Formally étale morphism

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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. B is formally étale if for all discrete A-algebras C, all nilpotent ideals J of C, and all continuous A-homomorphisms u : BC/J, there exists a unique continuous A-algebra map v : BC such that u = pv, where p : CC/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, let f : XY be a morphism of schemes, Z be an affine Y-scheme, J be a nilpotent sheaf of ideals on Z, and i : Z0Z be the closed immersion determined by J. Then f is formally étale if for every Y-morphism g : Z0X, there exists a unique Y-morphism s : ZX 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 : XY and g : YZ 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 : AB is formally étale if and only if for every prime Q of B, the induced map ABQ is formally étale.[9] Consequently, f is formally étale if and only if for every prime Q of B, the map APBQ is formally étale, where P = f−1(Q).

Examples[edit]

  • Localizations are formally étale.
  • Finite separable field extensions are formally étale.

See also[edit]

Notes[edit]

  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

References[edit]