THE MORAVA K-THEORY AND BROWN-PETERSON COHOMOLOGY OF SPACES RELATED TO BP TAKUJI KASHIWABARA AND W. STEPHEN WILSON Abstract.We calculate the Morava K-theory of the spaces in the Omega spectra for BP. They fit into an exotic array of short and long exact sequences of Hopf algebras. We apply this to calculate the p-adically co* *mpleted Brown-Peterson cohomology, as well as all of the intermediary cohomology theories, E, of these spaces. We give two descriptions of the answer, bo* *th of which turn out to be surprisingly nice. One part of our first descriptio* *n is just the image in the E cohomology of the corresponding space in the Ome* *ga spectrum for BP, which is as big as it could possibly be and which we sh* *ow how to calculate. The other part is just the E cohomology of several cop* *ies of Eilenberg-MacLane spaces, something which is already known. Our second description is inductive and gives us a new way of looking at the Brown- Peterson cohomology of Eilenberg-Mac Lane spaces. The Brown-Comenetz dual of BP shows up in our calculations and so we take up the study of this spectrum as well. It was already known that the Morava K-theory of * *the spaces in the Omega spectrum for the Brown-Comenetz dual of BP made it look like a product of Eilenberg-Mac Lane spaces and we find, somewha* *t to our surprise, that the same is true for the BP cohomology. In order to s* *tate our answers we set up the foundations for the category of completed Hopf algebras. November 22, 2000 1.Introduction The purpose of this paper is to understand, (in particular, to calculate) var* *ious generalized cohomology theories of the spaces in the Omega spectra for BP , where BP is the spectrum with coefficient ring BP * ' Z(p)[v1; v2; : :;:vq]; and the degree of vi is 2(pi - 1), see [Wil75] and [JW73 ]. Recall BP * ' Z(p)[v1; v2; : :]:, and let Im = (p; v1; : :;:vm-1 ). Most of our paper is s* *pent working with Morava K-theory but our main application is easy to state and gives various cohomologies of the spaces BP__r. We usually work with the p- adically completed version of BP cohomology, BPp^*(-), so that we can avoid the problems associated with phantom maps. We consider this theory, the theories P (m)*(-) with P (m)* ' BP *=Im [JW75 ], and the theories E(m; n)*(-), with E(m; n)* ' v-1nBP *=Im where 0 m n, 0 < n [RWY98 ]. When m = 0 we always mean the p-adically complete version of the theory (unless explicitly stated otherwise) and we can think of P (0) as BPp^. Note that when m = n, E(n; n) = K(n), the n-th Morava K-theory. We let E denote any of these theories. For all of the spaces we consider, we show that there is an E-cohomology K"unne* *th isomorphism (using a completed tensor product). This is not normally the case. Since all of our spaces are also homotopy commutative H-spaces we have all of t* *he 1 2 Takuji Kashiwabara and W. Stephen Wilson maps associated with Hopf algebras (replacing tensor product with completed ten- sor product). The topology (on the cohomology groups) prevents this from being an abelian category but we can still talk about kernels, cokernels and short ex* *act sequences of completed Hopf algebras. We let {G_*} denote the spaces in the Omega spectrum for a spectrum G. Let g(q) = 2(pq+1- 1)=(p - 1) = 2(1 + p + p2 + . .+.pq) throughout the paper where p is the prime associated withPE and the spectrum BP . Let I = (i1; i2; : :;* *:iq) with ik 0 and let d(I) = 2ik(pk - 1). We give two quite different descriptio* *ns of E*(BP__*) and we need to define a special map for one of them. We have: q Y q-1 Y BP__r_____- BP__r-|v_|____- BP__r-|v |-|v: | 0_r-|vi|to BP__r-|vi we use 1|-|vi2| the trivial map unless i is i1 or i2. If i = i1 we use (-1)i2vi2and if i = i2 w* *e use (-1)1+i1vi1. The main application of our work is the following. Recall that our kernel and cokernel are as completed Hopf algebras. Theorem 1.1. Let r = g(q) + k and kI = k - d(I). Let E be any of BPp^*(-), P (m)*(-), or E(m; n)*(-) where 0 m n and 0 < n. Then E*(BP__r) is P (m)* flat for the category of P (m)*P (m) modules which are finitely presented over P (m)* and algebraically determined by its isomorphism to both: Description 1: "O E*(K(Z (p); q + 2 + kI)) d(I)q ____r+|vi|; and, when k > 0, Description 2: O E*(K(Z(p); q + 2 + k))d 8 9 < [O *q-1 "O = Coker : E*(BP__r-|v)|---- E*(BP__r-|v |-|v):| 0 0. They show it to be equivalent to vi being monomorphic on Morava K-theory and Brown-Peterson cohomology 3 M=(vm ; : :;:vi-1)M for all m i (with v0 = p). Thus our E cohomology behaves quite nicely. There is no condition on k in the first description. When k 0 the first term* * with the Eilenberg-Mac Lane spaces goes away and we are left with only the second te* *rm which we knew anyway because BP__rhas no torsion and splits off of BP__rin t* *his range. For arbitrary k, this second term is just the image E*(BP__r) ! E*(BP* *__r) and we have b v*i[O (1.2) E*(BP__r) -! E*(BP__r) ____- E*(BP__r+|vi|) i>q is "exact," as completed Hopf algebras, at the middle term. Because BP__ris a space with no torsion, its E cohomology is as nice as can be and our second part is contained in it. As for the actual evaluation of the kernel, in principle, t* *he work of [RW77 ] gives all necessary information. The formulas in [BJW95 ] make that principle a reality. This second part of our first description might reasonabl* *y be expected, or at least hoped for, because it makes some sense. In fact, this sho* *ws that it is as big as possible because the composition vi (1.3) BP__r+|vi|__-BP_r____-BP__r is trivial for i > q. The first part of our first description is, however, a surprise. It is nice b* *ecause we know the E cohomology of these Eilenberg-Mac Lane spaces since they are com- pletely described in [RWY98 , Theorem 1.14], where, together with [Wil99a] and [Kasb ], the ability to go from Morava K-theory to Brown-Peterson cohomology was developed. This first part of this answer is intimately tied up with the Br* *own- Comenetz dual of BP . Unstably it is a finite Postnikov system which maps to BP__rand carries all of the homotopy of the first part of our result. The Mo* *rava K-theory of such a space always splits up in this manner, [HRW98 ], but one do* *esn't get the same for the Brown-Peterson cohomology and the others in general. In th* *is special case, it does split and although none of the homotopy from this space s* *hows up in BP__r, much of its BP cohomology does. The homotopy of the first part shows up in a second space which BP__rmaps to and which realizes the sec- ond part of our first description. The point here is that there is some interes* *ting topology underlying the first description. The second description is an inductive description. For k 0 we have known the answer (second part of first description) for decades. The cokernel part co* *mes complete with a description of the maps v*iand v*iis trivial on the Eilenberg- Mac Lane part. Related, we have a description of the cohomology of Eilenberg- Mac Lane spaces. Corollary 1.4.For k 0, and E as in Theorem 1.1, (i) E*(K(Z(p); q + 2 + k)) ' E*(BP__g(q)+k)=(v*1; v*2; : :;:v*q): (ii) E*(K(Z=(pc); q + 1 + k)) ' E*(BP__g(q)+k)=(pc*; v*1; v*2; : :;:v*q): 4 Takuji Kashiwabara and W. Stephen Wilson This generalizes the k = 0 version of this from [RWY98 , Theorem 1.14] where the use of v1; v2; : :;:vq-1 was found to be unnecessary. For k > 0 they become necessary. Remark 1.5.If we are looking only at the theories E = E(m; n), then, since K(n)*(K(Z=(pc); q + 1 + k)) is finitely generated and free over K(n)*, the proo* *fs in [RWY98 ] give us that E*(K(Z=(pc); q + 1 + k)) is also finitely generated and * *free over E*. Although free algebraically, the topology on each summand can be quite different. To simplify our notation a bit we remind the reader that g(q)P= 2(pq+1- 1)=(p -=1)2(1+p+p2+. .+.pq) and we let gffi(q) = g(q)-(q+1) = q P q i P q i=0(|vi| - 1) and gv(q) = g(q) - 2(q + 1) = i=02(p - 1) = i=0|vi|. These numbers are used throughout. The spaces in the Omega spectrum for the Brown-Comenetz dual of BP , IBP , arise naturally in our study and so we turn our attention towards them now. The connection to our work is the existence of a stable cofibration: (1.6) -gffi(q)IBP -! BP -! LqBP : Ln is Bousfield localization, [Bou79 ], with respect to the theory E(n) (our E(* *0; n), from [JW73 , Remark 5.13, p. 347]). This is explicit in Mahowald and Rezk's wor* *k, [MR ]. In our preliminary Section 3, we review Ravenel's functors, Nn and Ln on the stable category, [Rav84 ], Mahowald and Rezk's work, Mahowald and Sadofsky's work, [MS95 ], and generalize this to fit our needs. In traditional notation, (1.7) ss*IBP ' gv(q)BP *=(p1 ; v11; v12; : :;:v1q): Rephrased, the homotopy group ss-jIBP is a finite number of copies of Q=Z(p* *). The number of copies of Q=Z(p)is the same as the Z(p)rank of ssjBP , which is zero, by "sparseness", unless 2(p - 1) divides j. Strictly for degree reasons w* *e get a split short exact sequence: (1.8) 0 -! ss*BP -! ss*LqBP -! ss*-gffi(q)+1IBP -! 0: For any space X (all of our spaces are infinite loop spaces), we let X(s)deno* *te the s-connected cover of X and let X[s]denote the corresponding space with the same homotopy groups as X up to degree s so that we have a fibration X(s)! X ! X[s]. We also let X(t;s]be X(t)[s]= X[s](t). Observe that ssk(X(t;s]) = ssk(X) if k 2* * (t; s] and 0 otherwise. Note that for r > s, BP__ris the same as BP__(s)r. Because of this connection, equation (1.6), we have an interest in the Brown- Comenetz dual of BP , in particular, with the spaces IBP__rand IBP__(s* *)r. Note that since the top homotopy group of IBP__ris in degree r, it is a fini* *te Postnikov system. As such, its Morava K-theory is the same as if it were a prod* *uct of Eilenberg-Mac Lane spaces with the same homotopy, [HRW98 ], and these are understood by [RW80 ]. What was unanticipated is that the same is true for all * *of the other theories we use. The homology, H*(-; Z(p)), of these spaces is not of* * finite type over Z(p). Although that does not really present a serious problem for us * *it is perhaps easier to use a finite type approximation, i.e. the fibre of multiplica* *tion by pc, which gives us a stable triangle: c) pc (1.9) IZ=(p BP -! IBP ____-IBP : Morava K-theory and Brown-Peterson cohomology 5 This new spectrum is just the Brown-Comenetz dual of BP modulo pc.c This gives us a short exact sequence on homotopy groups. Not only is IZ=(p_)BP__ra finite Postnikov system but all of the homotopy groups are finite, i.e. we repl* *ace each Q=Z(p)by a Z=(pc). Much is known about our cohomology theories applied to finite Postnikov sys- tems. Theorem 1.10. Let E be as in Theorem 1.1 and let X be a homotopy commutative H space which has a finite Postnikov system. Then E* - E*(X(s)) - E*(X) - E*(X[s]) - E* is a short exact sequence of completed Hopf algebras. In particular, the images of the E*(X[s]) filter E*(X) with quotients given by the short exact sequences * *of completed Hopf algebras: E* - E*(K(sss(X); s)) - E*(X[s]) - E*(X[s-1]) - E*: Since our category is not abelian we need to explain what we mean by short ex* *act: the surjection is the cokernel and the injection is the kernel (see 6.7). For E* * = K(n) this result is in [HRW98 ]. For our more general E the injections and surjecti* *ons are shown in [RWY98 ] as well as the (algebraic) cokernel part. This follows f* *rom a general theorem which tells how Hopf algebra kernels in Morava homology K- theory give rise to cokernels in E cohomology, a theorem used over and over aga* *in in applications in [RWY98 ]. Results about the cokernel were expanded in [Wil9* *9a] and [Kasb ] but applications in the three papers never called for a general the* *orem about kernels. Here we show how Hopf algebra cokernels in Morava homology K- theory give rise to kernels in E cohomology. This is essential for the first de* *scription in Theorem 1.1 and, together with the completed Hopf algebra language developed here, allows us to state the above theorem. In [HRW98 ] it was further proved * *that the above short exact sequences all split for Morava K-theory, i.e. the middle * *term is the completed tensor product of the two end terms. It seems highly unlikely that this is true for the more general E although to be honest we do not have a counter example. Our intuition against such a splitting is also very much again* *st the existence of Theorem 1.10 and so is suspect. We are, however, able to prove such a splitting in all the cases of interest to us in this paper. Theorem 1.11. For r 0, q 0, and c > 0, let E be as in Theorem 1.1, then the filtration of Theorem 1.10 for IZ=(pc)BP__r(and also IZ=(pc)BP__(s)r) spl* *its, i.e. c) [O * Z=(pc) E*(IZ=(p_BP__r)' E (K(ssi(I_____BP__r); i)) 0ir O ' " E*(K(Z=(pc); r - d(I))) 0d(I)r as completed Hopf algebras. (For the s-connected case the first tensor product* * is over s < i r and the second over 0 d(I) < r - s.) The p-adic completion of E is not necessary for this result since all of the homotopy groups are finite. Remark 1.12.This result can be expanded to IBP__rand IBP__r. First note that the last two spaces are the same when 2(pq+1 - 1) > r. Now we observe that our 6 Takuji Kashiwabara and W. Stephen Wilson space is the direct limit of the spaces IZ=(pc)BP__r. Since these spaces are* * tor- sion spaces the Brown-Peterson cohomology is the same as the p-adically complete Brown-Peterson cohomology and so their inverse limit is the same. Likewise, the* *ir lim1s must be the same but this is zero for the p-adically completed version so* * it is zero for the non-completed version as well. c) BP *(IBP__r) ' BPp^*(IBP__r) ' lim0BP *(IZ=(p_BP__r) and a similar splitting follows. There are no odd degree elements. This theorem helps in Description 1 of Theorem 1.1 but it is really a side in* *terest for us in this paper. In fact, it is quite easy to prove from [RWY98 ] and to * *make it easy on readers only interested in the rather appealing Brown-Comenetz dual we have separated the proof of this (and the proof of the next result) out in its * *own Section 7 and made it reasonably self contained. Related to the last theorem, and perhaps also of more general interest, is a splitting for another class of spectra. Using the Baas-Sullivan theory of manifolds with singularities, [Baa73] (and * *now [EKMM96 ]), we can construct spectra which have a finite number of homotopy groups. Let I = (i0; i1; : :;:iq) or (i1; : :;:iq) with ik > 0 for all k q. * *The Baas-Sullivan theory gives us BP module spectra , BP I, with homotopy BP I*' BP *=(pi0; vi11; : :;:viqq) (with no pi0if i0 is not defined). Theorem 1.13. Let E be as in Theorem 1.1. Then the filtration of Theorem 1.10 splits for the spaces in the Omega spectrum for BP I. The p-adic completion * *is not necessary if i0 is defined. The BP cohomology is trivial stably because it is for Eilenberg-Mac Lane spac* *es. We say an algebraic object is completely algebraically determined if we can g* *ive a purely algebraic construction of it. For example, in [RW77 ] the Hopf ring E*BP* *__* was constructed algebraically for all complex oriented homology theories E*(-). This includes the E of present interest. We note that when r is even E*BP__ris a polynomial algebra on even degree generators and concentrated in even degrees. When r is odd, E*BP__ris an exterior algebra on odd degree generators. Further- more, because BP__rsplits off of BP__rwhen r g(q) ([Wil75], [BJW95 ], and [BWa ]), E*BP__ris completely algebraically determined in this case. All one needs to do is set the [vi] = [0] for i > q in E*BP__*. The proper way to say t* *his now is to use the Goerss-Hunton-Turner generalized tensor product, [Goe99 ] and [HT98 ], to write __ E*[BP *] E*[BP*]E*BP__r' E*BP__r when r g(q). By duality, since these are all free, we get an algebraic deter- mination of E*(BP__*). The Morava K-theory of Eilenberg-Mac Lane spaces is completely algebraically determined in [RW80 , Corollaries 11.3 and 12.2] and t* *hen again in [RWY98 , Proposition 1.16] by (1.26). The paper, [HRW98 , Theorem 2.