On a multiplicative-additive Galois invariant and wildly ramified extensions (Q1907858)

Let \(N/K\) be a finite Galois extension of number fields with Galois group \(G\). By a theorem of E. Noether, if \(N/K\) is at most tamely ramified the ring of integers, \({\mathcal O}_N\), is a locally free \(\mathbb{Z} G\)-module and so defines a class, \(({\mathcal O}_N)- [K: \mathbb{Q} ](\mathbb{Z} G)\), in the locally free class group \(\text{Cl} (\mathbb{Z} G)\). In [Ann. Math., II. Ser. 121, 351-376 (1985; Zbl 0567.12010)] \textit{T. Chinburg} defined an invariant \(\Omega (N/K, 2)\in \text{Cl} (\mathbb{Z} G)\) for any extension \(N/K\) and proved that in the case that \(N/K\) was at most tamely ramified it is equal to the class defined by \({\mathcal O}_N\). In this paper, the author generalizes Chinburg's result by proving that, for any \(N/K\), \[ \Omega (N/K, 2)= ({\mathcal O}_N)- [K: \mathbb{Q}](\mathbb{Z} G)\in \text{Cl}^S (\mathbb{Z} G), \] where \(S\) is a finite set of rational prime numbers containing the set of rational primes above which there is wild ramification in \(N/K\) and \(\text{Cl}^S (\mathbb{Z} G)\) is the torsion subgroup of the Grothendieck group \(K^S_0 (\mathbb{Z} G)\) of finitely-generated \(\mathbb{Z} G\)-modules which have no \(\mathbb{Z}\)-torsion and are projective outside of \(S\). The proof modifies the arguments of (loc. cit.) in order to deduce this more general case from the tame case.