Decomposition theorem for good moduli morphisms

TL;DR

This paper explains that good moduli space morphisms, despite not being separated, behave similarly to proper morphisms in sheaf operations, enabling key theorems like decomposition and base change to apply, aiding cohomological studies.

Contribution

It demonstrates that good moduli space morphisms satisfy properties akin to proper morphisms, allowing important theorems to hold in this context.

Findings

Decomposition theorem applies to good moduli morphisms.

Base change theorem holds for these morphisms.

Applications to cohomology of moduli spaces.

Abstract

In this short note, we will explain that the good moduli space morphisms behave as if they are proper when we consider sheaf operations, though they are not separated. For example, the decomposition theorem and the base change theorem hold for these morphisms, which have applications to the cohomological study of moduli spaces.