| Summary: | 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.
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