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EXOTIC MULTIPLICATIONS on MORAVA K-THEORIES and THEIR LIFTINGS Andrew Baker Manchester University Introduction. the Purpose of T

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EXOTIC MULTIPLICATIONS on MORAVA K-THEORIES and THEIR LIFTINGS Andrew Baker Manchester University Introduction. the Purpose of T EXOTIC MULTIPLICATIONS ON MORAVA K-THEORIES AND THEIR LIFTINGS Andrew Baker Manchester University Abstract. For each prime p and integer n satisfying 0 < n < , there is a ring spectrum K(n) called the n th Morava K-theory at p. We discuss exotic multiplications∞ upon K(n) and their liftings to certain characteristic zero spectra E[(n). Introduction. The purpose of this paper is to describe exotic multiplications on Morava’s spectrum K(n) and certain “liftings” to spectra whose coefficient rings are of characteristic 0. Many of the results we describe are probably familiar to other topologists and indeed it seems likely that they date back to foundational work of Jack Morava in unpublished preprints, not now easily available. A published source for some of this is the paper of Urs W¨urgler[12]. We only give sketches of the proofs, most of which are straightforward modifications of existing arguments or to be found in [12]. For all background information and much notation that we take for granted, the reader is referred to [1] and [7]. I would like to express my thanks to the organisers of the Luminy Conference for providing such an enjoyable event. Convention: Throughout this paper we assume that p is an odd prime. 1 Exotic Morava K-theories. Morava K-theory is usually defined§ to be a multiplicative complex oriented cohomology theory K(n)∗( ) which has for its coefficient ring F 1 K(n) = p[vn, vn− ] ∗ where vn K(n)2pn 2, and is canonically complex oriented by a morphism of ring spectra ∈ − σK(n) : BP K(n) −→ Key words and phrases. Morava K-theory, ring spectrum. The author would like to acknowledge the support of the Science and Engineering Research Council, and the Universities of Manchester and Bern whilst this research was in undertaken. Published in Ast´erisque 191 (1990), 35–43. [Version 9: 1/09/1999] Typeset by -TEX AMS 1 2 ANDREW BAKER which on coefficients induces the ring homomorphism σK(n) : BP K(n) ∗ ∗ −→ ∗ K(n) vn if k = n, σ (vk) = ∗ 0 otherwise. Here we have BP = Z(p)[vk : k > 1] with vk BP2pk 2 being the k th Araki generator, defined using the formal group∗ sum ∈ − BP k [p] X = v Xp . BP X k 06k As a homomorphism of graded rings, we can regard σK(n) as a quotient homomorphism ∗ K(n) 1 1 σ : vn− BP vn− BP / n = K(n) ∗ ∗ −→ ∗ M ∼ ∗ 6 1 1 where n = (vk : 0 k = n) / vn− BP is a maximal graded ideal of the ring vn− BP . Thus we can interpretM K(n) as a6 (graded) residue∗ field for this maximal ideal. ∗ ∗ Clearly this ideal n is not the only such maximal ideal and we can reasonably look at other examples and askM if the associated quotient (graded) fields occur as coefficient rings for cohomology theories is an analogous fashion. Notice that n contains the invariant prime ideal K(nM) In = (vk : 0 6 k 6 n 1) and the formal group law F therefore has height n. One way to construct K(n)-theory− is by using Landweber’s Exact Functor Theorem (LEFT) [6] in its modulo In version [14]; this allows us to make the definition K(n)∗( ) = K(n) P (n) P (n)∗() ∗ ⊗ ∗ on the category of finite CW spectra CWf , where P (n) is the spectrum for which P (n) = BP /In. ∗ ∗ 1 We thus concentrate on maximal ideals 0 / vn− BP containing the ideal In. We then have M ∗ 1 Theorem (1.1). Let 0 / vn− BP be a maximal (graded) ideal containing In. Then there is M ∗ f a unique multiplicative cohomology theory K( 0)∗(), defined on CW , for which there is a multiplicative natural isomorphism M K( 0)∗() = K( 0) P (n) P (n)∗() M ∼ M ∗ ⊗ ∗ 1 and where K( 0) = vn− BP / 0 is the coefficient ring. M ∗ ∗ M The proof is immediate using LEFT. Of course, if there is an isomorphism of graded rings, K( 0) = K( 00) , then we need to M ∗ ∼ M ∗ decide if the two theories arising from 0 and 00 can be naturally equivalent. M M 1 Theorem (1.2). Let 0, 00 / vn− BP be graded maximal ideals containing In and let M M ∗ f : K( 0) K( 00) be an isomorphism of graded rings. Then there is a natural isomor- M ∗ −→ M ∗ phism of multiplicative cohomology theories on CWf , f : K( 0)∗() K( 00)∗() M −→ M 1 