The Stacks project

Remark 87.19.12 (Warning). The discussion in Lemmas 87.19.8, 87.19.9, and 87.19.10 is sharp in the following two senses:

  1. If $A$ and $B$ are weakly admissible rings and $\varphi : A \to B$ is a continuous map, then $\text{Spf}(\varphi ) : \text{Spf}(B) \to \text{Spf}(A)$ is in general not representable.

  2. If $f : Y \to X$ is a representable morphism of affine formal algebraic spaces and $X = \text{Spf}(A)$ is McQuillan, then it does not follow that $Y$ is McQuillan.

An example for (1) is to take $A = k$ a field (with discrete topology) and $B = k[[t]]$ with the $t$-adic topology. An example for (2) is given in Examples, Section 110.75.

Comments (2)

Comment #1950 by Brian Conrad on

This warning is written in a manner that is too cryptic. It is better to tell the reader straight up what the issue is: the open ideals build in the proof of Lemma 14.10 might fail to be a cofinal system of open neighborhoods of 0 in .

Comment #2004 by on

OK, I split the warning into two parts and I explain how to get an example for each (but the second is awful). See here or wait till the website is updated later this week.

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Your email address will not be published.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 0AN7. Beware of the difference between the letter 'O' and the digit '0'.



View Remark 87.19.12 as pdf