TITLE: Definition of étale for rings
QUESTION [11 upvotes]: 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 ?

REPLY [3 votes]: If f is a map of local rings
$$f:A\rightarrow B$$ is étale iff it is flat and unramified (check out Bhargav Bhatt's notes at the stacks project 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".