Hypersurface support and prime ideal spectra for stable categories

Negron, Cris; Pevtsova, Julia

doi:10.48550/arxiv.2101.00141

Cited by 3 publications

(14 citation statements)

References 51 publications

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“…Now Theorem 9.1.1 yields the the following: Proposition 9.3.1. [40, Theorem 10.3] For each finite group scheme Ω, Spc mod(k[Ω]) ρ ⇄ Proj H • (k[Ω 0 ], k) π are inverse homeomorphisms and the mapΦ W C : ThickId mod(k[Ω]) → X sp Proj(H • (k[Ω 0 ], k)) πis an isomorphism of ordered monoids.This result recovers a theorem of Negron and Pevtsova[40, Theorem 10.3] who employed their hypersurface support theory. The example shows how it is obtained through a uniform approach based on Theorem D and Theorem 9.1.1.…”

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confidence: 85%

On the spectrum and support theory of a finite tensor category

Nakano¹,

Vashaw²,

Yakimov³

2021

Preprint

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Finite tensor categories (FTCs) 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 T.In this paper we introduce the new notion of the categorical center C • T of the cohomology ring R • T of an FTC, 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 • T of the cohomology ring R • T of an FTC, T. More specifically, we construct a continuous map from the noncommutative Balmer spectrum of an FTC, T, to the Proj of the categorical center C • T , and prove that this map is surjective under a weaker finite generation assumption for 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 T are classified by the specialization closed subsets of Proj C • T . We conjecture that both results hold for all FTCs. Many examples are presented that demonstrate how in important cases C • T arises as a fixed point subring of R • T and how the two-sided thick ideals of T 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.

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supporting

confidence: 85%

On the spectrum and support theory of a finite tensor category

Nakano¹,

Vashaw²,

Yakimov³

2021

Preprint

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“…The proof of Proposition 16.1 is a fairly straightforward application of our analysis of N -support, in particular Theorems 13.1 and 14.1, in conjunction with results from [89]. The proof is provided in Subection 16.7.…”

Section: Covering the Balmer Spectrummentioning

confidence: 94%

“…For the quantum Borel outside of type A, one simply wants to know that cohomological support for u(B q ) classifies thick ideals in the associated stable category. More directly, these "specific conjectures" claim that cohomological support for the small quantum Borel is a lavish support theory, in the language of [89]. Such a result was proved for the quantum Borel in type A in [89], and is expected to hold in general (see Section 17.3).…”

Section: Introductionmentioning

confidence: 92%

“…More directly, these "specific conjectures" claim that cohomological support for the small quantum Borel is a lavish support theory, in the language of [89]. Such a result was proved for the quantum Borel in type A in [89], and is expected to hold in general (see Section 17.3).…”

Section: Introductionmentioning

confidence: 92%

“…In total, we achieve the approximate calculation of the Balmer spectrum P( N ) → Spec(stab u(G q )) appearing in Theorem 17.1 via an application of the naturality result of Theorem 14.1, the analysis of the quantum Borel provided in [89] (where the type A bottleneck occurs), and a projectivity test for infinite-dimensional quantum group representations via the Borels, which we now recall.…”

Section: Introductionmentioning

confidence: 99%

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Support theory for the small quantum group and the Springer resolution

Negron¹,

Pevtsova²

2022

Preprint

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We consider the small quantum group u(Gq), for an almost-simple algebraic group G over the complex numbers and a root of unity q of sufficiently large order. We show that the Balmer spectrum for the small quantum group in type A admits a continuous surjection P( N ) → Spec(stab u(Gq)) from the (projectivized) Springer resolution. This surjection is shown to be a homeomorphism over a dense open subset in the spectrum. In type A 1 we calculate the Balmer spectrum precisely, where it is shown to be the projectivized nilpotent cone. Our results extend to arbitrary Dynkin type provided certain conjectures hold for the small quantum Borel. At the conclusion of the paper we touch on relations with geometric representation theory and logarithmic TQFTs, as represented in works of Arkhipov-Bezrukavnikov-Ginzburg and Schweigert-Woike respectively.

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Hypersurface Support for Noncommutative Complete Intersections

Negron¹,

Pevtsova²

2022

Nagoya Math. J.

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We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.

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Hypersurface support and prime ideal spectra for stable categories

Cited by 3 publications

References 51 publications

On the spectrum and support theory of a finite tensor category

On the spectrum and support theory of a finite tensor category

Support theory for the small quantum group and the Springer resolution

Hypersurface Support for Noncommutative Complete Intersections

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