• $f$ is locally of finite type and formally étale.
• $f$ is flat and the relative sheaf of differentials vanishes.
• $f$ is smooth of relative dimension zero.
• $f$ locally looks like $A[x_1,\ldots,x_n]/(p_1,\ldots,p_n)$ 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.

Bibliography

1

Jean Dieudonné, A Panorama of Pure Mathematics, Academic Press, 1982.

2

Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977 (GTM 52).