The Stacks project

Lemma 85.18.2. 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

  1. 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 taut map $B \to A$ of $\textit{WAdm}^{count}$,

  2. there exists a covering $\{ Y_ j \to Y\} $ as in Definition 85.7.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 85.7.1 such that each $X_{ji} \to Y_ j$ corresponds to a taut ring map in $\textit{WAdm}^{count}$,

  3. there exist a covering $\{ X_ i \to X\} $ as in Definition 85.7.1 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 taut ring map in $\textit{WAdm}^{count}$, and

  4. $f$ is representable by algebraic spaces.

Proof. The property of a map in $\textit{WAdm}^{count}$ being “taut” is a local property by Lemma 85.18.1. Thus Lemma 85.17.3 exactly tells us that (1), (2), and (3) are equivalent. On the other hand, by Lemma 85.15.10 being “taut” on maps in $\textit{WAdm}^{count}$ corresponds exactly to being “representable by algebraic spaces” for the corresponding morphisms of countably indexed affine formal algebraic spaces. Thus the implication (1) $\Rightarrow $ (2) of Lemma 85.15.4 shows that (4) implies (1) of the current lemma. Similarly, the implication (4) $\Rightarrow $ (1) of Lemma 85.15.4 shows that (2) implies (4) of the current lemma. $\square$


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