Finite Localizations

This short note is a response to the articles [7] of Doug Ravenel and [3] of Mark Mahowald and Hal Sadofsky. I give cleaner and more general constructions of the “telescopic” or “finite localization,” which they write Ln. I prefer to call this the finite E(n)-localization and write L f n for it, because, as I shall show, it can be characterized in exactly the same terms as the Bousfield localization, but with the addition of a finiteness assumption. If B is a finite E(n− 1)-acyclic spectrum with a vn-self-map φ : B−→Σ−qB [2], then LnB is the mapping telescope of B; so L f n is a generalization of this construction in that it can be applied to any spectrum X. By the same method, a finite localization LfA can be defined for any set A of homotopy types of finite spectra. Of particular interest is the case in which A is the set of finite E-acyclic spectra for some spectrum E, and in this case we will write LfE for the corresponding finite localization. The construction of this localization is simpler than that of the Bousfield homology localization— one can work entirely in the homotopy category, and a countable telescope suffices for the construction. It turns out to be easy to show that LfA is always “smashing” (i.e., the natural map X −→X ∧ LfAS is an equivalence) and coincides with Bousfield localization with respect to the spectrum LfAS . For any spectrum E, there is a canonical map LfEX −→LEX. The “telescope conjecture” for E (advertised for E = E(n) by Ravenel in [4]) is the assertion that this map is an equivalence. It is equivalent to require that any E-acyclic spectrum has an exhaustive filtration whose associated quotients are wedges of finite E-acyclic spectra. This structural feature is exactly what Bousfield checks for K-theory, and by virtue of Ravenel’s computation [5] we now know that it fails for E(2) at large primes. It would be be very interesting to have invariants vanishing on finite E(n)-acyclics and compatible with wedges and cofibrations, but not vanishing on all E(n)-acyclics.