étale morphism
\PMlinkescapephrase
one way
Definition 1
A morphism of schemes $f\mathrm{:}X\mathrm{\to}Y$ is étale if it is flat and unramified.
This is the appropriate generalization^{} of “local homeomorphism” from topology^{} or “local isomorphism” from real differential geometry. Equivalently, $f$ is étale if and only if any of the following conditions hold:

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$f$ is locally of finite type and formally étale.

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$f$ is flat and the relative sheaf of differentials^{} vanishes.

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$f$ is smooth of relative dimension zero.

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$f$ locally looks like $A[{x}_{1},\mathrm{\dots},{x}_{n}]/({p}_{1},\mathrm{\dots},{p}_{n})$ where the Jacobian vanishes.
A morphism $f:X\to Y$ of varieties^{} over an algebraically closed field is étale at a point $x\in X$ if it induces an isomorphism^{} between the completed local rings ${\widehat{\mathcal{O}}}_{x}$ and ${\widehat{\mathcal{O}}}_{f(x)}$. If $X$ and $Y$ are over an arbitrary field $k$, then the required condition becomes that $k(x)$ is a separable^{} algebraic extension^{} of $k(y)$, where $y=f(x)$, and $f$ induces an isomorphism between ${\widehat{\mathcal{O}}}_{y}{\otimes}_{k(y)}k(x)$ and ${\widehat{\mathcal{O}}}_{x}$.
A morphism $f$ of nonsingular varieties over an algebraically closed field is étale if and only if $f$ induces an isomorphism on the tangent spaces^{}. In the differentiable^{} category, the implicit function theorem implies that such a function is actually an isomorphism on some small neighborhood. On schemes, of course, the Zariski topology^{} is too coarse for this to be the case. One way to define a finer “topology”, making the scheme into a site, is by using étale maps.
The word étale comes from French, where it can be used to describe a calm or slack sea.
References
 1 Jean Dieudonné, A Panorama of Pure Mathematics, Academic Press, 1982.
 2 Robin Hartshorne, Algebraic Geometry^{}, Springer–Verlag, 1977 (GTM 52).
Title  étale morphism 
Canonical name  etaleMorphism 
Date of creation  20130322 14:08:40 
Last modified on  20130322 14:08:40 
Owner  mps (409) 
Last modified by  mps (409) 
Numerical id  14 
Author  mps (409) 
Entry type  Definition 
Classification  msc 14F20 
Classification  msc 14A15 
Synonym  étale 
Related topic  site 
Related topic  Site 
Related topic  FlatMorphism 
Related topic  EtaleFundamentalGroup 
Related topic  EtaleCohomology 
Related topic  CoveringSpace 