* = Z(p)[v1; : :;:vq]: There are spectra E(k; n) with coefficient rings E(k; n)* ' v-1nBP*=Ik with similar long exact sequences. A special case, when k = n > 0, is the nth Morava K-theory, K(n)*(X), with K(n)* ' Fp[vn; v-1n]. Before we state our main theorem we have a result which makes the statements easier to make. Throughout this paper we assume all of our spaces to be of the homotopy type of CW complexes with H*(X; Z(p)) of finite type. We will say that a graded object (such as the generalized cohomology of a spa* *ce) is even dimensional if it is concentrated in even degrees. Theorem 1.2. If K(n)*(X), X a space, is even dimensional for an infinite number of n, then K(n)*(X) is even dimensional for all n > 0. ____________ The first author was partially supported by the National Science Foundation. 1 2 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA We say X has even Morava K-theory if K(n)*(X) is even dimensional for all n > 0. We use the weaker sounding assumption to prove our results, but when all is said and done, the proofs show it is equivalent to having even Morava K-theo* *ry. We get simple, but interesting, corollaries: Corollary 1.3.If K(q)*(X) has a non-zero odd degree element for some q > 0 then K(n)*(X) has an odd degree element for all but a finite number of n. This does not apply to the usual extension of Morava K-theories to include K(0)*(X) = H*(X; Q) and there are examples (X = K(Z; 2n + 1), [RW80 ]) where this is non-zero in odd degrees but X has even Morava K-theory. Originally, this led us to worry a lot about the possibility of bad low Morava K-theories with K- theory "stabilizing" to even degrees. We thought that our results or proofs wou* *ld need exotic types of completion. However, such examples cannot exist. Because K(0) does not fit the pattern we must sometimes go to the p-adic completion of BP for our results. Proposition 1.4.If X and Y have even Morava K-theory, then so does X x Y . This follows from the K"unneth isomorphism for Morava K-theories and shows that the class of spaces for which our main results hold is closed under finite* * prod- ucts. We can now state our main theorem. To avoid unnecessary repetition we have: Definition 1.5.Let P (0) be BP if lim1BP *(Xm ) = 0 for each space X under discussion, and the p-adic completion of BP , BPp^, if any of the spaces do not* * have this property. Likewise, if we have chosen P (0) to be BPp^then we chose E(0; n) to be the p-adic completion as well. Remark 1.6.We make this definition so we always have the inverse limit giving t* *he cohomology, lim0P (0)*(Xm ) ' P (0)*(X): Definition 1.7.We say a P (k)*-module is Landweber flat if it is a flat P (k)*- module for the category of P (k)*(P (k))-modules which are finitely presented o* *ver P (k)*. Theorem 1.8. Let k 0. If a space X has even Morava K-theory then P (k)*(X) is even dimensional and is Landweber flat. Note that our results are strictly unstable. There are counter-examples if X * *is a spectrum and not a space. Note that this includes the case of BP when there are no phantom maps and BPp^if there are. Flatness, for BP , in the sense of our theorem has been explored by Peter Lan* *dwe- ber in [Lan76] where he proves his exact functor theorem. He shows that flat me* *ans that vn-multiplication on M=InM is always injective. So, we get BP flatness from him by proving the following: Theorem 1.9. Let a space X have even Morava K-theory. For k 0 we have short exact sequences (where v0 = p): 0 -! P (k)*(X) -vk!P (k)*(X) -! P (k + 1)*(X) -! 0 and P (k)*(X) is even dimensional. BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 3 Remark 1.10.Flatness was further studied by Zen-ichi Yosimura in [Yos76] and Nobuaki Yagita in [Yag76]. Their papers show that the above result implies that P (k)*(X) is Landweber flat. This follows from Landweber's results once it is k* *nown that P (k)*(P (k)) is a BP *(BP )-module from [Yag77]. Taken together we see that P (q)*(X) Landweber flat is equivalent to having the short exact sequences* * of Theorem 1.9 for all k q which in turn implies that P (k)*(X) is Landweber flat for all k q. A K"unneth isomorphism follows if one space is nice, and it must be a space as this is an unstable result. Theorem 1.11. Let k 0 and let X be a space with P (k)*(X) Landweber flat, e.g. if X is a space with even Morava K-theory. We have a K"unneth isomorphism: P (k)*(X x Y ) ' P (k)*(X)b P(k)*P (k)*(Y ): This generalizes early work of Peter Landweber. In [Lan70a] he has it for spe* *cial X and in [Lan76] he has it for Y finite without the completion. This K"unneth i* *so- morphism expands the number of spaces we have "computed" the BP -cohomology for quite dramatically. Recall that our spaces are all CW complexes of finite type and that P (0) is chosen according to Definition 1.5. There are similar isomorphisms for the theo* *ries E(k; n)*(-) if K(n)*(X) is even. Remark 1.12.By this K"unneth isomorphism, if X is an H-space with even Morava K-theory then P (k)*(X) has all the structure of a Hopf algebra. Although it is reasonable to ask for even Morava K-theory if you want all of these theories to be even dimensional, Landweber flatness is the really interes* *ting property and it should have nothing to do with even Morava K-theory. It seems some sort of fluke that there are so many examples of spaces with even Morava K-theory around. Presumably such spaces have a significantly deeper reason for having even Morava K-theory than their association with flatness. This is just * *the first nontrivial place the general phenomenon of a class of spaces having Landw* *eber flat Brown-Peterson cohomology has shown up. Having observed it here one would expect to see it frequently in the future in a more general setting. That futu* *re has arrived in the paper [Kasb ] by T. Kashiwabara. For example, we can see that P (k)*(QS2n) is Landweber flat but we can not see that P (k)*(QS2n+1) is. Kashiwabara can. He pushes this type of work much further than we have gone. Our results are a simultaneous generalization of previous observations on the* *se two rather different concepts: even Morava K-theory and flatness. First, if X is a finite complex then the Atiyah-Hirzebruch spectral sequence must collapse for K(n)*(X) when the dimension of the space is less than 2(pn - 1), i.e., for all * *big n. If K(n)*(X) is even dimensional for such an n and X, the mod p cohomology must also be even dimensional, which implies that there is no torsion and the integr* *al cohomology is even dimensional. It then follows that BP *(X) is free over BP *a* *nd is even dimensional. Our theorem generalizes this to infinite complexes. Second, over twenty years ago Peter Landweber, [Lan70a], computed the Brown-Peterson cohomology (at the time he worked with complex cobordism) of BG where G is a finitely generated abelian group and showed it was flat and even dimensional. This is a special case of our result applied to the first Eilenberg-Mac Lane sp* *ace, K(G; 1). The K(G; n) have many similar properties. 4 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Our results have nothing new to say about finite complexes. Infinite complexes can have many a subtle unpleasant property in cohomology. This, and other facto* *rs, motivated J. Frank Adams to steer people in the direction of homology rather than cohomology, [Ada74 ] [Ada69 ]. What we are observing is that things are not as bad as they seemed and that looking at cohomology can be rewarding. In particular it is turning out to be easier to compute and describe the cohomology than the homology in several examples. Landweber's example for BG where G is abelian should have showed the way. It was much later that BP*(BG), for G an elementary p-group, was computed ([JW85 ] ) and little progress has been made on more complicated abelian groups. Likewise, BP *(BO) was computed in a reasonable fashion ([Wil84]) before BP*(BO) was properly understood ([Yan95]). We can now add all Eilenberg-Mac Lane spaces to the list of spaces whose Brown- Peterson cohomology is completely described but whose Brown-Peterson homology is still a mystery. Although for many spaces that fit our hypothesis we do not have more detailed descriptions of the cohomology, our result is still way ahead of anything we can produce for homology. A brief description of our proof is now in order. First we show Proposition 4* *.12 that any given nontrivial element of P (k)*(X) maps nontrivially to E(k; n)*(X)* * if n is big enough. (P (k)* and E(k; n)* were defined in the opening paragraph.) S* *ec- ond, we show Lemma 5.1 that if K(n)*(X) is even dimensional then E(k; n)*(X) is also. (This allows us to "compute" E(k; n)*(X) for all spaces with K(n)*(X) even degree.) Thus, if X has even Morava K-theory, then P (k)*(X) is also even dimen- sional. This is proved using the Atiyah-Hirzebruch spectral sequence. Because a* *ll of our spaces are infinite complexes, there are technicalities to worry about. * *For example, we must show that for k > 0 there are no phantom maps in E(k; n)*(X). This, and more, is achieved using a generalization of Quillen's theorem saying * *that P (k)*(X), X a space, has only non-negative dimensional generators and a gener- alization of the Landweber exact functor theorem, which says that tensoring with E(k; n)* is exact in the category of finitely presented P (k)*(P (k))-modules, * *i.e. that E(k; n)* is Landweber flat. In the process of proof some subtle differences between cohomology and homolo* *gy for infinite complexes come to the surface. The Morava structure theorem for complex cobordism (see [JW75 ]) allows one to use the Morava K-theory, K(n)*(X), to compute the vn-torsion free part of P (n)*(X). Not so in the cohomology of infinite complexes. In fact, in all of our examples, all elements of P (n)*(X)* * are vn-torsion free, but only some show up in the Morava K-theory. This partial failure of the Morava structure theorem is compensated for by the lack of infin* *ite divisibility by vk in E(k; n)*(X), k < n, whereas in E(k; n)*(X) it is commonpl* *ace and shows up in the proof of the Conner-Floyd conjecture of [RW80 ]. In that proof it was important that the Morava structure theorem detected all of the vn torsion free part of homology and that one could have infinite divisibility as * *well. In the present work we can live without the Morava K-theory detecting all of the vn torsion free part but we must be able to eliminate infinite divisibility. The Morava structure theorem still tells us that we can recover K(n)*(X) from P (n)*(X) by using K(n)*(X) ' K(n)*b P(n)*P (n)*(X): BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 5 for infinite complexes, except that now this doesn't pick up all of the vn tors* *ion free part. Since P (n)*(X) is determined by P (0)*(X) for our special spaces with ev* *en Morava K-theory, we have a result which was first suggested in papers of Tezuka and Yagita, [TY89 ] and [TY90 ], and later in a paper of A. Kono and N. Yagita, [KY93 ]: K(n)*(X) ' K(n)*b P(0)*P (0)*(X): In fact we can replace K(n) with E(k; n). We started this project with the belief that the time had come to seriously attack the Brown-Peterson cohomology of Eilenberg-Mac Lane spaces. We tried many approaches, including the Adams spectral sequence, before we found the present one. Calculations led us to believe that it was possible everything was even dimensional; motivating our study even more. Since we began with Eilenberg- Mac Lane spaces we have a measure of satisfaction that these spaces all satisfy our conditions ([RW80 ]), and we are even more pleased that we can describe the* *ir BP-cohomology completely. We will give an algebraic construction of the Brown- Peterson cohomology of Eilenberg-Mac Lane spaces. There is a BP -module spectrum BP , [JW73 ] and [Wil75], with ss*(BP) = Z(p)[v1; : :;:vq] and for each q > 0 there is a stable cofibre sequence q-1) vq 2pq-1 (1.13) 2(p BP-! BP-! BP-! BP: This gives rise to corresponding fibrations in the -spectra for the BP, {BP* *__*}. The following is also true using P (n) cohomology in place of BP or BPp^. The K"unneth isomorphism, Theorem 1.11, gives us the Brown-Peterson cohomology for all (abelian) Eilenberg-Mac Lane spaces. Theorem 1.14. Let g(q) = 2(pq+1 - 1)=(p - 1). Then BPp^*(K(Z(p); q + 2)) is isomorphic to: BPp^*(BP__g(q))=(v*1; : :;:v*q) ' BPp^*(BP__g(q))=(v*q) and BP *(K(Z=(pi); q + 1)) is isomorphic to: BP *(BP__g(q))=(pi*; v*1; : :;:v*q) ' BP *(BP__g(q))=(pi*; v*q) for q > 0. For q = 0 delete the v*qfrom the ideal. We need the p-adic completion for K(Z(p); n), n > 2, because there are phantom maps for these spaces. However, we don't need it for finite groups. Remark 1.15.Because all of this comes from spaces and maps of spaces we have much more here than just the BP *module structure. In fact, these things are as good as Hopf algebras and the structure maps are included in what is known. Fur- thermore, everything is completely understood as unstable modules over BP *(BP_* *_*) (or BPp^*(BPp^_*)) from [BJW95 ]. Later we will give a set of algebra generato* *rs. The q = 0 version of the theorem was known to Stong and presumably others, in the 1960s. Landweber, in [Lan70a], showed these q = 0 cases were flat and th* *en calculated the result for products of these spaces. 