The Stacks project

Lemma 76.13.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : H \to G$ be transformations of functors. Consider the fibre product diagram

\[ \xymatrix{ H \times _{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^ a \\ H \ar[r]^ b & G } \]

  1. If $a$ is formally smooth, then the base change $a'$ is formally smooth.

  2. If $a$ is formally étale, then the base change $a'$ is formally étale.

  3. If $a$ is formally unramified, then the base change $a'$ is formally unramified.

Proof. This is formal. $\square$

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