The Endomorphism Ring of a Localized Coherent Functor

Abstract

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 S0⊆Ab(Λ) is the Serre subcategory of the finite length objects of Ab(Λ) andA∈Ab(Λ), it is proved that the endomorphism ring EndAb(Λ)/S0AS0of the localized objectAS0is 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.