Let $A$ be a representation-finite algebra and $M$ an indecomposable module with finite projective dimension $g >0$.

Question 1: Do we have $dim(Ext_A^g(M, \tau_g(M)))=1$? Here $\tau_g(M)=\tau ( \Omega^{g-1}(M))$.

 

Question 2: Could this even be true when $A$ is not represenation-finite? Probably not, but the computer did not find a counterexample yet.

Let $A$ be a representation-finite algebra and $M$ an indecomposable module with finite projective dimension $g >0$.

Question 1: Do we have $dim(Ext_A^g(M, \tau_g(M)))=1$? Here $\tau_g(M)=\tau ( \Omega^{g-1}(M))$.

Question 2: Could this even be true when $A$ is not represenation-finite? Probably not, but the computer did not find a counterexample yet.