On Adams Operations on the Higher K -Theory of Group Rings

Bernhard Köck

Abstract. Using Quillen’s universal transformation we verify some (standard) properties of Adams operations on the higher K -theory of projective modules over group rings. Furthermore, we rather explicitly describe Adams operations on the Whitehead group K_{1}(CΓ) associated with the group ring CΓ of a finite group Γ over an algebraically closed field of characteristic 0.

Introduction

Let Γ be a finite group, p a prime number, and L a finite extension of Q_{p}. Let O_{L}Γ denote the group ring associated with Γ and the ring of integers O_{L} in L and let K_{0}T (O_{L}Γ) denote the Grothendieck group of finite O_{L}Γ - modules possessing a O_{L}Γ -free resolution of length 1. This Grothendieck group is a fundamental object in the Galois module theory à la Fröhlich. In particular, it enters the picture in studying the projective class group of the group ring of Γ over the ring of integers in a number field.

The best way to describe K_{0}T (O_{L}Γ) in a little bit more familiar terms is the exact sequence

K_{1}(O_{L}Γ) → K_{1}(LΓ) → K_{0}T (O_{L}Γ) → 0

which comes from a localization sequence in K -theory (e. g. see sequence (1.3) of [T] on p. 2). It leads to the so-called Hom-description

K_{0}T (O_{L}Γ) ≅ \frac{Hom_{G}(K_{0}(L̄Γ), L̄^{×})}{Det((O_{L}Γ)^{×})}

of K_{0}T (O_{L}Γ) ; here G = Gal(L̄/L) denotes the absolute Galois group, K_{0}(L̄Γ) is the classical ring of virtual characters of Γ , and Det is the determinant map (e. g. see §2 of Chapter 1 in [T] for a precise definition of Det and Theorem 3.2 of [T] on p. 10 for a proof of the Hom-description). Using Taylor’s group logarithm techniques Cassou-Noguès and Taylor have shown that the k -th Adams operation ψ^{k} on K_{0}(L̄Γ) induces an operation on K_{0}T (O_{L}Γ) , if L/Q_{p} is non-ramified (e. g. see Theorem 1.2 of [T] on p. 98). The aim of this paper is to give a more or less satisfactory explanation of this operation in K -theoretical terms using power operations on modules.

For this we remark that we have constructed an Adams operation ψ^{k} on K_{1}(LΓ) using exterior power operations (see §3 of [Ko1]) and that, if p does not divide k, we have constructed an Adams operation ψ^{k} on K_{1}(O_{L}Γ) using generalizations of Atiyah’s cyclic power operations and shuffle products in higher K -theory (see section 3 of [Ko2]). In this paper we show that the homomorphism K_{1}(O_{L}Γ) → K_{1}(LΓ) in the above sequence commutes with ψ^{k} (see Corollary c) of Proposition 1). Hence ψ^{k} induces an operation ψ^{k} on K_{0}T (O_{L}Γ) , if p does not divide k. The question whether a p -th Adams operation ψ^{p} on K_{1}(O_{L}Γ) can be canonically defined remains open.

Furthermore we show that, via the well-known Hom-description

K_{1}(L̄Γ) ≅ Hom(K_{0}(L̄Γ), L̄^{×})

of K_{1}(L̄Γ) , the k -th exterior power operation λ^{k} on K_{1}(L̄Γ) (defined in §3 of [Ko1]) corresponds to the homomorphism on the right hand side induced by (−1)^{k−1}ψ̂^{k} where ψ̂^{k} is the adjoint of ψ^{k} on K_{0}(L̄Γ) with respect to the classical character pairing (see Theorem 1). One easily deduces from this that the k -th Adams operation ψ^{k} on K_{1}(L̄Γ) corresponds to the homomorphism on the right hand side induced by k · ψ̂^{k} (see Corollary 1 of Theorem 1).

If k is coprime to the order of Γ , the adjoint homomorphism ψ̂^{k} equals ψ^{k′} where k′ is a natural number which is an inverse of k modulo the order of Γ (see the proof of formula (1.7) of [T] on p. 101). This suggests that, up to a sign, the operation on K_{0}T (O_{L}Γ) defined by Cassou-Noguès and Taylor should be called an exterior power operation rather than an Adams operation. Since the canonical base change homomorphism K_{1}(LΓ) → K_{1}(L̄Γ) is injective (see Proposition 2.8 of [Que] on p. 247) the results explained above prove the more precise fact that, via the above Hom-description, the operation ψ^{k} on K_{0}T (O_{L}Γ) corresponds to the homomorphism on the right hand side induced by k · ψ^{k′}, if k is coprime to p and to the order of Γ . Thus it differs from the operation defined by Cassou-Noguès and Taylor by passing from k to k′ and by the factor k.

Finally we rather explicitly describe the K_{0}(L̄Γ) -module structure (see Theorem 2) and the Grothendieck filtration on K_{1}(L̄Γ) (see Proposition 6).

As a byproduct of the explicit description (of the K_{0}(L̄Γ) -module structure and) of the Adams operation on K_{1}(L̄Γ) we strengthen the induction formula (6.2) of [Ko3] in the situation considered in this paper (see Theorem 3).

