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.

36 pages; the proof of the existence of quasi-finite flat presentations has been excluded (now in 1005.2171); quasi-separatedness assumptions dropped; some parts have been reorganized; title changed from "Existence of quotients by finite groups and coarse moduli spaces"

References in corpus (6)

• Noetherian approximation of algebraic spaces and stacks
• Submersions and effective descent of etale morphisms
• Étale dévissage, descent and pushouts of stacks
• Representability of Hilbert schemes and Hilbert stacks of points
• Families of zero cycles and divided powers: I. Representability
• The space of generically étale families

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• Submersions and effective descent of etale morphisms
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• The Hilbert Stack
• Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves
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• Canonical reduction of stabilizers for Artin stacks with good moduli spaces
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• Good reduction and cyclic covers
• Families of zero cycles and divided powers: I. Representability
• Comparison of the Categories of Motives defined by Voevodsky and Nori
• Addendum: Étale dévissage, descent and pushouts of stacks
• Real topological Hochschild homology of schemes
• Computing isogenies from modular equations in genus two
• Nagata compactification for algebraic spaces
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• Relative orbifold Pandharipande-Thomas theory and the degeneration formula
• Arithmetic Satake compactifications and algebraic Drinfeld modular forms
• Moduli of parabolic sheaves and filtered Kronecker modules
• A generalization of Luna's fundamental lemma for stacks with good moduli spaces
• Stable points on algebraic stacks
• The moduli space of marked supersingular Enriques surfaces
• Building Data for Stacky Covers
• On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $G{L}_n$