Nilpotent local class field theory by

LXXXIII.1 (1998)

Nilpotent local class field theory

by

Helmut Koch (Berlin), Susanne Kukkuk (Detmold) and John Labute (Montreal)

1. Introduction. Let G be any profinite group and A an abelian profinite group. Let

L(G) =

∞

M

n=1

G

/G

be the graded Lie algebra associated with G by means of the lower central series (G

)

and let L(A) = L

n=1

L

(A) be the universal graded Lie algebra associated with A (see §2 for exact definitions). Any homomorphism ϕ of A into G/G

gives rise to a homomorphism ϕ

of L(A) into L(G).

In this paper we study the special situation where A is the profinite completion b K

of the multiplicative group K

of a local field K, i.e. a field which is complete with respect to a discrete valuation with finite residue class field. The group G is the absolute Galois group G

of K and ϕ is the Artin isomorphism of b K

onto G

/G

.

The surjectivity of ϕ implies the same for ϕ

. The goal of this paper is the determination of the kernel of ϕ

. This is equivalent to the determination of the kernel of the component homomorphisms

ϕ

(l) : L(A(l)) → L(G

(l)),

where l is any prime and B(l) is the maximal pro-l quotient of a profinite group B. The difficult case occurs when l = p, the residual characteristic of K. If K is of characteristic p, or if K is of characteristic zero and does not contain a primitive pth root of unity, this kernel is zero. So we assume that K is of characteristic zero and contains a primitive p

th root of unity ζ with κ chosen largest possible. In this case G

(p) is a Demushkin group so that the cup-product

H

(G

(p), Z/p

Z) × H

(G

(p), Z/p

Z) → H

(G

(p), Z/p

Z) = Z/p

Z

1991 Mathematics Subject Classification: 11S20, 11S31.

[45]

is non-degenerate. We now assume that p is odd. In this case, the form is alternating and so we obtain by duality an element in

G

/(G

)

∧ G

/(G

)

. Using the Artin isomorphism, this determines an element

τ ∈ L

(A(p)) ⊗ Z/p

Z

which is determined by G

up to a unit of Z/p

Z. Our main result is the following theorem:

Theorem 1.1. The kernel of ϕ

(p) is the ideal of L(A(p)) generated by the elements of the form [ad(λ)(ζ), ad(λ)(τ )], where λ is an element of the enveloping algebra of L(A(p)).

E.-W. Zink [Zi1], [Zi2] studied ϕ

: L

( b K

) → L

(G

) and showed that ϕ

is an isomorphism. His main interest in [Zi1], [Zi2] concerns the filtration (L

(G

)

)

of L

(G

) = G

/G

induced by the ramification groups G

of G

and the inverse image of this filtration in L( b K

). His results were augmented by Cram [Cr] and Kaufhold [Ka]. But the results of these three authors are far from the goal of giving an independent description of {ϕ

(L

(G

)

) | r ∈ R

}. There is of course a corresponding question for (ϕ

(L

(G

)

))

, but it will not be considered here.

The present paper originated from the thesis of the second author [Ku], directed by the first, and assisted by important suggestions of the third author. Section 5 was added by the third author.

2. Lie algebras. In this section we introduce the necessary definitions and facts about groups and related Lie algebras.

2.1. Let k be a commutative, associative ring with unity and let A be a k-module. Let T (A) be the non-associative tensor algebra of A considered as a k-module, i.e.

T (A) :=

∞

M

n=1

T

(A),

T

(A) := A, T

(A) := A ⊗

A, T

(A) := M

p+q=n

T

⊗

T

.

Then we define the Lie algebra L(A) as the factor algebra of T (A) by the ideal of T (A) generated by all elements of the form

a ⊗ a, (a ⊗ b) ⊗ c + (b ⊗ c) ⊗ a + (c ⊗ a) ⊗ b,

with a, b, c ∈ T (A). Since this ideal is homogeneous, we have L(A) =

∞

M

n=1

L

(A), L

(A) := (T

(A) + I(A))/I(A), and so L is a graded Lie algebra over k.

If ϕ : A → B is a homomorphism of k-modules, then to ϕ corresponds a homomorphism L(ϕ) of L(A) into L(B) so that L is a covariant functor from the category of k-modules to the category of graded Lie algebras over k.

Moreover, if L = L

n=1

L

is any graded Lie algebra over k, there is a unique homomorphism ψ of L(L

) into L such that

ψ(a) = a for a ∈ L

.

In the next section we apply this construction with k = Z to extend it to the case where A is a profinite abelian group. If A is finitely generated, we recover the above construction with k = b Z, the total profinite completion of Z.

2.2. Now let A be a profinite abelian group and U the filtration of A given by the set of open subgroups of A. We define L

(A) as the projective limit of the groups L

(A/U ) with U ∈ U. Then A and L

(A) are b Z-modules.

