Remark 35.11.6. With the assumptions (1) and $p$ smooth in Lemma 35.11.5 it is not automatically the case that $X \to Y$ is smooth. A counter example is $S = \mathop{\mathrm{Spec}}(k)$, $X = \mathop{\mathrm{Spec}}(k[s])$, $Y = \mathop{\mathrm{Spec}}(k[t])$ and $f$ given by $t \mapsto s^2$. But see also Lemma 35.11.7 for some information on the structure of $f$.

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