Symmetric Powers of Galois Modules on Dedekind Schemes

by

Bernhard Köck

Abstract. We prove a certain Riemann-Roch type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou-Noguès-Taylor operations. Mathematics Subject Classification 2000. 11R33; 19A31; 19B28; 11R29; 13F05.

Key words. Symmetric power operation, Adams operation, Grothendieck group, Bass-Whitehead group, locally free classgroup, Dedekind scheme, equivariant Adams-Riemann-Roch formula.

Introduction

Let G be a finite group and E a number field. Let O_{E} denote the ring of integers in E, Y := Spec(O_{E}), and

Cl(O_{Y} G) := ker(rank : K_{0}(O_{E}G) → ℤ)

the locally free classgroup associated with E and G. For any k ≥ 1, Cassou-Noguès and Taylor have constructed a certain endomorphism ψ_{k}^{CNT} of Cl(O_{Y} G) which, via Fröhlich’s Hom-description of Cl(O_{Y} G), is dual to the k-th Adams operation on the classical ring of virtual characters of G (see [CT]). Now, let gcd(k, ord(G)) = 1 and let k′ ∈ ℕ be an inverse of k modulo ord(G). In the paper [K 3], we have shown that then the endomorphism ψ_{k′}^{CNT} is a simply definable symmetric power operation σ^{k}.

Now, let F/E be a finite tame Galois extension with Galois group G. Let f : X := Spec(O_{F}) → Y denote the corresponding G-morphism and let f_{∗} be the homomorphism

f_{∗} : K_{0}(G, X) → Cl(O_{Y} G), [E] ↦ [f_{∗}(E)] − rank(E) · [O_{Y} G],

from the Grothendieck group K_{0}(G, X) of all locally free O_{X}-modules with (semilinear) G-action to Cl(O_{Y} G). Furthermore, let D denote the different of F/E and ψ^{k} the k-th Adams operation on K_{0}(G, X). The paper [BC] by Burns and Chinburg together with the identification of ψ_{k′}^{CNT} with σ^{k} mentioned above then implies the following Riemann-Roch type formula for all x ∈ K_{0}(G, X):

(1) σ^{k}(f_{∗}(x)) = f_{∗} (∑_{i=0}^{k′−1}[D^{−ik}] · ψ^{k}(x)) in Cl(O_{Y} G)/Ind_{1}^{G}Cl(O_{Y})

(see Theorem 5.6 and Theorem 3.7 in [K 3]).

We now assume that Y is an arbitrary Dedekind scheme (i.e., Noetherian, regular, irreducible, and dim(Y) = 1) and that X is the normalization of Y in a finite Galois extension F of the function field E of Y with Galois group G. We again assume that the corresponding G-morphism f : X → Y is tamely ramified. Similarly to the number field case, we define the locally free classgroup Cl(O_{Y} G) (see section 2 or [AB]), the symmetric power operation σ^{k} on Cl(O_{Y} G) (see sections 1 and 2), and the homomorphism f_{∗} : K_{0}(G, X) → Cl(O_{Y} G) (see section 3). The object of this paper is to study the following natural question. Does the formula (1) still hold in this more general situation?

First of all, we mention that the paper [BC] also implies that the formula (1) holds if Y is a projective smooth curve over a finite field L and the characteristic of L does not divide the order of G (see Theorem 3.5(b)). In this semisimple function field case, a Hom-description of Cl(O_{Y} G) again exists and the operation σ^{k} is dual to the Adams operation ψ^{k′} as in the number field case (see Theorem 2.10). In particular, Fröhlich’s techniques can be applied as in the number field case (see [BC]).

In this paper, we moreover obtain the following results whose proof however requires completely different methods since there is no Hom-description of Cl(O_{Y} G) available in general.

Theorem A. The formula (1) holds if one of the following assumptions is satisfied:

(a) k = 1.

(b) The group G is Abelian and f : X → Y is unramified.

Theorem B. The formula (1) holds after passing from Cl(O_{Y} G)/Ind_{1}^{G}Cl(O_{Y}) to K̂_{0}(G, Y)[k^{−1}]/(Ind_{1}^{G}K_{0}(Y))K̂_{0}(G, Y)[k^{−1}] via the Cartan homomorphism. Here, K_{0}(G, Y) denotes the Grothendieck group of all locally free O_{Y} -modules with G-action and K̂_{0}(G, Y)[k^{−1}] denotes the I-adic completion of K_{0}(G, Y)[k^{−1}] where I is the augmentation ideal of K_{0}(G, Y)[k^{−1}].

The proof of Theorem A in the case k = 1 relies on the results of the paper [C] by Chase (see Proposition 3.2). Note that, despite the fact σ^{k} = id for k = 1, the formula (1) is non-trivial since k′ may be an arbitrary natural number in the coset 1 + ord(G)ℤ. If G is Abelian and f : X → Y is unramified, the proof of Theorem A relies on the following two facts (see Theorem 3.5). Firstly, applying the operation σ^{k} to the element [Q] − [P] in Cl(O_{Y} G) is the same as pulling back the G-action on P and Q along the automorphism G → G, g ↦ g^{k} (see Theorem 2.7). Secondly, the map H^{1}(Y, G) → Cl(O_{Y} G) which maps a principal G-bundle f : X → Y to the class [f_{∗}(O_{X})]−[O_{Y} G] is a homomorphism (by Theorem 5 in the paper [W] by Waterhouse). Theorem B follows from the equivariant Adams-Riemann-Roch theorem (see [K 2]) and the case k = 1 of Theorem A (see Theorem 3.3). Moreover, in the semisimple function field case mentioned above, the formula (1) modulo torsion can be deduced from Theorem B if the order of G is a power of a prime (see Remark 3.6).

Acknowledgments. I would like to thank David Burns for many very helpful discussions and for his hospitality during my stay at the King’s College in London. In particular, he has drawn my attention to the paper [C] which is fundamental for the first case in Theorem A and also for Theorem B. Furthermore, I would like to thank the referee for several useful comments.

§1 Symmetric Power Operations on K_{0}-, K_{1}-, and Relative Grothendieck Groups

Let X be a Noetherian scheme and G a finite group which, in this section, is assumed to act trivially on X.

First, we introduce the category of locally projective modules over the group ring O_{X}G. Then, we (purely algebraically) construct symmetric power operations on the Grothendieck group K_{0}(O_{X}G) and the Bass group K_{1}^{det}(O_{X}G) associated with this category. While these constructions are more or less obvious generalizations of the constructions in section 1 of [K 3] (for K_{0} and K_{1}), the subsequent construction of symmetric power operations on relative Grothendieck groups (in the sense of [B]) is new. We furthermore show that these operations are compatible with the maps in the localization sequence. Finally, we present some cases in which the relative Grothendieck groups can be identified with Grothendieck groups of certain torsion modules.

By a (quasi-)coherent O_{X}G-module we mean a (quasi-)coherent O_{X}-module P together with an action of G on P by O_{X}-homomorphisms. Homomorphisms and exact sequences of quasi-coherent O_{X}G-modules are defined in the obvious way. We call a coherent O_{X}G-module P locally projective iff the stalk P_{x} is a projective O_{X,x}Gmodule for all x ∈ X. Let K_{0}(O_{X}G) denote the Grothendieck group of all locally projective O_{X}G-modules.

Remark 1.1. If X = Spec(A) is affine, then P is locally projective if and only if P := H^{0}(X, P) is a finitely generated projective AG-module. (Easy to prove.)

We are now going to construct the above-mentioned symmetric power operations. As in section 1 of [K 3], it is convenient to introduce the following categories. For any i ≥ 1, let M_{i} denote the smallest full subcategory of the Abelian category of all coherent O_{X}G-modules which is closed under extensions and kernels of O_{X}G-epimorphisms and which contains all the modules of the form Sym_{O_{X}}^{i_{1}}(P_{1}) ⊗_{O_{X}} … ⊗_{O_{X}} Sym_{O_{X}}^{i_{r}}(P_{r}) where P_{1}, …, P_{r} are locally projective coherent O_{X}G-modules, i_{1}, …, i_{r} are natural numbers with i_{1} + … + i_{r} = i, and G acts diagonally. So, M_{1} is the category of all locally projective coherent O_{X}G-modules. By Proposition 1.1 in [K 3], the category M_{i} is contained in M_{1} if gcd(i, ord(G)) is invertible on X. It is easy to see that, for all i, j ≥ 1, the functor

M_{i} × M_{j} → M_{i+j}, (P, Q) ↦ P ⊗_{O_{X}} Q,

is well-defined and bi-exact (cf. Lemma 1.2 in [K 3]). In particular, we obtain products K_{0}(M_{i}) × K_{0}(M_{j}) → K_{0}(M_{i+j}), i, j ≥ 1, and the set 1 + ∏_{i≥1} K_{0}(M_{i})t^{i} consisting of all power series 1 + ∑_{i≥1} a_{i}t^{i} with a_{i} ∈ K_{0}(M_{i}) forms an Abelian group with respect to multiplication of power series. As usual, one shows that the association [P] ↦ ∑_{i≥0}[Sym_{O_{X}}^{i}(P)]t^{i} can be extended to a well-defined homomorphism

σ : K_{0}(O_{X}G) → 1 + ∏_{i≥1} K_{0}(M_{i})t^{i}

(see §1 of Chapter V in [FL] and Lemma 1.3 in [K 3]). The i-th component of this homomorphism is denoted by σ^{i}. We have for all x, y ∈ K_{0}(O_{X}G):

σ^{i}(x − y) = ∑_{{a≥0,b_{1},...,bu≥1 \atop a+b_{1}+...+bu=i}}(−1)^{u}σ^{a}(x)σ^{b_{1}}(y) ⋯ σ^{b_{u}}(y)

= ∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+...+bu=i}}(−1)^{u}(σ^{a}(x) − σ^{a}(y))σ^{b_{1}}(y) ⋯ σ^{b_{u}}(y)

in K_{0}(M_{i}) (cf. section 2 in [G 2]). If gcd(i, ord(G)) is invertible on X, let σ^{i} also denote the composition

K_{0}(O_{X}G) ⟶^{σ^{i}} K_{0}(M_{i}) ⟶^{can} K_{0}(O_{X}G).

The map σ^{i} is called i-th symmetric power operation.

Now, let K_{0}(ℤ, M_{i}) denote the Grothendieck group of all pairs (P, α) where P is an object of M_{i} and α is an O_{X}G-automorphism of P. We put K_{0}(ℤ, O_{X}G) := K_{0}(ℤ, M_{1}). As above, the association ((P, α), (Q, β)) ↦ (P ⊗_{O_{X}} Q, α ⊗_{O_{X}} β) induces a multiplication map

K_{0}(ℤ, M_{i}) × K_{0}(ℤ, M_{j}) → K_{0}(ℤ, M_{i+j})

(for all i, j ≥ 1) and the association (P, α) ↦ ∑_{i≥0}(Sym_{O_{X}}^{i}(P), Sym_{O_{X}}^{i}(α))t^{i} induces a homomorphism

σ : K_{0}(ℤ, O_{X}G) → 1 + ∏_{i≥1} K_{0}(ℤ, M_{i})t^{i}.

By restricting, we obtain symmetric power operations

σ^{i} : K̃_{0}(ℤ, O_{X}G) = K̃_{0}(ℤ, M_{1}) → K̃_{0}(ℤ, M_{i}), i ≥ 1,

between the reduced Grothendieck groups

K̃_{0}(ℤ, M_{i}) := ker(K_{0}(ℤ, M_{i}) → K_{0}(M_{i}), [P, α] ↦ [P]), i ≥ 1.

We denote the factor group of K_{0}(ℤ, M_{i}) modulo the subgroup generated by the relations of the form [P, αβ] − [P, α] − [P, β] by K_{1}^{det}(M_{i}). (Note that in particular [P, id] is in the group of relations.) If X = Spec(A) is affine, the group K_{1}^{det}(O_{X}G) = K_{1}^{det}(M_{1}) coincides with the usual Bass-Whitehead group K_{1}(AG) of the group ring AG (by Remark 1.1). In the sequel, we consider K_{1}^{det}(M_{i}) as the factor group of K̃_{0}(ℤ, M_{i}) modulo the subgroup I_{i} generated by the relations of the form [P, αβ] − [P, α] − [P, β] + [P, id]. Since

([P, αβ] − [P, α] − [P, β] + [P, id]) · [Q, γ]

= ([P ⊗ Q, αβ ⊗ γ] − [P ⊗ Q, α ⊗ γ] − [P ⊗ Q, β ⊗ id] + [P ⊗ Q, id ⊗ id])

− ([P ⊗ Q, β ⊗ γ] − [P ⊗ Q, id ⊗ γ] − [P ⊗ Q, β ⊗ id] + [P ⊗ Q, id ⊗ id]) ,

the group I_{i}K_{0}(ℤ, M_{j}) is contained in I_{i+j} and we obtain a multiplication map

K_{1}^{det}(M_{i}) × K_{1}^{det}(M_{j}) = K̃_{0}(ℤ, M_{i})/I_{i} × K̃_{0}(ℤ, M_{j})/I_{j} → K_{1}^{det}(M_{i+j})

(for all i, j ≥ 1) which is obviously trivial, i.e., the product of any two power series ∑_{i≥0} x_{i}t^{i}, ∑_{i≥0} y_{i}t^{i} in 1 + ∏_{i≥1} K_{1}^{det}(M_{i})t^{i} is 1 + ∑_{i≥1}(x_{i} + y_{i})t_{i}.

Lemma 1.2. The homomorphism σ : K̃_{0}(ℤ, O_{X}G) → 1 + ∏_{i≥1} K̃_{0}(ℤ, M_{i})t^{i} induces a homomorphism σ : K_{1}^{det}(O_{X}G) → 1 + ∏_{i≥1} K_{1}^{det}(M_{i})t^{i}. Each component σ^{i} : K_{1}^{det}(O_{X}G) → K_{1}^{det}(M_{i}) of σ is a homomorphism.

Proof. Let P ∈ M_{1} and α, β ∈ Aut_{O_{X}G}(P). We write S for Sym. Then, for all a ≥ 1, the element

[S^{a}(P ⊕ P), S^{a}(αβ ⊕ id)] − [S^{a}(P ⊕ P), S^{a}(α ⊕ β)]

= ∑_{c=0}^{a} ([S^{c}(P) ⊗ S^{a−c}(P), S^{c}(αβ) ⊗ S^{a−c}(id)] − [S^{c}(P) ⊗ S^{a−c}(P), S^{c}(α) ⊗ id]

−[S^{c}(P) ⊗ S^{a−c}(P), S^{c}(β) ⊗ id] + [S^{c}(P) ⊗ S^{a−c}(P), id ⊗ id])

− ∑_{c=0}^{a} ([S^{c}(P) ⊗ S^{a−c}(P), S^{c}(α) ⊗ S^{a−c}(β)] − [S^{c}(P) ⊗ S^{a−c}(P), S^{c}(α) ⊗ id]

−[S^{c}(P) ⊗ S^{a−c}(P), id ⊗ S^{a−c}(β)] + [S^{c}(P) ⊗ S^{a−c}(P), id ⊗ id])

is contained in I_{a}. Since

σ^{i}(x − y) = ∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+...+bu=i}}(−1)^{u}(σ^{a}(x) − σ^{a}(y))σ^{b_{1}}(y) ⋯ σ^{b_{u}}(y)

(for all x, y ∈ K_{0}(ℤ, M_{1})), this implies that the element

σ^{i} ([P, αβ] − [P, α] − [P, β] + [P, id]) = σ^{i} ([P ⊕ P, αβ ⊕ id] − [P ⊕ P, α ⊕ β]) is contained in I_{i}, as was to be shown. For all x, y ∈ K_{1}^{det}(M_{1}), we have

σ(x + y) = σ(x) · σ(y) = 1 + ∑_{i≥1}(σ^{i}(x) + σ^{i}(y))t^{i} in 1 + ∏_{i≥1} K_{1}^{det}(M_{i})t^{i};

thus, σ^{i} is a homomorphism for all i ≥ 1.

Now, let j : U → X be a morphism between Noetherian schemes. Similarly to §5 of Chapter VII in [B], let K_{0}(co(j_{i}^{∗})) denote the Grothendieck group of all triples (P, α, Q) where P and Q are objects in M_{i} and α : j^{∗}(P) → j^{∗}(Q) is an O_{U}Gisomorphism. As above, the association

((P, α, Q), (P′, α′, Q′)) ↦ (P ⊗_{O_{X}} P′, α ⊗_{O_{U}} α′, Q ⊗_{O_{X}} Q′)

induces, for all i, i′ ≥ 1, a multiplication map

K_{0}(co(j_{i}^{∗})) × K_{0}(co(j_{i′}^{∗})) → K_{0}(co(j_{i+i′}^{∗}))

and the association (P, α, Q) ↦ ∑_{i≥0}(Sym_{O_{X}}^{i}(P), Sym_{O_{U}}^{i} (α), Sym_{O_{X}}^{i}(Q))t^{i} induces a homomorphism

σ : K_{0}(co(j_{1}^{∗})) → 1 + ∏_{i≥1} K_{0}(co(j_{i}^{∗}))t^{i}.

By restricting, we obtain symmetric power operations

σ^{i} : K̃_{0}(co(j_{1}^{∗})) → K̃_{0}(co(j_{i}^{∗})), i ≥ 1,

between the reduced Grothendieck groups

K̃_{0}(co(j_{i}^{∗})) := ker(K_{0}(co(j_{i}^{∗})) → K_{0}(M_{i}), [P, α, Q] ↦ [P]).

Let K_{0}(j_{i}^{∗}) denote the factor group of K_{0}(co(j_{i}^{∗})) modulo the subgroup generated by the relations of the form [P, βα, R] − [P, α, Q] − [Q, β, R] (see also Proposition (5.1) on p. 370 in [B]). In the sequel, we consider K_{0}(j_{i}^{∗}) as the factor group of K̃_{0}(co(j_{i}^{∗})) modulo the subgroup I_{i} generated by the elements of the form [P, βα, R]−[P, α, Q]− [Q, β, R] + [Q, id, Q]. As above, one easily sees that I_{i}K_{0}(co(j_{i′}^{∗})) is contained in I_{i+i′} and we obtain a multiplication map

K_{0}(j_{i}^{∗}) × K_{0}(j_{i′}^{∗}) → K_{0}(j_{i+i′}^{∗})

for all i, i′ ≥ 1 which however (in contrast to K_{1}^{det}) seems not to be trivial in general.

Lemma 1.3. The homomorphism σ : K̃_{0}(co(j_{1}^{∗})) → 1 + ∏_{i≥1} K̃_{0}(co(j_{i}^{∗}))t^{i} induces a homomorphism σ : K_{0}(j_{1}^{∗}) → 1 + ∏_{i≥1} K_{0}(j_{i}^{∗})t^{i}.

Proof. Similarly to Lemma 1.2.

The association [P, α, Q] ↦ [Q] − [P] obviously defines a homomorphism

ν_{i} : K_{0}(j_{i}^{∗}) → K_{0}(M_{i})

for all i ≥ 1. (Side Remark: If we would have chosen the map [P, α, Q] ↦ [Q] in the definition of the reduced Grothendieck group, then we would have to replace ν_{i} by −ν_{i} in the following lemma.)

Lemma 1.4. The multiplication maps are compatible with the homomorphisms ν_{i}, i ≥ 1. The same holds for the symmetric power operations σ^{i}, i ≥ 1; i.e., the following diagram commutes for all i ≥ 1:

\xymatrix{K_{0}(j_{1}^{∗}) \ar[r]^{ν_{1}} \ar[d]^{σ^{i}} & K_{0}(M_{1}) \ar[d]^{σ^{i}} \\ K_{0}(j_{i}^{∗}) \ar[r]^{ν_{i}} & K_{0}(M_{i}).}

Proof. We only prove the assertion for σ^{i}. Let P, Q, R ∈ M_{1} and let α : j^{∗}(P) ˜→ j^{∗}(Q) and β : j^{∗}(Q) ˜→ j^{∗}(R) be O_{U}G-isomorphisms. We again write S for Sym. Then we have in K_{0}(M_{i}):

ν_{i}σ^{i}(P, α, Q) = ν_{i}σ^{i}((P, α, Q) − (P, id, P))

= ν_{i}( ∑_{{a≥0,b_{1},...,bu≥1 \atop a+b_{1}+...+bu=i}}(−1)^{u}(S^{a}(P) ⊗ S^{b_{1}}(P) ⊗ … ⊗ S^{b_{u}}(P), S^{a}(α) ⊗ id ⊗ … ⊗ id,

S^{a}(Q) ⊗ S^{b_{1}}(P) ⊗ … ⊗ S^{b_{u}}(P)))
= ∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+...+bu=i}}(−1)^{u} ([S^{a}(Q)] − [S^{a}(P)]) · [S^{b_{1}}(P) ⊗ … ⊗ S^{b_{u}}(P)]

= σ^{i}([Q] − [P]) = σ^{i}ν_{1}(P, α, Q).

We now assume that U = Spec(F) is affine. Then, by Proposition (2.1) on p. 393 in [B], the association (⊕^{m} F G, α) ↦ (⊕^{m} O_{X}G, α, ⊕^{m} O_{X}G) induces a connecting homomorphism

∂ : K_{1}(F G) → K_{0}(j_{1}^{∗})

with ν_{1} ∘ ∂ = 0.

Lemma 1.5. Let gcd(i, ord(G)) be invertible on X. Then we have:

σ^{i} ∘ ∂ = ∂ ∘ σ^{i} in Hom(K_{1}(F G), K_{0}(j_{1}^{∗})).

The multiplication maps are compatible with ∂ (in the obvious sense), too. In particular, the multiplication on Image(∂) is trivial and the operation σ^{i} is a homomorphism on Image(∂).

Proof. Easy.

Proposition 1.6. The following sequence is exact:

K_{1}(F G) →^{∂} K_{0}(j_{1}^{∗}) →^{ν_{1}} K_{0}(O_{X}G) →^{j^{∗}} K_{0}(F G).

Proof. Apply Theorem (2.2)(b) on p. 396 in [B].

Now, let H denote the category of all coherent O_{X}G-modules V which allow a resolution of length ≤ 1 by locally projective coherent O_{X}G-modules and for which j^{∗}(V) = 0 holds. Furthermore, let K_{0}T (O_{X}G) denote the Grothendieck group of H. By mapping the class [V] of a coherent O_{X}G-module V with the resolution 0 → P →^{α} Q → V → 0 and with j^{∗}(V) = 0 to the element (P, j^{∗}(α), Q) in K_{0}(j_{1}^{∗}), we obviously obtain a homomorphism

ψ : K_{0}T (O_{X}G) → K_{0}(j_{1}^{∗}).

Proposition 1.7. The homomorphism ψ is bijective in the following cases:

(a) X = Spec(A) is affine, F is the localization A_{S} of A by a multiplicative set S of non-zero-divisors in A, and j : U = Spec(F) → X = Spec(A) is the canonical morphism.

(b) The morphism j : U = Spec(F) → X is an open immersion and the ideal I of the complement Y := X\U is locally generated by a non-zero-divisor.

(c) X is a Dedekind scheme (i.e., Noetherian, regular, irreducible, and dim(X) = 1), F is the function field of X and j : U = Spec(F) → X is the canonical morphism.

Proof. The assertion (a) follows from (the proof of) Theorem (5.8) on p. 429 in [B]. In the case (b), we construct an inverse map as follows: Let (P, α, Q) be a generator of K_{0}(j_{1}^{∗}). Then, the image of the composition

α̃ : P ⟶^{can} j_{∗}j^{∗}(P) ⟶^{j_{∗}(α)} j_{∗}j^{∗}(Q) = ∪_{n≥0}I^{−n}Q

(see Lemma 2 on p. 231 in [G 1] for the last equality) is contained in I^{−n}Q for some n ≥ 0. We put

ϕ(P, α, Q) := [coker(P ↪^{α̃} I^{−n}Q)] − [coker(Q ↪^{can} I^{−n}Q)] ∈ K_{0}T (O_{X}G).

As in loc. cit., one easily checks that the association (P, α, Q) ↦ ϕ(P, α, Q) induces a well-defined map ϕ : K_{0}(j_{1}^{∗}) → K_{0}T (O_{X}G) which is an inverse of ψ. In the case (c), we construct an inverse map as follows. Let (P, α, Q) be a generator of K_{0}(j_{1}^{∗}). The isomorphism α : j^{∗}(P) ˜→ j^{∗}(Q) can be extended to an isomorphism P|_{U} ˜→ Q|_{U} where U is an open subset of X. The ideal I of the complement Y := X\U is then locally generated by a non-zero-divisor. We now define ϕ(P, α, Q) as in the case (b). As in loc. cit., one again easily checks that the association (P, α, Q) ↦ ϕ(P, α, Q) induces a well-defined map ϕ : K_{0}(j_{1}^{∗}) → K_{0}T (O_{X}G) which is an inverse of ψ.

Remark 1.8. We assume that one of the conditions (a), (b), (c) of Proposition 1.7 holds.

(a) The K-theory space of the exact category H is homotopy equivalent to the homotopy fibre of the canonical continuous map from the K-theory space of M_{1} to the K-theory space of the exact category consisting of all finitely generated projective F G-modules (see [G 1] and [AB]). Hence, we have a long exact (localization) sequence

… → K_{1}(F G) → K_{0}T (O_{X}G) → K_{0}(O_{X}G) → K_{0}(F G).

The end of this sequence can be identified with the exact sequence in Proposition 1.6 by virtue of Proposition 1.7.

(b) If gcd(i, ord(G)) is invertible on X, we obtain a symmetric power operation σ^{i} : K_{0}T (O_{X}G) → K_{0}T (O_{X}G) by virtue of the isomorphism ψ. It maps the class [V] of a coherent O_{X}G-module V in H with the resolution 0 → P →^{α} Q → V → 0 to the element

∑_{{a,b_{1},...,bu≥1 \atop a+b_{1}+...+bu=i}}(−1)^{u}[coker(Sym^{a}(P) ⊗ Sym^{b_{1}}(P) ⊗ … ⊗ Sym^{b_{u}}(P)

[figure: Sym^{a}(α)⊗id⊗...⊗id]Sym^{a}(Q) ⊗ Sym^{b_{1}}(P) ⊗ … ⊗ Sym^{b_{u}}(P))].

(Note that the contribution of a = 0 would be 0 to this sum.) Alternatively, the operation σ^{i} on K_{0}T (O_{X}G) can also be constructed as follows. Let E denote the exact category of all short exact sequences 0 → P → Q → V → 0 with P, Q ∈ M_{1} and V ∈ H. Then, we have a canonical isomorphism

K_{0}T (O_{X}G) = K_{0}(H) ≅ ker(K_{0}(E) → K_{0}(O_{X}G), [0 → P → Q → V → 0] ↦ [P]).

The association

[0 → P →^{α} Q → V → 0] ↦ [0 → Sym^{i}(P) ⟶^{Sym^{i}(α)} Sym^{i}(Q) → coker(Sym^{i}(α)) → 0]

induces an operation σ^{i} on K_{0}(E) as usual. It is then easy to check that its restriction to K_{0}T (O_{X}G) coincides with the operation σ^{i} constructed above. Moreover, the latter construction can be extended to all higher K-groups K_{q}(H), q ≥ 0, by using the methods of [G 2]. On the other hand, we have a symmetric power operation σ^{i} on the K-theory space of M_{1} and on the K-theory space of the category consisting of all finitely generated projective modules (see section 1 in [K 3]), hence also on the homotopy fibre mentioned in (a) and finally on K_{q}(H), q ≥ 0. It seems to be plausible that these two constructions of σ^{i} on K_{q}(H), q ≥ 0, coincide. I hope to say more on this in a future paper.

§2 Symmetric Power Operations on Locally Free Classgroups of Dedekind Schemes

Let X be a Dedekind scheme (i.e., Noetherian, regular, irreducible and dim(X) ≤ 1) with function field F, and let G be a finite group.

First, we recall the definition of the locally free classgroup Cl(O_{X}G) (see [AB] or [BC]). Using the tools developed in section 1 and a theorem of Swan, we then show that the locally free classgroup coincides with the analogously defined locally projective classgroup and that the operations σ^{i}, i ≥ 1, constructed in section 1 are homomorphisms on Cl(O_{X}G). Furthermore, we prove the following concrete interpretations of the operations σ^{i}, i ≥ 1, on Cl(O_{X}G). Firstly, if G is Abelian and gcd(i, ord(G)) = 1, then pulling back the action of G on locally free O_{X}G-modules along the automorphism G → G, g ↦ g^{i}, induces the operation σ^{i} on Cl(O_{X}G). Secondly, if X is a smooth curve over an (algebraically closed or) finite field L such that the characteristic of L does not divide the order of G, then the identification of the locally free with the locally projective classgroup allows us a simple module theoretic description of the isomorphism between Cl(O_{X}G) and Hom_{Galois}(K_{0}(L̄G), Cl(X̄)) (developed in [AB]), and the operation σ^{i} on Cl(O_{X}G) is dual to the adjoint Adams operation ψ̂^{i} on K_{0}(L̄G) with respect to this isomorphism. The proof of the latter result presented here can also be applied in the number field case and then simplifies the proof of Theorem 3.7 in [K 3].

A coherent O_{X}G-module P is called locally free over O_{X}G iff the stalk P_{x} is a free O_{X,x}G-module for all x ∈ X. By Proposition (30.17) on p. 627 in [CR], this is equivalent to the condition that P_{x} ⊗_{O_{X,x}} Ô_{X,x} is a free Ô_{X,x}G-module for all closed points x ∈ X. (Here, Ô_{X,x} denotes the 𝔪_{x}-adic completion of O_{X,x} and 𝔪_{x} the maximal ideal in O_{X,x}.) Let K_{0}^{lf}(O_{X}G) denote the Grothendieck group of all coherent O_{X}G-modules which are locally free over O_{X}G.

Remark 2.1. Let X = Spec(A) be affine. Then we also write K_{0}^{lf}(AG) for K_{0}^{lf}(O_{X}G). This is the Grothendieck group considered for instance in [F 1]. If A is a local Dedekind domain, then the rank (over AG) induces an isomorphism K_{0}^{lf}(AG) ˜→ ℤ. If char(A) = 0 and and no prime divisor of ord(G) is a unit in A, then any finitely generated projective AG-module is already locally free by Swan’s theorem (see Theorem (32.11) on p. 676 in [CR]). The same holds if p = char(A) > 0 and G is a p-group since then the group rings O_{X,x}G, x ∈ X, are local rings. We will prove in Proposition 2.4 that the locally free classgroup defined below always coincides with the analogously defined locally projective classgroup.

Definition 2.2. The group

Cl(O_{X}G) := ker(K_{0}^{lf}(O_{X}G) ⟶^{can} K_{0}^{lf}(F G) ≅ ℤ)

is called the locally free classgroup associated with X and G.

Let K_{0}T (O_{X}G) (resp., K_{0}^{lf}T (O_{X}G)) denote the Grothendieck group of all coherent O_{X}G-modules which are O_{X}-torsion modules and which allow a resolution of length ≤ 1 by locally projective (resp., locally free) O_{X}G-modules. The notation K_{0}T (O_{X}G) obviously agrees with the notation introduced in section 1 (if j : U = Spec(F) → X is the canonical morphism).

Lemma 2.3. The canonical homomorphisms

K_{0}T (O_{X}G) → ⊕_{x∈X closed} K_{0}T (O_{X,x}G) and K_{0}^{lf}T (O_{X}G) → ⊕_{x∈X closed} K_{0}^{lf}T (O_{X,x}G)

are bijective.

Proof. Let x be a closed point of X and V a finitely generated O_{X,x}G-module which is O_{X,x}-torsion and which allows an O_{X,x}G-projective (resp., O_{X,x}G-free) resolution 0 → P → Q →^{ε} V → 0. Let i : Spec(O_{X,x}) ↪ X denote the inclusion. It suffices to show that i_{∗}(V) has a (global) locally projective (resp., locally free) resolution of length ≤ 1. If P and Q are O_{X,x}G-free, i.e., if they are isomorphic to ⊕^{m} O_{X,x}G for some

m ≥ 0, then the composition ε̃: ⊕^{m} O_{X}G ⟶^{can} i_{∗}(⊕^{m}(O_{X,x}G)) ⟶^{i_{∗}(ε)} i_{∗}(V) is surjective and ker(ε̃) is a locally free O_{X}G-module, i.e., i_{∗}(V) has a locally free resolution of length 1. If P and Q are only projective over O_{X,x}G, we choose a (non-equivariant) surjective homomorphism E → i_{∗}(V) with a locally free O_{X}-module E. Then, the induced homomorphism ε̃: O_{X}G ⊗_{O_{X}} E → i_{∗}(V) is an equivariant surjection and the coherent O_{X}G-module ker(ε) is locally projective by Schanuel’s Lemma, i.e., i_{∗}(V) has a locally projective resolution of length 1.

Proposition 2.4. The canonical homomorphism K_{0}^{lf}(O_{X}G) → K_{0}(O_{X}G) induces an isomorphism

Cl(O_{X}G) ˜→ ker(K_{0}(O_{X}G) ⟶^{can} K_{0}(F G)).

Proof. We have a natural commutative diagram of groups

[figure: K_{1}(F G); K_{0}^{lf}T (O_{X}G); K_{0}^{lf}(O_{X}G); K_{0}^{lf}(F G); K_{1}(F G); K_{0}T (O_{X}G); K_{0}(O_{X}G); K_{0}(F G);]

here, the lower row is the exact localization sequence constructed in Proposition 1.6 and Proposition 1.7; the maps in the upper row are defined as in the lower row; one can prove as in section 1 or as in Theorem 1(ii) on p. 3 in [F 2] that also the upper sequence is exact. Thus, it suffices to prove that the map K_{0}^{lf}T (O_{X}G) → K_{0}T (O_{X}G) is bijective. By Lemma 2.3, it furthermore suffices to prove that the map K_{0}^{lf}T (O_{X,x}G) → K_{0}T (O_{X,x}G) is bijective for all closed points x ∈ X. We have a natural commutative diagram of groups

[figure: K_{1}(O_{X,x}G); K_{1}(F G); K_{0}^{lf}T (O_{X,x}G); 0; K_{1}(O_{X,x}G); K_{1}(F G); K_{0}T (O_{X,x}G); K_{0}(O_{X,x}G); K_{0}(F G)]

with exact rows (e.g., see Theorem 1(ii) on p. 3 in [F 2]). Furthermore, the map K_{0}(O_{X,x}G) → K_{0}(F G) is injective by a theorem of Swan (see Theorem (32.1) on p. 671 in [CR]). This proves Proposition 2.4.

Let K_{0}(G, X) denote the Grothendieck group of all coherent O_{X}G-modules which are locally free as O_{X}-modules.

Corollary 2.5. If ord(G) is invertible on X, the Cartan homomorphism K_{0}^{lf}(O_{X}G) → K_{0}(G, O_{X}) induces an isomorphism

Cl(O_{X}G) ˜→ ker (K_{0}(G, X) ⟶^{can} K_{0}(G, F) ≅ K_{0}(F G)) .

Proof. This immediately follows from Proposition 2.4 and the fact that a finitely generated O_{X,x}G-module is projective over O_{X,x}G if and only if it is projective over O_{X,x}.

Now, we fix i ∈ ℕ such that gcd(i, ord(G)) is invertible on X. By section 1, we have a symmetric power operation σ^{i} : K_{0}(O_{X}G) → K_{0}(O_{X}G). By restricting, we obtain an operation σ^{i} on ker(K_{0}(O_{X}G) → K_{0}(F G)) ≅ Cl(O_{X}G). In the same way, we obtain a multiplication map on Cl(O_{X}G).

Proposition 2.6. The multiplication on Cl(O_{X}G) is trivial and the operation σ^{i} on Cl(O_{X}G) is a homomorphism.

Proof. Since the canonical homomorphism K_{0}T (O_{X}G) → Cl(O_{X}G) is surjective, it suffices to show the corresponding assertions for K_{0}T (O_{X}G) (by Lemma 1.4). By Lemma 2.3, we may furthermore assume that X = Spec(A) where A is a local Dedekind domain. Then, the connecting homomorphism ∂ : K_{1}(F G) → K_{0}T (O_{X}G) is surjective (see the proof of Proposition 2.4), and Proposition 2.6 follows from Lemma 1.5.

Theorem 2.7. Let G be Abelian and gcd(i, ord(G)) = 1. We fix i′ ∈ ℕ such that ii′ ≡ 1 mod e(G) where e(G) denotes the exponent of G. Let ϕ_{i′} denote both the O_{X}-algebra automorphism O_{X}G → O_{X}G given by [g] ↦ [g^{i′}] and the automorphism of K_{0}(O_{X}G) or Cl(O_{X}G) induced by the association [P] ↦ [O_{X}G ⊗_{O_{X}G} P] (where O_{X}G is considered as an O_{X}G-algebra via ϕ_{i′}). Then we have:

σ^{i} = ϕ_{i′} on Cl(O_{X}G).

Proof. As in Proposition 2.6, it suffices to show the corresponding assertion for K_{1}(F G) where ϕ_{i′} on K_{1}(F G) is defined analogously. Since F G is semilocal and commutative, the canonical homomorphism (F G)^{×} → K_{1}(F G) is bijective (see Corollary (9.2) on p. 267 in [B]). Under this isomorphism, the automorphism ϕ_{i′} corresponds to the restriction of the (analogously defined) automorphism ϕ_{i′} of F G. Thus it suffices to show that the following diagram commutes:

\xymatrix{(F G)^{×} \ar[r]^{∼} \ar[d]^{ϕ_{i′}} & K_{1}(F G) \ar[d]^{σ^{i}} \\ (F G)^{×} \ar[r]^{∼} & K_{1}(F G).}

Now, let W be a local domain of characteristic 0 whose residue class field is isomorphic to F. (If char(F) = 0, we may choose F itself for W. If p = char(F) > 0, the ring of infinite Witt vectors over F associated with the prime p is such a ring.) Since the group ring W G is semilocal and commutative, the canonical map (W G)^{×} → K_{1}(F G) is bijective (see loc. cit.) and the canonical homomorphism (W G)^{×} → (F G)^{×} is surjective. Thus it suffices to show that the following diagram commutes:

\xymatrix{(W G)^{×} \ar[r]^{∼} \ar[d]^{ϕ_{i′}} & K_{1}(W G) \ar[d]^{σ^{i}} \\ (W G)^{×} \ar[r]^{∼} & K_{1}(W G).}

In a similar way, we conclude that it suffices to show that the corresponding diagram commutes if W is replaced by the quotient field Q of W and finally by the algebraic closure Q̄ of Q. In the latter case, the commutativity follows from Theorem 1.6(d) in [K 3], Theorem 3.3 in [K 1], and Lemma 3.6(b) in [K 3]. This ends the proof of Theorem 2.7.

Remark 2.8. Let gcd(i, ord(G)) = 1. Theorem 2.7 implies in particular that σ^{i+e(G)} = σ^{i} on Cl(O_{X}G) if G is Abelian. This also holds if X = Spec(O_{F}) where O_{F} is the ring of integers in a number field F (see Corollary 3.8 in [K 3]) or if X is a smooth curve over a finite field (this follows from Theorem 2.10). It is not clear to me whether this is true in general.

Now, let L be an algebraically closed field such that char(L) does not divide ord(G), and let p : X → Spec(L) be an irreducible smooth curve over L. Then, for any finitely generated LG-module V, the pull-back p^{∗}(V) is a locally projective coherent O_{X}Gmodule. Furthermore, for any locally projective coherent O_{X}G-module P, P′, the O_{X}-module Hom_{O_{X}}(P, P′) ≅ P^{∨} ⊗_{O_{X}} P is again a locally projective O_{X}G-module. Finally, for any locally projective O_{X}G-module P, the O_{X}-module P^{G} of G-fixed elements is locally free since ord(G) is invertible on X. Thus, we obtain a well-defined homomorphism

\xymatrix{K_{0}(O_{X}G) \ar[r] & Hom(K_{0}(LG), K_{0}(X)) \\ [P] \ar@{|->}[r] & ([V] ↦ [Hom_{O_{X}G}(p^{∗}(V), P)]).}

This homomorphism is bijective (see the proof of Proposition (2.2) on p. 133 in [S]) and induces an isomorphism

Cl(O_{X}G) ˜→ Hom(K_{0}(LG), Cl(X)) (2)

by Proposition 2.4.

Let ψ^{i} denote the i-th Adams operation on K_{0}(LG). In the sequel, we will identify K_{0}(LG) with the ring of virtual characters of G. Then ψ^{i} maps a character χ to the character G → L, g ↦ χ(g^{i}). Let ψ̂^{i} denote the adjoint operation (with respect to the usual character pairing). Note that the assumption char(L) |̸ ord(G) implies that gcd(i, ord(G)) is invertible on X for all i ∈ ℕ. To avoid further definitions we will use the somewhat complicated notation Hom(f, B) (and similar notations) in the usual functorial way (for any Abelian group B and any homomorphism f : A → A′).

Theorem 2.9. Under the isomorphism (2), the operation σ^{i} on Cl(O_{X}G) corresponds to the endomorphism Hom(ψ̂^{i}, Cl(X)) of Hom(K_{0}(LG), Cl(X)).

Proof. By Theorem 3.3 on p. 145 in [K 1] and Theorem 1.6(d)(ii) in [K 3], the operation σ^{i} on K_{1}(F G) (constructed e.g. in section 1) corresponds to the endomorphism Hom(ψ̂^{i}, K_{1}(F)) of Hom(K_{0}(LG), K_{1}(F)) under the isomorphism

\xymatrix{K_{1}(F G) \ar[r]^{∼} & Hom(K_{0}(LG), K_{1}(F)) \\ (P, α) \ar@{|->}[r] & ([V] ↦ (Hom_{F G}(F ⊗_{L} V, P), Hom_{F G}(F ⊗_{L} V, α))).}

For any closed point x ∈ X, the association [M] ↦ ([V] ↦ [Hom_{O_{X,x}G}(O_{X,x}⊗_{L}V, M)]) induces an isomorphism K_{0}T (O_{X,x}G) ˜→ Hom(K_{0}(LG), K_{0}T (O_{X,x})) (both sides are isomorphic to K_{0}(LG)!) such that the following diagram commutes:

\xymatrix{K_{1}(F G) \ar@{->>}[r]^{∂} \ar[d]^{≀} & K_{0}T (O_{X,x}G) \ar[d]^{≀} \\ Hom(K_{0}(LG), K_{1}(F)) \ar@{->>}[r]^{Hom(K_{0}(LG),∂)} & Hom(K_{0}(LG), K_{0}T (O_{X,x})).}

Hence, by Lemma 1.5, the operation σ^{i} on K_{0}T (O_{X,x}G) corresponds to the endomorphism Hom(ψ̂^{i}, K_{0}T (O_{X,x})) of Hom(K_{0}(LG), K_{0}T (O_{X,x})). Under the isomorphism of Lemma 2.3, the operation σ^{i} on K_{0}T (O_{X}G) obviously corresponds to the endomorphism ⊕_{x∈X closed} σ^{i} of ⊕_{x∈X closed} K_{0}T (O_{X,x}G). Thus, under the isomorphism

K_{0}T (O_{X}G) ≅ Hom(K_{0}(LG), K_{0}T (O_{X})), [M] ↦ ([V] ↦ [Hom_{O_{X}G}(p^{∗}(V), M)]), the operation σ^{i} on K_{0}T (O_{X}G) corresponds to the endomorphism Hom(ψ̂^{i}, K_{0}T (O_{X})) of Hom(K_{0}(LG), K_{0}T (O_{X})). Furthermore, the following diagram obviously commutes:

\xymatrix{K_{0}T (O_{X}G) \ar[r]^{can} \ar[d]^{≀} & K_{0}(O_{X}G) \ar[d]^{≀} \\ Hom(K_{0}(LG), K_{0}T (O_{X})) \ar[r]^{can} & Hom(K_{0}(LG), K_{0}(X)).}

Now, Theorem 2.9 follows from Lemma 1.4 and Proposition 1.6.

Now, let L be a finite field with char(L) |̸ ord(G) and p : X → Spec(L) an irreducible smooth curve over L. Let L̄ denote an algebraic closure of L and p̄: X̄ := X ×_{L} L̄ → Spec(L̄) the corresponding curve over L̄. Then, the composition of the canonical map K_{0}(O_{X}G) → K_{0}(OX̄G) with the isomorphism K_{0}(OX̄G) ≅ Hom(K_{0}(L̄G), K_{0}(X̄)) constructed above obviously induces a homomorphism

K_{0}(O_{X}G) → Hom_{Gal(L̄/L)}(K_{0}(L̄G), K_{0}(X̄)).

Theorem 2.10. This homomorphism is bijective. In particular, we obtain an isomorphism

Cl(O_{X}G) ˜→ Hom_{Gal(L̄/L)}(K_{0}(L̄G), Cl(X̄)).

Under this isomorphism, the operation σ^{i} on Cl(O_{X}G) corresponds to the endomorphism Hom_{Gal(L̄/L)}(ψ̂^{i}, Cl(X̄)) of Hom_{Gal(L̄/L}(K_{0}(L̄G), Cl(X̄)).

Proof. The bijectivity can be shown as in section 6 of [AB] using Morita equivalence and the Galois descent property K_{0}(X×_{L}L′) ≅ K_{0}(X̄)^{Gal(L̄/L′)} (for any finite extension L ⊆ L′ ⊂ L̄ of L). Proposition 2.4 then yields the Hom-description of the classgroup. The last assertion immediately follows from Theorem 2.9.

§3 Equivariant Riemann-Roch Type Formulas for Tame Extensions of Dedekind Schemes

The aim of this section is to prove Theorem A and Theorem B presented in the introduction.

Let Y be a Dedekind scheme and G a finite group of order n. Let Ind_{1}^{G} : Cl(O_{Y}) → Cl(O_{Y} G) and Ind_{1}^{G} : K_{0}T (O_{Y}) → K_{0}^{lf}T (O_{Y} G) denote the induction maps. The following lemma generalizes Lemma 2.6 on p. 933 in [BC].

Lemma 3.1. The image of the natural multiplication maps

K_{0}T (O_{Y}) × K_{0}^{lf}(O_{Y} G) → K_{0}^{lf}T (O_{Y} G) and Cl(O_{Y}) × K_{0}^{lf}(O_{Y} G) → Cl(O_{Y} G)

is contained in Ind_{1}^{G}K_{0}T (O_{Y}) resp. Ind_{1}^{G}Cl(O_{Y}).

Proof. The assertion for the first map is clear. The assertion for the second map follows from this since the natural map K_{0}T (O_{Y}) → Cl(O_{Y}) is surjective.

Now, let F/E be a finite Galois extension of the function field E of Y with Galois group G. Let X denote the normalization of Y in F. Then X is a Dedekind scheme endowed with a natural G-action and the corresponding G-morphism f : X → Y is finite (see the proof of Theorem (8.1) on p. 47 in [N]). We assume that f is tamely ramified. As in Lemma 5.5 in [K 3], one easily shows that then, for any locally free coherent O_{X}-module E with (semilinear) G-action, the direct image f_{∗}(E) is a locally free coherent O_{Y} G-module in the sense of section 2. Let K_{0}(G, X) denote the Grothendieck group of all such modules E. Thus, we have a homomorphism

f_{∗} : K_{0}(G, X) → K_{0}^{lf}(O_{Y} G), [E] ↦ [f_{∗}(E)].

The different D := D_{X/Y} := Ann_{O_{X}}(Ω_{X/Y}^{1} ) is a G-stable ideal in O_{X}, hence a module E as above. The following proposition generalizes formula (2.8) on p. 933 in [BC].

Proposition 3.2. For all x ∈ K_{0}(G, X) we have:

f_{∗} (x · ∑_{i=0}^{n−1}[D^{−i}]) = 0 in K_{0}^{lf}(O_{Y} G)/(Ind_{1}^{G}Cl(O_{Y}) ⊕ nℤ[O_{Y} G]).

Proof. We may assume that x = [E] where E is a module as above. Let r := rank_{O_{X}}(E). Then we have:

∑_{i=0}^{n−1} ([f_{∗}(E ⊗ D^{−i})] − r[O_{Y} G])

= n([f_{∗}(E)] − r[O_{Y} G]) + ∑_{i=1}^{n−1} ([f_{∗}(E ⊗ D^{−i})] − [f_{∗}(E)]) in Cl(O_{Y} G).

In the sequel, let M ↦ M^{t} denote the forgetful functor from the category of O_{Y} Gmodules to the category of O_{Y} -modules. (We will consider M^{t} also as an O_{Y} G-module with trivial G-action.) Then, the elements [f_{∗}(O_{X})^{t}] − n[O_{Y}] and [f_{∗}(E)^{t}] − nr[O_{Y}] are contained in Cl(O_{Y}). Hence, we have by Lemma 3.1:

n([f_{∗}(E)]−r[O_{Y} G]) = [f_{∗}(O_{X})^{t}⊗f_{∗}(E)]−[f_{∗}(E)^{t}⊗O_{Y} G] in Cl(O_{Y} G)/Ind_{1}^{G}Cl(O_{Y}).

The homomorphism

f_{∗}(O_{X})^{t} ⊗ f_{∗}(E) → f_{∗}(E)^{t} ⊗ O_{Y} G, a ⊗ b ↦ ∑_{g∈G} ag(b) ⊗ [g^{−1}],

of O_{Y} G-modules is generically bijective since F/E is a Galois extension and any finitely generated module over the twisted group ring F #G is isomorphic to ⊕^{m} F for some m ≥ 0. In particular, this map is a monomorphism and the cokernel R_{X/Y} (E) is an O_{Y} G-torsion module. Hence, it suffices to show that we have:

[R_{X/Y} (E)] = ∑_{i=1}^{n−1}[f_{∗}(E ⊗ D^{−i}/O_{X})] in K_{0}^{lf}T (O_{Y} G)/Ind_{1}^{G}K_{0}T (O_{Y}).

(The notation E ⊗ D^{−i}/O_{X} means E ⊗ (D^{−i}/O_{X}), of course. Similar simplifying notations will be used also below.) By Lemma 2.3, it furthermore suffices to show that we have

[R_{X/Y} (E)_{y}] = ∑_{i=1}^{n−1}[f_{∗}(E ⊗ D^{−i}/O_{X})_{y}] in K_{0}^{lf}T (O_{Y,y}G)/Ind_{1}^{G}K_{0}T (O_{Y,y})

for all closed points y ∈ Y. We now fix y ∈ Y and x ∈ X with f(x) = y. Let G_{x} := {g ∈ G : xg = x} denote the decomposition group of x. Furthermore, let f′ : X′ := Spec(Ô_{X,x}) → Spec(Ô_{Y,y}) =: Y′ denote the induced G_{x}-morphism whereˆ denotes completion. We identify the category of coherent torsion modules on Y′ with the category of coherent torsion modules on Y supported in y. An easy generalization of Corollary 3.11(b) on p. 239 in [C] shows that R_{X/Y} (E)_{y} is isomorphic to the direct sum of [G : G_{x}] copies of Ind_{G_{x}}^{G}R_{X′/Y′}(Ê_{x}). Furthermore, it is clear that f_{∗}(E ⊗ D_{X/Y}^{−i} /O_{X})_{y} is isomorphic to Ind_{G_{x}}^{G}f′_{∗}(Ê_{x} ⊗ D_{X′/Y′}^{−i}/O_{X′}) for all i ≥ 0. For i ≡ j mod ord(G_{x}), we finally have

[f′_{∗}(Ê_{x} ⊗ D_{X′/Y′}^{−i}/O_{X′})] = [f′_{∗}(Ê_{x} ⊗ D_{X′/Y′}^{−j}/O_{X′})] in K_{0}^{lf}T (O_{Y′}G_{x})/Ind_{1}^{G_{x}}K_{0}T (O_{Y′})

since the ideal D_{X′/Y′}^{ord(G_{x})} of O_{X′} can be written as (f′)^{∗}(𝔞) with some ideal 𝔞 in O_{Y′} and since, for any locally free coherent O_{Y′}G-module P, we have

[P/𝔞P] = [O/𝔞 ⊗ P] = 0 in K_{0}^{lf}(O_{Y′}G_{x})/Ind_{1}^{G_{x}}K_{0}T (O_{Y′})

by Lemma 3.1. Thus it suffices to prove that

[R_{X′/Y′}(Ê_{x})] = ∑_{i=1}^{ord(G_{x})−1}[f′_{∗}(Ê_{x} ⊗ D_{X′/Y′}^{−i}/O_{X′})] in K_{0}T (O_{Y′}G_{x})/Ind_{1}^{G_{x}}K_{0}T (O_{Y′}).

We now write G for G_{x}, X for X′, E for Ê_{x}, and so on. Let Δ ⊆ G denote the inertia group, e the order of Δ, 𝔓 the ideal in O_{X} which corresponds to the closed point in X, and χ the Δ-module 𝔓/𝔓^{2}. We decompose f : X → Y into X →^{g} Z →^{h} Y where Z := Spec(Γ(X, O_{X})^{Δ}); i.e., the function field of Z is the inertia field of F/E. Since K_{0}(G, X) is generated by the classes of fractional G-stable ideals in O_{X} (see Lemma 5.5(c) in [K 3]), we may assume that E = 𝔓^{j} for some j ∈ ℤ. An easy generalization of Corollary 3.8 on p. 236 and Theorem 2.8 on p. 222 in [C] shows that we have the following isomorphisms:

R_{X/Y} (𝔓^{j}) ≅ Ind_{Δ}^{G}h_{∗}(R_{X/Z}(𝔓^{j}))
≅ Ind_{Δ}^{G}h_{∗} ( ⊕_{i=1}^{e−1} g_{∗}((𝔓^{j}/𝔓^{j+i})^{t} ⊗ χ^{j+i}))
≅ Ind_{Δ}^{G}f_{∗} ( ⊕_{i=1}^{e−1}(𝔓^{j}/𝔓^{j+i})^{t} ⊗ χ^{j+i}) .

Thus we have:

[R_{X/Y} (𝔓^{j})] = ∑_{i=1}^{e−1} i[Ind_{Δ}^{G}f_{∗}(χ^{j+i})] in K_{0}T (O_{Y} G).

Since D = 𝔓^{e−1} and 𝔓^{e} = f^{∗}(𝔭) (where 𝔭 is the ideal in O_{Y} which corresponds to the closed point in Y), we can conclude as above using Lemma 3.1:

∑_{i=1}^{n−1}[f_{∗}(𝔓^{j} ⊗ D^{−i}/O_{X})] = \frac{n}{e} ∑_{i=1}^{e−1}[f_{∗}(𝔓^{j} ⊗ D^{−i}/O_{X})] = \frac{n}{e} ∑_{i=1}^{e−1}[f_{∗}(𝔓^{j+i}/𝔓^{j+e})]

= \frac{n}{e} ∑_{i=1}^{e−1} i[f_{∗}(𝔓^{j+i}/𝔓^{j+i+1})] in K_{0}T (O_{Y} G)/Ind_{1}^{G}K_{0}T (O_{Y}).

(For the second equality note that D^{−i}/O_{X} = 𝔓^{i(1−e)}/O_{X} is isomorphic to 𝔓^{i}/𝔓^{ie} which has a filtration with quotients 𝔓^{i}/𝔓^{e} and 𝔓^{e}/𝔓^{ie}.) Thus it suffices to prove

that the O_{Y} G-modules Ind_{Δ}^{G}f_{∗}(χ^{i}) and ⊕^{n/e} f_{∗}(𝔓^{i}/𝔓^{i+1}) are isomorphic for all i ∈ ℤ. For this, we consider the O_{Y} G-homomorphism

\xymatrix{h_{∗}(O_{Z})^{t} ⊗ f_{∗}(𝔓^{i}/𝔓^{i+1}) \ar[r] & Maps_{Δ}(G, f_{∗}(𝔓^{i}/𝔓^{i+1})) \\ a ⊗ b \ar@{|->}[r] & (g ↦ ag(b)).}

This homomorphism is bijective since h is unramified (e.g., see pp. 214-215 in [C]).

Furthermore, the left hand side is obviously isomorphic to ⊕^{n/e} f_{∗}(𝔓^{i}/𝔓^{i+1}) and the right hand side is isomorphic to Ind_{Δ}^{G}f_{∗}(χ^{i}). So, Proposition 3.2 is proved.

Now, let k ∈ ℕ with gcd(k, n) = 1 and k′ ∈ ℕ with kk′ ≡ 1 mod n. Let σ^{k} denote the k-th symmetric power operation on K_{0}(G, Y) and ψ^{k} the k-th Adams operation on K_{0}(G, Y) or K_{0}(G, X) (e.g., see section 1 in [K 3]). The composition of the map f_{∗} : K_{0}(G, X) → K_{0}^{lf}(O_{Y} G) with the Cartan homomorphism K_{0}^{lf}(O_{Y} G) → K_{0}(G, Y) is denoted by f_{∗} again. Finally, let K̂_{0}(G, Y)[k^{−1}] denote the J-adic completion of K_{0}(G, Y)[k^{−1}] where J := ker(K_{0}(G, Y) ⟶^{rank} ℤ)[k^{−1}] is the augmentation ideal in K_{0}(G, Y)[k^{−1}].

Theorem 3.3. For all x ∈ K_{0}(G, X) we have:

σ^{k}(f_{∗}(x) − rank(x) · [O_{Y} G]) = f_{∗} (∑_{i=0}^{k′−1}[D^{−ik}] · ψ^{k}(x))

in K̂_{0}(G, Y)[k^{−1}]/(Ind_{1}^{G}K_{0}(Y))K̂_{0}(G, Y)[k^{−1}].

Proof. Let

f̂_{∗} : K̂_{0}(G, X)[k^{−1}] := K_{0}(G, X) ⊗_{K_{0}(G,Y)} K̂_{0}(G, Y)[k^{−1}] → K̂_{0}(G, Y)[k^{−1}]

denote the homomorphism which is induced by f_{∗} : K_{0}(G, X) → K_{0}(G, Y), and let θ^{k}(D^{−1}) := 1 + [D^{−1}] + … + [D^{−(k−1)}] ∈ K_{0}(G, X) denote the Bott element. As in Theorem 5.4 in [K 3], one easily deduces the following assertion from the equivariant Adams-Riemann-Roch theorem (see Theorem (4.5) in [K 2]): The element θ^{k}(D^{−1}) is invertible in K̂_{0}(G, X)[k^{−1}] and we have

ψ^{k}(f_{∗}(x)) = f̂_{∗}(k · θ^{k}(D^{−1})^{−1} · ψ^{k}(x)) in K̂_{0}(G, Y)[k^{−1}]

for all x ∈ K_{0}(G, X). Furthermore, we have:

θ^{k}(D^{−1}) · (∑_{i=0}^{k′−1}[D^{−ik}]) = ∑_{j=0}^{k−1} ∑_{i=0}^{k′−1}[D^{−(j+ik)}] = ∑_{i=0}^{kk′−1}[D^{−i}] = [O_{X}] + ∑_{i=1}^{kk′−1}[D^{−i}]

in K_{0}(G, X). Thus, we have:

θ^{k}(D^{−1})^{−1} = ∑_{i=0}^{k′−1}[D^{−ik}] − θ^{k}(D^{−1})^{−1} ∑_{i=1}^{kk′−1}[D^{−i}] in K̂_{0}(G, X)[k^{−1}].

Hence, we obtain the equality

ψ^{k}(f_{∗}(x)) = k · f̂_{∗} ((∑_{i=0}^{k′−1}[D^{−ik}] − θ^{k}(D^{−1})^{−1} · ∑_{i=1}^{kk′−1}[D^{−i}]) · ψ^{k}(x))

= k · f_{∗} (∑_{i=0}^{k′−1}[D^{−ik}] · ψ^{k}(x)) in K̂_{0}(G, Y)[k^{−1}]/(Ind_{1}^{G}K_{0}(Y))K̂_{0}(G, Y)[k^{−1}]

by Proposition 3.2. Since we have ψ^{k} = k · σ^{k} on Cl(O_{Y} G) (by Proposition 2.6) and ψ^{k}([O_{Y} G]) = [O_{Y} G] (by Theorem 1.6(e) in [K 3]), this implies Theorem 3.3.

Note that the formula of Theorem 3.3 lives within the somewhat complicated group K̂_{0}(G, Y)[k^{−1}]/(Ind_{1}^{G}K_{0}(Y))K̂_{0}(G, Y)[k^{−1}]. The next proposition computes this group in a special case.

Proposition 3.4. Let L be an algebraically closed field, Y a projective smooth irreducible curve over L, and n = ord(G) a power of a prime l ≠ char(L). Let I denote the augmentation ideal in K_{0}(LG). Then we have:

K̂_{0}(G, Y)[k^{−1}] ≅ K_{0}(Y)[k^{−1}] ⊕ I ⊗ ℤ_{l} ⊕ I ⊗ ℤ_{l};

under this isomorphism, the extended ideal (Ind_{1}^{G}K_{0}(Y))K̂_{0}(G, Y)[k^{−1}] corresponds to the subgroup {(ny, ([ℤG]−n)⊗rank(y), ([ℤG]−n)⊗deg det(y)) : y ∈ K_{0}(Y)[k^{−1}]} of K_{0}(Y)[k^{−1}] ⊕ I ⊗ ℤ_{l} ⊕ I ⊗ ℤ_{l}.

Proof. The canonical map K_{0}(LG) ⊗ K_{0}(Y) → K_{0}(G, Y) is an isomorphism by Proposition (2.2) on p. 133 in [S]. Since the augmentation ideal of K_{0}(Y) is nilpotent (e.g., by Proposition 2.6) and the I-adic topology on I coincides with the l-adic topology (see Proposition 1.1 on p. 277 in [AT]), the completion K̂_{0}(G, Y)[k^{−1}] is isomorphic to the direct sum of K_{0}(Y)[k^{−1}] and the l-adic completion of I ⊗K_{0}(Y)[k^{−1}]. Furthermore, we have K_{0}(Y) ≅ ℤ ⊕ ℤ ⊕ Pic^{0}(Y) where Pic^{0}(Y) denotes the group of line bundles on Y of degree 0. Since Pic^{0}(Y) is an l-divisible group (see item (iv) on p. 42 in [M]), the l-adic completion of I ⊗ K_{0}(Y)[k^{−1}] is isomorphic to I ⊗ ℤ_{l} ⊕ I ⊗ ℤ_{l}. Thus, we have

K̂_{0}(G, Y)[k^{−1}] ≅ K_{0}(Y)[k^{−1}] ⊕ I ⊗ ℤ_{l} ⊕ I ⊗ ℤ_{l}.

Under the isomorphism K_{0}(G, Y) ≅ K_{0}(LG)⊗K_{0}(Y), the ideal Ind_{1}^{G}K_{0}(Y) of the ring K_{0}(G, Y) corresponds to the ideal Ind_{1}^{G}K_{0}(L)⊗K_{0}(Y) (≅ K_{0}(Y)) of K_{0}(LG)⊗K_{0}(Y) which is generated by the element [ℤG] ⊗ 1 = n ⊗ 1 + ([ℤG] − n) ⊗ 1. One easily deduces the second assertion of Proposition 3.4 from this. (Note that [ℤG] · x = 0 for all x ∈ I.)

Now, let f_{∗} : K_{0}(G, X) → Cl(O_{Y} G) denote the composition of f_{∗} : K_{0}(G, X) → K_{0}^{lf}(O_{Y} G) with the canonical projection K_{0}^{lf}(O_{Y} G) ≅ Cl(O_{Y} G)⊕ℤ[O_{Y} G] → Cl(O_{Y} G).

Theorem 3.5. Suppose that one of the following conditions holds:

(a) Y = Spec(O_{E}) where O_{E} is the ring of integers in a number field E.

(b) Y is an irreducible projective smooth curve over a finite field L and gcd(char(L), n) = 1.

(c) The group G is Abelian and f : X → Y is unramified.

(d) k = 1. Then we have for all x ∈ K_{0}(G, X):

σ^{k}(f_{∗}(x)) = f_{∗} (∑_{i=0}^{k′−1}[D^{−ik}] · ψ^{k}(x)) in Cl(O_{Y} G)/Ind_{1}^{G}Cl(O_{Y}).

Proof. In the case (a), Theorem 3.5 can be deduced from Corollary 2.7 on p. 933 in [BC] using Theorem 3.7 and Lemma 5.5 in [K 3] (see also the proof of Theorem 5.6 in [K 3]). The same can be done in the case (b) by using Lemma 3.6(a) in [K 3] and Theorem 2.9 (in place of Theorem 3.7 in [K 3]) and an obvious generalization of Lemma 5.5(c) in [K 3]. (For completeness sake, we mention that it is easy to check that the additional assumptions in Theorem 2.1 on p. 932 in [BC] about the absolute discriminant or the characteristic of E are not necessary for Corollary 2.7 on p. 933 in [BC].) We now prove Theorem 3.5 in the case (c), i.e., we want to show the formula

(3) σ^{k}(f_{∗}(x)) = k′ · f_{∗}(ψ^{k}(x)) in Cl(O_{Y} G)/Ind_{1}^{G}Cl(O_{Y})

for all x ∈ K_{0}(G, X). First, we show that it suffices to prove the formula (3) for x = 1 = [O_{X}]. Indeed, for an arbitrary x ∈ K_{0}(G, X), there is a y ∈ K_{0}(Y) ⊆ K_{0}(G, Y) such that x = f^{∗}(y) (e.g., see Theorem 1(B) on p. 112 in [M]). Furthermore, we have σ^{k}(Ind_{1}^{G}Cl(O_{Y})) ⊆ Ind_{1}^{G}Cl(O_{Y}). This follows from Proposition 1.1 in [K 3] as there is a polynomial Q_{k} ∈ ℤ[X_{1}, …, X_{k}; Y_{1}, …, Y_{k}] which is homogeneous of weight k in both sets of variables such that

σ^{k}(z · [O_{Y} G]) = Q_{k} (σ^{1}(z), …, σ^{k}(z); [Sym^{1}(O_{Y} G)], …, [Sym^{k}(O_{Y} G)]) in Cl(O_{Y} G)

for all z ∈ Cl(O_{Y}) (by Theorem 2.2 in [K 3]). Thus we have in Cl(O_{Y} G)/Ind_{1}^{G}Cl(O_{Y}):

σ^{k}(f_{∗}(x)) = σ^{k}(f_{∗}(f^{∗}(y))) = σ^{k}(y · f_{∗}(1)) (Projection formula)
= σ^{k}(rank(y) · f_{∗}(1)) (Lemma 3.1)
= rank(y) · σ^{k}(f_{∗}(1)) (Proposition 2.6)
= rank(y) · k′ · f_{∗}(1) (by assumption)
= k′ · ψ^{k}(y) · f_{∗}(1) (Lemma 3.1)
= k′ · f_{∗}(ψ^{k}(f^{∗}(y))) = k′ · f_{∗}(ψ^{k}(x)) (Projection formula).

We now prove formula (3) for x = 1. Since f is unramified, the scheme X is a principal G-bundle over Y (see Proposition 2.6 on p. 115 in [SGA 1]). There is a well-known natural bijection between the set of all principal G-bundles over Y and the cohomology group H^{1}(Y, G). We write [X] for the corresponding element in H^{1}(Y, G). We define a new principal G-bundle X_{k′} over Y as follows: X_{k′} = X as Y -schemes and the new action ∗ of G on X_{k′} is given by x ∗ g := xg^{k} for “x ∈ X” and g ∈ G. Then, it is easy to check that the association X ↦ X_{k′} corresponds to the multiplication with k′ on H^{1}(Y, G). Let cl : H^{1}(Y, G) → Cl(O_{Y} G) denote the map which maps a principal G-bundle f : X → Y to the class [f_{∗}(O_{X})] − [O_{Y} G]. This map is a homomorphism by Theorem 5 and the subsequent remarks on p. 189 in [W] and by Proposition 3.9 in [AB]. Thus we have:

σ^{k}(f_{∗}([O_{X}])) = ϕ_{k′}(cl([X])) (Theorem 2.7)
= cl([X_{k′}]) = cl(k′ · [X]) = k′ · cl([X]) = k′ · f_{∗}([O_{X}]).

in Cl(O_{Y} G), as was to be shown. In the case (d), Theorem 3.5 immediately follows from Proposition 3.2.

Remark 3.6. If one of the conditions (a), (b), (c), (d) of Theorem 3.5 is satisfied, then Theorem 3.3 follows from Theorem 3.5 by passing from Cl(O_{Y} G) ⊂ K_{0}(O_{Y} G) to K_{0}(G, Y) via Cartan homomorphism and finally by passing from K_{0}(G, Y) to the completion K̂_{0}(G, Y)[k^{−1}] of K_{0}(G, Y)[k^{−1}]. In particular, in the case (a), the formula of Theorem 3.3 is substantially weaker than the formula in Theorem 3.5, as already the passage from K_{0}(O_{Y} G) to K_{0}(G, Y) loses much information. On the other hand, in the case (b), the formula of Theorem 3.5 modulo torsion follows from the formula in Theorem 3.3 if n is a power of a prime. This can be proved as follows. The Cartan homomorphism K_{0}(O_{Y} G) → K_{0}(G, Y) is bijective since n is invertible on Y. Furthermore, the canonical map Cl(O_{Y} G)/Ind_{1}^{G}Cl(O_{Y}) ⊆ K_{0}(O_{Y} G)/Ind_{1}^{G}K_{0}(Y) → K_{0}(OȲ G)/Ind_{1}^{G}K_{0}(Ȳ) is injective by Theorem 2.10. (Here, Ȳ denotes the curve Y ×_{L}L̄ over the algebraic closure L̄ of L.) Hence, it suffices to prove the formula

σ^{k}(f̄_{∗}(x)) = f̄_{∗} (∑_{i=0}^{k′−1}[D_{X̄/Ȳ}^{−ik}] · ψ^{k}(x)) in K_{0}(G, Ȳ)_{ℚ}/Ind_{1}^{G}K_{0}(Ȳ)_{ℚ} (4)

for all x ∈ K_{0}(G, X̄). Furthermore, we have K_{0}(G, Ȳ)_{ℚ} ≅ K_{0}(L̄G)_{ℚ} ⊗ K_{0}(Ȳ)_{ℚ} and K_{0}(G, Ȳ)_{ℚ}/Ind_{1}^{G}K_{0}(Ȳ)_{ℚ} ≅ I ⊗ K_{0}(Ȳ)_{ℚ} ≅ I_{ℚ} ⊕ I_{ℚ} (see the proof of Proposition 3.4). On the other hand, (K̂_{0}(G, Ȳ)[k^{−1}]/(Ind_{1}^{G}K_{0}(Ȳ))K̂_{0}(G, Ȳ)[k^{−1}])_{ℚ} is isomorphic to

I ⊗ℚ_{l} ⊕I ⊗ℚ_{l} by Proposition 3.4. Hence, the canonical map K_{0}(G, Ȳ)_{ℚ}/Ind_{1}^{G}K_{0}(Ȳ)_{ℚ} → (K̂_{0}(G, Ȳ)[k^{−1}]/(Ind_{1}^{G}K_{0}(Ȳ))K̂_{0}(G, Ȳ)[k^{−1}])_{ℚ} is injective, and formula (4) follows

from Theorem 3.3.

Remark 3.7.

(a) If one of the conditions (a), (b), or (c) holds, Theorem 3.5 can be slightly strengthened: It suffices to assume that k′ is an inverse modulo the exponent of G (see [BC] and Theorem 2.7, respectively). It is not clear to me whether this is true also in the case (d).

(b) Let Y be an irreducible smooth projective curve over a finite field L. Then, the case (c) is particularly interesting as complementary case of the semisimple case which is assumed in the case (b). Indeed, if G is an (Abelian) char(L)-group, then the tameness condition already implies that f is unramified.

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Mathematisches Institut II der Universität Karlsruhe, D-76128 Karlsruhe, Germany; e-mail: Bernhard.Koeck@math.uni-karlsruhe.de.