== gi(x) = gi(a) = gi(a)=: |___| The following is the main techinical result behind the proof of Theorem 2.7. Theorem 2.11. The indecomposables functor Q : HV ! MV has a right ad- joint S*. Furthermore the natural map M ! QS*(M) in MV is an isomorphism. Note that Theorem 2.7 follows immediately. The one natural map M ! QS*(M) is an isomorphism by this result; the other natural map S*(QH) ! H is an isomorphism because of Proposition 2.1 and Lemma 2.5, and the composite ~= * QH -! QS (QH) ! QH forces QS*(QH) ! QH to be an isomorphism. Proof. We begin by constructing S*(M) for M finitely generated in each degree. Define functors 0; 1 : MV -! MV as follows. 0(M)n = Mn Mn=p Mn=p2 . . . where is is understood that Mk = 0 if k is a fraction. Define V on 0(M) by V (x0; x1; x2; . .).= (x1; x2; . .).: 14 PAUL G. GOERSS There is a natural injection j : M ! 0(M) given by j(x) = (x; V x; V 2x; . .).: Define 1(M) by the formula 1(M)n = Mn=p Mn=p2 . . . and V again defined by projection. There is a natural map d : 0(M) ! 1(M) given by d(x0; x1; x2; . .).= (x1 - V x0; x2 - V x1; . .).: This map is surjective and the sequence 0 ! M -j!0(M) -d!1(M) ! 0 is short exact in MV . In fact, in each degree it is split short exact. Further* *more, choosing a set of generators (over Zp) for Mn for each n defines an isomorphism 0(M) ~=ffQCWkff(1): The number of times each positive integer appears as a kffis finite. There is a similar isomorphism for 1(M): 1(M) ~=fiQCWkfi(1): Define S*(0(M)) = ffCWkff(1): Note that in any given degree this tensor product is the tensor product of fini* *tely many groups. Similary define S*(1(M)) and note that Proposition 2.10 implies that there is a map in HV f : S*(0(M)) -! S*(1(M)) with Qf = d. Define S*(M) by the pull-back diagram in HV S*(M) _____//S*(0(M)) | | | |f fflffl| fflffl| Zp_______//S*(1(M)): Note that S*(M) is the usual Hopf algebra kernel of f. We must now examine this construction. The first thing to do is to show QS*(M) ! QS*(0(M)) is an injection. In fact there is a diagram 0 ____//_P S*(M)___//_P S*(0(M)) | | | | |fflffl fflffl| QS*(M) ____//_QS*(0(M)) where the vertical maps are the canonical maps from the primitives to indecom- posables. Since these are Hopf algebras in HV, Lemma 2.1 implies that these maps are injections with torsion cokernel. Hence QS*(M) ! QS*(0(M)) is one-to-one. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 15 Next let H be any object in HV. Then there is a diagram with columns exact 0 0 | | | | fflffl| fflffl| Hom HV(H; S*(M)) ________//_HomMV(QH; QS*(M)) | | | | fflffl| ~= fflffl| Hom HV(H; S*(0(M))) _____//HomMV (QH; QS*(0(M))) | | | | fflffl| ~= fflffl| Hom HV(H; S*(1(M))) _____//HomMV (QH; QS*(1(M))) Note that, for example Hom MV (QH; QS*(0(M))) ~=Hom MV (QH; 0(M)) and the horizontal maps labelled as isomorphims are so by Lemma 2.10. This diagram implies that Hom HV(H; S*(M)) ~=Hom MV (QH; QS*(M)): Furthermore, if we set H = CWn(k) for various n and k, Lemma 2.9 implies that M ~=QS*(M). These two equations together yield the desired adjunction formula Hom HV(H; S*(M)) ~=Hom MV (QH; M): To complete the proof, we must extend S* to M which are not finitely generated * *in each degree. To do this write such a general M as a filtered colimit of sub-obj* *ects Mi which are finitely generated in each degree then define S*(M) = colimS*(Mi): Note that this colimit can be calculated in graded Zp modules. If H 2 HV is fin* *itely generated as an algebra Hom HV (H; S*(M))~=colimHom HV (H; S*(Mi)) ~=colimHom MV (QH; Mi) ~=Hom MV (QH; M); as needed. In particular, setting H = CWn(k) for various n and k, Lemma 2.9 implies that M ~=QS*(M). Finally, if H is general, Lemma 2.5 implies that H is the filtered colimit of its finitely generated sub-objects which are finitely g* *enerated as algebras. Then, using an argument similar to that given at the beginning of Proposition 2.10, it follows that Hom HV(H; S*(M)) ~=Hom MV (QH; M): |___| Example 2.12. (The Husemoller Splitting [14]). Consider the Hopf algebra H*BU ~=Zp[a1; a2; a3; : :]: 16 PAUL G. GOERSS with the degree of ai equal to 2i and X ak = ai aj: i+j=k This represents the functor on graded Zp algebras (A) = (1 + tA[[t]])0 where A[[t]] is the graded power series on A with deg(t) = -2. (A) becomes a group under power series multiplication. If f(t) = aiti2 (A), then, modulo p, f(t)p apitpi from which it follows that on H*(BU; Fp), [p]ai= (ai=p)p or ai= ai=p where ai=p= 0 if i=p is a fraction. Now define : H*BU ! H*BU to be the unique Hopf algebra map inducing * : (A) ! (A) where *f(t) = f(tp) = aitpi. Then (ai) = ai=pand is a lift of the Frobenius. Thus, from Proposition 2.2 one has M QH*BU ~= QCW2n(1) p-n in MV , hence one has the Husemoller splitting O H*BU ~= CW2n(1): p-n To be fair, these arguments are not so different that than the original ones. 3. The Implications of a lift of the Frobenius The main purpose of this section is to prove Lemma 2.1: that a Hopf algebra with a lift of Verschiebung which is also finitely generated as an algebra is a* * poly- nomial algebra. This requires as examination of the dual Hopf algebra, and here* * I am very indebted to ideas of Bousfield [1]. To keep the record straight note that some finiteness hypothesis is necessar* *y; for example, if H is the primitively generated symmetric algebra over Zp on Qp with trivial lift of the Verschiebung, then H is not a polynomial algebra. The following is the algebra input. It is a consequence of Nakayama's Lemma. Lemma 3.1. Let f : M ! N be a homomorphism of finitely generated Zp modules. Then f is an isomorphism if and only if Fp f : Fp M ! Fp N is an isomorphism. This result has the following immediate corollary. Proposition 3.2. Let A is a graded, connected torsion-free Zp algebra, and suppose A is finitely generated as an algebra. Then A is a polynomial algebra i* *f and only of Fp A is a polynomial algebra. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 17 If H is a finitely generated torsion-free Hopf algebra over Zp, then we may * *apply Borel's structure theorem to Fp H and conclude that there is an isomorphism of algebras O ni Fp H ~= Fp[ai]=(ap ) i where ni is an integer 1 ni 1. Thus Fp H is a polynomial algebra if and only if the Frobenius is injective. Equivalently, one need only show that t* *he Verschiebung on the dual Hopf algebra (Fp H)* is surjective. This will be proved below; see Corollary 3.9. Because we are considering the dual Fp Hopf algebra (Fp H)* we begin by considering the dual Zp Hopf algebra H* = Hom Zp(H; Zp). Since H has a lift of the Verschiebung : H ! H, the dual H* has a lift of the Frobenius OE = *. We now give a short investigation into the category of such objects. Let AF be the category of torsion-free graded connected Zp algebra equipped with a lift of the Frobenius. Let M be the category of torsion-free, positively* * graded Zp modules. Lemma 3.3. The augmentation ideal functor I : AF ! M has a left adjoint S'. Furthermore, for all finitely M 2 M, S'(M) is isomorphic to a polynomial algebra. Proof. First suppose M is finitely generated. Choose a homogeneous set of generators {yi} for M. If A 2 AF and f : M ! IA is given, the Dwork Lemma 1.1 supplies a unique map (3.1) gi: Zp[x0; x1; x2; : :]:! A so that wn 7! 'nfyi. Thus one obtains a unique map in AF O g : S'(M) = Zp[x0; x1; x2; : :]:! A i so that the evident composite M ! IS'(M) ! IA is f. The clause stipulating that S'(M) is a polynomial algebra follows from Equation 3.1. To finish the argument, note that the construction of S'(M) is natural in M, at least where so far defined; that is, for finitely generated M. For general * *M, write M = colimffMffwhere Mff M runs over the diagram of finitely generated sub-modules of M. Define S'(M) = colimffS'(Mff); since the diagram is filtered * *the colimit in AF is isomorphic to the colimit as graded modules. One easily checks_ S'(M) has the requisite universal property. |__| Now let HF be the category of Hopf algebra with a lift of the Frobenius and * *CA the category of graded, connected, torsion free coalgebras over Zp. Let J : CA * *! M be the "coaugmentation coideal functor"; that is, JC = coker(Zp ! C). Proposition 3.4. The forgetful functor HF ! CA has a left adjoint F . Fur- thermore, for if C 2 CA if finitely generated as a Zp module in each degree, th* *en F (C) is a polynomial algebra; indeed, as algebras F (C) ~=S'(JC): 18 PAUL G. GOERSS Proof. Define F (C) to be the algebra S'(JC) with the diagonal induced by completing the following diagram using the universal property of S': C ____________//_C C | | | | fflffl| fflffl| S'(JC) _ _ _//S'(JC) S'(JC): Then one easily checks F (C) 2 HF fulfills the conclusions of the result. * * |___| If A is any torsion-free Zp algebra equipped with a lift ' of the Frobenius * *define a function : A ! A by the formula '(x) = xp + p(x) Compare [1] for the following result. Lemma 3.5. 1) For all x and y in A, (xy) = (x)yp + xp(y) + p(x)(y) 2) For all x and y in A, p-1X1 (x + y) = (x) + (y) - _ p xp-iyi: i=1p i Proof. These are both consequences of the fact that ' is an algebra map. |* *___| Note that these formulas imply that if A is equipped with an augmentation A ! Zp, then the augmentation ideal is closed under . The operation may clarify the structure of free objects in AF. Compare [1] Lemma 3.6. Suppose M is a free Zp module in each degree. Let {yi} M be a set of generators for M. Then there is an isomorphism of Zp algebras S'(M) ~=Zp[nyi| n 0]: Proof. By the construction of Lemma 3.3, it is sufficient to consider the c* *ase where M has one generator in degree m. Then S'(M) ~=Zp[x0; x1; x2; : :]:= CWm (1) in AF, where 'wn = wn+1. Thus one need only show xn xn+1 modulo p and indecomposables. The formula wn+1 = 'wn = wpn+ pwn and the formulas of Lemma 3.5 imply (pnxn) pnxn+1 modulo pn+1 and decomposables. Since pn = '(pn) = ppn + p(pn), Lemma 3.5.1 implies pnxn pnxn+1 modulo pn+1 and decomposables, as required. |___| HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 19 Next let H 2 HF be a Hopf algebra with a lift of the Frobenius and let : Fp H ! Fp H be the Verschiebung. If x 2 H, let {x} denote the class of x in Fp H ~=H=pH. Lemma 3.7. For all x 2 H, there is a congruence {(x)} {x} modulo the decomposables in Fp H. Proof. First suppose H is a polynomial algebra. Then FpH is a polynomial algebra. We will use the formula (y)p = [p](y), where [p] is p times the identi* *ty map in the abelian group of Hopf algebra endomorphisms of Fp H. Thus it is sufficient to show [p]{(x)} {x}p modulo decomposables. Since [p] is a morphism of Hopf algebras and commutes with the lift of the Frobenius, one has [p](x) = ([p]x). Now [p]x = px + z where z 2 IH2. Hence Xp 1 [p]x = (px) + (z) - _ p (px)p-izi: i=1p i The last term is zero modulo p. Lemma 3.5.1 implies that (IH)2 IH2. Hence [p]x (px) modulo p and indecomposables. But (px) = (p)xp + pp(x) + p(p)(x) p = p_-_p_pxp + pp(x) + (p - pp)(x) xp mod p: For the general case of H, fix x 2 H and let C H be a coalgebra so that x 2 C and so that C is finitely generated over Zp. Consider the induced map F (C) ! H given by Corollary 3.4. Then F (C) is a polynomial algebra, and this_ result follows from the naturality of and . |__| This has the following immediate consequence: Proposition 3.8. Let H be a graded, connected torsion-free Hopf algebra over Zp equipped with a lift of the Frobenius. The the Verschiebung : Fp H -! Fp H is surjective. Proof. The previous result shows that is surjective on indecomposables._ This implies is surjective. |__| For completeness we now add: Corollary 3.9. Let H 2 HV be a graded, connected, torsion-free Hopf algebra over Zp equipped with a lift of the Frobenius. If H is finitely generated as an* * algebra, then H is a polynomial algebra. Proof. Combining Lemma 3.2 with the remarks following that result, we need only show that the Verschiebung is surjective on (Fp H)*. This follows from th* *e __ previous result, and the fact that H* has a lift of the Frobenius. * * |__| 20 PAUL G. GOERSS 4. Dieudonne theory. Positively graded bicommutative Hopf algebras over a perfect field form an abelian category with a set of projective generators. As such this category is * *equiv- alent to a category of modules over some ring. Dieudonne theory says which mod- ules over which ring. In this section we use the results of the previous sectio* *ns to elucidate the case k = Fp. Here the main classification result is due to Schoel* *ler [23]; the main advance is that we gave a formula of Theorem 4.8 for computation* *s. Let HA be the category of graded, connected, bicommutative Hopf algebras over Fp. We describe a good set of generators for this category. In this contex* *t, a set of objects {Aff} is a set of generators if every object is a quotient of a * *coproduct of the Aff. Let n > 0 be a positive integer, n = pkm where (p; m) = 1. Consider the torsion free Hopf algebra CWm (k) = Zp[x0; x1; : :;:xk] of Definition 1.6. It h* *as the Witt vector diagonal. Define H(n) = Fp CWm (k) ~=Fp[x0; x1; : :;:xk]: There are many proofs of the following result. A very conceptual argument, due * *to Fabien Morel, can be found in [9]. Lemma 4.1. The Hopf algebras H(n) are projective in HA and form a set of generators. Furthermore is : H(n) ! H(n) is the Verschiebung, then (xi) = xi-1: Here and elsewhere in this document, one takes x-1 to be zero. Of all the morphisms in HA between the various H(n) we emphasize two. The first is in the inclusion v : H(n) = Fp[x0; : :;:xk] ! Fp[x0; : :;:xk+1] = H(pn): For the second, we note that Proposition 1.8.2 implies that there is a unique m* *ap of Hopf algebras f : H(pn) ! H(n) so that the following diagram commutes H(pn) __f__//_H(n) III | III |v [p]II$$Ifflffl| H(pn): Note that, by construction, vf = [p] and that Proposition 1.8.2 also implies th* *at fv = [p]. Definition 4.2.If H 2 HA, the Dieudonne module D*H is the graded abelian group {DnH}n1 with DnH = Hom HA(H(n); H) and homomorphisms F = f* : DnH ! DpnH and V = v* : DpnH ! DnH: HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 21 It follows from the remarks on v and f above that F V = V F = p. Also Proposition 1.8.2 implies that the order of the identity in Hom HA (H(n); H(n))* * is ps+1 if n = psk with (p; k) = 1; hence, we also have ps+1DnH = 0. This suggests the following definition. Definition 4.3.Let D be the category of graded modules M equipped with operators F and V so that F V = V F = p and ps+1Mpsk= 0 if (k; p) = 1. This is the category of Dieudonne modules. Thus we have a functor D* : HA ! D. Some familiar functors on Hopf algebras can be recovered from the Dieudonne module. Let be the "doubling" functor on graded modules; that is, (M)pn = Mn and (M)k = 0 of p does not divide k. Then, for example, if M is a Dieudonne module, V defines a homomorphism of graded modules V : M ! (M). Lemma 4.4. 1) Let H 2 HA, then there is an exact sequence of graded abelian groups 0 ! P H ! D*H -V!D*H: 2) There is also an exact sequence D*H -F!D*H ! QH ! 0: Proof. For 1), notice there is a short exact sequence of Hopf algebras Fp ! H(n) -v!H(pn) ! Fp[xk+1] ! Fp where v defines V . Part 2) is proved in a similar manner, after the introduction of some auxili* *ary_ technology. See [10], for example. |__| Proposition 4.5. Let f : H ! K be a morphism in HA. If the induced homomorphism D*f : D*H ! D*K is an isomorphism, then f is an isomorphism. Proof. From Lemma 4.4, we have that both Qf : QH ! QK and P f : P H ! __ P K are isomorphisms. Hence f is an isomorphism. |__| A crucial calculational result is the following. Proposition 4.6. The homomorphism D* : Hom HA(H(n); H(m)) ! Hom D(D*H(n); D*H(m)) is an isomorphism. This is a consequence of a more general calculation, which we give below in Theorem 4.8. See Corollary 4.11. An immediate consequence of Lemma 4.1 and Proposition 4.6 is the following result, which is Schoeller's Theorem [23]. The* * proof follows from standard techniques in abelian category theory. See, for example, * *the proof of Mitchell's Theorem in [7]. Theorem 4.7. The functor D* : HA ! D has a right adjoint U and the pair (D*; U) form an equivalence of categories. As mentioned, Proposition 4.6 follows from a much more general calculation. To set the stage, let R be the graded ring R = Zp[V; F ]=(V F - p) 22 PAUL G. GOERSS where the degree of V is -1 and the degree of F is +1. Then an object M 2 D* may be regarded as a graded R module, provided one adopts the convention that for a 2 R and x 2 M, (4.1) deg(ax) = pdeg(a)x: With this convention, D* is exactly the category of positively graded R modules, provided we agree that ax = 0 if deg(ax) is a fraction. Similarly, we have a ca* *tegory of positively graded Zp[V ] modules M, with deg(V ) = -1 and the same convention on degrees. Let us call this category DV . The category MV of Definition 2.6 * *is the full subcategory of DV with torsion free objects. There is a forgetful func* *tor D ! DV and it has a left adjoint given by M 7! R Zp[V ]M: Now let H 2 HV be a Hopf algebra with a lift of the Verschiebung. We wish to give a formula to calculate D*(FpH). Since H is determined by QH 2 MV DV , we'd like this formula to be functorial in QH. We can define a natural homorphism j : QH ! D*(Fp H) as follows. Write n = pkm with (p; m) = 1. Then Theorem 2.7 and Proposition 2.9.2 supply an isomorphism (QH)n ! Hom HV(CWm (k); H) and the morphism V : (QH)pn ! (QH)n is induced by the inclusion CWm (k) CWm (k + 1). In degree n define j to be the composition (QH)n ~=Hom HV(CWm (k); H) ! Hom HA(H(n); Fp H) = Dn(Fp H): Then j is a morphism in DV and, hence, it extends to a natural map of Dieudonne modules "H : R Zp[V ]QH ! D*(Fp H): The following is the main result of this section. Theorem 4.8. The map "H is a natural isomorphism of Dieudonne modules for all H 2 HV. This will be proved below after some preliminary calculations. Lemma 4.9. Let M 2 DV be torsion-free and R = Zp[V; F ]=(V F - p) the Dieudonne ring. Then for all s > 0 Tor sZp[V(]R; M) = 0: Proof. Because Tor commutes with filtered colimits, we may assume M is finitely generated. Then, if M(n) M is the sub-module of elements in degree n or less, one has exact sequences 0 ! M(n - 1) ! M(n) ! M(n)=M(n - 1) ! 0 in MV and M(n)=M(n - 1) is isomorphic to a direct sum of modules, which Zp[n], meaning a single copy of Zp in degree n. If Tor sZp[V(]R; Zp[n]) = 0 for all n,* * then a simple induction argument finishes the proof. Define a complex of right Zp[V ] modules M @ M " 0 ! ys Zp[V ] -! xs Zp[V ] -! R ! 0 s0 s0 HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 23 with "(xs 1) = F sand @(ys 1) = xs p - xs+1 V . This is a projective resolution of R as a right Zp[V ] module. Tensoring with Zp[n] yields a complex M @ M " 0 ! ys Zp[n] -! xs Zp[n] -! R Zp[V ]Zp[n] ! 0: s0 s0 One calculates that @(ys 1) = xs p. Hence @ is an injection and the result_ follows. |__| We also need a calculation. Lemma 4.10. Let Fp[x] be the primitively generated Hopf algebra on an element of degree n and f : H(n) ! Fp[x] the unique map of Hopf algebras so that f(xk) * *= x and f(xi) = 0 for i < s. Then D*Fp[x] is the free module over Fp[F ] on f. Proof. Since the Verschiebung is zero on Fp[x], Lemma 4.1 and the definition of V on D*(Fp[x]) show that V = 0 on D*(Fp[x]). The result now follows_from Lemma 4.4.1. |__| We now give the proof of Theorem 4.8. Proof. Since both the source and target of "H commute with filtered colimits we may assume that H is finitely generated and hence a polynomial algebra. (See Proposition 2.1). Since QH has a finite filtration 0 = Ms . . .M1 M0 = QH so that Mk=Mk+1 is a direct sum modules of the form Zp[n], Theorem 2.7 implies that H has a finite filtration Zp = Hs . . .H1 H0 = H so that QHk = Mk and ZpHk+1Hk is a primitively generated polynomial algebra with indecomposables isomorphic to Mk=Mk+1. Both the source and target of "H are exact on this filtration (here we use Lemma 4.9) and we are reduced to the case where H is a primitively generated polynomial algebra. Since both the source and target of "H commute with coproducts we may assume H = Zp[x] with deg(x) = n. Then it is a matter of direct calculation. The module QH ~=Zp[n] ~= Hom HV(CWm (k); H) and we can choose as generator of the latter group the map CWm (k) = Zp[x0; x1; : :;:xk] ! Zp[x] = H sending xk to x and xi to 0 if i < k. Let f 2 Dn(Fp H) be the reduction. Now R Zp[V ]QH is the free module over Fp[F ] ~=R Zp[V ]Zp on f. Hence the result_ follows from Lemma 4.10 |__| Now write n = pkm with (m; p) = 1 and let n 2 DnH(n) be the identity. Corollary 4.11. Let H(n) = Fp[x0; x1; : :;:xk] be one of the projective gen- erators of HA. Then for all Dieudonne modules M, then the natural map Hom D*(D*H(n); M) ! Mn given by sending f to f(n) is an isomorphism. Furthermore D*H(n) ~=R Zp[V ]K(n) where K(n) is the Zp[V ] module with K(n)m = 0 unless m = pik, 0 i s, K(n)pis~=Z=pi+1Z and V is onto. 24 PAUL G. GOERSS Proof. This follows from Theorem 4.8 and Proposition 2.9. |___| Example 4.12. Let's calculate the Dieudonne module of H*BU = H*(BU; Fp). We could use Example 2.12 and Theorem 4.8, but here's another way more suited to a later application. Consider the coalgebra H*(CP 1; Zp). This is dual to the algebra H*(CP 1; Zp) ~= Zp[x], which has a lift of the Frobenius given by x 7! xp. Thus H*(CP 1; Zp) has a lift of the Verschiebung ; indeed, if we defi* *ne fii 2 H2i(CP 1; Zp) to be dual to xi, then (fii) = fip=iwhere fip=i= 0 if p=i * *is a fraction. Now H*(BU; Zp) = S*(H*(CP 1; Zp)) = Zp[b1; b2; . .]