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.

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.