Homotopy groups of homotopy fixed point spectra associated to En Ethan S. Devinatz* Department of Mathematics, University of Washington, Seattle, Washington, USA Email: devinatz@math.washington.edu URL: http://www.math.washington.edu/~devinatz/ Abstract We compute the mod (p) homotopy groups of the continuous homotopy fixed point spectrum EhH22for p > 2, where En is the Landweber exact spectrum whose coefficient ring is the ring of functions on the Lubin-Tate moduli space * *of lifts of the height n Honda formal group law over Fpn, and Hn is the subgroup W Fxpno Gal(Fpn=Fp) of the extended Morava stabilizer group Gn. We examine some consequences of this related to Brown-Comenetz duality and to finiteness properties of homotopy groups of K(n)*-local spectra. We also indicate a plan for computing ss*(EhHnn^ V (n - 2)), where V (n - 2) is an En*-local Toda complex. AMS Classification numbers Primary: 55Q10, 55T25 Secondary: Keywords: Brown-Peterson homology, Morava stabilizer group, K(n)*-local homotopy theory ____________________________* Partially supported by a grant from the NSF 1 Introduction Let En denote the Landweber exact spectrum with coefficient ring En* = W Fpn[[u1, . .,.un-1]][u, u-1], where W Fpn denotes the ring of Witt vectors with coefficients in the field Fpn of pn elements, and whose BP*-algebra structure map r : BP* ! En* is given by 8 i < uiu1-p i < n r(vi) = u1-pn i = n , : 0 i > n where vi2 BP* is the ith Hazewinkel generator. In particular, each ui has de- gree 0 and u has degree -2. En is a commutative ring spectrum, and Morava theory tells us that the group of ring automorphisms of En is isomorphic to the profinite group Gn = Sn o Gal, where Sn denotes the group of (not necessarily strict) isomorphisms of the height n Honda formal group law over Fpn, and Gal is the Galois group of Fpn=Fp. A priori, Gn acts on En only in the sta- ble category, but Hopkins and Miller (later improved by Goerss and Hopkins) proved that this can be made an honest action in an appropriate point set cat- egory of spectra (see [9] and [13]). "Continuous homotopy fixed point spectra" may also be constructed [7]: if G is a closed subgroup of Gn, the continuous homotopy G fixed point spectrum will be denoted by EhGn; if G is finite, this spectrum agrees with the ordinary homotopy fixed point spectrum. Moreover, EhGnn' LK(n)S0, the K(n)*-localization of S0, EhGnhas the expected functo- rial properties, and there is a strongly convergent "continuous homotopy fixed point spectral sequence" H*c(G, E*nX) ) (EhGn)*X for any spectrum X . (H*c(G, E*nX) denotes the continuous cohomology of G with coefficients in the profinite G-module E*nX .) The hope of this paper is to make some headway towards the computa- tion of ss*EhGn, for G a closed subgroup of Gn. At first sight, this program seems impossible: the formulas for the action of (most elements of) Gn on En* are extremely complicated (see [6]), making the computation of H*c(G, En*) apparently inaccessible. However, H*c(Gn, N) = Ext*Mapc(G,En*)(En*, N), where (En*, Map c(G, En*)) is the complete Hopf algebroid defined using the action of G on En* (see for example [5]). Since Map c(G, En*) is a quotient of Map c(Gn, En*) = En* bBP* BP*BP b BP*En* E^n*En, 2 one may try to use the Hopf algebroid structure maps in BP*BP together with several Bockstein spectral sequences to go from, for example, H*c(G, En*=In) to