Summary: Let \(\Gamma\) be a component of the Auslander-Reiten quiver of an Artin algebra containing projective and injective modules. Assume that the length of any path from an injective in \(\Gamma\) to a projective in \(\Gamma\) is bounded by some fixed number. We prove here that such component has no oriented cycles and is generalized standard, so containing only finitely many \(\tau_\Lambda\)-orbits. Components of the Auslander-Reiten quiver \(\Gamma_\Lambda\) of an Artin algebra \(\Lambda\) containing paths in \(\text{ind\,}\Lambda\) from injective to projective modules have appeared naturally in some classes of algebras such as quasitilted [D. Happel, I. Reiten, and S. O. Smalø, Mem. Am. Math. Soc. 575 (1996; Zbl 0849.16011)] or shod [the authors, Manuscr. Math. 100, No. 1, 1-11 (1999; Zbl 0966.16001)]. In both cases, these paths can be refined to paths of irreducible maps and any such path has either none hooks (in the case of a quasitilted algebra) or at most two of them (in the case of a shod algebra).
Here, we are interested in studying the components of \(\Gamma_\Lambda\) such that there exists a number \(m_0\) such that any path from an injective to a projective lying on it has at most \(m_0\) hooks. We shall see that this is equivalent to the existence of a number \(n_0\) such that any path in \(\text{ind\,}\Lambda\) from an injective to a projective lying on it has length at most \(n_0\) (Theorem 4.1). Such a component does not have oriented cycles (Corollary 3.4) and it is generalized standard (Theorem 4.3), hence containing only finitely many \(\tau_\Lambda\)-orbits.
In fact, when considering the existence of oriented cycles in such components, we prove the following more general result. If \(\Gamma\) is a component of \(\Gamma_\Lambda\) such that the number of hooks in any path of irreducible maps from an injective in \(\Gamma\) to a projective in \(\Gamma\) is bounded, then \(\Gamma\) has no oriented cycles (Theorem 3.1). This generalizes S. Liu’s main result [in Commun. Algebra 29, No. 2, 687-694 (2001; Zbl 0991.16010)], where he showed that any component \(\Gamma\) of \(\Gamma_\Lambda\) such that any path of irreducible maps from an injective in \(\Gamma\) to a projective in \(\Gamma\) is sectional (that is, with no hooks) has no oriented cycles. Observe that S. Li [Commun. Algebra 28, No. 10, 4635-4645 (2000; Zbl 0978.16016)] has also considered such components \(\Gamma\) and has shown that the above condition on the paths from injectives to projectives characterizes the existence of a section in \(\Gamma\). The proof of Theorem 3.1, however, follows closely some ideas contained in [F. U. Coelho and A. Skowroński, Fundam. Math. 149, No. 1, 67-82 (1996; Zbl 0848.16012)], also generalizing results proven there for quasitilted algebras.
In another paper [the authors, J. Algebra 265, No. 1, 379-403 (2003; Zbl 1062.16018)], we make use of the results proven here to study the class of algebras such that the length of any path from an indecomposable injective module to an indecomposable projective module is bounded by some number \(n_0\). This class of algebras contains the quasitilted and the shod algebras.