Auslander-Reiten components determined by their composition factors. (Q2846838)

Let \(A\) be an Artin algebra and \(\text{mod\,}A\) the category of finitely generated right \(A\)-modules. It is an interesting open problem to determine criteria for two indecomposable modules \(M\) and \(N\) in \(\text{mod\,}A\) with the same composition factors to be isomorphic. In this paper, the authors study the related problem of finding criteria for a component \(\mathcal C\) of the Auslander-Reiten quiver \(\Gamma_A\) of \(A\) to be uniquely determined by the composition factors of its indecomposable modules.NEWLINENEWLINE A component quiver \(\Sigma_A\) of \(A\) is the quiver whose vertices are the components \(\mathcal C\) of \(\Gamma_A\), and two components \(\mathcal C\) and \(\mathcal D\) of \(\Gamma_A\) are linked in \(\Sigma_A\) by an arrow \(\mathcal C\to\mathcal D\) if \(\text{rad}^\infty_A(X,Y)\neq 0\) for some modules \(X\in\mathcal C\) and \(Y\in\mathcal D\), where \(\text{rad}^\infty_A\) denotes the infinite radical of \(\text{mod\,}A\). A short cycle in \(\Sigma_A\) is a cycle \(\mathcal C\to\mathcal D\to\mathcal C\), where possibly \(\mathcal C=\mathcal D\).NEWLINENEWLINE In the first main result, the authors have shown that, if \(A\) is an Artin algebra, and \(\mathcal C\) and \(\mathcal D\) are two components of \(\Gamma_A\) with the same composition factors, such that \(\mathcal C\) is not a stable tube of rank one and does not lie on a short cycle in \(\Sigma_A\), then \(\mathcal C=\mathcal D\). It is pointed out that the assumption on \(\mathcal C\) not being a stable tube of rank one is essential for the validity of this theorem, and the second main result of the paper clarifies the situation when \(\mathcal C\) is a stable tube of rank one and \(\mathcal C\) and \(\mathcal D\) have the same composition factors.