The group $\mathrm{TK}_1$ of graded and valued division algebras

Abstract

For a division algebra $D$, let $K_1(D) = D^*/[D^*, D^*]$ and let $\operatorname{TK}_1(D)$ be the torsion subgroup of the abelian group $K_1(D)$. We study this torsion group for graded and valued division algebras, in parallel with the known theory of $\operatorname{SK}_1$. For a graded division algebra $E$ finite-dimensional over its center, we give exact sequences describing $\operatorname{TK}_1(E)$ in terms of $E_0$, the grade group~$\Gamma_E$, and the conjugation action of $E^*$ on $E_0$. These yield explicit formulas for $\operatorname{TK}_1(E)$ in the unramified, totally ramified, and semiramified cases. For a tame valued division algebra $D$ over its Henselian-valued center $K$, we identify the obstruction group $\mathbf H$ to a congruence theorem for $\TK(D)$. We show that if the residue field~$\overline K$ of the valuation on $K$ has characteristic $p > 0$, then $\mathbf H \cong\mu_K[p]$, the $p$-primary component of the group $\mu_K$ of roots of unity in $K$; but if $\operatorname{char}(\overline K)=0$, then $\mathbf H=1$. We further prove a short exact sequence $$ 1\,\longrightarrow \,\mathbf H\, \longrightarrow \,\operatorname{TK}_1(D)\, \longrightarrow\, \operatorname{TK}_1(\gr(D))\, \longrightarrow \,1, $$ where $\gr(D)$ is the associated graded division algebra determined by the valuation on $D$ obtained from the valuation on $K$. We also prove a stability theorem for a graded division algebra $E$ with quotient division ring~$q(E)$, i.e., $$ \operatorname{TK}_1(E)\,\cong \,\operatorname{TK}_1(q(E)), $$ together with a new proof of the corresponding stability theorem for $\operatorname{SK}_1$. As applications, we obtain graded analogues of Motiee's primary decomposition and scalar-extension results for torsion Whitehead groups.