Existence and properties of geometric quotients

TL;DR

This paper investigates conditions under which geometric quotients of groupoids and stacks exist and are categorical, providing explicit local descriptions and extending classical theorems to more general algebraic spaces and stacks.

Contribution

It introduces a natural topological condition for categorical quotients, proves the existence of quotients for finite flat groupoids, and offers an improved, more general version of Keel and Mori's theorem.

Findings

Established a topological criterion for categorical geometric quotients

Proved existence of quotients for finite flat groupoids

Provided explicit local descriptions of quotients

Abstract

In this paper, we study quotients of groupoids and coarse moduli spaces of stacks in a general setting. Geometric quotients are not always categorical, but we present a natural topological condition under which a geometric quotient is categorical. We also show the existence of geometric quotients of finite flat groupoids and give explicit local descriptions. Exploiting similar methods, we give an easy proof of the existence of quotients of flat groupoids with finite stabilizers. As the proofs do not use noetherian methods and are valid for general algebraic spaces and algebraic stacks, we obtain a slightly improved version of Keel and Mori's theorem.