Mixed Tate Motives with Finite Coefficients and the Koszulity Conjecture
Leonid Positselski, Institute for Advanced Study, <posic@ias.edu>
The Beilinson--Lichtenbaum conjecture describes the Ext spaces
between the Tate motives with finite coefficients Z/m(i) in terms
of Galois cohomology of the basic field (the conjecture connecting
Milnor K-theory with the Galois cohomology is a part of this).
Unlike in the classical rational coefficients situation, the most
natural candidate for the category of "mixed" Tate motives with
finite coefficients is never an abelian category, but is endowed
with a structure of an exact category in Quillen's sense. This
exact category is equivalent to a certain category of filtered
modules over the Galois group; the triangulated category of
Tate motives is equivalent to the derived category of this exact
category if and only if the the Galois cohomology algebra has the
so-called Koszul property (assuming that certain roots of unity
are present). On the other hand, if the Milnor algebra modulo m
is Koszul, then the Beilinson--Lichtenbaum conjecture follows,
according to the result of an earlier paper by A.Vishik and me
(and the Suslin-Voevodsky theorem).