section vs retraction

Thanks to Laurent Moret-Bailly
https://stacks.math.columbia.edu/tag/09AZ#comment-7764

proetale.tex

@@ -1039,8 +1039,8 @@ \section{Identifying local rings versus ind-Zariski}
 Let $A$ be a ring. The following are equivalent
 \begin{enumerate}
 \item every faithfully flat ring map $A \to B$ identifying local rings
-has a section,
-\item every faithfully flat ind-Zariski ring map $A \to B$ has a section, and
+has a retraction,
+\item every faithfully flat ind-Zariski ring map $A \to B$ has a retraction, and
 \item $A$ satisfies
 \begin{enumerate}
 \item $\Spec(A)$ is w-local, and
@@ -1058,7 +1058,7 @@ \section{Identifying local rings versus ind-Zariski}
 We will use without further mention the fact that a flat map
 $A \to B$ is faithfully flat if and only if every closed point
 of $\Spec(A)$ is in the image of $\Spec(B) \to \Spec(A)$.
-We will show that $A \to B$ has a section.
+We will show that $A \to B$ has a retraction.
 
 \medskip\noindent
 Let $I \subset A$ be an ideal such that $V(I) \subset \Spec(A)$ is
@@ -1085,14 +1085,14 @@ \section{Identifying local rings versus ind-Zariski}
 
 \medskip\noindent
 Assume (1) or equivalently (2). Let $A \to A_w$ be the ring map constructed in
-Lemma \ref{lemma-make-w-local}. By (1) there is a section $A_w \to A$.
+Lemma \ref{lemma-make-w-local}. By (1) there is a retraction $A_w \to A$.
 Thus $\Spec(A)$ is homeomorphic to a closed subset of $\Spec(A_w)$. By
 Lemma \ref{lemma-closed-subspace-w-local} we see (3)(a) holds.
 Finally, let $T \to \pi_0(A)$ be a surjective map with $T$ an
 extremally disconnected, quasi-compact, Hausdorff topological space
 (Topology, Lemma \ref{topology-lemma-existence-projective-cover}).
 Choose $A \to B$ as in Lemma \ref{lemma-construct-profinite}
-adapted to $T \to \pi_0(\Spec(A))$. By (1) there is a section
+adapted to $T \to \pi_0(\Spec(A))$. By (1) there is a retraction
 $B \to A$. Thus we see that $T = \pi_0(\Spec(B)) \to \pi_0(\Spec(A))$
 has a section. A formal categorical argument, using
 Topology, Proposition
@@ -1357,7 +1357,7 @@ \section{Constructing ind-\'etale algebras}
 \begin{lemma}
 \label{lemma-first-construction}
 Given a ring $A$ there exists a faithfully flat ind-\'etale $A$-algebra $C$
-such that every faithfully flat \'etale ring map $C \to B$ has a section.
+such that every faithfully flat \'etale ring map $C \to B$ has a retraction.
 \end{lemma}
 
 \begin{proof}
@@ -1423,18 +1423,27 @@ \section{Constructing ind-\'etale algebras}
 \begin{lemma}
 \label{lemma-have-sections-quotient}
 Let $A$ be a ring such that every faithfully flat \'etale ring map
-$A \to B$ has a section. Then the same is true for every quotient ring
+$A \to B$ has a retraction. Then the same is true for every quotient ring
 $A/I$.
 \end{lemma}
 
 \begin{proof}
-Omitted.
+Let $A/I \to \overline{B}$ be faithfully flat \'etale. By Algebra, Lemma
+\ref{algebra-lemma-lift-etale} we can write $\overline{B} = B/IB$ for
+some \'etale ring map $A \to B'$. The image $U$ of $\Spec(B) \to \Spec(A)$
+is open and contains $V(I)$. Hence the complement $Z = \Spec(A) \setminus U$
+is quasi-compact and disjoint from $V(I)$. Hence
+$Z \subset D(f_1) \cup \ldots \cup D(f_r)$ for some $r \geq 0$
+and $f_i \in I$. Then $A \to B' = B \times \prod A_{f_i}$
+is faithfully flat \'etale and $\overline{B} = B'/IB'$.
+Hence the retraction $B' \to A$ to $A \to B'$, induces
+a retraction to $A/I \to \overline{B}$.
 \end{proof}
 
