Towards the finiteness of π* L K (n) S0

Let G be a closed subgroup of the nth Morava stabilizer group S-n, n >= 2, and let E-n(hG) denote the continuous homotopy fixed point spectrum of Devinatz and Hopkins. If G = < z >, 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 < z >) -homology of any K(n - 2)(*) -acyclic finite spectrum annihilated by p is of essentially finite rank. We also show that the units in E-n* 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 >= 5, 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(p)h(U x F(x)) is of essentially finite rank. (c) 2008 Elsevier Inc. All rights reserved.