The Chevalley–Shephard–Todd theorem for finite linearly reductive group schemes

Abstract

We generalize the classical Chevalley–Shephard–Todd theorem to the case of finite linearly reductive group schemes. As an application, we prove that every scheme $X$ which is étale-locally the quotient of a smooth scheme by a finite linearly reductive group scheme is the coarse space of a smooth tame Artin stack (as defined by Abramovich, Olsson, and Vistoli), whose stacky structure is supported on the singular locus of $X$.