Étale morphism

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In algebraic geometry, an étale morphism (French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation. This 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.

The word étale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.[1]

Definition

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Let   be a ring homomorphism. This makes   an  -algebra. Choose a monic polynomial   in   and a polynomial   in   such that the derivative   of   is a unit in  . We say that   is standard étale if   and   can be chosen so that   is isomorphic as an  -algebra to   and   is the canonical map.

Let   be a morphism of schemes. We say that   is étale if and only if it has any of the following equivalent properties:

  1.   is flat and unramified.[2]
  2.   is a smooth morphism and unramified.[2]
  3.   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  .[2]
  4.   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  .[2]
  5.   is a smooth morphism of relative dimension zero.[3]
  6.   is a smooth morphism and a locally quasi-finite morphism.[4]
  7.   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.[5]
  8.   is locally of finite presentation and is formally étale.[2]
  9.   is locally of finite presentation and is formally étale for maps from local rings, that is:
    Let   be a local ring and   be an ideal of   such that  . Set   and  , and let   be the canonical closed immersion. Let   denote the closed point of  . Let   and   be morphisms such that  . Then there exists a unique  -morphism   such that  .[6]

Assume that   is locally noetherian and f is locally of finite type. For   in  , let   and let   be the induced map on completed local rings. Then the following are equivalent:

  1.   is étale.
  2. For every   in  , the induced map on completed local rings is formally étale for the adic topology.[7]
  3. For every   in  ,   is a free  -module and the fiber   is a field which is a finite separable field extension of the residue field  .[7] (Here   is the maximal ideal of  .)
  4.   is formally étale for maps of local rings with the following additional properties. The local ring   may be assumed Artinian. If   is the maximal ideal of  , then   may be assumed to satisfy  . Finally, the morphism on residue fields   may be assumed to be an isomorphism.[8]

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.[7]

Examples

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Any open immersion is étale because it is locally an isomorphism.

Covering spaces form examples of étale morphisms. For example, if   is an integer invertible in the ring   then

 

is a degree   étale morphism.

Any ramified covering   has an unramified locus

 

which is étale.

Morphisms

 

induced by finite separable field extensions are étale — they form arithmetic covering spaces with group of deck transformations given by  .

Any ring homomorphism of the form  , where all the   are polynomials, and where the Jacobian determinant   is a unit in  , is étale. For example the morphism   is etale and corresponds to a degree   covering space of   with the group   of deck transformations.

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.

Let   be a dominant morphism of finite type with X, Y locally noetherian, irreducible and Y normal. If f is unramified, then it is étale.[9]

For a field K, any K-algebra A is necessarily flat. Therefore, A is an etale algebra if and only if it is unramified, which is also equivalent to

 

where   is the separable closure of the field K and the right hand side is a finite direct sum, all of whose summands are  . This characterization of etale K-algebras is a stepping stone in reinterpreting classical Galois theory (see Grothendieck's Galois theory).

Properties

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  • Étale morphisms are preserved under composition and base change.
  • Étale morphisms are local on the source and on the base. In other words,   is étale if and only if for each covering of   by open subschemes the restriction of   to each of the open subschemes of the covering is étale, and also if and only if for each cover of   by open subschemes the induced morphisms   is étale for each subscheme   of the covering. In particular, it is possible to test the property of being étale on open affines  .
  • The product of a finite family of étale morphisms is étale.
  • Given a finite family of morphisms  , 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. In particular, if   and   are étale over  , then any  -morphism between   and   is étale.
  • Quasi-compact étale morphisms are quasi-finite.
  • A morphism   is an open immersion if and only if it is étale and radicial.[10]
  • If   is étale and surjective, then   (finite or otherwise).

Inverse function theorem

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Étale morphisms

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 yY, 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 morphism 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   is étale and finite, then for any point y lying in Y, there is an étale morphism VY containing y in its image (V can be thought of as an étale open neighborhood of y), such that when we base change f to V, then   (the first member would be the pre-image of V by f if V were a Zariski open neighborhood) is a finite disjoint union of open subsets isomorphic to V. In other words, étale-locally in Y, the morphism f is a topological finite cover.

For a smooth morphism   of relative dimension n, étale-locally in X and in Y, f is an open immersion into an affine space  . This is the étale analogue version of the structure theorem on submersions.

See also

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References

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  1. ^ fr: Trésor de la langue française informatisé, "étale" article
  2. ^ a b c d e EGA IV4, Corollaire 17.6.2.
  3. ^ EGA IV4, Corollaire 17.10.2.
  4. ^ EGA IV4, Corollaire 17.6.2 and Corollaire 17.10.2.
  5. ^ Milne, Étale cohomology, Theorem 3.14.
  6. ^ EGA IV4, Corollaire 17.14.1.
  7. ^ a b c EGA IV4, Proposition 17.6.3
  8. ^ EGA IV4, Proposition 17.14.2
  9. ^ SGA1, Exposé I, 9.11
  10. ^ EGA IV4, Théorème 17.9.1.

Bibliography

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