Lemma 87.29.9 (0ANW)—The Stacks project

Lemma 87.29.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent

for every commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to a morphism of $\textit{WAdm}^{count}$ which is taut and topologically of finite type,
there exists a covering $\{ Y_ j \to Y\} $ as in Definition 87.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 87.11.1 such that each $X_{ji} \to Y_ j$ corresponds to a morphism of $\textit{WAdm}^{count}$ which is taut and topologically of finite type,
there exist a covering $\{ X_ i \to X\} $ as in Definition 87.11.1 and for each $i$ a factorization $X_ i \to Y_ i \to Y$ where $Y_ i$ is an affine formal algebraic space, $Y_ i \to Y$ is representable by algebraic spaces and étale, and $X_ i \to Y_ i$ corresponds to a morphism of $\textit{WAdm}^{count}$ which is, taut and topologically of finite type, and
$f$ is locally of finite type.

Proof. By Lemma 87.29.6 the property $P(\varphi )=$“$\varphi $ is taut and topologically of finite type” is local on $\text{WAdm}^{count}$. Hence by Lemma 87.21.3 we see that conditions (1), (2), and (3) are equivalent. On the other hand, by Lemma 87.29.8 the condition $P$ on morphisms of $\textit{WAdm}^{count}$ corresponds exactly to morphisms of countably indexed, affine formal algebraic spaces being locally of finite type. Thus the implication (1) $\Rightarrow $ (3) of Lemma 87.24.6 shows that (4) implies (1) of the current lemma. Similarly, the implication (4) $\Rightarrow $ (1) of Lemma 87.24.6 shows that (2) implies (4) of the current lemma. $\square$

The Stacks project

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