GALOIS INVARIANTS OF K_{1}-GROUPS OF IWASAWA ALGEBRAS.

DMITRIY IZYCHEV AND OTMAR VENJAKOB

Introduction

Let S be an arbitrary ring with unit, then K_{1}(S) is defined by the exact sequence

1 → E(S) → GL(S) → K_{1}(S) → 1,

where GL(S) = lim_{→} GL_{n}(S) denotes the general linear group of S and E(S) denotes the subgroup of elementary matrices over S.

In [9, Rem. 3.4.6] Fukaya and Kato formulate the following expectation, which plays an important role in the definition of so called ε-isomorphisms concerning the local Iwasawa theory of ε-constants: For an adic ring Λ in the sense of 1.4.1 in (loc. cit.) the sequence

\xymatrix{0 \ar[r] & K_{1}(Λ) \ar[r] & K_{1}(Λ̃) \ar[r]^{1−φ_{∗}} & K_{1}(Λ̃) \ar[r] & 0}

should be exact, where Λ̃ := ℤ̂_{p}^{ur} ˆ⊗_{ℤ_{p}}Λ denotes the completed tensor product and φ denotes the Frobenius morphism acting on the ring of integers ℤ̂_{p}^{ur} of the p-adic completion ℚ̂_{p}^{ur} of the maximal unramified extension of ℚ_{p}. For any finite group G and Λ := ℤ_{p}[G] this amounts to the statement

i_{∗} : K_{1}(ℤ_{p}[G]) ≅ K_{1}(ℤ̂_{p}^{ur}[G])^{φ=id},

where i_{∗} is induced by the inclusion i : ℤ_{p} → ℤ̂_{p}^{ur}, and the Frobenius map φ acts coefficientwise on ℤ̂_{p}^{ur}[G] and hence on the K_{1}-group.

The original motivation of this note was to show Fukaya and Kato’s expectation in this specific case. A more general question would rather be whether the following statement

(0.1)  i_{∗} : K_{1}(S^{Δ}[G]) ≅ K_{1}(S[G])^{Δ},

Date: June 8, 2010.

1991 Mathematics Subject Classification. 19B28, 11S23.

Key words and phrases. algebraic K-group, group algebra, Iwasawa algebra, descent, Whitehead group.

We acknowledge support by the DAAD, ERC and DFG.

holds whenever S is a ring and Δ is a group acting on S by ring automorphisms. But surprisingly, it turns out, that neither of the above statements does hold in general (see 3.2). In this paper we restrict our attention to the case, where Δ is the Galois group of some algebraic field extension related to the extension of a specific class of rings S/S^{Δ} contained in the completion ℂ_{p} of \overline{ℚ_{p}}. We obtain partial results toward (a corrected version of) (0.1), see Theorem 2.45. In particular, we show that in general the following sequence is exact

\xymatrix{1 \ar[r] & SK_{1}(ℤ_{p}[G]) \ar[r] & K_{1}(ℤ_{p}[G]) \ar[r]^{i_{∗}} & K_{1}(ℤ̂_{p}^{ur}[G])^{φ=id} \ar[r] & 1}

and induces an isomorphism for the rational K-groups

K_{1}(ℤ_{p}[G])_{ℚ} ≅ K_{1}(ℤ̂_{p}^{ur}[G])_{ℚ}^{φ=id}.

If S is a finite algebraic extension of ℤ_{p} and SK_{1}(S[G]) = 1, the isomorphism (0.1) reduces to Galois descent of the determinantal image:

i_{∗} : Det(S^{Δ}[G]) ≅ Det(S[G])^{Δ}

as has been proved by M. Taylor in the case where S is unramified. But the case of infinite extensions of ℤ_{p} and infinite groups Δ seems not to be covered in the literature, not even by the fairly general recent treatment [4], where only finite group actions are considered, as was pointed out to us by M. Taylor. Actually one has to check that the techniques of integral group logarithms extend to this situation, either by extending Taylor’s original definition as pursued in (loc. cit.) or by using Snaith’ version in [24] - both in the case of p-groups- and then use standard induction techniques to reduce the general case of finite groups to it, as e.g. in [8]. Both approaches work and for the convenience of the reader we show that the methods of [4] extends easily to our setting recalling the main steps of their proof, but noting that for ramified extensions Snaith’s construction might be better adapted.

The reason for the defect in (0.1) relies on the surprising vanishing

SK_{1}(ℤ̂_{p}^{ur}[G]) = 1

for all finite groups G, in particular SK_{1} - in contrast to the Det-part - does not have good Galois descent in general, see Corollary 2.44 for a more precise statement.

The paper is organized as follows: In the first section we recall for the convenience of the reader Galois descent results for group rings with coefficients in local or global fields using Fröhlich’s Hom-description. In the second section, the heart of the paper, we first concentrate on descent results for the Det-part. In particular we obtain a rather general result not only for finite groups, but also for compact p-adic Lie groups and their Iwasawa algebras, which turns out to be quite useful in number theory, see [2]. Then we deal with the SK_{1}-part recalling and generalizing results from [15]. Altogether both parts lead to the desired descent description for K_{1}. Finally, in the last section we derive similar descent results over the corresponding residue class fields.

We are very grateful to Victor Snaith and Martin Taylor for various discussions and for informing us about the state of the art regarding the descent problem.

1. The case of “local” and “number” fields.

The goal of this section is to prove the following theorem which is certainly known to experts but for lack of a reference we treat it here, because it forms the prototype for the descent results in the integral cases later.

Theorem 1.1. Let L be a finite Galois extension of ℚ_{p} and M be either an arbitrary (possibly infinite) Galois extension M^{0} of ℚ_{p} or the p-adic completion of a Galois extension M^{0} of finite absolute ramification index over ℚ_{p}, so that ℚ_{p} ⊆ L ⊆ M ⊆ ℂ_{p}. Further, let Δ = Gal(M^{0}/L), such that M^{Δ} = L (see Remark 1.3). Let Γ be a finite group. Then

(1.1)  i_{∗} : K_{1}(L[Γ]) ≅ K_{1}(M[Γ])^{Δ},

where Δ acts on the K_{1}-group coefficientwise.

For this we need the following

Proposition 1.2. Let N be either an arbitrary (possibly infinite) algebraic extension of ℚ_{p} or the completion of an algebraic extension of finite absolute ramification index over ℚ_{p}. Let Γ be a finite group. Then

(i) The map i_{∗} : K_{1}(N[Γ]) → K_{1}(\overline{N}[Γ]) is injective, (ii) If N is a finite Galois extension of ℚ_{p} and G_{N} = Gal(\overline{N}/N) is the absolute Galois group, then

i_{∗} : K_{1}(N[Γ]) ≅ K_{1}(\overline{N}[Γ])^{G_{N}} .

Proof. The first statement is well known for the local fields, i.e. finite extensions of ℚ_{p}, and more generally for the perfect discrete valued fields (see Proposition 2.8 in [18]; Theorem 1 and the Remark after Theorem 2 in [13]). The infinite algebraic extensions can always be written as a direct limit of their finite subextensions. Since the direct limit is exact on the category of abelian groups and K_{1} commutes with the direct limit (see [21], Exercise 2.1.9), (i) is true for infinite algebraic extensions. This completes the proof of (i).

Let R_{Γ} = R_{Γ}(\overline{N}) denote the Grothendieck group of finitely generated \overline{N}[Γ]- modules. Alternatively R_{Γ} will be viewed as the group of virtual \overline{N}-valued characters of Γ. Since \overline{N} is algebraically closed, R_{Γ} is a free abelian group on the irreducible characters.

Using the Wedderburn’s decomposition of \overline{N}[Γ] we get an isomorphism of G_{N}- modules

(1.2)  K_{1}(\overline{N}[Γ]) ≅ ∏_{χ} \overline{N}^{×} ≅ Hom(R_{Γ}, \overline{N}^{×}),

where χ are irreducible \overline{N}-valued characters and the action of G_{N} on the Homgroup is given by the actions on R_{Γ} and on \overline{N}^{×} in the standard way

f^{g}(χ) = (f(χ^{g^{−1}}))^{g}, ∀f ∈ Hom(R_{Γ}, \overline{N}^{×}), g ∈ G_{N}, χ ∈ R_{Γ}.

From [25] (part 1, paragraph 2) we obtain the corresponding Hom-description for K_{1}(N[Γ])

(1.3)  K_{1}(N[Γ]) ≅ Hom_{G_{N}} (R_{Γ}, \overline{N}^{×}).

Then the second statement is obvious, as i_{∗} is a Galois homomorphism. □

Remark 1.3. We fix an embedding \overline{ℚ_{p}} ↪ ℂ_{p}. With the previous notation we have by the theorem of Ax-Sen-Tate

M^{Δ} = (M^{0})^{Δ} = L and O_{M}^{Δ} = (O_{M^{0}})^{Δ} = O_{L}.

In the following we will always use this theorem for the case of completions.

Now let L and M be as in Theorem 1.1. We introduce a commutative diagram

(1.4)  \begin{aligned}K_{1}(L[Γ]) & → & K_{1}(\overline{L}[Γ]) & ˜→ & ∏_{χ} \overline{L}^{×} \\ ↓i_{∗} & & ↓ & & ↓ \\ K_{1}(M[Γ]) & → & K_{1}(\overline{M}[Γ]) & ˜→ & ∏_{ψ} \overline{M}^{×}, \end{aligned}

where χ and ψ are irreducible characters of Γ with values in \overline{L}^{×} and \overline{M}^{×} respectively. The rows are injective by the Proposition 1.2. The right hand side column is injective as \overline{L}^{×} ⊆ \overline{M}^{×} and each ψ being a character of a finite group is the composition of one of the characters χ with the inclusion map i : \overline{L} → \overline{M}. It follows, that the left hand side column is also injective. Note also, that

R_{Γ}(\overline{L}) ≅ R_{Γ}(\overline{M}),

which we take as an identification henceforth.

Taking invariants under the action of Δ, which is a left exact functor, we obtain the first inclusion of Theorem 1.1

(1.5)  K_{1}(L[Γ]) ⊆ (K_{1}(M[Γ]))^{Δ}.

