7
$\begingroup$

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 ?

$\endgroup$
12
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.