Galois Codescent For Motivic Tame Kernels

Abstract

Let $L/F$ 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 $H_{\mathcal{M}}^2(o_L,\mathbf{Z}(i){)}_G\to H_{\mathcal{M}}^2(o_F,\mathbf{Z}(i))$. 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 $H_{\mathcal{M}}^2(F,\mathbf{Z}(i))\to H_{\mathcal{M}}^2(L,\mathbf{Z}(i){)}^G$.

References in corpus (1)

• Tate kernels, etale K-theory and the Gross kernel