Galois structure of \(S\)-units (Q2801719)

The abstract of the article is: ``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, \(\mu\)) the \(G\)-module of \(S\)-units of \(K\) (respectively, roots of unity). Previous work using the \textit{Tate sequence} of \(E\) and the \textit{Chinburg class} \(\Omega_m\) has shown that the stable isomorphism class of \(E\) is determined by the data \(\Delta S,\mu,\Omega_m\), and a special character \(\varepsilon\) of \(H^2(G,\mathrm{Hom}(\Delta S,\mu))\). This paper explains how to build a \(G\)-module \(M\) from this data that is stably isomorphic to \(E \oplus \mathbb Z G^n\), for some integer \(n\).''NEWLINENEWLINEThe purpose of this paper is to specify the stable isomorphism class of the \(G\)-module \(E\) in a much more explicit way than in [\textit{K. W. Gruenberg} and the second author, Am. J. Math. 119, No. 5, 953--983 (1997; Zbl 0884.11045), Theorem B] which is a \textit{prerequisite to the understanding of this article.} See also [\textit{J. Tate}, Nagoya Math. J. 27, 709--719 (1966; Zbl 0146.06501); Les conjectures de Stark sur les fonctions \(L\) d'Artin en \(s=0\). Boston-Basel-Stuttgart: Birkhäuser (1984; Zbl 0545.12009)].