The Stacks project

Lemma 87.10.3. Let $X$ be an affine formal algebraic space over a scheme $S$.

  1. If $X$ is Noetherian, then $X$ is adic*.

  2. If $X$ is adic*, then $X$ is adic.

  3. If $X$ is adic, then $X$ is weakly adic.

  4. If $X$ is weakly adic, then $X$ is classical.

  5. If $X$ is weakly adic, then $X$ is countably indexed.

  6. If $X$ is countably indexed, then $X$ is McQuillan.

Proof. Statements (1), (2), (3), and (4) follow by writing $X = \text{Spf}(A)$ and where $A$ is a weakly admissible (hence complete) linearly topologized ring and using the implications between the various types of such rings discussed in Section 87.7.

Proof of (5). By definition there exists a weakly adic topological ring $A$ such that $X = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/I)$ where the colimit is over the ideals of definition of $A$. As $A$ is weakly adic, there exits in particular a countable fundamental system $I_\lambda $ of open ideals, see Definition 87.7.1. Then $X = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/I_ n)$ by definition of $\text{Spf}(A)$. Thus $X$ is countably indexed.

Proof of (6). Write $X = \mathop{\mathrm{colim}}\nolimits X_ n$ for some system $X_1 \to X_2 \to X_3 \to \ldots $ of thickenings of affine schemes over $S$. Then

\[ A = \mathop{\mathrm{lim}}\nolimits \Gamma (X_ n, \mathcal{O}_{X_ n}) \]

surjects onto each $\Gamma (X_ n, \mathcal{O}_{X_ n})$ because the transition maps are surjections as the morphisms $X_ n \to X_{n + 1}$ are closed immersions. Hence $X$ is McQuillan. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AIK. Beware of the difference between the letter 'O' and the digit '0'.