Adelic resolution for homology sheaves

TL;DR

This paper introduces a new adelic resolution for homology sheaves, generalizing classical constructions, which is contravariant and multiplicative, enabling new insights into Chow groups and $K$-cohomology.

Contribution

It constructs a novel adelic complex for sheaves of $K$-groups, providing a flasque resolution that is contravariant and multiplicative, unlike the classical Gersten resolution.

Findings

Adelic complex provides a flasque resolution for homology sheaves.

The natural morphism to the Gersten complex is a quasi-isomorphism.

Reproves intersection product in Chow groups using $K$-cohomology.

Abstract

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.