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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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
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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