### Abstract

In this note we study a certain formal group law over a complete discrete valuation ring F[u_{n}-1] of characteristic p > 0 which is of height n over the closed point and of height n - 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law H_{n-1} of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S_{n-1} of H _{n-1}. We show that the automorphism group S_{n} of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of S_{n-1} to the cohomology of S_{n} with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

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

Pages (from-to) | 1037-1077 |

Number of pages | 41 |

Journal | American Journal of Mathematics |

Volume | 125 |

Issue number | 5 |

Publication status | Published - Oct 2003 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On degeneration of one-dimensional formal group laws and applications to stable homotopy theory.** / Torii, Takeshi.

Research output: Contribution to journal › Article

*American Journal of Mathematics*, vol. 125, no. 5, pp. 1037-1077.

}

TY - JOUR

T1 - On degeneration of one-dimensional formal group laws and applications to stable homotopy theory

AU - Torii, Takeshi

PY - 2003/10

Y1 - 2003/10

N2 - In this note we study a certain formal group law over a complete discrete valuation ring F[un-1] of characteristic p > 0 which is of height n over the closed point and of height n - 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law Hn-1 of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group Sn-1 of H n-1. We show that the automorphism group Sn of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of Sn-1 to the cohomology of Sn with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

AB - In this note we study a certain formal group law over a complete discrete valuation ring F[un-1] of characteristic p > 0 which is of height n over the closed point and of height n - 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law Hn-1 of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group Sn-1 of H n-1. We show that the automorphism group Sn of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of Sn-1 to the cohomology of Sn with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0141652966&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141652966&partnerID=8YFLogxK

M3 - Article

VL - 125

SP - 1037

EP - 1077

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 5

ER -