H*c(G, En*). As usual, In is the maximal ideal (p, u1, . .,.un-1) in En*. Let Hn = W FxpnoGal Gn, where W Fxpnis the subgroup of Sn consisting of the diagonal matricesi(see x1), andjlet M(p) denote the mod (p) Moore spectrum. We compute ss* EhH22^ M(p) for all primes p > 2 (Theorem 3.8). Of course, ss*LK(2)M(p) is known ([14], [15]) for p > 2, so it is unclear if our computation yields any new homotopy information. Our computation is, how- ever, much simpler and already indicates the necessity of "p-adic suspensions" in the Gross-Hopkins work on Brown-Comenetz duality (Remark 3.9). More- over, we believe that computations such as ss* EhHnn^ V (n - 2) _ recall that the Toda complex V (n - 2) exists En*-locally whenever p is sufficiently large compared to n _ should be accessible to more skilled calculators. Even when a complete calculation of ss*EhGnis unattainable, partial infor- mation can lead to interesting consequences. For example, it is a long-standing conjecture that ss*LK(n)S0 is a module of finite type over the p-adic integers Zp. (This conjecture is known to be true for n = 1 and, if p 3, for n = 2 [16], [17].) By a thick subcategory argument _ see [3] for a discussion of this in the En*-local category _ if ss*LK(n)X is of finite type for some X in the En*-local category, then ss*LK(n)Y is of finite type for any finite Y such th* *at {m n : K(m)*Y 6= 0} {m n : K(m)*X 6= 0}. This in turn only requires that we prove that ss*(EhGn^ X) is of finite type for some closed subgroup G of Gn for which there exists a chain G = K0 C K1 C . .C.Kt= Gn of closed subgroups. Indeed, assume inductively that ss*(EhKin^ X) is of finite type. Then, since Ki+1=Ki is a p-adic analytic profinite group (see [8, Theorem 9.6]), we have that H*c Ki+1=Ki, ss*(EhKin^ Y )is also of finite type. (This follows from the fact that any p-adic analytic profinite group is of type p-F P1 in the language of [19].) But, in [7], we constructed a strongly convergent spectral sequence i j H*c Ki+1=Ki, ss*(EhKin^ X) ) ss*(EhKi+1n^ X) and showed that its E1 term has a horizontal vanishing line. This implies that ss*(EhKi+1n^ X) is of finite type and hence by induction so is ss*LK(n)X = ss*(EhGnn^ X). These considerations are unfortunately not applicableito Gj= Hn, since the normalizer of Hn in Gn is Hn, and, moreover, ss* EhH22^ M(p) is not even 3 of finite type. Yet it is, in some sense, "almost" of finite type (see x4), alt* *hough the significance of this property is not clear. 1 H*c(Hn, En*=In) and its Hopf algebroid description Recall that the group Sn may be described in several ways. If n denotes the height n Honda formalPgroup law over Fpn, then Sn consists of all formal power series of the form in 0bixpi with each bi2 Fpn and bi6= 0. The ring of endomorphisms of n may also be described as the ring obtained by adjoining an indeterminate S _ which corresponds to the endomorphism f(x) = xp _ to W Fpn along with the relations Sn = p and Sw = woeS , wherePoe : W Fpn ! W Fpn denotes the frobenius automorphism. The automorphism in 0bixpi P n-1 corresponds to the element i=0 aiSi with X ai= e(bi+nk)pk, k 0 where e(b) is the multiplicative representative of b in W Fpn. The subgroup W Fxpnof Sn is then the group of automorphisms with ai= 0 for all i > 0. In terms of matrices, Sn is the subgroup of GLn(W Fpn) consisting of matrices of the form 2 3 a0 pan-1 pan-2 . . . pa1 66 aoe-11 aoe-10 paoe-1n-1. . .paoe-127 66 .. .. .. 