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. Prominent ...

and

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a morphism is called formally étale if it has a lifting property 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 In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...

, 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 of a ...

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 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) mean ...

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 r ...

of a

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

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

Open immersion Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...

s are formally étale. *The property of being formally étale is preserved under composites, base change, and

fibered product In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is of ...

s. *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 Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...

separable ''A''-algebra ''B'' is formally étale.

Notes

References