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.

Arbarello, de Concini, and Kac have constructed a central extension of the ideles group on a smooth projective algebraic curve C. We show that this central extension induces the theta-bundle on the class group of degree g − 1 divisors on C, where g is the genus of the curve C. The other result of the paper is the relation between the product of the norms of the tame symbols over all points of the curve, considered as a pairing on the ideles group, and the Poincaré biextension of the Jacobian of C. As an application we get a new proof of the adelic formula for the Weil pairing. * The author was partially supported by RFFI grants 04-01-00613 and 05-01-00455.

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