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タイトル: Brauer groups, Mackey and Tambara functors on profinite groups, and 2-dimensional homological algebra
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
著者(別言語): 中岡, 宏行
発行日: 2009年3月23日
抄録: 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.
内容記述: 報告番号: 甲24983 ; 学位授与年月日: 2009-03-23 ; 学位の種別: 課程博士 ; 学位の種類: 博士(数理科学) ; 学位記番号: 博数理第338号 ; 研究科・専攻: 数理科学研究科数理科学専攻
URI: http://hdl.handle.net/2261/28162
出現カテゴリ:021 博士論文
12120 博士論文(数理科学専攻)

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