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.[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 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.[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 → BQ is formally étale.[9] Consequently, f is formally étale if and only if for every prime Q of B, the map AP → BQ 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.[10]
gollark: Oh, I thought you meant world of warcraft and was *very* confused.
gollark: Mysterious.
gollark: no .
gollark: No, rewrite heavpoot's game.
gollark: I was kind of tempted to do that honestly.

See also

Notes
  1. EGA 0IV, Définition 19.10.2.
  2. EGA 0IV, 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. mathoverflow.net question
  10. Ford (2017, Corollary 4.7.3)

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.
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