* *1], gives a complete algebraic determination of the Morava K-theory of all of the f* *inite Postnikov systems which are homotopy commutative H-spaces. The E cohomology of Eilenberg-Mac Lane spaces was algebraically determined in [RWY98 , Theorem 1.14]. Our terminology is a bit of a misnomer when we are working with cohomolo* *gy Morava K-theory and Brown-Peterson cohomology 7 groups because they come with a topology on them. When we say "algebraically determined" in this paper, we mean we have all of the structure, including the topology. In [RWY98 ], cohomologies were algebraically determined, but the top* *ol- ogy was not proven to be determined. Our work in this paper is a significant improvement and allows us to upgrade the concept of "algebraically determined" to include the topology. This is done mainly by our introduction of completed Hopf algebras and our theorems about them. This work should have been made more prominent in this paper. However, the depths of the problems created by the topology, and our solutions to them, came late in the game. Consequently th* *is work is buried in section 6. The results of [RWY98 ], [Wil99a], [Kasb ], and this paper have made calcula* *ting our E*(-) cohomology groups (including BP *(-)) possible for lots of examples as corollaries of calculating the Morava K-theory (homology) of related spaces. Morava K-theory is relatively easy to work with because it has a K"unneth isomo* *r- phism and because the category of Hopf algebras we work in is abelian. Most of our work in this paper is done with Morava K-theory (homology) and our other results are applications of these calculations. Whenever possible in this paper we will use K*(-) to denote the Morava K- theory, K(n)*(-), in order to suppress the n from our notation, i.e. K* = K(n)** * ' Fp[vn; v-1n]. Much of the notation used in this paper is quite unpleasant. As* * we inherit some of it from the literature we are not entirely to blame. Our main computation is really to show the following result for Morava K-theo* *ry. Note that the first space is trivial for r g(q). Theorem 1.14. For q + 2 < r the fibration IBP__(q+1)r-gffi(q)-! BP__r-! LqBP__(q+2)r gives rise to a split short exact sequence of Hopf algebras in Morava K-theory K* -! K*IBP__(q+1)r-gffi(q)-! K*BP__r-! K*LqBP__(q+2)r-! K*: The first term is always even degree and because it is a finite Postnikov syste* *m splits up further. The last term is completely algebraically determined by the Goerss- Hunton-Turner generalized tensor product: K*LqBP__(q+2)r' image{K*BP__r! K*BP__r} __ ' K*[BP *] K*[BP*]K*BP__r and so is in even degrees if r is even. This splitting gives a completely alge* *braic determination for K*BP__r: When n q + 1 the first term is trivial and we get the nice __ K(n)*[BP *] K(n)*[BP*]K(n)*BP__r' K(n)*BP__r The proof of the above is a bit involved and requires a number of variations * *on the Koszul complex. First, we start with the usual resolution of Z(p)over BP * ' Z(p)[v1; v2; : :;:vq] We define P i M X s2(p s- 1) KZ BP*j' BP *: 0*j-! KZ BP*j-1 summandP by summand using (-1)1+tvit to map from the summand P s2(pis- 1) BP* BP* BP * in KZ j to the similar summand in KZ j-1 with no it. This realizes the graded version of the Koszul complex. Its homology is a Z(p)concentrated in degree zero of the zeroth homology. KZ BP**is a finite resolution of Z(p)by BP * with all maps split over Z(p). This construction c* *an now be mimicked with the spectrum BP to obtain KZ BP*. The homotopy groups of KZ BP*give the corresponding alebraic resolution, KZ BP**. If * *we take the Omega spectrum for all of the spectra in KZ BP*we get unstable ver- BP_ sions KZ * *. The minus signs in the maps must be interpreted as the H-space BP_2(q+1)+k inverse. The indexing which we will use frequently is KZ j which is ju* *st Y BP__P2(pis-1)+2(q+1)+k: 0_2(q+1)+k (1.15) KZ q ' BP__g(q)+k which is convenient for us. Since we are working with BP module spectra we can make similar definitions for Koszul complexes KZ LqBP*and KZ IBP*. We can also make this unstable (s) IBP IBP(s) with KZ LqBP_*, KZ LqBP_*, KZ ______*, and KZ ______*, etc. We can then apply K*(-) to all of these complexes, noting that -1 becomes the Hopf algebra conjugation, and we have KZ K*LqBP_*, etc. In this case and in all others, w* *hen we have a sequence of K*-Hopf algebras, all of the undefined ones are assumed to be the trivial Hopf algebra, K*. In addition, we have Koszul complexes for Z=(p* *c), KZ=(pc)Z(-)*, for all of the above. We just index over 0 i0 < i1 < : :<:ij q and let the map v0 be pc. Note that the length of the complex is now q + 1 and * *we give up our splittings. Because vivj = vjvi, the composition of any two maps, KZ (-)j+1-! KZ (-)j-! KZ (-)j-1 is always trivial which gives sense to the term "complex." Since the category * *of Hopf algebras we work in is abelian, [Bou96b ], [Bou96a ], [HRW98 ], and [SW98* * ], we can talk about exactness and homology of complexes of Morava K-theory Hopf algebras. Recall the stable cofibration n-1) vn (1.16) 2(p BP -! BP ! BP gives rise to the (BP module) boundary map n-1 BP -ffi!2p BP : Iterating these we have Morava K-theory and Brown-Peterson cohomology 9 (1.17) ffi : K(Q=Z(p); 2(0) + q + 1) ! K(Z(p); 2(1) + q) ' 2(1)+qBP <0> ! 2)+q-2 g(q) 2(1+p)+q-1BP <1> ! 2(1+p+p BP <2> ! . .!. BP : We abuse notation a bit and allow ffi to be any iterated boundary map. Which it* * is will be uniquely determined by the source and target. We will call EKZ BP*the extended Koszul complex when we tack on a q + 1 term to KZ BP*, i.e. (1.18) EKZ BPq+1= K(Z(p); -q) -ffi! g(q)-2(q+1)BP = KZ BPq= EKZ BPq: Because ffi is a BP module map and the vi act trivially on Eilenberg-Mac Lane spaces, this is still a complex. (Note we are referring here to viacting on the* * spaces, BP_ not on BP cohomology.) We get corresponding unstable versions, EKZ * *and K*BP_ Z=(pc)BP EKZ * *. We have a similar extended Koszul complex, EK Z* , with corresponding unstable versions. Here, c)BP c EK Z=(pZq+2 = K(Z=(p ); -(q + 1)): We also have EKZ IBP*with EKZ IBPq+1= K(Q=Z(p); gv(q)) and of course ther* *e is a Z=(pc) version as well. All of these definitions are leading up to our next t* *heorem, the proof of which is thoroughly linked to the proof of our short exact sequenc* *e, Theorem 1.14, which is just the EKZ q part of the following with r = gv(q) + t. Theorem 1.19. For t > 2(q + 1) there is a short exact sequence of long exact sequences with all maps split as algebra maps: K*IBP_(q+1)t-g (q)K*BP K* -! EKZ * -! EffiKZ*_____t K*LqBP_(q+2) -! KZ * t - ! K* For the term on the right, this Koszul complex inductively determines the last term K*LqBP__(q+2)t+gv(q)which is also determined by the generalized tensor * *pruduct in Theorem 1.14. There is yet another way to view our results. In particular this gives an- other perspective on the once mysterious way the Morava K-theory of Eilenberg- Mac Lane spaces shows up in our answers. We have used the Goerss-Hunton-Turner generalized tensor product to describe our answer. Hunton and Turner go further with this and define the derived funct* *ors which they call CTor where CTor0 is just this new tensor product. In their pape* *r, [HT98 ], they do precisely the algebra we need. This is developed even further * *in [Kasa]. Our CTor is still a Hopf algebra in our category and so it splits into* * an exterior algebra part and an even degree part, [HRW98 ]. A more detailed state* *ment will follow in the final section, but for now we will just observe: Theorem 1.20. Let Ej be the exterior algebra part of CTorj, then *] * K*BP__r' CTorK*[BP*(K*[BP ]; K*BP__r+*)=(Ej, j > 0): 10 Takuji Kashiwabara and W. Stephen Wilson We can calculate this by taking a Koszul resolution of BP * by BP*, going to the `ring-rings' K*[BP *] and K*[BP *], taking the generalized tensor pro* *duct and computing the homology. Thus it is easy to see that *] * BP K*BP_ (1.21) CTor K*[BP*(K*[BP ]; K*BP__*) ' H*(K *Z* *): The notation here accurately suggests the use of a Koszul type resolution of BP* * * by BP* and similar versions on spectra and spaces. We show it is equivalent to K*BP_ calculating the homology of KBP*Z* * for i big. For q = 0 it is enough * *to do the homology of KZ K*BP**, something we have already studied. We use this and the obvious exact sequences to compute the general case. Such a result is not completely unexpected. Letting F (j) = KBPZBPj and letting G(0) = BP we can define G(j + 1) inductively (stably) where G(j + 1* *) ! F (j) ! G(j) is a stable triangle. If, and this is a big if since it doesn't ha* *ppen, we were lucky enough to get a long exact sequence in Morava K-theory on the spaces in the Omega spectra (1.22) . .!.K*G(j_+_1)_i! K*F_(j)_i! K*G(j)_i! K*G(j_+_1)_i+1! . . . then we would have a spectral squence: *] * (1.23) CTor K*[BP*(K*[BP ]; K*BP__r+*) ) K*BP__r: Our theory says that if we ignore the exterior part of CTorj, j > 0, we get the correct answer. The space LqBP__(q+1)r, r > q + 1, plays a crucial role in our study becau* *se we can show the map LqBP__r-! LqBP__(q+1)rgives a surjection in ALL Morava K-theories. The space LqBP__(q+1)rthus arises for the age old reason "becaus* *e it works." After this paper was submitted, some communication with Pete Bousfield helped give us a partial, but not complete, step towards understanding where th* *is space comes from. Theorem 1.24 (with A.K. Bousfield).When r > q, we have a homotopy equiva- lence: LqBP__r' LqBP__(q)r: Pete Bousfield made a very general conjecture which included the above theore* *m. He had every step of the proof for this special case except for showing the map* * is an isomorphism on the q-th Morava K-theory. (The map, in general, is neither surjective nor injective for higher Morava K-theories.) This isomorphism is eas* *y to see from our work here. We had hoped he could put this result in his paper, [Bo* *u ], so we could just quote it, but he claimed it didn't fit. This is not satisfacto* *ry for us because it still differs by one homotopy group from the space important to u* *s. However, it is a lot closer than we were before. A word about motivation is probably appropriate here. Several times while working on other projects, the question of the Morava K-theory of spaces in the Omega spectrum for BP has come up as possibly useful. Since the answers were not known the questions have generally gone away. However, they did serve to get us somewhat interested. With the discovery that many spaces have computable Landweber flat Brown-Peterson cohomology, [RWY98 ], [Kas98], and [Kasb ], the Morava K-theory and Brown-Peterson cohomology 11 question naturally arose for the spaces BP__r. In particular there are the s* *pectra and maps: (1.25) S0 ! T (1) ! . .!.T (q) ! . .B.P . .!.BP ! . .!.BP <1> ! K(Z(p)): The T (q) are Ravenel's ring spectra, from [Rav84 ], [Rav85 ] and [Rav86 ], whi* *ch were so important in the proof of his conjectures (from [Rav84 ]) in [DHS88 ]. The BP cohomology of the spaces in the Omega spectrum for BP has been known explicitly since [RW77 ]. In [RWY98 ] the Eilenberg-Mac Lane spaces (on the right) were d* *one. Using [RWY98 ], [Kas98] did all of the even spaces for S0 and T (q). His work * *in [Kas98], together with either [Wil99a] or [Kasb ] gets all of the odd spaces as* * well. This paper completes our knowledge of the BP cohomology of the spaces in the above sequence. Two examples motivated us further. The first was the simple case of the fibra* *tion (see [RWY98 , section 2.6]) F -! BSU(2m-1)- ! BSU: Here F is a finite Postnikov system. The Morava K-theory of this fibration gives rise to a short exact sequence of Hopf algebras. As m gets bigger, the Morava K- theory continues to see all of the space BSU and also all of the missing homoto* *py groups! When we use K(1) it doesn't see F at all. We found this intriguing and pursued its generalization. Turning this example into BP_<1>_kand replacing F with the fiber of the rationalization of F , this example is our q = 1 case. T* *his example was almost more confusing than helpful since it was not at all clear how to generalize it. It is only after the fact that we see that the E(1) localizat* *ion of bu is the correct object to have here instead of BSU on the right, which works * *in this case but which does not generalize. The Koszul complexes came about from the second example. In [RWY98 , Propo- sition 1.16], the exact sequence (1.26)K* -! K*K(Z(p); q + 2) -! K*BP_g(q)vq*-!K*BP_g(q)-2(pq-1)-! . . . was proven as well as the more likely exact sequence: (1.27) vi* O K* -! K*K(Z(p); q + 2) -! K*BP_g(q)___- K*BP_g(q)-2(pi-1)-! . . . 