WEIL PAIRINGS AND MORAVA K-THEORY M. ANDO AND N. P. STRICKLAND 1. Introduction In [5], M. Hopkins and the present authors showed that for every elliptic spe* *ctrum E, there is a canonical map oeE : MU<6> -!E of ring spectra, called the oe-orientation, which generalises the Witten genus * *[11] and gives an interesting interpretation of its modular invariance; for more background and motivation, s* *ee the introduction to [5]. A central part of the proof of the above result is the computation of the ring E0* *BU<6> (where BU<6> is the 5-connected cover of BU) or equivalently the description of the scheme spec(E0B* *U<6>). For this description we do not need E to be an elliptic spectrum, but merely a commutative 2-periodi* *c ring spectrum whose odd-dimensional homotopy groups are trivial. However, in order to cover this l* *evel of generality, it was necessary to give a rather computational proof. In the present work, we give an alternative argument that is valid when E is * *a two-periodic Morava K-theory spectrum (with associated formal group GK say). This proof is more con* *ceptual, and it makes much closer contact with previous work by algebraic geometers on the theory of * *biextensions [8, 2]. In their language, the result is that spec(K0BU<6>) is the scheme C3(GK ) of rigid cubic* *al structures on the trivial Gm -torsor over GK ; this is formulated more explicitly as Theorem 1.4. Although the details of the argument given here were worked out and written u* *p by the authors listed, we should emphasise that the conceptual basis is part of a large ongoing projec* *t which is joint work with M. Hopkins. We are grateful to him for allowing us to distribute this fragment of * *our enterprise. Notation. Throughout this paper we fix a fixed prime number p > 0 and an intege* *r h > 0. We fix as well a 2-periodic Morava K-theory K with ground field k def=K0(pt) of characteristic* * p and formal group GK of height h. We let n denote a (varying) power of p, and we write P for CP1 and P* *[n] for BZ=n. We write = n for the multiplication map H=n ^ H=n -!H=n of Eilenberg-MacLane spectra, o* *r for the map P[n]2= K(Z=n; 1) x K(Z=n; 1) -!K(Z=n; 2) derived from this. We also write fi = fin for the map H=n -! H that induces the* * Bockstein operation Hk(X; Z=n) -!Hk+1(X; Z), or for the maps P[n] = K(Z=n; 1) -!P = K(Z; 2) K(Z=n; 2) -!K(Z; 3) derived from this. ___________ Date: July 1998. Supported by the NSF. 1 2 ANDO AND STRICKLAND By a k-group we shall mean an affine commutative group scheme over k. These f* *orm an abelian category (the opposite via specof the category of abelian Hopf algebras over k). A map o* *f k-groups G = specA specf----!specB = H is a monomorphism if f : B ! A is a surjective map of rings; it is an epimorphi* *sm if equivalently f is faithfully flat or injective [4, III,x3, n. 7]. If G is a p-divisible formal gr* *oup over k, then its n-torsion is a k-group which we shall denote G[n]. We will need to consider various torsors [4, III,x4] and biextensions [8, 2];* * these will always be assumed to have the multiplicative group Gm as the fibre. We identify Gm-torsors with line* * bundles in the usual way. If L and M are two torsors over X then their product will be denoted L M. It will* * be convenient to make use of the following "punctual" notation: if f : Y ! X is a morphism in some ca* *tegory with a notion of torsor, and L ! X is a torsor, then we shall describe the torsor f*L ! Y as the* * torsor whose fibre at a generic point y of Y is (f*L)y = Lf(y): Similarly a map of torsors g : M=Y ! L=X covering f may be described as a map My g-!Lf(y): All formal groups will be assumed commutative. A formal group G over a ring R* * defines a functor from the category of pro-Artin R-algebras to groups, and a "point" of G will mean a * *pro-Artin R-algebra T and an element g 2 G(T). We will often omit the ring T from the notation, and even * *write g 2 G. Recall that if G and H are formal groups, then a biextension structure on a t* *orsor B -!G x H equips the torsor B |gxH with the structure of a central extension of H by Gm; its identit* *y element will be g1. Similarly for h 2 H the torsor B |Gxh becomes a central extension of G by Gm whose identi* *ty will be denoted 1h. We use the notations oLand oRrespectively for the maps Lo Bg1;h Bg2;h-!Bg1+g2;h Ro Bg;h1 Bg;h2-!Bg;h1+h2 which give the multiplication in the two central extensions. Now let o : G x G ! G x G be the map exchanging the two factors. A symmetric * *biextension of G is a biextension B -!G x G equipped with a map of biextensions : B -!o*B; whose restriction to the diagonal is the canonical isomorphism of torsors [2, p* *. 