Rational torus-equivariant stable homotopy theory II: the algebra of the standard model
aa r X i v : . [ m a t h . A T ] A ug RATIONAL TORUS-EQUIVARIANT STABLE HOMOTOPY II: ALGEBRAOF THE STANDARD MODEL
J. P. C. GREENLEES
Contents
1. Introduction 12. The algebraic model. 23. Inflation maps and localization maps. 54. Homological dimension. 65. A larger diagram of rings and modules. 106. Change of groups functors 137. The associated extended functor 168. Torsion functors 18References 231.
Introduction
The purpose of this paper is to prove a number of purely algebraic results about thecategory A ( G ) constructed in [8] to model rational G -equivariant cohomology theories, where G is a torus of rank r ≥
0. In more detail, the paper [8] introduced an abelian category A ( G ), and a homology functor π A∗ : G - spectra −→ A ( G ), and showed that they can beused to give an Adams spectral sequenceExt ∗ , ∗A ( G ) ( π A∗ ( X ) , π A∗ ( Y )) ⇒ [ X, Y ] G ∗ convergent for any rational G -spectra X and Y . Furthermore, the category A ( G ) was shownto be of injective dimension ≤ r , so that the spectral sequence converges in a finite numberof steps.This is already be enough to motivate an algebraic study of the abelian category A ( G ),but in joint work with Shipley [11], it is shown that the Adams spectral sequence can belifted to a Quillen equivalence G - spectra / Q ≃ DG − A ( G ) . This means that A ( G ) is not only a means for calculation but also it is a means of construc-tion. Accordingly, phenonmena discovered in A ( G ) are realized in G -spectra.The purpose of the present paper is to give two basic results about A ( G ). Firstly, we show(Theorem 4.1) that A ( G ) has injective dimension precisely r . Secondly we give construc-tions of certain torsion functors which allow us to construct certain right adjoints by firstworking in a larger category and then applying the torsion functor; the existence (but notthe construction) of the right adjoint is used in [11]. Along the way, we have an opportunityto prove a flatness result, to describe algebraic counterparts of some basic change of groups djunctions, and to introduce terminology which fits well with the more elaborate structuresused in [11].It is intended to return to the study of A ( G ) elsewhere, giving a more systematic studyof A ( G ), highlighting its similarities to categories of sheaves over projective varieties andexplaining the local-to-global principles alluded to in [8].2. The algebraic model.
In this section we recall relevant results from [8] which constructs an abelian category A ( G ) modelling the category of G -spectra and an Adams spectral sequence based on it. Thestructures from that analysis will be relevant to much of what we do here.2.A. Definition of the category.
First we must construct the the category A ( G ). For thepurposes of this paper we view this as a category of modules over a diagram of rings.The diagram of rings is modelled on the partially ordered set ConnSub ( G ) of connectedsubgroups of G . To start with we consider the single ring O F = Y F ∈F H ∗ ( BG/F ) , where the product is over the family F of finite subgroups of G . To specify the value of thering at a connected subgroup K , we use Euler classes: indeed if V is a representation of G we may defined c ( V ) ∈ O F by taking its components c ( V )( F ) = c H ( V F ) ∈ H ∗ ( BG/F ) tobe classical Euler class for ordinary homology.The diagram O of rings is defined by O ( K ) = E − K O F where E K = { c ( V ) | V K = 0 } ⊆ O F is the multiplicative set of Euler classes of K -essentialrepresentations.The category A ( G ) is a category of modules M over the diagram O of rings. Thus thevalue M ( K ) is a module over E − K O F , and if L ⊆ K , the structure map β KL : M ( L ) −→ M ( K )is a map of modules over the map E − L O F −→ E − K O F of rings.The category A ( G ) is a certain non-full subcategory of the category of O -modules. Thereare two requirements. Firstly they must be quasi-coherent , in that they are determined bytheir value at the trivial subgroup 1 by the formula M ( K ) = E − K M (1) . The second condition involves the relation between G and its quotients. Choosing aparticular connected subgroup K , we consider the relationship between the group G withthe collection F of its finite subgroups and the quotient group G/K with the collection F /K of its finite subgroups. For G we have the ring O F and for G/K we have the ring O F /K = Y ˜ K ∈F /K H ∗ ( BG/ ˜ K ) here we have identified finite subgroups of G/K with their inverse images in G , i.e., withsubgroups ˜ K of G having identity component K . There is an inflation mapinf : O F /K −→ O F whose F th component is the inflation map for the quotient G/F −→ G/ ( F K ). The secondcondition is that the object should be extended , in the sense that for each connected subgroup K there is a specified isomorphism M ( K ) = E − K O F ⊗ O F /K φ K M for some O F /K -module φ K M . These identifications should be compatible in the evident waywhen we have inclusions of connected subgroups.2.B. Spheres.
