分次赋值除环的群$\mathrm{TK}_1$
The group $\mathrm{TK}_1$ of graded and valued division algebras
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中文总结 AI 辅助
研究分次赋值除环的挠群$\mathrm{TK}_1$，给出描述$\mathrm{TK}_1(E)$的正合序列及显式公式，确定阻碍群$\mathbf{H}$，证明短正合序列和稳定性定理，还得到挠怀特黑德群相关结果的分次类似物。
中文摘要 AI 辅助
对于一个除环$D$，令$K_1(D)=D^*/[D^*,D^*]$，并令$\mathrm{TK}_1(D)$为阿贝尔群$K_1(D)$的挠子群。我们研究分次赋值除环的这个挠群，与已知的$\mathrm{SK}_1$理论并行。对于在其中心上有限维的分次除环$E$，我们给出了用$E_0$、分次群$\Gamma_E$以及$E^*$对$E_0$的共轭作用来描述$\mathrm{TK}_1(E)$的正合序列。这些给出了在非分歧、完全分歧和半分歧情形下$\mathrm{TK}_1(E)$的显式公式。对于在其亨赛尔赋值中心$K$上的驯顺赋值除环$D$，我们确定了关于$\mathrm{TK}(D)$的同余定理的阻碍群$\mathbf{H}$。我们表明，如果$K$上赋值的剩余域$\overline{K}$的特征$p>0$，那么$\mathbf{H}\cong\mu_K[p]$，即$K$中单位根群$\mu_K$的$p$ - 主分量；但如果$\mathrm{char}(\overline{K}) = 0$，那么$\mathbf{H}=1$。我们进一步证明了一个短正合序列$1\,\longrightarrow \,\mathbf{H}\, \longrightarrow \,\mathrm{TK}_1(D)\, \longrightarrow\, \mathrm{TK}_1(\mathrm{gr}(D))\, \longrightarrow \,1$，其中$\mathrm{gr}(D)$是由$D$上从$K$上的赋值得到的赋值所确定的相关分次除环。我们还证明了对于具有商除环$q(E)$的分次除环$E$的一个稳定性定理，即$\mathrm{TK}_1(E)\,\cong \,\mathrm{TK}_1(q(E))$，以及$\mathrm{SK}_1$的相应稳定性定理的一个新证明。作为应用，我们得到挠怀特黑德群的莫蒂定理的主分解和标量扩张结果的分次类似物。
英文摘要
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~$Γ_E$, 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 $\mathbf H \congμ_K[p]$, the $p$-primary component of the group $μ_K$ 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.
