Title: Towards the finiteness of the homotopy groups of the K(n)-localization
of S^0.
Author: Ethan S. Devinatz
E-mail: devinatz@math.washington.edu
Abstract: Let G be a closed subgroup of the nth Morava stabilizer group
S_n, n>1, and let E_n^{hG} denote the continuous homotopy fixed point
spectrum of Devinatz and Hopkins. If G=, the subgroup topologically
generated by an element z in the p-Sylow subgroup S_n^0 of S_n, and z is
non-torsion in the quotient of S_n^0 by its center, we prove that the
E_n^{h}-homology of any K(n-2)-acyclic finite spectrum annihilated by
p is of essentially finite rank. (The definition of essentially finite
rank is given in the paper.) We also show that the units in the
coefficient ring of E_n which are fixed by z are just the units in the
Witt vectors with coefficients in the field of p^n elements. If n=2 and
p>3, we show that, if G is a closed subgroup of S_n^0 not contained in
the center, then G contains an open subnormal subgroup U such that the
mod(p) homotopy of E_n^{hV} is of essentially finite rank, where V is
the product of U with the units in the field of p elements.