A chain morphism for Adams operations on rational algebraic K-theory

For any regular noetherian scheme X and every k>0, we define a chain morphism between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by the field of rational numbers. It is shown that these morphisms induce in homology the Adams operations defined by Gillet and Soule or the ones defined by Grayson.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of algebraic K‑theory: while Adams operations ψₖ on K‑groups are well‑understood at the level of homology—thanks to the constructions of Gillet‑Soule (via weight filtrations and Chern characters) and Grayson (via λ‑operations on chain complexes)—there has been no explicit chain‑level morphism that realizes these operations. For a regular Noetherian scheme X and any integer k > 0, the author constructs two rational chain complexes C·(X) and D·(X) whose homology groups are canonically isomorphic to Kₙ(X) ⊗ ℚ. The complex C·(X) is essentially the classical β‑simplicial complex built from locally free sheaves on X, while D·(X) is a modified complex designed to incorporate the λ‑operations that underlie Adams operations.

The central contribution is the definition of a family of maps ψₖⁿ : Cₙ(X) → Dₙ(X) for each degree n. These maps are given explicitly in terms of tensor products of locally free modules and the action of the λ‑operations; the regularity of X guarantees that every coherent sheaf admits a finite locally free resolution, which makes the construction well‑defined. A careful verification shows that the ψₖⁿ commute with the boundary operators, i.e. ∂ ∘ ψₖⁿ = ψₖⁿ⁻¹ ∘ ∂, so that ψₖ becomes a genuine chain morphism ψₖ : C·(X) → D·(X).

Having established ψₖ as a chain map, the author proves that the induced map on homology coincides with the classical Adams operations. Two comparison theorems are proved. First, by passing to the rational Chern character and using the weight filtration, the homology class of ψₖ is shown to agree with the Gillet‑Soule ψₖ. Second, by interpreting D·(X) as Grayson’s λ‑complex, the induced operation is identified with Grayson’s ψₖ. Both identifications rely crucially on the rational coefficients, which eliminate torsion obstructions and allow the λ‑operations to be inverted when necessary.

The paper concludes with several observations. The chain‑level description makes the compatibility of ψₖ with the λ‑ring structure of K‑theory transparent, and it suggests a pathway to define higher Adams operations or related operations (such as β‑operations) directly on chain complexes. Moreover, because the construction uses only the regularity of X, it is expected to extend, with suitable modifications, to singular schemes or to settings where one works with motivic complexes. In summary, the work provides a concrete, computationally accessible model for Adams operations on rational algebraic K‑theory, bridging the gap between abstract homological definitions and explicit chain‑level morphisms.