Torsion subgroups of Whitehead groups of graded division algebras

Technical Summary: Torsion Subgroups of Whitehead Groups of Graded Division Algebras

Problem Statement
The paper addresses the computation of the torsion subgroup, denoted $TK_1(E)$, of the Whitehead group $K_1(E) = E^*/[E^*, E^*]$ for a graded division algebra $E$ that is finite-dimensional over its center $T$. While the reduced Whitehead group $SK_1(E)$ (the subgroup of elements with reduced norm 1 modulo commutators) has been extensively studied, particularly by Hazrat and Wadsworth in the graded setting and by Ershov and Platonov in the valued setting, the structure of the full torsion subgroup $TK_1(E)$ remains less explored. The authors aim to establish formulas for $TK_1(E)$ in specific structural cases (unramified, totally ramified, and semiramified) and utilize these results to compute the torsion subgroup $TK_1(D)$ for valued division algebras $D$ over henselian fields.

Methodology
The authors employ a two-pronged approach combining graded ring theory and valuation theory:

1. Graded Division Algebra Analysis: The paper first investigates $TK_1(E)$ for a graded division algebra $E = \bigoplus_{\gamma \in \Gamma} E_\gamma$. The methodology relies on the structural properties of the graded components, specifically the degree zero component $E_0$ and the value group $\Gamma_E$. Key tools include:

   • The construction of exact sequences relating $TK_1(E)$ to the torsion subgroup of the multiplicative group of the center, $\tau(T^*)$, and the reduced Whitehead group $SK_1(E)$.
   • The use of Galois cohomology, specifically the $-1$-Tate cohomology group $\hat{H}^{-1}(G, E_0^*)$, where $G = \text{Gal}(E_0/T_0)$.
   • The analysis of commutator subgroups $[E^*, E^*]$ and their relationship with the degree zero subring $E_0$.

2. Connection to Valued Division Algebras: To apply these graded results to classical valued division algebras $D$ over a henselian field $F$, the authors utilize the associated graded division algebra $\text{gr}(D)$. The core of this methodology is establishing an isomorphism between $TK_1(D)$ and $TK_1(\text{gr}(D))$ under specific "strongly tame" conditions. This involves:

   • Proving a "Congruence Theorem" (Theorem 4.7) which asserts that for a strongly tame division algebra $D$, the intersection of the torsion subgroup of the Whitehead group with the principal units $(1 + M_D)$ is trivial (contained within the commutator subgroup), provided the torsion group contains no elements of order $\text{char}(\bar{F})$.
   • Utilizing Hensel's Lemma and properties of the reduced norm map to lift results from the graded ring $\text{gr}(D)$ back to the valued ring $D$.

Key Contributions and Results

• Main Theorem for Graded Algebras (Theorem 1.1): The authors provide a comprehensive structural description of $TK_1(E)$ via a commutative diagram involving exact sequences. This theorem relates $TK_1(E)$ to the torsion subgroup of the center's multiplicative group and the Galois group of the degree zero component.

  • Corollary 1.2 derives explicit formulas for $TK_1(E)$ based on the ramification type of $E$:
    • Unramified: $TK_1(E) \cong TK_1(E_0)$.
    • Totally Ramified: $TK_1(E) \cong \tau(T_0^*)/\mu_e(T_0)$, where $e$ is the exponent of the value group quotient.
    • Semiramified: $TK_1(E)$ fits into an exact sequence involving the exterior square of the Galois group $G = \text{Gal}(E_0/T_0)$ and the Tate cohomology group $\hat{H}^{-1}(G, E_0^*)$.

• Isomorphism Theorem for Valued Algebras (Theorem 1.3): The paper establishes that if $D$ is a strongly tame division algebra over a henselian field $F$, and $TK_1(D)$ contains no element of order $\text{char}(\bar{F})$, then $TK_1(D) \cong TK_1(\text{gr}(D))$. This result serves as a bridge, allowing the graded formulas to be applied directly to valued division algebras.

• Computational Formulas for Valued Algebras (Corollary 1.4): By combining Theorem 1.1 and Theorem 1.3, the authors provide explicit descriptions of $TK_1(D)$ for unramified, totally ramified, and semiramified valued division algebras.

  • For nicely semiramified algebras, the paper shows that $TK_1(D)$ fits into an exact sequence where the kernel is isomorphic to $SK_1(D)$ (which is identified with $\hat{H}^{-1}(G, D^*)$) and the cokernel is related to the norm group of the residue field.

• Examples and Applications: The paper constructs specific examples using the "T-construction" (iterated symbol algebras over iterated Laurent series fields) to demonstrate the computation of $TK_1(D)$. These examples illustrate cases where $TK_1(D)$ is an extension of a finite cyclic group by a locally cyclic group.

Significance and Claims
The paper claims to extend the systematic study of $SK_1$ to the broader torsion subgroup $TK_1$ within the graded setting. The authors position their work as a natural progression from the foundational studies of Hazrat and Wadsworth on $SK_1$.

The significance of the results lies in:

1. Unification: Providing a unified framework to compute torsion subgroups for both graded and valued division algebras.
2. Generalization: The results regarding the realization of torsion abelian groups as $TK_1(D)$ for prime index algebras are presented as a graded version of a theorem by Motiee. The authors note that their graded approach offers an alternative method to obtain Motiee's result and potentially extend it.
3. Structural Clarity: The derivation of exact sequences and isomorphisms for specific ramification types clarifies the relationship between the arithmetic of the center, the value group, and the torsion properties of the Whitehead group.

The authors remain modest regarding the scope, noting that the isomorphism $TK_1(D) \cong TK_1(\text{gr}(D))$ relies on the "strongly tame" hypothesis and the absence of elements of order $\text{char}(\bar{F})$. They explicitly state that it is currently unknown whether these results hold under the weaker "tame" hypothesis. Furthermore, while they establish that $TK_1(E)$ is a locally cyclic group for graded division algebras of prime index, they pose the open question of whether every locally cyclic group can be realized as $TK_1(E)$ for such an algebra, framing this as a graded analogue to Motiee's classification.