INFINITE SUBGROUPS OF MORAVA STABILIZER GROUPS V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS Abstract. In this note we discuss certain infinite subgroups of the Morav* *a stabi- lizer groups and outline some applications in homotopy theory. 1. Description of the main result and its applications First we discuss a theorem about the structure of the group of proper units o* *f the maximal order in a certain class of cyclic division algebras over a local field* *. This theorem states that such a group contains a free product of finite cyclic subgr* *oups which is dense in the subgroup of elements of norm one. We believe this result * *should be of interest not only to topologists. The importance of the above kind of groups for homotopy theory stems from Morava's work [11] in which he outlined a general algebraic way of studying the homotopy category. The major ingredient in this outline is the chromatic filtra* *tion of the homotopy category [13], [14]. The terms of this filtration can be descr* *ibed by the Bousfield localization functors LK(n): for more details see [6]. An alg* *ebraic description of these functors can be given using certain p-adic Lie groups (the* * so-called Morava stabilizer groups Sn) which are defined for each prime p and each positi* *ve integer n. The group Sn can be described as the group of units in the maximal order of the cyclic division algebra over Qp of index n with Hasse invariant 1_* *n. Each group Sn comes equipped with a representation n in the group of automorphisms of a certain ring En. The cohomology of Sn with coefficients in this representati* *on is related (via a spectral sequence which often collapses) to the homotopy groups * *of the Bousfield localization LK(n) of the sphere, which in turn approximate the homot* *opy groups of spheres. The group Sn has another algebraic characterization. Sn may be viewed as the group of automorphisms of a certain p-typical formal group F of height n over t* *he algebraic closure of a finite field of p elements. The representation n is defi* *ned by the canonical action of Sn on the moduli space of deformations of F , which is * *En. Among the applications of the above theorem to homotopy theory are the follow- ing. For any prime p the theorem we prove implies an interesting description of* * the Morava stabilizer group Sp-1 and gives a conceptual explanation of some of Rave* *nel's 1 2 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS computations of the cohomology of this group with trivial coefficients. It also* * sug- gests an approach to the computation of the cohomology of the Morava stabilizer group S2 for the prime 3. This has been done in the joint paper by V. Gorbounov, S. Siegel and P. Symonds [8]. This last result is important for the following reasons. The functor LK(1)is * *com- pletely understood for all primes. The functor LK(2)is currently a topic of in* *tense study. It appears to be closely related to the theory of elliptic curves. The p* *-typical formal group F , which has the Morava group S2 as its ring of automorphisms, is actually the formal group of a supersingular elliptic curve for the primes 2 an* *d 3. The work of a number of people in algebraic geometry (see for example [9]) sugg* *ests a close relation between the cohomology of S2 for the primes 2 and 3 with coeffic* *ients in the representation n and algebraic properties of the moduli scheme of elliptic * *curves. A way to describe this relation in topological terms was suggested recently b* *y M. Hopkins and H. Miller. They constructed the so-called higher real K-theories. L* *et us describe these briefly. They proved that the Morava group Sn acts in a "nice" w* *ay on a certain spectrum En, so one can talk about the homotopy fixed point spectrum * *EH of this action for any discrete subgroup H of Sn. A description of the finite s* *ubgroups of Sn is known due to S. Amitsur and T. Hewett [1, 5]. Following M. Hopkins and* * H. Miller, take H to be a maximal finite subgroup of Sp-1 and call EH the spectrum* * of the higher real K-theory for the given n and the prime p. They denote it by EOp* *-1. For the primes 2 and 3 these maximal finite subgroups happen to be the groups of automorphisms of appropriate supersingular elliptic curves over algebraically c* *losed fields of characteristics 2 and 3, respectively. Thus the construction of EO2 a* *t p = 2; 3 suggests that this spectrum is related to the moduli scheme of elliptic curves.