Existence and properties of geometric 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.