An imbedding property of \(K\)-groups
Let \(A\subset B\) be two rings that satisfy that any system of polynomial equations in a finite number of variables and coefficients in \(A\) which has a solution in \(B\), has also a solution in \(A\). The author proves the following theorem for such rings: For any two rings \(A\subset B\) with the above property, we have that the natural maps \(KV_{n}(A)\to KV_{n}(B),\quad KV_{n}(A)/\ell^{k}\to KV_{n}(B)/\ell^{k}, \text{ and } KV_{n}(A;\mathbb{Z}/\ell^{k})\to KV_{n}(B;\mathbb{Z}/\ell^{k})\) are imbeddings for all \(n\geq 1\) and \(k\in \mathbb{N}\); where \(KV_{n}(\;)\) denotes Karoubi-Villamayor \(K\)-theory. An important example for which the above condition holds is the following: let \(R\) be a local ring with maximal ideal \(\mathcal{P}\), and let \(P_{\mathcal{P}}\) be the \(\mathcal{P}\)-completion of \(R\) and \(R^{h}_{\mathcal{P}}\) the henselization of \(R\) at \(\mathcal{P}\). Let \(R_{0}\) be a field or an excellent discrete valuation ring and \(R\) a local \(R_{0}\)-algebra essentially of finite type over \(R_{0}\). Then the rings \(R^{h}_{\mathcal{P}}\subset R_{\mathcal{P}}\) satisfy the hypothesis of the theorem hence the conclusion applies in this situation for \(\ell\) a prime number. If we assume further that \(R\) is regular then the conclusion holds for Quillen \(K\)-theory.
scientific article; zbMATH DE number 5124471
scientific article