Symmetric Powers of Galois Modules on Dedekind Schemes

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. Introduction Let G be a finite group and E a number field. Let OE denote the ring of integers in E, Y := Spec(OE), and Cl(OYG) := ker(rank : K0(OEG) → 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 of Cl(OYG) which, via Fröhlich’s Hom-description of Cl(OYG), 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 k ∈ N an inverse of k modulo ord(G). In the paper [K 3], we have shown that then the endomorphism ψ k′ is a simply definable symmetric power operation σ . Now, let F/E be a finite tame Galois extension with Galois group G. Let f : X := Spec(OF ) → Y denote the corresponding G-morphism and f∗ the homomorphism f∗ : K0(G,X) → Cl(OYG), [E ] 7→ [f∗(E)]− rank(E) · [OYG], from the Grothendieck group K0(G,X) of all locally free OX-modules with (semilinear) G-action to Cl(OYG). Furthermore, let D denote the different of F/E and ψ k the k-th Adams operation on K0(G,X). The paper [BC] by Burns and Chinburg