. 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 BPshows 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 BPmade 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 BPis 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):| 00. 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-jIBPis 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 BPmodulo 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 , BPI, with homotopy BPI*' 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 BPI. 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 BPto 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) IBPIBP(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)BPEKZ * *. We have a similar extended Koszul complex, EK Z* , with corresponding unstable versions. Here, c)BPc 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*BPK* -! 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 *] * BPK*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) = BPwe 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 BPhas 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)-! . . . 0spac* *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! BPand 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_