On the other hand, from the diagram (1.4) and the isomorphism (1.2) we deduce

(K_{1}(M[Γ]))^{Δ} = (K_{1}(M[Γ]))^{G_{L}} ⊆ (K_{1}(\overline{M}[Γ]))^{G_{L}} ≅
Hom_{G_{L}}(R_{Γ}(\overline{M}), \overline{M}^{×}) ≅ Hom_{G_{L}}(R_{Γ}(\overline{L}), \overline{L}^{×}).

The last isomorphism is given by Remark 1.3.

By the Proposition 1.2 (ii) the last Hom-group is equal to K_{1}(L[Γ]), thus we get the second inclusion of Theorem 1.1, and hence the isomorphism (1.1).

Remark 1.4. Unfortunately we cannot prove Theorem 1.1 in full generality, i.e., for M being the completion of an arbitrary Galois extension of ℚ_{p}. For instance, it is not known to us, whether the theorem holds for M being the completion of the infinite purely ramified extension ℚ_{p}(μ_{p^{∞}}) of ℚ_{p}.

Remark 1.5. The proof above also works for “number” fields, i.e. algebraic (possible infinite) extensions of ℚ. We just have to replace ℚ_{p} by ℚ in Theorem 1.1. Then, letting L be a finite Galois extension of ℚ and M be an arbitrary (possible infinite) Galois extension of ℚ, we follow the proof of Theorem 1.1 using the same arguments and references to get

i_{∗} : K_{1}(L[Γ]) ≅ K_{1}(M[Γ])^{Δ}.

The only difference is, that the elements of Hom-groups in the proof are to be totally positive on all symplectic representations.

2. The case of rings of integers of “local” fields.

Let G be a finite group. If S is an integral domain of characteristic zero with field of fractions L, then \overline{L} will denote a chosen algebraic closure of L. We have a map induced by base extension \overline{L} ⊗_{S} −

Det : K_{1}(S[G]) → K_{1}(\overline{L}[G]) = ∏_{χ}\overline{L}^{×} ≅ Hom(R_{G}, \overline{L}^{×}),

where the direct product extends over the irreducible \overline{L}-valued characters of G. Since the map from K_{1}(L[G]) to K_{1}(\overline{L}[G]) induced by \overline{L} ⊗_{L} − is injective (see Proposition 1.2 (i)), the Det-map factorizes over K_{1}(L[G]). We write SK_{1}(S[G]) for Ker(Det), so that we have an exact sequence

(2.1)  1 → SK_{1}(S[G]) → K_{1}(S[G]) → Det(K_{1}(S[G])) → 1.

Therefore we shall consider the two parts of K_{1}, namely the Det-part and the SK_{1}-part, separately.

Remark 2.1. From the exact sequence (2.1) and the fact, that K_{1} commutes with direct limits (see [21], Exercise 2.1.9), we deduce, that Det and SK_{1} also commute with direct limits.

2.1. The Det-part. We keep the notation of the introduction. In this subsection let S = O_{L}, where L is either an arbitrary (possible infinite) Galois extension L^{0} of finite absolute ramification index over ℚ_{p} or the completion of such L^{0}. Then S is a Noetherian local ring, i.e. S has a unique maximal left ideal, and S[G] is semilocal, i.e. the quotient S[G]/rad(S[G]) of the ring by its Jacobson radical is left Artinian (see Proposition 20.6 in [11]). We have the following Proposition 2.2. Let Λ be a semilocal ring (for example S[G]). The maps

Λ^{×} = GL_{1}(Λ) ↪ GL(Λ) ↠ K_{1}(Λ)

induce an equality

Det(Λ^{×}) = Det(GL(Λ)) = Det(K_{1}(Λ)).

Proof. See Theorem 40.31 in [7]. □

Conjecture 2.3. Let S = O_{L} and G be as above. Let Δ be an open subgroup of Gal(L^{0}/ℚ_{p}) acting coefficientwise on S[G], and hence on Det-groups. Then

i_{∗} : Det(S^{Δ}[G]^{×}) ≅ Det(S[G]^{×})^{Δ}.

The proof of Conjecture 2.3 proceeds in two steps. At present we can prove step 1 and thus Conjecture 2.3 only under further assumptions on S (see Theorem 2.21). We first do the proof for finite extensions and completions of infinite extensions, since S is p-adically complete in these cases, and then we generalize the statement to infinite algebraic extensions using direct limits (see Remark 2.20).

Remark 2.4. The map i_{∗} in the conjecture is always a monomorphism, as the following diagram commutes and respects the action of Δ

Det(S^{Δ}[G]^{×}) ↪ K_{1}(L^{Δ}[G])
↓i_{∗} ↓i_{∗}
Det(S[G]^{×}) ↪ K_{1}(L[G]),

and the right hand side map is injective (see section 1).

2.1.1. Step 1. The p-group case. Let G be a finite p-group. For this step we generalize the ideas of [4] to the case of an infinite group Δ. We want to do the proof in full generality of [4], thus we have to assume, that S is an arbitrary unitary ring satisfying the following conditions:

(i) S is an integral domain, which is torsion free as an abelian group, (ii) the natural map S → lim_{←} S/p^{n}S is an isomorphism, so that S is p-adically complete, (iii) S supports a lift of Frobenius, that is to say an endomorphism F : S → S with the property that for all s ∈ S

F (s) ≡ s^{p} mod pS.

Remark 2.5. Proposition 2.2 holds also for S satisfying the conditions (i)-(iii) above and G being a finite p-group.

Proof. See Theorem 1.2 in [4]. □

Now we are ready to formulate the main theorem of step 1.

Theorem 2.6. Let G be a finite p-group. Let S be an arbitrary unitary ring satisfying the conditions (i)-(iii) and Δ be a group acting on S by the ring automorphisms, such that R = S^{Δ} also satisfies the conditions (i)-(iii). We do not suppose, that the lift of Frobenius F_{R} is compatible with the lift of Frobenius F_{S}, so that F_{S} |_{R} need not equal F_{R}. Then we have the isomorphism

i_{∗} : Det(R[G]^{×}) ≅ Det(S[G]^{×})^{Δ},

where Δ acts on Det-groups coefficientwise.

Proof. Since Δ acts on S by the ring automorphisms, we have the equality (S^{×})^{Δ} = R^{×} and for any finitely generated free S-module M = ⨁_{i}Se_{i}, on which

Δ acts coefficientwise, M^{Δ} is the finitely generated free R-module M^{Δ} = ⨁_{i}Re_{i}.

S and R both satisfy the Hypothesis in [4], thus the proof of the theorem is identical with the proof of Theorem 4.1 in [4]. Note that this proof does not depend on the condition, whether Δ is a finite group or not; it only uses the equalities above. □

Remark 2.7. In particular, Theorem 2.6 holds for S = O_{L}, where L is either a finite unramified extension L^{0} of ℚ_{p} or the completion of an infinite unramified extension L^{0} of ℚ_{p} - in other words for the ring of Witt vectors W (κ) of any algebraic extension κ of 𝔽_{p} -, and for Δ being an open subgroup of Gal(L^{0}/ℚ_{p}).

2.1.2. Step 2. Reduction of the general case to the p-group case. In contrast to the first step the second one can be proved in the full generality of Conjecture 2.3 for a given axiomatic setting. So let S and G be as in the conjecture (for infinite Galois extensions see Remark 2.20, so we assume, that S is p-adically complete). Let Δ be an open subgroup of Gal(L^{0}/ℚ_{p}), such that R = S^{Δ} is the ring of integers of a finite extension of ℚ_{p}. Then S is a local, Noetherian, normal ring satisfying

(i) S is an integral domain, which is torsion free as an abelian group, (ii) the natural map S → lim_{←} S/p^{n}S is an isomorphism, so that S is p-adically complete, (iii) S supports a lift of Frobenius, that is to say an endomorphism F : S → S with the property that for all s ∈ S

F (s) ≡ s^{p} mod 𝔐,

where 𝔐 is the maximal ideal of S.

Note that with S also R satisfies (i)-(iii).

Our reduction step is based on the following axioms to be satisfied for every finite p-group G (S, R, Δ being as above); actually they were essential ingredients of step 1 in the unramified case and we will show that Conjecture 2.3 is true for any class of rings satisfying them:

(1) There exists a homomorphism ν defined using the lift of Frobenius on S

ν : Det(1 + I(S[G])) → L[C_{G}],

such that L = ν ∘ Det (for the definition of L see pp. 12-13 in [4]). Here I(S[G]) denotes the augmentation ideal of the group ring S[G] and C_{G} denotes the set of conjugacy classes of G.

(2) Let ν′ denote the restriction of the homomorphism ν to Det(1+A(S[G])), where A(S[G]) is the kernel of the natural map from S[G] to S[G^{ab}], then ν′ is an isomorphism

Det(1 + A(S[G])) ˜ ⟶ pϕ(A(S[G])),

and ϕ : L[G] → L[C_{G}] denotes the L-linear map obtained by mapping each element of G to its conjugacy class.

(3) We have the exact sequence

0 → ϕ(A(S[G])) ⟶^{(ν′)^{−1}∘(p·)} Det(S[G]^{×}) → S[G^{ab}]^{×} → 1.

(4) We have the isomorphism

i_{∗} : Det(S^{Δ}[G]^{×}) ≅ Det(S[G]^{×})^{Δ},

where Δ acts coefficientwise on Det-groups.

From now we assume these axioms and describe the reduction step in this hypothetical generality, in the hope to prove Conjecture 2.3 some day in full generality.

Most ideas and techniques of the proof are contained in [4]. Thus, we only have to relax the condition (iii) on the ring in [4]. So, let S be as above. The proof will now proceed in different stages, restricting G first to special types of groups.

ℚ-p-elementary groups.

We begin with an algebraic result which we shall require later on this stage.

Suppose, that O_{K} is the ring of integers of a finite unramified extension K of ℚ_{p}. Let A denote the ring S ⊗_{ℤ_{p}} O_{K} and let M be the ring of fractions of A. Since M is a separable L-algebra, it can be written as a finite product of field extensions M_{i} of L:

M = ∏_{i=1}^{n} M_{i}.

Since K is a finite unramified extension of ℚ_{p}, we know that A is etale over S and hence is normal (see e.g. page 27 in [12]). If A_{i} is the normalization of S in M_{i}, then

A = ∏_{i=1}^{n} A_{i}.

