ONE DIMENSIONAL FORMAL GROUP LAWS OF HEIGHT N AND N 1. Takeshi Torii

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1 ONE DIMENSIONAL FORMAL GROUP LAWS OF HEIGHT N AND N 1 by Takeshi Torii A dissertation submitted to the Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy Baltimore, Maryland May, 2001 c Takeshi Torii 2001 All rights reserved

2 Abstract By Morava s point of view on the stable homotopy category, the quotient in some sense associated to the filtration related to the height of formal group laws is studied by the category of modules over the function ring of the deformation space of the Honda group law of height n with the lift of the action of the automorphism group on the closed fibre through the Adams- Novikov spectral sequence. The next step to understand the stable homotopy category may be to solve the extensions. It may be necessary to know the relation between the formal group laws of height n and n 1. In this thesis we study a certain formal group law over a complete discrete valuation ring which is of height n over the closed fibre and of height n 1 over the generic fibre. We show that there is a Galois extension of the quotient field of the discrete valuation ring with the Galois group isomorphic to the Morava stabilizer group S n 1 of height n 1. The action of the Morava stabilizer group S n of height n on the quotient field lifts to the action on the Galois extension which commutes with the action of the Galois group. Furthermore, there is an S n S n 1 equivariant morphism from the lift of the formal group law over the Galois extension to the Honda group law of height n 1 on which S n acts trivially. Then, by a kind of correspondence, we construct a ring homomorphism from the cohomology of S n 1 to the cohomology of S n with the coefficients in the quotient field. Advisor: Professor Jack Morava ii

3 Acknowledgments The author would like to thank the faculty and staff of the Johns Hopkins University for their support. I especially would like to thank Professor Morava for the discussion on the subject. iii

4 Contents 1 Introduction 1 2 Deformation of formal group laws 4 3 Homomorphisms of formal group laws 9 4 Isomorphisms between F n and H n Action of G on L 22 6 Vanishing of some cohomology 25 7 Normalized cochain complex 28 8 Inflation maps 31 9 Cohomology of G n 1 (F) in F[w ±1 ] Cohomology of G n in F((u n 1 ))[u ±1 ] Cohomology of G(i) Ring homomorphism 42 References 45 Vita 48 iv

5 1 Introduction The Hopf algebroid MU MU is interpreted in terms of the one dimensional formal group laws. In [11] Morava investigated the category C of p-local comodules over MU MU by using the filtration C = C 0 C 1 C n where n corresponds to the height of formal group laws over p-local ring. Then he related the quotient category C n /C n+1 to the category of modules over the function ring of the deformation space of the standard formal group law H n of height n with the action of the automorphism group S n of H n. Motivated by Morava s work, Miller, Ravenel and Wilson [10] established the frame work on organizing systematically the periodic phenomena on the E 2 -term of the Adams-Novikov spectral sequence based on the cobordism theory M U. Then Ravenel [12] formulated his conjectures on the reflection of the algebraic structure on the Adams-Novikov E 2 -term on the actual stable homotopy category. Devinatz, Hopkins and Smith [4, 6] solved all the Ravenel conjectures except for the telescope conjecture. By these works, we get a filtration of full subcategories in the stable homotopy category C of p-local finite spectra C = C 0 C 1 C n where n is related to the height of formal group laws. By Morava s point of view on the stable homotopy category, the quotient in some sense associated to the above filtration is studied by the category of modules over the function ring of the deformation space of H n with the lift of the action of S n on the 1

6 closed fibre through the Adams-Novikov spectral sequence. The next step to understand the stable homotopy category may be to solve the extensions. It may be necessary to know the relation between the formal group laws of height n and n 1. In this thesis we study a certain formal group law over a complete discrete valuation ring which is of height n over the closed fibre and of height n 1 over the generic fibre. Let F be a finite field which contains the finite fields F p n and F p n 1. There is the Honda group law H n of height n over the field F. The Morava stabilizer group S n is the automorphism group of H n. There is a universal deformation F n of H n. The formal group law F n is defined over the formal power series ring W F[u 1,..., u n 1 ] where W F is the ring of Witt vectors in F. Then the action of S n Γ on H n lifts to the action on F n which induces a continuous action of S n Γ on W F[u 1,..., u n 1 ] where Γ is the Galois group of F over the prime field F p. Since the ideal generated by p, u 1,..., u n 2 is invariant under the action of S n Γ, there is an induced action of S n Γ on the quotient ring F[u n 1 ]. We denote by K the quotient field F((u n 1 )). We consider that the formal group law F n is defined over F[u n 1 ]. Then the formal group law F n is of height n on the closed fibre F and of height n 1 on the generic fibre K. By the result of Lazard [7], the formal group laws over the separably closed field of characteristic p > 0 are classified up to isomorphism by their height. Hence there is an isomorphism between F n and the Honda group law H n 1 of height n 1 over the separable closure K sep. In [1] Ando, Morava and Sadofsky showed that there is a unique isomorphism between F n and H n 1 over K sep which satisfies certain conditions motivated from a geometric point of view. We would like to consider the above situation 2

7 with the action of the Morava stabilizer group S n. Let Φ be an isomorphism between F n and H n 1 over the separable closure K sep. Let L be an extension of K obtained by adjoining all the coefficients of Φ. Hence we have a morphism of the formal group laws from (F n, L) to (H n 1, F). The main theorem of this note is as follows. Theorem 1.1 (Theorem 4.9). The group (S n S n 1 ) Γ acts on (F n, L) where the action of S n Γ is a lift of the action on (F n, K) and the subgroup S n 1 Γ is identified with the Galois group of the extension L/F p ((u n 1 )). If we consider that the group (S n S n 1 ) Γ acts on (H n 1, F) where the subgroup S n acts trivially, then there is a (S n S n 1 ) Γ equivariant morphism from (H n 1, F) to (F n, L). In geometric terms Spec(F[u n 1 ]) is an S n invariant 1-dimensional subspace of the formal deformation space of the Honda group law H n. Let U be the punctured disk Spec(K) Spec(F). Then there is a Galois covering of U with the Galois group isomorphic to S n 1. The action of S n lifts to the Galois covering which commutes with the action of the Galois group. Furthermore, if we consider that the product group S n S n 1 acts on H n 1 where the action of S n is trivial, then there is a S n S n 1 equivariant morphism from the lift of F n on the Galois covering to H n 1 on the point Spec(F). By a kind of correspondence using Theorem 1.1, we construct a ring homomorphism from the cohomology of S n 1 to the cohomology of S n with the coefficients in K[u ±1 ]: H (S n 1 ; F[w ±1 ]) Γ H (S n ; K[u ±1 ]) Γ where w satisfies w (pn 1 1) = v n 1 (cf. Theorem 12.3). 3