. where we write bi for the image of fii. Theorem 4.8 now implies that D*H*BU ~=R Zp[V ]"H*(CP 1; Zp) with V (fii) = fip=i. In particular, if we also write bifor the image of fiiin * *D2iH*BU, then D2iH*BU ~=Z=p(i)Z generated by bi where (i) = k if i = pkj with (j; p) = 1. Furthermore, V bi= bi* *=p, and this forces F (bi) = pbpi. Part II: Bilinear Maps, Pairings, and Ring Objects 5. Bilinear maps and the tensor product of Hopf algebras First we establish a categorical framework for tensor products, then special* *ize to our main interest_the category of Hopf algebras over over a field. Much of t* *he ideas about tensor products in the category of coalgebras can be found in [13]. Let C be a category and A C a sub-category of abelian objects in C. Thus, for all A 2 A, FA = Hom C(.; A) : Cop ! Ab is a functor to abelian groups. Morphisms f : A ! B in A induce natural trans- formations FA ! FB of group-valued functors. We will assume, for simplicity, that 5.1.1)both C and A have all limits and colimits; 5.1.2)the forgetful functor A ! C has a left adjoint S(.). Definition 5.2.Let A, B, and C be objects in A. A morphism ' : AxB ! C in C is a bilinear map if for all X 2 C, the induced map FA (X) x FB (X) ! FC (X) is a natural bilinear map of abelian groups. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 25 This is equivalent to demanding that the following diagrams commute: 1xmB A x B x B _________________//_A x B (1xTx1)(Ax1)|| |'| fflffl| fflffl| A x B x APxPB t9C9t PPPP tttt 'x'PPPPP'' ttttmC C x C mAx1 A x A x B__________________//A x B (1xTx1)(1xB)|| |'| fflffl| |fflffl A x B x APxPB t9C9t PPPP tttt 'x'PPPPP'' ttttmC C x C Here mA : A x A ! A is the multiplication, A is the diagonal, and T is the twist map interchanging factors. We now define tenor products. The symbol "" will be reserved for the tensor product of modules over rings and, hence, for the product of coalgebras. Definition 5.3.A tensor product of A, B 2 A is an initial bilinear map " : A x B ! A B in A. Specifically, if ' : A x B ! C is any bilinear map there is a unique morphism : A B ! C in A making the diagram A x BF______"_____//A B FFF wwww 'FFFF www F## --ww C commute in C. Remark 5.4. If tensor products exist they are unique up to isomorphism in A. Also, there is then a natural transformation of group-valued functors (FA FB )(X) = FA (X) FB (X) ! FA B(X): This need not be an isomorphism: consider the case where C is the category of s* *ets, A the category of abelian groups and A = B = Z. The assumptions made above make it easy to show tensor products exist. See [13]. Proposition 5.5. Under the assumptions of (5.1) any two objects A, B 2 A have a tensor product A B in A. Proof. This is an adaptation of the case where A is the category of R-modul* *es for some ring R. Define A B to be the colimit of the diagram in A fA fB S(A x A x B) S(A x B) S(A x B x B) gA gB 26 PAUL G. GOERSS where fA is adjoint to A x A x B mAx1!A x B ! S(A x B) and gA is adjoint to A x A x B ! A x B x A x B ! S(A x B) x S(A x B) m!S(A x B) where the first map is (1 x T x 1)(1 x B ). The morphism fB and gB are defined similarly. Let " : A x B ! A B be defined by the composite A x B ! S(A x B) ! A B; it is bilinear by construction. Also if ' : A x B ! C is any bilinear map, the induced morphism S(A x B) ! C factors uniquely through A B. |___| We now begin to specialize to the case where A is a category of bicommutative Hopf algebras over a commutative ring k and C is a category of coalgebras. In t* *his case a bilinear map is a morphism of coalgebras ' : H1 H2 ! K where H1, H2, and K are Hopf algebras. It is convenient to write xOy for '(xy). Lemma 5.6. For Hopf algebras the following formulas hold. 1. For all x y 2 H1 and z 2 H2 X xy O z = (x O zi)(y O z0i) i where z = zi z0i. A similar formula holds for x 2 H1, y, z 2 H2. 2. If 1 2 H1 is the unit of H1 regarded as an algebra, and x 2 H2, then 1 O x = "(x)1, where " : H2 ! k is the augmentation. Proof. Part 1.) merely rewrites in formulas what it means to be bilinear. F* *or 2.), it is sufficient to show k H2 ~=k. More generally, given A and C as in 5.* *1, let * 2 A be the terminal object. Then for all A 2 A, * A ~=*. To see this notice that if * x A ! C is a bilinear map, the induced bilinear map F*(X) x FA (X) ! FC (X) is the trivial map, since it factors through F*(X) FA (X) and F*(X) = 0. Now consider the pair ("; 1A ) 2 F*(A) x FA (A) where " : A ! * is the unique map. This maps to 0 in FC (A) so there is a diagr* *am A ______//_* ~=|| |j| fflffl| fflffl| * x A ____//_C where j is the unit map in C from * to C. This implies that the unique map __ * x A ! * is the initial bilinear map, as required. |_* *_| HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 27 For the next few results we will stipulate only that C is a category of coco* *m- mutative coalgebras over a ring k and that A is a category of bicommutative Hopf algebras over k. We will abbreviate this by saying A is a category of Hopf alge* *bras and C is a category of coalgebras. Lemma 5.7. Let A be a category of Hopf algebras and C a category of coalge- bras. Then for any diagram in A, the natural map colimff(Aff B) ! (colimffAff) B is an isomorphism. Proof. This result is true in much more general contexts; however, it does require that the abelian category A satisfy certain axioms which can be uncover* *ed by examining the argument. First consider pushouts. Let A1 f1A12f2!A2 be a diagram of Hopf algebras. Then the push-out in A is A1 A12A2; that is, the algebra A1 A2 modulo the ideal generated by elements of the form (5.1) f1(a)x y - x f2(a)y: Suppose one has a diagram f21 A12 B _____//A2 B f11 || |'2| fflffl| fflffl| A1 B _'1____//_C where '1 and '2 are bilinear. Define a bilinear map ' : (A1 A12A2) B ! C by X (5.2) '(x y b) = '1(x bi)'2(y ci) i where B b = bi ci. The associativity of the diagonal, the formula 5.2, and Lemma 5.6.1 imply ' is well-defined. Now take C = A1 B A12 B A2 B and '1 and '2 the evident bilinear maps. Then there is a resulting bilinear map ' : (A1 A12A2) B ! A1 B A12 B A2 B: We leave it as an exercise to show ' has the requisite universal property to pr* *ove the result in this case. Since the result is true for push-outs, it is true for finite coproducts and* * co- equalizers. I claim it is true for all coproducts. In fact, I claim it is true * *for filtered colimits, so the claim about coproducts follows from the fact that any coproduct is the filtered colimit of its finite sub-coproducts. To see that it is true fo* *r filtered colimits, one uses the construction given in the proof of Proposition 5.5 and t* *he fact that, in this case, the left adjoint S commutes with filtered colimits in the s* *ense that for any diagram of coalgebras, the natural map in C colimffCS(Cff) ! S(colimffCCff) ~=colimffAS(Cff) is an isomorphism; that is, the forgetful functor from A to C makes filtered co* *limits. 28 PAUL G. GOERSS Finally the result is true for all colimits because there is, for any diagra* *m Aff in A, a coequalizer diagram a a Aff Aff! colimffAff: ff!fi ff Hence the general result follows from the result on coproducts and coequalizers* *. |___| We next calculate the tensor product of free objects. Corollary 5.8. Let A be a category of Hopf algebras and K 2 A. Then the functor A 7! A given by H 7! H K has a right adjoint. Proof. This is a consequence of the special adjoint functor theorem [15]_and the fact that . K commutes with all colimits. |__| It is appropriate to call this functor hom (K; .) so that one has a formula (5.3) Hom A (H1 K; H2) ~=Hom A(H1; hom(K; H2)): It would interesting to have a concrete description of this homomorphism object and its Dieudonne module. 6. Bilinear pairings for Hopf Algebras with a lift of the Verschiebung. We now examine the case where A = HV, the category of connected, torsion- free Hopf algebras over Zp equipped with a lift of the Verschiebung, and C = CV, the category of connected, torsion free coalgebras over Zp, also equipped with * *a lift of the Verschiebung. These objects were discussed in some detail in section 2. * *The forgetful functor HV ! CV has left adjoint S*: if C 2 CV, S*C is the symmetric algebra on JC = ker{" : C ! Zp} with diagonal induced from C and lift of the Verschiebung given by extending the left on C. Thus HV has the pairing . The indecomposables functor Q defines a functor from HV to the category MV of graded Zp[V ] modules M which are torsion-free as Zp modules. If x 2 Mk, then V x 2 Mk=p, wwhere we mean V x = 0 if p does not divide k. In Theorem 2.11 it w* *as noted that this indecomosables functor has a right adjoint S*(.) and in Theorem 2.7 we showed that these two functors give an equivalence of categories. Now let H1, H2, and K be in HV and ' : H1 H2 ! K be a bilinear map in CV. Then, because ' is a morphism in CV, ' commutes with the lifts of the Verschiebung by definition. Lemma 5.6.1 now implies that ' ind* *uces a pairing (6.1) Q' : QH1 QH2 ! QK where, writing x O y for Q'(x y), one must have V (x O y) = V x O V y: Thus we should give QH1 QH2 the structure of an object in MV by defining V (x y) = V x V y. Now let H1 = S*(M) and H2 = S*(N) be two objects in HV. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 29 Proposition 6.1. There is a natural isomorphism in HV S*(M) S*(N) ~=S*(M N): Proof. We define a bilinear pairing S*(M) S*(N) ! S*(M N), and then we will show it has the correct universal property. First note that if C is a coalgebra with a lift of the Verschiebung, one can* * regard the coaugmentation ideal JC as an object in MV and the functor J : CV ! MV has as right adjoint S*(-), where we forget the algebra structure. To see this,* * let S* : CV ! HV be left adjoint to the forgetful functor. Then Hom CV(C; S*(M)) ~= Hom HV (S*(C); S*(M)) ~= Hom MV (QS*(C); M) ~= Hom MV (JC; M): With this in hand, let OE : S*(M) S*(N) ! K = S*(QK) be any bilinear map in CV. The adjoint J(S*(M) S*(N)) ! QK factors, by Lemma 5.6, as J(S*(M) S*(N)) ! J(S*(M)) J(S*(N)) ! M N QOE-!QK where QOE is an in equation 6.1. This supplies a factoring of OE *QOE S*(M) S*(N) -j!S*(M N) S-! S*(QK) ~=K: If we can show the first map is bilinear, we'll be done. To do this, we give an* *other construction of j. Note there is an obvious bilinear pairing Hom MV (JC; M) x Hom MV (JC; N) ! Hom MV (JC; M N) sending a pair (f; g) to the composition (6.2) JC J!J(C C) ! JC JC fg!M N Thus we get a bilinear pairing (6.3) Hom CV(C; S*(M)) x Hom CV(C; S*(N)) ! Hom CV(C; S*(M N)) and, hence, a bilinear pairing j0: S*(M) S*(N) ! S*(M N): Note that j0is obtained by applying 6.3 to the two projections S*(M) S*(N) ! S*(M) and S*(M) S*(N) ! S*(N). Hence the morphism of coalgebras j0 is adjoint to the morphism in MV J(S*(M) S*(N)) ! JS*(M) JS*(N) ! M N where the last map is projection onto the indecomposables. Since j and j0 are * * __ adjoint to the same map, and since j0 is bilinear, j must be bilinear. * * |__| 30 PAUL G. GOERSS The universal bilinear map j : S*(M) S*(N) ! S*(M N) induces a map, via equation 6.1 Qj : QS*(M) QS*(N) ! QS*(M N) which fits into the following diagram: QS*(M) QS*(N) ____//_QS*(M N) | | | | fflffl| fflffl| M N _____=______//_M N: The vertical maps are induced by the isomorphism of functors QS* ! 