K_{1} OF CERTAIN IWASAWA ALGEBRAS, AFTER KAKDE

PETER SCHNEIDER AND OTMAR VENJAKOB

This paper contains a detailed exposition of the content of section five in Kakde’s paper [Kak]. We proceed in a slightly more axiomatic way to pin down the exact requirements on the p-adic Lie group under consideration. We also make use of our conceptual theory of the completed localization of an Iwasawa algebra as developed in [SV1]. This simplifies some of the arguments. Otherwise, with the exception of the notation at certain places, we follow Kakde’s paper.

Let G be a one dimensional compact p-adic Lie group and Λ(G) its Iwasawa algebra. The main purpose is to establish a description of the K-group K_{1}(Λ(G)) in terms of the groups of units Λ(U)^{×} for a suitable system of abelian subquotients U of G. The strategy for achieving this relies on the following commutative diagram

\xymatrix{1 \ar[r] & μ(O) × G^{ab} \ar[d]_{=} \ar[r]^{ι} & K′_{1}(Λ(G)) \ar[d]_{θ} \ar[r]^{L} & O[[Conj(G)]] \ar[d]_{β}^{≅} \ar[r]^{ω} & G^{ab} \ar[d]_{=} \ar[r] & 1 \\ 1 \ar[r] & μ(O) × G^{ab} \ar[r]^{θ∘ι} & Φ \ar[r]^{L} & Ψ \ar[r]^{ω∘β^{−1}} & G^{ab} \ar[r] & 1}

which to explain in detail is not necessary at this point. We only mention that O[[Conj(G)]] is an additive version of K_{1}(Λ(G)), L is the integral logarithm of Oliver and Taylor, and Φ and Ψ are the description we want to achieve and its additive version, respectively. The upper row is exact by work of Oliver (and Fukaya/Kato). In a first step (section 3) the somewhat easier additive isomorphism β will be established. Then it will be a major point to define the map L which makes the middle square commutative. Finally it will be shown that the lower row is exact as well. For both see section 4. Once all of this is done it is a formal consequence that the map θ is an isomorphism. Some of this will be generalized to the completed localization B(G) of Λ(G), additively in section 3 and multiplicatively in section 6. The latter requires an extension of the integral logarithm to the B(G)-setting (section 5).

1. Skew Laurent series

All Iwasawa algebras under consideration will have coefficients in the ring of integers O of a finite extension of ℚ_{p}. We fix a prime element π ∈ O. Let G be any compact p-adic Lie group. We assume that G contains a closed normal subgroup H such that G/H ≅ ℤ_{p}.

Date: June 15, 2011.

In order to give a description of Λ(G) relative to Λ(H) we choose a closed subgroup Γ ⊆ G such that Γ ⟶^{≅} G/H and we pick a topological generator γ of Γ (in particular, G = H ⋊ Γ). We then have the ring automorphism

σ : Λ(H) ⟶ Λ(H)
a ⟼ γaγ^{−1}

as well as the left σ-derivation δ := σ −id on Λ(H). We recall that the latter means that

δ(ab) = δ(a)b + σ(a)δ(b) for any a, b ∈ Λ(H) .

Proposition 1.1. (Venjakob) We have the isomorphism

Λ(H)[[t; σ, δ]] = {∑_{i≥0} a_{i}t^{i} : a_{i} ∈ Λ(H)} ⟶^{≅} Λ(G)

sending t to γ−1 between the (σ, δ)-skew power series ring over Λ(H) and the Iwasawa algebra Λ(G); the multiplication on the left hand side is determined by the rule

ta = σ(a)t + δ(a) for any a ∈ Λ(H)

and by continuity.

According to [CFKSV] the set

S := S(G) := {f ∈ Λ(G) : Λ(G)/Λ(G)f is finitely generated over Λ(H)}

is an Ore set in Λ(G) consisting of regular elements. We then may form the localization A(G) := Λ(G)_{S} as well as its Jac(Λ(H))-adic completion

B(G) := { \atop Λ(G)_{S}} .

Theorem 1.2. ([SV1] Thm. 4.7 and Prop. 2.26 (i))

i. The isomorphism in Prop. 1.1 extends to an isomorphism between Λ(H) ≪ t; σ, δ]] := {∑_{i∈ℤ} a_{i}t^{i} : a_{i} ∈ Λ(H), lim_{i→∞} a_{i} = 0 in Λ(H)} and B(G).

ii. B(G) is noetherian and pseudocompact with

Jac(B(G)) = {∑_{i∈Z} a_{i}t^{i} ∈ B(G) : a_{i} ∈ Jac(Λ(H))} .

iii. B(G) is flat over A(G) and hence over Λ(G).

We point out that the commutation rules in the ring Λ(H) ≪ t; σ, δ]] are considerably more complicated. For example, one has

at^{−1} = ∑_{i<0} t^{i}σδ^{−i−1}(a) for any a ∈ Λ(H)

involving an infinite Laurent series.

For later applications we note that, for finite H, the pseudocompact topology on B(G) is the π-adic one.

Let U ⊆ G be an open subgroup (equipped with H ∩ U). Proposition 1.3. ([SV1] Prop. 4.5) We have

A(G) = A(U) ⊗_{Λ(U)} Λ(G) and B(G) = B(U) ⊗_{Λ(U)} Λ(G)

as bimodules; in particular, A(G), resp. B(G), is free as an A(U)-, resp. a B(U)-, module of rank equal to [G : U].

In the special case where H is finite we find an open subgroup Γ_{0} ⊆ Γ which centralizes H. Hence Γ_{0} is open in G and lies in the center of G. It follows that

– A(G) is an A(Γ_{0})-algebra which is finitely generated free as a module, and

– B(G) is a B(Γ_{0})-algebra which also is finitely generated free as a

module. It is a special case of Thm. 1.2 that B(Γ_{0}) is a complete discrete valuation ring with prime element π. Using the Weierstrass preparation theorem one sees directly that A(Γ_{0}) is the localisation of Λ(Γ_{0}) = O[[t]] in the height one prime ideal πΛ(Γ_{0}) and hence is a discrete valuation ring with completion B(Γ_{0}).

2. Additive commutators

For any ring A and any set X we let A[X] denote the free A-module over the basis X. By [A, A] we denote the additive subgroup of A generated by all additive commutators [a_{1}, a_{2}] with a_{1}, a_{2} ∈ A. We note that, if A_{0} is the center of A, then [A, A] is an A_{0}-submodule of A. Finally, for any group G we denote by Conj(G) the set of all conjugacy classes [g]_{G} of elements g ∈ G. Lemma 2.1. For any finite group G we have

O[G]/[O[G], O[G]] ⟶^{≅} O[Conj(G)]

as O-modules. Proof. We consider the surjective O-module homomorphism

O[G] ⟶ O[Conj(G)]
g ⟼ [g]_{G} .

Because of gh − hg = gh − h(gh)h^{−1} it has [O[G], O[G]] in its kernel. On the other hand let ∑_{g∈G} a_{g}g be any element in the kernel. It is the sum of the elements ∑_{g∈[h]_{G}} a_{g}g in the kernel. In particular, ∑_{g∈[h]_{G}} a_{g} = 0. Hence

∑_{g∈[h]_{G}} a_{g}g = ∑_{g∈[h]_{G}} a_{g}(g − h) = ∑_{g∈G/G_{h}} a_{ghg^{−1}}(ghg^{−1} − h)

= ∑_{g∈G/G_{h}} a_{ghg^{−1}}((gh)g^{−1} − g^{−1}(gh)) ∈ [O[G], O[G]]

(where G_{h} denotes the centralizer of h in G). □

For our Lie group G we put

O[[Conj(G)]] := \underleftarrow{lim} O[Conj(G/N)]

where N runs over all open normal subgroups of G. We then obtain in the projective limit an O-module isomorphism

Λ(G)/\overline{[Λ(G), Λ(G)]} ⟶^{≅} O[[Conj(G)]] .

If G_{0} ⊆ G denotes the center then this, in an obvious way, is a Λ(G_{0})-module isomorphism. Suppose that G_{0} is open in G and put G := G/G_{0}. Then G_{0} acts by multiplication on Conj(G) with finitely many orbits. In fact,

G_{0}\ Conj(G) ⟶^{≃} Conj(G)
[g]_{G} ⟼ [gG_{0}]_{G}

is a bijection. Hence, noncanonically,

O[[Conj(G)]] ≅ Λ(G_{0})[Conj(G)]

as Λ(G_{0})-modules.

Lemma 2.2. If G_{0} is open in G then we have:

i. [Λ(G), Λ(G)] is closed in Λ(G); ii. B(G)/[B(G), B(G)] = B(G_{0}) ⊗_{Λ(G_{0})} O[[Conj(G)]].

Proof. i. With Λ(G) also [Λ(G), Λ(G)] is a finitely generated Λ(G_{0})-module. Hence [Λ(G), Λ(G)] is the image of a continuous Λ(G_{0})-linear map Λ(G)^{m} ⟶ Λ(G) between compact modules.

ii. By Prop. 1.3 we have B(G) = B(G_{0}) ⊗_{Λ(G_{0})} Λ(G). It follows that [B(G), B(G)] = B(G_{0}) · [Λ(G), Λ(G)]. □

3. The additive theory

Let U ⊆ G be any open subgroup. On the one hand we let N(U) denote the normalizer of U in G, and we put W (U) := N(U)/U. On the other hand we note that, by [DDMS] Thm. 8.3.2 and Prop. 1.19, the commutator subgroup [U, U] is closed in U. Hence

U^{ab} := U/[U, U]

is a commutative Lie group. We let pr_{U^{ab}}^{U} : O[[Conj(U)]] ↠ O[[Conj(U^{ab})]] = Λ(U^{ab}) denote the canonical surjection induced by the obvious projection U → U^{ab}.

Let U ⊆ V ⊆ G be open subgroups. Any g ∈ V acts by left multiplication on V/U. We note:

– Any set of representatives for the left cosets of U in V also is a basis of Λ(V) as a right Λ(U)-module.

– We have gxU = xU if and only if x^{−1}gx ∈ U.

Since gx = x(x^{−1}gx) the trace map in this situation therefore is given by

tr_{U}^{V} : O[[Conj(V)]] ⟶ O[[Conj(U)]]
[g]_{V} ⟼ ∑_{x∈V/U,x^{−1}gx∈U}[x^{−1}gx]_{U} .

It is Λ(G_{0} ∩ U)-linear. If [V, V] ⊆ U, we shall also need the two maps

\xymatrix{ & Λ(V^{ab}) \ar[d]^{τ_{U}^{V} :=tr_{U/[V,V ]}^{V^{ab}}} \\ Λ(U^{ab}) \ar[r]^{π_{U}^{V}:=pr_{U/[V,V ]}^{U^{ab}}} & Λ(U/[V, V]),}

where pr_{U/[V,V]}^{U^{ab}} is the obvious canonical surjection.

Remark 3.1. i. If U is normal in V we have

tr_{U}^{V} ([g]_{V}) = \begin{cases}∑_{x∈V/U}[xgx^{−1}]_{U} & if g ∈ U, \\ 0 & if g ∉ U. \end{cases}

ii. If V is abelian we have

tr_{U}^{V} (g) = \begin{cases}[V : U]g & if g ∈ U, \\ 0 & if g ∉ U. \end{cases}

iii. If [V, V] ⊆ U then

τ_{U}^{V} (g[V, V]) = \begin{cases}[V : U]g[V, V] & if g ∈ U, \\ 0 & if g ∉ U. \end{cases}

Suppose that U is abelian. Then:

– For any g ∈ W (U) we have the well defined ring automorphism

σ_{U,g} : Λ(U) ⟶ Λ(U)
f ⟼ gfg^{−1}

as well as the Λ(G_{0} ∩ U)-linear endomorphism

σ_{U} := ∑_{g∈W (U)} σ_{U,g} .

– The image of σ_{U} is an ideal in the subring Λ(U)^{W (U)} := {f ∈ Λ(U) :

σ_{U,g}(f) = f for any g ∈ W (U)}.

In a later section we will need, more generally, for two open subgroups U ⊆ V ⊆ G such that U is normal in V the Λ(G_{0} ∩ U)-linear map

σ_{U}^{V} : Λ(U^{ab}) ⟶ Λ(U^{ab})
f ⟼ ∑_{g∈V/U} gfg^{−1} .

