Étale morphism


In algebraic geometry, an étale morphism 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.

Definition

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 G-unramified.
  2. is a smooth morphism and unramified.
  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.
  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.
  5. is a smooth morphism of relative dimension zero.
  6. is a smooth morphism and a locally quasi-finite morphism.
  7. is locally of finite presentation and is locally a standard étale morphism, that is,
  8. :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.
  9. is locally of finite presentation and is formally étale.
  10. is locally of finite presentation and is formally étale for maps from local rings, that is:
  11. :Let A be a local ring and J be an ideal of A such that. Set and, and let be the canonical closed immersion. Let z denote the closed point of Z0. Let and be morphisms such that. Then there exists a unique Y-morphism such that.
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.
  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.
  4. f is formally étale for maps of local rings with the following additional properties. The local ring A may be assumed Artinian. If m is the maximal ideal of A, then J may be assumed to satisfy. Finally, the morphism on residue fields may be assumed to be an isomorphism.
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

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.
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.

Properties

Étale morphisms
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
to the y-axis. This morphism is étale at every point except the origin, because the differential is given by 2x, which does not vanish at these points.
However, there is no 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 V → Y containing y in its image, such that when we base change f to V, then 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.