6 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Some explanation is called for. The ideal is generated by the images of the maps in Brown-Peterson cohomology induced from the maps of spaces in the - spectrum which come from the stable maps described above in 1.13. There is a map which induces this isomorphism. It comes from the iterated boundary maps of 1.13. Unstably, the boundary map is: BP__j! BP_ _j+2pk-1: and the iteration is: K(Z=(pi); q + 1) ! K(Z(p); q + 2) ! BP_<1>_q+2p+1! . .!.BP_ _g(q): This is the same map used by Hopkins-Ravenel [HR92 ] to prove that suspension spectra are harmonic. The reason this is a satisfactory answer for us is that everything is "known" about BP *(BP__g(q)), BP *(BP__g(q)-2(pq-1)), and the map v*qbetween them. This is because BP__g(q)splits off of BP__g(q)and all of the spaces BP__*are* * well understood from [RW77 ]. In particular, in that paper we give an algebraic cons* *truc- tion for BP*(BP__k). BP*(BP__k) is a well defined quotient of this construct* *ion for k g(q) (just set all [vi] = 0 for i > q where [vi] is defined in BP0(BP__-* *2(pi-1)) using vi 2 ss2(pi-1)(BP ) ' [pt; BP__-2(pi-1)]). Since the spaces BP__k, k * *g(q) are all torsion free for ordinary homology, we know that they are BP* free and the Brown-Peterson cohomology is just the BP *dual. Likewise, the maps are just the dual maps. This theorem gives insights into H. Tamanoi's results, [Tam83b ], [Tam83a ](an announcement with no proofs), (see [Yag86, Theorem 3.3]), and vice versa. H. Tamanoi has only recently written up his work in [Tam97 ]. Not only do we know the BP -homology of the above spaces and use it to descri* *be the BP -cohomology of the Eilenberg-Mac Lane spaces, but the same can be done for Morava K-theory. Although in principle the maps are all known, in practice it can be difficult to compute them. Our main technical proposition about these spaces which allows us to go up to our BP cohomology answer is (see [HRW97 ] and [SW ] for the category of K(n)*-Hopf algebras): Proposition 1.16.Let g(q) = 2(pq+1- 1)=(p - 1). There is an exact sequence in the category of K(n)*-Hopf algebras: K(n)* ! K(n)*(K(Z(p); q + 2)) ! K(n)*(BP_g(q)) vq*-!K(n)*(BP_g(q)-2(pq* *-1)): In order to translate this Morava K-theory information into information about Brown-Peterson cohomology we have to have some general results about exact- ness for Morava K-theory implying exactness for BP . We have theorems about injectivity, surjectivity and just enough exactness for our purposes: Theorem 1.17. Let spaces Xi, i = 1; 2, have even Morava K-theory. If f : X1 -! X2 has f* : K(n)*(X2) -! K(n)*(X1) surjective (injective) for all n > 0, then f* : P (k)*(X2) -! P (k)*(X1) is also surjective (injective), for k 0. Theorem 1.18. Let spaces Xi, i = 1; 2; 3, have even Morava K-theory. If X1 f1-! X2 f2-!X3 has f2 O f1 ' 0 and gives rise to exact sequences (as K(n)* modules) * f* 0 - K(n)*(X1) f1-K(n)*(X2) -2 K(n)*(X3) BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 7 for all n > 0 then for all n 0 we get exact sequences: * f* 0 - P (n)*(X1) f1-P (n)*(X2) -2 P (n)*(X3): Theorem 1.19. Let spaces Xi, i = 1; 2; 3, have even Morava K-theory. Assume that X1 f1-!X2 f2-!X3 has f2Of1 ' 0 and all spaces are H-spaces and all maps are H-space maps. Assume also that this gives exact sequences of bicommutative Hopf algebras for all n >* * 0: f1* f2* K(n)* -! K(n)*(X1) ---------! K(n)*(X2) ---------! K(n)*(X3): Then, P (n)*(X1) ' P (n)*(X2)=(f*2) for all n 0. The above theorem will be used repeatedly for our examples. What is surprising is that, more often than not, our spaces do not come from fibrations. Although we cannot give the level of detail for most of our examples that we give for Eilenberg-Mac Lane spaces, there are some general statements which we can make about generators and relations. Note that these results do not depend on having even Morava K-theory but only on being Landweber flat which even Morava K-theory implies. Theorem 1.20. Let a space X have P (n)*(X) Landweber flat for n 0, then there is a set Tn in P (n)*(X) such that the elements Tn satisfy (a)-(c). Let * *Rn be the set of relations on the elements Tn in P (n)*(X). Then any set Tn which satisfies (a)-(c) also satisfies (d)-(g). (a) They generate P (n)*(X) topologically as a P (n)*-module, (b) are all essential to generate, and (c) are almost all in F s, the sthskeletal filtration of the Atiyah-Hirzebruch* * spectral sequence, for each s 0. (d) These elements, Tn, reduce to a set, Tq, in P (q)*(X), q > n, with the same properties. (e) all relations must be infinite sums, in particular, the elements of Tn are* * lin- early independent over P (n)*, (f) any relation, in Rq, q > n, on the reduced set Tq in P (q)*(X) comes from Rn, and (g) anyPrelation whose coefficients all map to zero in P (q)* can be written q-1 i=nviri, with ri in Rn. The last statement is a nice generalization of "regular" in Landweber's paper [Lan70a]. It is clear that the image of the set Tn in K(q)*(X) generates. From * *the next result we see that every element in Tn must show up in some Morava K-theory (or else it would be unnecessary). In fact, it follows that every generator mus* *t be detected by an infinite number of the Morava K-theories. It seems reasonable, b* *ut we were unable to prove, that if a generator shows up in K(q)*(X), then it also shows up in K(q + 1)*(X). Such is the case for Eilenberg-Mac Lane spaces. Our next result says that if elements generate the Morava K-theories then they actu* *ally generate everything. This is a strong result which allows us to prove our exact* *ness theorems and go on to attack Eilenberg-Mac Lane spaces. 8 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Theorem 1.21. Let a space X have P (n)*(X) Landweber flat. Let n 0 and let Tn P (n)*(X) be such that (a) the elements of Tn are almost all in F s, the sthskeletal filtration of th* *e Atiyah- Hirzebruch spectral sequence and (b) for each q n (q > 0), K(q)*(X) is generated topologically as a K(q)*-modu* *le by the image of Tn. Then Tn generates P (n)*(X) topologically as a P (n)*-module. Remark 1.22.If Tn is multiplicatively generated by a finite subset Gn of elemen* *ts of positive skeletal filtration, then Tn satisfies (a). Corollary 1.23.Let X and Tn be as in Theorem 1.21 with all of the elements of Tn essential. Then Tn satisfies the conditions of Theorem 1.20. Corollary 1.24.Let X and Tn be as in Theorem 1.20. Every t 2 Tn maps non- trivially to K(q)*(X) for an infinite number of q > n. We see structure in many examples of Brown-Peterson cohomology where there was not known to be structure before. We consider what we have done as just a start. The problem of computing these examples more completely is a problem that remains, but now with more than a glimmer of hope that the answers we will find will be nice. We hope this work will inspire others to tackle these expli* *cit computations. We leave people with the question: If these things are so nice, w* *hat are they? Section 2 elaborates on our examples. In Section 3 we organize the preliminar* *ies needed in the rest of the paper. After that we have Section 4 on the Atiyah- Hirzebruch spectral sequence for our theories. Then we move on to assume even Morava K-theory and deduce the main result in Section 5 (Theorems 1.2, 1.8, and 1.9). We then do our work with generators and relations, Section 6 (Theorems 1.* *20 and 1.21, and Corollaries 1.23 and 1.24), followed by our work with exactness in Section 7 (Theorems 1.17, 1.18, and 1.19). In Section 8 we work out the details* * of the Eilenberg-Mac Lane example (Theorem 1.14 and Proposition 1.16). Our final section, Section 9, deals with the K"unneth isomorphism (Theorem 1.11). We would like to thank the Japan-United States Mathematics Institute (JAMI) at The Johns Hopkins University for making it possible for the authors to get together during the fall of 1991 for this work. The second author wishes to tha* *nk the Johns Hopkins University for leave support during the time of this work and John Sheppard and the University of Zimbabwe for computer support during the first writing of the paper in the spring of 1992. The authors would like to thank Michael Boardman, David Johnson, Jack Morava, Hal Sadofsky, Katsumi Shimomura, and Neil Strickland for various conversations about the material. We would also like to acknowledge the influence of the work* * of Hirotaka Tamanoi, which led us to believe this project was possible and which we should have paid more attention to years ago. In addition, we thank the referee* * for his careful reading and his many suggestions. 2. Examples Before listing our examples, we need the following easy result. Proposition 2.0.1.Let F -i! E -f! B BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 9 be a fibration in which K(n)*(F ) and H*(B) (= H*(B; Z=(p))) are concentrated in even dimensions. Then K(n)*(i) is one-to-one, K(n)*(f) is onto, and K(n)*(E) is even degree. Moreover, if the fibration is one of loop spaces, then K(n)* -! K(n)*(F ) -i*!K(n)*(E) -f*!K(n)*(B) -! K(n)* is a short exact sequence of Hopf algebras. Corollary 2.0.2.If we have a fibration of double loop spaces as in Proposition 2.0.1, F has even Morava K-theory and H*(B) is even, then P (n)*(F ) ' P (n)*(E)=(f*): Proof of Proposition.The Atiyah-Hirzebruch-Serre spectral sequence converging to K(n)*(E) with E2 = H*(B; K(n)*(F )) collapses since it is concentrated in even dimensions. The result follows. * * |___| Proof of Corollary.The double loops implies bicommutative Hopf algebras so this_ follows from Proposition 2.0.1 using Theorem 1.19. |__| 2.1. Finite Postnikov systems. All Eilenberg-Mac Lane spaces for abelian groups not having the circle, S1, as a homotopy factor, and all products of such space* *s sat- isfy the conditions of our theorems. The first from [RW80 ] and the products fr* *om Proposition 1.4. Furthermore, we can describe the Brown-Peterson cohomology of all of these spaces explicitly. See the introduction and Section 8 of the paper* *. This is our main example. In Hopkins-Ravenel-Wilson, [HRW97 ], we show that the loop space of a finite Postnikov system has even Morava K-theory, provided that it does not have an S1 as a factor. If F is such a space, but with double loops replacing loops, th* *en K(n)*(F ) is isomorphic as a Hopf algebra to K(n)*(E), where E is the product of Eilenberg-Mac Lane spaces having the same homotopy as F . In other words, k-invariants are not seen by Morava K-theory. 2.2. Classifying spaces of compact Lie groups. In [HKR92 ] it was conjec- tured that the Morava K-theory of a finite group, i.e., K(n)*(BG), should be ev* *en dimensional. If this conjecture had been true, then our result would have appli* *ed to all finite groups. It is true for most groups whose Morava K-theory is known. However, the conjecture is false, [Kri97], [Lee]. As it is, our result applies * *only to those groups with even Morava K-theory which have had their Morava K-theory computed. That list starts with finite abelian groups. Their Brown-Peterson co- homology was known, in detail, to Landweber [Lan70a] who also knew of their flatness. Perhaps next on the list, in terms of interest, are the symmetric gro* *ups. Hopkins-Kuhn-Ravenel, [HKR92 ] and [HKR ], and Hunton, [Hun90 ], independently proved that the Morava K-theory of these groups is even. These would be good ex- amples to understand explicitly. Hopkins, Kuhn, and Ravenel give other examples where the result is known. The result is known for groups G with rankpG 2. For groups with rankpG = 2 all but one case is done by Tezuka and Yagita in [TY89 ] and [TY90 ]. All cases, including the missing one, are done in [Yag93]. M. Tana* *be also has an interesting class of examples in [Tan95]. The result about symmetric groups mentioned above is a consequence of the following statement. If the conjecture is true for a finite group G, then it is* * also 10 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA true for the wreath product Z=(p)o G. By this we mean the evident semidirect product in the split group extension Gp -! Z=(p)o G -! Z=(p) in which Z=(p)acts on Gp by permuting the factors cyclically. The proofs given * *by Hunton and Hopkins-Kuhn-Ravenel differ slightly in the assumption made about K(n)*(BG). The latter assume that it is additively generated by images under the transfer map of Euler classes of representations of subgroups H G, while Hunton assumes that there is a map f : BG ! Y with K(n)*(f) onto and K(n)*(Y ) a finitely generated power series ring. He calls such a map a `unitary like embed* *ding' because in the case where G is abelian, Y can be taken to be BU(m) for a suitab* *le unitary group U(m). We can improve on the wreath product result of [HKR92 ] as follows. For a gr* *oup G let TreK(n)(G) denote the subalgebra of K(n)*(BG) generated by transferred Euler classes of irreducible representations of subgroups of G, and similarly f* *or TreBP(G) and TreP(n)(G). (In [HKR92 , Cor. 8.3] it is shown that the module generated by transferred Euler classes of all representations of subgroups of G* * is the same as this algebra.) We let Tr*denote the transfer and e(ae) the Euler cl* *ass of a representation ae. Then Tr*(e(ae)) stands for the transferred Euler class* * of a representation. We will say that a group G is good if K(n)*(BG) = TreK(n)(G): for all n. We know that finite abelian groups and groups G with rankpG 2 are good. A group G is good if its p-Sylow subgroup is. In [HKR92 ] it was shown t* *hat W = Z=(p)o G is good if G is. The following result is a consequence of Theorem 1.21. Corollary 2.2.1.Let G be a finite group which is good in the sense above, and l* *et W = Z=(p)o G. Then, with P (0) = BP , P (n)*(BG) = TreP(n)(G); and P (n)*(BW ) = TreP(n)(W ): Proof. First note that for X = BG or BW , we know from [BM68 ] and [Lan72] that lim1BP *(Xm ) = 0, so P (0) is BP as in Definition 1.5. Now let Tn P (n)*(BG) be the subalgebra generated by the set of transferred Euler classes of irreducible representations of subgroups of G. There are fini* *tely many such classes, so Tn is multiplicatively generated by a finite set as requi* *red by Remark 1.22 so the statements about the cohomology of BG follow from Theorem 1.21. Let Tn0 P (n)*(BW ) be similarly defined. Since W is good, Tn0also satisfies the hypotheses of Theorem 1.21 because of Remark 1.22 and the statements_about the cohomology of BW follow. |__| We want to give a more detailed description. Theorem 2.2.2. Let G be good and BP *(BG) BP* Z=(p)' Z=(p){b }, that is, the b are BP *generators. Then BP *(B(Z=(p)o G)) BP* Z=(p)' 0 0 Z=(p){P (); oe(1; : :;:p)ys; ys | s 0; s > 0; 9i6= j} BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 11 where oe(1; : :;:p) = Tr*(b1 . . .bp ); with Tr*: BP *(BGp) ! BP *(B(Z=(p)o G)); y = ss*("y) with ss* : BP *(BZ=(p)) ' BP *[["y]]=[p]("y) ! BP *(B(Z=(p)o G)); P () = Tr*(e(a^e)) if b = Tr*(e(ae )) for some representation ae of H G, and ^aeis the representation of Z=(p)o H with ^ae|Hp = ae . . .ae . Proof. The exact sequence 1 -! Gp -! Z=(p)o G -! Z=(p)-! 