The only objects we will need to make explicit are spheres of virtual repre-sentations. We begin with spheres of genuine representations.It is convenient to use suspensions, so we recall the definition. If V is an n -dimensionalcomplex representation we divide F into n +1 sets F , F , · · · , F n where F ∈ F i if dim C ( V F ) = i . All but one of these sets is finite. Now, for an O F -module M , the suspension Σ V M isdefined by breaking it into summands corresponding to the partition of finite sets, andsuspending by the appropriate amount on each one:Σ V M = n M i =0 Σ i e F i M. Turning to spheres, we write S V both for the usual one-point compactification and for theassociated object of A ( G ). To start with we haveΦ K S V = S V K . At the identity subgroup we have S V (1) = Σ V S (1) = Σ V O F , this shows us that φ K S V = Σ V K O F /K . We would like to consider the fundamental class ι V = Σ V ∈ Σ V O F = S V (1) . However, we note that this is not a class in a single degree. It is a finite sum of homogeneouselements, namely the idempotent summands corresponding to the partition of F into finitesubgroups F with dim C ( V F ) constant. Bearing this abuse of notation in mind, ι V behaveslike a generator of the O F -module S V (1) (its homogeneous summands generate). We notethat the element 1 ⊗ ι V K ∈ E − K O F ⊗ O F /K Σ V K O F /K = S V ( K ) , acts as a generator in the same sense.The structure maps have the effect e ( V − V K ) β K ( ι V ) = 1 ⊗ ι V K , so that in general we have β KL ( ι V L ) = e ( V L − V K ) − ⊗ ι V K in E − K O F /L ⊗ O F /K Σ V K O F /K . he contents of this section apply without essential change to the case when V = V − V is a virtual representation. It is only necessary to use multiplicativity to define e ( V − V ) = e ( V ) /e ( V ), and to note that the inverses of Euler classes make sense wherever they havebeen used.2.C. Maps out of spheres.
Next we need to understand maps out of spheres, so we considera map θ : S V −→ Y . Lemma 2.1.
Maps θ : S V −→ Y correspond to systems of elements x K ∈ Σ − V K φ K Y for allconnected subgroups, with the property that if L ⊆ Ke ( V L − V K ) β KL ( x L ) = 1 ⊗ x K . The correspondence is specified by θ ( K )(1 ⊗ ι V K ) = 1 ⊗ x K . Proof :
Since the domain is the suspension of a free module, maps Σ V K O F /K −→ φ K Y are uniquely specified by elements x K . For compatibility, note that whenever we have acontainment L ⊆ K of connected subgroups we have a commutative square e ( V L − V K ) − ⊗ ι V K θ β KL ( x L ) ↑ ↑ ι V L θ x L . (cid:3) Remark 2.2.
