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Let $A$ be a finite dimensional local Frobenius algebra with simple module $k$ and an indecomposable non-projective module $M$ (that is also finite dimensional).

Question:

Is there an example of such an $M$ such that every map $f: \Omega^n(M) \rightarrow k \rightarrow \Omega^{n-1}(M)$ is zero in the stable module category of $A$ for all $n \in \mathbb{Z}$?

Note that $\operatorname{Ext}_A^1(M,M) \cong \underline{\operatorname{Hom}_A}(\Omega^{n}(M),\Omega^{n-1}(M))$ and it seems to be unknown whether this can be zero.

$\operatorname{Ext}_A^1(M,M)$ is for example always non-zero for radical cube zero algebras or Hopf algebras.

(I reformulated the question as suggested by a comment of Jeremy Rickard)

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    $\begingroup$ You want this for every $f$? In other words, you want an $M$ so that all compositions of the forms $M\to k\to\Omega^{-1}M$ and $\Omega M\to k\to M$ are zero in the stable module category? Also, you ask for an easy example: do you know a difficult example? $\endgroup$
    – Jeremy Rickard
    Commented Nov 29, 2018 at 9:34
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    $\begingroup$ @JeremyRickard yes every $f$, Ill add it. I do not know an example. Every example is welcome, Im just a little afraid of verifing very complicated examples. $\endgroup$
    – Mare
    Commented Nov 29, 2018 at 9:38
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    $\begingroup$ @JeremyRickard Your formulation of the question seems actually to be better/more compact. Maybe even asking the question for all $n \in \mathbb{Z}$ and $\Omega^{n}(M) \rightarrow k \rightarrow \Omega^{n-1}(M)$ might be interesting. $\endgroup$
    – Mare
    Commented Nov 29, 2018 at 9:54

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Let $A$ be the six-dimensional algebra $\langle x,y\mid x^2=y^2=0, xyx=yxy\rangle$, and $M$ the three dimensional uniserial (right) module $A/yA$.

Then $\Omega M\cong M$, and so $\Omega^n M\cong M$ for all $n\in\mathbb{Z}$.

Up to scalar multiplication, there is a unique map $M\to M$ that factors through $k$, and this also factors through the free module, and so is zero in the stable module category.

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  • $\begingroup$ Thanks. It doesnt look so easy to come up with such an example as the algebra is quite exotic. Do you also know an example of a module that is not 1-periodic? Being 1-periodic immediately implies $Ext_A^1(M,M) \neq 0$ so the question is not so interesting for 1-periodic modules. $\endgroup$
    – Mare
    Commented Nov 29, 2018 at 21:38
  • $\begingroup$ I think I was able to construct similar examples that are not necessarily 1-periodic. $\endgroup$
    – Mare
    Commented Nov 29, 2018 at 22:35

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