1 induces the spectral sequence H*(BZ=(p); K(n)*(Gp)) =) K(n)*(B(Z=(p)o G)): In [HKR92 , between 8.3 and 8.6], the differentials are computed and they get a similar theorem as ours for K(n) except that there are some restrictions. One m* *ust use a subset of the b's, s pn - 1 and s0 pn - 1. All of the elements in the statement of the theorem can be defined for BP cohomology and we now see that their reductions generate all of the Morava K-theories. Our result follows fro* *m __ Theorem 1.21 and we see that all of these elements are necessary as well. * * |__| Now let X = BG, where G is a compact Lie group. From Buhstaber-Mischenko, [BM68 ], and Landweber, [Lan72], it is known that lim1BP *(Xm ) = 0. In [KY93 ], Kono-Yagita conjecture that BP *(X) is even degree and flat in our sense. They go on to prove this for O(n), SO(2n + 1), P U(3) and F4. The Brown-Peterson cohomology of BO, BO(n) and MO(n) was computed in [Wil84]. Remark 2.2.3.Our results show that the Hopkins-Kuhn-Ravenel conjecture about finite groups is equivalent to the Kono-Yagita conjecture (for finite groups). * *Since the first is false, so is the second. 2.3. The sphere spectrum. The evenly indexed spaces in the -spectrum for the sphere, QS2n, have even Morava K-theory as they are the limit of spaces which h* *ave even Morava K-theory. (This follows from Hunton's theorem about the Morava K- theory of wreath products [Hun90 ].) We want to thank Takuji Kashiwabara for bringing this example to our attention. Kashiwabara has, since we proved our basic theorems, pushed this example to its ideal conclusion in [Kasa]. There, * *he shows that if E is a bouquet of BP spectra and there is a map, f, from BP to E such that * 0 - BP *(S0) - BP *(BP ) f- BP *(E); is an exact sequence, then there is an exact sequence of K(n)*-Hopf algebras K(n)* -! K(n)*(QS2k) -! K(n)*(BP__2k) (f2k)*-!K(n)*(E_2k) which, by Theorem 1.19, gives BP *(QS2k) as a quotient, BP *(BP__2k)=(f*2k): Note that these maps and spaces do not form a fibration. It is easy to come up with a spectrum E and a map fromiBP . A minimal one, for example, is just to have E be the wedge of 2(p-1)pBP for each i 0. The maps just cover the generators for BP *(BP ). In principle, this gives complete 12 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA information not just for BP *(QS2k) but for K(n)*(QS2k). Everything about the map from BP to E is known and one can use the techniques developed in [BJW95 ]. Kashiwabara has recently pushed his work even further, see [Kasb ]. 2.4. Image of J and related spaces. We thank Stewart Priddy and Fred Cohen for tutorials which allowed us to include this example. We will outline a computation of K(n)*(J) for an odd prime p, where J is the fibre of k-1 BU(p)---------! BU(p) for a suitable choice of the integer k, namely it must be congruent to a primit* *ive (p - 1)throot of unity mod p but not mod p2. It is also known that if k is a po* *wer of some prime other than p, then the fibre of the map above is the p-localizati* *on of algebraic K-theory of the field Fk [Qui72]. In any case this space is known to * *be a direct limit of the classifying spaces of finite groups studied by Tanabe in [T* *an95] ([Qui72]). He shows that each of them has even Morava K-theory. In particular J has even Morava K-theory, so the theorems of this paper apply to it. Moreover, the fibration k-1 J ---------! BU(p)---------! BU(p) gives a short exact sequence of Hopf algebras in Morava K-theory by Proposition 2.0.1. Corollary 2.0.2 then gives us the result that BP *(J) ' BP *(BU(p))=(( k - 1)*) because there are no lim1problems (J is torsion and BU(p)has no torsion). This discussion could be made self contained, and thus not dependent on Tanab* *e, by showing that ( k - 1)*is surjective as a map from K(n)*(BU(p)) to itself. A simple argument then shows that K(n)*(J) is even. As it is, Tanabe, with the collapsing of the spectral sequence, gives us this surjectivity. The effect of the map k - 1 in BP -cohomology can be computed with the help of the formal group law and the splitting principle. We have BP *(BU) = BP *[[c1; c2; . .].]; the power series ring on the Chern classes. Consider the formal expression X c(t) = citi wherec0 = 1: i0 Under the splitting principle we can write Y c(t) = (1 + xjt); j which should be understood to mean that ci is the ithelementary symmetric func- tion in the xj. Then we have Y ( k)*(c(t)) = (1 + [k]BP* (xjt)); j where [k]BP* (x) denotes the k-series for the formal group law associated with * *BP * and this gives the action of ( k)* on our generator for BP *(CP 1). The express* *ion on the right is symmetric in the xj, so the coefficient of ti is a certain symm* *etric polynomial (with coefficients in BP *) in the xj, so it can be written in terms* * of the BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 13 elementary symmetric functions. Then ( k)*(ci) is the corresponding polynomial in the Chern classes. In a similar sense we have Y 1 + [k]BP* (xjt) ( k - 1)*(c(t)) = _____________ : j 1 + xjt 2.5. BO. The object of this section is to recover the second author's computati* *on of BP *(BO) [Wil84] using the results of this paper. It is shown there that BP *(B* *O) is a certain quotient of BP *(BU) = BP *[[c1; c2; . .].] (the power series ring on the Chern classes of the universal complex vector bun* *dle) under the map Bi : BO ! BU induced by the complexification map i : O ! U. Let c*idenote the ith Chern class of the conjugate of the universal bundle [MS7* *4 , page 167]. Then the result of [Wil84] which we want to reprove is BP *(BO) = BP *(BU)=(ci- c*i: i > 0): We do not know if similar methods can be used to recover BP *(BO(m)). For more on BO(m) the reader should see [Kri97, Section 5]. We first observe that BO does not have a lim1problem so that we really can use BP and not BPp^. In general this is done by Landweber in [Lan72] but in this ca* *se it it quite easy to see because the rational cohomology of BU surjects to BO. T* *he only way there can be an infinite number of differentials in the Atiyah-Hirzebr* *uch spectral sequence, giving a phantom class, is if there are an infinite number of differentials on one of the integral classes of BO. Some multiple is in the ima* *ge of the spectral sequence from BU and that spectral sequence collapses, so this can* *not happen. Next we observe that BO has no odd prime torsion so we have nothing to prove to get the result at odd primes. For p = 2 we begin by showing that BO has even Morava K-theory. K(n)*(BO) can be computed as follows. We know that (the mod 2 homology) H*(BO) = P (b1; b2; . .). where bi 2 Hi(BO) is the image of the generator of Hi(RP 1). The action of the Milnor primitive Qn is given by ae n+1 Qn(bi) = bi+1-2n+10ifoitisheveneandriwi2se. It follows that in the Atiyah-Hirzebruch spectral sequence for K(n)*(BO) we have ae n+1 d2n+1-1(bi) = vnbi+1-2n+10ifoitisheveneandriwi2se, so we have E2n+1= K(n)*[b2; b4; . .b.2n+1-2] K(n)*[b22i:i 2n]: It follows that K(n)*(BO) and K(n)*(BO) are even dimensional. Recall that we are trying to show that the map Bi : BO ! BU induces a surjection in BP -cohomology. Bott periodicity ([Bot59]; see also Milnor's trea* *tment in [Mil63, x24]) gives us a fibre sequence Z x BO -Bi!Z x BU -j! Sp=U = Sp: 14 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Delooping this gives 2Sp -! U -j! Sp where j is the usual inclusion map. Delooping twice more gives Sp -f!U -! 2O where f is the usual inclusion map. Now Sp has even dimensional homology. To see this, recall [Whi78 , xVII.4] that H*(Sp; Z) = E(x3; x7; . .;.x4m+3; . .). with|x4m+3| = 4m + 3: The Eilenberg-Moore spectral sequence for the homology of its loop space collap* *ses, giving H*(Sp; Z) = P (x2; x6; . .;.x4m+2; . .). with|x4m+2| = 4m + 2: Likewise, the bar spectral sequence collapses giving E0H*(BSp; Z) = (x4; x8; . .;.x4m; . .). with|x4m| = 4m where means the divided power Hopf algebra. Hence Proposition 2.0.1 and Corollary 2.0.2 apply to the fibration BO -Bi!BU -j! Sp and we have BP *(BO) ' BP *(BU)=((j)*) which is not quite what we want yet. It came from a short exact sequence of Hopf algebras from Proposition 2.0.1 K(n)* ! K(n)*(BO) ! K(n)*(BU) ! K(n)*(Sp) ! K(n)* Now consider the fibration Sp=U _____wBU ______wBSpBj ||||||||||||||||||||||||| | |||||||||||||||||||||||||||| |||||||||||||||||||||||||||| |||||||||||||||||||||||||||| |||||||||||||||||||||||||||| ||||f |||||||||||| Sp _____wSU All three spaces have even dimensional homology, so by Proposition 2.0.1 we have another short exact sequence of Hopf algebras K(n)* ! K(n)*(Sp) ! K(n)*(BU) ! K(n)*(BSp) ! K(n)* Thus we get a diagram (fj) Z x BO _____ZwxBBUi|___________________Zwx|BU ||||||||||||| | ||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | ||||||||||||| | | | j f |||||||| U _________Spw _________Uw and we can splice together the two exact sequences to get an exact sequence K(n)* -! K(n)*(BO) -! K(n)*(BU) -! K(n)*(BU): BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 15 This no longer comes from a fibration but the result of [Wil84] can be recove* *red by using Theorem 1.19 after we have identified the self-map (fj) on BU as the one inducing the difference between the universal complex bundle and its conjug* *ate. To do this, consider the composite U(2m) -j!Sp(2m) -f!U(4m): To study this we suppose that we have inclusions R2m C2m H2m = C2m + jC2m where the quaternion j 2 H has its usual meaning. Then for M 2 U(2m) Sp(2m) and a; b 2 C2m we have ___ M(a + jb) = Ma + jM b; note here that conjugation in U(2m) is well defined since we have chosen a real subspace of C2m. It follows that the map fj sends M to M _0_ 0 M 2 U(4m): It follows that (fj) = 1 c where c : U ! U is the conjugation map. All that remains is to show c = -Bc. Restricting to U(1) ' Z, we see that c induces multiplication by -1 in ss0. To evaluate c on the 0-component BU ' SU, recall [Mil63, Theorem 23.3] that this equivalence is derived from a certain map Gm (C2m) -g!SU(2m) where Gm (C2m) is the Grassmannian of complex m-planes in C2m. (Bott proves the complex case of his theorem by showing that the map g is an equivalence thr* *ough a range of dimensions that increases with m.) The map is defined by associating to each point in Gm (C2m) a path in SU(2m) from I to -I as follows. Choose a basis of C2m such that the m-dimensional subspace in question is spanned by the first m basis elements. We parametrize the path by 2 [0; ss] with i 7! e0 I e-0iI 2 SU(2m); where I here denotes the identity element in U(m). In other words (independently of the choice of basis) we send to a unitary transformation having eigenvalue * *ei on the given subspace and e-i on its complement. Again we note that there is a well defined conjugation map c on SU(2m), given our choice of a real subspace of C2m. Applying it does two things. First it con- jugates the basis, replacing each subspace by its conjugate. It also conjugates* * the coefficient ei , so that with respect to the conjugated basis the map above bec* *omes -i 7! e 0 I e0iI 2 SU(2m): Thus the direction of the path gets reversed, which effectively replaces the co* *nju- gated subspace by its unitary complement. It follows that the conjugation map c on SU(2m) restricts on the subspace Gm (C2m) to the map which sends each complex m-plane through the origin in C2m to the complement of its conjugate. Passing to the limit as m ! 1, we see that Z x BU -c! Z x BU 16 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA is the map inducing the Whitney inverse of the conjugate universal bundle as re- quired. 2.6. Connective covers of BU and related spaces and spectra. Let BU<2m> denote the (2m - 1)-connected cover of BU for m 2 and consider the fibration F -! BU<2m> -! BSU: Then K(n)*(F ) is even dimensional for n > 0 by [HRW97 ], as is H*(BSU). Thus Proposition 2.0.1 applies and we conclude that K(n)*(BU<2m>) is even dimen- sional. We also know that K(0)*(BU<2m>) (the rational homology of BU<2m>) is even dimensional. Thus the results of this paper give information about BPp^*(BU<2m>), and similarly for MU<2m>, the associated Thom spectrum. Localizing at an odd prime, we can say the same about the fibration F 0-! BO<4m> -! BSO; so we can say a lot about BP *(BO<4m>) and BP *(MO<4m>). 3.Preliminaries We need a large selection of theories to state and prove our results. First, * *there is the Brown-Peterson cohomology, BP *(-), associated with a prime, p. Some basic references for BP are Brown-Peterson, [BP66 ], Adams, [Ada74 ], Quillen, [Qui69* *], Ravenel, [Rav86 ], and Wilson, [Wil82]. Next, we need the p-adic completion of * *BP , BPp^, defined by (3.1) BPp^= lim0(BP ^ M(pi)) where M(pi) is the mod pi Moore spectrum. The coefficient ring for BP , BP *, is Z(p)[v1; v2; : :]:where the degree of vn is -2(pn - 1). The coefficient rin* *g for BPp^is just the p-adic completion of this. Either of these theories can be labe* *led P (0). Next, we need the theories introduced by Morava, P (n). Their coefficient rings are BP *=In where In = (p; v1; : :;:vn-1). For references, see Johnson-Wi* *lson, [JW75 ], W"urgler, [W"ur77], and Yagita, [Yag77]. Thanks to their construction * *us- ing Baas-Sullivan singularities, [Baa73], [BM71 ], they come equipped with stab* *le cofibrations: n-1) vn (3.2) 2(p P (n) ---------! P (n) ---------! P (n + 1); which give us long exact sequences in cohomology. Note that P (0) can be either BP or BPp^in this cofibration (let v0 = p). Letting BP*be Z(p)[v1; : :;:vn], theories, E(k; n), can be constructed, using Baas-Sullivan singularities and lo* *caliza- tion, which have coefficients v-1nBP *=Ik, and similar stable cofibrations. * *These spectra are discussed by Baker-W"urgler in [BW89 , page 523] and in [Hun92 ]. * *The earliest reference to these theories is probably in [Yos76, Prop. 4.6] (where t* *hey go by a different name). The theories without localization play a prominent role * *in [Yos76], [Yag76] and [BW ]. For k = 0 these theories are usually denoted by E(* *n). They come in two flavors; regular and p-adically complete. They have been studi* *ed in [JW73 ], [Lan76], [Rav84 ], and others. It is proven in [JW73 , Remark 5.13* *, p. 347], and later follows from the Landweber exact functor theorem of [Lan76], th* *at E(n)*(X) = E(n)* BP* BP*(X): BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 17 A similar result for k > 0 (with BP* replaced by P (k)*) was proved in [Yag76]. As a special case, when k = n > 0, we have the nth Morava K-theory, K(n), see [JW75 ], [Hop87 ], [Rav86 ], [Rav92 ], [W"ur91] and [Yag80]. One of our main tools is the Atiyah-Hirzebruch spectral sequence (3.3) E*;*r) G*(X) where (3.4) Es;t2' Hs(X; Gt) which we will denote E*;*r, E*;*r(X), or E*;*r(G*(X)), depending on the context. The differential, dr, has bidegree (r; 1-r). When G is one of our connected spe* *ctra this is a fourth quadrant spectral sequence. If not, it is a first and fourth q* *uadrants spectral sequence. Let F s= ker(G*(X) ! G*(Xs-1)) where Xs-1 is the s - 1 skeleton of X. Then we have F s=F s+1' Es;*1and F 1 gives the phantom maps. The spectral sequence really converges to G*(X)=F 1 so it will be important f* *or us to be able to show that F 1 is zero in our cases. By Milnor's theorem, [Mil6* *2]: Theorem 3.5 (Milnor). There is a short exact sequence 0 ! lim1G*-1(Xm ) ! G*(X) ! lim0G*(Xm ) ! 