The Chevalley-Shephard-Todd Theorem for Finite Linearly Reductive Group Schemes

TL;DR

This paper extends the Chevalley-Shephard-Todd theorem to finite linearly reductive group schemes and demonstrates that certain quotient schemes can be realized as coarse spaces of smooth tame Artin stacks with controlled stacky structures.

Contribution

It generalizes a classical theorem to a broader class of group schemes and links quotient schemes to smooth tame Artin stacks with specific properties.

Findings

Generalization of Chevalley-Shephard-Todd theorem to linearly reductive group schemes

Characterization of schemes as coarse spaces of smooth tame Artin stacks

Identification of stacky structures supported on singular loci

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 etale 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.