* * The homotopy groups of the Bousfield localization of the sphere, on the other hand,* * are related to the homotopy groups of EO2. This is how the theory of moduli schemes for elliptic curves should merge into homotopy theory. Using our results about infinite subgroups of the Morava group we describe an infinite subgroup L of S2 such that H < L < S2. The spectrum EL is hopefully a better approximation to LK(2). On the other hand, it is plausible that the grou* *p L is the group of automorphisms of an algebraic surface. An appropriate candidate* * is a K3 surface. If this is the case then one has a chance to use information abou* *t the moduli space of K3 surfaces for further study of the Bousfield localization fun* *ctor LK(2). 2. Structure of p-adic division algebras We consider a cyclic algebra D over Qp of index n with Hasse invariant 1_n. W* *e give a brief description here: for more details see [15]. Let W be the totally unram* *ified extension of Qp of degree n (so W ~=Qp [i] where i is a (pn - 1)-st root of uni* *ty). The Galois group of the extension W=Qp is a cyclic group of order n. It is gene* *rated by the Frobenius homomorphism oe. We form the crossed product algebra of W and 3 Gal(W=Qp ). This amounts to introducing a variable S which commutes with W according to the formula: wS = Swoe; w 2 W and Sn belongs to Qpx . To define D we set Sn = p. D is a division algebra over* * Qp of rank n2. It has a faithful representation : D ! GLn(W): If n-1X a = aiSi i=0 then (a) is the following matrix: 8 j Let det: GLn(W) ! Qpx be the determinant map. Let O be the maximal order in D : it is generated by S and the integers of W.* * We are interested in three subgroups of O. The first is the group of units of O, * *which we denote by Sn. The second is the subgroup of strict units in Sn. It consists * *of the elements a 2 Sn such that a 1 mod S. We denote it by S0n. The third is the ker* *nel of the reduced norm, nrd = detO, restricted to S0n. We denote this subgroup by * *Sl. Both S0nand Sl are pro-p-groups. Let us recall some facts about the generators and cohomology of pro-p-groups.* * A set of elements g1; : :;:gn 2 G is said to generate G if the subgroup that they* * generate algebraically is dense in the pro-p topology. Let G* denote the minimal closed * *normal subgroup that contains all the commutators and all the p-th powers. Theorem 2.1. [10], [18, Prop I-25] Let G be a pro-p-group and let g1; : :;:gn * *be a subset of G. Then g1; : :;:gn generate G if and only if they generate G=G*. For the pro-p-group Sl the map Sl ! Sl=Sl* admits the following elegant descr* *ip- tion due to [16, Thms 7, 21]. Let Hidenote the subgroup of Sl consisting of ele* *ments congruent to 1 modulo Si. Note that H1 = Sl and H2 = Sl* = [Sl; Sl]. For any field k let k+ denote its additive group. Then Sl=(S) ~=F+pn~=W=(p). Any element a 2 Sl can be written uniquely as 1 + a1S where a1 2 O, and the quotient of Sl * *by its commutator is isomorphic to F+pn, via the map ae: Sl ! Fpn+ given by the fo* *rmula ae(a) a1modS 2 F+pn Assume now that n is equal to p-1. It is known [15] that D contains any exten* *sion of Qp of degree p - 1. In particular it contains a p-th root of unity, say z. S* *ince Qp does not contain p-th roots of unity different from 1 we conclude that nrd z = * *1, so z belongs to Sl. It is also known that D contains a (p - 1)-st root of -p, * *e.g. 4 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS ! = i(p-1)=2S. The following fact from number theory will be useful for us lat* *er: Qp(!) ~=Qp (z) [19]. Our main theorem is Theorem 2.2. There is a p-th root of unity ff 2 Sl such that i (1) {ff! , 1 i p-1} generate an infinite subgroup G of Sl, which is isomorp* *hic to a free product of p - 1 copies of Z=p : Z=p * . .