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
theory. Voevodsky has produced a proof of the so-called "Milnor
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
is a giant step in this area. Perhaps as interesting as the results are the
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
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''
Its resolution would permit the proof of the odd primary
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
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
Quillen-Lichtenbaum conjectures originally. The key object in the
argument is the so called ``homotopy fixed point set'' of the action of the
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,
denotes the K-theory spectrum of F, a ring spectrum. Carlsson's new
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
ring of the group G. If the model is correct, it will provide a good
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.