Earliest/most standard reference for derived categories of hereditary algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T18:03:00Z http://mathoverflow.net/feeds/question/87830 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87830/earliest-most-standard-reference-for-derived-categories-of-hereditary-algebras Earliest/most standard reference for derived categories of hereditary algebras David Speyer 2012-02-07T19:28:57Z 2012-07-16T16:33:41Z <p>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.</p> <p>I would like to know a standard reference for this. Ideally, I would like to know the original source.</p> <p>Right now, the only sources I know are lecture notes, such as Section 2.5 of <a href="http://www.math.jussieu.fr/~keller/ictp2006/lecturenotes/keller.pdf" rel="nofollow">Keller's notes</a> or Theorem 2.1 in <a href="http://www.math.jussieu.fr/~keller/ictp2006/lecturenotes/lenzing1.pdf" rel="nofollow">Lenzing's</a>.</p> <p>Thanks!</p> http://mathoverflow.net/questions/87830/earliest-most-standard-reference-for-derived-categories-of-hereditary-algebras/87836#87836 Answer by Mariano Suárez-Alvarez for Earliest/most standard reference for derived categories of hereditary algebras Mariano Suárez-Alvarez 2012-02-07T19:50:31Z 2012-02-07T19:50:31Z <p>Dieter Happel's <a href="http://www.ams.org/mathscinet-getitem?mr=935124" rel="nofollow">book</a> <em>Triangulated Categories in the representation theory of finite dimensional algebras</em> is a pretty canonical source, and it includes the result you mention.</p> <p>(That particular result might be folkloric, though)</p> http://mathoverflow.net/questions/87830/earliest-most-standard-reference-for-derived-categories-of-hereditary-algebras/102366#102366 Answer by Rasmus for Earliest/most standard reference for derived categories of hereditary algebras Rasmus 2012-07-16T16:33:41Z 2012-07-16T16:33:41Z <p>In Krause's <a href="http://www.math.uic.edu/~bshipley/krause.chicago.pdf" rel="nofollow"><em>Derived categories, resolutions, and Brown representability</em></a>, the general version of this result for hereditary abelian categories is proved (Section 1.6).</p>