The Stacks project

Lemma 75.13.5. Let $S$ be a scheme. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a representable transformation of functors.

  1. If $a$ is smooth then $a$ is formally smooth.

  2. If $a$ is étale, then $a$ is formally étale.

  3. If $a$ is unramified, then $a$ is formally unramified.

Proof. Consider a solid commutative diagram

\[ \xymatrix{ F \ar[d]_ a & T \ar[d]^ i \ar[l] \\ G & T' \ar[l] \ar@{-->}[lu] } \]

as in Definition 75.13.1. Then $F \times _ G T'$ is a scheme smooth (resp. étale, resp. unramified) over $T'$. Hence by More on Morphisms, Lemma 37.11.7 (resp. Lemma 37.8.9, resp. Lemma 37.6.8) we can fill in (resp. uniquely fill in, resp. fill in at most one way) the dotted arrow in the diagram

\[ \xymatrix{ F \times _ G T' \ar[d] & T \ar[d]^ i \ar[l] \\ T' & T' \ar[l] \ar@{-->}[lu] } \]

an hence we also obtain the corresponding assertion in the first diagram. $\square$


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