On functoriality and the tensor product property in noncommutative tensor-triangular geometry

TL;DR

This paper investigates the properties of support theories in monoidal triangulated categories, establishing a universal prime spectrum that is functorial and supports the tensor product property, with applications to crossed product categories.

Contribution

It introduces the complete prime spectrum as a universal support data, providing criteria for injectivity and surjectivity of induced maps, and characterizes the universal support theory for braided categories.

Findings

Complete prime spectrum is universal among support data with tensor product property.

Criteria established for injectivity and surjectivity of maps on prime spectra.

Determined the spectrum for crossed product categories.

Abstract

Two pertinent questions for any support theory of a monoidal triangulated category are whether it is functorial and if the tensor product property holds. To this end, we consider the complete prime spectrum of an essentially small monoidal triangulated category, which we show is universal among support data satisfying the tensor product property, even if it is empty. The complete prime spectrum is functorial and parametrizes radical thick tensor-ideals, a noncommutative analogue of Balmer's reconstruction theorem. We give criteria for when induced maps on complete prime spectra are injective or surjective, and determine the complete prime spectrum for crossed product categories. Finally, we determine the universal functorial support theory for monoidal triangulated categories coinciding with the Balmer spectrum on braided monoidal triangulated categories.