GROSS-HOPKINS DUALITY N. P. STRICKLAND In [8] Hopkins and Gross state a theorem revealing a profound relationship be* *tween two different kinds of duality in stable homotopy theory. A proof of a related but weaker res* *ult is given in [3], and we understand that Sadofsky is preparing a proof that works in general. Her* *e we present a proof that seems rather different and complementary to Sadofsky's. We thank I-C* *hiau Huang for help with Proposition 18, and John Greenlees for helpful discussions. We first indicate the context of the Hopkins-Gross theorem. Cohomological dua* *lity theorems have been studied in a number of contexts; they typically say that Hk(X*) = Hd-k(X)_ for some class of objects X with some notion of duality X $ X* and some type of* * cohomology groups Hk(X) with some notion of duality A $ A_ and some integer d. For example* *, if M is a compact smooth oriented manifold of dimension d we have a Poincare duality isom* *orphism Hk(M; Q) = Hom (Hd-k(M; Q); Q) (so here we just have M* = M). For another example, let S be a smooth complex * *projective variety of dimension d, and let d be the sheaf of top-dimensional differential * *forms. Then for any coherent sheaf F on S we have a Serre duality isomorphism Hk(S; Hom(F; d)) = Hom (Hd-k(S; F ); C): This can be seen as a special case of the Grothendieck duality theorem for a pr* *oper morphism [7], which is formulated in terms of functors between derived categories. There is a* * well-known analogy between Boardman's stable homotopy category of spectra and the derived category* * of sheaves over a scheme, and, inspired by this, Neeman has used tools from homotopy theory to * *prove the main facts about Grothendieck duality [20]. This leads one to expect that there shou* *ld be a kind of duality theorem in the stable homotopy category itself. However, experience sug* *gests that such theorems require finiteness conditions which are not satisfied in the category * *of spectra. The subcategory of K(n)-local spectra [21, 12] has much better finiteness propertie* *s and thus seems a better place to look for duality phenomena. The Hopkins-Gross theorem is a ki* *nd of analog of Serre duality in the K(n)-local stable homotopy category. We next explain the result in question in more detail; our formulation will b* *e compared with various other possible formulations in Remark 21. Fix a prime p and an integer * *n > 0 and let Ln be the Bousfield localisation functor [21] with respect to E(n). Let MS be the * *fibre of the natural map LnS -!Ln-1S, and let bI= IMS be its Brown-Comenetz dual, which is character* *ised by the existence of a natural isomorphism [X; bI] = Hom (ss0(MX); Q=Z) = Hom (ss0(MS ^ X); Q=Z): This lies in the K(n)-local category K, which is a symmetric monoidal category * *with unit bS= LK(n)S and smash product Xb^Y = LK(n)(X ^ Y ). The formal properties of the ca* *tegory K are studied in detail in [12]. If X 2 K we write bIX = F (X; bI) and DX = F (X;* * bS); these are the natural analogues of the Brown-Comenetz dual and the Spanier-Whitehead dual* * of X in the K(n)-local context. The following result is contained in [8, Theorem 6]; it is * *convenient for us to state it separately. Proposition 1 (Hopkins-Gross).The spectrum bIis invertible: there exists a spec* *trum bI-12 K with bIb^bI-1= bS. It follows that bIX = F (X; bI) = bIb^DX for all X 2 K. ___________ Date: November 17, 2000. 1 2 N. P. STRICKLAND Proof.The original Hopkins-Gross proof has never appeared, but a proof of the f* *irst claim was given in [12, Theorem 10.2(e)]. The second follows purely formally from the fi* *rst: the functor bIb^(-): K -!K is an equivalence, so we have [Y; bIX] = [Xb^Y; bI] = [bI-1b^Xb^Y; S] = [bI-1b^Y; DX] = [Y; bIb^D* *X]: Yoneda's lemma completes the argument. * * |___| Our next problem is to study these dualities using a suitable cohomology theo* *ry, which we call Morava E-theory. It is a 2-periodic Landweber exact theory with coefficient ring E* = W [[u1; : :;:un-1]][u; u-1] where |uk| = 0 and |u| = 2 and W is the Witt ring of the finite field Fq= Fpn. * *We make this a BP*-algebra by the map