77 66 . . . 777 , 4 ... ... ... paoe-(n-2)n-15 aoe-(n-1)n-1aoe-(n-1)n-2aoe-(n-1)n-3.a.o.e-(n-1)0 and W Fxpnis the subgroup of diagonal matrices in Sn. Now let S0nbe the p-Sylow subgroup of Sn consisting of strict automor- phisms of n. There is a split extension S0n! Sn ! Fxpn; P n-1 __ the map Sn ! Fxpnis given by i=0aiSi 7! a0, and the splitting sends a 2 Fpn to e(a) 2 W Fxpn Sn. This map also gives us a splitting of the short exact sequence 0 ! W F0pn! W Fxpn! Fxpn! 0, and hence an isomorphism W Fxpn! W F0pnx Fxpn. Since the order of Fxpnis prime to p, it follows that x H*c(W Fxpn, N) -! H*c(W F0pn, N)Fpn 4 x whenever N is a discrete Zp [W Fpn] -module. If, in addition, N is a W Fpn- module and Hn-module in such a way that oe(c) = coeoe(n) for all c 2 W Fpn and n 2 N , then it follows from [1, Lemma 5.4] that Hi Gal , H*c(W Fxpn, N)= 0 for all i > 0, and hence H*c(Hn, N) -! H*c(W Fxpn, N)Gal. Now Sn acts on En*=In = Fpn[u, u-1] via Fpn-algebra homomorphisms, and the action on u is given by `n-1 ' P i __ aiS (u) = a0u, (1.1) i=0 where, once again, __a0is the mod (p) reduction of a0. From this it follows that H*c(W Fxpn, Fpn[u, u-1]) = Fpn[vn, v-1n] FpnH*c(W F0pn, Fpn). Moreover, since Gal acts trivially on vn, H*c(Hn, Fpn[u, u-1]) = Fp[vn, v-1n] H*c(W F0pn, Fpn)Gal. It is also easy to compute H*c(W F0pn, Fpn)Gal. Let gi 2 Hom c(W F0pn, Fpn) be defined by _ ! P j pi oei gi 1 + e(cj)p = c1 = c1 , (1.2) j 1 0 i n - 1. Since the Galois automorphisms id, oe, . .,.oen-1 are linearly independent over Fpn, so are the gi's. Now, and for the rest of this section, assume that p > 2. Then Znp W Fpn- ! W F0pn(via the map sending x 2 P pjxj W Fpn to exp(px) = 1 + j 1____j!2 W F0pn), so that H*c(W F0pn, Fpn) is the exterior algebra over Fpn on n generators in H1c(W F0pn, Fpn). This implies that these generators may be taken to be g0, g1, . .,.gn-1. Each gi is Galois invariant, so H*c(Hn, Fpn[u, u-1]) = Fp[vn, v-1n] E(g0, g1, . .,.gn-1).(1.3) Next consider the complete Hopf algebroid En*, Map c(W F0pn, En*) (En*, * *n). We explicitly identify n=In n as a quotient of E^n*En=InE^n*En and give cobar representatives for gi2 H1,0c(W F0pn, Fpn[u, u-1]) = Ext1,0 n=In(nFpn[u, u-1], Fpn[u, u-1]). First recall that the maps jL, jR : En* ! Map c(Gn, En*) are given by jR (x)(s) = x, jL(x)(s) = s-1x. Since W F0pn Gn acts trivially on W Fpn 5 fi fi En*, it follows that jR fifi = jLfifi in n, so that n is a Hopf algebra WFpn WFpn over W Fpn and is a quotient of W Fpn Zp Map c(Sn, En*)Gal= W Fpn Zp (EGaln*bBP*BP*BP b BP*EGaln*) = W Fpn Zp (EGaln)^*EGaln, where EGaln is the Landweber exact spectrum with coefficient ring Zp[[u1, . .,.un-1]][u, u-1]. Now let u jR (u) and w jL(u) in (EGaln)^*EGaln. By 1.1, we have that u = w in n=In n. Moreover, the image of tj 2 BP*BP in (EGaln)^*EGaln_ also denoted tj _ satisfies 0 1 X i i tj@ n bixp A= u1-p b-10bi mod In(EGaln)^*EGaln i 0 (see [1, Proposition 2.11]), and thus n=In n = Fpn[u, u-1][tn, t2n, . .].=Jn, (1.4) with n pn-1 pn p2n-1 pn pjn-1 Jn = (tpn - vn tn, t2n - vn t2n, . .,.tjn - vn tjn, . .).. Finally, let g = v-1ntn 2 n. These considerations imply that gpi 2 n=In n is a cobar representative for gi2 H1(W F0pn, Fpn). 