0 spac* *es in the above case have no torsion and so it is the "easy" case. The last sequen* *ce still holds when delooped and so still works for BP cohomology, Corollary 1.4. * *The sequence (1.27)is the beginning point for all of our calculations here and is u* *sed repeatedly in this paper. Finally, we had proven many of the Morava K-theory exact sequences in this paper purely algebraically where the only spaces involved were BP__*. Because of the appearance of the Morava K-theory of Eilenberg-Mac Lane spaces we had a strong feeling that there had to be topology underlying it and that the topology 12 Takuji Kashiwabara and W. Stephen Wilson would be interesting. Furthermore, we needed the topology to go from Morava K-theory results to our results about Brown-Peterson cohomology. We came up with our own version of topology but with the help of Mark Mahowald, Douglas Ravenel, Charles Rezk, and Hal Sadofsky, were able to get the more interesting * *Ln localizations and Brown-Comenetz duals involved. We wish to thank them for their help in this matter. In addition, the second author wishes to thank the Centre * *de Recerca Matematica in Barcelona, Spain, and the Department of Mathematics at Kyoto University in Kyoto, Japan, for their ideal work environments during the writing of this paper. The work of Richard Kramer, [Kra90] and [BKW99 ], and the short exact squence K(n)* ! K(n)*K(Z=(p); r - 2pq + 1) ! K(n)*k(q)_r! K(n)*k(q)_r-2(pq-1)! K(n)* when r 2pq + q suggests that similar things happen with the spectra P (n; q) of [RWY98 ], [BWa ], and [BWb ]. We found ourselves easily discouraged at the th* *ought of any more of this. At one time potential bases were written down for much of this work. However, they are very difficult to prove. The present "coordinate free" version is much* * nicer anyway. Our next section is a more detailed statement of results. Section 3 does the preliminary work we need for our proofs. Section 4 proves our core results from which almost everything else follows. We finish up many of the rest of the proo* *fs in Section 5 except for the lifting of our results to BP . Those results are pr* *oven in Section 6 (where we also set up completed Hopf algebras) except for our isolati* *on of the results on the E cohomology of the Brown-Comenetz dual and the Baas- Sullivan spectra in Section 7 which are put here for ease of access to readers * *only interested in them. Section 8 discusses our results on CTor and our final Secti* *on 9 proves Theorem 1.24. Acknowledgment. We would like to thank the referee for his careful reading and many suggestions. 2.Detailed Statement of Results For q < s < r - 1 we consider diagram (2.1)in which all rows and columns are fibration sequences. Theorem 2.2. If we apply Morava K-theory, K*(-), to diagram (2.1)for q < s < r - 1, then we get a series of short exact sequences of Hopf algebras. (i)The following short exact sequence is split as Hopf algebras. Each space i* *s a finite Postnikov system and the Morava K-theory is naturally isomorphic to that of a product of Eilenberg-Mac Lane spaces with the same homotopy and thus is completely algebraically determined and concentrated entirely in e* *ven degrees. K* -! K*IBP_(s)r-gffi(q)-! K*IBP_(q)r-gffi(q)-! K*IBP_(q;s]r-gf* *fi(q)-! K* (ii)The following short exact sequence is split as Hopf algebras. K* -! K*IBP_(s)r-gffi(q)-! K*BP_r-! K*LqBP_(s+1)r-! K*: Morava K-theory and Brown-Peterson cohomology 13 IBP__(q;s]r-gffi(q) | | | | |? IBP__(s)r-gffi(q)_-BP__r ______- LqBP__(s+1)r | | | | | | (2.1) | ' | | | | | |? |? |? IBP__(q)r-gffi(q)_-BP__r ______- LqBP__(q+1)r | | | | | | | | | | | | |? |? |? IBP__(q;s]r-gffi(q)_- * ______- IBP__(q+1;s+1]r+1-gffi(q) When s = q + 1 the last term is completely algebraically determined by the Goerss-Hunton-Turner generalized tensor product: K*LqBP__(q+2)r' image{K*BP__r! K*BP__r} __ ' K*[BP *] K*[BP*]K*BP__r and so is in even degrees if r is even. This splitting gives a completely * *algebraic determination for K*BP__r: When n q + 1 (and s = q + 1) the first term is trivial and we get the nice __ K(n)*[BP *] K(n)*[BP*]K(n)*BP__r' K(n)*BP__r: (iii)The following short exact sequence is split as algebras. K* -! K*IBP_(q)r-gffi(q)-! K*BP_r-! K*LqBP_(q+1)r-! K*: When r is even, the last term is a polynomial algebra on even degree gener* *ators and so K*BP__ris also concentrated in even degrees. When r is odd, the last term is an exterior algebra on odd degree generators. This last term * *is completely algebraically determined. (iv)There is a short exact sequence of Hopf algebras which is split as algebra* *s. K* ! K*IBP_(q;s]r-gffi(q)! K*LqBP_(s+1)r! K*LqBP_(q+1)r! K*: When s = q + 1 this is: K* -! K*K(ssq+1IBP__r-gffi(q); q + 1) -! __ (q+1) K*[BP *] K*[BP*]K*BP__r-! K*LqBP__r ! K*: Part (ii) of this for s = q + 1 is Theorem 1.14 in the Introduction. We have another exact sequence which does not come from the diagram. We re- call that there are maps BP ! BP and X ! LqX and that Lq-1LqX ' Lq-1X. 14 Takuji Kashiwabara and W. Stephen Wilson Theorem 2.3. For r > q + 1 > 1 there is a four term exact sequence of Hopf algebras, vq* (q+1) K* -! A(q; r) -! K*LqBP__(q+1)r+2(pq-1)-!K*LqBP__r - ! __ K*Lq-1BP__(q+1)r' K*[BP *] K*[BP*]K*BP__r: -! K*; where A(q; r) is trivial, thus giving us a short exact sequence, except when r + 2(pq - 1) = g(q) - 1 + 2(p - 1)t; t 0: In this case A(q; r) is an exterior algebra with TorA(q;r)(K*; K*) an associated graded object for K*K(ssq+1IBP__r-gffi(q-1); q + 1): In particular, note that when r is even, vq*injects. Theorem 1.19 is just one of many Koszul complex type theorems we have. It is the s = q + 1 case of part (iii)below. Theorem 2.4. (i)There is a short exact sequence of long exact sequences of Hopf algebras w* *ith all maps split as Hopf algebra maps: K*IBP_(s) K*IBP_ K*IBP_[s] K* ! EKZ * * ! EKZ * *! EKZ * * ! K* (ii)There is a short exact sequence of long exact sequences of Hopf algebras: c)K*IBP_(s) Z=(pc)K*IBP_ K* -! EKZ=(p Z* * -! EK Z* * c)K*IBP_[s] -! EKZ=(p Z* * -! K*: (iii)For 2(q + 1) + k = t, k 0, and q s < t - (q + 1) there is a short exact sequence of long exact sequences with all maps split as algebra maps: K*IBP_(s)t-g (q) K* -! EKZ * -!ffi K*BP_ K*LqBP_(s+1) EKZ * t-! KZ * t - ! K* The right side long exact sequence gives an inductive algebraic determinat* *ion of the final term, i.e. K*LqBP__(s)g(q)+k. (iv)For 2(q + 1) + k = t, k 0, and q s < t - (q + 1) there is a short exact sequence of long exact sequences: c)K*IBPt-gffi(q)_(s) K* -! EKZ=(p Z* * -! c)K*BP_ Z=(pc)K*LqBP_(s+1) EKZ=(p Z* t -! K Z* t -! K* Morava K-theory and Brown-Peterson cohomology 15 (v) For 2(q + 1) + k = t, k 0, and q < s < t - (q + 1) there is a short exact sequence of long exact sequences which is split as algebras: K*IBP_(q;s]t-g (q) K* -! KZ * -! ffi K*LqBP_(s+1) K*LqBP_(q+1) KZ * t - ! KZ * t - ! K*: The restrictions on s in the theorem are just to avoid degenerate cases. This* *, in K*BP_ (iii), calculates the homology of the complex EKZ * twhen t 2(q + 1). We return to this in Corollary 8.10 where we compute the homology for t < 2(q + 1) as well. The homology of the other Koszul complexes follows from this. For r g(q), the spaces BP__rsplit off of the BP spaces and have no torsion and so their Morava K-theory is described by [RW77 ]. We have given one algebra* *ic determination of K*BP__rfor r > g(q) above in Theorem 2.2 (ii)(the s = q + 1 version). The Koszul complexes give us yet another way to do this (the s = q + k version of Theorem 2.2 (ii)). For the extended complexes the only thing missed * *is the extension itself, which happens to split off in this range anyway. This spl* *itting is part of the following corollary. Let k > 0 and recall the identifications and i* *terated boundary map K*BP_2(q+1)+kffi (2.5) K*BP__g(q)+k= KZ q -! K*BP_2(q+k+1) K*BP__g(q+k)= KZ q+k : An algebraic determination of K*BP__g(q)+kfollows from the corollary which splits off the extension. Corollary 2.6.For k > 0 and r = g(q)+k, we have the Hopf algebra decomposition O (q+k+1) K*K(Q=Z(p); q + 1 + k) K*LqBP__r ' K*BP__r coming from the exact sequence K*BP_2(q+1)+k K* -! KZ q -! K*BP_2(q+k+1)O K*BP_2(q+1)+k KZ q+k KZ q-1 K*BP_2(q+k+1)O K*BP_2(q+1)+k -! KZ q+k-1 KZ q-2 -! . . . which inductively algebraically determines K*BP__g(q)+kas well as the maps v* *i*. Remark 2.7.This follows immediately from Theorem 2.4 (iii). However, we need a little more than this for future reference. We need that the composition of t* *he two geometric maps in the corollary are trivial. The second one is easy since i* *t is just the Koszul complex. In the first one, the maps can be taken in any order s* *ince they commute. We need to show that viO ffi ' ffi O vi is trivial, but the delta* * from the lower space on the right side of this equation must be trivial because it f* *actors through the range of the splitting Theorem 3.29. The final part of the extended Koszul complex (the middle term of Theorem 2.4 (iii)) gives another algebraic determination of the Morava K-theory of Eilenber* *g- Mac Lane spaces. 16 Takuji Kashiwabara and W. Stephen Wilson Corollary 2.8.For k 0, v0 = pc, c 1 the exact sequence of Hopf algebras, K* -! K*K(Z=(pc); q + 1 + k) -ffi*!K*BP__g(q)+k vi*_-O K*BP__g(q)+k-|vi|-! . .;. algebraically determines K*K(Z=(pc); q + 1 + k). The k = 0 case of this was done in [RWY98 ] and we rely on it extensively in* * this paper. The fact that the extension splits off K*BP__rwhen r > g(q) can be general* *ized significantly. The algebraic splitting of TheoremP2.2 (ii)comes from geometric maps. Let vI = vi11vi22: :v:iqqwith d(I) = |vI| = 2(pj - 1)ij. Let r = g(q) +* * k, k > 0. The top homotopy group of IBP__(s)r-gffi(q); s > q, is in degree r + * *(q + 1) - g(q) = (q + 1) + k and the bottom is in degree s + 1. The Q=Z(p)summands in homotopy can be indexed over 1=vI, d(I) k - (s - q) with the 1=vI summand in degree (q+1)+k-d(I) = (q+1)+kI where kI > 0. By this definition of kI we have k - kI = d(I) and we see that kI > 0 because s > q and so the lowest homotopy group is always in degree greater than s, i.e. definitely greater than q + 1. W* *e can map all of our spaces with a vI followed by an iterated ffi to get: I ffi IBP__(s)r-gffi(q)-! BP__r=g(q)+kv-!BP__g(q)+kI-! BP_2(q+k +1) BP__g(q+kI)' KZ q+kI I: Note that with this last space we are in the range of torsion free spaces and a* *re at the first exact extended Koszul complex. The image of the Morava K-theory of our left hand space is just the image of the Morava K-theory of our chosen Q=Z(* *p) summand which is precisely the kernel of the Morava K-theory of the last space mapped to the (q + kI - 1)-th term of its Koszul complex. Thus we get: Theorem 2.9. Let r = g(q) + k, k > 0, kI = k - d(I) and q < s < (q + 1) + k. We have the Hopf algebra decomposition O K*BP_r' K*LqBP_(s+1)r K*K(Q=Z(p); (q + 1) + kI) d(I)k-(s-q) coming from the maps BP__r-! Y BP_2(q+k +1) LqBP__(s+1)r KZ q+kI I d(I)k-(s-q) Y BP_2(q+k +1) -! * KZ q+kI-1 I: d(I)k-(s-q) Morava K-theory and Brown-Peterson cohomology 17 which gives the exact sequence K* -! K*BP_r-! O K*BP_2(q+k +1) K*LqBP_(s+1)r KZ q+kI :I d(I)k-(s-q) O K*BP_2(q+k +1) -! K*(pt) KZ q+KI-1 I-! . .:. d(I)k-(s-q) This is a more geometric version of our Theorem 2.2 (ii)and gives us the Hopf algebra splitting there. The interesting cases are the two extremes. On the o* *ne extreme, for s = q + 1 this is the version of Theorem 2.2 (ii)which uses the Go* *erss- Hunton-Turner tensor product and which is particularly useful for the proof of Theorem 1.1, Description 1. The other extreme, with s = k + q (i.e. Corollary 2* *.6), will gives us Description 2. When we started this project an obvious approach was to look at the spectral sequences which come from the fibration sequence (2.10) . .-.! BP__r-1-! BP__r+2(pq-1)-! BP__r-! BP__r-! . .:. This approach turned out to not be productive. However, it is a sequence which should be understood now that we know more. The Morava K-theory does not give an exact sequence but we can measure how far it deviates from that by taking its homology. In fact, it turns out to be exact mostly and its homology is moderate* *ly well understood. Theorem 2.11. Taking the homology of the complex we obtain by taking the Morava K-theory of the sequence of fibrations (2.10)we find that it is an exact sequen* *ce everywhere but at K*BP__r-1-! K*BP__r+2(pq-1)-! K*BP__r: The homology at this point is trivial except when r +2(pq-1) = g(q)-1+2(p-1)t, t 0. In this case the homology is just the A(q; r) of Theorem 2.3 . Having come this far it is reasonable to ask what the Morava K-theory of the spaces in the Omega spectrum for LqBP are. This is easy for us to answer for the even spaces but not so easy, and not pursued, for the odd spaces. Theorem 2.12. There is a short exact sequence of Hopf algebras: K* -! K*LqBP__(q+1)2r-! K*LqBP__2r-! K*IBP__[q+1]2r+1-gffi(q)-! K*: 3.Preliminaries Remark 3.1 (Morava homology Hopf algebras).We will denote by K*(-) the Morava K-theory, K(n)*(-). It has a K"unneth isomorphism which makes it par- ticularly amenable to calculations. We use Hopf algebras over K*, see [Bou96b ], [Bou96a ], [HRW98 ], and [SW98 ]. The Hopf algebras we use form an abelian cat* *e- gory so a short exact sequence of complexes gives rise to a long exact sequence* * in homology. In particular, if two of them are long exact, i.e. have trivial homol* *ogy, then so does the third. When we just have a map of two short exact sequences this degenerates into the snake lemma giving a six term exact sequence relating* * the kernels, H1(-), with the cokernels, H0(-). 