most recent 30 from http://mathoverflow.net 2013-05-25T10:10:28Z http://mathoverflow.net/feeds/question/82398 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82398/hereditary-algebras Antonio 2011-12-01T18:37:34Z 2011-12-02T07:22:09Z

<p>I have the following problem: If $\Lambda$ is a hereditary, basic and connected algebra and $e$ is an idempotent of $\Lambda$, how can I prove that $e\Lambda e$ is also hereditary?</p>

http://mathoverflow.net/questions/82398/hereditary-algebras/82419#82419 Benjamin Steinberg 2011-12-02T01:39:27Z 2011-12-02T07:22:09Z

<p>If $\Lambda$ is split basic, then by Gabriel's theorem it is isomorphic to $\Bbbk Q$ where $Q$ is a finite acyclic quiver. Up to isomorphism you can assume $e$ is the sum of empty paths running over some subset $X$ of vertices. Then $e\Lambda e$ is isomorphic to the path algebra on the full (i.e. induced) subquiver on the vertex set $X$. Thus it is hereditary. </p>