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In the Stacks project and in a book of Brian Conrad the "main" derived category of a scheme is the one of $\mathcal{O}_X$-modules. I would like to understand whether $D(\mathcal{O}_X)$ is better than $D(Qcoh(X))$ in any sense? Note here that if $X$ is Noetherian then $D(Qcoh(X))$ is essentially a full subcategory of $D(\mathcal{O}_X)$; see https://stacks.math.columbia.edu/tag/09T4. Moreover, subcategories of this type appear to be respected by the "connecting functors".

So, do I miss something "serious" here, or is it just easier to expose the theory using $\mathcal{O}_X$-modules? Also, does there exist a "substitute" for the Stacks project (where $\mathcal{O}_X$-modules are not mentioned in the main formulations)? Moreover, I would like to find a complete proof of the countability lemma https://stacks.math.columbia.edu/tag/0G0V somewhere in the literature (to avoid giving my own proof in my paper).

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    $\begingroup$ First of all, there are important $\mathcal{O}_X$-modules that are not quasi-coherent, e.g., the sheaf of total rings of fractions for certain schemes $X$ (e.g., a square-zero extension of a nodal curve where the nilradical is the structure sheaf of the node). Second, the category of quasi-coherent sheaves is not stable under products, yet the category of $\mathcal{O}_X$-modules is stable under products. So there are some categorical constructions that, a priori, might leave the category of quasi-coherent sheaves (usually boundedness hypotheses on $D(Qcoh(X))$ fix this also). $\endgroup$
    – Jason Starr
    Commented Mar 28, 2023 at 17:13
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    $\begingroup$ Think about the case where X is affine. In fact there is a unique sheaf of ∞-categories on the big (Zariski, say) site whose sections over an affine Spec(A) give the derived ∞-category D(A) of A-modules, and that sheaf is the one sending a scheme X to the ∞-category D_qc(X) of complexes of O_X-modules with quasi-coherent cohomology. So for a general scheme, neither D(O_X) nor D(Qcoh(X)) is the "correct" thing to consider, but instead D_qc(X) is the right way to globalize the derived category of a commutative ring. I am pretty sure the Stacks Project also discusses D_qc(X). $\endgroup$
    – crystalline
    Commented Mar 28, 2023 at 18:29
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    $\begingroup$ @JasonStarr Gabber showed that the category of quasi-coherent sheaves has arbitrary limits and colimits ; it is Grothendieck abelian. $\endgroup$
    – crystalline
    Commented Mar 28, 2023 at 18:32
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    $\begingroup$ @crystalline The limits in the subcategory of quasi-coherent sheaves may disagree with the limit of those same sheaves in the category of $\mathcal{O}_X$-modules. That is what I should have written. $\endgroup$
    – Jason Starr
    Commented Mar 28, 2023 at 21:36

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