8]. Finally, if L is a torsor over G, let (L) be the torsor over G x G whose fibr* *e at (g; h) is (L)g;hdef=Lg+h_L: g Lh A cubical structure on L is a particular sort of symmetric biextension structur* *e on (L) [2, 2.2]. A rigid torsor over G is a torsor L equipped with a section of the fibre L0 o* *ver the identity. If L ! G is a rigid torsor, then a rigid cubical structure on L is a cubical structure w* *hose various identity sections coincide with the sections produced from the rigid structure. WEIL PAIRINGS AND MORAVA K-THEORY * * 3 If R is a k-algebra, then C3(G)(R) will denote the group of rigid cubical str* *uctures on the trivial torsor over GR. Such a thing is just a map f : G3R-!Gm such that f(0; 0; 0) = 1 f(g1; g2; g3) is symmetricgin1; g2; andg3 (1* *.1) f(g1; g2; g3)f(g0+ g1; g2; g3)-1f(g0; g1+ g2; g3)f(g0; g1; g3)-* *1 = 1: A choice of coordinate on G determines an isomorphism OGR ~=R[[x]] and a formal* * group law F over k. In these terms, an element of C3(G)(R) amounts to a power series u(x1; x2; x3) wit* *h coefficients in R, such that u(0; 0; 0) = 1 u(x1; x2; x3) is symmetric in x1, x2, and x3 u(x1; x2; x3)u(x0+F x1; x2; x3)-1u(x0; x1+F x2; x3)u(x0; x1; x3)-* *1 = 1: It is then not hard to check (see [5]) that there is an initial example of a * *ring equipped with such a power series, and so C3(G) is represented by a k-group, which we shall also denote by* * C3(G). BU<6> and Cubical Structures. Let BU<6> be the 5-connected cover of BU. The map P3 -!BU Q3 classifying the virtual bundle i(1 - Li) lifts to a map Q 3(1-L ) P3 --i----i!BU<6>: From it we obtain a map K0(P3) -!K0BU<6> (1* *.2) whose adjoint 32 K0(P3)^ K0BU<6> ~=O(G3K)K0BU<6> (1* *.3) is a rigid cubical structure on the trivial torsor over (GK )K0BU<6>. It is cla* *ssified by a map specK0BU<6> 3-!C3(GK ); which we also denote by 3. The purpose of this note is to give a geometric proo* *f of Theorem 1.4.3 induces an isomorphism of k-groups specK0(BU<6>) ~=C3(GK ): In [5], M. Hopkins and the authors prove the analogue of Theorem 1.4 for any * *even-periodic ring spectrum. The purpose of this paper is to give a proof inspired by the study of cubical s* *tructures by algebraic geometers in for example [8] and [2]. The main ingredient is a description of the K-homology of the fibration K(Z; 3) fl-!BU<6> -!BSU (1* *.5) in terms of the geometry of cubical structures. Before proceeding with the main* * text, we give a brief account of this model. Some of the results we cite are fairly involved, for example [2,* * Corollary 7.13]. However none of these more difficult results is required by the arguments in this paper; wit* *h the Atiyah-Hirzebruch-Serre spectral sequence and Ravanel and Wilson's calculation [10] of K0K(Z; 3) in han* *d, one may proceed in a more elementary fashion. 4 ANDO AND STRICKLAND Mumford [8] shows that if G is a p-divisible formal group over k, then the fu* *nctor of isomorphism classes of symmetric biextensions of G is represented by the scheme W(G) of Weil pairin* *gs for G. Associating to a cubical structure the Weil pairing of its underlying symmetric biextension give* *s a map C3(G) e*-!W(G): (1* *.6) The kernel of e* consists of cubical structures whose underlying symmetric bi* *extension is trivial. For a k-algebra R let C2(G)(R) be the group of maps f : G2R-!Gm such that f(0; 0) = 1 f(g1; g2) = f(g2; g1) (1* *.7) f(g1; g2)f(g0+ g1; g2)-1f(g0; g1+ g2)f(g0; g1)-1 = 1: To give such an f is precisely to give the structure of a commutative central e* *xtension on the trivial torsor over GR. Remarks similar to those for C3(G) show that C2(G) is represented by a* * k-group. There is a map of k-groups C2(G) ffi-!C3(G) (see (7.1)), and Proposition 2.11 of [2] shows that C2(G) ffi-!C3(G) e*-!W(G) (1* *.8) is exact. In fact, it is a short exact sequence. The kernel of ffi consists of symmetri* *c bilinear maps from G to Gm, and it is easy to see (Lemma 7.3) that there is only one. One way to see that e* ** is surjective in the category of k-groups is to remark that the map G 2-!G is an isogeny, and in that case Breen [2, 7.13] shows that every symmetric biex* *tension of G by Gm may be refined, locally in the flat topology, to a cubical structure. We shall not nee* *d to be more precise, as the surjectivity of e* is a pleasant consequence (Corollary 7.6) of our comparison * *to the topological situation. In any case, the sequence (1.8)is our model for the fibration (1.5). We shall* * show that there are isomor- phisms b*:specK0K(Z; 3) ~=W(GK ) 2: specK0BSU ~=C2(GK ) such that the diagram specK0BSU ----! specK0BU<6> specK0fl-----!specK0K(Z; 3) ? ? ? 2?y~= ?y3 ~=?yb* (1* *.9) C2(GK ) ---ffi-! C3(GK ) -e*---! W(GK ) commutes (up to a possible sign). That 3 is an isomorphism follows by the five-* *lemma. See Remark 4.4 for discussion of the sign. The paper is organised as follows. In Section 2 we recall Ravenel and Wilson'* *s calculation of K0K(Z; 3) [10]. This gives the identification b* between specK0K(Z; 3) and W(GK ). In Sec* *tion 3 we give an explicit formula for the map e*. This makes it possible to show in Section 5 that the ri* *ght-hand square of (1.9) commutes. Section 6 describes the isomorphism 2, and shows that the left-hand s* *quare of (1.9)commutes. In Section 7 we show that ffi is injective, and finish the proof of Theorem 1.4. WEIL PAIRINGS AND MORAVA K-THEORY * * 5 2. Ravenel and Wilson's calculation of K0K(Z; 3) In this section we express some calculations of Ravenel and Wilson [10] in th* *e language of schemes. Definition 2.1.An en-pairing for G over a k-algebra R is a map f : G[n]2R-!(Gm )R of schemes over spec(R) such that f(g1+ g2; h)= f(g1; h)f(g2; h) f(g; h1+ h2)= f(g; h1)f(g; h2) f(g; g)= 1: The group of en pairings for G over R will be denoted Wn(G)(R). This functor of* * R is represented by a scheme Wn(G) (as one sees easily after introducing a coordinate). Remark 2.2. More generally, we say that a map f satisfying the first two condit* *ions is biexponential, and that f is alternating if it satisfies the third condition. We say that f is wea* *kly alternating if f(g; h)f(h; g) = 1; by considering f(g +h; g +h) we see that alternating biexponential maps are wea* *kly alternating. Conversely, if f is weakly alternating then f(g; g)2 = 1 so f(2g; 2g) = f(g; g)4 = 1. If p * *is odd then the map g 7! 2g is an isomorphism and we see that f is alternating. Proposition 2.3.There is a canonical isomorphism Wn(GK ) = spec(K0K(Z=n; 2)). Proof.Let : P[n]2 = K(Z=n; 1)2 -! K(Z=n; 2) be the map induced by the cup pro* *duct in ordinary cohomology. This gives a map m : spf(K0P[n])2 -! spf(K0K(Z=n; 2)) of formal s* *chemes. As the cup product H1 x H1 -!H2 is weakly alternating and additive in both variables, we s* *ee that m is a weakly alternating, biadditive map of formal group schemes. We claim that m is actually alternating in the strong sense. We may assume th* *at p = 2, for otherwise there is nothing to prove. It is enough to check that the composite g def=(P[n] diagonal-----!P[n]2-! K(Z=n; 2)) induces the trivial map on eK0P[n]. If we compose with the Bockstein map fi : K* *(Z=n; 2) -!K(Z; 3), we get a map in [K(Z=n; 1); K(Z; 3)] = H3B(Z=n) = 0. Moreover, fi fits in a fibration K(Z=n; 1) = P[n] -!P -[n]!P -ae!K(Z=n; 2) fi-!K(Z; 3): We deduce that g factors through ae, so it is enough to check that ae is trivia* *l in reduced K-homology, or that [n] induces an epimorphism in K-homology, or that [n] induces a monomorphism in* * K-cohomology. This is true by well-known calculation, because [n]*(x) = xnh. We next claim that spf(K0K(Z=n; 2)) is the universal example of a formal grou* *p scheme equipped with an alternating biadditive map to it from GK [n]2. To see this, assume for the m* *oment that p > 2. In that case, Ravenel and Wilson show [10, 11.3] that K0(K(Z=n; *)) is the free Hopf ri* *ng generated over k[Fp] by the bicommutative Hopf algebra K0K(Z=n; 1) = K0P[n]. A Hopf ring is just a * *graded-commutative ring object in the category of cocommutative coalgebras over k, and this catego* *ry is equivalent to that of formal schemes over k by [3, p. 12]. Thus, spf(K0K(Z=n; *)) is the free graded-* *commutative formal ring scheme generated over the constant scheme Fp by the formal group scheme spf(K0K* *(Z=n; 1)) = GK [n]. Moreover, such free objects are constructed in the obvious way: one can define * *colimits and tensor products of formal group schemes, so one can define k(A) def=Ak =k (where k acts with si* *gns) and these objects are the homogeneous pieces of the free ring scheme. These facts are implicit i* *n the literature on Hopf rings, and an explicit treatment has recently been given by Hunton and Turner [* *6]. We conclude that spf(K0K(Z=n; 2)) = 2GK [n] as claimed. 6 ANDO AND STRICKLAND For the case p = 2 we quote [7, Appendix]. They prove thatPK0K(Z=n; *) is th* *e free Hopf ring on K0K(Z=n; 1) modulo the relation that the squaring map a 7! a0O a00is trivial.* * Equivalently, we see that spf(K0K(Z=n; *)) is the free graded-commutative formal ring scheme on spf(* *K0K(Z=n; 1)) modulo the relation that is strongly alternating. It follows again that spf(K0K(Z=n; 2)) * *has the required universal property. We next use Cartier duality (see for example [3, p. 27]), which amounts to th* *e fact that spec(K0K(Z=n; 2)) = Hom(spf(K0K(Z=n; 2)); Gm): Note that K(Z=n; 2) is connected so the augmentation ideal in K0K(Z=n; 2) is to* *pologically nilpotent, so spf(K0K(Z=n; 2)) is a connected formal neighbourhood of the identity element. T* *hus any map spf(K0K(Z=n; 2)) -!Gm that sends 0 into bGmsends everything into bGm, so spec(K0K(Z=n; 2)) = Hom(spf(* *K0K(Z=n; 2)); bGm)._Our description of spf(K0K(Z=n; 2)) immediately identifies this with Wn(GK ). * * |__| Remark 2.4. Ravenel and Wilson's more explicit calculations show that K0K(Z=n; * *2) has finite dimension over k so we have only used the simplest and most classical version of Cartier * *duality. In Lemma 6.2 we will use a version that applies in the infinite-dimensional case; this is treated in* * [3, p. 27]. Using Proposition 2.3, we have a map b = bn: specK0(K(Z; 3)) specK0fin------!specK0(K(Z=n; 2))) ~=Wn(G* *K ): We also have a commutative diagram as follows: K(Z=pn; 1)2pm----!K(Z=pn;-2)fipn---!K(Z; 3) ?? ? ? y ?y ?yp (2* *.5) K(Z=n; 1)2-n---!K(Z=n; 2)--fin--!K(Z; 3): When we apply the functor spf(K0(-)), the map K(Z=pn; 1) -!K(Z=n; 1) becomes th* *e map p: GK [pn] -! GK [n]. This implies that when a is a point of spec(K0K(Z; 3)) and g, h are poi* *nts of GK [pn] we have bpn(a)(g; h)p = bn(a)(pg; ph): (2* *.6) Definition 2.7.If G is a formal group over k and R is a k-algebra, then a Weil * *pairing on G over R is a collection f* of en-pairings fn 2 Wn(G)(R) such that fpn(g; h)p = fn(pg; ph): For a k-algebra R,Qlet W(G)(R) be the group of Weil pairings on GR. This is cle* *arly represented by a closed subscheme W(G) n Wn(G). Remark 2.8. Let (fn) be a collection of weakly alternating biexponential maps G* *[n] x G[n] -!Gm such that fpn(x; y)p = fn(px; py). If