THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1 WITH COEFFICIENTS IN Ep-1 PETER SYMONDS Abstract. We calculate the Tate-Farrell cohomology of the Morava stabiliz* *er group Sp-1 with coefficients in the moduli space Ep-1for odd primes p. 1. Introduction We present a calculation motivated by homotopy theory, although our methods a* *re alge- braic and involve the Tate cohomology of a profinite group with compact coeffic* *ients. As a reference to the background in homotopy theory we suggest [4, 5]. For the Ta* *te-Farrell cohomology of profinite groups with coefficients in compact module we refer to * *[12], although most of the results are analogues of one for discrete groups, for which see [2]. Let p be an odd prime and n 2 N and let R be the ring of integers of the unra* *mified extension of ^Qpof degree n, so k ~=Fpn. Let Ø be the Frobenius automorphism an* *d Gal= <Ø> the Galois group. Let Sn denote the (full) nth Morava stabilizer group: this is* * the group of units in the R-algebra M generated by S subject to the relations Sn = p and rS * *= SØ(r) for r 2 R. The Galois group Gal acts on Sn simply by Ø(rSi) = Ø(r)S. Thus Sn is virtually a pro-p group of virtual cohomological dimension n2 and * *type FP 1. Ifn n denotes the commutative one-dimensional p-typical formal group law with* * p-series xp , then Sn is isomorphic to the group of automorphisms of n over Fp. It ther* *efore acts on the ring of functions on the Lubin-Tate moduli space of ?-isomorphism classes o* *f lifts of n, which is En,0= R[[u1, . .,.un-1]], a profinite RSn-module. We denote the catego* *ry of such modules by CR(Sn). There is also an action of Sn on a graded version En,*= En,0* *[u 1]. This is graded by the power of u, normalized so that u has degree -2 (called the int* *ernal degree). This combines with the action of Gal on En,*via its action on the coefficient* *s to give an action of the semi-direct product Sn o Gal on En,*, and so each En,r2 CR(SnLo G* *al). We would like to calculate the ring H*(Sn, En,*)Gal, by which we mean r,sHr* *(Sn o Gal, En,s), since this is the initial term of a spectral sequence which converg* *es to ß*LK(n), the homotopy groups of the localization of the sphere spectrum at the nth Morav* *a K-theory (all at the prime p). What we will actually do is to calculate the Tate-Farrell* * cohomology in the case n = p - 1: this is equal to the ordinary cohomology in degrees grea* *ter than n2. Theorem 1.1. For odd p and n = p - 1 ^H*(Sn, En,*)Gal= ^H*(G, En,*)Gal (x0, . .,.xn-1) = ^H*(Sn, ^Zp) (ff) Fp[ 1] = Fp[ 1, fi 1] (ff, x0, . .,.xn-1). The generators will be defined in the course of the calculation. 1 THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT* *S IN Ep-12 L * * L Remark. It would be natural to regard En,*as ^ sEn,s, the sum in CR(Sn), but Hr* *(Sn, ^ sEn,s) ~= sHr(Sn, En,s). Since only the homogeneous parts appear in the spectral sequenc* *e, the dif- ference is immaterial, but we conform to the conventional usage. We will need the following corollary of [12] 7.3 and the remark following it.* * It is what we would expect from the theory for discrete groups in [2]. Similar results for pr* *ofinite groups but with discrete coefficients also appear in [11] and [10]. Theorem 1.2. Let G be a profinite group of finite virtual cohomological dimensi* *on over R. Suppose that G has no subgroup isomorphic to Z=p x Z=p and only a finite num* *ber of conjugacy classes of subgroups isomorphic to Z=p, which we denote by S(p). Let* * M be a module in CR(G). Then the Tate-Farrell cohomology satisfies M H^*(G, M) ~= H^*(NG(P ), M). P2S(p) 2. Trivial Coefficients From now on n = p - 1. The group Sn contains an element a of order p and * * is a maximal finite p-subgroup, unique up to conjugacy. The centralizer C = CSn(a) c* *orresponds to the units in the ring of cyclotomic integers ^Zp[a] M so, written additive* *ly, has