Classifications of exact structures and Cohen–Macaulay-finite algebras

Enomoto, Haruhisa

doi:10.1016/j.aim.2018.07.022

Cited by 27 publications

(37 citation statements)

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“…In addition, since we have Ext 1 F = F ⊂ E = Ext 1 E and s F coincides with the restriction of s, we obtain F ⊂ E. By abuse of notation, for an exact structure E on a skeletally small additive category C, we put def E := def E E , that is, def E is the subcategory of Mod C consisting of those M for which there is an E-deflation B y − → C wich M ∼ = Coker C(−, y). This category was used in [3] to classify all possible exact structures. By Proposition 2.9, it is a Serre subcategory of the abelian category coh E, and is hence an abelian category.…”

Section: Smentioning

confidence: 99%

“…In [3], it was assumed that C is idempotent complete, but we do not need this assumption in this paper. Actually, Corollary 4.3 cannot be used to classify the exact structures on C without explicit knowledge of E max or def E max , hence we cannot deduce the main result of [3].…”

Section: Remark 44mentioning

confidence: 99%

“…The poset of exact structures was also studied in [1]. This corollary was recently obtained by [4,Theorem 2.8] under the assumption of idempotent completeness, whose proof uses results in [3]. Our proof uses Theorem B, is a rather simple, and does not use any results in [3].…”

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confidence: 93%

“…For exact categories, the author gave in [3] a classification of possible exact structures on a given idempotent complete additive category C. More precisely, for an exact structure E on an idempotent complete additive category C, we can associate the category of defects def E, which is a subcategory of the category Mod C of C-modules. Then it was shown in [3] that this gives a bijection between the set of possible exact structures on C and the set of Serre subcategories of the category of finitely generated C-modules satisfying an additional 2-regularity condition.…”

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confidence: 99%

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Classifying Substructures of Extriangulated Categories via Serre Subcategories

Enomoto

2021

Appl Categor Struct

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We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a similar way to the author's classification of exact structures of a given additive category. More precisely, for an extriangulated category, possible substructures are in bijection with Serre subcategories of an abelian category consisting of defects of conflations. As a byproduct, we prove that for a given skeletally small additive category, the poset of exact structures on it is isomorphic to the poset of Serre subcategories of some abelian category. IntroductionRecently, Nakaoka and Palu [9] introduced an extriangulated category as a simultaneous generalization of exact categories and triangulated categories. An extriangulated category consists of a triple (C, E, s), where C is an additive category, E is an additive bifunctor Communicated by Henning Krause.

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Section: Smentioning

confidence: 99%

Section: Remark 44mentioning

confidence: 99%

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confidence: 93%

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confidence: 99%

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Classifying Substructures of Extriangulated Categories via Serre Subcategories

Enomoto

2021

Appl Categor Struct

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“…It has been shown in [DRSS] that these two concepts coincide, that is, the additive closed subbifunctors correspond to exact structures. Recently, exact structures have become focus of work by several authors, like [En18] who classifies exact structures on a given Krull-Schmidt No matter which formalism one chooses, the iterated application of reductions leads to more and more complicated categories. We propose a different approach in this paper, that is: Keep the objects of the original category, but change its exact structure.…”

Section: Introductionmentioning

confidence: 99%

Reduction of exact structures

Brüstle¹,

Hassoun²,

Langford³

et al. 2020

Journal of Pure and Applied Algebra

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Examples of exact categories in representation theory are given by the category of ∆−filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset (Ex(A), ⊂) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean.We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A, E). We investigate the Gabriel-Roiter measure for (A, E), and further study how these notions change when the exact structure varies. category of finite type, using Auslander algebra, or [INP18] where the more general concept of extriangulated structures is studied.While every exact category (A, E) can be embedded into a module category, notions like length or simple object cannot be borrowed from such an embedding. The first goal of this paper is to give an intrinsic definition relative to the class of morphisms in E, thus, in section 3, we call an object E−simple if it does not admit proper monomorphisms that belong to the class E. And we say that X is anThis change of definition requires to work out a number of notions and results that are granted in abelian categories, such as the notion of simple objects, artinian and finite objects or the length of an object. It turns out that in general not all the desired properties can be guaranteed. We also define, in section 3, the notion of the quiver of an exact category Q(A, E).The motivation for studying reductions of exact structures stems from the matrix reduction technique. The method of matrix reduction has been applied successfully by the Kiev school to solve various important problems in representation theory, like the Brauer-Thrall conjectures, or to show the tame-wild dichotomy. While the basic technique is elementary, the formalism of matrix reductions is somewhat complicated. Various models have been proposed to formalise matrix reductions: poset representations or bimodule problems cover only some cases. For the general case, one needs to study bocs representations, as introduced by Roiter in [Ro79], or iterated quotients of bimodule problems as in [Brü].This result allows to show that the length function l E of a finite essentially small exact category (A, E) is a measure for the poset ObjA in the sense of Gabriel [Gab]. We further show that most of the work of Krause [Kr11] on the Gabriel-Roiter measure for abelian length categories can be generalized to the context of exact categories: For the partially ordered set (indA, ⊂ E ) equipped with the length function l E , we define the Gabriel-Roiter measure as a morphism of partially ordered sets which refines the length function l E , see Theorem 7...

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The Jordan-Hölder property and Grothendieck monoids of exact categories

Enomoto

2022

Advances in Mathematics

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Classifications of exact structures and Cohen–Macaulay-finite algebras

Cited by 27 publications

References 40 publications

Classifying Substructures of Extriangulated Categories via Serre Subcategories

Classifying Substructures of Extriangulated Categories via Serre Subcategories

Reduction of exact structures

The Jordan-Hölder property and Grothendieck monoids of exact categories

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