p > 2 then any weakly alternating map is alte* *rnating. If p = 2 then fn(2x; 2x) = f2n(x; x)2 and this is 1 because f2nis weakly alternating. If G ha* *s finite height then the map 2 : G[2n] -!G[n] is faithfully flat and thus an epimorphism, so fn(y; y) = 1 fo* *r all y, so f is alternating. Thus, it does not matter whether we specify alternating or weakly alternating m* *aps in the definition of a Weil pairing. WEIL PAIRINGS AND MORAVA K-THEORY * * 7 Remark 2.9. If G is p-divisible and f is an epn-pairing then it is not hard to * *check that there is a unique map f0: G[n]2 -!Gm such that f0(pg; ph) = f(g; h)p, and that this is an en-pai* *ring. This construction gives a map q : Wpn(G) -!Wn(G) and it is clear that W(G) = invlimrWpr(G). One c* *an also check that * * 0 under our identification spec(K0K(Z=pn; 2)) = Wn(GK ), the Bockstein map K(Z=n;* * 2) fi-!K(Z=pn; 2) has spec(K0fi0) = q. The maps bn clearly fit together to give a map b*: spec(K0K(Z; 3)) -!W(GK ). Proposition 2.10.The map b*: spec(K0K(Z; 3)) -!W(GK ) is an isomorphism. Proof.Ravenel and Wilson show that the map K0K(Z; 3) -!colimrK0K(Z=pr; 2): is an isomorphism. * * |___| 3.The Weil pairing of a cubical structure Mumford [8] shows that a cubical structure gives rise in a functorial way to * *a Weil pairing; see also [2]. Thus, there is a canonical map C3(G) e*-!W(G): In this section we give an explicit formula for e*. Lemmas 3.2 and 3.3 are stan* *dard facts about biextensions; see for example [2, Chapter 4]. If B -!G x H is any biextension, there is a canonical isomorphism (n x 1)*B ~=(1 x n)*B (3* *.1) of torsors over G x H, given by (n x 1)*B -[n]L--Bn [n]R--!(1 x n)*B; where Bn denotes the torsor over G x H whose fibre at (g; h) is the n-fold prod* *uct Bg;h Bg;h : : :Bg;h: Lemma 3.2. The isomorphism (3.1)is an isomorphism of biextensions. * * |___| Lemma 3.3. Any biextension B -!G x H has a canonical trivialisation when restri* *cted to 0 x H or G x 0, given by G 3 g7! g1 H 3 h7! 1h: * *|___| Combining Lemmas 3.2 and 3.3 gives an automorphism 1_~=(n x 1)*B ~=(1 x n)*B ~=1_ of the trivial biextension over G[n] x H[n], or equivalently a biexponential map G[n] x H[n] -!Gm : Now suppose that G = H; that L ! G is a torsor; and t is a trivialisation of * *L. Let s(g; h) def=t(g_+th)(g)t(h) 8 ANDO AND STRICKLAND be the resulting trivialisation of (L). A cubical structure on L gives rise to * *a function u: G3-! Gm satisfying the equations (1.1)and such that the biextension structure on (L) is* * given by the formulae s(g1; g3) oLs(g2;=g3)u(g1; g2; g3)s(g1+ g2; g3) s(g1; g2) oRs(g1;=g3)u(g1; g2; g3)s(g1; g2+ g3): Proposition 3.4.The en pairing associated to u is given by the explicit formula n-1Yu(g; jg; h) en(g; h) = en(u)(g; h) = ________: (3* *.5) j=1u(g; jh; h) Proof.The isomorphism Bn ~=(n x 1)*B over G x G is given by the formula 0 1 n-1Y s(g;_h)_:_:s:(g;-h)z_____"7! @ u(g; jg; h)As ng; h ;(3* *.6) n terms j=1 as we see by induction on n. If ng = 0 then s(ng; h) = s(0; h) = 1h: Following the recipe for en given above yields formula (3.5). * * |___| Remark 3.7. Note that the formula (3.5)makes sense on G2, and so en(u) may be r* *egarded as a function G2! Gm. Proposition (3.4)implies that this function gives a root of unity when * *evaluated on G[n]2. It is particularly transparent in formula (3.6)that Lemma 3.8. en(g; g) = 1 en(g; h) = en(h; g)-1: * *|___| Formula (3.5)also shows directly that Lemma 3.9. epn(g; h)p = en(pg; ph): (3.* *10) Proof.Consider the section s(g; h)np2of Bnp2. The axioms of a biextension guar* *antee that all ways of multiplying these to an element of B over (npg; ph) are the same. It follows th* *at there is an equation 2 p2 3 3 2 np23 p2 3 * * 3 np-1Y p-1Y p-1Y p-1Y n-1Y 4 u(g; jg; h)54u(npg; jh; h)5= 4 u(g; jg; h)54 u(pg; jh; h)54u(pg; jpg;* * ph)5 j=1 j=1 j=1 j=1 j=1 Similarly, all ways of multiplying these to an element of B over (pg; nph) are * *the same, so there is an equation 2 p23 3 2 np23 p2 3 * * 3 np-1Y p-1Y p-1Y p-1Y n-1Y 4 u(g; jh; h)54 u(g; jg; nph)5= 4 u(g; jg; h)54 u(pg; jh; h)54u(pg; jph;* * ph)5: j=1 j=1 j=1 j=1 j=1 WEIL PAIRINGS AND MORAVA K-THEORY * * 9 Dividing the first by the second, we get 2 3p2 3 np-1Y p-1Y n-1Y 4 u(g;_jg;_h)54 u(npg;_jh;_h)5= u(pg;_jpg;_ph) j=1 u(g; jh; h)j=1u(g; jg; nph) j=1u(pg; jph; ph) Setting npg = 0 = nph gives the result. * * |___| From Lemmas 3.8 and 3.9, one has Proposition 3.11.There is a unique map e*: C3(G) -!W(G) whose image in Wn(G) is en. * * |___| 4.The bundle associated to the en-pairing Let V be a virtual complex bundle over a space X, with a chosen lifting v : X* * -!BU<6> of the classifying map X -!ZxBU. Given a class c 2 K0BU<6> we obtain an element v*c 2 K0X which we* * can think of as a characteristic class of v (or by abuse, of V ). Alternatively, the map v*: K0X * *-!K0BU<6> can be regarded as an element u 2 K0X bK0BU<6>, which we call the total characteristic class of* * V . Note that if L1, L2 and L3 are line bundles over X then each virtual bundle 1* * - Lihas a unique lifting to BU, which is the second spaceQin the connective complex K-theory spectrum ku, w* *hich is a ring spectrum. This gives a natural lift of 3i(1 - Li) to the sixth space of the ku spectrum,* * which is