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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Moduli stack of principal bundles ： ウィキペディア英語版 Moduli stack of principal bundles In algebraic geometry, given a smooth projective curve ''X'' over a finite field $\backslash mathbf\_q$ and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by $\backslash operatorname\_G(X)$, is an algebraic stack given by:〔http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf〕 for any $\backslash mathbf\_q$-algebra ''R'', :$\backslash operatorname\_G(X)(R)\; =$ the category of principal ''G''-bundles over the relative curve $X\; \backslash times\_\; \backslash operatornameR$. In particular, the category of $\backslash mathbf\_q$-points of $\backslash operatorname\_G(X)$, that is, $\backslash operatorname\_G(X)(\backslash mathbf\_q)$, is the category of ''G''-bundles over ''X''. Similarly, $\backslash operatorname\_G(X)$ can also be defined when the curve ''X'' is over the field of complex numbers. Roughly, in the complex case, one can define $\backslash operatorname\_G(X)$ as the quotient stack of the space of holomorphic connections on ''X'' by the gauge group. Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of $\backslash operatorname\_G(X)$. In the finite field case, it is not common to define the homotopy type of $\backslash operatorname\_G(X)$. But one can still define a (smooth) cohomology and homology of $\backslash operatorname\_G(X)$. == Basic properties == It is known that $\backslash operatorname\_G(X)$ is a smooth stack of dimension $(g(X)\; -\; 1)\; \backslash dim\; G$ where $g(X)$ is the genus of ''X''. It is not of finite type but of locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf. the Harder–Narasimhan stratification.) If ''G'' is a split reductive group, then the set of connected components $\backslash pi\_0(\backslash operatorname\_G(X))$ is in a natural bijection with the fundamental group $\backslash pi\_1(G)$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Moduli stack of principal bundles」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.