Quotients by Groupoids

Abstract

We show that if a flat group scheme acts properly, with finite stabilizers, on an algebraic space, then a quotient exists as a separated algebraic space. More generally we show any flat groupid for which the family of stabilizers is finite has a uniform geometric, uniform categorical quotient in the category of algebraic spaces. Our argument is elementary and essentially self contained.

AMSTeX

Cited by in corpus (86)

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• Étale dévissage, descent and pushouts of stacks
• On the moduli of Kahler-Einstein Fano manifolds
• Affineness and chromatic homotopy theory
• Functorial destackification of tame stacks with abelian stabilisers
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• Geometricity for derived categories of algebraic stacks
• Chamber Structure and Wallcrossing in The ADHM Theory of Curves I
• Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic
• Uniqueness of K-polystable degenerations of Fano varieties
• Tropical geometry of moduli spaces of weighted stable curves
• The category of toric stacks
• Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal
• Stable maps and Hurwitz schemes in mixed characteristic
• Equivariant intersection theory
• K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems
• Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves
• Finiteness theorems for the Picard objects of an algebraic stack
• Nonexistence of semiorthogonal decompositions and sections of the canonical bundle
• Constructing quotients of algebraic varieties by linear algebraic group actions
• A modular description of ${\mathcal{X}}_0(n)$
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• Towards a classification of modular compactifications of the moduli space of curves
• Parity and symmetry in intersection and ordinary cohomology
• Quotient stacks and equivariant étale cohomology algebras: Quillen's theory revisited
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• Compactifications of the universal Jacobian over curves with marked points
• A predicted distribution for Galois groups of maximal unramified extensions
• Lagrangian tens of planes, Enriques surfaces and holomorphic symplectic fourfolds
• A complete degeneration of the moduli of $G$-bundles on a curve
• The Balmer spectrum of a tame stack
• On the moduli of degree 4 Del Pezzo surfaces
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• Modular compactifications of ${\mathcal{M}}_{2,n}$ with Gorenstein singularities
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• A Theory of Stacky Fans
• Addendum: Étale dévissage, descent and pushouts of stacks
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• Moduli of twisted sheaves
• Functorial resolution of tame quotient singularities in positive characteristic
• The circle quantum group and the infinite root stack of a curve (with an appendix by Tatsuki Kuwagaki)
• Mori Dream Stacks
• Motivic integration over wild Deligne-Mumford stacks
• Moduli of elliptic curves via twisted stable maps
• G-marked moduli spaces
• Additive invariants of orbifolds
• Nagata compactification for algebraic spaces
• Computing isogenies from modular equations in genus two
• Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture
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• Basepoint freeness for big line bundles in positive characteristic, with applications to M_g and to 3-fold MMP
• Augmented Teichmuller Spaces and Orbifolds
• The space of generically étale families
• An embedding problem for finite local torsors over twisted curves
• Arithmetic geometry of the moduli stack of Weierstrass fibrations over ${\mathbb{P}}^1$
• Artin algebraization and quotient stacks
• A compactification of the moduli space of multiple-spin curves
• Higher theta series for unitary groups over function fields
• Unitary representations of the fundamental group of Orbifolds
• Full exceptional collections for anticanonical log del Pezzo surfaces
• Lannes's $T$-functor and equivariant Chow rings
• Weak del Pezzo surfaces with irregularity
• Quotients of functors of Artin rings
• Geometry of Weighted Homogeneous Spaces
• Using stacks to impose tangency conditions on curves
• Quot Functors for Deligne-Mumford Stacks
• Moduli stacks of (phi,Gamma)-modules: a survey
• A compactification of the moduli scheme of abelian varieties
• Brauer groups and quotient stacks
• Uniformity of stably integral points on principally polarized abelian surfaces
• A ratio of integration between quotients in geometric invariant theory