In the following algebra means always b Z-algebra. The product of a, b ∈ L(A) is denoted by [a, b]. The functor L is a covariant functor from the category of profinite abelian groups to the category of profinite graded Lie algebras, i.e., graded Lie algebras (over b Z) whose homogeneous components are profinite.

2.3. Let L = L

n=1

L

be any profinite graded Lie algebra. Then we have a natural homomorphism ψ of L(L

) into L with ψ(a) = a for a ∈ L

. 2.4. The proof of our main result (Theorem 1.1) is based on the com- parison of various filtrations of a profinite group G.

A filtration of G is a sequence of descending closed subgroups G

(i ≥ 1) such that the following conditions are fulfilled:

(i) G

= G,

(ii) [G

, G

] ⊆ G

for i, j ∈ N,

where [G

, G

] denotes the closed subgroup of G generated by the commu- tators

(g, h) := g

h

gh for g ∈ G

, h ∈ G

.

The most interesting filtration is the descending central series (G

), which is defined by induction:

G

:= G, G

:= [G, G

].

One proves by induction that (G

) is a filtration of G using the following

well known rules for commutators (see e.g. [Hl], 10.2), where x

means y

xy:

(h, g) = (g, h)

, (1)

h

= h(h, g), (2)

(f, gh) = (f, h)(f, g)((f, g), h), (3)

(f g, h) = (f, h)((f, h), g)(g, h), (4)

(f

, (g, h))(g

, (h, f ))(h

, (f, g)) = 1, (5)

for f, g, h ∈ G.

We associate with a filtered group G a graded Lie algebra L(G) as follows.

By definition, the groups G

are normal subgroups of G. We put L

(G) := G

/G

and [g, h] := (g, h)

for g ∈ G

, h ∈ G

. It is easy to see that this definition does not depend on the choice of g and h in the classes g ∈ L

(G) and h ∈ L

(G) and that it defines the structure of a profinite graded Lie algebra on

L(G) :=

∞

M

n=1

L

(G) by (1)–(5).

2.5. We now restrict ourselves to the special situation of a free pro- p-group F , where p denotes a prime number (see [Se2] for the definition of F ).

Theorem 2.1. Let L(F ) be the Lie algebra associated with the descend- ing central series of F . The natural map ψ : L(F/F

) to L(F ) is an isomorphism of graded Lie algebras over Z

.

P r o o f. Let F be the free pro-p-group with generator system {s

| i ∈ I}

and let S be any finite subset of I. Furthermore, let F

be the factor group of F with generator system S. Then F is the projective limit of the groups F

and L

(F/F

) (resp. L(F )) is the projective limit of the profinite groups L

(F

/F

) (resp. L

(F

)). Hence, it is sufficient to prove the theorem for free pro-p-groups F with finite generator rank N .

Let s

, . . . , s

be the free generator system of F and let x

be the class of s

in F/F

. Then F/F

is the free Z

-module with generators x

, . . . , x

and hence L(F/F

) is the free Z

-Lie algebra with generators x

, . . . , x

.

On the other hand, L(F ) as well is the free Z

-Lie algebra with generators

x

, . . . , x

as follows from the argument of [Wi1] applied to the embedding

of F into the completed group algebra Z

(see §2.7). We have

rk

L

(F/F

) = rk

L

(F ) = 1 n

X

d|n

µ(n/d)N

, where µ denotes the M¨ obius function.

This completes the proof of Theorem 2.1 since ψ is surjective.

2.6. The special filtrations (G

) of a pro-p-group G with the property G

⊆ G

are called p-filtrations.

If (G

) is a p-filtration of G, then L(G) is an F

-Lie algebra with an extra homogeneous operator π of degree 1 defined by

π(gG

) = g

G

, i = 1, 2, . . . Using induction over s one proves

(gh)

≡ g

h

(g, h)

(mod G

) for g ∈ G

, h ∈ G

. This shows that π is linear for p > 2 and for i > 1 if p = 2. If p = 2 and a, b ∈ L

(G) one has

π(a + b) = πa + πb + [a, b].

Using (2), one proves by induction over s that

(g

, h) ≡ (g, h)

((g, h), g)

(mod G

) for g ∈ G

, h ∈ G

. This shows that

π[a, b] = [πa, b]

if a ∈ L

(G), b ∈ L

(G) and p > 2 or if i > 1. Altogether we see that L(G) is a graded F

[π]-Lie algebra in the case where p > 2 and (G

) (n ≥ 1) is a p-filtration. If p = 2 then L

(G) := L

n=2

L

(G) is a graded F

[π]-Lie algebra.

2.7. Let F be a free pro-p-group with generators s

, . . . , s

. Beside the filtration (F

)

we need more general filtrations called κ-filtrations, and corresponding p-filtrations called (κ, p)-filtrations. They were introduced in [Lz], II.3.2, in much greater generality, but we restrict ourselves to what will be necessary for our paper.

