The connected components of the Auslander-Reiten quiver of a tilted algebra
The result on connected Auslander-Reiten components of a tilted algebra is stated in (3.7): Let \(A\) be a hereditary connected Artin algebra, \(T\) a tilting module and \(B= \text{End}_ A (T)\). If \(\Gamma\) is the connected component of the Auslander-Reiten quiver \(\Gamma(B)\) of \(B\), not preprojective, preinjective or the connecting component, then it is a quasiserial component or obtained from a quasiserial component by ray insertions or coray insertions. The author deduces this theorem from results in more general situations which are of independent interest. If \(A\) is an Artin algebra he calls a full subcategory \({\mathcal C}\) of \(\text{mod } A\) an Auslander-Smalø subcategory, if (1) \({\mathcal C}\) is either closed under submodules or factor modulus, (2) \({\mathcal C}\) has relative almost split sequences and (3) \({\mathcal C}\) contains only finitely many indecomposable Ext-injective and Ext-projective modules. The central results in this part are (2.7, 2.8): If \({\mathcal C}\) is an Auslander-Smalø subcategory and \(\Gamma\) a connected component in the (relative) Auslander-Reiten quiver \(\Gamma ({\mathcal C})\) of \({\mathcal C}\), then \(\Gamma\) either is of type \(\mathbb{Z} A_ \infty\) or obtained from \(\mathbb{Z} A_ \infty\) by ray insertions (coray insertions), provided some technical conditions on \({\mathcal C}\) hold. In the next section this result is specialized to the case where \({\mathcal C}\) is the class of torsion free modules \({\mathcal F} (T)\) for a splitting tilting module \(T\) in \(A\)-mod, where \(A\) is an Artin algebra. Hence it holds in the torsion class \({\mathcal X} (T)\) induced by \(T\) in \(\text{End}_ A (T)\)-mod, by the tilting functor. By specialization to hereditary Artin algebras, the author finally obtains the result quoted at the beginning.
scientific article; zbMATH DE number 554255