LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

J.P.C. GREENLEES AND J.P. MAY Abstract. Let G be a nite extension of a torus. Working with highly structured ring and module spectra, let M be any module over M U ; examples include all of the standard homotopical M U -modules, such as the Brown-Peterson and Morava K-theory spectra. We shall prove localization and completion theorems for the computation of M (BG) and M (BG). The G-spectrum M UG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum SG , and there is an M UG module MG whose underlying M U -module is M . This allows the use of topological analogues of constructions in commutative algebra. The computation of M (BG) and M (BG) is expressed in terms of spectral sequences whose respective E2 terms are computable in terms of local cohomology and local G homology groups that are constructed from the coecient ring M U and its G . The central feature of the proof is a new norm map in equivariant module M stable homotopy theory, the construction of which involves the new concept of a global I -functor with smash product.

Contents 1. Introduction and statements of results 2. The strategy of proof 3. Constructing suciently large nitely generated ideals 4. The idea and properties of norm maps 5. Global I -functors with smash product 6. The passage to spectra 7. Wreath products and the denition of the norm map 8. The proof of the double coset formula 9. The norm map on sums and its double coset formula 10. The Thom classes of Thom spectra 11. The proof of Lemma 3.4 References 1 6 9 11 13 16 17 20 22 25 26 27

1. Introduction and statements of results Completion theorems relate the nonequivariant cohomology of classifying spaces to algebraic completions of associated equivariant cohomology theories. They are at the heart of equivariant stable homotopy theory and its nonequivariant applications.
The authors are grateful to Gustavo Comeza a, Jim McClure, and Neil Strickland for helpful n conversations. The rst author thanks the University of Chicago for its hospitality and the Nueld Foundation for its support. The second author acknowledges support from the NSF..
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J.P.C. GREENLEES AND J.P. MAY

The most celebrated result of this kind is the Atiyah-Segal completion theorem [1]. For any compact Lie group G, it computes K(BG) as the completion of the representation ring R(G) at its augmentation ideal. A more recent such result is the Segal conjecture [3]. For any nite group G, it computes the cohomotopy (BG) as the completion of the equivariant cohomotopy G at the augmentation ideal of the Burnside ring A(G). Unlike the Atiyah-Segal completion theorem, in which the representation ring is under good algebraic control, the Segal conjecture relates two sequences of groups that are largely unknown and dicult to compute. Shortly after the Atiyah-Segal completion theorem appeared, Landweber [31] and others raised the problem of whether an analog might hold for complex cobordism. It was seen almost immediately that the appropriate equivariant form of complex cobordism to consider was the stabilized version, M UG , introduced by tom Dieck [8]. Shortly after the question was raised, Ler [36] sketched a proof of the o following result. A complete argument has been given by Comezaa and May [6]. n Theorem 1.1 (Ler). If G is a compact Abelian Lie group, then o (M U )J M U (BG), =
G
G

where JG is the augmentation ideal of M UG .

Here M U (BG) is completely understood [30, 35, 36], and the result is not dicult because the Euler classes of the irreducible complex representations of G, which of course are all 1-dimensional, are under good control. There has been no further progress in over twenty years. In fact, in his 1992 survey of equivariant stable homotopy theory [4], Carlsson stated the problem as follows: Formulate a conjecture about M U (BG), for G a nite group. Landweber [31] had noted that the problem of studying M U (BG) seemed to be even harder than the problem of studying M U (BG). In [15], the rst author introduced a new approach to the Atiyah-Segal completion theorem (for nite groups), in which he deduced it from what we now understand to be a kind of localization theorem giving a computation of K (BG) in terms of local cohomology. When such a localization theorem holds in homology, it is a considerably stronger result than the implied completion theorem in cohomology. For example, the localization theorem for stable homotopy theory is false, although the completion theorem for stable cohomotopy is true. We refer the reader to [21, 6-8] and [22] for a general discussion of localization theorems in equivariant homology and completion theorems in equivariant cohomology. We shall here prove theorems of this kind for stabilized equivariant complex cobordism. Our results were announced in [9], and an outline of the proofs has appeared in [23]. To make sense of the approach of [15], one must work in a suciently precise context of highly structured ring and module spectra that one can mimic constructions in commutative algebra topologically. The theory developed by Elmendorf, Kriz, Mandell, and the second author [11] provides these essential foundations. That paper was written nonequivariantly but, as stated in a metatheorem in its introduction and explained in more detail in [12], all of its theory applies verbatim to G-spectra for any compact Lie group G; see also [10, 21, 13]. In the language of [11], stabilized equivariant cobordism is represented by a commutative algebra M UG over the equivariant sphere G-spectrum SG . The underlying nonequivariant S-algebra of M UG is M U . In earlier language, this means that M UG is an

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

E ring G-spectrum with underlying nonequivariant E ring spectrum M U . We understand SG -algebras to be commutative from now on. A considerable virtue of the kind of localization theorem that we have in mind is that, when it applies to an SG -algebra RG with underlying nonequivariant Salgebra R, it automatically implies localization and completion theorems for the computation of M (BG) and M (BG) for the underlying R-module M of any split RG -module MG . (The notion of a split G-spectrum is dened and discussed in [34, II.84], [19, 0], and [21, 3].) This is an especially happy feature of our work since M UG is split and every M U -module M is the underlying nonequivariant spectrum of a certain split M UG -module MG = M UG M U M [38]. Therefore, by [11, V4], our work applies to all of the standard M U -modules that are constructed from M U by quotienting out the ideal generated by a regular sequence of elements of M U and localizing by inverting some other elements. In particular, it applies to the Brown-Peterson spectra BP , the Morava K-theory spectra K(n), and the Johnson-Wilson spectra E(n). There is a long and extensive history of explicit calculations of groups M (BG) and M (BG) in special cases. Some of the relevant authors are: Landweber; Johnson, Wilson, and Yan; Tezuka and Yagita; Bahri, Bendersky, Davis, and Gilkey; Hopkins, Kuhn, and Ravenel; Hunton; and Kriz. See [31, 30, 32, 2] for M U , [28, 29, 40, 41, 42, 43] for BP , and [26, 27, 25] for K(n). Our theorem gives a general conceptual framework into which all such computations must t. As we shall make precise shortly, the theorem shows that these nonequivariant homology and cohomology groups are isomorphic to the equivariant homotopy groups of certain homotopical JG -power torsion M UG -modules JG (MG ) and homotopical completion M UG -modules (MG )G , where JG is the augmentation ideal J G of M U . There result spectral sequences for the computation of these homotopy groups in terms of local cohomology groups and local homology groups that G G can be computed from knowledge of the ring M U and its module M . Thus the theorem establishes a close connection between the geometrically dened equivariant cobordism groups and the homology and cohomology of classifying spaces with coecients in M U -modules. This is entirely satisfactory on a conceptual level. However, like the Segal conjecture, our theorem relates two sequences of groups that are largely unknown and dicult to compute. Thus, on the computational level, it merely points the direction towards further study. Explicit computations will require better understanding G of M U than is now available. We recall an old and probably false conjecture.
G Conjecture 1.2. M U is M U -free on generators of even degree.

The conjecture is true when G is Abelian, as was announced by Ler [35, 36] o and proven in detail by Comezaa [5]. Little is known for non-Abelian groups. n Since our work is based on the importation of techniques of commutative algebra into equivariant stable homotopy theory, we briey recall the relevant algebraic constructions; see [20] for details and discussion. Let R be a graded commutative ring and let I = (1 , . . . , n ) be a nitely generated ideal in R. Dene K (I) to be the tensor product of the graded cochain complexes K (i ) = (R R[1/i ]),

J.P.C. GREENLEES AND J.P. MAY

where R and R[1/i ] lie in homological degrees 0 and 1. Up to quasi-isomorphism, K (I) depends only on the radical of I. For a graded R-module M , dene
s,t HI (R; M ) = H s,t (K (I) M ),

where s indicates the homological degree and t the internal grading. Such local cohomology groups were rst dened by Grothendieck [24]. It is easy to see that 0 HI (R; M ) is the submodule I (M ) = {m M |I N m = 0 for some N } of I-power torsion elements of M . If R is Noetherian it is not hard to prove directly that the functor HI (R; ) is eaceable and hence that local cohomology calculates the right derived functors of I () [24]. It is clear that the local cohomology groups vanish above degree n, but in the Noetherian case Grothendiecks vanishing theorem shows the powerful fact that they are zero above the Krull dimension of R. One key fact that we shall use is that if I then HI (R; M )[1/] = 0; this is a restatement of the easily proven fact that K (I)[1/] is exact [20, 1.1]. We abbreviate HI (R) = HI (R; R). These algebraic local cohomology groups are relevant to topological homology groups. There are dual local homology groups which, to the best of our knowledge, were rst introduced in [17, 18]. Replacing K (I) by a quasi-isomorphic R-free chain complex K (I), dene
I Hs,t (R; M ) = Hs,t (Hom(K (R), M )).

