Definition 86.9.7. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is McQuillan if $X$ satisfies the equivalent conditions of Lemma 86.9.6. Let $A$ be the weakly admissible topological ring associated to $X$. We say

1. $X$ is classical if $X$ is McQuillan and $A$ is admissible (More on Algebra, Definition 15.36.1),

2. $X$ is weakly adic if $X$ is McQuillan and $A$ is weakly adic (Definition 86.7.1),

3. $X$ is adic if $X$ is McQuillan and $A$ is adic (More on Algebra, Definition 15.36.1),

4. $X$ is adic* if $X$ is McQuillan, $A$ is adic, and $A$ has a finitely generated ideal of definition, and

5. $X$ is Noetherian if $X$ is McQuillan and $A$ is both Noetherian and adic.

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