For the definitions of these filtrations we consider the completed group

algebra A := Z

[[F ]], which is isomorphic to the ring Z

[[X

, . . . , X

]] of

associative formal power series in the variables X

, . . . , X

with coefficients

in Z

. The isomorphism α is defined by α(s

) = 1 + X

([Se1]). In the

following we identify A and Z

[[X

, . . . , X

]] by means of α. The restriction

of α to F yields the Magnus representation of F .

For any natural number κ we define a valuation v of A in the sense of Lazard ([Lz], I.2.2) by means of

v  X

i1,...,ik

a

X

. . . X



= inf

i1,...,ik

{b

} with

b

= ν

(a

) + (i

+ . . . + i

)κ,

where ν

denotes the p-adic (exponential) valuation of Z

. Then v defines a filtration (A

) of A with

A

:= {u ∈ A | v(u) ≥ i}.

We define the (κ, p)-filtration of F by

F b

:= {x ∈ F | v(x − 1) ≥ i}.

The associated Lie algebra b L = P

n=1

L b

is an F

[π]-Lie algebra if p > 2 or κ > 1. In what follows, we will assume that p > 2.

In the same way one can define the filtration ( e F

) by means of the valuation w of A which is given by

w

 X

i1,...,ik

a

X

. . . X



= inf

i1,...,ik

{c

} with

c

= (i

+ . . . + i

)κ.

We define a filtration (B

) of A:

B

:= {u ∈ A | w(u) ≥ i}.

Then

F e

:= {x ∈ F | w(x − 1) ≥ i}.

We denote the associated Lie algebra by e L = P

n=1

L e

. The Lie algebra e L is a free Lie algebra over Z

on the images of s

, . . . , s

in L

= e F

/ e F

. Let L be the Lie subalgebra of b L generated by σ

:= s

F b

, i = 1, . . . , N , and let

L

:= b L

∩ L, n = 1, 2, . . . Then L

= {0} if n 6≡ 0 (mod κ).

We have the following structure theorem for b L:

Theorem 2.2. L is the free F

-Lie algebra with generators σ

, . . . , σ

and b L is the free F

[π]-Lie algebra with generators σ

, . . . , σ

.

P r o o f. This result is well known. It is proved in [Lz], II.3.2, and goes

already back to A. Skopin ([Sk]). In fact, the assertions follow easily from

the embedding of F in the algebra A and the theorem of Witt about Lie polynomials in A ([Wi1]).

2.8. We want to compare the κ- and the (κ, p)-filtration of the free pro-p-group F . For this purpose we introduce filtrations in e L

and b L

. In L e

our filtration is simply e L

:= p

L e

, h ≥ 1.

Proposition 2.3.

p

L e

= ( e F

∩ b F

) e F

/ e F

= ( e F

∩ b F

F e

)/ e F

.

P r o o f. An element in p

L e

has the form x

F e

with x ∈ e F

. Therefore, x

∈ b F

. Let now y be an element of e F

∩ b F

. We want to show that y e F

is in p

L e

.

We assume that n = κm with m ∈ N. Then y ≡ 1 + y

(mod B

),

where y

is a homogeneous polynomial of degree m in A. Furthermore, y ∈ b F

if and only if y

∈ A

. This is possible only if each coefficient of the polynomial y

is divisible by p

. Hence y has the form

y ≡ 1 + p

z

(mod B

)

with z

∈ B

. By the theorem of Witt ([Wi1]), z

is a Lie polynomial in A. Hence, there is a z ∈ e F

such that z ≡ 1 + z

(mod B

) and this implies z

F e

= y e F

∈ p

L e

.

By Theorem 2.2 the group b L

has the form

L b

=

n−1

M

m=0

π

L

.

We define a filtration ( b L

)

of b L

by

L b

:=

n−h

M

m=0

π

L

.

Proposition 2.4.

L b

= ( b F

∩ e F

) b F

/ b F

= ( b F

∩ e F

F b

)/ b F

. The proof of this proposition is a variation of the proof of Theorem 2.2.

Now we define the following maps ω

from e L

onto π

L

, which allow

us to compare e L with b L:

ω

: e L

= ( e F

∩ b F

) e F

/ e F

→ ( e F

∩ b F

) e F

F b

/ e F

F b

∼=

→ ( e F

F b

∩ b F

)/( e F

F b

∩ b F

)

∼=

→ b L

/ b L

→ π

L

,

where the arrows denote the corresponding natural maps.

Proposition 2.5. ker ω

= e L

. P r o o f. By definition

ker ω

= ( e F

∩ b F

F e

)/ e F

= e L

.

3. The Artin map. We first recall some facts from class field theory (see e.g. [We]).

3.1. A local field is a finite extension of the field Q

of rational p-adic numbers (case of characteristic 0) or a finite extension of the field F

((x)) of power series in the variable x over the field F

with p elements (case of characteristic p). A global field is a finite extension of the field Q of rational numbers (case of characteristic 0) or a finite extension of the field F

(x) of rational functions with coefficients in F

.