*.Z=p (2) G is dense in Sl. Remark 2.3. By similar methods we can show that if n = (p - 1)pr then Sl conta* *ins a free product of n copies of Z=p(r+1)which is dense. If n = (p - 1)pra with (a* *; p) = 1 then all the torsion elements are contained in Ha. (If (p - 1) 6 |n then Sl is * *torsion free.) The proof of the main theorem is in two parts. First we show that "genericall* *y" a pair of subgroups of D generates a free product, and then we show that Sl is ge* *nerated by torsion elements. 3. Free products in the group of units of an algebra We say that two subgroups G and H of a group X generate a projective free pro* *duct if all the relations in X come either from G or H or from the fact that G* *\Z (X) and H \ Z(X) must be central (where Z(X) denotes the centre of X). Equivalently, the images of G and H in X=Z (X) generate a free product. Theorem 3.1. Let k be a field, A a finitely generated k-algebra and G and H two countable subgroups of Ax . Suppose that there is a field extension K of k and* * an element x 2 A k Kx such that G and Hx generate a projective free product in A kKx . Then the set of x 2 Ax for which G and Hx do not generate a free product all lie within a countable family of subvarieties of A. Remark 3.2. A is naturally an affine space over k. By subvariety of A we mean * *the zeros of some non trivial polynomials in the coordinates with coefficients in k. The hypotheses of the theorem remain in force for the corollaries. Corollary 3.3. If k is uncountable then the set of x which do yield a projectiv* *e free product is not empty. Corollary 3.4. If k is either R or C or a p-adic field then the set of X which * *do yield a projective free product is dense (in the k-topology). Its complement ha* *s Haar measure 0. Corollary 3.5. Suppose that R is a discrete valuation ring with countable resid* *ue class field and quotient field k, and is an R-order in A. Let p denote a gener* *ator of the prime ideal of R. Then for any positive integer r there is an x 2 1 + pr* * x which yields a projective free product. 5 Proof.The set of reduced words in the projective free product of G and H is a countable set. Each word is of the form w = g1h1g2h2. ...Let r(w) be the number of occurences of an element of H. Define a function OEw :A k Kx ! A k Kx by OEw(x) = g1hx1g2hx2: :.:We are interested in the equation OEw(x) = 1, which exp* *resses a relation in the group . By hypothesis this equation is not satisfied * *by all x 2 A k Kx unless w represents 1 in the projective free product. It is now convenient to regard A as embedded in an algebra M of matrices over* * k. Let adj(x) denote the classical adjoint matrix of x (so adj(x) = det(x)x-1) and* * define h^x= adj(x)hx, ^OEw(x) = g1h^x1g2h^x2... .The equation OEw(x) = 1 can be rewrit* *ten as ^OEw(x) = det(x)r(w), but the latter can be regarded as a system of polynomi* *al equations (no denominators involving the determinant) over k on the affine space M k K. By hypothesis this system is non trivial on A k K so defines a subvariety Aw of A for all non-identity w. If x 2 Ax is not contained in any Aw then, by construction, G and Hx generate* * a projective free product. __|_ | Proof of corollaries: 3.3: Suppose the contrary. Then A is the union of the subvarieties Aw and det* *= 0, contradicting Lemma 1.6 below. 3.4: This follows from the k-analytic structure [2]. 3.5: Let {ai}i2Ibe a set of representatives of the residue classes of modulo* * pr. If the subvarieties Aw and det = 0 cover 1 + pr, then the subvarieties aiprAw; * *i 2 I; r 2 Z and det= 0 cover A. This leads to a contradiction as in the proof of 3* *.4. Lemma 3.6. If k is an uncountable field, then An(k) is not the union of a cou* *ntable family of subvarieties. Proof.Use induction on n : the case n = 1 is clear since k is uncountable. It* * is sufficient to assume that each subvariety is given by just one polynomial. Now * *the equation of a hyperplane divides any polynomial which is trivial when restricte* *d to it, so a non-trivial polynomial can only be trivial on finitely