I have a question about derived equivalences:
Let $k$ be a field and $A$ and $B$ two finite dimensional algebras over $k$. Let $F : D^-(A) \to D^-(B)$ be an equivalence of triangulated categories. Let $K^b(proj(A))$ (resp. $K^b(proj(B))$ ) be the subcategory consisting of the bounded complexes of finitely generated projective $A$-modules (resp. $B$-modules).
It is known that the equivalence $F$ restricts to an equivalence between these two categories (because they correspond to the categories of compact objects).
It is probably obvious, but I was wondering if the functor $F$ has to send a bounded complex of finitely generated injective modules to a bounded complex of finitely generated injective modules ?
In particular, is there a way to characterize the category of bounded complexes of finitely generated injective modules in the derived category $D^-(A)$ ?