Lemma 87.10.5. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space. The following are equivalent

1. $X$ is Noetherian,

2. $X$ is adic* and for every closed immersion $T \to X$ with $T$ a scheme, $T$ is Noetherian,

3. $X$ is adic* and for some choice of $X = \mathop{\mathrm{colim}}\nolimits X_\lambda$ as in Definition 87.9.1 the schemes $X_\lambda$ are Noetherian, and

4. $X$ is weakly adic and for some choice $X = \mathop{\mathrm{colim}}\nolimits X_\lambda$ as in Definition 87.9.1 the schemes $X_\lambda$ are Noetherian.

Proof. Assume $X$ is Noetherian. Then $X = \text{Spf}(A)$ where $A$ is a Noetherian adic ring. Let $T \to X$ be a closed immersion where $T$ is a scheme. By Lemma 87.9.6 we see that $T$ is affine and that $T \to \mathop{\mathrm{Spec}}(A)$ is a closed immersion. Since $A$ is Noetherian, we see that $T$ is Noetherian. In this way we see that (1) $\Rightarrow$ (2).

The implications (2) $\Rightarrow$ (3) and (2) $\Rightarrow$ (4) are immediate (see Lemma 87.10.3).

To prove (3) $\Rightarrow$ (1) write $X = \text{Spf}(A)$ for some adic ring $A$ with finitely generated ideal of definition $I$. We are also given that the rings $A/I_\lambda$ are Noetherian for some fundamental system of open ideals $I_\lambda$. Since $I$ is open, we can find a $\lambda$ such that $I_\lambda \subset I$. Then $A/I$ is Noetherian and we conclude that $A$ is Noetherian by Algebra, Lemma 10.97.5.

To prove (4) $\Rightarrow$ (3) write $X = \text{Spf}(A)$ for some weakly adic ring $A$. Then $A$ is admissible and has an ideal of definition $I$ and the closure $I_2$ of $I^2$ is open, see Lemma 87.7.2. We are also given that the rings $A/I_\lambda$ are Noetherian for some fundamental system of open ideals $I_\lambda$. Choose a $\lambda$ such that $I_\lambda \subset I_2$. Then $A/I_2$ is Noetherian as a quotient of $A/I_\lambda$. Hence $I/I_2$ is a finite $A$-module. Hence $A$ is an adic ring with a finitely generated ideal of definition by Lemma 87.7.4. Thus $X$ is adic* and (3) holds. $\square$

Comment #1560 by Matthew Emerton on

I think statement (3) means "for any closed immersion" rather than "for some closed immersion", but I was a bit confused when I first read it, because of the slight ambiguity in the phrase "a closed immersion". If my interpretation is correct, I might suggest replacing "a" by "any", just for the sake of maximal clarity.

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