Abelian extensions of an absolutely unramified local field with general residue field
By analyzing the dual group \(H^ 1(K,\mathbb Q/\mathbb Z)\) of the maximal abelian extension of a Henselian discrete valuation field \(K\) in which an odd prime \(p\) serves as a prime element, the feature of abelian extensions of \(K\) is completely determined. Observe that there is no restriction imposed on the residue field \(F\) of \(K\). The main part of the paper deals with the construction of a certain surjective homomorphism \[ \phi_ n: H^ q(K,\mathbb Z/p^ n(q-1))\to W_ n\Omega_ F^{q-1}, \] where \(W_ n\Omega_ F^{\cdot}\) is the de Rham-Witt complex and where \(\mathbb Z/p^ n(q-1)\) means the Tate twist. Restricted to \(q=1\) there results an exact sequence \[ 0\to H^ 1(K,\mathbb Q_ p/\mathbb Z_ p)_{nr}\to H^ 1(K,\mathbb Q_ p/\mathbb Z_ p)\to \lim_{\vec V}W_ n(F)\to 0, \] where \(nr\) abbreviates ``non-ramified'' and \(V\) stands for ``Verschiebung'' of Witt vectors over \(F\). As an additional consequence, the author achieves several results on the Brauer group of \(K\) (compare also [\textit{H. Miki}, J. Fac. Sci., Univ. Tokyo, Sect IA 21, 377--394 (1974; Zbl 0301.12003)]).
scientific article; zbMATH DE number 4089675
scientific article