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 > 1 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 in degree r>0 \'etale cohomology Res_{G}^{(r)}: H^{r}(S_{\et},G)--> H^{r}(S'_{\et},G) to a certain quotient of the kernel of the mod n corestriction map in degree r-1, namely Cores_{G}^{(r-1)}/n: H^{r-1}(S'_{\et},G)/n\to H^{r-1}\lbe(S_{\et},G)/n. When r=1 and f is a Galois covering with Galois group D, our main theorem relates Ker Res_{G}^{(1)}=H^{1}(D,G(S')) to the subgroup of G(S') of those sections whose S'/S-norm lies in G(S)^{n}. We also include applications to the capitulation problem for Neron-Raynaud class groups of invertible tori and Tate-Shafarevich groups of abelian varieties.