Artin algebraization and quotient stacks

TL;DR

This paper provides an expository overview of Artin's approximation and algebraization theorems, and demonstrates that under certain conditions, algebraic stacks are etale locally quotient stacks near points with linearly reductive stabilizers.

Contribution

It offers a detailed exposition of Artin's theorems using desingularization techniques and establishes conditions under which algebraic stacks are locally quotient stacks.

Findings

Artin's approximation and algebraization theorems are explained using desingularization.

Algebraic stacks are etale locally quotient stacks near points with linearly reductive stabilizers.

The paper clarifies the local structure of algebraic stacks in the context of quotient stacks.

Abstract

This article contains a slightly expanded version of the lectures given by the author at the summer school "Algebraic stacks and related topics" in Mainz, Germany from August 31 to September 4, 2015. The content of these lectures is purely expository and consists of two main goals. First, we provide a treatment of Artin's approximation and algebraization theorems following the ideas of Conrad and de Jong which rely on a deep desingularization result due to Neron and Popescu. Second, we prove that under suitable hypotheses, algebraic stacks are etale locally quotients stacks in a neighborhood of a point with a linearly reductive stabilizer.