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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 categorical quotient ： ウィキペディア英語版 categorical quotient In algebraic geometry, given a category ''C'', a categorical quotient of an object ''X'' with action of a group ''G'' is a morphism $\backslash pi:\; X\; \backslash to\; Y$ that :(i) is invariant; i.e., $\backslash pi\; \backslash circ\; \backslash sigma\; =\; \backslash pi\; \backslash circ\; p\_2$ where $\backslash sigma:\; G\; \backslash times\; X\; \backslash to\; X$ is the given group action and ''p''2 is the projection. :(ii) satisfies the universal property: any morphism $X\; \backslash to\; Z$ satisfying (i) uniquely factors through $\backslash pi$. One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes. Note $\backslash pi$ need not be surjective. Also, if it exists, a categorical quotient is unique up to a canonical isomorphism. In practice, one takes ''C'' to be the category of varieties or the category of schemes over a fixed scheme. A categorical quotient $\backslash pi$ is a universal categorical quotient if it is stable under base change: for any $Y\text{'}\; \backslash to\; Y$, $\backslash pi\text{'}:\; X\text{'}\; =\; X\; \backslash times\_Y\; Y\text{'}\; \backslash to\; Y\text{'}$ is a categorical quotient. A basic result is that geometric quotients (e.g., $G/H$) and GIT quotients (e.g., $X/\backslash !/G$) are categorical quotients. == References == * Mumford, David; Fogarty, J.; Kirwan, F. ''Geometric invariant theory''. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN 3-540-56963-4 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「categorical quotient」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.