Let K be a local or global field. Then one has a formation module A

associated with K, which is the multiplicative group K

if K is a local field, and the idele class group of K if K is a global field. Furthermore, let A b

be the profinite completion of A

. Then the Artin map is a canonical map from A

into the Galois group of the maximal abelian extension K

of K, which induces an isomorphism φ

from b A

onto G(K

/K). In the following we call φ

the Artin map.

3.2. Let K be a fixed separable algebraic closure of K and let G

be the Galois group of K/K. By 2.1–2.4, the map φ

induces a homomorphism of Lie algebras from L( b A

) onto L(G

), which will be denoted by φ

as well. We let φ

be the component of degree n of φ

. Then φ

is the usual Artin map. We call φ

the Artin map of L( b A

).

3.3. Let G

be the Galois group of the maximal nilpotent extension

of K in K. Then the kernel of the projection G

→ G

is equal to the

intersection of the groups G

for n ≥ 1. Therefore, one has a natural

isomorphism of L(G

) onto L(G

). Since G

is canonically isomorphic to

the product of its l-components G

(l), this implies that L(G

) is canoni-

cally isomorphic to the direct product of the Lie algebras L(G

(l)), where

l runs through all primes. Similarly, the decomposition of b A

into the di- rect product of its l-components b A

(l) yields a canonical decomposition of L( b A

) as the product of the Lie algebras L( b A

(l)). The study of the Artin map φ

therefore reduces to the study of its l-components

φ

(l) : L( b A

(l)) → L(G

(l))

as l varies over all primes. The map φ

(p) and its p-component φ

(p) : L( b A

(p)) → L(G

(p))

with

A b

(p) := Y

l6=p

A b

(l), G

(p) := Y

l6=p

G

(l) are the subjects of our further investigations.

3.4. We now restrict ourselves to the case where K is a local field of residue characteristic p. We denote the ring of integers of K by O

and the maximal ideal of O

by p. Hence A

= K

and b A

is the direct product of a group (π) generated as topological group by a fixed prime element π, the group µ

of roots of unity in K of order dividing q − 1, where q is the number of elements in the residue field, and of the group 1 + p of principal units in K. The group (π) is isomorphic to b Z, the total completion of Z, the group µ

is cyclic of order q − 1 and the group 1 + p is a pro-p-group, where p denotes the residue characteristic of K. The group 1 + p is the direct product of a finite cyclic group and a free abelian pro-p-group.

The surjectivity of φ

implies the surjectivity of φ

(p) and φ

(p). The main goal of this paper is the determination of the kernel of φ

(p) and φ

(p).

3.5. In this section we consider φ

(p). We introduce the following notations: A profinite group G will be called a p-group if G is pro-nilpotent and all finite factor groups of G have order prime to p. Corresponding by a p-extension of K is a Galois extension of K with Galois group being a p-group.

Proposition 3.1. Let σ be an extension of the Frobenius automorphism of the maximal unramified p-extension of K and let τ be a topological gen- erator of the inertia group of G

(p). Then G

(p) is generated as p-group by σ and τ and has one generating relation

(6) (σ, τ )τ

= 1.

Let σ and τ be the images of σ and τ in L

(G

(p)) = G

(p)/G

(p)

.

If n ≥ 2, then L

(G

(p)) is a cyclic group of order q − 1 with generator

(7) [σ, [σ, . . . , [σ, τ ] . . .]].

P r o o f. The structure of the group G

(p) is well known (see e.g. [Ko1], p. 95). The relation (6) implies that any element of the form

[a

, [a

, . . . , [a

, a

] . . .]] ∈ L

(G

(p))

with a

∈ {σ, τ } is equal to 0 if at least for two of the a

, . . . , a

one has a

= τ . It follows that

(8) [σ, [σ, . . . , [σ, τ ] . . .]] = τ

G

(p)

is a generator of L

(G

(p)) and has order q − 1.

For the next proposition we introduce some further notation.

If α ∈ K

we denote by α the image of α under the map K

→ b K

→ b K

(p).

Let µ

be the group of roots of unity of order dividing q − 1 and let ζ be a generator of µ

. Furthermore, let π be a prime element of K. Then the pro-p-group K

(p) is generated by ζ and π. The elements of K

(p) are uniquely represented in the form ζ

π

with µ = 0, . . . , q − 2, ν ∈ Z

, i.e., K

(p) ∼ = µ

× Z

. We denote by M the derived algebra of L(K

(p)) and by N the ideal of L(K

(p)) generated by all the elements of the form

[ζ, ad(π)

ζ] (n ≥ 1).

Furthermore, let F = L(Z/(q − 1)Z ⊕ Z/(q − 1)). With these notations we have the following proposition.

Proposition 3.2. As a graded Lie algebra, M is isomorphic to the derived algebra of F and the kernel of the map φ

(p) is N . Furthermore,

M

= N

⊕ Z/(q − 1)Z · ad(π)

(ζ) (n ≥ 1).