many hyperplane* *s. This enables us to choose a hyperplane on which none of the polynomials are trivial,* * and we apply the induction hypothesis to it. __|_ | For our applications we need to construct projective free products in some ex* *tension field. Our main tool for this purpose is Nagao's theorem [12], [17], that for a* *ny field K, GL2(K[t]) = GL2(K) *B(K)B(K[t]), where B denotes the upper triangular matrices. In fact we can quotient out the centre Kx to obtain P GL2(K[t]) = P GL2(K) *PB(* *K) P B(K[t]).Thus if G is a subgroup of GL2(K) and H is a subgroup of B(K[t]) such that G \ B(K) Kx and H \ B(K) Kx then G and H generate a projective free product [[17], ch I, 1.1]. 6 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS We are particularly interested in the division algebra D of section 2, and it* *s maximal order O. Recall that O contains an element ! such that !p-1 = -p, which we shall keep fixed. Let a 2 O be any primitive p-th root of unity. Proposition 3.7. There is a conjugate ff of a in Ox arbitrarily close to a such* * that and generate a projective free product in Ox . p-1 Corollary 3.8. Ox contains the free product * * ::: * . Proof of proposition. By Corollary 3.5 applied to and , we only need t* *o find such a projective free product over some extension field K of Qp. Let K = Q(; ;* * j), where is a primitive p-th root of unity, is a primitive (p - 1)-st root of un* *ity, and j is a (p - 1)-st root of -p. K is a splitting field for A, so A Qp K ~=Mp-1(K)* *, the (p-1)x(p-1) matrices over K, and both a and ! are diagonalisable in Mp-1(K). It will suffice to find two matrices x and y such that x (resp. y) has the same ei* *genvalues as a (resp. !) and also <; > is a projective free product. For then a = x* *u and -1v v ! = yv for some u; v 2 GLp-1(K) and <; > = <; > is a projectiv* *e free product. Now a has eigenvalues i; 1 i p - 1, and S has eigenvalues jj; 0 j p - 2. Let ! ! 0 1 0 0 00c x00= 0 2 ; c = 1 1 ; x = x : Then ! r 0 0 x0r= 2r- r 2r ; so \ B(K) = 1: Let ! ! 1 0 1 t 0 00d y00= j 0 ; d = 0 1 ; y = y : Then ! 1 t(1 - r) 0 0p-1 y0r= jr 0 r ; so \ B(K) = =

: So by Nagao's theorem, as explained above, and generate a projectiv* *e free product in GL2(K[t]). Now extend x0 and y0 to (p - 1) x (p - 1) matrices x and y by adding the other eigenvalues of a or S along the diagonal. Finally, K[t] ca* *n be embedded in a p-adic completion of K by sending t to some transcendental over K. 7 4. Generation by torsion elements The isomorphism class of a crossed product algebra such as D depends only on * *the p-adic valuation of Sp-1. We could instead have used a generator T with T p-1= * *-p. p-1_ (In fact this new algebra is isomorphic to D by sending T to ! = i 2S. It will* * be convenient in this section to use this description of D and the corresponding m* *ap ae : Sl ! F+pp-1. (If a = 1 + a1T; a1 2 O, then aeT(a) = a1 mod T .) Note that 1-p_ ! oe aeT(a) = ae(a)i 2 and aeT(a ) = aeT(a) . In order to prove the Main Theorem we need to showpthat-we1can find a p-th root of unity a 2 Sl of such that a; a!; : :;:a! generate Sl=Sl*, i.e. such* * that p-1 aeT(a); ::; aeT(a! ) form a basis for Fpp-1over Fp. In other words, in view o* *f the the formula aeT(a!) = aeT(a)oeabove, aeT(a) and its conjugates should form a normal* * basis of Fpp-1=Fp. Lemma 4.1. The image under aeT : Sl ! Fpp-1of the p-th roots of unity consist* *s of the elements of the form up-1. (These are the elements of norm 1.) Proof.As mentioned in Section 2, Qp(T ) contains a pth root of unity z. Now aeT* *(z) 2 Fp, and since aeT(zr) = raeT(z), we may assume that aeT(z) = 1. It is a consequ* *ence of the Skolem-Noether Theorem that any p-th root of unity in D is of the form zt; * *t 2 Dx . We may write t = T ru(1 + vT ); u 2 W \ O, v 2 O. Then aeT(zt) = up-1 mod p. _* *_|_ | Lemma 4.2. The extension Fpp-1=Fp has a normal basis of the the form up-1. Proof.The characteristic polynomial of oe factorizes over Fp, so Fpp-1decomposes over Fp into eigenspaces for oe. If r is a generator of the multiplicative grou* *p Fxp, then the eigenvectors are er; e2r; ::; ep-1rof