1 e vn− BP / 0 vn− BP / 00 extending f if and only if the the formal group laws f F ∗ M and F ∗ M are strictly ∗ isomorphic over the ring K( 00) . M ∗ The main observation required to prove this result is that these two formal group laws are associated to two complex orientations induced by the composite of the morphisms of ring spectra 1 BP v− BP K( 00). −→ n −→ M EXOTIC MULTIPLICATIONS ON MORAVA K-THEORIES AND THEIR LIFTINGS 3 Corollary (1.3). The theories K( 0)∗() and K( 00)∗() are representable by ring spectra M M K( 0) and K( 00), unique up to canonical equivalence in the stable category. Moreover, K( 00) M M 1 M vn− BP / 0 and K( 00) are equivalent as ring spectra if and only if the formal group laws f F ∗ M M1 ∗ vn− BP / 00 and F ∗ M are strictly isomorphic over the ring K( 00) . M ∗ Let us now consider such ring spectra K( 0) where K( 0) ∼= K(n) as graded rings. By a result from [12] (see also [5]) these are preciselyM the ring spectraM having∗ the∗ homotopy type of K(n) (not necessarily multiplicatively). Thus, such ring spectra are classified to within equivalence as ring spectra by the set of maximal ideals 0 modulo strict isomorphism of the associated formal M group laws over K( 0) . We call the multiplicative cohomology theory associated to such a ring spectrum an exoticM Morava∗ K-theory. Let us consider such a spectrum K( 0), where K( 0) ∼= K(n) as graded rings. Then we have the following modification of a resultM of [13], M ∗ ∗ Theorem (1.4). As an algebra over K(n) , we have ∗ > K( 0) (K( 0)) = K( 0) (tk0 : k 1) ΛK(n) (a0, . , an0 1) M ∗ M ∼ M ∗ ⊗ ∗ − k k where t0 = 2p 2, a0 = 2p 1, and there are polynomial relations of the form | k| − | k| − n p (pk 1)/(p 1) tk0 vn − − tk0 = hk(t10 , . , tk0 1) − − over K(n) . ∗ The symbol ΛK(n) denotes an exterior algebra over K(n) on the indicated generators. To prove this result,∗ we rework the proof for the case of ∗K(n) (see [13], [7]) and define the generators tk0 by using the identity K( 0) s r+s K( 0) r r+s M (v t p Xp ) = M (t v p Xp ) X s0 r0 X r0 s0 r>0 r>0 s>n s>n where K( 0) s M p [p] K( ) X = (vs0 X ). F M0 X s>n The exterior generators are similarly derived. We can interpret the algebra > K( 0) (tk0 : k 1) M ∗ K( 0) as representing the strict automorphisms of the group law F M , in a way analogous to the case of K(n) (see [7]). 2 Liftings of exotic Morava K-theories. Recall that there is a ring§ spectrum E(n) for which E(n)∗() = E(n) BP BP ∗() ∼ ∗ ⊗ ∗ on CWf . Here we have 1 > E(n) = vn− BP /(vn+k : k 1). ∗ ∗ 4 ANDREW BAKER We showed in joint work with Urs W¨urgler(see [4]) that the Noetherian completion E[(n) of E(n), characterised by the formula [ ∗ k E(n) ( ) = lim E(n)∗()/InE(n)∗() ←−k f \1 1 on CW , is a summand of the Artinian completion vn− BP of vn− BP and indeed there is a product splitting \ v 1BP Σ2d(v)E[(n) n− Y ' v of (topological) ring spectra for v ranging over a suitable indexing set and d a suitable numerical function. The algebra underlying the proof is intimately related to liftings of Lubin–Tate group laws, i.e. group laws over Fp algebras classified by homomorphisms from K(n) . In this section ∗ we describe the analogous situation for liftings of exotic Morava K-theories of the form K( 0) as in 1. M § Now if K( 0) = K(n) as a ring, then the natural homomorphism ∗ ∼ ∗ M 1 θ : vn− BP K( 0) M0 −→ M ∗ given by r(k/n) ckvn if n k θ (vk) = ( | M0 0 otherwise for k > n and integers ck. Here, the numerical function r is given by (pmn 1) r(m) = − . (pn 1) − and we set ck = 0 = r(k/n) whenever k is not divisible by n or k = 0. Now consider an ideal of the form r(k/n) 1 J = (vk ckvn + gk : k > n) 0 / vn− BP − ⊂ M ∗ and satisfying J + In = 0. M Here gk 0 are certain elements chosen so that the last condition holds. Set E(J) = 1 ∈ M ∗ vn− BP /J. We now∗ define a cohomology theory E(J)∗( ) = E(J) BP BP ∗() ∗ ⊗ ∗ on CWf . This is a cohomology theory by Landweber’s Exact Functor Theorem, and is moreover multiplicative and canonically complex oriented by the obvious natural transformation BP ∗() E(J)∗(). −→ Furthermore there is a canonical multiplicative natural transformation E(J)∗() K( 0)∗(). −→ M We can form the Noetherian completion [ ∗ k E(J) ( ) = lim E(J)∗()/InE(J)∗() ←−k 1 1\ and also the Artinian completion of vn− BP with respect to the maximal ideal 0, vn− BP ( 0) (see [4]).
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