Filtration-Closed Auslander-Reiten Components for Wild Hereditary Algebras

Let H=kQ be a finite-dimensional connected wild hereditary pathalgebra, over some field k. Denote by H-reg the category offinite-dimensional regular H-modules, that is, the categoryof modules M with for all integers m, where _H denotes the Auslander–Reiten translation.Call a filtration of a regular H-module M a regular filtration if all subquotients M_i/M_i+1are regular. Call a regular filtration (*) a regular compositionseries if it is strictly decreasing and has no proper refinement.A regular component C in the Auslander–Reiten quiver (H) of H-mod is called filtration closed if, for each M addC, the additive closure of C, and each regular filtration (*)of M, all the subquotients M_i/M_i+1 are also in add C. We showthat most wild hereditary algebras have filtration-closed Auslander–Reitencomponents. Moreover, we deduce from this that there are alsoalmost serial components, that is regular components C, suchthat any indecomposable XC has a unique regular compositionseries. This composition series coincides with the Auslander–Reitenfiltration of X, given by the maximal chain of irreducible monosending at X. 1991 Mathematics Subject Classification: 16G70,16G20, 16G60, 16E30.