# Auslander-Solberg algebras from non-rigid modules

Let $$A$$ be a Nakayama algebra and $$M$$ be the direct sum of all indecomposable $$A$$-modules $$N$$ with $$Ext_A^1(N,N) \neq 0$$.

The following is suggested by computer experiments with QPA:

Question: Is $$B=End_A(M)$$ an Auslander-Solberg algebra?

Recall that an algebra $$B$$ is an Auslander-Solberg algebra in case we have $$Gordim(B) \leq 2 \leq domdim(B)$$ and they can be equivalently characterised as the algebras with the Gorenstein projective modules being an abelian category, see https://link.springer.com/article/10.1007/s10468-013-9448-5 .

In Combinatorial problem on periodic dyck paths from homological algebra it was proved that with $$N$$ indecomposable non-rigid, also $$\Omega^1(N)$$ has the same property. Thus we can associate to each Nakayama algebra a graph with points the indecomposable non-rigid modules and arrows $$N \rightarrow \Omega^1(N)$$. In case the above question has a positive answer, it might suggest that this graph has some nice properties. Besides that I have no real approch to this question since $$M$$ is not even a generator or cogenerator, which is the usual setting for Auslander-Solberg algebras.