# Derived category of a quotient

Le $$A$$ be an abelian category and $$B$$ a Serre subcategory of $$A$$. Under which conditions do we have $$D^*(A/B) \simeq D^*(A)/D^*(B),$$ where $$D^*$$ stands for the derived category with $$* = +,-,b$$ or nothing ?

What it is true is that $$D^*(A/B) \simeq D^*(A)/D_B^*(A),$$ where the category $$D_B^*(A)$$ is the subcategory of $$D^*(A)$$ whose homologies lie in $$B$$. In general it may not agree with $$D^*(B)$$.
Notice that $$D_B^*(A)$$ is identified with the full subcategory of $$D^*(A)$$ formed by objects whose image under the induced functor $$D^*(A) \longrightarrow D^*(A/B)$$ vanishes.