Lemma 32.10.2. Let I be a directed set. Let (S_ i, f_{ii'}) be an inverse system of schemes over I. Assume
all the morphisms f_{ii'} : S_ i \to S_{i'} are affine,
all the schemes S_ i are quasi-compact and quasi-separated.
Let S = \mathop{\mathrm{lim}}\nolimits _ i S_ i. Then we have the following:
For any sheaf of \mathcal{O}_ S-modules \mathcal{F} of finite presentation there exists an index i \in I and a sheaf of \mathcal{O}_{S_ i}-modules of finite presentation \mathcal{F}_ i such that \mathcal{F} \cong f_ i^*\mathcal{F}_ i.
Suppose given an index i \in I, sheaves of \mathcal{O}_{S_ i}-modules \mathcal{F}_ i, \mathcal{G}_ i of finite presentation and a morphism \varphi : f_ i^*\mathcal{F}_ i \to f_ i^*\mathcal{G}_ i over S. Then there exists an index i' \geq i and a morphism \varphi _{i'} : f_{i'i}^*\mathcal{F}_ i \to f_{i'i}^*\mathcal{G}_ i whose base change to S is \varphi .
Suppose given an index i \in I, sheaves of \mathcal{O}_{S_ i}-modules \mathcal{F}_ i, \mathcal{G}_ i of finite presentation and a pair of morphisms \varphi _ i, \psi _ i : \mathcal{F}_ i \to \mathcal{G}_ i. Assume that the base changes are equal: f_ i^*\varphi _ i = f_ i^*\psi _ i. Then there exists an index i' \geq i such that f_{i'i}^*\varphi _ i = f_{i'i}^*\psi _ i.
In other words, the category of modules of finite presentation over S is the colimit over I of the categories modules of finite presentation over S_ i.
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