Lemma 2.8. Let F denote the lift of Frobenius on A given by the tensor product of the lift of Frobenius on S with the Frobenius automorphism of O_{K}; then F (A_{i}) ⊂ A_{i}.

Proof. Let {e_{i}} denote the system of primitive orthogonal idempotents associated to the above product decomposition of A. As F is a ℤ_{p}-algebra endomorphism, we know, that {F (e_{i})} is a system of orthogonal idempotents with

1 = ∑_{i=1}^{n} F (e_{i})

and so this system corresponds to a decomposition of the commutative algebra A into n components. Since the decomposition of Noetherian commutative algebras into indecomposable algebras is unique, we must have F (e_{i}) = e_{π(i)} for some permutation π of {1, ..., n}. It will suffice to show, that the permutation π is the identity. Suppose for contradiction, that for some i, we have π(i) = j ≠ i. We know by definition, that

F (e_{i}) ≡ e_{i}^{p} = e_{i} (mod 𝔐A)

and so

e_{i} ≡ F (e_{i}) · e_{i} ≡ e_{j} · e_{i} = 0 (mod 𝔐A).

However, by Theorem 6.7 (page 123) in [6] we know, that, since 𝔐A is contained in the radical of A, e_{i} mod 𝔐A must be a primitive idempotent of A/𝔐A, and so we have the desired contradiction. □

Suppose G is a ℚ-p-elementary group, i.e. G may be written as a semidirect product C ⋊ P, where C is a cyclic normal subgroup of order s, which is prime to p, and where P is a p-group. We decompose the commutative group ring ℤ_{p}[C] according as the divisors m of s

(2.2)  ℤ_{p}[C] = ∏_{m|s} ℤ_{p}[m],

where ℤ_{p}[m] is the semilocal ring

ℤ_{p}[m] = ℤ[ζ_{m}] ⊗_{ℤ} ℤ_{p},

and where ζ_{m} is a primitive mth root of unity in \overline{ℚ_{p}}. For brevity we set S[m] = S ⊗_{ℤ_{p}} ℤ_{p}[m], and we note, that by Lemma 2.8 S[m] decomposes as a product of rings satisfying (i)-(iii), where the Frobenius is given by the restriction of the tensor product of the lift of Frobenius on S and the Frobenius automorphism of ℤ_{p}[m] to each component.

For each m the conjugation action of P on C induces a homomorphism

α_{m} : P → Aut ⟨ζ_{m}⟩

and we let H_{m} = Ker(α_{m}) and A_{m} = Im(α_{m}).

Tensoring the decomposition (2.2) with − ⊗_{ℤ_{p}[C]} S[G] affords a decomposition of S-algebras

(2.3)  S[G] = ∏_{m} S[m] ∘ P,

where S[m] ∘ P is the free S[m]-module on the set of elements of P with the following multiplication s_{1}p_{1} · s_{2}p_{2} = s_{1}s_{2}^{p_{1}}p_{1}p_{2}, P acting on S[m] through A_{m}. S[m] ∘ P is also called the twisted group ring. We shall study the determinant group Det(GL(S[G])) by studying the various subgroups Det(GL(S[m] ∘ P)). Note that the twisted group ring S[m] ∘ P contains the standard group ring S[m][H_{m}]. We therefore have the inclusion map i : S[m][H_{m}] → S[m] ∘ P. We also have a restriction map defined by choosing a transversal {a_{i}} of P/H_{m}. This induces a restriction homomorphism

res : GL_{n}(S[m] ∘ P) → GL_{n|A_{m}|}(S[m][H_{m}]).

By Proposition 2.2 we know, that Det(GL(S[m][H_{m}])) = Det(S[m][H_{m}]^{×}), and so we have defined the composition:

(2.4) r_{m} : Det(GL_{n}(S[m] ∘ P)) → GL_{n|A_{m}|}(S[m][H_{m}]) → Det(S[m][H_{m}]^{×}).

Since for π ∈ P, x ∈ (S[m] ∘ P)^{×}, we know, that Det(πxπ^{−1}) = Det(x), whence

r_{m} : Det(GL_{n}(S[m] ∘ P)) → Det(S[m][H_{m}]^{×})^{A_{m}}.

Here A_{m} acts via α_{m} on S[m] and by conjugation on H_{m}. From (3.8) on page 69 in [25] we know, that r_{m} is injective. Note for future reference, that for x ∈ S[m][H_{m}]^{×}, i(x) is mapped by restriction to the diagonal matrix diag(x^{a_{i}}); thus we write Det(x) for the usual element of Det(S[m][H_{m}]^{×}) whereas Det(i(x)) denotes an element of Det((S[m] ∘ P)^{×}). These two determinants are related by the identity

r_{m}(Det(i(x))) = ∏_{a∈A_{m}} Det(x^{a}) = N_{A_{m}}(Det(x)).

Next we describe Det((S[m] ∘ P)^{×}), and more generally Det(GL(S[m] ∘ P)), and the maps i and r_{m} in terms of character functions. In Lemma 2.14 we shall see, that every irreducible character of G may be written in the form Ind_{H_{m}}^{G}(ϕ_{m}) for some m, where ϕ_{m} is an abelian character of H_{m} with the property, that the restriction of ϕ_{m} to C has order m. With this notation the elements Det(i(x)) in Det((S[m] ∘ P)^{×}) are character functions on such Ind_{H_{m}}^{G}(ϕ_{m}) with

Det(i(x))(Ind_{H_{m}}^{G}(ϕ_{m})) = r_{m}(Det(i(x)))(ϕ_{m}) = ∏_{a∈A_{m}} Det(x^{a})(ϕ_{m}).

It is also instructive to see the above in the context of K-theory. We then have induction and restriction maps

\begin{aligned}ind & : & K_{1}(S[m][H_{m}]) & ⇆ & K_{1}(S[m] ∘ P) & : & r_{m} \\ ind & : & Det(S[m][H_{m}]^{×}) & ⇆ & Det((S[m] ∘ P)^{×}) & : & r_{m}. \end{aligned}

Similarly we have the corresponding maps on the representation rings

K_{0}(\overline{ℚ_{p}}[m][H_{m}]) ⇆ K_{0}(\overline{ℚ_{p}}[m] ∘ P)

and by Mackey theory the induction map i = Ind_{H_{m}}^{P} maps the ring K_{0}(\overline{ℚ_{p}}[m][H_{m}]) onto K_{0}(\overline{ℚ_{p}}[m] ∘ P) (see page 68 in [25]).

Let I_{H_{m}}, resp. I_{P}, denote the augmentation ideal of S[m][H_{m}], resp. the two sided S[m] ∘ P -ideal generated by I_{H_{m}}. The main result on this stage is to show:

Theorem 2.9. The map r_{m} defined in (6) gives an isomorphism

r_{m} : Det(GL(S[m] ∘ P)) = Det((S[m] ∘ P)^{×}) → Det(S[m][H_{m}]^{×})^{A_{m}}.

Proof. We have seen, that r_{m} is injective on Det(GL(S[m] ∘ P)); we now show, that r_{m} maps Det((S[m] ∘ P)^{×}) onto Det(S[m][H_{m}]^{×})^{A_{m}}.

First we put P̃_{m} = P/[H_{m}, H_{m}], then the ring S[m] ∘ P̃_{m} is isomorphic to the ring of |A_{m}| × |A_{m}| matrices over (S[m][H_{m}^{ab}])^{A_{m}}. To see it, we note, that by Lemma 2.8 S[m] = ∏_{i} S_{i}, where S_{i} are complete local rings. Further, by Lemma 8.2 on page 613 in [26] S_{i} ∘ P̃_{m} is an Azumaya algebra over some complete local ring B_{i}. Thus, it represents a class in the Brauer group of B_{i}, which is the quotient group of the group of Azumaya algebras by the subgroup of full matrix algebras. But by Theorem 6.5 in [1] the Brauer group of B_{i} is isomorphic to the Brauer group of the residue class field b_{i} of B_{i}, and thus is trivial, as b_{i} is an algebraic extension of 𝔽_{p} and the Brauer group of a quasi-algebraically closed field is trivial (see Theorem 6.5.4, Theorem 6.5.7 and Proposition 6.5.8 in [14]). Hence we see, that r_{m} induces an isomorphism

(2.5)  Det((S[m] ∘ P̃_{m})^{×}) ≅ ((S[m][H_{m}^{ab}])^{A_{m}})^{×}.

From the axioms (2) and (3) above (which trivially extends to products of rings, since formation of determinants commutes with ring products) and using (2.5), we have a commutative diagram with exact top row: (2.6)

Det(1 + A(S[m][H_{m}]))^{A_{m}} ↪ Det(S[m][H_{m}]^{×})^{A_{m}} → ((S[m][H_{m}^{ab}])^{A_{m}})^{×}
↑  ↑r_{m}  ↑≀
Det(i(1 + A(S[m][H_{m}]))) ↪ Det((S[m] ∘ P)^{×}) ↠ Det((S[m] ∘ P̃_{m})^{×}).

It will therefore suffice to show

r_{m}(Det(i(1 + A(S[m][H_{m}])))) ⊇ Det(1 + A(S[m][H_{m}]))^{A_{m}}

and this follows from the commutative diagram

Det(1 + A(S[m][H_{m}])) ˜ ⟶^{ν} ϕ(A(S[H_{m}])) ⊗_{S} S[m]
↓r_{m}  ↓tr_{A_{m}}
Det(1 + A(S[m][H_{m}]))^{A_{m}} ˜ ⟶^{ν^{Am}} (ϕ(A(S[H_{m}])) ⊗_{S} S[m])^{A_{m}}.

Recall that F is the tensor product of the lift of Frobenius on S with the Frobenius automorphism of ℚ_{p}(ζ_{m})/ℚ_{p}. Note also that A_{m} acts on S[m] = S ⊗ ℤ_{p}[m] via the second factor; so, because G is a ℚ-p-elementary, the action of A_{m} on ⟨χ(G)⟩ factors through Gal(ℚ_{p}(ζ_{m})/ℚ_{p}); this guarantees, that the actions of F and A_{m} commute; hence ν is an isomorphism of A_{m}-modules, and this gives the bottom row in the above diagram.

Since S[m] is a free S[A_{m}]-module, it follows, that ϕ(A(S[H_{m}])) ⊗_{S} S[m] is a projective S[A_{m}]-module (with diagonal action); and so tr_{A_{m}}, and therefore r_{m}, is surjective. □

Finally we show:

Theorem 2.10. Let G be a finite ℚ-p-elementary group and let S, Δ and R be as above. Further, let Δ act coefficientwise on Det(S[G]^{×}), then

Det(S[G]^{×})^{Δ} = Det(R[G]^{×}).

Proof. By (2.3) together with Theorem 2.9

Det(S[G]^{×})^{Δ} = ⨁_{m} Det((S[m] ∘ P)^{×})^{Δ} = ⨁_{m}(Det(S[m][H_{m}]^{×})^{A_{m}})^{Δ}.

Recall that Δ acts via the first term in S[m] ∘ P = S ⊗_{R} (R[m] ∘ P) and that A_{m} acts via the second term; hence the actions of Δ and A_{m} commute on S[m][H_{m}] = (S ⊗_{R} R[m])[H_{m}]; hence we see, that

Det(S[G]^{×})^{Δ} = ⨁_{m}(Det(S[m][H_{m}]^{×}))^{A_{m}×Δ} = ⨁_{m}(Det(S[m][H_{m}]^{×})^{Δ})^{A_{m}}

and so by the fact 4 and Theorem 2.9 together with (2.3) we have equalities

Det(S[G]^{×})^{Δ} = ⨁_{m}(Det(R[m][H_{m}]^{×}))^{A_{m}} = ⨁_{m} Det((R[m] ∘ P)^{×}) = Det(R[G]^{×}).

□

Application. We conclude this stage by considering the implications of the above result for an arbitrary finite group G. From 12.6 in [23] we know, that we can find an integer l prime to p, ℚ-p-elementary subgroups H_{i} of G, integers n_{i}, and θ_{i} ∈ K_{0}(ℚ_{p}[H_{i}]), such that

l · 1_{G} = ∑_{i} n_{i} · Ind_{H_{i}}^{G}(θ_{i}).

Thus, given Det(x) ∈ Det(GL(S[G]))^{Δ}, then by the Frobenius structure of the module GL(S[G]) over K_{0}(ℚ_{p}[H_{i}]) (see §5.a in [4]) we have

Det(x)^{l} = ∏_{i} (Ind_{H_{i}}^{G}(θ_{i} · Res_{G}^{H_{i}}(Det(x))))^{n_{i}}.

However, Theorem 2.10 above implies that

θ_{i} · Res_{G}^{H_{i}}(Det(x)) ∈ Det(S[H_{i}]^{×})^{Δ} = Det(R[H_{i}]^{×}).

Thus we have shown

Theorem 2.11. For any finite group G each element in the quotient group

Det(GL(S[G]))^{Δ}/Det(GL(R[G]))

has finite order, which is prime to p.

ℚ-l-elementary groups.

We consider a prime l ≠ p and a ℚ-l-elementary group G, i.e G may be written as (C × C′) ⋊ L, where C is a cyclic p-group, C′ is a cyclic group of order prime to pl and L is an l-group. On this stage we show:

Theorem 2.12. If G is a ℚ-l-elementary group, then

Det(S[G]^{×})^{Δ} = Det(R[G]^{×}).

Then, reasoning as in the Application on the previous stage, we can immediately deduce

Theorem 2.13. For an arbitrary finite group G each element in the quotient group

Det(GL(S[G]))^{Δ}/Det(GL(R[G]))

has finite order, which is prime to l.

This, together with Theorem 2.11 and Proposition 2.2, will then establish Conjecture 2.3.

Prior to proving Theorem 2.12, we first need to recall four preparatory results:

Lemma 2.14. Each irreducible character χ of G can be written in the form Ind_{Ω}^{G}ϕ, where ϕ is an abelian character of a subgroup Ω, which contains C × C′.

Proof. See 8.2 in [23] □

Proposition 2.15. Let O denote the ring of integers of the finite extension of ℚ_{p} generated by the values of all characters of G, let 𝔪 denote the maximal ideal of O, and let P denote the S ⊗_{ℤ_{p}} O-ideal generated by 𝔪. With the notation of the previous lemma, we write ϕ = ϕ′ϕ_{p}, where ϕ′ (resp. ϕ_{p}) has order prime to p (resp. p-power order); and we put χ′ = Ind_{Ω}^{G}ϕ′. Then for r ∈ GL(S[G]) we have the congruence

Det(r)(χ − χ′) ≡ 1 (mod P).

Proof. It is an easy generalization of Lemma 1.3 on page 35 in [25]. □

Proposition 2.16. Put G′ = G/C. Then ℤ_{p}[G′] is a split maximal ℤ_{p}-order, i.e. it is a product of matrix rings

ℤ_{p}[G′] = ∏_{i} M_{n_{i}}(O_{i})

over (local) rings of integers O_{i}. Thus we have the equalities

Det(GL(S[G′])) = ∏_{i}(S ⊗_{ℤ_{p}} O_{i})^{×} = Det(S[G′]^{×})

and

Det(S[G′]^{×})^{Δ} = ∏_{i} Det((S ⊗ O_{i})^{×})^{Δ} = ∏_{i}(S ⊗ O_{i})^{×Δ} =

∏_{i}(R ⊗ O_{i})^{×} = ∏_{i} Det((R ⊗ O_{i})^{×}) = Det(R[G′]^{×}).

Proof. See Theorem 41.1 and Theorem 41.7 in [19]. □

Lemma 2.17. With the previous notation, the map S[G]^{×} → S[G′]^{×} is surjective.

Proof. Consider the canonical projection

A = S[G] ↠^{φ} S[G′] = B.

Firstly, we have to prove, that Ker(φ) is contained in the radical of A. We know, that Ker(φ) = I(C) ⊂ A, where I(C) is generated by 1 − σ, σ ∈ C. Thus there is a positive integer r, such that (Ker(φ))^{p^{r}} ⊂ 𝔐, where 𝔐 is the maximal ideal of S. A positive power of an ideal is contained in the radical of a ring if and only if the ideal is contained in the radical (the image of such ideal would be a nilpotent ideal of A/rad(A), hence it is contained in rad(A/rad(A)) = (0), see [19], §6). Hence it is enough to prove, that 𝔐 is contained in rad(A). But this follows from the more general lemma below (see Lemma 2.18), as the radical of the ring A is the intersection of all maximal left ideals of A (see Theorem 6.3 in [19]).

Secondly, by the Theorem 6.10 in [19] and the first step of the proof φ induces an isomorphism

φ̄ : A/rad(A) ˜→B/rad(B),

as φ(rad(A)) ⊂ rad(B).

Finally, let x ∈ B^{×}. We denote its image in (B/rad(B))^{×} by x̄. Since φ is surjective, we can lift x to an element y ∈ A. The image of y in A/rad(A) denoted ȳ is then mapped under φ̄ onto x̄, thus it is contained in (A/rad(A))^{×} and hence y ∈ A^{×}, as 1 + rad(A) ⊂ A^{×} (see Theorem 6.5 in [19]). □

Lemma 2.18. Let S be a discrete valuation ring, but not a field. Let 𝔐 denote its unique maximal (left) ideal. Assume, that we have an inclusion of the rings i : S → A, where A is an arbitrary ring, then i(𝔐) · S is contained in every maximal non-zero left ideal M of A.

Proof. The preimage i^{−1}(M) of M is a non-zero prime left ideal of S, thus it is a maximal left ideal, hence it coincides with 𝔐. Now applying i we get i(𝔐) · S = i(i^{−1}(M)) · S ⊆ M. □

Proof of Theorem 2.12. Suppose, that we are given Det(z) ∈ Det(S[G]^{×})^{Δ} and let z′ denote the image of z in S[G′]. Then by Proposition 2.16 we know, that we can find x′ ∈ R[G′]^{×} with Det(x′) = Det(z′); moreover, by Lemma 2.17 we can find x ∈ R[G]^{×} with image x′ in R[G′]^{×}. Thus, to conclude, it will be sufficient to show, that Det(zx^{−1}) is in Det(R[G]^{×}). However, by construction, Det(zx^{−1}) is trivial on characters inflated from G′, and so by Proposition 2.15 we see that

Det(zx^{−1})(χ) = Det(zx^{−1})(χ − χ′) ≡ 1 (mod P), for all χ.

Hence Det(zx^{−1}) is in Hom(R_{G}, 1 + P), where R_{G} denote the group of virtual \overline{L}-valued characters of G. Moreover, since Det(zx^{−1}) is invariant under the action of Δ, Det(zx^{−1}) ∈ Hom_{Δ}(R_{G}, 1 + P). In the case, where R is the ring of integers of a finite extension of ℚ_{p}, the last Hom-group is isomorphic to the finite product of groups of higher principal units U^{(1)} of finite extensions of ℚ_{p}, and hence is a pro-p-group. Thus Det(zx^{−1}) is a pro-p-element of Det(S[G]^{×})^{Δ}. But by Theorem 2.11 we know, that Det(zx^{−1}) has image in the quotient group Det(S[G]^{×})^{Δ}/Det(R[G]^{×}) of finite order, which is prime to p. Therefore we may deduce, that Det(zx^{−1}) is in Det(R[G]^{×}). □

2.1.3. Conjecture 2.3: proved cases and one generalization. Because of our restrictions in the p-group case we have proved Conjecture 2.3 only for S = W (κ) the ring of Witt vectors of an algebraic extension κ of 𝔽_{p}. There are some possible generalizations of this result as will be explained now.

Remark 2.19. Using the reduction step described on the page 92 in [25], we can prove Conjecture 2.3 for S = O_{L}, where L is the completion of a tamely ramified extension L^{0} of finite absolute ramification index over ℚ_{p}, i.e., finite tamely ramified extension of the ring of Witt vectors of an algebraic extension κ of 𝔽_{p}, and Δ being an open subgroup of Gal(L^{0}/ℚ_{p}) containing the inertia group.

Remark 2.20. The infinite Galois extensions can always be written as a direct limit (or simply union) of their finite subextensions, and so their rings of integers, too. Thus, if we have proved Conjecture 2.3 for some class of finite Galois extensions of ℚ_{p}, we can obtain it also for infinite extensions in the same ind-class, as the Det-map commutes with direct limits (see Remark 2.1).

Explicitly, let S be the ring of integers of an infinite extension L of ℚ_{p}. Let L = ⋃_{i}L_{i} and S = ⋃_{i}S_{i}, where L_{i} are finite extensions and S_{i} their rings of

integers, and we have the statement of Conjecture 2.3 for all S_{i}. Further, let Δ be an open subgroup of Gal(L/ℚ_{p}), then

Det(S[G]^{×})^{Δ} = Det(⋃_{i}S_{i}[G]^{×})^{Δ} = (⋃_{i}Det(S_{i}[G]^{×}))^{Δ} = ⋃_{i}(Det(S_{i}[G]^{×})^{Δ}) =

⋃_{i}(Det(S_{i}^{Δ}[G]^{×})) = Det(⋃_{i}S_{i}^{Δ}[G]^{×}) = Det(S^{Δ}[G]^{×}),

where Δ acts on S_{i} through the corresponding quotient group and the union commutes with such defined Δ-action. The maps between Det-groups induced by inclusions of rings are inclusions by Theorem 13 in [8].

For example, Conjecture 2.3 holds for S = O_{L}, where L is the maximal unramified extension of ℚ_{p} and Δ is an open subgroup of Gal(L/ℚ_{p}) or using the previous remark L is the maximal tamely ramified extension of ℚ_{p} and Δ is an open subgroup of Gal(L/ℚ_{p}) containing the inertia group.

Remarks 2.7, 2.19 and 2.20 imply

Theorem 2.21. Let G be a finite group. Let S = O_{L}, where L is either an arbitrary (possibly infinite) tamely ramified extension L^{0} of ℚ_{p} (type 1) or the completion of a tamely ramified extension L^{0} of finite absolute ramification index over ℚ_{p} (type 2), and let Δ be an open subgroup of Gal(L^{0}/ℚ_{p}) containing the inertia group. Then

i_{∗} : Det(S^{Δ}[G]^{×}) ≅ Det(S[G]^{×})^{Δ}.

We conclude this subsection with the following result generalising Theorem 2.21 to the case of compact p-adic Lie groups and their Iwasawa algebras. Let G be a compact p-adic Lie group, let S = O_{L} be as in the theorem and denote by R = S^{Δ} the ring of integers of a finite unramified extension K over ℚ_{p}. We denote by Δ the Galois group Gal(L^{0}/K) and write

Λ_{S}(G) := S[[G]] (resp. Λ_{R}(G) := R[[G]])

for the Iwasawa algebra of G with coefficients in S (resp. in R). Note that it is a Noetherian pseudocompact ring (resp. a Noetherian compact ring). For the notion of pseudocompact rings and algebras see [3]. It is well known, that Λ_{R}(G) is a semilocal ring (see Proposition 5.2.16 in [14]), using the idea in the proof of Lemma 2.17 we deduce, that the same is true for Λ_{S}(G).

In the following we will use Froehlich’s Hom-description as it has been adapted to Iwasawa theory by Ritter and Weiss in [20]. We have the following commutative diagram

K_{1}(Λ_{S}(G)) ⟶^{Det} Hom_{G_{L^{0}}}(R_{G}, O_{ℂ_{p}}^{×})
↑  ⋃
K_{1}(Λ_{R}(G)) ⟶^{Det} Hom_{G_{K}}(R_{G}, O_{ℂ_{p}}^{×}),

where G_{L^{0}} = Gal(\overline{L}/L^{0}), G_{L} = Gal(\overline{L}/L) and R_{G} as before is the free abelian group on the isomorphism classes of irreducible \overline{ℚ_{p}}-valued Artin representations of G.

Now we are ready to formulate

Theorem 2.22. With the notation as above we have

Det(K_{1}(Λ_{S}(G)))^{Δ} = Det(K_{1}(Λ_{R}(G))),

where Δ acts on the K_{1}-groups coefficientwise.

Proof. From the diagram

Λ_{S}(G)^{×} ⟶^{Det} Hom_{G_{L^{0}}}(R_{G}, O_{ℂ_{p}}^{×})
⋃  ⋃
Λ_{R}(G)^{×} ⟶^{Det} Hom_{G_{K}}(R_{G}, O_{ℂ_{p}}^{×}),

which is commutative by the construction, and from Proposition 2.2 we get the first obvious inclusion

Det(Λ_{R}(G)^{×}) = Det(K_{1}(Λ_{R}(G))) ⊆ Det(K_{1}(Λ_{S}(G)))^{Δ} = Det(Λ_{S}(G)^{×})^{Δ}.

For the opposite inclusion we use Theorem 2.21, then we only have to show, how the general case can be reduced to the case of finite groups. To this end write G = \underleftarrow{lim}_{n} G_{n} as inverse limit of finite groups. By Theorem 2.21 we have compatible

continuous maps

R[G_{n}]^{×} ↠^{Det} Det(K_{1}(Λ_{S}(G_{n})))^{Δ} ↪ Hom_{G_{L^{0}}}(R_{G_{n}}, O_{ℂ_{p}}^{×})^{Δ},

where the topology on Hom_{G_{L^{0}}}(R_{G_{n}}, O_{ℂ_{p}}^{×}) is induced from the valuation topology on ℂ_{p}. Taking the inverse limit yields, by the compactness of Λ_{R}(G)^{×} = \underleftarrow{lim}_{n}(R/π^{n}[G_{n}])^{×} and by letting R_{G} = \underrightarrow{lim}_{n} R_{G_{n}}, a factorization of the homomor-

phism Det into

Λ_{R}(G)^{×} ↠^{Det} (\underleftarrow{lim}_{n} Det(K_{1}(Λ_{S}(G_{n}))))^{Δ} ↪ Hom_{G_{K}}(R_{G}, O_{ℂ_{p}}^{×}).

The claim follows, because denoting by

res_{n} : Hom_{G_{L^{0}}}(R_{G}, O_{ℂ_{p}}^{×}) → Hom_{G_{L^{0}}}(R_{G_{n}}, O_{ℂ_{p}}^{×})

the restriction we obtain from the universal mapping property for

\underleftarrow{lim}_{n} Hom_{G_{L^{0}}}(R_{G_{n}}, O_{ℂ_{p}}^{×}) ≅ Hom_{G_{L^{0}}}(R_{G}, O_{ℂ_{p}}^{×})

inclusions

Det(K_{1}(Λ_{S}(G))) ⊆ \underleftarrow{lim}_{n} Im(res_{n} ∘ Det) ⊆ \underleftarrow{lim}_{n} Det(K_{1}(Λ_{S}(G_{n}))),

whence

Det(K_{1}(Λ_{S}(G)))^{Δ} ⊆ Det(Λ_{R}(G)^{×}) = Det(K_{1}(Λ_{R}(G))).
□

For applications of the theorem above in number theory see [2].

End of the Det-part.

2.2. The SK_{1}-part. From [15] we have the following

Theorem 2.23. Let R be the ring of integers in any finite extension K of ℚ_{p}. Then for any p-group G, there is an isomorphism

Θ_{RG} : SK_{1}(R[G]) ˜→H_{2}(G)/H_{2}^{ab}(G),

where H_{2}^{ab}(G) = Im[ ∑ \{H_{2}(H) : H ⊆ G, H abelian\} \underrightarrow{∑ Ind} H_{2}(G)]

If L ⊇ K is a finite extension, and if S ⊆ L is the ring of integers , then

(i) i_{∗} : SK_{1}(R[G]) → SK_{1}(S[G]) (induced by inclusion) is an isomorphism, if L/K is totally ramified; and (ii) trf : SK_{1}(S[G]) → SK_{1}(R[G]) (the transfer) is an isomorphism, if L/K is unramified.

Proof. See proof of Theorem 8.7 in [15]. □

We see, that SK_{1}(S[G]) is independent of S. Note also, that i_{∗} and trf are Galois homomorphisms, hence SK_{1}(S[G]) has trivial Galois action. In order to treat infinite algebraic extensions of ℚ_{p} and the analogous descent statement of the introduction for SK_{1}-groups we have to describe the maps i_{∗} induced by inclusions, as they appear in the direct limits (see Remark 2.20 and Remark 2.1).

Now we assume that K is unramified over ℚ_{p}, then from Proposition 21 (i) in [16], which is also valid for i_{∗} with the same argument, we have commutative squares

SK_{1}(R[G]) ⟶^{Θ_{RG}} H_{2}(G)/H_{2}^{ab}(G)
↓i_{∗}  ↓?
SK_{1}(S[G]) ⟶^{Θ_{SG}} H_{2}(G)/H_{2}^{ab}(G),

and

SK_{1}(S[G]) ⟶^{Θ_{SG}} H_{2}(G)/H_{2}^{ab}(G)
↓trf  ↓id
SK_{1}(R[G]) ⟶^{Θ_{RG}} H_{2}(G)/H_{2}^{ab}(G)

where Θ_{RG}, Θ_{SG} and trf are isomorphisms. To describe i_{∗} and ? we need the following

Lemma 2.24. With the previous notation the map

trf ∘ i_{∗} : K_{1}(R[G]) → K_{1}(S[G]) → K_{1}(R[G])

is multiplication by n = [L : K].

Proof. By Proposition 1.18 in [15] the composite trf ∘ i_{∗} is induced by tensoring over R[G] with S[G] ≅ S ⊗_{R} R[G] regarded as an (R[G], R[G])-bimodule, where the bimodule structure on S ⊗_{R} R[G] is given through the second factor in the natural way. Since S is a free R-modules of rank n, S ⊗_{R} R[G] ≅ R[G]^{n} as (R[G], R[G])-bimodules, and so trf∘i_{∗} is multiplication by n on K_{1}(R[G]) (written additively). □

The map trf ∘i_{∗} on K_{1}(R[G]) corresponds via Θ_{RG} to the map H_{2}(G)/H_{2}^{ab}(G) ⟶^{n·} H_{2}(G)/H_{2}^{ab}(G), and since trf corresponds to the identity map, i_{∗} also corresponds to the multiplication by n. From the Theorem 3.14 in [15] we know, that SK_{1}(R[G]) (hence H_{2}(G)/H_{2}^{ab}(G)) is a finite p-group, so that we have proved the

Theorem 2.25. With the notation as above i_{∗} is an isomorphism in the following two cases

(i) if L/K is totally ramified, (ii) if L/K is unramified and p ∤ n. In this case i_{∗} corresponds via Θ_{RG} to the multiplication by n on the finite p-group (written additively).