18 Takuji Kashiwabara and W. Stephen Wilson Remark 3.2 (Finite Postnikov systems).All of our spaces are infinite loop spaces and all of our maps are infinite loop maps. The spaces IBP__*; as well as th* *eir X(s)and X[s]versions, are all finite Postinikov systems, i.e. they have only a * *finite number of non-trivial homotopy groups. The work in [HRW98 ] tells us that the Morava K-theory is the same as if it was a product of Eilenberg-Mac Lane spaces with the same homotopy. This splitting is natural and so when such spaces give a short (or long) exact sequence on homotopy we get a short exact (or long) exact sequence of Hopf algebras. If the maps on homotopy are split then the Hopf alge* *bra maps are too. Remark 3.3 (The Goerss-Hunton-Turner tensor product (GHT)).We will use the Goerss-Hunton-Turner generalized tensor product, [Goe99 ] and [HT98 ] __ K*[BP *] K*[BP*]K*BP__* and abuse the language slightly by writing __ K*[BP *] K*[BP*]K*BP__r when we mean the r-th part of it. In this case it amounts to setting the element [vs] = [0-2(ps-1)],s > q, where this last is the Hopf algebra unit in the same * *space as [vs]. We make frequent use of the bar spectral sequence . It will be helpful to wri* *te down some of the things we use over and over again. Theorem 3.4 (Folk). Given a fibration of infinite loop spaces and maps, F -i! E -! B; (i)then there is a spectral squence of Hopf algebras TorK*F*;*(K*E; K*) ) K*B (ii)with E2 term isomorphic to TorKer*i*;*(K*; K*) Cokeri* with Coker i* ' TorK*F0;*(K*E; K*) ' K*E K*F K*. (iii)If i* is injective we get a short exact sequence of Hopf algebras: K* -! K*F -i*!K*E -! K*B -! K*: (iv)K*K(Q=Z(p); i) is even degree and the spectral sequence TorK*K(Q=Z(p);i)(K*; K*) ) K*K(Q=Z(p); i + 1) for the fibration K(Q=Z(p); i) -! * -! K(Q=Z(p); i + 1) is all in even degrees and collapses. (v) Torof an exterior algebra is a divided power algebra and Tor of a polynomi* *al algebra is an exterior algebra. Proof.See [HRW98 , pp. 144-5] and [RW80 , pp. 704-705] for a discussion of this spectral sequence. Part (iii)follows from the previous part. Part (iv)follows_f* *rom [RW80 , Theorem 12.3]. Part (v)is standard. |__| Morava K-theory and Brown-Peterson cohomology 19 Remark 3.5 (The Brown-Comenetz dual).In [Rav84 , Section 5], Ravenel induc- tively constructs functors, Nn and Mn on the stable category with the stable co* *fi- bration: (3.6) NnX -! MnX -! Nn+1X using N0X = X and LnNnX = MnX where Ln is Bousfield's localization with respect to the theory E(n), [Bou79 ]. In SectionP5 he defines Cn as the fibre * *of X ! LnX and in Theorem 5.10 he shows that NnX = nCn-1X giving us the stable cofibration (3.7) -n-1Nn+1X -! X -! LnX for all X. He then goes on, Theorem 6.1, to calculate the homotopy groups of all of these functors when X = BP . We are interested in the same results for X = BP and it appears that the same proof works in this case when n q. The proof is a bit scanty and seriously nested. However, improvements in technology since that paper have made this much easier. Special thanks to Doug Ravenel and Hal Sadofsky for helping us work through this and to Mark Hovey who would have done it if they hadn't. As part of the calculation of the homotopy groups, Rave* *nel shows that NnBP and MnBP are BP module spectra, something we need for our case as well. This is now easy due to the smash product theorem, [Rav92 , Theor* *em 7.5.6], which says LnX ' X ^ LnS0. If X is a BP module spectrum, then all one has to do is take the fibration (3.8) CnS0 -! S0 -! LnS0 for the sphere and smash it with BP ^X ! X. This gives the BP module structures we need. Specializing to BP , we want our homotopy groups to be (3.9) ss*NnBP ' Nn BP * ss*MnBP ' Mn BP * where we start with N0BP * = BP * and define Mn BP * = v-1nNn BP * and Nn+1BP * inductively using the short exact sequence: (3.10) 0 ! Nn BP * ! Mn BP * ! Nn+1BP * ! 0 for n q. To do this it is enough to show: Lemma 3.11. If X is a BP module spectrum with ss*X all In torsion, then ss*LnX ' v-1nss*X. Proof.Theorem 1 of [Rav87 ] states that if BP*X is all In torsion, then BP*LnX is just v-1nBP*X. The fact that ss*X is In torsion implies BP*X is also In tors* *ion. Since we have maps Y ! BP ^ Y ! Y for Y = X and LnX this result follows __ from the BP homology result. |__| We are assured that the identification of Nq+1BP with the Brown-Comenetz dual of BP can be done directly from here but since we have a concrete refe* *rence in the literature we will use it. In [MR , Corollary 9.3] Mahowald and Rezk con* *struct a stable cofibration (using the p-adic completion of BP ): (3.12) -gffi(q)IBP -! BP -! LqBP where IBP is the Brown-Comenetz dual and the identification (3.13) -gv(q)Nq+1BP ' IBP 20 Takuji Kashiwabara and W. Stephen Wilson follows. When we look at the spaces in the Omega spectra for the spectra above we run into some problems, namely, the spaces for IBP and LqBP are not of fini* *te type. This presents no problems as long as we are working with Morava K-theory, but when we try to lift our results to the other theories such as BP , then we * *rely on the work of [RWY98 ] which requires finite type. We present two ways of working around this problem. The first, and easiest ca* *me to us last because we did not know about the Brown-Comenetz dual identification. However, using it, we can take the dual of the sequence (3.14) BP -! M0BP ( = p-1BP )-! N1BP to get another stable cofibration (3.15) IM0BP -! IBP -! IN1BP : The map (3.16) IM0BP -! IBP -! gffi(q)BP is trivial and so the map IBP -! gffi(q)BP factors through IN1BP . We can now define L0qBP and we have maps -gffi(q)IBP -! BP -! LqBP (3.17) | | | |? |? |? -gffi(q)+1IN1BP -! BP -! L0qBP : The spaces in our new Omega spectra now have lots of p-adics in them but we can handle that since their homology is of finite type over the p-adics. Unstably, * *the above maps induce isomorphisms on the Morava K-theories. We want to explore an alternative approach. This is what we had done before we learned about the Brown-Comenetz dual identification. It leads to a collecti* *on of finite Postnikov systems each with BP cohomology which splits as if it were * *just a product of Eilenberg-Mac Lane spaces. Furthermore, it allows us to identify t* *he iterated boundary map we use with the map of the top homotopy group in (3.12). We doubt if this is original but we don't know of any reference to anyone else * *doing it this way. There is something similar in [MS95 ]. We use the Baas-Sullivan theory of manifolds with singularities, [Baa73], to construct spectra. Let In = (i0; i1; : :;:in-1) with ik > 0 for all k. The Ba* *as- Sullivan theory gives us BP module spectra and maps of BP module spectra in| vinn (3.18) |vn BP In ______-BP In -! BP In+1 for n < q and where BP In*' BP *=(pi0; vi11; : :;:vin-1n-1). Ravenel's sp* *ectrum Nn+1BP is just the direct limit of this taken over the various In+1 but we * *must Morava K-theory and Brown-Peterson cohomology 21 prove that. We do this by first constructing maps of stable BP module cofibrati* *ons. vinn |vinn|BP In______- BP In ______- BP In+1 | | | (3.19) |j1 |j2 |j3 | | | |? |? |? f NnBP ______-MnBP ______- Nn+1BP : We understand that some large desuspension has to be applied to the top row to make sense of this. We can, by induction, define our maps. Since MnBP is known to be localization with respect to vn we can define j2 = v-innO f O j1. S* *ince we know all of the homotopy groups the rest follows and we have an injection of homotopy from the top row to the bottom row. Taking an appropriate limit over the Iq+1 we see (3.20) limIBP Iq+1' Nq+1BP : q+1 Iterated boundary maps give the analog of (3.21) -(q+1)Nq+1BP -! BP To see that this is the iterated boundary map defined in the introduction, just* * let i1 = i2 = : :=:1. We can take the cofibre of our map for use in all of our theo* *rems. However, we need something of finite type or we wouldn't bother with this at all. We modify our construction by using I0n= (i1; i2; : :;:in-1), ik > 0 for a* *ll k. We construct BP I0njust as above observing that we have a stable cofibration pi0 (3.22) BP I0n_____-BP I0n____-BP In which gives us the boundary map (3.23) -1BP In -! BP I0n: This, in turn, has a boundary map to some suspension of BP . Take the limit * *of this for n = q + 1 and we get (3.24) -1Nq+1BP -! N0q+1BP ( -gv(q)I0BP ): whose cofibre is rational. Consequently, we have a Morava K-theory isomorphism between the spaces in the Omega spectrum for these two. The stable cofibre of (3.25) -qN0q+1BP -! BP is our replacement, L0qBP , for LqBP . In the Omega spectrum we have to replace s connectivity with s + 1 connectivity of our new spaces. We get Morava K-theory isomorphisms of short exact sequences: IBP__(s)r-gffi(q)_- BP__r ______- LqBP__(s+1)r | | | (3.26) | |' | | | | |? |? |? I0BP__(s+1)r-gffi(q)+1_-BP__r_____-L0qBP__(s+2)r: We can replace spaces accordingly with new spaces with finite type. 22 Takuji Kashiwabara and W. Stephen Wilson Remark 3.27 (Another useful fibration).We find it convenient, but not necessary, to have vq (3.28) |vq|LqBP ____-LqBP -! Lq-1BP be a fibration and it is. Ravenel points out that it is enough to show that the* * map LqBP -! Lq-1BP is an equivalence and has kindly given us the proof of this fact. Since both are BP module spectra it is enough to show that * *the fibre, MqBP is BP acyclic. Since BP ^ MqBP ' BP ^MqBP which is Bousfield equivalent to BP ^ K(q) which is contractible because K(q)*BP is trivial. We rely heavily on the theorem which splits the spaces in the BP Omega spec- trum: Theorem 3.29 ( [Wil75], see also [BJW95 ] and [BWb ]). (i)For r g(q), BP__rsplits off of BP__r. (ii)For r g(q - 1) = g(q) - |vq| - 2, BP__r' BP__rx BP__r+|vq| Remark 3.30.We need to point out that although the Morava K-theory at p = 2 is not a communtative ring theory these results all still hold. For details, s* *ee [JW85 , appendix] and [Wil84, pages 1030-31]. When we go to lift the results to* * E cohomology, the same comments apply when we are working modulo 2. However, due to the work of Strickland, the theory E(n) is a commutative ring spectrum if the higher vs which are killed are chosen carefully enough [Str]. Indeed, since* * these theories (P (0) and E(n)) surject to all of the other theories, their commutati* *vity is forced. 4. The main computation We begin with the rather simple results on finite Postnikov systems. Proof of Theorem 2.2 (i).This follows automatically from the discussion Remark_ 3.2 about [HRW98 ]. |__| Proof of Theorem 2.4 (i) and (ii)I.f we can show that the maps which give rise to these Koszul complexes in (i)are split exact on homotopy, then the results of [HRW98 ] will give the exactness and splittings as Hopf algebras, Remark 3.2. * *This is not a hard result. One way to see it is to observe that the Koszul complex KZ ss*IBP*, is given by P -gv(q) KZ ss*q-j IBP' Hom (KZ BP*j; Q=Z(p)): Since Q=Z(p)is injective, our split exactness is preserved. There is a little w* *orry about the extension but that is easy. Since this is split exact degree by degr* *ee all of the cases of interest follow. (ii) is done similarly just using_exactne* *ss on homotopy. |__| We prove several of our main results at once and get the others from them or using similar proofs. We start with the following double induction. Theorem 4.1. Morava K-theory and Brown-Peterson cohomology 23 (i)For k > q + 1 - g(q) there is a short exact sequence of K*-Hopf algebras which is split as algebras. The right hand term is polynomial if k is even* * and exterior if k is odd. K* -! K*IBP__(q)k+q+1-! K*BP__g(q)+k -! K*LqBP__(q+1)g(q)+k-! K*: K*BP_2(q+1)+k (ii)For k 0 the complex EKZ * is exact in the category of K*- Hopf algebras. K*LqBP(q+1)_2(q+1)+k (iii)For k 0 the complex KZ * is exact in the category of K*-Hopf algebras and splits as algebras. (iv)The bar spectral sequence for the Morava K-theory of the fibration BP__g(q)+k-1-! * -! BP__g(q)+k collapses. (v) For k 0 we have a short exact sequence of long exact sequences with all maps split as algebra maps: K*IBP_(q)k+2(q+1)-g (q)K*BP_2(q+1)+k K* ! EKZ * ! EKffiZ* K*LqBP_(q+1)2(q+1)+k ! KZ * ! K*: Proof.The proof is by double induction. When q = 0 all of the statements are trivial or vacuous except for (iv) which was proven in [RW80 , Theorem 12.3, p. 743]. For q > 0 and k < 0, (i) is true because the first space in the fibration is * *trivial. (iv) is true for k 0 by [Wil75]. We prove (ii) for k = 0 assuming both (i) and (ii) for q-1. According to Theo* *rem 3.29 we have homotopy equivalences BP__g(q)-ff-|vq|~=BP_g(q)-ffx BP__g(q)-ff-|vq| if ff 2. The map vq BP__g(q)-ff-! BP__g(q)-ff-|vq| induces a homotopy equivalence on the first factor and thus an isomorphism of H* *opf algebras. Filtering out by these maps we are left with showing that the followi* *ng sequence of spaces induces a long exact sequence of Hopf algebras: vq * ! K(Z(p); q + 2) ! BP__g(q)__-BP__g(q)-|vq|! Y Y BP__g(q)-|v |-|v-|! BP__g(q)-|v |-|v |-|v-!|. .:. i_g(q)__-BP__g(q)-|vq| induces an exact sequence of Hopf algebras in Morava K-theory, (1.26). That lea* *ves only exactness at Y K*BP__g(q)-|vq|and K* BP__g(q)-|v |-|v | i q i to prove. The kernel at Y K* BP__g(q)-|v |-|v | i q i is known to be the cokernel of K*K(Z(p); q + 3) -! K*BP__g(q)-|vq| by induction on q using (ii). To complete our proof we will show that this is p* *recisely the cokernel of vq* K*BP__g(q)__-K*BP__g(q)-|vq| as well. Consider the bar spectral sequence associated to the fibration BP__g(q)! BP__g(q)-|vq|! BP__g(q)-|vq|: By [RWY98 , section 8.3.2], the kernel of K*(vq) is K*K(Z(p); q + 2). Thus we have, by Theorem 3.4 (ii), E2 ~=E1 ~=Coker(K*(vq)) T orK*K(Z(p);q+2)(K*; K*); which collapses because it is even degree. This T or part is the K*K(Z(p); q + * *3) factor in K*BP__g(q)-|vq|, and the rest is just the cokernel discussed a* *bove. This concludes our proof. We prove (i) and (iii) for k = 0. The map K*K(Q=Z(p); q + 1) -! K*K(Z(p); q + 2) is an isomorphism. Recall that IBP__(q)q+1= K(Q=Z(p); q + 1). This, togeth* *er with the injection of (1.26)and Theorem 3.4 (iii) give the short exact portion * *of (i). The exactness (ii) is just (iii) for k = 0 spliced together with the short* * exact sequence (i). The term K*LqBP__(q+1)g(q)is the last term in the Koszul compl* *ex (iii) (for k = 0) and therefore injects into a polynomial algebra. By [Bou96b , Theor* *em B.7], which says a subHopf algebra of a polynomial Hopf algebra is polynomial too, it must be polynomial. Since K*LqBP__(q+1)g(q)is polynomial, we get (i)* * with the splitting for k = 0 from [Bou96b , Theorem B.9], which says that a short ex* *act sequence of Hopf algebras which ends with a polynomial algebra is split as alge* *bras. Combining these two results of Bousfield's we see that all the maps in (iii) mu* *st split. There is little content to (v) for k = 0. Morava K-theory and Brown-Peterson cohomology 25 We now do our induction on k. Assume all parts of the theorem f* *or (k - 1) 0. We show (ii) and (iv) simultaneously. Taking the bar spectral se- K*BP_2(q+1)+k-1 quence on all of EKZ * we have a spectral sequence converging to K*BP_2(q+1)+k EKZ * . By induction we know that all of it collapses except possibly the q-th term, i.e. (iv) for k. Furthermore, everything in sight which we use* * in our induction is split as algebras, see (v). Because all of the maps split as a* *lgebras before we take Tor, we have exactness for the E2 term of the bar spectral seque* *nce K*BP_2(q+1)+k-1 on EKZ * after we take Tor. Since we know that all but one of K*BP_2(q+1)+k these collapses, we get exactness for EKZ * except at the point K*BP_2(q+1)+k K*K(Z(p); q + 2 + k) ! K*BP__g(q)+k! EKZ q-1 K*BP_2(q+1)+k recalling that K*BP__g(q)+k= EKZ q . We have exactness as Hopf algebras on the T or level of the spectral sequence so if we can show that the * *spectral sequence collapses, i.e. (iv), then we will also have our exactness (ii). We wa* *nt to remind the reader that our induction is on k. We are assuming that we know everything for k - 1. We could start this induction because we have already sho* *wn k = 0. In particular, we know the Morava K-theory of all spaces involved for k * *- 1 as algebras, which is all it takes to determine Tor. We want to emphasize that * *we are not doing induction on degrees. If k is even then everything (all of the T* *or groups) is in even degrees and the spectral sequence collapses. If k is odd th* *en all generators must be either odd degree in the first filtration, in which case* * they have no differentials, or in even degrees, in which case their target must be a* *n odd degree element in the first filtration. By the exactness of the spectral seque* *nce, and the fact that the first term above is even degree, we see that all of these* * odd elements must map to the next term which is part of a collapsing spectral seque* *nce and therefore these elements cannot be targets of differentials. This concludes* * the proof for (ii) and (iv). We now do the induction for (i) and (v). The left hand term of (v) is already known to be long exact and to split as Hopf algebras by Theorem 2.4 (i). We have just proven the center term, (ii), is exact. By induction, we have the injectio* *n from the left to the center except for the q-th term of the complex. So, in the diag* *ram (4.2), the vertical columns are exact, the top horizontal arrow is an isomorphi* *sm and the third horizontal arrow is an injection. This is enough to force an inje* *ction on the second horizontal arrow. By Theorem 3.4 (iii), we get our exact sequence (i). We now have a short exact sequence for (v) with the first two sequences long exact. This forces the third* * to be long exact and gives us (iii). Because all of the terms in (iii) are either pol* *ynomial or exterior, Bousfield's results tell us that all maps are split as algebra maps. * *Likewise the splitting for (i) follows too. All of the splittings together give us the s* *plittings for (v). __ |__| We have proven our basic results about short and long exact sequences now, namely, we have proven Theorem 2.2 (i)and (iii), and Theorem 2.4 (i), (ii), and the s = q version of (iii). We still have a number of such sequences to verify.