From now on we assume that

(H1)  G_{0} is open in G,

and we fix an open central subgroup Z ⊆ G. Let S(G, Z) denote the set of all subgroups Z ⊆ U ⊆ G. For U ∈ S(G, Z) we define

β_{U} := pr_{U^{ab}}^{U} ∘ tr_{U}^{G} .

Our main interest in this section lies in the Λ(Z)-linear map

β := β_{Z}^{G} : O[[Conj(G)]] ⟶ ∏_{U∈S(G,Z)} Λ(U^{ab})
f ⟼ (β_{U}(f))_{U} .

We exhibit three conditions any element (a_{U})_{U} in the image of β has to satisfy.

1. Let U ⊆ V be in S(G, Z) such that [V, V] ⊆ U. In particular, U is normal in V. We claim that the diagram

\xymatrix{O[[Conj(G)]] \ar[d]_{β_{U}} \ar[r]^{β_{V}} & Λ(V^{ab}) \ar[d]^{τ_{U}^{V}} \\ Λ(U^{ab}) \ar[r]^{π_{U}^{V}} & O[[U/[V, V]]]}

commutes. By the transitivity of traces this reduces to the commutativity of the diagram

\xymatrix{O[[Conj(V)]] \ar[d]_{_{trV U}} \ar[r]^{^{pr{V \atop V ab}}} & Λ(V^{ab}) \ar[d]^{_{tr{V ab \atop U/[V,V ]}}} \\ O[[Conj(U)]] \ar[r]^{pr_{U/[V,V ]}^{Uab} ∘ pr_{Uab}^{U}} & Λ(U/[V, V]).}

It suffices to check this on classes [g]_{V} for g ∈ V. Indeed, using Remark 3.1.i/iii we compute

tr_{U/[V,V]}^{V^{ab}} ∘ pr_{V^{ab}}^{V}([g]_{V}) = [V : U]g[V, V]

= ∑_{x∈V/U} xgx^{−1}[V, V]

= pr_{U/[V,V]}^{U^{ab}} ∘ pr_{U^{ab}}^{U}( ∑_{x∈V/U}[xgx^{−1}]_{U})

= pr_{U/[V,V]}^{U^{ab}} ∘ pr_{U^{ab}}^{U} ∘ tr_{U}^{V} ([g]_{V})

for g ∈ U; if g ∉ U then both sides are equal to zero. Hence:

(A1) τ_{U}^{V} (a_{V}) = π_{U}^{V} (a_{U})  for any U ⊆ V in S(G, Z) with [V, V] ⊆ U.

2) For any open subgroup U ⊆ G and any g ∈ G the diagram

\xymatrix{ & O[[Conj(G)]] \ar[dl]_{_{tr{G \atop U}}} \ar[dr]^{_{tr{G \atop gUg^{−1}}}} \\ O[[Conj(U)]] \ar[rr]^{g.g^{−1}} &  & O[[Conj(gUg^{−1})]]}

is commutative. This, in particular, implies

(A2)  a_{gUg^{−1}} = ga_{U}g^{−1}  for any U ∈ S(G, Z) and g ∈ G.

3) Let U ∈ S(G, Z). By the transitivity of traces we have

β_{U} = pr_{U^{ab}}^{U} ∘ tr_{U}^{G} = pr_{U^{ab}}^{U} ∘ tr_{U}^{N(U)} ∘ tr_{N(U)}^{G}

and hence

im(β_{U}) ⊆ im(pr_{U^{ab}}^{U} ∘ tr_{U}^{N(U)}) .

But Remark 3.1.i implies

pr_{U^{ab}}^{U} ∘ tr_{U}^{N(U)}([g]_{N(U)}) = \begin{cases}σ_{U}^{N(U)}(g[U, U]) & if g ∈ U, \\ 0 & if g ∉ U. \end{cases}

It follows that im(β_{U}) ⊆ im(σ_{U}^{N(U)}). We conclude:

(A3)  a_{U} ∈ im(σ_{U}^{N(U)})  for any U ∈ S(G, Z).

Let

Ψ := Ψ_{Z}^{G} ⊆ ∏_{U∈S(G,Z)} Λ(U^{ab})

denote the subgroup of all elements (a_{U})_{U} which satisfy (A1), (A2), and (A3). So far we have shown the following.

Lemma 3.2. im(β) ⊆ Ψ.

Our goal is to show that β induces an isomorphism O[[Conj(G)]] ≅ Ψ. For this purpose an important technical role is played by the subset C(G, Z) of all U ∈ S(G, Z) such that U/Z is cyclic. Let U ∈ C(G, Z). Then U can be generated by Z and at most one additional element, and hence is abelian. We introduce the Λ(Z)-linear maps

η_{U} : Λ(U) = ⊕_{g∈U/Z}Λ(Z)g ⟶ Λ(U)

with

η_{U}|Λ(Z)g := \begin{cases}id & if g generates U/Z, \\ 0 & otherwise \end{cases}

and

δ_{U} : Λ(U) ⟶ O[[Conj(G)]] ⊗ ℚ
g ⟼ \frac{1}{[G : U]}[g]_{G} .

Let

pr_{cyc} : ∏_{U∈S(G,Z)} Λ(U^{ab}) ⟶ ∏_{U∈C(G,Z)} Λ(U)

denote the obvious projection map and define the Λ(Z)-linear map

δ : ∏_{U∈C(G,Z)} Λ(U) ⟶ O[[Conj(G)]] ⊗ ℚ

(a_{U})_{U} ⟼ ∑_{U} δ_{U}(η_{U}(a_{U})) .

Lemma 3.3. δ ∘ pr_{cyc} ∘β = id.

Proof. Let g ∈ G and put U_{g} := < g, Z > ∈ C(G, Z). We compute

δ ∘ pr_{cyc} ∘β([g]_{G}) = ∑_{U∈C(G,Z)} δ_{U} ∘ η_{U} ∘ tr_{U}^{G}([g]_{G})

= ∑_{U} ∑_{x∈G/U,x^{−1}gx∈U} δ_{U} ∘ η_{U}(x^{−1}gx)

= ∑_{U} ∑_{x∈G/U,x^{−1}U_{g}x=U} \frac{1}{[G : U]}[g]_{G}

= ∑_{U conjugate to U_{g}} \frac{1}{[G : N(U_{g})]}[g]_{G}

= [g]_{G} .

□

It follows that the map β is injective.

At this point we need further assumptions:

(H2)  G is a pro-p group.

There is a system of representatives R ⊆ G for the cosets in G/Z which contains 1 and consists of full G-orbits.

(H3)

(In the one dimensional case G = H ⋊ γ^{ℤ_{p}} with Z = γ^{p^{e}ℤ_{p}} we may take R := {H × {γ^{j} : 0 ≤ j < p^{e}}.) For U ∈ S(G, Z) we put R_{U} := R ∩ U. We also introduce the subgroup

Ψ_{cyc} ⊆ ∏_{U∈C(G,Z)} Λ(U)

of all elements which satisfy (A1), (A2), and (A3) for all (pairs of) subgroups in C(G, Z).

Lemma 3.4. δ|Ψ_{cyc} is injective with δ(Ψ_{cyc}) ⊆ O[[Conj(G)]].

Proof. Let (a_{U})_{U} ∈ Ψ_{cyc}. We first assume that

δ((a_{U})_{U}) = ∑_{U∈C(G,Z)} δ_{U}(η_{U}(a_{U})) = 0 .

Step 1: We show that δ_{U}(η_{U}(a_{U})) = 0 for any U ∈ C(G, Z). By definition δ_{U}(η_{U}(a_{U})) is supported on ⋃_{g∈G} gUg^{−1}. So, if U_{1}, …, U_{m} are representatives for the G-orbits in C(G, Z), then we obtain

∑_{g∈G/N(U_{i})} δ_{gU_{i}g^{−1}}(η_{gU_{i}g^{−1}}(a_{gU_{i}g^{−1}})) = 0 for any 1 ≤ i ≤ m.

But (A2) implies that

δ_{gU_{i}g^{−1}}(η_{gU_{i}g^{−1}}(a_{gU_{i}g^{−1}})) = gδ_{U_{i}}(η_{U_{i}}(a_{U_{i}}))g^{−1} = δ_{U_{i}}(η_{U_{i}}(a_{U_{i}}))

for any 1 ≤ i ≤ m and g ∈ G. Step 2: We show that a_{U} = 0 for any U ∈ C(G, Z). Since

Λ(U) = ⊕_{h∈R_{U}} Λ(Z)h

we may write

a_{U} = ∑_{h∈R_{U}} a_{h}h with a_{h} ∈ Λ(Z).

For g ∈ N(U) we have

a_{U} = a_{gUg^{−1}} = ga_{U}g^{−1} = ∑_{h∈R_{U}} a_{h}ghg^{−1} = ∑_{h∈R_{U}} a_{g^{−1}hg}h

by (A2) (the last identity is the point where we need the assumption (H3)) and hence a_{g^{−1}hg} = a_{h}. It follows that

a_{U} = ∑_{ξ∈N(U)\R_{U}} a_{ξ}(∑_{h∈ξ} h) with a_{ξ} ∈ Λ(Z).

Choosing elements h_{ξ} ∈ ξ we then have

0 = δ_{U}(η_{U}(a_{U})) = ∑_{ξ∈N(U)\R_{U} , U=<h_{ξ},Z>} a_{ξ} \frac{|ξ|}{[G : U]}[h_{ξ}]_{G} .

If h_{ξ} generates U/Z then ξ is the intersection with R_{U} of a full G-orbit in R. It follows that a_{ξ} = 0 in this case. This shows first of all that a_{Z} = 0. Secondly, for U ≠ Z we obtain

a_{U} = ∑_{h∈R_{U′}} a_{h}h

where U′ ∈ C(G, Z) is such that U′/Z is the unique subgroup of index p in the cyclic p-group U/Z. On the other hand we deduce from (A1) that

a_{U′} = tr_{U′}^{U}(a_{U}) = pa_{U} .

The claim now follows by induction.

This establishes the asserted injectivity. For the claim about the image we note that, by (A3), we have

a_{U} = σ_{U}(b_{U}) with b_{U} ∈ Λ(U)

for any U ∈ C(G, Z). Hence

δ_{U}(η_{U}(a_{U})) = δ_{U}(η_{U}( ∑_{g∈W (U)} gb_{U}g^{−1})) = [N(U) : U]δ_{U}(η_{U}(b_{U})) .

Using (A2) we further obtain that

∑_{g∈G/N(U)} δ_{gUg^{−1}}(η_{gUg^{−1}}(a_{gUg^{−1}})) = [G : N(U)]δ_{U}(η_{U}(a_{U}))
= [G : U]δ_{U}(η_{U}(b_{U}))

lies in O[[Conj(G)]]. □

Lemma 3.5. The map pr_{cyc} restricts to an injection Ψ ↪ Ψ_{cyc}.

Proof. It is clear that pr restricts to a map Ψ → Ψ_{cyc}. To establish its injectivity we argue by contradiction. We assume that (a_{U})_{U} ∈ Ψ satisfies a_{U} = 0 for all U in C(G, Z) and that there exists a V in S(G, Z) \ C(G, Z) such that a_{V} ≠ 0. Moreover, we may and do assume that V has minimal order |V/Z| with this property. Let R ⊆ V denote a set of representatives for the cosets in V/Z[V, V]; we write h̄ ∈ V^{ab} for the image of h ∈ R. We have

a_{V} = ∑_{h∈R} a_{h}h̄ with a_{h} ∈ im(Λ(Z) → Λ(V^{ab})).

To achieve the contradiction we will show that all coefficients a_{h} have to vanish. Fix h_{0} ∈ R. Since V/Z is not cyclic (but is a p-group) its abelianization (V/Z)^{ab} is not cyclic either by [Hup] III.7.1.c. Hence we find a normal subgroup Z ⊆ U ⊴ V of index p such that h_{0} ∈ U. In particular, [V, V] ⊆ U. By the minimality of V we have a_{U} = 0. Then

0 = τ_{U}^{V} (a_{V}) = ∑_{h∈R} a_{h}τ_{U}^{V} (h̄) = p ∑_{h∈R∩U} a_{h}h̄

by (A1) and Remark 3.1.iii. It follows that a_{h} = 0 for any h ∈ R ∩ U and, in particular, a_{h_{0}} = 0. □

The last four lemmas together imply the following fact.

Theorem 3.6. All three maps in the commutative diagram

\xymatrix{ & Ψ \ar[dd]^{_{pr_{cyc}}}_{≅} \\ O[[Conj(G)]] \ar@{-->}[ur]^{β}_{≅} \ar@{-->}[dr]_{pr_{cyc} ∘β}^{≅} \\  & Ψ_{cyc}}

are isomorphisms.

Remark 3.7. Let Ψ_{ℚ_{p}} denote the subset of ∏_{U∈S(G,Z)} Λ(U^{ab}) ⊗_{ℤ_{p}} ℚ_{p} consisting of those tuples (a_{U})_{U} which satisfy the natural analogues of (A1) and (A2). Note that any such tuple automatically satisfies a_{U} ∈ im(σ_{U}^{N(U)})⊗_{ℤ_{p}}ℚ_{p} for all U in S(G, Z), because due to (A2) one has σ_{U}^{N(U)}(a_{U}) = |W (U)|a_{U}. Hence Ψ_{ℚ_{p}} can be identified with

Ψ ⊗_{ℤ_{p}} ℚ_{p} ⊆ ∏_{U∈S(G,Z)} Λ(U^{ab}) ⊗_{ℤ_{p}} ℚ_{p} .

Therefore, by Lemma 3.5 the projection pr_{cyc} ⊗_{ℤ_{p}}ℚ_{p} induces again an injection

Ψ_{ℚ_{p}} ↪ ∏_{U∈C(G,Z)} Λ(U) ⊗_{ℤ_{p}} ℚ_{p} .

Now assume that (a_{U})_{U} ∈ Ψ_{ℚ_{p}} satisfies a_{U} ∈ im(σ_{U}) ⊆ Λ(U) for all U in C(G, Z). Then (a_{U})_{U} belongs already to Ψ, in particular all a_{U} are integral! Indeed, by the above injectivity one has

(a_{U})_{U} = β(pr_{cyc} ∘β)^{−1}(pr_{cyc} ⊗_{ℤ_{p}}ℚ_{p})((a_{U})_{U}) ∈ Ψ .

Remark 3.8. For any pair W ⊆ V of subgroups of index p in C(G, Z) we have:

i. im(σ_{W}) ⊆ p im(σ_{V}); ii. Let (a_{U})_{U} ∈ ∏_{U∈C(G,Z)} Λ(U) be an element which satisfies (A1);

then: a. a_{V} = η_{V} (a_{V}) + \frac{1}{p}a_{W}; b. (a_{U})_{U} satisfies (A3) if and only if (η_{U}(a_{U}))_{U} satisfies (A3).

Proof. i. Using that obviously N(V) ⊆ N(W) we compute

σ_{W} (f) = ∑_{g∈N(W)/W} gfg^{−1} = ∑_{g∈N(V)/W} ∑_{h∈R} ghfh^{−1}g^{−1}

= p ∑_{g∈N(V)/V} g(∑_{h∈R} hfh^{−1})g^{−1} = pσ_{V} (∑_{h∈R} hfh^{−1})

for any f ∈ Λ(W), where R ⊆ N(W) is a fixed set of representatives for the right cosets of N(V) in N(W).

ii. By assumption we have tr_{W}^{V} (a_{V}) = a_{W}. If we write a_{V} = ∑_{h∈R_{V}} a_{h}h with a_{h} ∈ Λ(Z) then, using Remark 3.1.ii, we obtain

a_{W} = tr_{W}^{V} (a_{V}) = ∑_{h∈R_{V}} a_{h} tr_{W}^{V} (h) = p ∑_{h∈R_{W}} a_{h}h = p(a_{V} − η_{V} (a_{V})) .

Since η_{Z} = id the assertion in b. follows by induction from i. and a. □

Using Prop. 1.3 we may extend B(Z)-linearly all the maps which we have considered above. In particular, also using Lemma 2.2 we have the B(Z)- linear map

β_{B} := β_{B(Z)}^{B(G)} : B(G)/[B(G), B(G)] ⟶ ∏_{U∈S(G,Z)} B(U^{ab})
f ⟼ ((id_{B(Z)} ⊗β_{U})(f))_{U} .

In the right hand side we have the subgroup Ψ_{B} of elements which satisfy the obvious analogs of the conditions (A1) - (A3). Either by repeating the above arguments or by simply using that, according to Thm. 1.2.iii, B(Z) is flat over Λ(Z) we deduce the following consequence.

Theorem 3.9. β_{B} : B(G)/[B(G), B(G)] ⟶^{≅} Ψ_{B} is an isomorphism.

4. The multiplicative theory, part 1

We suppose G to satisfy (H1) and (H2), and we continue to fix an open central subgroup Z ⊆ G. For any open subgroups U ⊆ V ⊆ G we have the norm map

N_{U}^{V} : K_{1}(Λ(V)) ⟶ K_{1}(Λ(U)) .

We recall that it is induced by the exact functor which sends a finitely generated projective Λ(V)-module P to P viewed as a finitely generated projective Λ(U)-module. We introduce the composed homomorphisms

θ_{U} : K_{1}(Λ(G)) ⟶^{N_{U}^{G}} K_{1}(Λ(U)) ⟶ K_{1}(Λ(U^{ab})) = Λ(U^{ab})^{×}

where the right hand arrow is induced by the canonical surjection U ↠ U^{ab}. If U is abelian then θ_{U} = N_{U}^{G}, of course. The central object of the multiplicative theory is the homomorphism

θ := θ_{Z}^{G} : K_{1}(Λ(G)) ⟶ ∏_{U∈S(G,Z)} Λ(U^{ab})^{×}
x ⟼ (θ_{U}(x))_{U} .

Similarly as in the previous section we begin by exhibiting four conditions (M1) – (M4) which any element (x_{U})_{U} = θ(x) satisfies.

1) Let Z ⊆ U ⊆ V ⊆ G be open subgroups such that [V, V] ⊆ U. As in the last section we use the ring homomorphism

π_{U}^{V} : Λ(U^{ab}) ⟶ Λ(U/[V, V]) .

Furthermore, corresponding to the inclusion of groups U/[V, V] ↪ V^{ab} we have the norm map

ν_{U}^{V} := N_{U/[V,V]}^{V^{ab}} : Λ(V^{ab})^{×} ⟶ Λ(U/[V, V])^{×} .

We claim that the diagram

\xymatrix{K_{1}(Λ(G)) \ar[d]_{θ_{U}} \ar[r]^{θ_{V}} & Λ(V^{ab})^{×} \ar[d]^{ν_{U}^{V}} \\ Λ(U^{ab})^{×} \ar[r]^{π_{U}^{V}} & Λ(U/[V, V])^{×}}

is commutative, which implies that

(M1) ν_{U}^{V} (x_{V}) = π_{U}^{V} (x_{U})  for any U ⊆ V in S(G, Z) with [V, V] ⊆ U

holds true. We enlarge the above diagram to

\xymatrix{K_{1}(Λ(G)) \ar[d]_{N_{U}^{G}} \ar[dr]^{N_{V}^{G}} \\ K_{1}(Λ(U)) \ar[d] & K_{1}(Λ(V)) \ar[l]^{N_{U}^{V}} \ar[dd] \\ K_{1}(Λ(U^{ab})) \ar[d] \\ K_{1}(Λ(U/[V, V])) & K_{1}(Λ(V^{ab})) \ar[l]^{N_{U/[V,V ]}^{V^{ab}}}}

where the undecorated perpendicular arrows are induced by the obvious ring homomorphisms. The upper triangle is commutative by the transitivity of norm maps. The lower square is commutative because of the identity

Λ(U/[V, V]) ⊗_{Λ(U)} Λ(V) = Λ(V^{ab}) .

We point out that if V is abelian then (M1) simplifies to the condition

(M1a)  N_{U}^{V} (x_{V}) = x_{U} .

2) For any open subgroup U ⊆ G and any g ∈ G the diagram

\xymatrix{ & K_{1}(Λ(G)) \ar[dl]_{θ_{U}} \ar[dr]^{θ_{gUg^{−1}}} \\ Λ(U^{ab})^{×} \ar[rr]^{g.g^{−1}} &  & Λ((gUg^{−1})^{ab})^{×}}

is commutative. This implies

(M2)  x_{gUg^{−1}} = gx_{U}g^{−1}  for any U ∈ S(G, Z) and g ∈ G.

3) For the next condition we need the additional assumptions that

(H4)  O is absolutely unramified and p ≠ 2.

Let ϕ : O ⟶ O denote the Frobenius automorphism.

We first recall the construction of the Verlagerung. Let U ⊆ V ⊆ G be any two open subgroups, and let R ⊆ V be a set of representatives of the left cosets in V/U. Then R also is a basis of Λ(V) as a right Λ(U)-module. For g ∈ V and h ∈ R we write

gh = h_{g}c_{g,h} with h_{g} ∈ R and c_{g,h} ∈ U.

The matrix M_{g} of left multiplication by g on Λ(V) with respect to the basis R is the product of the permutation matrix describing the permutation action of g on the coset space V/U and the diagonal matrix with diagonal entries {c_{g,h}}_{h∈R}. The map

(ver)  ver_{U}^{V} :  V^{ab} ⟶ U^{ab}
g[V, V] ⟼ ∏_{h∈R} c_{g,h}[U, U]

is a well defined group homomorphism called the transfer map or Verlagerung (cf. [Hup] IV.1.4). In the trivial case where U is central in V we have ver_{U}^{V} (g) = g^{[V:U]} for any g ∈ V.

In our case where V is a pro-p group with p ≠ 2 we have:

– N_{U}^{V} (g) can be computed as the determinant of M_{g} in K_{1}(Λ(U)) = Λ(U)^{×}/[Λ(U)^{×}, Λ(U)^{×}].

– The determinant of the permutation matrix factor of M_{g} is equal to

one. Hence

(1)  N_{U}^{V} (g) ≡ ver_{U}^{V} (g) mod [Λ(U)^{×}, Λ(U)^{×}] .

In other words we have the commutative diagram

\xymatrix{V^{ab} \ar[d]_{_{verV U}} \ar@{^{(}->}[r] & K_{1}(Λ(V)) \ar[d]^{N_{U}^{V}} \\ U^{ab} \ar@{^{(}->}[r] & K_{1}(Λ(U)).}

In the following we extend ver_{U}^{V} to the unique ring homomorphism

ver_{U}^{V} : Λ(V^{ab}) ⟶ Λ(U^{ab})

such that ver_{U}^{V} |O = ϕ.

We also need to introduce, for any open subgroup U ⊆ G, the unique ℤ_{p}-linear continuous map

φ_{U} : O[[Conj(U)]] ⟶ O[[Conj(U)]]

such that

φ_{U}|O = ϕ and φ_{U}([g]_{U}) = [g^{p}]_{U} for any g ∈ U.

If U is abelian then φ_{U} is a ring endomorphism of Λ(U) with the property that modulo p it coincides with the map f ⟼ f^{p}.

We now assert:

(M3) ver_{U}^{V} (x_{V}) − x_{U} ∈ im(σ_{U}^{V})  for all U ⊆ V in S(G, Z) such that [V : U] = p.

Note that in this condition U automatically is normal in V. One easily checks that ver_{Z}^{U}(im(σ_{U}^{V})) ⊆ pΛ(Z). Hence (M3) inductively implies that ver_{Z}^{U}(x_{U}) = x_{Z} mod pΛ(Z) for any U ∈ S(G, Z). But ver_{Z}^{U} is just the |U/Z|- power map on the group elements. We therefore deduce:

(M3a)  x_{U}^{|U/Z|} ≡ x_{Z} mod pΛ(Z)  for all U in S(G, Z).

