  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 that it is contravariant and
multiplicative in contrast to the Gersten resolution. In particular, this
allows to reprove that the intersection in Chow groups coincides up to sign
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 certain indices.