8 2 Deformation of formal group laws Let R be a ring with a maximal ideal I such that the residue field k = R/I is of characteristic p > 0. Let G be a formal group law over k of height n <. In this section we recall the deformation theory of formal group laws by Lubin and Tate [8]. For a formal power series f(x) over a ring R and a ring homomorphism α : R S, we denote by α f(x) the formal power series over S obtained by the ring homomorphism α. We say that a local ring A with the maximal ideal m is complete if the canonical homomorphism A lim A/m i is an isomorphism. For a local homomorphism α between local rings, we denote by α the induced homomorphism on the residue fields. Let A be a complete Noetherian local R-algebra with the maximal ideal m such that IA m. We denote by ι the canonical inclusion of residue fields k A/m induced by the R-algebra structure. A deformation of G to A is a formal group law G over A such that ι G = π G where π : A A/m is a canonical projection. Let G 1 and G 2 be two deformations of G to A. We define a -isomorphism between G 1 and G 2 as an isomorphism ũ : G 1 G 2 over A such that π ũ is the identity map between π G1 = ι G = π G2. Lemma 2.1 (cf. [8]). There is at most one -isomorphism between G 1 and G 2. We denote by D(R) the category of complete Noetherian local R-algebras with morphisms as local R-algebra homomorphisms. Let DEF(A) be the set of all -isomorphism classes of the deformations of G to A. Then DEF defines a functor from D(R) to the category of sets. 4

9 Let R[t 1,..., t n 1 ] be a formal power series ring over R with n 1 variables. If R is a complete Noetherian local ring, then R[t 1,..., t n 1 ] is an object of D(R). There is a one-to-one correspondence between a local R-algebra homomorphism from R[t 1,..., t n 1 ] to A and an (n 1)-tuple (a 1,..., a n 1 ) of elements of the maximal ideal m of A. Lubin and Tate showed that there is a formal group law F (t 1,..., t n 1 ) over R[t 1,..., t n 1 ] which satisfies the following conditions. 1. π F (0,..., 0)(x, y) = G(x, y) where π : R k is the projection. 2. For each i (1 i n 1), F (0,..., 0, t i,..., t n 1 )(x, y) x + y + t i C p i(x, y) mod deg (p i + 1) where C p i(x, y) = (x pi + y pi (x + y) pi )/p. We say that a formal group law F (t 1,..., t n 1 ) satisfying the above conditions is a universal deformation of G by the following theorem. Theorem 2.2 (Lubin and Tate [8]). Let A be an object of D(R). For every deformation G of G to A, there is a unique local R-algebra homomorphism α : R[t 1,..., t n 1 ] A such that α F (t 1,..., t n 1 ) is -isomorphic to G. Hence, if R is a Noetherian local ring, the functor DEF is represented by R[t 1,..., t n 1 ]: DEF(A) = Hom D(R) (R[t 1,..., t n 1 ], A) and F (t 1,..., t n 1 ) is a universal object. We suppose that R is a complete Noetherian local ring. We abbreviate R[t 1,..., t n 1 ] to R[t] and F (t 1,..., t n 1 ) to F (t). For a formal group law 5

10 F (X, Y ) and an invertible power series f(x) over a same ring, we denote by F f (X, Y ) the formal group law f(f (f 1 (X), f 1 (Y ))). Let f(x) be an automorphism of G over k. For any power series f (X) in R[X ] such that π f f (X) = u(x), the formal group law F (t) e (X, Y ) is a deformation of G over R[t]. Hence we obtain a unique local R-algebra homomorphism α : R[t] R[t] such that α F (t) is -isomorphic to F (t) ef. If g (X) is another power series in R[X ] such that π g (X) = f(x), then we get a local R-algebra homomorphism α : R[t] R[t] such that α F (t) is -isomorphic to F (t) eg. But g ( f 1 f (X)) is a -isomorphism from F (t) e to F (t) eg. Hence we see that α = α. Let f be a unique -isomorphism from F (t) ef to α F (t). Then f(x) = f ( f (X)) is an isomorphism from F (t) to α F (t) such that π f(x) = f(x). We see that a homomorphism f(x) from F (t) to α F (t) such that π f(x) = f(x) is unique since there is at most one -isomorphism between deformations. Hence we obtain the following proposition. Proposition 2.3. We suppose that R is a complete Noetherian local ring. For every automorphism f(x) of G over k, there is a unique pair (α, f) of a local R-algebra automorphism α of R[t] and an isomorphism f(x) from F (t) to α F (t) such that π f(x) = f(x). For a ring R, we denote by W R the ring of Witt vectors with coefficients in R. If k is a perfect field of characteristic p > 0, then W k is a complete discrete valuation ring of characteristic 0 with the residue field k. In W k we can take p as a uniformizer. The ring W k for a perfect field of characteristic p > 0 has the following universal property. Lemma 2.4 (cf. II. 5. [14]). Let A be a complete local ring with the maximal ideal m. For a ring homomorphism α from a perfect field k of charac- 6

11 teristic p > 0 to the residue field A/m, there is a unique local homomorphism from W k to A such that the induced homomorphism on the residue field coincides with α. Let G be a formal group law of height n < over the perfect field k of characteristic p > 0. We denote by W k the category with objects as pairs (A, α) where A is a complete Noetherian local ring with maximal ideal m and α is a homomorphism from k to the residue field A/m. The morphisms from (A, α) to (B, β) consist of local homomorphism γ from A to B such that γ α = β. For an object (A, α) of W k, a deformation of G to (A, α) is a formal group law G over A such that α G = π G where π : A A/m is a canonical projection. A -isomorphism f(x) between deformations G 1 and G 2 of G to (A, α) is an isomorphism of formal group laws over A such that π f(x) = X. Let Def(A, α) be the set of all -isomorphism classes of deformations of G to (A, α). Then Def defines a functor from W k to the category of sets. By Lemma 2.4, we see that the category W k is isomorphic to the category D(W k). Then we obtain the following theorem by Theorem 2.2. We abbreviate the object (W k [t 1,..., t n 1 ], id) of W k to W k [t 1,..., t n 1 ]. Theorem 2.5. The functor Def is represented by W k [t 1,..., t n 1 ]: Def(A, α) = Hom Wk (W k [t 1,..., t n 1 ], (A, α)). Let F (X, Y ) be a universal deformation on W k [t]. For an automorphism α : k k and an isomorphism f : G α G, we take any power series f (X) in W k [X ] such that π f f (X) = f(x). Then F e (X, Y ) is a deformation of G to (W k [t], α). Hence we get a local homomorphism β : W k [t] W k [t] such that β f F (X, Y ) is -isomorphic to F e (X, Y ) and β = α. Let f (X) be 7

12 f a unique -isomorphism from F e (X, Y ) to β F (X, Y ). Then f ( f (X)) is a unique homomorphism from F (X, Y ) to β F (X, Y ) such that π f ( f (X)) = f(x). Hence we get the following proposition. Proposition 2.6. Let G be a height n < formal group law over a perfect field k of characteristic p > 0. Let F be a universal deformation on W k [t]. For every pair (α, f) of an automorphism of field α : k k and an isomorphism f : G α G, there is a unique pair (β, f) of a continuous automorphism β : W k [t] W k [t] and an isomorphism f : F β F such that β = α and π f(x) = f(x). Let F be a finite field which contains the finite filed F p n with p n elements. We note that F is perfect. The ring of Witt vectors W F is an unramified extension of the p-adic integer ring Z p. We consider the height n Honda formal group law H n defined over F. The formal group law H n is p-typical with the p-series [p] H n (X) = X pn. Let E n be a formal power series ring over W F with (n 1) variables: E n = W F[u 1,..., u n 1 ]. The ring E n is a complete Noetherian local ring with residue field F. There is a p-typical formal group law F n defined over E n with the p-series [p] F n (X) = px + Fn u 1 X p + Fn u 2 X p2 + Fn + Fn u n 1 X pn 1 + Fn X pn. The formal group law F n is a deformation of H n to (E n, id). 8

