Chromatic splitting for the K(2)-local sphere at p=2

Agnès Beaudry; Paul G. Goerss; Hans Werner Henn

Chromatic splitting for the K(2)-local sphere at p=2

Agnès Beaudry, Paul G. Goerss, Hans Werner Henn

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We calculate the homotopy type of L₁L_K(2)S⁰and L_K(1)L_K(2)S⁰at the prime 2, where L_K(n)is localization with respect to Morava K-theory and L₁localization with respect to 2-local K theory. In L₁L_K(2)S⁰we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology H^∗(G₂,E₀) where G₂is the Morava stabilizer group and E₀= W[[u₁]] is the ring of functions on the height 2 Lubin-Tate space. We show that the inclusion of the constants W → E₀induces an isomorphism on group cohomology, a radical simplification.

Original language	English (US)
Journal	Unknown Journal
State	Published - Dec 21 2017

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title = "Chromatic splitting for the K(2)-local sphere at p=2",

abstract = "We calculate the homotopy type of L1LK(2)S0and LK(1)LK(2)S0at the prime 2, where LK(n)is localization with respect to Morava K-theory and L1localization with respect to 2-local K theory. In L1LK(2)S0we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology H∗(G2,E0) where G2is the Morava stabilizer group and E0= W[[u1]] is the ring of functions on the height 2 Lubin-Tate space. We show that the inclusion of the constants W → E0induces an isomorphism on group cohomology, a radical simplification.",

author = "Agn{\`e}s Beaudry and Goerss, {Paul G.} and Henn, {Hans Werner}",

year = "2017",

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AU - Beaudry, Agnès

AU - Goerss, Paul G.

AU - Henn, Hans Werner

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N2 - We calculate the homotopy type of L1LK(2)S0and LK(1)LK(2)S0at the prime 2, where LK(n)is localization with respect to Morava K-theory and L1localization with respect to 2-local K theory. In L1LK(2)S0we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology H∗(G2,E0) where G2is the Morava stabilizer group and E0= W[[u1]] is the ring of functions on the height 2 Lubin-Tate space. We show that the inclusion of the constants W → E0induces an isomorphism on group cohomology, a radical simplification.

AB - We calculate the homotopy type of L1LK(2)S0and LK(1)LK(2)S0at the prime 2, where LK(n)is localization with respect to Morava K-theory and L1localization with respect to 2-local K theory. In L1LK(2)S0we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology H∗(G2,E0) where G2is the Morava stabilizer group and E0= W[[u1]] is the ring of functions on the height 2 Lubin-Tate space. We show that the inclusion of the constants W → E0induces an isomorphism on group cohomology, a radical simplification.

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