It is useful to be able to refer to patterns of this sort. Thus a footprint of x ∈ Y (1) is given by the pattern of its images under the basing maps. More precisely, it isa function defined on connected subgroups, and if β K ( x ) = Σ i λ i ⊗ y K,i , the value of the footprint at K is the set { λ i } of elements of E − K O F . Of course the expressionis not unique, so an element will have many footprints, and it is instructive to consider whatfootprints look like and when there are canonical footprints for elements. The footprint issomewhat analogous to the divisor of a function on an algebraic variety.It is worth noting that maps out of spheres are determined by their value at 1. Lemma 2.3.
A map θ is determined by the value θ ( ι V ) ∈ Y (1) . Proof :
Note that S V ( K ) = E − K O F ⊗ O F /K Σ V K O F /K . Under the restriction maps e ( V − V K ) ι V maps to ι V K . Since e ( V − V K ) is a unit in the range, ι V maps to e ( V − V K ) − ⊗ ι V K .Accordingly, θ (1 ⊗ ι V K ) = e ( V − V K ) − ⊗ β ( θ ( ι V )) . (cid:3) It is convenient to write Y ( V ) for the image of the evaluation mapHom( S − V , Y ) −→ Y (1) . . Inflation maps and localization maps.
In this section we discuss two maps between rings that arise in the structure of objects of A ( G ). When we form E − K O F ⊗ O F /K N we are concerned with the O F /K -module structureof E − K O F . We therefore need to discuss the inflation mapinf = inf GG/K : O F /K −→ O F from G/K to G , and then consider the localization. Proposition 3.1.
For any connected subgroup K , both of the maps O F /K inf −→ O F l −→ E − K O F are split monomorphisms of O F /K -modules, and hence in particular they are flat. We will treat the two maps in the following two subsections.3.A.
Inflation maps.
We begin by describing the maps in more detail. For this we needthe map q : F −→ F /K taking the image of a finite subgroup of G in G/K . First note thatthe condition q ( F ) = ˜ K/K amounts to
F K = ˜ K . In particular, ˜ K ⊇ F so that there is amap G/F −→ G/ ˜ K inducing inflationinf : H ∗ ( BG/ ˜ K ) −→ H ∗ ( BG/F ) . Indeed, this makes H ∗ ( BG/F ) into a polynomial algebra over H ∗ ( BG/ ˜ K ), and hence it freeas a module. We can find free module generators by choosing a splitting G/F −→ K/ ( K ∩ F )of the inclusion, and using the image of H ∗ ( BK/ ( K ∩ F )).Since we are working over the rationals, this is isomorphic to the case F = 1, wherenotation is less cluttered. We have a short exact sequence0 −→ H ∗ ( BG/K ) −→ H ∗ ( BG ) −→ H ∗ ( BK ) −→ G ∼ = G/K × K to show H ∗ ( BG ) ∼ = H ∗ ( BG/K ) ⊗ H ∗ ( BK ) . Taking all finite subgroups with q ( F ) = ˜ K we obtain a map˜∆ ˜ K : H ∗ ( BG/ ˜ K ) −→ Y F K = ˜ K H ∗ ( BG/F ) . The codomain is a product of free modules over H ∗ ( BG/ ˜ K ). The entire map q ∗ : O F /K −→O F is the product of the maps ˜∆ ˜ K over subgroups ˜ K with identity component K . Thecodomain is a product of projective modules over O F /K . Lemma 3.2.
For each subgroup ˜ K with identity component K , the map ˜∆ ˜ K is a splitmonomorphism of free H ∗ ( BG/ ˜ K ) -modules, and in particular it is flat. Proof:
To start with, we note that since any vector space has a basis, the product Q i Q isactually a sum of copies of Q . Now consider the polynomial ring P = H ∗ ( BG/ ˜ K ). Since P is finite dimensional in each degree and cohomologically bounded below, the natural map "Y i Σ n i Q ⊗ Q P −→ Y i Σ n i P s an isomorphism provided the suspensions n i are all cohomologically positive and withfinitely many in each degree. Each of the P -modules H ∗ ( BG/F ) can be written in the form H ∗ ( BG/F ) = M j Σ n j P ∼ = Y j Σ n j P. so we see that Q F K = ˜ K H ∗ ( BG/F ) is a free P -module. (cid:3) Localization maps.
