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This question is strongly related to this question. However it seems to me sufficiently distinct to warrant asking it separately.

Let $X$ be a quasi-compact, quasi-separated scheme. When is the ∞-category $\mathcal{D}_{\ge0}(X)$ of connective sheaves of complexes compactly generated?

Equivalently, when is every connective element of $\mathcal{D}(X)$ expressible as a filtered colimit of connective perfect complexes? (It is not hard to show that the compact objects of $\mathcal{D}_{\ge0}(X)$ are exactly the connective perfect complexes).

Note that I am using homological grading, so $\mathcal{D}_{\ge0}(X)$ is a colocalization of $\mathcal{D}(X)$ and not a localization.

Thanks to corollary C.6.3.3 in Lurie's Spectral Algebraic Geometry it's enough to show that every connective complex over $X$ receives a nonzero map from a connective perfect complex. For example, every connective complex $C$ such that $R\Gamma(C)$ is not concentrated in negative degree satisfied the condition. In particular the statement is true if $X$ has an ample family of line bundles

This is different from this question, because despite the similarities the derived category of a scheme is not the same thing as the derived category of the underlying ringed space. For example, if $U\subseteq X$ is an open subscheme, the restriction map $j^*:\mathcal{D}(X)\to \mathcal{D}(U)$ does not have a left adjoint.

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    $\begingroup$ Fantastic question Denis! I wondered about exactly this in and around Question 8.16 in the paper arxiv.org/abs/1812.01526. I would love to know the answer! $\endgroup$ – Benjamin Antieau Jul 8 at 12:46
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    $\begingroup$ True for any qcqs spectral algebraic space. SAG 9.6.1.2. $\endgroup$ – Jacob Lurie Jul 8 at 13:55
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    $\begingroup$ @JacobLurie Oh that's great, thank you! If you write that as an answer, I'll accept it (if you are not interested, I can write it as a CW answer and accept it in order to mark the question as solved) $\endgroup$ – Denis Nardin Jul 8 at 14:57
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This is true more generally for every qcqs spectral algebraic space (assuming that D(X) has the meaning that I think it does). This is proven in (the current version of) Spectral Algebraic Geometry as 9.6.1.2. Probably there is some more classical reference in the case of schemes (it follows by a mild variation on Thomason's argument), but I'm not sure where to find it.

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