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 ?



Derived category of a quotient
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.


