Suppose I have a morphism f:X&rarr;Y such that the relative sheaf of differentials &Omega;<sub>X/Y</sub> is locally free. Does it follow that f is smooth?

The answer is no, but for a silly reason. You could have some non-reducedness (Spec(k[e]/(e^2)) over Spec(k) has a free sheaf of differentials, but isn't smooth). But what if you add the hypothesis that the rank of &Omega;<sub>X/Y</sub> is dim(X)-dim(Y)?