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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 Base change ： ウィキペディア英語版 Base change In mathematics, especially in algebraic geometry, base change refers to a number of similar theorems concerning the cohomology of sheaves on algebro-geometric objects such as varieties or schemes. The situation of a base change theorem typically is as follows: given two maps of, say, schemes, $f:\; Y\; \backslash rightarrow\; X$, $g:\; X\text{'}\; \backslash rightarrow\; X$, let $f\text{'}$ and $g\text{'}$ be the projections from the fiber product $Y\text{'}\; :=\; Y\; \backslash times\_X\; X\text{'}$ to $X\text{'}$ and $Y$, respectively. Moreover, let a sheaf $\backslash mathcal\; F$ on X' be given. Then, there is a natural map (obtained by means of adjunction; see base change map) :$f^$ * R^i g_ * \mathcal F \rightarrow R^i g'_ * f'^ * \mathcal F. Depending on the type of sheaf, and on the type of the morphisms ''g'' and ''f'', this map is an isomorphism (of sheaves on ''Y'') in some cases. Here $R^i\; g\_$ * \mathcal F denotes the higher direct image of $\backslash mathcal\; F$ under ''g''. As the stalk of this sheaf at a point on ''Y'' is closely related to the cohomology of the fiber of the point under ''g'', this statement is paraphrased by saying that "cohomology commutes with base extension". ==Flat base change for quasi-coherent sheaves== The base change holds for a quasi-coherent sheaf $\backslash mathcal\; F$ (on $X\text{'}$), provided that the map ''f'' is flat (together with a number of technical conditions: ''g'' needs to be a separated morphism of finite type, the schemes involved need to be Noetherian). 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「Base change」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.