The Stacks project

Remark 87.29.4. Let $A$ be a weakly admissible topological ring and let $(I_\lambda )$ be a fundamental system of weak ideals of definition. Let $X = \text{Spf}(A)$, in other words, $X$ is a McQuillan affine formal algebraic space. Let $f : Y \to X$ be a morphism of affine formal algebraic spaces. In general it will not be true that $Y$ is McQuillan. More specifically, we can ask the following questions:

1. Assume that $f : Y \to X$ is a closed immersion. Then $Y$ is McQuillan and $f$ corresponds to a continuous map $\varphi : A \to B$ of weakly admissible topological rings which is taut, whose kernel $K \subset A$ is a closed ideal, and whose image $\varphi (A)$ is dense in $B$, see Lemma 87.27.5. What conditions on $A$ guarantee that $B = (A/K)^\wedge $ as in Example 87.27.6?

2. What conditions on $A$ guarantee that closed immersions $f : Y \to X$ correspond to quotients $A/K$ of $A$ by closed ideals, in other words, the corresponding continuous map $\varphi $ is surjective and open?

3. Suppose that $f : Y \to X$ is of finite type. Then we get $Y = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(B_\lambda )$ where $(B_\lambda )$ is an object of $\mathcal{C}$ by Lemma 87.29.3. In this case it is true that there exists a fixed integer $r$ such that $B_\lambda $ is generated by $r$ elements over $A/I_\lambda $ for all $\lambda $ (the argument is essentially already given in the proof of (1) $\Rightarrow $ (2) in Lemma 87.29.2). However, it is not clear that the projections $\mathop{\mathrm{lim}}\nolimits B_\lambda \to B_\lambda $ are surjective, i.e., it is not clear that $Y$ is McQuillan. Is there an example where $Y$ is not McQuillan?

4. Suppose that $f : Y \to X$ is of finite type and $Y$ is McQuillan. Then $f$ corresponds to a continuous map $\varphi : A \to B$ of weakly admissible topological rings. In fact $\varphi $ is taut and $B$ is topologically of finite type over $A$, see Lemma 87.29.2. In other words, $f$ factors as

   \[ Y \longrightarrow \mathbf{A}^ r_ X \longrightarrow X \]

   where the first arrow is a closed immersion of McQuillan affine formal algebraic spaces. However, then questions (1) and (2) are in force for $Y \to \mathbf{A}^ r_ X$.

Below we will answer these questions when $X$ is countably indexed, i.e., when $A$ has a countable fundamental system of open ideals. If you have answers to these questions in greater generality, or if you have counter examples, please email stacks.project@gmail.com.