Formally étale morphism

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.

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.^[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, 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.^[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 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.^[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 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.^[9] 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.

Notes

^ EGA 0_IV, Définition 19.10.2.
^ EGA 0_IV, Définition 19.10.2.
^ EGA IV₄, Définition 17.1.1.
^ EGA IV₄, Remarques 17.1.2 (iv).
^ EGA IV₄, proposition 17.1.3 (i).
^ EGA IV₄, proposition 17.1.3 (ii)–(iv).
^ EGA IV₄, proposition 17.1.4 and corollaire 17.1.5.
^ EGA IV₄, proposition 17.1.6.
^ mathoverflow.net question

References

[1] EGA 0_IV, Définition 19.10.2.

[2] EGA 0_IV, Définition 19.10.2.

[3] EGA IV₄, Définition 17.1.1.

[4] EGA IV₄, Remarques 17.1.2 (iv).

[5] EGA IV₄, proposition 17.1.3 (i).

[6] EGA IV₄, proposition 17.1.3 (ii)–(iv).

[7] EGA IV₄, proposition 17.1.4 and corollaire 17.1.5.

[8] EGA IV₄, proposition 17.1.6.

[9] thoverflow.net question

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