Left sections and the left part of an artin algebra

Abstract

Let A be an artin algebra. We are interested in studying the representation theory of A, thus the category modA of finitely generated right A-modules. For this purpose, we fix a full subcategory indA of modA having as objects exactly one representative from each isomorphism class of indecomposable modules. Following Happel, Reiten and Smalo [19], we define the left part LA to be the full subcategory of indA with objects those modules whose predecessors have projective dimension at most one. The right part is defined dually. These classes, whose definition suggests the interplay between homological properties of an algebra and representation theoretic ones, were heavily investigated and applied (see, for instance, the survey [4]). The initial motivation for this paper comes from the observations, made in [5, 2, 1], that the left part of an arbitrary artin algebra closely resembles that of a tilted algebra. Tilted algebras, introduced by Happel and Ringel in [20], are among the most important and best understood classes of algebras. Many criteria allow to recognise whether a given algebra is tilted or not. Most of them revolve around the existence of a combinatorial configuration, called “complete slice” or “section” inside the module category, see [20, 15, 22, 13, 28, 29, 7]. Perhaps the most efficient is the Liu-Skowronski criterion: they define (combinatorially) a so-called section in an Auslander-Reiten component and prove that, if there exists a section satisfying reasonable algebraic conditions, then the algebra is tilted (see [26, 30] or [9, (Chapter VIII)]). Surprisingly, however, as is shown in [1], none of the known criteria seems to apply directly to the tilted algebras arising from the study of the left part. The first aim of this paper is to derive a more suitable version of the Liu-Skowronski criterion, easier to apply in our case. For this purpose, we define a notion of left section in a translation quiver by weakening one of the Liu-Skowronski axioms for section (see (2.1)). Several known results for sections carry over to left sections, sometimes in a restricted form (see, for instance, (2.2) and (3.2)). We thus obtain our first main theorem.