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! $\endgroup$ Commented Jul 8, 2019 at 12:46
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    $\begingroup$ True for any qcqs spectral algebraic space. SAG $\endgroup$ Commented Jul 8, 2019 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) $\endgroup$ Commented Jul 8, 2019 at 14:57

2 Answers 2


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 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.


For qcqs algebraic spaces (non-spectral), this is lemma 2.7 in Algebraization and Tannaka Duality by Bhatt.


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