Generators versus projective generators in abelian categories

Abstract

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