# Can one check formal smoothness using only one-variable Artin rings?

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 $n$.

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I misunderstood the original question: I didn't see the "over a field k" part, so I thought you were asking if it was sufficient to use square-zero extensions of the form k[t]/t^n --> k[t]/t^m where k could be an arbitrary field. –  Anton Geraschenko Oct 10 '09 at 20:26
I meant to let the k in the Artin rings vary so edited the question. –  David Zureick-Brown Oct 11 '09 at 20:21
Is there more left to do in this question? –  Greg Kuperberg Jan 23 '10 at 2:57