The first cohomology group of the generalized Morava stabilizer algebra

Nakai, Hirofumi; Ravenel, Douglas

doi:10.1090/S0002-9939-02-06718-7

The first cohomology group of the generalized Morava stabilizer algebra
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by Hirofumi Nakai and Douglas C. Ravenel

Proc. Amer. Math. Soc. 131 (2003), 1629-1639

DOI: https://doi.org/10.1090/S0002-9939-02-06718-7

Published electronically: September 19, 2002

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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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