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.EGA 0IV, Définition 19.10.2.
Formally étale is equivalent to formally smooth plus formally unramified.EGA 0IV, Définition 19.10.2.
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 .EGA IV4, Définition 17.1.1.
It is equivalent to let Z be any Y-scheme and let J be a locally nilpotent sheaf of ideals on Z.EGA IV4, Remarques 17.1.2 (iv).
Properties Open immersions are formally étale.EGA IV4, proposition 17.1.3 (i). The property of being formally étale is preserved under composites, base change, and fibered products.EGA IV4, proposition 17.1.3 (ii)–(iv). If and are morphisms of schemes, g is formally unramified, and gf is formally étale, then f is formally étale. In particular, if g is formally étale, then f is formally étale if and only if gf is.EGA IV4, proposition 17.1.4 and corollaire 17.1.5. The property of being formally étale is local on the source and target.EGA IV4, proposition 17.1.6. The property of being formally étale can be checked on stalks. One can show that a morphism of rings is formally étale if and only if for every prime Q of B, the induced map is formally étale.mathoverflow.net question Consequently, f is formally étale if and only if for every prime Q of B, the map is formally étale, where .
Examples Localizations are formally étale. Finite separable field extensions are formally étale. More generally, any (commutative) flat separable A-algebra B is formally étale.
See also Formally unramified Formally smooth Étale morphism
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