Modules with irrational slope over tubular algebras

TL;DR

This paper investigates the structure of modules over tubular algebras with irrational slopes, revealing infinite width in the lattice of pp formulas and the existence of superdecomposable pure-injective modules for countable cases.

Contribution

It establishes that for irrational slopes, the lattice of pp formulas has infinite width and constructs superdecomposable pure-injective modules in countable tubular algebras.

Findings

Infinite width of pp formula lattice for irrational slopes

Existence of superdecomposable pure-injective modules in countable cases

Structural insights into modules over tubular algebras

Abstract

Let $A$ be a tubular algebra and let $r$ be a positive irrational. Let ${\mathcal D}_r$ be the definable subcategory of $A$-modules of slope $r$. Then the width of the lattice of pp formulas for ${\mathcal D}_r$ is $\infty$. It follows that if $A$ is countable then there is a superdecomposable pure-injective module of slope $r$.