On generalized standard Auslander-Reiten components having only finitely many non-directing modules.
Let \(A\) be an Artin algebra, \(\text{mod\,}A\) denotes the category of all finitely generated right \(R\)-modules, and \(\Gamma(\text{mod\,}A)\) the Auslander-Reiten quiver of \(A\). A component \(\Gamma\) of \(\Gamma(\text{mod\,}A)\) is called semiregular if it does not contain both a projective module and an injective module, and is called generalized standard if \(\text{rad}^\infty(X,Y)=0\) for all modules \(X\) and \(Y\) in \(\Gamma\). \(\Gamma\) 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 \(X_0,X_n\in\Gamma\), the \(X_i\) belong to \(\Gamma\) for \(i=1,2,\dots,n-1\). The author defines a component \(\Gamma\) of \(\Gamma(\text{mod\,}A)\) to be quasi-directed if it is generalized standard and contains only finitely many modules lying on cycles (of not necessarily irreducible morphisms). The author obtains complete characterizations of semiregular connected components of \(\Gamma(\text{mod\,}A)\) which are quasi-directed, and of non-semiregular connected components of \(\Gamma(\text{mod\,}A)\) which are quasi-directed and convex. Several applications and examples are given, related to classes of algebras studied extensively in the literature, including quasi-tilted algebras, laura algebras, or left supported algebras.
categories of finitely generated right modules
scientific article; zbMATH DE number 2104451
scientific article