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 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, 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.

It is equivalent to let Z be any Y-scheme and let J be a locally nilpotent sheaf of ideals on Z.

Properties

• Open immersions are formally étale.
• The property of being formally étale is preserved under composites, base change, and fibered products.
• If f : X → Y and g : Y → Z 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.
• The property of being formally étale is local on the source and target.
• The property of being formally étale can be checked on stalks. One can show that a morphism of rings f : A → B is formally étale if and only if for every prime Q of B, the induced map A → B_Q is formally étale. Consequently, f is formally étale if and only if for every prime Q of B, the map A_P → B_Q is formally étale, where P = f⁻¹(Q).

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

Notes

References

• Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1, MR 3618889
• Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20. doi:10.1007/bf02684747. MR 0173675.
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32. doi:10.1007/bf02732123. MR 0238860.