_g(q)) and that the cokernel is polynomial. The f* *irst term is cofree from [RW80 ] so our theorem applies and we see that: Corollary 1.2. As Hopf algebras, K(n)*(K(Z, q+2)) splits off of K(n)*(BP__g(* *q)). This includes the p = 2 case of K(Z, 3) ! BU<6>. In [KW01 ] it is shown that if you deloop the map in the corollary then the Hopf algebra splits off but they were unable to show this particular case which is essential for the computation* * of BP *(K(Z, q + 2)) in [RWY98 ]. In [Kas98], Kashiwabara showed that K(n)*(QS2q) is cofree and injects into K(n)*(BP__2q) with cokernel polynomial. Thus: Corollary 1.3. K(n)*(QS2q) splits off of K(n)*(BP__2q) as Hopf algebras. Actually, Kashiwabara does the same for the spectra T (j) between S0 and BP . In [KW01 ] the Morava K-theories of the spaces in the Omega spectrum for BPwere computed. There are several descriptions of the results. In one ver* *sion a fibration is found X2r! BP__2r! Y2r which gives a short exact sequence in Morava K-theory and where the Morava K- theory of the left term is cofree and the right term is polynomial. Our theorem applies: Corollary 1.4. The short exact sequence of [KW01 , Theorem 2.2 (iii), p. 159] * *is split as Hopf algebras. Just a few more examples: Corollary 1.5. Let p = 2, the short exact sequences in Morava K-theories coming from the fibrations K(Z, 3) ! BO__4! BSp, [KLW , Theorem 2.26(vi)], Sp=U ! BU ! BSp, [RWY98 , p. 162], and Sp=SU ! BSU ! BSp, [KLW , Theorem 2.36], all split as Hopf algebras. 2. Morava Dieudonn'e Modules In the following we let R be the ring Z[V, F ]=(V F - p). Define the category D, of Dieudonn'e modules to consist of Z=(pn - 1)-graded groups M* with an R- module structure where F : Mt- ! Mpt(the Frobenius) and V : Mt- ! Mt=p(the Verschiebung). The morphisms in D are graded R-module maps. Let MD be the category of Morava Dieudonn'emodules given by the full sub- category of D consisting of objects where every element is V -torsion. SPLITTINGS OF BICOMMUTATIVE HOPF ALGEBRAS 3 We also consider the category EC of evenly graded Hopf algebras known as Morava Hopf algebras, see above. Let W be the polynomial ring Z(p)[x1, x2, x3, . .].. Then W admits the struct* *ure of a bicommutative Hopf algebra such that the Witt vectors n pn-1 n wn = xp0 + p x1 + . .+.p xn are primitive (see [SW98 ].) For s 0, and t 2 Z=(pn - 1) define the graded sub Hopf algebra of W Fp W (s, t) = Fp[x0, x1, . .,.xs] by setting |xi| = pi2t. Notice that W (s, t) belongs to EC . For each algebra A* * in EC we define a Morava Dieudonn'e-module by mt(A) = lim-!HomEC(W (s, t=ps), A) where the maps are induced by the sequence W (0, t) - W (1, t=p) - . .-. W (s, t=ps) - . ... where the map W (s, t=p) -! W (s - 1, t) takes the primitive generator in degree 2t=p to zero. The action of V and F are as described in [SW98 ]. Theorem 2.1. [SW98 ] The functor m* : EC -! MD induces an equivalence of categories. The Morava Dieudonn'emodule for a Morava Hopf algebra may be represented as a picture. The picture below shows the Morava Dieudonn'emodule of Vt= W (0, t). The Frobenius acts injectively as shown, and the Verschiebung is trivial. Each * *circle represents a copy of Fp. O__VF=0_//O_VF=0_//_O_VF=0//_O__VF=0//_O . . . Now assume that X is a connected, homotopy commutative H-space such that K(n)*(X) is evenly graded. Then setting vn = 1, we get an object of EC . We shall identify K(n)*(X) with the corresponding Morava Hopf algebra. Examples of spaces X as described above are given by Eilenberg-MacLane spaces [SW98 ]. The Morava Dieudonn'emodules for some Eilenberg MacLane spaces are: m*(K(n)*K(Z=p, n)) = Z=p is concentrated in degree (pn+1 - 1)=(p - 1), where V operates trivially and F * *(1) = (-1)n-1. Moreover, m*(K(n)*K(Z, n + 1)) = Zp1 concentrated in degree (pn+1 - 1)=(p - 1), where V operates by multiplication by (-1)n-1p and F by multiplication by (-1)n-1. This module may be pictured as follows: 4 NITU KITCHLOO, GERD LAURES AND W. STEPHEN WILSON _____________________________________________* *_______________ _Odd__F________________________________________* *_______________________________FF_______ V _|__________________________ _|_____________________________________________* *________________________________ _Odd__F________________________________________* *_______________________________FF_______ V _|__________________________ _|_____________________________________________* *________________________________ _Odd__F________________________________________* *_______________________________GG______ _|___________________________________ V _|_______________________________ .. . A circle represents a copy of Fp and a vertical line is a non trivial extensi* *on. Hence in the picture the Verschiebung coincides with the multiplication by p map (up to an isomorphism) whereas the Frobenius is an isomorphism. We now give some examples of short exact sequences of Hopf algebras which spl* *it both as algebras and as coalgebras but do not split as Hopf algebras. There are three reasons for doing this: (1) to show that we are actually proving somethin* *g, (2) to show our result is not readily generalizable, and (3) because these type* *s of examples are not much found in the literature and are very easy to lose track o* *f. We will do our examples in the standard graded category. The same examples will do for the other two categories either by forgetting the grading or making it peri* *odic. Example 2.2 (A finite example). We construct our example as a Dieudonn'e mod- ule. We want a short exact sequence A ! B ! C. Let A be three copies of Z=(p) on generators a, b, and c, of degrees 2k, 2pk, and 2p2k respectively with V (b)* * = a, V (a) = V (c) = 0, F (b) = c, and F (c) = F (a) = 0. Let B be four copies of Z=* *(p) on generators d, e, f, and g of degrees 2k, 2pk, 2pk and 2p2k respectively with V (e) = d, F (f) = g and F and V zero on all the other generators. The map A ! B is given by a ! d, b ! e + f, and c ! g. The cokernel is just a Z=(p)in degree 2pk with trivial F and V . It is easy to see that the algebra structure,* * given by F , splits, and that the coalgebra structure, given by V , splits but that t* *he Hopf algebra structure doesn't split. Pictorially this is OOO OOO |F| F|| | | O _______// OO ______//_O ' fflffl V ''' V fflffl '' fflffl O O Of course this example can be dualized by changing the direction of all the arr* *ows and interchanging F and V . Example 2.3 (A polynomial example). We thank A.K. Bousfield for this example. This example is easy to describe in terms of Hopf algebras. We let A be W (0, p* *t) and B be W (0, t) W (1, pt). C is the cokernel of the map which takes x0 to xp0 1-1* * x0 (where the three x0 are in W (0, pt), W (0, t), and W (1, pt) respectively). Th* *is splits as coalgebras and algebras but not as Hopf algebras. All of the Hopf algebras a* *re SPLITTINGS OF BICOMMUTATIVE HOPF ALGEBRAS 5 polynomial. This may be easier to see with the Dieudonn'e modules .. . V f.fOO V f..fOO . ffff| ffff| .. ......|F| ... F|| OO OO OO | OO | F| F |F | OOO F | OOO | | |~~~~ |~~~~ OOO OOO OOO|F OOO F| | | F| F |F |~ O F |~O O| ___//_ O|OOO~V~~| O|~V~~OO F || F || O _______//_O where each circle represents a Z=(p)and the compositions V F = F V give extensi* *ons with multiplication by p. Example 2.4 (A cofree example). To get an example where each Hopf algebra is cofree just dualize the preceding polynomial example by reversing all the arrows and interchanging F and V . Example 2.5 (Another type of example). All of our examples lead to another kind of example as well. We can take the middle terms of all of the above non-s* *plit short exact sequences and compare them to the middle terms of the split short exact sequences with the same ends. This gives examples of Hopf algebras which are isomorphic as coalgebras and as algebras but not as Hopf algebras. We get t* *wo examples for finite Hopf algebras, an example where both Hopf algebras are poly- nomial and an example where both are cofree. A previous theorem with a similar flavor along this line is the result of [RW74 ] which states that the Hopf alge* *bra structure of graded bicommutative Hopf algebras over Z(p)which are polynomial and cofree is uniquely determined. 3. Extensions of Morava Dieudonn'e modules Notice that D is an abelian category, and MD is an abelian subcategory of D. Hence equivalence classes of extension of length n form a group ExtnMD(A, B) for Morava Dieudonn'emodules A, B. In D the extension groups can be computed by projective resolutions of the range or injective resolutions of the target. In* * MD projectives are hard to come by but there is a canonical isomorphism (3.1) Ext1MD(A, B) ~= Ext1D(A, B) since all extensions of Morava Dieudonn'emodules in D are Morava Dieudonn'e modules. Definition 3.1. A Morava Dieudonn'emodule is called cofree if V is surjective. This corresponds to Hopf algebras which are cofree as coalgebras. Proposition 3.2. Let Vt denote the polynomial algebra with one primitive gener- ator in degree t and let C be a Morava Dieudonn'emodule. Then we have æ V ExtiMD (m(Vt), C) = CtC = ker(V ) fori = 0 t=p=V= coker(V ) fori = 1 where Ctis the homogeneous part of C in degree t. In particular, Ext1MD(m(Vt), * *C) vanishes for all cofree C. 6 NITU KITCHLOO, GERD LAURES AND W. STEPHEN WILSON Proof.As explained in (3.1), it is sufficient to prove the proposition in the c* *ategory D. Consider the following projective resolution of m(Vt) in D ffV 0 -! R xt=p -! R-! m(Vt) -! 