Chow motives and cycle modules

亚历山大·格罗森迪克(Alexander Grothendieck，1928-2014)是1966年菲尔兹奖章(与来宝价格类似)的获得者，他提出了周星驰的动机。这些对象及其推广，例如弗拉基米尔·沃沃茨基的三角化动机范畴，在代数和代数几何中扮演着重要的角色。从Vladimir Voevodsky对Milnor猜想的证明开始，该猜想使用Markus Rost对范数二次曲面的Chow动机的计算，动机和密切相关的代数圈首先侵入二次型理论，然后是代数群论。所谓的Rost幂零原理，作为分解动机的主要工具之一，在这里扮演着重要的角色，它已经被证明是射影齐次变种，例如二次曲面。猜想这一原理对所有光滑射影簇都成立，并已被我自己证明用于特征为0的域上的曲面和某些三重曲面。在这些证明中，Markus Rost的循环模是不可缺少的，特别是作为计算Chow群的工具。进一步发展循环模的Rost理论，特别是着眼于领域扩展是本研究项目的目的之一。本文的目的不仅是找到一种证明高维变种的Rost幂零原理的方法，而且也是为了计算未分枝上同调。在他们寻找光滑方案上向量丛的Euler类理论的过程中，Jean Barge和Fabien Morel发现/引入了Milnor-Witt K-群。后来，Fabien Morel发展了一个类似于Rost的循环模理论的理论，然而Milnor-Witt K-群扮演了Milnor K-理论的角色。该项目的另一个目标是计算某些变种的未分支的Milnor-Witt K-群，以及未分支的Witt群。后者着眼于描述域的Witt群到该域上的二次函数域的Witt群的限制/基变化映射的核的描述。

Chow motives have been introduced by Alexander Grothendieck (1928-2014), winner of the fields Medal (analog of Noble Price) in 1966. These objects and their generalizations as for instance Vladimir Voevodsky's triangulated categories of motives play a significant role in algebra and algebraic geometry. Starting with Vladimir Voevodsky's proof of the Milnor conjecture which uses Markus Rost's computation of the Chow motive of a norm quadric, motives and the closely related algebraic cycles invaded first the theory of quadratic forms and later the theory of algebraic groups. The so called Rost nilpotence principle which has been verified for projective homogeneous varieties as for instance quadrics plays here an important role, as one of the main tools to decompose motives. It is conjectured that this principle is true for all smooth projective varieties, and has been proven by myself for surfaces and certain threefolds over fields of characteristic 0. In these proofs the cycle modules of Markus Rost, which in particular have been introduced as a tool to compute Chow groups, are indispensable.******The further development of Rost theory of cycle modules in particular with a view towards field extensions is one objective of this research project. The aim is here not only to find a method to prove the Rost nilpotence principle for higher dimensional varieties, but also to compute unramified cohomology.******In their search for a theory of Euler classes for vector bundles over smooth schemes Jean Barge and Fabien Morel discovered/introduced Milnor-Witt K-groups. Later Fabien Morel developed a theory which is similar to Rost's cycle module theory, where however Milnor-Witt K-groups play the role of Milnor K-theory. The other objective of the project is the computation of unramified Milnor-Witt K-groups of certain varieties, and also of unramified Witt groups. The latter with a view toward a description of the kernel of the restriction/base change map of the Witt group of a field to the Witt group of the function field of a quadric over this field.