Let $R = \oplus_{i\geq0} R_i$, $R_0 = k$ ($k$ a field) be a positively-graded commutative noetherian algebra, regarded as an augmented dg-algebra with zero differential.
Depending on one's interest, one may form $\mathrm{D}^b(R\text{-grmod})$, the derived category of graded $R$-modules with finitely generated, bounded cohomology. Alternatively, one may form $\mathrm{D}^b(R\text{-dgmod})$, the derived category of dg-modules with locally finite, bounded cohomology.
Is there a way to compare these two categories? I am far from an expert, but to me it seems that $\mathrm{D}^b(R\text{-grmod}) \subset \mathrm{D}^b(R\text{-dgmod})$ (perhaps the inclusion is not rigorously true but intuitively true). However I am unable to find any such relationship in the literature.
If the context helps, I am interested in the case where $R$ is Koszul.
