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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 Quotient by an equivalence relation ： ウィキペディア英語版 Quotient by an equivalence relation In mathematics, given a category ''C'', a quotient of an object ''X'' by an equivalence relation $f:R\; \backslash to\; X\; \backslash times\; X$ is a coequalizer for the pair of maps :$R\; \backslash overset\backslash to\; X\; \backslash times\; X\; \backslash overset\backslash to\; X,\; \backslash ,\; i\; =\; 1,2,$ where ''R'' is an object in ''C'' and "''f'' is an equivalence relation" means that, for any object ''T'' in ''C'', the image (which is a set) of $f:\; R(T)\; =\; \backslash operatorname(T,\; R)\; \backslash to\; X(T)\; \backslash times\; X(T)$ is an equivalence relation; that is, $(x,\; y)$ is in it if and only if $(y,\; x)$ is in it, etc. The basic case in practice is when ''C'' is the category of all schemes over some scheme ''S''. But the notion is flexible and one can also take ''C'' to be the category of sheaves. == Examples == *Let ''X'' be a set and consider some equivalence relation on it. Let ''Q'' be the set of all equivalence classes in ''X''. Then the map $q:\; X\; \backslash to\; Q$ that sends an element ''x'' to an equivalence class to which ''x'' belong is a quotient. *In the above example, ''Q'' is a subset of the power set ''H'' of ''X''. In algebraic geometry, one might replace ''H'' by a Hilbert scheme or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative Picard scheme of a flat projective scheme ''X''〔One also needs to assume the geometric fibers are integral schemes; Mumford's example shows the "integral" cannot be omitted.〕 as a quotient ''Q'' (of the scheme ''Z'' parametrizing relative effective divisors on ''X'') that is a closed scheme of a Hilbert scheme ''H''. The quotient map $q:\; Z\; \backslash to\; Q$ can then be thought of as a relative version of the Abel map. 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「Quotient by an equivalence relation」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.