# Formal smoothness implies local freeness of the sheaf of relative differentials

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.

• One can read EGA without being able to read a French menu or book for French 3-year-olds ("math French" requires no knowledge of French grammar or spelling and almost no vocabulary if one knows English). Without finiteness hypotheses on $f$, a more apt property than "locally free" is "projective on an affine open cover" (see 3.1.4(3) in Raynaud-Gruson for deep results on this notion), which holds for $\Omega^1_{X/Y}$ for formally smooth $f$ by EGA 0$_{\rm{IV}}$ 20.4.9 with discrete topologies (whose proof is very easy!): math.harvard.edu/~gaitsgde/Schemes_2009/EGA-IVa.pdf – nfdc23 Aug 1 '17 at 3:34
• Thanks for the digital copy of EGA which I didn't know the existence of! – Anonymous Coward Aug 1 '17 at 4:13
• Regarding EGA $0_IV$ 20.4.9: the proof seems to rely on theorem 19.5.3, which in itself seems just as nontrivial. I will try to read this tomorrow. About finiteness: just to make sure I understand, from the above it follows that for a smooth morphism (i.e.: formally smooth + essentially finite type) $f:X\rightarrow Y$ with $Y$ Noetherian $\Omega _{X/Y}$ is locally free of finite rank? (because then $\Omega _{X/Y}$ will be coherent and projective on an affine open cover which since $X$ is Noetherian implies locally free?) – Anonymous Coward Aug 1 '17 at 4:26
• The real technical headaches in the proof of 19.5.3 have to do with the topological aspects when $(0)$ isn't an ideal of definition. I shouldn't have written "very easy", but for the discrete topology the proof of 19.5.3 is an ordinary long proof, not ultimately so overwhelming. For $Y$ noetherian and $f$ smooth, the fact that $\Omega^1_{X/Y}$ is a vector bundle can be proved without going through this: you can assume $Y$ is artin local with algebraically closed residue field, and then enough to prove completed local rings at closed points are power series rings, which really is not hard. – nfdc23 Aug 1 '17 at 4:45