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

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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

''A''-algebras ''C'', all

nilpotent ideal In mathematics, more specifically ring theory, an ideal ''I'' of a ring ''R'' is said to be a nilpotent ideal if there exists a natural number ''k'' such that ''I'k'' = 0. By ''I'k'', it is meant the additive subgroup generated by the set ...

s ''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 In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) mea ...

Formally étale morphisms of schemes

Since the

structure sheaf In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a 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 is formally étale if for every affine ''Y''-scheme ''Z'', every nilpotent sheaf of ideals ''J'' on ''Z'' with be the closed immersion determined by ''J'', and 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

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s 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 .

Examples

* Localizations are formally étale. *Finite separable field extensions are formally étale. More generally, any (commutative) flat separable ''A''-algebra ''B'' is formally étale.

Notes

References