Laura algebras and quasi-directed components

Abstract

Abstract. Using a notion of distance between indecomposable modules we deduce new characterizations of laura algebras and quasi-directed Auslander-Reiten components. Afterwards, we investigate the infinite radical of Artin algebras and show that there exist infinitely many non-directing modules between two indecomposable modules X and Y if rad ∞ A (X, Y ) = 0. We draw as inference that a convex component is quasi-directed if and only if it is almost directed.1. Introduction. The aim of the representation theory of Artin algebras is to study an algebra A by means of the category mod A of finitely generated right A-modules, which often turns out to be easier to handle. A key example arises in [12] where Happel, Reiten and Smalø show that an algebra A is quasitilted, that is, A is the endomorphism ring of a tilting object in a locally finite hereditary abelian category, if and only if A has global dimension at most two and any indecomposable A-module lies either in L A or in R A . Recall that L A is the full subcategory of mod A consisting of all indecomposable A-modules whose predecessors have projective dimension at most one and R A is defined dually.Following this example, the module category, and particularly L A and R A , gained in importance, and new classes of algebras, defined by the homological properties of their indecomposable modules, appeared: the shod algebras [7], the weakly shod algebras [9] and the laura algebras [1,20]; each of them generalizing the previous ones. We refer the reader to [2] for a complete review on these classes of algebras.Laura algebras have been introduced independently by Assem and Coelho [1] and Reiten and Skowroński [20] as a generalization of the representation-finite algebras and the weakly shod algebras. Their nice properties have made them rather interesting and hugely investigated; see, for instance, [1,20,29,2,3,11]. In particular, it is shown in [29] that an algebra A is laura if and only if it is quasitilted or its Auslander-Reiten quiver contains a faithful quasi-directed component, which is, in this case, convex. In [30],

From Trisections in Module Categories to Quasi-Directed Components

“…By hypothesis, Γ is convex and almost directed. By [14] (4.2.3), it is quasi-directed. This proves (a) The last statement follows from (3.2).…”

“…Applying [14] (4.2.3), we get that Γ is quasi-directed. In particular, Γ has only finitely many τ Λ -orbits.…”

“…They are components containing at most finitely many modules lying on oriented cycles and which moreover are generalised standard (in the sense of [23]). These components and their connection to laura algebras have been studied in [27,14]. Our objective in this paper is to give simple new categorical characterisations of quasi-directed components.…”

Coverings of laura algebras: The standard case

“…Conversely, any convex quasi-directed component occurs in this way [43]. The techniques used for the study of laura algebras were applied in [27] to obtain useful results on the infinite radical of the module category. Their representation dimension is at most three and this is a class of algebras with possibly infinite global dimension which satisfies the finitistic dimension conjecture [9].…”

Almost laura algebras

“…Among them, laura algebras have been introduced independently by Assem and Coelho [3] and Reiten and Skowroński [33] as a generalization of representation-finite algebras and weakly shod algebras. Their nice properties have made them rather interesting and widely investigated (see [4,7,20,25,39,40], for instance). The aim of this paper is to introduce a new class of algebras, called almost laura, determined by the behavior of the infinite radical of mod A and generalizing laura algebras.…”