The kernel of the reciprocity map of varieties over local fields

For a smooth and proper variety X over a local field K with residue characteristic p the reciprocity map rho(X) : SK1 (X) -> pi(ab)(1) (X) is a well-defined map from its class group to the corresponding abelianized etale fundamental group of X. In this paper we show that its kernel is the direct sum of a finite group and a group which is l-divisible for all primes l not equal p. This result generalizes the work of Shuji Saito and Uwe Jannsen for curves and surfaces to arbitrary dimensions.