Let $A \subseteq B$ be commutative noetherian rings. I have found the following claim: "Separability implies smoothness".

However, it seems to me that, at least with my definitions, this claim is false: If $A \subseteq B$ is separable, then (assuming the claim is true) $A \subseteq B$ is smooth, then by a result of Grothendieck, $A \subseteq B$ is flat, but this seems too much (especially in view of Examples on pages 95-97 and Corollary 9).

I will really appreciate if one can either post a sketch of proof/tell me where I can find a proof, or a counterexample.

Actually, I have already posted this question here, but got no comments thus far.