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
[edit]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
[edit]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
[edit]- 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
[edit]- Localizations are formally étale.
- Finite separable field extensions are formally étale. More generally, any (commutative) flat separable A-algebra B is formally étale.[10]
See also
[edit]Notes
[edit]- ^ EGA 0IV, Définition 19.10.2.
- ^ EGA 0IV, 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
- ^ Ford (2017, Corollary 4.7.3)
References
[edit]- 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.