4 Corrected the question by changing the field k to k'.

Let f:X -> 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[t]/t^nk'[t]/t^n, where k' is also a field?

Considering cuspidal curves one can show that you do at least need arbitrarily large n.

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Let f:X -> Y be a morphism of schemes over a field k. Can one check that f is Formally formally smooth using only Artin rings of the form k[t]/t^n?

Considering cuspidal curves one can show that you do at least need arbitrarily large n.

2 edited title; deleted 12 characters in body

# Can one check formal smoothness onusingonly one-variable Artin rings?

Let f:X -> Y be a morphism of schemes over a field k. Can one check that f is Formally smooth using only infinitesmal extensions Artin rings of the form k[t]/t^n?

Considering cuspidal curves one can show that you do at least need arbitrarily large n.

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