Abelian subcategories of triangulated categories induced by simple minded systems

Jørgensen, Peter

doi:10.48550/arxiv.2010.11799

Cited by 4 publications

(5 citation statements)

References 5 publications

(5 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 3.20. In Corollary 3.19, when C is a triangulated category, it is just the main theorem of [D] and it was also rediscovered in [J,Proposition 2.5].…”

Section: Resultsmentioning

confidence: 99%

$n$-exact categories arising from $n$-exangulated categories

He¹,

Zhou²

2021

Preprint

View full text Add to dashboard Cite

Let C be a Krull-Schmidt n-exangulated category and A be an n-extension closed subcategory of C . Then A inherits the n-exangulated structure from the given n-exangulated category in a natural way. This construction gives n-exangulated categories which are not n-exact categories in the sense of Jasso nor (n + 2)-angulated categories in the sense of Geiss-Keller-Oppermann in general. Furthermore, we also give a sufficient condition on when an n-exangulated category A is an n-exact category. These results generalize work by Klapproth and Zhou.

show abstract

“…Remark 3.20. In Corollary 3.19, when C is a triangulated category, it is just the main theorem of [D] and it was also rediscovered in [J,Proposition 2.5].…”

Section: Resultsmentioning

confidence: 99%

$n$-exact categories arising from $n$-exangulated categories

He¹,

Zhou²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…4.2]. Note that the proof only requires the assumptions made here, not the stronger blanket assumptions of [14]. Proof.…”

Section: Lemmas On Proper Abelian Subcategoriesmentioning

confidence: 96%

“…Indeed, the heart of a t-structure has all negative self extensions equal to zero, but this property fails for large classes of proper abelian subcategories arising in practice. For instance, [14,Introduction] provides an explanation of this in the setting of negative cluster categories as developed in [4], [5], [6], [7], [11], [13].…”

Section: Introductionmentioning

confidence: 99%

Proper abelian subcategories of triangulated categories and their tilting theory

Jorgensen

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In the theory of triangulated categories, we propose to replace hearts of t-structures by proper abelian subcategories, which may be plentiful even when hearts are not. For instance, this happens in negative cluster categories.In support of our proposal, we show that proper abelian subcategories with a few vanishing negative self extensions permit a tilting theory which is a direct generalisation of Happel-Reiten-Smalø tilting of hearts.

show abstract

“…As a byproduct, we give the following criterion for an extension-closed subcategory of a triangulated category to be an exact category, which is proved in [Jø,Proposition 2.5] and [D, Theorem] in different ways.…”

Section: Extriangulated Categories With Negative First Extensionsmentioning

confidence: 99%

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

Adachi¹,

Enomoto²,

Tsukamoto³

2021

Preprint

View full text Add to dashboard Cite

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

show abstract

Abelian subcategories of triangulated categories induced by simple minded systems

Cited by 4 publications

References 5 publications

$n$-exact categories arising from $n$-exangulated categories

$n$-exact categories arising from $n$-exangulated categories

Proper abelian subcategories of triangulated categories and their tilting theory

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

Contact Info

Product

Resources

About