eigenvalues r; r2; ::; rp-1 respectiv* *ely, and they form a basis. Note that oeer = ep-1rby definition of oe, and oeer = rer by defi* *nition of er, so ep-1r= r. The group algebra Fp contains idempotents which project p-1 ontoPeach eigenspace, thus x; xoe; ::; xoe form a basis if and only if, when w* *e write x = p-1i=1ieir, no i is 0. Now ! ! p - 1 p - 1 2 p-1 (1 + er)p-1 = 1 + er + er + :: + er : 1 2 Since ep-1r= r, this shows that (1 + er)p-1 and its conjugates form a normal ba* *sis provided that 1 + r 6= 0 ie. p 6= 3. If p = 3 there are four elements of norm 1* *, so they are not all contained in F3. __|_ | Remark 4.3. If we had used S instead of T , then the pth roots of unity would * *have had image under ae of norm -1. By Corollary 3.8, there is anpx-21Sl such that ff = x-1ax gives rise to a fre* *e product G = * * : :*: and ae(ff) = ae(a). The inclusion of G in Sl * *induces an 8 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS isomorphism G=G* ~=Sl=Sl*. By proposition 2.1 this implies that G is dense in S* *l. This proves 2.2 completely. Also, since the cohomology of G is generated by H 1and its Bocksteins, we have the following easy proposition about the cohomology on continuous cochains of S* *l. Proposition 4.4. There is an epimorphism H *c(Sl; Fp) ! H*(G; Fp): 5. The proof of the main theorem for p=3 We give now an elementary proof of the main theorem for the primepp_= 3. We will be able to give precise formulas for the embedding of Q3 [ 31] into Sl whi* *ch we believe will be useful for computing the cohomology of Sl. In this case S2 * *is a subgroup of units of the maximal order of the division algebra (W=Q3 ; oe; 3), * *where oe, the Frobenius, is the generator of Gal(W=Q3 ). Since W is an unramified extension of degree 2 of Q3 it can be constructed by adjoining to Q3 a fourth root of unity, i. Since Q3 contains a square root of -* *2, we can set p ___(i + 1) i = -2 ______; 2 and it is easy to check that i is an eighth root of unity such that i2 = -i and p ___(1 - i) iS = i3 = -2 ______: 2 Proposition 5.1. In the above notation a cube root of unity z 2 D is given by t* *he formula 1 p___(i + 1) 1 i z = -__+ -2 ______ S = -__+ __S: 2 4 2 2 Proof. This is a simple computation in number theory.pAs_it was mentioned abo* *ve S2 contains a square root of -3, e.g. ! = iS and Q3[ -3 ] = Q3[z]. Therefore (5.2) z = a + b! where a; b 2 Q3. Using the fact that z3 = 1 we immediately obtain that a = -1_2, b = +_1_2 p __ Remark 5.3. The above method of constructing an embedding of Qp [ p1] into Sp-1 theoretically works for any prime p, but the coefficients in an expansion simil* *ar to 5.2 are not rational numbers. Since Q3 does not contain a primitive cube root of unity, the norm of z is eq* *ual to 1 and we conclude that z 2 Sl. Theorem 5.4. The subgroup G of Sl generated by z and zS is isomorphic to a free product of two copies of Z=3. Moreover G is dense in Sl. 9 Proof.We will use the matrix representation of this cyclic algebra which was d* *e- scribed above. z and zS are two matrices of order three with entries in W: 0 1 p ___(i-1)1 0 1 p ___(i+1)1 -_2 -3 -2 ____4 -_2 -3 -2 ____4 z = B@p___ CA; zS = B@p___ CA: -2 (i+1)_4 -1_2 -2 (i-1)_4 -1_2 Because of the form of the entries, we see that we can also consider these ma* *trices as elements of SL2(C). The subgroup they generate in SL2(C) is isomorphic to the subgroup they generate in SL2(W). Conjugating the two generators by the matrix ! 1 p0__ 0 3 we can get them to lie in SU(2). Using now the homomorphism SU(2) ! SO(3) we reduce ourselves to two matrices in SO(3). Using the formulas in [3] one se* *es that these are two rotations by 120O about two perpendicular axes. They generate a subgroup of SO(3) which is isomorphic to Z=3 * Z=3 due to the following theor* *em proved by F. Hausdorff [4]. Theorem 5.5. Consider a half turn rotation g (so g2 = 1) and a one third rotat* *ion h (so h3 = 1) of R3, the angle between axes being ss=4. Then g and h generate * *in SO(3) a group isomorphic to Z=2 * Z=3. The subgroup we are interested in is the one generated by h and hg. So the