The Stacks project

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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 097P. Beware of the difference between the letter 'O' and the digit '0'.