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Recall that for a scheme $X$, it has the resolution property if every coherent sheaf $E$ on $X$, is the quotient of a finite locally free $\mathcal{O}_X$-module.

On the other hand, Ben-Zvi-Nadler-Francis proved in : https://arxiv.org/pdf/0805.0157.pdf that for a separated quasi-compact scheme,one has $\mathit{QCoh}(X)=\mathit{Ind}(\mathit{Perf}(X))$, where $\mathit{QCoh}(X)$ is the $\infty$-category of quasi-coherent sheaves and $\mathit{Perf}(X)$ the category of perfect complexes. In particular, for a coherent sheaf $E$, one can write it as a colimit of perfect complexes $E_{\alpha}$. In which case, can we use this result to deduce that $X$ has the resolution property? Do we have a counter-example of a separated quasi-compact scheme that does not have the resolution property?

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    $\begingroup$ Why the mention of affine stacks in the title? The question doesn't feature that restriction. $\endgroup$
    – Jon Pridham
    Commented Sep 17, 2023 at 17:19

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A quasi compact and quasi-separated scheme has its derived category of sheaves of modules with quasi-coherent cohomology generated by perfect complexes. This is actually a theorem of Bondal and Van de Berg Generators and representability of functors in commutative and noncommutative geometry. However, there are non-separated schemes that, despite the previous theorem, do not posses the resolution property, see the paper by Totaro The resolution property for schemes and stacks for a discussion of the subtleties of the issue.

The proof by Bondal-Van den Bergh uses Thomason-Neeman localization: a perfect complex over a dense open subset extends. This does not hold for vector bundles in general. This is where the difference between compact generation and generation by vector bundles lies.

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    $\begingroup$ are you sure you mean cosmology? $\endgroup$
    – Matthieu Romagny
    Commented Sep 17, 2023 at 17:07
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    $\begingroup$ @MatthieuRomagny Sure! 😂 We want to look deeper into the galaxies of quasi-coherent sheaves... No, seriously, thank you for catching the typo. $\endgroup$
    – Leo Alonso
    Commented Sep 17, 2023 at 20:03
  • $\begingroup$ It doesn't answer the question. I considered separated schemes. $\endgroup$
    – prochet
    Commented Sep 17, 2023 at 21:24
  • $\begingroup$ Moreover, the definition of compactly generated in Bondal-Vande Berg is in terms of the right orthogonal of the category of perfect complexes which in general is a different condition than asking that QCoh= Ind(Perf) $\endgroup$
    – prochet
    Commented Sep 17, 2023 at 21:31
  • $\begingroup$ @prochet Being the category of quasi-coherent sheaves a Grothendicek category the fact that the perfect complexes has right orthogonal zero is equivalent to the fact that the smallest triangulated category that contains them is the full derived category. $\endgroup$
    – Leo Alonso
    Commented Sep 18, 2023 at 7:07

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