Is there a selfinjective finite dimensional algebra $A$ with a non-projective module $M$ such that $Ext^{i}(M,M)=0$ for all $i=1,2,...,2n-1$ when $n$ denotes the number of simple module of that algebra? Of course a positive answer to this would prove the Tachikawa conjecture, but maybe there is an easy example of such a module.

Is there a selfinjective finite dimensional algebra $A$ with a non-projective module $M$ such that $Ext^{i}(M,M)=0$ for all $i=1,2,...,2n-1$ when $n$ denotes the number of simple module of that algebra? Of course a nonexistence showing answer to this would prove the Tachikawa conjecture, but maybe there is an easy example of such a module.