A∞ Structures on Some Spectra Related to Morava K-theories

Let p denote an odd prime. We show that the spectrum [ E(n), the In-adic completion of Johnson and Wilson’s E(n), admits a unique topological A∞ structure compatible with its canonical ring spectrum structure. Furthermore, the canonical morphism of ring spectra [ E(n) −→ K(n) admits an A∞ structure whichever of the uncountably many A∞ structures of A. Robinson is imposed upon K(n), the n th Morava K-theory at the prime p. We construct an inverse system of A∞ module spectra over [ E(n) · · · −→ E(n)/I n −→ E(n)/I n −→ · · · −→ E(n)/In = K(n) for which holim ←− k E(n)/I n ' [ E(n). §0 Introduction. Recently, A. Robinson has described a theory of A∞ ring spectra, their module spectra and the associated derived categories (see [9], [10], [11], [12]). As a special case, in [12] he showed that at an odd prime p the n th Morava K-theory spectrum K(n) admits uncountably many distinct A∞ structures compatible with its canonical multiplication. The principal result of the present work is to show that Ê(n), the (Noetherian) In-adic completion of the spectrum E(n) defined by D. C. Johnson and W. S. Wilson, admits a unique topological A∞ structure compatible with its canonical ring spectrum structure; moreover, the canonical morphism of ring spectra Ê(n) −→ K(n) can be given the structure of an A∞ morphism whichever of Robinson’s A∞ structures we take. As an application, we construct an inverse system of A∞ module spectra over Ê(n) · · · −→ E(n)/I n −→ E(n)/I n −→ · · · −→ E(n)/In = K(n)