We now turn to the localization map O F −→ E − K O F , viewing itas a map of O F /K -modules via inflation.If W is any representation with W G = 0 the suspension Σ w O F is the projective O F -moduleobtained by suspending the F th factor by dim C ( W F ). We may compose q ∗ with the map O F −→ Σ w O F , so that the codomain is again a product of projective O F /K -modules.If W ⊆ W are two representations with W G = W G = 0 the inclusion Σ w O F −→ Σ w O F is multiplication by c dim C ( W F ) − dim C ( W F ) on the F th factor. If W K = W K = 0, using thetensor product decomposition H ∗ ( BG ) = H ∗ ( BG/K ) ⊗ H ∗ ( BK ) we see that the resultingmap H ∗ ( BG ) −→ Σ w − w H ∗ ( BG ) is split mono as a map of free H ∗ ( BG/K )-modules.Allowing quotients by finite subgroups and combining these, we see that Σ w O F −→ Σ w O F is split mono as a map of O F /K -modules. Passing to limits over representations W with W K = 0, we see that O F −→ E − K O F is split mono as a map of O F /K -modules as required.4. Homological dimension.
The purpose of this section is to establish the exact injective dimension of A ( G ). Theorem 4.1.
The category A ( G ) has injective dimension r . Since the category of torsion H ∗ ( BG )-modules has injective dimension r and sits as anabelian subcategory of A ( G ), it follows that any object of A ( G ) concentrated at 1 hasinjective dimension ≤ r , and the torsion module Q shows that there are objects of injectivedimension r . It remains to show that arbitrary objects of A ( G ) have injective dimension ≤ r .In [8, 5.3] we showed that A ( G ) is of injective dimension ≤ r , and deferred the proof ofthe exact bound. The proof is reminiscent of arguments with sheaves on a variety: although A ( G ) does not have enough projectives, the spheres provide a class of objects analagous totwists of the structure sheaf. One expects every object to be a quotient of extensions ofthese, whilst on the other hand, we can show they have injective dimension ≤ r by explicitconstruction. Remark 4.2.
One can also imagine a proof as in the rank 1 case [6, 5.5.2], where one uses thefact that not only is the category of torsion modules over Q [ c ] of injective dimension 1, butso is the category of all modules. The author has not managed to implement this argumentin general, because it involves isolating certain properties of injective resolutions. On the ther hand it would give valuable insights. For example at the level of objects supported indimension 0, one would need to identify a suitable subcategory of all modules which includesall the H ∗ ( BG )-modules that occur as e M (1) for an object of A e ( G ) and has products. Onewould need to understand injective resolutions in this category, and it would be convenientif the modules E − K H ∗ ( BG ) ⊗ H ∗ ( BG/K ) H ∗ ( BG/K ) were injective in the subcategory.4.A.
Injective dimension.
We first explain the strategy, which is to find a large enoughclass of objects known to have injective dimension ≤ r .We say that M is detected in dimension d if M ( H ) = 0 for H of dimension > d and if L ⊆ K with K of dimension d then M ( L ) −→ M ( K ) is monomorphism. Definition 4.3.
We say that a set of objects P forms a set of injective dimension estimators if the following conditions hold(1) Every object of P has injective dimension ≤ r (2) If X is an object of A ( G ) detected in dimension d , there is a map M i P i −→ X where P i is an object of P and the cokernel has support in dimension ≤ d − P is enough to give a proof of Theorem 4.1. Lemma 4.4.
