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 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.

💡 Research Summary

The paper introduces an “adelic complex” for any abelian sheaf on a scheme and shows that, when the sheaf arises from a homology theory satisfying a natural set of axioms (as K‑theory does), this complex furnishes a flasque resolution. The construction proceeds by taking, for each integer p, the direct sum over all chains of points of length p (i.e., sequences of specializations of length p) of the local sections of the sheaf on the corresponding open subsets. The differential is defined by alternating sums of restriction maps along inclusions of chains, exactly mirroring the classical adelic construction for ideles but now in arbitrary cohomological degree.

The key axioms required of the underlying homology theory are: (i) functoriality with respect to regular embeddings, (ii) a localization exactness property (the stalk at a point is the colimit of sections over neighborhoods), and (iii) continuity with respect to filtered colimits of chains. Under these hypotheses the adelic complex A⁎(X,𝔽) is shown to be flasque: every section on a Zariski open can be lifted to a global section in the appropriate degree, which forces higher cohomology of A⁎ to vanish.

A natural morphism ϕ: A⁎ → G⁎ to the classical Gersten complex is constructed by sending an adelic element to its image in the corresponding residue group. The paper proves that ϕ is a quasi‑isomorphism. The proof uses a careful analysis of “auxiliary maps” that correct the shift between chain length and cohomological degree, together with the exactness properties of the homology theory. Consequently, the adelic complex provides an alternative resolution of the same sheaf, but with markedly different functorial behavior.

Unlike the Gersten complex, which is covariant (it works well for push‑forwards but not for pull‑backs), the adelic complex is contravariant: a morphism of schemes f: Y → X induces a pull‑back f⁎ on adelic complexes that commutes with differentials. Moreover, the adelic construction is multiplicative: given two sheaves 𝔽₁, 𝔽₂, there is a natural product A⁎(X,𝔽₁) ⊗ A⁎(X,𝔽₂) → A⁎(X,𝔽₁⊗𝔽₂) compatible with differentials. This multiplicative structure allows the author to re‑derive the well‑known fact that the intersection product in Chow groups coincides, up to the sign (−1)^{rs}, with the product in K‑cohomology. The sign appears precisely from the Koszul rule when swapping the degrees of the two factors in the adelic complex.

A particularly striking application is the expression of the Weil pairing on the n‑torsion of a Jacobian as a Massey triple product in K‑cohomology. By choosing appropriate degrees (H¹(𝔾ₘ) and H²(μₙ)) inside the adelic complex, the author defines a Massey product ⟨a,b,c⟩ and shows that it reproduces the classical Weil pairing eₙ(a,b). This demonstrates that the adelic resolution is robust enough to host higher cohomological operations, something not readily accessible via the Gersten resolution.

Overall, the paper establishes that the adelic complex is a flasque, contravariant, and multiplicative resolution for homology sheaves, quasi‑isomorphic to the Gersten complex, and capable of encoding both classical intersection theory and higher operations such as Massey products. The results open the door to systematic use of adelic methods in K‑theory, motivic cohomology, and other homology‑type theories, especially where functorial pull‑backs and product structures are essential.