The endomorphism ring of a localized coherent functor

Let C be a commutative artinian ring and Lambda an artin C-algebra. The category of coherent additive functors A: mod-Lambda --> Ab on the finitely presented right Lambda-modules will be denoted by Ab(Lambda). This category is equivalent to the free abelian category over the ring Lambda. If L-0 subset of or equal to Ab(Lambda) is the Serre subcategory of the finite length objects of Ab(Lambda) and A is an element of Ab(Lambda), it is proved that the endomorphism ring End(Ab(Lambda)/L0) A(L0) of the localized object A(L0) is a locally artin C-algebra. This is used to show that the Krull-Gabriel dimension of the category Ab(Lambda) cannot equal 1. In particular, this holds for finite rings. (C) 1997 Academic Press.