1. Thus we have proved Corollary 6.2. If H1 and H2 are two Hopf algebras in HV, the universal bilinear map H1 H2 ! H1 H2 induces an isomorphism in MV QH1 QH2 ! Q(H1 H2): In the next result we are concerned with free Hopf algebras. If C 2 CV is a torsion-free coalgebra with a lift of the Verschiebung, let S*(C) be the free H* *opf algebra in HV on C. It is the symmetric algebra on the coaugmentation ideal of C, equipped with the coproduct and the lift of the Verschiebung induced from C. Given C1 and C2 in CV let C1 ^ C2 be the algebraic smash product of C1 and C2. This is defined as follows. First C1 _ C2 is defined by the push-out diagram Zp x Zp_____//C1 x C2 | | | | fflffl| fflffl| Zp _______//_C1 _ C2 where Zpx Zp ! Zp is addition. Then C1^ C2 is defined by the push-out diagram C1 _ C2_____//C1 C2 |"| || fflffl| fflffl| Zp _______//C1 ^ C2 If S*(C1) S*(C2) ! K is a bilinear map in HV, then the composite C1 C2 ! S*(C1) S*(C2) ! K factors, in CV, through C1 ^ C2, by Lemma 5.6. Thus one gets a map in HV S*(C1 ^ C2) ! K: Corollary 6.3. The induced map in HV S*(C1 ^ C2) ! S*(C1) S*(C2) is an isomporphism. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 31 Proof. There is a natural isomorphism QS*(C) ~=JC, where JC is the coaug- mentation ideal of C. The result now follows from Corollary 6.2 and Theorem_ 2.7. |__| 7. Universal bilinear maps for Hopf algebras over Fp. Let HA be the category of graded, connected Hopf algebras over Fp. The purpose of this section is to calculate the Dieudonne module of the target of t* *he universal bilinear map H K ! H K as a functor of D*H and D*K in HA. In summary, there will be an induced pairing D*H D*K ! D*(H K) which, while not an isomorphism, can be modified in a simple way to produce an isomorphism. The exact result is below in Theorem 7.7. The first observation is this. Let H and K be objects in HV, the category of graded connected torsion-free Hopf algebras over the Zp equipped with a lift of* * the Verschiebung. Let H K ! H K be the universal bilinear map in that category. Lemma 7.1. Suppose H and K in HV are finitely generated as Zp modules in each degree. Then the induced map (Fp H) (Fp K) ! Fp (H K) is an isomorphism. Proof. By construction (FpH) (FpK) is a quotient of of the symmetric algebra S*((Fp H) (Fp K)). See Proposition 5.5. This fact and Lemma 5.6.1) imply that if {xi} is a homogeneous basis of Q(Fp H) ~=Fp QH and {yj} is a homogeneous basis of Q(Fp K), then {xiO yj} spans Q((Fp H) (Fp K)). But Corollary 6.1 implies that the elements xi O yj are linearly independent in Q(Fp (H K)). Hence Q((Fp H) (Fp K)) ! Q(Fp (H K)) is an isomorphism. Since H K is a polynomial algebra by Proposition 2.1,_the_ result follows. |__| Remark 7.2. The finite type hypothesis can be removed by a filtered colimit argument, or by an application of Theorem 7.7 below. In the following result, we are primarily concerned with generators H(n) of HA. Recall from Equation 4.1 that H(n) ~=Fp CWm (k) ~=Fp[x0; : :;:xk] where n = pkm with (p; m) = 1. For simplicity write G(n) = QCWm (k): Corollary 7.3. Suppose H and K in HV are finitely generated as Zp modules in each degree. Then there is an isomorphism R Zp[V ](QH QK) ! D*((Fp H) (Fp K)): In particular if H(n) 2 HA are the projective generators, R Zp[V ](G(n) G(m)) ! D*(H(n) H(m)) is an isomorphism. 32 PAUL G. GOERSS Proof. The first isomorphism follows from Theorem 4.8 and Corollary 6.2.__ The second isomorphism follows from the first. |__| We now can define the pairing D*H D*K ! D*(H K), using the method of universal examples. By Proposition 2.9.2, if M 2 MV , then Hom MV (G(n); M) ~= Mn. Let n 2 G(n)n correspond to the identity. By abuse of notation, also write n for the reduction of this class in D*H(n) ~=R Zp[V ]G(n): Then Corollary 4.11 says the function Hom D (D*H(n); M) ! Mn given f 7! f(n) is an isomorphism. In particular, if H 2 HA, the natural isomor- phism Hom HA (H(n); H) ~=DnH is defined by f 7! D*f(n). Now let n m 2 D*(H(n) H(m)) ~=R Fp[V(]G(n) G(m)) be the evident class. If H1 H2 ! K is a bilinear pairing of objects in HA, we get a pairing (7.1) O : DnH1 x Dm H2 ! Dn+m K as follows. If x 2 DnH1 and y 2 Dm H2 we get a diagram (7.2) H(n) H(m) _____//H(n) H(m) fxfy || |g| fflffl| fflffl| H1 H2 ____________//_K where D*fx(n) = x and D*fy(m ) = y, and g is the unique Hopf algebra map filling the diagram. Then (7.3) x O y = D*g(n m ): Notice that element n O m 2 Dn+m [H(n) H(m)] is represented by the map g in the above commutative diagram. Lemma 7.4. This pairing is bilinear and induces a pairing of graded modules O : D*H1 D*H2 ! D*K: Proof. It is sufficient to examine the universal example (7.4) (H(n) H(n)) H(m) ! (H(n) H(n)) H(m): If 1n+ i2n2 Dn(H(n) H(m)) ~=DnH(n) DnH(n), we need (7.5) (1n+ 2n) O m = 1nO m + 2nO m : But the bilinear map of 7.4 is the reduction modulo p of a bilinear map of obje* *cts in HV. Indeed, write n = pjs and m = pkt where (p; s) = 1 = (p; t). Then the pairing of 7:4 is the reduction of pairing (CWs(j) CWs(j)) CWt(k) ! (CWs(j) CWs(j)) CWt(k) which induces a pairing on indecomposables (which is an isomorphism) O : (G(n) G(n)) G(m) ! Q[(CWs(j) CWs(j)) CWt(k)]: Here the formula 7.6 is obvious. |___| HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 33 Because the isomorphism QCWs(j) QCWt(k) ! Q(CWs(j) CWt(k)) is an isomorphism in MV , similar methods imply Lemma 7.5. If H1 H2 ! K is a bilinear pairing of Hopf algebras over Fp, then the pairing O : D*H1 D*H2 ! D*K has the property that V (x O y) = V x O V y: The operator F behaves differently. In fact, if F is supposed to reflect the Frobenius (.)p and V the Verschiebung , then one calculates xp O y = (x O y)p using Lemma 5.6.1. Then one might expect Lemma 7.6. The pairing O : D*H1 D*H2 ! D*K has the property that F x O y = F (x O V y) and x O F y = F (V x O y): Proof. Again, one need only calculate F n O m = F (n O V m ) in D*(H(n) H(m)). To do this, write m = pkt and n = pjs, where (s; p) = 1 = (t; p). Let CWs(1) = Zp[x0; x1; : :]:and CWt(1) = Zp[y0; y1; : :]:be the Hopf algebras of Definition 1.6. Then CWs(j) = Zp[x0; x1; : :;:xj] CWs(1) and, similarly, CWt(k) CWt(1). Now ae r QCWs(1)i~= Zp0 io=tphserwise and xr (we confuse xr with its image in the indecomposables) generates the modu* *le of indecomposables QCWs(1)prs. Also V xr = xr-1, by Lemma 4.9. Finally QCWs(j) CWs(1) is a split injection of Zp modules and n = xj. Notice that H(n) H(m) ! (FpCWs(1)) (FpCWt(1)) is an injection since QCWs(j) QCWt(k) ! QCWs(1) QCWt(1) is a split injection and both source and target are polynomial algebras. Thus D*H(n) H(m) ! D*[(Fp CWs(1)) (Fp CWt(1))] is an injection. Now one calculates F n O m = F xk O yj= F V xk+1 O yj = pxk+1 O yj = F V (xk+1 O yj) = F (xk O V yj) = F (n O V m ): |___| 34 PAUL G. GOERSS These results prompt the following definitions. A function f : M1 x M2 ! N of graded abelian groups will be called a graded pairing if deg(f(x; y)) = deg(x) + deg(y): A graded pairing f : M1 x M2 ! N of Dieudonne modules will be called bilinear in D if it is bilinear as a pairing of graded abelian groups and V f(x; y)= f(V x; V y) F f(x; V y)= f(F x; y) f(x; F y) = F f(V x; y): Then the content of Lemmas 7.4-7.6 is that a bilinear map of Hopf algebras over Fp induces a bilinear pairing in D D*H1 x D*H2 ! D*K: A universal bilinear pairing in D, written j : M1 x M2 ! M1 D M2 is an initial bilinear pairing in D out of M1x M2, in the obvious sense. If it exists, it is * *unique. We will show it exists and give a concrete description. Indeed, for the purposes of the next few paragraphs, define a Dieudonne modu* *le M D N as follows. For the graded tensor product M N, then this is a Zp[V ] module with V (x y) = V x V y: Our grading conventions are the exponential conventions given in Equation 4.1. * *Let R = Zp[F; V ] and (7.6) M D N = R Zp[V ](M N)=K where K is the sub-Dieudonne module generated by the relations F x V y = 1 F x y F V x y = 1 x F y: If M1 x M2 ! N is any bilinear pairing in D, there is a morphism of Dieudonne modules M1 D M2 ! N. Also the function j : M1 x M2 ! M1 D M2 given by (x; y) 7! x y is a bilinear pairing in D. One now easily checks that * *j has the requisite universal property, so D has universal bilinear maps. The following is the main result of this section. Theorem 7.7. Let H and K be Hopf algebras over Fp. Then the induced map of Dieudonne modules D*H D D*K ! D*(H K) is an isomorphism. Proof. We reduce to a special case. Let S* : CA ! HA be left adjoint to the forgetful functor. Then the Hopf algebras S*(Fp C) with C 2 CV finitely generated in each degree generate HA. This follows from the fact that the Hopf algebras H(n) = FpCWm (k) generate HA, so the Hopf algebras FpH, H 2 HV with H finitely generated in each degree generate HA and HV is, in turn, genera* *ted by Hopf algebras of the form S*(C) with C 2 CV, finitely generated in each degr* *ee. See Lemma 2.5. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 35 Thus we may write equations colimffS*(Fp Cff) ~=H and colimfiS*(Fp Cfi) ~=K for suitable diagrams, where Cffand Cfiare objects in CV. Since both the source and target of the natural map D*H D D*K ! D*(H K) commute with colimits (see Lemma 5.7), we may assume that H = S*(Fp C1) and K = S*(Fp C2), with C1 and C2 in CV, and finitely generated in each degree. To prove the result in this case, Lemma 7.1 implies there is a natural isomo* *r- phism Fp [S*(C1) S*(C2)] ! S*(Fp C1) S*(Fp C2): Then Corollary 6.3 and Theorem 4.8 complete the calculation: D*(S*(Fp C1) S*(Fp C2)) ~=D*(Fp [S*(C1) S*(C2)]) ~=D*(Fp S*(C1 ^ C2)) ~=R Zp[V ](JC1 JC2) ~=[R Zp[V ](JC1)] D [R Zp[V ](JC2)]: |___| The pairings on HA and D on D are symmetric and the isomorphism of Theorem 7.7 reflects the symmetry. Note that if t : H1 H2 ! H2 H1 is the switch map, the composite H1 H2- t!H2 H1 ! H2 H1 is bilinear and induces an isomorphism ~= t : H2 H1- ! H2 H1: The following can be proved by reducing to the case of universal examples, as in Lemma 7.4. Lemma 7.8. Let H1 and H2 be Hopf algebras in HA. Then the following dia- gram commutes: D*H1 D D*H2 __"__//D*(H1 H2) t|| D*(t)|| fflffl| " fflffl| D*H2 D D*H1 _____//D*(H2 H1): A similar statement holds for HV. Example 7.9. Suppose M, N are Dieudonne modules and V is surjective on M and N. Then one can define the structure of a Dieudonne module on M N as follows. First V (x y) = V x V y. Second, if x 2 M, write x = V z. Then one defines F (x y) = z F y: If V z = V z0, then z0= z + w where pw = 0. Write y = V y0. Then z0 F y = z F y + w F V y0= z F y 36 PAUL G. GOERSS so F is well-defined. One easily checks V F = F V = p. Now consider the inclusi* *on j : M N ! M D N: To start, this is only a morphism of Zp[V ] modules. However, (7.7) j(F (x y)) = 1 z F y = F x y = F j(x y) so this is a morphism of Dieudonne modules. I claim it is an isomorphism. To see this we use the universal property of M D N. Suppose f : M x N ! K is a bilinear pairing in D. Then we need only show there is a unique map of Dieudonne modules g : M N ! K making the obvious diagram commute. It is required, then, that g(xy) = f(x; y). It follows that g commutes with V . That g commutes with F follows exactly as in Equation 7.7 8.Symmetric monoidal structures. The categories of bicommutative Hopf algebras considered in the previous two sections are nearly symmetric monoidal categories; what is missing is a unit ob* *ject e so that e H ~=H e ~=H. In order to supply this object we must extend the category somewhat. Let H be a non-negatively graded Hopf algebra over a commutative ring k. Then H0 H_the elements of degree 0_form a Hopf algebra over k. With an eye to topological applications we insist that H0 be a group ring. Put another way,* * let X(H) = Hom CAk(k; H) where CAk is the category of non-negatively graded coalgebras over k. Then X(H) is an abelian group and evaluation at 1 2 k, defines an injection X(H) ! H and hence an injection of Hopf algebras k[X(H)] ! H where k[X(H)] is the group algebra. We will insist that k[X(H)] ! H0 be an isomorphism, and say H is group-like in degree 0. More generally, if C 2 CAk, we can define X(C) = Hom CAk(k; C) and get an injection k[X(C)] ! C0, where k[X(C)], the free k module on X(C), is now only a coalgebra. If this is an isomorphism, we will say C is set-like in d* *egree 0. Finally, if H is a non-negatively graded Hopf algebra over k, let Hc = k H0 H be the connected component of the identity. Then there is a natural isomorphism (8.1) H ! Hck H0: Such a splitting fails for coalgebras. The map H ! Hc is an isomorphism if and only if H is connected. We begin with a category of torsion-free Hopf algebras over Zp. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 37 Definition 8.1.Let HV+ be the category of torsion-free bicommutative Hopf algebras i) group-like in degree 0; ii) are equipped with a Hopf algebra map lifting the Verschiebung; and iii) = 1 in degree 0. The last hypothesis is innocuous as for all H 2 HV+ , the Verschiebung : Fp H ! Fp H is the identity in degree zero. The underlying coalgebra category will be called CV+ ; it consists of torsion-free coalgebras C over Zp equipped * *with a coalgebra map : C ! C lifting the Verschiebung. Again we ask that = 1 in degree 0. Lemma 8.2. The forgetful functor HV+ ! CV+ has a left adjoint F . Proof. Let C 2 CV+ and S*C be the symmetric algebra on C endowed with the obvious diagonal. This is not yet a Hopf algebra. In fact, let X = X(C). Th* *en there is an isomorphism S*(C)0 ~=Zp[NX] where NX is the free abelian monoid on X. Set F C = Zp[ZX] ZpNX S*(C): Note that F C is an appropriate group completion of S*C. |___| To classify HV+ as a category of modules, we extend the category MV of section 2 as follows: define a new category M+Vto be the category of graded abe* *lian groups M equipped with a shift map V : Mpn ! Mn so that Mn is a torsion free Zp module if n 1 and V = 1 : M0 ! M0. If M 2 M+Vthen the elements Mc M of positive degree form a sub-object and the natural isomorphism M ~=Mc x M0 defines an equivalence of categories M+V~=MV x Ab: Here Ab is the category of abelian groups. This will reflect the isomorphism of Equation 8.1 We next define a functor Q+ : HV+ ! M+Vby the equation Q+ (H) = QHcx X(H): In degree n, this functor is representable. If n > 0, let n = pks with (n; s) =* * 1 and CWs(k) the Witt vector Hopf algebra of Definition 1.6. Then we have, by Theorem 2.7 and Proposition 2.9, that [Q+ (H)]n ~=QHc ~=Hom HV(CWs(k); Hc) ~=HomHV+ (CWs(k); H): If n = 0, then [Q+ (H)]0 = X(H) ~=Hom HV+(Zp[Z]; H): The splitting of Equation 8.1 and Theorem 2.7 immediately imply Proposition 8.3. The functor Q+ defines an equivalence of categories Q+ : HV+ ! M+V. Now we turn to bilinear maps. Propositions 5.5 and Lemma 8.2 imply that the category HV+ has universal bilinear maps H K ! H K. Here is the first result on these. 38 PAUL G. GOERSS Lemma 8.4. 1.) Let A and B be abelian groups. Then in HV+ Zp[A] Zp[B] ~=Zp[A Z B]: 2.) Let H 2 HV+ be connected and A an abelian group. Then Zp[A] H is connected and Q(Zp[A] H) ~=[A Z QH]=T where T A Z QH is the torsion subgroup. 3.) For all H 2 HV+ there is a natural isomorphism Zp[Z] H ~=H: Proof. Part 1.) is immediate from the universal property and the requirement that objects on HV+ are group-like in degree 0. We next prove 3.) Let o 2 Z Zp[Z] be the generator. Write Z multiplicativel* *y. If f : Zp[Z] H ! K is any bilinear map, let h : H ! K be the map h(x) = f(o x) = o O x. This is a Hopf algebra map since o is group-like. See Lemma 5.6 Also f is determined by h and bilinearity. The claim is that there is a bilinea* *r map g making the following diagram commute Zp[Z] H _g__//_H uuu f || uuuu fflffl|hzzuuu K: If so, the result will follow. If x 2 H write the n-fold diagonal of x nx = xi1 . . .xin: Then g is defined by bilinearity and 8 0 g(on O x) = :j"(x) n = 0 O(xi1) . .O.(xin)n < 0 Here O : H ! H is the canonical anti-automorphism arising from the fact, that as a group object, H must support inverses. One easily checks g is bilinear. For part 2.), we first prove Zp[A] H is connected. Given any bilinear map Zp[A] H ! K we get a diagram of bilinear maps Zp[A] H _'____//K | | | | fflffl| fflffl| Zp[A] Zp_____//K0; hence, a diagram of Hopf algebras __' Zp[A] H _____//K "|| || fflffl| fflffl| Zp ________//_K0 since Zp[A] Zp ! Zp is an isomorphism. Hence __'factors through Kc. Now let ' be universal bilinear map Zp[A] H ! Zp[A] H; then __'= 1, and Zp[A] H is connected. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 39 To prove the assertion on indecomposables, note that part 3.) implies the re* *sult for A = Z. For general A, take a free resolution F1 ! F0 ! A ! 0: Then since Q(Zp[.] H) commutes with all colimits, one gets a short exact seque* *nce in MV F1 Z QH ! F0 Z QH ! Q(Zp[A] H) ! 0: Since all objects in MV are torsion free, the result follows. * * |___| This last result suggests how to define and analyze universal bilinear maps * *in M+V. A bilinear pairing in MV f : M1 x M2 ! N is a bilinear map of graded abelian groups (that is, degf(x; y) = deg(x) + deg(* *y)) so that V f(x; y) = f(V x; V y): There is a universal bilinear map M x N ! M M N. Write M ~=M0 Mc where M0 and Mc are the elements of degree 0 and positive degree respectively. Then M M N is nearly M N modulo torsion in positive degrees: M M N ~=M0 N0 M0 Nc=T1 Mc N0=T2 McZpNc; where T1 and T2 are the torsion sub-modules. Lemma 8.4 and Corollary 6.2 now imply Proposition 8.5. Let H K ! H K be the universal bilinear pairing in HV+ and Q+ : HV+ ! M+Vthe equivalence of categories. Then there is a bilinear pairing in M+V Q+ H x Q+ K ! Q+ (H K) and the induced map in MV Q+ H MQ+ K ! Q+ (H K) is an isomorphism. For the following result, we will say that a functor F : (C; ; e) ! (C0; 0; * *e0) is an equivalance of categories with symmetric monoidal structure if F is an equiv* *a- lence of categories F (e) ~=e0and there are natural isomorphisms F (X) 0F (Y ) ~=F (X Y ): Corollary 8.6. 1.) The category of Hopf algebras HV+ with pairing is a symmetric monoidal category with unit e = Zp[Z]. 2.) The category M+Vwith pairing M is a symmetric monoidal category with unit e = Z, concentrated in degree 0. 3.) Q+ : HV+ ! M+Vis an equivalence of categories with symmetric monoidal structure. Similar results hold over Fp. Let HA+ be the category of bicommutative Hopf algebras over Fp group-like in degree 0, and let CA+pbe the category of cocommu- tative coalgebras over Fp which are set-like in degree 0. Then: Lemma 8.7. The forgetful functor HA+ ! CA+ has a left adjoint. The corresponding category of Dieudonne modules is easily defined. 40 PAUL G. GOERSS Definition 8.8.An object M 2 D+ is a non-negatively graded abelian group equipped with operators V : Mpn ! Mn, F : Mn ! Mpn so that 1. V F = F V = p; 2. Mc 2 D; and 3. V = 1 : M0 ! M0. Notice, as in Equation 4.1, M 2 D+ may be regarded as a Zp[V; F ]=(V F - p) module. Define H(n) 2 HA+ by H(0) = Fp[Z] and H(n) = Fp[x0; x1; : :;:xs] = CWm (k) for n = pkm with (m; p) = 1. Then one has a functor D* : HA+ ! D+ given by DnH = Hom HA+ (H(n); H) with V and F given as in Definition 4.2 (for n > 0) or Defintion 8.8 for n = 0. Proposition 8.9. The functor D* : HA+ ! D+ is an equivalence of cate- gories. Lemma 8.7 and Proposition 5.5 imply HA+ has universal bilinear pairings H K ! H K. Just as in Lemma 8.4 one has: Lemma 8.10. 