13 Lemma 2.7. The formal group law F n is a universal deformation of H n. Proof. We show that F n satisfies the conditions of a universal deformation. Let E n,i be the formal power series ring over W F with variables u i,..., u n 1. Let p i : E n E n,i be a local W F-algebra homomorphism given by p i (u j ) = 0 for j = 1,..., i 1 and p i (u j ) = u j for j = i,..., n 1. Then p i F n is a p-typical formal group law over E n,i with the p-series [p] p i Fn (X) px + u i X pi mod deg (p i + 1). If p i F n (X, Y ) X +Y +bc k (X, Y ) mod deg k+1, then we have [n] p i Fn (X) nx + b (n nk ) λ X k mod deg (k + 1) where λ = p if k is a power of p and λ = 1 otherwise. In particular, we get [p] p i F n (X) px + b (p pk ) X k mod deg (k + 1). λ This implies that k = p i and b(1 p pi 1 ) = u i. Let t i = u i /(1 p pi 1 ). Then we have E n = W F[t 1,..., t n 1 ] E n,i = W F[t i,..., t n 1 ] The W F-algebra homomorphism E n E n,i given by t j 0 for 1 j < i and t j t j for i j < n is p i. Hence we get This completes the proof. p i F n = X + Y + t i C p i(x, Y ) mod deg (p i + 1). 3 Homomorphisms of formal group laws In this section we recall a generalization of a homomorphism between formal group laws over possibly different ground rings considered by several 9

14 authors (cf. [19]). Let R 1 and R 2 be two commutative rings. Let F 1 (resp. F 2 ) be a formal group law over R 1 (resp. R 2 ). We understand that a homomorphism from F 1 to F 2 is a pair (α, f) of a ring homomorphism α : R 2 R 1 and a homomorphism f : F 1 α F 2 in the usual sense. The composition of two homomorphisms (α, f) : F 1 F 2 and (β, g) : F 2 F 3 is defined as (α β, α g f) : F 1 F 3 : F 1 f α α F g 2 α (β F 3 ) = (α β) F 3. A homomorphism (α, f) : F 1 F 2 is an isomorphism if there exists a homomorphism (β, g) : F 2 F 1 such that (α, f) (β, g) = (id, id) and (β, g) (α, f) = (id, id). Then a homomorphism (α, f) : F 1 F 2 is an isomorphism if and only if α is a ring isomorphism and f is an isomorphism in usual sense. Let F be a finite field which contains the finite fields F p n with p n elements. Let H n be the height n Honda formal group law over F. The formal group law H n is a p-typical formal group law with the p-series [p] H n (X) = X pn. Let S n be the Morava stabilizer group. This is an automorphism group of H n over F in usual sense. We denote by G n (t) the automorphism group of H n over F in the generalized sense. Lemma 3.1. The automorphism group G n (t) is isomorphic to the semidirect product Gal(F/F p ) S n. 10

15 Proof. An automorphism of H n consists of a ring isomorphism α : F F and an isomorphism of formal group laws f : H n α H n. Then α Gal(F/F p ). Since H n is defined over the prime field F p, α H n = H n. Hence we get f S n. We regard S n as the subset of the power series ring F[X ]. Then the action of the Galois group Gal(F/F p ) induces an action on S n. The semidirect product Gal(F/F p ) S n with respect to this action is isomorphic to the automorphism group of H n over F. Let R be a complete Noetherian local ring with residue field k of characteristic p > 0. Let G be a formal group law over k of height n <. By, there is a universal deformation F of G over R[t 1,..., t n 1 ]. We understand that an automorphism of G is a pair (α, f) of a local R-algebra automorphism and an isomorphism f : F α F in usual sense. We denote by Aut R (F ) the automorphism group of F in generalized sense. Let aut(g) be the automorphism group of G in usual sense. There is a natural homomorphism from Aut R (F ) to aut(g). Lemma 3.2. The natural homomorphism Aut R (F ) aut(g) is an isomorphism. Proof. This follows from Proposition 2.3. Let G be a height n < formal group law over a perfect field k of characteristic p > 0. There is a universal deformation F over W k [t 1,..., t n ]. We understand that an automorphism of F is a pair (α, f) of a continuous automorphism of W k [t] and an isomorphism f : F α F in usual sense. We denote by Aut(G) (resp. Aut(F )) the automorphism group of G (resp. F ) in generalized sense. There is a natural homomorphism from Aut(F ) to Aut(G). 11

16 Lemma 3.3. The natural homomorphism Aut(F ) Aut(G) is an isomorphism. Proof. This follows from Proposition 2.6. Let E n be the complete local ring W F[u 1,..., u n 1 ] where W F is the ring of Witt vectors over F. We take a deformation F n of H n to (E n, id) as the p-typical formal group law with p-series [p] F n (X) = px + Fn u 1 X p + Fn + Fn u n 1 X pn 1 + Fn X pn. By Lemma 2.7, F n is a universal deformation of H n. Let G n (t) be the automorphism group of F n and let G W n F (F) be the subgroup of G n (F) which consists of the automorphism (α, f) such that α is a W F-algebra homomorphism. We note that there is a natural homomorphism G n (F) G n (F) and this induces a homomorphism G W F n (F) S n. Corollary 3.4. The natural homomorphisms G n (F) G n (F) and G W n F (F) S n are isomorphisms. 4 Isomorphisms between F n and H n 1 In this section we investigate a relation between two formal group laws F n and H n 1 over the field F((u n 1 )). In particular, we see that the Morava stabilizer group S n 1 is realized as the Galois group of the minimum extension over F((u n 1 )) on which an isomorphism between F n and H n 1 is defined. Let n 2. Let F be a finite field. In this section we assume that F contains the finite fields F p n and F p n 1. Let k = F((u n 1 )) be the quotient field 12

17 of the formal power series ring F[u n 1 ]. There is a W F-algebra homomorphism θ : E n k given by θ(u i ) = 0 for i = 1,..., n 2 and θ(u n 1 ) = u n 1. Then we get a p-typical formal group law θ F n. We abbreviate θ F n to F n. The formal group law F n is a p-typical with the p-series [p] Fn (X) = u n 1 X pn 1 + Fn X pn. Let H n 1 be the height n 1 Honda formal group law over k. Then H n 1 is a p-typical formal group law with the p-series [p] H n 1 (X) = X pn 1. Let K be a separable closure of k. Then there is an isomorphism between F n and H n 1 over K, since the height of F n is n 1 (cf. Appendix 2, [13]). We fix an isomorphism Φ from F n to H n 1. Since Φ is a homomorphism between p-typical formal group laws, Φ has a following form: Φ(X) = i 0 H n 1 Φ i X pi. Let L i = k(φ 0, Φ 1,..., Φ i ) for i 1 and L = i 1 L i. Lemma 4.1. The extension L i /k is totally ramified of degree (p n 1 1)p i(n 1) for i 0. Corollary 4.2. The extension L/k is totally ramified. Proof. Since Φ(X) is a homomorphism from F n to H n 1, we have Φ([p] Fn (X)) = [p] H n 1 (Φ(X)). The left hand side is Φ(u n 1 X pn 1 ) + Hn 1 Φ(X pn ) = i 0 H n 1 p Φ i u i n 1 X pn+i 1 + Hn 1 i 0 H n 1 Φ i X pn+i. 13

