Étale morphism

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In algebraic geometry, a field of mathematics, an étale morphism is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.

1 Definition
2 Examples of étale morphisms
3 Properties of étale morphisms
4 Etymology
5 References

[edit] Definition

Let $\phi : R \to S$ be a ring homomorphism. This makes $S$ an $R$ -algebra. Choose a monic polynomial $f$ in $R [x]$ and a polynomial $g$ in $R [x]$ such that the derivative $f'$ of $f$ is a unit in the localization $R [x] g$ . We say that $φ$ is standard étale if $f$ and $g$ can be chosen so that $S$ is isomorphic as an $R$ -algebra to $(R [x] / f R [x]) g$ . Geometrically, this represents $φ$ as an open subset of a covering space.

Let $f : X \to Y$ be a morphism of schemes. We say that $f$ is étale if it has any of the following equivalent properties:

$f$ is flat and unramified.
$f$ is a smooth morphism of relative dimension zero.
$f$ is locally of finite presentation and is locally a standard étale morphism, that is,

For every $x$ in $X$ , let $y = f (x)$ . Then there is an open affine neighborhood $Spec R$ of $y$ and an open affine neighborhood $Spec S$ of $x$ such that $f (Spec S)$ is contained in $Spec R$ and such that the ring homomorphism $R \to S$ induced by $f$ is standard étale.
$f$ is locally of finite presentation and is formally étale with respect to the discrete topology, that is,

Suppose that $Z$ is a scheme having a sheaf of ideals $I$ such that $I 2 = 0$ . Let $Z 0 = Spec (O Z / I)$ , and let $r : Z_0 \to Z$ be the induced map. Suppose further that there are morphisms $g : Z_0 \to X$ and $h : Z \to Y$ such that $h r = f g$ . Then there exists a unique morphism $s : Z \to X$ such that $s r = g$ and $f s = h$ .
For every $x$ in $X$ , the induced map on completed local rings $\hat{\mathcal O}_{Y,f(x)} \to \hat{\mathcal O}_{X,x}$ is an isomorphism.

The equivalence of these properties is difficult and relies heavily on Zariski's main theorem.

[edit] Examples of étale morphisms

Any open immersion is an étale map, by the description of étale maps in terms of standard étale maps.

Finite separable field extensions are étale.

Any ring homomorphism of the form $R \to S=R[x_1,\ldots,x_n]_g/(f_1,\ldots, f_n)$ , where all the $f i$ are monic polynomials, and where the Jacobian determinant $\det(\partial f_i/\partial x_j)$ is a unit in $S$ , is étale.

Expanding upon the previous example, suppose that we have a morphism $f$ of smooth complex algebraic varieties. Since $f$ is given by equations, we can interpret it as a map of complex manifolds. Whenever the Jacobian of $f$ is nonzero, $f$ is a local isomorphism of complex manifolds by the implicit function theorem. By the previous example, having non-zero Jacobian is the same as being étale.

[edit] Properties of étale morphisms

Étale morphisms are preserved under composition and base change. If $X$ and $X'$ are étale over $Y$ , then any $Y$ -map between $X$ and $X'$ is étale. Étale morphisms are local on the base.

Given a family of maps $\{f_\alpha : X_\alpha \to Y\}$ , the disjoint union $\coprod f_\alpha : \coprod X_\alpha \to Y$ is étale if and only if each $f α$ is étale.

[edit] Etymology

The word étale is French, and it can have two distinct meanings, both of which are applicable to étale morphisms. One meaning is "spread out". The other, more common in poetry, describes the appearance of a calm sea under a full moon.