sending vk to upk-1uk, where u0 = p and un = 1 and uk = * *0 for k > n. This gives rise to a Landweber exact formal group law and thus a multiplicative* * cohomology theory. We write E for the representing spectrum, which is K(n)-local, and is a* * wedge of finitely many suspended copies of the spectrum [E(n). If X 2 K then we put E_tX = sst(E* *b^X); this is the natural version of E-homology to use in a K(n)-local context. Our goal * *is to describe E_*bI. The proposition above is actually equivalent (by [9, Theorem 1.3]) to th* *e statement that E_*bI' E* as E*-modules (up to suspension). However, this is not sufficient for* * the applications; we need pin down the action of the group of multiplicative automorphisms of E * *(a version of the Morava stabiliser group) as well as the E*-module structure. This provides the * *input for various Adams-type spectral sequences. To explain the answer, we recall that there is a* * canonical map det: -! Zxpcalled the reduced determinant. If M is an E0-module with compatible* * action of (in short, an E0--module) then we write M[det] for the same E0-module with the* * -action twisted by det. Theorem 2 (Hopkins-Gross).There is a natural -equivariant isomorphism E_tbI= En-n2+t[det]: The proof of this result falls naturally into two parts. One part is the alg* *ebraic analysis of equivariant vector bundles on the Lubin-Tate deformation space, which is outlin* *ed by Hopkins and Gross in [8], with full details provided by the same authors in [6]. The o* *ther part is a topological argument to make contact with Morava E-theory. The theorem as state* *d implies a description of E_tbI(Z) when Z is a finite complex of type n, and Devinatz [3] * *has given a proof (following a sketch by Hopkins) that this description is valid when p > (n2+ n * *+ 2)=2. The rest of the paper will constitute our proof of the full theorem. We first need some information about the the structure of the group . We wil* *l assume that the reader is familiar with the general idea of the relationship between d* *ivision algebras, automorphisms of formal groups and cohomology operations. We therefore give jus* *t enough detail to pin down our group among the various possible versions of the Morava stabil* *iser group. The original reference for most of these ideas is [19], and [2] is a good source fo* *r many technical points. Definition 3.Let OE be the unique automorphism of W such that OE(a) = ap (mod p* *) for all a 2 W ; note that OEn = 1. Let D be the noncommutative ring obtained from W by * *adjoining an element s satisfying sa = OE(a)s (for a 2 W ) and sn = p; thus Q D is the cent* *ral division algebra over Qpof rank n2 and invariant 1=n. There is a unique way to extend OE to an a* *utomorphism of D with OE(s) = s, and we still have OEn = 1. This gives an action of the cyclic g* *roup C = on D and thus allows us to form the semidirect product Dx n C. Proposition 4.There is a natural isomorphism ' Dx n C. Proof.Write 0 = Dx n C. Let F0 be the unique p-typical formal group law over F* *q with [p](x) = xq, and let G0 be the formal group scheme over S0 := spec(Fq) associat* *ed to F0. If ! 2 Fq = W=p then there is a unique lift ^!2 W such that ^!q= ^!, and there is * *also an endomorphism (!) of G0 given by x 7! !x. We also have an endomorphism oe of G0 * *given by x 7! xp. It is well-known that there is a unique ring map :D -! End(G0) such t* *hat (s) = oe GROSS-HOPKINS DUALITY 3 and (^!) = (!) for all ! 2 Fq, and moreover that is an isomorphism. On the oth* *er hand, we can let C act on S0 via the Frobenius automorphism OE: S0 -!S0. As the coeffici* *ents of F0 lie in Fp, there is a natural identification (OEk)*G0 = G0. Using this, it is easy to * *identify 0with the group of pairs (ff0; fi0), where fi0:S0 -!S0 is an isomorphism of schemes and f* *f0:G0 -!fi*0G0 is an isomorphism of formal groups over S0. Now put S = spf(E0) and G = spf(E0CP 1) so that G is a formal group over S an* *d is the universal deformation of G0 in the sense of Lubin and Tate [17] (see also [24, * *Section 6] for an account in the present language.) Let 00denote the group of pairs (ff; fi) wher* *e fi :S -! S and ff: G -!fi*G. As S0 is the subscheme of S defined by the unique maximal ideal i* *n E0 = OS we see that fi(S0) = S0. As G0 = S0xS G, we also see that ff(G0) = G0, so we get a* * homomorphism 00-!0sending (ff; fi) to (ff|S0; fi|G0); deformation theory tells us that this * *is an isomorphism. The general theory of Landweber-exact ring spectra [14] gives an isomorphism 00* *-!Aut(E) = ;__ see [24, Section 8.7] for an account in the present language. * * |__| For the next proposition, we note that the topological ring Zp[[]] has both a* * left and a right action of , and these actions are continuous and commute with each other. We ca* *n use the left action to define continuous cohomology groups H*(; Zp[[]]), and the right actio* *n gives an action of on these cohomology groups. Proposition 5.We have Hn2(; Zp[[]]) = Zp, and the natural action of on this mo* *dule is trivial. Moreover, we have Hk(; Zp[[]]) = 0 for k 6= n2. Proof.First, we note that is a p-adic analytic group of dimension n2over Zp. D* *uality phenomena in the cohomology of profinite groups have been studied for a long time [23, 15* *], but the more recent paper [25] is the most convenient reference for the particular points th* *at we need. Write U = 1 + pD < Dx < , which is a torsion-free open subgroup of finite index in . * *It follows from [25, Corollary 5.1.6] that U is a Poincare duality group of dimension n2 i* *n the sense used in that paper, which means precisely that U has cohomological dimension n2 and Hn2* *(U; Zp[[U]]) = Zpand the other cohomology groups are trival. As U has finite index in , Shapir* *o's lemma gives an isomorphism H*(; Zp[[]]) = H*(U; Zp[[U]]); this proves the proposition excep* *t for the fact that acts trivially. There is a unique way to let act on D such that the subg* *roup Dx acts by conjugation and the subgroup C acts via OE. On the other hand, acts on U by co* *njugation and thus on the Qp-Lie algebra L of U. There is an evident -equivariant isomorphism* * L = Q D. Using the results of [25, Section 5] we get a -equivariant isomorphism 2 n2 n2 n2 Q Hn (; Zp[[]]) = Q H (U; Zp[[U]]) = L = Q D: (This is just a tiny extension of an argument of Lazard, which could be applied* * directly if were torsion-free.) We write addet(fl) for the determinant of the action of fl 2 on* * D; it is now enough to check that addet= 1. Suppose a 2 Dx , so a acts on D by x 7! axa-1.2Let K be the subfield of Q D * *generated over Qp by a. Put d = dimQpK, so that Q D ' Kn =das left K-modules. Using this* *, we see that the determinant of left multiplication by a is just NK=Qp(a)n2=d. By a sim* *ilar argument, the determinant of right multiplication by a-1 is NK=Qp(a)-n2=d. It follows that th* *e conjugation map has determinant one, so that addet(Dx ) = 1. Moreover, the action of OE 2 C on * *Q D is the same_ as the action of s 2 Q D by conjugation, which has determinant one by the same* * argument. |__| We next discuss some useful generalities about E-module spectra. As in [12, A* *ppendix A], we say that an E0-module M is pro-free if it is the completion at In of a free mod* *ule. Lemma 6. Let M be a K(n)-local E-module spectrum. Then the following are equiva* *lent: (a)ss1M = 0 and ss0M is pro-free as an E0-module. (b)M is a coproduct in K of copies of E. (c)M is a retract of a product of copies of E. Moreover, if M is the category of E-modules for which these conditions hold and* * M0 is the category of pro-free E-modules then the functor ss0:M -!M0 is an equivalence. 