Good Moduli Spaces in Derived Algebraic Geometry

TL;DR

This paper extends the classical theory of good moduli spaces to derived Artin stacks, establishing foundational results and applications like derived étale slice theorems and desingularization procedures.

Contribution

It develops a comprehensive theory of good moduli spaces in derived algebraic geometry, generalizing classical results and demonstrating their applicability in the derived context.

Findings

Many classical properties carry over to the derived setting.

Under natural assumptions, derived theory reduces to classical theory.

Derived versions of étale slice theorem and desingularization are established.

Abstract

We develop a theory of good moduli spaces for derived Artin stacks, which naturally generalizes the classical theory of good moduli spaces introduced by Alper. As such, many of the fundamental results and properties regarding good moduli spaces for classical Artin stacks carry over to the derived context. In fact, under natural assumptions, often satisfied in practice, we show that the derived theory essentially reduces to the classical theory. As applications, we establish derived versions of the \'{e}tale slice theorem for good moduli spaces and the partial desingularization procedure of good moduli spaces.