Suppose we have a not-short exact sequence 0 -> A -> B -> 0 in some abelian category. Now let us apply right exact functor F: F(A) -> F(B) -> 0. So could I consider a left derived functor H to obtain the sequence ...-> H0(F(A)) -> H0(F(B)) -> F(A) -> F(B) -> 0?
I haven't seen this done with exact sequences with two non zero objects but was wondering if the concept would still hold.