# 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. 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.

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, a morphism of schemes f : XY is formally étale if for every affine Y-scheme Z, every nilpotent sheaf of ideals J on Z with i : Z0Z be the closed immersion determined by J, and every Y-morphism g : Z0X, there exists a unique Y-morphism s : ZX such that g = si.

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 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.
• 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 f : AB is formally étale if and only if for every prime Q of B, the induced map ABQ is formally étale. 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).