2 The Bockstein spectral sequence Fix a prime p and integer n 2, and let N be a complete (En*, Map (Hn, En*))- comodule. Write x oGal H*N H*c(Hn, N) = H*c(W F0pn, N)Fpn x oGal = Ext * n(En*, N)Fpn . The Bockstein spectral sequence we will use is defined by the exact couple vn-1 H*(En*=(p, u1,P. .,.un-2))oo___ H*(En*=(p,7u1,7. .,.un-2)) (2.1) PPPP nnnnn PPPP nnnn PPP'' nnn H*(En*=In). 6 Truncated, this spectral sequence is isomorphic to the spectral sequence of the unrolled exact couple 022oo____H*(En*=In)_Eoo_____H*(En*=(p,_.<.,.un-2,Pu2n-1))oo___.<.6.6DD 22 yyyy EEEE nnnnnn PPPPPP 22 yy EE nnvn-1n PPPP v2 ssss2yy ""E nnn P(( n-1 H*(En*=In) H*(En*=In) H*(En*=In) Since H*.*(En*=(p, u1, . .,.un-2, ukn-1)) is finite in each bidegree, this spec* *tral sequence converges strongly to H*(En*=(p, . .,.un-2)) = limH*(En*=(p, u1, . .,.un-2, ujn-1)). j 3 Computation of ss*(EhH2 ^ M (p)) In this section, we specialize the above spectral sequence to the case n = 2, p > 2. Write ~ 2= 2=p 2, and let g = v-12t2 2 ~ 2as in x1. We will need the following congruences for our calculation of the different* *ials in the Bockstein spectral sequence. 2 p2 p+1 Lemma 3.1 v1-p2t2 = t2 mod v1 ~ 2. Proof Begin with the formula (see [12, Theorem A2.2.5]) X F i X pi tijR (vj)p = Fvitj i,j 0 i,j 0 in BP*BP=pBP*BP , where F is the universal p-typical formal group law on BP*. Up through power series degree p4 we then have 2 p3 v1 +F vp1t1 +F vp1t2 +F v1 t3 +F jR (v3) +F t1jR (v2)p +F t2jR (v2)p2 2 p2 (3.2) = v1 +F v1tp1+F v1tp2+F v1tp3+F v2 +F v2tp1+F v2t2 in E^2*E2=pE^2*E2. But jR (v2) = v2 + v1tp1- vp1t1 in BP*BP=pBP*BP , and therefore, since t1 2 v1~ 2, jR (v2) = v2 mod vp+11~ 2. t3 is also in v1~ 2; h* *ence, mod vp+11~ 2, equality reduces to 2 p p2 vp2t1 +F vp2t2 = v1t2 +F v2t2 . The desired result follows immediately from this equation. __|_| Lemma 3.3 t1 = v1gp - vp+21v-12g + vp+21v-12gp mod v2p+31~ 2. 7 Proof From [12, Corollary 4.3.21], 2 p p p2 p 1+p2 p2 1+p p p jR (v3) = v3+v2tp1+v1t2-v2t1-v1 t2-v1t1 +v1 t1 +v1w1(v2, v1t1, -v1t1) in BP*BP=pBP*BP , where w1(x, y, z) 1_p[xp+ yp+ zp - (x + y + z)p]. Hence 0 = v1tp2- vp2t1 - v21vp-12tp1+ vp+11vp-12t1 mod v2p+31~,2(3.4) and thus t1 = v-p2v1tp2 mod vp+21~ 2. Plug this relation for t1 back into the last two terms of 3.4 to get 2+p-1 p2 p+2 -1 p 2p+3 vp2t1 = v1tp2- vp+21v-p2 t2 + v1 v2 t2 mod v1 ~ 2. By the previous lemma, 2+p-1 p2 p+2 p-2 2p+3 vp+21v-p2 t2 = v1 v2 t2 mod v1 ~ 2. We then get the desired result. __|_| The next propositions will allow us to compute the Bockstein differentials * *on vk22 H0(Fp2[u, u-1]). In what follows, we will often suppress left multiplicati* *on by powers of v2 on one side of an equation, since the appropriate power can always be determined by examining gradings. For example, we might write the conclusion of the previous lemma as t1 = v1gp - vp+21g + vp+21gp mod v2p+31~.2 Proposition 3.5 In ~ 2=v3p+31~ 2, 2 p 2(1+p) p s - 1 2(1+p) p2 p 2 jR (v2)s - vs2= s[v1+p1(gp - g ) + v1 (g - g ) + _____v1 (g - g ) ]. 2 Proof Compute in ~ 2=v3p+31~:2 jR (v2)s - vs2= vs2[(v-12jR (v2))s - 1] = vs2[(1 + v-12v1tp1- v-12vp1t1)s - 1] s - 1 p p 2 = s[(v1tp1- vp1t1) + _____(v1t1 - v1t1) ] 2 2 p+1 p 2(p+1) p s - 1 2(p+1) p2 p 2 = s[vp+11gp - v1 g + v1 (g - g ) + _____v1 (g - g ) ] 2 by the previous lemma. __|_| 8 Proposition 3.6 Suppose that p 6 |s and k 0. Then there exists zspk2 E2* k such that zspk= vsp2 mod I2 and k+pk-1+...+1)(p+1)p (pk+pk-1+...