More ghost modules I

TL;DR

This paper precisely defines ghost modules in representation theory, explores their properties, introduces new types called extension ghosts, and discusses their role in visualizing invariants of group rings.

Contribution

It provides a formal definition of ghost modules, introduces extension ghosts, and analyzes their properties and applications in representation theory.

Findings

Defined ghost modules with examples

Proved properties of ghosts and pictures

Introduced extension ghosts for further applications

Abstract

Ghost modules were introduced in [I3] without definitions or proofs. We also introduced stability diagrams or "relative pictures" for torsion classes and torsion-free classes for representations of Dynkin quivers. Modules which were not in the chosen class reappeared as "ghosts", in fact one missing module produced two ghosts. In this short paper, we give a precise definition of ghost modules. We give several examples and prove basic properties of ghosts and pictures for torsion and torsion-free classes. We also introduce a third kind of ghost which we call "extension ghosts". In the next paper we will explain how these new ghosts can be used to visualize the computation of other invariants of $K_3$ of group rings.