Operations on Locally Free Classgroups

by

Bernhard Köck

Abstract. Let Γ be a finite group and K a number field. We show that the operation ψ_{k} defined by Cassou-Noguès and Taylor on the locally free classgroup Cl(O_{K}Γ) is a symmetric power operation if gcd(k, ord(Γ)) = 1. Using the equivariant Adams-Riemann-Roch theorem, we furthermore give a geometric interpretation of a formula established by Burns and Chinburg for these operations.

Introduction

Exterior or symmetric powers of modules play an important role in algebraic K-theory, in representation theory of finite groups, in number theory, or in algebraic geometry. The object of this paper is to work out their central role in a fascinating interplay of these four subjects we are now going to describe.

The k-th Adams operation ψ^{k} on the classical ring K_{0}(ℂΓ) of virtual characters of a finite group Γ can be defined in the following two ways: In terms of modules, ψ^{k} is given by

ψ^{k}(M) = N_{k}(Λ_{ℂ}^{1}(M), …, Λ_{ℂ}^{k}(M));

i.e., the exterior powers Λ_{ℂ}^{1}(M), …, Λ_{ℂ}^{k}(M) of the representation M of Γ are plugged into the k-th Newton polynomial N_{k} = t_{1}^{k} + ⋯ + t_{k}^{k} considered as a polynomial in the elementary symmetric functions of t_{1}, …, t_{k}. In terms of characters, ψ^{k} is given by

ψ^{k}(χ)(γ) = χ(γ^{k})

(for γ ∈ Γ and χ ∈ K_{0}(ℂΓ) a character).

Now let K be a number field. Let K_{0}(O_{K}Γ) denote the Grothendieck group of all projective modules over the group ring O_{K}Γ and let

Cl(O_{K}Γ) = ker(rank : K_{0}(O_{K}Γ) → ℤ)

denote the locally free classgroup associated with K and Γ. Fröhlich has established an isomorphism between Cl(O_{K}Γ) and a certain residue class group of the group

Hom_{Gal(K̄/K)}(K_{0}(K̄Γ), I(K̄))

consisting of Galois-invariant homomorphisms from K_{0}(K̄Γ) = K_{0}(ℂΓ) to the group I(K̄) of ideles of K̄ (e.g., see Theorem 1 on p. 20 in [F1]). Using Taylor’s group logarithm techniques, Cassou-Noguès and Taylor have shown that, under certain additional assumptions, the endomorphism Hom(ψ^{k}, I(K̄)) of Hom_{Gal(K̄/K)}(K_{0}(K̄Γ), I(K̄)) induces an endomorphism on Cl(O_{K}Γ) (e.g., see Chapter 9 in [T]). This endomorphism will be denoted by ψ_{k}^{CNT} in the following considerations. In the paper [BC], Burns and Chinburg have studied the question whether there is an algebraic description of ψ_{k}^{CNT}, for instance in terms of power operations on O_{K}Γ -modules. Their main result essentially is a formula for ψ_{k}^{CNT}(𝔄) where 𝔄 is an arbitrary Γ -stable fractional ideal in a tame Galois extension N/K with Galois group Γ. Their formula expresses ψ_{k}^{CNT}(𝔄) in terms of powers of the different 𝔇_{N/K} and of powers of the ideal 𝔄 (see Corollary 2.4 in [BC]).

The central result of this paper is the identification of the endomorphism ψ_{k}^{CNT} with a symmetric power operation on K_{0}(O_{K}Γ) in the case gcd(k, ord(Γ)) = 1. The aim of this paper finally is to give a geometric interpretation of the formula of Burns and Chinburg, namely as an equivariant Adams-Riemann-Roch formula for the Γ -morphism Spec(O_{N}) → Spec(O_{K}).

The following fact (see Proposition 1.1) is the fundamental observation for our central result: Let R be a commutative ring and M a projective module over the group ring RΓ; if gcd(k, ord(Γ)) is invertible in R, then the k-th exterior power Λ_{R}^{k}(M) and the k-th symmetric power Sym_{R}^{k}(M) are projective over RΓ again; here, as usual, Γ acts on the exterior or symmetric power diagonally.

A slightly more general form of this observation will enable us to define a k-th symmetric power operation σ^{k} on the Grothendieck group K_{0}(RΓ) and, more generally, on Quillen’s higher K-groups K_{q}(RΓ), q ≥ 0, (see section 1). An obvious generalization of the moduletheoretic definition of ψ^{k} on K_{0}(ℂΓ) explained above furthermore yields an (additive) k-th Adams operation ψ^{k} on K_{q}(RΓ), q ≥ 0, if gcd(k, ord(Γ)) is invertible in R (see section 1). In Theorem 1.6, we will prove several standard properties of these operations such as e.g. the compatibility with the Cartan homomorphism. Moreover, Theorem 1.6 contains a proof of the identity

ψ^{k}([RΓ]) = [RΓ]

conjectured in (1.12) in Chapter 9 of [T].

In section 2, we will show that ψ^{k} is multiplicative. Since there is no splitting principle available, this cannot be proved as e.g. in [FL]. Instead, we will use the universal form of the Cauchy decomposition and of the Pieri formula (established by Akin, Buchsbaum, and Weyman in characteristic-free representation theory) in order to express the exterior power Λ^{i}(M ⊗ N) of the tensor product of two modules as a universal polynomial in the exterior powers of M and N. It remains an open problem how to prove that ψ^{k} ∘ ψ^{l} = ψ^{kl} on K_{0}(RΓ) (here, l denotes another natural number such that gcd(l, ord(Γ)) is invertible in R).

We now assume that gcd(k, ord(Γ)) = 1 and fix a k′ ∈ ℕ with kk′ ≡ 1 mod ord(Γ). By the constructions explained above, we have operations σ^{k} and ψ^{k} on K_{0}(O_{K}Γ) and, in particular, on Cl(O_{K}Γ) by restricting. The precise form of our central result now is:

ψ_{k′}^{CNT} = σ^{k} and k · ψ_{k′}^{CNT} = ψ^{k} on Cl(O_{K}Γ)

(see Theorem 3.7). This result in particular answers the question (raised in (1.12) in Chapter 9 of [T]) how the endomorphism ψ_{k}^{CNT} behaves with respect to the Cartan homomorphism (see Corollary 3.9).

Now let N/K be a tame Galois extension with Galois group Γ. In section 4, we will essentially establish the following simple representation of the cotangential element T_{f}^{∨} associated with the Γ -morphism f : Spec(O_{N}) → Spec(O_{K}):

T_{f}^{∨} = [𝔇_{N/K}^{−1}] − [O_{N}].

Here, as above, 𝔇_{N/K} denotes the different associated with N/K.

This representation of T_{f}^{∨} implies a particularly simple shape of the Bott multiplier and then of the equivariant Adams-Riemann-Roch formula for f (see Theorem 5.4). On the other hand, we will reformulate the formula of Burns and Chinburg mentioned above using the central result Theorem 3.7 (see Theorem 5.6). It will then turn out that, more or less, the formula of Burns and Chinburg is a strengthening of the equivariant Adams-Riemann-Roch formula.

Acknowledgments. First of all, I would like to thank C. Greither for inviting me to stay with him at Université Laval in Québec for a few weeks in September/October 1996. In discussions during this visit, he made the observation Proposition 1.1 which has later turned out to be the most important tool in understanding the operations on locally free classgroups defined by Cassou-Noguès and Taylor. Moreover, he has contributed several algebraic arguments which have considerably improved the paper. It is a pleasure for me to thank him for all this assistance. Furthermore, I would like to thank J. Weyman for the reference to the universal form of the Pieri formula and for long e-mail discussions about the plethysm problem. Finally, I would like to thank D. Burns who asked me some time ago how to geometrically interpret the formula he developed jointly with T. Chinburg.

Notations. Let Γ be a finite group and let R be a commutative Γ -ring, i.e., a commutative ring together with an action of Γ by ring automorphisms. The twisted group ring associated with Γ and R is denoted by R#Γ. If Γ acts on R trivially, we simply write RΓ for R#Γ. We denote the Grothendieck group of all finitely generated projective R#Γ -modules (of all finitely generated R-projective R#Γ -modules, of all finitely generated R#Γ -modules) by K_{0}(R#Γ), (by K_{0}(Γ, R), and by K′_{0}(R#Γ), respectively). We have a canonical homomorphism c : K_{0}(R#Γ) → K_{0}(Γ, R) which is called the Cartan homomorphism. If the order of the group Γ is invertible in R, c is bijective (e.g., see the proof of Corollary 2.2c) in [Ko3]) and K_{0}(R#Γ) will be identified with K_{0}(Γ, R). Furthermore, we have a canonical homomorphism K_{0}(Γ, R) → K′_{0}(R#Γ). If R is regular (e.g., if R is a field or a ring of integers in a number field), this homomorphism is bijective (e.g., see Satz (2.1) in [Ko1]) and K_{0}(Γ, R) will be identified with K′_{0}(R#Γ). Thus, if R is a field of characteristic zero, all three Grothendieck groups coincide. The corresponding higher K-groups will be denoted by K_{q}(R#Γ), K_{q}(Γ, R), K′_{q}(R#Γ), q ≥ 0, (see [Q]). The corresponding identifications hold for higher K-groups as well. For K_{1}(R#Γ), we will use also the following two descriptions (e.g., see [B]): By the third theorem on p. 228 in [Gr1], K_{1}(R#Γ) is isomorphic to the abelianized infinite linear group GL_{∞}(R#Γ)^{ab}; furthermore, K_{1}(R#Γ) is the free Abelian group over all pairs (M, α), where M is a f. g. projective R#Γ -module and α is an R#Γ -automorphism of M, modulo the relations defined on p. 348 in [B]. The notation K_{0}T (−) will always stand for a Grothendieck group of torsion modules.

§1 Power Operations for Projective Modules over Group Rings

Let Γ be a finite group, R a commutative Γ -ring and k a natural number such that gcd(k, ord(Γ)) is invertible in R.

The aim of this section is to construct a symmetric power operation σ^{k}, an exterior power operation λ^{k}, and an Adams operation ψ^{k} on Quillen’s higher K-groups K_{q}(R#Γ), q ≥ 0, in particular on the Grothendieck group K_{0}(R#Γ) of all f. g. projective R#Γ -modules. Furthermore, we will prove several standard properties of these operations and we will show the equality ψ^{k}([R#Γ]) = [R#Γ] if gcd(k, ord(Γ)) = 1. The following proposition is the fundamental observation for this equality and for the whole section.

Proposition 1.1. Let k_{1}, …, k_{r} ∈ ℕ such that gcd(k_{1}, …, k_{r}, ord(Γ)) is invertible in R. (For instance, this holds if k_{1} + … + k_{r} = k or if at least one k_{i} is 1.) Let M_{1}, …, M_{r} be projective R#Γ -modules. Then the R#Γ -module

P := Sym^{k_{1}}(M_{1}) ⊗ ⋯ ⊗ Sym^{k_{r}}(M_{r})

is projective over R#Γ, too. (Here, all tensor products and all symmetric powers in P are formed over R and Γ acts on P diagonally.) If moreover gcd(k_{1}, …, k_{r}, ord(Γ)) = 1 and M_{1}, …, M_{r} are free over R#Γ, then P is free over R#Γ, too.

Proof. First, let r = 1. We write k for k_{1}. Let M be a free R#Γ -module and B ⊂ M an R-basis of M which carries a free action of Γ. For instance, the basis consisting of the Γ -images of an R#Γ -basis of M is such a basis. Let B^{k}/Σ_{k} denote the set of unordered k-tuples in B. For any H = (b_{1}, …, b_{k}) ∈ B^{k}/Σ_{k}, let b_{H} := ∏_{j=1}^{k} b_{j} denote the corresponding standard basis element of Sym^{k}(M). Then we obviously have γ(b_{H}) = b_{γ(H)} for all γ ∈ Γ, where the action of Γ on B^{k}/Σ_{k} is defined in the obvious way. For any H ∈ B^{k}/Σ_{k}, the stabilizer Stab(H) := {γ ∈ Γ : γ(H) = H} clearly acts on the sets H(n) := {b ∈ B : b occurs n times in H}, n ≥ 1, and this action is free by assumption. Thus, the order ord(Stab(H)) divides the number #H(n) of elements in H(n) for all n. Hence, it divides k = ∑_{n=1}^{∞} n · (#H(n)), too. Now, let r be arbitrary and M_{1}, …, M_{r} free R#Γ -modules. Furthermore, let B_{1}, …, B_{r} be bases of M_{1}, …, M_{r} as above. Then, for any H = (H^{1}, …, H^{r}) ∈ B_{1}^{k_{1}}/Σ_{k_{1}} × ⋯ × B_{r}^{k_{r}}/Σ_{k_{r}}, the element b_{H} := b_{H^{1}} ⊗ ⋯ ⊗ b_{H^{r}} is a standard basis element of P with γ(b_{H}) = b_{γ(H)}. The order ord(Stab(H)) of the stabilizer of H divides the order of the stabilizer of H^{i} for all i. Then, by the first part of the proof, it divides gcd(k_{1}, …, k_{r}, ord(Γ)), too. Hence, ord(Stab(H)) is invertible in R. Thus, the canonical R#Γ -linear surjection R#Γ → R#[Γ/Stab(H)], γ ↦ [γStab(H)], can be split by the well-defined R#Γ -linear map

R#[Γ/Stab(H)] → R#Γ, [γStab(H)] ↦ \frac{1}{ord(Stab(H))} ∑_{γ′∈Stab(H)}[γγ′].

This proves that the R#Γ -module R#[Γ/Stab(H)] is projective over R#Γ. The R#Γ - submodule generated by b_{H} obviously is isomorphic to R#[Γ/Stab(H)]. Thus, it is R#Γ - projective, too. Being a direct sum of such modules, also P is projective. This shows the first assertion in Proposition 1.1 for free R#Γ -modules M_{1}, …, M_{r} and, by passing to direct summands, also for projective R#Γ -modules. If finally gcd(k_{1}, …, k_{r}, ord(Γ)) = 1, then Stab(H) is trivial. Hence, P is a free R#Γ - module.

