Mathematical Sciences: Motivic Cohomology with Finite Coefficients

9625658 近年来，（混合）动机基本理论的建构取得了一些进展。 PI现在拥有许多技术，这些技术使得至少有可能接近一些“硬”动机。根据它们处理的是有理系数还是有限系数，这些代数可以分为两类。在这个建议中描述的研究的目标是了解有限系数的“动机世界”的结构。 结果表明，通常的"动机t结构"直觉在有限系数的情况下不起作用。其中一个可能的替代品，它似乎是"色"的直觉，这是非常成功地采用代数拓扑在过去十年。这种方法被用在最近PI的预印本中，它表明，如果我们有一个很好的定义，代数几何类似物的更高的摩拉瓦K-理论，可以解决一些关于有限系数动机的标准代数。 构建这些理论似乎是一个不平凡但相对简单的问题。所采取的步骤包括描述motivic上同调的上同调运算，通过Thom谱的代数几何模拟构造代数上边，利用motivic亚当斯谱序列计算其"系数环"，最后由代数上边构造代数Morava K-理论. 这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。 在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中也得到了应用。

9625658 Recently some progress has been achieved in the construction of the basic theory of (mixed) motives. The PI is in possession now of a number of techniques which make it possible to approach at least some of the "hard" motivic conjectures. These conjectures can be divided into two classes depending on whether they deal with rational or finite coefficients. The goal of the research described in this proposal is to understand the structure of the "Motivic world" for finite coefficients. It turns out that the usual "motivic t-structure" intuition does not work in the finite coefficients case. One of the possible replacements for it seems to be the "chromatic" intuition which was very successfully employed in algebraic topology in the last decade. This approach was used in the recent PI's preprints where it was shown that a number of standard conjectures about motives with finite coefficients could be solved if we had a good definition of algebro-geometrical analogs of the higher Morava K-theories. To construct these theories seems to be a nontrivial but relatively straight forward problem. The steps to be taken include the description of the cohomological operations in motivic cohomology, construction of algebraic cobordisms through an algebro-geometrical analog of Thom spectrum, computation of its "coefficients ring" using "motivic" Adams spectral sequence and finally the formal construction of algebraic Morava K-theories from the algebraic cobordisms. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics .