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2-categorical covering theory for algebras

Research Project

Project/Area Number 25610003
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionShizuoka University

Principal Investigator

Asashiba Hideto  静岡大学, 理学部, 教授 (70175165)

Co-Investigator(Renkei-kenkyūsha) IYAMA Osamu  名古屋大学, 大学院多元数理科学研究科, 教授 (70347532)
TAMAKI Dai  信州大学, 理学部, 教授 (10252058)
Project Period (FY) 2013-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords被覆 / グロタンディーク構成 / スマッシュ積 / 導来同値 / 2圏 / 軌道圏 / 両側加群 / 線型圏 / 森田型特異同値 / ラックス関手 / 線形圏
Outline of Final Research Achievements

We fix a commutative ring k, a small k-category I and a group G. 1. We denote by k-Catb the bicategory of all small k-categories with bimodules over them as 1-morphisms. We gave a natural definitions of "module category" Mod X and "derived category" D(Mod X) of a lax functor X:I-> Catb as lax functors, and defined a notion of derived equivalences between lax functors X. We proved that if lax functors X, X':I -> k-Catb are derived equivalent, then so are their Grothendieck constructions Gr(X) and Gr(X'). 2. We established a covering theory for bimodules over small k-categories. 3. For G-graded small k-categories R and S that are derived equivalent, we gave a sufficient condition for the smash products R#G and S#G to be derived equivalent.

Report

(4 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (39 results)

All 2016 2015 2014 2013 Other

All Journal Article (4 results) (of which Peer Reviewed: 3 results,  Acknowledgement Compliant: 1 results) Presentation (30 results) (of which Int'l Joint Research: 7 results,  Invited: 25 results) Remarks (5 results)

  • [Journal Article] Cohen-Montgomery duality of bimodules and its applications to equivalences given by bimodules2016

    • Author(s)
      Asashiba, Hideto
    • Journal Title

      Proceedings of the 48 Symposium on Ring Theory and Representation Theory

      Volume: 48

    • Related Report
      2015 Annual Research Report
  • [Journal Article] A Generalization of Gabriel's Galois Covering Functors II: 2-Categorical Cohen-Montgomery Duality2016

    • Author(s)
      Asashiba, Hideto
    • Journal Title

      Applied Categorical Structures

      Volume: 未定 Issue: 2 Pages: 155-186

    • DOI

      10.1007/s10485-015-9416-9

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type2015

    • Author(s)
      浅芝 秀人,木村真弓
    • Journal Title

      Algebra and Discrete Mathematics Jounal

      Volume: 未定

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type2014

    • Author(s)
      Hideto Asashiba and Mayumi Kimura
    • Journal Title

      Algebra and Discrete Mathematics Journal

      Volume: 印刷中

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Presentation] Cohen-Montgomery duality for bimodules2016

    • Author(s)
      浅芝 秀人
    • Organizer
      代数学セミナー
    • Place of Presentation
      精華大学・中国
    • Year and Date
      2016-05-05
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Cohen-Montgomery duality for bimodules2016

    • Author(s)
      浅芝 秀人
    • Organizer
      北京交通大学フォーラム
    • Place of Presentation
      北京交通大学・中国
    • Year and Date
      2016-05-04
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] A simple application of a 2-categorical covering theory to a construction of triangulated orbit categories2016

    • Author(s)
      浅芝 秀人
    • Organizer
      代数学セミナー
    • Place of Presentation
      北京師範大学・中国
    • Year and Date
      2016-04-29
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gluing of derived equivalences with bimodules2016

    • Author(s)
      浅芝 秀人
    • Organizer
      代数学セミナー
    • Place of Presentation
      首都師範大学・中国
    • Year and Date
      2016-04-28
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Derived equivalences of actions of a category2016

    • Author(s)
      浅芝 秀人
    • Organizer
      代数学セミナー
    • Place of Presentation
      北京交通大学・中国
    • Year and Date
      2016-04-26
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gluing of derived equivalences by functors2016

    • Author(s)
      浅芝 秀人
    • Organizer
      代数学セミナー
    • Place of Presentation
      北京交通大学・中国
    • Year and Date
      2016-04-26
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The smash product of Brauer graphs with group weights and Brauer permutations2016

