What is the least restrictive finiteness assumptions guaranteeing that for a formally smooth morphism of schemes $f:X\rightarrow Y$, the sheaf of relative differentials $\Omega _{X/Y}$ would be locally free? What is the least restrictive finiteness assumptions under which one can give a relatively easy/short proof of this fact? What changes (if anything) in the answers to these questions if we assume $X$ and $Y$ to be affine and test formal smoothness only against affine schemes?

I'm asking these questions for two reasons:

(1) I can not read french, nor do I have a copy of EGA $0_\text{IV}$, and:

(2) About 50 years have passed since the writing of EGA $0_\text{IV}$ and I'm sure that some results have been generalized and easier proofs have been found.

Anything (especially readable texts) on this matter would be great.