If there is a set P of injective dimension estimators then any qce module is ofinjective dimension ≤ r . Proof:
To start with, we show by induction on d that if M has support of dimension ≤ d then injdim( M ) ≤ r .If d = 0 then M is a sum over finite subgroups F of torsion modules M F over H ∗ ( BG/F ),and where each E − H M F = 0 for all connected subgroups H [8, 4.5]. It follows that M F canbe embedded in a sum of copies of realizable injective modules H ∗ ( BG/F ). This establishesthe base of the induction. The case when d = r will give the statement of the lemma.We now suppose d > < d have injective dimension ≤ r . For the inductive step we have a subsidiaryinduction. We suppose by downwards induction on s that it has been shown that all modulessupported in dimension ≤ d have injective dimension ≤ s . This holds for s = 2 r by [8, 5.3].We show that if s > r then s can be reduced, so that in at most r steps we will have obtainedthe required estimate.Suppose then that M ′ is supported in dimensions ≤ d . First we observe that it suffices todeal with the quotient M detected in dimension d ( M may be constructed as the image of M ′ in the sum of modules f H ( M ( H )) in the resolution in [8, 5.3]). Indeed, we have an exactsequence 0 −→ T −→ M ′ −→ M −→
0. Since T is supported in dimension ≤ d −
1, it hasinjective dimension ≤ r by induction, so if M has injective dimension ≤ s then so does M ′ .Since P is a set of injective dimension estimators, we may construct a map P −→ M ,with P a sum of elements of P so that the cokernel supported in dimension ≤ d −
1. Sincesums of realizable injectives are injective by [8, 5.2], we find injdim( P ) ≤ r . Factorizing p gives two short exact sequences0 −→ K −→ P −→ P −→ −→ P −→ M −→ M −→ . y the subsidiary induction injdim( K ) ≤ s , and since injdim( P ) ≤ r ≤ s − P ) ≤ s −
1. Turning to the second short exactsequence, since M is supported in dimension ≤ d −
1, it has injective dimension ≤ r ≤ s − M ) ≤ s − (cid:3) It remains to show that a set of injective dimension estimators exists. We will find a largeenough class to reduce dimension of support (i.e., to give Condition 2) in Subsection 4.B,and then in Subsection 4.C show that they have injective dimension ≤ r (i.e., that theysatisfy Condition 1).4.B. Sufficiency of spheres.
We show that there are enough virtual spheres to satisfy thesecond condition for a set of injective dimension estimators (4.3).
Proposition 4.5. If M is detected in dimension d then there is a map P −→ M and P awedge of spheres with cokernel supported in dimension ≤ d − . Proof:
It suffices to show that for any dimension d subgroup H and any x ∈ φ H M we canhit e ( U ) ⊗ x for some U with U H = 0. By Lemma 2.1 it suffices to argue that x = x H is partof a family of elements x K ∈ φ K M so that for K ⊆ H we have β K ( x ) = e ( U/U K ) ⊗ x K .To start with we find U and x . Indeed, the cokernel of M (1) −→ M ( H ) is E H -torsion,so we may find U with U H = 0 and x ∈ M (1) with β H ( x ) = e ( U ) ⊗ x . This automaticallygives β K ( x ), but we must argue that this takes the required form.For notational simplicity we write U = U ⊕ U ′ where U = U K and ( U ′ ) K = 0. Nowconsider K ⊆ H , and use the fact that the cokernel of φ K M −→ E − H/K φ K M = E − H/K O F /K ⊗ O F /H φ H M is E H/K -torsion. This shows that there is a representation V of G/K with V H = 0 and y K ∈ φ K M with β HK ( y K ) = e ( V ) e ( U ) ⊗ x H . Next, we use the fact that the cokernel of M (1) −→ M ( K ) is E K -torsion to choose x ′ ∈ M (1)and a representation V ′ with ( V ′ ) K = 0 with β K ( x ′ ) = e ( U ′ ) e ( V ′ ) ⊗ y K . Accordingly β H ( x ′ ) = e ( U ′ ) e ( U ) e ( V ′ ) e ( V ) ⊗ x H . Since the structure maps are monomorphic, we conclude e ( V ) e ( V ′ ) x = x ′ , and applying β K we obtain e ( V ′ ) e ( V ) β K ( x ) = β K ( x ′ ) = e ( V ′ ) e ( U ′ ) ⊗ y K . Cancelling e ( V ′ ) (which is a unit since ( V ′ ) K = 0), we find e ( V ) β K ( x ) = e ( U ′ ) ⊗ y K . Now e ( V ) is inflated from O F /K , so that if β K ( x ) = P i λ i ⊗ x K,i the left hand side is P i λ i ⊗ e ( V ) x K,i . Finally, we wish to conclude β K ( x ) = e ( U ′ ) ⊗ ˆ y K s required, where e ( V )ˆ y K = y K . For this we use the fact that the square φ K M e ( V ) −→ φ K M ↓ ↓E − K O F ⊗ O F /K φ K M ⊗ e ( V ) −→ E − K O F ⊗ O F /K φ K M is a pullback. This in turn follows since O F /K −→ E − K O F is a split monomorphisms byProposition 3.1. Indeed, we have the following elementary lemma Lemma 4.6. If M −→ M ′ is a monomorphism of R -modules and R −→ L is a splitmonomorphism of R -modules, then M −→ M ′ ↓ ↓ L ⊗ R M −→ L ⊗ R M ′ is a pullback. (cid:3) This completes the proof that there are sufficiently many virtual spheres. (cid:3)
Injective dimension of spheres.
Finally we show that spheres satisfy the first con-dition for a set of injective dimension estimators (4.3).
Proposition 4.7.
Any sphere is of injective dimension ≤ r . Proof:
In effect we will show that the Cousin complex gives an injective resolution. Althoughthis is a purely algebraic result, the complex is realizable. Indeed, the resolution for S isrealizable by maps of spaces, and resolutions of other speheres are obtained by suspension.In fact these resolutions are discussed in [8, Section 12], and the resolution of S is thespecial case of [8, Proposition 12.3] for the family of all subgroups. More precisely, it isstated that a certain sequence of spaces S −→ _ H ∈F r − E h H i −→ _ H ∈F r − E h H i −→ · · · −→ _ H ∈F E h H i induces an exact sequence in π A∗ . Here F i denotes the set of subgroups of dimension i , and π A∗ ( E h H i ) is injective for all subgroups H by [8, 10.2]. Each of the maps comes from acofibre sequence E ( F We will now reformulate the definition of A ( G ) slightly. In effect we are just making someof the structure more explicit. This is convenient for certain constructions, and also ensuresthe notation is consistent with that of the topological situation of [11], where the additionalnotation is essential.5.A. The diagram of connected quotients. The structure of our model is that in A ( G )we include a model for K -fixed point objects whenever K is a non-trivial connected sub-group. This leads us to consider the poset ConnSub ( G ), Because the subgroup K indexes G/K -equivariant information it is clearer to rename the objects and consider the poset ConnQuot ( G ) of quotients of G by connected subgroups, where the maps are the quotientmaps.Because this is fundamental, we standardize the display so that ConnQuot ( G ) is arrangedhorizontally (i.e., in the x -direction), with quotients decreasing in size from G/ G/G at the right. When G is the circle ConnQuot ( G ) = { G/ −→ G/G } but when G is a torus of rank ≥ G , butinfinitely many objects at every other level. This makes it harder to draw ConnQuot ( G ),so we will usually draw a diagram by choosing one or two representatives from each level, sothat in rank r , the diagram ConnSub ( G ) is illustrated by G/ −→ G/H −→ G/H −→ · · · −→ G/H r − −→ G/G where H i is an i -dimensional subtorus of G .5.B. The diagram of quotient pairs. The following diagram is what