18 The right hand side is i 0 H n 1 [p] H n 1 (Φ i X pi ) = i 0 H n 1 Φ i p n 1 X pn+i 1. By comparing the coefficient of X pn 1 p, we obtain Φ 0 u n 1 = Φ n 1 0. Since Φ(X) is an isomorphism, Φ 0 0. Hence p Φ n u n 1 = 0. This is an Eisenstein polynomial. Therefore L 0 /k is totally ramified of degree p n 1 1. In particular, Φ 0 is a prime element of L 0. We assume that L i 1 /k is totally ramified of degree (p n 1 1)p (i 1)(n 1) and Φ i 1 is a prime element of L i 1. By comparing the coefficient of X pn+i 1, we have p Φ i u i p n 1 + f(φ 0,..., Φ i 1 ) = Φ n 1 i where f(φ 0,..., Φ i 1 ) is an element of the integer ring O Li 1 such that f Φ i 1 mod (Φ 2 i 1 ). Hence this is an Eisenstein polynomial. Therefore L i /L i 1 is totally ramified of degree p n 1 and Φ i is a prime element of L i. By induction, we get the lemma. We recall that G n (F) is an automorphism group of F n over E n in the generalized sense. For g G n (F), we denote by α(g) the corresponding continuous automorphism of E n and by t(g) the corresponding isomorphism from F n to α(g) F n. An automorphism α(g) induces a continuous automorphism of k. We abbreviate the induced automorphism of k by α(g). We note that this is a right action of Gn (F) on k. For an element g G n (F), we denote by g a continuous automorphism of the separable closure K which is any extension of the automorphism α(g). 14

19 Lemma 4.3. For any g and i, L i eg = L i. In particular, L i /F p ((u n 1 )) is a Galois extension. Corollary 4.4. For any extension g of α(g), L eg = L. In particular, L/F p ((u n 1 )) is a Galois extension. Proof. We have an isomorphism F n By applying g, we get an isomorphism g F n Φ H n 1. eg Φ g H n 1. Note that g F n = α(g) F n (resp. g H n 1 = H n 1 ), since F n is defined over k (resp. F p ). By a commutative diagram: F n Φ H n 1 t(g) α(g) F n g Φ h(g,eg) H n 1, we have Φ eg (t(g)(x)) = h(g, g)(φ(x)). Here t(g)(x) = i 0 g F n t i (g)x pi. and t i are continuous functions from S n to the integer ring O k of k for all i 0. The automorphism h(g, g) : H n 1 H n 1 is an element of the Morava stabilizer group S n 1. The power series h(g, g)(x) has a following form: h(g, g)(x) = i 0 H n 1 h i (g, g)x pi 15

20 where h i (g, g) F p n 1. Then the left hand side is H n 1 Φ eg j t i (g) pj X pi+j. i,j 0 The right hand side is H n 1 p h j (g, g)φ j i X pi+j. i,j 0 By comparing the coefficient of X, we obtain Since h 0 (g, g) F p n 1, we get Φ 0 eg t 0 (g) = h 0 (g, g)φ 0. Φ 0 eg = h 0 (g, g)φ 0 t 0 (g) 1 k(φ 0 ) = L 0. We assume that Φ 0 eg,..., Φ i 1 eg L i 1. Then by comparing the coefficient of X pi, we obtain Hence we get This completes the proof. Φ i eg t 0 (g) pi h 0 (g, g)φ i L i 1. Φ i eg L i 1 (Φ i ) = L i. We suppose that a Galois group acts on the field on the right. For σ Gal(L/F p ((u n 1 ))), we consider the following diagram: F id σ n = F n Φ H n 1 Φσ h (σ) H n 1, 16

21 We note that F σ n = F n since F n is defined over F p ((u n 1 )). This diagram defines a map h : Gal(L/F p ((u n 1 ))) G n 1 (F). Lemma 4.5. The map h : Gal(L/F p ((u n 1 ))) G n 1 (F) is a homomorphism. Proof. For σ Gal(L/F p ((u n 1 ))), we have a commutative diagram: F n = F n Φ σ Φ σ σ H n 1 h (σ ) σ H n 1. Then we get a commutative diagram: F n = F n = F n Φ Φ σ σ H n 1 This means h is a homomorphism. Φσ h (σ) h H (σ ) σ n 1 H n 1. The Morava stabilizer group S n 1 is the automorphism group of H n 1 over the algebraic closure F p in usual sense. We denote an element h S n 1 by h = h 0 + h 1 T + h 2 T 2 + where h i W F p n 1, h pn 1 i h 0 0. Then h corresponds to the automorphism = h i for i 0 and h(x) = i 0 H n 1 π(h i )X pi where π : W F p n 1 F p n 1 is the projection. Let S (0) n 1 = S n 1. We define the subgroups S (i) n 1 for i 1 by S (i) n 1 = {h S n 1 h 0 = 1, h 1 = 0,..., h i 1 = 0}. 17

22 Then S (i+1) n 1 is a normal subgroup of S n 1 and the quotient group S n 1 /S (i+1) n 1 is finite of order (p n 1 1)p (n 1)i for i 0. The canonical homomorphism S n 1 lim S n 1 /S (i+1) n 1 is an isomorphism. Hence S n 1 and G n 1 (F) = Gal(F/F p ) S n 1 are profinite groups. Theorem 4.6. The map h : Gal(L/F p ((u n 1 ))) G n 1 (F) is an isomorphism. Proof. There is a commutative diagram of exact sequences: 1 Gal(L/k) Gal(L/F p ((u n 1 ))) Gal(k/F p ((u n 1 ))) 1 h 1 S n 1 G n 1 (F) Gal(F/F p ) 1. Since k/f p ((u n 1 )) is an unramified extension, the right vertical arrow is an isomorphism. Hence it is sufficient to show that the left vertical arrow h : Gal(L/k) S n 1 is an isomorphism. If σ Gal(L/L i ), then Φ σ (X) Φ(X) mod degree p i+1. Hence h (σ)(x) X mod degree p i+1. This shows that h (Gal(L/L i )) S (i+1) n 1. Then h induces a homomorphism: h : Gal(L i /k) S n 1 /S (i+1) n 1. If h (σ) = e for σ Gal(L/k), then we have h (σ)(x) X mod degree p i+1. This implies Φ σ (X) X mod degree p i+1. Therefore h is a monomorphism. Since the degree of L i over k is equal to the order of S n 1 /S (i+1) n 1, the homomorphism h is an isomorphism for all i. Therefore the homomorphism h is an isomorphism. Let Θ be the set of all isomorphism from F n to H n 1 over K. For Φ Θ, 18