If L/K is unramified and p|n, then i_{∗} still corresponds via Θ_{RG} to the multiplication by n on the finite p-group, which is neither surjective nor injective, as finite p-groups always have p-torsion elements.

Corollary 2.26. The groups SK_{1}(R[G]) and SK_{1}(S[G]) are always isomorphic (as abstract groups with the (trivial) action of Gal(L/K)), but the statement of the introduction for SK_{1}-groups, i.e.

(2.7)  i_{∗} : SK_{1}(R[G]) ≅ SK_{1}(S[G])^{Δ} (Δ = Gal(L/K)),

holds only in the cases (i) and (ii) of Theorem 2.25.

Corollary 2.27. Let M be an infinite algebraic extension of ℚ_{p} and let M^{0} be the maximal unramified extension of ℚ_{p} contained in M. We write M as the direct limit (union) of its finite subextensions and use Remark 2.1 and the theorem above to get the following result: If p^{∞} divides [M^{0} : ℚ_{p}] (as supernatural numbers), then SK_{1}(O_{M}[G]) = 1 for every finite p-group G.

Remark 2.28. From Corollary 2.27 we can obtain the generalization of Corollary 2.26 for infinite extensions: Let L be an infinite algebraic extension of ℚ_{p} and let K = L^{Δ}, where Δ is an open subgroup of Gal(L/ℚ_{p}). Then the statement (2.7) holds only in the cases (i) and (ii) of Theorem 2.25, here p ∤ n as supernatural numbers. In the case, where p^{∞} divides [L^{0} : ℚ_{p}] and SK_{1}(R[G]) ≠ 1, SK_{1}(R[G]) and SK_{1}(S[G]) are not isomorphic even as abstract groups. See [15, Example 8.11] for an example of non-trivial SK_{1}(ℤ_{p}[G]).

Now we generalize our results to the case of an arbitrary finite group G. Note, that Theorem 3.14 in [15] and Lemma 2.24 still hold in this case. From [17] we have the

Theorem 2.29. Let R be the ring of integers in any finite extension K of ℚ_{p} and let G be a finite group. Let L ⊇ K be a finite extension, and let S ⊆ L be the ring of integers , then (ii) i_{∗} : SK_{1}(R[G]) → SK_{1}(S[G]) is an isomorphism, if L/K is totally ramified; (iii) trf : SK_{1}(S[G]) → SK_{1}(R[G]) is onto, if L/K is unramified.

Proof. See proof of Theorem 1 in [17]. □

We need some more notation. For any finite group G, and any fixed prime p, G_{r} will denote the set of p-regular elements in G, i.e., elements of order prime to p. H_{n}(G, R(G_{r})) denotes the homology group induced by the conjugation action of G on R(G_{r}).

When R is the ring of integers in a finite unramified extension of ℚ_{p}, then Φ denotes the automorphism of H_{n}(G, R(G_{r})) induced by the map Φ(∑_{i} r_{i}g_{i}) = ∑_{i} φ(r_{i})g_{i}^{p} on coefficients. We set H_{n}(G, R(G_{r}))_{Φ} = H_{n}(G, R(G_{r}))/(1 − Φ). In analogy with the p-group case, we define

H_{2}^{ab}(G, R(G_{r}))_{Φ} = Im[ ∑ \{H_{2}(H, R(H_{r})) : H ⊆ G, H abelian\} \underrightarrow{∑ Ind}
\underrightarrow{∑ Ind} H_{2}(G, R(G_{r}))_{Φ}].

We use the notation above to formulate the

Theorem 2.30. Let R be the ring of integers in a finite unramified extension of ℚ_{p}. Then, for any finite group G, there is an isomorphism

Θ_{G} : SK_{1}(R[G]) ˜→(R/(1 − φ)R) ⊗_{ℤ_{p}} H_{2}(G, ℤ_{p}(G_{r}))_{Φ}/H_{2}^{ab}(G, ℤ_{p}(G_{r}))_{Φ}.

Proof. See proof of Theorem 12.10 in [15] and use the facts

H_{2}(G, R(G_{r}))_{Φ} ≅ (R/(1 − φ)R) ⊗_{ℤ_{p}} H_{2}(G, ℤ_{p}(G_{r}))_{Φ}

and

H_{2}^{ab}(G, R(G_{r}))_{Φ} ≅ (R/(1 − φ)R) ⊗_{ℤ_{p}} H_{2}^{ab}(G, ℤ_{p}(G_{r}))_{Φ}.
□

Arguing as in the p-group case we deduce the following

Theorem 2.31. With the notation as above i_{∗} is

(i) an isomorphism, if L/K is totally ramified; (ii) a monomorphism, if L/K is unramified and p ∤ n.

Remark 2.32. In the case (ii) we cannot say whether i_{∗} is surjective or not, since, in general, trf is only an epimorphism and not an isomorphism.

Corollary 2.33. The statement (2.7) holds in the cases (i)-(ii) of Theorem 2.31.

Proof. The statement is obvious in the case (i), since i_{∗} is a Galois isomorphism. For (ii) we use the isomorphism of Theorem 2.30 for S and R noting that

R/(1 − φ)R ≅ ℤ_{p} ≅ S/(1 − φ)

in this situattion, hence SK_{1}(S[G]) is isomorphic to SK_{1}(R[G]) (as an abstract finite group). Since i_{∗} is a Galois monomorphism in the case (ii), we get the statement. □

Remark 2.34. Corollary 2.33 generalizes immediately to the case of infinite algebraic extensions L.

To study more general rings (for example completions of infinite extensions of ℚ_{p}) we need generalizations of Oliver’s results on SK_{1} to such rings as are announced to appear in [5] in a very general setting. Meanwhile we outline an ad hoc description sufficient for our purposes. We just note that the same arguments as used below also should work for any ring R as in the beginning of subsection 2.1.1 and satisfying the surjectivity of 1 − F.

Let p be an odd prime number. For the rest of this section we assume that R is the ring of Witt vectors of a p-closed algebraic extension κ of 𝔽_{p}, i.e. κ does not allow any extension of degree p. The main example we have in mind being ℤ̂_{p}^{ur} = W (\overline{𝔽_{p}}). We note, that such a ring satisfies Hypothesis in [4] and is a discrete valuation ring. We write 𝔪 for its maximal ideal pR and start with a crucial (certainly well-known) observation:

Lemma 2.35. We have an exact sequence

\xymatrix{0 \ar[r] & ℤ_{p} \ar[r] & R \ar[r]^{1−φ} & R \ar[r] & 0,}

where φ denotes the Frobenius endomorphism of R.

Proof. By Artin-Schreier theory and the p-closeness of κ we have the obvious exact sequence

\xymatrix{0 \ar[r] & 𝔽_{p} \ar[r] & κ \ar[r]^{1−φ} & κ \ar[r] & 0,}

because (1 − φ)(x) = x − x^{p}. Inductively, one shows that, for all n ≥ 1, also

\xymatrix{0 \ar[r] & ℤ_{p}/p^{n}ℤ_{p} \ar[r] & R/p^{n}R \ar[r]^{1−φ} & R/p^{n}R \ar[r] & 0}

is exact. Thus for any given r = (r_{n})_{n} ∈ proj lim_{n} R/p^{n}R = R the sets S_{n} := {s_{n} ∈ R/p^{n}R|(1−φ)(s_{n}) = r_{n}} are finite and form an inverse system, whence S := proj lim_{n} S_{n} is non-empty and any s ∈ S satisfies (1 − φ)(s) = r by construction.

□

Let G be a finite p-group. The split exact sequence

\xymatrix{1 \ar[r] & I(R[G]) \ar[r] & R[G] \ar[r] & R \ar[r] & 1}

induces isomorphisms

K_{1}(R[G]) ≅ K_{1}(R[G], I(R[G])) ⊕ R^{×}

and

R[C_{G}] ≅ ϕ(I(R[G])) ⊕ R,

where ϕ : R[G] ↠ R[C_{G}] denotes the canonical map, C_{G} denoting the conjugacy classes of G. By Log(1 − x) we denote the logarithm series. Then the map \frac{1}{p}L = ϕ(\frac{1}{p}(p − Ψ)(Log(1 − x))) defined on page 13 of [4] induces by [4, cor. 3.3, thm. 3.17] and the lemma above a surjective map

Γ_{I(R[G])} : K_{1}(R[G], I(R[G])) ↠ ϕ(I(R[G])).

We use a generalization of Theorem 2.8 in [15] (for ideals contained in the Jacobson radical) in order to show, that this map is actually a group homomorphism, which together with the surjective homomorphism

Γ_{R} : R^{×} ↠ R,

which sends x ∈ 1 + 𝔪 to \frac{1}{p}(p − φ)Log(x) and x ∈ κ^{×} to zero (note that Log(1 + pR) = pR and that p − φ is an isomorphism of R), defines a surjective group homomorphism

Γ_{R[G]} = Γ_{I(R[G])} ⊕ Γ_{R} : K_{1}(R[G]) ↠ R[C_{G}],

which factorizes over

Γ_{Wh(R[G])} : Wh(R[G]) := K_{1}(R[G])/(G^{ab} × μ_{R}) ↠ R[C_{G}].

Setting

SK′_{1}(R[G]) := ker(Γ_{Wh(R[G])})

we obtain the following exact sequence

(2.8)  \xymatrix{1 \ar[r] & SK′_{1}(R[G]) \ar[r] & Wh(R[G]) \ar[r]^{Γ_{Wh(R[G])}} & R[C_{G}] \ar[r] & 1.}

The relation between SK′_{1}(R[G]) and the original SK_{1}(R[G]) will be cleared below.

Our goal is to prove the following

Theorem 2.36. Let G be a p-group. Then SK′_{1}(R[G]) = 1. In particular,

Wh(R[G]) ≅ R[C_{G}]

is torsion free and a Hausdorff topological group (the second group being a pseudocompact R-module).

