Towards the finiteness of π*LK(n)S0

Abstract

Let G be a closed subgroup of the nth Morava stabilizer group S n , n 2, and let E hG n 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 0 n of S n , and z is non-torsion in the quotient of S 0 n by its center, we prove that the E h z n -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 0 n not contained in the center, then G contains an open subnormal subgroup U such that the mod(p) homotopy of E h(U ×F × p ) n is of essentially finite rank.