Algebraic varieties, K-groups and the Hasse principle, by Cristian D. Gonzalez-Aviles

Let k be a number field. We show that the Galois cohomology group H^1(k,K_2\overline{k}(X)/H^0(\overline{X},\Cal K_2)) is finite for a large class of smooth projective geometrically integral algebraic k-varieties X. This class includes all such varieties having a torsion-free geometric Neron-Severi group whence, in particular, the above group is finite if X is an abelian variety. This result, which generalizes a well-known theorem of W. Raskind for curves, is proved using U. Jannsen's Hasse principle. We go on to prove new finiteness results for CH^2_tors. In addition, we generalize a well-known exact sequence of P. Salberger from the case of rational ruled surfaces to more general ruled surfaces, thereby complementing results of J.-L. Colliot-Thelene. Finally, we use Jannsen's theorem alluded to above in conjunction with a result of B.Kahn to obtain a Brauer-Hasse-Noether Theorem for K_2.

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