ISOTROPIC MOTIVES

Vishik, Alexander

Abstract

In this article we introduce the local versions of the Voevodsky category of motives with Fp coefficients over a field k, parameterized 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 Fp-coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this Conjecture in various cases.

Citation

Vishik, A. (2020). ISOTROPIC MOTIVES. Journal of the Institute of Mathematics of Jussieu, 21(4), 1271-1330. https://doi.org/10.1017/S1474748020000560

• Journal Article Type: Article
• Acceptance Date: Aug 12, 2020
• Online Publication Date: Dec 22, 2020
• Publication Date: Dec 22, 2020
• Deposit Date: Aug 18, 2020
• Publicly Available Date: Jun 23, 2021
• Journal: Journal of the Institute of Mathematics of Jussieu
• Print ISSN: 1474-7480
• Electronic ISSN: 1475-3030
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 21
• Issue: 4
• Pages: 1271-1330
• DOI: https://doi.org/10.1017/S1474748020000560
• Public URL: https://nottingham-repository.worktribe.com/output/3024327
• Publisher URL: https://www.cambridge.org/core/journals/journal-of-the-institute-of-mathematics-of-jussieu/article/abs/isotropic-motives/6D99EC8AA2B06FA0FE69C6571BAA6A0E