Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts

TL;DR

This paper develops a variant of Artin algebraization for pairs with applications to local structure theorems of stacks, derived stacks, and Ferrand pushouts, answering a question of Temkin-Tyomkin.

Contribution

It introduces a new Artin algebraization variant for pairs with applications to local structure theorems and Ferrand pushouts, advancing the understanding of stacks and their neighborhoods.

Findings

Existence of étale, smooth, or syntomic neighborhoods of closed substacks

Local structure theorems for stacks and derived stacks

Positive solution to the existence of Ferrand pushouts

Abstract

We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of \'etale, smooth, or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin-Tyomkin.