Let $ A$ be a unital dg-algebra over a base field $ k$ . We consider the category of (unbounded) right $ A$ -dg-modules with morphisms closed degree $ 0$ maps. We denote this category by dg-mod-$ A$ . We could consider homotopic morphisms and get the homotopy category [dg-mod-$ A$ ]. Moreover we could invert quasi-isomorphisms in [dg-mod-$ A$ ] and obtain its derived category. Let us denote this category by $ D(A)$ .

On the other hand, we could consider the dg-category DG-MOD-$ A$ : its objects are right $ A$ -dg-modules and its morphisms are chain complexes of maps between dg-modules.

We notice that there is a general construction of the derived category of a dg-category $ \mathcal{C}$ : We first consider the dg-category of right modules over $ \mathcal{C}$ , which is the dg-category of contravariant dg-functors from $ \mathcal{C}$ to Ch$ (k)$ . Let us denote this dg-category by DGM-$ \mathcal{C}$ . We also consider the full dg-subcategory Acycl-$ \mathcal{C}$ , which consists of dg-functors $ \mathcal{F}:\mathcal{C}\to \text{Ch}(k)$ such that $ \mathcal{F}(c)$ is acyclic for any object $ c\in \mathcal{C}$ . Then the derived category of $ \mathcal{C}$ is given by the Verdier quotient $ $ [\text{DGM}-\mathcal{C}]/[\text{Acycl}-\mathcal{C}] $ $ where $ [\cdot]$ is the homotopy category of dg-categories. We denote the derived category of $ \mathcal{C}$ by $ D(\mathcal{C})$ . We could show that the Yoneda functor induces a fully faithful functor $ [\mathcal{C}]\to D(\mathcal{C})$ .

Now go back to the dg-algebra $ A$ and the dg-category DG-MOD-$ A$ .

My question is: is the derived category $ D(A)$ equivalent to the derived category $ D(\text{DG-MOD}-A)$ ? Why?