Formally étale morphism explained

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.^[1]

Formally étale is equivalent to formally smooth plus formally unramified.^[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 .^[3]

It is equivalent to let Z be any Y-scheme and let J be a locally nilpotent sheaf of ideals on Z.^[4]

Properties

• Open immersions are formally étale.^[5]
• The property of being formally étale is preserved under composites, base change, and fibered products.^[6]
• 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.^[7]
• The property of being formally étale is local on the source and target.^[8]
• 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.^[9] 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

Notes and References

1. [#Reference-EGAIV1|EGA 0_IV]
2. [#Reference-EGAIV1|EGA 0_IV]
3. [#Reference-EGAIV4|EGA IV₄]
4. [#Reference-EGAIV4|EGA IV₄]
5. [#Reference-EGAIV4|EGA IV₄]
6. [#Reference-EGAIV4|EGA IV₄]
7. [#Reference-EGAIV4|EGA IV₄]
8. [#Reference-EGAIV4|EGA IV₄]
9. https://mathoverflow.net/q/16132 mathoverflow.net question