* * We do this mostly from the ones we know or with similar techniques. 26 Takuji Kashiwabara and W. Stephen Wilson K* K* | | | | | | |? |? ' K*K(Q=Z(p); q + 1 + k)______- K*K(Z(p); q + 2 + k) | | | | | | |? |? (4.2) K*IBP__(q)q+1+k ______- K*BP__g(q)+k | | | | | | |? |? K*IBP_(q)q+1+k-g_(q)| K*BP_2(q+1)+k EKZ q-1 |v____- EKZ q-1 | | | | | | |? |? 5. Proofs of odds and ends Proof of the short exact sequence part of Theorem 2.2W(ii).e map the three term sequence of (ii) to the known short exact sequence (iii). We know that the left hand term of (ii) injects to the left hand (iii) and the middle term is an* * iso- morphism. This forces the injection of the left hand term of (ii) into the midd* *le. By Theorem 3.4 (iii) we have the short exact sequence of (ii). The splitting and identification with the Goerss-Hunton-Turner generalized tensor product_will_co* *me later. |__| Proof of Theorem 2.2 (iv)and Theorem 2.4 (v).Now we have a map of short exact sequences of Theorem 2.2 (ii) to (iii) and we get a six term exact sequence rel* *ating the kernels with the cokernels (the snake lemma, see Remark 3.1) except that si* *nce the middle term gave an isomorphism we have an isomorphism from the kernel of the right hand map to the cokernel of the left hand map. This gives an algebraic version of (iv) but not one that we know comes from the geometric map! When r < g(q) the left hand term is trivial and the other two are isomorphic. For r * *= g(q), (iv) is just the sequence (iii). When s r + gffi(q) this sequence is just (iii* *). To get the correct version coming from the geometric map we have to go back and use our previous little trick with the Koszul complexes. We can turn all three terms in* *to the non-extended Koszul complexes of Theorem 2.4 (v). For r > g(q) we want our short exact sequence to be the q-th term of our Koszul complexes. By induction we have short exact on all the lower terms of the complex. We also have split exact on the homotopy of the left terms so that Koszul complex is long exact. We have already covered the case above when we are in the range when the extension* * is there. The right hand side is already known to be long exact from Theorem 4.1 (* *v), s = q (Theorem 4.1 (iii)). The left hand term of (iv) injects into the (q -1)-t* *h term of its Koszul complex which in turn injects into the middle. This forces inject* *ion of (iv) and thus the short exact sequence (iv) coming from the geometry. Because Morava K-theory and Brown-Peterson cohomology 27 we have a short exact sequence of complexes with both left and right long exact* *,_ the middle must be long exact too, Theorem 2.4 (v). |__| Proof of Theorem 2.4 (iii).The s = q case is done already. The short exact se- quence of Theorem 2.2 (ii)gives Theorem 2.4 (iii)as a short exact sequence of * *__ complexes. However, all three terms are already known to be long exact. |* *__| Proof of Theorem 2.4 (iv).We want to prove the Z=(pc) version of the exact- ness of the Koszul complex now, the middle term of Theorem 2.4 (iv), i.e. for c)K*BP_2(q+1)+k EK Z=(pZ* . There is an obvious injection of Koszul complexes K*BP_2(q+1)+k c K*BP_2(q+1)+k KZ* -! KZ=(p )Z* : Note that these are the unextended versions. The quotient is just another copy K*BP_2(q+1)+k of KZ* shifted by one in the Koszul degree. We get a long exact sequence in the homology of the short exact sequence of Koszul complexes. We know that the homology of the first one is just K*K(Z(p); q + 2 + k) in homolog* *ical degree q + 1. The homology of the quotient is just K*K(Z(p); q + 2 + k) in homo- logical degree q + 2. The boundary homomorphism is easily seen to be pc*which is known to be surjective, [RW80 , Corollary 13.1], with kernel K*K(Z=(pc); q + 1 * *+ k) in homological degree q + 2. Thus we have the proper Hopf algebra for the exten- K*BP_2(q+1)+k sion of KZ=(pc)Z* which it takes to make it exact. We get the fact that it comes from the geometric map from the commutativity of the diagrams geometrically. We get the short exactness from Theorem 2.2 (iii) plus the extension. We alre* *ady know the first two are long exact so the short exactness gives us the third_one* * is long exact too. |__| Remark 5.1.We wish to mention here that for any of the Koszul complexes which are long exact and which are not extended, the last term is completely algebrai* *cally determined inductively. In particular this gives us part of the statement of Th* *eorem 2.4 (iii). Proof of the GHT tensor product identification of Theorem 2.2 (ii), Theorem 2.3, and Theorem 2.11. Most of the remaining Morava K-theory results rely on an understanding of the various spectral sequences associated with the sequence of fibrations (2.10). We can analyze them one at a time using the fact that we alr* *eady know all of the answers. Since we need to use it soon we take a break to prove the vq injection for r * *even in Theorem 2.3. When r g(q - 1) (and even) we have the splitting Theorem 3.29 and get the injection easily. when r > g(q) and even we use the Koszul long exact sequence on the right hand side of Theorem 4.1 (v). We know that the last term injects in the second to last term but we know by induction that all of the terms in that product inject into the preceeding one except the one which our l* *ast term maps to by vq, so it must be an injection too. (This argument fails for r * *odd because when r = g(q - 1) + 1 we are neither in the range of the splitting nor * *in the range of the Koszul complexes.) Let r0= r + 2(pq - 1) in the commuting diagram (5.2)of short exact sequences of Hopf algebras. 28 Takuji Kashiwabara and W. Stephen Wilson K*! K*IBP_(q)r0-gffi(q)!K*BP_r0!K*LqBP_(q+1)r0!K* |vq* |vq* |vq* |? |? |? K*! K*IBP_(q)r-gffi(q)!K*BP_r! K*LqBP_(q+1)r! K* | | | |? |? |? (5.2) K*! K*IBP_(q+1)r-gffi(q-1)!K*BP_r!K*Lq-1BP_(q+2)r!K* | | | |? |? |? K*! K*IBP_(q+1)r0-gffi(q)+1!K*BP_r0+1!K*LqBP_(q+2)r0+1!K* |vq* |vq* |vq* |? |? |? K*! K*IBP_(q+1)r-gffi(q)+1!K*BP_r+1!K*LqBP_(q+2)r+1!K* All of the horizontal and vertical maps are induced by the corresponding fibr* *a- tions. The right one comes from Remark 3.27. We are interested in the bar spect* *ral sequence of the middle vertical column and so we take three rows at a time. The bar spectral sequence of the left hand side is always even degree and collapses, Theorem 3.4 (iv). It maps to the bar spectral sequence for the middle and so th* *ere can be no differentials on any of these elements by naturality. We start with the first three rows. We know the top left vertical map is surj* *ec- tive on homotopy and so it is surjective as Hopf algebras since the left side i* *s all finite Postnikov systems. The kernel of the homotopy is just the homotopy of the delooping of the lower left space. The bar spectral sequence for the left vert* *ical fibration is in even degrees and therefore collapses. Our concern of course is * *with the bar spectral sequence for the vertical fibration in the middle. We start wi* *th r even. We know from above that the right top vertical map is injective and that everything in sight is even degree. Thus, the E2 term of the spectral sequence from Theorem 3.4 (iii)is all even degree and collapses. This gives exactness at K*BP__rfor Theorem 2.11 if r is even. If r is odd, then the top two on the r* *ight hand side are known to be exterior Hopf algebras and the short exact sequences they are part of split as Hopf algebras. If there is a kernel, and sometimes th* *ere is, then it too must be exterior as is the cokernel. The E2 term of the spectr* *al sequence has two parts as usual, see Theorem 3.4 (ii). One part is the Tor of t* *he kernel. This comes in two parts, one part from the left hand side which we have already noticed has trivial differentials because the spectral sequence for the* * left hand side collapses. The second part will come from whatever kernel there is on the right hand side. Since it is exterior the Torwill be a divided power algebr* *a all concentrated in even degrees. We have no obvious restrictions on differentials * *on this part. We also have the cokernal part of E2. This will be exterior and if w* *e can show that all of these elements exist in K*BP__rthen there will be no pl* *ace for differentials to land. These exterior classes are in the zeroth filtration * *so they have no differentials and the even stuff can only land on an odd class and thes* *e are our only odd classes. So, our goal is to show that these exterior elements in t* *he cokernel all survive. Then we must identify the exterior kernel part if there i* *s one. Morava K-theory and Brown-Peterson cohomology 29 Our proof here uses some techniques which we have not yet had to invoke for t* *his paper and we doubt the necessity of having to do this now but we need to prove * *the result. For r even we found the right vertical maps to give a short exact seque* *nce. The Milnor-Moore theorem about exactness of indecomposables still holds in the category of Hopf algebras we are working in, namely we get an exact sequence: (5.3) . .-.! QK*LqBP__(q+1)r0-! QK*LqBP__(q+1)r -! QK*Lq-1BP__(q+2)r-! 0: There are some dramatic differences however and for the presently perplexed we should point some of these out. The top two terms of our vertical short exact sequence on the right side are both polynomial algebras (recall r is temporarily even). The bottom one is split as algebras and is part polynomial and sometimes has the Morava K-theory of an Eilenberg-Mac Lane space in it (to be identified soon). So, we can have a non-trivial extension here. All iterated p-powers in t* *he Morava K-theory of Eilenberg-Mac Lane spaces either go to zero or become cyclic* *al, see [RW80 ]. One way to achieve this as a quotient of two polynomial algebras w* *ould be to take a sequence of generators mapped as follows: x0 ! yp0; x1 ! y0 + yp1; x2 ! y1 + yp2; etc. Note that there are no indecomposables for the Morava K-theory of Q=Z(p) Eilenberg-Mac Lane spaces! An easy example of this is to consider a little Hopf algebra generated by x with xp = x. Its module of indecomposables is trivial. S* *o, Milnor-Moore's exact sequence is not as useful for us as it is in the graded ca* *se. A serious failing is that a surjection of indecomposables does not imply a surjec* *tion of Hopf algebras (map the trivial Hopf algebra to the example just given). However, we are still going to extract the information we need from the exact sequence f* *or the indecomposables. The polynomial part of the lower right hand part splits of* *f, as algebras, from both the lower right and the middle right terms. Let us call * *this P1. Thus the middle term is just the tensor product of two polynomial algebras,* * P1 and P2, one isomorphic to the polynomial part of the lower right term and the o* *ther in the middle of a short exact sequence starting with the upper right polynomial algebra, P3, and ending with the Morava K-theory of an Eilenberg-Mac Lane space part of the lower right. Since the indecomposable module of the last one is tri* *vial, the exact sequence gives a surjection from the indecomposables of the top right onto the indecomposables of the part of the middle right term involved in our s* *hort exact sequence, QP3 ! QP2 ! 