Welcome to Haruhisa ENOMOTO's Homepage!

My website will be moved to this page, and this page will NOT be updated from now on (2021-01-11).

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.


  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.