Affine weakly regular tensor triangulated categories

Ivo Dell’Ambrogio and Donald Stanley

Vol. 285 (2016), No. 1, 93–109

DOI: 10.2140/pjm.2016.285.93

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.

Keywords

tensor triangulated category, thick subcategory, localizing subcategory, spectrum

Mathematical Subject Classification 2010

Primary: 18E30

Secondary: 55P42, 55U35

Milestones

Received: 16 November 2015

Revised: 9 May 2016

Accepted: 9 May 2016

Published: 27 September 2016