Descent problems for K-theory of fields of Galois cohomological dimension one, by J.F. Jardine

Suppose that $\ell$ is prime bigger than 3, and suppose that $k$ is a field of characteristic not $\ell$ which has only $\ell$-primary algebraic extensions and has Galois cohomological dimension one with respect to $\ell$-torsion sheaves.

It is shown that the Lichtenbaum-Quillen conjecture is true for $k$ if and only if all even Postnikov sections $P_{2n}K/\ell$ of the mod $\ell$ K-theory spectrum $K/\ell$ satisfy descent with respect to Galois groups isomorphic to the group $\Bbb{Z}_{\ell}$ of $\ell$-adic integers. This result is proved in two ways: the first uses the lower central series for a finitely generated free pro $\ell$-group, while the second arises by first proving that the Lichtenbaum-Quillen conjecture for $k$ is equivalent to a finite descent property for all $P_{2n}K/\ell$.

This paper has been removed from these archives because it has appeared in Proceedings, Conference on Algebraic K-theory, Poznan, 1995, Contemporary Math. 199 (1996), 125-138.

J.F. Jardine <>