Lemma 86.20.6. Let $S$ be a scheme. Let

\[ P \in \left\{ \begin{matrix} countably\ indexed,
\\ countably\ indexed\ and\ classical,
\\ weakly\ adic,\ adic*,\ Noetherian
\end{matrix} \right\} \]

Let $X$ be a formal algebraic space over $S$. The following are equivalent

if $Y$ is an affine formal algebraic space and $f : Y \to X$ is representable by algebraic spaces and étale, then $Y$ has property $P$,

for some $\{ X_ i \to X\} _{i \in I}$ as in Definition 86.11.1 each $X_ i$ has property $P$.

**Proof.**
It is clear that (1) implies (2). Assume (2) and let $Y \to X$ be as in (1). Since the fibre products $X_ i \times _ X Y$ are formal algebraic spaces (Lemma 86.15.2) we can pick coverings $\{ X_{ij} \to X_ i \times _ X Y\} $ as in Definition 86.11.1. Since $Y$ is quasi-compact, there exist $(i_1, j_1), \ldots , (i_ n, j_ n)$ such that

\[ X_{i_1 j_1} \amalg \ldots \amalg X_{i_ n j_ n} \longrightarrow Y \]

is surjective and étale. Then $X_{i_ kj_ k} \to X_{i_ k}$ is representable by algebraic spaces and étale hence $X_{i_ kj_ k}$ has property $P$ by Lemma 86.19.10. Then $X_{i_1 j_1} \amalg \ldots \amalg X_{i_ n j_ n}$ is an affine formal algebraic space with property $P$ (small detail omitted on finite disjoint unions of affine formal algebraic spaces). Hence we conclude by applying one of Lemmas 86.20.1, 86.20.2, 86.20.3, 86.20.4, and 86.20.5.
$\square$

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