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

Abstract

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 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 S n−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.

Comparison of Morava E-theories

“…Note that there is an isomorphism G ∼ = Γ (S n × S n+1 ). Then G n+1 ∼ = Γ S n+1 and G n = Γ S n are subgroups of G. In [16] we have shown the following theorem. …”

“…2, we review the Lubin-Tate's deformation theory of formal group laws and the results of [16]. In Sect.…”

“…We recall the following lemma on Henselian rings. In [16,Sect. 2.3] we constructed a sequence of finite Galois extensions of F((u n )):…”

“…Hence the ring We denote the action of G n+1 on the formal group law ( F n+1 , B[[u There are isomorphisms S n ∼ = Gal(L/F((u n ))) and G n ∼ = Gal(L/F p ((u n ))) through the action of G n on L (cf. [2,16]). …”

HKR characters, $p$-divisible groups and the generalized Chern character

“…Then the attaching map of X (1) to X (0) is important to understand the relationship between the K(n)-local category and the K(n + 1)-local category. The connection between extensions of anétale p-divisible group of height 1 by a p-divisible formal group and stable homotopy theory has been discussed in [2, §5]; see also [10,30,31]. The p-divisible group F n+1 (∞) B over Spf B 0 n can be written as an extension of anétale p-divisible group of height 1 by its identity component.…”