# étale morphism

one way

###### Definition 1

A morphism of schemes $f:X\to Y$ 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:

• $f$ 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.

## References

• 1 Jean Dieudonné, A Panorama of Pure Mathematics, Academic Press, 1982.
• 2 Robin Hartshorne, , Springer–Verlag, 1977 (GTM 52).
 Title étale morphism Canonical name etaleMorphism Date of creation 2013-03-22 14:08:40 Last modified on 2013-03-22 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