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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 Function field (scheme theory) ： ウィキペディア英語版 Function field (scheme theory) The sheaf of rational functions ''KX'' of a scheme ''X'' is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of varieties, such a sheaf associates to each open set ''U'' the ring of all rational functions on that open set; in other words, ''KX''(''U'') is the set of fractions of regular functions on ''U''. Despite its name, ''KX'' does not always give a field for a general scheme ''X''. == Simple cases == In the simplest cases, the definition of ''KX'' is straightforward. If ''X'' is an (irreducible) affine algebraic variety, and if ''U'' is an open subset of ''X'', then ''KX''(''U'') will be the field of fractions of the ring of regular functions on ''U''. Because ''X'' is affine, the ring of regular functions on ''U'' will be a localization of the global sections of ''X'', and consequently ''KX'' will be the constant sheaf whose value is the fraction field of the global sections of ''X''. If ''X'' is integral but not affine, then any non-empty affine open set will be dense in ''X''. This means there is not enough room for a regular function to do anything interesting outside of ''U'', and consequently the behavior of the rational functions on ''U'' should determine the behavior of the rational functions on ''X''. In fact, the fraction fields of the rings of regular functions on any open set will be the same, so we define, for any ''U'', ''KX''(''U'') to be the common fraction field of any ring of regular functions on any open affine subset of ''X''. Alternatively, one can define the function field in this case to be the local ring of the generic point. 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「Function field (scheme theory)」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.