Bockstein Operations in Morava K-Theories

The Morava Ä> theories K (n)* ( ) (for p a prime and 0 < n < oo) form an important collection of multiplicative cohomology theories. In this paper we construct families of Operations reminiscent of the classical Bockstein operations in ordinary mod p cohomology and which are related both to the formal group theoretic ideas of Jack Morava and also the more recently discovered A^ structures on K (n) and related spectra E(n). 1980 Mathematics Subject Classification (1985 Revision): 55N22. §0 Introduction The Morava Ä^-theories K(ri)* () (for p a prime and 0 < n < oo) form a collection of multiplicative cohomology theories whose central röle 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 E (n). Given the underlying algebra involved in both of these areas we are tempted to speculate that there are äs yet undiscovered connections with crystalline cohomology and the de Rham complex — however, in the present work we merely hint at this. In § l we give an exposition of the theory of liftings of Lubin-Tate formal group laws from algebras over K(ri)^ to local Artinian rings A for which the maximal ideal m satisfies m = 0. In particular, such deformations give rise to automorphisms of The first named author would like to acknowledge the support of the Science and Engineering Research Council, Plnstitute des Hautes fitudes, and the Universities of Manchester and Bern whilst this research was in progress. 544 A. Baker, U. Würgler 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 Ä^-theory. In §2 we define our family of operations Qk: K («)*() -> K(n)* + 2 p k ~ l ( ) for 0 < k < n — l, and verify that they are K(n)^. derivations, determined upon the spaces CP°° and the skeleton L = (BZ/p)~\ 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 E (n) of the form Qi'. Efnj/IÜ -> TM + ( ) where v = VQ° · · · I? £LV E /„ O E(n)*' These are determined by their effect on both CP°° and certain skeleta of the classifying spaces B^jp. Closely related to these operations is a tower of A^ module spectra over E (n) * 4K(n) = E(n)/I„ 4E(n)/I 4. 4£(*)//* and for each spectrum X an associated spectral sequence with E1 (X) a direct sum of copies of K(n)* (X\ d t a direct sum of our generalised Bocksteins and converging to £(n)*(X)', thus, the functors K(n)*() and £(«)*() determine each other. The authors would like to thank Mike Hopkins, John Hunton, Nick Kühn, Jack Morava and Nigel Ray for helpful discussions on the material in this paper. §1 Liftings of height n Lubin-Tate formal group laws Let p be an odd prime and 0 < n < oo. Recall that the n th Morava £theory at the prime p is a complex oriented multiplicative cohomology theory K(n)*() whose coefficient ring is the graded field where vn e K(ri)2(pn _ 1}. The spectrum K(n) representing this theory is a commutative ring spectrum. Moreover, the canonical orientation class x e K(n) (C P) has an associated formal group law F over K(n)^ whose /? -series is given by and is therefore a height n Lubin-Tate group law [6]. In general, a height n Lubin-Tate group law over a (graded) Fp algebra A^ is a formal group law Fover A^ which has /j-series of the from with u a unit in ; the canonical /? -typification of F then has the same /? -series äs Fand is induced by a homomorphism (of ¥p algebras) K(n)+ -» A^ sending vn to u. (For those unfamiliar with Lubin-Tate theory this will suffice äs a definition.) Bocksteins in Morava '-Theories 545 The f ll algebra of stable operations in ^T(«)-theory was determined in [17], [15] and [16]; its most convenient description is s the K(n)^ dual of the cooperation algebra K(n)^(K(n)) whose structure we now recall. Theorem (1.1). There is an isomorphism of K(n)^. algebras Κ(ή)^(Κ(ή)) £ Σ(η\ ®K(n)^K(n)Si 0 < * < « 1) where Z(«)s|e = ^(/2)sie(/i: l < i) subject to relations and with \t{\ = 2(p* 1), \at\ = 2p 1. In fact, the subalgebra £(«)* can be identified with the image of the K(ri) homology of a spectrum E (n); in the literature, it is also frequently denoted K(n\ K(n), however this is likely to be confused with K(n)^ (K(n)) and so we avoid such notation. Recall from [10] the ring spectrum E(n) for which there are canonical morphisms of ring spectra BP > E(n) -? —+ K(n). Proposition (1.2). The map Q^: K(n)^(E(n)) -> K(n)^(K(n)) is a monomorphism of )sjc algebras with image equal t o Corollary (1.3). There is an isomorphism of K(n)^ modules K(n)*(E(n)) s Now K(n)^ (K(ri)} actually has the further structure of a Hopf algebra over K(n)^ (see [1] and [9]). The coproduct is determined by our next result. We use the symbol ]T to denote summation with respect to the formal group law F. Proposition (1.4). For a suitable choice of the generators tit a{ we have jpJC(n) /? K(n) χ ^(tk)x)= Σ (t^tfx^) 0 < k 0 < i and