There is a tri-graded universal coecient spectral sequence that converges to these groups; ignoring the internal grading t, which is unchanged by the dierentials, it converges in total degree s = (p + q) and satises
p,q q E2 = Extp (HI (R), M ) and R p+r,qr+1 p,q . dr : Er Er I There is a natural epimorphism H0 (R; M ) M whose kernel is a certain lim1 I group. It is not hard to check from the denition that if R is Noetherian and M I is free or nitely generated then H0 (R; M ) MI , and one may also prove that = in these cases the higher local homology groups are zero. It follows that, at least I when R is Noetherian, H (R; M ) calculates the left derived functors of the (not necessarily right exact) I-adic completion functor. These algebraic local homology groups are relevant to topological cohomology groups. Now, returning to topology, let RG be an SG -algebra and MG be an RG -module; we always understand algebras and modules in the highly structured sense of [11]. We understand G-spectra to be indexed on a complete G-universe U , which implies that our equivariant homology and cohomology theories are RO(G)-graded. However, we restrict attention to integer degrees except where explicitly stated othn G G erwise. We write En = EG for the nth homotopy group n (E) = [SG , EG ]G of a n G-spectrum EG . G For Rk , let RG [1/] be the telescope of iterates

RG k RG 2k RG of multiplication by and let K() be the ber of the canonical map RG G RG [1/]. For a nitely generated ideal I = (1 , . . . , n ) in R , let K(I) be

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

the smash product over RG of the RG -modules K(i ). Up to equivalence of RG modules, K(I) depends only on the radical of I. Dene I (MG ) = K(I) RG MG and (MG ) = FRG (K(I), MG ). I There is a spectral sequence converging to I (MG )G (in total degree p + q), with
p,q 2 G G Ep,q = HI (R ; M ) and r r dr : Ep,q Epr,q+r1 ,

and there is a spectral sequence converging to ((MG ) ) (in total degree p + q) I G with p,q I p,q p+r,qr+1 E2 = Hp,q (RG ; MG ) and dr : Er Er . Now take I to be a nitely generated ideal contained in the augmentation ideal G G JG = Ker(R R ). Note that the ring R need not be Noetherian and the G augmentation ideal need not be nitely generated. In particular, M U is not Noetherian and its augmentation ideal is not nitely generated, even when G is nite. Since R[1/] is nonequivariantly contractible for JG , the canonical map K(I) RG is an equivalence of underlying spectra and so induces an equivalence upon smashing with EG+ , where EG+ is the union of EG and a G-xed disjoint basepoint. Inverting this equivalence and using the projection EG+ S 0 , we obtain a canonical map of RG -modules : EG+ RG K(I). For an RG -module MG , induces maps of RG -modules EG+ MG and (MG ) = FRG (K(I), MG ) FRG (EG+ RG , MG ) I F (EG+ , MG ), EG+ RG RG MG K(I) RG MG = I (MG )

both of which will be equivalences for all RG -modules MG if is an equivalence. We can now state our completion theorem for modules over M UG . Theorem 1.3. Let G be nite or a nite extension of a torus. Then, for any suciently large nitely generated ideal I JG , : EG+ M UG K(I) is an equivalence. Therefore, EG+ MG I (MG ) and (MG ) F (EG+ , MG ) I

are equivalences for any M UG -module MG . It is reasonable to dene K(JG ) to be K(I) for any suciently large I and to dene JG (MG ) and (MG )G similarly. The theorem implies that these M UG J modules are independent of the choice of I. Our main interest is in nite groups. However, the fact that the result holds for a nite extension of a torus and therefore for the normalizer of a maximal torus in an arbitrary compact Lie group suggests the following generalization. There should be an appropriate transfer argument, but we have not succeeded in nding one. Conjecture 1.4. The theorem remains true for any compact Lie group G.

J.P.C. GREENLEES AND J.P. MAY

It is valuable to obtain a completion theorem about EG+ G X for a general based G-space X, obtaining the motivating result about BG (which referred to unreduced homology and cohomology) by taking X to be S 0 . For this purpose, we replace MG by MG X in the rst equivalence and by F (X, MG ) in the second. We write M and M for the reduced homology and cohomology of based spaces. If MG is split with underlying nonequivariant M U -module M , then M (EG+ G X) G ((EG+ X) Ad(G) MG ) =

and

G M (EG+ G X) = (F (EG+ X, MG )), where Ad(G) is the adjoint representation of G [34, II.8.4]. Thus the theorem has the following immediate consequence. Theorem 1.5. Assume the hypotheses of the theorem and assume that MG is split. Then M (EG+ G X) I (Ad(G) MG X)G = and M (EG+ G X) (F (X, MG )) ) = I G for any based G-space X. Implicitly replacing X by its suspension G-spectrum, we are entitled to the following spectral sequences. Corollary 1.6. There is a homological spectral sequence that converges from
p,q G G 2 (M U ; M (Ad(G) X)) Ep,q = HI

to M (EG+ G X). There is a cohomological spectral sequence that converges from

p,q I E2 = Hp,q (M UG ; MG (X))

to M (EG+ G X). Combining Lers theorem with ours, we see that the topology forces the folo lowing algebraic conclusion. A direct proof would be out of reach at present. Corollary 1.7. If G is a compact Abelian Lie group and I is suciently large, then I H0 (M UG ) ((M UG )) (M UG ) = I I G = and I Hp (M UG ) = 0 if p = 0. 2. The strategy of proof For clarity, we shall emphasize the general strategy of proof, focusing on M UG only where necessary. Let G be a compact Lie group and let SG be the sphere G-spectrum. We assume given a commutative SG -algebra RG with underlying nonequivariant commutative S-algebra R. For a (closed) subgroup H of G, let resG denote the restriction H
RG = RG (S 0 ) RG (G/H+ ) = RH ;

it is induced by the projection G/H+ S 0 . Let JH denote the augmentation ideal in RH , namely the kernel of resH : RH R . In [21, 7.5], we explained a 1 G general localization and completion theorem for the calculation of M (EG+ X)

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

and MG (EG+ X) for any split RG -module MG and G-spectrum X. In that # theorem, we assumed that G is nite, each RO(G)-graded theory RH has Thom isomorphisms, and each RH is Noetherian and satises

resG (JG ) = H

JH .

In fact, we did not discuss the verication of this last property in [21], and we remarked that its verication can be the the main technical obstruction to the implementation of our strategy when we work more generally with compact Lie groups and non-Noetherian coecient rings. See [16] for a discussion of this point in the Noetherian case. The algebraic and topological constructions in our general approach demand that we work with nitely generated subideals of JG , and the last property then makes little sense. Thus we need to modify our strategy of proof. We begin work by describing our modied strategy. Thom isomorphisms and Euler classes are essential to the strategy, and we begin with these. Here we must consider RO(G)-graded cohomology groups, and we # use the notation RG (X) for the RO(G)-graded cohomology of a G-spectrum X to # distinguish it from the Z-graded part RG (X). In particular, we write RG for the RO(G)-graded coecient ring. For a real representation V of G. The inclusion eV : S 0 S V induces a map
# # # e# : RG (S V ) RG (S 0 ) = RG . V 0 Let 1 RG be the identity element; it is represented by the unit : SG RG . V Its suspension V 1 is an element of RG (S V ), and we dene V V e(V ) = e# (V 1) RG (S 0 ) = RG . V

We need to be able to shift these classes into integer degrees.

# Denition 2.1. The theory RG has Thom isomorphisms if, for each complex representation V of G, there is a natural isomorphism # # V : RG (X S |V | ) RG (X S V ) # of RG -modules, where |V | is the real dimension of V . We may view V as giving isomorphisms W |V | W V : RG (X) RG V (X)

for W RO(G). Taking W = V + |V | and X = S 0 , we dene (V ) = V (e(V )) RG .

# Remark 2.2. Let (V ) = V (1) RG . Since V is an isomorphism of RG # modules, (V ) is a unit in RG , and we may as well insist that V (x) = x(V ) for all x R# (X S |V | ) and all G-spectra X. That is, we take our Thom isomorphisms to be given by right multiplication by Thom classes. In particular, (V ) = e(V )(V ). |V |V |V |

Remark 2.3. If V contains a trivial representation, so that V G = 0, then eV is null homotopic and therefore e(V ) = 0 and (V ) = 0.
G Now let I be a given nitely generated subideal of JG . For H G, let rH (I) G denote the resulting subideal resH (I) RH of JH .

J.P.C. GREENLEES AND J.P. MAY

# Denition 2.4. Assume that each RH has Thom isomorphisms. The ideal I in RG is suciently large at H if there is a non-zero complex representation V of H

such that V H = 0 and the Euler class (V ) RH is in the radical ideal I is suciently large if it is suciently large at all H G.

|V |

G rH (I). The

As explained in the introduction, we have a canonical map of RG -modules : EG+ RG K(I), and our goal is to prove that it is an equivalence. The essential point of our strategy is the following result, which reduces the problem to the construction of a suciently large nitely generated subideal I of JG .
# Theorem 2.5. Assume that RH has Thom isomorphisms for all H G. If I is a suciently large nitely generated subideal of JG , then

: EG+ RG K(I) is an equivalence. Therefore, EG+ MG I (MG ) and (MG ) F (EG+ , MG ) I are equivalences for any RG -module MG . Proof. Let EG be the cober of the projection EG+ S 0 . Then the cober of is equivalent to EG K(I), and we must prove that this is contractible. Using the G transitivity of restriction maps to see that rH (I) is a large enough subideal of RH , we see that the hypotheses of the theorem are inherited by any subgroup. Using the fact that there is no innite descending chain of compact Lie groups, we see that we may assume that the theorem holds for H P, where P is the family G of proper subgroups of G. Thus EH K(rH (I)) is contractible for H P, and clearly G (EG K(I))|H = EH K(rH (I)). From here, the proof is just like that of [20, 7.5]. We have that G/H+ EG K(I) is contractible for H P. We take EP to be the colimit of spheres S V , where V runs through the complex representations such that V G = {0} in a complete complex universe U . Since EP/S 0 is triangulable as a G-CW complex whose cells have proper orbit type, the induction hypothesis implies that (EP/S 0 ) EG K(I) is contractible. By the cober sequence S 0 EP EP/S 0 and the fact that EP S 0 EP EG is an equivalence, it suces to show that EP K(I) is and our translation contractible. For any RG -module MG , the construction of EP G of Euler classes to integer gradings imply directly that (EP MG ) is the localG 1 ization (MG )[ (V ) ] obtained by inverting the Euler classes (V ) (see [20, 3.20]). With MG = K(I), we have a spectral sequence that converges from the lo G G cal cohomology groups HI (R ) to (K(I)). Localizing by inverting the (V ), we G obtain a spectral sequence that converges from the localization HI (R )[ (V )1 ] G to (EP K(I)). As we pointed out earlier, the local cohomology of a ring at an ideal vanishes when it is localized by inverting an element in that ideal. Thus

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

our assumption that I is suciently large at G ensures that the E 2 -term of our spectral sequence is zero. 3. Constructing sufficiently large finitely generated ideals The idea is to obtain enough elements of JG to give a good approximation to it. For those groups G that act freely on a nite product of unit spheres of representations, such as the nite p-groups, there are enough representations that we can simply use nitely many Euler classes (V ). However, even for general nite groups, the ideal generated by the (V ) is usually not suciently large. We need to add in other elements, and we shall do so by exploiting norm, or multiplicative transfer, maps that are analogous to Evens norm maps in the cohomology of groups [14, Ch. 5]. This is a new construction in equivariant stable homotopy theory and should have other applications. However, we shall state three theorems and two lemmas that explain our strategy of proof before specifying in Denition 3.6 what it means for a theory to have norm maps. Similarly, the theorems are stated in terms of natural Thom isomorphisms, and we shall specify the relevant naturality conditions in Denition 3.7. We assume given a toral group G, namely an extension of the form 1 T G F 1, where T is a torus and F is a nite group. Most of our work will be necessary even when T is trivial and we are dealing only with the nite group F .
# Theorem 3.1. Let G be toral. If, for each H G, RH has natural Thom iso morphisms and RH has norm maps, then JG contains a suciently large nitely generated subideal.

We shall dene the notion of a global I functor with smash product, abbreviated G I -FSP, in Section 5. A G I -FSP T has an associated SG -algebra R(T )G for every compact Lie group G; regarded as an SH -algebra for H G, R(T )G is canonically isomorphic to R(T )H . The real work in this paper is the proof of the following theorem, which is the subject of Sections 69. Theorem 3.2. Let T be a G I -FSP T such that R(T )# has natural Thom isoG morphisms for every compact Lie group G. Then every R(T ) has norm maps. G The application to Thom spectra is justied by the following result, which is proven in Example 5.8 and Section 10. Theorem 3.3. There is a G I -FSP T U such that R(T U )G = M UG for every # compact Lie group G, and every M UG has natural Thom isomorphisms. The previous two results show that Theorem 3.1 applies to M UG . The proof of Theorem 3.1 depends on two lemmas. The rst, whose proof will be deferred to Section 11, is an exercise in the representation theory of Lie groups that has nothing to do with the hypotheses on our theories. For H G, we write resG : R(G) R(H) H for the restriction homomorphism. When H has nite index in G, we write indG : R(H) R(G) H for the induction homomorphism. Recall that indG V = C[G] C[H] V . H

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J.P.C. GREENLEES AND J.P. MAY

Lemma 3.4. There are non-zero complex representations V1 , , Vs of T such that T acts freely on the product of the unit spheres of the representations resG indG Vi . T T Lemma 3.5. Assume the hypotheses of Theorem 3.1 and let F be a subgroup of F with inverse image G in G. Then there is an element (F ) of JG such that resG ((F )) = (V )w , G where V is the reduced regular complex representation of F regarded by pullback as a representation of G and w is the order of W G = N G /G . The proof will be given at the end of the section. Proof of Theorem 3.1. We claim that the ideal I = ((indG V1 ), , (indG Vs )) + ((F )|F F ) T T is suciently large. If H is a subgroup of G that intersects T non-trivially, then, by Lemma 3.4, (resG indG Vi )HT = {0} for some i and therefore (indG Vi )H = {0}. Since T T T
G (resG indG Vi ) = resG ((indG Vi )) rH (I), H T H T

this shows that I is suciently large at H in this case. If H is a subgroup of G that intersects T trivially, as is always the case when G is nite, then H maps isomorphically to its image F in F . If G is the inverse image of F in G and V is the reduced regular complex representation of F regarded as a representation of G , then resG (V ) is the reduced regular complex representation H of H and (resG (V ))H = 0. By Lemma 3.5, we have resG ((F )) = (V )w and H G therefore
G (resG (V ))w = resG ((V )w ) = resG resG ((F )) = resG ((F )) rH (I). H H H H G

This shows that I is suciently large at H in this case. We need conjugation isomorphisms to describe the properties of norm maps and to prove the lemmas. For g G and H G, let gH = gHg 1 and cg : gH H be the conjugation isomorphism. For a representation V of H, let g V be the pullback of V along cg . For H G, we have a natural restriction homomorphism
H G resG : R (X) R (X) H

on based G-spaces X. For g G, we also have a natural isomorphism

H cg : R (X) RH (gX),
g

where X is a based H-space and gX denotes X regarded as a gH-space by pullback along cg . To give content to the proof of Lemma 3.5, we must explain our hypothesis G that R also has norm maps. We give a crude and perhaps unilluminating denition that prescribes exactly what we shall use in the proof. A description closer to the motivating example of group cohomology will be given in the next section.
G Denition 3.6. We say that R has norm maps if, for a subgroup H of nite H index n in G and an element y Rr , where r 0 is even, there is an element n

normG (1 + y) H
i=0

G Rri

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

11

H that satises the following properties; here 1 = 1H R0 denotes the identity element. (i) normG (1 + y) = 1 + y. G (ii) normG (1) = 1. H (iii) [The double coset formula]

resG normG (1 + y) = H K
g

H normK gHK resgHK cg (1 + y),

where K is any subgroup of G and g runs through a set of double coset representatives for K\G/H. We must also explain what it means for Thom isomorphisms to be natural.
# Denition 3.7. Let RH have Thom isomorphisms VH given by right multiplication by Thom classes (VH ) for all H G and all complex representations VH of H. We say that the Thom isomorphisms are natural if the following three conditions hold. (i) Compatibility under restriction: (VG )|H = (VG |H ). |V |gVH for (ii) Compatibility under conjugation: (g VH ) = cg ((VH )) in RH H g G. (iii) Multiplicativity: (VH )(VH ) = (VH VH ).

