ON NON-SEMIREGULAR COMPONENTS CONTAINING PATHS FROM INJECTIVE TO PROJECTIVE MODULES

ABSTRACT

Let 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 to a projective in 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 -orbits. Components of the Auslander-Reiten quiver of an Artin algebra containing paths in ind from injective to projective modules have appeared naturally in some classes of algebras such as quasitilted[7] or shod.[4] 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). See below for definitions.Here, we are interested in studying the components of such that there exists a number such that any path from an injective to a projective lying on it has at most hooks. We shall see that this is equivalent to the existence of a number such that any path in ind from an injective to a projective lying on it has length at most (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 -orbits.In fact, when considering the existence of oriented cycles in such components, we shall prove the following more general result. If is a component of such that the number of hooks in any path of irreducible maps from an injective in to a projective in is bounded, then has no oriented cycles (Theorem 3.1). This generalizes Liu's main result in,[11] where he shows that any component of such that any path of irreducible maps from an injective in to a projective in is sectional (that is, with no hooks) has no oriented cycles. Observe that Li[8] has also considered such components and has shown that the above condition on the paths from injectives to projectives characterizes the existence of a section in . The proof of Theorem 3.1, however, follows closely some ideas contained in[6], also generalizing results proven there for quasitilted algebras.

In a forthcoming paper,[5] we shall 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 . This class of algebras contains the quasitilted and the shod algebras.

These results were proven in an exchange project between Brazil and Uruguay. Both authors acknowledge financial support given by CNPq and FAPESP, Brazil and PEDECIBA, Uruguay.

Acknowledgments