Localization with respect to certain periodic homology theories. (English) Zbl 0586.55003

Ravenel, Douglas C.

Am. J. Math. 106, 351-414 (1984).

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

Keywords:

localization with respect to homology; Bousfield classes of spectra; Brown-Peterson spectra; Morava K-theory; Thom spectrum; complex cobordism