In commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

 and algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, a morphism is called formally étale if has a lifting property that is analogous to being a local diffeomorphism

Local diffeomorphism

In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a function between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below....

.

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 , there exists a unique continuous A-algebra map such that , where is the canonical projection.

Formally étale is equivalent to formally smooth plus formally unramified.

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 be a morphism of schemes, Z be an affine Y-scheme, J be a nilpotent sheaf of ideals on Z, and be the closed immersion determined by J. Then f is formally étale if for every Y-morphism , there exists a unique Y-morphism such that .

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

Properties

• Open immersions are formally étale.
• The property of being formally étale is preserved under composites, base change, and fibered products.
• 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.
• The property of being formally étale is local on the source and target.
• 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. Consequently, f is formally étale if and only if for every prime Q of B, the map is formally étale, where .
• A formally étale morphism is flat
  Flat morphism
  In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,is a flat map for all P in X...
  
  .

Examples

• Localizations
  Localization of a ring
  In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
  
  are formally étale.
• Finite separable field extensions are formally étale.