Galois structure ofS-units

Abstract

Let K/k be a finite Galois extension of number fields with Galois group G, S a large G-stable set of primes of K, and E (respectively µ µ µ) the G-module of S-units of K, (resp. roots of unity). Previous work using the Tate sequence of E and the Chinburg class Ω m has shown that the stable isomorphism class of E is determined by the data ∆S, µ µ µ, Ω m , and a special character ε of H 2 G, Hom(∆S, µ µ µ) . This paper explains how to build a G-module M from this data which is stably isomorphic to E ⊕ ZG n , for some integer n.Let K/k be a finite Galois extension of number fields with Galois group G and let S be a finite G-stable set of primes of K containing all archimedean primes. Assume that S is large in the sense that it contains all ramified primes of K/k and that the S-class group of K is trivial. Let E denote the G-module of S-units of K and µ µ µ the roots of unity in K. The purpose of this paper is to specify the stable isomorphism class of the G-module E in a much more explicit way than in Theorem B of [7].More precisely, and continuing in the notation of [7], we recall that [11], [12] defines a canonical 2-extension class of G-modules, represented by Tate sequenceswith A a finitely generated cohomologically trivial ZG-module, B a finitely generated projective ZG-module and ∆S the kernel of the G-map ZS → Z which sends every element of S to 1.