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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)

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The first cohomology group of the generalized Morava stabilizer algebra
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by Hirofumi Nakai and Douglas C. Ravenel PDF
Proc. Amer. Math. Soc. 131 (2003), 1629-1639 Request permission

Abstract:

There exists a $p$-local spectrum $T(m)$ with $BP_{*}(T(m))$= $\!BP_{*}[t_{1},\dots ,t_{m}]$. Its Adams-Novikov $E_2$-term is isomorphic to \begin{equation*} \text {Ext}_{\Gamma (m+1)}(BP_*,BP_*), \end{equation*} where \begin{equation*} \Gamma (m+1) = BP_{*} (BP)/ \left (t_{1},\dots ,t_{m}\right ) = BP_{*}[t_{m+1},t_{m+2},\dots ]. \end{equation*} In this paper we determine the groups \begin{equation*} \text {Ext}^{1}_{\Gamma (m+1)} (BP_{*},v_{n}^{-1}BP_{*}/I_{n}) \end{equation*} for all $m,n>0$. Its rank ranges from $n+1$ to $n^{2}$ depending on the value of $m$.
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Additional Information
  • Hirofumi Nakai
  • Affiliation: Oshima National College of Maritime Technology, 1091-1 komatsu Oshima-cho Oshima-gun, Yamaguchi 742-2193, Japan
  • Email: nakai@c.oshima-k.ac.jp
  • Douglas C. Ravenel
  • Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
  • Email: drav@math.rochester.edu
  • Received by editor(s): June 14, 2001
  • Received by editor(s) in revised form: December 19, 2001
  • Published electronically: September 19, 2002
  • Additional Notes: The second author acknowledges support from NSF grant DMS-9802516
  • Communicated by: Paul Goerss
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 1629-1639
  • MSC (2000): Primary 55P42, 55T15; Secondary 14L05, 20Jxx
  • DOI: https://doi.org/10.1090/S0002-9939-02-06718-7
  • MathSciNet review: 1950296