On the map between homology of henselization and completion of some local rings (Q1295614)

The authors prove the following theorem: Let \(R_0\) be a field or an excellent discrete valuation ring. Let \(R\) be a domain which is a local \(R_0\)-algebra essentially of finite type over \(R_0\). Let \({\mathfrak p}\) be the maximal ideal of \(R\). Then the natural map \(H_n(Sl(R^h_{\mathfrak p});\mathbb{Z})\to H_n(Sl(R_{\mathfrak p});\mathbb{Z})\) is injective. Here \(R_{\mathfrak p}\) denotes the completion of \(R\) at \({\mathfrak p}\), and \(R^h_{\mathfrak p}\) denotes the henselization of \(R\) at \({\mathfrak p}\). Their method of proof is to apply the Artin approximation theorem to the bar complex which calculates the homology. They deduce the proposition: The natural map \(H_n(Sl(\mathbb{A}_f^h);\mathbb{Z})\to H_n(Sl(\mathbb{A}_f);\mathbb{Z})\) is injective. Here, \(\mathbb{A}_f\) denotes the finite adele ring of a number field \(F\), and \(\mathbb{A}^h_f\) the henselian adele ring of \(F\) \((\mathbb{A}^h_f=\{(a_v)\in\prod_vF^h_v:a_v\in{\mathcal O}_v\) for almost all \(v\})\). Combining this with a result of \textit{D. Arlettaz} [J. Pure Appl. Algebra 71, No. 1, 1-12 (1991; Zbl 0728.19002)] on the Hurewicz homomorphism they obtain corollary 1: The kernel of the natural map \(K_n(\mathbb{A}_f^h)\to K_n(\mathbb{A}_f)\) is torsion. It has exponent dividing a number \(R_{n-1}\) which depends solely on \(n\). \textit{G. Banaszak} and \textit{P. Zelewski} prove a related result [J. Pure Appl. Algebra 120, No. 2, 161-165 (1997; Zbl 0892.19002)], but under the additional assumption that \(R\) is regular.