On the spectrum and support theory of a finite tensor category

Abstract

Finite tensor categories (FTCs) are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras, which play a key role in many areas of mathematics and mathematical physics. There are two fundamentally different support theories for them: a cohomological one and a universal one based on the noncommutative Balmer spectra of their stable (triangulated) categories . In this paper we introduce the key notion of the categorical center of the cohomology ring of an FTC, . This enables us to put forward a complete and detailed program to investigate the relationship between the two support theories, based on of the cohomology ring of an FTC, . Our main result is the construction of a continuous map from the noncommutative Balmer spectrum of an arbitrary FTC, , to the of the categorical center and a theorem that this map is surjective under a weaker finite generation assumption for than the one conjectured by Etingof–Ostrik. We conjecture that, for all FTCs, (i) the map is a homeomorphism and (ii) the two-sided thick ideals of are classified by the specialization closed subsets of . We verify parts of the conjecture under stronger assumptions on the category . Many examples are presented that demonstrate how in important cases arises as a fixed point subring of and how the two-sided thick ideals of are determined in a uniform fashion (while previous methods dealt on a case-by-case basis with case specific methods). The majority of our results are proved in the greater generality of monoidal triangulated categories and versions of them for Tate cohomology are also presented.