Étale morphism

Étale morphism

In algebraic geometry, a field of mathematics, an étale morphism (pronunciation IPA|) 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.

Definition

Let phi : R o 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 phi is "standard étale" if f and g can be chosen so that S is isomorphic as an R-algebra to (R [x] /fR [x] )_g. Geometrically, this represents phi as an open subset of a covering space.

Let f : X o 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 kappa(y).
# f is flat, locally of finite presentation, and for every y in Y and every algebraic closure k' of the residue field kappa(y), the geometric fiber f^{-1}(y) otimes_{kappa(y)} k' is the disjoint union of points, each of which is isomorphic to mbox{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 mbox{Spec } R of y and an open affine neighborhood mbox{Spec } S of x such that f(mbox{Spec } S) is contained in mbox{Spec } R and such that the ring homomorphism R o 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 = mbox{Spec }(O_Z/I), and let r : Z_0 o Z be the induced map. Suppose further that there are morphisms g : Z_0 o X and h : Z o Y such that hr = fg. Then there exists a unique morphism s : Z o X such that sr=g and fs=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 mbox{Spec } R of y and an open affine neighborhood mbox{Spec } S of x such that f(mbox{Spec } S) is contained in mbox{Spec } R. Write f^{#} for the induced homomorphism R o 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 o A_0 be the quotient map. Suppose further that there are homomorphisms g : S o A_0 and h : R o A such that rh = gf^{#}. Then there exists a unique morphism s : S o A such that rs = 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} o 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} o 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 kappa(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) o kappa(x) are isomorphisms, or if kappa(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.

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 o 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.

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 o Y}, the disjoint union coprod f_alpha : coprod X_alpha o Y is étale if and only if each f_alpha is étale.
* Letf : X o Y and g : Y o Z, and assume that g is unramified and gf 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 o Y is étale, then dim X = dim Y.
* A morphism f : X o Y is an open immersion if and only if it is étale and radicial.

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"="x"2to the "y"-axis. This map is étale at every point except the origin (0, 0), because the differential is given by 2"x", 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.

Etymology

The word étale is a French adjective, which means calm, immobile, something left to settle, for example the surface of a sea. It is related to the verb étaler, which has (among others) the meaning of spreading out, for example dough, on a surface.

References

* | year=1977
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1964
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Première partie
journal = Publications Mathématiques de l'IHÉS
volume = 20
pages = 5–259
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1964__20_

*
*
*
*cite book |author=J. S. Milne |title=Étale cohomology |publisher=Princeton University Press |location=Princeton, N.J |year=1980 |pages= |isbn=0-691-08238-3 |oclc= |doi=


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