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 110.74.

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.

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