Motivic cohomology of equicharacteristic schemes

TL;DR

This paper develops a new motivic cohomology theory for equicharacteristic schemes, connecting it to algebraic K-theory, étale cohomology, and algebraic cycles, with applications to singularities and non-reduced schemes.

Contribution

It introduces a motivic cohomology framework for schemes of equal characteristic, utilizing trace methods and cyclic homology, extending classical theories to singular and non-reduced schemes.

Findings

Recovers classical motivic cohomology on smooth varieties.

Establishes a spectral sequence relating motivic cohomology to K-theory.

Shows the theory satisfies projective bundle formula and pro cdh descent.

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 \'etale 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 Soul\'e 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.