Noncommutative tensor triangular geometry: classification via noetherian spectra

TL;DR

This paper establishes a classification of thick two-sided semiprime ideals in monoidal triangulated categories with noetherian spectra through Thomason subsets of their non-commutative spectrum, providing a new geometric perspective.

Contribution

It introduces a bijection between Thomason subsets and semiprime ideals in non-commutative spectra, generalizing previous approaches and showing the spectrum is a universal spectral space.

Findings

Bijection between Thomason subsets and semiprime ideals.

Spectrum is a spectral space under noetherian assumption.

Provides an alternative framework to existing classifications.

Abstract

Given a monoidal triangulated category $T$ with noetherian spectrum, we show that there is an order preserving bijection between the collection of all Thomason subsets of the non-commutative spectrum $\mathrm{Spc}(T)$ and the collection of all thick two-sided semiprime ideals of $T$. This provides an alternative to the hypotheses of Nakano, Vashaw and Yakimov, as well as the recent approach via completely prime ideals of Mallick and Ray. By assuming the spectrum is noetherian, we show that it is indeed a spectral space, and that it is universal among all such spaces classifying the ideals in question.