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On any ringed space $(X,\mathcal{O}_X)$ we can define quasi-coherent $\mathcal{O}_X$-modules: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x\in X$ there exists an open neighborhood $x\in U\subset X$ such that $\mathcal{F}|_U$ is isomorphic to the cokernel of a map

$$ \bigoplus_{j\in J}\mathcal{O}_U \to \bigoplus_{i\in I}\mathcal{O}_U. $$ See Stack Project Section 17.10.

We know that in general an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules is not necessarily quasi-coherent. See Stack Project Example 17.10.9. On the other hand, if $X$ is a scheme, then any direct sum of quasi-coherent $\mathcal{O}_X$-modules is still quasi-coherent. See Stack Project Section 26.24.

We can consider quasi-coherent $\mathcal{O}_X$-modules over ringed spaces which are not schemes. For example, $X$ is a complex manifold and $\mathcal{O}_X$ is the sheaf of holomorphic functions.

My question is: for a complex manifold $X$ with $\mathcal{O}_X$ the sheaf of holomorphic functions, is an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules still quasi-coherent?

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    $\begingroup$ That is not true. Gabber constructs a (nonzero) quasi-coherent sheaf $\mathcal{F}(p,q)$ on the open unit disk $X$ in the complex plane that has vanishing global sections on every open disk containing both $p$ and $q$. Now let $(p_n)$ and $(q_n)$ be sequences of points in the punctured unit disk that limit to zero. The direct sum of the sheaves $\mathcal{F}(p_n,q_n)$ is globally generated on no open disk containing zero. $\endgroup$
    – Jason Starr
    Commented Aug 19, 2023 at 11:21
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    $\begingroup$ Just to clarify, Gabber constructs a quasi-coherent sheaf that is isomorphic to a direct sum of countably many copies of the structure sheaf on either $X\setminus \{p\}$ or $X\setminus\{q\}$. $\endgroup$
    – Jason Starr
    Commented Aug 19, 2023 at 16:29

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