Local properties of good moduli spaces

TL;DR

This paper investigates the local structure of Artin stacks and their good moduli spaces, establishing conditions under which local and étale charts admit good moduli spaces, enhancing understanding of their local properties.

Contribution

It provides new criteria for the existence of good moduli spaces near certain points and relates local properties to étale charts, advancing the theory of Artin stacks.

Findings

Good moduli spaces exist near closed points with linearly reductive stabilizers.

Conditions are given for deducing good moduli spaces from étale charts.

Local properties of Artin stacks are characterized in relation to their moduli spaces.

Abstract

We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. We also give conditions for when the existence of good moduli spaces can be deduced from the existence of etale charts admitting good moduli spaces.