The Stacks project

Lemma 28.23.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\kappa $ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_ t)_{t \in T}$ of $\kappa $-generated $\mathcal{O}_ X$-modules such that every $\kappa $-generated $\mathcal{O}_ X$-module is isomorphic to one of the $\mathcal{F}_ t$.

Proof. There is a set of coverings of $X$ (provided we disallow repeats). Suppose $X = \bigcup U_ i$ is a covering and suppose $\mathcal{F}_ i$ is an $\mathcal{O}_{U_ i}$-module. Then there is a set of isomorphism classes of $\mathcal{O}_ X$-modules $\mathcal{F}$ with the property that $\mathcal{F}|_{U_ i} \cong \mathcal{F}_ i$ since there is a set of glueing maps. This reduces us to proving there is a set of (isomorphism classes of) quotients $\oplus _{k \in \kappa } \mathcal{O}_ X \to \mathcal{F}$ for any ringed space $X$. This is clear. $\square$

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