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.