K-motives of algebraic varieties

A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory as well as bivariant motivic kohomology groups are defined and studied. We use Grayson’s machinery to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic kohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of K-motives constructed in the paper. It is also shown that ordinary algebraic K-theory is represented by the K-motive of the point.

💡 Research Summary

The paper develops a new framework for “K‑motives” by combining the language of spectral categories with module theory, thereby extending the motivic picture beyond Voevodsky’s triangulated category of effective motives (DM_eff). The authors start by encoding transfers between smooth schemes X and Y as symmetric spectra O(X,Y), which serve as morphism objects in a spectral category O. Three principal spectral categories are constructed: O_K, O_{K⊕}, and O_{KGr}. O_K is built from Waldhausen’s K‑theory spectra of the exact categories P(X,Y) of coherent O_{X×Y}‑modules satisfying finiteness and local freeness conditions; O_{K⊕} uses the direct‑sum version, while O_{KGr} incorporates Grayson’s K‑theory spectra.

A key technical condition introduced is “motivically excisive” for a spectral category. Roughly, this requires that the transfer spectra respect A¹‑homotopy invariance and Nisnevich descent, mirroring the axioms that underlie Voevodsky’s motives. When O satisfies this condition, the category of O‑modules (presheaves of symmetric spectra equipped with O‑transfers) admits two Quillen model structures—projective and flat—both of which are left proper, cellular, weakly finitely generated, and symmetric monoidal with respect to the smash product. By Bousfield localization with respect to the usual stable equivalences, the authors obtain stable model structures, and the associated homotopy categories are denoted SH_mot O. These triangulated categories play the role of a “motivic” category attached to the chosen transfer data.

The central object of study is the K‑motive M_K(X) of a smooth scheme X over a base field F. It is defined as the image of the free O_K‑module O_K(–,X) in SH_mot O_K. The authors prove that ordinary algebraic K‑theory is represented by the K‑motive of the point: for any integer i, K_i(X) ≅ SH_mot O_K(M_K(X)