Extensions of covariantly finite subcategories revisited

He, Jing

doi:10.21136/cmj.2018.0338-17

“…Moreover, from Lemma 3.10 and the claim above, it follows that F(Θ) = * t i=1 X . Hence this result follows from [H,Theorem 3.3] and its dual.…”

Section: Filtered Objectsmentioning

confidence: 70%

“…The proof we give here uses the extriangulated version of Gentle-Todorov's theorem due to He [H,Theorem 3.3]. More precisely, let (C, E, s) be an extriangulated category with enough projectives, X and Y be two covariantly finite subcategories of C. Then X * Y is a covariantly finite subcategory of C. Theorem 3.11.…”

Section: Therefore We Obtain λmentioning

confidence: 99%

Tilting subcategories in extriangulated categories

Zhu

¹

,

Zhuang

²

2020

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Let R be an artin ring and Θ = {Θ(1), Θ(2), · · · , Θ(n)} be a family of objects in an artin extriangulated R-category (C, E, s) such that E(Θ(j), Θ(i)) = 0 for all j ≥ i. In this paper, we show that the class P(Θ) of the Θ-projective objects is a precovering class and the class I(Θ) of the Θ-injective objects is a preenveloping one in C. Furthermore, if C has enough projectives and enough injectives, we show that the subcategory F(Θ) of Θ-filtered objects is functorially finite in C. As an appliacation, this generalizes the works by Ringel in a module category case and Mendoza-Santiago in a triangulated category case.

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“…. By Proposition 3.17(2), X is covariantly finite in C. Since C has enough projectives, it follows from [H,Theorem 3.3] that T 1 * X is covariantly finite in C, and hence it is functorially finite in C. Similarly, we can show that Y * F 2 is functorially finite in C. Hence Ψ is well-defined. This finishes the proof.…”

Section: 2mentioning

confidence: 78%

Intervals of $s$-torsion pairs in extriangulated categories with negative first extensions

Adachi¹,

Enomoto²,

Tsukamoto³

2021

Preprint

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As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalizes hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: One is the bijection induced by HRS-tilt of t-structures on triangulated categories. The other is Asai-Pfeifer's and Tattar's bijections for torsion pairs in an abelian category, which is related to τ -tilting reduction and brick labeling.Recently, inspired by τ -tilting theory [AIR], the study of the poset structures of torsion pairs in abelian categories has been developed by various authors [IRTT, BCZ, DIRRT, AP]. Let t 1 := (T 1 , F 1 ) and t 2 := (T 2 , F 2 ) be torsion pairs in an abelian category A. We define t 1 ≤ t 2 if it satisfies T 1 ⊆ T 2 , and in this case, [t 1 , t 2 ] denotes the interval in the poset of torsion pairs in A consisting of t with t 1 ≤ t ≤ t 2 . We call the subcategorywhich tells us "the difference" between t 1 and t 2 . As for this heart, the following isomorphism was established in [AP] (when the heart is a wide subcategory) and [T].Theorem 1.2. [T, Theorem A] Let A be an abelian category and [t 1 , t 2 ] an interval in the poset of torsion pairs in A. Then there exists a poset isomorphism between [t 1 , t 2 ] and the poset of torsion pairs inThis isomorphism originally appeared in the context of τ -tilting reduction in [Ja], and is a useful tool to study various subcategories of module categories, e.g. [DIRRT, AP, ES].The aim of this paper is to show that two poset isomorphisms in Theorem 1.1 and Theorem 1.2 are consequences of a more general poset isomorphism in extriangulated categories. We

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Intervals of s-torsion pairs in extriangulated categories with negative first extensions

Adachi

¹

,

Enomoto²,

Tsukamoto³

2022

Math. Proc. Camb. Phil. Soc.

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As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to $\tau$ -tilting reduction and brick labelling.

show abstract

Extensions of covariantly finite subcategories revisited

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Cited by 7 publications

References 9 publications

Tilting subcategories in extriangulated categories

Tilting subcategories in extriangulated categories

Intervals of $s$-torsion pairs in extriangulated categories with negative first extensions

Intervals of s-torsion pairs in extriangulated categories with negative first extensions

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