0: Since the term on the right of Milnor's theorem is what the Atiyah-Hirzebruch spectral sequence converges to, the triviality of F 1 is equivalent to the lim1* *term being zero. Remark 3.6.One way to show the lim1term is zero is by using the Mittag-Leffler condition. In our case, we have a sequence of subgroups Im{Gn(Xm+i ) ! Gn(Xm )}: If they stabilize for big i and all n we say the Mittag-Leffler condition is sa* *tisfied. In this case, the lim1term in Milnor's theorem is zero. See [Ada74 ] for more d* *etails. Various assumptions on the G or X can give us the Mittag-Leffler condition: (i)If Es;t2is always finite. This can happen if one of Gs or Ht(X) is always * *finite and the other is finitely generated. This is the case for several of our t* *heories; e.g. P (n) and K(n), n > 0. (ii)If Es;t*always has only a finite number of nontrivial differentials on it.* * We show that this is the case for E(k; n) when 0 < k < n, which is a bit surp* *rising and our most difficult technical lemma. Remark 3.7.When the Mittag-Leffler condition is not satisfied for BP then we need to move to our p-completion (because the lim1term is not zero, [Lan70b]). The Mittag-Leffler condition is still not satisfied when we p-complete! Howeve* *r, because of the compactness of the p-adics we do have the lim1term is zero (see [Ada74 ]). Thus, by our choice of P (0), we always have P (0)*(X) ' lim0P (0)*(Xm ): The skeletal filtration of G*(X) associated with the Atiyah-Hirzebruch spectr* *al sequence also gives a topology on G*(X) which is nontrivial if X is an infinite complex. Since all of our spaces are infinite complexes this topology is always there. For a detail reference on such things, see [Boa95]. We will not need much about the topology here except that when we look at subgroups generated by a set of elements, we will mean topologically generated, i.e., the closure of the* * literal 18 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA subgroup. When G is a p-adic completion then we need a slightly different topol* *ogy. Here we use finite complexes which are torsion in the sense that the identity m* *ap is stably torsion. An open neighborhood of zero is the kernel of G*(-) when we map a torsion finite complex to X. If X is a finite complex and G is BPp^then the topology is just the p-adic completion. Then, since these are finitely gene* *rated they are compact and our lim1P (0)*(Xm ) is always zero as above. When we look at the p-adic completion of E(n) we lose our compactness but since E(n)*(Xm ) is E(n)* P (0)*(Xm ) we still have our lim1zero. We need three theorems which generalize known results. First we need a gener- alization of Quillen's theorem. Theorem 3.8 ([Qui71] for n = 0 and [BW ] for nF>o0).r X a finite complex, P (n)*(X), n 0, is generated by non-negative degree elements. Quillen proved the n = 0 version of this in [Qui71]. A second proof for Quill* *en's result, n = 0 of this, was given in [Wil75]. More recently, Quillen's result fo* *llows from abstract information about unstable BP operations, [BJW95 ]. When we started this project we believed the result could be proven for n > 0 using Qui* *llen's approach. By the time we discovered that this was not the case, there were two proofs, analogous to those in [Wil75] and [BJW95 ], following from the splitti* *ng theorem of [BW ] and are included there. All of our proofs go through for p = 2. Normally, there can be problems with this case because most of our theories that are mod 2 do not have a commutative multiplication on them. However, in our case they are always even dimensional. In [W"ur77], W"urgler computes the obstruction to commutativity and shows that * *it factors through odd degrees and is thus of no concern to us. Where it could bot* *her us because we do have odd degree elements, is in the Atiyah-Hirzebruch spectral sequence, but that is commutative by itself so it is no problem. The rest of t* *he arguments are no problem. The next result that we need is a generalization of the Landweber exact funct* *or theorem, [Lan76]. Theorem 3.9 ([Lan76] for k = 0. [Yos76] and [Yag76] for kL>e0).t In;k be the ideal (vk; : :;:vn-1) in P (k)*. M is Landweber flat, i.e. flat for the c* *ate- gory of finitely presented P (k)*(P (k))-modules, if vn multiplication is injec* *tive on M=In;kM for all n k. Note that this result is not a cohomological version of the Landweber ex- act functor theorem, but merely an algebraic statement about finitely presented P (k)*(P (k))-modules. We also need a generalization of the Landweber filtration theorem. Theorem 3.10 ([Lan73] for k = 0. [Yos76] and [Yag76] for kL>e0).t In;kbe the ideal (vk; : :;:vn-1) in P (k)*. Let M be a finitely presented P (k)*(P (k))-mo* *dule. There exists a finite filtration of M by P (k)*(P (k))-modules, M = M0 M1 . . .Mj = {0}; where Ms=Ms+1 = P (k)*=Ins;k: Theorem 3.11 (Boardman-Johnson-Wilson). Let M be an unstable BP module which is bounded above and of finite type, then there is a (finite) unstable La* *ndweber BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 19 filtration, as in Theorem 3.10, with the generators of the quotient modules all* * in non-negative degrees. Remark 3.12.This is from Theorem 20.11 of [BJW95 ]. The bounds on the degrees of the quotient modules are much more refined in [BJW95 ] than we give here. It is also stated quite differently in [BJW95 ]. There it is assumed that M is fi* *nitely presented, but the assumption is never used. In fact, finitely generated need n* *ot be assumed because of the algebraic version of Quillen's theorem, [BJW95 , 20.* *3], which says the generators are all in non-negative degrees. The proof of this th* *eo- rem inductively constructs cyclic submodules whose generators are in non-negati* *ve degrees, thus reducing the size of the non-negative part with each step. Finite* * pre- sentation follows from the finiteness of the filtration. In [BW ], a P (k) ver* *sion of this theorem is proven. For our X of finite type, BP *(X)=F sis an unstable module, including the case of s = 1. We show later, Corollary 4.8, that F 1 = 0 for P (k), k > 0. Our work relies heavily on the unstable Quillen-Boardman-Wilson result. The unstable Landweber filtration is a stronger statement. Our interest in it is the followi* *ng. Corollary 3.13.Let X be a space, then BP *(X)=F s+1 BP *(Xs) is a BP *(BP ) module which is finitely presented over BP *. Proof. This is an unstable module which fits the hypothesis of Theorem 3.11. * *|___| Remark 3.14.In fact, we prove this corollary ourselves in Lemma 6.1 for P (k) except when k = 0 and P (0) is the p-adic completion of BP . For the p-adic completion we must resort to the unstable Landweber filtration. The only place * *in this paper where we need this result is when we prove the K"unneth isomorphism 1.11 for k = 0 when X has lim1BP *(Xi) non-zero. Remark 3.15.Since P (k)* is a coherent ring, all of our finitely presented modu* *les are coherent, and, of course, coherent implies finitely presented, see [Smi69]. Remark 3.16.The unstable Landweber filtration answers the cohomology version of an old question of Landweber's from [Lan71, Problem 4]. It shows that for X of finite type, BP *(X)=F 1 is pseudo-coherent, i.e. every finitely generated submodule is finitely presented. This is still not known for homology despite b* *eing bounded below, unlike the cohomology which is not bounded in either direction. Likewise, similar theorems are true for P (k). 4. The Atiyah-Hirzebruch spectral sequence In this section we develop the Atiyah-Hirzebruch spectral sequence for the th* *ings which we need. In particular, we accomplish two main goals. First, we show that there are no phantom maps in G*(X) for all of the theories that we are concerned with except BP . This simplifies our life considerably. Originally it seemed * *that some sort of exotic completions would be necessary to state our theorem, but be- cause of this lack of phantom maps in general, the only place we have to go to completion is occasionally with BP where we have to resort only to p-adic compl* *e- tion. This lack of phantom maps is just the same as having all elements of G*(X) represented in the Atiyah-Hirzebruch spectra sequence; i.e., having the infini* *te filtration, F 1, equal to zero. We must eliminate the phantom maps so that in t* *he next section we can show K(n)*(X) even implies E(k; n)*(X) is also even. With 20 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA the advantage of hindsight, an alternative route to these results might be to u* *se Yosimura's work [Yos88]. Second, we show that for any given element of P (k)*(X* *), k 0, there is some N such that the element maps nontrivially to E(k; n)*(X) for every n > N. Our proof is somewhat technical and uses the Atiyah-Hirzebruch spectral sequence extensively. However, to see what is going on is not so diffi* *cult. Consider the k = 0 case where we are working with BP and E(0; n) = E(n), the localization of BP . From [Wil75] we know that BP_ _isplits off of BP__ifor i 2(pn+1 - 1)=(p - 1). It is easy to see that it splits off of E(n)_ias well. * *Any 0 6= x 2 BP i(X) must reduce to a non-trivial element in BP i(X) for some la* *rge n. Thus we can see, quite geometrically by looking at the classifying spaces, t* *he result we want. No such splitting was around for k > 0 when we wanted to gener- alize this. Our proof depends heavily on the theorem that P (n)*(X) is generated by non-negative degree elements for finite complexes. Since this proof, a split* *ting theorem has been found, [BW ], which would allow us to prove the result by loo* *king at the representing spaces for the cohomology theories. This gives an alternati* *ve approach to this part of the proof as well. Unless otherwise stated, let E*;*r(X) ) P (n)*(X), n 0, be the Atiyah- Hirzebruch spectral sequence with X a space. Although we do not need X to be a space for the first few lemmas we will assume it anyway. X could just as well b* *e a (-1)-connected spectra. We will point out when having X a space becomes neces- sary. Let R be either the integers localized at p or the p-adic integers, depen* *ding on which P (0) we are using for a given X. We define L(0; N) ' R[v1; v2; : :;:vN-1 ] P (0)* and L(n; N) ' Z=(p)[vn; : :;:vN-1 ] P (n)* for n > 0. Lemma 4.1. Let E*;*r(X) ) P (n)*(X), X a space, with n 0. For each r and s there is a number N = N(s; r) such that there is a finitely generated L L(n; N* *)- module, Asr= As;*r, generated in nonpositive degrees (second degree) and satisf* *ying Es;*r' AsrL P (n)*: Proof. The proof is by induction on r. For r = 2 we can take As2= Hs(X; Z=(p)) and N = n. (For n = 0 we use Hs(X; R) and N = 1.) Assume the case r. Then we have integers N(s; r) as in the lemma for all s. Let {yi(s)} be a (finite) s* *et of generators of Asrwith (second) degree |yi(s)|. Now fix an s. Write M = max {N(k; r): 0 k s + r} S = L(n; M) d = min{|yi(k)|: 0 k s + r; i > 0}: Then, for n > 0, there is a bigraded S-module Bqr= Bq;*rgenerated by {yi(q)} for 0 q s + r, such that Eq;*r' BqrS P (n)*; 0 q s + r; with 0 |yi(q)| d for all 0 q s + r. Consider the differential X dr(yi(s0)) = ci;j(s0+ r)yj(s0+ r); ci;j(s0+ r) 2 P (n)* j BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 21 for 0 s0 s. Then, 1 - r + |yi(s0)| = |ci;j(s0+ r)| + |yj(s0+ r)|; so ci;j(s0+ r) 2 L(n; M0) where M0 is the smallest number such that -2(pM0-1 - 1) 1 - r + d. Take Sr+1 = L(n; M0). Then dr induces an Sr+1 map dr : BsrSrSr+1 -! Bs+rrSrSr+1 such that ker(dr : Es;*r! Es+r;*r) = ker(dr|BsrSrSr+1) Sr+1P (n)*: Similarly, consider dr : Es-r;*r! Es;*r. Then Es;*r+1' As Sr+1P (n)* for some Sr+1-module As = As;*. As is a subquotient of the finitely generated Sr+1-module BsrSr Sr+1. Hence As is finitely generated as an Sr+1-module since Sr+1 is Noetherian. Take N = N(s; r + 1) as M0 and Asr+1as As. This completes_the_ induction. |__| Recall Xm is the m-skeleton of X and i : Xm ! X is the inclusion. Lemma 4.2. Let E*;*r(X) ) P (n)*(X), X a space, n 0. For each r, Es;*r(X) ' Es;*r(Xm ) for all0 s m - r(r - 1)=2: Proof. Since Es;*2(X) ' Es;*2(Xm ) for s m - 1 and |dr| = (r; 1 - r) this_fol* *lows by an easy induction. |__| Lemma 4.3. Let E*;*r(X) ) P (n)*(X), X a space, n 0. For all m (s + 1)s=2 + s and all r, i* : Es;*r(X) -! Es;*r(Xm ) is injective. Proof. By Lemma 4.2 we have an isomorphism for r s + 1 when 0 s m - (s + 1)s=2, i.e., m (s + 1)s=2 + s. Then, because |dr| = (r; 1 - r), Es;t* *ris not in the image of dr for r > s so we have Es;*s+1(X)____Es;*s+1(Xm ) u u | | | | | | |y |y * Es;*r(X)______Es;*r(Xmw)iuu | | | | | | |y |y * Es;*1(X)______Es;*1(Xmw)i for r > s. This implies the result. |___| The next lemma uses the Boardman-Wilson version (Theorem 3.8) of Quillen's theorem for P (n)*(-), and is the main technical lemma which makes the Atiyah- Hirzebruch spectral sequence approach work. At this stage it becomes essential that we are working with spaces and not spectra. 