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

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.