Let $A$ be a finite dimensional symmetric algebra over a field (we can also assume that it is connected).
Call a non-projective indecomposable module $M$ strange in case $Ext^i(M,M)=0$ for all but finitely many $i$.
Is it true that strange $M$ have complexity equal to one, which means that the terms in a minimal projective resolution have bounded dimension?
In fact, in all my example $\Omega^{i}(M)$ even had the same dimension as $M$ for all $i \in \mathbb{Z}$.



Strange modules part II
Let $M$ be one of the examples you know. Then $M\otimes\_kM$ is a strange module for $A\otimes\_kA$, but does not have complexity one.


