The Stacks project

Lemma 85.6.3. 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 countably indexed.

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

Proof. Parts (1) and (2) are immediate from the definitions.

Proof of (3). By definition there exists an 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 adic, there exits an ideal $I$ such that $\{ I^ n\} $ forms a fundamental system of neighbourhoods of $0$. Then each $I^ n$ is an ideal of definition and $X = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/I^ n)$. Thus $X$ is countably indexed.

Proof of (4). 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. $\square$

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