The Stacks project

Lemma 75.13.6. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. Assume that $a$ is representable, surjective, and étale.

  1. If $b$ is formally smooth, then $b \circ a$ is formally smooth.

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

  3. If $b$ is formally unramified, then $b \circ a$ is formally unramified.

Conversely, consider a solid commutative diagram

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

with $T'$ an affine scheme over $S$ and $i : T \to T'$ a closed immersion defined by an ideal of square zero.

  1. If $b \circ a$ is formally smooth, then for every $t \in T$ there exists an étale morphism of affines $U' \to T'$ and a morphism $U' \to G$ such that

    \[ \xymatrix{ G \ar[d]_ b & T \ar[l] & T \times _{T'} U' \ar[d] \ar[l]\\ H & T' \ar[l] & U' \ar[llu] \ar[l] } \]

    commutes and $t$ is in the image of $U' \to T'$.

  2. If $b \circ a$ is formally unramified, then there exists at most one dotted arrow in the diagram above, i.e., $b$ is formally unramified.

  3. If $b \circ a$ is formally étale, then there exists exactly one dotted arrow in the diagram above, i.e., $b$ is formally étale.

Proof. Assume $b$ is formally smooth (resp. formally étale, resp. formally unramified). Since an étale morphism is both smooth and unramified we see that $a$ is representable and smooth (resp. étale, resp. unramified). Hence parts (1), (2) and (3) follow from a combination of Lemma 75.13.5 and Lemma 75.13.3.

Assume that $b \circ a$ is formally smooth. Consider a diagram as in the statement of the lemma. Let $W = F \times _ G T$. By assumption $W$ is a scheme surjective étale over $T$. By Étale Morphisms, Theorem 41.15.2 there exists a scheme $W'$ étale over $T'$ such that $W = T \times _{T'} W'$. Choose an affine open subscheme $U' \subset W'$ such that $t$ is in the image of $U' \to T'$. Because $b \circ a$ is formally smooth we see that the exist morphisms $U' \to F$ such that

\[ \xymatrix{ F \ar[d]_{b \circ a} & W \ar[l] & T \times _{T'} U' \ar[d] \ar[l]\\ H & T' \ar[l] & U' \ar[llu] \ar[l] } \]

commutes. Taking the composition $U' \to F \to G$ gives a map as in part (5) of the lemma.

Assume that $f, g : T' \to G$ are two dotted arrows fitting into the diagram of the lemma. Let $W = F \times _ G T$. By assumption $W$ is a scheme surjective étale over $T$. By Étale Morphisms, Theorem 41.15.2 there exists a scheme $W'$ étale over $T'$ such that $W = T \times _{T'} W'$. Since $a$ is formally étale the compositions

\[ W' \to T' \xrightarrow {f} G \quad \text{and}\quad W' \to T' \xrightarrow {g} G \]

lift to morphisms $f', g' : W' \to F$ (lift on affine opens and glue by uniqueness). Now if $b \circ a : F \to H$ is formally unramified, then $f' = g'$ and hence $f = g$ as $W' \to T'$ is an étale covering. This proves part (6) of the lemma.

Assume that $b \circ a$ is formally étale. Then by part (4) we can étale locally on $T'$ find a dotted arrow fitting into the diagram and by part (5) this dotted arrow is unique. Hence we may glue the local solutions to get assertion (6). Some details omitted. $\square$


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