The Stacks project

Lemma 95.5.2. Let $(X, \mathcal{O}_ X)$ be a ringed space.

  1. The category of finite type $\mathcal{O}_ X$-modules has a set of isomorphism classes.

  2. The category of finite type quasi-coherent $\mathcal{O}_ X$-modules has a set of isomorphism classes.

Proof. Part (2) follows from part (1) as the category in (2) is a full subcategory of the category in (1). Consider any open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$. Denote $j_ i : U_ i \to X$ the inclusion maps. Consider any map $r : I \to \mathbf{N}$. If $\mathcal{F}$ is an $\mathcal{O}_ X$-module whose restriction to $U_ i$ is generated by at most $r(i)$ sections from $\mathcal{F}(U_ i)$, then $\mathcal{F}$ is a quotient of the sheaf

\[ \mathcal{H}_{\mathcal{U}, r} = \bigoplus \nolimits _{i \in I} j_{i, !}\mathcal{O}_{U_ i}^{\oplus r(i)} \]

By definition, if $\mathcal{F}$ is of finite type, then there exists some open covering with $\mathcal{U}$ whose index set is $I = X$ such that this condition is true. Hence it suffices to show that there is a set of possible choices for $\mathcal{U}$ (obvious), a set of possible choices for $r : I \to \mathbf{N}$ (obvious), and a set of possible quotient modules of $\mathcal{H}_{\mathcal{U}, r}$ for each $\mathcal{U}$ and $r$. In other words, it suffices to show that given an $\mathcal{O}_ X$-module $\mathcal{H}$ there is at most a set of isomorphism classes of quotients. This last assertion becomes obvious by thinking of the kernels of a quotient map $\mathcal{H} \to \mathcal{F}$ as being parametrized by a subset of the power set of $\prod _{U \subset X\text{ open}} \mathcal{H}(U)$. $\square$

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