statement of 2.2 is an easy corollary of the above theorem. To prove that G is dense in Sl we will prove that z and zS are topological ge* *nerators of Sl. According to 2.1 we need to show that ae(z) and ae(zS) are generators of Sl=Sl* ~=F9+: Direct computation using the formula from 5.1 shows that ae(z) = i ae(zS) = i3 Since i and i3 generate the additive group of F9 the theorem is proven. __|_ | 10 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS 6.Applications We start by describing a certain E1 spectrum En. The ring of its coefficient* *s is isomorphic to WFpn[[u1; : :u:n-1]][u; u-1]: One can obtain such a spectrum by completing the Ravenel - Wilson spectrum E(n) in the appropriate way. M. Hopkins and H. Miller proved that the Morava stabili* *zer group Sn acts by E1 maps on En. This implies that for any discrete subgroup H of Sn one can form a spectrum EHn of homotopy fixed points of the action of H on En. This spectrum gives an approximation of the Bousfield localization of the s* *phere spectrum with respect to Morava K-theory K(n). The latter is denoted by LK(n)S0. The computation of the homotopy groups of the spectra LK(n)S0 can be done usi* *ng the homotopy fixed point spectral sequence which starts with the continuous Gal* *ois cohomology of Sn, H*c(Sn; En;*)Gal. The spectra EHn for different subgroups H o* *f Sn approximate LK(n)S0 and therefore can be useful as a substitution for LK(n)S0. * *The homotopy groups of EHn can be computed in a similar manner to those of LK(n)S0 using the homotopy fixed point spectral sequence which starts with H*(H; En;*). Now let p be an odd prime and n = p - 1. The authors of [7] consider the maxi* *mal finite subgroup H of Sp-1, which is unique up to isomorphism, and is isomorphic in this case to Z=pZ=(p - 1)2 [5] . They call the spectrum EHp-1the "higher re* *al K-theory" and denote it by EOp-1. The reason for this is that, at the prime 2, * *S1 is just the group of units in Z2 and so its maximal finite subgroup is Z=2. The* * KO spectrum is the homotopy fixed point spectrum of this Z=2 action on the spectrum of complex K-theory completed at 2, which is E1. Now we are ready to outline the applications of the main theorem. 7. The spectrum EL for p = 3 We deal here with the case p = 3 and n = 2. The maximal finite subgroup H of S2 in this case is isomorphic to Z=3Z=4. It is not hard to see that this gr* *oup is generated by z and i2, for example. The infinite subgroup G of Sl defines an in* *finite subgroup L of S2 which we will describe now. Note that S conjugates z in the sa* *me way as the 8-th root of unity i. Indeed 1 i S 1 i3 zS = (-__+ __S) = -__+ __S 2 2 2 2 On the other hand 1 i 1 i11 1 i3 zi = i7(-__+ __S)i = -__+ ___S = -__+ __S 2 2 2 2 2 2 2 i2 2 However zS = z and z = z . 11 We define the group L to be the semi-direct product (Z=3 * Z=3)Z=8 where Z=8 is the group generated by i and the free product is generated by z and zi. So we have the following sequence of subgroups of S2: S2 > (Z=3 * Z=3)Z=8 > Z=3Z=4; and therefore the map of spectra EL ! EH . As we pointed out in the introduction the spectrum EL should be a better approximation to the spectrum LK(2)S0. 8.Cohomology Let us describe the computation of H*(S02; F3) from [13]. First we need some background. There is a profinite Hopf algebra S(n) such that S(n)* Fpn~= Fpn[S0n]: It was shown in [13] that Ext*S(n)(Fp) Fpn~= H*c(S0n; Fpn): Let p = 3, n = 2 and consider R = E(h10; h11) P (b10; b11)=I where (8.1) I = (h10h11; b210+ b211; h10b10- h11b11; h11b10+ h10b11): Theorem 8.2. [13] H*(S(2); F3) is isomorphic as an algebra to E(i1; i2) R, wh* *ere R is as defined above, i1 is an element of H1(S(2); F3) determined by the deter* *minant map, and i2 is a certain four-fold Massey product which belongs to H2(S(2); F3). Here is the interpretation of a part of this computation in terms of the infi* *nite subgroups described above. Proposition 8.3. After extending the coefficients to F9, R becomes isomorphic to H*(Z=3 * Z=3; F9). Proof.Let i 2 F9 be an element such that i2 = -1. 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