Algebraic K-Theory and Motivic Cohomology

Northwestern University, Evanston IL

Investigators

Abstract

Suslin proposes to investigate topics in motivic cohomology theory. These topics are of principal importance for the development of the cohomology theory of schemes and each has seen progress during the recent years in the works of Suslin and his collaborators. Firstly Suslin plans to eliminate the perfectness assumption in the basic requirements of motivic cohomology theory and to define the corresponding tensor triangulated category of motivic complexes over arbitrary fields. The second topic is an attempt to get a new and clearer proof of the main duality theorem which would work for schemes over arbitrary fields (and not only over fields of characteristic zero as the current proof does). Next Suslin plans jointly with A. Merkurjev to generalize the previous computation of the motivic cohomology of Severi-Brauer varieties to the case of generic splitting fields of higher symbols. Probably the most interesting and important part of the grant proposal is an attempt to compare the algebraic cobordism theory developed by Morel and Levin with the algebraic part of the cohomology theory constructed by Voevodsky. Here the plan is to use the machinery of framed sheaves developed by Voevodsky. The main objective of mathematics is to provide an accurate picture to the physical world or at least an appropriate approximation of that picture. From this point of view algebraic varieties are of principal importance, first they are relatively easy to understand since they are just defined by polynomial equations, next they usually give a rather accurate approximation to other shapes, most importantly they do appear naturally in quite a lot of subjects from theoretical physics to coding theory. That's why algebraic geometry - the theory of algebraic varieties is so important for the development and applications of mathematics. This project is devoted to the study of certain fundamental problems of motivic cohomology theory - a relatively new and very quickly developing branch of algebraic geometry. Geometry is blended with algebra and topology in this part of mathematics, ideas and methods to be used come equally from all these directions. As part of the broader impact of this grant proposal let me point out that I intend to involve graduate students into the work over some parts of this grant proposal thus allowing them to get into a fast developing and quite important field of mathematics.