is needed to indexthe relevant information in our context. Definition 5.1. The diagram Q (G) of quotient pairs of G is the partially ordered set withobjects ( G/K ) G/L for L ⊆ K ⊆ G , and with two types of morphisms. The horizontal morphisms h HK : ( G/K ) G/L −→ ( G/H ) G/L for L ⊆ K ⊆ H ⊆ G and the vertical morphisms v KL : ( G/H ) G/K −→ ( G/H ) G/L for L ⊆ K ⊆ H ⊆ G. We will refer to the terms ( G/H ) G/L with rank( G/L ) = d as the rank d row , and to thosewith rank( G/H ) = d as the rank d column . The terms G/H = ( G/H ) G/ form the bottomrow and the terms ( G/L ) G/L form the leading diagonal . Remark 5.2. The notation is generally chosen to be compatible with the slightly largerdiagrams that are required in [11]. However, we have simplified it slightly: the poset Q (G)is the localization-inflation diagram LI ( G ) of [11], and the object ( G/K ) G/L is there denoted( G/K, c ) G/L . y way of illustration, suppose G is of rank 2. The part of a diagram R : Q (G) −→ C including only one circle subgroup K would then take the form R ( G/G ) G/G (cid:15) (cid:15) R ( G/K ) G/K / / (cid:15) (cid:15) R ( G/G ) G/K (cid:15) (cid:15) R ( G/ G/ / / R ( G/K ) G/ / / R ( G/G ) G/ We think of the bottom row as consisting of the basic information, and the higher rows as pro-viding additional structure. The omission of brackets to write R ( G/K ) G/L = R (( G/K ) G/L )is convenient and follows conventions common in equivariant topology.5.C. The structure ring. One particular diagram will be of special significance for us. Definition 5.3. The structure diagram for G is the diagram of rings defined by R ( G/K ) G/L := E − K/L O F /L . Since V K = 0 implies V H = 0, we see that E H/L ⊇ E K/L , so it is legitimate to take thehorizontal maps to be localizations h HK : E − K/L O F /L −→ E − H/L O F /L . To define the vertical maps, we begin with the inflation map inf G/LG/K : O F /K −→ O F /L ,and then observe that if V is a representation of G/K with V H = 0, it may be regardedas a representation of G/L , and Euler classes correspond in the sense that inf( e G/K ( V )) = e G/L ( V ). We therefore obtain a map v HK : E − H/K O F /K −→ E − H/L O F /L . Illustrating this for a group G of rank 2 in the usual way, we obtain O F /G (cid:15) (cid:15) O F /K / / (cid:15) (cid:15) E − G/K O F /K (cid:15) (cid:15) O F / / E − K O F / / E − G O F At the top right, of course O F /G = Q , but clarifies the formalism to use the more complicatednotation. .D. Diagrams of rings and modules. Given a diagram shape D , we may consider adiagram R : D −→ C of rings in a category C . We may then consider the category of R -modules. These will be diagrams M : D −→ C in which M ( x ) is an R ( x )-module and forevery morphism a : x −→ y in D , the map M ( a ) : M ( x ) −→ M ( y ) is a module map overthe ring map R ( a ) : R ( x ) −→ R ( y ).It is sometimes convenient to use extension of scalars to obtain the map f M ( a ) : R ( y ) ⊗ R ( x ) M ( x ) = R ( a ) ∗ M ( x ) −→ M ( y )of modules over the single ring R ( y ).5.E. The category of R -modules. In discussing modules, we need to refer to the structuremaps for rings, so for an R -module M , if L ⊆ K ⊆ H ⊆ G , we generically write α HK : M ( G/H ) G/K −→ M ( G/H ) G/L for the vertical map, and˜ α HK : E − H/L O F /L ⊗ O F /K M ( G/H ) G/K = ( v HK ) ∗ M ( G/H ) G/K −→ M ( G/H ) G/L for the associated map of O F /L -modules. Similarly, we generically write β HK : M ( G/K ) G/L −→ M ( G/H ) G/L for the horizontal map, and˜ β HK : E − H/L M ( G/K ) G/L = ( h HK ) ∗ M ( G/K ) G/L −→ M ( G/H ) G/L