On wild algebras and super-decomposable pure-injective modules

TL;DR

This paper proves that for finite-dimensional wild algebras over algebraically closed fields, the lattice of pointed modules has undefined width, implying the existence of super-decomposable pure-injective modules, and introduces a new family of modules called independent pairs of dense chains.

Contribution

It introduces the concept of independent pairs of dense chains of pointed modules, strengthening previous results on the structure of modules over wild algebras.

Findings

The width of the lattice of pointed modules is undefined for wild algebras.

Existence of super-decomposable pure-injective modules over such algebras.

Introduction of independent pairs of dense chains of pointed modules.

Abstract

Assume that $k$ is an algebraically closed field and $A$ is a finite-dimensional wild $k$-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointed $A$-modules is undefined and hence there exists a super-decomposable pure-injective $A$-module, if the base field $k$ is countable. Here we give a proof of a stronger result. Namely, we show that there exists a special family of pointed $A$-modules, called an independent pair of dense chains of pointed modules. This also yields that the width of the lattice of all pointed $A$-modules is undefined.