The Stacks project

Lemma 87.19.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent:

1. the morphism $f$ is representable by algebraic spaces,

2. there exists a commutative diagram

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

   where $U$, $V$ are formal algebraic spaces, the vertical arrows are representable by algebraic spaces, $U \to X$ is surjective étale, and $U \to V$ is representable by algebraic spaces,

3. for any commutative diagram

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

   where $U$, $V$ are formal algebraic spaces and the vertical arrows are representable by algebraic spaces, the morphism $U \to V$ is representable by algebraic spaces,

4. 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 $X_{ji} \to Y_ j$ is representable by algebraic spaces for each $j$ and $i$,

5. 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, such that $X_ i \to Y_ i$ is representable by algebraic spaces, and

6. add more here.