The Endomorphism Ring of a Localized Coherent Functor Original Research Article

LetCbe a commutative artinian ring and Λ an artinC-algebra. The category of coherent additive functorsA: mod-Λ → Ab on the finitely presented right Λ-modules will be denoted by Ab(Λ). This category is equivalent to the free abelian category over the ring Λ. If image0 subset of or equal to Ab(Λ) is the Serre subcategory of the finite length objects of Ab(Λ) andA set membership, variant Ab(Λ), it is proved that the endomorphism ring EndAb(Λ)/image0 Aimage0of the localized objectAimage0is a locally artinC-algebra. This is used to show that the Krull-Gabriel dimension of the category Ab(Λ) cannot equal 1. In particular, this holds for finite rings.