Exact categories

Bühler, Theo

doi:10.1016/j.exmath.2009.04.004

Cited by 330 publications

(474 citation statements)

References 28 publications

Supporting

Mentioning

461

Contrasting

Unclassified

Order By: Relevance

“…Our main reference for this theory is the refined treatment presented in [18, Appendix A] and the systematic study of [7]. We will also use the basic facts about Frobenius categories.…”

Section: Reminders On Dg Categories and Their Orbit Categoriesmentioning

confidence: 99%

On Frobenius (Completed) Orbit Categories

Chávez

2017

Algebr Represent Theor

View full text Add to dashboard Cite

Abstract. Let E be a Frobenius category, P its subcategory of projective objects and F : E → E an exact automorphism. We prove that there is a fully faithful functor from the orbit category E/F into gpr(P/F ), the category of finitely-generated Gorensteinprojective modules over P/F . We give sufficient conditions to ensure that the essential image of E/F is an extension-closed subcategory of gpr(P/F ). If E is in addition KrullSchmidt, we give sufficient conditions to ensure that the completed orbit category E /F is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.

show abstract

“…Our main reference for this theory is the refined treatment presented in [18, Appendix A] and the systematic study of [7]. We will also use the basic facts about Frobenius categories.…”

Section: Reminders On Dg Categories and Their Orbit Categoriesmentioning

confidence: 99%

On Frobenius (Completed) Orbit Categories

Chávez

2017

Algebr Represent Theor

View full text Add to dashboard Cite

show abstract

“…The concept is originally due to Quillen, but the common reference for a simple axiomatic description is [39,Appendix A] and an extensive treatment is given in [19].…”

Section: Exact Categoriesmentioning

confidence: 99%

“…In fact, one easily sees that Proposition 1.2(i) holds in the following more general form: Lemma 1.3. [19,Lemma 10.20] Let G be an exact category and F be a full subcategory which is closed under extensions. Then F together with all conflations in G whose all terms belong to F , is an exact category.…”

Section: Exact Categoriesmentioning

confidence: 99%

See 1 more Smart Citation

On compactly generated torsion pairs and the classification of co-$t$-structures for commutative noetherian rings

Šťovíček

Pospíšil

2015

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We classify compactly generated co-t-structures on the derived category of a commutative noetherian ring. In order to accomplish that, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the literature) in triangulated categories that resembles Bousfield localization theory. Finally, we show that the category of perfect complexes over a connected commutative noetherian ring admits only the trivial co-t-structures and (de)suspensions of the canonical co-t-structure and use this to describe all silting objects in the category.

show abstract

“…The theory of exact categories was developed by Bühler, Fu, Gillespie, Hovey, Keller, Krause, Neeman,Šťovíček and possibly others, see [4,8,10,13,14,15,16,20,21], and so on. In addition, Auslander and Solberg developed in [1,2] the theory of relative homological algebra with respect to a subfunctor F of Ext…”

Section: Introductionmentioning

confidence: 99%

Applications of exact structures in abelian categories

WANG¹,

Li²,

Huang³

2016

Publ. Math. Debrecen

View full text Add to dashboard Cite

In an abelian category A with small Ext groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors F of Ext 1 A (−, −) such that A has enough F-projectives and enough F-injectives and Quillen exact structures E with enough E -projectives and enough E -injectives. In this case, we get a strengthened version of the translation of the Wakamatsu lemma to the exact context, and also prove that subcategories which are E -resolving and epimorphic precovering with kernels in their right E -orthogonal class and subcategories which are E -coresolving and monomorphic preenveloping with cokernels in their left E -orthogonal class are determined by each other. Then we apply these results to construct some (pre)enveloping and (pre)covering classes and complete hereditary E -cotorsion pairs in the module category.

show abstract

Exact categories

Cited by 330 publications

References 28 publications

On Frobenius (Completed) Orbit Categories

On Frobenius (Completed) Orbit Categories

On compactly generated torsion pairs and the classification of co-$t$-structures for commutative noetherian rings

Applications of exact structures in abelian categories

Contact Info

Product

Resources

About