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?
