Generators versus projective generators in abelian categories

Charles Paquette

Published 2017-10-19Version 1

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 $\mathcal{A}$.