4 N. P. STRICKLAND W Proof.Suppose (b) holds, so M = ffE say.WHere the coproduct is the K(n)-locali* *sation of the ordinary wedge, which means that M = lim -ffE=J, where J runs over a suitable f* *amily of open J ideals in E0. In this second expression, itWmakes noLdifference whether the wed* *ge is taken in K or in the categoryLof all spectra, so ss* ffE=J = ffE*=J, and it follows eas* *ily that ss0M is the completion of ffE0 and ss1M = 0. Thus (b))(a). Conversely,Wif (a) holds, choose a topological basisW{eff} for ss0M and use i* *tLto construct a map f : ffE -! M of E-modules in the usual way. Then ss0 ffE is the completion of* * ffE0 and so ss*(f) is an isomorphism, so (b) holds. UsingQthis we find that for any K(n)* *-local E-module N, the group of E-module maps M -! N is just ffss0N = Hom E0(ss0M; ss0N). If * *we write M for the category of E-modules satisfying (a) and (b), it is now clear that ss0:* *M -! M0 is an equivalence. We know from [12, Appendix A] that M0 is closed under arbitrary products and * *retracts, and that any pro-free E0-module is a retract of a product of copies of E0. Given th* *is,_we can easily deduce that (a),(c). * *|__| Definition 7.We say that an E-module spectrum M is pro-free if it satisfies the* * conditions of the lemma. Corollary 8.If M is pro-free and {Xff} is the diagram of small spectra over X t* *hen [X; M] = lim[-Xff; M]. ff Proof.If M = E then this holds by a well-known compactness argument based on th* *e fact that E0Y is finite for all small Y . As any M can be written as a retract of a produ* *ct of_copies_of E, the claim follows in general. * * |__| Lemma 9. Let M and N be pro-free E-modules, and use the E-module structure on M* * to make M^bN an E-module. Then M^bN is pro-free. Proof.It is easy to reduce to the case M = N = E. By the argument of [12, Prop* *osition 8.4(f)] it suffices to check that (E=In)*E is concentrated in even degrees. We * *know by standard calculations that (E=In)*BP = E*[tk | k > 0]=In (with |tk| = 2(pk - 1)) and tha* *t (E=In)*E_= (E=In)*BP BP* E* by Landweber exactness, and the claim follows. * * |__| Lemma 10. Let C(; E0) be the ring of continuous functions from (with its profi* *nite topology) to E0 (with its In-adic topology). Then C(; E0) is pro-free. Proof.Write bFq= {a 2 W | aq = a}; the reduction map bFq-! Fq isPwell-known to * *be an isomorphism. For any d 2 Dx there is a unique way to write d = k0 ok(d)sk wit* *h ok(d) 2 bFq and o0(d) 6= 0. This defines functions ok:Dx -! E0, and ok is constant on the c* *osets of 1 + sk+1D and thus is locally constant. Lagrange interpolation shows that the evident map* * from Fq[o]=(oq-o) to the ring of all functions Fq-! Fq is surjective, and thus an isomorphism by * *dimension count. Our maps oigive a bijection (o0; : :;:ok): Dx =(1 + sk+1D) -!Fxqx Fkq: Putting these facts together, we see that the ring of functions from Dx =(1 + s* *k+1D) to E0 is generated over E0 by the functions o0; : :;:ok subject only to the relations oq* *i= oiand oq-10= 1. The direct limit of these rings as k tends to 1 is the ring of all locally cons* *tant functions from to E0, which is thus isomorphic to E0[ok | k 0]=(oq-10- 1; oqk- ok): Now let ei:C -! E0 be the characteristic function of {OEi} (for i = 0; : :;:n -* * 1), so the ring of functions from C to E0 is just E[e0; : :;:en-1]=(eiej- ffiijei). Recall that = Dx n C; this can be identified with Dx x C as a set, so the ri* *ng of locally constant functions from to E0 is just the tensor product of the rings for Dx a* *nd C, which we now see is a free E0-module. The ring of all continuous functions is the comple* *tion of the_ring of locally constant functions, and thus is pro-free. * * |__| GROSS-HOPKINS DUALITY 5 We next want to identify E_0E with C(; E0). This is in some sense well-known,* * but the details are difficult to extract from the literature in a convenient form, so at the su* *ggestion of the referee we indicate a proof. It will first be helpful to have a more coherent view of the topologies on ou* *r various algebraic objects. Recall that a K(n)-local spectrum W is small if it is a retract of the* * K(n)-localisation of a finite spectrum of type n. For any X; Y 2 K we define a topology on [X; Y* * ] whose basic neighbourhoods of 0 are the kernels of maps u*:[X; Y ] -! [W; Y ] such that W i* *s small and u: W -! X. We call this the "natural topology"; its formal properties are deve* *loped in [12, Section 11]. It is shown there that the natural topology on E0 is the same as t* *he In-adic topology, which is easily seen to be the same as the profinite topology. Lemma 11. The natural topology on [E; E] is also the same as the profinite to* *pology. Proof.Consider a map u: W -! E and the