The Balmer spectrum of a tame stack

TL;DR

This paper classifies the thick tensor ideals of perfect complexes on certain algebraic stacks and computes their Balmer spectra, especially focusing on tame stacks and their coarse spaces.

Contribution

It provides a classification of thick tensor ideals and computes the Balmer spectrum for tame stacks, linking it to their coarse spaces.

Findings

Classification of thick tensor ideals for quasi-compact algebraic stacks.

Explicit computation of Balmer spectrum for tame stacks.

Isomorphism of Balmer spectra between a stack and its coarse space.

Abstract

Let $X$ be a quasi-compact algebraic stack with quasi-finite and separated diagonal. We classify the thick $\otimes$-ideals of $\mathsf{D}_{\mathrm{qc}}(X)^c$. If $X$ is tame, then we also compute the Balmer spectrum of the $\otimes$-triangulated category of perfect complexes on $X$. In addition, if $X$ admits a coarse space $X_{\mathrm{cs}}$, then we prove that the Balmer spectra of $X$ and $X_{\mathrm{cs}}$ are naturally isomorphic.