Correspondences between valued division algebras and graded division algebras
Let \(D\) be a finite-dimensional central division algebra over a field \(F\) with a Henselian valuation \(v_F\). The valuation \(v_F\) extends to a (Schilling) valuation \(v_D\) on \(D\), and there is an associated graded ring \(GD=\bigoplus_{\gamma\in\Gamma_D}GD_\gamma\), where \(\Gamma_D\) is the value group of \(v_D\) and, for \(\gamma\in\Gamma_D\), \(GD_\gamma\) is the quotient of \(\{x\in D\mid v(x)\geq\gamma\}\) by \(\{x\in D\mid v(x)>\gamma\}\). Since the homogeneous elements are all units, this graded ring is a graded division ring. The main theme of this paper is to investigate the relationship between \(D\) and \(GD\), which is particularly close when \(D\) is tame, i.e. when its \(p\)-primary component is split by an unramified extension of \(F\), where \(p\) is the characteristic of the residue field. In that case, \(GD\) is a central graded division algebra over \(GF\) of the same dimension as \(D\). (This had already been observed by \textit{M. Boulagouaz} [Commun. Algebra 23, No. 11, 4275-4300 (1995; Zbl 0844.16011)].) The authors show that the map \([D]\mapsto[GD]\) yields a Schur-index-preserving isomorphism from the tame part of the Brauer group of \(F\) to the graded Brauer group of \(GF\), which is functorial with respect to field extensions and corestrictions. Using this map, the authors establish a correspondence between \(F\)-subalgebras of \(D\) and graded \(GF\)-subalgebras of \(GD\): they show that if \(K\) is a tame valued field extension of \(F\) and \(D\) and \(A\) are tame division algebras with center \(F\), then \(K\) (resp. \(A\)) embeds in \(D\) if and only if \(GK\) (resp. \(GA\)) embeds in \(GD\). In the first sections, some properties of graded central simple algebras over a graded field with torsion-free grade group are proved. The authors particularly emphasize those properties which have analogues for tame division algebras over a Henselian valued field. They also give a cohomological description of the graded Brauer group of a graded field using graded crossed products.
scientific article; zbMATH DE number 1356293
scientific article