Bousfield Localization Functors and Hopkins’ Chromatic Splitting Conjecture

This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be p-local for a prime p throughout this paper. Recall that if E is a spectrum, a spectrum X is E-acyclic if E ∧X is null. A spectrum is E-local if every map from an E-acyclic spectrum to it is null. A map X → Y is an E-equivalence if it induces an isomorphism on E∗, or equivalently, if the fibre is E-acyclic. In [Bou79], Bousfield shows that there is a functor called E-localization, which takes a spectrum X to an E-local spectrum LEX, and a natural transformation X → LEX which is an E-isomorphism. Studying LEX is studying that part of homotopy theory which E sees. These localization functors have been very important in homotopy theory. Ravenel [Rav84] showed, among other things, that finite spectra are local with respect to the wedge of all the Morava K-theories ∨ n<∞K(n). This gave a conceptual proof of the fact that there are no non-trivial maps from the EilenbergMacLane spectrum HFp to a finite spectrum X. Hopkins and Ravenel later extended this to the chromatic convergence theorem [Rav92]. If we denote, as usual, the localization with respect to the first n + 1 Morava K-theories K(0) ∨ · · · ∨ K(n) by Ln, the chromatic convergence theorem says that for finite X, the tower . . . πiLnX → πiLn−1X . . . → πiL0X is pro-isomorphic to the constant tower {πiX}. In particular, X is the inverse limit of the LnX.