P r o o f. The natural projection b K

(p) → b K

(p)/ b K

(p)

induces a surjective homomorphism φ

of graded Lie algebras. Since b K

(p)/ b K

(p)

is a free Z/(q − 1)Z-module of rank 2 it follows that the restriction of φ

to M is an isomorphism and L( b K

(p)/ b K

(p)

) is the free graded Lie algebra with two generators over the ring Z/(q − 1)Z. Furthermore, we can choose σ and τ in Proposition 3.1 such that

φ

(p)(π) = σ, φ

(p)(ζ) = τ .

Proposition 3.1 implies that for n ≥ 2 the group L

( b K

(p)) is the direct sum of (ker φ

(p))

and the cyclic group of order q − 1 generated by

[π, [π, . . . , [π, ζ] . . .]].

This proves Proposition 3.2.

4. The map φ

(p). It remains to consider φ

(p). This is the main

goal of the paper. We restrict ourselves to the case p 6= 2.

The structure of G

(p) is well known (see e.g. [La0], [Ko1], pp. 96–105):

If Char K = p, or if K does not contain the pth roots of unity, then G

(p) is a free pro-p-group and φ

(p) is an isomorphism.

Proposition 4.1. Let K be a local field of characteristic p or of char- acteristic 0 and not containing the pth roots of unity. Then φ

(p) is an isomorphism of L( b K

(p)) onto L(G

(p)).

Now let K be a local field of characteristic 0 which contains the pth roots of unity. Then b K

(p) is isomorphic to µ

× Z

, where N = [K : Q

] + 2 and κ is the natural number such that µ

⊂ K but µ

6⊂ K. Then G

(p) is a Demushkin group and so is a group with N generators s

, . . . , s

and one generating relation r. One can choose s

, . . . , s

such that

r = s

κ

1

(s

, s

)(s

, s

) . . . (s

, s

).

In the following we identify G

(p) with F/R, where F is the free pro-p- group with generators s

, . . . , s

and R is the closed normal subgroup of F generated by r. The projection F → G

(p) induces a surjective homomor- phism

θ : L(F ) → L(G

(p)).

We let ψ be the unique homomorphism of L(F ) onto L( b K

(p)) such that θ = φ

(p)ψ.

We first study θ. With the identification G

(p) = F/R, R = (r)

this study is a question of group theory. We introduce the following nota- tions:

R

:= R ∩ F

,

N

(R) := R

F

/F

, N (R) :=

∞

X

n=1

N

(R).

Proposition 4.2. N

(R) is the kernel of θ

: L

(F ) → L

(F/R).

P r o o f. We have

L

(F/R) = (F/R)

/(F/R)

∼ = F

R/F

R ∼ = F

/F

(F

∩ R).

Hence, ker θ

= F

(F

∩ R)/F

.

Let U be the enveloping algebra of L(F ) ([Se1]). Since the Z

-Lie algebra L(F ) is a free algebra generated by

{σ

:= s

F

| i = 1, . . . , N }

we can identify U with the ring of polynomials in the non-commutative indeterminants σ

, . . . , σ

with coefficients in Z

. The ring U operates on L(F ) by adjoint action such that

ad(α)β = [α, β],

ad(λ

λ

)α = ad(λ

)ad(λ

)α and

ad(λ

+ λ

)α = ad(λ

)α + ad(λ

)α for α, β ∈ L(F ), λ

, λ

∈ U .

We put

t := (s

, s

) . . . (s

, s

), τ := tF

∈ L

(F ).

Let N

(R) be the ideal of N (R) generated by the elements (9) p

σ

, [ad(λ)σ

, ad(λ)τ ] (λ ∈ U ).

Then N

(R) is generated as a Z

-module by the element p

σ

together with the elements

(10) [ad(λ)σ

, ad(λ)τ ],

(11) [ad(λ)σ

, ad(µ)τ ] + [ad(µ)σ

, ad(λ)τ ]

with λ, µ homogeneous elements of U . The goal of this section is the proof of the following theorem:

Theorem 4.3. N (R) = N

(R).

Corollary 4.4. The subalgebra of L(G

(p)) generated by σ

, . . . , σ

is a free Lie algebra over Z

on these generators.

The corollary follows immediately from the fact that N (R) is a subset of the ideal of L generated by σ

.

To prove the theorem we first show that N

(R) ⊆ N (R). Firstly, rF

= s

F

= p

σ

,

and, to show that the elements of the form (10), (11) lie in N (R), we may assume that

λ = σ

. . . σ

and µ = σ

. . . σ

. Then

[ad(λ)σ

, ad(λ)τ ] = ((s

, . . . , (s

, s

) . . .), (s

, . . . , (s

, t) . . .))F

. Since r = s

t, we have

(s

, t) ∈ (s

, s

r)F

R = (s

, s

)F

R

and

(s

, s

)F

= (s

, s

)

((s

, s

), s

)

F

= (s

, s

)

((s

, s

), r)

F

. We get

((s

, . . . , (s

, s

) . . .), (s

, . . . , (s

, t) . . .))F

∈ RF

∩ F

and this implies

[ad(λ)σ

, ad(λ)τ ] ∈ N

(R),

which shows that elements of the form (10) belong to N (R). In a similar manner one shows that the elements of the form (11) also belong to N (R).

