Relations for Grothendieck groups and representation-finiteness

Enomoto, Haruhisa

doi:10.1016/j.jalgebra.2019.07.032

Cited by 13 publications

(15 citation statements)

References 14 publications

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“…We recall the necessary definitions in Section 2.3. We formulate and prove explicitly the following statement, again essentially by combining some results of Enomoto [28,29].…”

Section: Introductionmentioning

confidence: 99%

“…Enomoto [28] classified all subcategories of Mod A that arise as categories of defects of exact structures on A. By using some properties of these categories proved by Enomoto in [28,29], we slightly improve on his classification and provide a simpler description of the lattice of exact structures.…”

Section: Introductionmentioning

confidence: 99%

“…He calls an exact category E admissible if any object in the category eff E has finite length. In [29], he further investigated the relation between the admissiblity of an exact category and the structure of its almost split, or Auslander-Reiten conflations. We recall the necessary definitions in Section 2.3.…”

Section: Introductionmentioning

confidence: 99%

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Exact structures and degeneration of Hall algebras

Fang¹,

Gorsky²

2020

Preprint

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We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra H(E) of an exact category E is the Hall algebra of a smaller exact structure E ′ < E on the same additive category A. When E is admissible in the sense of Enomoto, for any E ′ < E satisfying suitable finiteness conditions, we prove that H(E ′ ) is a degeneration of H(E) of this kind.In the additively finite case, all such degree functions form a simplicial cone whose face lattice reflects properties of the lattice of exact structures. For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group, as well as the associated polyhedral structure studied by Fourier, Reineke and the first author.Along the way, we give minor improvements to certain results of Enomoto and Brüstle-Langford-Hassoun-Roy concerning the classification of exact structures on an additive category. We prove that for each idempotent complete additive category A, there exists an abelian category whose lattice of Serre subcategories is isomorphic to the lattice of exact structures on A. We show that every Krull-Schmidt category admits a unique maximal admissible exact structure and that the lattice of smaller exact structures of an admissible exact structure is Boolean.

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“…We recall the necessary definitions in Section 2.3. We formulate and prove explicitly the following statement, again essentially by combining some results of Enomoto [28,29].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exact structures and degeneration of Hall algebras

Fang¹,

Gorsky²

2020

Preprint

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“…A converse to this theorem is given by Auslander for artin algebras [1] and by Hiramatsu in the case of a Gorenstein ring with an isolated singularity [13,Theorem 1.2], where the latter is extended by Kobayashi [15,Theorem 1.2]. Results of the type described above were recently generalized to the setup of exact categories by Enomoto [9] and to certain extriangulated categories by Padrol, Palu, Pilaud and Plamondon [16].…”

Section: Introductionmentioning

confidence: 99%

Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

Haugland¹

2021

Algebr Represent Theor

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We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander–Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.

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“…Let R be a commutative Cohen-Macaulay complete ring. Auslander and Reiten in [4] and Hiramatsu in [15] studied this question for the subcategory of Cohen-Macaulay R-modules of mod R. Enomoto studied this question in the context of Quillen's exact categories [13]. Let C be an essentially small abelian Krull-Schmidt category.…”

Section: Introductionmentioning

confidence: 99%

Relations for Grothendieck groups of $n$-cluster tilting subcategories

Diyanatnezhad¹,

Nasr-Isfahani²

2021

Preprint

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Let Λ be an artin algebra and M be an n-cluster tilting subcategory of mod Λ. We show that M has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of M if and only if every effaceable functor M → Ab has finite length. As a consequence we show that if mod Λ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of Λ.

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Relations for Grothendieck groups and representation-finiteness

Cited by 13 publications

References 14 publications

Exact structures and degeneration of Hall algebras

Exact structures and degeneration of Hall algebras

Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

Relations for Grothendieck groups of $n$-cluster tilting subcategories

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