In the appendix we give a new proof of Queyrut’s Hom-description (see section 3 of [Que]) of the Grothendieck group K_{0}(Γ, l) associated with finitely generated lΓ -modules where l is a field of positive characteristic.

Acknowledgements. The idea for the results of this paper was born during the preparation of a lecture for the K -theory conference in Poznań in September 1995. I would like to thank G. Banaszak, W. Gajda, and P. Krason for perfectly organizing an interesting meeting and for giving me the opportunity to report on my results.

1. Quillen’s Universal Transformation

The purpose of this section is to introduce some notations used throughout this paper and to recall Quillen’s universal transformation introduced in Hiller’s paper [Hi].

For any (abstract) group G and any (not necessarily commutative) G -ring A let K_{q}(G, A) denote Quillen’s q -th K -group associated with the exact category of finitely generated, projective A -modules with semilinear G - action. (If no action of G on A is given, the trivial action is meant and the action of G on modules is then assumed to be linear.) If G is the trivial group, we put K_{q}(A) := K_{q}(G, A) . Furthermore let

K̃_{0}(G, A) := ker(K_{0}(G, A) ⟶^{can} K_{0}(A))

be the reduced Grothendieck group. We obviously have K_{0}(G, A) = K_{0}(A)⊕ K̃_{0}(G, A) , if G acts trivially on A.

For any ring A let BGL(A)^{+} be the plus construction associated with the classifying space BGL(A) of the general linear group GL(A) = ∪_{n≥0}GL_{n}(A) (e. g. see [L]). For any CW-complex X let [X, BGL(A)^{+}] denote the set of free homotopy classes of free continuous maps from X to BGL(A)^{+} . This is the same as the set of pointed homotopy classes of pointed continuous maps from X to BGL(A)^{+} since BGL(A)^{+} is a connected H-space (see Theorem 9 of [S] on p. 384). Now Quillen’s natural transformation

q(−) : K̃_{0}(π_{1}(−), A) → [−, BGL(A)^{+}]

of functors from the category of connected finite CW-complexes to the category of groups (or more generally to the category of pointed sets) is defined as follows (see section 1 of [Hi]): Let X be a connected finite CW-complex and let ρ : π_{1}(X) → Aut(P) be a representation of the fundamental group π_{1}(X) of X on a finitely generated, projective A -module P . We choose a projective A -module P′ such that P ⊕ P′ is free over A, say of rank n. Then q(X)(ρ) is defined to be the homotopy class of the composition of the maps

X ⟶ Bπ_{1}(X) ⟶^{B(ρ⊕1)} BGL_{n}(A) ⟶^{can} BGL(A) ⟶^{can} BGL(A)^{+}.

Here X → Bπ_{1}(X) is the canonical 2-coskeleton. It is shown in Proposition 1.1 of [Hi] on p. 243 that q(X)(ρ) does not depend on the chosen module P′ and that the association ρ ↦ q(X)(ρ) induces a well-defined map q(X) : K̃_{0}(π_{1}(X), A) → [X, BGL(A)^{+}] .

Furthermore the natural transformation q(−) is universal in the following sense (see Corollary 2.3 of [Hi] on p. 246): For any H-space Z and for any natural transformation K̃_{0}(π_{1}(−), A) → [−, Z] there is a unique natural transformation [−, BGL(A)^{+}] → [−, Z] such that the following diagram commutes:

K̃_{0}(π_{1}(−), A)  ⟶^{q(−)}  [−, BGL(A)^{+}]
↘  ↙
[−, Z].

Note that the ring A need not to be commutative neither for the construction of q(−) nor for the proof of the universality of q(−) . In particular we may take a group ring for A.

2. On Adams Operations on K_{q}(R#Γ)

Let Γ be a group, R a commutative (noetherian) Γ -ring, and k a natural number which is invertible in R. Let A := R#Γ denote the associated twisted group ring. Using generalizations of Atiyah’s cyclic power operations, shuffle products in higher K -theory, and Grayson’s construction of power operations on higher K -theory we have constructed an Adams operation ψ^{k} on K_{q}(R#Γ) , q ≥ 0 , in [Ko2]. The object of this section is to carry over some properties of ψ^{k} proved in section 2 of [Ko2] for K_{0} to higher K -theory using Quillen’s universal transformation recalled in section 1.

Let l be a prime which is invertible in R. For any a ∈ ZZ/lZZ and for any R -module V let V [a]_{l} be the cyclic l -th power of V with eigenvalue ζ_{l}^{a}. It is defined as follows (see [Ko2]): Let S = R[ζ_{l}] = R[T]/(1 + … + T^{l−1}) be the l -th cyclotomic extension of R, c : V^{⊗l} → V^{⊗l}, v_{0} ⊗ … ⊗ v_{l−1} ↦ v_{l−1} ⊗ v_{0} ⊗… ⊗ v_{l−2} , the cyclic permutation on V^{⊗l}, and G := (ZZ/lZZ)^{×} the group of invertible elements in ZZ/lZZ acting on S as Galois group as usual and on V^{⊗l} via

σ(v_{0} ⊗… ⊗ v_{l−1}) := v_{σ^{−1}·0} ⊗… ⊗ v_{σ^{−1}·(l−1)} (for σ ∈ G, v_{0},…, v_{l−1} ∈ V).

