Cofinite Submodule Closed Categories and the Weyl group

TL;DR

This paper establishes a field-independent bijection between Weyl groups and cofinite submodule closed subcategories in hereditary Artin algebras, providing a new construction method for modules containing preinjectives.

Contribution

It introduces a novel, field-independent proof of the bijection and a simple construction method for modules containing preinjective modules.

Findings

Bijection between Weyl groups and cofinite submodule closed subcategories

Field-independent proof extending previous results

New construction method for modules with preinjective submodules

Abstract

We consider hereditary Artin algebras over arbitrary fields and prove that there is a natural bijection between the Weyl groups and the sets of full additive cofinite submodule closed subcategories of the module categories. While Oppermann, Reiten and Thomas have shown this for algebraically closed fields and finite fields, we give a different method of proof that holds independently of the field. In particular, we show a relatively simple way to construct all modules that contain a given preinjective module as a submodule.