0. We have some understanding of the behavior of the generators of the top two terms on the right row. We know that the bar spectral sequence for their delooping collapses. We know this because Torof a p* *oly- nomial algebra is just an exterior algebra with generators in filtration 1 and * *so there can be no differentials. For the bottom term we have to worry a little about the Eilenberg-Mac Lane part but we know that differentials on those elements are al* *so trivial because they come from our finite Postnikov systems which always collap* *se. Thus, the Tor of the middle term is the tensor product of two exterior algebras. One maps isomorphically to the corresponding exterior algebra in the lower right hand term because of the splitting off of the polynomial algebra. Because of our surjection of indecomposables from P3 to P2, we get that the exterior generators P2 creates when we take Tor are all in the image of our map from the top. The point here is that we have shown that our exterior cokernel all survives and so* * the 30 Takuji Kashiwabara and W. Stephen Wilson spectral sequence we care about collapses, and, we have also shown that we have exactness at the middle term, K*BP__rfor r odd. Assuming we have a kernel for r odd, then it is easy to identify as having to* * have a Tor which gives the Eilenberg-Mac Lane part of the lower right hand term. At first glance, looking at all our results in Theorem 2.2, it appears that it sho* *uld have all of the Morava K-theory of the q and q + 1 homotopy of IBP__(q-1)r-gf* *fi(q-1), i.e. of a two stage Postnikov system. However, we know from sparseness that the* *re are never two non-trivial homotopy groups in adjacent degrees. In fact there are other restrictions and we can now see that it must be in degree q + 1 or not at all. The only degrees that the homotopy can be in are: r - gffi(q - 1) - 2t(q +* * 1). Recalling that r is odd, mod 2 this is (1 + q). Thus we can have (q + 1) degree homotopy but not q. This particular spectral sequence argument, which shows that the cokernel of vq*always injects, gives us the identification of the Goerss-Hunton-Turner tens* *or product in Theorem 2.2 (ii), which was the last remaining thing to prove in that theorem. We also have finished the proof of Theorem 2.3. We pursue our calculations with these short exact sequences in order to prove Theorem 2.11. We move now to the next spectral sequence, for the fibration: BP__r-! BP__r-! BP__r0+1: Going back to our diagram (5.2)we consider now rows two through four. We have already analyzed the maps needed for the spectral sequence. The upper left map is zero and we see right off that the spectral sequence for it collapses. L* *et us first just look at the case for r even. The kernel of the right hand map is kno* *wn to be K*LqBP__(q+1)r0which is known to be polynomial. We also know the map is surjective. Thus we have the Torpart comes in two parts. One is from the fin* *ite Postnikov system and we know that it has no differentials in it by naturality f* *rom the left hand spectral sequence. The second is from the polynomial part and it gives an exterior part and so has no differentials and there is nothing which c* *an be a source of differentials which can hit it. Thus the spectral sequence collapse* *s and we see we have exactness at the middle term. Now we let r be odd. We know that the kernel on the right is exterior so the E2 is concentrated in even degrees and the spectral sequence collapses giving us exactness again at the middle space. We are ready for our third and final spectral sequence. This comes from our middle vertical term of the last three rows of the big diagram and we only cons* *ider these last three rows. This time the upper left vertical map injects and so the* * left spectral sequence collapses and is even degree as usual. Let's start with r odd* * this time. The kernel of the right hand top map is exterior and perhaps some Eilenbe* *rg- Mac Lane stuff. In any case it has even degree Torso the spectral sequence coll* *apses and we get exactness at the middle term. For r even, the upper right corner is two parts, (1) polynomial and (2) the (q + 1) Eilenberg-Mac Lane part if there is any and the map to the next term is trivial. Because we are now working with odd degree spaces in the second and third rows, we recognize from our previous calculation that these spaces are re* *ally equivalent to the q connected one on the left and (q + 1) connected one on the right. Tor of the kernel is all even degree and in E2 we end up with too much o* *dd Morava K-theory and Brown-Peterson cohomology 31 degree stuff because we know we do not have injectivity of the middle right hand side to the lower right hand side. We also know the kernel as A(q; r + 1). Th* *is must all be hit by differentials then, i.e. this spectral sequence does not col* *lapse when A(q; r + 1) 6= 0. There can be no differentials coming from elements on the left because their spectral sequence collapses. We must then look at Tor of the kernel on the right. First, there is the polynomial part which gives rise to ex* *terior elements. These must all survive. We know they survive in the spectral sequence for delooping and so by naturality any differential which hit them in the spect* *ral sequence for BP__r+1would have to map to a differential which hit them in the spectral sequence for BP__r+1. However, the Tor of the Eilenberg-Mac Lane part, K*K(ssq+1IBP__(q-1)r-gffi(q-1); q + 1) fits nowhere in our known a* *nswer. We know how this Torlooks from [RW80 , Theorem 12.3]. It is a divided power algebra on transpotence elements in the second filtration. Since it must disappear the * *only possibility is a d2 which takes these transpotence elements isomorphically to t* *he exterior generators of A(q; r + 1), thus killing off all of A(q; r + 1) tensore* *d with the Torof this Eilenberg-Mac Lane part. Exactness fails here. This finishes our description of the behavior of the spectral sequences and a* *lso of our proof of Theorem 2.11. It is now easy to see why this approach to the origi* *nal_ problem failed us so badly. |__| Proof of Theorem 2.12.We look at the bar spectral sequence for the fibration (5.4) IBP__[q]r+1-gffi(q)-! LqBP__(q+1)r-! LqBP__r: The first map is trivial in Morava K-theory. At first glance this is not so obv* *ious. We know the target to be either an exterior algebra on odd generators or a poly* *no- mial algebra on even generators. The Hopf algebra splittings of [HRW98 ] tell * *us that if it is exterior then there are no maps. To do the even case you have to know * *some- thing explicit about the Morava K-theory of the Eilenberg-Mac Lane spaces. The Dieudonne modules are written down in [SW98 ] and they are p-divisible groups. Polynomial algebras have no p-divisible elements in their Dieudonne modules so there can be no maps. Thus the E2 term of the bar spectral sequence is O [q+1] K*LqBP__r' K*LqBP__(q+1)rK*IBP__r+1-gffi(q): When r is even this is even degree and so collapses. It doesn't always collapse* *_when r is odd. |__| 6.The BP results Remark 6.1 (The K"unneth isomorphism).We use freely two previous results throug* *h- out. If E*(X) and E*(Y ) are both Landweber flat, then O E*(X x Y ) ' E*(X)d E*(Y ) [RWY98 , Theorem 1.11] and is also Landweber flat [Wil99a, Theorem 1.8]. In [RWY98 , Theorem 1.19] it was proven that if you had maps of H-spaces f1 f2 (6.2) X1 ______-X2______-X3 32 Takuji Kashiwabara and W. Stephen Wilson with the compositon trivial and all spaces having even Morava K-theory and givi* *ng an exact sequence of bicommutative Hopf algebras for all Morava K-theories (6.3) K* -! K*X1 -! K*X2 -! K*X3 -! then we got (6.4) E*(X1) ' E*(X2)=(f*2): The statement in [RWY98 ] is only for E = P (m) but the proof starts assuming * *the result for E = K(n) and then proves it for all of our E on the way to P (m). Another theorem is that spaces with even Morava K-theory have Landweber flat E cohomology concentrated in even degrees, [RWY98 , Theorem 1.8]. After this paper it was noticed that the theorem of (6.4)was true even if you drop the even Morava K-theory assumption and replaced it with the weaker Landweber flat assumption, [Kasb ] and [Wil99a]. In fact one does not need even this assumption on the space X1 to conclude the result and the fact that it too is Landweber fl* *at. This is made explicit in [Wil99a] and is implicit in [Kasb ]. What is really pr* *oven is that E*(-) for the sequence of spaces Xi is coexact in the category of algeb* *ras. This coexactness is just the definition of a cokernel, or, equivalently, (6.4). In order to prove Theorem 1.1 we need a dual version of (6.4), something not to be found in any of [RWY98 ], [Wil99a] or [Kasb ], caused principally, we su* *ppose, by never having had a need for it before. Remark 6.5 (Completed Hopf algebras).We need to remind the reader that E*(X) always comes equipped with a topology on it which is an important part of the structure. If X and Y both have Landweber flat Brown-Peterson cohomology, then so does their product, and it is just the completed tensor product of the * *two, Remark 6.1. This means, in particular, that if X is an H-space, we get a "compl* *eted coalgebra" structure O (6.6) E*(X) -! E*(X)d E*(X): The dual result would then say something about kernels in this category. Howeve* *r, if we combine the two structures, topologized algebra and completed coalgebra we g* *et "completed Hopf algebras" and our cokernel of algebras and kernel of coalgebras become cokernel and kernel of completed Hopf algebras. If we begin with our completed coalgebras and note that the completed tensor product for this catego* *ry is the product, then the Hopf algebra maps give us a group object in the catego* *ry, i.e. a completed Hopf algebra. Our category of completed Hopf algebras is not abelian but we define a short exact sequence anyway. In our category we say f g (6.7) A _____-B _____-C is short exact if f is an injection, g is a surjection, A is the kernel of g, a* *nd C is the cokernel of f. We need to state our theorems explicitly. We assume our spaces are all of fin* *ite type, i.e. H*(-; Z(p)) is of finite type over Z(p). Theorem 6.8. Let E be as in Theorem 1.1. Given a sequence of maps as in (6.2) with the composition trivial and an exact sequence of bicommutative Hopf algebr* *as (6.3)for all Morava K-theories with the last two spaces having Landweber flat E cohomology, then the first does too and E*(X1) is the completed Hopf algebra cokernel of f*2. Morava K-theory and Brown-Peterson cohomology 33 As noted above, this was known to be an algebra cokernel. We will put the topology in it so it really is the cokernel in the correct category. The dual result then is: Theorem 6.9. Let E be as in Theorem 1.1. Given a sequence of maps as in (6.2) with the composition trivial and an exact sequence of bicommutative Hopf algebr* *as -! K*X1 -! K*X2 -! K*X3 -! K* for all Morava K-theories with the first two spaces having Landweber flat E coh* *o- mology, then the third does too and E*(X3) is the completed Hopf algebra kernel* * of f*1. Corollary 6.10.Let E be as in Theorem 1.1. Given a sequence of maps as in (6.2)with the composition trivial and a short exact sequence of bicommutative H* *opf algebras K* -! K*X1 -! K*X2 -! K*X3 -! K* for all Morava K-theories with the middle space having Landweber flat E cohomol- ogy, then the other two spaces have Landweber flat E cohomology and we get a sh* *ort exact sequence of completed Hopf algebras E* - E*X1 - E*X2 - E*X3 - E*: Proof.This follows at once from Theorems 6.8 and 6.9 after we get the flatness * *for the two ends. This flatness follows at once from the injection and the surjecti* *on,_ see [Wil99a] or [Kasb ]. |__| The proof of Theorem 1.10 now follows immediately from the Corollary and the corresponding result from [HRW98 ] on Morava K-theory. It would be nice to be able to say that the proof of this is just dual to the* * proof of the dual theorem, but that isn't quite true and so we must delve into some d* *etail here to show the difference and how to patch it up. The proof is dual once the following dual to [RWY98 , Theorem 1.18] has been proven and so this is all we need to do. Proposition 6.11.Let E be as in Theorem 1.1. Given a sequence of maps as in (6.2)with the composition trivial and a short exact sequence of K(n)* modules f* *or all Morava K-theories: K(n)*(X1) K(n)*(X2) K(n)*(X3) 0; if the first two spaces have Landweber flat E cohomology then we get an exact sequence of E* modules: E*(X1) E*(X2) E*(X3) 0 and E*(X3) is also Landweber flat. Proof.The proof really differs little from being the dual proof of the related * *theo- rem. We start off with the cofibre of f2, C(f2), but in order to get our short * *exact sequences on our cohomology theories, we have to suspend everything. The maps C(f2) -! X2 -! X3 now get a short exact sequence on all the cohomology theories used in the proof of the dual, in particular, all E*(-). If we stabilize, then we see that, by o* *ur assumptions, our map X1 ! X2 must factor through -1C(f2). The map X1 ! 34 Takuji Kashiwabara and W. Stephen Wilson -1C(f2) gives an injection on Morava cohomology and so on E cohomology. We find ourselves working with spectra using theorems which only work for spaces. However, these are suspension spectra and so the cohomology theories work the way they are supposed to. The only thing stable is the map and we use it just to get an algebraic factorization. All else, except this little trick making thing* *s_stable, is the same (i.e. dual). |__| Proof of Theorem 1.1.First we recall the maps and isomorphisms of (3.26). These are all BP module spectra and so we can replace our old spaces with our new spa* *ces throughout the paper for all of our Morava K-theory results, in particular, our Theorem 2.6 (where we need to replace K(Q=Z(p); q+1+k) with K(Z(p); q+2+k)). Remark 2.7 gives us our final hypothesis for Theorem 6.8 and our Description 2 follows by induction and Remark 6.1. Description 1 is a little different. The composition of the two maps in Theor* *em 2.9 is not zero so we have to do something different. We have the short exact sequence from Theorem 1.14 with the spaces replaced so we have finite type: K* -! K*I0BP__(q+2)r+1-gffi(q)-! K*BP__r-! K*L0qBP__(q+3)r-! K*: This gives us, by Corollary 6.10, a short exact sequence of completed Hopf alge* *bras: E* - E*(I0BP__(q+2)r+1-gffi(q)) - E*(BP__r) - E*(L0qBP__(q+3)r) -* * E*: We have, by our identification of the Goerss-Hunton-Turner generalized tensor product, an exact sequence: O vi* (q+3) (6.12) . .