Then V [a]_{l} is defined to be the G -fixed module associated with the ζ_{l}^{a} - eigenspace of the endomorphism 1 ⊗ c of S ⊗_{R} V^{⊗l} (see section 2 of [Ko2]). If V carries a (semilinear) Γ -action, then V [a]_{l} obviously does as well. If V is furthermore R -projective or R#Γ -projective, the same holds for V [a]_{l} (see Corollary b) of Proposition 1 in [Ko2]).

The main result of section 3 in [Ko2] is that the association V ↦ V [a]_{l} yields an operation [a]_{l} on K_{q}(R#Γ) and on K_{q}(Γ, R) for any q ≥ 0 . The l -th Adams operation on K_{q}(R#Γ) is then defined to be ψ^{l} := [0]_{l} − [1]_{l}. Now let X be a connected finite CW-complex. Applying this construction to the group π_{1}(X) × Γ in place of Γ we in particular obtain an operation [a]_{l} on K_{0}(A[π_{1}(X)]) = K_{0}(R#(π_{1}(X) × Γ)) and on K_{0}(π_{1}(X) × Γ, R) . Similarly one can define an operation [a]_{l} on K̃_{0}(π_{1}(X), A) . By the universality of the natural transformation q(−) this operation induces a natural transformation [a]_{l} on [−, BGL(A)^{+}] . In particular, we have once more defined an operation [a]_{l} on K_{q}(R#Γ) = K_{q}(A) = [S^{q}, BGL(A)^{+}] for any q ≥ 1 .

Proposition 1. The operation [a]_{l} on K_{q}(R#Γ) , q ≥ 1 , defined in this way agrees with the operation [a]_{l} defined in section 3 of [Ko2].

Proof. This can be proved similarly to section 9 of [Gr].

Corollary. Let q ≥ 0 .

a) For all x ∈ K_{0}(R#Γ) and y ∈ K_{q}(R#Γ) we have:

ψ^{l}(x · y) = ψ^{l}(x) · ψ^{l}(y) in K_{q}(R#Γ).

b) Let l′ be another prime which is invertible in R. Then we have for all x ∈ K_{q}(R#Γ) :

ψ^{l}(ψ^{l′}(x)) = ψ^{l′}(ψ^{l}(x)) in K_{q}(R#Γ).

c) If Γ is finite and the group order ord(Γ) is invertible in R, then ψ^{l} commutes with the Cartan homomorphism c : K_{q}(R#Γ) → K_{q}(Γ, R) .

Proof. These assertions have already been proved in section 2 of [Ko2], if q = 0 . So let q ≥ 1 .

a) Tensoring with a projective A -module over R transforms projective A - modules into projective A -modules (see Lemma 3 of [Ko2]). Hence by the usual techniques (see [Q] or [Hi]) we obtain a K_{0}(A) -module structure on K_{q}(A) . For any x ∈ K_{0}(A) the map

K_{q}(A) → K_{q}(A), y ↦ ψ^{l}(x) · ψ^{l}(y),

is the S^{q} -level of a natural transformation [−, BGL(A)^{+}] → [−, BGL(A)^{+}] which makes the following diagram commutative:

z  K̃_{0}(π_{1}(−), A) ⟶^{q(−)} [−, BGL(A)^{+}]
↓  ↓  ↓
ψ^{l}(x) · ψ^{l}(z) K̃_{0}(π_{1}(−), A) ⟶^{q(−)} [−, BGL(A)^{+}].

Since ψ^{l}(x) · ψ^{l}(z) = ψ^{l}(x · z) for all z ∈ K̃_{0}(π_{1}(X), A) (cf. Proposition 5 of [Ko2]) the natural transformation y ↦ ψ^{l}(x · y) makes the above diagram commutative as well. Now the universality of q(−) proves assertion a).

b) Similarly to a) this follows from the fact that ψ^{l} ∘ ψ^{l′} = ψ^{l′} ∘ ψ^{l} on K̃_{0}(π_{1}(X), A) (cf. Proposition 6 in [Ko2]).

c) If Γ is finite and ord(Γ) is invertible in R, an R#Γ -module is finitely generated and projective over R, if and only if it is finitely generated and projective over R#Γ . (This is an easy generalization of Maschke’s theorem.) Hence the Cartan homomorphism is an isomorphism and we only have to show that the Adams operation ψ^{l} on K_{q}(Γ, R) defined via cyclic power operations agrees with the Adams operation ψ^{l} defined via the l - th Newton polynomial in the exterior power operations λ_{1},…, λ_{l}. Similarly to a) this follows from the corresponding fact for K̃_{0}(π_{1}(X), A) = K̃_{0}(π_{1}(X) × Γ, R) (cf. Proposition 4 of [Ko2]). (The same proof even shows that [a]_{l} = C_{l,a}(λ^{1},…, λ^{l}) on K_{q}(R#Γ) = K_{q}(Γ, R) where C_{l,a} is the polynomial defined in Lemma 6 of [Ko2]).

Remark.