Now we are going to construct the power operations mentioned above. For this, it is convenient to define the following exact subcategories M_{i}, i = 1, 2, … of the exact category M of all f. g. R-projective R#Γ -modules: Let M_{i} be the smallest full subcategory of M which is closed under extensions and kernels of R#Γ -epimorphisms and which contains all modules of the form Sym^{k_{1}}(M_{1}) ⊗ ⋯ ⊗ Sym^{k_{r}}(M_{r}), where M_{1}, …, M_{r} are f. g. projective R#Γ -modules and k_{1}, …, k_{r} are natural numbers with k_{1} + ⋯ + k_{r} = i. We call the modules in M_{i} modules of weight i. Obviously, M_{1} is the category of all f. g. projective R#Γ -modules. By Proposition 1.1, the category M_{k} is contained in M_{1}.

Lemma 1.2. For all i, j ∈ ℕ, the tensor product induces a functor

⊗ : M_{i} × M_{j} → M_{i+j}.

Proof. A module of weight i is called of level 0 iff it is of the form Sym^{k_{1}}(M_{1}) ⊗ ⋯ ⊗ Sym^{k_{r}}(M_{r}) as above. Inductively, a module of weight i is called of level ≤ m iff it is an extension or the kernel of an R#Γ -epimorphism of two modules of weight i and level ≤ m − 1. Then one easily shows by induction that the tensor product of a module of weight i and level ≤ m with a module of weight j and level ≤ n is a module of weight i + j and level ≤ m + n. Since any module of weight i or j is of finite level, this shows Lemma 1.2.

For any exact category P and for any i ≥ 1, let F_{i}(P) denote the category of all sequences M_{1} ↪ ⋯ ↪ M_{i} of admissible monomorphisms in P of length i. Here, a monomorphism M′ → M in P is called admissible iff it can be completed to a short exact sequence 0 → M′ → M → M′′ → 0 in P.

Lemma 1.3. For any sequence M_{1} ↪ ⋯ ↪ M_{i} in F_{i}(M_{1}), the R#Γ -module

M_{1} ⋯ M_{i} := Image(M_{1} ⊗ ⋯ ⊗ M_{i} ⟶^{can} Sym^{i}(M_{i}))

is contained in M_{i}.

Proof. In the canonical filtration

M_{1} ⋯ M_{i} ⊆ M_{1} ⋯ M_{i−2}M_{i}M_{i} ⊆ ⋯ ⊆ Sym^{i}(M_{i})

of Sym^{i}(M_{i}), each successive quotient is isomorphic to one of the modules M_{1} ⋯ M_{r−1} ⊗ Sym^{i−r+1} (\frac{M_{i}}{M_{r}} ), r = 1, …, i. Now, Lemma 1.3 follows from Lemma 1.2 by induction on i.

It is easy to prove that the exact categories M_{1}, M_{2}, … together with the tensor products ⊗ : M_{i} × M_{j} → M_{i+j}, i, j ∈ ℕ, (well-defined by Lemma 1.2) and the symmetric power operations F_{i}(M_{1}) → M_{i}, (M_{1} ↪ ⋯ ↪ M_{i}) ↦ M_{1} ⋯ M_{i}, (well-defined by Lemma 1.3) satisfy the axioms (E1) through (E5) in section 7 in [Gr2]. Unlike Grayson in section 7 in [Gr2], we cannot define operations F_{i}(M_{j}) → M_{ij} for j > 1, since it is not clear whether the i-th symmetric power Sym^{i}(M) of a module M of weight j is a module of weight ij. This problem is related to the so-called plethysm problem (see also Remark 2.6). Nevertheless, we may apply the construction of section 7 in [Gr2] for j = 1. It yields continuous maps

σ^{i} : |GM_{1}| → |G^{i}M_{i}|, i ≥ 1,

from the geometric realization of the G-construction associated with M_{1} to the geometric realization of the i-fold iterated G-construction associated with M_{i}. By passing to homotopy groups, we obtain symmetric power operations

σ^{i} : K_{q}(R#Γ) → K_{q}(M_{i}), q ≥ 0, i ≥ 1,

on Quillen’s K-groups. For i = k, we finally obtain a symmetric power operation

σ^{k} : K_{q}(R#Γ) → K_{q}(R#Γ)

on the higher K-groups K_{q}(R#Γ), q ≥ 0, since M_{k} is contained in M_{1} by Proposition 1.1.

Lemma 1.4. For any sequence M_{1} ↪ ⋯ ↪ M_{i} in F_{i}(M_{1}), the R#Γ -module

M_{1} ∧ … ∧ M_{i} := Image(M_{1} ⊗ ⋯ ⊗ M_{i} ⟶^{can} Λ^{i}(M_{i}))

is contained in M_{i}.

Proof. For any i ≥ 1 and any f. g. projective R#Γ -module M, the Koszul complex

0 → Λ^{i}(M) → M ⊗ Λ^{i−1}(M) → ⋯ → Sym^{i−1}(M) ⊗ M → Sym^{i}(M) → 0

is an exact sequence of R#Γ -modules. It follows from Lemma 1.2 by induction on i that Λ^{i}(M) is contained in M_{i}. Now, Lemma 1.4 can be deduced from this fact as in the proof of Lemma 1.3.

Remark 1.5. Lemma 1.4, Lemma 1.2, and Proposition 1.1 imply that Proposition 1.1 holds as well if some symmetric powers are replaced by exterior powers in the definition of P. This can of course also be shown directly as in Proposition 1.1 by introducing some signs. Moreover, it is more convenient to define the categories M_{i}, i ≥ 1, with symmetric powers than with exterior powers since it would otherwise not be clear whether symmetric powers (and, more generally, (co)Schur modules, see Proposition 2.1) are contained in the categories M_{i}, i ≥ 1.

As above, we now obtain exterior power operations

λ^{i} : |GM_{1}| → |G^{i}M_{i}|, i ≥ 1,

and the corresponding maps on the K-groups.

By Lemma 1.2, we furthermore have continuous maps

⊗ : |G^{i}M_{i}| × |G^{j}M_{j}| → |G^{i+j}M_{i+j}|, i, j ≥ 1,

(see also section 1 of [Ko2]). Let N_{i}(s_{1}, …, s_{i}) denote the i-th Newton polynomial t_{1}^{i} + ⋯ + t_{i}^{i} considered as a polynomial in the elementary symmetric functions s_{1}, …, s_{i} of t_{1}, …, t_{i}. Replacing s_{1}, …, s_{i} by the exterior power operations λ^{1}, …, λ^{i} defined above and the products in N_{i}(s_{1}, …, s_{i}) by the tensor products defined above, we finally obtain continuous maps

ψ^{i} : |GM_{1}| → |G^{i}M_{i}|, i ≥ 1,

and the corresponding maps on the K-groups as above. These maps are called Adams operations.

Applying Grayson’s construction to the exact category M consisting of all f. g. Rprojective R#Γ -modules (i.e., all categories M_{i}, i ≥ 1, in section 7 in [Gr2] are identical to M), we similarly obtain power operations

σ^{i}, λ^{i}, ψ^{i} : K_{q}(Γ, R) → K_{q}(Γ, R), i ≥ 1, q ≥ 0,

(see also §2 and §3 in [Ko1]).

Theorem 1.6.

(a) Let γ be one of the operations σ^{i}, λ^{i}, ψ^{i}, i ≥ 1. Then the following diagram commutes for all q ≥ 0:

K_{q}(R#Γ) ⟶^{c} K_{q}(Γ, R)
↓ γ  ↓ γ
K_{q}(M_{i}) ⟶^{can} K_{q}(Γ, R).

In particular, the operations σ^{k}, λ^{k}, and ψ^{k} on K_{q}(R#Γ) are compatible with the corresponding operations on K_{q}(Γ, R) with respect to the Cartan homomorphism (for all q ≥ 0).

(b) The operations σ^{i}, λ^{i}, ψ^{i}, i ≥ 1, commute with base change with respect to any homomorphism R → R′ of Γ -rings and they commute with the restriction map for any subgroup Γ′ of Γ.

(c) For all i ≥ 1 and for all M, N ∈ M_{1} we have in K_{0}(M_{i}):

(i) σ^{i}([M] − [N]) = ∑_{{a≥0, b_{1},...,bu≥1 \atop a+b_{1}+···+bu=i}} (−1)^{u}[Sym^{a}(M) ⊗ Sym^{b_{1}}(N) ⊗ ⋯ ⊗ Sym^{b_{u}}(N)].

(ii) The assertion (i) analogously holds for exterior powers.

(iii) ψ^{i}([M] + [N]) = ψ^{i}([M]) + ψ^{i}([N]).

(iv) ∑_{j=0}^{i}(−1)^{j}λ^{j}([M]) · σ^{i−j}([M]) = 0.

(d) For all i ≥ 1 and q ≥ 1, we have:

(i) The maps λ^{i}, σ^{i}, ψ^{i} : K_{q}(R#Γ) → K_{q}(M_{i}) are homomorphisms.

(ii) σ^{i} = (−1)^{i−1}λ^{i} : K_{q}(R#Γ) → K_{q}(M_{i}).

(iii) ψ^{i} = (−1)^{i−1}iλ^{i} : K_{q}(R#Γ) → K_{q}(M_{i}).

(e) If gcd(k, ord(Γ)) = 1, we have:

ψ^{k}([R#Γ]) = [R#Γ] in K_{0}(R#Γ).

(f) Let K̃_{0}(ℤ, R#Γ) := ker(K_{0}(ℤ, R#Γ) ⟶^{can} K_{0}(R#Γ)) denote the reduced Grothendieck group of all pairs (M, α), where M is a f. g. projective R#Γ -module and α is an R#Γ -automorphism of M. Let γ : K̃_{0}(ℤ, R#Γ) → K̃_{0}(ℤ, R#Γ) be one of the operations σ^{k}, λ^{k}, ψ^{k} defined similarly as above. Then the following diagram commutes:

K̃_{0}(ℤ, R#Γ) ⟶^{can} K_{1}(R#Γ)
↓ γ  ↓ γ
K̃_{0}(ℤ, R#Γ) ⟶^{can} K_{1}(R#Γ).

Proof. The assertions (a) and (b) follow from the functoriality of Grayson’s construction. The assertions (c)(i) and (c)(ii) are proved in section 8 in [Gr2]. The λ-series λ_{t}([M]) := ∑_{i≥0} λ^{i}([M])t^{i} satisfies the rule λ_{t}([M + N]) = λ_{t}([M]) · λ_{t}([N]) by (c)(ii); being the logarithmic derivative of the λ-series (see page 23 in [FL]), the ψ-series is additive. This proves assertion (c)(iii). The assertion (c)(iv) follows from the fact that the Koszul complex

0 → Λ^{i}(M) → M ⊗ Λ^{i−1}(M) → ⋯ → Sym^{i−1}(M) ⊗ M → Sym^{i}(M) → 0

is an exact sequence of R#Γ -modules. The assertion (d)(i) follows from the fact that maps on higher homotopy groups which are induced by continuous maps are homomorphisms. By the same argument, the products K_{q}(M_{i}) × K_{q}(M_{j}) → K_{q}(M_{i+j}) induced by the tensor product M_{i} × M_{j} → M_{i+j} are linear and hence trivial for q ≥ 1. This together with Newton’s recursion formula proves assertion (d)(iii). Assertion (d)(ii) can be proved as in Theorem 6.1 in [Ko2]. To show assertion (e), we express ψ^{k} as a polynomial in the symmetric power operations σ^{1}, …, σ^{k} using (c)(iv). Then all monomials in this polynomial are of weight k. Thus, by Proposition 1.1, there is an n ∈ ℕ such that ψ^{k}([R#Γ]) = n · [R#Γ]. We have n = 1 since rank_{R}(ψ^{k}([R#Γ])) = rank_{R}([R#Γ]). This proves assertion (e). The assertion (f) can be proved as in section 9 in [Gr2] (see also the proof of Theorem 3.3 in [Ko3]).

Remark 1.7.

(a) If not only gcd(k, ord(Γ)), but k is invertible in R, we have constructed an Adams operation ψ^{k} on K_{q}(R#Γ), q ≥ 0, in [Ko4] by using so-called cyclic powers. One should be able to prove that that operation agrees with the Adams operation defined in this section.

Since there is no splitting principle available for K_{0}(R#Γ), this is a problem already (and in fact mainly) for K_{0}(R#Γ). At the end of the paper [Ko4], some speculations are given from which a solution for this problem would follow. If R is a field and Γ acts on R trivially, the Cartan homomorphism c : K_{0}(RΓ) → K_{0}(Γ, R) is injective (see §5 in [Ke]). Since both operations ψ^{k} commute with c, they agree in this case. In particular, Proposition 1.1 yields a considerable simplification of §5 in [Ke].

(b) Compared with the construction of this section, the construction in [Ko4] has the disadvantage that it is only possible under the stronger assumption k ∈ R^{×}. However, it has the advantage that both the multiplicativity of ψ^{k} and the rule ψ^{k} ∘ ψ^{l} = ψ^{l} ∘ ψ^{k} can rather easily be proved and that the definition of ψ^{k}([M]) involves only two modules if k is a prime number.

Remark 1.8. Let Γ′ be a subgroup of Γ. In §6 of [Ko5], the following question has been raised: Under which conditions does the equality ψ^{k}([R#[Γ/Γ′]]) = [R#[Γ/Γ′]] hold in K_{0}(Γ, R). Using the equivariant Adams-Riemann-Roch formula, we have proved in §6 in [Ko5] that this equality holds in a certain completion of K_{0}(Γ, R)[k^{−1}] for all k ∈ ℕ. By Theorem 1.6(e), we now have that this equality holds already in K_{0}(Γ, R) if Γ′ is a normal subgroup and gcd(k, ord(Γ/Γ′)) = 1.

§2 Multiplicativity of Adams Operations

Let Γ be a finite group, R a commutative Γ -ring, and k ∈ ℕ such that gcd(k, ord(Γ)) is invertible in R.

