Lemma 87.33.2 (0AQG)—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 87.33.2 (cite)

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Lemma 87.33.2. Let $S$ be a scheme. Let $A$ be a weakly admissible topological $S$ -algebra such that $A/I$ is a local ring for some weak ideal of definition $I \subset A$ . Let $X$ be a 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. Let $\varphi : \text{Spf}(A) \to X$ be a morphism. Since $\mathop{\mathrm{Spec}}(A/I)$ is local we see that $\varphi$ maps $\mathop{\mathrm{Spec}}(A/I)$ into an affine open $U \subset X$ . However, this then implies that $\mathop{\mathrm{Spec}}(A/J)$ maps into $U$ for every ideal of definition $J$ . Hence we may apply Lemma 87.33.1 to see that $\varphi$ comes from a morphism $\mathop{\mathrm{Spec}}(A) \to X$ . This proves surjectivity of the map. We omit the proof of injectivity. $\square$

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