Preprojective Modules and Auslander-Reiten Components

Shiping Liu
2003 Communications in Algebra  
In [2] , Auslander and Smalø introduced and studied extensively preprojective modules and preinjective modules over an artin algebra. We now call a module hereditarily preprojective or hereditarily preinjective if its submodules are all preprojective or its quotient modules are all preinjective, respectively. In [4] , Coelho studied Auslander-Reiten components containing only hereditarily preprojective modules and gave a number of characterizations of such components. We shall study further
more » ... l study further these modules by using the description of shapes of semi-stable Auslander-Reiten components; see [6, 7] . Our results will imply the result of Coelho [4, (1.2)] and that of Auslander-Smalø [2, (9.16)]. As an application, moreover, we shall show that a stable Auslander-Reiten component with "few" stable maps in TrD-direction is of shape Z ZA ∞ . Preliminaries on Auslander-Reiten components Throughout this note, A denotes an artin algebra, mod A the category of finitely generated right A-modules, and rad ∞ (mod A) the infinite radical of mod A. Let Γ A be the Auslander-Reiten quiver of A which is defined in such a way that its vertices form a complete set of the representatives of isoclasses of the indecomposables of mod A. We denote by τ the Auslander-Reiten translation DTr. The reader is referred to [7] for notions not defined here. We first reformulate a result stated in [7, (2.3)] for later use. Its proof can be found in the proofs of [7, (2.2), (2.3)]. Proposition. Let Γ be a left stable component of Γ A with no τperiodic module. If Γ contains an oriented cycle, then every module in Γ admits at most two immediate successors in Γ and there exists an infinite sectional path with t > 0 and {N 1 , . . . , N s } a complete set of representatives of τ -orbits in Γ . 1.2. Lemma. Let X be a module in Γ A , not lying in any finite τ -periodic stable component. Then there exists some r ≥ 0 such that τ r X lies on an oriented cycle of Γ A of left stable modules if one of the following holds : (1) τ n X lies on an oriented cycle in Γ A for infinitely many n > 0. (2) A module is a predecessor of τ n X, for infinitely many n > 0, in Γ A . Proof. Assume that either (1) or (2) occurs. In particular, X is left stable. It suffices to consider the case where X is not τ -periodic. Then there exists s ≥ 0 such that the τ n X with n ≥ s lie in an infinite non τ -periodic left stable component Γ of Γ A . Suppose that Γ contains no oriented cycle. Then Γ admits a section ∆, and hence Γ is embedded in Z Z∆; see [7, (3.1), (3.4)]. For modules M, N in Γ , there exist at most finitely many n ≥ 0 such that N is predecessor in Γ of τ n M . Moreover, applying some power of τ , we may assume 1
doi:10.1081/agb-120024866 fatcat:ukufjzbsbvdfvik4tsd4a3dzhm