-.! K*BP__r+|vi|____-K*BP__r-! K*L0qBP__ - ! K*: i>q r This gives us the identification of the right hand term in Theorem 1.1, using T* *he- orem 6.9. We use Theorem 6.9 again to show O (q+3) (6.13) E*(BP__r) ' E*(I0BP__(q+2)r+1-gffi(q))d E*(L0qBP__r ): We just use the maps Y (q+2) (6.14) * x BP__r+|vi|-! I0BP__r+1-g (q)x BP__r-! BP__r: i>q ffi This is right exact in K*(-) so our completed Hopf algebra splitting follows. Thus we are only left with identifying E*(I0BP__(q+2)r+1-gffi(q)) with the* * left side of Theorem 1.1, Description 1. That will be done in the next section. * *|___| Proof of Corollary 1.4.In [RWY98 ] we use the theorem of (6.3)with the sequenc* *es (1.26)and (1.27), [RWY98 , Proposition 1.16], to calculate the Brown-Peterson cohomology of Eilenberg-Mac Lane spaces, [RWY98 , Theorem 1.14]. This sequence shows up as the extension and the last two terms of the lowest case of the Kosz* *ul complexes in the middle terms of Theorem 2.4 (iii)and (iv). The exactness for the higher cases sets us up for the use of Theorem 6.8 if we have flatness. We have already calculated the E cohomology of the spaces in the Omega spectrum __ for BP and we know they are flat so we are done. |__| Morava K-theory and Brown-Peterson cohomology 35 Remark 6.15 (The forgotten topology).When you look back to [RWY98 ], [Wil99a], and [Kasb ], the proof of (6.4)was just as algebras. Although there is nothing * *wrong with the statements and proofs in those papers, that is not really the category* * we are working in. We work with topologized algebras and so the cokernel of algebr* *as must have a topology on it and it is possible to put a topology on it so that i* *t is not the cokernel stated in our Theorem 6.8. Likewise for our Theorem 6.9. This * *is the sort of problem which prevents our category of completed Hopf algebras from being abelian. For an oversimplified example, let us look at the map R ! R0where R = Fp[vn] with the discrete topology and R0is Fp[vn] where we use the ideals (* *vkn) as open sets. This map is continuous, injective and surjective, but not an isom* *or- phism. If such things were to happen to us in our Theorems 6.8 and 6.9, then we really wouldn't have cokernels and kernels, something we want to use, for examp* *le in defining short exactness in our category of completed Hopf algebras. Injecti* *vity and surjectivity would be unaffected. So, straightening out the problems with t* *he topology is important for our results to work properly in the category of compl* *eted Hopf algebras. Thus, we have some nontrivial work yet to do on these theorems. Below when we complete the proofs, we find that for the injection we don't really need our exact sequence but for our surjection we do. That is because our oversimplified example above almost does exist. In particular, let us look at * *the surjection E(q; q + 1)*(BP__g(q)) -! E(q; q + 1)*(K(Z=(pc); q + 1)): By Remark 1.5 we know the right hand side is finitely generated and free. The left hand side has no torsion in standard homology and so any finitely generated submodule has the discrete topology on it. Thus every generator is also a gener* *ator of the q-th Morava K-theory whereas there is no q-th Morava K-theory for the Eilenberg-Mac Lane space. The vq part of the topology of the Eilenberg-Mac Lane spaces is always nontrivial, but it is always trivial for the other space, much* * like our oversimplified example (but not the same). It is worth the trouble to make sense out of this since we claim our result produces the correct topology. We need the third term to do this so that we are really working with a cokernel. What happe* *ns is, roughly, there is a sequence of generators {xi} in E(q; q + 1)*(BP__g(q)* *). x1 may map to a generator of E(q; q + 1)*(K(Z=(pc); q + 1)) which, although vq acts freely on it, doesn't support a Morava K-theory generator. So, there must be a generator y1 which comes and hits something like vt1qx1+ x2. And then another y2 must come and hit vt2qx2 + x3. This must go on indefinitely with the xi in high* *er and higher filtration. This produces the desired result of having our topology * *be both nontrivial and determined. The rest of this section is devoted to dealing with the topology in the proof* *s of Theorems 6.8 and 6.9. Completion of the proofs of Theorems 6.8 andW6.9.e first look at our new case, Theorem 6.9, as it is slightly easier but contains all of the ideas. There is * *only one place the topology was ignored which matters. In [RWY98 ], (only for even degree Morava K-theories), [Wil99a], and [Kasb ], it is proven that if X ! Y gi* *ves an injection in all Morava cohomology K-theories then it does also for all of o* *ur E cohomology theories of Theorem 1.1. What we didn't concern ourselves with in those papers was to show that our injection gives a homeomorphism of E*(Y ) to 36 Takuji Kashiwabara and W. Stephen Wilson the image with the inherited topology. If we can show this then the our topolog* *ical concerns for the health of Theorem 6.9 will be over. From [RWY98 , Corollary 4* *.8] we know that there are never any phantom maps for any of the E which we use, so we always have the topology on E*(-) complete and Hausdorf. The topology comes from our open sets F s+1E*(Z), the kernel of the map E*(Z) ! E*(Zs) where Zs is the s-skeleton of Z and we have E*(Z) ' lim0E*(Z)=F sE*(Z). To prove our result we must see that the image of F sE*(Y ) contains some F tE*(X) for some large t. An equivalent way of looking at this is that for every s ther* *e is a t such that if x 2 E*(Y )=F sE*(Y ) is nontrivial then there is a lift, x02 E* **(Y ) such that its image in E*(X)=F tE*(X) is nontrivial. Another equivalent way to say it is that there is a bound on how high the map can raise filtration on ele* *ments in E*(Y )=F sE*(Y ). We can use a couple of facts discussed in [RWY98 ]. Fir* *st, E*(Z)=F sE*(Z) is always finitely presented [RWY98 , Corollaries 3.13 and 4.8]. Second, it always has a finite Landweber filtration, so that the quotients look* * like E*=(vm ; : :;:vk) where k n, see [RWY98 , Theorem 3.10]. (We are using the E of Theorem 1.1 (and a few others in a minute) and when n isn't defined the last condition is k finite.) If n is defined then the Landweber filtration is just t* *he tensor product of E* with the Landweber filtration for P (m) (because all of our coeff* *icient rings are Landweber flat, [RWY98 , Corollary 4.8]). We begin by showing the result for Morava K-theory and anything which be- haves like Morava K-theory, namely theories with coefficient rings v-1mP (m)*, v-1mv-1nP (m)*, or v-1mE(m; n)*. In this case, E*(Y )=F sE*(Y ) is always a fin* *itely generated free module over E* by our Landweber filtration. None of our generato* *rs for Morava K-theory get mapped to the infinite filtration since there are no ph* *an- tom maps. Since we only have a finite number of generators it is easy to see we have our result. All of the other theories behave exactly the same, with the sa* *me generators. We now do a downward induction on m for E = E(m; n) and v-1nP (m). Recall from our previous papers that we have a short exact sequence: vm * * (6.16) 0 -! E(m; n)*(Y ) _____-E(m; n) (Y ) -! E(m + 1; n) (Y ) -! 0 which maps injectively into a similar short exact sequence for X. The same proo* *fs work for the theory v-1nP (m) as well. Inductively all of the elements in E(m + 1; n)*(Y )=F sare detected in E(m + 1; n)*(X)=F tfor some big t. If elements in E(m; n)*(Y )=F sare vm torsion, then since it is finitely generated, there can * *only be a finite number of them to worry about and since nothing goes to the infinite filtration we can easily handle a finite situation. Our problems only come up i* *f we have an element in E(m; n)*(Y )=F s(the same proof works for v-1nP (m)) which is vm torsion free, but which maps to an element in E(m; n)*(X)=F swhich is vm torsion and changes filtration indefinitely as you multiply more and more times* * by vm . This is precisely the situation we had in our oversimplified example in t* *he previous remark. In that example, if we invert vn, one ring, R, becomes K* and the other, R0, goes away, and we lose our injection on Morava K-theory. We must use our Morava K-theories in the same way here. If we have our vm torsion free element x 2 E(m; n)*(Y )=F s(where the topology is discrete) then it will still* * be nontrivial if we invert vm and we have already handled the case of v-1mE(m; n) * *and by naturality our x and the powers of vm times it cannot raise filtration any h* *igher than in v-1mE(m; n)*(X). Since we are finitely generated, there can only be a f* *inite Morava K-theory and Brown-Peterson cohomology 37 number of these torsion free elements to worry about so we can deal with them a* *ll at once. To get our result up to P (m) we need only use the fact that P (m)*(Y )=F sin* *jects into v-1nP (m)*(Y )=F sfor some high n. (The ring v-1nP (m)* is Landweber flat * *so we can tensor the Landweber filtration for P (m) with it to get the Landweber filtration of it. Since the Landweber filtration is finite we can find an n big* * enough so there is no vn torsion and our localization is injective.) Having already so* *lved this case we are done. We should make some noises about the m = 0 case here. When we are working over the p-adics, then there is more to the topology than just the skeletal fil* *tration and in this topology everything is p torsion modulo an open set so our arguments still work ([RWY98 , Just before Theorem 3.8]). When we do not need the p-adic* *s, then we do need the injection on rational homology, usually considered Morava's zeroth K-theory, a hypothesis we have normally not needed. Theorem 6.8 is a little more complicated because you wouldn't expect a surjec* *tion in Morava cohomology K-theory to imply the quotient topology is the same as the topology of what we are mapping onto. Our oversimplified example illustrates the point. For this we need extra information. We need the setup of (6.2)where we have not just a surjection but an exact sequence: (6.17) 0 - K*(X1 ) - K*(X2 ) - K*(X3 ) - . .:. This will prevent us from having our oversimplified example show up. If it trie* *s to, we can use X3 to make it right. Otherwise the proof is quite similar.__ |__| 7.The Brown-Comenetz dual We want this section to be as self contained as possible for readers interest* *ed in just this part of our work. The simplest case is the one we need the most, name* *ly E*(I0BP__r); where this is just an "integral" version of the Brown-Comenetz dual of BP . * *The proof is the same for the highly connected case and needs only a minor modifica* *tion for the Z=(pc) case. The finite spectra of Theorem 1.13 will still need a litt* *le discussion. Theorem 7.1. As completed Hopf algebras, O E*(I0BP__r) ' " E*(K(Z(p); r - d(I))) d(I)r and is Landweber flat. Proof.First, note that whenever we have r < 2(pq - 1) we can replace our space with the q - 1 version. We always do this when given a chance. Then, we study the map I0BP__r-! BP__g(q)+r-q-2from our section on preliminaries where we discuss the Brown-Comenetz dual, Remark 3.5. We know that r q + 2 by the previous remark. We can continue with iterated boundary maps (from the introduction) until we have (7.2) I0BP__r-! BP__g(q)+r-q-2-! BP__g(r-2): 38 Takuji Kashiwabara and W. Stephen Wilson Note that this is independent of q! Mapping K(Z(p); r) ! I0BP__rand compos- ing, we get a sequence vr-2 (7.3) K(Z(p); r) -! BP__g(r-2)___-BP__g(r-2)-2(pr-2-1): This is the sequence which gives rise to the exact sequence of Hopf algebras in Morava K-theory (1.26). For d(I) < r we have a map vI 0 0 0 (7.4) I0BP__r____-I_BP__r-d(I)= I_BP_r-d(I)-! BP__g(q0)+r-d(I)-q0-2-! BP_2_g(r-d(I)-2): Here we might have to replace I0BP__r-d(I)using a smaller q, i.e. q0. The ma* *ps xffivI (7.5) I0BP__r___- Y xvr-d(I)-2 BP_2_g(r-d(I)-2)________- d(I)r Y BP_2_g(r-d(I)-2)-2(pr-d(I)-2-1) d(I)r give us an exact sequence (7.6) K* -! K*I0BP__r-! O K*BP_2_g(r-d(I)-2)-! d(I)r O K*BP_2_g(r-d(I)-2)-2(pr-d(I)-2-1)-! . .:. d(I)r We are almost ready to use our Theorem 6.8. Since all of our spaces on the righ* *t are torsion free spaces with even homology, we only need the version from [RWY98 ]. There is one remaining thing to do. We must check that the composition of maps in (7.5)is trivial. Since the iterated boundary maps commute with multiplication by vi it is enough to check our generic map (7.2)composed with multiplication by vr-2 is trivial. By the commutativity with the boundary map, it is enough to see that the map (7.7) I0BP__r-2(pr-2-1)-! BP__g(r-2)-2(pr-2-1) is trivial. From Theorem 3.29 we know that (7.8) BP__g(q0)-! BP__g(q0+1)-1 is trivial and our map of concern factors through this. One has to worry a litt* *le about the low dimensional cases, but when q0 = 0 we don't have a composition__ because there is no v0 in what we are doing. |__| If we want to do the case of E*(I0BP__(s)r) the same proof works, we just * *don't have to use as many maps. The only difference in the proof for Theorem 1.11 in the introduction and this is we must also use the maps pc to get exactness from [RWY98 ]. Showing the composition is trivial is easy since pc on our space is * *trivial. The connected case is similar. Morava K-theory and Brown-Peterson cohomology 39 Proof of Theorem 1.13.FromcRemark 3.5 we know we have a map of BP I to either I0BP or IZ=(p )BP which is split injective on homotopy. We need * *only_ restrict to the maps in the above proof which correspond to the homotopy here. * * |__| 8. CTOR We have given our motivation for looking at CTor in the introduction and so we can get to work on the calculations immediately. We can start assuming equation (1.21)in the introduction. Proposition 8.1. K*BP_ BP K*BP_ dir limKBP*Z* * ' K *Z* *: Since homology respects direct limits it will be enough to compute K*BP_ H*(KBP*Z* * ) for big m. Proof.It isn't even clear that we have a map to begin with. We look at K*BP_ KBP*Z* r where r < g(q) + 2m - 2q and show that there is a map K*BP_ BP K*BP_ KBP*Z* r ! K *Z* r: P LetPI = (iq+1; iq+2; : :;:im ) where each ik is 0 or 1, `(I) = ik, and let d(* *I) = 2ik(pik- 1). Then K*BP_ O (8.2) KBP*Zj r ' K*BP__r+d(I): `(I)=j We have r + d(I) < g(q) + 2(m - q) + d(I) g(m). In this range we have an H-space splitting (Theorem 3.29) of all of the spaces and so we also get a Hopf algebra splitting and have maps K*BP__r+d(I)! K*BP__r+d(I). This is all we need. |___| We can use the above proposition to calculate with because of the following. Note that we always have a map (m q): K*BP BP K*BP_ (8.3) KBP*Z* r -! K *Z* r Proposition 8.4.For r g(q) + 2m - 2q, the map (8.3)induces an isomorphism on homology. Proof.Again, by the splitting Theorem 3.29 we can see what the kernel of the map (8.3)is. For each I with im+1 = 1 we have a copy of K*BP__r+d(I)and if im+1 = 0 we have: (8.5) O K*BP__r+d(I)' K*BP__r+d(I)+2(pim+1-1)K*BP__r+d(I) and our kernel is K*BP__r+d(I)+2(pim+1-1). This makes it easy to calcula* *te the homology since vm+1 maps the first kind of term with an im+1 = 1 isomorphi- cally to the second kind of term with the same I except with im+1 = 0. Thus the 40 Takuji Kashiwabara and W. Stephen Wilson homology is trivial and since we are taking the homology of a complex in a short exact sequence of complexes, the other two (i.e. the two in our proposition)_mu* *st have isomorphic homologies. |__| K*BP_2m+2 From Theorem 2.4 (iii), we have the homology of KZ * is given by the extension, i.e. K*BP_2m+2 Hi(KZ * ) ' K* i 6= m (8.6) ' K*K(Q=Z(p); m + 1) i = m: Before we proceed we must insert a previously unstated theorem. We would like K*BP_2m+1 to know the homology of the complex KZ * . Proposition 8.7. K*BP_2m+1 Hi(KZ * ) ' K* i 6= m ' A(m; g(m - 1) + 1) i = m where A(m; g(m - 1) + 1) is an exterior algebra with TorA(m;g(m-1)+1)(K*; K*) an associated graded object for K*K(Q=Z(p); m + 1). These A have already shown up in Theorems 2.3 and 2.11. To simplify notation, we will denote this special A, A(m; g(m - 1) + 1) by Am . It is the first nontr* *ivial A in the sense that A(m; i) = K* when i g(m - 1). Proof.Most of the work has already been done, it is just a matter of reinterpre* *ting it. In particular, we