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Is there an example of a presentable, stable, $k$-linear $\infty$-category which is dualizable but not compactly generated, where $k$ has characteristic zero, and which is $\text{QCoh}(X)$ (by which I mean the derived dg category of quasicoherent sheaves on $X$) for some prestack $X$?

Or, perhaps by removing some of the conditions above, e.g. being $\text{QCoh}(X)$, $k$ having characteristic zero, stability.

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  • $\begingroup$ What do you mean by $\mathrm{QCoh}(X)$? The derived ∞-category of $X$? The ∞-category of ind-pseudocoherent complexes? (I am more used to see that symbol denote the 1-category of quasicoherent sheaves) $\endgroup$ Mar 20, 2018 at 9:24
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    $\begingroup$ I think this is open even if you drop the $\infty$, see this paper where they have a partial result in that direction. $\endgroup$ Mar 20, 2018 at 9:47
  • $\begingroup$ @DenisNardin Yes, sorry, I mean the dg derived category of quasicoherent sheaves. $\endgroup$ Mar 21, 2018 at 0:59

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If $X$ is a locally compact topological space, then $\mathrm{Shv}(X, \mathrm{Mod}_{k} )$ is a presentable $k$-linear stable $\infty$-category which is dualizable (in fact, self-dual), but is rarely compactly generated (for example, this fails for $X = \mathbf{R}$).

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  • $\begingroup$ There’s a typo here somewhere. I’m guessing the first “compactly generated”? $\endgroup$ Mar 20, 2018 at 10:50
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    $\begingroup$ If I may, a reference for this is section 21.1.7 in math.harvard.edu/~lurie/papers/SAG-rootfile.pdf. Also worth noting is that a stable $k$-linear $\infty$-category is dualizable if and only if it is a retract of a compactly generated one (Theorem D.7.0.7 in op. cit.). $\endgroup$ Mar 20, 2018 at 12:05

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