ÉTALE MORPHISM

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.

Contents

Definition

Examples of étale morphisms

Properties of étale morphisms

Etymology

References

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 $f\text{'}$ 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 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 $mbox\{Spec\; \}\; R$ 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 $fs=h$.
The equivalence of these properties is difficult and relies heavily on Zariski's main theorem.
If all the extensions of residue fields of local rings are trivial, then the conditions above
are also equivalent to:

★ $f$ is locally of finite presentation and for every $x$ in $X$, the induced map on completed local rings $hat\{mathcal\; O\}\_\{Y,f(x)\}\; o\; hat\{mathcal\; O\}\_\{X,x\}$ 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.

★ If $X$ and $X\text{'}$ are étale over $Y$, then any $Y$-map between $X$ and $X\text{'}$ is étale.

★ Étale morphisms are local on the base.

★ Given a finite family of maps , the disjoint union is étale if and only if each is étale.

★ Quasi-compact étale morphisms are quasi-finite.

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.

References

★ Algebraic Geometry, Robin Hartshorne, , , Springer-Verlag, 1977, ISBN 0387902449

★ Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Première partie, , Alexandre, Grothendieck, Publications Mathématiques de l'IHÉS, 1964

★ Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie, , Alexandre, Grothendieck, Publications Mathématiques de l'IHÉS, 1967

★ Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques '3'), , Alexandre, Grothendieck, Société Mathématique de France, 2003, ISBN 2-85629-141-4

★ Étale cohomology, J. S. Milne, , , Princeton University Press, 1980,