Tame stacks in positive characteristic

TL;DR

This paper introduces tame algebraic stacks in positive characteristic, showing they are locally quotients by linearly reductive finite group schemes, which improves their structural understanding and potential applications.

Contribution

It defines tame algebraic stacks in positive characteristic and characterizes their local structure as quotients by linearly reductive finite group schemes.

Findings

Tame algebraic stacks include tame Deligne-Mumford stacks.

They are étale locally quotients by linearly reductive finite group schemes.

Complete characterization of finite flat linearly reductive schemes over any base.

Abstract

We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are \'etale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps.