Torsion in \(K_ 2\) of fields and 0-cycles on rational surfaces
Let k be a field which is algebraically closed in K. Using ''Hilbert's theorem 90 for \(K_ 2''\), proved by Merkurev-Suslin, the authors show, inter alia, that \(i: K_ 2(k)\to K_ 2\)(K) is injective in characteristic zero and that ker(i) is p-primary torsion in characteristic p. Amongst the applications much of these results are some finiteness results for \(A_ 0(X)\), the group of 0-cycles of degree zero on a rational surface X over suitably nice fields (e.g. k local). Suslin has also obtained these results and similar results have been obtained by J.- L. Colliot-Thélène, W. Raskin, J.-J. Sansuc and C. Soulé [see for example \textit{J.-L. Colliot-Thélène}, \textit{J.-J. Sansuc} and \textit{C. Soulé}, C. R. Acad. Sci., Paris, Sér. I 294, 749-752 (1982; Zbl 0572.14005)].
scientific article; zbMATH DE number 3916442