Chow motives and cycle modules

1966年菲尔兹奖章（类似于诺贝尔奖）赢家亚历山大·格罗滕迪克（1928-2014）提出了周恩来的动机。这些对象和他们的概括，例如弗拉基米尔Voevodsky的三角类的动机发挥了重要作用，代数和代数几何。从弗拉基米尔Voevodsky的证明米尔诺猜想使用马库斯罗斯特的计算周动机的规范二次，动机和密切相关的代数周期入侵第一理论的二次形式，后来理论的代数群。所谓的罗斯特零素原理已被验证为射影齐性品种，例如二次曲面在这里起着重要的作用，作为一个主要的工具来分解动机。这是澄清，这一原则是真实的所有光滑的射影品种，并已证明了我自己的曲面和某些三倍的领域的特征0。在这些证明中，Markus Rost的循环模是必不可少的，特别是作为计算Chow群的工具引入的循环模。进一步发展的罗斯特理论的循环模块，特别是着眼于领域的扩展是本研究项目的目标之一。本文的目的不仅是找到一种方法来证明高维簇的Rost零维原理，而且还要计算非分歧上同调。在他们寻找光滑格式上向量丛的欧拉类理论的过程中，Jean Barge和Fabien Morel发现/介绍了Milnor-Witt K-群。后来法比安莫雷尔制定了一个理论，这是类似于罗斯特的循环模理论，但米尔诺尔-维特K-群发挥作用的米尔诺尔K-理论。该项目的另一个目标是计算非分歧的Milnor-Witt K-群的某些品种，也是非分歧的Witt群。后者的一个视图对内核的描述的限制/基地的变化地图的维特群的一个领域的维特群的功能领域的二次曲面在这个领域。

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.