Algebraic K-theory

Researchers

Research topics

The first algebraic K-group that was defined, K0, was a group in which two modules over a ring could be equivalent "modulo short exact sequences", a generalization of the notion of dimension or rank. Since then higher algebraic K-groups have been defined, both for rings and for algebraic varieties. They are related to algebraic cycles, the homology of the general linear group, and form a universal cohomology theory (motivic cohomology). The algebraic K-theory of a variety over a number field has deep connections with the arithmetic of the variety.

Algebraic K-theory is a very active field, one of the most important recent developments being Voevodsky's work that earned him a Fields medal.

We describe the research areas in algebraic K-theory in the department below.

Algebraic K-theory of fields and curves (R. de Jeu) Quillen defined algebraic K-groups in great generality, but those groups are quite difficult to make explicit. Goncharov, stimulated by work of Zagier, conjectured how one could describe the K-groups of fields and those of curves, in terms of complexes of free abelian groups with certain relations. Parts of those conjectures have been proved so they (and classical results) can be used to find explicit examples of K-groups for fields and curves. In case the curve is defined over a number field then one can also try to apply this to the Beilinson conjectures mentioned below.

Basic reading:

The Beilinson and Lichtenbaum conjectures (H. Gangl, R. de Jeu) The Beilinson conjectures state that, for a projective variety over a number field, there should be a relation between the regulator of an algebraic K-group (a certain real number) and a value of a certain L-function: their quotient should be a rational number. Those conjectures have been proved in full only when the variety is a point, but there are also many results for other varieties. There is a p-adic analogue of the conjecture, where the regulator and the L-function are both p-adic. This is a very new area of research, with many open problems.

The Lichtenbaum conjecture gives an interpretation of the rational number in the Beilinson conjecture in terms of torsion in K-groups. This is analogous to the residue formula for the zeta function of a number field, which contains the order of the ideal class group and the number of roots of unity in the number field. Assuming the Lichtenbaum conjecture one can devise experiments that lead to predictions about the torsion in certain K-groups.

Basic reading:

Algebraic K-theory, algebraic cycles, and the homology of the general linear group (H. Gangl, R. de Jeu) There is another way of defining algebraic K-theory, through Spencer Bloch's higher Chow groups. This approach is more explicit, and can be linked explicitly with the homology of the general linear group, making this approach easier for obtaining explicit information about K-groups. For example, it is still a big challenge to compute the higher K-groups even for the ring of integers. This would be a practicable task (at least for Kn with n not divisible by 4) if one had a proof of Milnor's conjecture for all primes (Voevodsky proved it for the prime 2). Still, this approach gives quite a bit of information and allowed to prove that K5(Z)=Z and K6(Z)=0. As another application one can give an algorithm for finding a non-zero element in K3 for any imaginary quadratic number field. Finally, using the higher Chow groups one can give cycles related to polylogarithms and multiple polylogarithms that can be directly related to Goncharov's motivic iterated integrals.

Basic reading:

Galois module structure, and knot modules (S. Wilson) Those two topics have links with algebraic K-theory, but they are described in more detail on the algebraic number theory page.

Selected publications

Algebraic K-theory of fields and curves

The Beilinson and Lichtenbaum conjectures

Algebraic K-theory, algebraic cycles, and the homology of the general linear group

Postgraduate students

PhD projects

The Beilinson conjectures; algebraic K-theory of fields and curves; algebraic cycles and the homology of the general linear group (Herbert Gangl, Rob de Jeu) There are many projects possible in these directions with either Herbert Gangl or Rob de Jeu (or both), some of which cross over into or are closely linked with Arithmetic Algebraic Geometry. We mention:

Galois module structure, and knot modules (S. Wilson) For potential problems in Galois module structure and knot modules that have links with algebraic K-theory please look here.