Motivic cohomology of equicharacteristic schemes
Elden Elmanto, Matthew Morrow
TL;DR
This work constructs motivic cohomology Z(j)^{mot} for qcqs equicharacteristic schemes by harnessing trace methods from K-theory, blending KH, TC, and syntomic/derived de Rham data into a motivic filtration on K(X). It proves central features including Beilinson–Lichtenbaum-type comparisons to étale and syntomic cohomology, a projective-bundle formula, pro cdh descent, and a robust relation to algebraic cycles and zero-cycles on surfaces. The theory recovers classical motivic cohomology on smooth varieties and interfaces with lisse motivic cohomology via Nesterenko–Suslin-type isomorphisms, while also delivering strong vanishing results such as motivic Soulé–Weibel vanishing. Extending to derived schemes, the construction provides a broad, non–A^1-invariant framework that coherently interpolates between cycle-theoretic and homotopy-theoretic perspectives, with potential applications to arithmetic and singular geometry. The paper also situates its approach among recent non–A^1-invariant motivic theories and pro-cdh/topological techniques, and it establishes precise comparison maps with existing motivic, lisse, and cdh-local theories. Finally, it connects the motivic framework to concrete cycle-theoretic descriptions, including zero-cycles on surfaces, thereby enriching the algebraic-cycle viewpoint in singular settings.
Abstract
We construct a theory of motivic cohomology for quasi-compact, quasi-separated schemes of equal characteristic, which is related to non-connective algebraic $K$-theory via an Atiyah--Hirzebruch spectral sequence, and to étale cohomology in the range predicted by Beilinson and Lichtenbaum. On smooth varieties over a field our theory recovers classical motivic cohomology, defined for example via Bloch's cycle complex. Our construction uses trace methods and (topological) cyclic homology. As predicted by the behaviour of algebraic $K$-theory, the motivic cohomology is in general sensitive to singularities, including non-reduced structure, and is not $\mathbb{A}^1$-invariant. It nevertheless has good geometric properties, satisfying for example the projective bundle formula and pro cdh descent. Further properties of the theory include a Nesterenko--Suslin comparison isomorphism to Milnor $K$-theory, and a vanishing range which simultaneously refines Weibel's conjecture about negative $K$-theory and a vanishing result of Soulé for the Adams eigenspaces of higher algebraic $K$-groups. We also explore the relation of the theory to algebraic cycles, showing in particular that the Levine--Weibel Chow group of zero cycles on a surface arises as a motivic cohomology group.