 \begin{lemma}
 \label{lemma-have-sections-strictly-henselian}
 Let $A$ be a ring such that every faithfully flat \'etale ring map
-$A \to B$ has a section. Then every local ring of $A$ at a maximal
+$A \to B$ has a retraction. Then every local ring of $A$ at a maximal
 ideal is strictly henselian.
 \end{lemma}
 
@@ -1460,13 +1469,13 @@ \section{Constructing ind-\'etale algebras}
 Thus the complement $\Spec(A) \setminus U$ is closed
 and we can find $f \in A$, $f \not \in \mathfrak p$ such that
 $\Spec(A) = U \cup D(f)$. The ring map $A \to B \times A_f$
-is faithfully flat and \'etale, hence has a section
+is faithfully flat and \'etale, hence has a retraction
 $\sigma : B \times A_f \to A$ by assumption on $A$.
 Observe that $\sigma$ is \'etale, hence flat as a map between \'etale
 $A$-algebras (Algebra, Lemma \ref{algebra-lemma-map-between-etale}).
 Since $\mathfrak q$ is the only prime of $B \times A_f$ lying
 over $A$ we find that $A_\mathfrak p \to B_\mathfrak q$ has
-a section which is also flat. Thus
+a retraction which is also flat. Thus
 $A_\mathfrak p \to B_\mathfrak q \to A_\mathfrak p$
 are flat local ring maps whose composition is the identity. Since
 a flat local homomorphism of local rings is injective we conclude these
@@ -1476,10 +1485,10 @@ \section{Constructing ind-\'etale algebras}
 \begin{lemma}
 \label{lemma-have-sections-localize}
 Let $A$ be a ring such that every faithfully flat \'etale ring map
-$A \to B$ has a section. Let $Z \subset \Spec(A)$ be a closed subscheme.
+$A \to B$ has a retraction. Let $Z \subset \Spec(A)$ be a closed subscheme.
 Let $A \to A_Z^\sim$ be as constructed in Lemma \ref{lemma-localization}.
 Then every faithfully flat \'etale ring map $A_Z^\sim \to C$ has
-a section.
+a retraction.
 \end{lemma}
 
 \begin{proof}
@@ -1488,8 +1497,8 @@ \section{Constructing ind-\'etale algebras}
 The image $U' \subset \Spec(A)$ of $\Spec(B') \to \Spec(A)$
 is open and contains $V(I)$, hence we can find $f \in I$ such
 that $\Spec(A) = U' \cup D(f)$. Then $A \to B' \times A_f$
-is \'etale and faithfully flat. By assumption there is a section
-$B' \times A_f \to A$. Localizing we obtain the desired section
+is \'etale and faithfully flat. By assumption there is a retraction
+$B' \times A_f \to A$. Localizing we obtain the desired retraction
 $C \to A_Z^\sim$.
 \end{proof}
 
@@ -1847,13 +1856,13 @@ \section{Constructing w-contractible covers}
 \begin{definition}
 \label{definition-w-contractible}
 Let $A$ be a ring. We say $A$ is {\it w-contractible} if every
-faithfully flat weakly \'etale ring map $A \to B$ has a section.
+faithfully flat weakly \'etale ring map $A \to B$ has a retraction.
 \end{definition}
 
 \noindent
 We remark that by Proposition \ref{proposition-weakly-etale}
 an equivalent definition would be to ask that every faithfully
-flat, ind-\'etale ring map $A \to B$ has a section.
+flat, ind-\'etale ring map $A \to B$ has a retraction.
 Here is a key observation that will allow us to construct
 w-contractible rings.
 