(i) Conjecturally assertion c) is true even when the group order is not invertible in R. At the end of the paper [Ko2] some speculations are presented how one should be able to prove this in the general case. (ii) Let k ∈ IN be invertible in R. Using a factorization of k into prime factors we may define an Adams operation ψ^{k} on K_{q}(R#Γ) for any q ≥ 1 . By assertion b) this does not depend on the ordering of the prime factors. (iii) If p := char(R) is a prime, we may define an Adams operation ψ^{k} even for all k ∈ IN by defining ψ^{p} to be the base change homomorphism associated with the Frobenius endomorphism (see last Remark in [Ko2]). If Γ is finite and ord(Γ) is invertible in R, one can deduce from Proposition 7 of [Ko2] similarly to the proof of assertion a) that ψ^{p} agrees with the usual Adams operation on K_{q}(R#Γ) = K_{q}(Γ, R) .

3. On Adams Operations on K_{1}(CΓ)

Let Γ be a finite group. The aim of this section is to describe the Adams operations and the K_{0}(CΓ) -module structure on K_{1}(CΓ) (defined in [Ko1] or [Ko2]) in explicit terms. This will enable us to strengthen the induction formula (6.2) of [Ko3] in this situation. Computing the Grothendieck filtration on K_{1}(CΓ) explicitly we will furthermore show that, for any subgroup Γ′ of Γ , the induction map K_{1}(CΓ′) → K_{1}(CΓ) is continuous with respect to the Grothendieck filtrations as conjectured in (5.6) of [Ko3].

More generally, let C be an algebraically closed field such that the group order of Γ is invertible in C and let CΓ denote the group ring associated with Γ and C. Then K_{0}(CΓ) is the classical ring of virtual characters of Γ . It is a free abelian group with basis the set S of isomorphism classes of simple finitely generated CΓ -modules. Furthermore K_{0}(ZZ, CΓ) can be identified with the Grothendieck group associated with the category of pairs (M, α) consisting of a finitely generated CΓ -module M and a CΓ -automorphism α of M. Let ZZ[S × C^{×}] denote the free abelian group with basis S × C^{×}.

Proposition 2. The group homomorphism

Φ : ZZ[S × C^{×}] → K_{0}(ZZ, CΓ) given by (S, β) ↦ (S, β)

is bijective.

Proof. We define an inverse map as follows: Let (M, α) be a pair as above. Then there are natural numbers n_{S}, S ∈ S, such that M ≅ ⊕_{S∈S}S^{n_{S}} and there are matrices A_{S} ∈ GL_{n_{S}}(C) , S ∈ S, such that α corresponds to ⊕_{S∈S}A_{S} under M ≅ ⊕S^{n_{S}} . For A ∈ GL_{n}(C) and β ∈ C let m_{β}(A) := dim_{C}ker ((A − β · id)^{∞}) . Then the association

(M, α) ↦ ∑_{(S,β)∈S×C^{×}} m_{β}(A_{S}) · (S, β)

obviously defines a group homomorphism

Ψ : K_{0}(ZZ, CΓ) → ZZ[S × C^{×}]

such that Ψ ∘ Φ = id . Furthermore for all matrices A ∈ GL_{n}(C) there is an upper triangular matrix B ∈ GL_{n}(C) which is equivalent to A. In other words, the ZZ - representation (S^{n}, A) is isomorphic to the ZZ -representation (S^{n}, B) for any S ∈ S. Now

0 ⊂ (S × 0 ×… × 0, B) ⊂ (S × S × 0 ×… × 0, B) ⊂ … ⊂ (S^{n}, B)

is a filtration of (S^{n}, B) by ZZ -representations whose successive quotients are isomorphic to (S, β) for some β ∈ C^{×} and the the pair (S, β) occurs precisely m_{β}(B) = m_{β}(A) times. This shows that (S^{n}, A) = ∑_{β∈C^{×}} m_{β}(A)· (S, β) in K_{0}(ZZ, CΓ) and hence that Φ ∘ Ψ = id .

According to Proposition 2 we have K_{0}(ZZ, CΓ) ≅ K_{0}(CΓ)[C^{×}] . Henceforth we will write elements of K_{0}(ZZ, CΓ) also in the form ∑_{β∈C^{×}} z_{β}[β] with z_{β} ∈ K_{0}(CΓ) for all β ∈ C^{×}. Let

⟨ , ⟩ : K_{0}(CΓ) × K_{0}(CΓ) →  K_{0}(C) ˜→ ZZ
(P, M)  ↦ Hom_{CΓ} (P, M) ˆ= dim_{C}Hom_{CΓ} (P, M)

be the classical character pairing. It is a perfect symmetric pairing. Let ⟨ , ⟩ also denote the following pairing:

⟨ , ⟩ : K_{0}(CΓ) × K_{0}(ZZ, CΓ) →  K_{0}(ZZ, C) (P, (M, α))  ↦ (Hom_{CΓ} (P, M), Hom_{CΓ} (P, α)) .

We will furthermore write ⟨ , ⟩ for the composition of ⟨ , ⟩ with the natural map

K_{0}(ZZ, C) ⟶^{det} Pic(ZZ, C) ˜→ C^{×}
(N, γ)  ↦ (Λ_{C}^{top}N, Λ_{C}^{top}γ) ↦ det(γ|_{N}).