To show that N (R) ⊆ N

(R) we use a technique of [La3] consisting in the comparison of the κ- and (κ, p)-filtrations of F , where now κ is equal to the κ appearing in the defining relation r = s

(s

, s

) . . . (s

, s

) of G

(p).

We introduce the following notation as supplement to the notation in 2.7–2.8:

σ e

:= s

F e

∈ e L

, i = 1, . . . , m + 2, e τ := (s

, s

) . . . (s

, s

) e F

∈ e L

, N e

(R) := (R ∩ e F

) e F

/ e F

,

N (R) := e

∞

X

n=1

N e

(R).

Then e N

(R) is the ideal of e L generated by p

e σ

and ad(λ) σ e

∧ ad(λ) e τ for λ ∈ e U , where e U denotes the enveloping algebra of e L. Set

σ b

:= s

F b

∈ b L

, i = 1, . . . , m + 2, b τ := (s

, s

) . . . (s

, s

) b F

∈ b L

, N b

(R) := (R ∩ b F

) b F

/ b F

,

N (R) := b

∞

M

n=1

N b

(R).

The homogeneous component b N

(R) contains the element r b F

= π

σ b

+ b τ

and by Theorem 4

of [La1], b N (R) is even generated as an ideal of b L by π

σ b

+ b τ . This is the initial point of our proof.

Now we show e N

(R) = e N (R). The proof of e N

(R) ⊆ e N (R) is similar to

the proof of N

(R) ⊆ N (R).

Let U be the enveloping algebra of L. Then U can and will be iden- tified with the F

-subalgebra of the enveloping algebra b U of b L generated by b σ

, . . . , b σ

. Any non-zero homogeneous element λ of b L can be uniquely written in the form

(12) λ = λ

+ πλ

+ . . . + π

λ

with λ

, λ

, . . . , λ

∈ U and λ

6= 0. Since deg(λ

) ≡ 0 (mod κ) and deg(λ

)

= deg(λ

) + i, we have λ

= 0 if i 6≡ l (mod κ).

Let I be the ideal of L generated by σ b

and let N be the ideal of L generated by the elements of the form

(13) [ad(λ) σ b

, ad(λ) b τ ] with λ ∈ U .

Lemma 4.5.

(14) (15)

N b

(R) ∩ b L

⊆ (

π

N

+ b L

if j < κ, π

I

+ b L

if j ≥ κ.

P r o o f. Any element % of b N

(R) has the form ad(λ)(π

b σ

+ τ ) with λ b as above. If l = dκ + e with 0 ≤ e < κ, we have

% = π

ad(λ

) τ + b

d

X

j=1

π

(ad(λ

) b σ

+ ad(λ

) b τ ) + π

ad(λ

) b σ

. If ad(λ

) b σ

6= 0, we have

% ∈ π

I

+ b L

m

,

which yields the required result.

Now suppose that ad(λ

) b σ

= 0. Then λ

lies in the annihilator of b σ

. By [La4], Theorem 2, the annihilator of b σ

consists of the elements u ∈ U of the form

u = X

v∈U

a

(ad(v) σ b

)v (a

∈ U ).

Therefore λ

has this form. If l < κ, we have

% = π

X

v∈U

a

[ad(v) b σ

, ad(v) τ ] ∈ π b

N

as required. If l ≥ κ and ad(λ

) σ b

+ ad(λ

) τ 6= 0, we have b

% ∈ π

I

+ b L

as required. If ad(λ

) σ b

+ ad(λ

) b τ = 0 we get ad(λ

) b σ

= −ad(λ

) τ = − b X

v∈U

ad(a

(ad(v) σ b

)v) τ b

= X

v∈U

ad(a

(ad(v) b τ )v) b σ

since

ad((ad(v) σ b

)v) τ = ad(ad(v) b σ b

)ad(v) b τ = [ad(v) b σ

, ad(v) τ ] b

= −[ad(v) b τ , ad(v) b σ

] = −ad((ad(v) b τ )v) σ b

. Hence

λ

− X

v∈U

a

(ad(v) b τ )v is in the annihilator of b σ

. Therefore,

λ

∈ ann( σ b

) + ann( τ ). b

If ad(λ

) σ b

+ ad(λ

) b τ = 0 for 1 ≤ j ≤ d then, repeating the above argument, we get

λ

∈ ann( σ b

) + ann( τ ), b

which yields % ∈ π

N

. Otherwise, there is a j such that ad(λ

) σ b

+ ad(λ

) b τ 6= 0 and

% ∈ π

I

+ b L

m

.

