When is derived category of ringed space perfectly generated?

Question

Let $(X,\mathcal{O})$ be a ringed space. Also assume that $X$ is nice, e.g. locally compact, Hausdorff, some type of finite dimension, ...

We can then consider $\mathcal{D}(\mathcal{O}\text{-}Mod)$. When is this a perfectly generated category? Recall that a non-empty set of objects $S\subset Ob(\mathcal{C})$ for a category $\mathcal{C}$ with coproducts is said to perfectly generate $\mathcal{C}$ if

• $S$ is a generating set, i.e. if $Hom(s,C)=0$ for all $s\in S$, then $C=0$
• For a non-empty family of morphisms $(C_i\to D_i)$ such that $Hom(s,C_i)\to Hom(s,D_i)$ is surjective for each $i$ and each $s$, then $Hom(s,\bigoplus C_i)\to Hom(s,\bigoplus D_i)$ is surjective for each $s\in S$.

I suspect this has been studied before, so even just links to some sources would be highly appreciated!

Currently my conjecture is that the skyscraper sheaves of the stalks and all their shifts are a perfect generating set under nice geometric assumptions, but I am unable to verify that this is the case.

Answer 1

This can be found as Theorem 14.2.1 in Categories and Sheaves by Kashiwara and Schapira. One has to note that they use a slightly different notion for the second property.

• For a non-empty countable family of morphisms $(C_i\to D_i)$ such that $Hom(s,C_i)\to Hom(s,D_i)$ vanishes for each $i$ and $s$ then $Hom(s,\bigoplus C_i)\to Hom(s,\bigoplus D_i)$ vanishes for each $s$.

I will ignore the countability assumption as this seems to be mainly a matter of taste. The relation between 'surjective' and 'vanishes' is easy to see from completing $C_i\to D_i$ to a distinguished triangle and noting that $Hom(s,-)$ is homological.