étale morphism

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:

• $f$ 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 $A[x_1,\mathrm{\dots },x_n]/(p_1,\mathrm{\dots },p_n)$ where theJacobian vanishes.

A morphism $f:X\to Y$ of varieties over an algebraicallyclosed field is étale at a point $x\in X$ if it induces anisomorphism between the completed local rings ${\widehat{𝒪}}_x$and ${\widehat{𝒪}}_{f(x)}$. If $X$ and$Y$ are over an arbitrary field $k$, then the requiredcondition becomes that $k(x)$ is a separable algebraic extensionof $k(y)$, where $y=f(x)$, and $f$ induces an isomorphism between${\widehat{𝒪}}_y{\otimes }_{k(y)}k(x)$ and ${\widehat{𝒪}}_x$.

A morphism $f$ of nonsingular varieties over an algebraically closedfield is étale if and only if $f$ induces an isomorphism on the tangent spaces. In the differentiable category, the implicit function theoremimplies that such a function is actually an isomorphism on some smallneighborhood. On schemes, of course, the Zariski topology is toocoarse 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.