Proof of Lemma 3.5. Since the restriction of the reduced regular representation of F to any proper subgroup contains a trivial representation, the restriction of (V ) RG to a subgroup that maps to a proper subgroup of F is zero. In RG , the double coset formula gives (3.8) resG normG (1 + (V )) = G G
g

normG G resgG G cg (1 + (V )), gG G

where g runs through a set of double coset representatives for G \G/G . Taking V as in the statement of the lemma, compatibility under conjugation gives that cg (1 + (V )) = 1 + (g V ). Here g V is the reduced regular representation of gG . Clearly gG G is the inverse image in G of gF F . If gF F is a proper subgroup of F , then the restriction of (V ) to gG G is zero. Therefore all terms in the product on the right side of (3.8) are 1 except for those that are indexed on elements g N G . There is one such g for each element of W G = N G /G , and the term in the product that is indexed by each such g is just 1 + (V ). Therefore (3.8) reduces to (3.9) resG normG (1 + (V )) = (1 + (V ))w . G G

If V has real dimension r, then the summand of (1 + (V ))w in degree rw is (V )w . Since resG preserves the grading, we may take (F ) to be the summand G of degree rw in normG (1 + (V )). G 4. The idea and properties of norm maps We give an intuitive idea of the construction, leaving details and rigor to later sections. Let H be a subgroup of nite index n in a compact Lie group G. For based H-spaces X, we can give the smash power X n an action of G. Intuitively,

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J.P.C. GREENLEES AND J.P. MAY

this is done in exactly the same way that one induces up a representation of H to a representation of G, and the analogy will guide much of our work. To begin with, the norm map will be a natural function (4.1)
H G normG : R0 (X) R0 (X n ). H

Norm maps normG in the sense of Denition 3.6 will be obtained by taking X to H be the wedge S 0 S r , studying the decomposition of X n into wedge summands of G-spaces described in terms of representations, and using Thom isomorphisms to translate the result to integer gradings. The norm map normG will satisfy the H following properties. (4.2) (4.3) (4.4) normG is the identity function. G
H normG (1H ) = 1G , where 1H R0 (S 0 ) is the identity element. H H H normG (xy) = normG (x)normG (y) if x R0 (X) and y R0 (Y ). H H H

Here the product xy on the left is dened by use of the evident map (4.5)
H H H R0 (X) R0 (Y ) R0 (X Y )

and similarly on the right, where we must also use the isomorphism
G R0 (X n Y n ) R0 ((X Y )n ). = G

The most important property of the norm map will be the double coset formula (4.6) resG normG (x) = K H
g H normK gHK resgHK cg (x),
g

where K is any subgroup of G and g runs through a set of double coset representatives for K\G/H. Here, if gH K has index n(g) in gH, then n = n(g) and the product on the right is dened by use of the evident map (4.7)
g H R0 (X) K K R0 (X n(g) ) R0 (X n ).

is represented by an H-map x : SG RG X. There is An element of no diculty in using the product on RG to produce an H-map
x SG (SG )n (RG X)n (RG )n X n RG X n . = =
n

The essential point of our construction is that this may be done in such a way as to produce a G-map: this will be normG (x). Here and later, all powers are H understood to be smash powers. This is the basic idea, but carrying it out entails several diculties. Since our group actions involve permutations of smash powers, we cannot hope to control equivariance unless we are using a smash product that is strictly associative and commutative and a multiplication on RG that is strictly associative and commutative. Moreover, our G-actions come by restriction of actions of wreath products n H, and it turns out to be essential to work with (n H)-spectra. If we start just with a G-spectrum RG , then it is not clear how to proceed. Similarly, to make our spectra precise, we must specify appropriate universes on which to index them, and we nd that the norm map acts nontrivially on universes.

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

13

We explain why the last phenomenon occurs and at the same time explain the perhaps surprising restriction of our initial description of the norm map to degree zero. The point is that, for q 0 at least, the norm gives a map RH (X) = RH (X S q ) RG ((X S q )n ) = RG (X n (S q )n ).
q 0 0 0

However, the sphere (S ) is twisted: in fact, it is the sphere S V associated to the G representation V = indH (q), where q denotes the q-dimensional trivial representaG tion of H. Thus the target is RV (X n ), which is not in the integer graded part of the theory. Of course, if q is even and RG has Thom isomorphisms, then we can use them to translate to integer degrees and thus to obtain the translated norm map
H G Rq (X) Rnq (X n ).

q n

An elaboration of this idea to sums of elements will give the modied norm maps normG . We shall explain this elaboration of the denition after making sense of H the geometric construction of the norm map normG and proving its double coset H formula. 5. Global I -functors with smash product To deal with the diculties that we have indicated, we shall assume that RG arises from a GI -FSP, where FSP stands for functor with smash product. This is a global version of the notion of an I -prefunctor that was introduced in [37, IV.2.1]. The earlier notion was dened nonequivariantly but transcribes directly to a denition in which we restrict attention to a given compact Lie group G acting on everything in sight. The adjective global means that we allow G to range through all compact Lie groups G (or through all G in some suitably restricted class). Let G denote the category of compact Lie groups and their homomorphisms; for the purposes of the present theory, it would suce to restrict attention to injective homomorphisms, but our examples are dened on the larger category. Denition 5.1. Dene the global category GT of equivariant based spaces to have objects (G, X), where G is a compact Lie group and X is a based G-space. The morphisms are the pairs (, f ) : (G, X) (G , X ) where : G G is a homomorphism of Lie groups and f : X X is an -equivariant map, in the sense that f (gx) = (g)f (x) for all x X and g G. Topologize the set of maps (G, X) (G , X ) as a subspace of the evident product of mapping spaces and observe that composition is continuous. Denition 5.2. Dene the global category GI of nite dimensional equivariant inner product spaces to have objects (G, V ), where G is a compact Lie group and V is a nite dimensional inner product space with an action of G through linear isometries. The morphisms are the pairs (, f ) : (G, V ) (G , V ) where : G G is a homomorphism and f : V V is an -equivariant linear isomorphism. We often nd it convenient to work with complex rather than real inner product spaces. Our denitions apply equally well under either interpretation.

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J.P.C. GREENLEES AND J.P. MAY

Observe that each morphism (, f ) in GI factors as a composite (G, V ) (G, W ) (H, W ), where G acts through on W . We have a similar factorization of morphisms in GT . Observe too that we have forgetful functors GI G and GT G . We shall be interested in functors GI GT over G , that is, functors that preserve the group coordinate. For example, one-point compactication of inner product spaces gives such a functor, which we shall denote by S . As in this example, the space coordinate of our functors will be the identity on morphisms of the form (, id). For these reasons, we shall usually omit the group coordinate from the notation for functors. Denition 5.3. A GI -functor is a continuous functor T : GI GT over G , written (G, T V ) on objects (G, V ), such that T (, id) = (, id) : (G, T W ) (H, T W ) for a representation W of H and a homomorphism : G H. The following observation will be the germ of the denition of the norm map. Lemma 5.4. Let A = Aut(G, V ) be the group of automorphisms of (G, V ) in the category GI . For any GI -functor T , the group A G acts on the space T V . Proof. An element of A is a map (, f ) : (G, V ) (G, V ) such that f (gv) = (g)v for all v V and g G and is an isomorphism. The semi-direct product A G is the set A G with the multiplication ((, f ), g)((, ), h) = ((, f ), 1 (g)h), and G is contained in A G as the normal subgroup of elements (id, g). The action of A G on T V is specied by ((, f ), g)x = T (, f )(gx). This does dene an action since functoriality and equivariance imply that T (, f )(gT (, )(hx)) = (g)(h)T (, f )(x) = T (, f )( 1 (g)hx). Dene the direct sum functor : GI GI GI by (G, V ) (H, W ) = (G H, V W ). Dene the smash product functor : GT GT GT by (G, X) (H, Y ) = (G H, X Y ). These functors lie over the functor : G G G . Denition 5.5. A GI -FSP is a GI -functor together with a continuous natural unit transformation : S T of functors GI GT and a continuous natural product transformation : T T T of functors GI GI GT such that the composite
id T V T V S 0 T V T (0) T (V 0) T V = = (id,f ) (,id)

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

15

is the identity map and the following unity, associativity, and commutativity diagrams commute: / TV TW SV SW
=

S V W TV TW TZ  T V T (W Z) and TV TW
id

 / T (V W ), / T (V W ) T Z

id

 / T (V W Z), / T (V W )
T ( )

 TW TV

 / T (W V ).

Actually, the notion that we have just dened is that of a commutative GI -FSP; For the more general non-commutative notion, the commutativity diagram must be replaced by a weaker centrality of unit diagram. Observe that the space coordinate of each map T (, f ) is necessarily a homeomorphism since (, f ) = (, id) (id, f ) and f is an isomorphism. We record the following observation for later use. Remark 5.6. For objects (Hi , Vi ) of GI , 1 i n, and a permutation n , the associativity and commutativity diagrams imply the following commutative diagram: T V1 T Vn

/ T (V1 Vn )
T ()

 T V1 (1) T V1 (n)

/ T (V1 (1) V1 (n) ).

Here and in the commutativity axiom, we have suppressed the group coordinate from the notation and T () means T (, ): we must permute the groups in the same way that we permute the inner product spaces. Example 5.7. The sphere functor S is a GI -FSP with unit given by the identity maps of the S V and product given by the isomorphisms S V S W S V W . For = any GI -FSP T , the unit : S T is a map of GI -FSPs. Example 5.8. Let dim V = n and dene T V to be the one-point compactication of the canonical n-plane bundle EV over the Grassmann manifold Grn (V V ). An action of G on V induces an action of G that makes EV a G-bundle and TV a based G-space. Take V = V {0} as a canonical basepoint in Grn (V V ). The inclusion of the ber over the basepoint induces a map : S V T V . The canonical bundle map EV EW E(V W ) induces a map : T V T W T (V W ). With the evident denition of T on morphisms, T is a GI -functor. Actually, there are two variants: we write T O when dealing with real inner product spaces and T U when dealing with complex inner product spaces.

16

J.P.C. GREENLEES AND J.P. MAY

6. The passage to spectra It is useful to regard a GI -FSP as a GI -prespectrum with additional structure. Denition 6.1. A GI -prespectrum is a GI -functor T : GI GT together with a continuous natural transformation : T S T of functors GI GI T such that the composites
T V T V S 0 T (V 0) T V = =

are identity maps and each of the following diagrams commutes: T V SW SZ

= id

/ T (V W ) S Z

 T V S W Z

 / T (V W Z).

We say that a GI -prespectrum is an inclusion GI -prespectrum if each adjoint map : T V F (S W , T (V W )) is an inclusion. Lemma 6.2. If T is a GI -FSP, then T is a GI -prespectrum with respect to the composite maps : T V S W T V T W T (V W ). Now x a group G and a G-universe U , namely a countably innite dimensional inner product space that contains a trivial representation and contains each of its nite dimensional representations innitely often. We say that the universe U is complete if it contains all irreducible representations of G. A G-prespectrum indexed on U consists of based G-spaces T V for nite dimensional inner product spaces V U and a transitive system of structure G-maps : W V T V T W for V W , where W V is the orthogonal complement of V in W . A spectrum E is a prespectrum whose adjoint structure maps EV W V EW are homeomorphisms. There is a spectrication functor L from prespectra to spectra that is left adjoint to the evident forgetful functor. See [34, 21, 39] for the development of equivariant stable homotopy theory from this starting point. It is evident that a GI -prespectrum restricts to a G-prespectrum indexed on U for every G and U . Notations 6.3. Let T(G,U ) denote the G-prespectrum indexed on U associated to a GI -FSP T . Write R(T )(G,U ) for the spectrum LT(G,U ) associated to T(G,U ) . Let L (j) be the G-space of linear isometries U j U , with G acting by conjugation. As discussed in the cited sources and [10], L is a G-operad and is an E G-operad when U is complete. There is a notion of an L -prespectrum [37, IV.1.1] (amended slightly in [34, VII.2.4-2.6]). Exactly as in [37, IV.2.2], T(G,U ) is an L -prespectrum. The essential point is that if f : U j U is a linear isometry and Vi are indexing spaces in U , then we have maps j (f ) : T V1 T Vj T (V1 Vj ) T f (V1 Vj ). The notion of an L -prespectrum is dened in terms of just such maps.
Tf id

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

17

The notion of an L -spectrum E is dened more conceptually in terms of maps L (j) E j E. (In fact, the twisted half-smash product was not known when [37] was written.) However, by [37, IV.1.6] and, in current terms, [34, VII2], the functor L converts L -prespectra to L -spectra. We conclude that, for every G and every complete Guniverse U , R(T )(G,U ) is an L -spectrum and thus an E ring G-spectrum. Moreover, as explained in [11, 10], L -spectra functorially determine weakly equivalent commutative SG -algebras. While the constructions of [20, 21] on which our work is based depend on the fact that the G-spectra we are working with admit SG -algebra structures, we will not need to make explicit use of these structures in our study of the norm map. Remark 6.4. Let T be a global G I -functor, choose a complete G-universe UG for each G, and consider the G-spectra R(T )(G,UG ) as G varies. It is a routine exercise to verify that the collection R(T )(G,UG ) denes a G -spectrum, in the sense dened in [34, II.8.5]. In particular, it follows immediately from [34, II.8.6 and II.8.7] that R(T )(G,UG ) is a split G-spectrum for each G. This shows that our Thom G-spectra are split, as stated in the introduction. Remark 6.5. If U is a complete complex G-universe, we may regard it as a complete real G-universe by neglect of structure. This gives two variants of all denitions in sight, one in which we restrict attention to complex nite dimensional inner product spaces V in U and the other in which we allow all real nite dimensional inner product spaces. The resulting categories of G-spectra are canonically equivalent [34, I.2.4] because the adjoint structure maps of G-spectra are homeomorphisms and complex inner product spaces are conal among real ones. 7. Wreath products and the definition of the norm map Recall that the wreath product n H is the semi-direct product n H n , where n acts by permutations on H n ; explicitly, n H is the set n H n with the product (7.1) (, h1 , . . . , hn )(, k1 , . . . , kn ) = (, h 1 k1 , , h n kn ). We have the following evident actions of this group. We display them explicitly because of their centrality in our work. Lemma 7.2. If V is a representation of H, then the sum V n of n copies of V is a representation of n H with action given by (, h1 , . . . , hn )(v1 , . . . , vn ) = (h1 (1) v1 (1) , . . . , h1 (n) v1 (n) ). Lemma 7.3. If X is a based H-space, then the smash power X n is a (n space with action given by (, h1 , . . . , hn )(x1 . . . xn ) = h1 (1) x1 (1) . . . h1 (n) x1 (n) . This leads to the following crucial observation. Proposition 7.4. Let T be a GI -FSP. For an H-representation V , (T V )n and T (V n ) are n H-spaces and the map : (T V )n T (V n ) H)-

18

J.P.C. GREENLEES AND J.P. MAY

is (n H)-equivariant. If U is an H-universe, then U n is a (n and the maps dene a map of (n H)-prespectra indexed on U n : (T(H,U ) )n T(n R H,U n ) ,

H)-universe

where (T(H,U ) )n is the nth external smash power of T(H,U ) . If T = S , then is an isomorphism of prespectra. If n = n(i), where n(i) 1 and 1 i m, then the following diagram of prespectra commutes:
m n(i) i i=1 (T(H,U ) )

m i=1

T(n(i) R H,U n(i) )

(T(H,U ) )n

/ T(n R H,U n ) .

Proof. For n , (, ) is an automorphism of (H n , V n ) and thus n maps to A = Aut(H n , V n ). This induces a map from n H to A H n (which is an injection unless V = {0}). Now Lemma 5.4 restricts to give the action of n H on T (V n ). We see that is (n H)-equivariant by taking each (Hi , Vi ) to be (H, V ) in the diagram of Remark 5.6. We may index our prespectra on the conal family of indexing spaces of the form V n in U n , and the external smash product has V n th space (T V )n . The prespectrum level statements are now easily veried from the denition of a GI -FSP. We shall use the proposition to dene the norm map, but we rst need a bit of algebra. For the rest of the section, assume given a subgroup H of nite index n in a compact Lie group G. Choose coset representatives t1 , t2 , , tn for H in G, taking t1 = e, and dene the monomial representation (7.5) by the formula (7.6) (7.7) () = ((), h1 (), . . . , hn ()), ti = t()(i) hi (). where () and hi () are dened implicitly by the formula Lemma 7.8. The map is a homomorphism of groups. If is dened with respect to a second choice of coset representatives {ti }, then and dier by a conjugation in n H. Proof. The rst statement holds by the denition (7.1) of the product in n and the observation that ()ti = t()(i) hi () = t()(()(i)) h()(i) ()hi (). For the second statement, if ti = ti ki , then
1 ti = t()(i) hi ()ki = t()(i) k()(i) hi ()ki

: G n H

and therefore () = (1, k1 , . . . , kn )1 ()(1, k1 , . . . , kn ). The homomorphism is implicitly central to induction in representation theory, as the following lemma explains. Write W for a representation W of n H regarded as a representation of G by pullback along .

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

19

Lemma 7.9. If V is a representation of H, then V n indG V . = H Proof. Recall that indG V = C[G] C[H] V . The isomorphism is given by mapping H ti v on the left to v in the ith summand on the right, as is dictated by the case i = 1 and equivariance. Analogously, for a based n H-space Y , write Y for Y regarded as a G-space G by pullback along . In particular, ((S V )n ) S indH V . = Here, nally, is the denition of the norm map. Recall Notations 6.3. Denition 7.10. Let T be a GI -FSP, let X be a based H-space, and let U be a complete H-universe. An element x R(T )H (X) is given by a map of H0 spectra x : S(H,U ) R(T )(H,U ) X. Dene the norm of x to be the element of R(T )G ( X n ) given by the pullback along of the composite map of n H0 spectra indexed on U n displayed in the commutative diagram: S(
n

H,U n )

/ (S(H,U ) )n

xn

/ (R(T )(H,U ) X)n

=

 R(T )(
n

H,U n )

Xn o

id

 (R(T )(H,U ) )n X n .

