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For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
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When is derived category of ringed space perfectly generated?
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 …
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When is derived category of ringed space perfectly generated?
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 $\mathcal{D}(\mathcal{O}\text{-}Mod)$. …