TITLE: Derived category of a quotient
QUESTION [7 upvotes]: 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 ?

REPLY [12 votes]: 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.