Singularity categories of derived categories of hereditary algebras are derived categories

Kimura, Yuta

doi:10.1016/j.jpaa.2019.06.013

Cited by 7 publications

(4 citation statements)

References 20 publications

(16 reference statements)

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“…In [Ki,Lemma 2.5(a)], it was shown that mod 2 E is closed under summands in Mod E. Note that C 2 (E) is equal to the intersection of two subcategories of mod 2 E:…”

Section: 2mentioning

confidence: 99%

Classifications of exact structures and Cohen–Macaulay-finite algebras

Enomoto

2018

Advances in Mathematics

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We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the Grothendieck group of such a category is generated by AR conflations. Moreover, we obtain an explicit classification of (1) Gorensteinprojective-finite Iwanaga-Gorenstein algebras, (2) Cohen-Macaulay-finite orders, and more generally, (3) cotilting modules U with ⊥ U of finite type. In the appendix, we develop the AR theory of exact categories over a noetherian complete local ring, and relate the existence of AR conflations to the AR duality and dualizing varieties.

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“…In [Ki,Lemma 2.5(a)], it was shown that mod 2 E is closed under summands in Mod E. Note that C 2 (E) is equal to the intersection of two subcategories of mod 2 E:…”

Section: 2mentioning

confidence: 99%

Classifications of exact structures and Cohen–Macaulay-finite algebras

Enomoto

2018

Advances in Mathematics

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“…By applying (−)⊗ A A/I to a bimodule resolution of A, A/I has a finite projective resolution by finitely generated projective A-modules. This implies that A I has a projective resolution with finitely generated projective A-modules by Schanuel's lemma, using, for example [24,Lemma 2.5]. By symmetry also I A has a projective resolution with finitely generated projective right A-modules.…”

mentioning

confidence: 91%

On deformed preprojective algebras

Crawley-Boevey¹,

Kimura²

2021

Preprint

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Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry. In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed preprojective algebras for extended Dynkin quivers.

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“…The equivalence (4) is given in [48] (see also [34, 4.11]) for the very special case

\Lambda

is hereditary. We have two independent strategies to build the triangle equivalence above.…”

Section: Introductionmentioning

confidence: 99%

Yoneda algebras and their singularity categories

Hanihara

2022

Proceedings of London Math Soc

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For a finite dimensional algebra Λ of finite representation type and an additive generator 𝑀 for mod Λ, we investigate the properties of the Yoneda algebra Γ = ⨁ 𝑖⩾0 Ext 𝑖 Λ (𝑀, 𝑀). We show that Γ is graded coherent and Gorenstein of self-injective dimension at most 1, and the graded singularity category D ℤ sg (Γ) of Γ is triangle equivalent to the derived category of the stable Auslander algebra of Λ. These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category  of Λ as the additive closure of the shifts of the Λ-modules in the derived category D b (mod Λ). We show that  is coherent and Gorenstein of self-injective dimension at most 1, and the singularity category of  is triangle equivalent to the derived category D b (mod(mod Λ)) of the stable category mod Λ.To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an f-category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.

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Singularity categories of derived categories of hereditary algebras are derived categories

Cited by 7 publications

References 20 publications

Classifications of exact structures and Cohen–Macaulay-finite algebras

Classifications of exact structures and Cohen–Macaulay-finite algebras

On deformed preprojective algebras

Yoneda algebras and their singularity categories

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