23 we consider the following commutative diagram: F n = F n Φ H n 1 Φ h H n 1. Since the isomorphism h : H n 1 H n 1 is defined over F p n 1, we see that Φ is defined over L. Then the Galois group Gal(L/k) acts on Θ. Corollary 4.7. The action of Gal(L/k) on Θ is simply transitive. Proof. For Φ Θ, we get h S n 1 by the above commutative diagram. Let σ Gal(L/k) be the corresponding element under the isomorphism h : Gal(L/k) = S n 1. Then we have a commutative diagram: F n = F n Φ H n 1 Φσ h H n 1. By comparing two commutative diagrams, we see that Φ σ = Φ. This shows that the action is transitive. If σ Gal(L/k) satisfies Φ σ = Φ, then we have Φ σ i = Φ i for all i. Since L is generated by Φ i for i = 0, 1,... over k, we see that σ = id. This completes the proof Let Aut(k) be the group consisting of the automorphisms of the topological field k. We consider Aut(k) acts k on the right. There is a homomorphism G n (F) Aut(k) given by g α(g). Since L is algebraic over k, there is a unique valuation on L extending the valuation on k. We regard L as a 19

24 topological field by means of this valuation. Let Aut(L) be the automorphism group of L as a topological field. We denote by A(L/k) the subgroup of Aut(L) consisting of the automorphisms which preserve the subfield k: A(L/k) = {θ Aut(L) θ(k) = k}. Then we have a restriction homomorphism A(L/k) Aut(k). Let G = G n (F) Aut(k) A(L/k) be the fibre product: G p G n (F) A(L/k) Aut(k). By Lemma 4.3, the natural projection p : G G n (F) is surjective. It is clear that the kernel of p is the Galois group Gal(L/k). Hence we have an exact sequence: 1 Gal(L/k) G p G n (F) 1. Let G n 1 (L) be the automorphism group of H n 1 over L in generalized sense. By the same way as Lemma 3.1, we have an isomorphism G n 1 (L) = Aut(L) S n 1. Let A(L/k) S n 1 be the subgroup of G n 1 (L). For (g, g) G, we consider the following commutative diagram: F n Φ H n 1 t(g) α(g) F n Φeg h(g,eg) H n 1. This diagram defines a map f : G A(L/k) S n 1 by (g, g) ( g, h(g, g)). 20

25 Lemma 4.8. The map f : G A(L/k) S n 1 is a homomorphism. Proof. For (g, g ) G, we have a commutative diagram: α(g) F n Φ eg t(g ) α(g) α(g) α(g ) F n Φ eg eg H n 1 h(g,eg ) eg H n 1. Then we get a commutative diagram: F n Φ H n 1 t(g) α(g) t(g F ) α(g) n Φeg α(g) α(g ) F n Φ eg eg h(g,eg) h(g H,eg ) eg n 1 H n 1. This means that Hence f is a homomorphism. f((g, g ) (g, g)) = ( g g, h(g, g ) eg h(g, g)). There are homomorphisms A(L/k) Aut(k) Gal(F/F p ) where the first homomorphism is restriction and the second homomorphism is obtained by considering the induced automorphism on the residue field. These homomorphisms are compatible with the action on the Morava stabilizer group S n 1. Hence we get a homomorphism f : A(L/k) S n 1 G n 1 (F). There are homomorphisms G f f G n 1 (F) Gal(F/F p ). By Corollary 3.4, we have a natural isomorphism G n (F) = G n (F). We identify G n (F) and G n (F) by this isomorphism. Then we have homomorphisms G p G n (F) Gal(F/F p ). 21

26 We verify that the following diagram is commutative: f f G p G n (F) G n 1 (F) Gal(F/F p ). Then we get a commutative diagram of exact sequences: 1 Gal(L/k) G f f p G n (F) 1 1 S n 1 G n 1 (F) Gal(F/F p ) 1. The left vertical arrow is an isomorphism from Theorem 4.6. Hence we get the following Theorem. Theorem 4.9. There are isomorphisms G = G n (F) Gal(F/Fp ) G n 1 (F) = Gal(F/F p ) (S n S n 1 ). 5 Action of G on L By Theorem 4.9, G is a profinite group. There is an action of G on L by using the projection G A(L/k). In this section we show that the profinite group G acts on L continuously. Since L is algebraic over k, there is a unique valuation v on L extending the valuation on k. We regard L as a metric space by using v. We note that every g G preserves the valuation: v(x g ) = v(x) for all x L. In particular g G induces a homeomorphism g : L L. For any x L and m > 0, we define U(x, m) as an open neighbourhood of x given by U(x, m) = {y L v(y x) > m}. 22

27 Lemma 5.1. If G acts on L i continuously for all i, then G acts on L continuously. Proof. We take any x L and any m > 0. There exists i such that x L i. Since G acts on L i continuously, there exists an open neighbourhood V of the identity G such that x V U(x, m). Then for any y U(x, m) and h V, we have v(y hg x g ) = v((y hg x hg ) + (x hg x g )) min{v(y hg x hg ), v(x hg x g )} = min{v(y x), v(x h x)} > m. This shows that G acts continuously on L. Hence we consider the action of G on L i for fixed i. We denote by v the valuation of L i such that v (Φ i ) = 1. Let U i (x, m) be the open nighbourhood of x given by {y L i v (y x) > m}. Lemma 5.2. For any m > 0, there exists an open neighbourood V of the identity of G such that v (Φ g i Φ i ) > m for all g V. Proof. The action of (g, g) G is described by the commutative diagram F n Φ H n 1 t(g) α(g) F n Φeg h(g,eg) H n 1. Hence we have a relation i,j 0 Hn 1 Φ j eg t i (g) pj X pi+j = i,j 0 23 H n 1 h j Φ pj i X pi+j.

28 If v (t 0 (g) 1) > m, v (t 1 (g)) > m,..., v (t i (g)) > m and h 0 = 1, h 1 = 0,..., h i = 0, then v (Φ eg j Φ j) > m for j = 1,..., i. There exists an open subgroup S (j) n of S n such that v (t 0 (g) 1) > m, v (t 1 (g)) > m,..., v (t i (g)) > m for all g S n (j). Then the open subgroup Gal(F/F p ) (S n (j) G satisfies the condition. S (m+1) n 1 ) of Proposition 5.3. The profinite group G acts on the metric space L continuously. Proof. By Lemma 5.1, it is sufficient to prove that G acts on L i continuously for all i. Let x L i such that x = j=m x j Φ i where x j F for j m. For any m > 0, by Lemma 5.2, there exists an open neighbourhood V of the identity of G such that v (Φ g i Φ i) > m m + 1 for all g V. Let W = V S n S n 1. Since S n S n 1 is an open subgroup, W is an open set. Then we have x W U i (x, m). For any y U i (x, m) and any h W, v (y hg x g ) = v ((y hg x hg ) + (x hg x g )) j min{v (y hg x hg ), v (x hg x g )} = min{v (y x), v (x h x)} > m. This shows that G acts on L i continuously. 24

