The Stacks project

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.