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.

Lemma 87.17.3. 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 induced map $f_{red} : X_{red} \to Y_{red}$ between reductions (Lemma 87.12.1) is a quasi-compact morphism of algebraic spaces,

2. for every quasi-compact scheme $T$ and morphism $T \to Y$ the fibre product $X \times _ Y T$ is a quasi-compact formal algebraic space,

3. for every affine scheme $T$ and morphism $T \to Y$ the fibre product $X \times _ Y T$ is a quasi-compact formal algebraic space, and

4. there exists a covering $\{ Y_ j \to Y\} $ as in Definition 87.11.1 such that each $X \times _ Y Y_ j$ is a quasi-compact formal algebraic space.

Lemma 87.17.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces. Then $f$ is quasi-compact in the sense of Definition 87.17.4 if and only if $f$ is quasi-compact in the sense of Bootstrap, Definition 80.4.1.