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


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