# Welcome to Haruhisa ENOMOTO's Homepage!

In Japanese

I am a Ph.D. student (the 3rd year) at Graduate School of Mathematics, Nagoya University.
E-mail: m16009t [at] math.nagoya-u.ac.jp
Supervisor: Osamu Iyama

## Research Interests

Quillen's exact categories in the representation theory of algebras, Various subcategories of module (or abelian) categories.

## Published Papers

1. H. Enomoto, Classifying exact categories via Wakamatsu tilting, J. Algebra 485 (2017), 1-44. (arXiv, journal)
This paper studies the Morita type theorem for exact categories. The Ext-perpendicular category of cotilting modules are precisely exact category with progen & inj cogen & higher kernels.
Essentially I considered Auslander-Reiten's famous paper Applications of contravariantly finite subcategories in the context of exact categories.
2. H. Enomoto, Classifications of exact structures and Cohen-Macaulay-finite algebras, Adv. Math. 335 (2018), 838-877. (arXiv, journal)
My attempt to find an analogue of Auslander correspondence for exact categories, especially a kind of CM Auslander correspondence for Iwanaga-Gorenstein algebras.
For a given category, I classify possible exact structures by using functor category.
Also Auslander-Reiten theory for exact categories are studied.
3. H. Enomoto, Relations for Grothendieck groups and representation-finiteness, J. Algebra 539 (2019), Pages 152-176. (arXiv, journal)
My attempt to unify results on the relation between representation-finiteness and "AR=Ex" condition (relations of K_0 are generated by AR sequences), of course in the context of exact categories.

## Preprints

1. H. Enomoto, The Jordan-Holder property and Grothendieck monoids of exact categories, (arXiv:1908.05446)
I consider when an exact category satisfies the Jordan-Holder property, (JHP) (the uniquenss of decompositions of object into simple objects). I gave a characterization of it by using the new invariant "Grothendieck monoid." In many cases arising in the rep. theory of algebras, (JHP) is equivalent to "number of projectives = numer of simples". I investigated simples in torsion-free classes over type A quiver by using the symmetric group, and observed that Bruhat order appears!
2. H. Enomoto, Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras, (arXiv:2002.09205)
As a generalization of the above paper, we consider preprojective algebras and path algebras of Dynkin type and classify simples in their torsion-free classes (Buan-Iyama-Reiten-Scott's category C_w) by using its relation with the root system. I also give a method to find simples in a torsion-free class F by using maximal green sequence of F (or a path in the Hasse quiver from F to 0) via their brick labeling by DIRRT.
3. H. Enomoto, Schur's lemma for exact categories implies abelian, (arXiv:2002.09241)
In this short note, I generalize Ringel's bijection between semibricks and wide subcategories to exact categories. In particular, I proved the title.
4. H. Enomoto, Monobrick, a uniform approach to torsion-free classes and wide subcategories, (arXiv:2005.01626)
It is well known that semibricks are in bijection with wide (=extension-closed exact abelian) subcategory in a length abelian category, by considering simple objects in wide subcats. I continue studying simples in torsion-free classes, and find that similar classification can be achieved by weakening "semibrick" to "monobrick", a set of bricks in which every non-zero morphism is an injection. By this, I can prove that torsion-free classes are in bijection with "cofinally closed monobrick," a monobrick satisfying som poset theoretical condition, without any assumption on functorially finitenss, without using tau-tilting theory. This enables us to consider wide subcats and torsion-free classes simultaneously, and several results like a bijection between wide and torf, a finiteness of torf and bricks etc, can be proved poset theoretically or combinatorially.
5. H. Enomoto, Rigid modules and ICE-closed subcategories in quiver representations, (arXiv:2005.05536)
In the previous paper, monobricks are in bijection with "left Schur subcategories" of a length abelian category. A typical example is a subcategory of a subcategory closed under kernels, extensions and images. We consider this (dual) Image-Cokernel-Extension closed subcategories in the quiver representation. We found that this ICE-closed subcategories are in bijection with rigid modules (modules without self-extensions), which generalizes a bijection between torsion classes and support tilting modules due to Ingalls-Thomas. We also show that the number of ICE-closed subcats only depends on the underlying graph of Dynkin quiver. This paper contains an explicit formula for the number of ICE-closed subcats for each Dynkin type. For type A, this number coincides with the large Schroeder number. (Later I noticed that the main theorem can be better understood by considering exceptional sequences, and I added it.)
6. H. Enomoto, Classifying substructures of extriangulated categories via Serre subcategories, (arXiv:2005.13381)
For a given extriangulated category, I classify all possible substructures on it. More precisely, substructures are in bijection with Serre subcategories of the category of defects. As an application to exact categories, I proved that the lattice of exact structures on a given additive category is isomorphic to the lattice of Serre subcategories of some abelian category.
7. H. Enomoto, A. Sakai ICE-closed subcategories and wide $\tau$-tilting modules, (arXiv:2010.05433)
In the previous paper, I classified ICE-closed subcategories over hereditary algebras via partial tilting modules. We succeed in generalizing this for any algebras, introducing wide $\tau$-tilting modules, a $\tau$-tilting object in some functorially finite wide subcategory. To achieve this, we study ICE-closed subcategories via the hearts of intervals in the lattice of torsion classes (we borrow the word heart from the Tattar's paper, and is used in DIRRT, Asai-Pfeifer, and so on). We prove that every ICE-closed subcats are realized as hearts of some intervals, and characterize such intervals in a purely lattice-theoretic way. Using this, it follows that ICE-closed subcategories are precisely torsion classes of some wide subcategories. This enables us to use tau-tilting theory to classify ICE-closed subcategories. Moreover, we discuss how to compute wide tau-tilting modules from the support tau-tilting poset for the tau-tilting finite case.

## Some Notes

1. H. Enomoto, "Relative Auslander correspondence via exact categories", Master Thesis. (Slide, Talk video in VR )
Just a combination of published papers [1] and [2] with some additional Introduction.
2. H. Enomoto, "On categories of modules over locular categories" (Japanese), Bachelor Thesis.
I considered the correspondence between semiperfect ring, Krull-Schmidt categories and their indecomposable parts in a functorial way, and studied a kind of perfect ringoid.