# Isotropic motives

## TL;DR

This paper develops local versions of Voevodsky's motives over fields, simplifying the global category over flexible fields and introducing local invariants, with computations and conjectures relating local and numerical Chow motives.

## Contribution

It introduces local motivic categories parameterized by finitely-generated extensions, and demonstrates their simplicity and relation to topological counterparts over flexible fields.

## Key findings

- Computed local motivic cohomology of a point for p=2
- Studied local Chow motivic category and introduced local Chow groups
- Conjectured and proved cases where local Chow motives match numerical Chow motives

## Abstract

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