Finite generatation of Ext - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:18:24Z http://mathoverflow.net/feeds/question/27360 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27360/finite-generatation-of-ext Finite generatation of Ext ashpool 2010-06-07T15:09:18Z 2010-06-07T15:09:18Z <p>If \$A\$ is a Noetherian ring and \$M\$, \$N\$ are finitely generated modules over \$A\$, it is easy to see that \$\mbox{Ext}_{A}(M,N)\$ is finitely generated by taking a finitely generated projective resolution of \$M\$. But if I take an injective resolution of \$N\$ instead, it is not at all clear to me why \$\mbox{Ext} _{A}(M,N)\$ should be finitely generated. It is my understanding that injective hulls are in general not finitely generated. Does this mean that if I take \$\mbox{Hom}(M,-)\$ of the injective resolution and compute its cohomology I magically get a finitely generated module?</p>