Re-framing the classification of ideals in noncommutative tensor-triangular geometry
Timothy De Deyn, Sam K. Miller
TL;DR
The paper develops a lattice-theoretic foundation for noncommutative tensor-triangular geometry by recasting the Balmer spectrum in terms of spatial frames and Stone duality. It defines and analyzes the semiprime thick ⊗-ideals lattice T_s(K), proves Ts(K) is a spatial frame, and introduces the principal part t_s(K) to connect compact elements with principality under compact detection and principal closure. Under these conditions, the noncommutative Balmer spectrum Spc(K) behaves like a spectral space, providing a classification of semiprime thick ⊗-ideals via Thomason subsets of Spc(K) in favorable cases, and clarifies when such a classification fails. The framework is extended to group actions through crossed products and applied to answering Negron–Pevtsova’s question on one-sided ideals using cohomological support, with central generation and Noetherianity playing pivotal roles. These results unify lattice-theoretic and support-theoretic approaches to noncommutative tensor-triangular geometry and extend Balmer-type classifications beyond the commutative setting.
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.