Let $A$ be a symmetric algebra and define $\psi_M:=sup \{ i \geq 1 | Ext^{i}(M,M) \neq 0 \}$ (infinite if this Ext is nonzero infinitely often) for an indecomposable non-projective module $M$. Call $M$ $i$-special in case $\psi_M=i$ is finite. Let $N:=A \oplus X$ with $X$ containing an $i$-special module. Then the algebra $B:=End_A(N)$ should contain modules with $Ext^i(D(B),L)=0$ for all $i >0$ (called CMCM-modules) that are not Gorenstein injective. Such CMCM-modules are pretty rare and were found only 40 years after the definition of Gorenstein projective by Auslander and Bridger. I think such algebras $B$ are good candidates to challenge the homological conjectures.

Questions:

1.In a local algebra, can we have $\psi_M \geq 2$ and finite?

2. What are example of modules with $\psi_M=i$ for any $i \geq 2$? Is there a symmetric algebra with two non-projective indecomposable modules $M, N$ such that $\psi_M \neq \psi_N$ and both are finite?

3. I was able to construct such modules only over algebras over field containing elements that are not roots of unity. Thus my construction does not work for example for finite fields. Are there modules with finite $\psi_M$ when in the relation of the algebra $A$ there are just the field elements +1 or -1 involved and can one construct such modules over algebras over any field?