resulting map u*: -! E0W , which can als* *o be thought of (using the evident action of on E0W = [W; E]) as the map u 7! fl-1:* *u. The set E0W is finite by an easy thick subcategory argument [12, Theorem 8.5] so the st* *abiliser of u has finite index in . As the subgroup 1 + pD < is a finitely topologically ge* *nerated pro-p group of finite index in , we see from [4, Theorem 1.17] that every finite inde* *x subgroup of is open. It now follows easily that the map u*: -![W; E] is continuous when [W; E]* * is given the discrete topology. It follows in turn that if we give the profinite topology a* *nd E0E the natural topology then the inclusion map is continuous. We also know from [12, Proposit* *ion 11.5] that E0E is Hausdorff. A continuous bijection from a compact space to a Hausdorff sp* *ace_is always a homeomorphism, and the lemma follows. * * |__| Theorem 12. There is a natural isomorphism E_0E = C(; E0). Proof.Define a map OE: x E_0E -!E0 by b^fl mult (fl; a) = (S a-!Eb^E 1--!Eb^E ---! E): Using [12, Propositions 11.1 and 11.3] we see that is continuous, so we have * *an adjoint map # :E_0E -! C(; E0). We claim that this is an isomorphism. As both source and t* *arget are pro-free, it suffices to show that the reduction of # modulo In is an isomorph* *ism. Let K be the representing spectrum for the functor X 7! E*=InK(n)*K(n)*X; this is a wedeg of* * finitely many suspended copies of K(n), and it can be made into an E-algebra spectrum with K** * = E*=In. It is not hard to see that (E_0E)=In = (E=In)0E = K0E; and C(; E0)=In = C(; E0=In) = C(; Fq). To analyse K0E, let x be the standard p* *-typical coordinate on G and let F be the resulting formal group law over E0 _ see [22, * *Appendix 2] for details and useful formulae. A standard Landweber exactness argument shows * *that K0E is the universal example of a ring R equipped with maps Fq ff-!R -fi E0 and an iso* *morphism f :fi*F -! ff*F of formal group laws. As ff*F has height n we see that the same* * must be true of fi*F , so fi(In) = 0, so we can regard fi as a map Fq= E0=In -!R. The coefficie* *nts of F modulo In actually lie in Fp Fqand there is only one map Fp-! R so ff*F = fi*F ; we ju* *st write F for this formalPgroupklaw. Using the standard form for isomorphisms of p-typical FG* *L's we can write f(x) = Fk0tkxp , where t0 is invertible because f is an isomorphism. (Readers* * may be more familiar with the graded case where one gets strict isomorphisms with t0 = 1, b* *ut we are working with the degree zero part of two-periodicPtheoriesPand t0 need not be 1 in this* * context.) As f commutes with [p]F(x) = xq we have Fktkxpn+k= Fktqkxpn+kand thus tqk= tk. I* *n fact, this condition is sufficient for f to be a homomorphism of FGL's (see [22, Appendix * *2], for example) and we deduce that R = (Fq Fq)[tk | k 0]=(tq-10- 1; tqk- tk) = C(Dx ; Fq Fq): We next claim that Fq Fqcan be identified with the ring F (C; Fq) of functions * *from the Galois group C = to Fq. Indeed, we can define a map O: Fq Fq -! F (C; * *Fq) by O(a b)(oe) = aoe(b). The Fq-linear dual of this is the evident map Fq[C] -! En* *dFp(Fq) which 6 N. P. STRICKLAND is injective by Dedekind's lemma on the independence of automorphisms, and this* * bijective by dimension count. Thus O is an isomorphism, and we obtain an isomorphism R = C(D* *x n C; Fq). After some comparison of definitions we see that this is the same as the map #* *_:K0E_-!C(; Fq), as required. |* *__| Definition 13.We write Jk = E(k+1)= Eb^: :b:^E (with (k + 1) factors). We can u* *se the ring structure on E to assemble these objects into a cosimplicial spectrum and thus * *a cochain complex of spectra. We write C (k+1; E0X) for the set of continuous -equivariant maps k* *+1 -!E0X (where everything is topologised as in [12, Section 11]). Lemma 14. There is a natural isomorphism [X; Jk] = C (k+1; E0X), which respects* * the evident cosimplicial structures. Proof.Given fl_= (fl0; : :;:flk) 2 k+1, we define (fl_) = (E(k+1)fl0^:::^flk------!E(k+1)mult---!E): We then define OEX :[X; Jk] -! Map(k+1; E0X) by OEX (a)(fl_) = (fl_) O a: X -! * *E0. Note that if ffi 2 then (ffifl_) = ffi