Question

Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C_{\bullet}$ in $\mathcal{D}$, we have $C_{\bullet} \cong H_{\bullet}(C_{\bullet})$, where the right hand side is the complex whose $i$-th term is $H_i(C_{\bullet})$ and all of whose maps are zero.

I would like to know a standard reference for this. Ideally, I would like to know the original source.

Right now, the only sources I know are lecture notes, such as Section 2.5 of Keller's notes or Theorem 2.1 in Lenzing's.

Thanks!

Answer 1

Dieter Happel's book Triangulated Categories in the representation theory of finite dimensional algebras is a pretty canonical source, and it includes the result you mention.

(That particular result might be folkloric, though)

Answer 2

In Krause's Derived categories, resolutions, and Brown representability, the general version of this result for hereditary abelian categories is proved (Section 1.6).