TITLE: Ext in symmetric algebras and group algebras
QUESTION [6 upvotes]: 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 $M$.

REPLY [5 votes]: 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 $i=1,2$.