29 6 Vanishing of some cohomology Let G be a topological group and let M be a topological G-module. this section we define a cohomology group of G with the coefficients in M parameterized by a topological space. Then we consider a vanishing condition of this cohomology group. Let X be a topological space and let A be a subspace of X. We denote by (X, A) such a pair of topological spaces. We define a homogeneous n- cochain of G with the coefficients in M over a topological pair (X, A) to be a continuous map f from X G n+1 to M such that and f(x; σσ 0, σσ 1,..., σσ n ) = σ f(x; σ 0, σ 1,..., σ n ). f(a; σ 1,..., σ n ) = 0 if a A. We denote by C(X,A) n (G; M) the abelian group of all homogeneous n-cochains for G in M over (X, A). As usual the coboundary map d : C(X,A) n (G; M) C n+1 (X,A)(G; M) is given by n+1 df(x; σ 0,..., σ n+1 ) = ( 1) i f(x; σ 0,..., ˆσ i,..., σ n+1 ). i=0 Let H(X,A) (G; M) be the cohomology group of the cochain complex C (X,A) (G; M). For (X, A) =pt (one point space), we see that H pt(g; M) = H (G; M) is the continuous cohomology group of G with the coefficients in M. In the following of this section we assume that G is a finite group and regard it as a discrete group. Let C(X, A; M) be the abelian group of all 25 In

30 continuous functions f : X M such that f(a) = 0 for a A. Then a homogeneous n-cochain of G with the coefficients in M over (X, A) is naturally identified with a homogeneous n-cochain of G with the coefficients in C(X, A; M). In particular, we have a natural isomorphism H (X,A) (G; M) = H (G; C(X, A; M)). We recall some vanishing condition of the cohomology group of G (cf. Chap. I 6 [5]). A mean on M is an additive function I which associates with each map f : G M an element I(f) M such that 1. if f(σ) = m M for all σ G, then I(f) = m, 2. for all σ G, I(σ f) = σ I(f) where (σ f)(τ) = σ f(σ 1 τ). Proposition 6.1. If M is a G-module which admits a mean, then H n (G; M) = 0 for all n > 0. Proof. Let f be a homogeneous n-cocycle for G in M. For fixed σ 1,..., σ n, we consider the map φ(f; σ 1,..., σ n ) : σ f(σ, σ 1,..., σ n ). This map has a mean value (I n f)(σ 1,..., σ n ) M. Since φ(f; σσ 1,..., σσ n ) = σ φ(f; σ 1,..., σ n ), we get (I n f)(σσ 1,..., σσ n ) = σ (I n f)(σ 1,..., σ n ). Hence I n f is a homogeneous (n 1)-cochain for G in M. We show that d(i n f) = f. Since f is an n-cocycle, So we have 0 = df(x, σ 0, σ 1,..., σ n ) = f(σ 0,..., σ n ) n i=0 ( 1)i f(x, σ 0,..., σ i,..., σ n ). n ( 1) i φ(f; σ 0,..., σ i,..., σ n ) = f(σ 0,..., σ n ). i=0 26

31 By taking mean values of both sides, we get n ( 1) i (I n f)(σ 0,..., σ i,..., σ n ) = f(σ 0,..., σ n ). i=0 The left hand side is equal to d(i n f)(σ 0,..., σ n ). This completes the proof. Let R be a commutative ring. We denote by R[G] the group ring of G over R. Let g (g G) be a canonical base of R[G]. Lemma 6.2. The G-module R[G] admits a mean. Proof. A map f : G R[G] has a following form: f(g) = f g (g)g g G where f g : G R. We define I(f) as I(f) = g G f g (g)g. Then it is easy to verify that I is a mean. If R is a topological commutative ring, the group ring R[G] is naturally topological G-module. Let f : X G R[G] be a continuous function. Fixed x X, the function f(x, ) : G R[G], g f(x, g) has a mean value I(f)(x) R[G] by using the mean of the proof of Lemma 6.2. Lemma 6.3. The function I(f) : X R[G] is continuous. Proof. The function f : X G R[G] is a following form: f(x, g) = g G f(x, g) g g. 27

32 Then f is continuous if and only if f(, g) g : X R is continuous for all g, g G. On the other hand, the function I(f) = g G I(f) gg is continuous if and only if I(f) g : X R is continuous for all g G. By the proof of Lemma 6.2, I(f) g (x) = I(f(x, )) g = f(x, g) g. This complete the proof. Corollary 6.4. The G-module C(X, A; R[G]) admits a mean. Proof. Let f : G C(X, A; R[G]) be a map. The adjoint ad(f) : X G R[G] is a continuous map. By Lemma 6.3, we have I(ad(f)) C(X, A; M). Then the function I(ad( )) is a mean on C(X, A; M). Proposition 6.5. Let R be a topological commutative ring. If M = R[G] as topological G-modules, then H(X,A) n (G; M) = 0 for all n > 0. Proof. Since H (X,A) (G; M) = H (G; C(X, A; M)), this follows from Proposition 6.1 and Corollary 6.4. Remark 6.6. We can also define the cohomology H(X,A) (G; M) of G in M over a topological pair (X, A) by using a (normalized) nonhomogeneous cochain complex of G in M. If G is finite, we also have a natural isomorphism H (X,A)(G; M) = H (G; C(X, A; M)). Hence, under the same condition as Proposition 6.5, we have H (X,A) (G; M) = 0 for all n > 0. 7 Normalized cochain complex Let G be a topological group and let M be a topological G-module. A continuous n-cochain for G in M is a continuous function f : G n M. We 28

33 denote by C n = C n (G; M) the abelian group of all continuous n-cochains for G in M. The coboundary map d : C n C n+1 is given by df(γ 1,..., γ n+1 ) = γ 1 f(γ 2,..., γ n+1 ) + n i=1 ( 1)i f(γ 1,..., γ i γ i+1,, γ n+1 ) +( 1) n+1 f(γ 1,..., γ n ). A normalized continuous n-cochain for G in M is a continuous function f : G n M such that f(γ 1,..., γ n ) = 0 if γ i is equal to the identity e for some i (1 i n). We denote by A n the abelian group of all normalized continuous n-cochains for G in M. It is easy to verify that A is a subcochain complex of C. In this section we show that the natural cochain map A C induces isomorphisms on cohomology groups. We define a filtration of the cochain complex C. Let F p C n be a subgroup of C n consisting of all continuous n-cochains for G in M such that f(γ 1,..., γ n ) = 0 if γ i is equal to the identity e for some i (1 i p). Then we have a filtration of the abelian group C n : C n = F 0 C n F 1 C n F n 1 C n F n C n = F n+1 C n =. It is easy to see that d(f p C n ) F p C n+1. Hence F p C is a sub-cochain complex of C. We obtain a filtration of the cochain complex C : C = F 0 C F 1 C F p C. Note that p F p C = A. In the following we show that the inclusion F p C F p 1 C induces isomorphisms on cohomology groups. For 1 p n, we define an abelian group B p,n to be the set of all continuous functions f : G n 1 M such that f(γ 1,..., γ n 1 ) = 0 if γ i is 29

