The Stacks project

Lemma 32.10.2. Let $I$ be a directed set. Let $(S_ i, f_{ii'})$ be an inverse system of schemes over $I$. Assume

  1. all the morphisms $f_{ii'} : S_ i \to S_{i'}$ are affine,

  2. 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:

  1. 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 $\mathcal{F} \cong f_ i^*\mathcal{F}_ i$.

  2. 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 $.

  3. 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 $S_ i$.

Proof. We sketch two proofs, but we omit the details.

First proof. If $S$ and $S_ i$ are affine schemes, then this lemma is equivalent to Algebra, Lemma 10.127.6. In the general case, use Zariski glueing to deduce it from the affine case.

Second proof. We use

  1. there is an equivalence of categories between quasi-coherent $\mathcal{O}_ S$-modules and vector bundles over $S$, see Constructions, Section 27.6, and

  2. a vector bundle $\mathbf{V}(\mathcal{F}) \to S$ is of finite presentation over $S$ if and only if $\mathcal{F}$ is an $\mathcal{O}_ S$-module of finite presentation.

Having said this, we can use Lemma 32.10.1 to show that the category of vector bundles of finite presentation over $S$ is the colimit over $I$ of the categories of vector bundles over $S_ i$. $\square$

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