the form ^Znpx Z=p x Z=n. There is an element b of order n2 which normalizes , with b* *n generating the Z=n in the centralizer: let e be an integer such that b-1ab = be. Then G = * * is the maximal finite subgroup of order divisible by p. The subgroup N0 = NSnoGal() fits into a short exact sequence ^Znp! N0 ! T * *, where |T | = n3 and T fits into a short exact sequence ! T ! Gal. The second gene* *rator c of T can be chosen to centralize a. The action of T on ^Znpis via a cyclic quotient of order n generated by the i* *mage of b, and as a module for this it is free of rank 1 or, equivalently, a sum of rank 1 R-l* *attices, one for each possible eigenvalue. First we calculate the Tate-Farrell cohomology with trivial coefficients. We * *use to denote an exterior algebra over Fp. Proposition 2.1. For p odd and n = p - 1: H^*(Sn, ^Zp) = ^H*(G, ^Zp) (x0, . .,.xn-1) = Fp[fi 1] (x0, . .,.* *xn-1), H^*(Sn, Fp) = ^H*(G, Fp) (x0, . .,.xn-1) = Fp[fi 1] (ff, x0, . .,* *.xn-1), where |fi| = 2n, |xi| = 1 - 2i and |ff| = -1. Proof.By Theorem 1.2 we find that H^*(Sn, ^Zp) ~= H^*(N, ^Zp), where N = NSn(<* *a>) ~= (^Znpx ) o . Notice that ^H*(N, ^Zp) ~=^H*(^Znpx , ^Zp), since b ha* *s order coprime to p. By the Künneth Theorem in Tate-Farrell cohomology ([2] X 3 ex. 4), H^*(C, ^Zp* *) ~= H^*(, ^Zp) H*(^Znp, Fp). It is well known that H^*(, ^Zp) = Fp[i 1], where |i| = 2. To find the act* *ion of b on i use dimension shifting to see that H2(, R) ~=H1(, k) ~=Hom (, k). Then* * b acts on the latter by sending f to (x 7! bf(b-1xb)), so b(i) = ei. THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT* *S IN Ep-13 Now a basis y0, . .,.yn-1 of H1(^Znp, Fp) can be chosen so that b(yi) = eiyi.* * We finish by calculating the invariants under b using the last part of Lemma 4.1 and setting* * xi= i-i yi. The calculation for Fp coefficients is almost identical. 3. Coefficients in En Next we calculate ^H*(, En,*) following the method of Nave [7, 8], which i* *n turn is based on unpublished work of Hopkins and Miller. This is also treated in detail for t* *he prime 3 in [6]. First we need a change of basis. Lemma 3.1. ([7, 8]) There are elements z, z1, . .,.zn-1 2 En,0such that, where * *m denotes the ideal (p, u1, . .,.un-1) in En,0: (1) z cu mod (p, m2) for some c a unit in R, (2) zi ciuui mod (p, u1, . .,.ui-1, m2) for some ci a unit in R. (3) (1 + a + . .+.ap-1)z = 0, (4) b(z) = jz for j 2 R a primitive n2 root of unity such that jp-1 = e (mod* * p), (5) (a - 1)z = zn-1 and (a - 1)zi+1= zi for 1 i < n - 1. It follows from (1) and (2) that En,*= R[[z-1z1, . .,.z-1zn-1]][z 1]. Let V be the R-submodule of En,-2spanned by {z, z1, . .,.zn-1}. It follows fr* *om (3),(4) and (5) that V is an RG-submodule. Let ffi = p-1i=0ai(z): then a(ffi) = ffi an* *d b(ffi) = jpffi = ejffi. Consider the symmetric algebra S[V ] En,*. We claim that, as RG-modules, 8 >>0 r odd, >< 0 ffi-r R (proj)r = 2pr0 0, (y) R[V ]r = 0 >>ffi-r V (proj)r = 2(pr0- 1) 0, >: (proj) otherwise. and R[V ]r-2p= ffiR[V ]r (proj) forr < 0. Here (proj) indicates a projectiveLsummand. We will write this in the condense* *d form R[V ] = B (proj), where B = iffii(R V ). Recall that if G is a finite group of order not divisible by p2 and M 2 CR(G)* * is projective in CR then the isomorphism class of M is uniquely determined by its reduction m* *odulo p, k R M. This is true for a cyclic group of order p by the classification of RZ* *=p-lattices, (see [9], [3] 34.31), and this classification extends to CR(Z=p). The general c* *ase follows by a transfer argument. Thus we only have to check the claim over kG. But it is true over k from t* *he calculation of the symmetric algebra by Almkvist and Fossum [1]. The general case follows b* *ecause both 0 -r0 ffi-r R and ffi V are defined over G, and the quotients by them must still be* * projective over G since this depends only on the restriction to the Sylow