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

Abstract

For a division algebra $D$, let $K_1(D)=D^{\ast }/[D^{\ast },D^{\ast }]$ and let ${\mathrm{TK}}_1(D)$ 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 ${\mathrm{SK}}_1$. For a graded division algebra $E$ finite-dimensional over its center, we give exact sequences describing ${\mathrm{TK}}_1(E)$ in terms of $E_0$, the grade group~${\Gamma }_E$, and the conjugation action of $E^{\ast }$ on $E_0$. These yield explicit formulas for ${\mathrm{TK}}_1(E)$ 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 $\mathrm{char}(\overline{K})=0$, 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., $${\mathrm{TK}}_1(E)\cong {\mathrm{TK}}_1(q(E)),$$ together with a new proof of the corresponding stability theorem for ${\mathrm{SK}}_1$. As applications, we obtain graded analogues of Motiee's primary decomposition and scalar-extension results for torsion Whitehead groups.