22 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Lemma 4.4. Let E*;*r(X) ) P (n)*(X), X a space, n > 0 or if P (0) = BP , n = 0. For each s there is an m such that i* : Es;*1(X) ' Es;*1(Xm ) and dr(Es;*r(X)) = 0 for r > m, i.e., Es;*m+1(X) ' Es;*1(X). Remark 4.5.The result is not true if P (0) = BPp^because if the Mittag-Leffler condition does not hold for BP then it still fails for BPp^even though we have * *no lim1problems there. Proof. Since each group Es;tris finite, for n > 0, (if n = 0, each group0is0fin* *itely generated over Z(p)), we can find a T s such that dr restricted to {Esr;t|s0 s; t0 -s0} is zero for all r T . (For n = 0, if we had an infinite number of differentials then the Mittag-Leffler condition would not be satisfied. We see * *from [Lan70b] that the Mittag-Leffler condition is equivalent to the lim1 term being non-zero. This would contradict our choice0of P (0)0= BP .) From Lemma 4.2 and Lemma 4.3 we can find m s such that EsT;t(X) ' EsT;t(Xm ) for all s0 s and i* : Es0;tr(X) ! Es0;tr(Xm ) injects for all r and s0 s. Certainly Es;*m-s+1(X) ,! Es;*m-s+1(Xm ) ' Es;*1(Xm ): We want to show that Es;*m-s+1(X) ' Es;*1(X). Assume there is some 0 6= "x2 Es;tr(X), and r m - s + 1 with dr("x) 6= 0. We have i*("x) = "xm2 Es;tr(Xm ) ' Es;t1(Xm ): This is non-zero by our injectivity. Furthermore, t < -s because our original c* *hoice says dr = 0 if t -s. Thus the total degree of "xis negative. By Boardman-Wilso* *n's version of Quillen's Theorem for P (n)*(Xm ), Theorem 3.8, we know0there0are no* *n- negative degree P (n)* generators, gmi, for P (n)*(Xm )=F s+1in Es1;t(Xm0)0with s0 s and t0 -s0, which our starting assumption says is isomorphic to Es1;t(X). (The fact that X is a space rather than a connective spectrum is crucial at t* *his point of the proof. In the latter case, Theorem 3.8 would not give us the preci* *se control we need on the dimensions of these generators.) Thus P (n)*(X)=F s+1surjects to P (n)*(Xm )=F s+1in non-negative degrees. So we can choose generators {gi 2 P (n)*(X)} which reduce0to0the generators {gmi2 P (n)*(Xm )=F s+1}. This is a finite set because Es2;t, s0 s and t0 -s0, is fin* *ite (for n = 0, finitely generated over Z(p)). If we have our dr("x) 6= 0, then "xm= i*("x) 2 Es;tr(Xm ) = Es;t1(Xm ) is not in the image of i* : Es;t1(X) ! Es;t1(XmP). Let xm 2 P (n)*(XmP)=F s+1be an element represented by "xm. Then xm = 0v(i)gmi. Define z = v(i)gi 2 P (n)*(X). Then i*(z) = xm so "zmust be in Es1;twith s0< s. This is so because "xmis not in the image but the element it represents0is; therefore the element representing the element that hits it must be in Es1;twith s0< s. But, that mea* *ns "zmust go to zero in order to do its duty of changing filtrations to hit "xm. T* *his contradicts the injectivity of Lemma 4.3 and we have Es;*m-s+1(X) ' Es;*1(X). T* *he argument just given also shows that Es;*r(X) maps surjectively to Es;*r(Xm_) for r m - s + 1. |__| We have some quick corollaries now. BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 23 Corollary 4.6.Let X be a space and let n > 0 or if P (0) = BP , n = 0. For each r, there is an m such that P (n)*(X)=F r' P (n)*(Xm )=F r where F ris the rth filtration of the Atiyah-Hirzebruch spectral sequence. Corollary 4.7.Let X be a space and let n > 0 or if P (0) = BP , n = 0. P (n)*(X) is (topologically) generated by non-negative degree elements. Corollary 4.8.Let X be a space. Let n k 0 and n > 0. Let G be P (k), E(k; n), v-1nP (k), or K(n). Let E*;*r(G*(X)) ) G*(X) be the Atiyah-Hirzebruch spectral sequence. (For k = 0 we use the p-adically complete E(k; n) if we use * *it for P (0).) (a) Then F 1 = 0, every element in G*(X) is represented in E*;*1, G*(X) ' lim0G*(Xm ) ' lim0G*(X)=F m+1, and lim1G*(Xm ) = 0 = lim1G*(X)=F m+1. (b) Furthermore E*;*r(E(k; n)*(X))' E*;*r(P (k)*(X)) P(k)*E(k; n)* ' E*;*r(v-1nP (k)*(X)) v-1nP(k)*E(k; n)* and (c) E*;*r(v-1nP (k)*(X)) ' E*;*r(P (k)*(X)) P(k)*v-1nP (k)*; which is just localization. Remark 4.9.This is not true for spectra. For example, P (n)*(k(n)) = 0 but K(n)*(k(n)) is not, so the first line for k = n cannot hold. Proof. The statements of (a) are all equivalent so it is enough to show any one of them. For G = P (k), k > 0, we have that P (k)s is finite so Es;tris finite.* * By the Mittag-Leffler condition, Remark 3.6(i), we are done. For k = 0 we can use Remark 3.7 to see that F 1 = 0, the main purpose of our choice of P (0). There is nothing to prove in this case as it is really part of our assumptions. The o* *ther G will follow from the displayed tensor products (of (b) and (c)) and Lemma 4.4. Keep in mind that K(n) is just a special case of E(k; n), with n = k. All of the tensor products (of (b) and (c)) are true for E*;*2. Since the tensor product * *of (c) is just localization, and localization preserves exactness, we see that it * *is true. Es;*rfor P (k) is always a finitely generated P (k)*(P (k)) module so tensoring* * with E(k; n)* is exact by the generalized Landweber exact functor theorem for P (k)** *(-) which says that E(k; n)* is P (k)* flat in this situation, Theorem 3.9. This gi* *ves the first equivalence of (b). Lemma 4.4 gives the first equivalence (of (a)), (F 1 * *= 0), for E(k; n) (with k > 0) by Remark 3.6(ii). If k = 0 and P (0) = BP then this also follows. If P (0) = BPp^then E(0; n)* is a module over the p-adics and by compactness, see Remark 3.7, F 1 = 0. The second isomorphism of (b) follows__ from the first (of (b)) and (c). |__| Remark 4.10.Although this is a very technical result, it is an exciting one bec* *ause it removes most of our lim1problems. Since all of our theorems are about infini* *te complexes, worrying about phantom maps was a major concern which this corollary eliminates. For example, certain types of elements cannot exist. In the spect* *ral sequence for P (n)*(X) it is quite possible, as we shall see in the next sectio* *n (see 24 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Remark 5.6), to have an element whose filtration is raised every time you multi* *ply by vn, and which is never zero. Such an element, by the corollary, can never gi* *ve rise to an element in K(n)*(X) even though it is a torsion free element. Corollary 4.11.Let X be a space. Let 0 k < n. If x is infinitely divisible by * *vk (v0 = p) in E(k; n)*(X), then it is zero. Any such element would have to be a phantom map, and there are none. Proof.By Corollary 4.8 we have E(k; n)*(X) ' lim0E(k; n)*(Xm ) so if x is non- zero it maps nontrivially to some E(k; n)*(Xm ) and is still infinitely divisib* *le by vk there. This is the difficult step, reducing the proof to looking at a finite co* *mplex. To complete the proof we use the generalized Landweber filtration of Theorem 3.* *10. We have P (k)*(Xm ) = M = M0 M1 . . .Mj = {0}: By the generalized Landweber exact functor Theorem 3.9, E(k; n)* is exact. We c* *an tensor the filtration with E(k; n)* to get a filtration of E(k; n)*(Xm ) with s* *uccessive quotients given by E(k; n)*=Ins;k. If ns > n then this is zero. Eliminating the* * zero quotients we have a finite filtration E(k; n)*(Xm ) = M0 = M00 M01 . . .M0j0= {0} with M0 s=M0 s+1 = E(k; n)*=Ins;k, ns n. We need to show that there is no x 2 E(k; n)*(Xm ) which is infinitely divisible by vk. Assume that there is suc* *h an x. Find the maximum s with such an x 2 M0 s. This x must reduce non-trivially to M0 s=M0 s+1= E(k; n)*=Ins;kand still be infinitely divisible by vk here, whi* *ch is impossible. (If it were not infinitely divisible by vk then there would have* *_to be such an x 2 M0s+1, contradicting the assumption on s.) |__| Proposition 4.12.Let X be a space. Let k 0. Given 0 6= x 2 P (k)*(X), there exists an N such that x maps nontrivially to E(k; n)*(X) for all n N. Proof. Let k > 0. The element x is represented in the Atiyah-Hirzebruch spectral sequence by an element "xin Es;*1, which, by Lemma 4.4 is isomorphic to Es;*rfor some big r. For degree reasons we can pick an N such that "xis a P 0(N)*-genera* *tor where P 0(N)* is a subalgebra of P (n)* isomorphic to P (N)* and Es;*1is P 0(N)* free. Using Lemma 4.1 we can assure that N is big enough so that "xsurvives the tensor product of Corollary 4.8 to the spectral sequence for E(k; n)*(X) for n > N. In this spectral sequence we still have Es;*r' Es;*1so our element x maps nontrivially. For the case of P (0) we use Lemma 4.3 with r = 1. For m big enou* *gh we have an injection: Es;*1(X) -! Es;*1(Xm ): The right hand side is very nice and we can tensor it with our (p-adically comp* *lete)_ E(0; n)* for big n, forcing what we need, as above, for the left hand side. * * |__| 5. Even Morava K-theory We complete the proof of the main Theorem 1.8 in this section. Lemma 5.1. Let X be a space. Let 0 k n and n > 0. If K(n)*(X) is even dimensional, then E(k; n)*(X) is even dimensional and has no vk torsion (v0 = p* *). BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 25 Remark 5.2.For k = 0 this is E(n)*(X) or its p-adic completion if necessary. It follows from the proof that E(n)*(X) maps onto K(n)*(X). This was proven by Hunton in [Hun92 , Theorem 11], for finite*complexes. This is improved in [BW91* * ], p. 559, to give a surjection of [E(n)(X), the In-adic completion. As we only ne* *ed p-adic completion our result is somewhat stronger. Proof. We prove this by downward induction on k. Since E(n; n) = K(n) our in- duction is grounded by our assumption. By induction, assume that E(k +1; n)*(X) is even dimensional and has no vk+1 torsion. We have a long exact sequence from the cofibration analogous to 3.2: E(k; n)*(X)_____________________wE(k;vn)*(X)k 4 4 ffi 474 ae E(k + 1; n)*(X) Since E(k + 1; n)*(X) is even dimensional and ffi is an odd degree map, there a* *re two possible types of odd degree elements in E(k; n)*(X): (i)an element which never shows itself in E(k + 1; n)*(X) because it is infin* *itely divisible by vk and not vk torsion; (ii)an element which is infinitely divisible by vk but is vk torsion (the elem* *ent that vk kills comes by way of ffi). Either way the element is infinitely divisible by vk, which cannot happen by Corollary 4.11. Thus ffi is zero and all elements are even degree. If any even * *degree element were vk torsion, then it would have to be hit by ffi coming from an odd degree element, which doesn't exist by our induction assumption. Thus we get a * *__ short exact sequence and all elements are vk torsion free. |* *__| We can now prove Theorem 1.2. Lemma 5.3. Let X be a space. If K(n)*(X) is even dimensional for an infinite number of n, then P (k)*(X), k 0, and K(k)*(X), k > 0, are even dimensional. Proof. If 0 6= x 2 P (k)*(X) pick N as in Lemma 4.12 so x maps nontrivially into E(k; n)*(X) for n > N. Find some n > N for which K(n)*(X) is even dimensional. By Lemma 5.1, E(k; n)*(X) is even dimensional so x must be even dimensional as well. This concludes the proof for P (k)*(X). By Lemma 4.8, all elements of P (k)*(X) are represented in E*;*1(P (k)*(X)) which is even dimensi* *onal. Furthermore, E*;*1(K(k)*(X)) is just the tensor product with K(k)* so it too is* *_even dimensional, and it also represents elements. |__| Corollary 5.4.Let X be a space with even Morava K-theory. For k 0 we have the short exact sequence: 0 -! P (k)*(X) -vk!P (k)*(X) -! P (k + 1)*(X) -! 0: Proof. The three terms fit into a long exact sequence with odd degree connecting term, by 3.2. By Lemma 5.3 all terms are even dimensional so the boundary_ homomorphism must be zero. |__| Corollary 5.5.If X is a space with even Morava K-theory and k 0 then P (k)*(X) is even degree and is Landweber flat. 26 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Proof. P (k)*(X) is even dimensional by Lemma 5.3. To prove flatness we need only invoke the generalized Landweber exact functor theorem for P (k), Theorem_ 3.9, and Corollary 5.4. |__| This finishes the proof of Theorem 1.9 and Theorem 1.8 follows. Remark 5.6.This is a good time to insert a fundamental example which illustrates the phenomenon described in Remark 4.10. This is an old, well known example but it supplies useful guidance. Let X = BZ=(p). The mod p cohomology is E(e1) P (x2), so E2 of the spectral sequence is E(e1) P (x2) P (n)*: n The only nontrivial differential takes e1 to vnxp2 leaving E1 to be a copy of P* * (n)* for each xi2for i < pn and a copy of P (n + 1)* for each xi2for i pn. Tensoring this with K(n)* we get the correct answer for K(n)*(BZ=(p)); free on generators xi2for i < pn. However, we know from our corollary that there is no vn-torsion. If you take an element in P (n)*(BZ=(p)) which is representednbynxi2for+i1 pn and you multiply by vn, then it is represented by vn+1xi-p2+p . So, iterating the multiplication by vn continues to raise filtration and give a nontrivial el* *ement. However, it does not give rise to an element of K(n)*(BZ=(p)). Looking briefly at P (n)*(BZ=(p)) we see that E1 is free over P (n + 1)* on elements ffi in degree 2i - 1 for i > 0 and free over P (n)* on fii in degree 2* *i for 0 < i < pn. The relations on the ff come from the p-sequence. In particular, we have vn+1ffi+vnffi+pn-pn+1mod (vn; vn+1)2. We see that all of the ffiare infini* *tely divisible by vn in E(n; n + 1)*(X). Remark 5.7.If Hk(X; Z(p)) is finite for all k then the Mittag-Leffler condition* *, Re- mark 3.6, is satisfied and the Atiyah-Hirzebruch spectral sequence for BP *(X) * *con- verges giving lim1BP *(Xm ) = 0, so the results of Theorem 1.8 hold for BP *(X). As we shall see later, this is the case for X = K(Z=(pi); n). When Hk(X) is not finite we may have to resort to the p-adic completion of BP , such as with X = K(Z(p); n), n > 2, which is known to have phantom maps. 6. Generators and relations In this section we will prove Theorems 1.20 and 1.21, and Corollaries 1.23 and 1.24. Proof of Theorem 1.20. We know that the Atiyah-Hirzebruch spectral sequence converges and Es;tr(X) is a finitely generated module. By Boardman-Wilson's and Quillen's Theorem 3.8 the generators must be represented by elements with s+t * *0. Since t 0, there can only be a finite number of generators for P (n)*(X) repre- sented in Es;*1. Assume inductively that we have chosen a minimal number of gen- erators for P (n)*(X)=F s. Then pick a few more, if necessary, that are represe* *nted in Es;*1in order to get minimal generators of P (n)*(X)=F s+1. The construction* * of Tn with properties (a), (b) and (c) is now complete. We now show (d), that Tn reduces to a set Tq with the same properties. We do this inductively. Because we know, Remark 1.10, that P (n)*(X) surjects to P (n + 1)*(X) we get part (a) that Tn+1 generates. The map is a filtered map so part (c) follows. Part (b), that all elements remain essential, is really the o* *nly thing BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 27 left to prove. If some proper subset of Tn could be used to generate P (n + 1)** *(X) then we could write some t 2 Tn+1 in terms of the t's: X t = citi where ci2 P (n + 1)* can all be lifted to ci2 P (n)* and we use the same notati* *on for elements in P (n)*(X) and P (n + 1)*(X). From the exact sequence 0 -! P (n)*(X) vn-!P (n)*(X) -! P (n + 1)*(X) -! 