Equivalence between a derived subcategory and a subcategory of the derived category

Question

Let $\mathcal{A}$ be a subcategory of $\mathcal{C}$. Let $D(\mathcal{A})$ and $D(\mathcal{C})$ be the associated derived categories. We can define $D_\mathcal{A}(\mathcal{C}) = \{X \in \mathcal{C}\hspace{2pt} |\hspace{2pt} H^i(X) \in \mathcal{A}\}$. We have natural inclusions $$D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C}) \to D(\mathcal{C})$$

My question is this: When is $D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C})$ an equivalence of categories?

I have read that it is not always an equivalence, but that it often is. I have examples of when it is: Is there an easy example for when it isn't?

Answer 1

Take $C$ to be sheaves of abelian groups on the sphere, and let $A$ be the abelian subcategory of locally constant abelian groups. Then $A$ is equivalent to the category of abelian groups and so $Hom(\mathbb Z,\mathbb Z[2])$ is different depending on whether you take it in $D(A)$ or $D(C)$

Answer 2

This is more comment than answer, but for bounded derived categories we have the following: let $\mathcal{C}$ be an abelian category and $\mathcal{A} \subset \mathcal{C}$ be a Serre subcategory. Then, a sufficient condition for the natural functor $\mathrm{D^b}(\mathcal{A}) \to \mathrm{D}^{\mathrm{b}}_{\mathcal{A}}(\mathcal{C})$ to be an equivalence is that for every exact sequence $S_1:0 \to A \to B \to C \to 0$ in $\mathcal{C}$ with $A \in \mathcal{A}$, there is an exact sequence $S_2: 0 \to A \to B' \to C' \to 0$ in $\mathcal{A}$ and a map of exact sequences $S_1 \to S_2$ which is the identity on $A$.

This result can be found in Keller's ``On the cyclic homology of exact categories'', who in turn refers to SGA5.