Ext in symmetric algebras and group algebras 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:


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*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 $M$.
 A: 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 $i=1,2$.
A: Suppose $A$ is commutative local (necessarily artinian) with the only simple $k\neq A$  then $\psi_k=1$, so your statement 1 will say that $Ext^1_A(M,M)=0$ implies $M$ is free.  I stated it as a conjecture for complete intersections here (conjecture 9.1.3). Technically, it was stated as $Ext^1_A(M,N)=0$ implies $Ext^i_A(M,N)=0$ for all $i>0$, but when $M=N$ the latter condition is equivalent to $M$ being free. 
One could also ask if $Ext^1_A(M,M)=0$ implies $M$ is free, still assuming that $A$ is Gorenstein. It was stated as a question (9.1.4) in the same survey. 
As far as I know, both are open even for complete intersections unless $A$ is a hypersurface (which will be representation finite anyway).   