R e m a r k. Lemma 4.5 deals with the ideal N (R) of the graded b F

-algebra b L generated by π

b σ

+ b τ . It is easy to be seen that Lemma 4.5 is valid in the case p = 2 as well. This will be used in the proof of Theorem 5.1.

Corollary 4.6. b N (R) ∩ L = N .

We now consider the homomorphism ω

of e L

onto L

. By Proposi- tion 2.5 its kernel is p e L

. Furthermore, ω

maps e N

(R) onto L

∩ b N

(R)

= N

. Hence

(16) N (R) ⊆ e e N

(R) + p e L.

More generally, we prove by induction

(17) N (R) ⊆ e e N

(R) + p

L, e h = 0, 1, . . . , using the homomorphisms ω

.

Lemma 4.7. e N

(R) ∩ p

L e

= (R ∩ e F

∩ b F

) e F

/ e F

.

P r o o f. Let η ∈ e N

(R) ∩ p

L e

. Then η = yF

with y = u

v, u ∈ e F

, v ∈ e F

. Since e F

⊆ b F

, we have v ∈ e F

∩ b F

. Let l ≥ 1 be largest such that there exists s ∈ R ∩ e F

with vs ∈ F b

∩ e F

. Assume that δ < h and let δ be the image of vs in ( b F

∩ F e

) b F

/ b F

. Then

δ ∈ b N

(R) ∩ b L

for some integer m with 0 ≤ m ≤ l, which we can assume is maximal and 6= l. By 4.5, we have δ = δ

+ δ

where δ

∈ b L

and

δ

∈

 π

N

if l ≤ κ, π

I

+ π

N

if l > κ,

where I is the ideal of L generated by σ b

. It follows that there is an el- ement y

∈ b F

∩ e F

with δ

= y

F b

. But then vyy

= δ

contradicting the maximality of m.

Now, since

ω

((R ∩ e F

∩ b F

) e F

/ e F

)

= ((R ∩ e F

∩ b F

) b F

/ b F

) b L

/ b L

= ( b N

(R) ∩ b L

) + b L

/ b L

, we have

ω

( e N

(R) ∩ p

L e

) = ( b N

(R) ∩ b L

) + b L

/ b L

. Assume that we proved

N (R) ⊆ e e N

(R) + p

L e for a certain h. We want to show

N (R) ⊆ e e N

(R) + p

L. e

Let ξ ∈ e N

(R). Then there exists ξ

∈ e N

such that ξ

= ξ − ξ

∈ p

L e

. It follows that ξ

∈ e N

(R) ∩ p

L e

. By Lemma 4.7 we have

(18)

(19) ω

(ξ

) ∈

 π

N

if h < κ, π

I

+ π

N

if h ≥ κ.

Hence there exists δ ∈ e N

(R) such that ω

(ξ

) = ω

(δ), which implies that ξ

− δ ∈ p

L e

and hence that ξ − (δ + ξ

) ∈ p

which com- pletes the inductive step. It follows that e N (R) ⊆ e N

(R) and hence that N (R) = e e N

(R).

Since the grading of e L is only a rescaling of the grading of L it follows

immediately that N (R) = N

(R), i.e. Theorem 4.3.

Now we consider the map

ψ : L(F ) → L( b A

(p))

in connection with θ : L(F ) → L(G

(p)). Let s

R, 1 ≤ i ≤ m + 2, be a system of generators of F/R = G

(p) such that R = (r) with

r = s

[s

, s

] . . . [s

, s

].

Then (see e.g. [Ko1], Lemma 10.7) there are elements α

, . . . , α

in K

such that

 α

K

(p)/K



= s

RF

, and α

is a root of unity of order p

in K.

Since the kernel of ψ : L

(F ) → b K

(p) is generated by p

σ

, where σ

:= s

F

, the kernel of ψ, as an ideal of L(F ), is generated by p

σ

as well.

Combining our knowledge of θ and ψ we get the following result about the kernel of the Artin map.

Theorem 4.8. The kernel of the Artin map φ

(p) is generated as an ideal of L( b K

(p)) by the elements of the form

[ad(λ)α

, ad(λ)β] (λ ∈ U ), where U is the enveloping algebra of L( b K

(p)) and β = [α

, α

] + . . . + [α

, α

].

This yields Theorem 1.1 since, by Satz 7.23 of [Ko1], the image of β in (L

( b K

(p))) ⊗ Z/p

Z is equal to τ .

Corollary 4.9. The kernel of φ

(p) is p

-torsion and , modulo torsion, L( b K

(p)) is a free Lie algebra over Z

with basis the images of α

, . . . , α

. Actually, as we shall see in Section 5, the kernel of φ

(p) is a free Z/p

Z-module as is the torsion submodule of L(G

(p)). We shall, more- over, give formulae for the ranks of these free modules.