For the proof of (M3) consider as before the diagram

\xymatrix{K_{1}(Λ(G)) \ar[d]_{N_{U}^{G}} \ar[dr]^{N_{V}^{G}} \\ K_{1}(Λ(U)) \ar[d]^{_{pr{U \atop U^{ab}}}} & K_{1}(Λ(V)) \ar[l]^{N_{U}^{V}} \ar[d]^{_{pr{V \atop V^{ab}}}} \\ K_{1}(Λ(U^{ab})) & K_{1}(Λ(V^{ab}))} ,

from which we see that it suffices to show for all x ∈ Λ(V)^{×} the relation

ver_{U}^{V} (pr_{V^{ab}}^{V} x̄) − pr_{U^{ab}}^{U} N_{U}^{V} x̄ ∈ im(σ_{U}^{V})

holds, in which x̄ denotes the image of x in K_{1}(Λ(V)). We choose any g ∈ V \ U and use the decomposition

(2)  Λ(V) ≅ ⨁_{i=0}^{p−1} Λ(U)g^{i}

to write x = ∑ x_{k}g^{k}. In order to calculate N_{U}^{V} x̄ we denote by σ the endomorphism of Λ(U) induced by u ↦ gug^{−1} and consider the matrix

( \begin{matrix}x_{0} & x_{1} & … & x_{p−1} \\ σ(x_{p−1})g^{p} & σ(x_{0}) & … & σ(x_{p−2}) \\ σ^{2}(x_{p−2})g^{p} & σ^{2}(x_{p−1})g^{p} & … & σ^{2}(x_{p−3}) \\ ⋮ & ⋮ & ... & ⋮ \\ σ^{p−1}(x_{1})g^{p} & σ^{p−1}(x_{2})g^{p} & … & σ^{p−1}(x_{0}) \end{matrix} )

describing the Λ(U)-linear map which arises by multiplication with x from the right using the basis given in (2) and which thus represents N_{U}^{V} x̄ in K_{1}(Λ(U)). The image in Λ(U^{ab})^{×} therefore is represented by its determinant. Using the Leibniz rule we obtain

pr_{U^{ab}}^{U} N_{U}^{V} x̄ = ∑_{δ∈S} sign(δ)g^{pe_{δ}} ∏_{c∈C} σ^{c} pr_{U^{ab}}^{U} x_{κ(δ(c)−c)} ;

here S = S(ℤ/pℤ) denotes the symmetric group on (the group) C := ℤ/pℤ, κ(c) ∈ {0, 1, …, p − 1} is the respective representative of c, and e_{δ} is the number of c ∈ C such that κ(δ(c)) < κ(c). Note that, since g^{p} ∈ U, the endomorphism σ^{p} induces the identity on Λ(U^{ab}).

The above sum can be decomposed with respect to the following action of C on S. Let γ ∈ S denote the cycle of order p defined by γ(c) := c + 1. We put

C × S ⟶ S
(c, δ) ⟼ δ^{c} := γ^{c}δγ^{−c} .

Explicitly we have

δ^{c}(c′) := δ(c′ − c) + c for all c, c′ ∈ C.

This action obviously respects the signum of elements in S. Moreover, the function e_{δ} is constant on each C-orbit in S. We see that for every δ_{0} ∉ S^{C} the partial sum

∑_{δ∈Cδ_{0}} sign(δ)g^{pe_{δ}} ∏_{c∈C} σ^{c} pr_{U^{ab}}^{U} x_{κ(δ(c)−c)}

= sign(δ_{0})g^{pe_{δ_{0}}} ∑_{a∈C} ∏_{c∈C} σ^{c} pr_{U^{ab}}^{U} x_{κ(δ_{0}^{a}(c)−c)}

= ∑_{a∈C} σ^{a} (sign(δ_{0})g^{pe_{δ_{0}}} ∏_{b∈C} σ^{b} pr_{U^{ab}}^{U} x_{κ(δ_{0}(b)−b)})

belongs to im(σ_{U}^{V}) (the second identity comes from substituting b for c − a). On the other hand, if δ ∈ S^{C}, one checks immediately that δ(c) − c = δ(0) is constant for all c ∈ C, i.e., δ = γ^{δ(0)} = γ^{κ(δ(0))}, whence sign(δ) = 1 (since p ≠ 2). Furthermore e_{δ} = κ(δ(0)). As S decomposes disjointly into S^{C} and the orbits of order p, we altogether obtain modulo im(σ_{U}^{V})

(3)  pr_{U^{ab}}^{U} N_{U}^{V} x̄ ≡ ∑_{δ∈S^{C}} sign(δ)g^{pe_{δ}} ∏_{c∈C}^{p−1} σ^{c} pr_{U^{ab}}^{U} x_{κ(δ(c)−c)}

= ∑_{k=0}^{p−1} g^{pk} ∏_{i=0}^{p−1} σ^{i} pr_{U^{ab}}^{U} x_{k} .

Using the fact that ver_{U}^{V} is a ring homomorphism we are now able to determine modulo im(σ_{U}^{V}) :

ver_{U}^{V} (pr_{V^{ab}}^{V} x̄) = ∑_{k=0}^{p−1} ver_{U}^{V} (pr_{V^{ab}}^{V}(x_{k})) ver_{U}^{V} (pr_{V^{ab}}^{V}(g)^{k})

≡ ∑_{k=0}^{p−1} ∏_{i=0}^{p−1} σ^{i}(pr_{U^{ab}}^{U}(x_{k})) pr_{U^{ab}}^{U}(g^{p})^{k}

≡ pr_{U^{ab}}^{U} N_{U}^{V} x̄ .

Using the set of representatives R = {1, g, …, g^{p−1}} one checks from the definition that ver_{U}^{V} (pr_{V^{ab}}^{V}(g)) = pr_{U^{ab}}^{U}(g^{p}). The last congruence is (3) while the middle one can be seen as follows. First recall that im(σ_{U}^{V}) is a Λ(U^{ab})^{V/U}- ideal and that pr_{U^{ab}}^{U}(g^{p}) belongs to Λ(U^{ab})^{V/U}. Fix any k and write x_{k} = ∑_{h∈U/Z} a_{h}h with a_{h} ∈ Λ(Z). Using again the fact that ver_{U}^{V} is a ring homomorphism combined with formula (ver) (with c_{u,g^{i}} = σ^{−i}(u)) we obtain

ver_{U}^{V} (pr_{V^{ab}}^{V}(x_{k})) = pr_{U^{ab}}^{U} ( ∑_{h∈U/Z} φ_{Z}(a_{h}) ∏_{i=0}^{p−1} σ^{i}(h))

≡ pr_{U^{ab}}^{U} ( ∑_{h∈U/Z} a_{h}^{p} ∏_{i=0}^{p−1} σ^{i}(h))

as φ_{Z}(a_{h}) − a_{h}^{p} ∈ pΛ(Z), so that its projection to Λ(U^{ab}) lies in im(σ_{U}^{V}), and ∏_{i=0}^{p−1} σ^{i}(h) belongs to Λ(U)^{V/U} for every h ∈ V/U. Since moreover

∏_{i=0}^{p−1} ( ∑_{h∈U/Z} a_{h}σ^{i}(h)) − ∑_{h∈U/Z} a_{h}^{p} ∏_{i=0}^{p−1} σ^{i}(h)

is a sum of (mixed) terms of the form

∑_{i=0}^{p−1} σ^{i} (∏_{j=0}^{p−1}(a_{h_{j}}σ^{j}(h_{j})))

with h_{j} ∈ U/Z, such that {h_{j} : 0 ≤ j ≤ p − 1} consists of at least two elements, and hence belongs to im(σ_{U}^{V}), we can continue our above congruences by

≡ pr_{U^{ab}}^{U} ∏_{i=0}^{p−1} ( ∑_{h∈U/Z} a_{h}σ^{i}(h))

= ∏_{i=0}^{p−1} σ^{i}(pr_{U^{ab}}^{U}(x_{k}))

as desired.

4) The last condition requires further preparations. Let U ∈ C(G, Z). If U = Z we put

α_{Z} : Λ(Z)^{×} ⟶ 1 + pΛ(Z) ⊆ Λ(Z)^{×}
f ⟼ \frac{f^{p}}{φ_{Z}(f)} .

If U ≠ Z then we let U′ ∈ C(G, Z) be the unique subgroup of U such that [U : U′] = p. Since U is abelian we may consider the composed homomorphism

Λ(U)^{×} ⟶^{N_{U′}^{U}} Λ(U′)^{×} ⟶^{⊆} Λ(U)^{×} .

It is shown in [SV2] Prop. 2.3 (under more general circumstances) that modulo p this map coincides with the map f ⟼ f^{p}. It follows that

α_{U} : Λ(U)^{×} ⟶ 1 + pΛ(U) ⊆ Λ(U)^{×}
f ⟼ \frac{f^{p}}{N_{U′}^{U}(f)}

is a well defined homomorphism.

We also need, for any open subgroup U ⊆ G, the ring

Λ^{∞}(U) := \underleftarrow{lim} K[U/N]

where N runs over all open normal subgroups of U and where K denotes the field of fractions of O. We obviously have Λ(U) ⊆ Λ^{∞}(U). The usual logarithm series induces a homomorphism

log : K_{1}(Λ(U)) = Λ(U)^{×}/[Λ(U)^{×}, Λ(U)^{×}] ⟶

Λ^{∞}(U)^{ab} := Λ^{∞}(U)/\overline{[Λ^{∞}(U), Λ^{∞}(U)]} .

Our maps tr_{U}^{V}, η_{U}, and φ_{U} extend in an obvious way to maps, denoted by the same symbols, between the Λ^{∞}(U)^{ab}. As a consequence of the first step in the proof of [OT] Thm. 1.4 we have, for any open subgroups U ⊆ V ⊆ G, the commutative diagram

(4)  \xymatrix{K_{1}(Λ(V)) \ar[d]_{N_{U}^{V}} \ar[r]^{log} & Λ^{∞}(V)^{ab} \ar[d]^{_{tr_{V}_{U}}} \\ K_{1}(Λ(U)) \ar[r]^{log} & Λ^{∞}(U)^{ab}.}

Lemma 4.1. For any U ≠ Z in C(G, Z) the diagram

\xymatrix{Λ(U)^{×} \ar[d]_{α_{U}} \ar[r]^{log} & Λ^{∞}(U) \ar[d]^{pη_{U}} \\ Λ(U)^{×} \ar[r]^{log} & Λ^{∞}(U)}

is commutative.

Proof. Let U′ ∈ C(G, Z) be the unique subgroup of U of index p. Then

log(α_{U}(f)) = p log(f) − log(N_{U′}^{U}(f)) = p log(f) − tr_{U′}^{U}(log(f))
= (p id − tr_{U′}^{U})(log(f)) .

Hence we have to establish the identity

id −\frac{1}{p} tr_{U′}^{U} = η_{U} .

But, by definition, we have

η_{U}(g) = \begin{cases}g & if g ∈ U \ U′, \\ 0 & if g ∈ U′. \end{cases}

Remark 3.1.ii says that the same computation holds for the left hand side. □

For V ∈ S(G, Z) let P (V) denote the set of all W ≠ Z in C(G, Z) such that the unique subgroup Z ⊆ W′ ⊆ W with [W : W′] = p is contained in V. We now introduce the map

L = (L_{V})_{V} : [ ∏_{U∈S(G,Z)} Λ(U^{ab})^{×}]_{(M3a)} ⟶ ∏_{V ∈S(G,Z)} Λ(V^{ab}) ⊗_{ℤ_{p}} ℚ_{p}

defined by

L_{V} ((y_{U})_{U}) := \frac{1}{p^{2}|V/Z|} log (\frac{y_{V}^{p^{2}|V/Z|}}{φ_{Z}(y_{Z}^{p}) ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|}} )

where […]_{(M3a)} indicates the subgroup of all those elements which satisfy (M3a). The individual factors φ_{W} (α_{W} (y_{W})) appearing in the above definition lie in 1+pΛ(W′), since α_{W} (y_{W}) ∈ 1+pΛ(W), and hence can be viewed in 1 + pΛ(V^{ab}). Moreover using (M3a) we have

φ_{Z}(y_{Z}^{p}) ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|} ≡ φ_{Z}(y_{Z}^{p})

≡ y_{Z}^{p^{2}}
≡ y_{V}^{p^{2}|V/Z|} mod pΛ(V^{ab}) .

Thus the logarithms in the asserted map are defined and lie in pΛ(V^{ab}).

Lemma 4.2. If (y_{U})_{U} ∈ [∏_{U∈S(G,Z)} Λ(U^{ab})^{×}]_{(M3a)} satisfies the condition (M1), resp. (M2), then L((y_{U})_{U}) satisfies (A1), resp. (A2).

Proof. It is straightforward to check that (M2) implies (A2). Let therefore U ⊆ V be any two subgroups in S(G, Z) such that [V, V] ⊆ U. We note that

log(\frac{y_{V}^{p^{2}|V/Z|}}{φ_{Z}(y_{Z}^{p}) ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|}})

= log(\frac{y_{V}^{p^{2}|V/Z|}}{φ_{Z}(y_{Z}^{p})} ) − log( ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|}) .

