SK1 of graded division algebras

Hazrat, Roozbeh; Wadsworth, Adrian R.

doi:10.1007/s11856-011-0045-1

Cited by 18 publications

(13 citation statements)

References 29 publications

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“…In the non-graded setting, we denote a division algebra by D and its centre by K; this D is equipped with an involution τ , and we set F = K τ = {a ∈ K | τ (a) = a}. In the graded setting, we write E for a graded division algebra with centre T, and R = T τ , where τ is a graded involution on E. (This is consistent with the notation used in [8].) Depending on the context, we write τ (a) or a τ for the action of the involution on an element, and K τ for the set of elements of K invariant under τ .…”

Section: Preliminariesmentioning

confidence: 99%

“…The reduced Whitehead group, SK 1 (E), is defined as E (1) /E , where E (1) denotes the set of elements of E * with reduced norm 1, and E is the commutator subgroup [E * , E * ] of E * . This group was studied in detail in [8]. We shall be using the following facts which were established in that paper.…”

Section: Graded Division Algebrasmentioning

confidence: 99%

“…This may be due in part to the complexity of the formulas in Yanchevskiȋ's work, and the difficulty in following some of his arguments. This paper is a sequel to [8] where the reduced Whitehead group SK 1 for a graded division algebra was studied. Here we consider the reduced unitary Whitehead group of a graded division algebra with unitary graded involution.…”

Section: Introductionmentioning

confidence: 99%

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Unitary SK₁ of graded and valued division algebras

Hazrat

Wadsworth

2011

Proceedings of the London Mathematical Society

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Abstract. The reduced unitary Whitehead group SK1 of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary SK1 of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary SK1 of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary SK1 for much wider classes of division algebras over henselian fields.

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Section: Preliminariesmentioning

confidence: 99%

Section: Graded Division Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unitary SK₁ of graded and valued division algebras

Hazrat

Wadsworth

2011

Proceedings of the London Mathematical Society

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“…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].…”

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confidence: 99%

Unramified Scalar Extensions of Graded Division Algebras

Hwang¹

2014

Communications of the Korean Mathematical Society

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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

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“…This group has been studied extensively when A is a division algebra, or a graded division algebra (which is an Azumaya algebra) in literature (see [10,11] and [4]). …”

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confidence: 99%