BU<6>. Thus,Qif V is an arbitrary virtual bundle, then an expression for V as a sum of virtual bu* *ndles of the form 3i(1 - Li) gives rise to a lifting of V to BU<6>. Motivated by Proposition 3.4, we make the following definition. Definition 4.1.If L1, L2 are line bundles over a space X, we put n-1X n-1X dn(L1; L2) def= (1 - L1)(1 - Lj1)(1 - L2) - (1 - L1)(1 - Lj2)(1 - L2) 2 k* *u6X = [X; BU<6>]: j=1 j=1 Note that if we forget about liftings to BU<6> and just work with virtual bundl* *es we have 0 1 n-1X dn(L1; L2)= (1 - L1)(1 - L2) @ Lj1- Lj2A j=1 = (1 - L2)(1 - Ln1) - (1 - L1)(1 - Ln2) = -L2- Ln1+ Ln1L2+ L1+ Ln2- L1Ln2: The main ingredient in the proof of Theorem 1.4 is the following compatibilit* *y. Theorem 4.2.Let L1 and L2 be the obvious line bundles over P2. Then for each n,* * the following diagram commutes (up to a possible sign). P[n]2 (4* *.3) JJJ fi|| Jdn(L1;L2)JJJJ fflffl| J$$ K(Z; 3)_fl_//_BU<6> 10 ANDO AND STRICKLAND Remark 4.4. Let us say that a sequence of spectra X f-!Y -g!Z h-!X is a -fibration if either (f; g; h) is a fibration, or (f; g; -h) is a fibratio* *n. This definition is independent of all conventions such as which end of the unit interval we take as the basepoint* *, which side we write cone and suspension coordinates and so on. Cofibrations are -fibrations, and -fibrations* * are preserved by shifting, suspending and dualising. Suppose we have a diagram as follows, in which the sq* *uare commutes up to sign and the rows are -fibrations. X ---f-! Y --g--!Z --h--!X ? ? ? u?y w?y u ?y 0 g0 h0 X0--f--!Y 0----! Z0 ----! X0 Then there is a map v : Y -! Y 0making everything commute up to sign; moreover,* * if [Z; Y 0] = 0 then w is unique up to sign. These facts are easily deduced from the corresponding ones f* *or genuine fibrations. The sign ambiguity in Theorem 4.2 could be resolved by a careful analysis of conven* *tions to determine whether certain sequences are fibrations, or whether a sign needs to be changed to make* * this true. We have not felt it worthwhile to pursue these questions. At the end of this section we give a short proof of a weaker result (Lemma 4.* *6); the next section constitutes the rather harder proof of the theorem itself. First, however, we draw an impor* *tant corollary. Corollary 4.5.The diagram specK0(BU<6>)-specK0(fl)------!specK0(K(Z; 3)) ? ? 3 ?y ?yb* C3(GK ) --e*--! W(GK ) commutes (up to a possible sign). Proof.By the definition of W, it suffices to check that that bnspecK0(fl) = (en* *3)fflin Wn(GK ) for arbitrary n, where ffl = 1 and is independent of n. In fact the independence of n is auto* *matic: if v* and w* are two Weil pairings, and vn(g; h) = wn(g; h)ffl(n); then vpn(pg; ph) = vn(g; h)p = wn(g; h)pffl(n)= wpn(pg; ph)ffl(n); since p: GK ! GK is surjective, it follows that ffl(pn) = ffl(n). Moreover, Wn(GK ) is a subscheme of the scheme of maps GK [n]2-! Gm, so it su* *ffices to check that the two adjoint maps GK [n]2x spec(K0BU<6>) -!Gm are the same. From now on we fix n and write fi for fin and so on. Note that any element z 2 kU6Z = [Z; BU<6>] gives rise to a map z*: K0Z -!K0B* *U<6> or equivalently an element ^z2 K0Z bK0BU<6> which may be viewed as a map ^z: spf(K0Z) x spec(K0BU<6>) -!Gm : Q 3 This construction converts sums to products and is natural in Z. If z = i(1 * *- Li) 2 ku6P3 then ^z: G3Kx spec(K0BU<6>) -!Gm is just the composite G3Kx spec(K0BU<6>) -!G3Kx C3(GK ) E-!Gm; WEIL PAIRINGS AND MORAVA K-THEORY * * 11 where E(g1; g2; g3; f) = f(g1; g2; g3). It follows by naturality that if z = (1* * - L1)(1 - Lk1)(1 - L2) 2 ku6P2 then ^zcorresponds to the map (g1; g2; f) 7! f(g1; kg1; g2), and thus that dn(L* *1; L2)^ corresponds to the map n-1Yf(g ; kg ; g ) (g0; g1; f) 7! ____0____0_= 1en(f)(g0; g1): k=1f(g0; kg1; g1) This is what we get by going around the bottom left of the square in the statem* *ent of the corollary. A similar procedure converts elements w 2 H3Z to maps ^w: spf(K0Z) x spec(K0K* *(Z; 3)) -!Gm . If w = fi(a1a2) 2 H3P[n]2 then ^w: G[n]2x spec(K0K(Z; 3)) -!Gm is adjoint to bm: G[n]2-! spf(K0K(Z; 3)); essentially by the definition of b. By an obvious naturality statement we see t* *hat ^wO spec(K0(fl)) is adjoint to dfl*w, but in the case w = fi(a1a2) the theorem gives fl*w = dn(L1; L2). Th* *is implies_that_dfl*w= (dn(L1; L2))^, which is adjoint to (en3)1 . * * |__| We next give the promised crude version of Theorem 4.2. Lemma 4.6. There is a unique = n 2 Z=n such that the following diagram commute* *s. P[n]2 JJJ fi || dn(L1;L2)JJJJJ fflffl| J$$ K(Z; 3)i__//_BU<6> Proof.For brevity we put d def=dn(L1; L2): P[n]2-! BU<6>. As bundles we have d = (1 - L2)(1 - Ln1) - (1 - L1)(1 - Ln2); and we work over P[n]2so Ln1= Ln2= 1 so d = 0. This means that the map P[n]2-d!