    • Author(s)
      浅芝 秀人
    • Organizer
      Workshop on Brauer graph algebras
    • Place of Presentation
      Stuttgart大学・ドイツ
    • Year and Date
      2016-03-21
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] A simple application of a 2-categorical covering theory to a construction of triangulated orbit categories2016

    • Author(s)
      浅芝 秀人
    • Organizer
      代数学セミナー
    • Place of Presentation
      Stuttgart大学・ドイツ
    • Year and Date
      2016-03-08
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Cohen-Montgomery duality for bimodules and its application to stable equivalences of Morita type2015

    • Author(s)
      浅芝 秀人
    • Organizer
      表現論セミナー
    • Place of Presentation
      IPM-Isfahan, Iran
    • Year and Date
      2015-09-21
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Cohen-Montgomery duality for bimodules and its applications to equivalences given by bimodules2015

    • Author(s)
      浅芝 秀人
    • Organizer
      日本数学会秋期総合分科会
    • Place of Presentation
      京都産業大学(京都府京都市)
    • Year and Date
      2015-09-13
    • Related Report
      2015 Annual Research Report
  • [Presentation] Cohen-Montgomery duality for bimodules and its applications to equivalences given by bimodules2015

    • Author(s)
      浅芝 秀人
    • Organizer
      第48回環論および表現論シンポジウム
    • Place of Presentation
      名古屋大学(愛知県名古屋市)
    • Year and Date
      2015-09-07
    • Related Report
      2015 Annual Research Report
  • [Presentation] Cohen-Montgomery duality for bimodules and its applications to equivalences given by bimodules2015

    • Author(s)
      浅芝 秀人
    • Organizer
      第7回日中韓環論国際会議
    • Place of Presentation
      浙江工科大学・中国
    • Year and Date
      2015-07-02
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Cohen-Montgomery duality for bimodules and its applications to equivalences given by bimodules2015

    • Author(s)
      浅芝 秀人
    • Organizer
      ムルシア大学表現論セミナー
    • Place of Presentation
      Murcia, Murcia大学,スペイン
    • Year and Date
      2015-03-18
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Gluing derived equivalences together with bimodules2015

    • Author(s)
      浅芝 秀人
    • Organizer
      神楽坂代数セミナー
    • Place of Presentation
      東京都,東京理科大学
    • Year and Date
      2015-03-10
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Grothendieck constructions and derived equivalences of linear categories2015

    • Author(s)
      浅芝 秀人
    • Organizer
      東京理科大学院生向けセミナー
    • Place of Presentation
      東京都,東京理科大学
    • Year and Date
      2015-03-09 – 2015-03-10
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Cohen-Montgomery duality for bimodules and applications to Morita equivalences and stable equivalences of Morita type2015

    • Author(s)
      浅芝 秀人
    • Organizer
      シュトゥットガルト大学表現論セミナー
    • Place of Presentation
      Stuttgart, Stuttgart大学,ドイツ
    • Year and Date
      2015-02-17
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Infinite devissageによるRickard, Kellerの定理の証明とdg algebraの理論による片側tilting complexの両側tilting complexへの持ち上げ構成2014

    • Author(s)
      浅芝 秀人
    • Organizer
      第15回静岡代数学セミナー
    • Place of Presentation
      静岡市,静岡大学
    • Year and Date
      2014-12-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] DG algebras and Morita type theorem by Rickard and Gluing derived equivalences together with bimodules2014

    • Author(s)
      浅芝 秀人
    • Organizer
      Short Courses on DG algebras and Morita type theorem by Rickard and Gluing derived equivalences together with bimodules
    • Place of Presentation
      Isfahan, IPM-Isfahan, Iran
    • Year and Date
      2014-09-17 – 2014-09-18
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Tilted algebras and configurations of self-injective algebras of Dynkin type2014

    • Author(s)
      中島健,浅芝秀人
    • Organizer
      第47回環論および表現論シンポジウム
    • Place of Presentation
      大阪市,大阪市立大学
    • Year and Date
      2014-09-13
    • Related Report
      2014 Research-status Report
  • [Presentation] Gluing of derived equivalences with bimodules2014

