Let $A$ be a selfinjective algebra and for an indecomposable module $M$ define $\psi\_M:= \inf \{ i \geq 1 | Ext\_A^i(M,M) \neq 0 \}$.


Questions:


1. In case $A$ is symmetric, do we have $\psi\_M \leq max \{ \psi\_S | S $ is simple $\}$ for each indecomposable non-projective module $M$?
This should be true in case $A$ is representation-finite.
2. In case $A=kG$ is a group algebra over a field of characteristic $p$. Do we have even $\psi\_M \leq \psi\_K$ when $K$ is the trivial module and each indecomposable non-projective $M$ ? I can prove this for $p$-groups and in case $p$ does not divide the dimension of $M$.



Ext in symmetric algebras and group algebras
I think this example answers both questions.


Let $k$ have characteristic $3$, and let $G=C\_3\times S\_3$.


Then $kG$ has two simple modules, both one-dimensional, and for each simple module $S$, $\text{Ext}^1(S,S)$ is one-dimensional.


But if $M=kC\_3$, with $S\_3$ acting trivially, then $\text{Ext}^i(M,M)=0$ for $i=1,2$.