go back to the proof of Theorem 4.1 (ii) for k = 0. It wo* *rks just as well for k = -1 if we ignore the extension and use the fact that the ke* *rnel of vm* (8.8) K*BP__g(m)-1______-K*BP__g(m-1)+1 is just our Am . This last fact we know from the proof of the identification of* * the GHT tensor product in Section 5. The proof is quite degenerate in the case we__ need since this is the first nontrivial A. |_* *_| We have, using Propositions 8.1, 8.4, 8.7, and equation 8.6 completed the fol- lowing calculation. *] Corollary 8.9.CTorK*[BPm (K*[BP <0>*]; K*BP__i) ' K*K(Q=Z(p); m + 1) i = 2m + 2 Am i = 2m + 1 K* i 6= 2m + 1 or2m + 2: This completes the answer to a previously unasked question. Namely, we have K*BP_ already calculated the homology of KZ * r when r 2m+2 in Theorem 2.4. We can compute the homology for lower r now. We have, using Propositions 8.1, 8.4, and Corollary 8.9 completed the following calculation. Morava K-theory and Brown-Peterson cohomology 41 Corollary 8.10. K*BP_ Hj(KZ * r )' K*K(Q=Z(p); r - m - 1) j = m and r 2m + 2 ' K*K(Q=Z(p); j + 1) r = 2j + 2 < 2m + 2 ' Aj r = 2j + 1 < 2m + 2 ' K* r and j otherwise. *] Our goal is to calculate CTorK*[BP* (K*[BP *]; K*BP__*). We have done the q = 0 case now. By Propositions 8.1 and 8.4, and equation (1.21)in the introduc- tion, we see that *] * BP K*BP_ (8.11) CTorK*[BP*(K*[BP ]; K*BP__r) ' H*(K *Z* r) K*BP_ when r g(q). However, KBP*Z*_ r is just K*BP__rwhich we know to be CTor0, i.e. K*[BP *] K*[BP*]K*BP__r. So, we can begin our induction with the lemma: *] Lemma 8.12. For r g(q), CTorK*[BPj (K*[BP *]; K*BP__r) ' K*BP__r j = 0 __ ' K*[BP *] K*[BP*]K*BP__r and ' K* j > 0: The exterior algebras, A, which come into the calculations above are not of m* *uch interest to us. It makes calculations a lot easier to work modulo exterior alge* *bras for the higher CTor groups. The Morava K-theory of Eilenberg-Mac Lane spaces, the part we are interested in, never contains exterior generators as it is conc* *entrated in even degrees. These Hopf algebras are known to split into even degree parts * *and exterior Hopf algebras and there are never any maps between them, see [HRW98 ]. We are now ready to prove Theorem 1.20 from the Introduction. We always work modulo exterior algebras now in our CTorj when j > 0. We denote this by CTorE. Working modulo the exterior part simplifies Corollary 8.9 and gives the q = 0 version of Theorem 1.20 which grounds our induction. Theorem 8.13. Let r - g(q) - j > 0, the only possible positive degree nontrivi* *al CTorE groups are, *] * CTorEK*[BPr-g(q)-j(K*[BP ]; K*BP__r+r-g(q)-j) ' K*K(ssr-gffi(q)-jIBP_r-gffi(q); r - gffi(q) - j): Proof of Theorem 1.20.We compare the answer in Theorem 1.14 to what we have here. First, we are not working modulo exterior algebras in CTor0 and we have t* *he right side of Theorem 1.20 is just the GHT tensor product, which is our CTor0. The left side of Theorem 1.20 is precisely given by Theorem 8.13 above, which_is written just as in Theorem 1.20. |__| Proof of Theorem 8.13.We do this by induction on q having grounded our induc- tion with the q = 0 case in Corollary 8.9. Our induction proceeds using a B"ock* *stein 42 Takuji Kashiwabara and W. Stephen Wilson spectral sequence which comes from the exact sequence vq+1 BP * ______-BP *______-BP *: This goes over to [vq+1] K*____-K*[BP *] ______-K*[BP *]______-K*[BP *]____-K*: From this we get a long exact sequence in CTorE . (Moding out by our exterior algebras does not destroy exactness because the category of Hopf algebras split* *s.) Assuming by induction that we know the q version of our CTorE we can attempt to use our B"ockstein spectral sequence. First we must note that all elements * *of CTorE for the q + 1 case are [vq+1] torsion. This is because they must eventual* *ly land up in the groups of Lemma 8.12 which are all trivial. Now note that the j * *of CTorEj for the q case is the same number used for the Eilenberg-Mac Lane space. Thus, any differential in the B"ockstein spectral sequence must be a map of Mor* *ava K-theory of Eilenberg-Mac Lane spaces which are in different degrees. All such maps are trivial by [HRW98 ]. Thus our spectral sequence collapses. In princip* *le there are two possibilities. The first is that an element in the q case comes f* *rom the q + 1 case and the element it comes from is [vq+1] torsion free. However, we ha* *ve shown that there are no such elements. Thus, all elements must map nontrivially around the boundary. The image of each of these elements must then be infinitely [vq+1] divisible. Another way to say this is that the long exact sequence of CT* *orE relating the q + 1 case to the q case is short exact in that the reduction of C* *TorE for q + 1 to q is always trivial (remember that we are only dealing with CTorE * *jfor j > 0). The maps on CTorE precisely mimic the stable maps on homotopy which also give a short exact squence: vq+1 q+1 IBP -! IBP _____--2(p -1)IBP : The details are left to the reader. |___| 9. Unstable Bousfield localization In this section we prove Theorem 1.24. Proof of Theorem 1.24.We begin by showing that LqBP__(q)ris E(q) local. First we note that LqBP__ris E(q) local by [Bou82 , Proposition 1.3]. Next, we want LqBP__[q]rto be E(q) local. This follows from Bousfield's most recent [Bou ]* * where he shows that the E(q) localization of a connected p-local Postnikov H-space pr* *e- serves the j-th homotopy group for j < q + 1, divides the q + 1 group by its to* *rsion subgroup and rationalizes the higher groups. Then the fiber, the space we are concerned with, is also E(q) local by [Bou75 , Theorem 12.9]. All that remains to be shown is that our map LqBP__r-! LqBP__(q)ris an E(q) equivalence. This is true if it is a rational equivalence and an isomorphi* *sm on all Morava K-theories, K(n)*(-), 0 < n q, or, by [Bou99 ], a rational equivale* *nce and an isomorphism on K(q)*(-) (we cannot use [Wil99b] because our spaces are not of finite type). Since we know the homotopy of the spaces, the rational equivalence is obvious. From Theorem 2.2 (iii)we know LqBP__r-! LqBP__(q+1)rgives an isomor- phism on all K(n)*(-), n q. All we need now is just to show there is a K(n)*(-) Morava K-theory and Brown-Peterson cohomology 43 isomorphism for n q between the spaces: LqBP__(q+1)r-! LqBP__(q)r: To see this we look at the bar spectral sequence for the fibration: (9.1) K(G; q) -! LqBP__(q+1)r-! LqBP__(q)r: Here, G is the missing homotopy group (which by sparseness is frequently zero). We use Theorem 3.4 to get our isomorphism. K(n)*K(G; q), n < q, is trivial, so we easily get an isomorphism of the other two spaces. For n = q this is not tri* *vial. However, if we can show the map from it is trivial then since Tor of K(q)*K(G; * *q) is trivial by [RW80 ], the result follows from Theorem 3.4. All we need to do i* *s show that the map K(q)*K(G; q) -! K(q)*LqBP__(q+1)r is trivial. From Theorem 2.2 (iii)we know that this second Hopf algebra is eith* *er a polynomial algebra or an exterior algebra. Since our first Hopf algebra is ev* *en degree, we cannot have a nontrivial map to an exterior algebra. We also cannot * *have a map to a polynomial Hopf algebra but this is more difficult to see. The easy * *way for us is to note that the Dieudonne module ([SW98 ]) of a polynomial algebra h* *as no elements which are p-divisible. However, all of the elements in the Dieudonne module for K(q)*K(G; q) are p-divisible ([SW98 , Theorem 1.3]) (because G is a finite sum of Q=Z(p)) and therefore all maps are trivial from the first_Hopf_al* *gebra to the second. |__| References [Baa73] N. A. Baas. On bordism theory of manifolds with singularities. Math. S* *cand., 33:279- 302, 1973. [BJW95] J. M. Boardman, D. C. Johnson, and W. S. Wilson. Unstable operations i* *n generalized cohomology. In I. M. James, editor, The Handbook of Algebraic Topology* *, chapter 15, pages 687-828. Elsevier, 1995. [BKW99] J. M. Boardman, R. Kramer, and W. S. Wilson. The periodic Hopf ring of* * connective Morava K-theory. Forum Mathematicum, 11:761-767, 1999. [Bou] A. K. Bousfield. On the telescopic homotopy theory of spaces. In prepa* *ration. [Bou75] A. K. Bousfield. The localization of spaces with respect to homology. * *Topology, 14:133- 150, 1975. [Bou79] A. K. Bousfield. The localization of spectra with respect to homology.* * Topology, 18:257-281, 1979. [Bou82] A. K. Bousfield. K-localizations and K-equivalences of infinite loop s* *paces. Proc. London Math. Soc. (3), 44(2):291-311, 1982. [Bou96a] A. K. Bousfield. On -rings and the K-theory of infinite loop spaces. K* *-Theory, 10:1- 30, 1996. [Bou96b] A. K. Bousfield. On p-adic -rings and the K-theory of H-spaces. Mathem* *atische Zeitschrift, 223:483-519, 1996. [Bou99] A. K. Bousfield. On Morava K-equivalences of spaces. In J.-P. Meyer, J* *. Morava, and W. S. Wilson, editors, Homotopy invariant algebraic structures: a * *conference in honor of J. Michael Boardman, Contemporary Mathematics, Providence,* * Rhode Island, 1999. To appear. [BWa] J. M. Boardman and W. S. Wilson. k(n)-torsion-free H-spaces and P(n)-c* *ohomology. In preparation. [BWb] J. M. Boardman and W. S. Wilson. Unstable splittings related to Brown-* *Peterson cohomology. To appear. [DHS88] E. Devinatz, M. J. Hopkins, and J. H. Smith. Nilpotence and stable hom* *otopy theory. Annals of Mathematics, 128:207-242, 1988. [EKMM96] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules* * and Al- gebras in Stable Homotopy Theory. Number 47 in Amer. Math. Soc. Survey* *s and Monographs. American Mathematical Society, Providence, 1996. 44 Takuji Kashiwabara and W. Stephen Wilson [Goe99] P. G. Goerss. Hopf rings, Dieudonnemodules, and E*2S3. In J.-P. Meyer,* * J. Morava, and W. S. Wilson, editors, Homotopy invariant algebraic structures: a * *conference in honor of J. Michael Boardman, volume 239 of Contemporary Mathematics, * *pages 115-174, Providence, Rhode Island, 1999. American Mathematical Society. [HRW98] M. J. Hopkins, D. C. Ravenel, and W. S. Wilson. Morava Hopf algebras a* *nd spaces K(n) equivalent to finite Postnikov systems. In Paul S. Selick et. al.* *, editor, Stable and Unstable Homotopy, volume 19 of The Fields Institute for Research in M* *athematical Sciences Communications Series, pages 137-163, Providence, R.I., 1998.* * American Mathematical Society. [HT98] J. R. Hunton and P. R. Turner. Coalgebraic algebra. Journal of Pure an* *d Applied Algebra, 129:297-313, 1998. [JW73] D. C. Johnson and W. S. Wilson. Projective dimension and Brown-Peterso* *n homology. Topology, 12:327-353, 1973. [JW75] D. C. Johnson and W. S. Wilson. BP-operations and Morava's extraordina* *ry K- theories. Mathematische Zeitschrift, 144:55-75, 1975. [JW85] D. C. Johnson and W. S. Wilson. The Brown-Peterson homology of element* *ary p- groups. American Journal of Mathematics, 107:427-454, 1985. [Kasa] T. Kashiwabara. Homological algebra for coalgebraic modules and mod p * *K-theory of infinite loop spaces. Preprint. [Kasb] T. Kashiwabara. On Brown-Peterson cohomology of QX. Preprint. [Kas98] T. Kashiwabara. Brown-Peterson cohomology of 1 1 S2n. Quarterly Journa* *l of Mathematics, 49(195):345-362, 1998. [Kra90] R. Kramer. The periodic Hopf ring of connective Morava K-theory. PhD t* *hesis, Johns Hopkins University, 1990. [Lan76] P. S. Landweber. Homological properties of comodules over MU*(MU) and * *BP*(BP). American Journal of Mathematics, 98:591-610, 1976. [MR] M. E. Mahowald and C. Rezk. Brown-Comenetz duality and the Adams spect* *ral sequence. American Journal of Mathematics. To appear. [MS95] M. E. Mahowald and H. Sadofsky. vn telescopes and the Adams spectral s* *equence. Duke Mathematical Journal, 78(1):101-130, 1995. [Rav84] D. C. Ravenel. Localization with respect to certain periodic homology * *theories. Amer- ican Journal of Mathematics, 106:351-414, 1984. [Rav85] D. C. Ravenel. The Adams-Novikov E2-term for a complex with p cells. A* *merican Journal of Mathematics, 107:933-968, 1985. [Rav86] D. C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres* *. Aca- demic Press, New York, 1986. [Rav87] D. C. Ravenel. The geometric realization of the chromatic resolution. * *In W. Browder, editor, Algebraic topology and algebraic K-theory, pages 168-179, 1987. [Rav92] D. C. Ravenel. Nilpotence and periodicity in stable homotopy theory, v* *olume 128 of Annals of Mathematics Studies. Princeton University Press, Princeton, * *1992. [RW77] D. C. Ravenel and W. S. Wilson. The Hopf ring for complex cobordism. J* *ournal of Pure and Applied Algebra, 9:241-280, 1977. [RW80] D. C. Ravenel and W. S. Wilson. The Morava K-theories of Eilenberg-Mac* * Lane spaces and the Conner-Floyd conjecture. American Journal of Mathematics, 102:* *691-748, 1980. [RWY98] D. C. Ravenel, W. S. Wilson, and N. Yagita. Brown-Peterson cohomology * *from Morava K-theory. K-Theory, 15(2):149-199, 1998. [Str] N. Strickland. Products on MU -modules. Transactions of the American M* *athematical Society. To appear. [SW98] H. Sadofsky and W. S. Wilson. Commutative Morava homology Hopf algebra* *s. In M. E. Mahowald and S. Priddy, editors, Homotopy Theory in Algebraic To* *pology, volume 220 of Contemporary Mathematics, pages 367-373, Providence, Rho* *de Island, 1998. American Mathematical Society. [Wil75] W. S. Wilson. The -spectrum for Brown-Peterson cohomology, Part II. Am* *erican Journal of Mathematics, 97:101-123, 1975. [Wil84] W. S. Wilson. The Hopf ring for Morava K-theory. Publications of Resea* *rch Institute of Mathematical Sciences, Kyoto University, 20:1025-1036, 1984. Morava K-theory and Brown-Peterson cohomology 45 [Wil99a] W. S. Wilson. Brown-Peterson cohomology from Morava K-theory, II. K-Th* *eory, 17:95-101, 1999. [Wil99b] W. S. Wilson. K(n + 1) equivalence implies K(n) equivalence. In J.-P. * *Meyer, J. Morava, and W. S. Wilson, editors, Homotopy invariant algebraic str* *uctures: a conference in honor of J. Michael Boardman, volume 239 of Contemporary* * Math- ematics, pages 375-376, Providence, Rhode Island, 1999. American Mathe* *matical Society. [Yag76] N. Yagita. The exact functor theorem for BP*=In-theory. Proceedings of* * the Japan Academy, 52:1-3, 1976. [Yos76] Z. Yosimura. Projective dimension of Brown-Peterson homology with modu* *lo (p; v1; : :;:vn-1) coefficients. Osaka Journal of Mathematics, 13:289-* *309, 1976. Institut Fourier, Universite de Grenoble I, U.M.R. au C.N.R.S.,, B. P. 74, 38* *402 Saint-Martin-d'Heres CEDEX France E-mail address: Takuji.Kashiwabara@ujf-grenoble.fr Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 2121* *8, Department of Mathematics, Kyoto University, Kyoto 606-8502 Japan E-mail address: wsw@math.jhu.edu