The Stacks project

Situation 70.5.5. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be the limit of a directed inverse system of algebraic spaces over $S$ with affine transition morphisms (Lemma 70.4.1). We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$. We also choose an element $0 \in I$.