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:

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.

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 $M$.

shouldit be true? $\endgroup$shouldbe true. Explain why it should, as that can only improve your question — claims that something should or should not be unaccompanied with reasons are extremely vaporous in almost all contexts1. Also: itseemsto be true in those cases orisit true? $\endgroup$shouldbe true or that itseemsto be true, say why you think it is true, or why you expect it to be true, by mentioning the evidence you have and so on. Nothinghasto be true. Newcomers to the subject, specially, will appreciate that — deontological claims only contribute to make them excluded of some inaccesible lore that would allow them to know, like you do, why things should be true. $\endgroup$1more comment