34 equal to the identity e for some i (1 i < p). For p > n, we set B p,n = 0. There is an exact sequence for all p 1 and n: 0 F p C n F p 1 C n B p,n 0 where the right hand map is given by the restriction of the source G n 1 = G p 1 {e} G n p G n. There is a section of F p 1 C n B p,n obtained by the projection forgetting the pth component of G: G n = G p 1 G G n p G p 1 {e} G n p = G n 1. Since F p C is a sub-cochain complex of F p 1 C, there is a map d : B p,n B p,n+1 which makes B p, a cochain complex. The coboundary map d : B p,n B p,n+1 is given by the following form: (df)(γ 1,..., γ n ) = n 1 i=p ( 1)i f(γ 1,..., γ p 1, γ p,..., γ i γ i+1,..., γ n ) +( 1) n f(γ 1,..., γ p 1, γ p,..., γ n 1 ). Hence it is sufficient to show that all the cohomology groups of the cochain complex B p, vanish so as to prove that H n (F p C) H n (F p 1 C) are isomorphisms for all n. We define a map s : B p,n B p,n 1. For f B p,n, the function s(f) : G n 2 M is defined by s(f)(γ 1,..., γ n 2 ) = f(γ 1,..., γ p 1, e, γ p,..., γ n 2 ). Clearly s(f) B p,n 1. Note that if f B p,p, then s(f) = 0. We compute 30

35 d(s(f)) and s(d(f)). First we have d(s(f))(γ 1,..., γ n 1 ) = n 2 i=p ( 1)i s(f)(γ 1,..., γ p 1, γ p,..., Second we have, γ i γ i+1,..., γ n 1 ) +( 1) n 1 s(f)(γ 1,..., γ n 2 ) = n 2 i=p ( 1)i f(γ 1,..., γ p 1, e, γ p,...,, γ i γ i+1,..., γ n 1 ) +( 1) n 1 f(γ 1,..., γ p 1, e, γ p,..., γ n 2 ). s(d(f))(γ 1,..., γ n 1 ) = d(f)(γ 1,..., γ p 1, e, γ p,..., γ n 1 ) Hence we get = ( 1) p f(γ 1,..., γ p 1, γ p,..., γ n 1 ) + n 2 i=p ( 1)i+1 f(γ 1,..., γ p 1, e, γ p,...,, γ i γ i+1,..., γ n 1 ) +( 1) n f(γ 1,..., γ p 1, e, γ p,..., γ n 2 ). d(s(f)) + s(d(f)) = ( 1) p f. This shows that H n (B p, ) = 0 for all n. Theorem 7.1. The cochain map A C induces isomorphisms on cohomology groups H (A) = H (C). 8 Inflation maps Let G be a Hausdorff topological group and let K be a finite normal subgroup. We denote by H the quotient group G/K and π : G H the quotient map. 31

36 In this section we assume that there is a continuous section s : H G such that s(e) = e. Note that s is not necessarily group homomorphism. For example, if G is a profinite group, then there is such a section [15]. Let M be a topological G module. The fixed submodule M K is naturally a topological H module. In this section we study the inflation map H (H; M K ) H (G; M) under some conditions. G n A normalized continuous n-cochain for G in M is a continuous map f : M such that f(γ 1,..., γ n ) = 0 if γ i is equal to the identity e for some i (1 i n). We denote by A n = A n (G; M) the abelian group of all normalized continuous n-cochains for G in M. By definition, A 0 = M. The non-homogeneous coboundary map d : A n A n+1 is given by (df)(γ 1,..., γ n+1 ) = γ 1 f(γ 2,..., γ n+1 ) + n i=1 ( 1)i f(γ 1,..., γ i γ i+1,..., γ n+1 ) +( 1) n+1 f(γ 1,..., γ n ). By Theorem 7.1, the cohomology of A is the continuous cohomology H (G; M). We define a filtration of the cochain complex A. For j = 0, we set A n 0 = A n. For 0 < j n, F j A n is defined as a subgroup of A n consisting of f A n such that f : G n M factors through the continuous map f : G n j H j M. For j > n, we set F j A n = 0. Hence we get a filtration of A n : A n = F 0 A n F 1 A n F n A n F n+1 A n = 0. It is easy to verify that d(f j A n ) F j A n+1. Hence (F j A ) j 0 is a filtration of the cochain complex A : A = F 0 A F 1 A F n A. 32

37 Let N be the topological K module obtained from the topological G module M by the restriction of the action to K. In the following we assume that there is a topological commutative ring R such that the topological K module N is isomorphic to the group ring R[K] as topological K modules. Since K is discrete and finite, the normalized continuous n-cochain group A n (K; N) is naturally isomorphic to a direct product of finite many copies of N. We introduce a topology on A n (K; N) by using this isomorphism and the product topology. Let A j (G; A i (K; N)) be the abelian group of all normalized continuous j-cochains of G in A i (K; M). Note that a map from a topological space X to A i (K; M) is continuous if and only if the adjoint X K i M is continuous. We define a homomorphism r j : F j A i+j A j (H; A i (K; M)) by r j (f)(σ 1,..., σ j )(τ 1,..., τ i ) = f (τ 1,..., τ i, σ 1,..., σ j ) where f : G i H j M is a continuous map such that f(γ 1,..., γ n ) = f (γ 1,..., γ n j, πγ n j+1,..., πγ n ). It is easy to see that r j (f) = 0 if f F j+1 A i+j. Hence we get a homomorphism r j : F j A i+j /F j+1 A i+j A j (H; A i (K; M)). We note that r j : F j A j /F j+1 A j A j (H; M) is an isomorphism. Let d be the coboundary operator of F j A /F j+1 A. The coboundary operator of A (K; M) induces a homomorphism d K : A j (H; A (K; M)) A j (H; A +1 (K; M)). Then we obtain that d K r j = r j d. 33

38 Lemma 8.1. H(A j (H; A (K; M)), d K ) = A j (H; M K ) for all j. Proof. Let T n be a subspace of G n given by T n = n k=1g k 1 {e} G n k 1 G n. By Remark 6.6, H (G n,t n ) (K; M) = M K. This shows that the lemma holds. Lemma 8.2. r j : H j (F j A /F j+1 A ) = A j (H; M K ). Proof. Let f F j A j such that df F j+1 A j+1. Then d K (r j (f)) = 0. By Lemma 8.1, r j (f) A j (H; M K ). A j (H; M). We define f F j A j by f(γ 1,..., γ j ) = f(πγ 1,..., πγ j ). Conversely, let f A j (H; M K ) Then for any τ K, we easily see that df(γ 1 τ, γ 2,..., γ j+1 ) = df(γ 1, γ 2,..., γ j+1 ). Lemma 8.3. H n (F j A /F j+1 A ) = 0 for all n > j. Proof. Put i = n j 1 0. Let f F j A n such that df F j+1 A n+1. Since d K r j = r j d, we have d K (r j (f)) = 0. By Lemma 8.1, there is u A j (H; A i (K; M)) such that d K u = r j (f). We define a continuous function g : K i G j M by g(σ 1,..., σ i, γ 1,..., γ j ) = u(πγ 1,..., πγ j )(σ 1,..., σ i ). 34

