Lemma 84.26.2. 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

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

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 84.26.1 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. $\square$

There are also:

• 2 comment(s) on Section 84.26: Maps out of affine formal schemes

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).