On a~theorem of Hurewicz and $K$-theory of complete discrete valuation rings

I. A. Panin

Izvestiya. Mathematics , Tome 29 (1987) no. 1, pp. 119-131

Résumé

It is proved that for a complete discrete valuation ring $\mathfrak D$ of zero characteristic with residue field $k$ of positive characteristic $p$ and maximal ideal $\mathfrak M$, the natural homomorphism of $K$-groups with coefficients $$ K_i(\mathfrak D;\mathbf Z/p^n\mathbf Z)\to\varprojlim_iK_i(\mathfrak D/\mathfrak M^j;\mathbf Z/p^n\mathbf Z) $$ is an isomorphism for all positive $i$ and $n$. For the ring of integers $\mathfrak D$ in a local field $K/\mathbf Q_p$, the groups $K_i(\mathfrak D;\mathbf Z/p^n\mathbf Z)$ are finite. Bibliography: 13 titles.