On some modules with bounded syzygies 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:


*

*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? 
 A: 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.
A: For (4), the answer is yes.  For selfinjective algebras, the condition that the syzygies are non-zero and indecomposable is automatic for non-projective indecomposable M.  Thus "cool" reduces to the notion of "complexity one" in this case.  It is easy to see (using the horseshoe lemma for example) that the property of a module having complexity $\leq 1$ is preserved by extensions and taking direct summands.  Thus if all simples have complexity one, then so do all non-projective indecomposables.
Question (3), however, is wide open.   Over a selfinjective algebra, any simple module that is cool must also be periodic (at least over an algebraically closed field) [arXiv:1203.2408].  Thus (3) is equivalent to asking for a non-periodic selfinjective algebra for which every simple module is periodic.  I don't know of any such example yet, and believe it may have been conjectured that none exist (but I can't find a reference to this "conjecture" at the moment).
