Adelic resolution for homology sheaves

A generalization of the usual ideles group is proposed, namely, we construct certain adelic complexes for sheaves of K-groups on schemes. More generally, such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated to the presheaf of a homology theory with certain natural axioms, satisfied by K-theory. In this case it is proven that the adelic complex provides a flasque resolution for the above sheaf and that the natural morphism to the Gersten complex is a quasiisomorphism. The main advantage of the new adelic resolution is its multiplicativity property. In particular, this allows to reprove that the intersection in Chow groups coincides with the natural product in the corresponding K-cohomology groups. Also, we show that the Weil pairing can be expressed as a Massey triple product in K-cohomology groups with suitable indices.