In this MO question the author asks about the sufficiency of tangential criteria for determining formal smoothness/unramifiedness/étaleness, at least for varieties.
This comment states:
For varieties over $k$, tangential criteria OK in smooth case at $k$-points (as the complete local ring is power series ring over $k$).
As a complete novice in algebraic geometry seeking a good understanding of formally smooth/unramified/étale morphisms, I am not able to fill the gaps in the linked comment(s).
- What are some references for sufficiency of tangential criteria for smooth varieties?
- Why are tangential criteria not always sufficient? Particularly what are some examples where they are satisfied but the morphism in question is not formally smooth/unramified/étale?
I think my second question is related to this one, asking for examples of non-split square-zero extensions (in characteristic zero).