p-subgroup. Being pro* *jective they force the extension to split, and our claim is proved. If we invert ffi we obtain a dense subset R[V ][ffi-1] En,*. As an RG-modul* *e this still has the same form B (proj), by the second identity in y. In fact this form is pr* *eserved by completion: Proposition 3.2. As a sum of compact modules for RG, En,*= B (proj). THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT* *S IN Ep-14 Proof.Let (z-1z1, . .,.z-1zn-1) denote the ideal generated (topologically) by t* *he given ele- ments in En,0. It is easy to check that En,-2r= R[V ]-2r (z-1z1, . .,.z-1zn-1)r+1En,-2r, r > 0 and also R[V ]-2r= B-2r P-2r, r > 0 for some projective P-2r. Thus, for r + pt > 0, En,-2r= ffitEn,-2(r+pt) = ffitR[V ]-2(r+pt) ffit(z-1z1, . .,.z-1zn-1)r+1En,-2(r+pt) = B-2r P-2(r+pt) (z-1z1, . .,.z-1zn-1)r+1En,-2r. Now R[V ][ffi-1]-2r = B-2r lim-!P-2(r+pt)and En,2r= B-2r lim-P-2(r+pt)as t ! * *1. As a consequence, if lim-!P-2(r+pt)= iQi, as a sum of indecomposable projective R* *G-modules then En,2r= B2r iQi. We say that x 2 ^Hr(-, En,s) has bidegree |x| = (r, s). Corollary 3.3. ([7]) The Tate cohomology is given by H^*(, En,*) = k[ffi 1, i 1] ( ), where |ffi| = (0, -2p), |i| = (2, 0), |ff| = (1, -2) and b acts by b(ffi) = ejffi, b(i) = ei, b( ) = ej . As a consequence ^H*(G, En,*)= k[ 1, fi 1] (ff), ^H*(G, En,*)Gal= Fp[ 1, fi 1] (ff). where | | = (-2, 2n), |fi| = (2n, 0) and |ff| = (1, 2n). Proof.The first calculation is an easy consequence of 3.2 (we identify ffi with* * its image in H^0(, En,-2p)). H^*(, En,*)= ^H*(, B) M = H^*(, ffir(R V )) r2Z M = ffirk[i 1] ( ) (a well-known calculation) r2Z = k[ffi 1, i 1] ( ). The action of b on ffi is from the definition of ffi and e and that on i was fo* *und in the proof of 2.1. For the action on 2 H1(, V ) it is easy to verify that the quotient ma* *p V ! V= rad(V ) ~= kz induces an isomorphism on H1, so H1(, V ) ~= zH1(, k) as* * a - module, and this combines the action on z with that found in calculating the ac* *tion on i in 2.1. Thus ^H*(G, En,*) ~=^H*(, En,*)and the invariants can be calculated usi* *ng lemma 4.1 below. They are generated by fi = in, = ffi-ni-1 and their inverses and ff = * *ffi-1 . THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT* *S IN Ep-15 Finally, notice that c acts on the R-module ^Hr(G, En,s) according to the for* *mula c(`x) = Ø(`)c(x), ` 2 R, x 2 ^Hr(G, En,s). This cohomology group is either k or 0, so t* *he invariants under c are either Fp or 0. Since the generators , fi, ff can be replaced by * *any non-zero element of the ^Hr(G, En,s) that they appear in, we may assume that they are al* *l invariant under c and hence generate the invariants under c. Proof.of 1.1. As before we use Theorem 1.2 to see that ^H*(Sn, En,*)Gal~=H^*(N0* *, En,*). Recall that, for any short exact sequence of profinite groups of finite virtu* *al cohomological dimension I ! J ! K with K torsion-free, there is a spectral sequence H*(K, ^H** *(I, M)) ) H^*(J, M) ([2] X 3 ex. 5). Apply this to C = ^Znpx to obtain H*(^Znp, ^H*(, En,*)) ) ^H*(C, En,*)* *. If we fix both r and s then ^H*(, En,r) is either k or 0 so ^Znp, being a pro-p group, must* * act trivially. Thus the E2-term is isomorphic to ^H*(, En,*) H*(^Znp, Fp) ~=^H*(, En,*) * *(y0, . .,.yn-1). We claim that this spectral sequences collapses, so that H^*(C, En,*) ~= ^H*(* *, En,*) Fp(y0, . .,.yn-1). To see this notice that, from the proof of 3.3, that the m* *ap En,r! En,r=mEn,r~= k induces an injection on H^*(, -). The corresponding spectral* * sequence with coefficients k collapses, by the Künneth Theorem, so ours must too. Now compute the invariants under b using Lemma 4.1. The result is H^*(, E* *n,*) Fp(x0, . .,.xn-1), where the xiare as in 2.1. Finally, c acts only on the first factor, so taking the invariants under c ju* *st replaces H^*(G, En,*) by ^H*(G, En,*)Gal. 4.Invariants The following lemma is