+p+2)(p+1) dzspk jR (zspk)-zspk= cv(p1 (g -g) mod v ~ 2 for some c 2 Fxp. Proof Proceed by induction on k. If k = 0, let zs = vs2. Then by the preceding proposition, 2 p 2(p+1) jR (zs) - zs = svp+11(gp - g ) mod v1 ~ 2. But gp2 = g mod vp+11~ 2, so we get the desired result. Suppose now that zspk-1has been chosen. Then k+pk-1+...+p)(p+1)p2 p (pk+pk-1+...+p2+2p)(p+1) d(zspk-1)p = cv(p1 (g - g ) mod v1 ~ 2. k-pk-1-...-p+1 Next consider dv(s-1)p2 . Since the exponent of v2 is equal to 1 mod (p), we have that k-pk-1-...-p+1 1+p p2 p 2(1+p) p 3p+3 dv(s-1)p2 = v1 (g - g ) + v1 (g - g ) mod v1 ~ 2. Then take k+pk-1+...+p-1)(p+1)(s-1)pk-pk-1-...-p+1 zspk= (zspk-1)p - cv(p1 v2 . __|_| Corollary 3.7 Suppose that p 6 |s and k 0. Up to multiplication by a unit in Fp, k (s-1)pk-pk-1-...-p-1p d(pk+pk-1+...+1)(p+1)vsp2= v2 (g - g) in the Bockstein spectral sequence 2.1. In particular, vt2(gp - g) is a boundary for all t 2 Z. Now, essentially by their definition, g and gp are permanent cycles; more- over, H*(Fp2[u, u-1]) = Fp[v2, v-12] E(g + gp, g - gp). Using the preceding corollary, it's easy to read off the remaining differential* *s, yielding our main result. Theorem 3.8 If p 3 8 >>>:Q Fp[v1] 1 t2Z _____(vnt1){cti} i = 2 9 as Fp[v1]-modules, where i = g+gp, ct reduces to vt2(g-gp) 2 H1(Fp2[u, u-1]), and ae nt= p(+p1i+ pi-1+ . .+.1)(p + 1) tt6==-1(smod- (p)1)pi- pi-1- . .-.p - 1, s* * 6= 0 mod (p). By sparseness, sst-s(EhH22^ M(p)) Hs,t(Fp2[[u1]][u, u-1]). Remark 3.9 Let In denote the Brown-Comenetz dual of LnS0, the En*- localization of S0. In is characterized by ss0F (X, In) = [X, In]0 = Hom (ss0LnX, Q=Z(p)) for any spectrum X . In [10] (see also [18]), Gross and Hopkins establish a remarkable relationship between Brown-Comenetz and Spanier-Whitehead du- ality: they prove that if p is sufficiently large compared to n 2, and if X is a K(n - 1)*-acyclic finite complex with pEn*X = 0 and with vn self-map 2pN (pn-1)X ! X , then F (X, In) ' ffLnDX, (3.10) where ff is any integer with ff = 2pnN (pn - 1=p - 1) + n2 - n mod (2pN (pn - 1)). (As usual, DX denotes the Spanier-Whitehead dual of X .) There is, however, no integer ff for which 3.10 is satisfied for all X . This contrasts with the situation when n = 1: here we have I1 ' 2L1(S0p) (if p > 2), where S0p denotes S0 completed at p, and thus F (X, I1) ' 2L1DX whenever X is a rationally acyclic finite spectrum. Historically, it was Shimomura's calculation [14] of ss*L2M which shattered the hope that I2 might also be an integral suspension of L2(S0p). Our calculati* *on of ss*(EhH22^ M(p)) yields this result as well; a sketch of the proof follows. Suppose there existed an integer c with F (M(p, vk1), I2) ' cL2DM(p, vk1) (3.11) for a cofinal set of k, where M(p, vk1) denotes a finite spectrum with BP*M(p, * *vk1) = BP*=(p, vk1). In addition, we may assume that DM(p, vk1) ' -2k(p-1)-2M(p, vk1). Let E2*M(p, vk1)~ = Hom (E2*M(p, vk1), Q=Z(p)), and recall that 4(E2*M(p, vk1)~ ) E2*F (M(p, vk1), I2) 10 as modules over E2* and G2. (See [18, Proposition 17] or [2] for p 5; note, however, that we are using Strickland's definition of E2*M(p, vk1)~ .) Then 3.11 implies that E2*M(p, vk1)~ c-2k(p-1)-6E2*M(p, vk1), and, by the theory of Poincar'e pro-p groups (cf. [6, Sections 5, 6]), there is* * a map H2,6+2k(p-1)-c(E2*M(p, vk1)) ! Q=Z(p) such that Hi(E2*M(p, vk1)) H2-i(E2*M(p, vk1)) ! H2(E2*M(p, vk1)) ! Q=Z(p) is a perfect pairing. Hence there must exist, for each k, an element dk in H2,6+2(p-1)-c(E2*M(p, vk1)) such that vk-11dk 6= 0. But the computation of H2(Fp2[[u1]][u, u-1]) together with the exact sequence vk1 2 -1 2 k H2(Fp2[[u1]][u, u-1] -! H (Fp2[[u1]][u, u ]) ! H (E2*M(p, v1)) ! 