- Étale morphism
In
algebraic geometry , a field ofmathematics , 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 theimplicit function theorem , but because open sets in theZariski 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 thealgebraic fundamental group and theétale topology .Definition
Let be a
ring homomorphism . This makes an -algebra. Choose amonic polynomial in and a polynomial in such that thederivative of is a unit in the localization . We say that is "standard étale" if and can be chosen so that is isomorphic as an -algebra to . Geometrically, this represents as an open subset of acovering space .Let be a morphism of schemes. We say that is "étale" if it has any of the following equivalent properties:
# is flat and unramified.
# is flat, locally of finite presentation, and for every in , the fiber is the disjoint union of points, each of which is the spectrum of a finite separable field extension of the residue field .
# is flat, locally of finite presentation, and for every in and every algebraic closure of the residue field , the geometric fiber is the disjoint union of points, each of which is isomorphic to .
# is asmooth morphism of relative dimension zero.
# is a smooth morphism and aquasi-finite morphism .
# is locally of finite presentation and is locally a standard étale morphism, that is,
#:For every in , let . Then there is an open affine neighborhood of and an open affine neighborhood of such that is contained in and such that the ring homomorphism induced by is standard étale.
# is locally of finite presentation and is formally étale with respect to the discrete topology, that is,
#:Suppose that is a scheme having a sheaf of ideals such that . Let , and let be the induced map. Suppose further that there are morphisms and such that . Then there exists a unique morphism such that and .
# is locally of finite presentation and on open affines, is formally étale with respect to the discrete topology, that is,
#:Let be a point of and let . Choose an open affine neighborhood of and an open affine neighborhood of such that is contained in . Write for the induced homomorphism . Suppose that is a ring having an ideal such that . Let , and let be the quotient map. Suppose further that there are homomorphisms and such that . Then there exists a unique morphism such that and .
# is locally of finite presentation and on stalks, is formally étale with respect to the discrete topology, that is,
#:For every in , let . Then the induced morphism on local rings 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 is locally noetherian. For in , let and let be the induced map on completed local rings. Then the following are equivalent:
# is étale.
# For every in , the induced map on completed local rings is formally étale for the adic topology.
# For every in , is a free -module and the fiber is a field which is a finite separable field extension of the residue field . (Here is the maximal ideal of .)If in addition all the maps on residue fields are isomorphisms, or if is separably closed, then is étale if and only if
* For every in , 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 , where all the are polynomials, and where the
Jacobian determinant is a unit in , is étale.Expanding upon the previous example, suppose that we have a morphism of smooth complex algebraic varieties. Since is given by equations, we can interpret it as a map of complex manifolds. Whenever the Jacobian of is nonzero, 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 , the disjoint union is étale if and only if each is étale.
* Let and , and assume that is unramified and is étale. Then is étale.
* If and are étale over , then any -map between and is étale.
* Quasi-compact étale morphisms are quasi-finite.
* If is étale, then dim = dim .
* A morphism 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 correspondingtangent space s is an isomorphism. This is in turn precisely the condition needed to ensure that a map betweenmanifold s 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 theparabola :"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" isquasi-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 theverb 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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