2026/07/19 by Huynh Viet Khanh, Nguyen Duc Anh Khoa, Adrian R. Wadsworth
Mathematics · #math.RA #math.GR #math.KT
For a division algebra D, let K1(D) = D^*/[D^*, D^*] and let TK1(D) be the torsion subgroup of the abelian group K1(D). We study this torsion group for graded and valued division algebras, in parallel with the known theory of SK1. For a graded division algebra E finite-dimensional over its center, we give exact sequences describing TK1(E) in terms of E0, the grade group~ΓE, and the conjugation action of E^* on E0. These yield explicit formulas for TK1(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~ K of the valuation on K has characteristic p > 0, then \mathbf H ≅μK[p], the p-primary component of the group μK of roots of unity in K; but if char( K)=0, then \mathbf H=1. We further prove a short exact sequence 1 \longrightarrow \mathbf H \longrightarrow TK1(D) \longrightarrow TK1(\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., TK1(E) ≅ TK1(q(E)), together with a new proof of the corresponding stability theorem for SK1. As applications, we obtain graded analogues of Motiee's primary decomposition and scalar-extension results for torsion Whitehead groups.