The Stacks project

Lemma 85.6.5. 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 some choice of $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 85.5.1 the schemes $X_\lambda $ are Noetherian,

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

Proof. This follows from the fact that if $A$ is a ring complete with respect to a finitely generated ideal $I$, then $A$ is Noetherian if and only if $A/I$ is Noetherian, see Algebra, Lemma 10.97.5. Details omitted. $\square$

Comments (2)

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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