The aim of this section is to prove the multiplicativity of the Adams operation ψ^{k} on K_{0}(R#Γ) (and on the higher K-groups) defined in the previous section. Since there is no splitting principle available for K_{0}(R#Γ), this cannot be proved as e.g. in [FL]. In place of the splitting principle, we will use a filtration of the exterior power Λ^{i}(M ⊗ N) of the tensor product of two modules M and N which has been constructed by K. Akin, D. Buchsbaum, and J. Weyman. Each successive quotient in this filtration is isomorphic to a tensor product of (co)Schur modules (see [ABW]). Furthermore, we will use a universal form of the Pieri formula established by K. Akin and D. Buchsbaum in [AB] to express (co)Schur modules as polynomials in exterior powers. The results of this section will not be used in the subsequent sections.

Let λ = (λ_{1} ≥ λ_{2} ≥ …) ∈ ℕ^{∞} be a partition with 1 ≤ |λ| := λ_{1} + λ_{2} + ⋯ < ∞. The transposed partition (see p. 217 in [ABW]) is denoted by λ̃ = (λ̃_{1} ≥ λ̃_{2} ≥ …). For any f. g. free R-module F, the Schur module L_{λ}(F) is defined as the image of a certain canonical map

d_{λ}(F) : Λ^{λ_{1}}(F) ⊗ Λ^{λ_{2}}(F) ⊗ ⋯ → Symλ̃_{1}(F) ⊗ Symλ̃_{2}(F) ⊗ ⋯

(see pages 219 and 220 in [ABW]). By Theorem II.2.16 in [ABW], L_{λ}(F) is a f. g. free R-module and L_{λ} commutes with base change. For any f. g. projective R-module M, we define the Schur module L_{λ}(M) in the same way. By localization, we see that L_{λ}(M) is a f. g. projective R-module. If furthermore M is an R#Γ -module, then the map d_{λ}(M) obviously is R#Γ -linear and L_{λ}(M) is an R#Γ -module. The same holds for the coSchur module K_{λ}(M) defined on p. 220 in [ABW].

Let s_{λ} denote the Schur polynomial associated with λ̃ (see I §3 in [McD]). It will be viewed as a polynomial in the elementary symmetric functions. It is homogeneous of weight |λ|. Thus, for any f. g. projective R#Γ -module M,

s_{λ}(M) := s_{λ}([Λ^{1}(M)], [Λ^{2}(M)], …)

is a well-defined element of K_{0}(M_{|λ|}) by Lemma 1.2 and Lemma 1.4. (Here, the category M_{|λ|} is the category of modules of weight |λ| defined in section 1.)

Proposition 2.1. Let M be a f. g. projective R#Γ -module. Then the Schur module L_{λ}(M) and the coSchur module K_{λ}(M) are contained in the category M_{|λ|}. Furthermore, we have:

[L_{λ}(M)] = s_{λ}(M) and [K_{λ}(M)] = sλ̃(M) in K_{0}(M_{|λ|}).

Proof. First, we show that the Schur module L_{λ}(M) is contained in M_{|λ|}. For this, we use induction on λ with respect to lexicographic order (i.e., λ < μ iff there is an r ∈ ℕ such that λ_{1} = μ_{1}, …, λ_{r} = μ_{r} and λ_{r+1} < μ_{r+1}). For λ = (1, 0, 0, …), the assertion is trivial. For the induction step, let μ denote that partition which arises from λ by omitting the last column and let p denote the size of the last column in λ (i.e., p := λ̃_{l(λ̃)} and μ_{i} := λ_{i} − 1 for i = 1, …, p and μ_{i} := λ_{i} for i > p). By the universal form of the Pieri formula (see Theorem (3)(a) in section 3 in [AB]), we have for all f. g. free R-modules F: There is a filtration on the tensor product L_{μ}(F) ⊗ Sym^{p}(F) such that the successive quotients are isomorphic to Schur modules. More precisely, we have: If ν runs through the set

I := {ν a partition : μ_{i} ≤ ν_{i} ≤ μ_{i} + 1 for all i and |ν| = |λ|},

then L_{ν}(F) runs through the sequence of successive quotients. The partition λ is obviously the biggest partition in I with respect to lexicographic order; in particular, L_{λ}(F) is isomorphic to the smallest (nontrivial) module in the filtration of L_{μ}(F) ⊗ Sym^{p}(F) (see also section 2 in [AB]). By localization, one easily sees that the corresponding assertions hold for f. g. R-projective R#Γ -modules as well. By the induction hypothesis and by Lemma 1.2, the module L_{μ}(M) ⊗ Sym^{p}(M) and the modules L_{ν}(M), ν ∈ I\{λ}, are of weight |λ|. Thus, all filtration steps and, in particular, L_{λ}(M) are of weight |λ|. Using the corresponding theorem for coSchur modules, one similarly deduces from this that K_{|λ|}(M) is of weight |λ|. In order to show the equality [L_{λ}(M)] = s_{λ}(M), we replace the Pieri formula for L_{μ}(M)⊗ Sym^{p}(M) by the Pieri formula for L_{μ}(M) ⊗ Λ^{p}(M) (see Theorem (3)(b) in [AB]) and the lexicographic order by the transposed lexicographic order in the procedure above. Then, we immediately obtain a polynomial s′_{λ} such that [L_{λ}(M)] = s′_{λ}([Λ^{1}(M)], [Λ^{2}(M)], …) in K_{0}(M_{|λ|}). We have s′_{λ} = s_{λ} by the Pieri formula for Schur polynomials (cf. Appendix A in [FH]). Thus, we have shown the desired equality [L_{λ}(M)] = s_{λ}(M) in K_{0}(M_{|λ|}). The equality [K_{λ}(M)] = sλ̃(M) can be shown analogously.

For any i ≥ 1, let \overline{M_{i} ⊗ M_{i}} be the smallest subcategory of M which is closed under extensions and kernels of R#Γ -epimorphisms and which contains all modules of the form M ⊗ N, where M, N ∈ M_{i}. The category \overline{M_{i} ⊗ M_{i}} is obviously contained in M_{2i}. By Proposition 1.1, \overline{M_{1} ⊗ M_{1}} is contained in M_{1}. Thus, by Proposition 1.1, \overline{M_{k} ⊗ M_{k}} is contained in M_{1}. In particular, the tensor product defines a ring structure on K_{0}(R#Γ) (in general without 1). Let P_{i} be the universal polynomial defined on page 5 in [FL]. It is homogeneous of weight i in both sets of variables. Thus, for all f. g. projective R#Γ -modules M, N,

P_{i}(M, N) := P_{i}([Λ^{1}(M)], …, [Λ^{i}(M)]; [Λ^{1}(N)], …, [Λ^{i}(N)])

is a well-defined element of K_{0}(\overline{M_{i} ⊗ M_{i}}) by Lemma 1.2 and Lemma 1.4.

Theorem 2.2. For all f. g. projective R#Γ -modules M, N and all i ≥ 1, the exterior power Λ^{i}(M ⊗ N) is contained in \overline{M_{i} ⊗ M_{i}} and we have:

[Λ^{i}(M ⊗ N)] = P_{i}(M, N) in K_{0}(\overline{M_{i} ⊗ M_{i}}).

Proof. By Theorem III.2.4 in [ABW], there is a filtration on Λ^{i}(M ⊗ N) by R#Γ - submodules with the following property: If λ runs through the set of partitions of weight i, then the tensor product L_{λ}(M) ⊗ K_{λ}(M) runs through the sequence of successive quotients. (This is proved in [ABW] only in the non-equivariant situation and only for free R-modules M, N. The general assertion follows from this using the standard arguments already explained above.) Hence, by Proposition 2.1, all filtration steps and, in particular, Λ^{i}(M ⊗ N), are contained in \overline{M_{i} ⊗ M_{i}} and we have:

[Λ^{i}(M ⊗ N)] = ∑_{|λ|=i}[L_{λ}(M) ⊗ K_{λ}(N)]

= ∑_{|λ|=i} s_{λ}(M) · sλ̃(N)

= P_{i}(M, N) in K_{0}(\overline{M_{i} ⊗ M_{i}}).

Here, the last equality is formula (4.3’) on p. 35 in [McD].

Corollary 2.3. For all x, y ∈ K_{0}(R#Γ) and all i ≥ 1, we have

ψ^{i}(x · y) = ψ^{i}(x) · ψ^{i}(y) in K_{0}(\overline{M_{i} ⊗ M_{i}}).

In particular, ψ^{k} : K_{0}(R#Γ) → K_{0}(R#Γ) is a ring homomorphism.

Proof. It obviously suffices to show Corollary 2.3 for elements x = [M] and y = [N] where M, N are f. g. projective R#Γ -modules. In this case, Corollary 2.3 follows from Theorem 2.2 and the following identity of polynomials which is easy to prove:

N_{i}(P_{j}(X_{1}, …, X_{j}; Y_{1}, …, Y_{j}), j = 1, …, i) = N_{i}(X_{1}, …, X_{i}) · N_{i}(Y_{1}, …, Y_{i}).

The tensor product yields a K_{0}(R#Γ)-module structure on Quillen’s K-groups K_{q}(R#Γ), q ≥ 0, in the usual way (e.g., see §3 in [Q]).

Corollary 2.4. Let q ≥ 0. For all x ∈ K_{0}(R#Γ) and all y ∈ K_{q}(R#Γ), we have:

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

Proof. For q = 0, this has been proved in Corollary 2.3. So let q ≥ 1. For any arbitrary group G let K_{0}(G, R#Γ) denote the Grothendieck group of all representations of the group G on f. g. projective R#Γ -modules. As in section 1, one can define a k-th Adams operation ψ^{k} on K_{0}(G, R#Γ). Using Quillen’s universal transformation, we then obtain a k-th Adams operation ψ^{k} on K_{q}(R#Γ) as in section 2 of [Ko3]. As in section 9 of [Gr2], one easily shows that this Adams operation agrees with the Adams operation defined in section 1. Furthermore, one easily proves as in Corollary 2.3 that ψ^{k} on K_{0}(G, R#Γ) is multiplicative. Finally, as in Corollary 2.2a) in [Ko3], one deduces Corollary 2.4 from this using Quillen’s universal transformation.

Remark 2.5. If Γ acts on R trivially, the tensor product defines a K_{0}(Γ, R)-module structure on K_{0}(RΓ) by Lemma 2.1 in [Ko4]. One easily shows as above that the k-th Adams operation ψ^{k} : K_{0}(RΓ) → K_{0}(RΓ) is semi-linear with respect to the k-th Adams operation on K_{0}(Γ, R) defined e.g. in section 1.

Remark 2.6. Let l be another natural number such that gcd(l, ord(Γ)) is invertible in R. One should be able to prove similarly to Corollary 2.3 that ψ^{k} ∘ ψ^{l} = ψ^{kl} in End(K_{0}(R#Γ)). For this, one should be able to construct a “universal decomposition” of Λ^{k}(Λ^{l}(M)) similarly to the universal form of the Cauchy decomposition of Λ^{i}(M ⊗ N) used in the proof of Theorem 2.2. This problem is called the plethysm problem in the literature and is not yet solved. If R is a field and Γ acts on R trivially, then the Cartan homomorphism c : K_{0}(RΓ) → K_{0}(Γ, R) is injective (see §5 in [Ke]). Thus, the rule ψ^{k} ∘ ψ^{l} = ψ^{kl} for K_{0}(RΓ) follows from the corresponding rule for K_{0}(Γ, R) in this case. If R is the ring of integers in a number field and Γ acts on R trivially, the rule ψ^{k} ∘ ψ^{l} = ψ^{kl} follows from Theorem 1.3 (1) on p. 99 in [T] and Theorem 3.7 which is the main result of the next section.

§3 Power Operations on Locally Free Classgroups

Let K be a number field, Γ a finite group, and k ∈ ℕ such that k is coprime to the exponent e(Γ) of Γ. Furthermore, we fix a k′ ∈ ℕ with kk′ ≡ 1 mod e(Γ). Let O_{K} denote the ring of integers in K and Cl(O_{K}Γ) := ker(rank : K_{0}(O_{K}Γ) → ℤ) the locally free classgroup associated with K and Γ.

In this section, we will prove that the endomorphism ψ_{k′}^{CNT} of Cl(O_{K}Γ) defined by Ph. Cassou-Noguès and M. Taylor in [CNT] coincides with the restriction of the symmetric power operation σ^{k} on K_{0}(O_{K}Γ) defined in section 1.

To begin with, we recall some basic facts about the locally free classgroup (e.g., see §1 in Chapter 1 in [T]). Let K_{0}T (O_{K}Γ) denote the Grothendieck group of all O_{K}-torsion modules over O_{K}Γ which have a O_{K}Γ -projective resolution of finite length (and then also of length 1 by Theorem 8 on p. 145 in [Se2]). Let ∂ : K_{1}(KΓ) → K_{0}T (O_{K}Γ) be the connecting homomorphism; it is uniquely determined by the following property: Let F be a f. g. free O_{K}Γ -module and let α ∈ GL(F ⊗K)∩End(F); then ∂((F ⊗K, α)) = coker(α : F → F). Furthermore, let i_{∗} : K_{0}T (O_{K}Γ) → K_{0}(O_{K}Γ) denote the resolution map: it maps a torsion module M with the O_{K}Γ -projective resolution 0 → P → Q → M → 0 to the element [P] − [Q] in K_{0}(O_{K}Γ). Then the sequence

K_{1}(KΓ) ⟶^{∂} K_{0}T (O_{K}Γ) ⟶^{i_{∗}} K_{0}(O_{K}Γ) ⟶^{rank} ℤ

is exact; in particular, we have:

Cl(O_{K}Γ) ≅ coker(∂ : K_{1}(KΓ) → K_{0}T (O_{K}Γ)).

