An adelic resolution for homology sheaves

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.

Notes on the biextension of Chow groups

“…When the paper was finished, the author discovered that a similar but more general construction of a resolution for sheaves on algebraic varieties was independently done in [4, Section 4. Let us briefly recall several notions and facts from [16]. A non-degenerate flag of length p on X is a sequence of schematic points η 0 .…”

“…In [16] the author proposed another way to construct resolutions for a certain class of abelian sheaves on smooth algebraic varieties, namely, the adelic resolution. This class of sheaves includes the sheaves K n .…”

“…Sections 2.1-2.3 contain the description of various geometric constructions that are necessary for definitions of biextensions of Chow groups. In particular, in Section 2.4 we recall several facts from [16] about adelic resolution for sheaves of K-groups.…”

“…Section 4.3 is devoted to the construction of the biextension in terms of K-cohomology groups and a pairing between sheaves of K-groups (Proposition 4.9). We give an explicit description of this biextension in terms of the adelic resolution for sheaves of K-groups introduced in [16]. In Section 4.4 we construct a filtration on the Picard category of virtual coherent sheaves on a variety (Definition 4.10) and we define the biextension of Chow groups in terms of the determinant of cohomology (Proposition 4.29).…”

Geometric and analytic structures on the higher adèles

“…Finally, instead of allowing just quasi-coherent sheaves as coefficients, one may also allow other sheaves as coefficients. See, for example, [13,25].…”