On the spectrum and support theory of a finite tensor category
Name
208_2023_2759_ReferencePDF.pdf
Size
603.77 KB
Format
Adobe PDF
Checksum (MD5)
2d9d3bb494e00e2f3405ca66b55d7f05
Author(s)
Nakano, Daniel K.
Vashaw, Kent B.
Yakimov, Milen T.
Date Issued
November 24, 2023
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Citation
Nakano, D.K., Vashaw, K.B. & Yakimov, M.T. On the spectrum and support theory of a finite tensor category. Math. Ann. 390, 205–254 (2024).
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00208-023-02759-8
