Division algebras over Henselian fields

Abstract

In 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.