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Concept
Étale morphism
Summary
In algebraic geometry, an étale morphism (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.
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:
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.
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