2024/04/22 更新

写真a

ナカオカ ヒロユキ
中岡 宏行
NAKAOKA Hiroyuki
所属
大学院多元数理科学研究科 多元数理科学専攻 基幹数理 准教授
大学院担当
大学院多元数理科学研究科
学部担当
理学部 数理学科
職名
准教授

学位 1

  1. 博士(数理科学) ( 2009年3月   東京大学 ) 

 

論文 5

  1. Auslander–Reiten theory in extriangulated categories 査読有り 国際共著

    Iyama O., Nakaoka H., Palu Y.

    Transactions of the American Mathematical Society Series B   11 巻   頁: 248 - 305   2024年1月

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    記述言語:英語   出版者・発行元:Transactions of the American Mathematical Society Series B  

    The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated cate-gories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of suf-ficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre du-ality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category C of an extriangu-lated category C is a τ-category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if C has enough projectives, almost split extensions and source morphisms. This gives various consequences on C, including Igusa– Todorov’s Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of C via the complete mesh category of the Auslander–Reiten species of C. Finally we prove that any locally finite symmetrizable τ-quiver (=val-ued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms.

    DOI: 10.1090/btran/159

    Scopus

  2. Localization of extriangulated categories 査読有り

    Nakaoka H., Ogawa Y., Sakai A.

    Journal of Algebra   611 巻   頁: 341 - 398   2022年12月

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    記述言語:英語   掲載種別:研究論文(学術雑誌)   出版者・発行元:Journal of Algebra  

    In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Indeed we give such a construction for a bit wider class of morphisms, so that it covers several other localizations appeared in the literature, such as Rump's localization of exact categories by biresolving subcategories, localizations of extriangulated categories by means of Hovey twin cotorsion pairs, and the localization of exact categories by two-sided admissibly percolating subcategories.

    DOI: 10.1016/j.jalgebra.2022.08.008

    Scopus

  3. n-Exangulated categories (II): Constructions from n-cluster tilting subcategories 査読有り 国際共著

    Herschend M., Liu Y., Nakaoka H.

    Journal of Algebra   594 巻   頁: 636 - 684   2022年3月

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    記述言語:英語   出版者・発行元:Journal of Algebra  

    In n-Exangulated Categories (I), we introduced the notion of n-exangulated categories for each positive integer n. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a common generalization of n-exact categories in the sense of Jasso and (n+2)-angulated categories in the sense of Geiss-Keller-Oppermann. In this second article we introduce the notion of n-cluster tilting subcategories for extriangulated categories with enough projectives and injectives and show that under certain conditions such n-cluster tilting subcategories are n-exangulated.

    DOI: 10.1016/j.jalgebra.2021.11.042

    Scopus

  4. Finite gentle repetitions of gentle algebras and their Avella-Alaminos-Geiss invariants 査読有り

    Nakaoka Hiroyuki

    COMMUNICATIONS IN ALGEBRA     2021年11月

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    記述言語:英語   出版者・発行元:Communications in Algebra  

    Among finite dimensional algebras over a field K, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras A, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of A in terms of some values of its Avella-Alaminos–Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebras (Formula presented.) indexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos–Geiss invariants of A to the dimensions of Hochschild cohomologies of (Formula presented.) On the way we will see that the Avella-Alaminos–Geiss invariants of (Formula presented.) are completely determined by those of A. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the last part of this article, we will see how the generalized Auslander–Platzeck–Reiten reflection can be related to the finite gentle repetition.

    DOI: 10.1080/00927872.2021.2008412

    Web of Science

    Scopus

  5. n-exangulated categories (I): Definitions and fundamental properties 査読有り 国際共著

    Herschend Martin, Liu Yu, Nakaoka Hiroyuki

    JOURNAL OF ALGEBRA   570 巻   頁: 531 - 586   2021年3月

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    記述言語:英語   出版者・発行元:Journal of Algebra  

    DOI: 10.1016/j.jalgebra.2020.11.017

    Web of Science

    Scopus

科研費 2

  1. 高次圏・豊穣圏的視点からの、完全圏と三角圏を包括する統一的ホモロジー代数の発展

    研究課題/研究課題番号:24K06645  2024年4月 - 2027年3月

    科学研究費助成事業  基盤研究(C)

    中岡 宏行

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    担当区分:研究代表者 

    配分額:2470000円 ( 直接経費:1900000円 、 間接経費:570000円 )

  2. 完全圏と三角圏におけるホモロジー代数の理論的発展

    研究課題/研究課題番号:17K18727  2017年6月 - 2020年3月

    科学研究費助成事業  挑戦的研究(萌芽)

    中岡 宏行

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    担当区分:研究代表者 

    配分額:3900000円 ( 直接経費:3000000円 、 間接経費:900000円 )

    Yann Palu氏とのextriangulated category(以下、ET圏)を定義した共同研究は改定後論文誌に掲載された。Yu Liu氏とのET圏の余ねじれ対のハート構成を調べた共同研究は改定後論文誌に掲載された。
    Martin Herschend氏・Yu Liu氏との共同研究で高次数版としてn-exangulated categoryという概念を定義した。現在査読待ち。Osamu Iyama氏・Yann Palu氏との共同研究ではAuslander-Reiten理論をET圏で考察するプレプリントを作成した。
    他に、gentle多元環の導来不変量に関する単著のプレプリントを作成した。
    ホモロジー代数の舞台として重要な完全圏・三角圏のクラスは定義としては互いにほぼ排他的であり、両者で定義される類似の概念が同じような性質を持つ場合であっても、別々に取り扱い、しばしば同じような議論を二度行う必要が生じていた。
    本研究では、これらの二つの圏のクラスをExt1-関手の言葉で同時に扱う概念としてYann Paluと共に導入したextriangulated category を用いて、統一的なホモロジー代数の理論整備を目指している。相対ホモロジー代数、Auslander-Reiten理論といった完全圏・三角圏で知られる事柄を実際に統一的に扱うことができた。また、高次数版の定義も考察した。