Let A be a finite dimensional algebra. Call an indecomposable module M cool in case $\Omega^{i}(M)$ is nonzero and indecomposable for every $i \geq 1$ and $dim(\Omega^{i}(M))$ is bounded. 


Questions:


1. In case every simple module is cool, is the algebra selfinjective?
Wrong by an answer of Jeremy Rickard.


2.In case every simple module is cool, is every indecomposable non-projective module cool? Wrong by an answer of Jeremy Rickard.


3.What are classes of selfinjective algebras such that every simple (or every indecomposable) module is cool and such that the algebra is not periodic?


4.In case every simple module is cool and the algebra is selfinjective, is every indecomposable non-projective module cool? 



On some modules with bounded syzygies
For (1), take a quiver with two vertices, an arrow $\alpha$ from vertex $1$ to vertex $2$, a loop $\beta$ at vertex $2$, and relations $\alpha\beta=0$ and $\beta^2=0$.


For (2), take a quiver with four vertices, arrows $\alpha\_i$ from vertex $i$ to vertex $4$ for $i=1,2,3$, a loop $\beta$ at vertex $4$, and relations $\alpha\_i\beta=0$ and $\beta^2=0$. There's an indecomposable representation $V$ with dimension vector $(1,1,1,1)$ which is uncool since $\Omega V\cong S\_4\oplus S\_4$ is decomposable.


