K(n)-localization of the K(n + 1)-local E_n+1-Adams spectral sequences

Takeshi Torii

Vol. 250 (2011), No. 2, 439–471

DOI: 10.2140/pjm.2011.250.439

Abstract

We construct a spectral sequence converging to the homotopy set of maps from a spectrum to the K(n)-localization of the K(n + 1)-local sphere. We also construct a map of spectral sequences from the K(n)-local E_n-Adams spectral sequence to the preceding one. Then we compare the map on E₂-terms with a map induced by the inflation maps of continuous cohomology groups for Morava stabilizer groups. As an application we show that ζ_n in π₋₁(L_K(n)S⁰) represented by the reduced norm map in the K(n)-local E_n-Adams spectral sequence has a nontrivial image under the map π_∗(L_K(n)S⁰) → π_∗(L_K(n)L_K(n+1)S⁰).

Keywords

Adams spectral sequence, K(n)-localization, Morava E-theory

Mathematical Subject Classification 2000

Primary: 55T25, 55P42

Secondary: 55Q51, 55N22, 55N20

Milestones

Received: 3 April 2009

Revised: 17 December 2009

Accepted: 4 February 2011

Published: 31 March 2011