5. The module structure of L(G

(p)) and ker φ

(p). Let L

= L

(x

, . . . , x

) be the free Lie algebra over Z on the elements x

, . . . , x

and let U be its enveloping algebra. Assume that N is even, let y = [x

, x

] + [x

, x

] + . . . + [x

, x

],

and let N be the ideal of L

generated by the elements of the form [ad(u)x

, ad(u)y] with u ∈ U . Then, by Theorem 4.8, L(G

(p)) is isomorphic to (L

/N ) ⊗ Z

modulo the ideal generated by p

x

.

Theorem 5.1. L

/N is a free Z-module.

P r o o f. Since the homogeneous components of L

/N are finitely gener- ated, it suffices to prove that, for each prime l, the homogeneous components of (L

/N ) ⊗ F

have ranks which are independent of l. Let L = L

⊗ F

and let N be the image of N in L. Then

L/N = (L

/N ) ⊗ F

.

Let b L = L ⊗ F

[π], where π is an indeterminate over F

of degree 1 and let N (R) as in Section 4 (κ = 1) be the ideal of b b L generated by πx

+ y, where x

is the image of x

in b L and

y = [x

, x

] + [x

, x

] + . . . + [x

, x

].

Then, since b L is the free Lie algebra over F

[π] on x

, . . . , x

, we have N (R) ∩ L = N by Corollary 4.6, which implies that b

L/N ∼ = (L + b N (R))/ b N (R).

By Lemma 4.5 the initial form π

λ

of a homogeneous element λ = π

λ

+ π

λ

+ . . . + λ

, λ

6= 0, of b N (R) is in the ideal of b L generated by x

. Hence

( b N (R) + L) ∩ π b L(x

, . . . , x

) = 0,

which implies that b L is the direct sum of b N (R) + L and π b L(x

, . . . , x

) and hence that

dim(L/N )

= dim( b L/ b N (R))

− dim(π b L(x

, . . . , x

))

= dim( b L/ b N (R))

− dim(b L(x

, . . . , x

))

+ dim(L(x

, . . . , x

))

.

Now, by Th´ eor` eme 3 of [La1], b L/ b N (R) is a free graded F

[π]-module and so L/ b b N (R) ∼ = (L/(y)) ⊗

F

[π]

as graded F

-modules. Now, by [La1], Th´ eor` eme 2, the Poincar´ e series of the enveloping algebra of L/(y) is

1

1 − N t + t

= 1

(1 − β

t)(1 − β

t) ,

where β

+ β

= N , β

β

= 1. If a

= dim

(L/(y))

, we have (by the Birkhoff–Witt Theorem)

Y

n≥1

1

(1 − t

)

= 1

1 − N t + t

,

which yields by a standard calculation (see [Se1], LA 4.5–4.6), a

= 1

n X

d|n

µ(n/d)(β

+ β

).

It follows that

dim( b L/ b N (R))

=

n

X

k=1

1 k

X

d|k

µ(k/d)(β

+ β

) and hence that

dim(L/N )

=

n

X

k=1

1 k

X

d|k

µ(k/d)(β

+ β

− (N − 1)

) (20)

+ 1 n

X

d|n

µ(n/d)(N − 1)

is independent of l.

Theorem 5.2. We have

ker φ

(p)

∼ = (Z/p

Z)

, L

(G

(p)) ∼ = (Z/p

Z)

⊕ Z

, where

b

=

n

X

k=1

1 k

X

d|k

µ(k/d)(β

+ β

− (N − 1)

),

c

=

n−1

X

k=1

1 k

X

d|k

µ(k/d)((N − 1)

− β

− β

) + 1 n

X

d|n

µ(n/d)(N

− β

− β

),

d

= 1 n

X

d|n

µ(n/d)(N − 1)

.

P r o o f. By Theorem 4.8 we have ker φ

(p) isomorphic to N ⊗ Z/p

Z, which gives the first isomorphism since the Z/p

Z-rank of N

⊗ Z/p

Z is the dimension of N

over F

, which in turn equals c

.

Again, by Theorem 4.8, the torsion submodule of L

(G

(p)) is isomor- phic to ((x

)/N ) ⊗ Z/p

Z, and L

(G

(p)) modulo torsion is isomorphic to the free Lie algebra over Z

on N − 1 generators. This yields the second isomorphism since the Z/p

Z-rank of ((x

)/N ) ⊗ Z/p

Z is the dimension of (x

)/N over F

, which in turn is equal to b

.

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Institut f¨ ur Mathematik Teichstr. 33

Lehrstuhl Zahlentheorie D-32756 Detmold

Humboldt-Universit¨ at zu Berlin Germany

J¨ agerstr. 10-11, D-10 117 Berlin Germany

E-mail: koch@mathematik.hu-berlin.de Department of Mathematics and Statistics McGill University

Burnside Hall

805 Sherbrooke Street West Montreal QC H3A 2K6, Canada E-mail: labute@galois.math.mcgill.ca

Received on 20.5.1997 (3184)