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Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is the cokernel of a homomorphism between free $\mathcal{O}_{U_i}$-modules. The global section functor $\mathsf{Qcoh}(X) \to \mathsf{Mod}(\Gamma(X,\mathcal{O}_X))$ has a left adjoint $M \mapsto \widetilde{M}$, the sheaf of modules associated to $M$. If $X$ has a fundamental system of quasi-compact neighborhoods, every quasi-coherent sheaf is locally associated to a module, and the converse is always true.

My first question is: Is there any detailed treatment of the categorical properties of $\mathsf{Qcoh}(X)$ which goes beyond the basics treated in the stacks project? More specifically, I wonder if $\mathsf{Qcoh}(X)$ is a cocomplete symmetric monoidal category, when the underlying space of $X$ has nice topological properties. This is well-known to be true when $X$ is a scheme, but what about the case that $X$ is, say, a smooth manifold equipped with the sheaf of smooth functions?

The only thing to check is that $\mathsf{Qcoh}(X)$ is stable under colimits computed in $\mathsf{Mod}(X)$. When $X$ has a fundamental system of quasi-compact neighborhoods, it is not hard to reduce this to directed colimits.

Question. Assuming that the underlying space of $X$ is nice enough, is $\mathsf{Qcoh}(X)$ closed under directed colimits?

There is a remark in the stacks project which says that this fails for general $X$ (without proof, can someone give an example?), but I am not really interested in weird spaces. Let me explain why the naive proof attempt doesn't work: Say we are given a sequence $F_0 \to F_1 \to \dotsc$ of quasi-coherent sheaves on $X$. Given $x \in X$, for every $n \in \mathbb{N}$ there is an open neighborhood $U_n$ of $x$ such that $F_n|_{U_n}$ has a presentation. The problem is that $\cap_n U_n$ is not open.

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Doesn't the example of Gabber mentioned in the comments to your earlier question… lead to counterexamples with countably infinite direct sums on the open unit disc in $\mathbf{C}$ viewed as a complex manifold? (Example 2.1.10 in loc. cit. uses Gabber's example to make such counterexamples in the rigid-analytic setting, and that seems to carry over without much change to the complex-analytic open unit disc.) – user30379 Feb 28 '13 at 15:53
Brian Conrad uses a different definition of quasi-coherent modules. – Martin Brandenburg Feb 28 '13 at 16:17
Gabber's example is a gluing two countably infinite direct sums of copies of the structure sheaf (on certain open subspaces), so those are quasi-coherent in your sense, and Example 2.1.10 in loc. cit. is a countably infinite direct sum of various instances of Gabber's example. So it seems like a counterexample to what you're asking for (upon adapting from the rigid-analytic to complex-analytic case, which looks like it should be rather straightforward to carry out); am I misunderstanding your objection? I don't think it matters what else is going on in that paper (terminology or otherwise). – user30379 Feb 28 '13 at 16:44

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