3
$\begingroup$

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!

$\endgroup$

2 Answers 2

6
$\begingroup$

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)

$\endgroup$
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.