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Localization with respect to certain periodic homology theories. (English) Zbl 0586.55003
Let $$<E>$$ denote the Bousfield class of the spectrum E. The author studies the Bousfield classes of various spectra associated with the Brown-Peterson spectra BP: the spectra K(n) (Morava K-theory) and certain spectra E(n) with the property $$<E(n)>=<v_ n^{-1}BP>$$. It is shown that $$<E(n)>=\bigvee^{n}_{i=0}<K(i)>$$. Let $$L_ n$$ be the localization functor with respect to E(n). There are natural transformations $$L_ n\to L_{n-1}$$, and the author describes a conjectured periodicity of the fibers of these maps.
The last section of the paper contains a number of conjectures which have already stimulated work in homotopy theory. For example, the conjecture that if X is a finite spectrum and MU is the Thom spectrum defining complex cobordism, and $$f: X\to \Sigma^ kX$$ is a map such that $$MU_*(f)=0$$, then f is nilpotent is now a theorem of E. Devinatz, M. Hopkins, and J. Smith.
Reviewer: A.Liulevicius

##### MSC:
 55P42 Stable homotopy theory, spectra 55P60 Localization and completion in homotopy theory
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