0 we can lift this to X X t = citi+ vn diti in P (n)*(X), contradicting (b). The result follows for Tq by induction. Proof of part (e). Let F Rn be the set of finitePlinear relations among the ele* *ments of Tn in P (n)*(X). A typical relation looks like icitiwhere ti2 Tn and ci2 P (n* *)*. We can write the ci in terms of monomials in the vk (where we let v0 = p for P (0)*). We can define the length of a monomialPas the sum of the powers of v's, i.e., for vI = vinnvin+1n+1:w:e:define l(vI) = ik. We now extend this definit* *ion to the elements of F Rn. We take the length of a relation to be the maximal length* * of a monomial occurring in any of its coefficients, ci. There is an obvious map fr* *om F Rn to F Rn+1. Because there are a finite number of coefficients, every eleme* *nt goes to zero after enough of these maps have been applied. Let us find a relati* *on, r, which has the minimal length as defined above. Let us assume that it is in F* * Rn and maps to zero in F Rn+1. We can do this because the length of a relation can never increase under these maps. Recall from Theorem 1.9 that we have a short exact sequence. Since each coefficient, ci, maps to zero in P (n + 1)* it must* * be divisible by vn. Thus we can divide the sum r by vn to get r=vn. This is a fini* *te sum with a smaller length than our minimal one so it must be a non-zero element. This cannot be true as it is a vn-torsion element in P (n)*(X) which by Theorem 1.9 is known to have no such torsion. Thus there are no finite relations anywhe* *re. ProofPof part (f). We prove this by downward induction. Let r 2 Rq be written citiwith ci2 P (q)*. LiftPeach cito P (q-1)* (and note that vq-1 does not div* *ide them). Then the element citi 2 P (q - 1)*(X) reduces to zero in P (q)*(X). If itPis not zero in P (q - 1)*(X), then it is divisible by vq-1 and we can wri* *te 0 = citi+ vq-1r0 where r0 can be written in terms of the ti. Thus we have a relation which reduces to our r. Proof of part (g). This follows from (f). |__* *_| We need a couple of lemmas to prove Theorem 1.21. Lemma 6.1. Let X be a space and q 0, then P (q)*(X)=F s+1is coherent. Proof. By Lemma 4.4 we have P (q)*(X)=F s+1' P (q)*(Xm )=F s+1 for some large m except when q = 0 and P (0) = BPp^. P (q)* is coherent. Since Xm is finite, P (q)*(Xm ) is coherent. Since P (q)*(Xm )=F s+1is the image of * *the map P (q)*(Xm ) -! P (q)*(Xs) we see that it is coherent. For the case of q = 0 and P (0) = BPp^we have_to re* *sort to Corollary 3.13. |__| 28 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Lemma 6.2. Let X be a space and let q > 0. Let J2 = (vq+1; vq+2; : :):. Let T * *be a set in v-1qP (q)*(X) such that (a) all but a finite number of the elements of T are in F s, the s filtration * *for the Atiyah-Hirzebruch spectral sequence and (b) the image of T in K(q)*(X) generates (topologically). Then the image of T generates v-1qP (q)*(X)=(F s+1+ JN2) for all s and all N. Proof. P (q)*(X)=F s+1is a finitely presented P (q)*(P (q))-module and, as such* *, it has a Landweber filtration, Theorem 3.10. When you localize at vq such a filtra* *tion becomes a finitely generated free module over v-1qP (q)*. It is then easy to se* *e that K(q)*(X)=F s+1' K(q)* v-1qP(q)*v-1qP (q)*(X)=F s+1: Pick a set of generators xi for v-1qP (q)*(X)=F s+1. The image of these xi must generate K(q)*(X)=F s+1; as does the image of T . Thus, modulo J2,Pthe xi mustP be in the submodule generated by T . We have (finite) sums xi= vsi;kqtk+ ci* *;jxj where ci;j2 J2. (Note that si;k2 Z.) Now, to show that the xi are in the image modulo J22we just substitute the equations for the xj into this. Iterate_to get* * the theorem modulo JN2. |__| Lemma 6.3. Let q > n 0. Let X be a space with P (n)*(X) Landweber flat. Let J2 = (vq+1; vq+2; : :): and J1 = (vn; : :;:vq-1): Let Tn be a set in v-1qP (n)*(X) such that (a) all but a finite number of the elements of Tn are in F s, the s filtration* * for the Atiyah-Hirzebruch spectral sequence and (b) the image of Tn in K(q)*(X) generates (topologically). Then the image of Tn generates v-1qP (n)*(X)=(F s+1+ JN1+ JN2) for all s and N. Proof. From the short exact sequences of Remark 1.10 we see we can localize with respect to vq to get v-1qP (q)*(X) ' v-1qP (q)*b v-1qP(n)*v-1qP (n)*(X) Thus, if we have x; y 2 v-1qPP(n)*(X) which reduce to the same element in v-1qP (q)*(X), then x = y + eiri where ei 2 J1 and the sum is possibly infi- nite. Fix the N and s of the Lemma. Let T be the image of Tn in v-1qP (q)*(X). We can pick generators, {yi}, for P (n)*(X) with0property (a) above by picking * *(a finite number of) generators for P (n)*(X)=F s +1, lifting them to P (n)*(X) and extending this choice by enlarging s0. See the first part of the proof of Theor* *em 1.20 above for more detail. Map these generators to a set of generators, {xi}, * *for v-1qP (n)*(X). Reduce these elements further to zi 2 v-1qP (q)*(X). By Lemma 6.2, we can write zi, in v-1qP (q)*(X)=(F s0+1+ JN2), in terms of the reduction* * of T . Taking the limit, we can write each X X zi= di;ktk + ci;jzj P P where di;k2 v-1qP (q)* and ci;j2 JN2. The two elements, xiand di;ktk+ ci;jxj both reduce to the sameXelement andXwe see fromXthe above that xi= di;ktk + ci;jxj+ ei;mxm BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 29 where ei;m 2 J1 and the sums are possibly infinite. Reduce this to v-1qP (n)*(X)=F s+1and the sums are now finite. As in the proof of Lemma 6.2, substitute this formula in for the xiand iterate in order to show that_the t's * *generate modulo JN1+ JN2 |__| Proof of Theorem 1.21. We prove our theorem by induction on s, i.e. we show that Tn generates P (n)*(X)=F s. Assume inductively that we have this for s. No* *w, if we have an x 2 P (n)*(X)=F s+1which is not in the submodule generated by Tn then we will derive a contradiction. Since there are only a finite number of th* *e Tn which are non-zero, the quotient of P (n)*(X)=F s+1(which is coherent by Lemma 6.1) by the submodule generated by Tn must be coherent, [Smi69]. We will show that is not the case. Our x must be represented in Es;*1. Pick an N such that we can see that our x is not in JN1+ JN2 for strictly dimensional reasons. (For the case n = 0 we have to modify this a little. Put a weight on p to act as a non-trivial degree so th* *at the previous statement still holds. Otherwise, we can just prove the result for n >* * 0 first and then lift it to n = 0 easily afterwards, a choice we can make to avoi* *d the use of Lemma 6.1 in the one case which depends on [BJW95 ].) Pick N0 such that vN0 acts freely on P (n)*(X)=F s+1by Lemmas 4.1 and 4.3. Thus P (n)*(X)=F s+1 injects to v-1qP (n)*(X)=F s+1for q N0. In the last group, by the previous lem* *ma, we can write x in terms of t's modulo JN1+ JN2. (We don't need to do this for every q N0, only for an infinite number of such q.) However, we may use negati* *ve powers of vq to do so. Since all sums are finite, we can multiply by some power* * of vq, say sq, so that vsqqx is in the image of the submodule generated by Tn modu* *lo JN1+JN2. This is true for all q N0. Thus we see that there are an infinite num* *ber of relations; one each with a term vsqqx in it, q > N0. Thus it is not coherent* * and_we have our contradiction. x must therefore be in the submodule generated by Tn. * *|__| Proof of Corollary 1.23.This is immediate. |___| Proof of Corollary 1.24.If for some t 2 Tn, t goes to zero in K(q)*(X) for q N for some large N, then t is not essential to generate P (N)*(X) by Theorem 1.21. However, Theorem 1.20 says the reduction of Tn to Tq retains property (b)_of Theorem 1.20. Contradiction. |__| Remark 6.4.For some of the most interesting examples which we "understand" completely, all of the generators reduce to mod p cohomology where they are sti* *ll independent. This is the case for QS2k and Eilenberg-Mac Lane spaces and it probably contributes a great deal to our being able to understand them. This is not always the case though. When all generators are of this sort, then they nev* *er change filtration when we map from the spectral sequence for P (k)*(X) to that for P (n)*(X), n > k. The filtration can change when we map from P (k)*(X) to K(n)*(X) though. Generators that do not map to mod p cohomology must behave quite differently. They must change filtration when we map from the spectral sequence for P (k)*(X) to that for P (n)*(X) if n is large enough because the l* *ocation in the spectral sequence, x 2 Es;t2, t < 0, is zero when t > -2(pn - 1). So, as* * n grows, the filtration of such a generator must keep changing and it never shows* * up in mod p cohomology. An example of this behavior was pointed out to us by Takuji Kashiwabara. The example is BSO(4) which was computed in [KY93 , Theorem 5.5]. 30 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA 7. Exactness Once again we want properties of Morava K-theories to imply similar properties for Brown-Peterson cohomology. We have four theorems to prove in this section: one for surjectivity, one for injectivity, and two for the exactness that we ne* *ed in our applications. Although we state our theorems with the assumption of even Morava K-theory and injectivity or surjectivity for all of the Morava K-theorie* *s, we can get by with only assuming these things for an infinite number of the Mor* *ava K-theories. The proofs are unchanged. The statements of the theorems are much cleaner this way and there are no examples that need our greater generality. In this section we give the proofs for Theorems 1.17, 1.18, and 1.19 from the introduction. Proof of surjectivity in Theorem 1.17.By Theorem 1.20 we can pick a set T0 which generates P (0)*(X2). We know that it reduces to generators for each P (n)*(X2) and thus also for all K(n)*(X2). Map these generators to P (n)*(X1). By natural* *ity and the fact that the Morava K-theories surject, we have that the image of T0 in the P (n)*(X1) satisfies the conditions of Theorem 1.21 and so we see that_the * *image generates. |__| To prove the theorem on injectivity we need a lemma. Lemma 7.1. Let X1 and X2 be spaces with even Morava K-theory. Let f : X1 -! X2. If f* : K(n)*(X2) -! K(n)*(X1) is injective, n > 0, then so is f* : E(k; n)*(X2) -! E(k; n)*(X1), 0 k n. Proof. The proof is by downward induction on k. By Lemma 5.1 we have short exact sequences: 0 -! E(k; n)*(Xi) -vk!E(k; n)*(Xi) -! E(k + 1; n)*(Xi) -! 0: Given 0 6= x 2 E(k; n)*(X2), we know it cannot be infinitely divisible by vk by Corollary 4.11. Find a y 2 E(k; n)*(X2) and a j such that x = vjky and y maps non-trivially to E(k + 1; n)*(X2). By our induction, E(k + 1; n)*(X2) injects * *to E(k +1; n)*(X1) so y must map non-trivially to E(k; n)*(X1). Since this group h* *as no vk torsion, x = vjky must map non-trivially. |__* *_| Proof of injectivity in Theorem 1.17. For k 0 and x 2 P (k)*(X2) we use Propos* *i- tion 4.12 to see that x maps non-trivially to some E(k; n)*(X2). By the injecti* *vity of K(n)*(-) and Lemma 7.1 we have that this group injects to E(k; n)*(X1). By the naturality of maps between all of the cohomology theories involved, we must_ have x mapping non-trivially to P (k)*(X1). |__| Proof of Theorem 1.18. Take the cofibre X1 f1!X2 r! C(f1). This gives rise to a long exact sequence in any cohomology theory. By Theorem 1.17 we have the surjectivity of f*1. By this surjectivity of f*1, we have a short exact sequence * r* 0 G*(X1) f1G*(X2) G*(C(f1)) 0: for all G = P (k). Since f2 O f1 ' 0 , f2 factors through C(f1). By the assumpt* *ion of exactness for all K(n) the map C(f1) X3 is surjective for all of the Morava K-theories. Thus, by Theorem 1.17 we have surjectivity for the P (k). We then* * __ just patch up our surjectivity with the short exact sequence to get the result.* * |__| BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 31 Proof of Theorem 1.19. For bicommutative Hopf algebras we have that the cokernel can be constructed using the tensor product and the kernel from the cotensor pr* *od- uct, see [HRW97 , Section 4] or, better yet, [Bou96 , Appendix, especially The* *orem 10.12], so that although this is a theorem about Hopf algebras, algebras play t* *he main role here. We want to reduce this theorem to Theorem 1.18. Define a map F by X2 ! (X2 x X3)=(X2 x *) by diag (I;f2) X2 ---------! X2 x X2 ---------! X2 x X3 ! (X2 x X3)=(X2 x *): We have F Of1 ' 0. Our exact sequence of Hopf algebras implies an exact sequence of K(n)* modules (this is from the cotensor product model for the kernel): 0 ! K(n)*(X1) ! K(n)*(X2) F*-!K(n)*((X2 x X3)=(X2 x *)) which dualizes to (the tensor product model for cokernel): * 0 K(n)*(X1) K(n)*(X2) F- K(n)*((X2 x X3)=(X2 x *)): The lim1 condition of Theorem 1.18 is satisfied for the product by Landweber, [Lan70a, Lemma 6], because X2 and X3 both satisfy the condition. By Theorem 1.18 we now have an exact sequence: * 0 P (n)*(X1) P (n)*(X2) F- P (n)*((X2 x X3)=(X2 x *)): Let I(-) be the augmentation ideal. Then P (n)*(X2) ^I(P (n)*(X3)) maps to the last module. We claim that this map is surjective. To see this, pick sequences of generators, {ti} and {si} for P (n)*(X2) and I(P (n)*(X3)) respect* *ively. The elements {ti sj} map to generators of K(n)*((X2 x X3)=(X2 x *)) because K(n)*(-) has a K"unneth isomorphism (and our X2 and X3 are very nice spaces of the sort we are studying). Mapping these elements over to P (n)*((X2x X3)=(X2x *)) we see that since they generate all of the Morava K-theories then they must* *, by Theorem 1.21, generate. Since the tensor product maps onto generators, it must_ be surjective. The result follows. |__| 8.Eilenberg-Mac Lane spaces In this section we give a purely algebraic construction for the P (0)* algebra which is isomorphic to the P (0) cohomology of an Eilenberg-Mac Lane space, and then, of course, we go on to show the isomorphism of Theorem 1.14. 