Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 47, No. 1, pp. 1-14 (2006)

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

Xuejun Guo and Aderemi Kuku

Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, The People's Republic of China, and The Abdus Salam International Center for Theoretical Physics, Trieste, Italy; Institute for Advanced Study, Princeton, NJ, USA

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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Electronic version published on: 9 May 2006. This page was last modified: 4 Nov 2009.

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