9 research outputs found
On the spectrum and support theory of a finite tensor category
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 for determining the exact relationship
between the two support theories, based on of
the cohomology ring of an FTC, .
More specifically, we construct a continuous map from the noncommutative
Balmer spectrum of an FTC, , to the of the categorical
center , and prove that this map is surjective
under a weaker finite generation assumption for 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 are classified by the specialization closed subsets of .
We conjecture that both results hold for all FTCs. 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. The majority of our results are proved in the greater generality of
monoidal triangulated categories.Comment: Appendix B has been revised from the prior version after considering
comments from Greg Stevenso
Catenarity in quantum nilpotent algebras
In this paper, it is established that quantum nilpotent algebras (also known
as CGL extensions) are catenary, i.e., all saturated chains of inclusions of
prime ideals between any two given prime ideals have the same
length. This is achieved by proving that the prime spectra of these algebras
have normal separation, and then establishing the mild homological conditions
necessary to apply a result of Lenagan and the first author. The work also
recovers the Tauvel height formula for quantum nilpotent algebras, a result
that was first obtained by Lenagan and the authors through a different
approach.Comment: 11 page