 
Summary: ON THE LOCAL QUOTIENT STRUCTURE OF ARTIN STACKS
JAROD ALPER
ABSTRACT. We show that near closed points with linearly reductive stabilizer, Artin stacks are for
mally locally quotient stacks by the stabilizer. We conjecture that the statement holds ´etale locally
and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a
point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space, gen
eralizing the results of Pinkham and Rim. We provide a generalization and stacktheoretic proof of
Luna's ´etale slice theorem which shows that GIT quotient stacks are ´etale locally quotients stacks by
the stabilizer.
1. INTRODUCTION
This paper is motivated by the question of whether an Artin stack is "locally" near a point a
quotient stack by the stabilizer at that point. While this question may appear quite technical in
nature, we hope that a positive answer would lead to intrinsic constructions of moduli schemes
parameterizing objects with infinite automorphisms (e.g. vector bundles on a curve) without the
use of classical geometric invariant theory.
We restrict ourselves to studying Artin stacks X over a base S near closed points X with
linearly reductive stabilizer.
We conjecture that this question has an affirmative answer in the ´etale topology. Precisely,
Conjecture 1. If X is an Artin stack finitely presented over an algebraic space S and X is a
closed point with linearly reductive stabilizer with image s S, then there exists an ´etale neighborhood
