Total Ramification

Abstract

AbstractA symplectic module (M,a) is a finite abelian group M together with a nondegenerate alternating pairing \(a\colon M\times M \to \mathbb {Q}/\mathbb {Z}\). Such pairings arise (i) on subgroups called armatures of A ×/F × when the algebra A with center F is a tensor product of symbol algebras, and (ii) on Γ D /Γ F when D is a valued division algebra totally ramified over its center F. In §7.1 we consider the set Symp(Ω) of all symplectic modules on finite subgroups of an abelian torsion group Ω; we describe the canonical operation on Symp(Ω) making it into a torsion abelian group. If Γ is a torsion-free abelian group, we prove in Th. 7.22 that \(\mathit{Symp}({\mathbb{T}}(\Gamma)) \cong {\mathbb{T}}(\wedge^{2} \Gamma)\), where \({\mathbb{T}}(\Gamma) = \Gamma\otimes_{\mathbb {Z}}(\mathbb {Q}/\mathbb {Z})\). In §7.2 we consider armatures on algebras and their homogeneous counterparts on graded algebras. We show how an armature on a central simple algebra over a valued field can be used to build a gauge on the algebra. In §7.3 and §7.4 the focus is on totally ramified graded and valued division algebras. If F is an inertially closed graded field (i.e., F 0 is separably closed), we prove a group isomorphism from Br(F) to the part of \(\mathit{Symp}({\mathbb{T}}(\Gamma_{\mathsf {F}}))\) of torsion prime to \(\operatorname {\mathit{char}}(\mathsf {F}_{0})\) mapping \({[\mathsf {D}] \mapsto (\Gamma_{\mathsf {D}}/\Gamma_{\mathsf {F}}, \overline{c}_{\mathsf {D}})}\), where \(\overline{c}_{\mathsf {D}}\) is the canonical pairing induced by commutators. For F not inertially closed, this leads to a description of Br(F)/Br in (F) as a subgroup of \(\mathit{Symp}({\mathbb{T}}( \Gamma_{\mathsf {F}})))\) determined by the roots of unity in F 0. The analogous result is proved for \(\operatorname {\mathit{Br}}_{t}(F)/\operatorname {\mathit{Br}}_{ \mathit{in}}(F)\) for a Henselian field F.KeywordsTotal BranchingSymplectic ModuleTorsion Abelian GroupGraded Division AlgebraHenselian FieldsThese 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.