Lemma 87.33.1 (0AQF)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.33: Maps out of affine formal schemes
Lemma 87.33.1 (cite)

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Lemma 87.33.1. Let $S$ be a scheme. Let $A$ be a weakly admissible topological $S$-algebra. Let $X$ be an affine scheme over $S$. Then the natural map

\[ \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(A), X) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _ S(\text{Spf}(A), X) \]

is bijective.

Proof. If $X$ is affine, say $X = \mathop{\mathrm{Spec}}(B)$, then we see from Lemma 87.9.10 that morphisms $\text{Spf}(A) \to \mathop{\mathrm{Spec}}(B)$ correspond to continuous $S$-algebra maps $B \to A$ where $B$ has the discrete topology. These are just $S$-algebra maps, which correspond to morphisms $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(B)$. $\square$

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