I. Introduction. Let N/K be a finite Galois extension of number fields with group G = Gal(N/K). Define O_N to be the integers of N. E. Noether proved that O_N is a projective G-module if and only if N/K is at most tamely ramified. One then has a class Ω_a (N/K) = (O_N - [K: Q] (Z [G]) in the Grothendieck group K_0 (Z [G]) of finitely generated G-modules of finite projective dimension. Motivated by work of A. Fr?hlich [8], H. Stark [16] and J. Tate [17], we began in [5] a unified theory of the G-structure of O_N and of the group U_(N,S) of S-units of N when S is a sufficiently large finite set of places of N stable under the action of G.
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