Lemma 85.6.5. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space. The following are equivalent

$X$ is Noetherian,

$X$ 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,

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

## Comments (2)

Comment #1560 by Matthew Emerton on

Comment #1579 by Johan on