翻訳と辞書 Words near each other ・ Theopompus (comic poet) ・ Theopompus of Sparta ・ Theopropus elegans ・ Theoprosopon ・ Theora ・ Theora (genus) ・ Theora Hamblett ・ Theora mesopotamica ・ Theorbo ・ Theorem ・ Theorem of Bertini ・ Theorem of corresponding states ・ Theorem of the cube ・ Theorem of the three geodesics ・ Theorem of three moments ・ Theorem on formal functions ・ Theorem on friends and strangers ・ Theorem prover ・ Theorem Proving System ・ Theorem Stencil ・ Theorema ・ Theorema (disambiguation) ・ Theorema Egregium ・ Theorems and definitions in linear algebra ・ Theoren Fleury ・ Theoretical Advanced Study Institute ・ Theoretical and Applied Climatology ・ Theoretical and Applied Genetics ・ Theoretical and experimental justification for the Schrödinger equation ・ Theoretical and Mathematical Physics Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Theorem on formal functions ： ウィキペディア英語版 Theorem on formal functions In algebraic geometry, the theorem on formal functions states the following: :Let $f:\; X\; \backslash to\; S$ be a proper morphism of noetherian schemes with a coherent sheaf $\backslash mathcal$ on ''X''. Let $S\_0$ be a closed subscheme of ''S'' defined by $\backslash mathcal$ and $\backslash widehat,\; \backslash widehat$ formal completions with respect to $X\_0\; =\; f^(S\_0)$ and $S\_0$. Then for each $p\; \backslash ge\; 0$ the canonical (continuous) map: ::$(R^p\; f\_$ * \mathcal)^\wedge \to \varprojlim_k R^p f_ * \mathcal_k :is an isomorphism of (topological) $\backslash mathcal\_\; \backslash otimes\_\; \backslash mathcal\_S/\}$. : *$\backslash mathcal\_k\; =\; \backslash mathcal\; \backslash otimes\_\; (\backslash mathcal\_S/)$ : *The canonical map is one obtained by passage to limit. The theorem is used to deduce some other important theorems: Stein factorization and a version of Zariski's main theorem that says that a proper birational morphism into a normal variety is an isomorphism. Some other corollaries (with the notations as above) are: Corollary: For any $s\; \backslash in\; S$, topologically, :$((R^p\; f\_$ * \mathcal)_s)^\wedge \simeq \varprojlim H^p(f^(s), \mathcal\otimes_ (\mathcal_s/\mathfrak_s^k)) where the completion on the left is with respect to $\backslash mathfrak\_s$. Corollary: Let ''r'' be such that $\backslash operatorname\; f^(s)\; \backslash le\; r$ for all $s\; \backslash in\; S$. Then :$R^i\; f\_$ * \mathcal = 0, \quad i > r. Corollay:〔The same argument as in the preceding corollary〕 For each $s\; \backslash in\; S$, there exists an open neighborhood ''U'' of ''s'' such that :$R^i\; f\_$ * \mathcal|_U = 0, \quad i > \operatorname f^(s). Corollary: If $f\_$ * \mathcal_X = \mathcal_S, then $f^(s)$ is connected for all $s\; \backslash in\; S$. The theorem also leads to the Grothendieck existence theorem, which gives an equivalence between the category of coherent sheaves on a scheme and the category of coherent sheaves on its formal completion (in particular, it yields algebralizability.) Finally, it is possible to weaken the hypothesis in the theorem; cf. Illusie. According to Illusie (pg. 204), the proof given in EGA III is due to Serre. The original proof (due to Grothendieck) was never published. == The construction of the canonical map == Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map. Let $i\text{'}:\; \backslash widehat\; \backslash to\; X,\; i:\; \backslash widehat\; \backslash to\; S$ be the canonical maps. Then we have the base change map of $\backslash mathcal\_\; \backslash to\; R^p\; \backslash widehat\_$ * (i'^ * \mathcal). where $\backslash widehat:\; \backslash widehat\; \backslash to\; \backslash widehat$ is induced by $f:\; X\; \backslash to\; S$. Since $\backslash mathcal$ is coherent, we can identify $i\text{'}^$ *\mathcal with $\backslash widehat$ is also coherent (as ''f'' is proper), doing the same identification, the above reads: :$(R^q\; f\_$ * \mathcal)^\wedge \to R^p \widehat_ * \widehat_X/\mathcal^) and $S\_n\; =\; (S\_0,\; \backslash mathcal\_S/\backslash mathcal^)$, one also obtains (after passing to limit): :$R^q\; \backslash widehat\_$ * \widehat_n where $\backslash mathcal\_n$ are as before. One can verify that the composition of the two maps is the same map in the lede. (cf. EGA III-1, section 4) 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「Theorem on formal functions」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.