This will allow us to split the subsequent calculations into two parts. We begin by computing

τ_{U}^{V} (\frac{1}{p^{2}|V/Z|} log(\frac{y_{V}^{p^{2}|V/Z|}}{φ_{Z}(y_{Z}^{p})} )) = \frac{1}{p^{2}|V/Z|} log(\frac{ν_{U}^{V} (y_{V})^{p^{2}|V/Z|}}{ν_{U}^{V} (φ_{Z}(y_{Z}^{p}))} )

= \frac{1}{p^{2}|V/Z|} log(\frac{π_{U}^{V} (y_{U})^{p^{2}|V/Z|}}{φ_{Z}(y_{Z}^{p})^{|V/U|}})

= π_{U}^{V} (\frac{1}{p^{2}|U/Z|} log(\frac{y_{U}^{p^{2}|U/Z|}}{φ_{Z}(y_{Z}^{p})} ))

where the first, resp. second, identity uses (4), resp. (M1). Next we compute

τ_{U}^{V} (\frac{1}{p^{2}|V/Z|} log( ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|}))

= \frac{1}{p^{2}|V/Z|} ∑_{W ∈P (V)} τ_{U}^{V} (log(φ_{W} (α_{W} (y_{W}))^{|W/Z|}))

= \frac{1}{p|V/Z|} ∑_{W ∈P (V)} τ_{U}^{V} φ_{W} η_{W} log(y_{W}^{|W/Z|})

where the last identity uses Lemma 4.1. Using the definitions and Remark 3.1.iii it is straightforward to check that

τ_{U}^{V} φ_{W} η_{W} = \begin{cases}[V : U]π_{U}^{V} φ_{W} η_{W} & if W′ ⊆ U, \\ 0, & otherwise \end{cases}

where, as before, W′ denotes the unique subgroup of W of index p and containing Z. We therefore may continue the above computation by

= \frac{1}{p|V/Z|} ∑_{W ∈P (U)}[V : U]π_{U}^{V} φ_{W} η_{W} log(y_{W}^{|W/Z|})

= π_{U}^{V} (\frac{1}{p^{2}|U/Z|} ∑_{W ∈P (U)} log(φ_{W} (α_{W} (y_{W}))^{|W/Z|}))

= π_{U}^{V} (\frac{1}{p^{2}|U/Z|} log( ∏_{W ∈P (U)} φ_{W} (α_{W} (y_{W}))^{|W/Z|}))

where the second identity again uses Lemma 4.1. This establishes that

τ_{U}^{V} (L_{V} ((y_{U})_{U})) = π_{U}^{V} L_{U}((y_{U})_{U}) ,

i. e., the condition (A1) for L_{V} ((y_{U})_{U}). □

Lemma 4.3. For any U ∈ S(G, Z) and any f ∈ Λ^{∞}(G)^{ab} we have:

β_{U}(φ_{G}(f)) = \frac{1}{[U:Z]}φ_{Z}(β_{Z}(f)) + ∑_{W ∈P (U)} \frac{[W:Z]}{[U:Z]}φ_{W} (η_{W} (β_{W} (f))) .

Proof. It suffices to consider elements of the form f = [g]_{G} for some g ∈ G. First let g ∉ Z. We compute

β_{U}(φ_{G}([g]_{G})) = pr_{U^{ab}}^{U}(tr_{U}^{G}([g^{p}]_{G})) = pr_{U^{ab}}^{U}(  ∑_{h∈G/U,h^{−1}g^{p}h∈U}[h^{−1}g^{p}h]_{U})

= ∑_{h∈G/U,h^{−1}g^{p}h∈U} h^{−1}g^{p}h[U, U]

= \frac{1}{[U:Z]} ∑_{h∈G/Z,h^{−1}g^{p}h∈U} h^{−1}g^{p}h[U, U]

= \frac{1}{[U:Z]} ∑_{W ∈P (U)} ∑_{h∈G/Z,W =<h^{−1}gh,Z>} h^{−1}g^{p}h[U, U]

= \frac{1}{[U:Z]} ∑_{W ∈P (U)} φ_{W} ( ∑_{h∈G/Z,W =<h^{−1}gh,Z>} h^{−1}gh)

= \frac{1}{[U:Z]} ∑_{W ∈P (U)} φ_{W} ([W : Z] ∑_{h∈G/W,W =<h^{−1}gh,Z>} h^{−1}gh)

= \frac{[W:Z]}{[U:Z]} ∑_{W ∈P (U)} φ_{W} (η_{W} (β_{W} ([g]_{G}))) .

We also have φ_{Z}(β_{Z}([g]_{G})) = 0 in this case. Now suppose that g ∈ Z. Then the last term in the above computation vanishes. But we have

β_{U}(φ_{G}([g]_{G})) = [G : U]g^{p}[U, U] = \frac{1}{[U : Z]}φ_{Z}(β_{Z}([g]_{G})) .

□

Oliver and Taylor (cf. [Oli] Chap. 6 or [CR] §54) have shown that

L = L_{G} : K_{1}(Λ(G)) ⟶ O[[Conj(G)]]
x ⟼ log(x) − \frac{1}{p}φ_{G}(log(x))

is a well defined homomorphism which makes the diagram

\xymatrix{K_{1}(Λ(G)) \ar[r]^{log} \ar[dr]_{L} & Λ^{∞}(G)^{ab} \ar[r]^{id − \frac{1}{p} φ_{G}} & Λ^{∞}(G)^{ab} \\  & O[[Conj(G)]] \ar[ur]_{can}}

commutative. We observe that, since Λ^{∞}(G)^{ab} also can be viewed as the projective limit \underleftarrow{lim} K[Conj(G/N)], the right oblique arrow is injective.

Proposition 4.4. The diagram

\xymatrix{K_{1}(Λ(G)) \ar[d]_{θ} \ar[r]^{L} & O[[Conj(G)]] \ar[d]^{β} \\ [∏_{U∈S(G,Z)} Λ(U^{ab})^{×}]_{(M3a)} \ar[r]^{L} & ∏_{U∈S(G,Z)} Λ(U^{ab}) ⊗_{ℤ_{p}} ℚ_{p}}

is commutative.

Proof. Let x ∈ K_{1}(Λ(G)) and put (x_{U})_{U} := θ(x). Using (4) we obtain

β_{V} (log(x)) = log(θ_{V} (x)) = log(x_{V})

for any V ∈ S(G, Z). Using this identity together with Lemmas 4.1 and 4.3 we compute

β_{U}(L(x)) = β_{U}(log(x)) − \frac{1}{p}β_{U}(φ_{G}(log(x)))

= log(x_{U}) − \frac{1}{p[U:Z]}φ_{Z}(β_{Z}(log(x))) − ∑_{W ∈P (U)} \frac{[W:Z]}{p[U:Z]}φ_{W} (η_{W} (β_{W} (log(x))))

= log(x_{U}) − \frac{1}{p[U:Z]}φ_{Z}(log(x_{Z})) − ∑_{W ∈P (U)} \frac{[W:Z]}{p[U:Z]}φ_{W} (η_{W} (log(x_{W})))

= log(x_{U}) − \frac{1}{p[U:Z]}φ_{Z}(log(x_{Z})) − ∑_{W ∈P (U)} \frac{[W:Z]}{p^{2}[U:Z]}φ_{W} (log(α_{W} (x_{W})))

= \frac{1}{p^{2}[U:Z]}( log(x_{U}^{p^{2}[U:Z]}) − log(φ_{Z}(x_{Z}^{p})) − ∑_{W ∈P (U)} log(φ_{W} (α_{W} (x_{W}))^{[W:Z]}))

= L((x_{U})_{U}) = L(θ(x)) .

□

For any V ∈ C(G, Z) we let P_{c}(V) be the set of all W ∈ C(G, Z) such that V ⊆ W and [W : V] = p. We claim that

(M4) α_{V} (x_{V}) − ∏_{W ∈P_{c}(V)} φ_{W} (α_{W} (x_{W})) ∈ p im(σ_{V}) for any V ∈ C(G, Z).

holds true. We repeat that the image of the homomorphism φ_{W}, by construction, is contained in the subring Λ(V). Hence the left hand side of (M4) indeed lies in Λ(V).

Lemma 4.5. Let (y_{U})_{U} ∈ ∏_{U∈C(G,Z)} Λ(U)^{×} and V ∈ C(G, Z); then: i. We have

η_{V} (\frac{1}{p^{2}|V/Z|} log (\frac{y_{V}^{p^{2}|V/Z|}}{φ_{Z}(y_{Z}^{p}) ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|}} )) =

\frac{1}{p} log (\frac{α_{V} (y_{V})}{∏_{W ∈P_{c}(V)} φ_{W} (α_{W} (y_{W}))});

ii. if (y_{U})_{U} satisfies (M2) then the following assertions are equivalent: a. α_{V} (y_{V}) − ∏_{W ∈P_{c}(V)} φ_{W} (α_{W} (y_{W})) ∈ p im(σ_{V});

b. \frac{α_{V} (y_{V} )}{∏_{W ∈Pc(V )} φ_{W} (α_{W} (y_{W}))} ∈ 1 + p im(σ_{V});

c. log(\frac{α_{V} (y_{V} )}{∏_{W ∈Pc(V )} φ_{W} (α_{W} (y_{W}))}) ∈ p im(σ_{V}).

Proof. i. We check that

η_{V} (\frac{1}{|V/Z|} log(\frac{y_{V}^{p|V/Z|}}{φ_{Z}(y_{Z})})) = log(α_{V} (y_{V}))

and

η_{V} (\frac{1}{p|V/Z|} log( ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|})) =

log( ∏_{W ∈P_{c}(V)} φ_{W} (α_{W} (y_{W})))

hold true. In case V = Z we have η_{Z} = id_{Λ(Z)}, α_{Z}(y_{Z}) = \frac{y_{Z}^{p}}{φ_{Z}(y_{Z})} and P (Z) = P_{c}(Z) which makes the two identities obvious. If V ≠ Z we may use Lemma 4.1 and further reduce to the identities

\frac{1}{p|V/Z|} log(\frac{α_{V} (y_{V})^{p|V/Z|}}{α_{V} (φ_{Z}(y_{Z}))} ) = log(α_{V} (y_{V}))

and

∑_{W ∈P (V)} \frac{|W/Z|}{p|V/Z|}η_{V} (φ_{W} (η_{W} (log(y_{W})))) = ∑_{W ∈P_{c}(V)} φ_{W} (η_{W} (log(y_{W}))) .

The first one follows from α_{V} ∘ φ_{Z} = 1 and the second one from

η_{V} ∘ φ_{W} ∘ η_{W} = \begin{cases}φ_{W} ∘ η_{W} & if W ∈ P_{c}(V), \\ 0 & if W ∈ P (V) \ P_{c}(V). \end{cases}

These latter equations are straightforward from the definitions.

ii. We begin by observing that α_{V} (y_{V}) and ∏_{W ∈P_{c}(V)} φ_{W} (α_{W} (y_{W})) both are units in 1 + pΛ(V). Hence the fraction in b. lies in 1 + pΛ(V) so that its logarithm in c. is defined and contained in pΛ(V) (recall that we assume p ≠ 2). Since p im(σ_{V}) is an ideal in Λ(V)^{W (V)} it suffices, for the equivalence of a. and b., to see that ∏_{W ∈P_{c}(V)} φ_{W} (α_{W} (y_{W})) ∈ Λ(V)^{W (V)}. But this follows from (M2) since any g ∈ W (V) permutes the elements of the set P_{c}(V). For the equivalence of b. and c. it suffices to show that the isomorphisms

\xymatrix{1 + pΛ(V) \ar[r]^{log} & pΛ(V) \ar[l]^{exp}}

restrict to isomorphisms

\xymatrix{1 + p im(σ_{V}) \ar[r]^{log} & p im(σ_{V}) . \ar[l]^{exp}}

Consider

log(1 + pσ_{V} (f)) = ∑_{i≥1}(−1)^{i−1} \frac{p^{i}σ_{V} (f)^{i}}{i}

and

exp(pσ_{V} (f)) = 1 + ∑_{i≥1} \frac{p^{i}σ_{V} (f)^{i}}{i!} .

Since im(σ_{V}) is an ideal in Λ(V)^{W (V)} we have σ_{V} (f)^{i} ∈ im(σ_{V}) for any i ≥ 1. Moreover, \frac{p^{i}}{i!} ∈ pℤ_{p}. Hence each summand in the above two series lies in p im(σ_{V}). As the latter is closed in Λ(V) we conclude that

log(1 + pσ_{V} (f)) ∈ p im(σ_{V}) and exp(pσ_{V} (f)) ∈ 1 + p im(σ_{V}) .

