Unitary SK\(_1\) for a graded division ring and its quotient division ring. (Q435926)

In this article the authors study the group \(\text{SK}_1(D,\tau)\) of a division algebra with unitary involution. If NEWLINE\[NEWLINE\Sigma_\tau'(D)=\{a\in D^*:\text{Nrd}_D(a)=\tau(\text{Nrd}_D(a))\}NEWLINE\]NEWLINE and \(\Sigma_\tau(D)\) is the subgroup generated by those \(a\in D^*\) with \(\tau(a)=a\), then \(\text{SK}_1(D,\tau)=\Sigma_\tau'(D)/\Sigma_\tau(D)\).NEWLINENEWLINE Suppose that \(E=\bigoplus_{\gamma\in\Gamma_E}E_\gamma\) is a graded division algebra graded by a torsion-free Abelian group \(\Gamma_E\). A unitary graded involution \(\tau\) is a ring antiautomorphism with \(\tau^2=\text{id}\) and \(\tau(E_\gamma)=E_\gamma\) for each \(\gamma\in\Gamma_E\), and for which \(\tau\) is not the identity on \(Z=Z(E)\). Such an \(E\) has a reduced norm, and one can define \(\text{SK}_1(E,\tau)\) in an analogous way to above. Such an \(E\) has a division ring of quotients \(q(E)\), which is isomorphic to \(E\otimes_Zq(Z)\), and the involution extends in a canonical way to a unitary involution \(\tau\) on \(q(E)\).NEWLINENEWLINE The main result of this paper is that \(\text{SK}_1(E,\tau)\cong\text{SK}_1(q(E),\tau)\).NEWLINENEWLINE This complements the main result of \textit{R.\ Hazrat} and \textit{A. R.\ Wadsworth}, [Proc. Lond. Math. Soc. (3) 103, No. 3, 508-534 (2011; Zbl 1243.16020)]. In that paper, let \(F\) be a Henselian valued field and let \(D\) be an \(F\)-central division algebra. The valuation on \(F\) extends uniquely to a valuation on \(D\). Suppose that \(\tau\) is a valuation on \(D\) compatible with the valuation \(v\) (that is, \(v\circ\tau=v\)). Then \(\tau\) induces an involution \(\widetilde\tau\) on the graded division algebra \(\text{gr}(D)\), graded by the value group \(\Gamma_D\). If \(F^\tau\) is Henselian and \(F/F^\tau\) is tamely ramified, then \(\text{SK}_1(D,\tau)\cong\text{SK}_1(\text{gr}(D),\widetilde\tau)\).