* *BU<6> f-!BSU -g!BU is null. The fibre of g is S1 = K(Z; 1) and [P[n]2; K(Z; 1)] = H1P[n]2= 0 so we* * see that fd = 0. The fibre of f is K(Z; 3) fl-!BU<6> so there is a map d0: P[n]2 -!K(Z; 3) with d = fld0. * *The fibre of fl is SU, which is a retract of U = S1 x SU, and [P[n]2; U] = KU1P[n]2 = 0 by well-known calcul* *ations, so d0is unique. Further easy calculations show that [P[n]2; K(Z; 3)] = H3(P[n]2) = Z=n, generat* *ed by fi._Thus there is a unique 2 Z=n such that d0= fi as claimed. * * |__| 5.The proof of Theorem 4.2 We shall obtain the commutativity of (4.3)by looping down the corresponding d* *iagram of spectra, which we shall see commutes. So let H def=the integral Eilenberg-MacLane spectrum H=n def=the cofibreHofn-!H ku def=the connective complex K-theory spectrum B def=the sphere bundle of Ln over CP1 P def=the disk bundle of Ln over CP1 12 ANDO AND STRICKLAND Let us write j for the inclusion j : B -!P; which can be identified up to homotopy with the map P[n] -!P or the map fi : K(* *Z=n; 1) ! K(Z; 2). Let us label as q and ffi the maps in the cofibration sequence B j-!P -q!T = P=B ffi-!B: Then T is the Thom space of Lm over P. Let us use the notations y: P -!2H x: P -!2ku v: S2 -!ku oe: ku -!H ae: H -!H=n to denote i.the ordinary Euler class of L ii.the ku Euler class corresponding to the map (1 - L): P ! BU iii.the ku class corresponding to the reduced Hopf bundle (1 - H) over S2. iv.the ring map such that oex = y : P -!2H. v. the reduction map Note that there are -fibration sequences 2ku v-!kuoe-!H fl-!3ku H n-!H -ae!H=n fi-!H: The fibration 1.5 is obtained by looping down the first of these. The classes x and y determine complex orientations for ku and H. Let u 2 ku2(* *T) and w 2 H2(T) be the resulting Thom classes. Given a space X, a spectrum E and a class z 2 EkX = [1 X; kE], we write 1 (z)* * for the adjoint map X -!1 kE of spaces; this is of course a mild abuse. We will need the following simple lemma. Lemma 5.1. Suppose we have a diagram as follows, in which the left and right sq* *uares commute up to sign, the rows are -fibrations, and [Z; Y 0] = 0. X ---f-! Y --g--!Z --h--!X ? ? ? ? u?y v?y w?y u ?y 0 g0 h0 X0--f--!Y 0----! Z0 ----! X0 Then the central square also commutes up to sign. Proof.As in Remark 4.4, there is a map v0: Y -!Y 0making the whole diagram comm* *ute up to sign. After changing the sign of v in necessary, we have vf = v0f = f0u. Thus (v - v0)f = 0* *, so v - v0= rg for some_ r: Z -!Y 0, but [Z; Y 0] = 0 so v = v0. Thus, the central square comutes up to * *sign. |__| WEIL PAIRINGS AND MORAVA K-THEORY * * 13 Modelling fi. The obvious generator a 2 H1(B; Z=n) corresponds to a map a makin* *g the diagram B --j--! P ---q-! T --ffi--!B ? ? ? ? a?y ?yy ?yw ?ya (5* *.2) H=n --fin--!2H--n--!2H --ae--!2H=n commute up to sign. (The first two squares commute by well-known calculations, * *and [B; 2H] = H1B = 0 so the third square commutes up to sign by Lemma 5.1.) Let fi(a1a2) be the map making the diagram B(2) -a^a---!(H=n)(2) ? ? fi(a1a2)?y ?y 3H --fi-- 2H=n commute. The map 1 fi(a1a2) is a model for fi. Modelling dn(L1; L2). For each k 0 let [k](z) be the polynomial k [k](z) def=1_-_(1_-_vz)v2 ku*[z]: As Thom classes restrict to Euler classes on the zero-section, we have q*w = [n](x) 2 ku2(P): Let xi2 ku2(P(2)) be the Euler class of Lifor i = 1; 2. If n-1X d def=x1x2 [k](x1) - [k](x2)) 2 ku6(P(2)) < ku6(P2); k=1 then from Definition 4.1 it is clear that 1 d = dn(L1; L2). Combining these definitions and observations yields Lemma 5.3. To prove Theorem 4.2, it suffices to show that the diagram B(2) -j^j---!P(2) ? ? fi(a1a2)?y ?yd (5* *.4) 3H --fl--!6ku commutes up to sign. * * |___| Let us write r and for the indicated maps in the cofibrations B(2)j^j--!P(2)r-!P(2)=B(2)-!B(2): 14 ANDO AND STRICKLAND Lemma 5.5. There is a commutative diagram of the form _B^ffi_ B(2)OooOeeKB ^ TKK (5* *.6) KKKK KK ffi^B|| KKKB1||K Kj^TKKKK | K fflffl| K%%K T ^ B__B2//_KP(2)=B(2)//_PO^OT KK eeA2KKK KKK A | KKK |P^q KKK 1| r KKK | T^j KK%%fflffl|K | T ^ Pooq^P__P(2) in which the linear sequences are cofibrations. Proof.The map A1 is just the projection P_^_P_-!P_^_P_= P_^ P = T ^ P; B ^ B B ^ P B and similarly for A2. The map B1 is obtained from the inclusion B ^ P -! P ^ P * *by collapsing out the subcomplex B ^ B to get a map B ^ T = B_^_P_B-^!BP_^BP^;B and similarly for B2. From these definitions we see directly that all parts of* * the diagram not involving commute, and that the middle row and column are cofibrations. To see that B1 =* * B ^ ffi, recall the naturality of connecting maps and examine the following diagram. (Equivalently,* * one can think of this as an instance of the octahedral axiom.) B(2)--B^j--!B ^ P----! B ^ T -B^ffi---!B(2) ? ? ? ? =?y ?yj^B ?yB1 ?y= B(2)--j^j--!P(2)----! P(2)=B(2)----!B(2) ?? ? ? y ?y ?yA1 * ----! T ^ P _______T ^ P * *|___| Now let f : P(2)=B(2)-!4ku be the map f def=(u ^ x)A1- (x ^ u)A2 where : ku ^ ku ! ku is the multiplication. Lemma 5.7. To prove Theorem 4.2, it suffices to show that the diagram P(2)=B(2)----!B(2) ? ? f?y ?yfi(a1a2) (5* *.8) 4ku --oe--!4H commutes up to sign. * * |___| WEIL PAIRINGS AND MORAVA K-THEORY * * 15 Proof.Consider the following diagram. P(2)=B(2)----!B(2) -j^j---!P(2)-r---!2P(2)=B(2) ? ? ? ? f?y ?yfi(a1a2) ?yd ?yf (5* *.9) 4ku ---oe-!4H --fl--!7ku --v--! 