Proof. The proof proceeds by induction on the order of G. If G is trivial, it is well-known that the SK′_{1}(R) = 1, because the kernel of Γ_{R} is just μ_{R}. Now assume G to be non-trivial. Then there exists a central element z ∈ G of order p. We set Ḡ := G/ < z > and write α : G ↠ Ḡ for the canonical projection. Consider the following commutative diagram with exact rows

\xymatrix{ &  & 0 \ar[d] & 0 \ar[d] \\  &  & ker(Wh(α)) \ar[d] \ar[r] & (1 − z)R[C_{G}] \ar[d] \ar[r] & 0 \\ 0 \ar[r] & SK′_{1}(R[G]) \ar[d]_{SK′_{1}(α)} \ar[r] & Wh(R[G]) \ar[d]_{Wh(α)} \ar[r]^{Γ_{Wh}} & R[C_{G}] \ar[d]_{H_{0}(α)} \ar[r] & 0 \\ 0 \ar[r] & SK′_{1}(R[Ḡ]) \ar[r] & Wh(R[Ḡ]) \ar[r]^{Γ_{Wh}} & R[CḠ] \ar[r] & 0,}

in which also the right column is exact by [4, lem. 3.9]. Here I_{z} denotes the ideal (1 − z)R[G].

An immediate generalization of [15, prop. 6.4] to our setting tells us that the logarithm induces an exact sequence

\xymatrix{1 \ar[r] & < z > \ar[r] & K_{1}(R[G], I_{z}) \ar[r]^{log} & H_{0}(G, I_{z}) \ar[r] & 0}

(note that τ in (loc. cit.) has to be replaced by the trivial map, because 1 − φ in Lem. 6.3 is surjective on κ; also [15, thm. 2.8] (for ideals contained in the Jacobson radical) needed for the proof generalizes immediately to our setting, because by [11, (20.4)] S := M_{n}(R[G]) is also semilocal and satisfies J(S)^{N} ⊆ pS for N sufficiently big).

Thus we obtain a commutative diagram with exact rows

\xymatrix{0 \ar[r] & B \ar[d] \ar[r] & K_{1}(R[G], I_{z})/ < z > \ar[d]_{log}^{≅} \ar[r] & ker(Wh(α)) \ar[d]_{Γ_{R[G]}} \ar[r] & 0 \\ 0 \ar[r] & H_{0}(G, (1 − z)R[Ω]) \ar[r] & H_{0}(G, (1 − z)R[G]) \ar[r] & (1 − z)R[C_{G}] \ar[r] & 0.}

By the Snake-lemma we see that

ker(SK′_{1}(α)) = ker(Wh(α)) ∩ SK′_{1}(R[G])
= H_{0}(G, (1 − z)R[Ω])/logB
= R[Ω]/ψ^{−1}(logB),

where ψ : R[Ω] ↠ H_{0}(G, (1 − z)R[Ω]) is induced by multiplication with (1 − z). Here, following [15] we write

Ω = {g ∈ G|g is conjugate to zg}

and note that our last term in the above equation corresponds to D/C in the proof of [15, thm. 7.1]. Hence, copying literally the same arguments and noting again that 1 − φ is surjective on R, we see from (c) on p. 176 in (loc. cit.) that C = ψ^{−1}(logB) = R[Ω]. In other words, ker(SK′_{1}(α)) is trivial. Since, by our induction hypothesis also SK′_{1}(R[Ḡ]) vanishes, the theorem is proved. □

Let L^{0} be the unique unramified algebraic extension of ℚ_{p} with residue field κ.

Corollary 2.37. For any open subgroup Δ ⊆ Gal(L^{0}/ℚ_{p}), there is an exact sequence

\xymatrix{1 \ar[r] & SK_{1}(R^{Δ}[G]) \ar[r] & K_{1}(R^{Δ}[G]) \ar[r] & K_{1}(R[G])^{Δ} \ar[r] & 1,}

and isomorphisms

H^{1}(Δ, μ_{R}) ≅ H^{1}(Δ, K_{1}(R[G]))

of continuous cochain cohomology groups and coinvariants

(μ_{R})_{Δ} ≅ K_{1}(R[G])_{Δ},

respectively.

Proof. Taking Δ-invariants of the exact sequence (of topological Hausdorff modules, cp. Corollary 2.38 for K_{1})

\xymatrix{1 \ar[r] & G^{ab} × μ_{R} \ar[r] & K_{1}(R[G]) \ar[r]^{Γ_{R[G]}} & R[C_{G}] \ar[r] & 0}

and noting the Galois invariance of Γ_{R[G]} (if we choose the arithmetic Frobenius in its definition) we obtain the following commutative diagram with exact rows (the first of which is the standard exact sequence as proved in [15])

\xymatrix{1 \ar[r] & G^{ab} × μ_{R^{Δ}} × SK_{1}(R^{Δ}[G]) \ar[d] \ar[r] & K_{1}(R^{Δ}[G]) \ar[d] \ar[r] & R^{Δ}[C_{G}] \ar[d]^{≅} \ar[r] & G^{ab} \ar@{^{(}->}[d] \ar[r] & 1 \\ 1 \ar[r] & G^{ab} × μ_{R}^{Δ} \ar[r] & K_{1}(R[G])^{Δ} \ar[r] & R[C_{G}]^{Δ} \ar[r] & H^{1}(Δ, μ_{R}) × G^{ab} \ar[r] & …}

from which the claim follows using [14, prop. 1.6.13]. Note that

H^{1}(Δ, R[C_{G}]) ≅ proj lim_{n} H^{1}(Δ, R/p^{n}R)^{#C_{G}}
≅ proj lim_{n}(R/p^{n}R)_{Δ}^{#C_{G}} = 0

by the straight forward generalization of [14, thm. 2.3.4] to pseudocompact modules, again [14, prop. 1.6.13] and Lemma 2.35. Alternatively, we may replace the long exact cohomology sequence above by the kernel/cokernel exact sequence arising from the Snake-lemma associated to multiplication by 1 − τ for any topological generator τ of Δ. □

Corollary 2.38. SK_{1}(R[G]) = 1, i.e., K_{1}(R[G]) ≅ Det(R[G]^{×}). In particular,

K_{1}(R[G]) ≅ proj lim_{n} K_{1}(R/p^{n}R[G]).

The last claim follows from the first one using [4, prop. 1.3]. For the proof of the first claim of the corollary consider the following diagram

(2.9)  \xymatrix{ &  & 1 \ar[d] \\  &  & SK_{1}(R[G]) \ar[d] \\ 1 \ar[r] & G^{ab} × μ_{R} \ar[r] & K_{1}(R[G]) \ar[d]_{Det} \ar[r]^{Γ_{R[G]}} & R[C_{G}] \ar@{^{(}->}[d]_{Tr} \ar[r] & 1, \\  &  & Hom(R_{G}, ℂ_{p}^{×}) \ar[r]^{Γ_{Hom}} & Hom(R_{G}, ℂ_{p})}

where Γ_{Hom} is defined as follows: Choose any continuous lift F : ℂ_{p} → ℂ_{p} of the absolute Frobenius automorphism and denote by log : ℂ^{×} → ℂ_{p} the usual p-adic logarithm. We write ψ^{p} and ψ_{p} for the pth Adams operator, which is characterized by

tr(g, ψ^{p}ρ) = tr(g^{p}, ρ) for all g ∈ G,

and its adjoint, respectively. Also the rule f(ρ) ↦ F (f(ρ^{F^{−1}})) induces an operator F̃ on Hom(R_{G}, ℂ_{p}^{×}) and Hom(R_{G}, ℂ_{p}), which commutes obviously with ψ_{p}. Now we set

Γ_{Hom}(f) = \frac{1}{p}(p − F̃ψ_{p})(log ∘f).

Finally, Tr, the additive analog of Det, is induced by

Tr(λ)(ρ) = tr(ρ(λ)),

where ρ : R[G] → M_{n}(ℂ_{p}) is the linear extension of ρ : G → GL_{n}(ℂ_{p}) keeping the same notation. One easily checks that

(2.10)  Det(F (λ))(ρ) = F̃(Det(λ))(ρ) = F Det(λ)(ρ^{F^{−1}})

and

(2.11)  Tr(Ψ(λ))(ρ) = F̃(Tr(ψ_{p}λ))(ρ) = F Tr(λ)(ψ_{p}ρ^{F^{−1}})

The Corollary will follow immediately from the following

Lemma 2.39. The diagram (2.9) commutes and Det restricted to G^{ab} × μ_{R} is injective.

Proof. The injectivity being well-known we only check the commutativity similarly as in [24, prop. 4.3.25]. Since K_{1}(R[G], I(R[G])) and K_{1}(R) generate K_{1}(R[G]) as has been observed above, it suffices to check this individually on each direct summand. The case of K_{1}(R) being similar but easier, we assume that a belongs to K_{1}(R[G], I(R[G])) and calculate using the definitions, (2.11), the continuity of ρ, the fact that log transforms det into tr and (2.10):

(Tr ∘ Γ(a))(ρ) = Tr(\frac{1}{p}(p − Ψ) log(a))(ρ)
= Tr(log(a))(ρ) − \frac{1}{p}F Tr(log(a)(ψ_{p}ρ^{F^{−1}})
= tr(log ρ(a)) − \frac{1}{p}F tr log(ψ_{p}ρ^{F^{−1}}(a))
= log(det(ρ(a))) − \frac{1}{p}F log(det(ψ_{p}ρ^{F^{−1}}(a))
= \frac{1}{p}\{p log Det(a)(ρ) − F log Det(a)(ψ_{p}ρ^{F^{−1}})\}
= \frac{1}{p}(p − F̃ψ_{p}) log Det(a)(ρ)
= (Γ_{Hom} ∘ Det(a))(ρ).
□

Remark 2.40. The case p = 2 should be treated separately, since Log(R^{×}) = Log(1 + 2R) = 4R + 2(1 − φ)R ≅ 2R, so that we have to replace R[C_{G}] by 2R[C_{G}] in the exact sequence (2.8), or, if we keep R[C_{G}] in (2.8), then we get a finite cokernel, which we denote by μ = ⟨−1⟩. Doing required corrections we can proof Corollaries 2.37 and 2.38 also in this case. Note, that the finite cokernel μ also appears in the first row of the commutative diagram in the proof of Corollary 2.37 (see [15, thm. 6.6]).

Lemma 2.41. The statement (i) of Theorem 2.23 is true also for R being the ring of Witt vectors of a p-closed algebraic extension κ of 𝔽_{p}.

