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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 Geometric quotient ： ウィキペディア英語版 Geometric quotient In algebraic geometry, a geometric quotient of an algebraic variety ''X'' with the action of an algebraic group ''G'' is a morphism of varieties $\backslash pi:\; X\; \backslash to\; Y$ such that :(i) For each ''y'' in ''Y'', the fiber $\backslash pi^(y)$ is an orbit of ''G''. :(ii) The topology of ''Y'' is the quotient topology: a subset $U\; \backslash subset\; Y$ is open if and only if $\backslash pi^(U)$ is open. :(iii) For any open subset $U\; \backslash subset\; Y$, $\backslash pi^:\; k()\; \backslash to\; k()^G$ is an isomorphism. (Here, ''k'' is the base field.) The notion appears in geometric invariant theory. (i), (ii) say that ''Y'' is an orbit space of ''X'' in topology. (iii) may also be phrased as an isomorphism of sheaves $\backslash mathcal\_Y\; \backslash simeq\; (\backslash pi\_$ *\mathcal_X)^G. In particular, if ''X'' is irreducible, then so is ''Y'' and $k(Y)\; =\; k(X)^G$: rational functions on ''Y'' may be viewed as invariant rational functions on ''X'' (i.e., rational-invariants of ''X''). For example, if ''H'' is a closed subgroup of ''G'', then $G/H$ is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same). == Relation to other quotients == A geometric quotient is a categorical quotient. This is proved in Mumford's geometric invariant theory. A geometric quotient is precisely a good quotient whose fibers are orbits of the group. 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「Geometric 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.