The Block-Kato conjecture

A joint project by AIM, NSF, and Stanford University

During the last three years, the work of V. Voevodsky has revolutionized algebraic K- theory. Voevodsky has produced a proof of the so-called "Milnor conjecture", which identifies the Milnor K-theory of a field in terms of Galois cohomology. This work has also allowed the resolution of the Quillen Lichtenbaum conjectures for number fields at the prime 2. These problems have been outstanding for over twenty years, and their resolution is a giant step in this area. Perhaps as interesting as the results are the methods which Voevodsky and Suslin-Voevodsky have developed in order to attack the problem, in particular the A1-homotopy theory. This is a theory for algebraic vareities over an arbitrary base field which is quite analogous to stable homotopy theory for topological spaces. In the case of topological spaces, the spheres play a crucial role. Taking smash products with spheres is the stabilization procedure which provides the key simplification from unstable homotopy theory. In the case of varieties, there are actually two spheres. One is the affine line with a point removed, the other is the projective line. These two varieties have quite different properties, particularly from the point of view of K-theory, and their introduction provides a bigraded theory which is used in resolving the Milnor conjecture. This theory has also shed light on questions about motivic cohomology. It certainly appears that the surface has only been scratched in the study of this theory. It appears likely that it will have applications in many directions within algebraic geometry.

The odd primary analogue to the Milnor conjecture is called the ``Bloch-Kato'' conjecture. Its resolution would permit the proof of the odd primary Quillen-Lichtenbaum conjecture for number fields, and Voevodsky has outlined a program to prove it as well, by analogy with his methods for the Milnor conjecture. What remains to be done is the construction of varieties with appropriate topological behavior, within the A1-theory. M. Rost has been working on these constructions, and the it appears likely that the proof of this conjecture is now within range.

Recent work of G. Carlsson has proposed another approach to the study of the K-theory of fields, more along the lines of the standard descent arguments which motivated the Quillen-Lichtenbaum conjectures originally. The key object in the standard descent argument is the so called ``homotopy fixed point set'' of the action of the absolute Galois group G of a field F on the algebraic closure of F. In terms of the modern language of ring spectra and module spectra, one can view this action as a module structure on the K- theory spectrum of the algebraic closure of F over the KF group ring of G, where KF denotes the K-theory spectrum of F, a ring spectrum. Carlsson's new approach involves the introduction of a larger ring spectrum of operators, including the KF-group ring, which act on the K-theory of the algebraic closure of F. Descent definitely works over this ring spectrum of operators, and the question becomes how to analyze it. Carlsson proposes a model for this ring of operators in terms of the homological algebra of the representation ring of the group G. If the model is correct, it will provide a good homotopy theoretic model for the K-theory of a field, not only the K-groups. The model appears to be correct for a number of fields whose absolute Galois group is topologically cyclic. If this model is correct, it would shed additional light on the relationship of the K-theory of a field with constructions over the Galois group.

Results of the Project

This project involved the participation of 3 senior investigators, G. Carlsson, M. Rost,and V. Voevodsky, and two postdoctoral researchers, D. Sinha and D. Karageuezian. The project involved interaction between all members of the group at regular intervals (3times per week) during which various ideas related to the project were presented inlecture form.The goal of the project was to find the relationship between the Suslin-Voevodsky"motivic" approach to the conjectures of Quillen-Lichtenbaum and Bloch-Kato. This was achieved. Specifically, the following conclusions were drawn, as a result of theseinteractions.(A) The results predicted by the two approaches agree for free and free abelian profiniteabsolute Galois groups. That is, the homotopy groups of the model spectrum for algebraic K-theory constructed by Carlsson agree with the homotopy groups comingout of the motivic approach for these absolute Galois groups.(B) There is a map of spectral sequences from the Carlsson descent spectral sequence tothe Bloch Lichtenbaum spectral sequence, which should be an isomorphism ofspectral sequences if Carlsson's "ascent conjecture" is valid.(C) The Bloch-Kato conjecture can be interpreted as an isomorphism between certain derived functors over the representation ring of the absolute Galois group withcohomology groups of this group.(D) Carlsson's ascent conjecture holds for free and free abelian absolute Galois groups.

Other conclusions: The project made it clear that Carlsson's model, which involves a so-called "derived completion" of the representation ring of the absolute Galois group, mustfor non-abelian absolute Galois groups be interpreted in the context of Mackey functors.That is, the representation ring must be viewed as a ring object in the category of Mackeyfunctors (a "Green functor") rather than just as a ring, when the Galois groups are non-abelian. This suggests that it is imperative to develop the homological algebra in thiscategory (building on work of G. Lewis) to be able prove the conjecture in full generality. The development of equivariant stable homotopy theory in the motivic context is also ahigh priority, and will be addressed at a workshop which will be held at Stanford inAugust of 2000, with support from AIM, Stanford University, and the NSF.