Capitulation, ambiguous classes and the cohomology of the units

Abstract

This paper presents results on both the kernel and cokernel of the S-

On Néron-Raynaud class groups of tori and the capitulation problem

“…S.-I.Katayama [14], Theorem 2, and V.Voskresenskii [31], Chapter 7, §20, obtained results for the class group of the standard model of P = R K/F (G m,K )/G m,F which bear some resemblance to the case T = G m,F of Corollary 6.4. Specializing Corollary 6.4 to T = G m, F , we obtain the following result on the classical S-capitulation map j K/F, S which supplements those obtained in [10]. Corollary 6.6.…”

“…The following immediate corollary of the proposition generalizes [10], Theorem 2.4 (note that, if S ⊃ B, then T K has multiplicative reduction over U since T has multiplicative reduction over U ): Theorem 4.6. Assume that S ⊃ B.…”

“…

We discuss the Capitulation Problem for Néron-Raynaud class groups of tori over global fields F and obtain generalizations of the main results of [10]. We also show that short exact sequences of F -tori induce long exact sequences involving the corresponding Néron-Raynaud class groups.

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“…where the map involved is induced by the projection Φ v (k(v)) → C T, Fv . Applying the snake lemma to the diagram which is derived from (4.8) by replacing both instances of T (F v ) there by their images in their target groups, we obtain the following generalization of [10], Lemma 2.3.…”

Čech cohomology and the capitulation kernel

“…An ideal class in C F, Σ is said to capitulate in K if it lies in Ker j K/F, Σ , which is often called the capitulation kernel associated to the pair (K/F, Σ). It was shown in [5,Theorem 2.3] that Ker j K/F, Σ is naturally isomorphic to the kernel of the canonical localization map in ∆-cohomology…”