On the degree of irreducible morphisms.
Let \(A\) be an Artin algebra and \(\text{mod\,}A\) denote the category of all finitely generated right \(R\)-modules. The notion of the left degree \(d_l(f)\) (or right degree \(d_r(f)\)) of an irreducible morphism \(f\colon X\to Y\) in \(\text{mod\,}A\) was introduced by \textit{S. Liu} [J. Lond. Math. Soc., II. Ser. 45, No. 1, 32-54 (1992; Zbl 0703.16010)], and has been very useful in the study of components of the Auslander-Reiten quiver of Artin algebras. Recall that a component \(\Gamma\) of the Auslander-Reiten quiver of \(A\) is said to be with length if paths with the same origin and end vertices in \(\Gamma\) have equal length. \(\Gamma\) is called generalized standard if \(\text{rad}^\infty(X,Y)=0\) for all \(X,Y\in\Gamma\), and \(\Gamma\) is convex if for every chain \(X_0\to X_1\to\cdots\to X_{n-1}\to X_n\) of non-zero non-isomorphisms between indecomposable modules in \(\text{mod\,}A\), with \(X_0,X_n\in\Gamma\), each \(X_i\) belongs to \(\Gamma\) for \(i=1,2,\dots,n-1\). In this paper, the authors study the degree of irreducible morphisms in generalized standard convex components with length of the Auslander-Reiten quiver of \(A\), and give criteria for this degree to be finite or infinite. It is shown that \(d_l(f)=\infty\) if and only if \(fg=0\) for each nonzero morphism \(g\colon M\to X\) with \(M\) in the component. Another main result states that \(d_l(f)\) is finite if and only if \(\text{Ker}(f)\) belongs to the component. The results obtained are applied to irreducible morphisms over Artin algebras of finite representation type, and over tame hereditary algebras.
scientific article; zbMATH DE number 2134424
scientific article