\'Etale d\'evissage, descent and pushouts of stacks

TL;DR

This paper establishes the existence of pushouts in algebraic stacks for étale morphisms and open immersions, enabling a gluing approach for sheaves and a de9vissage method for certain morphisms.

Contribution

It introduces a new construction of pushouts in algebraic stacks for étale morphisms and open immersions, facilitating gluing techniques and de9vissage methods for complex morphisms.

Findings

Pushouts of étale morphisms and open immersions exist in algebraic stacks.

Quasi-coherent sheaves on these pushouts can be described by simple gluing.

A de9vissage method for representable étale and quasi-finite flat morphisms is outlined.

Abstract

We show that the pushout of an \'etale 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 d\'evissage method for representable \'etale morphisms using such pushouts. We also give a variant of the d\'evissage method for representable quasi-finite flat morphisms.