On Artin algebras with almost all indecomposable modules of projective or injective dimension at most one. (Q1856359)

For an Artin algebra \(\Lambda\), denote by \({\mathcal L}_\Lambda\) (and by \({\mathcal R}_\Lambda\)), the category of all indecomposable \(\Lambda\)-modules all whose predecessors (or successors, respectively) have projective (or injective, respectively) dimension at most one. Such categories were first studied by \textit{D. Happel}, \textit{I. Reiten} and \textit{S. O. Smalø} [Mem. Am. Math. Soc. 575 (1996; Zbl 0849.16011)], where the authors introduced the class of quasitilted algebras and, more recently, lead to the description of further classes of algebras, e.g. the shod algebras [\textit{F. U. Coelho} and \textit{M. A. Lanzilotta}, Manuscr. Math. 100, No. 1, 1-11 (1999; Zbl 0966.16001)], which are characterized by the property that the union \({\mathcal L}_\Lambda\cup{\mathcal R}_\Lambda\) contains all indecomposable \(\Lambda\)-modules. In the paper under review, the author gives a description of all algebras such that \({\mathcal L}_\Lambda\cup{\mathcal R}_\Lambda\) contains all but finitely many (up to isomorphism) indecomposable \(\Lambda\)-modules and derives some nice consequences. In a paralel and independent work, such class of algebras was also studied by \textit{I. Assem} and \textit{F. U. Coelho} [J. Algebra 269, No. 2, 456-479 (2003; Zbl 1045.16006)].