The Stacks project

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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