8.1. Preliminaries. From [RW77 ] we have a completely algebraic construction for the Hopf ring E*(BP__*) whenever E is a complex orientable generalized homology theory. Because the answer is a free E* module we have duality and have also given a construction for E*(BP__*). In particular, we can use K(n), BP , and BP* *p^ for E. The nice properties all come from the fact that H*(BP__*; Z(p)) has no torsion, [Wil73]. For the evenly indexed spaces this is even dimensional and is* * a bi- polynomial Hopf algebra (i.e. both it and its dual are polynomial algebras) and* * for the odd spaces it is an exterior algebra. The same is true for the cohomology. * *These properties lift to E*(BP__*), and, by duality, give completed exterior algebras* * (for odd spaces) and power series algebras (for evenly indexed spaces) for cohomolog* *y. We really need information about the BP_ _*and we can derive it from the above using: 32 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Theorem 8.1.1 ([Wil75]).Let g(q) = 2(pq+1 - 1)=(p - 1). For k g(q), the standard maps, BP__k! BP__kand BP__k! BP_ _k, split. For k < g(q) this splitting is as H-spaces. The second splitting splits the fibration: vq+1 BP__k+2(pq+1-1)---------!BP_ k-! BP_ _k to give a homotopy equivalence: BP__k' BP_ _kx BP__k+2(pq+1-1): It now follows that K(n)*(BP_ _2k) is a polynomial algebra for 2k < g(q) and is even dimensional for 2k = g(q). This is also true for BP*(-). For our computations we need the bar spectral sequence (see [RW80 , pages 704- 5] and [HRW97 , Section 2]). In our cases all of our maps are of infinite loop* * spaces and we only need it for Morava K-theory. Theorem 8.1.2 (Bar spectral sequence).Let F ! E ! B be a fibration of infi- nite loop spaces, then we have a spectral sequence of Hopf algebras, converging* * to K(n)*(B), with E2 term: TorK(n)*(F)(K(n)*(E); K(n)*): Next we need to know how this behaves in a special case that has already been computed. The following was proved in [RW80 , Theorem 12.3, p. 743]. Theorem 8.1.3. For the path space fibration: K(Z(p); q + 1) ! P K(Z(p); q + 1) ! K(Z(p); q + 2) the bar spectral sequence for K(n)*(-) is even dimensional and collapses. Looking at the statement of Theorem 1.14 we see that for each type of Eilenbe* *rg- Mac Lane space we really have two statements. For them to both be true we must have that the ideals (v*q) and (v*1; v*2; : :;:v*q) are equal. We will prove th* *e theorem for the ideal (v*q). Since this is contained in the "bigger" ideal, it is enoug* *h to show that our map of P (0)*(K(Z(p); q + 2)) P (0)*(BP__g(q))=(v*q) factors through P (0)*(BP__g(q))=(v*1; v*2; : :;:v*q): The stable cofibration sequence 1.13 is one of BP module spectra, [JW73 ]. Thus, all of the boundary maps used to define our map K(Z(p); q + 2) ! BP__g(q) commute with multiplication by vj. Since the map K(Z(p); q + 2) -vj!K(Z(p); q + 2 - 2(pj- 1)) is homotopically trivial, we have what we need. We admit that the equality of the two ideals was quite a surprise to us which* * we did not notice until late in the game. BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 33 8.2. Construction. In [RW77 ], a completely algebraic construction for the Hopf ring, E*(BP__*) is given. By the splitting above, we know that the algebraic co* *n- struction of E*(BP__k) maps surjectively to E*(BP__k) for k g(q) and that it factors through the quotient given by setting all [vi] = [0-2(pi-1)] for i > q * *(see [RW77 ]). There is a minor concern that maybe there could be some other relation in order to get injectivity. However, this is not the case. Note in the lemma t* *hat when we mod out by I(q), we are setting elements in it equal to the [0i], not 0, although because everything is in positive degrees it is the same. Lemma 8.2.1. Let ER*(BP__*) be the algebraic construction for E*(BP__*) from [RW77 ]. If we mod out by I(q) = ([vq+1]; [vq+2]; : :):we have ER*(BP__k)=I(q)* * ' E*(BP__k) for 0 < k g(q). We should point out that neither the statement nor proof of Theorem 1.14 de- pends on this lemma. The theorem is given strictly in terms of spaces and we do need the splitting 8.1.1. The attraction of the theorem to us is this lemma bec* *ause it gives us a purely algebraic construction for everything in the theorem. Proof. The map E*(BP__i) [vq]-!E*(BP__i-2(pq-1)) is just the induced algebraic * *map coming from multiplication by vq. Each of the spectra BPis a BP module spectra ([JW73 ]) so the maps between spectra commute with the maps of vq. We prove our lemma with a multiple induction. It is enough to prove our lemma for mod p homology because all of our spaces are torsion free and everything is therefore E* free. Our main induction is on j - k in Hj(BP_~~_k). Our second induction is downward induction on s. To ground our first induction, there is nothing to prove if j = k (and k g(s)). To ground our second induction, we see that Hj(BP__i) ' Hj(BP_~~~~_i) for j - i < 2(ps+1- 1) because these spaces a* *re homotopy equivalent in this range. For a fixed j - k we must pick s such that j - k < 2(ps+1 - 1). Then we can start the second induction to prove our result for this degree. To do the second induction we need only observe that the split fibration in Theorem 8.1.1 must give rise to a short exact sequence of Hopf alg* *ebras_ where the first map is just [vq+1]O multiplication. |* *__| One can go further with this and write E*(BP_~~_2k) as a power series ring on generators dual to the primitives E*(BP__2k) for 2k g(q), which can be writ* *ten down directly from [RW77 ] as was done in [Sin76]. We will discuss this more af* *ter the proof. In principle, [RW77 ] tells you how to compute the map [vq]O BP*(BP__g(q)) ---------! BP*(BP__g(q)-2(pq-1)): What that really means is that you can train a computer to do it, because, in practice, it is very difficult although complete information is available. Beca* *use ev- erything is BP* free you can take the duals and the dual map and again, everyth* *ing is, in principle, computable. It is certain that BP *(BP__g(q))=(v*q) is a well-defined algebraic construct, as is its p-adic completion. Likewise for BP *(BP__g(q))=(pi*; v*q) 34 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA which comes from the product map (pi; vq) BP__g(q)! BP__g(q)x BP__g(q)-2(pq-1): 8.3. Proof for K(Z(p); q + 2). We can now prove Theorem 1.14 for the integral spaces. It is known that BP *(K(Z(p); q)), q > 2, has phantom maps. This follows from [AH68 ] where they show this for complex K-theory and [Lan72] where it is shown that the situation for complex cobordism is the same as that for complex K-theory. We must work, therefore, with P (0) = BPp^. In fact, because of The- orem 1.19, all we must prove is Proposition 1.16. We just let K(Z(p); q + 2) = X1, BP__g(q)= X2, f1 the iterated boundary map given in the introduction, BP__g(q)-2(pq-1)= X3 and f2 the map coming from vq. Observe that the com- position of the two maps is indeed null homotopic because the first map factors through the boundary map BP__g(q-1)+1-! BP_ _g(q) which is just the inclusion of the fibre of the map f2. Note that this is an ex* *ample where the spaces do not form a fibration. The proof breaks up into two pieces. First, we need to show injectivity of the map K(n)*(K(Z(p); q + 2)) -! K(n)*(BP__g(q)); which we do using the Steenrod algebra and a bit of Hopf algebra machinery. Sec* *ond we need to show that the cokernel of this map injects to the third Hopf algebra: K(n)*(BP__g(q))==K(n)*(K(Z(p); q + 2)) -! K(n)*(BP__g(q)-2(pq-1)): We will do this using the bar spectral sequence a few times. 8.3.1. Proof of injectivity. This is going to reduce to a calculation over the * *Steenrod algebra. With apologies to the reader, understanding this proof will also requi* *re an intimacy with the Morava K-theory of Eilenberg-Mac Lane spaces from [RW80 ]. From that paper we know [RW80 , Corollary 12.2, p. 742] that lim-!K(n)*(K(Z=(pi); q + 1)) ' K(n)*(K(Z(p); q + 2)): i Furthermore, we know ([RW80 , Theorem 11.1(b), p. 734]) that the very first space in this limit, K(Z=(p); q + 1), picks up all of the Hopf algebra primitiv* *es for K(n)*(K(Z(p); q + 2)). To get our injection for this last space we want to * *just show that the primitives P K(n)*(K(Z=(p); q + 1)); and thus also for K(n)*(K(Z(p); q + 2)), inject to those for K(n)*(BP__g(q))* *. An injection on primitives automatically gives an injection on K(n)*(K(Z(p); q + 2* *)), see, for example, [HRW97 , Lemma 4.2]. This calculation is probably contained in H. Tamanoi's Master's Thesis, [Tam8* *3b ], and should have been deduced by us from [Yag86]. It is certainly contained in [Tam97 ]. Those proofs are in cohomology and we work in homology but the results are the same. H. Tamanoi computes the image of the map: BP *(K(Z(p); q + 2)) ! H*(K(Z(p); q + 2); Z=(p)): BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 35 In the proof one sees that these elements all come from BP *(BP__g(q)). As * *it turns out, these elements generate. At any rate, it is Tamanoi who first made t* *he connection between BP *(K(Z(p); q +2)) and BP *(BP__g(q)) more than ten years ago! We now assume a working knowledge of [RW80 ]. Let A* be the dual of the Steenrod algebra. We have the usual map from H*(K(Z=(p); q + 1); Z=(p)) to A* which is an A* comodule map. K(n)*(K(Z=(p); 1)) has elements a(i), 0 i < n, in degrees 2piwhich are represented in the Atiyah-Hirzebruch spectral sequence * *by the elements used to define the oi of the Steenrod algebra ([RW80 , Theorem 5.7* *]). Under the usual map Oq+1 K(n)*(K(Z=(p); 1)) -! K(n)*(K(Z=(p); q + 1)) all elements a(i0)O a(i1)O . .O.a(iq) with 0 i0 < i1 < i2. .<.iq < n are nontrivial ([RW80 , Theorem 9.2]). They are therefore represented by elements which map to oi0oi1oi2. .o.iq in the Steenrod algebra. The elements which are primitive are those with i0 = 0, ([RW80 , Theorem 9.2]). It is not important, but note that there are only a fin* *ite number of these elements. Define a subvector space, E(q; n), of A* with basis oi1oi2. .o.ij with 0 i1 < i2. .<.ij < n with j q in A*. This is clearly a subcomodule of A* over A* and we can take its quotient, A*=E(q; n) which is now a comod- ule over A*. Note that the above set of elements of H*(K(Z=(p); q + 1); Z=(p)) which survive to primitives in the Atiyah-Hirzebruch spectral sequence maps in- jectively to a subcomodule (over A*) of A*=E(q; n); call it E(q). We have our map, K(Z=(p); q + 1) ! BP__g(q)which induces a map of A* comodules in mod p homology. All of the elements in H*(BP__g(q); Z=(p)) survive in the Atiya* *h- Hirzebruch spectral sequence to the Morava K-theory because the space has no torsion. Thus, it is enough to show that our elements which represent primitives map nontrivially and independently to H*(BP__g(q); Z=(p)). In cohomology, the iterated boundary map, K(Z=(p); q + 1) ! BP__g(q); takes the fundamental class in H*(BP__g(q); Z=(p)) to Q0Q1. .Q.qtimes the fu* *n- damental class in H*(K(Z=(p); q + 1); Z=(p)), see [Wil75]. This tells us two th* *ings we need to know. First, it says our map is trivial on E(q; n) because Qi is du* *al to oi and there are q or fewer o but q + 1 Q. So, we get a map of A* comodules, E(q) ! H*(BP__g(q); Z=(p)). Second, it says our map is non-zero on the lowest dimensional element o0o1. .o.q: All we have to do now is show that this element forces an injection of E(q). 36 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Recall that the coproduct on oi is X pj oi 1 + i-j oj: 0ji We can ignore the first term in computing the comodule expansion on oI because it will lead to a product of the o on the right in E(q; n). Because oI always h* *as o0 in it (recall these are the primitives), there is only one term we can use from* * its coproduct, 1 o0. Recall also that o2j= 0, so we cannot use the o0 term of any * *of the other o's. Thus, modulo E(q; n) we have X pJ (oI) = I-J o0oJ; 0_g(q-1)) ' K(n)*(K(Z(p); q + 1)) P Aq-1 where P Aq-1 is a polynomial algebra. This induction is trivial to ground; just use K(n)*(BP_<0>_2) = K(n)*(K(Z(p); 2)) which we know because we know K(n)*(CP 1). Here the polynomial part is vacu- ous. The first step in our induction is to compute K(n)*(BP_ _g(q-1)+1). To * *do this we use the bar spectral sequence 8.1.2 with E2 term: TorK(n)*(BP _g(q-1))(K(n)*; K(n)*): BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 37 By induction and the K"unneth isomorphism, we see that this breaks into two par* *ts: TorK(n)*(K(Z(p);q+1))(K(n)*; K(n)*) TorPAq-1(K(n)*; K(n)*) where TorPAq-1(K(n)*; K(n)*) ' EAq-1 where EAq-1 is an exterior algebra on the homology suspension of the generators of the polynomial algebra P Aq-1. These generators lie in the first filtration * *in the spectral sequence so all differentials on them are trivial. We also know how to compute TorK(n)*(K(Z(p);q+1))(K(n)*; K(n)*) from Theorem 8.1.3. Furthermore, we have maps of fibrations: K(Z(p); q + 1)_______pt:w_______K(Z(p);wq + 2) | ||||||||||||||||||| | ||||||||||||||||| | ||||||||||||| | | | ||||||||||||||||| |u ||||||||||||| | |u BP_ _g(q-1)_____pt:w____wBP_ _g(q-1)+1: By naturality, we have no differentials on this part of the bar spectral sequ* *ence we are using to compute K(n)*(BP_ _g(q-1)+1). Since there can be no differen* *tials on the exterior part, we see that the spectral sequence collapses and, as algeb* *ras, we have: K(n)*(BP_ _g(q-1)+1) ' K(n)*(K(Z(p); q + 2)) EAq-1: All of the algebra extension problems in the K(n)*(K(Z(p); q + 2)) part have be* *en solved by naturality. In case there is any question about this algebra splittin* *g as a tensor product recall that EAq-1 is a free commutative algebra (if our prime * *is odd) on odd degree elements. We certainly have a short exact sequence with EAq-1 the quotient. Because it is free we can split it. If p = 2 we must observe that* * the generators of P Aq-1 come from BP*(BP__g(q-1)) and thus, so do the generators of EAq-1 come from BP*(BP__g(q-1)+1). Since they are exterior generators in BP there can be no extension problems where we are working. That ends the proof of the first step of the induction and we can move on to * *the next (and final) step. We will study the bar spectral sequence for the fibratio* *n: BP_ _g(q-1)+1! BP_ _g(q)! BP__g(q)-2(pq-1): We know quite a lot about things already. (i)We know that K(n)*(BP__g(q)-2(pq-1)) is a polynomial algebra. (ii)We know that K(n)*(BP__g(q)) is even dimensional. (iii)We have just "computed" K(n)*(BP__g(q-1)+1): 38 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA (iv)We know that we have the injection part of our desired exact sequence K(n)* ! K(n)*(K(Z(p); q + 2)) ! K(n)*(BP_ _g(q)): Because of (ii) we see that the map K(n)*(BP__g(q-1)+1) ! K(n)*(BP_ _g(q)) must take EAq-1 to zero. All of this allows us to simplify our computation of t* *he E2 term of the bar spectral sequence converging to K(n)*(BP__g(q)-2(pq-1)): The E2 term starts out as TorK(n)*(BP_g(q-1)+1)(K(n)*(BP_ _g(q)); K(n)*) and simplifies, by [Smi70, Theorem 2.4, p. 67], to K(n)*(BP__g(q))==K(n)*(K(Z(p); q + 2)) TorEAq-1(K(n)*; K(n)*) where the Toris just a divided power Hopf algebra. In particular, it is even di* *men- sional, as is the first part; thus this spectral sequence collapses. We can now just read off our answers. The quotient Hopf algebra is just the coker which we wanted to inject into K(n)*(BP__g(q)-2(pq-1)) and the map is just the edge homomorphism. This gives us the desired injection. However, to complete our induction we must show that this cokernel is polynomial. It is a s* *ub- Hopf algebra of a polynomial Hopf algebra and so it must be polynomial as well (this follows immediately from [Bou , Theorem B.7]). Now we have a short exact sequence of Hopf algebras K(n)*(K(Z(p); q + 2)) ! K(n)*(BP__g(q)) ! P Aq: Because P Aq is free we see that this splits as algebras and we have completed our induction ([Bou , Proposition B.9]). We thank S. Halperin, J. Moore and F. Peterson for help solving the above Hopf algebra problems before we found the paper by Bousfield. 