On the spectrum and support theory of a finite tensor category

TL;DR

This paper explores the relationship between two support theories for finite tensor categories, introducing the categorical center of their cohomology ring to unify and classify their structural properties.

Contribution

It introduces the categorical center of the cohomology ring of a finite tensor category and establishes a detailed program linking the two support theories, including surjectivity and homeomorphism results.

Findings

Constructed a continuous map from the noncommutative Balmer spectrum to the Proj of the categorical center.

Proved surjectivity of the map under weaker finite generation assumptions.

Under stronger assumptions, established the map as a homeomorphism and classified thick ideals.

Abstract

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.