If we take it as understood that G acts on U n through , then the composite dening normG (x) may be rewritten more simply as H S(G,U n ) (7.11)
normG (x) H 1

/ (S(H,U ) )n

xn

/ (R(T )(H,U ) X)n 

=

R(T )(G,U n ) X n o

id

(R(T )(H,U ) )n X n .

Observe that the G-universe U n is complete; for example, if G is nite, this holds because the regular representation of H induces up to the regular representation of G. Strictly speaking, if we start with H-spectra dened in xed complete Huniverses UH for all H, then we must choose an isomorphism UG UH to transfer = n the norm to a map of spectra indexed on UG . This is a standard procedure that must be applied to various of the maps that we shall construct; compare Remark 6.4. Property (4.2) of the norm is obvious. Property (4.3) is an easy consequence of the unity and associativity diagrams in the denition of a GI -FSP. Property (4.4) also follows easily from the denition of a GI -FSP, once we make precise how to interpret the product (4.5). Thus suppose given H-universes U and U and maps x : S(H,U ) R(T )(H,U ) and y : S(H,U ) R(T )(H,U ) . For the present purpose, U and U could be the same, but we will want to allow them to be dierent in the next section. We then dene xy to be the composite displayed in the diagram (7.12) S(G,U U
xy ) 1

/ S(G,U ) S(G,U

xy )

/ R(T )(G,U ) X R(T )(G,U ) Y  R(T )(G,U ) R(T )(G,U ) X Y

=

R(T )(G,U U ) X Y o

id

20

J.P.C. GREENLEES AND J.P. MAY

Of course, when U = U , we can internalize this external multiplication by use of a linear isometry f : U U U ; it is then obvious that it agrees with the standard homotopical denition of such a product. The external form makes the verication of (4.4) transparent. 8. The proof of the double coset formula The proof of the double coset formula requires a precise combinatorial discussion of double cosets. We again suppose given a subgroup H of nite index n in a compact Lie group G. We x coset representatives tj and use them to dene the monomial representation , as in the previous section. We suppose given a second subgroup K of G and we choose representatives g1 , , gm for the double cosets K\G/H. We shall choose the gi s from among the tj s, in a manner to be specied. The gi give a decomposition of the nite K-set G/H as
m gi i=1

K/ H K
i=1

Kgi H = K\G/H.

Explicitly, the ith component is the isomorphism K/gi H K Kgi H that sends k(giH K) to kgi H. Let the ith double coset have n(i) elements, dene q(i) = n(1) + + n(i 1), and label the gi and tj so that gi = tq(i)+1 and the tq(i)+r , 1 r n(i), run through the coset representatives of G/H that are in the ith double coset Kgi H. Thus
n(i) =

Kgi H =
r=1

tq(i)+r H.

Dene (8.1)
1 si,r = tq(i)+r gi K.

Thus the si,r are coset representatives for K/giH K that map to our chosen coset representatives tq(i)+r in Kgi H. With this choice, we dene homomorphisms (8.2) by the formula (8.3) where i () and (8.4)
r ()

i : K n(i) i () = (i (),

gi

H K

1 (), . . . , n(i) ()),

are dened implicitly by the formula si,r = si,i ()(r) r ().

Lemma 8.5. With these choices of and the i , the permutations () for K decompose as block sums 1 () m () with i () n(i) , and i () = i () and
1 hq(i)+r () = gi r ()gi .

Proof. The rst statement holds since must permute the tq(i)+1 , , tq(i)+n(i) among themselves. For the second statement, we obtain
1 1 tq(i)+r gi = tq(i)+i ()(r) gi r ()

by inserting the formula (8.1) into (8.4), while (7.7) gives tq(i)+r = tq(i)+i ()(r) hq(i)+r ().

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

21

The lemma can be rewritten in terms of homomorphisms, giving a description of the restriction of to K in terms of the i . Lemma 8.6. The restriction of to K maps into diagram commutes: G O
i

n(i) / n O

H, and the following H

K NN NNN NNN N (1 ,...,m ) NNN & n(i) i

(1 ,...,m )

/ i n(i) l6 lll lll Q lll i id R cgi lll gi H K,

1 where i () = (i (), hq(i)+1 (), . . . , hq(i)+n(i) ()) and cgi (gi hgi ) = h.

We can now prove (4.6). For a map x : S(H,U ) R(T )(H,U ) X, resG normG (x) K H is the composite map of K-spectra indexed on U n that is displayed in the following diagram, where K acts on U n and X n through the restriction of to K: S(K,U n ) 
1 n / S(H,U ) xn

/ (R(T )(H,U ) X)n 

=

R(T )(K,U n ) X n o

id

(R(T )(H,U ) )n X n .

Dene i (x) to be the composite displayed in the following diagram, in which K acts through i on U n(i) and X n(i) : S(K,U n(i) ) (8.7)
i (x) 1

/ (S(H,U ) )n(i)

xn(i)

/ (R(T )(H,U ) X)n(i)  (R(T )(H,U ) )n(i) X n(i) .

=

R(T )(K,U n(i) ) X n(i) o

id

In view of the transitivity diagram for given in Proposition 7.4, applied to both R(T ) = LT and S, we see immediately that the following diagram commutes: S(K,U n )
resG normG (x) K H 1

m i=1

S(K,U n(i) )

i i (x)

m i=1 (R(T )(K,U n(i) ) =

X n(i) )

 R(T )(K,U n ) X n o

id

m i=1

R(T )(K,U n(i) ) ) (

m i=1

X n(i) ).

Comparing with (7.12), we see that precisely such a diagram makes sense of the iterated product (4.7), and we conclude that resG normG (x) is the product of the K H i (x). Abbreviating notation (as in (4.6)), let g = gi . To complete the proof of (4.6), we need only show that
H i (x) = normK gHK resgHK cg (x).
g

This means that i (x) coincides with the composite displayed in the following diagram, in which gX denotes the H-space X regarded as a gH-space by pullback

22

J.P.C. GREENLEES AND J.P. MAY

along cg : gH H, g U denotes the H-universe U regarded as a gH-universe by pullback along cg , and K acts through i on g U n(i) and gX n(i) : (8.8) S(K,g U n(i) ) 
1

/ (S(gHK,g U ) )n(i)

cg (x)n(i)

/ (R(T )(gHK,g U ) gX)n(i) 

n(i) =

R(T )(K,g U n(i) ) gX n(i) o

id

R(T )(gHK,g U ) gX n(i) .

It is immediate from the denition of a GI -functor that R(T )(gH,g U ) = gR(T )(H,U ) , where gR(T )(H,U ) denotes R(T )(H,U ) regarded as a gH-spectrum by pullback along cg . The same is true for S, and cg (x) = g x is just the map x regarded as a gH-map by pullback along cg . By Lemma 8.6, i = cg i : K n(i) H. Therefore X n(i) regarded as a K-space via i is identical to gX n(i) regarded as a K-space via i and U n(i) regarded as a K-universe via i is identical to g U n(i) regarded as a K-universe via i . Except that we have used that i takes values in n(i) gH K to restrict the group action in some of the terms of (8.8), we see that the diagrams (8.7) and (8.8) display one and the same map. 9. The norm map on sums and its double coset formula Consider normG (x + y), where x R(T )H and y R(T )H for even integers q r H q 0 and r 0. Here we are considering the case X = S q S r of the norm map that we dened in Denition 7.10. For based H-spaces X and Y , we have (9.1) as n (9.2) (X Y )n =
n i=0 (n

H)

(i ni )

X i Y ni

H-spaces. For any subgroup K G, we therefore have (X Y )n =

n

K
i=0

((

i ni )

H)K

(X i Y ni )

as K-spaces, where runs through a set of double coset representatives for K \ (n H) / ((i ni ) H).

Taking K = G, we see that, in general, the norm of the sum of elements of R(T )H (X) and R(T )H (Y ) is an element of 0 0 (9.3) R(T )G ((X Y )n ) = 0
n i=0

R(T )0

((i ni )

H)G

( (X i Y ni )).

Now return to our elements x R(T )H and y R(T )H . We are thinking q r of x = 1. In order to obtain the norm of Denition 3.6, we must transform the element normG (x + y) to an element of R(T )G , and we must do so in a fashion H that makes sense of and validates the double coset formula of Denition 3.6. This is where we use the assumption in Theorem 3.2 that each R(T )# has natural Thom G isomorphisms. By working on the level of n H as long as possible, we shall

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

23

circumvent any need to deal with the complexities displayed in formulas (9.2) and (9.3). Let Vn denote Rn with its permutation action by n and regard Vn as a n Hrepresentation with trivial action by H. Thus Vn indG R. Writing V q for the = H q sum of q copies of V , we have (S q )n = S Vn as a n H-space. Therefore (9.1) gives (9.4) as n
R

(S q S r )n =

n i=0 (n

H)

(i ni )

S Vi

r Vni

H-spaces. Dene a translated induction map

H

i : R(T ) n

((n

H)

(i ni )

S Vi

r Vni

) R(T ) n

(S qi+r(ni) )

by commutativity of the following diagram: (9.5) R(T ) n

((n

H)
i

(i ni )

S Vi

r Vni

/ R(T )(i ni )

(S Vi

r Vni

V q V r
R

R(T ) n

 H (S qi+r(ni) ) o

n ind( i

H ni )

R(T )
H

(i ni )

R

ni

(S qi+r(ni) ),

where we have written indG for the ordinary transfer homomorphism associated to H H G. In particular, restricting to degree zero, this gives i : R(T )0 n

((n

H)

(i ni )

S Vi

r Vni

n ) R(T )qir(ni) .

