Examples of complexes of modules for wich homomorphisms "homological" implies "homotopic" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:10:56Z http://mathoverflow.net/feeds/question/84042 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84042/examples-of-complexes-of-modules-for-wich-homomorphisms-homological-implies-ho Examples of complexes of modules for wich homomorphisms "homological" implies "homotopic" One_math_boy 2011-12-21T20:56:59Z 2012-12-08T12:48:58Z <p>Let $V$ and $V^\prime$ - complexes of modules over ring $A$, and $f, g$ - homomorphisms $V\rightarrow V^\prime$. </p> <blockquote> <p>I am interested in various conditions on $A, V, V^\prime$: ($f$ and $g$ are homological) $\Rightarrow$ ($f$ and $g$ are homotopic).</p> </blockquote> <p>(I knew one example: $A$ - hereditary algebra and $V, V^\prime$ - complexes of projective modules, bounded from the right. But recently I understood that in this case it's not true that ($f$ and $g$ are homological) $\Rightarrow$ ($f$ and $g$ are homotopic))</p> http://mathoverflow.net/questions/84042/examples-of-complexes-of-modules-for-wich-homomorphisms-homological-implies-ho/84053#84053 Answer by Fernando Muro for Examples of complexes of modules for wich homomorphisms "homological" implies "homotopic" Fernando Muro 2011-12-21T22:59:39Z 2011-12-21T23:46:20Z <p>This question relates to a very complicated problem known as <strong>Freyd's generating hypothesis</strong>. The problem was first posed for the stable homotopy category but it can be extended to more general triangulated categories, such as the derived category $D(R)$ of a ring $R$. In this context the hypothesis (which may or may not be satisfied, depending on $R$) says that your $\Rightarrow$ is satisfied whenever the complexes are (quasi-isomorphic to) bounded complexes of f.g. projectives. Keir H. Lockridge proved (JPAA, 2007) that for $R$ commutative this is true if and only if $R$ is von Neumann regular. You can look at this problem in other contexts (e.g. modular representation theory) to get more examples and counterexamples.</p>