Example 87.24.7. Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over $S$. Let $A \to A'$ be a finite type ring map. Then

is a weakly admissible ring and the corresponding morphism $\text{Spf}((A')^\wedge ) \to \text{Spf}(A)$ is representable, see Example 87.19.11. If $T \to \text{Spf}(A)$ is a morphism where $T$ is a quasi-compact scheme, then this factors through $\mathop{\mathrm{Spec}}(A/I)$ for some weak ideal of definition $I \subset A$ (Lemma 87.9.4). Then $T \times _{\text{Spf}(A)} \text{Spf}((A')^\wedge )$ is equal to $T \times _{\mathop{\mathrm{Spec}}(A/I)} \mathop{\mathrm{Spec}}(A'/IA')$ and we see that $\text{Spf}((A')^\wedge ) \to \text{Spf}(A)$ is of finite type.

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