Galois Codescent For Motivic Tame Kernels

Let $L\mathrm{/}F$ be a finite Galois extension of number fields with an arbitrary Galois group $G$. We give an explicit description of the kernel of the natural map on motivic tame kernels $H^2_{$\mathcal{M}$}(o_L, {\bf Z}(i)){G} {\rightarrow} H^2{$\mathcal{M}$}(o_F, {\bf Z}(i))$. Using the link between motivic cohomology and $K$-theory, we deduce genus formulae for all even $K$-groups $K_{2i-2}(o_F)$ of the ring of integers. As a by-product, we also obtain lower bounds for the order of the kernel and cokernel of the functorial map $H^2_{$\mathcal{M}$}(F, {\bf Z}(i)) \rightarrow H^2_{$\mathcal{M}$}( L, {\bf Z}(i) )^{G}$.