Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post [When do chain complexes decompose as a direct sum?](https://mathoverflow.net/questions/32854/when-do-chain-complexes-decompose-as-a-direct-sum)) that any chain complex over $R$ is equivalent to its cohomology as an object of the derived category of $R$-module. How does one prove such a thing? (Either a sketch of the argument or a reference would be appreciated).



Chain complexes split in the derived category over rings of global dimension 1
One reference is H. Krause, "Derived categories, resolutions, and Brown representability", Contemporary Math. vol.436, AMS, 2007, p.101-139 or <https://arxiv.org/abs/math/0511047> , Section 1.6.


Another possible reference is L. Positselski, O.M. Schnürer, "Unbounded derived categories of small and big modules: Is the natural functor fully faithful?", J. Pure Appl. Algebra 225 (2021) Paper No. 106722 or <https://arxiv.org/abs/2003.11261> , Appendix A.


