TITLE: Which pullbacks preserve the sheaf of differentials?
QUESTION [7 upvotes]: Let $f : X \to Y$ be a morphism of $S$-schemes. Assume that the canonical morphism $\alpha : f^* \Omega^1_{Y/S} \to \Omega^1_{X/S}$ is an isomorphism. What can we say about $f$? Is $f$ formally étale? Of course we may assume that $S$ is affine.
Background: It is known that $\alpha$ is an isomorphism when $f$ is formally étale (EGA IV, 17.2.4). One can show that $\alpha$ is an epimorphism if and only if $f$ is formally unramified (EGA IV, 17.2.2), and $\alpha$ is a monomorphism if  $f$ is smooth (EGA IV, 17.2.3). So my question is if the converse also holds.

REPLY [3 votes]: As per Mr. Brandenburg's request:
This comes from Lei Fu's Etale Cohomology (it's proposition 2.5.3)

Theorem: Let $S$ be a scheme, $X$ and $Y$ two $S$-schemes, and $f:X\to Y$ a morphism of $S$-schemes locally of finite presentation. If $f$ is etale, then the canonical map $f^\ast\Omega_{Y/S}\to\Omega_{X/S}$ is an isomorphism. The converse holds if $X$ and $Y$ are smooth over $S$.