Proof. The injectivity is obvious SK_{1}(R[G]) being trivial (see Corollary 2.38) and the surjectivity follows from the generalized Proposition 15 in [16]. □

Corollary 2.38 and the lemma above imply

Corollary 2.42. SK_{1}(S[G]) = 1 for any totally ramified integral extension S of R, where R is the ring of Witt vectors of a p-closed algebraic extension κ of 𝔽_{p}.

Finally, we want to generalize Corollaries 2.27 and 2.42 to the case of an arbitrary finite group G. For this we need

Theorem 2.43. Fix a prime p and a Dedekind domain R with field of fractions K, such that ℚ_{p} ⊆ K ⊆ ℂ_{p}. For any finite group G, let g_{1}, …, g_{k} be K-conjugacy class representatives for elements in G of order n_{i} = ord(g_{i}) prime to p, and set

N_{i} = N_{G}^{K}(g_{i}) = {x ∈ G : xg_{i}x^{−1} = g_{i}^{a}, some a ∈ Gal(K(ζ_{n_{i}})/K)}

and Z_{i} = C_{G}(g_{i}). Then SK_{1}(R[G]) is computable for induction from p-elementary subgroups of G and there is an isomorphism

SK_{1}(R[G]) ≅ ⨁_{i=1}^{k} H_{0}(N_{i}/Z_{i}; \underrightarrow{lim}_{H∈P(Z_{i})}SK_{1}(R[ζ_{n_{i}}][H])),

where P(Z_{i}) is the set of p-subgroups of Z_{i} and R[ζ_{n_{i}}] denotes the integral closure of R in K(ζ_{n_{i}}).

Proof. See [15, thm. 11.8 and thm. 12.5 ] □

Corollary 2.44. Let G be an arbitrary finite group. Let S be O_{M}, with M as in Corollary 2.27, but of finite absolute ramification index over ℚ_{p} (resp. S be a finite totally ramified extension of R, where R is the ring of Witt vectors of a p-closed algebraic extension κ of 𝔽_{p}), then SK_{1}(S[G]) = 1.

Proof. Use Corollaries 2.27, 2.42 and Theorem 2.43. Note also, that R[ζ_{n_{i}}] is a finite unramified extensions of R. □

End of the SK_{1}-part.

For a finite group G and S being the ring of integers of either an arbitrary tamely ramified extension L of ℚ_{p} (type 1) or the completion of a tamely ramified extension L of finite absolute ramification index over ℚ_{p}, whose residue field is pclosed (type 2). Let R = S^{Δ} be the fixed ring of S, where Δ is an open subgroup of Gal(L/ℚ_{p}) containing the inertia group. We write K and C for the kernel and cokernel of the map

i_{∗} : SK_{1}(R[G]) → SK_{1}(S[G])^{Δ},

respectively. Note that they are finite p-primary abelian groups. By K_{1}(R[G])_{ℚ} we denote rational K-groups K_{1}(R[G]) ⊗_{ℤ} ℚ. The Snake-lemma, Theorem 2.21 and the SK_{1}-part imply immediately

Theorem 2.45. Let p^{n} be the p-part of [L : L^{Δ}] (0 ≤ n ≤ ∞).

(1) The following sequence is exact

\xymatrix{1 \ar[r] & K \ar[r] & K_{1}(R[G]) \ar[r]^{i_{∗}} & K_{1}(S[G])^{Δ} \ar[r] & C \ar[r] & 1}

and induces

K_{1}(R[G])_{ℚ} ≅ K_{1}(S[G])_{ℚ}^{Δ}.

(2) If S is of type 1 and n = 0, then

K = 1 and C = 1.

(3) If S is either of type 2 or of type 1 with n = ∞, then we have

K ≅ SK_{1}(R[G]) and C = 1.

(4) Let G be a p-group and S be of type 1 with 0 < n < ∞, then

K ≅ SK_{1}(R[G])[p^{n}] and C ≅ SK_{1}(R[G])/p^{n}.

3. The case of residue class fields.

Let λ be an arbitrary (not necessary finite) Galois extension of 𝔽_{p} and G be a finite group. Let ϕ denote the Frobenius automorphism on λ, which takes x ∈ λ to x^{p}, then Gal(λ/𝔽_{p}) = \overline{⟨ϕ⟩}. Moreover, if 𝔽_{p^{n}} ⊂ λ, then 𝔽_{p^{n}} = λ\overline{⟨ϕ^{n}⟩}. We fix such an n, set κ = 𝔽_{p^{n}} and Δ = \overline{⟨ϕ^{n}⟩}. We are going to prove the following

Theorem 3.1. With the notation as above, we have an exact sequence

\xymatrix{1 \ar[r] & K \ar[r] & K_{1}(κ[G]) \ar[r]^{i_{∗}} & K_{1}(λ[G])^{Δ} \ar[r] & C \ar[r] & 1,}

which induces

K_{1}(κ[G])_{ℚ} ≅ K_{1}(λ[G])_{ℚ}^{Δ},

where K and C are as in Proposition 2.45 for R and S the unique unramified extensions of ℤ_{p} lifting κ and λ, respectively. As usual ϕ (resp. ϕ^{n}) acts on the K_{1}-groups coefficientwise.

Proof. From [22] we have an exact sequence

(3.1)  0 → ℤ_{p}[C_{G}] → K_{1}(ℤ_{p}[G]) → K_{1}(𝔽_{p}[G]) → 1,

where ℤ_{p}[C_{G}] is a finitely generated free ℤ_{p}-module over the set of conjugacy classes in G.

With the same argument as in [22], we can obtain (3.1) for finite unramified extensions of ℚ_{p} and their rings of integers, and, since K_{1} commutes with the direct limit, also for infinite unramified extensions. Using this fact we get the following commutative diagram with exact rows

1 ↓ K ↓

0 → R[C_{G}] → K_{1}(R[G]) → K_{1}(κ[G]) → 1
↓ ↓ ↓
0 → S[C_{G}]^{Δ} → K_{1}(S[G])^{Δ} → K_{1}(λ[G])^{Δ} → H^{1}(Δ, S[C_{G}]),

↓ C ↓ 1

where the bottom row is the part of the long exact sequence in the group cohomology associated to the short exact sequence of Δ-modules, and Δ acts coefficientwise.

The left hand side vertical map is an isomorphism, as R[C_{G}] and S[C_{G}] are finitely generated free R- and S-modules respectively. The middle column is exact by Prop. 2.45. Thus, to prove the theorem, it is enough to show, that H^{1}(Δ, S[C_{G}]) = 1.

From [10] we know, that H^{1}(Gal(M_{1}/M_{2}), O_{M_{1}}) = 1 for every finite unramified extension M_{1} of ℚ_{p}. Since Δ can be written as the inverse limit of finite groups corresponding to the finite unramified subextensions of S, and S[C_{G}] as a Δmodule is isomorphic to the direct sum of copies of S, we get the statement above by using standard properties of the cohomology of groups. □

References

[1] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97, 1960, pp. 367-409.

[2] T. Bouganis and O. Venjakob, On the non-commutative main conjecture for elliptic curves with complex multiplication, preprint, 2010.

[3] A. Brumer, Pseudocompact algebras, profinite groups and class formations, Journal of Algebra 4, 1966, pp. 442-470.

[4] T. Chinburg, G. Pappas and M. J. Taylor, K-1 of a p-adic group ring I. The determinantal image, Journal of Algebra, Manuscript Draft, 2009.

[5] T. Chinburg, G. Pappas and M. J. Taylor, K-1 of a p-adic group ring II. The determinantal kernel SK-1, to appear.

[6] C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol 1, John Wiley and Sons, New York, 1987.

[7] C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. 2, John Wiley and Sons, New York, 1987.

[8] A. Froehlich, Galois module structure of algebraic integers, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 1, 1983.

[9] T. Fukaya and K. Kato, A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory, preprint, 2004.

[10] Y. Furuta and Y. Sawada, On the Galois cohomology group of the ring of integers in a global field and its Adele ring, Nagoya Mathematical Journal Vol. 32, 1968, pp. 247-252.

[11] T.Y. Lam, A first course in noncommutative rings, Springer, Graduate Texts in Mathematics Vol. 131, second edition, 2001.

[12] J. Milne, Etale cohomology, Princeton: Princeton University Press, 1980.

[13] T. Nakayama und Y. Matsushima, Ueber die multiplikative Gruppe einer p-adischen Divisionsalgebra, Proc. Imp. Acad. Tokyo 19, 1943, pp. 622-628.

[14] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields, Springer, A Series of Comprehensive Studies in Mathematics Vol. 323, second edition, 2008.

[15] R. Oliver, Whitehead groups of finite groups, London Mathematical Society, Lecture Note Series 132, 1988.

[16] R. Oliver, SK-1 for finite group rings II, Math. Scand. 47, 1980, pp. 195-231.

[17] R. Oliver, SK-1 for finite group rings III, Algebraic K-theory, Evanston, 1980.

[18] J. Queyrut, S-Groupes des Classes d’un Ordre Arithmétique, Journal of Algebra 76, 1982, pp. 234-260.

[19] I. Reiner, Maximal orders, Academic Press, 1975.

[20] J. Ritter and A. Weiss, Toward equivariant Iwasawa theory, Springer, manuscripta math. 109, 2002, pp. 131–146.

[21] J. Rosenberg, Algebraic K-theory and Its Applications, Springer, Graduate Texts in Mathematics 147, 1994.

[22] P. Schneider, O. Venjakob, A splitting for K_{1} of completed group rings, preprint, 2010.

[23] J.-P. Serre, Linear representations of finite groups, Springer, Graduate Texts in Mathematics 42, 1977.

[24] V.P. Snaith, Explicit Brauer Induction (with applications to algebra and number theory), Cambridge studies in advanced mathematics 40, Cambridge University Press (1994).

[25] M. Taylor, Classgroups of Group Rings, London Mathematical Society, Lecture Note Series 91, 1984.

[26] C.T.C. Wall, Norms of units in group rings, Proc. L.M.S (3) 29, 1974, pp. 593-632.

Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany.

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

Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany.

E-mail address: venjakob@mathi.uni-heidelberg.de
URL: http://www.mathi.uni-heidelberg.de/~otmar/