A long homology localization tower

Let R be fixed as a subring of the rational numbers or a finite field of the form Z/pT], p prime. The purpose of this paper is to give a new description of the R-homology localization XR of a space X [1]. The main ingredient is an inverse limit construction for XR (complementary to Bousfield's direct limit construction [1, 11.5]) which is obtained by transfinitely iterating the R-nilpotent completion process of [3]. Thus one immediate benefit is a clearer understanding of the relationship between XR and the R-nilpotent completion R=X of X. A space X is said to be R-Bousfield if XR is homotopy equivalent to X. The possibility of two constructions for XR is suggested by the fact that the natural map X----~ XR has two universal properties: (i) X--* XR is terminal, up to homotopy, in the category of all maps X ~ Y which induce isomorphisms H , ( X ; R ) ~ H , ( Y ; R). (ii) X ~ XR is initial, up to homotopy, in the category of all maps X--~ Y which have an R-Bousfield target space Y. In order to exploit property (ii) effectively, it is necessary to study the