Spectra of tensor triangulated categories over category algebras

TL;DR

This paper computes the spectra of tensor triangulated categories derived from category algebras over finite EI categories and transporter categories, aiding classification of tensor ideal subcategories.

Contribution

It provides explicit computations of spectra for these categories, extending Balmer’s spectrum theory to non-rigid tensor triangulated categories.

Findings

Computed spectrum of D^b(kC-mod) for finite EI categories.

Determined spectrum of stable Cohen-Macaulay modules over Gorenstein category algebras.

Classified tensor ideal thick subcategories using the spectra.

Abstract

Let C be a finite EI category and k be a field. We consider the category algebra kC. Suppose K(C)=D^b(kC-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 C=G*P is a finite transporter category, the category algebra becomes Gorenstein so we can define the stable module category \CM k(G*P), of maximal Cohen-Macaulay modules, as a quotient category of K(G*P). Since \CM k(G*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, despite the fact that the previously mentioned tensor categories are not rigid.