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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 Étale morphism ： ウィキペディア英語版 Étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology. The word ''étale'' is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.〔'':fr: Trésor de la langue française informatisé'', ("étale" article )〕 == Definition == Let $\backslash phi\; :\; R\; \backslash to\; S$ be a ring homomorphism. This makes $S$ an $R$-algebra. Choose a monic polynomial $f$ in $R()$ and a polynomial $g$ in $R()$ such that the derivative $f\text{'}$ of $f$ is a unit in $(R()/fR())\_g$. We say that $\backslash phi$ is ''standard étale'' if $f$ and $g$ can be chosen so that $S$ is isomorphic as an $R$-algebra to $(R()/fR())\_g$ and $\backslash phi$ is the canonical map. Let $f\; :\; X\; \backslash to\; Y$ be a morphism of schemes. We say that $f$ is ''étale'' if it has any of the following equivalent properties: # $f$ is flat and unramified.〔EGA IV4, Corollaire 17.6.2.〕 # $f$ is a smooth morphism and unramified.〔 # $f$ is flat, locally of finite presentation, and for every $y$ in $Y$, the fiber $f^(y)$ is the disjoint union of points, each of which is the spectrum of a finite separable field extension of the residue field $\backslash kappa(y)$.〔 # $f$ is flat, locally of finite presentation, and for every $y$ in $Y$ and every algebraic closure $k\text{'}$ of the residue field $\backslash kappa(y)$, the geometric fiber $f^(y)\; \backslash otimes\_\; k\text{'}$ is the disjoint union of points, each of which is isomorphic to $\backslash mbox\; k\text{'}$.〔 # $f$ is a smooth morphism of relative dimension zero.〔EGA IV4, Corollaire 17.10.2.〕 # $f$ is a smooth morphism and a locally quasi-finite morphism.〔EGA IV4, Corollaire 17.6.2 and Corollaire 17.10.2.〕 # $f$ is locally of finite presentation and is locally a standard étale morphism, that is, #:For every $x$ in $X$, let $y\; =\; f(x)$. Then there is an open affine neighborhood of $y$ and an open affine neighborhood of $x$ such that is contained in and such that the ring homomorphism induced by $f$ is standard étale.〔Milne, ''Étale cohomology'', Theorem 3.14.〕 # $f$ is locally of finite presentation and is formally étale.〔 # $f$ is locally of finite presentation and is formally étale for maps from local rings, that is: #:Let ''A'' be a local ring and ''J'' be an ideal of ''A'' such that . Set and , and let be the canonical closed immersion. Let ''z'' denote the closed point of ''Z''0. Let and be morphisms such that . Then there exists a unique ''Y''-morphism such that .〔EGA IV4, Corollaire 17.14.1.〕 Assume that $Y$ is locally noetherian and ''f'' is locally of finite type. For $x$ in $X$, let $y\; =\; f(x)$ and let $\backslash hat\_\; \backslash to\; \backslash hat\_$ be the induced map on completed local rings. Then the following are equivalent: # $f$ is étale. # For every $x$ in $X$, the induced map on completed local rings is formally étale for the adic topology.〔EGA IV4, Proposition 17.6.3〕 # For every $x$ in $X$, $\backslash hat\_$ is a free $\backslash hat\_$-module and the fiber $\backslash hat\_/m\_y\backslash hat\_$ is a field which is a finite separable field extension of the residue field $\backslash kappa(y)$.〔 (Here $m\_y$ is the maximal ideal of $\backslash hat\_$.) # ''f'' is formally étale for maps of local rings with the following additional properties. The local ring ''A'' may be assumed Artinian. If ''m'' is the maximal ideal of ''A'', then ''J'' may be assumed to satisfy . Finally, the morphism on residue fields may be assumed to be an isomorphism.〔EGA IV4, Proposition 17.14.2〕 If in addition all the maps on residue fields $\backslash kappa(y)\; \backslash to\; \backslash kappa(x)$ are isomorphisms, or if $\backslash kappa(y)$ is separably closed, then $f$ is étale if and only if for every $x$ in $X$, the induced map on completed local rings is an isomorphism.〔 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「Étale 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.