Then for all x, z ∈ K_{0}(CΓ) and β ∈ C^{×} we obviously have ⟨x, z[β]⟩ = ⟨x, z⟩[β] in K_{0}(ZZ, C) respectively ⟨x, z[β]⟩ = β^{⟨x,z⟩} in C^{×}.

For any k ≥ 1 let λ^{k} and ψ^{k} denote the k -th exterior power operation and k -th Adams operation, respectively, on K_{0}(ZZ, CΓ) or K_{0}(CΓ) or K_{0}(ZZ, C) or K_{1}(CΓ) (see [Ko1] and section 2). Then for all z ∈ K_{0}(CΓ) and β ∈ C^{×} we obviously have

λ^{k}(z[β]) = λ^{k}(z)[β^{k}] and ψ^{k}(z[β]) = ψ^{k}(z)[β^{k}] in K_{0}(ZZ, CΓ).

Since the classical character pairing is perfect there is an adjoint homomorphism ψ̂^{k} on K_{0}(CΓ) associated with ψ^{k}. If k is coprime to ord(Γ) , we have ψ̂^{k} = ψ^{k′} where k′ is a natural number such that k · k′ ≡ 1 mod ord(Γ) (see the proof of formula (1.7) of [T] on p. 101). If G is abelian, one can easily show that ψ̂^{k} is induced by the association M ↦ CΓ ⊗_{CΓ} M where CΓ is considered as an CΓ -algebra via the k -multiplication on Γ .

Proposition 3. For all x ∈ K_{0}(CΓ) and y ∈ K̃_{0}(ZZ, CΓ) we have

⟨x, λ^{k}(y)⟩ = ⟨(−1)^{k−1}ψ̂^{k}(x), y⟩ in C^{×}.

Proof. Since for all z_{1} , z_{2} ∈ K_{0}(CΓ) and β_{1} , β_{2} ∈ C^{×} we have

⟨x, (z_{1}[β_{1}]−z_{1}[1])·(z_{2}[β_{2}]−z_{2}[1])⟩ = (β_{1}β_{2})^{⟨x,z_{1}·z_{2}⟩} ·β_{1}^{−⟨x,z_{1}·z_{2}⟩} ·β_{2}^{−⟨x,z_{1}·z_{2}⟩} = 1

both sides of the above formula are linear in y. Hence we may assume that y = z[β] − z[1] with some z ∈ K_{0}(CΓ) and β ∈ C^{×}. Using the equation λ′_{t}(z) · λ_{t}(−z) = \frac{d}{dt} log λ_{t}(z) = ∑_{j=1}^{∞}(−1)^{k−1}ψ^{k}(z)t^{k−1} of power series (see p. 23 of [FL]) we then obtain that

⟨x, λ^{k}(y)⟩ = ⟨x, λ^{k}(z)[β^{k}] + λ^{k−1}(z)λ^{1}(−z)[β^{k−1}] + … + λ^{k}(−z)[1]⟩

= β^{k⟨x,λ^{k}(z)⟩} · β^{(k−1)⟨x,λ^{k−1}(z)λ^{1}(−z)⟩} ·. . . · β^{⟨x,λ^{1}(z)λ^{k−1}(−z)⟩}
= β^{⟨x,kλ^{k}(z)+(k−1)λ^{k−1}(z)λ^{1}(−z)+...+λ^{1}(z)λ^{k−1}(−z)⟩}
= β^{⟨x,(−1)^{k−1}ψ^{k}(z)⟩}
= β^{⟨(−1)^{k−1}ψ̂^{k}(x),z⟩}
= ⟨(−1)^{k−1}ψ̂^{k}(x), y⟩

as was to be shown.

Now we will use the following description of the Whitehead group K_{1}(CΓ) = [S^{1}, BGL(CΓ)^{+}] : It is the factor group of the free abelian group with basis the isomorphism classes of pairs (M, α) as above modulo the relations defined e. g. on p. 348 of [Ba]. Furthermore, by Proposition 2.2 of [Que] on p. 244, the map

K_{1}(CΓ) → Hom_{ZZ}(K_{0}(CΓ), C^{×})
(M, α) ↦ (P ↦ det(Hom_{CΓ} (P, α)|_{Hom_{CΓ} (P,M)}))

is a well-defined group isomorphism.

Theorem 1. The following diagram commutes:

K_{1}(CΓ) ˜→ Hom_{ZZ}(K_{0}(CΓ), C^{×})
λ^{k} ↓  ↓ Hom((−1)^{k−1}ψ̂^{k}, C^{×})
K_{1}(CΓ) ˜→ Hom_{ZZ}(K_{0}(CΓ), C^{×}).

Proof. We consider the following diagram:

y  ↦  (x ↦ ⟨x, y⟩) K̃_{0}(ZZ, CΓ)  → Hom_{ZZ}(K_{0}(CΓ), C^{×}) ∥  ∥ K̃_{0}(ZZ, CΓ)  →^{D}  K_{1}(CΓ) ↘ ∥  ∥  GL(CΓ)^{ab} ↙ K̃_{0}(π_{1}(S^{1}), CΓ) →^{q(S^{1})}  [S^{1}, BGL(CΓ)^{+}] .

