Andrei Suslin

Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients

In this paper we show that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of (smooth) varieties with finite coefficients is equivalent to the Bloch-Kato Conjecture, relating Milnor K-theory to Galois cohomology. The latter conjecture is known to be true in weight 2 for all primes [M-S] and in all weights for the prime 2 [V 3].

Algebraic K-theory and the norm-residue homomorphism

Recent results on the structure of the group K2 of a field and its connections with the Brauer group are presented. The K-groups of Severi-Brauer varieties and simple algebras are computed. A proof is given of Milnors conjecture that for any field F and natural number n > 1 there is the isomorphismRn,F:K2(F)/nK2(F)→∼nBr(F). Algebrogeometric applications of the main results are presented.

General linear and functor cohomology over finite fields

In recent years, there has been considerable success in computing Extgroups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories [F-L-S], [F-S]. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p. The reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of Ext-groups. For example, the existence of nonzero Ext-groups of positive degree is equivalent to the existence of objects which are not “direct sums” of simple objects. Indeed, a knowledge of Ext-groups provides considerable knowledge of compound objects. In the study of modular representation theory of finite Chevalley groups such as GLn(Fq), Ext-groups play an even more central role: it has been shown in [CPS] that a knowledge of certain Ext-groups is sufficient to prove Lusztig’s Conjecture concerning the dimension and characters of irreducible representations. We consider two different categories of functors, the category F(Fq) of all functors from finite dimensional Fq-vector spaces to Fq-vector spaces, where Fq is the finite field of cardinality q, and the category P(Fq) of strict polynomial functors of finite degree as defined in [F-S]. The category P(Fq) presents several advantages over the category F(Fq) from the point of view of computing Extgroups. These are the accessibility of injectives and projectives, the existence of a base change, and an even easier access to Ext-groups of tensor products. This explains the usefulness of our comparison in Theorem 3.10 of Ext-groups in the category P(Fq) with Ext-groups in the category F(Fq). Weaker forms of this theorem have been known to us since 1995 and to S. Betley independently

Algebraic K-theory

One gives a survey of the fundamental methods and results of the algebraic K-theory obtained in the past decade. One presents the basic constructions of the K-theory of rings and of the K-theory of exact categories. A special attention is given to the K-theory of schemes.

Algebraic K-theory and motivic cohomology

The general idea of motivic cohomology as a universal cohomology theory on the category of schemes goes back to Grothendieck. But it was not until 1982 that this general idea got a precise form. Around that time Beilinson formulated his famous conjectures.

The Beilinson spectral sequence for theK-theory of the field of real numbers

We construct the Beilinson spectral sequence in the case of the field of real numbers. This sequence connects the homotopy groups of the classifying spaces BU and BO for this situation. We give a proof the Bott periodicity theorem.