). We establish a local slice theorem for unramified actions, prove interesting lifting properties for linearly reductive group schemes, and establish a slice theorem for actions by commutative group schemes inducing tame quotient stacks. We show that these actions are induced from an action of an extension of the inertia group on a finitely presented flat neighborhood. Finally, we consider the notion of tame action and its relation to tame quotient stacks.