Formally étale morphism


In commutative algebra and algebraic geometry, a morphism is called formally étale if it has a lifting property that is analogous to being a local diffeomorphism.

Formally étale homomorphisms of rings

Let A be a topological ring, and let B be a topological A-algebra. Then B is formally étale if for all discrete A-algebras C, all nilpotent ideals J of C, and all continuous A-homomorphisms, there exists a unique continuous A-algebra map such that, where is the canonical projection.
Formally étale is equivalent to formally smooth plus formally unramified.

Formally étale morphisms of schemes

Since the structure sheaf of a scheme naturally carries only the discrete topology, the notion of formally étale for schemes is analogous to formally étale for the discrete topology for rings. That is, a morphism of schemes is formally étale if for every affine Y-scheme Z, every nilpotent sheaf of ideals J on Z with be the closed immersion determined by J, and every Y-morphism, there exists a unique Y-morphism such that.
It is equivalent to let Z be any Y-scheme and let J be a locally nilpotent sheaf of ideals on Z.

Properties