### 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.

Cristian D. Gonzalez-Aviles <gonzomat2002@yahoo.es>