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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 Smooth morphism ： ウィキペディア英語版 Smooth morphism In algebraic geometry, a morphism $f:X\; \backslash to\; S$ between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point $\backslash overline\; \backslash to\; S$ the fiber $X\_\}$ is regular. (iii) means that for any $s\; \backslash in\; S$ the fiber $f^(s)$ is a nonsingular variety. Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If ''S'' is the spectrum of a field and ''f'' is of finite type, then one recovers the definition of a nonsingular variety. There are many equivalent definitions of a smooth morphism. Let $f:\; X\; \backslash to\; S$ be locally of finite presentation. Then the following are equivalent. # ''f'' is smooth. # ''f'' is formally smooth (see below). # ''f'' is flat and the sheaf of relative differentials $\backslash Omega\_$ is locally free of rank equal to the relative dimension of $X/S$. # For any $s\; \backslash in\; S$, there exists a neighborhood $\backslash operatornameB$ of ''s'' and a neighborhood $\backslash operatornameA$ of $f(s)$ such that $B\; =\; A(\backslash dots,\; t\_n)/(P\_1,\; \backslash dots,\; P\_m)$ and the ideal generated by the ''m''-by-''m'' minors of $(\backslash partial\; P\_i/\backslash partial\; t\_j)$ is ''B''. # Locally, ''f'' factors into $X\; \backslash overset\backslash to\; \backslash mathbb^n\_S\; \backslash to\; S$ where ''g'' is étale. # Locally, ''f'' factors into $X\; \backslash overset\backslash to\; \backslash mathbb^n\_S\; \backslash to\; \backslash mathbb^\_S\; \backslash to\; \backslash cdots\; \backslash to\; \backslash mathbb^1\_S\; \backslash to\; S$ where ''g'' is étale. A morphism of finite type is étale if and only if it is smooth and quasi-finite. A smooth morphism is stable under base change and composition. A smooth morphism is locally of finite presentation. A smooth morphism is universally locally acyclic. == Formally smooth morphism == One can define smoothness without reference to geometry. We say that an ''S''-scheme ''X'' is formally smooth if for any affine ''S''-scheme ''T'' and a subscheme $T\_0$ of ''T'' given by a nilpotent ideal, $X(T)\; \backslash to\; X(T\_0)$ is surjective where we wrote $X(T)\; =\; \backslash operatorname\_S(T,\; X)$. Then a morphism locally of finite type is smooth if and only if it is formally smooth. In the definition of "formally smooth", if we replace surjective by "bijective" (resp. "injective"), then we get the definition of formally étale (resp. formally unramified). 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「Smooth morphism」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.