1.) Let A and B be abelian group. Then in HA+ Fp[A] Fp[B] ~=Fp[A Z B]: 2.) Let H 2 HA+ be connected and A an abelian group. Then Fp[A] H is connected and D*(Zp[A] H) ~=A D*H: 3.) For all H 2 HA+ , there is a natural isomorphism Fp[Z] H ~=H. To characterize bilinear pairings via Dieudonne modules, we introduce the no- tion of a D+ bilinear map. It is a mild generalization of the notion introduced* * in section 7. A pairing f : M x N ! K of objects in D+ is a bilinear pairing in D+ if it is a bilinear pairing of graded abelian groups and f(V x; V y)= V f(x; y) f(F x; y) = F f(x; V y)andf(x; F y) = F f(V x; y): There is a universal bilinear pairing M x N ! M D N: In fact, we can write M D+ N ~=R0 Z[V ](M N)=K where R0 = Z[V; F ]=(V F - p) and K is the submodule generated by F (x V y) - 1 (F x y) and F (V x y) - 1 (x F y): This mimics the construction M D N of the previous section. In fact if we write M ~=M0 Mc and N ~=N0 Nc where the 0 and c indicate elements of degree 0 and positive degree respectively, we have Lemma 8.11. There is a natural isomorphism in D+ M D+ N ~=M0 Z N0 M0 Z Nc McZ N0 Mc D Nc: HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 41 Proof. One has M D+ N ~=M0 D+ N0 M0 D+ Nc Mc D+ N0 Mc D+ Nc: Now Mc D+ Nc ~=Mc D Nc, so one must identify the other three summands. I claim M0Z Nc ! M0 D+ Nc is an isomorphism. To see this note that M0Z Nc has a structure of a Dieudonne module with V (x y) = V x V y = x V y and F (x y) = x F y. The bilinear pairing M0x Nc ! M0Z Nc is a bilinear pairing in D+ and any bilinear pairing in D+ factors through this one. Hence M0 Z Nc has the required universal property. The other summands are handled the_same way. |__| Now let H1 H2 ! K be a bilinear pairing in HA+ . The one gets an induced pairing : D*H1 x D*H2 ! D*K: Indeed, there are universal maps H(n + m) ! H(n) H(m) given by Equation 7.2 for n > 0 and m > 0, and by Lemma 8.10.3 for n = 0 or m = 0. If f : H(n) ! H1 and g : H(m) ! H2 represent x 2 DnH1 and y 2 Dm H2 respectively, then (x; y) is represented by H(n + m) ! H(n) H(m) f! gH1 H2 ! K: Lemma 8.12. This pairing is a bilinear pairing in D+ . Proof. If n > 0 and m > 0, this follows from Lemmas 7.4-7.6. If n = 0 (or m = 0), one only has to check the universal examples. Thus, for example, the pairing is bilinear because one has a diagram Fp[Z] H(n) __1__//(Fp[Z] Fp[Z]) H(n) ~=|| |~=| fflffl| fflffl| H(n) _____________//H(n) H(n) using Lemma 8.10.3 (or, rather, the proof of that statement_see Lemma 8.4.3) and the fact that commutes with coproducts. That (V x; V y) = V (x; y) follows from the diagram (8.2) Fp[Z] H(n) _1__v//_Fp[Z] H(pn) |~=| |~=| fflffl| v fflffl| H(n) ____________//_H(pn) where v induces V (see Definition 4.2). This implies (F x; y) = F (x; V y) beca* *use F x = px and (px; y) = p(x; y) = F V (x; y) = F (x; V y): Finally (x; F y) = F (V x; y) = F (x; y) by using a diagram similar to of Equat* *ion_ 8.2, using f : H(pn) ! H(n) (which defines F ) in place of the morphism v. * *|__| That completed, note that Lemmas 8.10, 8.11, and Theorem 7.7 will now imply the following result. The proof is the same as for Proposition 8.5. 42 PAUL G. GOERSS Proposition 8.13. Let H K ! H L be the universal bilinear pairing in HA+ . Then the induced bilinear pairing in D+ D*H x D*K ! D*(H K) induces an isomorphism D*H D+ D*K ! D*(H K): We now can write down the analog of Corollary 8.6. Corollary 8.14. The categories HA+ and D+ are symmetric monoidal cat- egories and the equivalence of categories D* : HA+ ! D+ is an equivalence of categories with symmetric monoidal sturcture. Part III: Hopf rings associated to homology theories 9. Skew commutatative Hopf algebras If X is a double loop space, the homology Hopf algebra H*X = H*(X; Fp) is not commutative, but skew commutative, meaning that if x 2 Hm X and y 2 HnX, then (9.1) xy = (-1)mn yx: Similary, the homology coalgebra of a space Y is skew cocommuative. This turns out to be only a mild variation on the commutative case considered up to now, a* *nd this section adapts Dieudonne theory and the theory of bilinear pairings to this new situation. To make the definitions precise, we work with non-negatively graded vector spaces over the field Fp, p > 2. The case p = 2 is the commutative case. Then t* *he signed twist map of the tensor product of two such objects t : V W -! W V is given on homogeneous elements by t(x y) = (-1)mn y x, with x 2 Vm and y 2 Wn. A skew commutative coalgebra over Fp is a coassociative coalgebra C, set-like in degree zero, so that the following diagram commutes C _____//FC C FF FFF |t FF##Ffflffl|| C C: The category of such will be written CA . Similary, a skew commutative Hopf algebra H is a Hopf algebra in the category of skew commutative coalgebras so t* *hat the multiplication satisfies the formula of Equation 9.1 The category of such H* *opf algebras will be written HA . The reason this adaptation is simple is the following result, known as the s* *plit- ting principle. See [18]. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 43 Proposition 9.1. Let H be a skew commutative Hopf algebra. Then there is a natural isomorphism in HA H ~=Hev Hodd where Hev is concentrated in even degrees and Hoddis an exterior algebra on pri* *m- itive generators in odd degrees. Notice that a skew commutative Hopf algebra concentrated in even degrees is, in fact, commutative. In effect, Proposition 9.1 implies that there is a na* *tural equivence of categories (9.2) HA ' HA+ x V* where V* is the category of non-negatively graded Fp vector spaces and HA+ is t* *he category of commutative Hopf algebras which are group-like in degree zero. The splitting principle immediately translates into a Dieudonne theory for t* *his situation. Definition 9.2.The category D of skew commutative Dieudonne modules has, as objects, non-negatively graded abelian groups M equipped with homomor- phisms F : M2n ! M2pn and V : M2pn! M2n so that 1. F V = V F = p; 2. V = 1 : M0 ! M0; 3. if n = pkm > 0 and (m; p) = 1, then pk+1M2n = 0; and 4. pM2n+1 = 0 for all n 0. Note that the category of skew commutative Dieudonne modules can be re- garded as a category of modules over the ring R0 = Z[V; F ]=(V F - p), subject to the exponential grading conventions of Equation 4.1 and the requirement that V x = F x = 0 if x is of odd degree. Proposition 9.1, the assumption that our Hopf algebras are group-like in deg* *ree zero, and Schoeller's Theorem 4.7 now immediately imply: Proposition 9.3. There is an equivalence of categories D* : HA -! D : The functor H 7! DnH is representable, just as in the commutative case. Indeed, if n = 2pkm, with (p; m) = 1, then DnH ~=Hom HA (H(n); H) ~=Hom HA(H(n); Hev) where H(n) = Fp CW2m(k) is the Witt vector Hopf algebra of Definition 4.2. A similar remark holds in degree zero with H(0) = Fp[Z]. If n = 2m + 1, then let (Fp[n]) be the exterior algebra on a single primitive generator of degree n. Th* *en DnH ~=Hom HA ((Fp[n]); H) ~=Hom Fp(Fp[n]); P H) ~=(P H)n: Here P H are the primitives. If we write H (2n) = H(2n) and H (2m + 1) = (Fp[2m + 1]), then these formulas combine to read (9.3) DnH ~=Hom HA (H (n); H): 44 PAUL G. GOERSS We now come to bilinear pairings. The forgetful functor from HA ! CA has a left adjoint; it is the free skew-commutative algebra functor suitably gr* *oup- completed in degree zero. Compare Lemma 8.2. As a result, Proposition 5.5 impli* *es that HA has bilinear pairings. We'd like to compute these via Dieudonne module* *s. The following is the crucial result. Let S (.) denote the free skew commutati* *ve algebra functor. If W is a vector spaces, the S (W ) can be made into a primit* *ively generated skew commutative Hopf algebra. Proposition 9.4. Let (V ) be the primitively generated exterior algebra on a vector space V concentrated in odd degrees. If H 2 HA is connected, then there is a natural ismorphism of Hopf algebras (V ) H ~=S (V QH): Proof. Because (.) H commutes with colimits, by Lemma 5.7, we may assume that V = Fp[n] for some odd integer n. Let x 2 Fp[n] a generator. If OE : (Fp[n]) H ! K is any bilinear map, then the formulas of Lemma 5.6 imply that for all y and z * *in the augmentation ideal of H, OE(x yz) = x O yz = 0: Furthermore, for any y, the element x O y 2 K is primitive. In fact, there is a factoring of OE (Fp[n]) H _q___//Fp[n] QH PPP PPPP | OEPPPPP((Pfflffl|| P K where q is the composite (Fp[n]) H ! I(Fp[n]) IH ! Q(Fp[n]) QH ~=Fp[n] QH: The vertical map extends to a Hopf algebra map S (Fp[n] QH) ! K and one checks that (Fp[n]) H -q!Fp[n] QH ! S (Fp[n] QH) is bilinear. The result follows from the uniqueness of the universal bilinear_m* *aps. |__| Exterior algebras and group rings behave in the expected way. The proofs are the same as those of Lemma 8.4. Lemma 9.5. Let (V ) a primitively generated exterior algebra on a vector sp* *ace concentrated in odd degrees. 1. There is a natural isomorphism Fp[Z] (V ) ~=(V ); 2. For all abelian groups A, Fp[A] (V ) ~=(A V ). Proposition 9.4 and Lemma 9.5 also allow one to come to term with H (V ). In fact, if t : H1H2 ! H2H1 is the signed switch map, then t is an isomorphism of Hopf algebras and the composition H1 H2- t!H2 H1 ! H2 H1 HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 45 is bilinear and induces an ismorphism ~= t : H1 H2- ! H2 H1: We now notice that the bilinear pairings defined on the various categories of Dieudonne modules of the previous sections extends to D as well. Regarding an object in D as a R0 = Z[V; F ]=(V F - p) module, we define, for M; N 2 D M D M = R0 Z[V ](M N)=K where K is the submodule generated by F (x V y) - 1 (F x y) and F (V x y) - 1 (x F y): This mimics the construction M D N of the two previous sections. The tech- niques of section 7 now imply that for all H1; H2 2 HA there is a natural map * *of Dieudonne modules " : D*H1 D D*H2 ! D*(H1 H2): The expected result is the following: Theorem 9.6. This natural map " : D*H1 D D*H2 ! D*(H1 H2): is an isomorphism in D . Proof. As in the proof of Proposition 8.13, one splits up the bilinear pair* *ing in D and reduces the result to previous calculations. If M 2 D , then there is a natural splitting in D M ~=Mev Modd where Mev and Moddare the elements of even and off degrees respectively. This reflects the splitting principle of Proposition 9.1. From this it follows that M D N ~=MevD+ NevModdNev=F NevMev=F MevNoddModdNodd: The result now follows from the splitting principle Proposition 9.1, the analag* *ous result for HA+ Proposition 8.13, Proposition 9.4, Lemma 9.5, and Lemma 4.4 which says that for H 2 HA, there is a natural isomorphism QH ~=D*H=F D*H: |___| It is worth pointing out that the natural isomorphism of the previous result reflects the skew symmetric nature of the category HA . In the following let t stand for any of the signed switch maps; that is if the degree of x is m and the degree of y in n, then t(x y) = (-1)nm (y x): Also let t also be the induced isomorphism of Hopf algebras t : H1 H2 ! H2 H1: For the following, compare Lemma 7.8. 