Galois structure of S-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, $\boldsymbol {\mu } )$ the $G$-module of $S$-units of $K$ (respectively, roots of unity). Previous work using the Tate sequence of $E$ and the Chinburg class $\Omega _m$ has shown that the stable isomorphism class of $E$ is determined by the data$\Delta S,\boldsymbol {\mu }, \Omega _m,$ and a special character $\varepsilon $ of $H^2(G,{\rm Hom}(\Delta S,\boldsymbol {\mu } )).$ 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 $n.$