We also write i for the corresponding map of G-equivariant homology groups obr tained by pullback along . Observe that Viq and Vni are not representations of G, so that we must start with (9.1) and not (9.2) in order for the Thom isomorphisms that we use here to make sense. Note too that we require Thom isomorphisms for n H and its subgroups, not just for G and its subgroups. Use of the i allows us to redene the norm of sums in a Z-graded form. We have normG (x + y) R(T )G ( (S q S r )n ). H 0 It is the restriction to G of an element of R(T )0 n

((S q S r )n ) =

n i=0

R(T )0 n
n

((n

H)

(i ni )

S Vi

r Vni

).

We write normG (x + y)i for the component in the ith summand and dene H (9.6) normG (x + y) = H
i=0

i (normG (x + y)i ). H

The double coset formula is still valid for these modied norm maps. Proposition 9.7. For elements x R(T )H and y R(T )H , q r resG normG (x + y) = H K
g H normK gHK resgHK cg (x + y),
g

where K is any subgroup of G and {g} runs through a set of double coset representatives for K\G/H.

24

J.P.C. GREENLEES AND J.P. MAY

Proof. To simplify the notation, we restrict attention to the case q = 0, which is 0 the case of interest. Since Vi0 = {0}, i acts trivially on S 0 = S Vi . Reverse the roles of i and n i in the notations above and let i n be the subgroup of permutations that x the rst n i letters. Recall that if gH K has index n(g) in gH, then n = n(g). Fix an ordering of the gs and write {n(g)} for g n(g) regarded as a subgroup of n . The left side of the equation in the statement is the restriction to K of an element of
n i=0
n R(T )ri

The right side is the product over {g} of the restrictions to K of elements of
n(g)
n(g) R(T )ra(g)

a(g)=0

In view of Lemma 8.6, we see that the relevant products are obtained by adding up restrictions to K of products
g

R(T )0 n(g)

(S ra(g) )

/ R(T ){n(g)} 0

(S ri ),

where 0 a(g) n(g) and a(g) = i. For such a sequence {a(g)}, let {a(g)} = a(g) regarded as a subgroup of i . The original double coset formula (4.6) g made use of the restriction to K of the product
g

R(T )0 n(g)

((S 0 S r )n(g) )

/ R(T ){n(g)} 0

((S 0 S r )n ).

Under the wedge decomposition (9.1) and change of groups isomorphisms like those in the top line of (9.5), this product agrees with the sum over sequences {a(G)} of the product maps
a(g) g R(T )0

(S Va(g) )

/ R(T ){a(g)} 0

(S Vi ).

We claim that the following diagram commutes for each such sequence {a(g)}, where the left horizontal arrows are products and the right horizontal arrows are restrictions:
a(g) g R(T )0

(S Va(g) )

/ R(T ){a(g)} 0 

(S Vi ) o
r

R(T )0 i  R

(S Vi )
i

V r

a(g)

V r

V r

R(T )0 a(g)
n(g) a(g)

(S ra(g) )

/ R(T ){a(g)} 0

(S ri ) o
R R
H H

R(T )0 i  R

(S ri )

R R

ind

H H

n(g) g R(T )0

(S ra(g) )

 R / R(T ){n(g)} H (S ri ) o 0

ind

{n(g)} {a(g)}

indnR
i

H H

R(T )0 n

(S ri )

The top two squares commute since our Thom isomorphisms are multiplicative and compatible under restriction. The bottom two squares commute since transfer commutes with products and restriction by [34, IV.4.4 and IV.5.2]. It follows directly that the present version of the double coset formula follows from the original version.

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

25

Taking x = 1, we obtain norm maps as specied in Denition 3.6. 10. The Thom classes of Thom spectra In this section, we write T for the Thom G I -functor T U of Example 5.8. It determines a G -prespectrum by neglect of structure. The structure maps specied in Lemma 6.2 are cobrations between CW complexes, hence their adjoints are cobrations and therefore inclusions [33]. We restrict attention to complex G-universes U and complex inner product spaces, and we think of R as the underlying real inner product space of U G C ; compare Remark 6.5. For each compact Lie group = G and G-universe U , we obtain an inclusion G-prespectrum T(G,U ) indexed on U . We write M U(G,U ) for its associated G-spectrum. We have seen in Section 6 that these are E ring G-spectra and so determine weakly equivalent S(G,U ) -algebras. All that remains to complete the proof of Theorem 3.3 and thus of Theorem 1.3 is to construct Thom classes (10.1)
2nV 2n (V ) M U(G,U ) M U(G,U ) (S V ), =

where V is a complex representation of complex dimension n, and prove their naturality. We take U to be a complete complex G-universe, and we may assume that V is a nite dimensional subspace of U . Let T(G,U ) (V ) be the Thom complex associated to the canonical complex n-plane G-bundle over the Grassmannian Grn (V U ) of n-planes in V U . For V W , let : T(G,U ) V S W V T(G,U ) W be the map of Thom complexes induced by the evident map from the sum of the canonical n-plane bundle over Grn (V U ) and the trivial bundle W V over the point {W V } to the canonical q-plane bundle over Grn (W U ), where dim(W ) = q. The inclusion V V V U induces a G-map T V T(G,U ) V , and these maps together dene a map of inclusion G-prespectra i : T(G,U ) T(G,U ) . A standard and easy comparison of colimits of homotopy groups shows that the associated map M U(G,U ) M U(G,U ) of G-spectra is a spacewise G-equivalence and thus a weak equivalence. We also have evident unit and product maps : S V T(G,U ) V and : T(G,U ) V T(G,U ) W T(G,U U ) (V W ) that are compatible with the unit and product maps of T . Of course, the advantage of T(G,U ) and its associated G-spectrum M U(G,U ) is that the Grassmannians Grn (V U ) are classifying spaces for complex n-plane G-bundles. The inclusion of V in U may be viewed as a map of G-bundles from the trivial bundle over a point to the universal n-plane bundle over Grn (Cn U ). On passage to Thom complexes, it gives a map t(V ) : S V T(G,U ) Cn .

26

J.P.C. GREENLEES AND J.P. MAY

Composing with the natural map to M U(G,U ) Cn , we see that t(V ) represents an element (V ) as in (10.1). It is a Thom class, as is standard (e.g. [8] or [7, 2.1]) and can be veried in various ways. Perhaps the simplest is to dene t1 (V ) : S 2n T(G,U ) V by reversing the roles of V and Cn . Composing with the natural map to M U(G,U ) V ,
V 2n we see that t1 (V ) represents an element 1 (V ) M U(G,U ) . Via , the smash 1 product of t(V ) and t (V ) induces a G-map

S V +2n T(G,U U ) (V Cn ) that is homotopic to the map obtained by including V Cn as the base plane. The latter map is part of the unit map , and a standard unravelling of denitions shows that (V ) and 1 (V ) are inverse units of the RO(G)-graded ring M U G M UG . = # This completes the verication that each M UG has Thom classes. To show their compatibility under restriction, consider a G-space V in a G-universe U and observe that (10.2) t(V )|H = t(V |H ) : S V T(H,U ) Cn , hence (V )|H = (V |H ). To show their compatibility under conjugation, consider an H-space V in an Huniverse U and an element g G, write gU for the universe U regarded as a gH universe by pullback along cg : gH H, and observe that (10.3) t(g V ) = c (t(V )) : S g
g

T(gH,gU ) Cn , hence (g V ) = cg ((V )).

To prove their multiplicativity, recall the external form of our basic products displayed in (7.12). The following immediate observation gives an external multiplicativity formula from which the internal one of Denition 3.7 follows. Lemma 10.4. Let U be a G-universe and U be a G -universe, and let V U and V U . Then the following diagram commutes: SV SV
= t(V )t(V )

/ T(G,U ) V T(G ,U ) V 

 S V V
t(V V )

/ T(GG ,U U ) (V V ).

