Let R be a commutative Noetherian ring and I an ideal of R. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the niteness of generalized local cohomology modules. Let M be a nitely generated R-module and N a representable R-module. We introduce the notions of the representation dimension rI (M;N) and artinianness dimension aI (M;N) of M;N with respect to I by rI (M;N) = inffi 2 N0 : HI i (M;N) is not representableg and aI (M;N) = inffi 2 N0 : HI i (M;N) is not artiniang and we show that aI (M;N) = rI (M;N) = inffrIRp (Mp;p N) : p 2 Spec(R)g inffaIRp (Mp;p N) : p 2 Spec(R)g. Also, in the case where R is semi-local and N a semi discrete linearly compact R-module such that N= T t>0 ItN is artinian we prove that inffi : HI i (M;N) is not minimaxg=inffrIRp (Mp;p N) : p 2 Spec(R)nMax(R)g: 1. Introduction
