Gunnar Carlsson

Abstract:   Some thirty years ago, D. Quillen introduced the higher K-groups of rings. These groups are defined as the homotopy groups of a topological space, whose homotopy type carries information about various topological, geometric, and arithmetic objects. For instance, one can obtain information about automorphism groups of manifolds when one considers the K-theory of group rings. There is also the expectation that when the algebraic K-groups of fields are understand, they will carry information about arithmetic and algebraic geometric questions.

We do not yet understand the K-theory of fields, although there has been a great deal of interesting progress. One approach has been through a "descent" procedure, using the (known, by a theorem of Suslin) K-theory of algebraically closed fields together with the action of the absolute Galois group of a field on the K-theory of its algebraic closure. This approach produces a spectral sequence that conjecturally converges to the K-theory of the field. This convergence conjecture is the Quillen-Lichtenbaum conjecture. If true, it would provide information about the K-groups in dimensions larger than the cohomological dimension of the field. In the last few years, there has been spectacular progress by Voevodsky and Suslin-Voevodsky using so-called "motivic" techniques. This method allows explicit computation in a number of cases, and permits the proof of a long-standing conjecture of Milnor on a portion of the K-groups. The relationship of this motivic theory with properties of the Galois group is not as direct as one would like.

The purpose of this talk is to outline a third procedure, which produces a conjectural model for the homotopy type of the algebraic K-theory of the space representing the algebraic K-theory of the field. This model is built out of linear representations of the absolute Galois group of the field. If true, it would compute all the algebraic K-groups of the field in terms of an object called the "derived representation ring" of the absolute Galois group. We will describe this model, show how it relates to K-theory, and relate properties of the absolute Galois group with the well known conjectures in the field, namely Quillen-Lichtenbaum's conjecture and the conjecture of Bloch-Kato.