The kernel of the reciprocity map of varieties over local fields

For a smooth and proper variety over a local field with residue characteristic the reciprocity map ρ : SK ) → π ) is a well-defined map from its class group to the corresponding abelianized étale fundamental group of . 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 ≠ . This result generalizes the work of Shuji Saito and Uwe Jannsen for curves and surfaces to arbitrary dimensions.