□

Prop. 4.4 implies that (L_{V} ((x_{U})_{U}))_{V} satisfies (A1) and that

L_{V} ((x_{U})_{U}) ∈ im(σ_{V}) for any V ∈ C(G, Z).

Using Remark 3.8.ii.b and Lemma 4.5.i we deduce that

pη_{V} (L_{V} ((x_{U})_{U})) = log(\frac{α_{V} (x_{V})}{∏_{W ∈P_{c}(V)} φ_{W} (α_{W} (x_{W}))}) ∈ p im(σ_{V})

for any V ∈ C(G, Z). Hence Lemma 4.5.ii implies that (M4) holds true for (x_{U})_{U} = θ(x).

Let Φ := Φ_{Z}^{G} ⊆ ∏_{Z⊆U⊆G} Λ(U^{ab})^{×} be the subgroup of all elements (x_{U})_{U} which satisfy, for all U ⊆ V in S(G, Z), the conditions

ν_{U}^{V} (x_{V}) = π_{U}^{V} (x_{U}) if [V, V] ⊆ U, (M1)

x_{gUg^{−1}} = gx_{U}g^{−1} for any g ∈ G, (M2)

(M3) ver_{U}^{V} (x_{V}) − x_{U} ∈ im(σ_{U}^{V}) if [V : U] = p, and

α_{U}(x_{U}) − ∏_{W ∈P_{c}(U)} φ_{W} (α_{W} (x_{W})) ∈ p im(σ_{U}) if U ∈ C(G, Z). (M4)

So far we have established that

im(θ) ⊆ Φ .

Lemma 4.6. L(Φ) ⊆ Ψ.

Proof. Let (y_{U})_{U} ∈ Φ and put (a_{V})_{V} := L((y_{U})_{U}). We already know from Lemma 4.2 that (a_{V})_{V} ∈ Ψ_{ℚ_{p}}. Lemma 4.5 and (M4) imply that

η_{V} (a_{V}) ∈ im(σ_{V}) for any V ∈ C(G, Z).

At this point we have to go back to Remark 3.8.ii.a and observe that the identity there equally holds for elements in ∏_{V ∈C(G,Z)} Λ(V)⊗_{ℤ_{p}}ℚ_{p} satisfying (A1). We check by induction with respect to the order of V/Z that

a_{V} ∈ im(σ_{V}) for any V ∈ C(G, Z).

For V = Z we have η_{Z} = id and the claim is trivial. Let V ≠ Z and let Z ⊆ W ⊆ V be the unique subgroup of index p. by the induction hypothesis we have a_{W} ∈ im(σ_{W}). Remark 3.8.i then implies that \frac{1}{p}a_{W} ∈ im(σ_{V}). Hence the identity a_{V} = η_{V} (a_{V}) + \frac{1}{p}a_{W} shows that a_{V} ∈ im(σ_{V}). We now may apply Remark 3.7 to see that (a_{V})_{V} ∈ Ψ. □

It follows that we have the commutative diagram

\xymatrix{K_{1}(Λ(G)) \ar[d]_{θ} \ar[r]^{L} & O[[Conj(G)]] \ar[d]^{β}_{≅} \\ Φ \ar[r]^{L} & Ψ}

where the right perpendicular arrow, assuming also (H3), is an isomorphism by Thm. 3.6. We define

SK_{1}(Λ(G)) := ker (K_{1}(Λ(G)) ⟶ K_{1}(Λ^{∞}(G)))

and

K′_{1}(Λ(G)) := K_{1}(Λ(G))/SK_{1}(Λ(G)) .

For any open subgroup U ⊆ G we consider the diagram

\xymatrix{K_{1}(Λ(G)) \ar[d]_{N_{U}^{G}} \ar[r] & K_{1}(Λ^{∞}(G)) \ar[d]^{N_{U}^{G}} \\ K_{1}(Λ(U)) \ar[d] \ar[r] & K_{1}(Λ^{∞}(U)) \ar[d] \\ K_{1}(Λ(U^{ab})) \ar[r] & K_{1}(Λ^{∞}(U^{ab})) \\ Λ(U^{ab})^{×} \ar[u]^{≅} \ar[r] & Λ^{∞}(U^{ab})^{×}. \ar[u]_{≅}}

The uppermost square commutes since

Λ(G) ⊗_{Λ(U)} Λ^{∞}(U) = Λ^{∞}(G) .

The two lower squares commute for trivial reasons. The indicated isomorphism on the right hand side is a special case of [SV2] Prop. 3.1. Since the lowermost horizontal arrow visibly is injective we conclude that

SK_{1}(Λ(G)) ⊆ ker(θ_{U}) for any Z ⊆ U ⊆ G.

Hence θ factorizes through a homomorphism

θ : K′_{1}(Λ(G)) ⟶ Φ .

According to [SV2] Cor. 3.2 we have

SK_{1}(Λ(G)) = \underleftarrow{lim} SK_{1}(O[G/N]) .

It therefore follows from [Oli] Thm. 6.6 and Thm. 7.3 that SK_{1}(Λ(G)) also lies in the kernel of the integral logarithm L and that, more precisely, we have the exact sequence

(5)  1 → μ(O) × G^{ab} ⟶^{ι} K′_{1}(Λ(G)) ⟶^{L} O[[Conj(G)]] ⟶^{ω} G^{ab} → 1

where μ(O) ⊆ O^{×} denotes the subgroup of all roots of unity, ι is the obvious map, and ω is given by

ω(a[g]_{G}) := g^{Tr_{O/ℤp}(a)}[G, G] for any a ∈ O and g ∈ G.

Let us now contemplate the commutative diagram

\xymatrix{1 \ar[r] & μ(O) × G^{ab} \ar[d]_{=} \ar[r]^{ι} & K′_{1}(Λ(G)) \ar[d]_{θ} \ar[r]^{L} & O[[Conj(G)]] \ar[d]_{β}^{≅} \ar[r]^{ω} & G^{ab} \ar[d]_{=} \ar[r] & 1 \\ 1 \ar[r] & μ(O) × G^{ab} \ar[r]^{θ∘ι} & Φ \ar[r]^{L} & Ψ \ar[r]^{ω∘β^{−1}} & G^{ab} \ar[r] & 1.}

Our goal is to show that the lower row is exact as well, which then implies that θ is an isomorphism. Clearly θ_{G} ∘ι : μ(O)×G^{ab} ⟶ Λ(G^{ab})^{×} is injective. Hence θ ∘ ι is injective, and satisfies im(θ ∘ ι) ⊆ ker(L). For trivial reasons ω ∘ β^{−1} is surjective with im(L) ⊇ ker(ω ∘ β^{−1}). It therefore remains to establish the following two facts:

a. ker(L|Φ) ⊆ im(θ ∘ ι);
b. im(L|Φ) ⊆ ker(ω ∘ β^{−1}).

But first of all we observe that, if G is abelian, then θ_{G} and consequently by (M1a), also θ : K_{1}(Λ(G)) ⟶^{≅} Φ are isomorphisms. In particular, a. and b. hold in this case.

Lemma 4.7. For (y_{U})_{U} ∈ Φ we have L_{G}((y_{U})_{U}) = \frac{1}{p} log(\frac{y_{G}^{p}}{φ_{G^{ab}}(y_{G})}).

Proof. As a consequence of Lemmas 3.3 and 3.4 any (a_{U})_{U} ∈ Ψ satisfies

a_{G} = ∑_{V ∈C(G,Z)} \frac{1}{[G:V]}η_{V} (a_{V}) in Λ(G^{ab}).

Hence

L_{G}((y_{U})_{U}) = ∑_{V ∈C(G,Z)} \frac{1}{[G:V]}η_{V} (L_{V} ((y_{U})_{U})) in Λ(G^{ab}),

and, inserting the definition and using Lemma 4.5.i, we obtain

L_{G}((y_{U})_{U}) = ∑_{V ∈C(G,Z)} \frac{1}{p|G/V |} log (_{\frac{α_{V} (y_{V} )}{∏_{W ∈Pc(V )} φ_{W} (α_{W} (y_{W}))}})

= \frac{1}{p|G/Z|} log (_{\frac{α_{Z}(y_{Z})}{∏_{|W/Z|=p} φ_{W} (α_{W} (y_{W}))}})

+ ∑_{V ∈C(G,Z),V ≠Z} \frac{1}{p|G/V |} log (_{\frac{α_{V} (y_{V} )}{∏_{W ∈Pc(V )} φ_{W} (α_{W} (y_{W}))}})

= \frac{1}{p^{2}|G/Z|} log (_{\frac{α_{Z}(y_{Z})^{p}}{∏_{|W/Z|=p} φ_{W} (α_{W} (y_{W}))^{p}}} · ∏_{V ∈C(G,Z),V ≠Z} _{\frac{α_{V} (y_{V} )^{p|V/Z|}}{∏_{W ∈Pc(V )} φ_{W} (α_{W} (y_{W}))^{p|V/Z|}}} )

= \frac{1}{p^{2}|G/Z|} log (\frac{∏_{W ∈C(G,Z)} α_{W} (y_{W})^{p|W/Z|}}{∏_{W ∈C(G,Z),W ≠Z} φ_{W} (α_{W} (y_{W}))^{|W/Z|}} )

emphasizing that the computation takes place in Λ(G^{ab}). Comparing this to the definition

L_{G}((y_{U})_{U}) = \frac{1}{p^{2}|G/Z|} log (\frac{y_{G}^{p^{2}|G/Z|}}{φ_{Z}(y_{Z}^{p}) ∏_{W ∈C(G,Z),W ≠Z} φ_{W} (α_{W} (y_{W}))^{|W/Z|}} )

leads to the identity

log \frac{y_{G}^{p^{2}|G/Z|}}{φ_{Z}(y_{Z}^{p})} = log ∏_{W ∈C(G,Z)} α_{W} (y_{W})^{p|W/Z|}

= log (\frac{y_{Z}^{p^{2}}}{φ_{Z}(y_{Z}^{p})} · ∏_{W ∈C(G,Z),W ≠Z} α_{W} (y_{W})^{p|W/Z|})

and hence to

log y_{G}^{p|G/Z|} = log (y_{Z}^{p} · ∏_{W ∈C(G,Z),W ≠Z} α_{W} (y_{W})^{|W/Z|})

where, we repeat, all factors in the argument of log are viewed in Λ(G^{ab}). We therefore may apply the ring homomorphism φ_{G^{ab}}, and we get

log φ_{G^{ab}}(y_{G})^{p|G/Z|} = log (φ_{Z}(y_{Z}^{p}) · ∏_{W ∈C(G,Z),W ≠Z} φ_{W} (α_{W} (y_{W}))^{|W/Z|}) .

By inserting this back into the definition we finally arrive at

L_{G}((y_{U})_{U}) = \frac{1}{p^{2}|G/Z|} log (\frac{y_{G}^{p^{2}|G/Z|}}{φ_{G^{ab}}(y_{G})^{p|G/Z|}} ) = \frac{1}{p} log(\frac{y_{G}^{p}}{φ_{G^{ab}}(y_{G})}) .
□

Lemma 4.8. b. is satisfied.

Proof. Let (y_{U})_{U} ∈ Φ. Using Lemma 4.7 we compute

(ω ∘ β^{−1})(L((y_{U})_{U})) = ω(L_{G}((y_{U})_{U})) = \frac{1}{p} log(\frac{y_{G}^{p}}{φ_{G^{ab}}(y_{G})})
= ω(L_{G^{ab}}(y_{G})) = 1 ,

where the last identity holds by the exact sequence (5) for the group G^{ab}. □

Lemma 4.9. a. is satisfied.

Proof. Let (x_{U})_{U} be in ker(L|Φ). This in particular means by Lemma 4.5.i that

log (\frac{α_{U}(x_{U})}{∏_{V ∈P_{c}(U)} φ_{V} (α_{V} (x_{V}))}) = 0

for any U ∈ C(G, Z). Since the logarithm is taken of an element in 1+pΛ(U) we deduce that

α_{U}(x_{U}) = ∏_{V ∈P_{c}(U)} φ_{V} (α_{V} (x_{V}))

for any U ∈ C(G, Z). If U maximal in C(G, Z) then P_{c}(U) is empty and hence α_{U}(x_{U}) = 1. By downward induction we obtain

(6)  α_{U}(x_{U}) = 1  for any U ∈ C(G, Z).

