Étale dévissage, descent and pushouts of stacks

2010/05/31 by David Rydh · 40 citations
Arts and Humanities · Mathematics · #Algebraic Geometry and Number Theory #Descent (aeronautics) #French Literature and Poetry #Geography #Mathematics #North African History and Literature #math.AG #msc:14A20 #msc:14F20 #msc:18A30 #msc:18F20

paper · pdf · doi:10.1016/j.jalgebra.2011.01.006

published in Journal of Algebra 331(1), 194-223 (Elsevier BV) · 34 pages; added more references; final version

openalex publication_date 2011/02/04 · arxiv created 2014/09/18 · arxiv updated 2014/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Première partie
• Éléments de géométrie algébrique : II. Étude globale élémentaire de quelques classes de morphismes
• Algebraization of Formal Moduli: II. Existence of Modifications
• Crit�res de platitude et de projectivit�

Cited by

• Addendum: Étale dévissage, descent and pushouts of stacks
• Compact approximation and descent for algebraic stacks

Related

• Constructing colimits by gluing vector bundles
• Uniqueness of dg-lifts via restriction to injective objects
• Gluing restricted nerves of ∞-categories
• The pro-étale topology for algebraic stacks
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