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ISOTROPIC MOTIVES

Part of: Cycles and subschemes (Co)homology theory $K$-theory in geometry

Published online by Cambridge University Press:  22 December 2020

Alexander Vishik Open the ORCID record for Alexander Vishik [Opens in a new window]
Alexander Vishik*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK ([email protected])
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Abstract

In this article we introduce the local versions of the Voevodsky category of motives with $\mathbb{F} _p$ -coefficients over a field k, parametrized by finitely generated extensions of k. We introduce the so-called flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy ‘local’ invariants from which one can read motivic information. We compute the local motivic cohomology of a point for $p=2$ and study the local Chow motivic category. We introduce local Chow groups and conjecture that over flexible fields these should coincide with Chow groups modulo numerical equivalence with $\mathbb{F} _p$ -coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this conjecture in various cases.

Keywords

motivemotivic cohomologyChow groupsnumerical equivalence of cyclesMilnor's operations

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C25: Algebraic cycles 19E15: Algebraic cycles and motivic cohomology 14F42: Motivic cohomology; motivic homotopy theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 4 , July 2022 , pp. 1271 - 1330
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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