0 shows that this is impossible. 4 Some remarks on finiteness In this section, we work in the En*-local stable category, so that by a finite spectrum, we mean an object of the thick subcategory generated by LnS0. Let G be a closed subgroup of Gn, n 1. Proposition 4.1 Let Y be a K(n-1)*-acyclic finite spectrum. Then ss*(EhGn^ Y ) is of finite type (as a graded abelian group). Proof The proof is just as we argued in the Introduction: use the strongly convergent spectral sequence H**c(G, En*Y ) ) ss*(EhGn^ Y ) whose E1 term has a horizontal vanishing line. Since En*Y is of finite type, so is H**c(G, En*Y ). The horizontal vanishing line then implies that ss*(EhGn^ Y ) is also of finite type. __|_| Now suppose n 2 and X is a K(n-2)*-acyclic finite spectrum with vn-1 self-map . Let X( k) denote the cofiber of k : k| |X ! X , and let X( 1 ) denote the cofiber of X ! -1X , so that X( 1 ) = holim -k| |X( k). There !k are also canonical maps X( k) ! X( k-1) and X ! holimX( k). We will k need the following well-known result (cf. [11, Section 2]). 11 Lemma 4.2 If Z is any (En*-local) spectrum, the map Z ^ X ! holimZ ^ k X( k) is the K(n)*-localization of Z ^ X . Proposition 4.3 -1ss*(EhGn^ X) is countable if and only if ss*(EhGn^ X) is of finite type. Proof Proposition 4.1 implies that ss*(EhGn^ X( 1 )) is countable, and there- fore -1ss*(EhGn^ X) is countable if and only if ss*(EhGn^ X) is countable. But EhGn^ X ' holimEhGn^ X( k); it therefore again follows from Proposition 4.1 k that ssi(EhGn^ X) is profinite and is thus countable if and only if it's finite* *. __|_| Remark 4.4 The chromatic splitting conjecture (see [11]) actually identifies -1(EhGnn^ X) = -1LK(n)X as Ln-1X _ -1Ln-1X . Although ss*(EhH22^ M(p)) is not of finite type, this proposition suggests to us the sense in which it is "almost" of finite type. The details follow. We will consider graded modules over the graded ring Fp[ ], where has positive even (unless p = 2) degree, satisfying the following two conditions: i. M is complete in the sense that M = limM= iM . i ii. M= M is an Fp vector space of finite type. Proposition 4.5 Let X and be as above and suppose that p : X ! X is trivial. Then ss*(EhGn^ X) is an Fp[ ]-module satisfying conditions i and ii. Proof Since _ss*(EhGn^_X)_ hG k ,! ss*(En ^ X( )), kss*(EhGn^ X) we have the requisite finiteness. Moreover, it follows from the commutative diagram hG ^ X( k)) ss*(EhGn^ X) _____//_limkss*(En | n77n | nnnn | nnnn fflffl|)nn hG^X) lim _ss*(En____ kss*(EhG^X) k n that ss*(EhGn^ X) is complete. __|_| 12 In [20, Proposition 4.10], Torii shows that such a module M may be written as Y Y M nffFp[ ] x mfiFp[ ]=( ifi). (4.6) ff fi If M is of finite type, then the nff's are bounded below and Y M -1M -1 nffFp[ ] = nffFp[ , -1]. ff ff Q In general, the torsion submoduleQT of M is a submodule of fi mfiFp[ ]=( ifi); its closure T~is equal to fi mfiFp[ ]=( ifi). Let us say that M is essentially of finite rank if there are only a finite number of ff in the decomposition 4.6; that is, if and only if M=T~ is a finitely generated Fp[ ]-module. Our main theorem shows that ss*(EhH22^ M(p)) is essentially of finite rank for p > 2. 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