Direct sum decompositions in Grothendieck categories

Abstract

Throughout this paper, A will denote a Grothendieck category, i.e. an abelian category with generators and exact direct limits.Our main theorem gives a sufficient condition for an object to decompose into a direct sum of indecomposahle objects.This theorem will then be applied to obtain decompositions of injective objects in focally noetherian categories and of projective modules over perfect rings.Some applications will also be given to relative splitting problems, i.e. splitting by E-proper subobjects where ~ is a proper class in the sense of relative homological algebra.In a preliminary version of this paper (cited in [12]), A was assumed to be locally finitely generated.I am grateful to J.-E.Roos for pointing out that the results may be extended to AB 6 categories.