2018 Fiscal Year Final Research Report
Diversified research of ring theory and representation theory with derived categories as its center
| Project/Area Number |
25287001
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Partial Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Shizuoka University |
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Principal Investigator |
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| Research Collaborator |
Iyama Osamu
Kawata Shigeto
Sato Masahisa
Yamaura Kota
Hoshino Mitsuo
Miyachi Jun-ichi
Mori Izuru
Aihara Takuma
Minamoto Hiroyuki
Koshitani Shigeo
Kunugi Naoko
Sanada Katsunori
Ueda Akira
Nishinaka Tsunekazu
Hida Akihiko
Kikumasa Isao
Mizuno Yuya
Adachi Takahide
Itaba Ayako
Yoshiwaki Michio
Nakashima Ken
Kimura Yuta
Kozakai Yuta
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| Project Period (FY) |
2013-04-01 – 2019-03-31
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| Keywords | 導来同値 / 導来圏 / 被覆 / クイバー / 傾複体 / 傾加群 / 自己入射多元環 |
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| Outline of Final Research Achievements |
The central purpose of representation theory of algebras is in the study of the module category of an algebra. To this end the study of its derived category became important. In our research we investigated (1) structures of the derived categories; (2) constructions of tilting objects; (3) derived equivalence classifications of self-injective algebras; (4) related topics. The following are examples of our results: (1) for representation-infinite hereditary algebra A the bounded derived category of A is stably equivalent to the repetitive category of stable module category of A. (2) All the tilting complexes over preprojective algebras of Dynkin type were given. (3) The Broue's conjecture for the 1st Conway group was solved. (4) For any group G the smash product of a G-graded category and G is presented by quiver with relations. Also we supported the symposium on ring theory and representation theory (heavily related to our research) and published its proceedings.
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| Free Research Field |
代数学
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| Academic Significance and Societal Importance of the Research Achievements |
多元環の表現論の中心課題は、環の加群圏の研究であるが、それにはその加群圏の導来圏の研究が重要である。本研究では，環の加群圏の導来圏の構造，傾対象の構成，導来同値分類について研究を行った。今回の研究成果により導来圏の構造に対する理解が深まり加群圏に関する様々な問題が解けるようになった。位相的データー解析によりすでにデータ解析に加群圏が応用され，さらに現在，導来圏も応用されつつあるため，今後その重要性はますます高くなることが期待される。
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