39 Set g 0 = g. We define a sequence of continuous functions g 1,..., g i such that g k is defined on G k K i k G j with its values in M and g k is an extension of g k 1 for all 1 k i. We write ρ t s = (ρ s,..., ρ t ) G t s+1, γ t s = (γ s,..., γ t ) G t s+1 and σ t s = (σ s,..., σ t ) K t s+1 for 1 s t. Let g 1 (ρ, σ i 2, γ j 1) = s(πρ) g(s(πρ) 1 ρ, σ i 2, γ j 1) f(s(πρ), s(πρ) 1 ρ, σ i 2, γ j 1). For k > 1, we define the g k s recursively by g k (ρ k 1, σ i k+1, γj 1) = g k 1 (ρ k 2 1, ρ k 1 s(πρ k ), s(πρ k ) 1 ρ k, σ i k+1, γj 1) +( 1) k f(ρ k 1 1, s(πρ k ), s(πρ k ) 1 ρ k, σ i k+1, γj 1). Then we can show that f dg i F j+1 A n as the proof of Theorem of [5]. Therefore we get an E 1 -term of the spectral sequence associated with the filtration (F j A ) j 0 of the cochain complex A E p,q 1 = A p (H; M K ) if q = 0, 0 if q 0. It is easy to verify that the differential d 1 is given by the coboundary map of the normalized continuous cochain complex A (H; M K ) of H in M K. Hence we get an E 2 -term E p,q 2 = H p (H; M K ) if q = 0, 0 if q 0. The spectral sequence collapses from E 2 -term and converges to the cohomology groups H (A ) = H (G; M). It is easy to verify that the edge homomorphism E p,0 2 H p (A ) is identified with the inflation map H p (H; M K ) H p (G; M). Hence we get the following theorem. 35

40 Theorem 8.4. Let G be a Hausdorff topological group, K a finite normal subgroup and H = G/K the quotient group. continuous section s : H G. We assume that there is a Let M be a topological G module such that M is isomorphic as topological K-modules to a group ring R[K] for some topological commutative ring R. Then the inflation map H (H; M K ) H (G; M) is an isomorphism H (H; M K ) = H (G; M). 9 Cohomology of G n 1 (F) in F[w ±1 ] Let F be a finite field containing the finite fields F p n and F p n 1. The profinite group G n 1 (F) is a semi-direct product Gal(F/F p ) S n 1. There is an action of G n 1 (F) on the graded field F[w ±1 ] where the degree of w is 2. In this section we study the cohomology of G n 1 (F) in the coefficients F[w ± ]. We recall that the group G n 1 (F) is the automorphism group of the Honda formal group law H n 1 over the field F in the generalized sense. The Morava stabilizer group S n 1 which is the automorphism group of H n 1 in usual sense is a normal subgroup of G n 1 (F) and its quotient group is the Galois group Gal(F/F p ). In fact we have an isomorphism G n 1 (F) = Gal(F/F p ) S n 1. The profinite group G n 1 (F) acts on the graded field F[w ±1 ] from the right as follows. For every h S n 1, h has a following expression h = h 0 + h 1 T + h 2 T 2 +, h i W F p n 1, h pn 1 i = h i, h

41 where T n 1 = p and T h i = h p i T for all i 0. The subgroup S n 1 of G n 1 (F) acts on F[w ±1 ] as F-algebra automorphisms by w h = π(h 0 ) 1 w, h S n 1 where π : W F p n 1 F p n 1 is the projection to the residue field. The subgroup Gal(F/F p ) acts on F[w ±1 ] by (aw n ) σ = a σ w n, σ Gal(F/F p ), a F, n Z. Then we obtain an action of G n 1 (F) on F[w ±1 ] compatible to the above actions of the subgroups S n 1 and Gal(F/F p ). We consider the cohomology of G n 1 (F) in F[w ±1 ]. Since we have an exact sequence 1 S n 1 G n 1 (F) Gal(F/F p ) 1 and the quotient group Gal(F/F p ) is finite, we have a Lyndon-Hochschild- Serre spectral sequence E, 2 = H (Gal(F/F p ); H (S n 1 ; F[w ±1 ])) = H (G n 1 (F); F[w ±1 ]). The cohomology H (S n 1 ; F[w ±1 ]) is Gal(F/F p ) module over F and the action of Gal(F/F p ) satisfies the relation (cm) σ = c σ m σ for all c F and m H (S n 1 ; F[w ±1 ]). By Lemma 5.4 of [2], the E 2 -term is given by E p,q 2 = H (S n 1 ; F[w ±1 ]) Gal(F/Fp), if q = 0, 0 if q 0. Hence the spectral sequence collapses from E 2 -term and we obtain an isomorphism H (G n 1 (F); F[w ±1 ]) = H (S n 1 ; F[w ±1 ]) Gal(F/Fp). 37

42 We denote by Γ(n 1) the Galois group of the extension F p n 1/F p. The subfield F p n 1[w ±1 ] F[w ±1 ] is stable under the action of S n 1. Hence we have H (S n 1 ; F[w ±1 ]) = H (S n 1 ; F p n 1[w ±1 ]) Fp n 1 F. This isomorphism implies an isomorphism H (S n 1 ; F[w ±1 ]) Gal(F/F p) = H (S n 1 ; F p n 1[w ±1 ]) Γ(n 1). Therefore we get the following proposition. Proposition 9.1. There is an isomorphism H (G n 1 (F); F[w ±1 ]) = H (S n 1 ; F p n 1[w ±1 ]) Γ(n 1). 10 Cohomology of G n in F((u n 1 ))[u ±1 ] The profinite group acts on the graded field F((u n 1 ))[u ±1 ] where the degree of u is 2. In this section we study the cohomology of G n in the coefficients F((u n 1 ))[u ±1 ]. Let k = F((u n 1 )). The profinite group G n = Gal(F/F p ) S n acts on the field k continuously from the right. Explicitly, the Morava stabilizer group S n acts on k as F -algebra automorphisms by u n 1 g = t 0 (g) (pn 1 1) u n 1, g S n where t 0 (g) is the leading coefficient of the homomorphism t(g) : F n α(g) F n, t(g)(x) = i 0 α(g) F n t i (g)x pi. 38

43 The Galois group Gal(F/F p ) acts on F naturally and on u n 1 trivially. We define an action of the profinite group G n on the graded field k[u ±1 ] extending the action on the degree 0 part given above. For g S n, we set u g = t 0 (g) 1 u and Gal(F/F p ) acts on u trivially. Then we see that this defines a continuous action of G n on k[u ±1 ] from the right. We consider the cohomology of G n in the coefficients k[u ±1 ]. Since there is an exact sequence 1 S n G n Gal(F/F p ) 1 and the quotient group Gal(F/F p ) is finite, we have a Lyndon-Hochschild- Serre spectral sequence E, 2 = H (Gal(F/F p ); H (S n ; k[u ±1 ])) = H (G n ; k[u ±1 ]). By the same reason as the case of H (S n 1 ; F[w ±1 ]), we see that the E 2 -term is given by E p,q 2 = H (S n ; k[u ±1 ]) Gal(F/F p), if q = 0, 0 if q 0 and the spectral sequence collapses from E 2 -term. Since the subfield F p n((u n 1 ))[u ±1 ] k[u ±1 ] is stable under the action of G n, we obtain the following proposition. Proposition There is an isomorphism H (G n ; k[u ±1 ]) = H (S n ; F p n((u n 1 ))[u ±1 ]) Gal(F p n/fp). 39