Motivic cohomology of equicharacteristic schemes
Abstract
We construct a theory of motivic cohomology for quasicompact, quasiseparated schemes of equal characteristic, which is related to nonconnective algebraic $K$theory via an AtiyahHirzebruch 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 nonreduced 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 NesterenkoSuslin 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 LevineWeibel Chow group of zero cycles on a surface arises as a motivic cohomology group.
 Publication:

arXiv eprints
 Pub Date:
 September 2023
 DOI:
 10.48550/arXiv.2309.08463
 arXiv:
 arXiv:2309.08463
 Bibcode:
 2023arXiv230908463E
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Algebraic Geometry