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Let $A$, $B$ two smooth $R$-algebras of finite type for a artinian local ring $R$.

Let $I$ an ideal such that $I^{2}=0$ and $\bar{R}=R/I$.

We assume that the map $Spec B/IB\rightarrow Spec A/IA$ is smooth,

do we have that $ Spec(B)\rightarrow Spec(A)$ is smooth?

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Under your assumptions, "smooth" is equivalent to "flat with smooth fibers". So the only problem is flatness. Use the flatness criterion by fibers (EGA IV, 11.3.10).

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