ICE-closed subcategories and wide $$\tau $$-tilting modules

Enomoto, Haruhisa; Sakai, Arashi

doi:10.1007/s00209-021-02796-6

Cited by 9 publications

(7 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Theorem 3.6(1), we have that [ES,Proposition 3.3]. Combining these facts with Lemma 3.7, we can prove the assertion as follows.…”

Section: Cmrmentioning

confidence: 64%

From the lattice of torsion classes to the posets of wide subcategories and ICE-closed subcategories

Enomoto¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we compute the posets of wide subcategories and ICE-closed subcategories from the lattice of torsion classes in an abelian length category in a purely latticetheoretical way, by using the kappa map in a completely semidistributive lattice. As for the poset of wide subcategories, we give two more simple constructions via a bijection between wide subcategories and torsion classes with canonical join representations. More precisely, for a completely semidistributive lattice, we give two poset structures on the set of elements with canonical join representations: the kappa order (defined using the extended kappa map of Barnard-Todorov-Zhu), and the core label order (generalizing the shard intersection order for congruence-uniform lattices). Then we show that these posets for the lattice of torsion classes coincide and are isomorphic to the poset of wide subcategories. As a byproduct, we give a simple description of the shard intersection order on a finite Coxeter group using the extended kappa map. Contents 1. Introduction 1 2. Preliminaries 4 2.1. The kappa map and the join-irreducible labeling 5 2.2. The kappa map in the lattice of torsion classes 7 3. Computing the posets of torsion hearts 9 3.1. Preliminaries on torsion hearts 9 3.2. Construction 10 3.3. Posets of subcategories as posets of join-irreducibles 11 3.4. The map j-label in terms of the join-irreducible labeling 12 4. The kappa order, the core label order, and wide A 13 4.1. Canonical join representation and the extended kappa map 13 4.2. Canonical join representation and widely generated torsion classes 16 4.3. The kappa order and wide subcategories 19 4.4. The core label order and wide subcategories 20 4.5. Combinatorial consequences 23 References 24

show abstract

“…By Theorem 3.6(1), we have that [ES,Proposition 3.3]. Combining these facts with Lemma 3.7, we can prove the assertion as follows.…”

Section: Cmrmentioning

confidence: 64%

From the lattice of torsion classes to the posets of wide subcategories and ICE-closed subcategories

Enomoto¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The terminology heart was introduced in [18]. Moreover, this notion plays a crucial role in the study of subcategories of abelian categories closed under images, cokernels and extensions in [8]. Now we state a main result of this paper.…”

Section: •2 Intervals and Hearts Of S-torsion Pairsmentioning

confidence: 99%

“…This isomorphism originally appeared in the context of τ -tilting reduction in [13], and is a useful tool to study various subcategories of module categories, e.g. [2,6,8].…”

Section: Introductionmentioning

confidence: 99%

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

Adachi

Enomoto²,

Tsukamoto³

2022

Math. Proc. Camb. Phil. Soc.

Self Cite

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 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

“…Indeed, one has the equality T = U * H [U ,T ] , in other words, the torsion class U mutates into another torsion class T by the brick S q . (See also the dual version of [5,Lemma 2.8]. )…”

Section: Introductionmentioning

confidence: 99%

The realization problem for generalized torsion theories

Yoshizawa

2022

Communications in Algebra

View full text Add to dashboard Cite

Understanding how torsion theories are described and constructed is crucial to the study of torsion theory. Mutations of torsion theories have been studied as a method of constructing another torsion theory from a given one. We have already obtained how to mutate ordinary torsion theories into generalized torsion theories associated with a Serre subcategory. The paper investigates when the generalized torsion theories give ordinary torsion theories and when ordinary torsion theories provide each other.

show abstract

ICE-closed subcategories and wide $$\tau $$-tilting modules

Cited by 9 publications

References 20 publications

From the lattice of torsion classes to the posets of wide subcategories and ICE-closed subcategories

From the lattice of torsion classes to the posets of wide subcategories and ICE-closed subcategories

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

The realization problem for generalized torsion theories

Contact Info

Product

Resources

About