commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

and

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a morphism is called formally étale if it has a

lifting property In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly give ...

that is analogous to being a

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

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''-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 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 ...

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 In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

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

* 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 Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...

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