46 PAUL G. GOERSS Lemma 9.7. Let H1 and H2 be Hopf algebras in HA . Then the following diagram commutes: D*H1 D D*H2 _"___//D*(H1 H2) t|| |D*(t)| fflffl| fflffl| D*H2 D D*H1 _"__//_D*(H2 H1): Proof. It is only necessary to check this for the universal examples H1 = H* *(n) and H2 = H(m) of Equation 9.3 (Compare the proof of Lemma 7.4.) If n and m are even this follows from Lemma 7.8. If either n or m is odd, the result_follo* *ws_ from Proposition 9.4 and Lemma 9.5. |__| 10. The Hopf ring of complex oriented cohomology theories. Let E* be a multiplicative cohomology theory represented by a ring spectrum E. Define spaces E(n) by the formula E(n) = 1 nE: The spaces E(n) are generalized Eilenberg-Mac Lane spaces in the sense that if X is a CW complex, then there is a natural isomorphism EnX ~=[X; E(n)]: The cup product pairings E(m) ^ E(n) ! E(m + n) induce bilinear maps H*E(m) H*E(n) ! H*E(m + n) and, hence, D*H*E = {D*H*E(n)}n2Z is a graded ring object in the category D of skew-commutative Dieudonne modules. This means that the ring multiplication satisfies the formulas of Lemmas 7.4-7.6. Such an object will be called a Dieud* *onne ring. Since D0H*E(n) ~=ss0E(n) ~=ss0nE ~=ss-n E ~=En; D*H*E is an E* algebra and the operators V and F act on E* as the identity and multiplication by p. We will call such an object an E* Dieudonne algebra. We will be particularly interested in homotopy commutative ring spectra E. This implies that that for all integers n and m there is a homotopy commutative diagram E(m) ^ E(n) ____//_E(m + n) |T| |Om;n| |fflffl fflffl| E(n) ^ E(m) ____//_E(m + n) where the horizontal maps are the cup product maps, T is topological switch map, and Om;n is Om;n = 1 [(-1)nm id] 2 [E(n + m); E(n + m)]: This observation and Lemma 9.7 immediately imply the following result. HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 47 Lemma 10.1. Let E be a homotopy commutative ring spectrum. Suppose x 2 DiH*E(m) and y 2 DjH*E(n). Then x O y = (-1)ij+mny O x 2 Di+jH*E(m + n): Now suppose E is complex oriented; thus, there is a chosen element x 2 E2CP 1 = [CP 1; E(2)] so that the composite S2 = CP 1,! CP 1!x E(2) represents the multiplicative identity in ss0E = ss2E(2). The map x induces a morphism of coalgebras H*CP 1 ! H*E(2); hence, by adjointness, a morphism of Hopf algebras H*BU ~=S(H*CP 1) ! H*E(2): Since H*BU is the reduction modulo p of a torsion-free Hopf algebra over Zp with a lift of the Verschiebung we have, as in Example 4.12, D*H*BU ~=R Zp[V ]QH*(BU; Zp): In fact, if fii 2 H2i(CP 1; Zp) is the standard generator, we obtain an induced element bi2 D2iH*BU under the composition eH*(CP 1; Zp) ! eH*(BU; Zp) ! R Zp[V ]QH*(BU; Zp) ~=D*H*BU; Furthermore, V bpi= bi. Note that in H*(CP 1 x CP 1; Zp) X (10.1) *fii= fijx fik j+k=i We will also use later that if m : CP 1 x CP 1 ! CP 1 is the multiplication, th* *en (10.2) m*(fijx fik) = j +jk fij+k: The complex orientation induces a map D*x* : D*H*BU ! D*H*E(2): Define bEi2 D2iH*E(2) by bEi= (D*x*)(bi): If there is no ambiguity, we may abuse notation and write bi for bEi. Now define an E* Dieudonne algebra with underlying E* algebra R0(E) = E*[b1; b2; : :]: with E* in Dieudonne-degree 0 and biin bidegree (2i; 2) where 2i is the Dieudon* *ne degree. We require V bpi= biand V = 1 on E*, and that V be multiplicative. This and the formulas of Lemma 5.6 determine the action of F . The existence of the elements bEidetermine a morphism of E* Dieudonne algebras R0(E) ! D*H*E: This brings us to the Ravenel-Wilson relation. Let x +F y 2 E*[[x; y]] 48 PAUL G. GOERSS be the formal group law for E* and let 1X b(t) = bEiti i=1 be the evident power series over the ring D*H*E. Since D*H*E is an E* algebra we can form the power series in two variables b(s) +F b(t) 2 (D*H*E)[[s; t]] and, in this context, the Ravenel-Wilson relation becomes: Proposition 10.2. Over D*H*E there is a formula b(s) +F b(t) = b(s + t): Proof. The argument here is not so different than the one in [21]; the idea* * of using the total unstable operation as an organizing principle is due to Neil St* *rick- land. For any CW complex X, there is a total unstable operation = X : E*X ! Hom CA(H*X; H*E) sending f 2 EnX = [X; E(n)] to f* : H*X ! H*E(n) this is continuous E* algebra homomorphism and natural in X. If m : CP 1 x CP 1 ! CP 1 is the H-space multiplication we then get a commutative diagram E*[[x]] ~=E*(CP 1)_____________//_HomCA(H*CP 1; H*E) E*m|| |Hom(m*;1)| fflffl| fflffl| E*[[x; y]] ~=E*(CP 1 x CP 1)___//HomCA(H*(CP 1 x CP 1); H*E) so (10.3) (Hom (m*; 1) O )(x) = ( O E*m)(x): Since is an E* algebra homomorphism (10.4) ( O E*m)(x) = (x +F y) = (x) +F (y) where (x) is the coalgebra map H*(CP 1 x CP 1) ss1!H*CP 1!x*H*E(2) and (y) uses ss2 instead of ss1. On the other hand, (Hom (m*; 1) O )(x) is the composite (10.5) H*(CP 1 x CP 1) m*!H*CP 1!x*H*E(2): Given any map of coalgebras f : H*(CP 1 x CP 1) ! H*E(n) it extends to a map of Hopf algebras f" : S(H*(CP 1 x CP 1)) ! H*E(n) HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 49 and the isomorphisms (10.6) Hom CA(H*(CP 1 x CP 1);H*E) ~=Hom HA(S(H*(CP 1 x CP 1)); H*E) ~=Hom D+(D*SH*(CP 1 x CP 1); D*H*E) are isomorphisms of E* algebras. Since S(H*(CP 1xCP 1)) is the mod p reduction of S(H*(CP 1xCP 1); Zp), which has a lift of the Frobenius, we can calculate, by Theorem 4.8 D*S(H*(CP 1 x CP 1)) ~= R Zp[V ]QS(H*(CP 1 x CP 1; Zp)) ~= R Zp[V ]eH*(CP 1 x CP 1; Zp): Let bi;j2 D2(i+j)S(H*(CP 1 x CP 1)) be the image of fiix fij. Then naturality and 10.2 implies that D*S(m*) : D*S(H*(CP 1 x CP 1)) ! D*S(H*CP 1) i j sends bi;jto i+jjbi+j. Similarly if ss1 is projection on the first factor D*S(ss1)* : D*S(H*(CP 1 x CP 1)) ! D*S(H*CP 1) is given by sending bi;jto biif j = 0 and 0 otherwise. There is an analogous fo* *rmula involving ss2. Now let f : H*(CP 1 x CP 1) ! H*E(2) be either of the maps of (10.3) and D*f" : D*S(H*(CP 1 x CP 1)) ! D*H*E(2) the induced map. Then D*f" extends to a map of E* Dieudonne algebras E*[bi;j] ! D*H*E which we will also call D*f". Let bEi;j= D*f"(bi;j) and consider the power seri* *es over D*H*E: b(s; t) = bEi;jsitj: Note i 1 or j 1. Using the expression for f given in (10.5) we have X X b(s; t) = bEi;jsitj= D*f"(bi;j)sitj X = D*x* O D*S(m*)(bi;j)sitj X i + j = D*x* j bi+jsitj X i + j = j bEi+jsitj = b(s + t): This rewrites b(s; t) in one way. Next, using the expression for f given in (1* *0.4) and that the isomorphisms of (10.6) are E* algebra maps we have b(s; t) = D*(x)"b(s; t) +F D*(y)"b(s; t): However, since (x) involves projection on the first factor X D*(x)"b(s; t)= D*(x)"(bi;j)sitj X = D*x* O D*S(ss1)*(bi;j)sitj X = D*x*(bi)si= b(s): 50 PAUL G. GOERSS Similarly D*(y)"b(s; t) = b(t), so b(s; t) = b(s)+F b(t). Combining the two exp* *res-_ sions for b(s; t) yields the result. |_* *_| Now let I E*[b1; b2; : :]:= R0(E) be the ideal of relations forced by requi* *ring that b(s + t) = b(s) +F b(t). Then we get an induced map (10.7) RE = R0(E)=I ! D*H*E of E* Dieudonne algebras. Under favorable circumstances this is almost an iso- morphism, but not quite. To see what's missing, note that R(E) is concentrated in bidegrees (s; t) with both s and t even. To account for odd degree groups we proceed as follows. If S1 = S0 is the stable 1-sphere, let e 2 D1H*1 S1 ~=[P H*1 S1]1 be the image of the generator of ss1S1 under the Hurewicz map. For any spectrum X, there is a bilinear pairing D*H*1 S1 D*H*1 -1X ! H*1 (S1 ^ -1X) = H*1 X and we can define a degree raising map (10.8) D*H*1 -1X ! Dn+1H*1 X by (10.9) x 7! e O x: Note that since V e1 = 0 we have, by Lemma 9.5 and Theorem 9.6 (10.10) V (e O x) = 0 = e O F x: Hence there is a factorization (by Lemma 4.4) (10.11) D*1 -1X __e__//_D*H*1OXO | | | | fflffl| oe | QH*1 -1X ______//P H*1 X The map labeled oe is the homology suspension induced by the evaluation ffl : 1 -1X ! 1 X: To see this note that there is a commutative diagram __ffl_//_1 S1 ^ 1 -1X X | | | |= fflffl| fflffl| 1 S1 ^ 1 -1X _____//1 X: So, at p > 2, there is a diagram of bilinear pairings of Hopf algebras (e) H*1 -1X _______//H*1 X | | | | fflffl| fflffl| H*1 S1 H*1 -1X _____//H*1 X HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 51 which induces a diagram (using Proposition 9.4) S(QH*1 X) ~=(e) H*1 -1X ____//_H*1 X | | | | fflffl| fflffl| H*1 S1 H*1 -1X __________//_H*1 X and the top map in this diagram is induced by the homology suspension. The argument at p = 2 is similar. If E is a ring spectrum and X = n+1E, then 1 n+1E = E(n + 1) and the natural map 1 S1 ^ 1 E(n) ! 1 E(n + 1) fits into a diagram 1 S1 ^ E(n) _____//E(n + 1) 1|j^1| |=| fflffl| fflffl| E(1) ^ E(n)_____//_E(n + 1) where j : S1 ! E is the suspension of the unit and the bottom map is the cup product pairing. Let eE 2 D1H*E(1) be the image of e under D*H*1 j. If E is complex oriented, then there is a diagram S1 ^ S1_____//1 S1 ^ 1 S1 ____//_E(1) ^ E(1) | | | | fflffl| x fflffl| S2 ___________//CP 1___________//_E(2) where x is the complex orientation. By definition of x the bottom composite must be the unit in ss2E(2) ~=ss0E. It follows that e2E= bE12 D2H*E(2) and, hence, we can extend 10.7 to a map of E* Dieudonne algebras (10.12) OE : RE [e]=(e2 - b1) ! D*H*E: The equation e2 = b1 also appears in [2] and [21]. The main theorems on Hopf rings [11, 13, 21] can now be rephrased as follows: Theorem 10.3. Suppose E* is a Landweber exact homology theory with coef- ficient ring E* concentrated in even degrees. Then OE : RE [e]=(e2 - b1) ! D*H*E is an isomorphism. There is another proof of this fact in the next section. It might be worth pointing out that the ring RE has an interpretation in the language of formal groups. If is an E* Dieudonne algebra, the formal group law over E* passes to a formal group law over via the ring homomorphism E* ! . This might be called the E* formal group law over . Then the set of E* Dieudonne algebra homomorphisms RE -! is in one-to-one correspondence with homomorphisms of the additive formal group law over to the E* formal group law. Compare Paul Turner's interpretation of QBP*BP [25]. 52 PAUL G. GOERSS Example 10.4. Suppose E = K is complex oriented K-theory; hence K* = Z[; -1] where 2 K-2. The formal group law for K is x +F y = x + y + xy; so the equation b(s + t) = b(s) +F b(t) implies bibj = i +jj bi+j in D*H*K. Thus multiplication by -1 induces isomorphisms -n : D*H*K(0) ! D*H*K(2n) -n : D*H*K(1) ! D*H*K(2n + 1) and, with the p-adic valuation, DiH*K(0) = DiH*(Z x BU) ae (j) ~= Z=(p + 1); i = 2j > 0 0; i = 2j + 1 The generator of D2jH*K(0) is bi. ae DiH*K(1) = DiH*U = 0Z=p ii==2j2j + 1: The generator of D2j+1H*K(1) is e if j = 0 and ebj if j > 0. Example 10.5. Because of the connection between the element e 2 D1H*E(1) and the homology suspension given in Equation 10.11, one can use Theorem 10.3 to give a description of H*E for certain E. Indeed, there is an isomorphism H*E ~=D*H*E[1_e] where D*H*E[1=e] in degree k is the colimit of . .!.Dn+kH*E(n) e!Dn+1+kH*E(n + 1) e!. . . Note that En = D0H*E(-n) maps to HnE; this is the image of the Hurewicz homomorphism E* ! H*E. If we let ai 2 H2iE be the image of bi+1 (note the shift in indices), then we get a surjective map E*[a0; a1; : :]:=(p; a0 - 1) ! H*E and the kernel is determined by the relation a(s + t) = a(s) +F a(t) P 1 where a = i=0aiti+1. In short, a is a strict isomorphism between F modulo p and the additive formal group law; that is, a is an exponential for F after red* *ucing mod p. Compare Corollary 4.1.9 of [20] HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 53 11. The role of E*2S3. If X is a spectrum, then H*1 X is a graded bicommutative Hopf algebra and one can study the functor X 7! DnH*1 X; n 0: The following was proved in [9]. Proposition 11.1. There is a spectrum B(n) and a natural surjection B(n)nX ! DnH*1 X which is an isomorphism if n 6 1 mod 2p. As the notation suggests, the spectra B(n) are the Brown-Gitler spectra. This is to say, if p = 2, (11.1) H*B(n) ~=A=A{O(Sqi) : 2i > n} or if p > 2, (11.2) H*B(n) ~=A=A{O(fifflP i) : 2pi + 2ffl > n}; and, furthermore, if B(n) ! HZ=pZ classifies the generator of H0B(n), then the induced map (11.3) B(n)nZ ! HnZ is surjective for all CW complexes Z. The group homomorphisms V : D2pnH*1 X ! D2nH*1 X and F : D2nH*1 X ! D2pnH*1 X are induced, respectively, by maps (11.4) OE : B(2pn) ! 