Self-duality and torsion Galois modules in number fields
Let K/k be a Galois extension of number fields with Galois group \(\Gamma\). If \({\mathcal O}\) is the ring of integers of K, and \(o\) the one of k, then the structure of \({\mathcal O}\) as \(o\Gamma\)-module is investigated by comparing \({\mathcal O}\) with its dual \(Hom_{o}({\mathcal O},o)={\mathcal O}^+\). The trace map \(tr_{K/k}: K\to k\) induces an \(o\Gamma\)-module isomorphism between \({\mathcal O}^+\) and the codifferent C(K/k) of K/k. The \(o\Gamma\)-module \(C(K/k)/{\mathcal O}=T(K,k),\) which is a torsion module, is described explicitly in \(K^{tor,S}(o\Gamma)\), the Grothendieck group of finitely generated torsion modules with finite homological dimension outside (a suitable) S. In the paper the problem is considered in the wildly ramified case. The tame case has been worked out by \textit{S. Chase} in [J. Algebra 91, 207-257 (1984)].
scientific article; zbMATH DE number 3861278