0 where R denotes the free R-module generated by one element xt in degree t. The name V is intended to be illustrative, but could be confusing. The map V ca* *r- ries the element xt=pto the element V (xt) where this second V is the Verschieb* *ung. It is easy to see that R is projective in D. Notice that Hom D(R , C) =* * Ct, where a morphism ' is identified with '(xt). Therefore, on mapping out of the above resolution into C, we get the cochain complex that computes the required Ext groups 0 - Ct=p-V Ct- 0. Theorem 3.3. Let C be cofree in MD , and let D be a polynomial algebra in EC . Then the group Ext1MD(m(D), C) is trivial. Proof.As before, it is sufficient to prove this in the category D. Consider the* * fol- lowing exhaustive filtration of m(D) given by the Morava Dieudonn'esubmodules F i= ker(V i: m(D) ! m(D)). We claim that the associated quotients Qi+1are sums of Morava Dieudonn'emodules of type m(Vt). To see this, observe that Qi+1 is isomorphic to the image of the map V i: F i+1-! m(D). and as such is isomorphic to a submodule of the kernel of V on m(D). Since D is polynomial, this image corresponds to m(U), where U is polynomial [Bou96 , Theorem B.7]. Since V acts trivially on m(U), it follows that this is a sum of copies of m(Vt). By induction and Proposition 3.2, we see with that Ext1D(F i, C) = 0. Now consider the short exact sequence M 1-oeM 0 -! F i-! F i-! m(D) -! 0 i i where oe is the inclusion map oe : F i F i+1. This yields a long exact sequen* *ce for Ext groups, which splits, by the definitions of limit and lim1, into short * *exact sequences: 0 -! lim1Extn-1D(F i, C) -! ExtnD(m(D), C) -! limExtnD(F i, C) -! 0. For n = 1, the term on the right vanishes by what we have shown above. Hence, to prove the theorem, we need to show that the lim1Hom D(F i, C) = 0. This follows if we can verify the Mittag-Leffler condition that requires the natural map Hom D(F i+1, C) -! Hom D(F i, C) to be surjective. The cokernel of the above map is Ext1D(Qi+1, C), which we know to be trivial, and therefore the proof is complete. SPLITTINGS OF BICOMMUTATIVE HOPF ALGEBRAS 7 References [Bou96]A. K. Bousfield, On p-adic ~-rings and the K-theory of H-spaces, Mathema* *tische Zeitschrift 223 (1996), 483-519. [HRW98]M. J. Hopkins, D. C. Ravenel, and W. S. Wilson, Morava Hopf algebras and* * spaces K(n) equivalent to finite Postnikov systems, Stable and Unstable Homotop* *y (Provi- dence, R.I.) (Paul S. Selick et. al., ed.), The Fields Institute for Res* *earch in Mathemat- ical Sciences Communications Series, vol. 19, American Mathematical Soci* *ety, 1998, pp. 137-163. [Kas98]T. Kashiwabara, Brown-Peterson cohomology of 1 1 S2n, Quarterly Journa* *l of Mathematics 49 (1998), no. 195, 345-362. [KL02] N. Kitchloo and G. Laures, Real structures and Morava K-theories, K-Theo* *ry 25 (2002), no. 3, 201-214. [KLW] N. Kitchloo, G. Laures, and W.S. Wilson, The Morava K-theory of spaces r* *elated to BO , Submitted. [KW01] T. Kashiwabara and W. S. Wilson, The Morava K-theory and Brown-Peterson * *coho- mology of spaces related to BP, Journal of Mathematics of Kyoto Universi* *ty 41 (2001), no. 1, 43-95. [RW74] D. C. Ravenel and W. S. Wilson, Bipolynomial Hopf algebras, Journal of P* *ure and Applied Algebra 4 (1974), 41-45. [RW80] D. C. Ravenel and W. S. Wilson, The Morava K-theories of Eilenberg-Mac L* *ane spaces and the Conner-Floyd conjecture, American Journal of Mathematics 102 (19* *80), 691- 748. [RWY98]D. C. Ravenel, W. S. Wilson, and N. Yagita, Brown-Peterson cohomology fr* *om Morava K-theory, K-Theory 15 (1998), no. 2, 149-199. [SW98] H. Sadofsky and W. S. Wilson, Commutative Morava homology Hopf algebras,* * Homo- topy Theory in Algebraic Topology (Providence, Rhode Island) (M. E. Maho* *wald and S. Priddy, eds.), Contemporary Mathematics, vol. 220, American Mathemati* *cal Society, 1998, pp. 367-373. Department of Mathematics Johns Hopkins University 3400 N. Charles Street Bal- timore, MD 21218, USA Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 288, * *D- 69120 Heidelberg, Germany E-mail address: wsw@math.jhu.edu, nitu@math.jhu.edu, gerd@laures.de