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

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.