ContemporaryMathematicsVolume 00, 0000
Bousfield Localization Functors and Hopkins'
Chromatic Splitting Conjecture
MARK HOVEY
July 1993
Introduction
This paper arosefrom attempting to understand Bousfield localization func-
tors in stable homotopy theory. All spectra will be p-local for a prime p throu*
*g-
hout this paper. Recall that if E is a spectrum, a spectrum X is E-acyclic if
E ^X is null. A spectrum is E-local if every map from anE -acyclicsp ectrum
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 sho*
*ws
that there is a functor called E-localization, which takes a spectrum X to an
E-local spectrum LE X, and a natural transformation X ! LE X 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, among other things, thatWfinite spectra are local with
respect to the wedge of all the Morava K-theories n<1 K(n): This gave a con-
ceptual proof of the fact that there are no non-trivial maps from the Eilenberg-
MacLane spectrum HFp to a finite spectrum X.
Hopkins andRavenel later extended this to the chromatic convergencetheo-
rem [Rav92 ]. If we denote, as usual, the localization withresp ect to the fir*
*st
n!+ 1 Morava K-theories K(0) _ _ K(n) by Ln, the chromatic convergence
theorem!says!that for finite X, the tower : :s:siLnX ! ssiLn1 X : :!:ssiL0X
is!pro-isomorphic!to the constant tower fssiXg: In particular, X is the inverse
limit!of the LnX.
!
!
1991Mathematics Subject Classification.55P42, 55P60, 55N22, 55N20.
This paper is infinal form and no version of it will be submitted elsewhere.
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