In particular, for U = Z we get

\frac{x_{Z}^{p}}{φ_{Z}(x_{Z})} = α_{Z}(x_{Z}) = 1 .

But Z is abelian. In this case the integral logarithm is

L_{Z} : K_{1}(Λ(Z)) ⟶ Λ(Z)
x ⟼ \frac{1}{p} log (\frac{x^{p}}{φ_{Z}(x)}) .

We see that x_{Z} ∈ ker(L_{Z}) = μ(O) × Z ⊆ K_{1}(Λ(Z) by (5). Next suppose that [U : Z] = p. Then, using (M1a), we get

x_{U}^{p} = N_{Z}^{U}(x_{U}) = x_{Z} ∈ μ(O) × Z ⊆ μ(O) × U ⊆ K_{1}(Λ(U)) .

From (5) for L_{U} we see that Λ(U)^{×}/(μ(O) × U) is torsion free. We conclude that x_{U} ∈ μ(O) × U. Again by induction it then follows that

x_{U} ∈ μ(O) × U ⊆ K_{1}(Λ(U)) for any U ∈ C(G, Z).

Similarly as above we have the relation

\frac{x_{U}^{p^{2}|U/Z|}}{∏_{W ∈P (U)} φ_{W} (α_{W} (x_{W}))^{|W/Z|}} = 1

for all U in S(G, Z). But by (6) the denominator is equal to one, i. e., x_{U}^{p^{2}|U/Z|} = 1. Again by (5) for L_{U^{ab}} we see that the only torsion of Λ(U^{ab})^{×} is contained in μ(O) × U^{ab}. Hence x_{U} = 1. We now have established that

(x_{U})_{U} ∈ ∏_{U∈S(G,Z)} (μ(O) × U^{ab}) ∩ Φ .

At this point we need the condition (M3). We claim that it implies

x_{U} = ver_{U}^{G}(x_{G}) for any U ∈ S(G, Z).

Indeed, let first U be of index p in G and assume that for g, h ∈ U^{ab} and ζ, ξ ∈ μ(O) the element ζg−ξh lies in im(σ_{U}^{G}). Then, under the augmentation map ϵ of Λ(U^{ab}) we have that ϵ(ζg − ξh) = ζ − ξ ∈ ϵ(im(σ_{U}^{G})) = |G/U|O is divisible by p. Since ζ and ξ are roots of unity in an unramified extension of ℤ_{p} this implies that ζ = ξ. As ζ is a unit in O it follows that g − h lies in im(σ_{U}^{G}) and therefore is, as any element in im(σ_{U}^{G}), invariant under the conjugation action of G/U. We choose a set of representatives R ⊆ U^{ab} of the cosets in U/Z[U, U] and write

g = z_{g}u_{g} , h = z_{h}u_{h}, and g − h = σ_{U}^{G}(∑_{u∈R} a_{u}u)

with u_{g}, u_{h} ∈ R, z_{g}, z_{h} ∈ Z[U, U]/[U, U], and a_{u} ∈ Λ(Z). On the one hand the invariance of g − h implies that u_{g} and u_{h} are both invariant as p ≠ 2. On the other hand any τ ∈ G/U induces a permutation τ of R such that τuτ^{−1} = b_{τ,u}τ(u) with b_{τ,u} ∈ Z for any u ∈ R. We compute

z_{g}u_{g} − z_{h}u_{h} = g − h = σ_{U}^{G}(g − h) = ∑_{u∈R} ∑_{τ∈G/U} a_{u}τuτ^{−1}

= ∑_{u∈R} ∑_{τ∈G/U} a_{u}b_{τ,u}τ(u)

= ∑_{u∈R}( ∑_{τ∈G/U} a_{τ^{−1}(u)}b_{τ,τ^{−1}(u)})u .

Since τ(u_{g}) = u_{g}, τ(u_{h}) = u_{h}, and b_{τ,u_{g}} = b_{τ,u_{h}} = 1 for any τ it follows that

z_{g} = [G : U]a_{u_{g}} = pa_{u_{g}} and z_{h} = [G : U]a_{u_{h}} = pa_{u_{h}}

provided u_{g} ≠ u_{h}. Since this is not possible we must have u := u_{g} = u_{h}. But then z_{g} − z_{h} = pa_{u} which implies z_{g} = z_{h} and hence g = h. This proves our claim for U of index p in G. The general case then follows by induction. This together with (1) shows that (x_{U})_{U} ∈ im(θ ∘ ι). □

Theorem 4.10. Assuming (H1) - (H4) the map θ : K′_{1}(Λ(G)) ⟶^{≅} Φ_{Z}^{G} is an isomorphism.

5. The integral logarithm for B(G)

In this section we will construct an extension of the integral logarithm L of Oliver and Taylor to K_{1}(B(G)). We assume (H2), (H4), and that H is finite, which implies (H1). The ring B(G) being local we have the surjection B(G)^{×} ↠ K_{1}(B(G)). The usual computation for the convergence of the logarithm series

log(x) = Log(1 + x) = ∑_{n≥1} \frac{(−1)^{n+1}}{n} x^{n}

shows that it induces a homomorphism

log : 1 + Jac(B(G)) ⟶ (B(G)/[B(G), B(G)]) ⊗_{ℤ_{p}} ℚ_{p} .

The additional point to note is that the ideal Jac(B(G))/pB(G) is nilpotent in B(G)/pB(G); in particular, the denominators appearing in the image of this map are bounded.

On the other hand, as φ_{Z}(S(Z)) ⊆ S(Z) we obtain a unique extension φ_{Z} : A(Z) ⟶ A(Z) of φ_{Z}. It respects Jac(A(Z)) and therefore further extends to a homomorphism of rings φ_{Z} : B(Z) ⟶ B(Z), which modulo p induces the endomorphism of B(Z)/pB(Z) ≅ O/(p)[[Z]] which sends f ↦ f^{p}. Furthermore, for any subgroup U ∈ S(G, Z) we may use the identification B(U)/[B(U), B(U)] = B(Z)⊗_{Λ(Z)}O[[Conj(U)]] from Lemma 2.2.ii to extend the earlier map φ_{U} to the map

φ_{U} : B(U)/[B(U), B(U)] ⟶ B(U)/[B(U), B(U)]
z ⊗ f ⟼ φ_{Z}(z) ⊗ φ_{U}(f) .

If U is abelian then φ_{U} is again a ring endomorphism of B(U) which modulo p coincides with the map f ↦ f^{p}.

Lemma 5.1. The group B(G)^{×}/[1+pB(G)]·B(Z′)^{×}, for any open subgroup Z′ ⊆ G_{0}, is annihilated by p^{ℓ} for a sufficiently large ℓ ∈ ℕ.

Proof. Let I_{B} := ker(B(G) ⟶ B(Γ)). We also fix a section σ : Γ → G of the projection map G → Γ. It induces a section B(Γ) ⟶^{≅} B(σ(Γ)) ⊆ B(G) of the ring homomorphism B(G) ⟶ B(Γ). Hence

(7)  B(G)^{×} = [1 + I_{B}] · B(σ(Γ))^{×} = [1 + Jac(B(G))] · B(σ(Γ))^{×} .

If Jac(B(G))^{p^{ℓ}} ⊆ pB(G) then [1 + Jac(B(G))]^{p^{ℓ}} ⊆ 1 + pB(G). On the other hand, the subgroup σ(Γ) ∩ Z′ is open and hence contains σ(Γ)^{p^{ℓ}} for some sufficiently large ℓ ∈ ℕ. It follows that φ_{σ(Γ)}^{ℓ}(σ(Γ)) ⊆ Z′ and therefore that

φ_{σ(Γ)}^{ℓ}(B(σ(Γ))^{×}) ⊆ B(Z′)^{×} .

Since φ_{σ(Γ)}^{ℓ}(x) ≡ x^{p^{ℓ}} mod 1 + pB(σ(Γ)) for any x ∈ B(σ(Γ))^{×} we conclude that

B(σ(Γ))^{×p^{ℓ}} ⊆ [1 + pB(σ(Γ))] · B(Z′)^{×} ⊆ [1 + pB(G)] · B(Z′)^{×} .

□

We now define the homomorphism

[1 + Jac(B(G))]B(G_{0})^{×} ⟶ (B(G)/[B(G), B(G)]) ⊗_{ℤ_{p}} ℚ_{p}
x ⟼ \begin{cases}log(x) − \frac{1}{p}φ_{G}(log(x)) & if x ∈ 1 + Jac(B(G)), \\ \frac{1}{p} log(\frac{x^{p}}{φ_{G_{0}}(x)}) & if x ∈ B(G_{0})^{×}. \end{cases}

This is well defined since:

– \frac{x^{p}}{φ_{G_{0}}(x)} ∈ 1 + pB(G_{0}) for x ∈ B(G_{0})^{×}.

– \frac{1}{p} log(\frac{x^{p}}{φ_{G_{0}}(x)}) = log(x)− \frac{1}{p}φ_{G}(log(x)) for x ∈ [1+Jac(B(G))]∩B(G_{0})^{×}.

As a consequence of Lemma 5.1 and the unique divisibility of the target this map extends uniquely to B(G)^{×} and induces a natural homomorphism

L_{B} := L_{B(G)} : K_{1}(B(G)) ⟶ (B(G)/[B(G), B(G)]) ⊗_{ℤ_{p}} ℚ_{p} .

Proposition 5.2. L_{B} has image in B(G)/[B(G), B(G)].

Proof. First of all we note that B(G)/[B(G), B(G)] by Lemma 2.2.ii is ptorsion free so that the statement makes sense.

For x ∈ B(σ(Γ))^{×} we have that L_{B}(x) is the image in B(G)/[B(G), B(G)] of

\frac{1}{p} log(\frac{x^{p}}{φ_{σ(Γ)}(x)}) ∈ \frac{1}{p} log(1 + pB(σ(Γ))) ⊆ B(σ(Γ)) .

Therefore (7) reduces us to proving that

log(x) − \frac{1}{p}φ_{G}(log(x)) ∈ B(G)/[B(G), B(G)] for any x ∈ 1 + Jac(B(G)).

This is done by arguments completely analogous to the case of the integral logarithm of Oliver and Taylor (cf. [Oli] Chap. 6 or [CR] §54). □

Lemma 5.3. If i : Λ(G) → B(G) denotes the inclusion map then the diagram

\xymatrix{K_{1}(Λ(G)) \ar[d]_{L_{G}} \ar[r]^{K_{1}(i)} & K_{1}(B(G)) \ar[d]^{L_{B(G)}} \\ O[[Conj(G)]] \ar[r]^{i} & B(G)/[B(G), B(G)]}

commutes.

Proof. If I := ker(Λ(G) ⟶ Λ(Γ)) then we have, as in the proof of Lemma 5.1, that Λ(G)^{×}/[1 + I] · Λ(G_{0})^{×} is annihilated by some p^{ℓ}. But

L_{B(G)}|[1 + I] = i ∘ L_{G}|[1 + I]

by the very definitions and

L_{B(G_{0})}|Λ(G_{0}) = L_{G_{0}}

since G_{0} is commutative. □

Next we define

SK_{1}(A(G)) := image of SK_{1}(Λ(G)) in K_{1}(A(G)),
SK_{1}(B(G)) := image of SK_{1}(Λ(G)) in K_{1}(B(G))

and

K′_{1}(A(G)) := K_{1}(A(G))/SK_{1}(A(G)),
K′_{1}(B(G)) := K_{1}(B(G))/SK_{1}(B(G)).

Lemma 5.3 implies that L_{B(G)} induces a homomorphism

L_{B} = L_{B(G)} : K′_{1}(B(G)) ⟶ B(G)/[B(G), B(G)] .

6. The multiplicative theory, part 2

Throughout this section we assume (H1) - (H4) and, in fact, that H is finite. We choose Z small enough so that it is torsion free (i. e., Z ≅ ℤ_{p}). We shall study the analogs

θ_{A} := (θ_{Z}^{G})_{A} : K_{1}(A(G)) ⟶ ∏_{U∈S(G,Z)} A(U^{ab})^{×}

and

θ_{B} := (θ_{Z}^{G})_{B} : K_{1}(B(G)) ⟶ ∏_{U∈S(G,Z)} B(U^{ab})^{×}

of θ, where we abbreviate

A := A(G) := Λ(G)_{S(G)}

and as before

B := B(G) := { \atop Λ(G)_{S}} .