    • Author(s)
      浅芝 秀人
    • Organizer
      第16回多元環の表現論国際会議
    • Place of Presentation
      三亜市,精華三亜国際数学フォーラム,中国
    • Year and Date
      2014-08-19
    • Related Report
      2014 Research-status Report
  • [Presentation] Gluing of derived equivalences with bimodules2014

    • Author(s)
      浅芝 秀人
    • Organizer
      名古屋大学環論表現論セミナー
    • Place of Presentation
      名古屋市,名古屋大学
    • Year and Date
      2014-07-18
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 2-categorical covering theory and derived equivalences2014

    • Author(s)
      浅芝 秀人
    • Organizer
      A Short Course on 2-Categoriecal Covering Theory and Derived Equivalences
    • Place of Presentation
      Isfahan, IPM-Isfahan, Iran
    • Year and Date
      2014-05-18 – 2014-05-19
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Gluing of derived equivalences along bimodules2014

    • Author(s)
      Hideto Asashiba
    • Organizer
      シュトゥットガルト大学表現論セミナー
    • Place of Presentation
      シュトゥットガルト大学(ドイツ)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Gluing of derived equivalences by Grothendieck constructions2013

    • Author(s)
      Hideto Asashiba
    • Organizer
      Recent Trends in Rings and Algebras
    • Place of Presentation
      ムルシア大学(スペイン)
    • Related Report
      2013 Research-status Report
  • [Presentation] Derived equivalences and 2-categorical covering theory (gluing of derived equivalences)2013

    • Author(s)
      Hideto Asashiba
    • Organizer
      ボン大学表現論セミナー
    • Place of Presentation
      ボン大学(ドイツ)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Lax functors of bicategories and derived equivalences with application to triangular matrix algebras2013

    • Author(s)
      Hideto Asashiba
    • Organizer
      ビーレフェルト大学表現論セミナー
    • Place of Presentation
      ビーレフェルト大学(ドイツ)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Colax actions of a category and gluing of derived equivalences2013

    • Author(s)
      Hideto Asashiba
    • Organizer
      イェーナー大学表現論セミナー
    • Place of Presentation
      イェーナー大学(ドイツ)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Grothendieck constructions of (co)lax functors and gluing of derived equivalences2013

    • Author(s)
      Hideto Asashiba
    • Organizer
      シュトゥットガルト大学表現論セミナー
    • Place of Presentation
      シュトゥットガルト大学(ドイツ)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Gluing of derived equivalences along bimodules2013

    • Author(s)
      Hideto Asashiba
    • Organizer
      ボン大学表現論セミナー
    • Place of Presentation
      ボン大学(ドイツ)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Gluing of derived equivalences along bimodules2013

    • Author(s)
      Hideto Asashiba
    • Organizer
      第13回静岡代数学セミナー
    • Place of Presentation
      静岡大学(静岡県)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Remarks] Hideto Asashiba's website at Shizuoka University

    • URL

      http://www.ipc.shizuoka.ac.jp/~shasash/

    • Related Report
      2015 Annual Research Report
  • [Remarks] Homepage of Hideto Asashiba at Shizuoka Universtiy

    • URL

      http://www.ipc.shizuoka.ac.jp/~shasash/

    • Related Report
      2014 Research-status Report
  • [Remarks] 静岡大学教員データベース

    • URL

      https://tdb.shizuoka.ac.jp/RDB/public?Default2.aspx?id=11037&1=0

    • Related Report
      2014 Research-status Report
  • [Remarks] Homepage of Hideto Asashiba at Shizuoka University

    • URL

      http://www.ipc.shizuoka.ac.jp/~shasash/

    • Related Report
      2013 Research-status Report
  • [Remarks] 静岡大学教員データベース

    • URL

      https://tdb.shizuoka.ac.jp/RDB/public/Default2.aspx?id=11037&l=0

    • Related Report
      2013 Research-status Report

URL: 

Published: 2014-07-25   Modified: 2019-07-29  

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