Definition of étale for rings

Question

Let $A \to B$ be a ring extension.

What is the definition of $B/A$ étale ?

When $A$ is a field, do we get a nice characterization ?

Answer 1

You say that a ring homomorphism $\phi: A \to B$ is étale (resp. smooth, unramified), or that $B$ is étale (resp. smooth, unramified) over $A$ is the following two conditions are satisfied:

• $A \to B$ is formally étale (resp. formally smooth, formally unramified): for every square-zero extension of $A$-algebras $R' \to R$ (meaning that the kernel $I$ satisfies $I^2 = 0$) the natural map $$\mathrm{Hom}_A(B, R') \to \mathrm{Hom}_{A}(B, R)$$ is bijective (resp. surjective, injective).
• $B$ is essentially of finite presentation over $A$: $A \to B$ factors as $A \to C \to B$, where $A \to C$ is of finite presentation and $C \to B$ is $C$-isomorphic to a localization morphism $C \to S^{-1}C$ for some multiplicatively closed subset $S \subset C$.

The second condition is just a finiteness condition; the meat of the concept is in the first one. Formal smoothness is often referred to as the infinitesimal lifting property. Geometrically speaking, it says that if the affine scheme $\mathrm{Spec} \ B$ is smooth over $\mathrm{Spec} \ A$, then any map from $\mathrm{Spec} \ R$ to $\mathrm{Spec} \ B$ lifts to any square-zero (and hence any infinitesimal) deformation $\mathrm{Spec} \ R'$. Moreover, if $\mathrm{Spec} \ B$ is étale over $\mathrm{Spec} \ A$ this lifting is unique.

Differential-geometrically, unramifiedness, smoothness and étaleness correspond to the tangent map of $\mathrm{Spec} \ \phi$ being injective, surjective and bijective, respectively. In particular, étale is the generalization to the algebraic case of the concept of local isomorphism.

There are two references you might want to consult. The first one, in which you can read all about the formal properties of these morphisms, is Iversen's "Generic Local Structure of the Morphisms in Commutative Algebra" (Springer LNM 310). The second one, Hartshorne's book "Deformation Theory" (Graduate Texts in Mathematics 257), will give you a lot of information about the geometry; section 4 of chapter 1 talks about the infinitesimal lifting property.

EDIT: The EGA definition of étale morphism of rings is slightly different from the above, in the sense that it requires finite presentation, not just locally of finite presentation: see the comments below.

Answer 2

Apparently $B$ should be finitely generated as a ring over $A$ and be a flat $A$-module, and the module of Kaehler differentials of $B$ over $A$ should vanish. When $A$ is a field, the characterization is that $B$ should be a finite direct sum of finite separable field extensions of $A$.

Answer 3

$\DeclareMathOperator{\Spec}{Spec}$ That should be read "$B$ is etale over $A$". This happens when the map from $A\to B$ is an etale ring map, which means that its dual map is an etale morphism of affine schems from $\Spec B \to \Spec A$, which is defined:

http://en.wikipedia.org/wiki/Etale_morphism

As with most things in ring theory, this condition is somewhat more trivial when $A$ is a field $k$. We get flatness since the only stalk of $\Spec A$ is $A$ ($\Spec A$ has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because $k$ is a field. If $k$ is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.

There's a more direct definition which says that the morphism $A\to B$ is a smooth ring map with relative dimension zero.

If you'd like to read a section on them in more generality, you can check out Stacks Project Tag 00U0.

I'm sure it's also in Hartshorne.

Answer 4

If $f$ is a map of local rings $$ f\colon A\rightarrow B $$ is étale iff it is flat and unramified (check out Bhargav Bhatt's notes link text). If $A$ is a field and $B$ is finite over $A$, then $f$ is étale iff $B$ is isomorphic to a finite product of separable field extensions of $A$ (see proposition I.3.1 of Milne's book Étale cohomology). More generally, for $f$ any ring homomorphism, check out definition II.1.1 of SGA 4.5 ($B$ is a finitely presented $A$-algebra and satisfies a Jacobian criterion is a possible definition. Or $B$ is a finitely presented $A$-algebra and $B$ is flat and the relative differentials are trivial). The definition comes down to "smooth of relative dimension 0".