elementary, but systematic use of it simplifies the in* *variant calcula- tions above. For example in the proof of 3.3, first calculate k[ffi 1, i 1]=* * (k[i 1] k[ 1]) and then (k[ffi 1, i 1] ( )). Lemma 4.1. Let H be a finite abelian group and let R be a commutative integral * *domain such that |H| is invertible in R and R contains a root of unity of order the ex* *ponent of H. Suppose that A and B are two RH-modules such that A is a graded-commutative R a* *lgebra and the action of H is compatible with this structure. Let H act on A R B diag* *onally. Let C be the set of isomorphism classes of homomorphisms from H to Rx. (This * *can be identified, perhaps not canonically, with the characters of H.) Then there are * *decompositions of RH-modules A = c2CAc and B = c2CBc, where Ac = {a 2 A|ha = c(h)a, h 2 H} a* *nd similarly for B. Let CA = {c 2 C|Ac 6= 0}. Suppose that for each c 2 CA there is a homogeneous element ac 2 Ac that is i* *nvertible in A. Then M (A B)H = AH ad-1 Bd, d2CA M (A B)c= AH ad-1 Bcd. d2CA Suppose that B is also a graded commutative R-algebra and that H acts compatibl* *y with this structure. Then A B is also a graded-commutative R-algebra in the usual way, * *and H acts as a group of automorphisms. THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT* *S IN Ep-16 (1) If, for each c 2 CA \ CB, there is a homogeneous element bc 2 Bc that is* * invertible in B, then (A B)H is a free AH BH -module with basis {ac-1 bc : c 2* * C0}. Furthermore if the monomials in c1, . .,.cr 2 CA \ CB yield all the c * *2 CA \ CB then (A B)H is generated as a ring by AH , BH and the ac-1i bci. (2) If B is generated as an R-algebra by d1, . .,.ds, where di 2 Bcdifor som* *e cdi 2 CA \ CB, then (A B)H is generated as a ring by AH and the ac-1d di. i If the di freely generate B as a graded-commutative R-algebra then the* * ac-1d di * * i freely generate (A B)H over AH . (So if B = R(d1, . .,.ds) then (A * * B)H = AH R R(ac-1d d1, . .,.ac-1 ds).) 1 ds Proof.This is left as an exercise for the reader. Notice that (A B)H = c2C0A* *c-1 Bc and Acac0= Acc0. References [1]Almkvist, G. and Fossum, R., Decomposition of symmetric and exterior powers* * of indecomposable Z=pZ- modules in characteristic p and relations to invariants, in S'eminaire d'Al* *g`ebre Paul Dubreil, Lecture Notes in Math. 641, Springer, Berlin (1978), 1-111. [2]Brown, K.S., Cohomology of Groups, Graduate Texts in Mathematics 87, Spring* *er-Verlag, Berlin 1982. [3]Curtis, C.W. and Reiner, I., Methods of Representation Theory, vol. 1, Wile* *y, New York (1981). [4]Devinatz, E.S., Morava's change of rings theorem, in The ~Cech Centennial, * *Contemp. Math. 181, Amer. Math. Soc., Providence (1995) 83-118. [5]Devinatz, E.S. and Hopkins, M.J., The action of the Morava stabilizer algeb* *ra on the Lubin-Tate moduli space of lifts, Amer. J. Math. 117 (1995) 669-710. [6]Gorbounov, V. and Symonds, P., Toward the homotopy groups of the higher rea* *l K-theory EO2, in Homotopy Theory via Algebraic Geometry and Group Representations, Contemp. * *Math. 220, Amer. Math. Soc., Providence (1998) 103-116. [7]Nave, L.S., The Cohomology of Finite Subgroups of Morava Stabilizer Groups * *and Smith-Toda Com- plexes, Ph.D. thesis, U. Washington (1999). [8]Nave, L.S., On the non-existence of Smith-Toda complexes, at http://hopf.ma* *th.purdue.edu (1998). [9]Reiner, I., Integral representations of cyclic groups of prime order, Proc.* * Amer. Math. Soc. 8 (1957), 142-146. [10]Ribes, L. and Zalesskii, P., Profinite Groups, Springer-Verlag, Berlin 2000. [11]Scheiderer, C., Farrell cohomology and Brown theorems for profinite groups,* * Manuscripta Math. 91 (1996), 247-281. [12]Symonds, P., Permutation resolutions for profinite groups* *, preprint at http://www.ma.umist.ac.uk/pas/preprints. Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, Engla* *nd E-mail address: Peter.Symonds@umist.ac.uk