Bousfield localization functors and Hopkins’ zeta 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 Eacyclic 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 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 Eilenberg-MacLane 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 conver-