A generalization of Luna's fundamental lemma for stacks with good moduli spaces

TL;DR

This paper extends Luna's fundamental lemma to a broader class of stacks with good moduli spaces, including non-smooth morphisms and complexes, generalizing previous results in the field.

Contribution

It provides a generalized version of Luna's fundamental lemma applicable to smooth and certain non-smooth morphisms between stacks with good moduli spaces, including coherent sheaves and complexes.

Findings

Generalization of Luna's fundamental lemma to smooth morphisms

Conditions for non-smooth morphisms

Extensions to coherent sheaves and complexes

Abstract

We generalize Luna's fundamental lemma to smooth morphisms between stacks with good moduli spaces. We also give a precise condition for when it holds for non-smooth morphisms and versions for coherent sheaves and complexes. This generalizes earlier results by Alper, Abramovich--Temkin, Edidin and Nevins.