Reduction in \(K\)-theory of some infinite extensions of number fields (Q1038631)

Let \(F\) be a number field with ring of integers \({\mathcal O}_F\). Let \(\nu\) be a prime ideal of \(F\). Let \(F(\mu_\infty)\) be the infinite extension \(F(\mu_\infty)= \bigcup_{m\geq 1}F(\mu_m)\). Let \(\widetilde\nu\) be a prime ideal of \(F(\mu_\infty)\) lying over \(\nu\). Let \(\kappa_{\widetilde\nu}\) be the residue fields of \(\nu\) and \(\widetilde\nu\) respectively (so that \(\kappa_{\widetilde\nu}\) is the algebraic closure of the finite field \(\kappa_\nu\)). Fix \(n\geq 1\) and let \(r_{\widetilde\nu}\) denote the natural reduction map from \(K_{2n+ 1}(F(\mu_\infty))\cong K_{2n+1}{\mathcal O}_{F(\mu_\infty)}\) to \(K_{2n+1}(\kappa_{\widetilde\nu})\). In this note the author gives a short and elegant proof of the following fact (Theorem 1.1): For any element \(t\in K_{2n+1}(\kappa_{\widetilde\nu})\), there exists a nontorsion element \(x\in K_{2n+1}(F(\mu_\infty))\) such that \(r_{\widetilde\nu})= t\). The proof uses Borel's \(K\)-theory rank computations and the Harris-Segal direct summands of the \(K\)-theory of algebraic integers.