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.

在本文中，我们介绍了 在域k上具有 $\mathbb{F} _p$ -系数的 Voevodsky动机类别的本地版本，由k的有限生成扩展参数化。我们介绍了所谓的灵活领域，其动机是保守的。我们证明，在灵活的领域中，构建的局部动机类别比全局类别简单得多，并且更容易让人联想到拓扑对应物。这提供了方便的“本地”不变量，人们可以从中读取动机信息。我们计算 $p=2$ 的点的局部动机上同调并研究局部 Chow 动机类别. 我们引入局部 Chow 群并推测在灵活场上这些应该与Chow 群模数值等价与 $\mathbb{F} _p$ -系数一致，这意味着局部 Chow 动机与数值 Chow动机一致。我们在各种情况下证明了这个猜想。