# When is the non-negative derived category compactly generated?

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.

• 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! Jul 8, 2019 at 12:46
• True for any qcqs spectral algebraic space. SAG 9.6.1.2. Jul 8, 2019 at 13:55
• @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) Jul 8, 2019 at 14:57