FILTRATION-CLOSED AUSLANDER–REITEN COMPONENTS FOR WILD HEREDITARY ALGEBRAS

Abstract

Let $H=k\cal Q$ be a finite-dimensional connected wild hereditary path algebra, over some field $k$. Denote by $H$-reg the category of finite-dimensional regular $H$-modules, that is, the category ofmodules $M$with $\tau_H^{-m}(\tau_H^m M) \cong M$ for all integers $m$, where $\tau_H$ denotes the Auslander--Reiten translation. Call a filtration\begin{equation}M = M_0\supset M_1\supset\ldots\supset M_r\supset M_{r+1}=0 \tag{$*$}\end{equation}of a regular $H$-module $M$ a {\em regular filtration}if all subquotients $M_i/M_{i+1}$ are regular. Call a regular filtration $(*)$ a {\em regular composition series} if it is strictly decreasing and has no properrefinement.A regular component $\cal C$ in the Auslander--Reiten quiver $\Gamma (H)$ of $H$-modis called {\em filtration closed} if, for each$M\in\text{add\,}\cal C$, the additive closure of $\cal C$, and each regular filtration $(*)$ of $M$, all the subquotients $M_i/M_{i+1}$ are also in$\text{add\,}\cal C$. We show that mostwild hereditary algebras have filtration-closedAuslander--Reiten components. Moreover,we deduce from this that there are also {\em almost serial} components, that is regular components $\cal C$, such that any indecomposable $X\in\cal C$ has a unique regular composition series.This composition series coincides with theAuslander--Reiten filtration of $X$, given by the maximal chain of irreducible monos ending at $X$. 1991 Mathematics Subject Classification: 16G70, 16G20, 16G60, 16E30.