In algebraic geometry, an étale morphism (pronounced: [eˈtal]) 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.
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 it has any of the following equivalent properties:
- is flat and unramified.
- is a smooth morphism 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 a smooth morphism of relative dimension zero.
- is a smooth morphism and a locally quasi-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 Spec R of and an open affine neighborhood Spec S of such that f(Spec S) is contained in Spec R and such that the ring homomorphism R → S induced by is standard étale.
- is locally of finite presentation and is formally étale.
- is locally of finite presentation and is formally étale for maps from local rings, that is:
- Let A be a local ring and J be an ideal of A such that J2 = 0. Set Z = Spec A and Z0 = Spec A/J, and let i : Z0 → Z be the canonical closed immersion. Let z denote the closed point of Z0. Let h : Z → Y and g0 : Z0 → X be morphisms such that f(g0(z)) = h(i(z)). Then there exists a unique Y-morphism g : Z → X such that gi = g0.
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:
- 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 .)
- 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 mJ = 0. Finally, the morphism on residue fields κ(y) → A / m 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 of étale morphisms
Any open immersion is étale because it is locally an isomorphism.
Morphisms induced by 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.
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 of étale morphisms
- 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.
- If is étale and surjective, then (finite or otherwise).
Étale morphisms and the inverse function theorem
As said in the introduction, 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 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 = 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 quasi-compact, then for any point y lying in f(X), there is an étale morphism V → Y 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.
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