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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 Proper base change theorem ： ウィキペディア英語版 Proper base change theorem :''There is also a proper base change theorem in topology. For that, see base change map.'' In algebraic geometry, there are at least two versions of proper base change theorems: one for ordinary cohomology and the other for étale cohomology. ==In ordinary cohomology== The proper base change theorem states the following: let $f:\; X\; \backslash to\; S$ be a proper morphism between noetherian schemes, and $\backslash mathcal$ ''S''-flat coherent sheaf on $X$. If $S\; =\; \backslash operatorname\; A$, then there is a finite complex $0\; \backslash to\; K^0\; \backslash to\; K^1\; \backslash to\; \backslash cdots\; \backslash to\; K^n\; \backslash to\; 0$ of finitely generated projective ''A''-modules and a natural isomorphism of functors :$H^p(X\; \backslash times\_S\; \backslash operatorname\; -,\; \backslash mathcal\; \backslash otimes\_A\; -)\; \backslash to\; H^p(K^\backslash bullet\; \backslash otimes\_A\; -),\; p\; \backslash ge\; 0$ on the category of $A$-algebras. There are several corollaries to the theorem, some of which are also referred to as proper base change theorems: (the higher direct image $R^p\; f\_$ * \mathcal is coherent since ''f'' is proper.) Corollary 1 (semicontinuity theorem): Let ''f'' and $\backslash mathcal$ as in the theorem (but ''S'' may not be affine). Then we have: *(i) For each $p\; \backslash ge\; 0$, the function $s\; \backslash mapsto\; \backslash dim\_\; H^p\; (X\_s,\; \backslash mathcal\_s):\; S\; \backslash to\; \backslash mathbb$ is upper semicontinuous. *(ii) The function $s\; \backslash mapsto\; \backslash chi(\backslash mathcal\_s)$ is locally constant, where $\backslash chi(\backslash mathcal)$ denotes the Euler characteristic. Corollary 2: Assume ''S'' is reduced and connected. Then for each $p\; \backslash ge\; 0$ the following are equivalent *(i) $s\; \backslash mapsto\; \backslash dim\_\; H^p\; (X\_s,\; \backslash mathcal\_s)$ is constant. *(ii) $R^p\; f\_$ * \mathcal is locally free and the natural map ::$R^p\; f\_$ * \mathcal \otimes_ k(s) \to H^p(X_s, \mathcal_s) :is an isomorphism for all $s\; \backslash in\; S$. :Furthermore, if these conditions hold, then the natural map ::$R^\; f\_$ * \mathcal \otimes_ k(s) \to H^(X_s, \mathcal_s) :is an isomorphism for all $s\; \backslash in\; S$. Corollary 3: Assume that for some ''p'' $H^p(X\_s,\; \backslash mathcal\_s)\; =\; 0$ for all $s\; \backslash in\; S$. Then the natural map ::$R^\; f\_$ * \mathcal \otimes_ k(s) \to H^(X_s, \mathcal_s) :is an isomorphism for all $s\; \backslash in\; S$. 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「Proper base change theorem」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.