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 is a consequence of 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, and show that rigid centrally generated monoidal-triangulated categories satisfy this property.