On the spectrum and support theory of a finite tensor category

Nakano, Daniel K.; Vashaw, Kent B.; Yakimov, Milen T.

doi:10.1007/s00208-023-02759-8

Cited by 3 publications

(7 citation statements)

References 44 publications

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“…by Lemma 3.2.2(a) and the fact that L I 2 (A) ∈ Loc(I). It follows L I 2 Γ I 1 (A) ∼ = L I 2 (A) in K. Now we also have the distinguished triangle [33,Lemma B.0.2]), we have an isomorphism Γ I 1 (A) ∼ = L I 2 Γ I 1 (A) ∼ = L I 2 (A) in the Verdier quotient K/ Loc(I 1 ∩ I 2 ); in this Verdier quotient, hence have the distinguished triangle…”

Section: A General Version Of Carlson's Theoremmentioning

confidence: 85%

“…Then we have a map Proj C • K η − → Spc K such that ρη = id, see[33, Proposition 8.1.1]. We obtain a version of the Carlson theorem for central cohomological support.Corollary 7.3.1.…”

mentioning

confidence: 87%

“…This section contains background material on monoidal triangulated categories, thick ideals, the noncommutative Balmer spectrum and localization functors. For details the reader is referred to [31,32,33].…”

Section: Background On Monoidal Triangulated Categoriesmentioning

confidence: 99%

“…see [33,Section 1.4]. The central cohomological support variety of an object A ∈ K is then defined to be…”

Section: 2mentioning

confidence: 99%

“…If A is an indecomposable object, then W C (A) is connected.Proof. We claim that ρ(V (A)) = W C (A): on one hand, we have ρ(V (A)) ⊆ W C (A) by [33, Corollary 6.2.5], and on the other hand, if p ∈ W C (A), then η(p) ∈ V (A) by[33, Section 8.1. ], and ρ(η(p)) = p. By Theorem 7.2.1, V (A) is connected; since W C (A) is a continuous image of V (A), it follows that W C (A) is connected as well.…”

mentioning

confidence: 99%

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Noncommutative tensor triangular geometry

Nakano¹,

Vashaw²,

Yakimov³

2022

American Journal of Mathematics

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In this paper the authors prove fundamental decomposition theorems pertaining to the internal structure of monoidal triangulated categories (M∆Cs). The tensor structure of an M∆C enables one to view these categories like (noncommutative) rings and to attempt to extend the key results for the latter to the categorical setting. The main theorem is an analogue of the Chinese Remainder Theorem involving the Verdier quotients for coprime thick ideals. This result is used to obtain orthogonal decompositions of the extended endomorphism rings of idempotent algebra objects of M∆Cs. The authors also provide topological characterizations on when an M∆C contains a pair of coprime proper thick ideals, and additionally, when the latter are complementary in the sense that their intersection is contained in the prime radical of the category.As an application of the aforementioned results, the authors establish for arbitrary M∆Cs a general version of Carlson's theorem on the connnectedness of supports for indecomposable objects. Examples of our results are given at the end of the paper for the derived category of schemes and for the stable module categories for finite group schemes.

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Section: A General Version Of Carlson's Theoremmentioning

confidence: 85%

mentioning

confidence: 87%

Section: Background On Monoidal Triangulated Categoriesmentioning

confidence: 99%

“…see [33,Section 1.4]. The central cohomological support variety of an object A ∈ K is then defined to be…”

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

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Noncommutative tensor triangular geometry

Nakano¹,

Vashaw²,

Yakimov³

2022

American Journal of Mathematics

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Support varieties without the tensor product property

Bergh,

Plavnik,

Witherspoon

2024

Bulletin of London Math Soc

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Balmer spectra and Drinfeld centers

Vashaw

2024

Alg. Number Th.

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The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category C to C extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson-Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general.

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On the spectrum and support theory of a finite tensor category

Cited by 3 publications

References 44 publications

Noncommutative tensor triangular geometry

Noncommutative tensor triangular geometry

Support varieties without the tensor product property

Balmer spectra and Drinfeld centers

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