2n(p-1)B(n) and (11.5) : 2n(p-1)B(n) ! B(2pn): The map OE is very familiar, as it is the one that fits into the "Mahowald exact sequence"_the cofibration sequence B(2pn - 1) -! B(2pn) -OE!2n(p-1)B(n); explored, at p = 2, in [16] and [3], and at odd primes in [5]. The map is less familiar_it is, for example, zero in cohomology. At p = 2, the maps V : D4n+2H*1 X ! D2n+1H*1 X and F : D2n+1H*1 X ! D4n+2H*1 X are induced by OE : B(4n + 2) ! 2n+1B(2n + 1) and : 2n+1B(2n + 1) ! B(4n + 2) respectively, but the maps are not uniquely determined. 54 PAUL G. GOERSS If X and Y are spectra, there is a natural bilinear pairing in HA H*1 X H*1 Y ! H*1 (X ^ Y ) induced by the map 1 X ^ 1 Y ! 1 (X ^ Y ) adjoint to the evaluation 1 (1 X ^ 1 Y ) ! X ^ Y . Thus, we get a pairing DnH*1 X Dm H*1 Y ! Dn+m H*1 (X ^ Y ): This yields pairings (11.6) : B(n) ^ B(m) ! B(n + m) which are uniquely determined if n + m 6 1 mod 2p and, in particular if n and m are both even. These pairings are also familiar, at least in cohomology, as * : H*B(n + m) ! H*B(n) H*B(m) sends the generator to the tensor product of the two generators. The ambiguity in the definition of when n+m 1 mod 2p can be removed by noting that there there are canonical maps B(n) ! B(n + 1) which is a weak equivalence if n is even. Thus if n is odd we could require : B(n) ^ B(m) ! B(n + m) to be the composite B(n) ^ B(m) -'!B(n - 1) ^ B(m) ! B(n + m - 1) -'!B(n + m): This said, the object B = {B(n)}n0 becomes a graded, commutative ring spec- trum. Now let E be a ring spectrum representing a cohomology theory E* with prod- ucts. Then EkB(n) ~=B(n)kE ~=B(n)nn-kE so there is a surjective homomorphism EkB(n) ! DnH*1 E(n - k): This induces a surjection (11.7) h : E*B ! D*H*E which skews degrees. Note that when k = 0, we get an isomorphism Ek ~=EkB(0) ~=D0H*E(-k) = E-k : This is the standard isomorphism E* ~=E-* and makes the homomorphism h is a morphism of E* ~=E-* modules. It is for this reason that we will speak of E* Dieudonne algebras in the sequel, rather than E* Dieudonne algebras. The properties of the map h of Equation 11.7 are recorded in the following sequence of results. Proposition 11.2. At primes p > 2, the bigraded E* module E*B is an E* Dieudonne algebra. At the prime 2, E*B has homomorphisms V : E*B(4n) ! E*B(2n) and F : E*B(2n) ! E*B(4n) satisfying the formulas of Lemmas 7.4 - 7.6 HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 55 Proposition 11.3. At odd primes, the map h : E*B ! D*H*E is a surjective homomorphism of E* Dieudonne algebras. At the prime 2, the map h respects the homomorphisms V and F . Remark 11.4. 1) At p = 2, D*H*1 E(.) has operators V : D4n+2H*1 E(.) ! D2n+1H*1 E(.) and F : D2n+1H*1 E(.) ! D4n+2H*1 E(.) which are not unambiguously defined in E*B. 2) The failure of the B(2n + 1)2n+1X ! D2n+1H*1 X to be an isomorphism can be measured by the operator e introduced in Equation 10.8. Since B(2n + 1) = B(2n), we have a diagram ~= B(2n)2n-1X ______//D2nH*1 -1X ~=|| |e| fflffl| fflffl| B(2n + 1)2n+1X______//D2n+1H*1 X and we see that these kernel of B(2n + 1)2n+1X ! D2n+1H*1 X is isomorphic to the kernel of e : D2nH*1 -1X ! D2n+1H*X: We now begin our analysis of specific ring spectra. The following result wi* *ll allow a careful examination of the kernel of E*B ! D*H*E in some cases, for example any Landweber exact theory with coefficients in even degrees. For the following compare [21] and [2]. Lemma 11.5. Suppose E is a ring spectrum with E* torsion free and concen- trated in even degrees. Suppose further that E*B(n) is concentrated in even deg* *rees. Then for all k, 1) H*E(2k) is concentrated in even degrees. 2) On H*E(2k), the Frobenius is injective and the Verschiebung surjective. 3) There is an isomorphism of primitively generated Hopf algebras H*E(2k + 1) ~=(QH*E(2k)): Proof. These will be proved together. The surjection 0 = E2n-2k-1B(2n) ! D2nH*E(2k + 1) shows D*H*E(2k + 1) is concentrated in odd degrees, and, hence, H*E(2k + 1) is an exterior algebra. Let E0(2n) E(2n) be the component of the basepoint. Then the Rothenberg-Steenrod spectral sequence, (QH*E(2k - 1)) ~=Tor H*E(2k-1)*(Fp; Fp) ) H*E0(2n) is a spectral sequence of Hopf algebras and, hence, must collapse. Since the Ve* *r- schiebung is surjective at E2 on this spectral sequence, it is on H*E0(2n) and * *hence on H*E(2n). 56 PAUL G. GOERSS Similarly, consider (11.8) Tor H*E(2k)s(Fp; Fp)t) Hs+tE(2k + 1): We work at p > 2. The argument for p = 2 is similar. Let QH*E(2k) denote the indecomposables and Q+ and Q- the even and odd degree sub-vector spaces of QH*E(2k). Let L1QH*E(2k) be the first derived functor of Q applied to H*E(2k). Then L1QH*E(2k) = L1Q is concentrated in degrees congruent to zero modulo 2p. Then Tor H*E(2k)*(Fp; Fp) ~=(Q+ ) (Q- ) (L1Q) where QH*E(2k) ~=Tor H*E(2k)1(Fp; Fp) and L1Q Tor H*E(2k)2(Fp; Fp) are the primitives. If Q- 6= 0 or (L1Q) 6= 0, the lowest degree non-zero class * *in Q- L1Q would produce a non-zero even degree class in H*E(2k + 1), a contradiction, so Q- = 0 = L1Q and QH*E(2k) = Q+ : The spectral sequence of Equation 11.8 now collapses, proving part 3. Since the module of indecomposables QH*E(2k) is in even degrees part 1 follows. Half of p* *art 2 has already been proved, and since L1QH*E(2k) = 0, the Frobenius is injective* *__ on H*E(2k). See [10]. In fact, the kernel of F in Lemma 4.4 is exactly L1Q. |* *__| Now let DE D*H*E be the Dieudonne ring DE = {D2mH*E(2n)}: If E* is concentrated in even degrees. Let e 2 D1H*E(1) be the suspension eleme* *nt of Equation 10.8 Proposition 11.6. Suppose E is a ring spectrum with E* torsion free and concentrated in even degrees. Suppose E*B(n) is concentrated is even degrees for all n. Then the natural map of E* Dieudonne algebras DE [e]=(e2 - b1) ! D*H*E is an isomorphism. Proof. Since D2m+1H*E(2n) = 0 (by Lemma 11.5.1), the induced map (DE [e]=(e2 - b1))*;2n! D*H*E(2n) is an isomorphism. The result will follow once we show eD*H*E(2n) = { e O x : x 2 D*H*E(2n) } ! D*H*E(2n + 1) is an isomorphism. Since e(F x) = F (V e O x) = 0, and D*H=F D*H ~=QH eD*H*E(2n) = QH*E(2n) and the result follows from Lemma 11.5. |___| HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 57 If E*B(n) is concentrated in even degrees for all n, then we may define E*B(ev) = {E*B(2n)} = {E2mB(2n)} and the homomorphism of E* Dieudonne algebras E*B ! D*H*E restricts to an isomorphism of E* Dieudonne algebras E*B(ev) ! DE : Calculating the source of this map is where E*2S3+comes in. Write 2S3+for the suspension spectrum of the space 2S3 with a disjoint basepoint. The May-Milgram filtration {Fk2S3} of 2S3 suspends to a filtration {Fk = Fk2S3+} of 2S3+. The Snaith splitting implies this stable filtration is trivial: 2S3+' _kFk=Fk-1. At a prime p, the filtration quotients are Brown- Gitler spectra. Specifically, at p = 2, Fk=Fk-1 ' kB(k) and if p > 2, 8 < 2n(p-1)B(2n); ifk = pn; (11.9) Fk=Fk-1 ' : 2n(p-1)+1B(2n + 1); ifk = pn + 1; * ifk 6 0; 1 mod p: This previous paragraph summarizes work of [3, 5, 12, 16, 24]. In short, the associated graded spectrum of 2S3+is a regraded version of the graded spectrum B = {B(n)}. It is convenient, for our purposes, to eliminate the odd Brown-Gitler spectr* *a. If we write 2S3<3> for the universal cover of 2S3 and Fk2S3<3> for the universal cover of Fk2S3, then {Fk2S3<3>} is a filtration of 2S3<3> and we get a filtrati* *on {Fk0= Fk2S3<3>+ } of the suspension spectrum. Let B0 be the associated graded spectrum. Then Equation 11.9 and a homology calculation shows that at any prime B0' {2k(p-1)B(2k)}: In short, B0is a regraded version of B(ev). In particular, there is a degree sh* *earing isomorphism of E* modules E*B(ev) ~=E*B0: We next observe that the loop space multiplication on 2S3<3> gives 2S3<3>+ the structure of a ring spectrum and, since the May-Milgram filtration is multiplic* *ative, gives B0 the structure of a graded ring spectrum. I don't know whether B(ev) ' B0 as graded ring spectra; this would settle Ravenel's Conjecture 3.4 ([19]), f* *or example. However, the standard calculations ([4]) show that (11.10) H*B(ev) ~=H*B0 as Fp algebras. We can use this fact and Ravenel's Adams spectral sequence calc* *u- lations ([19]) to calculate DBP D*H*BP : Let RE be the ring of Equation 10.7 58 PAUL G. GOERSS Theorem 11.7. The graded BP* modules BP*B(n) are concentrated in even degrees and the natural maps BP*B(ev) ! DBP RBP are isomorphisms of BP* Dieudonne algebras. Proof. Let I = (p; v1; v2; : :): BP* be maximal ideal. If M is an BP* module we can filter M by powers of I and form the associated graded object E0M. Then E0BP* ~=Ext**A(Fp; H*BP ) = Ext *E(Fp; Fp); where A is the dual Steenrod algebra and E = Fp *BP A. If X is a spectrum, then the Adams spectral sequence Ext *;*E(Fp; H*X) ) BP*X is a spectral sequence of E0BP* modules. The calculations of Theorem 3.3 of [19] combined with Theorem 3.14 of [21] (which is algebraic and precedes their computation of H*BP ) now imply E0BP*B0= E*;*1= Ext *;*E(Fp; H*B0) ~=E0RBP ; all as E0BP* algebras. As a result we have a commutative square of E0BP* algebr* *as ~= E0BP*B(ev)O_____//E0D*H*BPOOO ~=|| || | ~= | E0BP*B0 ________//E0RBP : This finishes the proof. |___| Landweber exactness now implies the following result, which finishes the pro* *of of Theorem 10.3 Corollary 11.8. Let E* be any Landweber exact theory with coefficients con- centrated in even degrees. Then E*B(n) is in even degrees and the natural maps E*B(ev) ! DE RE are isomorphisms of E* Dieudonne algebras. Proof. Note that Landweber exactness implies that E* is torsion free. Since B is p-local, M MU*B(ev) ~= (2nBP )*B(ev) (p;n)6=1 and the result follows from Theorem 11.7. For general E we have a commutative diagram ~= E* MU* MU*B(ev) _____//E* MU* RMU ~=RE ~=|| || fflffl| ~= fflffl| E*B(ev) _________________//DE : |___| HOPF RINGS, DIEUDONNE MODULES, AND E*2S3 59 Glossary Categories of Hopf Algebras HA: graded, connected bicommutative Hopf algebras over Fp: section 4 HA+ : graded, bicommutative Hopf algebras over Fp that are group-like in degree 0: section 8. HA : graded, skew-commutative Hopf algebra over Fp that are group-like in degree 0; for example, the homology of a double loop space: section 9. HV: graded, connected, torsion-free Hopf algebra over Zp equipped with a lift of the Verschiebung: section 2. 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