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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?

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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.

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  • $\begingroup$ Thank you. I added a 4. question, so I wait a little before accepting your answer. Actually in the examples I found I have always that the algebras are selfinjective. $\endgroup$
    – Mare
    Commented Feb 21, 2017 at 19:12
  • $\begingroup$ For interested readers: In (1), the first simple module $S_1$ has that $\Omega^{1}(S_1)$ is 1-periodic and the second simple module is 1-periodic itself. $\endgroup$
    – Mare
    Commented Feb 21, 2017 at 19:18
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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).

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  • $\begingroup$ Oh, right I knew that with the complexity once but forget about it... $\endgroup$
    – Mare
    Commented Feb 21, 2017 at 21:23
  • $\begingroup$ Taking the Nakayama (quiver) algebra with kupisch series ([3,3,3,3,3,3,3,2,1] and then its trivial extension, my Pc suggests it has a simple module with bounded Omegas but which is not periodic. It is over a finite field with characteristic 3, but I think that it does not matter what the field is for that example, since the relations are field independent. $\endgroup$
    – Mare
    Commented Feb 21, 2017 at 21:26
  • $\begingroup$ sorry, I think I did not wait long enough. It might be that the syzygies grow very slowly and I didnt wait long enough. So I think the dimensions are not bounded (but it really looked like that calculating syzygies till 100). $\endgroup$
    – Mare
    Commented Feb 21, 2017 at 22:07
  • $\begingroup$ Yes, the trivial extensions of this type of Nakayama algebra are an interesting class of examples. They tend to be periodic, but the periods of the simples are somewhat irregular. $\endgroup$
    – Alex Dugas
    Commented Feb 21, 2017 at 22:20

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