In the proof that an étale morphism induces an isomorphism on tangents, we use the fact that, if the morphism is unramified, then the induced map on cotangents is surjective. Then we conclude using flatness.
Something I can't get from the proof is the following: is it true that a flat (possibly not étale) morphism induces an injective morphism on cotangents? (And a surjective morphism on tangents?)
I know that an étale morphism preserves regularity "downwards", so it would not be surprising if my claim were true. But the proof of this last fact uses homological algebra, so I don't find it useful in order to proof the claim.
Thank you for your possible suggestions.
