Finite dimensional modules and perpendicular subcategories

TL;DR

This paper investigates the construction of finite dimensional modules within specific subcategories and demonstrates limitations in generating certain triangulated quotients of module categories.

Contribution

It introduces a method to construct finite dimensional modules in prescribed subcategories with non-generatable direct limits, revealing new insights into triangulated quotient categories.

Findings

Certain triangulated quotients are not generated by their finite dimensional subcategories.

Explicit constructions of modules with prescribed properties.

Limitations of generating triangulated quotients from finite dimensional modules.

Abstract

We explain how, under some hypotheses, one can construct a sequence of finite dimensional $kG$-modules that lie in certain prescribed additive subcategories, but whose direct limits do not. We use these to show that many of the triangulated quotients of $\Mod$ are not generated, as triangulated categories, by the corresponding quotient of $\mod$ considered as a full subcategory.