Isotropic Motives

Abstract In this article we introduce the local versions of the Voevodsky category of motives with 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 F ₚ - coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this conjecture in various cases.