Haruhisa Enomoto

• PostDoc Position at Osaka Prefecture University

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 w...

We study IE-closed subcategories of a module category, subcategories which are closed under taking Images and Extensions. We investigate the relation between IE-closed subcategories and torsion pairs, and characterize $\tau$-tilting finite algebras using IE-closed subcategories. For the hereditary case, we show that IE-closed subcategories can be c...

In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval \([\mathcal {U},\mathcal {T}]\) in the lattice of torsion classes, we associate a subcategory \(\mathcal {T} \cap \mathcal {U}^\perp \) called the heart. We show that every ICE-closed subcategory...

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...

We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a length exact category is abelian if and only if simple objects form a semibrick, that is, Schur's lemma holds.

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 categor...

For an element w of a simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory F ( w ) of the module category over the preprojective algebra of Dynkin type. This paper studies categorical properties of F ( w ) using the root system. We show that simple objects in F ( w ) bijectively correspond to Bruhat inversion roots of w, and obtai...

In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval $[\mathcal{U},\mathcal{T}]$ in the lattice of torsion classes, we associate a subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ called the heart. We show that every ICE-closed subcategory can be r...

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...

We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. We show that ICE-closed subcategories over the path algebra of Dynkin type are in bijection with basic rigid modules, that ICE-closed subcategories are precisely torsion classes in some wide s...

For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injecti...

For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory $\mathcal{F}(w)$ of a module category over a preprojective algebra of Dynkin type. This paper aims at studying categorical properties of $\mathcal{F}(w)$ via its connection with the root system. We show that by taking dimension vectors, simple objects i...

We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a length exact category is abelian if and only if simple objects form a semibrick, that is, the Schur's lemma ho...

We investigate the Jordan-H\"older property (JHP) in exact categories. First we introduce a new invariant of exact categories, the Grothendieck monoids, and show that (JHP) holds if and only if the Grothendieck monoid is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next we app...

For an exact category E, we study the Butler's condition “AR=Ex”: the relation of the Grothendieck group of E is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that E has finitely many indecomposables. This can be applied to functorially finite torsion(free) classes and contravariantly finite...

For an exact category $\mathcal{E}$, we study the Butler's condition "AR=Ex": the relation of the Grothendieck group of $\mathcal{E}$ is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that $\mathcal{E}$ has finitely many indecomposables. This can be applied to functorially finite torsion(free)...

We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the Grothendieck group of such a category is generated by AR conflations. Moreover, we obtain an explicit classification of...

Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category has enough injectives, the image of the embedding can be described in terms of Wakamatsu tilting (=semi-dualizing) subcategories. If moreover the exact category has...