Division Algebras over Henselian Fields

Abstract

AbstractIn this chapter we focus on the tame division algebras D with center a field F with Henselian valuation v. As usual, we approach this by first obtaining results for graded division algebras, then lifting back from \(\operatorname {\mathsf {gr}}(D)\) to D. This is facilitated by results in §8.1 on existence and uniqueness of lifts of tame subalgebras from \(\operatorname {\mathsf {gr}}(D)\) to D. In §8.2, we describe four fundamental canonical (up to conjugacy) subalgebras of D that reflect its valuative structure. The rest of the chapter is devoted to Brauer group factorizations of D corresponding to the noncanonical direct product decomposition of \(\operatorname {\mathit{Br}}_{t}(F)\) given in Cor. 7.85. The factor \(\operatorname {\mathit{Hom}}^{c}(\operatorname {\mathcal {G}}(\overline{F}), {\mathbb{T}}(\Gamma_{F}))\) is represented by a type of division algebra N called decomposably semiramified, defined in §8.3, and characterized by the property that N contains a maximal subfield inertial over F and another totally ramified over F. We show in §8.4 that every tame division algebra D is Brauer equivalent to some S⊗ F T where S is inertially split and T is tame and totally ramified over F. We show further that every inertially split division algebra S is Brauer equivalent to some I⊗ F N, where I is inertial over F and N is decomposably semiramified. The classes \([\,\overline{I}\,]\) for the I appearing in the I⊗ F N decompositions of S are shown to range over a single coset of \(\mathit{Dec}(Z(\overline{S})/\overline{F})\) in \(\operatorname {\mathit{Br}}(\overline{F})\), called the specialization coset of S. In the final subsection, §8.4.6, we summarize what happens in the special case that v is discrete of rank 1, where substantial simplifications occur.KeywordsGraded Division AlgebraHenselian FieldsBrauer EquivalenceHenselian ValuationSpecialization CosetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.