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The first cohomology group of the generalized Morava stabilizer algebra


Authors: Hirofumi Nakai and Douglas C. Ravenel
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
Published electronically: September 19, 2002
MathSciNet review: 1950296
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Abstract | References | Similar Articles | Additional Information

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{displaymath}\text{Ext}_{\Gamma(m+1)}(BP_*,BP_*), \end{displaymath}

where

\begin{displaymath}\Gamma (m+1) = BP_{*} (BP)/ \left(t_{1},\dots ,t_{m}\right) = BP_{*}[t_{m+1},t_{m+2},\dots ]. \end{displaymath}

In this paper we determine the groups

\begin{displaymath}\text{Ext}^{1}_{\Gamma (m+1)} (BP_{*},v_{n}^{-1}BP_{*}/I_{n}) \end{displaymath}

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

DOI: https://doi.org/10.1090/S0002-9939-02-06718-7
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
Article copyright: © Copyright 2002 American Mathematical Society

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