O (fl_): Jk -! E; this means that the map OEX (a)* *: k+1 -! E0X is -equivariant. The results of [12, Section 11] show that OEX (a) is continuous, * *so we can regard OEX as a map [X; Jk] -!C (k+1; E0X). Now let X be the dual of a generalised Moore spectrum [10, 12] of type S=I fo* *r some ideal I = (ua00; : :;:uan-1n-1) so that E0X = E0=I. We know from Theorem 12 and Lemm* *a 10 that (E=I)0E = C(; E0=I) and that this is a free module over E0=I so that (E=I)0E(k)= C(; E0=I)k = C(k; E0=I) = C (k+1; E0=I): After some comparison of definitions, we find that OEX is an isomorphism when X* * = D(S=I). Moreover, the construction M 7! C(k; M) gives an exact functor from finite disc* *rete Abelian groups to Abelian groups, so the construction X 7! C(k; E0X) gives a cohomology* * theory on the category of small K(n)-local spectra. By a thick subcategory argument, we d* *educe that OEX is an isomorphism when X is small. Now let X be a general K(n)-local spectrum, and* * let {Xff} be the diagram of small spectra over X. As Jk is a pro-free E-module we see from C* *orollary 8 that [X; Jk] = lim[-Xff; Jk]. We also see that C (k+1; E0X) = lim C (k+1; E0Xff), an* *d it follows ff -ff * * __ that OEX is an isomorphism as claimed. * * |__| Corollary 15.There is a strongly convergent spectral sequence Est1= [X; Js]t= C (s+1; EtX) =) [X; bS]t+s: Proof.The axiomatic treatment of Adams resolutions discussed in [18] can be tra* *nsferred to many other triangulated categories; this will certainly work for unital stable homot* *opy categories in the sense of [11], and thus for K. The spectra Js clearly form an E-Adams_resoluti* *on of bS, so we have a spectral sequence whose E1 page is as described. Let i: E -!Sbbe the fib* *re of the unit map j :bS-!E; it_is known that our spectral sequence is associated to the filtr* *ation of bSby the spectra Ys_:=_E(s). The map jb^1: E -! Eb^E is split by the product map : E ^ E* * -! E, so ib^1 = 0: Eb^E -! E. It follows that if Z lies in the thick subcategory generat* *ed by E then the map i(s)b^1: Ysb^Z -! Z is zero for s 0. However, we know from [12, Theorem 8* *.9] that bS lies in this thick subcategory, so i(s)= 0 for s 0. Using this and the definit* *ion of our spectral_ sequence, we see that it converges strongly to [X; bS]*. * * |__| Proposition 16.We have [E; Sn2-t] = Et as E0-modules, and thus DE = -n2E as E-m* *odule spectra. Moreover, this equivalence respects the evident actions of . Proof.We use the spectral sequence of Corollary 15. To analyse the E1 term, we* * claim that E0E = Zp[[]]b ZpE0 as left -modules, where we let act in the obvious way on Zp* *[[]] and GROSS-HOPKINS DUALITY 7 trivially on E0. To see this, define maps OE: E0 Zp[] -!E0E and :Zp[] E0 -!E* *0E by OE(a [fl])= (E fl-!E xa--!E) ([fl] a)= (E xa--!E fl-!E): Clearly fl0:OE(a [fl]) = OE(fl0(a) [fl0fl]) and fl0: ([fl] a) = ([fl0fl] a* *) and ([fl] a) = OE(fl(a) [fl]) (because fl is a ring map). By dualising Theorem 12 we see that OE exten* *ds to give an isomorphism E0bZpZp[[]] -!E0E, and it follows from the above formulae that ex* *tends to give an isomorphism Zp[[]]b ZpE0 -! E0E with the required equivariance. Next, using* * the obvious description of E0 = W [[u1; : :;:un-1]] in terms of monomials,Qwe see that E0 i* *s isomorphic as a topologicalQZp-module to a product of copies of Zp, say E0 = ffZp. It follow* *s that E0E = ffZp[[]] as -modules, and using this that Y C (k+1; E0E) = C (k+1; Zp[[]]) = C (k+1; Zp[[]])b ZpE0: ff These identifications can easily be transferred to nonzero degrees, and they * *respect the cosim- plicial structure, so it now follows from Proposition 5 that in our spectral se* *quence we have ( t 2 Es;t2= E ifs = n 0 otherwise. As the spectral sequence is strongly convergent, we have [E; S]r = Er-n2= En2-r* *as claimed. It is not hard2to check that this is an isomorphism of E0-modules, and it follows * *in the usual way that DE = -n E as E-module spectra. One can see from the construction that this is c* *ompatible_ with the action of . * *|__| Proposition 17.There is a natural isomorphism E_tbI= I(ssn2-tME). Proof.First, we have 2-t 2 I(MEt+n2bI)= [-n