Therefore (V )(V ) = (V V ). 11. The proof of Lemma 3.4 Let us say that a representation V of G detects a subgroup H if V = 0 but V H = 0. Then Lemma 3.4 can be interpreted as stating that there are nitely many complex representations Vi of the T such that every subgroup of T is detected by one of the induced representations indG Vi . Recall that F denotes the nite quotient T group G/T . Consider the irreducible representations V of T . They may be viewed as elements of T = Hom(T, S 1 ). Clearly V detects H unless H ker(V ). It is clear from the denition of induction that resG indG V = T T
f F f

V.

LOCALIZATION AND COMPLETION THEOREMS FOR M U -MODULE SPECTRA

27

Thus H is detected by indG V if and only if it is detected by all of the conjugate T representations f V , that is, if and only if, for all f F , H is not in the kernel of f V . Note that ker(f V ) = f (ker V ). Thus, for any list of irreducible representations V1 , , Vq , if Ki is the kernel of Vi , then each subgroup not detected by the Vi is a subgroup of
f1

K1 fqKq

for some list of elements fi of F . Let T have rank r. Proceeding inductively, we choose irreducible representations V1 , , Vr such that, for 1 q r, each displayed intersection of conjugated kernels has rank r q. We begin the induction by choosing any V1 . Certainly each f1K1 has rank r 1. Assume that V1 , , Vq have been chosen, where q < r. Thinking on the Lie algebra level, and noting that conjugations induce translations of Lie algebras, we see that it suces to choose the kernel Kq+1 of Vq+1 so that none of the (r q)-dimensional subspaces f1 (LK1 ) fq (LKq )
1 of LT is contained in any fq+1 (LKq+1 ). Translating by fq+1 , we see that each such condition excludes an (r q)-dimensional subspace of LT from lying in LKq+1 . Dually, consider the nitely many q-dimensional subspaces 1 { | (fq+1 (f1 (LK1 ) fq (LKq ))) = 0} (LT ) .

Since q < r, the set theoretic union of these subspaces cannot be dense in (LT ) . Therefore we can choose (LT ) such that is in none of these subspaces and such that the kernel of has rational basis with respect to the lattice on LT given by the kernel of the exponential. The rationality condition ensures that this kernel is the Lie algebra of the kernel Kq+1 of a representation Vq+1 : T S 1 whose induced map LT R of Lie algebras is . At the rth stage, all of the intersections
f1

K1 frKr

have dimension zero and are therefore nite. To detect the nitely many subgroups in these nitely many intersections, we need only detect their nonidentity elements g. However, if Vg is a representation whose kernel Kg does not contain all of the conjugates fg, then indG Vg detects g, so this is easily done. T References
[1] M.F. Atiyah and G.B. Segal. Equivariant K-theory and completion. J. Di. Geom. 3 (1969), 1-18. [2] A.Bahri, M.Bendersky, D.Davis and P.Gilkey. Complex bordism of groups with periodic cohomology. Trans. Amer. Math. Soc. 316 (1989), 673-687. [3] G. Carlsson. Equivariant stable homotopy and Segals Burnside ring conjecture. Ann. Math. 120 (1984), 189-224. [4] G. Carlsson. A survey of equivariant stable homotopy theory. Topology 31 (1992), 1-27. [5] G. Comeza a. Calculations in complex equivariant bordism. In J. P. May, et al. Equivariant n homotopy and cohomology theory. NSF-CBMS Regional Conference Series in Mathematics No. 91. 1996. [6] G. Comeza a and J. P. May. A completion theorem in complex cobordism. In J. P. May, et n al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Series in Mathematics No. 91. 1996. [7] S.R. Costenoble. The equivariant Conner-Floyd isomorphism. Trans. Amer. Math. Soc. 304, 801-818. [8] T. tom Dieck. Bordism of G-manifolds and integrality theorems. Topology 9 (1970), 345-358.

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[9] A.D. Elmendorf, J.P.C. Greenlees, I. Kriz, and J.P. May. Commutative algebra in stable homotopy theory and a completion theorem. Math. Res. Letters 1 (1994), 225-239. [10] A.D. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May. Modern foundations of stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James. North Holland. 1995. [11] A.D. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May. Rings, modules, and algebras in stable homotopy theory. Amer. Math. Soc. Surveys and Monographs Vol 47. 1996. [12] A.D. Elmendorf, L.G. Lewis, Jr., and J.P. May. Brave new equivariant foundations. In J. P. May, et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Series in Mathematics No. 91. 1996. [13] A.D. Elmendorf and J.P. May. Algebras over equivariant sphere spectra. J. Pure and Applied Algebra. To appear. [14] L. Evens. The cohomology of groups. Oxford Mathematical Monographs. 1991. [15] J.P.C. Greenlees. K homology of universal spaces and local cohomology of the representation ring. Topology 32 (1993), 295-308. [16] J.P.C. Greenlees. Augmentation ideals of equivariant cohomology rings. Preprint, 1996. [17] J.P.C. Greenlees and J.P. May. Completions of G-spectra at ideals of the Burnside ring. Proc. Adams Memorial Symposium, Volume II, CUP (1992), 145-178. [18] J.P.C. Greenlees and J.P. May. Derived functors of I-adic completion and local homology. J. Algebra 149 (1992), 438-453. [19] J.P.C. Greenlees and J.P. May. Generalized Tate cohomology. Memoir American Math. Soc. No. 543. 1995. [20] J.P.C. Greenlees and J.P. May. Completions in algebra and topology. Handbook of Algebraic Topology, edited by I.M. James. North Holland. 1995. [21] J.P.C. Greenlees and J.P. May. Equivariant stable homotopy theory. Handbook of Algebraic Topology, edited by I.M. James. North Holland. 1995. [22] J.P.C. Greenlees and J.P. May. Brave new equivariant algebra. In J. P. May, et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Series in Mathematics No. 91. 1996. [23] J.P.C. Greenlees and J.P. May. Localization and completion in complex bordism. In J. P. May, et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Series in Mathematics No. 91. 1996. [24] A. Grothendieck (notes by R.Hartshorne). Local cohomology. Lecture notes in maths. Springer Lecture Notes Volume 42. 1967. [25] I. Kriz. Morava K-theory of classifying spaces: some calculations. Topology. To appear. [26] M.J. Hopkins, N.J. Kuhn, and D.C. Ravenel. Morava K-theory of classifying spaces and generalized characters for nite groups. Springer Lecture Notes in Mathematics Volume 1509. 1992, pp 186-209. [27] J.R. Hunton. The Morava K-theory of wreath products. Math. Proc. Camb. Phil. Soc. 107 (1990), 309-318. [28] D.C. Johnson and W.S. Wilson. The Brown-Peterson homology of elementary p-groups. Amer. J. Math. 107 (1985), 427-453. [29] D.C Johnson, W.S. Wilson, and D.Y. Yan. Brown-Peterson homology of elementary pgroups.II. Topology and its Applications 59 (1994), 117-136. [30] P.S. Landweber. Coherence, atness and cobordism of classifying spaces. Proc. Adv. Study Inst. Alg. Top. Aarhus, Denmark, 1970, 257-269. [31] P.S. Landweber. Cobordism and classifying spaces. Proc. Symp. Pure Math. 22 (1971), 125129. [32] P.S. Landweber. Complex bordism of classifying spaces. Proc. Amer. Math. Soc. 27 (1971), 175-179. [33] L.G. Lewis, Jr. When is the natural map X X a cobration? Trans. Amer. Math. Soc. 273(1982), 147-155. [34] L.G. Lewis, Jr., J.P. May, and M. Steinberger (with contributions by J.E. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics Volume 1213. 1986. [35] P. Ler. Equivariant unitary bordism and classifying spaces. Proc. Int. Symp. Topology and o its applications, Budva, Yugoslavia, (1973), 158-160. [36] P. Ler. Bordismengruppen unitrer Torusmannigfaltigkeiten. Manuscripta Math. 12 o a (1974), 307-327.

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[37] J. P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). E ring spaces and E ring spectra. Springer Lecture Notes in Mathematics Volume 577. 1977. [38] J. P. May. Equivariant and nonequivariant module spectra. J. Pure and Applied Algebra. To appear. [39] J. P. May, et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Series in Mathematics No. 91. 1996. [40] M. Tezuka and N.Yagita. Cohomology of nite groups and Brown-Peterson cohomology I. Springer Lecture Notes in Mathematics Volume 1370. 1989, 396-408. [41] M. Tezuka and N. Yagita. Cohomology of nite groups and Brown-Peterson cohomology II. Springer Lecture Notes in Mathematics Volume 1418. 1990, 57-69. [42] N.Yagita. Equivariant BP cohomology of nite groups. Trans. Amer. Math. Soc. 317 (1990), 485-499. [43] N.Yagita. Cohomology for groups of rankp G = 2 and Brown-Peterson cohomology. J. Math. Soc Japan 45 (1993), 627-644. School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK E-mail address: j.greenlees@@sheffield.ac.uk Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA E-mail address: may@@math.uchicago.edu