Generators versus projective generators in abelian categories

TL;DR

This paper explores conditions under which abelian categories admit projective generators, establishing links to module categories and characterizing subcategories via Hom-orthogonal Schur objects.

Contribution

It proves that certain generator conditions imply the existence of projective generators and module category equivalences, extending to length categories and subcategory classifications.

Findings

Existence of a generator with right artinian endomorphism ring implies a projective generator.

In Grothendieck categories, this leads to equivalence with module categories.

In length categories, subcategories correspond to collections of Hom-orthogonal Schur objects.

Abstract

Let $\mathcal{A}$ be an essentially small abelian category. We prove that if $\mathcal{A}$ admits a generator $M$ with ${\rm End}_{\mathcal{A}}(M)$ right artinian, then $\mathcal{A}$ admits a projective generator. If $\mathcal{A}$ is further assumed to be Grothendieck, then this implies that $\mathcal{A}$ is equivalent to a module category. When $\mathcal{A}$ is Hom-finite over a field $k$, the existence of a generator is the same as the existence of a projective generator, and in case there is such a generator, $\mathcal{A}$ has to be equivalent to the category of finite dimensional right modules over a finite dimensional $k$-algebra. We also show that when $\mathcal{A}$ is a length category, then there is a one-to-one correspondence between exact abelian extension closed subcategories of $\mathcal{A}$ and collections of Hom-orthogonal Schur objects in $\mathcal{A}$.