Spectra of tensor triangulated categories over category algebras

Abstract

Let ${\mathcal{C}}$ be a finite EI category and k be a field. We consider the category algebra ${k\mathcal{C}}$. Suppose ${\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}$ is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When ${\mathcal{C}=G \propto \mathcal{P}}$ is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category ${\underline{\sf{CM}} k(G \propto \mathcal{P})}$, of maximal Cohen–Macaulay modules, as a quotient category of ${{\sf{K}}(G \propto \mathcal{P})}$. Since ${\underline{\sf{CM}} k(G\propto\mathcal{P})}$ is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.