Artin algebraization and 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.