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Let $A=KG$ be a group algebra for a finite group $G$. Let $S$ be a simple $A$-module. The extreme no loop conjecture predicts that $Ext_A^1(S,S) \neq 0$ implies $Ext_A^i(S,S) \neq 0$ for infinitely many $i$ for a general finite dimensional algebra. One can prove this for cocommutative Hopf algebra (hence also for group algebras).

Question: In case $A=KG$, does $Ext_A^1(S,S) \neq 0$ imply $Ext_A^i(S,S) \neq 0$ for all $i>0$?

I doubt that but I have no counterexample at the moment even for $S$ being just indecomposable instead of simple.

It is true for $KG$ being representation-finite,local or commutative. I have not much experience with other cases for group algebras but my computer at least suggested that it is also true for $k S_4$ with $k$ of characteristic 2.

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  • $\begingroup$ Take $K$ to be a field of characteristic 3, and $G=S_3$. Take $S=K$ with the trivial action. Then $Ext^1_A(S,S)=0$ because $Ext^1_A(S,S) = Hom_{Grp}(S_3,K)=0.$ On the other hand, by a result of Quillen we know that the ring $Ext^*_A(S,S)$ has Krull dimension 1, so it has infinitely many values of i for which $Ext^i_A(S,S)\neq 0$ $\endgroup$
    – Ehud Meir
    Commented Sep 11, 2019 at 12:44
  • $\begingroup$ @EhudMeir But we assume that Ext^1(S,S) is nonzero. $\endgroup$
    – Mare
    Commented Sep 11, 2019 at 12:56
  • $\begingroup$ Sorry, I misread the question. $\endgroup$
    – Ehud Meir
    Commented Sep 11, 2019 at 15:41

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