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.
is étale if it has any of the following equivalent properties: Assume that
be the induced map on completed local rings.
Then the following are equivalent: If in addition all the maps on residue fields
, the induced map on completed local rings is an isomorphism.
[7] Any open immersion is étale because it is locally an isomorphism.
Covering spaces form examples of étale morphisms.
Morphisms induced by finite separable field extensions are étale — they form arithmetic covering spaces with group of deck transformations given by
Expanding upon the previous example, suppose that we have a morphism
is given by equations, we can interpret it as a map of complex manifolds.
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.
be a dominant morphism of finite type with X, Y locally noetherian, irreducible and Y normal.
[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).
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.
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 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.
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.