Noncommutative tensor triangulated categories and coherent frames

TL;DR

This paper introduces a point-free, frame-theoretic approach to classify radical thick tensor ideals in noncommutative tensor triangulated categories, establishing a homeomorphism with the spectrum's open subsets.

Contribution

It develops a novel point-free framework for the spectrum of noncommutative tensor triangulated categories, unifying support data and ideal classification.

Findings

Classifies radical thick tensor ideals using frame theory.

Establishes a homeomorphism with the Hochster dual topology.

Recovers universal support data in the noncommutative setting.

Abstract

We develop a point-free approach for constructing the Nakano-Vashaw-Yakimov-Balmer spectrum of a noncommutative tensor triangulated category under some mild assumptions. In particular, we provide a conceptual way of classifying radical thick tensor ideals of a noncommutative tensor triangulated category using frame theoretic methods, recovering the universal support data in the process. We further show that there is a homeomorphism between the spectral space of radical thick tensor ideals of a noncommutative tensor triangulated category and the collection of open subsets of its spectrum in the Hochster dual topology.