nLab
etale morphism

This entry is about étale morphisms between schemes. The word étale map is preferred in the context of topology and differential geometry, see étalé space for the topological version.

Definition

Definition

A morphism of schemes is an étale morphism if the following equivalent conditions hold

1. it is

   1. smooth

   2. of relative dimension $0$.

2. it is

   1. flat

   2. unramified;

3. it is

   1. formally étale;

   2. locally of finite presentation

(A number of other equivalent definitions are listed at wikipedia.)

Definition

Jointly surjective collection of étale morphisms $\{U_i\to X\}$ is called an étale cover.

There is a weaker notion of a formally étale morphism.

Definition

A morphism is formally étale morphism if it is

• formally smooth (satisfying an infinitesimal lifting property)

• and formally unramified.

Definition

These are sheaf-like properties, which can be formalized in the language of Q-categories (monopresheaf and epipresheaf properties on the $Q$-category of nilpotent thickenings).

Properties

Closure properties

• A composite of étale morphism is étale.

• The property of being étale is preserved under pullbacks along any morphism of schemes.

• A smooth map of schemes is étale iff there is an étale cover of the base scheme by open subschemes such that the pullback of the projection to each of them is an open local isomorphism of locally ringed spaces (and in particular, the pullback of the projection morphism is an étale map of the corresponding underlying topological spaces).

  This disjointness picture of étale covers make them suitable for having nontrivial cohomology in situations where Zariski covers give vanishing cohomology.

Classes of examples

Proposition

Let $k$ be a field. A morphism of schemes $Y\to \mathrm{Spec}k$ is étale precisely if $Y$ is a coproduct $Y\simeq {\coprod }_i\mathrm{Spec}{k}_i$ for each $k_i$ a finite and separable field extension of $k$.

This appears for instance as de Jong, prop. 3.1 i).

Remark

Such étale morphisms are classified by the classical Galois theory of field extensions.

Proposition

A ring homomorphism $A\to B$ is étale precisely if $B\simeq R[x_1,\cdots ,x_n]/(f_1,\cdots ,f_n)$ where

• all the $f_i$ are polynomials;

• the Jacobian $\det (\frac{\partial f_i}{\partial x_j})$ is a unit in $S$.

This appears for instance as de Jong, prop. 3.1 ii).

As locally constant sheaves

Proposition

A sheaf $F$ on a scheme $X$ corresponds to an étale morphism $Y\to X$ precisely if there is an étale cover $\{U_i\to X\}$ such that each restriction

$$F{\mid }_{U_i}\simeq \mathrm{LConst}{K}_i$$F|_{U_i} \simeq LConst K_i

is isomorphic to a constant sheaf on a set $K_i$.

A proof is in (Deligne).

Examples

• A finite separable field extension $K\hookrightarrow L$ corresponds dually to an étale morphism $\mathrm{Spec}L\to \mathrm{Spec}K$. These are the morphisms classified by classical Galois theory.

Related concepts

• étale morphism, étale site, étale cohomology

• étale (∞,1)-site

References

The classical references are

• Pierre Deligne et al., Cohomologie étale , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.

• James Milne, Etale cohomology, Princeton Mathematical Series 33, 1980. xiii+323 pp.

Lecture notes are

• James Milne, Lectures on etale cohomology (pdf)

• Aise Johan de Jong, Étale cohomology (pdf)