### Abstract

Let n ≥ 1 and let p be any prime. Also, let E_{n} be the Lubin-Tate spectrum, G_{n} the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this paper show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_{n} ∧ X))^{hG}_{n}, which is formed with respect to the continuous action of G_{n} on L_{K(n)}(E_{n} ∧ X). In this paper, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π_{*}(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to π_{*}((L_{K(n)}(E_{n} ∧ X))^{hG}_{n}).

Original language | English |
---|---|

Pages (from-to) | 1097-1103 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 140 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Every K(n)-local spectrum is the homotopy fixed points of its Morava module.** / Davis, Daniel G.; Torii, Takeshi.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 140, no. 3, pp. 1097-1103. https://doi.org/10.1090/S0002-9939-2011-11189-4

}

TY - JOUR

T1 - Every K(n)-local spectrum is the homotopy fixed points of its Morava module

AU - Davis, Daniel G.

AU - Torii, Takeshi

PY - 2012

Y1 - 2012

N2 - Let n ≥ 1 and let p be any prime. Also, let En be the Lubin-Tate spectrum, Gn the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this paper show that if X is a finite spectrum, then the localization LK(n)(X) is equivalent to the homotopy fixed point spectrum (LK(n)(En ∧ X))hGn, which is formed with respect to the continuous action of Gn on LK(n)(En ∧ X). In this paper, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π*(LK(n)(X)) is isomorphic to the descent spectral sequence that abuts to π*((LK(n)(En ∧ X))hGn).

AB - Let n ≥ 1 and let p be any prime. Also, let En be the Lubin-Tate spectrum, Gn the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this paper show that if X is a finite spectrum, then the localization LK(n)(X) is equivalent to the homotopy fixed point spectrum (LK(n)(En ∧ X))hGn, which is formed with respect to the continuous action of Gn on LK(n)(En ∧ X). In this paper, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π*(LK(n)(X)) is isomorphic to the descent spectral sequence that abuts to π*((LK(n)(En ∧ X))hGn).

UR - http://www.scopus.com/inward/record.url?scp=82255191455&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82255191455&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2011-11189-4

DO - 10.1090/S0002-9939-2011-11189-4

M3 - Article

AN - SCOPUS:82255191455

VL - 140

SP - 1097

EP - 1103

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -