The norm map and the capitulation kernel

Abstract

Let f:S′→S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n≥2 and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. Under certain restrictions on f and G, we relate the kernel of the restriction map ResG(r+1):Hr+1(Se´t,G)→Hr+1(Se´t′,G) in étale cohomology, where r≥0, to a quotient of the kernel of the mod n corestriction map CoresG(r)/n:Hr(Se´t′,G)/n→Hr(Se´t,G)/n. When r=0 and f is a Galois covering with Galois group Δ, our main theorem relates KerResG(1)=H1(Δ,G(S′)) to the subgroup of G(S′) of sections whose (S′/S)-norm lies in G(S)n. Applications are given to the capitulation problem for Néron–Raynaud class groups of tori and Tate–Shafarevich groups of abelian varieties.