Non-Noetherian representation categories of generalized fields

TL;DR

This paper demonstrates that certain categories of modules over generalized fields exhibit non-Noetherian behavior, highlighting their unusual homological properties.

Contribution

It shows that categories of finitely generated projective modules over specific generalized fields are not locally Noetherian, revealing their atypical homological characteristics.

Findings

Categories of finitely generated projective modules are not locally Noetherian.

Generalized fields exhibit strange homological behavior.

Provides new examples of non-Noetherian module categories.

Abstract

In this paper, we show that the categories of finitely generated projective $\mathbb{B}$-modules and $\mathbb{F}_\infty$-modules with morphisms being (splittable) injections are not locally Noetherian. This provides another instance of the fact that these generalized fields have strange homological behavior.