Here the map D is defined in such a way that the upper square commutes. Then we obviously have D(S[β] − S[1]) = (S, β) in K_{1}(CΓ) for all S ∈ S and β ∈ C^{×}. Let S′ be a CΓ -module such that S ⊕ S′ is free over CΓ , say of rank n, and let α ∈ GL_{n}(CΓ) be the automorphism of (CΓ)^{n} corresponding to β ⊕ id_{S′}. Then by Theorem (1.2)(1) of [Ba] on p. 448 the isomorphism K_{1}(CΓ) ˜→ [S^{1}, BGL(CΓ)^{+}] maps the class of (S, β) to the homotopy class of the continuous map

S^{1} = BZZ →^{Bα} BGL_{n}(CΓ) →^{can} BGL(CΓ) →^{can} BGL(CΓ)^{+}.

Since q(S^{1}) maps the element S[β] − S[1] of K̃_{0}(π_{1}(S^{1}), CΓ) to the same homotopy class (see section 1) the lower square commutes. Hence the map D commutes with λ^{k} by definition of λ^{k} on K_{1}(CΓ) . Now Proposition 3 proves Theorem 1 since D is surjective.

Remark. If more generally C is a commutative ring such that ord(Γ) is invertible in C, the above arguments essentially show that we have λ^{k}(S, β) = ((−1)^{k−1}ψ^{k}(S), β^{k}) in K_{1}(CΓ) for any CΓ -module S and β ∈ C^{×}.

Corollary 1. The following diagram commutes:

K_{1}(CΓ) ˜→ Hom_{ZZ}(K_{0}(CΓ), C^{×})  f
ψ^{k} ↓  ↓  ↓
K_{1}(CΓ) ˜→ Hom_{ZZ}(K_{0}CΓ), C^{×}) μ^{k} ∘ f ∘ ψ̂^{k},

where μ^{k}(β) = β^{k} for β ∈ C^{×}.

Proof. Since the multiplication on K_{1}(CΓ) is defined to be trivial we have ψ^{k} = (−1)^{k−1}kλ^{k}. Thus Theorem 1 proves Corollary 1. Alternatively Corollary 1 follows from the following proposition similarly to the proof of Theorem 1.

Proposition 4. For all x ∈ K_{0}(CΓ) and y ∈ K_{0}(ZZ, CΓ) we have

⟨x, ψ^{k}(y)⟩ = ψ^{k}(⟨ψ̂^{k}(x), y⟩) in K_{0}(ZZ, C).

Proof. We may assume that y = z[β] with some z ∈ K_{0}(CΓ) and β ∈ C^{×}. Then we have

⟨x, ψ^{k}(y)⟩ = ⟨x, ψ^{k}(z)[β^{k}]⟩
= ⟨ψ̂^{k}(x), z⟩[β^{k}] = ψ^{k}(⟨ψ̂^{k}(x), z⟩[β]) = ψ^{k}(⟨ψ̂^{k}(x), y⟩).

Corollary 2. Via the isomorphism K_{1}(CΓ) ≅ Hom_{ZZ}(K_{0}(CΓ), C^{×}) the Grothendieck operation γ^{k} on K_{1}(CΓ) corresponds to the (additively written) homomorphism

∑_{i=0}^{k−1}(−1)^{k−i−1}\binom{k − 1}{i}Hom(ψ̂^{k−i}, C^{×})

= (−1)^{k−1}Hom(ψ̂^{k}, C^{×}) + (−1)^{k−2}(k − 1)Hom(ψ̂^{k−1}, C^{×}) + … + id

on Hom(K_{0}(CΓ), C^{×}) .

Proof. By definition (see p. 47 of [FL]) we have γ^{k} = ∑_{i=0}^{k−1} \binom{k−1}{i}λ^{k−i}. Thus Theorem 1 proves Corollary 2.

Let

∗ : K_{0}(CΓ) → K_{0}(CΓ), P ↦ P^{∗} := Hom_{C}(P, C),

denote the dualizing map.

Proposition 5. For all x, z ∈ K_{0}(CΓ) and y ∈ K_{0}(ZZ, CΓ) we have

⟨x, z · y⟩ = ⟨z^{∗} · x, y⟩ in K_{0}(ZZ, C).

Proof. This follows from the canonical isomorphisms

Hom_{CΓ} (P ⊗_{C} Q, R) ≅ Hom_{CΓ} (Q, Hom_{C}(P, R)) ≅ Hom_{CΓ} (Q, P^{∗} ⊗_{C} R)

(for all finitely generated CΓ -modules P, Q, R).

Theorem 2. For all z ∈ K_{0}(CΓ) the following diagram commutes:

y K_{1}(CΓ) ˜→ Hom_{ZZ}(K_{0}(CΓ), C^{×})  f
↓  ↓  ↓  ↓
z · y K_{1}(CΓ) ˜→ Hom_{ZZ}(K_{0}(CΓ), C^{×}) (x ↦ f(z^{∗} · x)).

Proof. This can be deduced from Proposition 5 similarly to the proof of Theorem 1.

