The Stacks project

Lemma 75.13.8. Let $S$ be a scheme. Let $F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. Assume $b$ is formally unramified.

  1. If $b \circ a$ is formally unramified then $a$ is formally unramified.

  2. If $b \circ a$ is formally étale then $a$ is formally étale.

  3. If $b \circ a$ is formally smooth then $a$ is formally smooth.

Proof. Let $T \subset T'$ be a closed immersion of affine schemes defined by an ideal of square zero. Let $g' : T' \to G$ and $f : T \to F$ be given such that $g'|_ T = a \circ f$. Because $b$ is formally unramified, there is a one to one correspondence between

\[ \{ f' : T' \to F \mid f = f'|_ T\text{ and }a \circ f' = g'\} \]

and

\[ \{ f' : T' \to F \mid f = f'|_ T\text{ and }b \circ a \circ f' = b \circ g'\} . \]

From this the lemma follows formally. $\square$


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