Unramified Scalar Extensions of Graded Division Algebras

Abstract

Abstract. Let E be a graded central division algebra (GCDA) over a grade field R. Let S be an unramified graded field extension of R. We describe the grading on the underlying GCDA E ′ of E ⊗ R S which is analogous to the valuation on a tame division algebra over Henselian valued field.Let D be a division algebra with a valuation. To this one associates a graded division algebra GD = γ∈ΓD GD γ , where Γ D is the value group of D and the summands GD γ arise from the filtration on D induced by the valuation (see [5] for details). As is illustrated in [5], even though computations in the graded setting are often easier than working directly with D, it seems that not much is lost in passage from D to its corresponding graded division algebra GD. This has provided motivation to systematically study this correspondence, notably by Boulagouaz [1], Hwang, Tignol and Wadsworth [4,5,8], and to compare certain functors defined on these objects, notably the Brauer group. Also, the graded method is effectively used to calculate the reduced Whitehead group SK 1 of a division algebra, first on the graded level and then specialise to the non-graded setting by Hazrat, Wadsworth, and Yanchevskiȋ [2,3,9].Let Γ be a torsion-free abelian group. A ring E is a graded division ring (with grade group in Γ) if E has additive subgroups E γ for γ ∈ Γ such that E = γ∈Γ E γ and E γ E δ ⊆ E γ+δ for all γ, δ ∈ Γ, and each E γ \ {0} lies in E * , the group of units of E. For background on graded division rings and proofs of their properties mentioned here, see [5]. The grade group of E is Γ E = {γ ∈ Γ | E γ = {0} }, a subgroup of Γ. For a ∈ E γ \ {0} we write deg(a) = γ. A significant property is that E