\'etale morphismEtaleMorphism2004-02-10 15:20:362006-02-09 01:34:55Definition/ay-tal''''''''/\'etale morphism% this is the default PlanetMath preamble. as your knowledge
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\begin{defn}
A morphism of schemes $f:X\to Y$ is \emph{\'etale} if it is flat and unramified.
\end{defn}
This is the appropriate generalization of ``local homeomorphism'' from topology or ``local isomorphism'' from real differential geometry.
Equivalently, $f$ is \'etale if and only if any of the following conditions hold:
\begin{itemize}
\item
$f$ is locally of finite type and formally \'etale.
\item
$f$ is flat and the relative sheaf of differentials vanishes.
\item
$f$ is smooth of relative dimension zero.
\item
$f$ locally looks like $A[x_1,\ldots,x_n]/(p_1,\ldots,p_n)$ where the
Jacobian vanishes.
\end{itemize}
A morphism $f:X\to Y$ of varieties over an algebraically
closed field is \'etale 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
%$\widehat{\mathcal{O}}_y$
$\widehat{\mathcal{O}}_y \otimes_{k(y)} k(x)$ and $\widehat{\mathcal{O}}_x$.
A morphism $f$ of nonsingular varieties over an algebraically closed
field is \'etale 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 \'etale maps.
The word \'etale comes from French, where it can be used to describe a calm or slack sea.
\begin{thebibliography}{9}
\bibitem{dieudonne} Jean Dieudonn\'{e}, {\em A Panorama of Pure Mathematics}, Academic Press, 1982.
\bibitem{hartshorne} Robin Hartshorne, {\em Algebraic
Geometry}, Springer--Verlag, 1977 (GTM {\bf 52}).
\end{thebibliography}