Noetherian approximation of algebraic spaces and stacks

Abstract

We show that every scheme/algebraic space/stack that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme/algebraic space/stack. More generally, we show that any stack which is etale-locally a global quotient stack can be approximated. Examples of applications are generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma to the non-noetherian setting. We also show that every quasi-compact algebraic stack with quasi-finite diagonal has a finite generically flat cover by a scheme.

39 pages; complete overhaul of paper; generalized results and simplified proofs (no groupoid-calculations); added more applications and appendices with standard results on constructible properties and limits for stacks; generalized Thm C (no finite presentation hypothesis); some minor changes in 2,1-2.8, 8.2, 8.8 and 8.9; final version

References in corpus (6)

• Existence and properties of geometric quotients
• Submersions and effective descent of etale morphisms
• Étale dévissage, descent and pushouts of stacks
• Relative Riemann-Zariski spaces
• Representability of Hilbert schemes and Hilbert stacks of points
• The canonical embedding of an unramified morphism in an étale morphism

Cited by in corpus (37)

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• The étale local structure of algebraic stacks
• Approximation of sheaves on algebraic stacks
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• A thick subcategory theorem for modules over certain ring spectra
• General Hilbert stacks and Quot schemes
• Virtual fundamental classes of derived stacks I
• An improvement on the base-change theorem and the functor $f^{!}$
• Enhanced adic formalism and perverse t-structures for higher Artin stacks
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• Experiments on the Brauer map in High Codimension
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• Addendum: Étale dévissage, descent and pushouts of stacks
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• The Nisnevich Motive of an Algebraic Stack
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• Quiver GIT for Varieties with Tilting Bundles
• Nagata compactification for algebraic spaces
• Localizations of tensor categories and fiber products of schemes
• A motivic change of variables formula for Artin stacks
• A generalization of Luna's fundamental lemma for stacks with good moduli spaces
• Good Hilbert functors
• Positivity, plethysm and hyperbolicity of Siegel varieties in positive characteristic
• The moduli space of marked supersingular Enriques surfaces
• The Derived Category of quasi-coherent sheaves on an Artin stack via model structures