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Let $X$ be a scheme. Is the category $QCoh(X)$ of quasi-coherent sheaves on $X$ locally presentable? If so, can we say anything about the $\kappa$ for which $QCoh(X)$ is locally $\kappa$-presentable? (e.g. is it always finitely presentable? Or related to the $\kappa$ of Gabber's result?)

I'm particularly interested in the case where $X$ is quasi-comact quasi-separated (qcqs).


In my searching for references, I've come across answers ranging from "we don't know", "when qcqs", to "always", and would appreciate some clarity.

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Zariski descent tells us that

$$\operatorname{QCoh}(X)=\lim_{U\subseteq X} \operatorname{QCoh}(U)$$

where $U$ ranges through all open affines and the limit is taken in the $(2,1)$-categorical sense. Since small limits of presentable categories are presentable and $\operatorname{QCoh}(\operatorname{Spec}R)=\operatorname{Mod}_R$ is presentable, we have that that $\operatorname{QCoh}(X)$ is presentable.

I'm not sure how to get bounds on the accessibility degree in general, though when $X$ is qcqs it is compactly generated (i.e. $\omega$-presentable) by the argument in this answer by Denis-Charles Cisinski.

PS: following recent trends in homotopy theory, I'm dropping the "locally" from "locally presentable'.

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