Modules of bounded length in Auslander-Reiten components (Q1102354)

Let \({\mathcal K}\) be a full additive subcategory of a length category \({\mathcal A}\), \(\tau_{{\mathcal K}}\) the Auslander-Reiten translation in \({\mathcal K}\) and \({\mathcal C}^ a \)regular component of \({\mathcal K}\), that is, a component whose Auslander-Reiten quiver \(\Gamma\) (\({\mathcal C})\) is stable. Assume that the following conditions are satisfied: (1) There are indecomposable objects \(C_ 1\), \(C_ 2\) in \({\mathcal C}\) and \(X_ 1\), \(X_ 2\) in \({\mathcal K}\setminus {\mathcal C}\) with \(Hom(X_ 1,C_ 1)\neq 0\) and \(Hom(C_ 2,X_ 2)\neq 0\); (2) There is a constant d such that for any indecomposable object X of \({\mathcal C}\), we have \(| \tau_{{\mathcal K}}X| \leq d| X|\). These conditions are satisfied in the case \({\mathcal C}\) is a stable component of \({\mathcal A}={\mathcal K}=A\)-mod (the category of left finite dimensional modules over an algebra A). The main result of the paper is as follows. Assume that there is a covering \(\pi\) : \(Z\Delta\to \Gamma ({\mathcal C})\), where \(\Delta\) is either a finite valued quiver without oriented cycles or else \(\Delta =A_{\infty}\), \(B_{\infty}\), \(C_{\infty}\) or \(D_{\infty}\). Then, for every \(\ell \in N\), there are only finitely many isomorphism classes of indecomposable objects in \({\mathcal C}\) which have length \(\ell\). Moreover, \(\pi\) is a finite covering. It was shown by \textit{W. W. Crawley- Boevey} [Proc. Lond. Math. Soc., III. Ser. 56, No.3, 451-483 (1988)] that, for a representation-infinite tame algebra A, every component of the Auslander-Reiten quiver of A contains only finitely many isomorphism classes of indecomposables of each dimension.