TITLE: Can one check formal smoothness using only one-variable Artin rings?
QUESTION [9 upvotes]: Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is also a field?
Considering cuspidal curves one can show that you do at least need arbitrarily large $n$.

REPLY [3 votes]: I'm not sure what you mean by "using" but I think the answer is no.  X can be nonempty and singular over Y with X(k) empty.