On the spectrum and support theory of a finite tensor category

Finite tensor categories (FTCs) \(\bf T\) 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 \(\underline{\bf T}\). In this paper we introduce the key notion of the categorical center \(C^\bullet_{\underline{\bf T}}\) of the cohomology ring \(R^\bullet_{\underline{\bf T}}\) of an FTC, \(\bf T\). This enables us to put forward a complete and detailed program for determining the exact relationship between the two support theories, based on \(C^\bullet_{\underline{\bf T}}\) of the cohomology ring \(R^\bullet_{\underline{\bf T}}\) of an FTC, \(\bf T\). More specifically, we construct a continuous map from the noncommutative Balmer spectrum of an FTC, \(\bf T\), to the \(\text{Proj}\) of the categorical center \(C^\bullet_{\underline{\bf T}}\), and prove that this map is surjective under a weaker finite generation assumption for \(\bf T\) than the one conjectured by Etingof-Ostrik. Under stronger assumptions, we prove that (i) the map is homeomorphism and (ii) the two-sided thick ideals of \(\underline{\bf T}\) are classified by the specialization closed subsets of \(\text{Proj} C^\bullet_{\underline{\bf T}}\). We conjecture that both results hold for all FTCs. Many examples are presented that demonstrate how in important cases \(C^\bullet_{\underline{\bf T}}\) arises as a fixed point subring of \(R^\bullet_{\underline{\bf T}}\) and how the two-sided thick ideals of \(\underline{\bf T}\) are determined in a uniform fashion. The majority of our results are proved in the greater generality of monoidal triangulated categories.