Chow motives and cycle modules

1966年获得菲尔兹奖（相当于诺贝尔奖）的亚历山大·格罗腾迪克（Alexander Grothendieck, 1928-2014）提出了周动机。这些对象和它们的推广例如Vladimir Voevodsky的动机的三角分类在代数和代数几何中扮演着重要的角色。从Vladimir Voevodsky利用Markus Rost对范数二次元的Chow动机的计算证明Milnor猜想开始，动机和密切相关的代数循环首先侵入了二次型理论，后来又侵入了代数群理论。所谓Rost零幂原理作为分解动机的主要工具之一，在这里起着重要的作用，该原理已被证明适用于投影齐次变量，例如二次函数。据推测，这一原理对所有光滑的射影变体都成立，并已由我本人对特征为0的域上的曲面和某些三折进行了证明。在这些证明中，Markus Rost的循环模块，特别是作为计算Chow群的工具而引入的循环模块，是不可或缺的。

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.