## 75.13 Formally smooth, étale, unramified transformations

Recall that a ring map $R \to A$ is called formally smooth, resp. formally étale, resp. formally unramified (see Algebra, Definition 10.138.1, resp. Definition 10.150.1, resp. Definition 10.148.1) if for every commutative solid diagram

$\xymatrix{ A \ar[r] \ar@{-->}[rd] & B/I \\ R \ar[r] \ar[u] & B \ar[u] }$

where $I \subset B$ is an ideal of square zero, there exists a, resp. exists a unique, resp. exists at most one dotted arrow which makes the diagram commute. This motivates the following analogue for morphisms of algebraic spaces, and more generally functors.

Definition 75.13.1. Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Consider commutative solid diagrams of the form

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

where $T$ and $T'$ are affine schemes and $i$ is a closed immersion defined by an ideal of square zero.

1. We say $a$ is formally smooth if given any solid diagram as above there exists a dotted arrow making the diagram commute1.

2. We say $a$ is formally étale if given any solid diagram as above there exists exactly one dotted arrow making the diagram commute.

3. We say $a$ is formally unramified if given any solid diagram as above there exists at most one dotted arrow making the diagram commute.

Lemma 75.13.2. Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Then $a$ is formally étale if and only if $a$ is both formally smooth and formally unramified.

Proof. Formal from the definition. $\square$

Lemma 75.13.3. Composition.

1. A composition of formally smooth transformations of functors is formally smooth.

2. A composition of formally étale transformations of functors is formally étale.

3. A composition of formally unramified transformations of functors is formally unramified.

Proof. This is formal. $\square$

Lemma 75.13.4. 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. $\square$

Lemma 75.13.5. Let $S$ be a scheme. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a representable transformation of functors.

1. If $a$ is smooth then $a$ is formally smooth.

2. If $a$ is étale, then $a$ is formally étale.

3. If $a$ is unramified, then $a$ is formally unramified.

Proof. Consider a solid commutative diagram

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

as in Definition 75.13.1. Then $F \times _ G T'$ is a scheme smooth (resp. étale, resp. unramified) over $T'$. Hence by More on Morphisms, Lemma 37.11.7 (resp. Lemma 37.8.9, resp. Lemma 37.6.8) we can fill in (resp. uniquely fill in, resp. fill in at most one way) the dotted arrow in the diagram

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

an hence we also obtain the corresponding assertion in the first diagram. $\square$

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. $\square$

Remark 75.13.7. It is tempting to think that in the situation of Lemma 75.13.6 we have “$b$ formally smooth” $\Leftrightarrow$ “$b \circ a$ formally smooth”. However, this is likely not true in general.

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. $\square$

[1] This is just one possible definition that one can make here. Another slightly weaker condition would be to require that the dotted arrow exists fppf locally on $T'$. This weaker notion has in some sense better formal properties.

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