The next theorem strengthens the induction formula (6.2) of [Ko3] in the situation considered in this section. For this let Γ′ be a subgroup of Γ and let

i_{∗} : K_{1}(CΓ′) → K_{1}(CΓ), (M, α) ↦ (CΓ ⊗_{CΓ′} M, 1 ⊗ α),

be the induction map (see also section 6 in [Ko3]).

Theorem 3. For all y ∈ K_{1}(CΓ′) we have

ψ^{k}i_{∗}(y) = i_{∗}ψ^{k}(y) in \{ \begin{matrix}K_{1}(CΓ), & if (k, ord(Γ)) = 1 \\ K̂_{1}(CΓ)[k^{−1}], & else. \end{matrix}

Here K̂_{1}(CΓ)[k^{−1}] denotes the F^{1}K_{0}(CΓ)[k^{−1}] -adic completion of K_{1}(CΓ)[k^{−1}] .

Proof. Let i^{∗} : K_{0}(CΓ) → K_{0}(CΓ′) be the restriction map. Then by Frobenius reciprocity the diagram

K_{1}(CΓ′) ˜ ⟶ Hom(K_{0}(CΓ′), C^{×})
i_{∗} ↓  ↓ Hom(i^{∗}, C^{×})
K_{1}(CΓ) ˜ ⟶ Hom(K_{0}(CΓ), C^{×})

commutes. If (k, ord(Γ)) = 1 we have ψ̂^{k} = ψ^{k′} where k′ is a natural number such that k · k′ ≡ 1 mod ord(Γ) . Hence i^{∗} commutes with ψ̂^{k} and thus i_{∗} ˆ= Hom(i^{∗}, C^{×}) commutes with ψ^{k} ˆ= Hom(ψ̂^{k}, C^{×}) as was to be shown. If (k, ord(Γ)) ≠ 1 , this is proved in Theorem (6.2) of [Ko3] using the equivariant Adams-Riemann-Roch theorem. Alternatively this can be proved as follows without using geometric arguments: Let I := F^{1}K_{0}(CΓ) . By Example (6.8) of [Ko3] we have ψ^{k}i_{∗}(x) − i_{∗}ψ^{k}(x) ∈ ∩_{n≥0}I[k^{−1}]^{n} for all x ∈ K_{0}(CΓ′) . Hence by Frobenius reciprocity we have f ∘ψ̂^{k}∘i^{∗}−f ∘i^{∗}∘ψ̂^{k} ∈ ∩_{n≥0}I^{n}Hom(K_{0}(CΓ), ZZ)[k^{−1}] for all f ∈ Hom(K_{0}(CΓ′), ZZ) . Here the multiplication of I^{n} ⊆ K_{0}(CΓ) on Hom(K_{0}(CΓ), ZZ) is defined as in Theorem 2. Writing a homomorphism f ∈ Hom(K_{0}(CΓ′), C^{×}) as the image of a map K_{0}(CΓ′) → ZZ[β_{1}] ⊕ … ⊕ ZZ[β_{r}] we deduce from this that f ∘ ψ̂^{k} ∘ i^{∗} − f ∘ i^{∗} ∘ ψ̂^{k} ∈ ∩_{n≥0}I^{n}Hom(K_{0}(CΓ), C^{×})[k^{−1}] . Now Corollary 1 of Theorem 1 and Theorem 2 prove Theorem 3.

Now we are going to describe the Grothendieck filtration on K_{1}(CΓ) . For this let K(CΓ) := K_{0}(CΓ) ⊕ K_{1}(CΓ) be equipped with the ring structure induced by the ring structure on K_{0}(CΓ) , by the K_{0}(CΓ) -module structure on K_{1}(CΓ) , and by the trivial multiplication on K_{1}(CΓ) . Then the exterior power operations on K_{0}(CΓ) and K_{1}(CΓ) make K(CΓ) a λ -ring (see §2 of [Ko1]). Let

F^{1}K(CΓ) := ker(K(CΓ) ⟶^{can} K_{0}(CΓ) ⟶^{dim} ZZ)

be the augmentation ideal and let (F^{n}K(CΓ))_{n≥0} be the associated Grothendieck filtration. Recall that F^{n}K(CΓ) is generated as abelian group by the elements

γ^{n_{1}}(z_{1}) ·. . . · γ^{n_{r}}(z_{r}), z_{1},…, z_{r} ∈ F^{1}K(CΓ), n_{1} + … + n_{r} ≥ n.

The ideal F^{n}K(CΓ) is obviously a homogeneous ideal, i. e. F^{n}K(CΓ) = F^{n}K_{0}(CΓ) ⊕ F^{n}K_{1}(CΓ) with a certain subgroup F^{n}K_{1}(CΓ) of K_{1}(CΓ) .

Proposition 6. We have

F^{n}K_{1}(CΓ) = \{ \begin{matrix}K_{1}(CΓ) & for n = 0, 1 \\ ker(K_{1}(CΓ) ⟶^{can} K_{1}(C)) & for n ≥ 2. \end{matrix}

In particular, the induction map i_{∗} : K_{1}(CΓ′) → K_{1}(CΓ) is continuous with respect to the Grothendieck filtrations as conjectured in (5.6) of [Ko3].

