Lower bounds for faithful, preinjective modules
Let A be a k-algebra and \(\tau\) its Auslander-Reiten translation. An interesting problem is to determine the faithful, indecomposable preinjective A-modules, i.e. modules of the form \(\tau^ nI\), with I injective. If A is the path algebra of a quiver \({\vec \Delta}\), there are only a finite number of indecomposable preinjectives which are not faithful. Hence, there is a minimum m(\({\vec \Delta})\) of the numbers n such that \(\tau^ nI\) is always faithful. The maximum of the m(\({\vec \Delta})'s\) for the different orientations of \(\Delta\) is denoted by m(\(\Delta)\). This paper solves the problem of determining m(\(\Delta)\) when \(\Delta\) is a tree of infinite representation type. If \(\Delta\) is a tame tree, it is shown that \(m(\Delta)=r(\Delta)+1\), where r(\(\Delta)\) is the replication number of \textit{R. Bautista} and \textit{S. Brenner} [see Proc. Lond. Math. Soc., III. Ser. 47, 429-462 (1983; Zbl 0505.16012)]. In the general cases m(\(\Delta)\) coincides with the diameter d(\(\Delta)\) of \(\Delta\), and it is always of the form \(m(\Delta)=d(\Delta)+c(\Delta)\) where c(\(\Delta)\) is a constant depending on certain structural properties of \(\Delta\). The paper is very clear and very well written.
scientific article; zbMATH DE number 3983425