8.4. Proof for K(Z=(pi); q + 1). This proof is only a slight modification of the previous proof. Our sequence of spaces is now: (pi;vq) K(Z=(pi); q + 1) -! BP__g(q)---------!BP__g(q)x BP__g(q)-2(pq-1) so we need an exact sequence of Hopf algebras: K(n)* | |u K(n)*(K(Z=(pi); q + 1)) | |u K(n)*(BP__g(q)) i;[v ]) (p* q|u K(n)*(BP__g(q)) K(n)*(BP__g(q)-2(pq-1)): BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 39 The last tensor product is just the product in the category of Hopf algebras and we have already computed the kernel of the map [vq] to the right side. It was j* *ust K(n)*(K(Z(p); q + 2)). All we have to do now is worry about the kernel of the map pi*restricted to this part. From [RW80 , Corollary 13.1, p. 745], we have an extension of Hopf algebras which solves that problem: K(n)* |u K(n)*(K(Z=(pi); q + 1)) |u K(n)*(K(Z(p); q + 2)) i p* |u K(n)*(K(Z(p); q + 2)); and we are almost done with this case. We want to use BP as opposed to BPp^in this case. Our assumptions in Theorem 1.19 require us to have lim1BP *(Xm ) = 0 for all spaces involved. We have this for the Eilenberg-Mac Lane space by Remark 5.7. Because the other spaces have no torsion we know that the Atiyah-Hirzebruch spectral sequence collapses and the lim1 for them is zero as well. 8.5. Generators and relations. If one really wants to use the construction given above to describe BPp^*(Z(p); q + 2) there are some simplifications which an intimacy with [RW77 ] can give you quite quickly but which the reader has been spared the necessity of knowing so far. We will just briefly describe here what can be done. Both BP *(BP__g(q)) and BP *(BP__g(g)-2(pq-1)) are power series rings on generators dual to the primitives in the BP homology. In [RW77 ], a basis for t* *he primitives is written down explicitly and one can see that most of them are map* *ped to basis elements for primitives in the second space. The consequence in the du* *al is that we do not need to worry about those primitives at all. The remaining primitives for BP*(BP__g(q)) are identified in [RW77 ] as 2 Opq b(0)O bOp(j1)O bOp(j2)O . .O.b(jq) where 0 j1 . . .jq. This element is in degree 2(1+pj1+1+pj2+2+. .+.pjq+q). Note that as q goes up, the degree of the generators goes up much faster. (Reca* *ll that the Brown-Peterson cohomology of the Eilenberg-Mac Lane spectra is trivial* *.) When reduced to K(n)*(BP__g(q)) we can see that each of these elements is in the image of K(n)*(K(Z(p); q + 2)) for some n large enough. If we take a bunch of dual generators, say cJ, we can see that BPp^*(K(Z(p); q + 2)) is a quotient* * of the power series algebra on the cJ. To see what the relations would be requires a good deal more work. For a slight check on reality, there is only one case he* *re that is degenerate enough to be familiar. Let q = 0 and we are talking about BPp^*(K(Z(p); 2)) and there is only one generator in degree 2. It may or may not be an interesting exercise to try to say more about the relations. These genera* *tors are those found by Tamanoi. He just did not know that he had found them all. 40 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Because of the splitting of Theorem 8.1.1 the map K(Z=(pi); q + 1) ---------! BP__g(q) is really a map to BP__g(q). Tamanoi ([Tam97 ]) calls this the fundamental cla* *ss and from the above it is easy to see that using stable BP operations, the al- gebra structure, and topological completion, this class generates everything in BP *(K(Z=(pi); q + 1)). 9. The K"unneth isomorphism This section is dedicated to the proof of Theorem 1.11, which turns out to be much more involved and much more general than we expected. We thank Michael Boardman, Dan Christensen, Michael Mandell, Peter May, Jean-Pierre Meyer, and Hal Sadofsky for some help with our general education about limits. We are also indebted to the paper by Jan-Erik Roos, [Roo61 ]. Although the result could be proven directly just for this situation, we pref* *er to bring to light some very nice mathematics which we were previously unaware of. * *It also makes our proof shorter. The main algebraic result we need is: Theorem 9.1 (Roos, Theorem 2, [Roo61 ]).Let A = {Aff} be a direct system of R-modules and M be an R-module with a finite projective resolution of finite ty* *pe, P*, then there are two spectral sequences which converge to the same thing: Ep;q2= Tor-p(limqAff; M) and Ep;q2= limpTor-q(Aff; M): There is some guidance for the proof in [Roo61 ] and much more in [Jen72] whe* *re this theorem appears as Theorem 4.4 with more discussion on pages 102-3. Dan Christensen helped us understand this approach, which gives much more insight into what is going on than our previous approach. Proof.In the case when we are indexed over the natural numbers and we have maps fi: Ai+1! Ai, then we are familiar with the exact sequence of Milnor from [Mil62], Y Y 0 ! lim0Ai! Ai- f! Ai! lim1Ai! 0; where the map f is given by f(a1; a2; : :):! (a1 - f1(a2); a2 - f2(a3); : :):: In this case, lim*A is just the homology of the complex Y f Y Ai! Ai: In the general case, there is a complex whose homology gives lim*A and whose terms are all (big) products of the Aff, see [Jen72, Theorem 1.1, page 32] and [Roo61 ]. Denote this complex by A*. We then get our two spectral sequences from the two standard filtrations of the bicomplex A* P*. Filtering first using P* we use only the differential on A*. Since each Pi i* *s a finitely generated projective R-module, taking the tensor product with A* and t* *hen taking the differential on A* gives us (lim*A) P*. Taking the second different* *ial to get our E2 term we have Tor*(lim*A; M), giving us our first spectral sequenc* *e. Filtering next using A* we useQonly the differential on P*.QBecause Piis a fi* *nitely generated projective R-module. ( Aff) P* is the same as Q(Aff P*) and so the homology is also the same. This is easy to evaluate as Tor*(Aff; M), since BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 41 products are exact. To take the next differential to get our E2 term we see that this is just the complex which gives lim*Tor*(Aff; M), giving our second spectr* *al sequence. Both spectral sequences converge (to the same thing) because of the finitenes* *s_ of the resolution P*. |__| This spectral sequence simplifies a great deal when limiis always zero for i * *> 1. This is always the case for us for multiple reasons. In particular, it is the c* *ase when we are indexed over the natural numbers, which we always are. It is also true w* *hen the ground ring for the algebra R is Z=(p), Z(p), or Zp, which is the case for * *us since R = P (n). Corollary 9.2 ([Roo61 ]).If all limi= 0, i > 1, and we have the conditions of Theorem 9.1, then 0 flfl 0 flflffl A AAC A lim1Tor1(Aff;fM)l Tor1(lim1Aff; M) flfl AAC flfflA A H flfl A AAC flflffl A (lim0Aff) M ___________wlim0(Aff M)fl AAC flfl A A flffl 0 0 where the diagonals give short exact sequences. Proof. This follows immediately from Theorem 9.1. |___| The commutativity of the tensor product and the inverse limit will be of prim* *ary importance to us. This result measures the failure to commute explicitly. We wi* *ll show that in the cases we care about, both terms having lim1will be zero, making the horizontal arrow an isomorphism. Built in to all of our assumptions is that lim1Affwill be zero, so the lim1Torterm is all we need to consider. At this sta* *ge we need to get more specific about our modules and rings. In all that follows our tensor products and our Torare over P (k)*. Lemma 9.3. Let the Affbe P (k)* modules which are bounded above and of finite type and let M be a P (k)*(P (k)) module which is finitely presented over P (k)* **. For each degree, Tor*1(Aff; M) is finite. Proof. For k > 0 everything in sight is a finite dimensional Z=(p)vector space * *and so our result follows immediately. We prove the k = 0 case by induction on the Landweber filtration, Theorem 3.10. We have a long exact sequence: : :!:Tor1(Aff; Mq+1) ! Tor1(Aff; Mq) ! Tor1(Aff; P (0)*=Inq) ! : : : where if nq > 0 we have P (0)*=Inq is finite in each degree and therefore so is Tor1(Aff; P (0)*=Inq) (this uses the assumptions on Aff). If nq = 0 then Tor1(Aff; P (0)*) = 0 because P (0)* is free. By induction Tor1(Aff; Mq+1) is * *fi- __ nite so since Tor1(Aff; Mq) is trapped between two finite groups, it too is fin* *ite. |__| 42 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA Corollary 9.4.Let {Ai} be a direct system, indexed over N, the natural num- bers, of P (k)* modules which are bounded above and of finite type, and let M be a P (k)*(P (k)) module which is finitely presented over P (k)*. We have lim1Tor*1(Ai; M) = 0. Proof. lim1of finite groups is always zero. |__* *_| Corollary 9.5.Let {Ai} be a direct system, indexed over N, of P (k)* modules which are bounded above and of finite type, and let M be a P (k)*(P (k)) module which is finitely presented over P (k)*. Assume that lim1Ai= 0. Then (lim0Ai) M -'! lim0(Ai M): Proof. This follows immediatedly from Corollaries 9.2 and 9.4. |_* *__| Remark 9.6.This result is what we need and it can be proven directly, but not as nicely. Note that the proof as we have given it really shows that Torn(lim0Ai; M) -'!lim0Torn(Ai; M): It is time to put some topology into the argument. Recall that we have lim0P (k)*(Zi) ' P (k)*(Z) and lim1P (k)*(Zi) = 0 for Z = X and Y . We have X2ix Y 2i (X x Y )2i Xix Y i so they give rise to two equivalent sequences and we have P (k)*(X x Y )' lim0P (k)*((X x Y )i) ' lim0P (k)*(Xix Y i) and lim1P (k)*(Xix Y i) = 0 by [Lan70a, Lemma 6] (If there are no phantom maps (for MU) for X and for Y then there are none for X x Y .) when P (0) = BP , by Remark 3.7 when we are p-adically complete, and by Corollary 4.8(a), (F 1 = 0), when k > 0. In particu* *lar, if Y is finite, we have lim1P (k)*(Xix Y ) = 0 and P (k)*(X x Y )' lim0P (k)*((X x Y )i) ' lim0P (k)*(Xix Y ): We need the following to proceed. It is possible these statements don't warra* *nt a proof but we are neophytes at this business. Lemma 9.7. Let X and Y be spaces and let P (k)*(X) be Landweber flat. Then P (k)*(X x Y )' lim0P (k)*(X x Y i) ' lim0(P (k)*(X) P (k)*(Y i)) and lim1P (k)*(X x Y i) = 0: BROWN-PETERSON COHOMOLOGY FROM MORAVA K-THEORY 43 Proof. Both P (k)*(X x -) and P (k)*(X) P (k)*(-) are cohomology theories for finite complexes. We have a map P (k)*(X) P (k)*(-) -! P (k)*(X x -) which is an isomorphism on a point. The usual arguments by induction on the number of cells gives us P (k)*(X) P (k)*(-) ' P (k)*(X x -) for finite complexes. This proves the vertical isomorphism. What we need now is either one of the two other statements since by Milnor, [Mil62], we have 0 ! lim1P (k)*(X x Y i) ! P (k)*(X x Y ) ! lim0P (k)*(X x Y i) ! 0: Comparing this with the other Milnor sequence, we have 0______wlim1P(k)*(X x Y_i)P(k)*(Xwx_Y_)lim0P(k)*(Xwx_Y_i)_w0 |u |u |u 0______lim1P(k)*((Xwx Y_)i)P(k)*(Xwx_Yl)im0P(k)*((Xwx_Y_)i)0_w and since the middle vertical arrow is an isomorphism and lim1P (k)*((XxY )i)_=* * 0, we must have lim1P (k)*(X x Y i) = 0. |__| By Corollary 4.8(a), we have P (k)*(Z)=F s+1-! P (k)*(Zs) induces (9.8) lim0P (k)*(Z)=F s+1-'!lim0P (k)*(Zs) ' P (k)*(Z); and (9.9) lim1P (k)*(Z)=F s+1= 0 = lim1P (k)*(Zs): Remark 9.10.In our proof below of the K"unneth isomorphism we use that P (k)*(X)=F i+1and P (k)*(Y i) are finitely presented and that P (k)*(Y )=F i+1* *is bounded above and of finite type. The first one is the only difficult one and * *it requires that X be a space and so this is an unstable result. When k > 0 or if P (0) = BP , then our proof of Lemma 6.1 is independent of [BJW95 ]. The only place we need the unstable Landweber Filtration is if P (0) = BPp^because lim1BP *(Xi) 6= 0. Proof of the K"unneth isomorphism, Theorem 1.11. 44 DOUGLAS C. RAVENEL, W. STEPHEN WILSON, AND NOBUAKI YAGITA P (k)*(X x Y ) ' lim0jP (k)*(X x Y j) by Lemma 9.7 ' lim0j(P (k)*(X) P (k)*(Y j)) by Lemma 9.7 ' lim0j((lim0iP (k)*(X)=F i+1) P (k)*(Y j))by Equation 9.8 ' lim0jlim0i(P (k)*(X)=F i+1 P (k)*(Y j)) by Proposition 9.5 ' lim0ilim0j(P (k)*(X)=F i+1 P (k)*(Y j)) ' lim0i(P (k)*(X)=F i+1 (lim0jP (k)*(Y j)))by Proposition 9.5 ' lim0i(P (k)*(X)=F i+1 (lim0jP (k)*(Y )=F j+1))by Equation 9.8 ' lim0ilim0j(P (k)*(X)=F i+1 P (k)*(Y )=F j+1)by Proposition 9.5 ' lim0i;j(P (k)*(X)=F i+1 P (k)*(Y )=F j+1) which, by definition, is P (k)*(X)b P(k)*P (k)*(Y ): |___| References [Ada69]J. F. Adams. Lectures on generalized cohomology, volume 99 of Lecture No* *tes in Math- ematics, pages 1-138. Springer-Verlag, 1969. [Ada74]J. F. Adams. Stable Homotopy and Generalised Homology. 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