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:
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?
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?
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?
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 Xwhere 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.
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