Auslander-Reiten components containing cones
Throughout this review \(R\) will be an Artinian ring and \(\mathcal A\) a locally finite Abelian \(R\)-category with the assumption that it has Auslander-Reiten sequences. If \(\mathcal A\) has projective objects, it is assumed that the indecomposable projective objects have a unique maximal subobject. \(\Gamma({\mathcal A})\) will denote the Auslander-Reiten quiver of \(\mathcal A\), which is a translation quiver. If \(\mathcal C\) is an Auslander-Reiten component of \(\Gamma({\mathcal A})\) and \(X\) is a vertex of \(\mathcal A\), we call \((X\to)\) a right cone if \((X\to)\) is isomorphic as translation quiver to some right cone \((x\to)\) of a translation quiver of type \(\mathbb{Z} A_\infty\). Left cones are defined dually. Regular Auslander-Reiten components with a right cone are of type \(\mathbb{Z} A_\infty\). It is an open problem whether the existence of \(\mathbb{Z} A_\infty\) components in the Auslander-Reiten quiver \(\Gamma(A)\) of a finite dimensional algebra \(A\), implies wildness of \(A\)-mod. This paper gives a criterion in \(\mathcal A\) for the existence of \(\mathbb{Z} A_\infty\) components which covers several situations. It is known that for finite-dimensional algebras, Auslander-Reiten components with cones are never generalized standard. The authors construct examples of infinite-dimensional algebras with Auslander-Reiten components of type \(\mathbb{Z} A_\infty\), which are standard components. They give a description of the Auslander-Reiten quiver of such algebras. The authors also apply their results to the following situations: (i) An Auslander-Reiten component of the Auslander-Reiten quiver \(\Gamma(A)\) of a finite dimensional self-injective algebra \(A\). (ii) A regular non periodic component in \(\Gamma({\mathcal A})\), where \(\mathcal A\) is an Abelian category with rank function. (iii) A regular non periodic component of an Abelian category \(\mathcal A\), considered additionally as a Noetherian category. (iv) Regular components in the Auslander-Reiten quiver \(\Gamma(A)\) of a connected tilted algebra \(A\).
scientific article; zbMATH DE number 1845360