5ku We first show that the right-hand square commutes. One composite is fr= (u ^ x)A1r - (x ^ u)A2r = (u ^ x)(q ^ P) - (x ^ u)(P ^ q) by (5.6) = [n](x1)x2- x1[n](x2): The other composite is n-1X vd= x1x2 (1 - vx1)k- (1 - vx2)k k=1 n 1 - (1 - vx2)n = x1x2 1_-_(1_-_vx1)_vx- ___________ 1 vx2 = x2[n](x1) - x1[n](x2); as required. We also have [P(2); 4H] = H3P(2)= 0. Thus, if we have fi(a1a2) O = oef we ca* *n apply Lemma 5.1 __ to see that e O (j ^ j) = fld, and the claim then follows by Lemma 5.3. * * |__| Lemma 5.10. We have noef = 0 = nfi(a1a2) in H4(P(2)=B(2)). Proof.It is clear that nfi(a1a2) = 0, so it suffices to check that noef = 0. No* *w by definition, oef = (w ^ y)A1- (y ^ w)A2: The commutative diagram P --q--! T ? ? y?y ?yw 2H --n--!2H gives the commutative triangles in the diagram B1 P(2)=B(2)__//_T5^ P 55 B2|| T^q||555 fflffl|q//^fflffl|w^ny555T P ^ T______UUUUTI^55TI UUUUUU w^wH55H ny^wUUUUUH55HH UUU*aeae5*U$$H 4H commute. The square commutes because either composite is just the projection P_^_P_-!___P_^_P___: B ^ B B ^ P [ P ^ B 16 ANDO AND STRICKLAND * * __ The difference between the two outermost composites is noef. * * |__| Lemma 5.11. The maps B*1: H4(P(2)=B(2))-!H4(B ^ T) B*2: H4(P(2)=B(2))-!H4(T ^ B) are such that Ker(B*1) \ Ker(B*2) is torsion-free. Proof.In the diagram B(2)----! (B ^ P) [ (P ^ B)----!(B ^ T) [ (T ^ B) ? ? ? =?y ?y ?yB1[B2 B(2)----! P(2) ----! P(2)=B(2) ?? ? ? y ?y ?y * ----! T(2) _______ T(2); the rows and columns are cofibrations, and so we have an exact sequence *B* H3(B ^ T) H3(T ^ B) -!H4(T(2)) -!H4(P(2)=B(2)) B1-2----!H4(B ^ T) H4(T* * ^ B): It is easy to check that the first term is torsion, and the second term is Z. T* *hus, the first_map must be zero, and the kernel of the third map must be Z. * * |__| Proof of Theorem 4.2.By Lemma 5.7, it suffices to check that the element z def=oef + fi(a1a2) 2 H4P(2)=B(2) is zero. By Lemma 5.10 we have nz = 0, so by Lemma 5.11, it suffices to check t* *hat B*1z = B*2z = 0. For B*1z we have fi(a1a2)B1= fi(a1a2)(B ^ ffi) by Lemma 5.5 = fi(a ^ a)(B ^ ffi) = fi(a ^ affi) = fi(a ^ aew) by diagram (5.2) = fi(a1aew2) = (y1j)w2 by diagram (5.2): On the other hand, oefB1= (w ^ y)A1B1- (y ^ w)A2B1 = (y ^ w)(T ^ j) by Lemma 5.5 = -(y1j)w2: Thus B*1z = fi(a1a2)B1+ oefB1= 0 as claimed. The case of B*2z is similar. * *|___| WEIL PAIRINGS AND MORAVA K-THEORY * * 17 6.Morava K-theory of BSU The Morava K-theory of BSU is accessible through the fibration BSU -!BU B-det--!P: (6* *.1) Note that the inclusion i : S1 = U(1) -!U gives Bi : P -!BU which is a non-addi* *tive splitting of B det, and the sum of Bi with the inclusion of BSU gives an equivalence P x BSU ~=BU. Lemma 6.2. The sequence specK0P -specK0B-det------!specK0BU -!specK0BSU is a short exact sequence of k-groups. Proof.The splitting gives a short exact sequence of formal group schemes 0 -!spf(K0BSU) -!spf(K0BU) -!spf(K0P) -!0; which splits nonadditively. Cartier duality is exact, so we have a short exact * *sequence as claimed. (Note however that Cartier duality is only functorial for homomorphisms, so spf(K0Bi)* * does not_induce a splitting of this sequence.) * * |__| The adjoint of the map K0P -K0(1-L)-----!K0BU is an element 1 of O(GK)K0BUwhose value at the origin is 1. Let C1(GK ) denote* * the scheme of such functions; 1 is classified by a map specK0BU 1-!C1(GK ): Lemma 6.3. 1 is an isomorphism of k-groups. Moreover there is an isomorphism specK0P ~=Hom(GK ; Gm) such that the diagram specK0P ----!~Hom (GK ; Gm) ? = ? specK0B det?y ?ynatural inclusion specK0BU --1--!~ C1(GK ) = commutes. Proof.We leave it to the reader to check that this is a coordinate-free version* * of the usual descriptions_of the Hopf algebras K0BU and K0P, as in for example [1] and [9, 3.4] * * |__| Recall from (1.7)that for a k-algebra R, C2(GK )(R) is defined to be the grou* *p of symmetric 2-cocycles on (GK )R with values in the multiplicative group Gm . (Such maps necessarily l* *and in bGm Gm, so Gm may be replaced by bGmif desired.) Moreover as a functor of R it is represented* * by a k-group which we denote C2(GK ). The adjoint of the map K0Q2i(1-Li) K0P2 --------! K0BSU 18 ANDO AND STRICKLAND is naturally an element 22 C2(GK )(K0BSU) (6* *.4) which is classified by a map specK0BSU 2-!C2(GK ): The purpose of this section is to give a proof of the following. In fact the an* *alogous statement is true for any even periodic ring spectrum [5]. Proposition 6.5.2 is an isomorphism of k-groups. There is a natural map ffi : C1(GK ) -!C2(GK ) given by the formula (ffif)(g1; g2) def=f(g1)f(g2)_f(g: 1+ g2) Lemma 6.6. The diagram specK0BU ----! specK0BSU ? ? 1?y ?y2 C1(GK ) --ffi--!C2(GK ) commutes. Proof.Let , ss1, and ss2 denote the product and two projection maps P2 -!P: Let L denote the tautological line bundle over P, let Lidef=ss*iL: In view of the descriptions of the maps 1 and 2, the lemma boils down to the eq* *uation (1 - L1)(1 - L2) = ss*1(1 - L) + ss*2(1 - L) - *(1 - L): * *|___| Proposition 6.7.The map of k-groups C1(GK ) ffi-!C2(GK ) is an epimorphism with kernel Hom(GK ; Gm). Remark 6.8. The argument which follows has the virtue of brevity, but its tone * *is not really in keeping with the coordinate-free arguments in the rest of the paper. There is a more "n* *atural" proof in [5], to give which would however require a substantial detour in the present paper. The proof of the proposition uses the Artin-Hasse exponential 8 9 1, then let S be a faithfully flat extension of R containing* * a solution b to the equation b + bp= a ifh = 1 bp= a ifh > 1: Here h is the height of GK so K = K(h). Let g be the specialisation to S