Since by Prop. 1.3 we have

A(·) = A(Z) ⊗_{Λ(Z)} Λ(·)  and  B(·) = B(Z) ⊗_{Λ(Z)} Λ(·)

the norm maps N_{U}^{V} extend naturally and one immediately checks that the analogs

(θ_{A})_{U} : K_{1}(A(G)) ⟶^{N_{U}^{G}} K_{1}(A(U)) ⟶ K_{1}(A(U^{ab})) = A(U^{ab})^{×}

and similarly (θ_{B})_{U} of θ_{U} are defined and induce a commutative diagram

\xymatrix{K_{1}(Λ(G) \ar[d] \ar[r]^{θ_{U}} & Λ(U^{ab})^{×} \ar@{^{(}->}[d] \\ K_{1}(A(G)) \ar[d] \ar[r]^{(θ_{A})_{U}} & A(U^{ab})^{×} \ar@{^{(}->}[d] \\ K_{1}(B(G)) \ar[r]^{(θ_{B})_{U}} & B(U^{ab})^{×}}

which implies an analogous diagram for θ, θ_{A}, and θ_{B}.

Let U ⊆ V be subgroups in S(G, Z). As Z ⊆ U^{ab} we have A(U^{ab}) = A(Z)⊗_{Λ(Z)} Λ(U^{ab}) and similarly for B(U^{ab}). It follows easily that the maps π_{U}^{V}, ν_{U}^{V}, and σ_{U}^{V} extend naturally to the rings A(·) and B(·). In particular, α_{Z} extends to a map α_{Z} : B(Z)^{×} → 1 + pB(Z) ⊆ B(Z)^{×} sending again f to \frac{f^{p}}{φ_{Z}(f)}. If U ∈ C(G, Z) differs from Z then the reasoning in the proof of [SV2] Prop. 2.3 still works and shows that the map

α_{U} : B(U)^{×} ⟶ B(U)^{×}
f ⟼ \frac{f^{p}}{N_{U′}^{U}(f)} ,

which extends the earlier such map, has image in 1 + pB(U).

If [V : U] = p^{n} then the Verlagerung satisfies

ver_{U}^{V} (zḡ) = z^{p^{n}} ver_{U}^{V} (ḡ) = φ_{Z}^{n}(z) ver_{U}^{V} (ḡ)

for any z ∈ Z and ḡ ∈ U^{ab}. Hence we may extend the earlier ring homomorphism ver_{U}^{V} : Λ(V^{ab}) ⟶ Λ(U^{ab}) to the ring homomorphism

B(V^{ab}) = B(Z) ⊗_{Λ(Z)} Λ(V^{ab}) ⟶ B(U^{ab}) = B(Z) ⊗_{Λ(Z)} Λ(U^{ab})
z ⊗ f ⟼ φ_{Z}^{n}(z) ⊗ ver_{U}^{V} (f) .

It maps A(V^{ab}) into A(U^{ab}).

We now may define the subsets

Φ_{A} = (Φ_{Z}^{G})_{A} ⊆ ∏_{U∈S(G,Z)} A(U^{ab})^{×}

and

Φ_{B} = (Φ_{Z}^{G})_{B} ⊆ ∏_{U∈S(G,Z)} B(U^{ab})^{×}

by the corresponding conditions (M1) - (M4), and we obtain the obvious inclusions

Φ ⊆ Φ_{A} ⊆ Φ_{B} .

For the purpose of proving the main conjecture the following result is essential.

Theorem 6.1. i. The image of θ_{A} is contained in Φ_{A}.

ii. Within ∏_{U∈S(G,Z)} A(U^{ab})^{×} we have

Φ_{A} ∩ ∏_{U∈S(G,Z)} Λ(U^{ab})^{×} = Φ = im(θ) .

Note that Thm. 6.1.ii follows trivially from the definition of Φ and Φ_{A} as well as from Thm. 4.10. While one checks that the image of θ_{A} satisfies (M1)

- (M3) for the same reasons as im(θ), condition (M4) would again require an (integral) logarithm, which due to convergence issues is not available for A(G). This is the only reason to consider also θ_{B} and Φ_{B} in the following! Thus we are going to prove the following variant

Theorem 6.2. i. The image of θ_{B} is contained in Φ_{B}.

ii. Within ∏_{U∈S(G,Z)} B(U^{ab})^{×} we have

Φ_{B} ∩ ∏_{U∈S(G,Z)} A(U^{ab})^{×} = Φ_{A} .

Again, Thm. 6.2.ii holds for trivial reasons. Moreover, Thm. 6.2.i clearly implies Thm. 6.1.i. It is straightforward to check (by the same arguments as before) that the image of θ_{B} satisfies the conditions (M1) - (M3). In particular, upon replacing (2) by

B(V) ≅ ⨁_{i=0}^{p−1} B(U)g^{i}

and using also

B(U) ≅ ⨁_{h∈U/Z} B(Z)h

the proof for (M3) is literally the same as before.

Before we discuss the condition (M4) we would like to point out the following conclusion.

Corollary 6.3. For any compact p-adic Lie group G and O the ring of integers of a finite unramified extension of ℚ_{p} the canonical maps

K′_{1}(Λ(G)) ↪ K′_{1}(A(G))  and  K′_{1}(Λ(G)) ↪ K′_{1}(B(G))

are injective.

Proof. Assuming first that G = G is pro-p of dimension one, the claim follows from the following commutative diagram

\xymatrix{K′_{1}(Λ(G)) \ar@{^{(}->}[d]_{θ} \ar[r] & K′_{1}(A(G)) \ar[d]^{θ_{A}} \\ Φ \ar@{^{(}->}[r] & Φ_{A}.}

In the general pro-p case we may express Λ(G) as an inverse limit of Iwasawa algebras of groups G of dimension one which by [FK] Prop. 1.5.1 and [SV2] Cor. 3.2 gives rise to a commutative diagram

\xymatrix{K′_{1}(Λ(G)) \ar[d] \ar[r]^{≅} & \underleftarrow{lim} K′_{1}(Λ(G)) \ar@{^{(}->}[d] \\ K′_{1}(A(G)) \ar[r] & \underleftarrow{lim} K′_{1}(A(G))}

and to a corresponding one for K′_{1}(B(G)). The claim follows. For general G one uses the same reduction steps as discussed in [Su]. □

In order to show that the image of θ_{B} satisfies (M4) we need the commutativity of the diagram

(8)  \xymatrix{K_{1}(B(G)) \ar[d]_{θ_{B}} \ar[r]^{L_{B}} & B(G)/[B(G), B(G)] \ar[d]^{β_{B}} \\ [∏_{U∈S(G,Z)} B(U^{ab})^{×}]_{(M3a)} \ar[r]^{L_{B}} & ∏_{U∈S(G,Z)} B(U^{ab}) ⊗_{ℤ_{p}} ℚ_{p},}

where […]_{(M3a)} indicates, as before, the subgroup of elements satisfying (M3a) and where L_{B} = (L_{B,V})_{V ∈S(G,Z)} is defined in the same manner as earlier the map L:

L_{B,V} ((y_{U})_{U}) := \frac{1}{p^{2}|V/Z|} log (\frac{y_{V}^{p^{2}|V/Z|}}{φ_{Z}(y_{Z}^{p}) ∏_{W ∈P (V)} φ_{W} (α_{W} (y_{W}))^{|W/Z|}} ) .

By Λ(Z)-linearity the map η_{U} extends to a B(Z)-linear map

η_{U} : B(U) → B(U) .

Lemma 6.4. i. For any open subgroups U ⊆ V ⊆ G, we have the commutative diagram

[figure: \frac{1+pB(V)}{[1+pB(V),B(V)^{×}]}; N_{U}^{V}; log; B(V)/[B(V), B(V)] ⊗_{ℤ_{p}} ℚ_{p}; _{trV U}; \frac{1+pB(U)}{[1+pB(U),B(U)^{×}]}; log; B(U)/[B(U), B(U)] ⊗_{ℤ_{p}} ℚ_{p}.]

ii. For any U ≠ Z in C(G, Z) the diagram

\xymatrix{1 + pB(U) \ar[d]_{α_{U}} \ar[r]^{log} & B(U) ⊗_{ℤ_{p}} ℚ_{p} \ar[d]^{pη_{U}} \\ 1 + pB(U) \ar[r]^{log} & B(U) ⊗_{ℤ_{p}} ℚ_{p}}

is commutative.

Proof. ii. follows formally from i. in the same way as Lemma 4.1 follows from (4). For i. one has to adapt the first step in the proof of [OT] Thm. 1.4. □

Remark 6.5. By B(Z)-(semi)linearity the identity in Lemma 4.3 remains valid for any f ∈ B(G)/[B(G), B(G)] ⊗_{ℤ_{p}} ℚ_{p} and any U ∈ S(G, Z).

To establish the commutativity of (8) we have to show that

L_{B}(θ_{B}(x)) = β_{B}(L_{B}(x)) for any x ∈ K_{1}(B(G))

holds true. Because of Lemma 5.1 it suffices to treat the two special cases x ∈ 1 + pB(G) and x ∈ B(Z)^{×}. In the first case the computation is formally the same as the computation in the proof of Prop. 4.4, now using (8), Lemma 6.4, and Remark 6.5 of course. Let therefore x ∈ B(Z)^{×}. Then N_{U}^{G}(x) = x^{|G/U|} for any U ∈ S(G, Z) and α_{W} (x) = \frac{x^{p}}{N_{W′}^{W}(x)} = \frac{x^{p}}{x^{p}} = 1 for any W ∈ P (U).

Hence we obtain θ_{B}(x) = (x^{|G/U|})_{U} and

L_{B}(θ_{B}(x)) = (\frac{1}{p^{2}|V/Z|} log (\frac{x^{p^{2}|G/Z|}}{φ_{Z}(x^{p|G/Z|}) ∏_{W ∈P (V)} φ_{W} (α_{W} (x))^{|G/Z|}} ))_{V}

= (\frac{|G/V |}{p} log (\frac{x^{p}}{φ_{Z}(x)}))_{V}

= (|G/V |L_{B}(x))_{V} = β_{B}(L_{B}(x)) .

To see that θ_{B}(x) satisfies (M4) we observe that due to Thm. 3.9 and the commutativity of (8) the element L_{B}(θ_{B}(x)) = β_{B}(L_{B}(x)) satisfies (A1) and (A3). We further note that the obvious analogs of Remark 3.8.ii.b and Lemma 4.5 remain true in the present setting. Hence we may argue exactly as we did earlier (before Lemma 4.6) for θ(x).

This finishes the proof of Theorems 6.2 and 6.1.

References

[CFKSV] Coates J., Fukaya T., Kato K., Venjakob O.: The GL_{2} main conjecture for elliptic curves without complex multiplication. Publ. Math. IHES 101, 163-208 (2005)

[CR] Curtis C., Reiner I.: Methods of representation theory, vol. II. New York: J. Wiley 1987

[DDMS] Dixon J.D., du Sautoy M.P.F., Mann A., Segal D.: Analytic Pro-p-Groups. Cambridge Univ. Press 1999

[FK] Fukaya T., Kato K.: A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory. Proc. St. Petersburg Math. Soc. 12, pp. 1-85, AMS Transl. Ser. 2, 219. Providence: American Math. Soc. 2006

[Hup] Huppert B.: Endliche Gruppen I. Springer 1967

[Kak] Kakde M.: The main conjecture of Iwasawa theory for totally real fields. Preprint 2011

[Oli] Oliver R.: Whitehead groups of finite groups. London Math. Soc. Lect. Notes 132. Cambridge Univ. Press 1988

[OT] Oliver R., Taylor L.R.: Logarithmic descriptions of Whitehead groups and class groups for p-groups. Memoirs AMS 392 (1988)

[SV1] Schneider P., Venjakob O.: Localizations and completions of skew power series rings. Amer. J. Math. 132, 1 - 36 (2010)

[SV2] Schneider P., Venjakob O.: A splitting for K_{1} of completed group rings. To appear in Comment. Math. Helvetici

[Su] Sujatha R.: Reductions of the main conjecture. This volume

Universität Münster, Mathematisches Institut, Einsteinstr. 62, 48291 Münster, Germany, http://www.uni-muenster.de/math/u/schneider/

   E-mail address: pschnei@uni-muenster.de

Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany, http://www.mathi.uni-heidelberg.de/˜ venjakob/

   E-mail address: venjakob@mathi.uni-heidelberg.de