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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 ?

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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.

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