The Balmer spectrum of certain Deligne-Mumford stacks

TL;DR

This paper characterizes the Balmer spectrum of perfect complexes on certain Deligne-Mumford stacks as the space of homogeneous prime ideals in the group cohomology ring, linking algebraic geometry and cohomology.

Contribution

It establishes a homeomorphism between the Balmer spectrum of perfect complexes on quotient stacks and the homogeneous prime spectrum of group cohomology.

Findings

Identifies the Balmer spectrum with the homogeneous prime spectrum of $H^*(G,A)$

Provides a geometric description of the spectrum for quotient stacks

Connects tensor triangulated categories with group cohomology

Abstract

We consider a Deligne-Mumford stack $X$ which is the quotient of an affine scheme $\operatorname{Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.