论文标题
在A-Motivic的共同组成部分
On the Integral Part of A-Motivic Cohomology
论文作者
论文摘要
在数量字段$ f $上定义的代数品种所附加的最深的算术不变性是其动机共同学的整体部分捕获的。当$ x $是一个平稳的投影品种时，本质上有两种定义它的方法：一种是通过$ k $ - 常规型号的理论，另一种是通过其$ \ ell $ - ad-adic实现的。两种方法都猜想是重合的。本文启动了积极特征全球领域的动机共同体研究的研究，此后将$ a $ a-Motivic共同体学命名为$ a-Motivic Coomology，其中经典的混合动机被混合的Anderson $ A $ MOTIVES取代。我们的主要目的是使用Gardeyn的最大模型概念$ a $ a $ a-Motives作为常规品种的类似物，以$ a $ a-动感共同体的整体部分设置型号版本的定义。我们的主要结果指出，该模型版本包含在$ \ ell $ -ADIC版本中。与数字字段设置中的预期相反，我们表明这两种方法一般不匹配。我们通过介绍混合Anderson $ A $ MOTIVES的受监管扩展的子模块来结束这项工作，我们希望这两种方法可以匹配，并解决了这一期望的一些特殊情况。
The deepest arithmetic invariants attached to an algebraic variety defined over a number field $F$ are conjecturally captured by the integral part of its motivic cohomology. There are essentially two ways of defining it when $X$ is a smooth projective variety: one is via the $K$-theory of a regular model, the other is through its $\ell$-adic realization. Both approaches are conjectured to coincide. This paper initiates the study of motivic cohomology for global fields of positive characteristic, hereafter named $A$-motivic cohomology, where classical mixed motives are replaced by mixed Anderson $A$-motives. Our main objective is to set the definitions of the model version and the $\ell$-adic version of the integral part of $A$-motivic cohomology, using Gardeyn's notion of maximal models of $A$-motives as the analogue of regular models of varieties. Our main result states that the model version is contained in the $\ell$-adic version. As opposed to what is expected in the number field setting, we show that the two approaches do not match in general. We conclude this work by introducing the submodule of regulated extensions of mixed Anderson $A$-motives, for which we expect the two approaches to match, and solve some particular cases of this expectation.
