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 $S_ i$.

## Comments (0)