Hodge Realizations of Motivic Cohomology

* 动机上同调的霍奇实现 * 让我们想象一下，你正在进行一次星星探险，去发现太阳系中不同行星上的文明。你第一次在一个星球上遇到一个文明，它具有“地球”的色彩，正如你随后在其他星球上遇到的文明所表明的那样，你的发现有一种深刻的“似曾相识”的感觉。在这一点上，人们可能会想象所有这些行星文明都只是“太阳系先进的宇宙文明”的“化身”，你既看不见也摸不着，但你本能地知道存在。在数学中，这是当前代数循环和动机上同调的世界。太阳系高级宇宙文明的类比正是动机上同调（一种宇宙上同调理论），它反映了所有上同调理论（类似于行星文明）的共同点。所有已知的motivic上同调的候选者都使用代数圈的基准作为构建块。调节器（或实现）类似于一个特定的行星文明。更精确地说，实现是从动机上同调到一个更“通俗”的上同调理论的映射。霍奇实现只是一类特殊的实现。我们理解动机上同调复杂性的唯一方法是通过它的实现。

***Hodge Realizations of Motivic Cohomology***Let us imagine for the moment that you are on a star trek expedition to discover civilizations on different planets in our solar system. You first encounter a civilization on one planet that has ``earthly'' overtones, and as your subsequent encounters of civilizations on other planets indicate, there is a sort of profound ``deja vu'' feeling about your discoveries. One might at this point, imagine that all such planetary civilizations are mere ``incarnations''of an ``advanced universal civilization of the solar system'', which you canneither see nor touch, but you instinctively know exists. In mathematics,this is the current world of algebraic cycles and motivic cohomology.The analog of the advanced universal civilization of the solar system isprecisely motivic cohomology (a universal cohomology theory), which is a reflectionof what all cohomology theories (analogous to planetary civilizations) havein common. All known candidates of motivic cohomology use the datum ofalgebraic cycles as building blocks. A regulator (or realization) is analogousto one particular planetary civilization. In more precise terms,a realization is a map from motivic cohomology to a more ``earthly'' cohomology theory.Hodge realizations are simply a particular class of realizations.Our only way to understand the complexity of motivic cohomology is via its realizations.