Wild kernels for higher $K$-theory of division and semi-simple algebras

Xuejun Guo and Aderemi Kuku

Abstract: Let $\Sigma$ be a semi-simple algebra over a number field $F$. In this paper, we prove that for all $n\geq 0$, the wild kernel $WK_n(\Sigma):=Ker(K_n(\Sigma)\longrightarrow\prod \limits_{{\rm finite}\ v} K_n(\Sigma_v))$ is contained in the torsion part of the image of the natural homomorphism $K_n(\Lambda)\longrightarrow K_n(\Sigma)$, where $\Lambda$ is a maximal order in $\Sigma$. In particular, $WK_n(\Sigma)$ is finite. In the process, we prove that if $\Lambda$ is a maximal order in a central division algebra $D$ over $F$, then the kernel of the reduction map $K_{2n-1}(\Lambda)\stackrel{\pi_v} {\longrightarrow}\prod\limits_{{\rm finite}\v} K_{2n-1} (d_v)$ is finite. In Section $3$ we investigate the connections between $WK_n(D)$ and $\mbox{div}(K_{n}(D))$ and prove that $\mbox{div} K_2(\Sigma)\subset WK_2(\Sigma)$; if the index of $D$ is square free, then $\mbox{div} (K_2(D))\simeq\mbox{div} (K_2(F))$ , $WK_2(F)\simeq WK_2(D)$ and $|WK_2(D)/{\mbox{div} (K_2(D))}|\leq 2$. Finally we prove that if $D$ is a central division algebra over $F$ with $[D:F]=m^{2}$, then (1) $\mbox{div}(K_n(D))_l=WK_n(D)_l$ for all odd primes $l$ and $n\leq 2$; (2) if $l$ does not divide $m$, then $\mbox{div}(K_3(D))_l=WK_3(D)_l=0$; (3) if $F=\mathbb{Q}$ and $l$ does not divide $m$, then $\mbox{div}(K_n(D))_l\subset WK_n(D)_l$ for all $n$.

Keywords: wild kernel, $K_n$ group, semi-simple algebra

Classification (MSC2000): 19C99, 19F27, 11S45

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