TY - JOUR

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

AU - Torii, Takeshi

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

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.

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U2 - 10.1353/ajm.2003.0036

DO - 10.1353/ajm.2003.0036

M3 - Article

AN - SCOPUS:0141652966

VL - 125

SP - 1037

EP - 1077

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 5

ER -