Derived categories as homotopy categories of model categories

Question

Given an abelian category A, is there a model structure on the category of complexes C(A) (or K(A) ("classical" homotopy category)) such that its homotopy category "is" the derived category D(A)?

Answer 1

Let $\mathbf{A}$ be an abelian (or Grothendieck) category, and consider the injective model structure on the category of cochain complexes $\mathrm{Ch}(\mathbf{A})$. Then the derived category $\mathcal{D}(\mathbf{A})$ is the homotopy category $\mathrm{Ho}(\mathrm{Ch}(\mathbf{A}))$ by inverting the weak equivalences.