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).


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)

  • 1
    $\begingroup$ 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 Nov 29 '18 at 9:34
  • 1
    $\begingroup$ @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 Nov 29 '18 at 9:38
  • 1
    $\begingroup$ @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 Nov 29 '18 at 9:54

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.

  • $\begingroup$ 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 Nov 29 '18 at 21:38
  • $\begingroup$ I think I was able to construct similar examples that are not necessarily 1-periodic. $\endgroup$ – Mare Nov 29 '18 at 22:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.