Proof. This is clear for n = 0, 1 . So let n ≥ 2 . Since F^{n}K_{1}(C) obviously vanishes for n ≥ 2 we have F^{n}K_{1}(CΓ) ⊆ ker(K_{1}(CΓ) → K_{1}(C)) =: K. Using the isomorphisms K ⊕ K_{1}(C) ≅ K_{1}(CΓ) ≅ Hom(K_{0}(CΓ), C^{×}) we see that the multiplication with ord(Γ) on K is surjective. Hence we have K = ord(Γ) · K ⊆ F^{2}K_{1}(CΓ) by Proposition (6.1) of [Ko3]. By induction on n we thus obtain the reverse inclusion

K = ord(Γ) · K = ord(Γ) · F^{n−1}K_{1}(CΓ) ⊆ F^{n}K_{1}(CΓ)

for all n ≥ 2 again by Proposition (6.1) of [Ko3]. Since the diagram

K_{1}(CΓ′) →  K_{1}(C)
i_{∗} ↓  ↓ [Γ : Γ′]
K_{1}(CΓ) →  K_{1}(C)

obviously commutes we finally obtain that i_{∗}(F^{n}K_{1}(CΓ′)) ⊆ F^{n}K_{1}(CΓ) for all n ≥ 0 . In particular, i_{∗} is continuous with respect to the Grothendieck filtrations.

Appendix

Let Γ be a finite group and let l be a finite field. In this appendix we will present a new proof for the Hom-description of the Grothendieck group K_{0}(Γ, l) given by Queyrut in section 3 of [Que].

To recall this Hom-description, let L be a local field of characteristic 0 with residue classfield l. Let L_{nr} be the maximal non-ramified extension in the algebraic closure L̄ of L. Then the residue classfield of L_{nr} (and L̄) is an algebraic closure l̄ of l. Let O_{L_{nr}} and OL̄ denote the ring of integers in L_{nr} and L̄, respectively. Let G_{L} := Gal(L̄/L) and G_{l} := Gal(l̄/l) denote the corresponding absolute Galois groups. We will identify G_{l} with Gal(L_{nr}/L) . Our proof of the Hom-description will be based on the following two facts:

(i) In the Swan triangle

K_{0}(l̄Γ)  \underrightarrow{e}  K_{0}(L̄Γ)
c ↘  ↙ d
K_{0}(Γ, l̄)

all homomorphisms are compatible with the obvious G_{L} -action (see [Se]).

(ii) The pairing

K_{0}(l̄Γ) × K_{0}(Γ, l̄) →  ZZ
(P, M)  ↦ diml̄Hom_{l̄Γ} (P, M)

induces an isomorphism of groups

K_{0}(Γ, l) ˜→ Hom_{G_{l}}(K_{0}(l̄Γ), ZZ)

via base extension (see Théorème 2.7 of [Que] on p. 247).

Let v : L̄^{×} → Q be the valuation normalized by v(π_{L}) = 1 where π_{L} is a prime element of L (and then of L_{nr} as well). We put

H(l, Γ) := {f ∈ Hom_{G_{L}}(K_{0}(L̄Γ), L̄^{×}) : f(Image(e)) ⊆ O× ; L̄}.

Theorem 4 (Hom-description of K_{0}(Γ, l) ). The homomorphism

Hom_{G_{L}}(K_{0}(L̄Γ), L̄^{×}) → Hom_{G_{l}}(K_{0}(l̄Γ), ZZ)
f  ↦ (y ↦ v ∘ f ∘ e(y))

induces an isomorphism

\frac{Hom_{G_{L}}(K_{0}(L̄Γ), L̄^{×})}{H(l, Γ)} ˜ ⟶ K_{0}(Γ, l).

Proof. Note that v ∘ f ∘ e(y) lies in ZZ since e(y) and hence f ∘ e(y) is fixed by the inertia group. The split exact sequence

0 → O_{L_{nr}}^{×} → L_{nr}^{×} →^{v} ZZ → 0

of G_{l} -modules induces the exact sequence

0 → Hom_{G_{l}}(K, O_{L_{nr}}^{×}) → Hom_{G_{l}}(K, L_{nr}^{×}) → Hom_{G_{l}}(K, ZZ) → 0

where K := K_{0}(l̄Γ) . Since K is fixed by the inertia group we have Hom_{G_{l}}(K, L_{nr}^{×}) = Hom_{G_{L}}(K, L̄^{×}) and Hom_{G_{l}}(K, O_{L_{nr}}^{×}) = Hom_{G_{L}}(K, O× ; L̄) . Furthermore the map e : K → K_{0}(L̄Γ) of the Swan triangle is a direct injection of G_{L} -modules (see [Se]). Hence the pull-back diagram

0 → H(l, Γ) → Hom_{G_{L}}(K_{0}(L̄Γ), L̄^{×}) → Hom_{G_{l}}(K, ZZ) → 0
↓  ↓ e  ∥
0 → Hom_{G_{L}}(K, O× ; L̄) →  Hom_{G_{L}}(K, L̄^{×})  → Hom_{G_{l}}(K, ZZ) → 0

proves Theorem 4.

References

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Math. Institut II der Universität Karlsruhe, D-76128 Karlsruhe, Germany

E-mail address: bk@ma2s2.mathematik.uni-karlsruhe.de