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A counterexample to a $BP$-analogue
of the chromatic splitting conjecture

Author: Ethan S. Devinatz
Journal: Proc. Amer. Math. Soc. 126 (1998), 907-911
MSC (1991): Primary 55N22, 55Q10
MathSciNet review: 1422861
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Abstract: We prove that, if $n\ge 2$, the $E(n-1)_\ast$-localization of the $K(n)_\ast$-localization map $BP_p\to L_{K(n)}BP$ is not a split monomorphism in the stable category by exhibiting spectra $Z$ for which the map $\pi _\ast(L_{n-1}(BP_p)\wedge Z)\to\pi _\ast(L_{n-1}(L_{K(n)}BP)\wedge Z)$ is not injective. If $p\ge \max\{\frac{1}{2}(n^2-2n +2), n+1\}$ and $n\geq 3$, we show that $Z$ may be taken to be a two-cell complex in the sense of $E(n-1)_\ast$-local homotopy theory. The question of whether the map $L_{n-1}(BP_p)\to L_{n-1}L_{K(n)}BP$ splits was asked by Hovey and is in some sense a $BP$-analogue of Hopkins' chromatic splitting conjecture.

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Ethan S. Devinatz
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195–4350

Received by editor(s): May 7, 1996
Received by editor(s) in revised form: August 30, 1996
Additional Notes: Partially supported by the National Science Foundation
Communicated by: Thomas Goodwillie
Article copyright: © Copyright 1998 American Mathematical Society