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.
Then 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.
[2] 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 f : X → Y is formally étale if for every affine Y-scheme Z, every nilpotent sheaf of ideals J on Z with i : Z0 → Z be the closed immersion determined by J, and every Y-morphism g : Z0 → X, there exists a unique Y-morphism s : Z → X such that g = si.
[3] It is equivalent to let Z be any Y-scheme and let J be a locally nilpotent sheaf of ideals on Z.