Lemma 85.16.4. Let $S$ be a scheme. Let $P \in \{ countably\ indexed, adic*, Noetherian\} $. 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 85.7.1 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 85.11.2) we can pick coverings $\{ X_{ij} \to X_ i \times _ X Y\} $ as in Definition 85.7.1. 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 85.15.9. 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 85.16.1, 85.16.2, and 85.16.3.
$\square$

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