Lemma 76.13.6. 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.
If b is formally smooth, then b \circ a is formally smooth.
If b is formally étale, then b \circ a is formally étale.
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.
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'.
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.
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 76.13.5 and Lemma 76.13.3.
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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