Definition 61.7.1. A ring map $A \to B$ is said to be ind-étale if $B$ can be written as a filtered colimit of étale $A$-algebras.
61.7 Ind-étale algebra
We start with a definition.
The category of ind-étale algebras is closed under a number of natural operations.
Lemma 61.7.2. Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes _ A A'$ be the base change of $B$. If $A \to B$ is ind-étale, then $A' \to B'$ is ind-étale.
Proof. This is Algebra, Lemma 10.154.1. $\square$
Lemma 61.7.3. Let $A \to B$ and $B \to C$ be ring maps. If $A \to B$ and $B \to C$ are ind-étale, then $A \to C$ is ind-étale.
Proof. This is Algebra, Lemma 10.154.2. $\square$
Lemma 61.7.4. A filtered colimit of ind-étale $A$-algebras is ind-étale over $A$.
Proof. This is Algebra, Lemma 10.154.3. $\square$
Lemma 61.7.5. Let $A$ be a ring. Let $B \to C$ be an $A$-algebra map of ind-étale $A$-algebras. Then $C$ is an ind-étale $B$-algebra.
Proof. This is Algebra, Lemma 10.154.5. $\square$
Lemma 61.7.6. Let $A \to B$ be ind-étale. Then $A \to B$ is weakly étale (More on Algebra, Definition 15.104.1).
Proof. This follows from More on Algebra, Lemma 15.104.14. $\square$
Lemma 61.7.7. Let $A$ be a ring and let $I \subset A$ be an ideal. The base change functor
\[ \text{ind-étale }A\text{-algebras} \longrightarrow \text{ind-étale }A/I\text{-algebras},\quad C \longmapsto C/IC \]
has a fully faithful right adjoint $v$. In particular, given an ind-étale $A/I$-algebra $\overline{C}$ there exists an ind-étale $A$-algebra $C = v(\overline{C})$ such that $\overline{C} = C/IC$.
Proof. Let $\overline{C}$ be an ind-étale $A/I$-algebra. Consider the category $\mathcal{C}$ of factorizations $A \to B \to \overline{C}$ where $A \to B$ is étale. (We ignore some set theoretical issues in this proof.) We will show that this category is directed and that $C = \mathop{\mathrm{colim}}\nolimits _\mathcal {C} B$ is an ind-étale $A$-algebra such that $\overline{C} = C/IC$.
We first prove that $\mathcal{C}$ is directed (Categories, Definition 4.19.1). The category is nonempty as $A \to A \to \overline{C}$ is an object. Suppose that $A \to B \to \overline{C}$ and $A \to B' \to \overline{C}$ are two objects of $\mathcal{C}$. Then $A \to B \otimes _ A B' \to \overline{C}$ is another (use Algebra, Lemma 10.143.3). Suppose that $f, g : B \to B'$ are two maps between objects $A \to B \to \overline{C}$ and $A \to B' \to \overline{C}$ of $\mathcal{C}$. Then a coequalizer is $A \to B' \otimes _{f, B, g} B' \to \overline{C}$. This is an object of $\mathcal{C}$ by Algebra, Lemmas 10.143.3 and 10.143.8. Thus the category $\mathcal{C}$ is directed.
Write $\overline{C} = \mathop{\mathrm{colim}}\nolimits \overline{B_ i}$ as a filtered colimit with $\overline{B_ i}$ étale over $A/I$. For every $i$ there exists $A \to B_ i$ étale with $\overline{B_ i} = B_ i/IB_ i$, see Algebra, Lemma 10.143.10. Thus $C \to \overline{C}$ is surjective. Since $C/IC \to \overline{C}$ is ind-étale (Lemma 61.7.5) we see that it is flat. Hence $\overline{C}$ is a localization of $C/IC$ at some multiplicative subset $S \subset C/IC$ (Algebra, Lemma 10.108.2). Take an $f \in C$ mapping to an element of $S \subset C/IC$. Choose $A \to B \to \overline{C}$ in $\mathcal{C}$ and $g \in B$ mapping to $f$ in the colimit. Then we see that $A \to B_ g \to \overline{C}$ is an object of $\mathcal{C}$ as well. Thus $f$ is an invertible element of $C$. It follows that $C/IC = \overline{C}$.
Next, we claim that for an ind-étale algebra $D$ over $A$ we have
\[ \mathop{\mathrm{Mor}}\nolimits _ A(D, C) = \mathop{\mathrm{Mor}}\nolimits _{A/I}(D/ID, \overline{C}) \]
Namely, let $D/ID \to \overline{C}$ be an $A/I$-algebra map. Write $D = \mathop{\mathrm{colim}}\nolimits _{i \in I} D_ i$ as a colimit over a directed set $I$ with $D_ i$ étale over $A$. By choice of $\mathcal{C}$ we obtain a transformation $I \to \mathcal{C}$ and hence a map $D \to C$ compatible with maps to $\overline{C}$. Whence the claim.
It follows that the functor $v$ defined by the rule
\[ \overline{C} \longmapsto v(\overline{C}) = \mathop{\mathrm{colim}}\nolimits _{A \to B \to \overline{C}} B \]
is a right adjoint to the base change functor $u$ as required by the lemma. The functor $v$ is fully faithful because $u \circ v = \text{id}$ by construction, see Categories, Lemma 4.24.4. $\square$
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