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Brauer groups, Mackey and Tambara functors on profinite groups, and 2-dimensional homological algebra
https://doi.org/10.15083/00004012
https://doi.org/10.15083/000040123f147cdd-6467-4932-8c19-c816c5078cab
| 名前 / ファイル | ライセンス | アクション |
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NakaokaH_21_3_PhD_a.pdf (12.0 MB)
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NakaokaH_21_3_PhD_b.pdf (52.4 kB)
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| Item type | 学位論文 / Thesis or Dissertation(1) | |||||
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| 公開日 | 2012-10-23 | |||||
| タイトル | ||||||
| タイトル | Brauer groups, Mackey and Tambara functors on profinite groups, and 2-dimensional homological algebra | |||||
| 言語 | ||||||
| 言語 | eng | |||||
| 資源タイプ | ||||||
| 資源 | http://purl.org/coar/resource_type/c_46ec | |||||
| タイプ | thesis | |||||
| ID登録 | ||||||
| ID登録 | 10.15083/00004012 | |||||
| ID登録タイプ | JaLC | |||||
| その他のタイトル | ||||||
| その他のタイトル | Brauer 群、プロ有限群上のMackey 及び丹原関手2次元ホモロジー代数 | |||||
| 著者 |
NAKAOKA, HIROYUKI
× NAKAOKA, HIROYUKI |
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| 著者別名 | ||||||
| 識別子Scheme | WEKO | |||||
| 識別子 | 9289 | |||||
| 姓名 | 中岡, 宏行 | |||||
| 著者所属 | ||||||
| 著者所属 | 東京大学大学院数理科学研究科 | |||||
| 著者所属 | ||||||
| 著者所属 | Graduate School of Mathematical Sciences, The University of Tokyo | |||||
| Abstract | ||||||
| 内容記述タイプ | Abstract | |||||
| 内容記述 | In this thesis, we investigated categories and functors related to Brauer groups. In 1986, E.T.Jacobson defined the Brauer ring B(E,D) for a finite Galois field extension E/D, whose unit group canonically contains the Brauer group of D. In Part 1, we investigate the structure of B (E,D). More generally, we determine the structure of the F-Burnside ring for any additive functor F. This reslut enables us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension C/R. For any finite etale covering of schemes, we can associate two homomorphisms of Brauer groups, namely the pull-back and the norm map. For any connected scheme X, if we take the Galois category C of finite etale coverings over X, we see these homomorphisms make Brauer groups into a bivariant functor (= Mackey functor) on C. As a corollary, restricting to a finite Galois covering of schemes, we obtain. a cohomological Mackey functor on its Galois group. This is a generalization of the result for rings by Ford [12]. The Tambara functor was defined by Tambara in the name of TNR-functor, to treat certain ring-valued Mackey functors on a finite group. Recently Brunrevealed the importance of Tambara functors in the Witt-Burnside construction. In Part 3, we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshida's generalized Burnside ring functor is the first example. Consequently, we can consider· a Tambara functor on any profinite group. In relation with the Witt-Burnside construction, we can give a Tambara-functor structure on Elliott's functor V M, which generalizes the completed Burnside ring functor of Dress and Siebeneicher. Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups SCG are investigated. In Part 4, we consider a 2-categorical analogue of an abelian category, in such a way that it contains SCG as an example. As the main theorem in this part, we construct a long cohomology 2-exact sequence from any extension of complexes in such a 2-category. Our axiomatic and self-dual definition will enable us to simplify various kind of arguments related to the 2-dimensional homological algebra, by analogy with abelian categories. | |||||
| 書誌情報 | 発行日 2009-03-23 | |||||
| 日本十進分類法 | ||||||
| 主題Scheme | NDC | |||||
| 主題 | 410 | |||||
| 学位名 | ||||||
| 学位名 | 博士(数理科学) | |||||
| 学位 | ||||||
| 値 | doctoral | |||||
| 学位分野 | ||||||
| Mathematical Sciences (数理科学) | ||||||
| 学位授与機関 | ||||||
| 学位授与機関名 | University of Tokyo (東京大学) | |||||
| 研究科・専攻 | ||||||
| Graduate School of Mathematical Sciences (数理科学研究科) | ||||||
| 学位授与年月日 | ||||||
| 学位授与年月日 | 2009-03-23 | |||||
| 学位授与番号 | ||||||
| 学位授与番号 | 甲第24983号 | |||||
| 学位記番号 | ||||||
| 博数理第338号 | ||||||
