**Описание:** |
From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors
Hp'q(-,A) :Sm/k^Ab
from smooth schemes over a given field к to abelian groups, indexed by integers p and q. The idea of motivic cohomology goes back to P. Deligne, A. Beilinson and S. Lichtenbaum.
Most of the known and expected properties of motivic cohomology (predicted in [ABS87] and [Lic84]) can be divided into two families. The first family concerns properties of motivic cohomology itself - there are theorems concerning homotopy invariance, Mayer-Vietoris and Gysin long exact sequences, projective bundles, etc. This family also contains conjectures such as the Beilinson-Soule vanishing conjecture (Hp'q = 0 for p < 0) and the Beilinson-Lichtenbaum conjecture, which can be interpreted as a partial etale descent property for motivic cohomology. The second family of properties relate motivic cohomology to other known invariants of algebraic varieties and rings. The power of motivic cohomology as a tool for proving results in algebra and algebraic geometry lies in the interaction of the results in these two families; specializing general theorems about motivic cohomology to the cases when they may be compared to classical invariants, one gets new results about these invariants.
The idea of these lectures was to define motivic cohomology and to give careful proofs for the elementary results in the second family. In this sense they are complimentary to the study of [VSF00], where the emphasis is on the properties of motivic cohomology itself. In the process, the structure of the proofs force us to deal with the main properties of motivic cohomology as well (such as homotopy invariance). As a result, these lectures cover a considerable portion of the material of [VSF00], but from a different point of view. |