### Abstract

We prove a higher chromatic analogue of Snaith’s theorem which identifies the K –theory spectrum as the localisation of the suspension spectrum of CP^{1} away from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of CP^{1} D K.Z; 2/. Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the K.n/–local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross–Hopkins dual of the sphere. Our main technical tool is a K.n/–local notion generalising complex orientation to higher Eilenberg–MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.

Original language | English (US) |
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Pages (from-to) | 1033-1093 |

Number of pages | 61 |

Journal | Geometry and Topology |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Mar 17 2017 |

### Keywords

- Chromatic homotopy theory
- Picard group
- Redshift conjecture
- Snaith theorem

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## Cite this

*Geometry and Topology*,

*21*(2), 1033-1093. https://doi.org/10.2140/gt.2017.21.1033