E; bIbS] 2-t = [-n E; bS] = E-t: We can deduce from the above by a thick subcategory argument that K*bIis finite* * in each degree and thus that bIis dualisable. (We could also have quoted this from [12]; the p* *roof given there is only a slight perturbation of what we've just done.) This implies that bIX = F * *(X; bI) = DXb^bIfor all X 2 K. In particular, we have 2 n2 Eb^bI= n DEb^bI= bIE; so that E_tbI= sst-n2bIE = I(ssn2-tME) as claimed. * * |___| We next need to recall the calculation of ss*ME and its connection with local* * cohomology. As usual, we define E*-modules E*=I1k and u-1kE*=I1k by the following recursive* * procedure: we start with E*=I10 = E*, we define u-1kE*=I1k by inverting the action of uk on E* **=I1k, and we define E*=I1k+1to be the cokernel of the evident inclusion E*=I1k -!u-1kE*=I1k.* * It is also well- known that there is a unique way to make act on all these modules that is comp* *atible with the E*-module structure and the short exact sequences E*=I1k-! u-1kE*=I1k-! E*=I1k+1: (The point is that inverting uk is the same as inverting vk, and for each finit* *ely generated sub- N module M* < E*=I1k there is some N such that vpk acts -equivariantly on M*; thi* *s determines the action on u-1kE*=I1k.) One can define E-module spectra E=I1k and u-1kE=I1k and cofibrations E=I1k-! u-1kE=I1k-! E=I1k+1 8 N. P. STRICKLAND such that everything has the obvious effect in homotopy; this can either be don* *e by Bousfield localisation [21] or by the theory of modules over highly structured ring spect* *ra [5, Section 6]. The former approach has the advantage that it is manifestly -equivariant. It i* *s easy to see from the definitions that M(u-1kE=I1k) = 0 so that M(E=I1k) = -1M(E=I1k+1), and* * also that M(E=I1n) = E=I1n; it follows that ME = -nE=I1n. All this is well-known and we r* *ecord it merely to fix the grading conventions. We next consider some parallel facts in local cohomology. Recall that if R is* * a Noetherian local ring with maximal ideal m and M is an R-module then the local cohomology groups* * Him(M) may be defined as lim-!ExtiR(R=mk; M); see [1] (for example) for an account of thes* *e groups. Clearly k Aut(R) acts naturally on H*m(R), and if any element u 2 m acts invertibly on M * *then Him(M) = 0. In particular, the element uk 2 In acts invertibly on u-1kE*=I1k, so H*In(u-1kE* **=I1k) = 0. Thus, the short exact sequence E*=I1k-! u-1kE*=I1k-! E*=I1k+1 gives rise to a -equivariant isomorphism HsIn(E*=I1k) ' Hs-1In(E*=I1k+1): It is also standard that H0In(E*=I1n) = E*=I1n and the other local cohomology g* *roups of this module vanish. We thus end up with a -equivariant isomorphism HnInE* = E*=I1n; and the other local cohomology groups of E* vanish. (This can be obtained more * *directly using an appropriate stable Koszul complex, but does not act on that complex and the* * whole question of equivariance is rather subtle from that point of view.) The next ingredient that we need is a certain "residue map" ae: E0=I1nE0 n-1 -!Qp=Zp: Here n-1 is the top exterior power of the module of K"ahler differentials for E* *0 relative to W ; it is freely generated over E0 by the element ffl := du1^ : :^:dun-1. The defin* *ition of ae involves the map o :W -! Zp that sends a 2 W to the trace of the map x 7! ax, considered* * as a Zp- linear endomorphism of W . This induces a map W=p1 -! Z=p1 which we also call o* *. We define o0:W -! Hom(W; Zp) by o0(a)(b) = o(ab); it is a standard fact of algebraic numb* *er theory that this is an isomorphism. Q Given a multiindex ff = (ff1; : :;:ffn-1) 2 Zn-1 we define uff= iuffii; we * *also write for the multiindex (1; : :;:1). The group E0=I1nE0 n-1 is a direct sum of copies of* * W [1=p]=W = W Qp=Zpindexed by monomials u--ffffl for which ffi 0 for all i. We put X ae( affu--ffffl) = o(a0): ff Proposition 18.The map ae is -invariant. Proof.This is an instance of the well-known principle of invariance of residues* *. Most of the formulae involved are very old, and were originally interpreted in a complex an* *alytic context; starting in the 1960's they were transferred into algebraic geometry [7, 16] bu* *t