In commutative algebra and algebraic geometry, a morphism is called formally étale if has a lifting property that is analogous to being a local diffeomorphism.
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Formally étale homomorphisms of rings
Let A be a topological ring, and let B be a topological A-algebra. B is formally étale if for all discrete A-algebras C, all nilpotent ideals J of C, and all continuous A-homomorphisms u : B → C/J, there exists a unique continuous A-algebra map v : B → C such that u = pv, where p : C → C/J 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, let f : X → Y be a morphism of schemes, Z be an affine Y-scheme, J be a nilpotent sheaf of ideals on Z, and i : Z0 → Z be the closed immersion determined by J. Then f is formally étale if for every Y-morphism g : Z0 → X, there exists a unique Y-morphism s : Z → X such that g = si.
It is equivalent to let Z be any Y-scheme and let J be a locally nilpotent sheaf of ideals on Z.