Lemma 76.13.8. 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04G5. Beware of the difference between the letter 'O' and the digit '0'.