The Stacks project

Lemma 61.8.4. Let $A$ be a ring such that every faithfully flat étale ring map $A \to B$ has a retraction. Then the same is true for every quotient ring $A/I$.

Proof. Let $A/I \to \overline{B}$ be faithfully flat étale. By Algebra, Lemma 10.143.10 we can write $\overline{B} = B/IB$ for some étale ring map $A \to B$. The image $U$ of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is open and contains $V(I)$. Hence the complement $Z = \mathop{\mathrm{Spec}}(A) \setminus U$ 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 étale and $\overline{B} = B'/IB'$. Hence the retraction $B' \to A$ to $A \to B'$, induces a retraction to $A/I \to \overline{B}$. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 61.8: Constructing ind-étale algebras

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Your email address will not be published.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 097U. Beware of the difference between the letter 'O' and the digit '0'.