of the* * reduction modulo p of s-1 A(bxp ) 2 Z(p)[b][[x]]: Using the formula x oFy = x + y + cph(x; y) + o[ph + 1]; it is easy to check that ffig(x; y) = 1 + acps(x; y) + o[ps+ 1]: So there is a faithfully flat extension S of R and a g 2 C1(GK ; Gm)(S) such th* *at f_= 1 + o[t + 1]: ffig By induction, one obtains R0and f0, and concludes that ffi is surjective. * * |___| In the proof of the preceding proposition we used the following algebraic res* *ult. 20 ANDO AND STRICKLAND Lemma 6.10. A map f : H ! G of k-groups is an epimorphism if and only if for e* *very k-algebra R and every R-valued point a 2 G(R), there is a faithfully flat R-algebra S and a* * point b 2 H(S) such that f(b) = aS. Proof.Since an epimorphism of k-groups is a map of k-groups which is faithfully* * flat, the only if direction is clear. For the other direction, consider the case R = OG and S = OH . The hy* *pothesis is that there is a faithfully flat R-algebra T and a factorisation "SOO " |* " f| """ | Too___R: In particular, f* is injective; it follows [4, III,x3, n. 7] that it is faithfu* *lly flat. |___| Proof of PropositionF6.5.rom Lemmas 6.2, 6.3, and 6.6 and Proposition 6.7, the * *diagram specK0P ----! specK0BU ----! specK0BSU ? ? ? ~=?y ~=?y ?y2 Hom (GK ; Gm)----! C1(GK ) --ffi--!C2(GK ) commutes, the rows are short exact sequences of k-groups, and the first two ver* *tical_arrows are isomorphisms. * *|__| 7. Proof of Theorem 1.4 There is a natural map C2(GK ) ffi-!C3(GK ) representing the natural transformation (ffif)(g1; g2; g3) def=f(g1+_g2;_g3)_f(g: (7* *.1) 1; g3)f(g2; * *g3) Lemma 7.2. The diagram specK0BSU ----! specK0BU<6> ? ? 2 ?y ?y3 C2(GK ) --ffi--!C3(GK ) commutes. Proof.The same as the proof of Lemma 6.6. * * |___| In contrast with the case of C1 -!C2, we have Lemma 7.3. C2(GK ) ffi-!C3(GK ) is injective in the category k-groups. WEIL PAIRINGS AND MORAVA K-THEORY * * 21 Proof.The kernel of ffi consists of symmetric, biexponential maps from G2Kto Gm* *. Let f be such a map. On GK [pr] x GK we have f(g; prh) = f(prg; h) = f(0; h) = 1, but GK is p-divisible* * so the map h 7! prh is an epimorphism (and remains so after taking the product with GK [pr]), so f(g; h) * *= 1 on GK [pr]xGK . Another consequence of the p-divisibility of GK is that GK = colimrGK [pr] (and again t* *his colimit_is preserved by products) so we conclude that f = 1. * * |__| Lemma 7.4. The fibration K(Z; 3) -!BU<6> -!BSU (7* *.5) gives rise to a short exact sequence of abelian Hopf algebras K0K(Z; 3) -!K0BU<6> -!K0BSU: Proof.The Atiyah-Hirzebruch spectral sequences for the K-homology of BSU and fo* *r the K-homology of__ the fibration (7.5)collapse, because they start in even bidegrees. * * |__| Proof of Theorem 1.4.According to Lemma 7.2 and Corollary 4.5, we have a commut* *ative diagram specK0BSU ----! specK0BU<6> ----!specK0K(Z; 3) ? ? ? 2 ?y~= 3 ?y b*?y~= C2(GK ) --ffi--!C3(GK ) -e*---! W: Proposition 6.5 and Proposition 2.10 give that the marked vertical arrows are i* *somorphisms. The top row is a short exact sequence of k-groups by Lemma 7.4, and ffi is injective by Lem* *ma 7.3. It_follows that e* is surjective and 3 is an isomorphism. * * |__| In the course of the proof we obtained the following result. This result foll* *ows from Corollary 7.13 of [2], which is proved by methods quite different from the topological methods of this* * paper. Corollary 7.6.The map of k-groups e*: C3(GK ) ! W is surjective. * * |___| References [1]J. Frank Adams. Stable homotopy and generalised homology. Univ. of Chicago * *Press, 1974. [2]Lawrence Breen. Fonctions th^eta et theoreme du cube, volume 980 of Lecture* * Notes in Mathematics. Springer, 1983. [3]Michel Demazure. Lectures on p-divisible groups. Springer-Verlag, Berlin, 1* *972. Lecture Notes in Mathematics, Vol. 302. [4]Michel Demazure and Pierre Gabriel. Groupes algebriques, tome I. North-Holl* *and, 1970. [5]Michael J. Hopkins, Matthew Ando, and Neil P. Strickland. Elliptic spectra,* * the Witten genus, and the theorem of the cube, 1998. Preprint. [6]John R. Hunton and Paul R. Turner. Coalgebraic algebra. Journal of Pure and* * Applied Algebra, 129(3):297-313, 1998. [7]David C. Johnson and W. Stephen Wilson. The Brown-Peterson homology of elem* *entary p-groups. American Journal of Mathematics, 102:427-454, 1982. [8]David Mumford. Biextensions of formal groups. In Arithmetic algebraic geome* *try (proceedings of Purdue conference). Harper, 1965. [9]Douglas C. Ravenel and W. Stephen Wilson. The Hopf ring for complex cobordi* *sm. J. Pure and Applied Algebra, 9, 1977. [10]Douglas C. Ravenel and W. Stephen Wilson. The Morava K-theory of Eilenberg-* *MacLane spaces and the Conner-Floyd conjecture. Amer. J. Math, 102, 1980. [11]Edward Witten. The index of the Dirac operator in loop space. In P. S. Land* *weber, editor, Elliptic Curves and Modular Forms in Algebraic Topology, volume 1326 of Lecture Notes in Mathematics, pa* *ges 161-181, New York, 1988. Springer- Verlag. 22 ANDO AND STRICKLAND Department of Mathematics, University of Virginia Current address: Department of Mathematics, The Johns Hopkins University E-mail address: ando@math.jhu.edu Trinity College, Cambridge CB2 1TQ, England E-mail address: n.strickland@dpmms.cam.ac.uk