The Stacks project

Situation 36.35.7. Let $S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i$ be a limit of a directed system of schemes with affine transition morphisms $g_{i'i} : S_{i'} \to S_ i$. We assume that $S_ i$ is quasi-compact and quasi-separated for all $i \in I$. We denote $g_ i : S \to S_ i$ the projection. We fix an element $0 \in I$ and a flat morphism of finite presentation $X_0 \to S_0$. We set $X_ i = S_ i \times _{S_0} X_0$ and $X = S \times _{S_0} X_0$ and we denote the transition morphisms $f_{i'i} : X_{i'} \to X_ i$ and $f_ i : X \to X_ i$ the projections.