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A scheme morphism that is both finite and étale.
A morphism of schemes is finite étale if it is both a finite morphism and an étale morphism.
Affine-locally, is finite étale exactly when is a finite, flat, finitely presented -algebra and . Equivalently, is a finite projective -module and is unramified over .
For a field extension ,
is finite étale exactly when is finite and separable.
A scheme is a locally ringed space that has an open cover for which every restricted locally ringed space is an affine scheme.
In other words, a scheme is assembled by gluing prime spectra of commutative rings, while simultaneously gluing the algebraic functions carried by their structure sheaves.
A morphism of schemes is finite if every affine open has affine inverse image
and is a finitely generated -module through the induced ring homomorphism .
It is enough to verify this condition on an affine open cover of . Thus finiteness is affine-local on the target, whereas the definition of a finite morphism is global.
For a field extension , the morphism is finite exactly when is finite-dimensional over , that is, when is a finite field extension.
A morphism of schemes is étale if it is simultaneously
These conditions are local on both source and target, so étaleness may be checked on affine open covers. For an affine morphism , this says that is a flat, finitely presented -algebra with .
For a finite field extension , the morphism is étale exactly when is separable. Étale morphisms are therefore the algebraic-geometric analogue of local diffeomorphisms.
For a homomorphism of commutative rings , the module of relative Kähler differentials is the -module equipped with an -derivation
that is universal: every -derivation from to a -module factors uniquely through a -linear map .
Let be an algebraic field extension.
Definition (separable extension). The extension is separable if every element is a separable element over .
When is finite, separability admits useful equivalent formulations. For example, if is an algebraic closure of , then is separable iff there are exactly distinct -embeddings . Separability behaves well in towers, as summarized in separability in towers.