Hereditary algebras - MathOverflow 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 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 Answer by Benjamin Steinberg for Hereditary algebras 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>