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.