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

Abstract

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. For example, we show that the Néron-Raynaud class group of any F -torus which is split by a metacyclic extension of F can be "resolved" in terms of classical ideal class groups of global fields.

Abelian class groups of reductive group schemes

“…Remark 3.17. The corollary extends the "good reduction" case of [GA2], Theorem 6.1, from K-tori to arbitrary connected reductive K-groups. Indeed, let T be a K-torus with multiplicative reduction over S which admits an invertible resolution 1 → F → R → T → 1, i.e., F is invertible and R is quasi-trivial (this is the case, for example, if T is split by a metacyclic extension of K, by [CTS1], Proposition 2, p.184).…”

“…be the class set of G. Since G is generically smooth, Ye.Nisnevich has shown that there exists a canonical bijection C(G) ≃ H 1 (S Nis , u * G), where u : S ét → S Nis is the canonical morphism of sites. See [GA2], Theorem 3.5.…”

On Néron class groups of abelian varieties

“…Now, although the arguments of [17], Chapter I, §2 (which are reproduced in [7], §3) are valid in principle only when M is affine, they admit a straightforward generalization to arbitrary M as above 1 . In particular, the pointed set C(M ) admits the following Nisnevich-cohomological interpretation (see [7], Theorem 3.5):…”

“…In particular, the pointed set C(M ) admits the following Nisnevich-cohomological interpretation (see [7], Theorem 3.5):…”

The norm map and the capitulation kernel

“…In Case (1) above, i.e., G F is an invertible F -torus, the group X 1 Σ (∆, G(K)) (which is isomorphic to a subgroup of H 1 (F, G) via the inflation map) is trivial since H 1 (F, G) is trivial by Hilbert's Theorem 90 [10,Lemma 4.8(a)]. In this case (6.5) reduces to a canonical isomorphism of abelian groups Ker j G F , K/F, Σ = Ker Res (1) G, S ′ /S .…”