The norm map and the capitulation kernel

2018/03/09 by Gonzalez-Aviles, Cristian D.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.03574

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 étale 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→ 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⁾=H¹(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.

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