Re-framing the classification of ideals in noncommutative tensor-triangular geometry

Abstract

We prove that, given the Balmer spectrum of any essentially small monoidal-triangulated category, one has a classification of semiprime thick tensor-ideals arising in terms of a "pseudo-Hochster-dual" of the noncommutative Balmer spectrum. This extends Balmer's classification of radical thick tensor-ideals to noncommutative tensor-triangular geometry. To achieve this, we utilize the notion of support data for lattices and frames, under which the classification follows via Stone duality. We also give a characterization for when the noncommutative Balmer spectrum behaves as it does in tensor-triangular geometry, that is, when it is a spectral space with quasi-compact opens given by complements of supports. Finally, we show that rigid centrally generated monoidal-triangulated categories satisfy this property, and we answer a question posed by Negron--Pevtsova regarding classification of one-sided tensor-ideals via cohomological support.