Affine weakly regular tensor triangulated categories

TL;DR

This paper establishes a homeomorphism between the Balmer spectrum of certain tensor triangulated categories and the Zariski spectrum of their graded central rings, leading to classification results and extensions of the telescope conjecture.

Contribution

It proves a new spectral homeomorphism under weak regularity conditions and extends classification and telescope conjecture results to a broader setting.

Findings

Homeomorphism between Balmer and Zariski spectra for certain categories

Classification of thick and localizing subcategories

Extension of the telescope conjecture

Abstract

We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.