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 Étale morphisms and the inverse function theorem
5 Etymology
6 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 flat, locally of finite presentation, and for every $y$ in $Y$ , the fiber $f - 1 (y)$ is the disjoint union of points, each of which is the spectrum of a finite separable field extension of the residue field $κ(y)$ .
$f$ is flat, locally of finite presentation, and for every $y$ in $Y$ and every algebraic closure $k'$ of the residue field $κ(y)$ , the geometric fiber $f^{-1}(y) \otimes_{\kappa(y)} k'$ is the disjoint union of points, each of which is isomorphic to $Spec k'$ .
$f$ is a smooth morphism of relative dimension zero.
$f$ is a smooth morphism and a quasi-finite morphism.
$f$ 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 $f s = h$ .
$f$ is locally of finite presentation and on open affines, $f$ is formally étale with respect to the discrete topology, that is,

Let $x$ be a point of $X$ and let $y = f (x)$ . Choose 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 $Spec R$ . Write $f^{\#}$ for the induced homomorphism $R \to S$ . Suppose that $A$ is a ring having an ideal $I$ such that $I 2 = 0$ . Let $A 0 = A / I$ , and let $r : A \to A_0$ be the quotient map. Suppose further that there are homomorphisms $g : S \to A_0$ and $h : R \to A$ such that $rh = gf^{\#}$ . Then there exists a unique morphism $s : S \to A$ such that $r s = g$ and $sf^{\#} = h$ .
$f$ is locally of finite presentation and on stalks, $f$ is formally étale with respect to the discrete topology, that is,

For every $x$ in $X$ , let $y = f (x)$ . Then the induced morphism on local rings $\mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}$ is formally étale with respect to the discrete topology.

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

Assume that $Y$ is locally noetherian. For $x$ in $X$ , let $y = f (x)$ and let $\hat{\mathcal O}_{Y,y} \to \hat{\mathcal O}_{X,x}$ be the induced map on completed local rings. Then the following are equivalent:

$f$ is étale.
For every $x$ in $X$ , the induced map on completed local rings is formally étale for the adic topology.
For every $x$ in $X$ , $\hat{\mathcal O}_{X,x}$ is a free $\hat{\mathcal O}_{Y,y}$ -module and the fiber $\hat{\mathcal O}_{X,x}/m_y$ is a field which is a finite separable field extension of the residue field $κ(y)$ . (Here $m y$ is the maximal ideal of $\hat{\mathcal O}_{Y,y}$ .)

If in addition all the maps on residue fields $\kappa(y) \to \kappa(x)$ are isomorphisms, or if $κ(y)$ is separably closed, then $f$ is étale if and only if

For every $x$ in $X$ , the induced map on completed local rings is an isomorphism.

[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 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.
Étale morphisms are local on the source and on the base.
The product of a finite family of étale maps is étale.
Given a finite 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.
Let $f : X \to Y$ and $g : Y \to Z$ , and assume that $g$ is unramified and $g f$ is étale. Then $f$ is étale.
If $X$ and $X'$ are étale over $Y$ , then any $Y$ -map between $X$ and $X'$ is étale.
Quasi-compact étale morphisms are quasi-finite.
If $f : X \to Y$ is étale, then dim $X$ = dim $Y$ .
A morphism $f : X \to Y$ is an open immersion if and only if it is étale and radicial.

[edit] Étale morphisms and the inverse function theorem

As said in the introduction, étale maps

f: X → Y

are the algebraic counterpart of local diffeomorphisms. More precisely, a morphism between smooth varieties is étale at a point iff the differential between the corresponding tangent spaces is an isomorphism. This is in turn precisely the condition needed to ensure that a map between manifolds is a local diffeomorphism, i.e. for any point y ∈ Y, there is an open neighborhood U of x such that the restriction of f to U is a diffeomorphism. This conclusion does not hold in algebraic geometry, because the topology is too coarse. For example, consider the projection f of the parabola

y=x²

to the y-axis. This map is étale at every point except the origin (0, 0), because the differential is given by 2x, which does not vanish at these points.

However, there is no (Zariski-)local inverse of f, just because the square root is not an algebraic map, not being given by polynomials. However, there is a remedy for this situation, using the étale topology. The precise statement is as follows: if f as above is smooth and surjective, Y is quasi-compact then there is a scheme X' , étale over X such that f has a section X' → Y. In other words, étale-locally, the map f does have a section.

[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.