@@ -1863,7 +1872,7 @@ \section{Constructing w-contractible covers}
 \begin{enumerate}
 \item $A$ is w-contractible,
 \item every faithfully flat, ind-\'etale ring map $A \to B$ has
-a section, and
+a retraction, and
 \item $A$ satisfies
 \begin{enumerate}
 \item $\Spec(A)$ is w-local,
@@ -1883,7 +1892,7 @@ \section{Constructing w-contractible covers}
 and ind-\'etale. We will use without further mention the fact that a flat map
 $A \to B$ is faithfully flat if and only if every closed point
 of $\Spec(A)$ is in the image of $\Spec(B) \to \Spec(A)$
-We will show that $A \to B$ has a section.
+We will show that $A \to B$ has a retraction.
 
 \medskip\noindent
 Let $I \subset A$ be an ideal such that $V(I) \subset \Spec(A)$ is
@@ -1895,7 +1904,7 @@ \section{Constructing w-contractible covers}
 closed points of $\Spec(B)$ is $V(IB)$. In this case $A \to B$
 identifies local rings by condition (3)(c) as it suffices to check
 this at maximal ideals of $B$ which lie over maximal ideals of $A$.
-Thus $A \to B$ has a section by
+Thus $A \to B$ has a retraction by
 Lemma \ref{lemma-w-local-extremally-disconnected}.
 
 \medskip\noindent
@@ -2067,7 +2076,7 @@ \section{Constructing w-contractible covers}
 A \longrightarrow A_{f_1} \times \ldots \times A_{f_r} \times B
 $$
 this is faithful flat and weakly \'etale. If $A$ is w-contractible,
-then there is a section $\sigma$. Consider the morphism
+then there is a retraction $\sigma$. Consider the morphism
 $$
 \Spec(A_Z^\sim) \to \Spec(A) \xrightarrow{\Spec(\sigma)}
 \coprod \Spec(A_{f_i}) \amalg \Spec(B)
@@ -2849,11 +2858,11 @@ \section{Weakly contractible objects}
 (Sites, Lemma \ref{sites-lemma-covering-surjective-after-sheafification}).
 Since $T$ is assumed weakly contractible, we see that there is an element
 $f \in h_V(T) = \Mor(T, V)$ whose image in $h_T(T)$ is $\text{id}_T$.
-Thus $A \to D$ has a section $\sigma : D \to A$.
+Thus $A \to D$ has a retraction $\sigma : D \to A$.
 Now if $A \to B$ is faithfully flat and weakly \'etale, then
 $D \to D \otimes_A B$ has the same properties, hence
-there is a section $D \otimes_A B \to D$ and combined
-with $\sigma$ we get a section $B \to D \otimes_A B \to D \to A$
+there is a retraction $D \otimes_A B \to D$ and combined
+with $\sigma$ we get a retraction $B \to D \otimes_A B \to D \to A$
 of $A \to B$. Thus $A$ is w-contractible and (1) holds.
 \end{proof}
 
@@ -4994,13 +5003,13 @@ \section{Closed immersions and pro-\'etale sites}
 Thus we have to show: If $A$ is a w-contractible ring, $I \subset A$
 an ideal contained in the Jacobson radical of $A$ and $A \to B \to A/I$
 is a factorization with $A \to B$ ind-\'etale, then there is
-a unique section $B \to A$ compatible with maps to $A/I$.
+a unique retraction $B \to A$ compatible with maps to $A/I$.
 Observe that $B/IB = A/I \times R$ as $A/I$-algebras.
 After replacing $B$ by a localization we may assume $B/IB = A/I$.
 Note that $\Spec(B) \to \Spec(A)$ is surjective as the image
 contains $V(I)$ and hence all closed points and is closed under
-specialization. Since $A$ is w-contractible there is a section $B \to A$.
-Since $B/IB = A/I$ this section is compatible with the map to $A/I$.
+specialization. Since $A$ is w-contractible there is a retraction $B \to A$.
+Since $B/IB = A/I$ this retraction is compatible with the map to $A/I$.
 We omit the proof of uniqueness (hint: use that $A$ and $B$ have
 isomorphic local rings at maximal ideals of $A$).
 \end{proof}