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 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 ${\widehat{𝒪}}_x$ and ${\widehat{𝒪}}_{f(x)}$. 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 ${\widehat{𝒪}}_y{\otimes }_{k(y)}k(x)$ and ${\widehat{𝒪}}_x$.

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.

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