only in the very recent paper [13] do they appear in the particular technical context that we ne* *ed. First note that W is the smallest closed subring of E0 containing all the (q - 1)'th roots of u* *nity in E0, so it is preserved by Aut(E0) and in particular by . Section 5 of the cited paper gives * *a canonical map res:Hn-1In(E0=p1 E0 n-1) -!W=p1 ; and the short exact sequence E0 -!E0[1=p] -!E0=p1 gives an isomorphism E0=I1nE0 n-1 = HnIn(n-1) = Hn-1In(E0=p1 E0 n-1); and o gives a canonical map W=p1 -! Qp=Zp. By putting all these together, we * *get a map ae0:E0=I1nE0n-1 -!Qp=Zp. All the constructions involved are functorial and thus* * -invariant. __ By examining the formulae in [13, Section 5] we see that ae = ae0. * * |__| GROSS-HOPKINS DUALITY 9 Proposition 19.There is a natural isomorphism I(sstME) = n-1E0 E-n-t. Proof.This is essentially the local duality theorem (see [13, Theorem 5.9] for * *example). We first translate the claim using the isomorphism ME = -nE=I1n; it now says that * *I(Et=I1n) = n-1 E-t. Using the evident isomorphism E-t = Hom E0(Et; E0) we can reduce to t* *he case t = 0. In that case the construction above gives a -invariant element ae 2 I(E0* *=I1n n-1) and thus a equivariant map OE: n-1 -!I(E0=I1n), defined by OE()(a) = ae(a). This s* *atisfies 0 1 ! X X X OE @ bfiufifflA affu--ff = o(affbff): fi ff ff As the map a b 7! o(ab) is a perfect pairing, it is easy to conclude that OE i* *s an isomorphism. |___| We next recall the theorem of Hopkins and Gross, which identifies n-1 in term* *s of certain modules whose -action is easier to understand. First, we write ! = E2, consider* *ed as an E0- module with compatible -action. To explain the notation, note that J := eE0CP 1* *is the group of formal functions on G that vanish at zero. Thus J=J2 is the cotangent space* * to G at zero, which is naturally isomorphic to the group of invariant one-forms on G, which i* *s conventionally denoted by !. On the other hand J=J2 is also naturally identified with eE0CP1= * *eE0S2 = E2, so the notation is consistent. Theorem 20 (Gross-Hopkins).There is a natural -equivariant isomorphism n-1 ' !n* * [det]. Proof.See [8, Corollary 3] (which relies heavily on [6]). * * |___| We can now give our proof of the main topological duality theorem. Proof of TheoremT2.his follows from Propositions 17 and 19 and Theorem 20 after* *_noting that !n = E2n. |__| Remark 21. We conclude by comparing our formulation of the theorem with various* * other pos- sibilities considered by other authors. Firstly, there are several alternative* * descriptions of the spectrum bI: we have Ib= IMS = LK(n)IS = LK(n)ILK(n)S: To see this, let X be a finite spectrum of type n, so that LK(n)X = MX = LnX, a* *nd smashing with X commutes with all the functors under consideration. The spectra listed a* *bove are all K(n)- local, and the canonical maps between them can be seen to become isomorphisms a* *fter smashing with X, and the claim follows. Next, there are several alternatives_to our spectrum E, most importantly the * *spectrum E0 constructed using the Witt ring of Fp rather than Fq. This is in some ways a m* *ore canonical choice, but it has the disadvantage that ss0E0is not compact (although it is li* *nearly compact) and Aut(E0) is not a p-adic analytic group. In any case, E0 is a pro-free E-module,* * so it is not hard to translate information between the two theories if desired. Finally, some other authors use the functor limE-00(S=I ^ X) in place of E_0X* *. Hopkins writes I this as K(X), and calls it the Morava module of X. One can show that K(X) = ss0(LK(n)(E0^ X)) = E00bE0E_0X for large classes of spectra X, for example all K(n)-locally dualisable spectra* *, or spectra for whichWK(n)*X is concentrated in even degrees. However, this equality can fail (* *for example when X = IS=I) and the formal properties of the functor E_0X seem preferable in gen* *eral. References [1]W. Bruns and J. Herzog. Cohen-Macaulay Rings, volume 39 of Cambridge Studies* * in Advanced Mathematics. Cambridge University Press, 1993. [2]E. S. Devinatz. 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