Decomposing torsion modules into 𝔭-primary torsion modules induces an isomorphism

K_{0}T (O_{K}Γ) ≅ ⊕_{𝔭∈Max(O_{K})}K_{0}T (O_{K_{𝔭}}Γ);

here, K_{𝔭} denotes the completion of K at 𝔭 and K_{0}T (O_{K_{𝔭}}Γ) is the Grothendieck group of all torsion modules over O_{K_{𝔭}}Γ which have a O_{K_{𝔭}}Γ -free resolution of length 1. Finally, we have canonical exact (localization) sequences

K_{1}(O_{K_{𝔭}}Γ) ⟶ K_{1}(K_{𝔭}Γ) ⟶^{∂} K_{0}T (O_{K_{𝔭}}Γ) ⟶ 0, 𝔭 ∈ Max(O_{K});

here, K_{1}(O_{K_{𝔭}}Γ) → K_{1}(K_{𝔭}Γ) is the canonical map and the local connecting homomorphism ∂ : K_{1}(K_{𝔭}Γ) → K_{0}T (O_{K_{𝔭}}Γ) is defined in the same way as the global one.

Now we fix a prime ideal 𝔭 ≠ 0 in O_{K}. Let Φ denote the composition

Φ : K_{1}(K_{𝔭}Γ) ⟶^{∂} K_{0}T (O_{K_{𝔭}}Γ) ⟶^{can} K_{0}T (O_{K}Γ) ⟶^{i_{∗}} K_{0}(O_{K}Γ).

Proposition 3.1. Let γ be one of the operations σ^{k}, λ^{k}, ψ^{k} defined in section 1. Then the following diagram commutes:

K_{1}(K_{𝔭}Γ) ⟶^{Φ} K_{0}(O_{K}Γ)
↓ γ  ↓ γ
K_{1}(K_{𝔭}Γ) ⟶^{Φ} K_{0}(O_{K}Γ).

We will give an algebraic proof and a topological proof of this proposition. The algebraic proof is elementary and explicit, but consists of lengthy computations. The topological proof is conceptual and uses higher K-theory. We start with a lemma which will be used in the algebraic proof.

Lemma 3.2. Let F be a f. g. free O_{K}Γ -module and α ∈ GL(F ⊗ K_{𝔭}). We write α = β/π with β ∈ End(F ⊗ O_{K_{𝔭}}) ∩ GL(F ⊗ K_{𝔭}) and π ∈ O_{K}\{0} and we put

C(β) := ker(F ↪ F ⊗ O_{K_{𝔭}}⟶ coker(β)).

Then we have:

(a) Φ((F ⊗ K_{𝔭}, α)) = [C(β)] − [F] in K_{0}(O_{K}Γ).

(b) For all k_{1}, …, k_{r} ∈ ℕ such that gcd(k_{1}, …, k_{r}, ord(Γ)) = 1 and for all β_{1}, …, β_{r} ∈ End(F ⊗ O_{K_{𝔭}}) ∩ GL(F ⊗ K_{𝔭}), the canonical O_{K}Γ -linear homomorphism

Sym^{k_{1}}(C(β_{1})) ⊗ ⋯ ⊗ Sym^{k_{r}}(C(β_{r})) → C((Sym^{k_{1}}(β_{1}) ⊗ ⋯ ⊗ Sym^{k_{r}}(β_{r}))

is bijective; here, Sym^{k_{1}}(β_{1}) ⊗ ⋯ ⊗ Sym^{k_{r}}(β_{r}) is considered as an endomorphism of Sym^{k_{1}}(F ⊗ O_{K_{𝔭}}) ⊗ ⋯ ⊗ Sym^{k_{r}}(F ⊗ O_{K_{𝔭}}) ≅ Sym^{k_{1}}(F) ⊗ ⋯ ⊗ Sym^{k_{r}}(F) ⊗ O_{K_{𝔭}} which is free over O_{K_{𝔭}}Γ by Proposition 1.1.

Proof. (a) In the commutative diagram

0 → C(β) →  F  → coker(β) → 0
↓  ↓  ∥
0 → F ⊗ O_{K_{𝔭}} →^{β} F ⊗ O_{K_{𝔭}} → coker(β) → 0,

the composition F ↪ F ⊗ O_{K_{𝔭}}⟶ coker(β) is surjective since (F ⊗ O_{K_{𝔭}})/F is a divisible Abelian group and coker(β) is a f. g. torsion group. Thus, the upper sequence is exact. Since coker(β) is cohomologically trivial, it has O_{K}Γ -projective dimension 1 (cf. Proposition 4.1 in [C]). Hence, the module C(β) is projective over O_{K}Γ and we have

Φ((F ⊗ K_{𝔭}, β)) = [C(β)] − [F] in K_{0}(O_{K}Γ).

For β = π, we obviously have C(β) ≅ F; consequently, Φ((F ⊗ K_{𝔭}, π)) = 0. This implies assertion (a).

(b) We write Sym^{k}(−) for Sym^{k_{1}}(−) ⊗ ⋯ ⊗ Sym^{k_{r}}(−). The canonical inclusion C(β) ↪ F ⊗ O_{K_{𝔭}} obviously induces an isomorphism C(β) ⊗ O_{K_{𝔭}} ˜→ F ⊗ O_{K_{𝔭}}. Thus, the canonical homomorphism Sym^{k}(C(β)) ⊗ O_{K_{𝔭}} → Sym^{k}(F ⊗ O_{K_{𝔭}}) is bijective. Hence, in the commutative diagram with exact rows

0 → Sym^{k}(C(β))  →^{i}  Sym^{k}(F)  →  coker(i)  → 0
↓  ∥  ↓
0 → C(Sym^{k}(β))  →  Sym^{k}(F)  → coker(Sym^{k}(β)) → 0
↓  ↓  ∥
0 → Sym^{k}(F ⊗ O_{K_{𝔭}}) →^{Sym^{k}(β)} Sym^{k}(F ⊗ O_{K_{𝔭}}) → coker(Sym^{k}(β)) → 0,

the right vertical composition is bijective. Then, the upper right vertical homomorphism is bijective, too. Hence, the upper left vertical homomorphism is bijective, as was to be shown.

Algebraic Proof of Proposition 3.1. We use additive notation for K_{1}(K_{𝔭}Γ). Let (F ⊗ K_{𝔭}, β/π) be an element of K_{1}(K_{𝔭}Γ) as in Lemma 3.2. Then the element

[(F ⊗ K_{𝔭}, β/π)] − [(F ⊗ K_{𝔭}, 1)] of K̃_{0}(ℤ, K_{𝔭}Γ)

is a preimage of (F ⊗ K_{𝔭}, β/π) under the canonical homomorphism K̃_{0}(ℤ, K_{𝔭}Γ) → K_{1}(K_{𝔭}Γ) (see also Theorem 1.6(f)). Using Theorem 1.6(c)(i) and (f), Lemma 3.2(a), Lemma 3.2(b), Theorem 1.6(c)(i), and Lemma 3.2(a) successively, we obtain:

Φ (σ^{k} ((F ⊗ K_{𝔭}, β/π))) =

= Φ( ∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+···+bu=k}}(−1)^{u}(Sym^{a}(F ⊗ K_{𝔭}) ⊗ Sym^{b_{1}}(F ⊗ K_{𝔭}) ⊗ ⋯ ⊗ Sym^{b_{u}}(F ⊗ K_{𝔭}),

Sym^{a}(β)/π^{a} ⊗ 1 ⊗ ⋯ ⊗ 1))

= ∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+···+bu=k}}(−1)^{u}([C(Sym^{a}(β) ⊗ 1 ⊗ ⋯ ⊗ 1)] −

[Sym^{a}(F) ⊗ Sym^{b_{1}}(F) ⊗ ⋯ ⊗ Sym^{b_{u}}(F)])

= ∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+···+bu=k}}(−1)^{u}([Sym^{a}(C(β)) ⊗ Sym^{b_{1}}(F) ⊗ ⋯ ⊗ Sym^{b_{u}}(F)] −

[Sym^{a}(F) ⊗ Sym^{b_{1}}(F) ⊗ ⋯ ⊗ Sym^{b_{u}}(F)])

= ∑_{{a≥0,b_{1},...,bu≥1 \atop a+b_{1}+···+bu=k}}(−1)^{u}[Sym^{a}(C(β)) ⊗ Sym^{b_{1}}(F) ⊗ ⋯ ⊗ Sym^{b_{u}}(F)]

= σ^{k} ([C(β)] − [F])

= σ^{k} (Φ ((F ⊗ K_{𝔭}, β/π))) .

This proves Proposition 3.1 for γ = σ^{k}. For γ = λ^{k} or γ = ψ^{k}, Proposition 3.1 can be shown analogously or can be deduced from the assertion for γ = σ^{k} using Theorem 1.6(c) and (d).

Topological Proof of Proposition 3.1. Let K_{0}T_{pr}(O_{K_{𝔭}}Γ) denote the Grothendieck group of all torsion modules over O_{K_{𝔭}}Γ which have a O_{K_{𝔭}}Γ -projective resolution of length 1. Then the canonical homomorphism K_{0}T (O_{K_{𝔭}}Γ) → K_{0}T_{pr}(O_{K_{𝔭}}Γ) is bijective. The injectivity has been proved in Theorem 1(ii) on p. 3 in [F2]. The surjectivity follows from the fact that, in the commutative diagram

K_{1}(K_{𝔭}Γ) →^{∂} K_{0}T (O_{K_{𝔭}}Γ)
∥  ↓
K_{1}(K_{𝔭}Γ) →^{∂} K_{0}T_{pr}(O_{K_{𝔭}}Γ) →^{i_{∗}} K_{0}(O_{K_{𝔭}}Γ) →^{can} K_{0}(K_{𝔭}Γ),

the lower sequence is exact (cf. Theorem 1(ii) on p. 3 in [F2]) and that the canonical homomorphism K_{0}(O_{K_{𝔭}}Γ) → K_{0}(K_{𝔭}Γ) is injective (cf. Theorem 6.1 in [Sw]). In the section “The localization theorem for projective modules” in [Gr1], it is shown that K_{0}T_{pr}(O_{K_{𝔭}}Γ) is isomorphic to the group of connected components of the homotopy fibre of the canonical continuous map from the K-theory space associated with O_{K_{𝔭}}Γ to the K-theory space associated with K_{𝔭}Γ. Thus, the continuous map σ^{k} on the K-theory space associated with O_{K_{𝔭}}Γ defined in section 1 induces a symmetric power operation σ^{k} on K_{0}T_{pr}(O_{K_{𝔭}}Γ) which in particular commutes with the connecting homomorphism ∂ : K_{1}(K_{𝔭}Γ) → K_{0}T_{pr}(O_{K_{𝔭}}Γ). Analogously, the Grothendieck group K_{0}T_{𝔭}(O_{K}Γ) of all 𝔭torsion modules over O_{K}Γ of O_{K}Γ -projective dimension 1 carries a symmetric power operation σ^{k} which commutes with the localization isomorphism K_{0}T_{𝔭}(O_{K}Γ) ˜→ K_{0}T_{pr}(O_{K_{𝔭}}Γ) and with the resolution map i_{∗} : K_{0}T_{𝔭}(O_{K}Γ) → K_{0}(O_{K}Γ). Since Φ coincides with the composition

K_{1}(K_{𝔭}Γ) →^{∂} K_{0}T_{pr}(O_{K_{𝔭}}Γ) ˜← K_{0}T_{𝔭}(O_{K}Γ) →^{i_{∗}} K_{0}(O_{K}Γ)

(cf. Proposition 2.5 on p. 183 in [Sh]), Proposition 3.1 is proved for σ^{k}. The assertions for λ^{k} and ψ^{k} can be deduced from this as in the algebraic proof.

Remark 3.3. Let X be a scheme and U an open subscheme of X. As in the proof above, one can define power operations on the homotopy fibre of the canonical continuous map from the K-theory space of X to the K-theory space of U. If U is affine and X\U is a divisor, it is shown in the section “The localization theorem … ” in [Gr1] that this homotopy fibre is homotopy equivalent to the K-theory space associated with the category H consisting of all quasi-coherent sheaves on X which are zero on U and which have a resolution by vector bundles on X of length 1. It is likely that one can use the ideas developed in the algebraic proof to describe these operations on the K-theory space of H simplicially, and, finally, on K_{0}(H) algebraically, i.e., in terms of modules. I hope to say more on this in a future paper.

Let M_{1}, M_{2}, … be the categories defined in section 1.

Proposition 3.4. For any 1 ≤ i ≤ k − 1, the composition

Cl(O_{K}Γ) × Cl(O_{K}Γ) ⟶^{σ^{i}×σ^{k−i}} K_{0}(M_{i}) × K_{0}(M_{k−i}) →^{⊗} K_{0}(O_{K}Γ)

is the zero map. In particular, the restriction of σ^{k} : K_{0}(O_{K}Γ) → K_{0}(O_{K}Γ) to the locally free classgroup Cl(O_{K}Γ) is a homomorphism. Analogous assertions hold for exterior power operations, too.

Proof. Let (x, y) ∈ Cl(O_{K}Γ) × Cl(O_{K}Γ). Let P_{1}, P_{2}, Q_{1}, Q_{2} be f. g. projective O_{K}Γ - modules such that x = [P_{1}] − [Q_{1}] and y = [P_{2}] − [Q_{2}]. By Theorem A in [Sw], we may assume that P_{1}, P_{2}, Q_{1}, Q_{2} are projective left ideals in O_{K}Γ. Then it suffices by Theorem 1.6(c)(i) to show Proposition 3.4 for elements x, y of the form x = [O_{K}Γ] − [I] and y = [O_{K}Γ] − [J] where I, J are projective left ideals in O_{K}Γ. Furthermore, we may assume by Theorem A in [Sw] that the annihilators of the torsion modules O_{K}Γ/I and O_{K}Γ/J are relatively prime. By Theorem 1.6(c)(i), we have in K_{0}(M_{i}):

σ^{i}(x) = ∑_{{a≥0,b_{1},...,bu≥1 \atop a+b_{1}+···+bu=i}}(−1)^{u} [Sym^{a}(O_{K}Γ) ⊗ Sym^{b_{1}}(I) ⊗ ⋯ ⊗ Sym^{b_{u}}(I)] =

= ∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+···+bu=i}}(−1)^{u}( [Sym^{a}(O_{K}Γ) ⊗ Sym^{b_{1}}(I) ⊗ ⋯ ⊗ Sym^{b_{u}}(I)] −

[Sym^{a}(I) ⊗ Sym^{b_{1}}(I) ⊗ ⋯ ⊗ Sym^{b_{u}}(I)] ).

For any a ≥ 1 and for any prime ideal 𝔭 ≠ 0 in O_{K} which does not lie in the support of O_{K}Γ/I, the localization of the canonical homomorphism

Sym^{a}(I) → Sym^{a}(O_{K}Γ)

at 𝔭 is bijective. In particular, Sym^{a}(I) → Sym^{a}(O_{K}Γ) is injective and the support of the torsion module Sym^{a}(O_{K}Γ)/Sym^{a}(I) is contained in the support of the torsion module O_{K}Γ/I. The same holds for y and J. Consequently, it suffices to show the following assertion: Let M_{1} ⊆ N_{1} and M_{2} ⊆ N_{2} be modules in M_{i} and M_{k−i}, respectively, such that N_{1}/M_{1} and N_{2}/M_{2} are torsion modules whose annihilators are relatively prime. Then we have:

([N_{1}] − [M_{1}]) · ([N_{2}] − [M_{2}]) = 0 in K_{0}(O_{K}Γ).

This immediately follows from the fact that, in the commutative diagram with exact rows

0 → M_{1} ⊗ M_{2} → N_{1} ⊗ M_{2} → N_{1}/M_{1} ⊗ M_{2} → 0
↓  ↓  ↓
0 → M_{1} ⊗ N_{2} → N_{1} ⊗ N_{2} → N_{1}/M_{1} ⊗ N_{2} → 0.

the right vertical homomorphism is bijective. The equality σ^{k}(x + y) = ∑_{i=0}^{k} σ^{i}(x) · σ^{k−i}(y) (for x, y ∈ Cl(O_{K}Γ)) now shows that σ^{k} : Cl(O_{K}Γ) → Cl(O_{K}Γ) is a homomorphism.

As for Proposition 3.1, we give a topological proof for Proposition 3.4 with a more conceptual argument.

Topological proof of Proposition 3.4 (Sketch). We will consider only the case k = 2, i = 1; i.e., we will only show that the multiplication

Cl(O_{K}Γ) × Cl(O_{K}Γ) → K_{0}(O_{K}Γ)

is trivial. The general statement can then be proved by generalizing the following argument as in the algebraic proof. As in the topological proof of Proposition 3.1, the tensor product induces a multiplication on the various K-theory spaces and, in particular, on the K-groups K_{1}(K_{𝔭}Γ), K_{0}T (O_{K}Γ), K_{0}(O_{K}Γ), … which are compatible with the maps in the local or global localization sequences. On K_{1}(K_{𝔭}Γ) (as on all higher K-groups), this multiplication is linear. Hence, it is trivial. Since the connecting homomorphism ∂ : K_{1}(K_{𝔭}Γ) → K_{0}T (O_{K_{𝔭}}Γ) is surjective, the multiplication on K_{0}T (O_{K_{𝔭}}Γ) is trivial, too. As in the algebraic proof, one shows furthermore that for all prime ideals 𝔭 ≠ 𝔮 in O_{K}, the multiplication K_{0}T_{𝔭}(O_{K}Γ) × K_{0}T_{𝔮}(O_{K}Γ) → K_{0}(O_{K}Γ) is trivial. Therefore, the multiplication K_{0}T (O_{K}Γ) × K_{0}T (O_{K}Γ) → K_{0}(O_{K}Γ) is trivial. This implies that the multiplication on Cl(O_{K}Γ) is trivial, as was to be shown.

Now let γ be one of the operations σ^{k}, λ^{k}, or ψ^{k} defined in section 1. For any prime ideal 𝔭 ≠ 0 in O_{K}, the operation γ commutes with the canonical homomorphism K_{1}(O_{K_{𝔭}}Γ) → K_{1}(K_{𝔭}Γ) by Theorem 1.6(b) and hence induces an operation γ on K_{0}T (O_{K_{𝔭}}Γ). Let γ also denote the direct sum of these operations on K_{0}T (O_{K}Γ) ≅ ⊕_{𝔭}K_{0}T (O_{K_{𝔭}}Γ). Then, γ commutes with the connecting homomorphism ∂ : K_{1}(KΓ) → K_{0}T (O_{K}Γ) by Theorem 1.6(b). Consequently, γ induces an operation on Cl(O_{K}Γ) ≅ coker(∂) denoted by γ again. On the other hand, the operation γ on K_{0}(O_{K}Γ) obviously induces an operation γ on Cl(O_{K}Γ) by restricting.

Corollary 3.5. These two definitions of the operation γ on Cl(O_{K}Γ) coincide.

Proof. This immediately follows from Propositions 3.1 and 3.4.

For any prime ideal 𝔭 ≠ 0 in O_{K}, let G_{𝔭} := Gal(K̄_{𝔭}/K_{𝔭}) denote the absolute Galois group. By Chapter 1 in [T], the pairing

K_{0}(K̄_{𝔭}Γ) × K_{1}(K̄_{𝔭}Γ) → K̄_{𝔭}^{×}
([P], (M, α))  ↦ detK̄_{𝔭}(Hom_{K̄_{𝔭}Γ} (P, α)|_{Hom_{K̄𝔭Γ} (P,M)})

(here, Hom_{K̄_{𝔭}Γ} (P, α) denotes the automorphism of Hom_{K̄_{𝔭}Γ} (P, M) induced by α) induces an isomorphism

K_{1}(K_{𝔭}Γ) ˜→ Hom_{G_{𝔭}}(K_{0}(K̄_{𝔭}Γ), K̄_{𝔭}^{×}).

Note that the determinant map used here coincides with the determinant map used in [T] (see Proposition 2.3 in [Que]). We identify the Grothendieck group K_{0}(K̄_{𝔭}Γ) with the classical ring of virtual characters of Γ. Let ⟨−, −⟩ : K_{0}(K̄_{𝔭}Γ) × K_{0}(K̄_{𝔭}Γ) → ℤ denote the classical character pairing.

Now let k be an arbitrary natural number. Let ψ̂^{k} ∈ End_{G_{𝔭}}(K_{0}(K̄_{𝔭}Γ)) denote the adjoint of the Adams operation ψ^{k} with respect to ⟨−, −⟩.

Lemma 3.6. (a) For all characters χ ∈ K_{0}(K̄_{𝔭}Γ) and all γ ∈ Γ, we have:

ψ̂^{k}(χ)(γ) = ∑_{τ^{k}=γ} χ(τ) in K̄_{𝔭}.

In particular, we have ψ̂^{k} = ψ^{k′} if gcd(k, ord(Γ)) = 1 and kk′ ≡ 1 mod e(Γ).

(b) Let Γ be an Abelian group. Let ϕ : K̄_{𝔭}Γ → K̄_{𝔭}Γ denote the ring homomorphism given by γ ↦ γ^{k}. Then we have for all f. g. K̄_{𝔭}Γ -modules M:

ψ̂^{k}([M]) = [(K̄_{𝔭}Γ)_{ϕ} ⊗_{K̄_{𝔭}Γ} M] in K_{0}(K̄_{𝔭}Γ).

Proof. (a) Let the class function ψ̃^{k}(χ) : Γ → ℂ be given by ψ̃^{k}(χ)(γ) = ∑_{τ^{k}=γ} χ(τ) for γ ∈ Γ. Then we have for all characters θ ∈ K_{0}(ℂΓ):

⟨ψ̃^{k}(χ), θ⟩ = \frac{1}{ord(Γ)} ∑_{γ∈Γ} ψ̃^{k}(χ)(γ)\overline{θ(γ)}

= \frac{1}{ord(Γ)} ∑_{γ∈Γ} ∑_{τ^{k}=γ} χ(τ)\overline{θ(γ)} = \frac{1}{ord(Γ)} ∑_{τ∈Γ} χ(τ)\overline{θ(τ^{k})}

= \frac{1}{ord(Γ)} ∑_{τ∈Γ} χ(τ)\overline{ψ^{k}(θ)(τ)} = ⟨χ, ψ^{k}(θ)⟩ = ⟨ψ̂^{k}(χ), θ⟩.

(Here, we have used Proposition (12.8) on p. 317 in [CR].) This shows the main assertion of Lemma 3.6(a). If gcd(k, ord(Γ)) = 1, then we have {τ ∈ Γ : τ^{k} = γ} = {γ^{k′}} for all γ ∈ Γ. This proves ψ̂^{k} = ψ^{k′} in this case.

(b) For all K̄_{𝔭}Γ -modules N, we have:

⟨(K̄_{𝔭}Γ)_{ϕ} ⊗_{K̄_{𝔭}Γ} M, N⟩ = dimK̄_{𝔭} Hom_{K̄_{𝔭}Γ} ((K̄_{𝔭}Γ)_{ϕ} ⊗_{K̄_{𝔭}Γ} M, N) =
= dimK̄_{𝔭} Hom_{K̄_{𝔭}Γ} (M, ψ^{k}(N)) = ⟨M, ψ^{k}(N)⟩ = ⟨ψ̂^{k}(M), N⟩.

This proves Lemma 3.6(b).

We now assume that the prime divisors of ord(Γ) are unramified in K. Furthermore, let k be odd if Γ has irreducible symplectic characters. Under these assumptions, Ph. Cassou-Noguès and M. Taylor have shown that the endomorphisms ψ_{k}^{CNT} := Hom(ψ^{k}, K̄_{𝔭}^{×}) of K_{1}(K̄_{𝔭}Γ) ≅ Hom_{G_{𝔭}}(K_{0}(K̄_{𝔭}Γ), K̄_{𝔭}^{×}), 𝔭 ∈ Max(O_{K}), induce an endomorphism of Cl(O_{K}Γ) denoted by ψ_{k}^{CNT} again (cf. Chapter 9 in [T]). In the paper [BC], D. Burns and T. Chinburg have raised the question whether there is an algebraic interpretation of this endomorphism. The following theorem gives an affirmative answer to this question in the case gcd(k, ord(Γ)) = 1. For this, let σ^{k} and ψ^{k} denote (the restriction of) the symmetric power operation and k-th Adams operation on Cl(O_{K}Γ) ⊂ K_{0}(O_{K}Γ) constructed in section 1.

Theorem 3.7. Let gcd(k, ord(Γ)) = 1 and k′ ∈ ℕ such that kk′ ≡ 1 mod e(Γ). Then we have:

ψ_{k′}^{CNT} = σ^{k} on Cl(O_{K}Γ)

and

k · ψ_{k′}^{CNT} ⊕ id = ψ^{k} on Cl(O_{K}Γ) ⊕ ℤ[O_{K}Γ] = K_{0}(O_{K}Γ).

Proof. By Theorem 3.3 in [Ko3], the operation λ^{k} on K_{1}(K̄_{𝔭}Γ) corresponds to the endomorphism (−1)^{k−1}Hom(ψ̂^{k}, K̄_{𝔭}^{×}) of Hom_{G_{𝔭}}(K_{0}(K̄_{𝔭}Γ), K̄_{𝔭}^{×}) via the isomorphism

K_{1}(K̄_{𝔭}Γ) ≅ Hom_{G_{𝔭}}(K_{0}(K̄_{𝔭}Γ), K̄_{𝔭}^{×}).

Thus, the operation σ^{k} corresponds to the endomorphism Hom(ψ^{k′}, K̄_{𝔭}^{×}) by Theorem 1.6(d)(ii) and Lemma 3.6(a). By construction (see Chapter 9 in [T]), ψ_{k′}^{CNT} on Cl(O_{K}Γ) is induced by Hom(ψ^{k′}, K̄_{𝔭}^{×}) on K_{1}(K̄_{𝔭}Γ), 𝔭 ∈ Max(O_{K}). Now the equality ψ_{k′}^{CNT} = σ^{k} on Cl(O_{K}Γ) follows from Corollary 3.5. The equality k · ψ_{k′}^{CNT} = ψ^{k} on Cl(O_{K}Γ) can be shown analogously. Finally, the equality ψ^{k} = id on ℤ[O_{K}Γ] ⊆ K_{0}(O_{K}Γ) has already been shown in Theorem 1.6(e).

Corollary 3.8. We have σ^{k} = σ^{k+e(Γ)} on Cl(O_{K}Γ).

Proof. This follows from Theorem 1.3(2) on p. 99 in [T] and Theorem 3.7.

The following corollary answers the question (1.12) in Chapter 9 of [T].

Corollary 3.9. Let c : Cl(O_{K}Γ) ↪ K_{0}(O_{K}Γ) → K_{0}(Γ, O_{K}) denote the Cartan homomorphism. Then the following diagrams commute:

Cl(O_{K}Γ) →^{c} K_{0}(Γ, O_{K})_{↓ ψ_{k′}^{CNT} ↓ σ^{k}}_{Cl(O_{K}Γ) →^{c} K_{0}(Γ, O_{K})} Cl(O_{K}Γ) ⊕ ℤ[O_{K}Γ]  →^{c} K_{0}(Γ, O_{K})

↓ k · ψ_{k′}^{CNT} ⊕ id  ↓ ψ^{k}
Cl(O_{K}Γ) ⊕ ℤ[O_{K}Γ]  →^{c} K_{0}(Γ, O_{K}).

Proof. This immediately follows from Theorem 1.6(a) and Theorem 3.7. In addition to the algebraic or topological arguments used in the proof of Theorem 3.7, we give a geometric proof of this corollary since Proposition 3.10 which will be used in this proof is of independent interest. By Theorem 1.3(3) on p. 99 in [T], the following diagram commutes for any prime ideal 𝔭 ≠ 0 in O_{K}:

K_{0}T (O_{K_{𝔭}}Γ) ⟶^{c} K_{0}T (Γ, O_{K_{𝔭}}) ≅ K_{0}(Γ, O_{K}/𝔭)
↓ ψ_{k′}^{CNT}  ↓ ψ^{k}
K_{0}T (O_{K_{𝔭}}Γ) ⟶^{c} K_{0}T (Γ, O_{K_{𝔭}}) ≅ K_{0}(Γ, O_{K}/𝔭).

Now Corollary 3.9 follows from the following Proposition 3.10 (and Theorem 1.6(e)) where, for the first diagram in Corollary 3.9, we in addition use the fact that, for all prime ideals 𝔭 ≠ 𝔮 in O_{K} and all x ∈ K_{0}(Γ, O_{K}/𝔭), y ∈ K_{0}(Γ, O_{K}/𝔮), the product i_{∗}(x)·i_{∗}(y) vanishes in K_{0}(Γ, O_{K}) (cf. the end of the (algebraic) proof of Proposition 3.4).

Proposition 3.10. For any prime ideal 𝔭 ≠ 0 in O_{K} and any j ≥ 0, the following diagrams commute:

\begin{matrix}K_{0}(Γ, O_{K}/𝔭) & →^{i_{∗}} & K_{0}(Γ, O_{K}) \\ ↓ ψ^{j} & & ↓ σ^{j} \\ K_{0}(Γ, O_{K}/𝔭) & →^{i_{∗}} & K_{0}(Γ, O_{K}) \end{matrix}  \begin{matrix}K_{0}(Γ, O_{K}/𝔭) & →^{i_{∗}} & K_{0}(Γ, O_{K}) \\ ↓ j · ψ^{j} & & ↓ ψ^{j} \\ K_{0}(Γ, O_{K}/𝔭) & →^{i_{∗}} & K_{0}(Γ, O_{K}). \end{matrix}

Here, i_{∗} denotes the composition of the canonical map K_{0}(Γ, O_{K}/𝔭) → K′_{0}(O_{K}Γ) with the isomorphism K′_{0}(O_{K}Γ) ≅ K_{0}(Γ, O_{K}) (cf. Notations).

Proof. The conormal module 𝔭/𝔭^{2} of the closed immersion i : Spec(O_{K}/𝔭) ↪ Spec(O_{K}) is trivial. Thus, the Bott multiplier θ^{j}(i) := 1 + [𝔭/𝔭^{2}] + ⋯ + [(𝔭/𝔭^{2})^{j−1}] associated with i equals j in K_{0}(Γ, O_{K}/𝔭). Now, the commutativity of the second diagram follows from the equivariant Adams-Riemann-Roch theorem without denominators (see Corollary (5.2) in [Ko1]). The commutativity of the first diagram follows from the (more general) equivariant Riemann-Roch theorem (see Satz (5.1) in [Ko1]) applied to the natural operation σ^{j}. For this, we have to show

σ^{j}(𝔭/𝔭^{2}, x) = ψ^{j}(x) in K_{0}(Γ, O_{K}/𝔭)

for all x ∈ K_{0}(Γ, O_{K}/𝔭). Here, σ^{j}(𝔭/𝔭^{2}, x) is defined as follows (see section 5 in [Ko1]): Let the polynomial ring K_{0}(Γ, O_{K}/𝔭)[N] be equipped with the unique (special) λ-ring structure such that K_{0}(Γ, O_{K}/𝔭) is a λ-subring and such that N is of λ-degree 1; then, σ^{j}(𝔭/𝔭^{2}, x) is defined to be the value of the polynomial σ^{j}(x · (1 − N))/(1 − N) at the place N = 1. As for exterior powers (see p. 5 in [FL]), there is a universal polynomial Q_{j} ∈ ℤ[X_{1}, …, X_{j}; Y_{1}, …, Y_{j}] such that

σ^{j}(x · (1 − N)) = Q_{j}(σ^{1}(x), …, σ^{j}(x); σ^{1}(1 − N), …, σ^{j}(1 − N)).

Since σ^{i}(1 − N) = 1 − N for all i ≥ 0 and since Q_{j} is homogeneous of weight j in both sets of variables, we have:

σ^{j}(𝔭/𝔭^{2}, x) = Q_{j}(σ^{1}(x), …, σ^{j}(x); 0, …, 0, 1) in K_{0}(Γ, O_{K}/𝔭).

If L_{1}, …, L_{n} are of modules of rank 1, M := L_{1} ⊕ ⋯ ⊕ L_{n}, and N is an arbitrary module, we have:

Q_{j} ([Sym^{1}(M)], …, [Sym^{j}(M)]; [Sym^{1}(N)], …, [Sym^{j}(N)]) =

= [Sym^{j}((L_{1} ⊕ ⋯ ⊕ L_{n}) ⊗ N)] = [Sym^{j}(L_{1} ⊗ N ⊕ ⋯ ⊕ L_{n} ⊗ N)]

= ∑_{b_{1}+···+b_{n}=j}[Sym^{b_{1}}(L_{1} ⊗ N) ⊗ ⋯ ⊗ Sym^{b_{n}}(L_{n} ⊗ N)]

= ∑_{b_{1}+···+b_{n}=j}[L_{1}^{⊗b_{1}} ⊗ ⋯ ⊗ L_{n}^{⊗b_{n}} ⊗ Sym^{b_{1}}(N) ⊗ ⋯ ⊗ Sym^{b_{n}}(N)].

Thus, for x = ∑_{i=1}^{n} l_{i} with elements l_{i} of λ-degree 1, we have:

Q_{j}(σ^{1}(x), …, σ^{j}(x); Y_{1}, …, Y_{j}) = ∑_{b_{1}+···+b_{n}=j} l_{1}^{b_{1}} ⋯ l_{n}^{b_{n}} · Y_{b_{1}} ⋯ Y_{b_{n}}

(with Y_{i} := 0 for i > j) and, hence,

Q_{j}(σ^{1}(x), …, σ^{j}(x); 0, …, 0, 1) = l_{1}^{j} + ⋯ + l_{n}^{j} = ψ^{j}(x).

Using the splitting principle (see section (2.5) in [Ko1]), we finally obtain the equality σ^{j}(𝔭/𝔭^{2}, x) = ψ^{j}(x) for an arbitrary x ∈ K_{0}(Γ, O_{K}/𝔭). This completes the proof of Proposition 3.10.

Remark 3.11. The construction of ψ_{k}^{CNT} in [T] only shows that, for any prime ideal 𝔭 ≠ 0 in O_{K}, the image of the canonical homomorphism

K_{1}(O_{K_{𝔭}}Γ) → K_{1}(K_{𝔭}Γ) ≅ Hom_{G_{𝔭}}(K_{0}(K̄_{𝔭}Γ), K̄_{𝔭}^{×})

is invariant under the endomorphism Hom(ψ^{k}, K̄_{𝔭}^{×}). If p := char(O_{K}/𝔭) does not divide gcd(k, ord(Γ)), the considerations above moreover show that there exists an endomorphism of K_{1}(O_{K_{𝔭}}Γ) which is compatible with Hom(ψ^{k}, K̄_{𝔭}^{×}) on K_{1}(K_{𝔭}Γ) and that the assumption “K_{𝔭} is absolutely unramified” used in [T] is not necessary for this. There remains open the question whether, in the case p | gcd(k, ord(Γ)), there exists an endomorphism of K_{1}(O_{K_{𝔭}}Γ) which is compatible with Hom(ψ^{k}, K̄_{𝔭}^{×}). This question has an affirmative answer in the following two (extreme) cases:

(a) Let Γ be an Abelian group. Let ϕ : O_{K_{𝔭}}Γ → O_{K_{𝔭}}Γ denote the ring homomorphism given by γ ↦ γ^{k}. Then ϕ induces an endomorphism on K_{1}(O_{K_{𝔭}}Γ) which is compatible with Hom(ψ^{k}, K̄_{𝔭}^{×}) on K_{1}(K_{𝔭}Γ). This is easy to prove or follows from formula (3.18) on p. 113 in [T] which holds for arbitrary Abelian groups and not only for p-groups.

(b) Let k be equal to the exponent of Γ. Let ϵ : O_{K_{𝔭}}Γ → O_{K_{𝔭}}Γ, ∑ a_{γ}[γ] ↦ ∑ a_{γ}[1], denote the augmentation homomorphism. Then ϵ induces an endomorphism on K_{1}(O_{K_{𝔭}}Γ) which is compatible with Hom(ψ^{k}, K̄_{𝔭}^{×}) on K_{1}(K_{𝔭}Γ), as one can easily show.

Remark 3.12. In the definition of ψ_{k}^{CNT}, it has been assumed that k is odd if Γ has irreducible symplectic characters. This assumption is obviously implied by the assumption gcd(k, ord(Γ)) = 1. Thus, the description of ψ_{k}^{CNT} given in this section does not yield any improvement into this direction. The following considerations about K_{1}(ℝΓ) will make clear why this assumption is necessary. Let j denote complex conjugation and let K_{0}(ℂΓ) = K_{0}^{ℝ}⊕K_{0}^{ℂ}⊕K_{0}^{ℍ} be the decomposition of the classical ring K_{0}(ℂΓ) of virtual characters introduced in I, section 12.6 in [Se1]. Then the subgroup K_{0}^{s} of K_{0}(ℂΓ) generated by symplectic characters has the following description:

K_{0}^{s} = 2K_{0}^{ℝ} ⊕ (1 + j)K_{0}^{ℂ} ⊕ K_{0}^{ℍ} = (1 + j)K_{0}(ℂΓ) + K_{0}^{ℍ}.

Furthermore, we have a canonical isomorphism

K_{1}(ℝΓ) ≅ Hom_{j}^{+}(K_{0}(ℂΓ), ℂ^{×})

where Hom_{j}^{+}(K_{0}(ℂΓ), ℂ^{×}) denotes the group of all Galois-invariant homomorphisms f : K_{0}(ℂΓ) → ℂ^{×} with f(K_{0}^{s}) ⊆ ℝ_{+}^{×} (see Chapter 1 in [T]). Now, the assertion that a Galois-invariant operation γ on K_{0}(ℂΓ) induces an operation on K_{1}(ℝΓ) is equivalent to the assertion that γ maps the subgroup K_{0}^{ℍ} into K_{0}^{s}. If Γ is the quaternion group Q_{8} and γ = ψ^{2}, the latter assertion is not true as one can easily show; i.e., the analogue of ψ_{2}^{CNT} cannot be defined on K_{1}(ℝQ_{8}) (and K_{1}(KQ_{8})). However, we have

ψ̂^{k}(K_{0}^{ℍ}) ⊆ K_{0}^{s}

for all k ≥ 1. This follows from the fact that Hom(ψ̂^{k}, ℂ^{×}) corresponds to the operation σ^{k} on K_{1}(ℝΓ) (see the proof of Theorem 3.7). Alternatively, this can be proved in the following elementary way: Let S ∈ K_{0}^{ℍ} be an irreducible ℂΓ -module. Then we have to show that, for all irreducible V ∈ K_{0}^{ℝ}, the number ⟨V, ψ̂^{k}(S)⟩ is even and that, for all irreducible V ∈ K_{0}^{ℂ}, the equality ⟨V, ψ̂^{k}(S)⟩ = ⟨j(V), ψ̂^{k}(S)⟩ holds. The first statement follows from the fact that, for all f. g. ℝΓ -modules W, the dimension ⟨ℂ ⊗ W, S⟩ = dim_{ℂ} Hom_{ℂΓ}(ℂ ⊗ W, S) is even since Hom_{ℝΓ}(W, S) is a vector space over ℍ. The second statement is clear since j fixes S.

Remark 3.13. Let Γ be an abelian group, 𝔭 ∈ Max(O_{K}), and ϕ_{k} : K_{𝔭}Γ → K_{𝔭}Γ given by γ ↦ γ^{k}. Pulling back the module structure along ϕ_{k} defines an endomorphism ϕ_{k}^{∗} on K_{1}(Γ, K_{𝔭}) = K_{1}(K_{𝔭}Γ). Then we have

ϕ_{k}^{∗} = σ^{k} on K_{1}(K_{𝔭}Γ)

for all k ≥ 1. This follows from Lemma 3.6(b) and the fact that the operation σ^{k} corresponds to Hom(ψ̂^{k}, K̄_{𝔭}^{×}) via the isomorphism K_{1}(K_{𝔭}Γ) ≅ Hom_{G_{𝔭}}(K_{0}(K̄_{𝔭}Γ), K̄_{𝔭}^{×}). Note that ϕ_{k}^{∗} cannot be defined on K_{1}(O_{K_{𝔭}}Γ) in general since a projective O_{K_{𝔭}}Γ -module need not to be projective again after pulling back the module structure along ϕ_{k}. Thus, it is unlikely that the endomorphism σ^{p} = Hom(ψ̂^{p}, K_{𝔭}^{×}) of K_{1}(K_{𝔭}Γ) induces an endomorphism of K_{0}T (O_{K_{𝔭}}Γ) (p := char(O_{K}/𝔭)) in general.

§4 The Galois Structure of the Module of Relative Differentials

Let N/K be a Galois extension of number fields with Galois group Γ = Gal(N/K). Let Ω := Ω_{O_{N} /O_{K}}^{1} denote the module of relative differentials, 𝔇 := Ann_{O_{N}} (Ω) the different, V := supp(Ω) the set of ramified primes in Spec(O_{N}), and 𝔍 the Γ -stable ideal ∏_{𝔓∈V} 𝔓 in O_{N}.

The aim of this section is to prove the following theorem which will be used in section 5 to compute the tangential element T_{f} associated with the morphism f : Spec(O_{N}) → Spec(O_{K}).

Theorem 4.1. We have:

[Ω] = [𝔍] − [𝔍𝔇] in K′_{0}(O_{N}#Γ).

Proof. The quotient module 𝔍/𝔍𝔇 decomposes into the direct sum of the Γ -modules 𝔍_{𝔭}/𝔍_{𝔭}𝔇_{𝔭}, 𝔭 ∈ f(V), over O_{N}. Thus, we have:

[𝔍] − [𝔍𝔇] = ∑_{𝔭∈f(V)}[𝔍_{𝔭}/𝔍_{𝔭}𝔇_{𝔭}] in K′_{0}(O_{N}#Γ).

Likewise, we have:

[Ω] = ∑_{𝔭∈f(V)}[Ω_{𝔭}] in K′_{0}(O_{N}#Γ).

Therefore, it suffices to show that, for all 𝔭 ∈ f(V), we have

[Ω_{𝔭}] = [𝔍_{𝔭}/𝔍_{𝔭}𝔇_{𝔭}]

in the Grothendieck group K_{0}T (Γ, O_{N,𝔭}) of all f. g. torsion modules over O_{N,𝔭}#Γ. Now we fix a 𝔭 ∈ f(V) and a 𝔓 ∈ V over 𝔭. Let Γ_{𝔓} ⊆ Γ denote the decomposition group of 𝔓. Since the canonical homomorphism K_{0}T (O_{N,𝔭}#Γ) → K_{0}T (O_{N,𝔓}#Γ_{𝔓}) is bijective, it suffices to show the equality

[Ω_{𝔓}] = [𝔓O_{N,𝔓}/𝔓𝔇_{𝔓}] in K_{0}T (Γ_{𝔓}, O_{N,𝔓}).

We now write Γ, O_{N}, 𝔓, Ω, and 𝔇 for Γ_{𝔓}, O_{N,𝔓}, 𝔓O_{N,𝔓}, Ω_{𝔓}, and 𝔇_{𝔓}, respectively. Let π ∈ 𝔓 be a prime element and let L ⊆ N denote the inertia field associated with 𝔓. We again write O_{L} for the localization of O_{L} in 𝔓 ∩ L. By Proposition 18 on p. 19 in [Se2], the O_{L}-algebra O_{N} is generated by π. Thus, the module Ω_{O_{N} /O_{L}} of relative differentials is generated by dπ. Since O_{L} is unramified over O_{K,𝔭}, the canonical homomorphism Ω_{O_{N} /O_{L}} → Ω is bijective. Hence, Ω is generated by dπ, too. Let the unit u_{γ} ∈ O_{N}^{×} be defined by γ(π) = u_{γ} · π for each γ ∈ Γ. Then, for all i ≥ 1, we have γ(π^{i}) = u_{γ}^{i} · π^{i} and, hence,

γ(π^{i} dπ) = u_{γ}^{i+1}π^{i} dπ + u_{γ}^{i}π^{i+1} du_{γ} in Ω.

Consequently, we have:

γ(\overline{π^{i} dπ}) = u_{γ}^{i+1}\overline{π^{i} dπ} in 𝔓^{i}Ω/𝔓^{i+1}Ω.

Thus, the sequence

\begin{aligned}0 & → & 𝔓^{i+2} & →^{can} & 𝔓^{i+1} & → & 𝔓^{i}Ω/𝔓^{i+1}Ω & → & 0 \\ & & & & a · π^{i+1} & ↦ & a · \overline{π^{i} dπ} \end{aligned}

(a ∈ O_{N}) is an exact sequence of Γ -modules over O_{N} for all i = 0, …, l − 1 (here, l denotes the length of Ω, i.e., 𝔇 = 𝔓^{l}). This implies the desired equality

[Ω] = ∑_{i=0}^{l−1}[𝔓^{i}Ω/𝔓^{i+1}Ω] = ∑_{i=1}^{l}[𝔓^{i}/𝔓^{i+1}] = [𝔓/𝔓𝔇] in K_{0}T (Γ, O_{N}).

Corollary 4.2. If N/K is a tame extension, we have

[Ω] = [𝔇^{−1}] − [O_{N}] in K′_{0}(O_{N}#Γ).

Proof. By Theorem (2.6) on p. 210 in [N], we have

𝔍𝔇 = ∏_{𝔓∈V} 𝔓^{e_{𝔓}}

where e_{𝔓} denotes the ramification index at 𝔓. Thus, for all 𝔭 ∈ f(V), the Γ -stable ideal 𝔍_{𝔭}𝔇_{𝔭} is equal to 𝔭O_{N,𝔭} and then isomorphic to O_{N,𝔭} as a Γ -module over O_{N,𝔭}. Thus, we have:

[Ω] = ∑_{𝔭∈f(V)}[Ω_{𝔭}] = ∑_{𝔭∈f(V)}[Ω_{𝔭} ⊗ (𝔍𝔇)_{𝔭}^{−1}] = [Ω ⊗ (𝔍𝔇)^{−1}] in K_{0}T (Γ, O_{N}).

By Theorem 4.1, we finally have:

[Ω] = [Ω ⊗ (𝔍𝔇)^{−1}] = ([𝔍] − [𝔍𝔇]) · ([𝔍𝔇]^{−1}) = [𝔇^{−1}] − [O_{N}] in K′_{0}(O_{N}#Γ).

The following Remark 4.3 and Example 4.4 contain situations where the equality [Ω] = [𝔇^{−1}] − [O_{N}] holds not only in the Grothendieck group K′_{0}(O_{N}#Γ), but Ω and 𝔇^{−1}/O_{N} are in fact equivariantly isomorphic.

Remark 4.3. Let 𝔓 ∈ Spec(O_{N}) be a tamely ramified prime ideal and let I ⊆ Γ denote the inertia group associated with 𝔓. Then the modules Ω_{𝔓} and 𝔇_{𝔓}^{−1}/O_{N,𝔓} are isomorphic as I-modules over O_{N,𝔓}.

Proof. Let L := N^{I} denote the inertia field associated with 𝔓, 𝔭 := 𝔓 ∩ L the prime below 𝔓, and e the ramification index at 𝔓. Furthermore, let Ô_{N,𝔓} and Ô_{L,𝔭} denote the respective completions. Then there is a prime element π ∈ Ô_{N,𝔓} such that π^{e} ∈ Ô_{L,𝔭} (e.g., see p. 26 in [F1]). As in the proof of Theorem 4.1, the module of relative differentials Ω_{𝔓} ≅ Ω_{O_{N,𝔓}/O_{L,𝔭}} ≅ Ω_{Ô_{N,𝔓}/Ô_{L,𝔭}} is generated by dπ. Let the e-th root of unity ζ_{γ} defined by γ(π) = ζ_{γ} · π for each γ ∈ I. Since ζ_{γ} is contained in Ô_{L,𝔭} (e.g., see p. 26 in [F1]), we have:

γ(dπ) = ζ_{γ} dπ + π dζ_{γ} = ζ_{γ} dπ in Ω_{𝔓}.

Hence, the sequence

\begin{aligned}0 & → & 𝔓^{e}Ô_{N,𝔓} & → & 𝔓Ô_{N,𝔓} & → & Ω_{𝔓} & → & 0 \\ & & & & a · π & ↦ & a · dπ \end{aligned}

(a ∈ O_{N}) is an exact sequence of I-modules over Ô_{N,𝔓}. The homomorphism 𝔇_{𝔓}^{−1} → 𝔓Ô_{N,𝔓}, a ↦ a · π^{e}, of I-modules over O_{N,𝔓} now induces the desired isomorphism

𝔇_{𝔓}^{−1}/O_{N,𝔓} ˜→ Ω_{𝔓}

of I-modules over O_{N,𝔓}.

Example 4.4. Let D be a square-free natural number, K the field of rational numbers, and N := ℚ(\sqrt{D}). Then the modules Ω and 𝔇^{−1}/O_{N} are isomorphic as Γ -modules over O_{N}.

Proof. We have

O_{N} = \{ \begin{matrix}ℤ[\sqrt{D}] ≅ ℤ[T]/(ϕ) & where ϕ = T^{2} − D, & if D ≢ 1 mod 4 \\ ℤ[(1 + \sqrt{D})/2] ≅ ℤ[T]/(ϕ) & where ϕ = T^{2} − T − (D − 1)/4, & if D ≡ 1 mod 4. \end{matrix}

Let t := \sqrt{D} or t := (1 + \sqrt{D})/2. By Satz (2.4) on p. 207 in [N], the different 𝔇_{N/K} is generated by ϕ′(t) which is equal to 2\sqrt{D} or \sqrt{D}. Let γ ∈ Γ = Gal(N/K) denote the non-trivial automorphism. Then we have γ(t) = −t or γ(t) = −t + 1 and hence

γ(dt) = −dt in Ω

in both cases. Consequently, the sequence

\begin{aligned}0 & → & O_{N} & →^{can} & 𝔇^{−1} & → & Ω & → & 0 \\ & & & & a · (ϕ′(t)^{−1}) & ↦ & a · dt \end{aligned}

(a ∈ O_{N}) is an exact sequence of Γ -modules over O_{N}. This proves Example 4.4.

§5 Equivariant Adams-Riemann-Roch Formulas for Tame Galois Extensions

Let N/K be a tame Galois extension of number fields with Galois group Γ = Gal(N/K). Let f : Spec(O_{N}) → Spec(O_{K}) denote the Γ -morphism associated with N/K. As in section 3, let ψ_{k}^{CNT} denote the endomorphism of the locally free classgroup Cl(O_{K}Γ) defined by Ph. Cassou-Noguès and M. Taylor in [CNT].

Any Γ -stable ideal 𝔄 in N defines an element (𝔄) ∈ Cl(O_{K}) in the obvious way (see below). In the paper [BC], D. Burns and T. Chinburg have established a formula for ψ_{k}^{CNT}((𝔄)). In this section, we will give a geometric interpretation of this formula: On the one hand, we will apply the equivariant Adams-Riemann-Roch formula of [Ko5] to the Γ -morphism f using Corollary 4.2. On the other hand, we will reformulate the formula of Burns and Chinburg using Theorem 3.7. It will then turn out, that, more or less, the formula of Burns and Chinburg is a strengthening of the equivariant Adams-Riemann-Roch formula in this situation.

Since O_{N} and O_{K} are regular rings, the morphism f is a local complete intersection morphism (see Remark on p. 86 in [FL]). Being a finite morphism, f in particular is a projective morphism. By Remark (3.5) in [Ko5], the morphism f then is Γ -projective in the sense of section 3 in [Ko5]. Let T_{f}^{∨} ∈ K_{0}(Γ, O_{N}) denote the equivariant cotangential element associated with f. It is defined as follows (see section 4 in [Ko5]): Let

Spec(O_{N}) ⟶^{i} 𝔸 = Spec(A) ⟶^{p} Spec(O_{K})

be a factorization of f into a closed Γ -immersion i and a smooth Γ -morphism p of constant relative dimension d. Then i is a regular embedding (see Remark on p. 86 in [FL]). Let I denote the Γ -stable ideal in A associated with i. Then T_{f}^{∨} is defined to be

T_{f}^{∨} := [Ω_{A/O_{K}}^{1} ⊗_{A} A/I] − [I/I^{2}] ∈ K_{0}(Γ, O_{N}).

Lemma 5.1. We have:

T_{f}^{∨} = [Ω_{O_{N} /O_{K}}^{1}] in K′_{0}(O_{N}#Γ).

Proof. We have a natural exact sequence

I/I^{2} → Ω_{A/O_{K}}^{1} ⊗_{A} A/I →^{ϵ} Ω_{O_{N} /O_{K}}^{1} → 0

of Γ -modules over O_{N} (e.g., see Theorem 58i) on p. 187 in [Ma]). Being a submodule of Ω_{A/O_{K}}^{1} ⊗ A/I, the kernel of ϵ is torsionfree; thus, it is O_{N}-projective. Furthermore, we obviously have rank_{O_{N}} (ker(ϵ)) = d = rank_{O_{N}} (I/I^{2}). Thus, the canonical surjection I/I^{2} → ker(ϵ) is an isomorphism. This implies that the left homomorphism in the sequence above is injective which proves Lemma 5.1.

Let 𝔇 denote the different associated with N/K.

Corollary 5.2. We have:

T_{f}^{∨} = [𝔇^{−1}] − [O_{N}] in K_{0}(Γ, O_{N}).

Proof. This immediately follows from Lemma 5.1 and Corollary 4.2 since K_{0}(Γ, O_{N}) ≅ K′_{0}(O_{N}#Γ) (see Notations).

Now, we fix a k ∈ ℕ. Let K̂_{0}(Γ, O_{K})[k^{−1}] denote the J-adic completion of K_{0}(Γ, O_{K})[k^{−1}] where J := ker(rank : K_{0}(Γ, O_{K}) → ℤ) is the augmentation ideal.

Proposition 5.3. The Bott element θ^{k}(𝔇^{−1}) := ∑_{i=0}^{k−1}[𝔇^{−1}] is invertible in

K_{0}(Γ, O_{N}) ⊗_{K_{0}(Γ,O_{K})} K̂_{0}(Γ, O_{K})[k^{−1}]

and we have:

θ^{k}(T_{f}^{∨})^{−1} = k · θ^{k}(𝔇^{−1})^{−1}

(see section 4 in [Ko5] for the definition of θ^{k}(T_{f}^{∨})^{−1}).

Proof. For any f. g. O_{N}-projective O_{N}#Γ -module P, let θ^{k}(P) ∈ K_{0}(Γ, O_{N}) denote the k-th Bott element associated with P (see section 4 in [Ko5]). By section 4 in [Ko5], there is a representation T_{f}^{∨} = [Ω] − [N] in K_{0}(Γ, O_{N}) such that θ^{k}(Ω) is invertible in K_{0}(Γ, O_{N}) ⊗_{K_{0}(Γ,O_{K})} K̂_{0}(Γ, O_{K})[k^{−1}]. By Corollary 5.2, we have θ^{k}(𝔇^{−1}) · θ^{k}(N) = k · θ^{k}(Ω). Thus, θ^{k}(𝔇^{−1}), θ^{k}(N), and θ^{k}(T_{f}^{∨}) are invertible and we have θ^{k}(T_{f}^{∨})^{−1} = k · θ^{k}(𝔇^{−1})^{−1}.

Pulling back the module structure along O_{K} → O_{N} defines a homomorphism

f_{∗} : K_{0}(Γ, O_{N}) → K_{0}(Γ, O_{K}).

Obviously, f_{∗} coincides with the push-forward homomorphism f_{∗} defined in section 3 of [Ko5]. The projection formula implies that f_{∗} induces a homomorphism

f̂_{∗} : K_{0}(Γ, O_{N}) ⊗_{K_{0}(Γ,O_{K})} K̂_{0}(Γ, O_{K})[k^{−1}] → K̂_{0}(Γ, O_{K})[k^{−1}].

Let ψ^{k} denote the k-th Adams operation on K_{0}(Γ, O_{K}) and K_{0}(Γ, O_{N}).

Theorem 5.4 (Equivariant Adams-Riemann-Roch formula applied to f). For all x ∈ K_{0}(Γ, O_{N}), we have:

ψ^{k}f_{∗}(x) = f̂_{∗}(k · θ^{k}(𝔇^{−1})^{−1} · ψ^{k}(x)) in K̂_{0}(Γ, O_{K})[k^{−1}].

Proof. Because of Proposition 5.3, this is the assertion of Theorem (4.5) in [Ko5] applied to f.

We are now going to reformulate the formula of Burns and Chinburg mentioned in the beginning of this section.

Lemma 5.5.

(a) Every f. g. O_{N}-projective O_{N}#Γ -module has a quotient which is isomorphic to a Γ -stable fractional ideal in N. In particular, any invertible O_{N}#Γ -module is isomorphic to such an ideal.

(b) Every f. g. O_{N}-projective O_{N}#Γ -module is projective as O_{K}Γ -module.

(c) The (additive) group K_{0}(Γ, O_{N}) is generated by the classes of Γ -stable fractional ideals in N.

Proof. Obviously, any f. g. O_{N}-projective O_{N}#Γ -module P is a Γ -stable lattice in P ⊗_{O_{N}} N. By Morita theory, the N#Γ -module P ⊗_{O_{N}} N is isomorphic to N^{rank(P)}. Thus, there is an N#Γ -submodule W of P ⊗_{O_{N}} N such that (P ⊗_{O_{N}} N)/W is isomorphic to N. Then, P/P ∩ W is isomorphic to a fractional ideal in N. This shows assertion (a). Using (a), one easily shows as in §3 of chapter I in [F1] that P is locally free over O_{K}Γ. By Theorem 1.1 on p. 1 in [T], this proves assertion (b). Assertion (c) follows from assertion (a) by induction on the rank of P.

By Lemma 5.5(b), we have a push-forward homomorphism f_{∗} : K_{0}(Γ, O_{N}) → K_{0}(O_{K}Γ) such that the following diagram commutes:

K_{0}(Γ, O_{N})
f_{∗} ↙  ↘ f_{∗}
K_{0}(O_{K}Γ)  ⟶^{c}  K_{0}(Γ, O_{K}).

Let Ind_{1}^{Γ} (Cl(O_{K})) denote the image of the classical classgroup Cl(O_{K}) ⊂ K_{0}(O_{K}) under the induction map Ind_{1}^{Γ} : K_{0}(O_{K}) → K_{0}(O_{K}Γ).

We now assume that the prime divisors of ord(Γ) are unramified in K and that k is coprime to the exponent e(Γ) of Γ. We fix a k′ ∈ ℕ such that kk′ ≡ 1 mod e(Γ).

Theorem 5.6 (Reformulation of a formula of Burns and Chinburg). We have for all x ∈ K_{0}(Γ, O_{N}):

ψ^{k}f_{∗}(x) = f_{∗} (k · ∑_{i=0}^{k′−1}[𝔇^{−ik}] · ψ^{k}(x)) in K_{0}(O_{K}Γ)/ (Ind_{1}^{Γ} (Cl(O_{K})) ⊕ e(Γ)ℤ[O_{K}Γ]) .

If the classical classgroup Cl(O_{K}) is trivial and if the rank of x is 0, then this formula already holds in K_{0}(O_{K}Γ).

Proof. Let ψ_{k′}^{CNT} denote the endomorphism of Cl(O_{K}Γ) defined by Cassou-Noguès and Taylor (e.g., see section 3). For any Γ -stable fractional ideal 𝔄 in N, (𝔄) := [f_{∗}(𝔄)] − [O_{K}Γ] is a well-defined element of Cl(O_{K}Γ) by Lemma 5.5(b). By Corollary 2.7 in [BC], we have:

ψ_{k′}^{CNT}((𝔄)) = ∑_{i=0}^{k′−1}(𝔇^{−ik} · 𝔄^{k}) in Cl(O_{K}Γ)/Ind_{1}^{Γ} (Cl(O_{K})).

Using Theorem 3.7, we obtain the equality

ψ^{k}([f_{∗}(𝔄)] − [O_{K}Γ]) = k · ∑_{i=0}^{k′−1}([f_{∗}(𝔇^{−ik} · 𝔄^{k})] − [O_{K}Γ]) in K_{0}(O_{K}Γ)/Ind_{1}^{Γ} (Cl(O_{K})).

Hence, we get the following equality in K_{0}(O_{K}Γ)/Ind_{1}^{Γ} (Cl(O_{K})) for all x ∈ K_{0}(Γ, O_{N}) (by Lemma 5.5(c)):

ψ^{k}f_{∗}(x) = f_{∗} (k · ∑_{i=0}^{k′−1}[𝔇^{−ik}] · ψ^{k}(x)) − kk′ · rank_{O_{N}} (x) · [O_{K}Γ] + rank_{O_{N}} (x) · ψ^{k}([O_{K}Γ]).

Now, Theorem 1.6(e) shows Theorem 5.6.

Using an idea of the proof of Theorem 2 in §4 in [CEPT], we show in the following lemma that the right side of the formula in Theorem 5.4 coincides with the right side of the formula in Theorem 5.6 (without f_{∗}) for elements of the form x = T_{f}^{∨} · y with y ∈ K_{0}(Γ, O_{N}).

Lemma 5.7. We assume that the O_{N}#Γ -modules 𝔇^{ord(Γ)} and O_{N} are isomorphic. (This certainly holds if the classical classgroup Cl(O_{K}) is trivial.) Then we have for all y ∈ K_{0}(Γ, O_{N}):

θ^{k}(𝔇^{−1})^{−1} · ψ^{k}(T_{f}^{∨} · y) = ∑_{i=0}^{k′−1}[𝔇^{−ik}] · ψ^{k}(T_{f}^{∨} · y) in K_{0}(Γ, O_{N}) ⊗_{K_{0}(Γ,O_{K})} K̂_{0}(Γ, O_{K})[k^{−1}].

Proof. We have:

θ^{k}(𝔇^{−1})^{−1} · ψ^{k}(T_{f}^{∨} · y) =

= (∑_{i=0}^{k−1}[𝔇^{−i}])^{−1} · ([𝔇^{−k}] − [O_{N}]) · ψ^{k}(y) (by Corollary 5.2)

= ([𝔇^{−1}] − [O_{N}]) · ψ^{k}(y) (geometric series)

= ([𝔇^{−kk′}] − [O_{N}]) · ψ^{k}(y) (since 𝔇^{ord(Γ)} ≅ O_{N} by assumption)

= (∑_{i=0}^{k′−1}[𝔇^{−ik}]) · ([𝔇^{−k}] − [O_{N}]) · ψ^{k}(y) (geometric series)

= ∑_{i=0}^{k′−1}[𝔇^{−ik}] · ψ^{k}(T_{f}^{∨} · y) (by Corollary 5.2).

Corollary 5.8. We assume that Cl(O_{K}) is trivial. Then the element f_{∗}(T_{f}^{∨}) in K_{0}(O_{K}Γ) or K_{0}(Γ, O_{K}) is an eigenvector of the Adams operation ψ^{k} with eigenvalue k.

Proof. This follows from Theorem 5.6 and the proof of Lemma 5.7.

In the following closing remark, we compare Theorems 5.4 and 5.6 and point out their respective advantages.

Remark 5.9. On the one hand, Theorem 5.4 is a special case of a general (geometric) equivariant Adams-Riemann-Roch formula which can be applied also in the non-tame case, which does not need the assumption “gcd(k, ord(Γ)) = 1”, and which holds without passing to residues modulo Ind_{1}^{Γ} (K_{0}(O_{K})). On the other hand, Theorem 5.6 holds already for K_{0}(O_{K}Γ) without passing to K̂_{0}(Γ, O_{K})[k^{−1}] or even K_{0}(Γ, O_{K}). In particular, the multiplier ∑_{i=0}^{k′−1}[𝔇^{−ik}] in Theorem 5.6 is definable already in K_{0}(Γ, O_{N}). In fact, Theorem 5.6 is not a reformulation but a weakening of the formula of Burns and Chinburg since we have multiplied their formula by the factor k in the proof of Theorem 5.6. Finally, choosing k = 1 and k′ = 1 + e(Γ) in Theorem 5.6 yields the (non-trivial!) formula

f_{∗} (∑_{i=1}^{e(Γ)}[𝔇^{−i}] · x) = 0 in K_{0}(O_{K}Γ) /(Ind_{1}^{Γ} (Cl(O_{K})) ⊕ e(Γ)ℤ[O_{K}Γ])

for all x ∈ K_{0}(Γ, O_{N}) (see also Remark 2.8 in [BC]). One can show that, after passing to K̂_{0}(Γ, O_{K})[k^{−1}] /(Ind_{1}^{Γ} (Cl(O_{K})) ⊕ e(Γ)ℤ[O_{K}Γ]), this formula together with the formula in Theorem 5.4 is in fact equivalent to the general form of the formula in Theorem 5.6. Moreover, the paper [BC] contains a formula for ψ_{k}^{CNT}((𝔄)) even if gcd(k, ord(Γ)) ≠ 1 (see Corollary 2.4 in [BC]). Since, however, there is no relation of ψ_{k}^{CNT} with Adams operations in this case (see Remarks 3.11 and 3.12), this formula cannot be compared with the Adams-Riemann-Roch formula.

References

[AB] K. Akin and D. A. Buchsbaum, Characteristic-free representation theory of the general linear group, Adv. Math. 58 (1985), 149-200.

[ABW] D. Akin, D. A. Buchsbaum, and J. Weyman, Schur functors and Schur complexes, Adv. Math. 44 (1982) 207-278.

[B] H. Bass, “Algebraic K-theory”, Math. Lecture Note Series (Benjamin, New York, 1968).

[BC] D. Burns and T. Chinburg, Adams operations and integral Hermitian-Galois representations, Amer. J. Math. 118 (1996), 925-962.

[CNT] Ph. Cassou-Noguès and M. J. Taylor, Opérations d’Adams et Groupe des classes d’Algèbre de groupe, J. Algebra 95 (1985), 125-152.

[C] T. Chinburg, Galois structure of de Rham cohomology of tame covers of schemes, Ann. of Math. 139 (1994), 443-490.

[CEPT] T. Chinburg, B. Erez, G. Pappas, and M. J. Taylor, Arithmetic equivariant Riemann-Roch theorems, preprint (Manchester Centre for Pure Mathematics, number 1993/16).

[CR] C. W. Curtis and I. Reiner, “Methods of representation theory with applications to finite groups and orders”, vol. I, Pure Appl. Math. (Wiley, New York, 1981).

[F1] A. Fröhlich, “Galois module structure of algebraic integers”, Ergeb. Math. Grenzgeb. (3) 1 (Springer, New York, 1983).

[F2] A. Fröhlich, “Classgroups and Hermitian modules”, Progr. Math. 48 (Birkhäuser, Boston, 1984).

[FH] W. Fulton and J. Harris, “Representation theory: a first course”, Grad. Texts in Math. 129 (Springer, New York, 1991).

[FL] W. Fulton and S. Lang, “Riemann-Roch algebra”, Grundlehren Math. Wiss. 277 (Springer, New York, 1985).

[Gr1] D. R. Grayson, Higher algebraic K-theory: II, in R. M. Stein (ed.), “Algebraic K-theory (Evanston, 1976)”, Lecture Notes in Math. 551 (Springer, New York, 1976), 217-240.

[Gr2] D. R. Grayson, Exterior power operations on higher K-theory, K-Theory 3 (1989), 247-260.

[Ke] M. Kervaire, Opérations d’Adams en Théorie des représentations linéaires des groupes finis, Enseign. Math. 22 (1976), 1-28.

[Ko1] B. Köck, Das Adams-Riemann-Roch-Theorem in der höheren äquivarianten K-Theorie, J. Reine Angew. Math. 421 (1991), 189-217.

[Ko2] B. Köck, Shuffle products in higher K-theory, J. Pure Appl. Algebra 92 (1994), 269-307.

[Ko3] B. Köck, On Adams operations on the higher K-theory of group rings, in: G. Banaszak et al. (eds.), “Algebraic K-theory (Poznań, 1995)”, Contemp. Math. 199 (Amer. Math. Soc., Providence, 1996), 139-150.

[Ko4] B. Köck, Adams operations for projective modules over group rings, Math. Proc. Cambridge Philos. Soc. 122 (1997), 55-71.

[Ko5] B. Köck, The Grothendieck-Riemann-Roch theorem for group scheme actions, Ann. Sci. École Norm. Sup. 31 (1998), 415-458.

[McD] I. G. Macdonald, “Symmetric functions and Hall polynomials”, Oxford Math. Monographs (Clarendon Press, Oxford, 1979).

[Ma] H. Matsumura, “Commutative algebra”, Math. Lecture Note Series (Benjamin, New York, 1970).

[N] J. Neukirch, “Algebraische Zahlentheorie” (Springer, Berlin, 1992).

[Que] J. Queyrut, S-groupes des classes d’un ordre arithmétique, J. Algebra 76 (1982), 234-260.

[Q] D. Quillen, Higher algebraic K-theory: I, in H. Bass (ed.), “Algebraic K-Theory I (Seattle, 1972)”, Lecture Notes in Math. 341 (Springer, New York, 1973), 85-147.

[Se1] J. P. Serre, “Lineare Darstellung endlicher Gruppen”, Logik und Grundlagen der Mathematik (Vieweg, Braunschweig, 1972).

[Se2] J. P. Serre, “Local fields”, Grad. Texts in Math. 67 (Springer, New York, 1979).

[Sh] C. Sherman, Connecting homomorphisms in localization sequences, in: G. Banaszak et al. (eds.), “Algebraic K-theory (Poznań, 1995)”, Contemp. Math. 199 (Amer. Math. Soc., Providence, 1996), 175-183.

[Sw] R. G. Swan, Induced representations and projective modules, Ann. of Math. 71 (1960), 552-578.

[T] M. Taylor, “Classgroups of group rings”, Lecture Notes Ser. 91 (Cambridge University Press, Cambridge, 1984).

Mathematisches Institut II Universität Karlsruhe D-76128 Karlsruhe Germany

e-mail: Bernhard.Koeck@math.uni-karlsruhe.de