Bockstein Operations in Morava K-theories

§0 Introduction. The Morava K-theories K(n)∗( ) (for p a prime and 0 < n < ∞) form a collection of multiplicative cohomology theories whose central rôle in homotopy theory is now well established. However, even though they have been intensively studied there are still many aspects of their structure which remain undeveloped. In this paper we give a construction for families of operations reminiscent of the classical Bockstein operations in ordinary mod p cohomology which is related both to the formal group theoretic techniques exploited by Jack Morava [8] and also the more recently discovered A∞ structures on K(n) and the related spectra Ê(n). Given the underlying algebra involved in both of these areas we are tempted to speculate that there are as yet undiscovered connections with crystalline cohomology and the de Rham complex–however, in the present work we merely hint at this. In §1 we give an exposition of the theory of liftings of Lubin-Tate formal group laws from algebras over K(n)∗ to local Artinian rings A∗ for which the maximal ideal m satisfies m = 0. In particular, such deformations give rise to automorphisms of the group scheme of p th roots of 0 for the lifted formal group law and in turn this produces a sequence of n derivations which are essentially our Bocksteins in K(n)-theory. In §2 we define our family of operations Qk : K(n)∗( ) −→ K(n)∗+2pk−1( ) for 0 6 k 6 n− 1, and verify that they are K(n)∗ derivations, determined upon the spaces CP∞ and the skeleton L = (BZ/p)[2pn−1]. This shows also that they agree with earlier constructions. In §3 we construct higher order Bocksteins which correspond to morphisms of A∞ module spectra over Ê(n) of the form