Galois invariants for S-units

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 primes which are archimedean or ramified over k and which generates the ideal class group. By Theorem B, the G-module E of S-units of K is determined, up to stable isomorphism, by (a) the [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]G-lattice ΔS, which is the kernel of the augmentation [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]S → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /], (b) the G-module μ of roots of unity in K, (c) a distinguished character ε of H²(G, Hom (ΔS, μ)), and (d) the Chinburg class Ω_m(K/k) in the locally free class group of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] G. Our method is a study of the homotopy properties of Tate sequences of K/k, hence it is essential that our Galois invariants be specified without prior knowledge of these sequences. This is clear for (a) and (b), while for (c) it follows from Theorem A, which expresses ε in terms of the local and global invariant maps of class field theory. And for (d), it would follow from Chinburg's conjecture that Ω_m is the root number class of K/k.