Summary: LOCAL PROPERTIES OF GOOD MODULI SPACES
ABSTRACT. We study the local properties of Artin stacks and their good moduli spaces, if
they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks
formally locally admit good moduli spaces. In particular, the geometric invariant theory is
developed for actions of linearly reductive group schemes on formal affine schemes. We
also give conditions for when the existence of good moduli spaces can be deduced from
the existence of ´etale charts admitting good moduli spaces.
We address the question on whether good moduli spaces for an Artin stack can be
constructed "locally." The main results of this paper are: (1) good moduli spaces exist for-
mally locally around points with linearly reductive stabilizer and (2) sufficient conditions
are given for the Zariski-local existence of good moduli spaces given ´etale-local existence.
We envision that these results may be of use to construct moduli schemes of Artin stacks
without the classical use of geometric invariant theory and semi-stability computations.
The notion of a good moduli space was introduced in [Alp08] to associate a scheme or
algebraic space to Artin stacks with nice geometric properties reminiscent of Mumford's
good GIT quotients. While good moduli spaces cannot be expected to distinguish be-
tween all points of the stack, they do parameterize points up to orbit closure equivalence.
See section 2 for the precise definition of a good moduli space and for a summary of its