Chow motives and cycle modules

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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.