Systems of submodules and an isomorphism problem for Auslander-Reiten quivers. (English) Zbl 1169.16011

Schmidmeier, Markus

Bull. Belg. Math. Soc. - Simon Stevin 15, No. 3, 523-546 (2008).

For representations of partially ordered sets \(\Omega\) over commutative local rings \(R\) of finite radical length, the dependence of Auslander-Reiten components on constants in \(R\) is considered. According to its \(R\)-module structure, a type is associated to every \(\Omega\)-representation. For example, if \(\mathcal L\) is a slice in a component \(\Gamma\) of the Auslander-Reiten quiver, it is shown that the types of the indecomposables in \(\Gamma\) are determined by the modules in \(\mathcal L\). Under a certain condition it is shown that the component \(\Gamma\) is determined by the structure of \(\mathcal L\).

Reviewer: Wolfgang Rump (Stuttgart)

MSC:

16G70	Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G20	Representations of quivers and partially ordered sets

Keywords:

Auslander-Reiten quivers; poset representations; finite partially ordered sets; translation quivers; connected components; uniserial rings; Birkhoff problem; chains of subgroups; relative homological algebra