[Paper Review] Unitary SK_1 for a Graded Division Ring and its Quotient Division Ring

Adrian R. Wadsworth, V. I. Yanchevskiĭ|arXiv (Cornell University)|Apr 9, 2011

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper establishes that the unitary SK₁ group of a graded division ring with a torsion-free abelian grade group and a degree-preserving unitary involution is isomorphic to the unitary SK₁ group of its quotient division ring. The result relies on structural properties of graded algebras and the behavior of reduced norms under quotient constructions, extending known isomorphisms from valued division algebras to the graded setting.

ABSTRACT

Let E be a graded division ring finite-dimensional over its center with torsion-free abelian grade group, and let q(E) be its quotient division ring. Let tau be a degree-preserving unitary involution on E. We prove that SK_1(E, tau) is isomorphic to SK_1(q(E), tau).

Motivation & Objective

• To establish an isomorphism between the unitary SK₁ groups of a graded division ring and its quotient division ring under a unitary graded involution.
• To extend known results on SK₁ for valued division algebras to the setting of graded division algebras via quotient rings.
• To provide a theoretical foundation for transferring SK₁ computations from the more tractable graded setting to the quotient division ring setting.
• To demonstrate that the Stability Theorem for unitary SK₁ follows as a corollary from the main isomorphism.
• To unify and generalize existing results on SK₁ in the context of twisted Laurent polynomial rings and iterated rational algebras.

Proposed method

• Utilizes the structure of a graded division algebra E with a torsion-free abelian grade group and a degree-preserving unitary involution τ.
• Constructs the quotient division ring q(E) = E ⊗_Z q(Z), where q(Z) is the quotient field of the center Z of E.
• Applies the reduced norm map Nrd_E: E → Z and defines Σ_τ(E) and Σ′_τ(E) as subgroups of E*, leading to SK₁(E,τ) = Σ′_τ(E)/Σ_τ(E).
• Extends the involution τ from E to q(E) canonically, preserving the unitary and degree-preserving properties.
• Employs the fact that E is semiramified and decomposably semiramified when the grade group is finitely generated, enabling explicit computation via iterated twisted Laurent polynomial structures.
• Relies on the isomorphism between SK₁ groups in the graded and quotient settings by analyzing the behavior of units and norms under localization and quotient formation.

Experimental results

Research questions

• RQ1 Does the unitary SK₁ group of a graded division ring remain isomorphic to that of its quotient division ring under a unitary graded involution?
• RQ2 Can results on SK₁ for graded division algebras be transferred to their quotient division rings via this isomorphism?
• RQ3 How does the isomorphism between SK₁(E,τ) and SK₁(q(E),τ) relate to the Stability Theorem in unitary K-theory?
• RQ4 What structural properties of graded division algebras and their quotients ensure the preservation of SK₁ under quotient construction?
• RQ5 To what extent can the isomorphism be used to simplify SK₁ computations in central simple algebras over rational function fields?

Key findings

• The main result establishes a canonical isomorphism: SK₁(E,τ) ≅ SK₁(q(E),τ) for a graded division ring E with torsion-free grade group and unitary graded involution τ.
• The isomorphism allows transfer of SK₁ computations from the more structured graded setting to the quotient division ring, which may be easier to analyze.
• The Stability Theorem for unitary SK₁ is shown to follow directly from the main isomorphism, as SK₁ remains unchanged under passage to rational extensions.
• In the case of finitely generated grade groups, E is isomorphic to an iterated twisted Laurent polynomial ring over E₀, and q(E) is the corresponding iterated twisted rational division algebra.
• For specific constructions such as tensor products of symbol algebras over rational function fields, the SK₁ group of the quotient division algebra is isomorphic to the SK₁ group of the associated graded algebra.
• The result generalizes earlier findings on SK₁ for valued division algebras via the associated graded ring construction, now extending to quotient rings of graded division algebras.