Étale dévissage, descent and pushouts of stacks

Abstract

We show that the pushout of an étale morphism and an open immersion exists in the category of algebraic stacks and show that such pushouts behave similarly to the gluing of two open substacks. For example, quasi-coherent sheaves on the pushout can be described by a simple gluing procedure. We then outline a powerful dévissage method for representable étale morphisms using such pushouts. We also give a variant of the dévissage method for representable quasi-finite flat morphisms.

34 pages; added more references; final version

References in corpus (3)

• Noetherian approximation of algebraic spaces and stacks
• Representability of Hilbert schemes and Hilbert stacks of points
• The canonical embedding of an unramified morphism in an étale morphism

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• Functorial destackification of tame stacks with abelian stabilisers
• Algebraic groups and compact generation of their derived categories of representations
• Hyperdescent and étale K-theory
• The étale local structure of algebraic stacks
• Approximation of sheaves on algebraic stacks
• An improvement on the base-change theorem and the functor $f^{!}$
• K-theory and G-theory of derived algebraic stacks
• Canonical reduction of stabilizers for Artin stacks with good moduli spaces
• Toroidal orbifolds, destackification, and Kummer blowings up
• Mayer-Vietoris squares in algebraic geometry
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• The Balmer spectrum of a tame stack
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• A generalization of Luna's fundamental lemma for stacks with good moduli spaces
• Separated quotients of Picard schemes
• Recasting results in equivariant geometry: affine cosets, observable subgroups and existence of good quotients