A counterexample to a $BP$-analogue of the chromatic splitting conjecture
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- by Ethan S. Devinatz PDF
- Proc. Amer. Math. Soc. 126 (1998), 907-911 Request permission
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.References
- E. S. Devinatz, The generating hypothesis revisited, to appear in Stable and Unstable Homotopy, Fields Institute Communications, Amer. Math. Soc., 1997.
- Mark Hovey, Bousfield localization functors and Hopkins’ chromatic splitting conjecture, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 225–250. MR 1320994, DOI 10.1090/conm/181/02036
- Peter S. Landweber, Homological properties of comodules over $M\textrm {U}_\ast (M\textrm {U})$ and BP$_\ast$(BP), Amer. J. Math. 98 (1976), no. 3, 591–610. MR 423332, DOI 10.2307/2373808
- Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), no. 3, 469–516. MR 458423, DOI 10.2307/1971064
- Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR 737778, DOI 10.2307/2374308
- Douglas C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, vol. 128, Princeton University Press, Princeton, NJ, 1992. Appendix C by Jeff Smith. MR 1192553
Additional Information
- Ethan S. Devinatz
- Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195–4350
- Email: devinatz@math.washington.edu
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 907-911
- MSC (1991): Primary 55N22, 55Q10
- DOI: https://doi.org/10.1090/S0002-9939-98-04104-5
- MathSciNet review: 1422861