The Stacks project

Lemma 87.17.1. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent

  1. the reduction of $X$ (Lemma 87.12.1) is a quasi-compact algebraic space,

  2. we can find $\{ X_ i \to X\} _{i \in I}$ as in Definition 87.11.1 with $I$ finite,

  3. there exists a morphism $Y \to X$ representable by algebraic spaces which is étale and surjective and where $Y$ is an affine formal algebraic space.

Proof. Omitted. $\square$

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View Lemma 87.17.1 as pdf