for the associated map of E − H/L O F /L -modules, which we refer to as the basing map after [6].In our case the horizontal maps are simply localizations, so all maps in the G/L -row canreasonably be viewed as O F /L -module maps. On the other hand, the vertical maps increasethe size of the rings, so it is convenient to replace the original diagram by the diagram inwhich all vertical maps in the G/L column have had scalars extended to E − L O F . We referto this as the ˜ α -diagram, and think of it as a diagram of O F -modules. Definition 5.4. If M is an R -module, we say that M is extended if whenever L ⊆ K ⊆ H the vertical map α HK is an extension of scalars along v HK : E − H/K O F /K −→ E − H/L O F /L , whichis to say that ˜ α HK : E − H/L O F /L ⊗ O F /K M ( G/H ) G/K ∼ = −→ M ( G/H ) G/L is an isomorphism of E − H/L O F /L -modules.If M is an R -module, we say that M is quasi-coherent if whenever L ⊆ K ⊆ H thehorizontal map β HK is an extension of scalars along h HK : E − K/L O F /L −→ E − H/L O F /L , which isto say that ˜ β HK : E − H/L M ( G/K ) G/L ∼ = −→ M ( G/H ) G/L is an isomorphism.We write qc- R -mod , e- R -mod and qce- R -mod for the full subcategories of R -modules withthe indicated properties. ext observe that the most significant part of the information in an extended object isdisplayed in its restriction to the leading diagonal. For example in our rank 2 example theytake the form M ( G/G ) G/G (cid:15) (cid:15) M ( G/K ) G/K / / (cid:15) (cid:15) E − G/K O F /K ⊗ O F /G M ( G/G ) G/G (cid:15) (cid:15) M ( G/ G/ / / E − K O F ⊗ O F /K M ( G/K ) G/K / / E − G O F ⊗ O F /G M ( G/G ) G/G We will typically abbreviate such a diagram by just writing the final row and abbreviating M φ ( G/K ) = M ( G/K ) G/K : M φ ( G/ / / E − K O F ⊗ O F /K M φ ( G/K ) / / E − G O F ⊗ O F /G M φ ( G/G ) , leaving it implicit that the particular decomposition as a tensor product is part of thestructure.5.F. The category A ( G ) as a diagram of modules over quotient pairs. Having thislanguage allows us a convenient way to encode the information in the category A ( G ). Indeed,there is a functor i : A ( G ) −→ R -moddefined by i ( M )( G/K ) G/L := E − K/L φ L M. It is straightforward to encode the quasi-coherence condition of A ( G ) in the horizontalmaps and the extendedness in the vertical maps. Lemma 5.5. The functor i takes quasicoherent objects of A ( G ) to quasicoherent objects of R -mod and extended objects of A ( G ) to extended objects of R -mod. The category A ( G ) isisomorphic to the category of qce R -modules. (cid:3) In view of this, we henceforth view i as being the inclusion of qce R -modules in all R -modules, which may be factored as A ( G ) = qce- R -mod j −→ e- R -mod k −→ R -mod . We will construct right adjoints Γ h to j in Section 8 and Γ v = k ! to k in Section 7. Thesubscripts h and v refer to the fact that either the horizontal or vertical structure maps havebeen forced to be extensions of scalars. Combining these, we obtain the right adjointΓ = Γ h Γ v to i . 6. Change of groups functors The two most important change of groups on spaces are fixed point functors and inflationfunctors. We describe counterparts to these in our algebraic context. .A. The functor Φ K . We begin with the counterpart of the geometric fixed point functor,which is the natural extension of the fixed point space functor to spectra. Indeed, the model A ( G ) is based around the geometric fixed point funcctor, so the algebraic counterpart ispainless to define and its good properties are built into the model.For emphasis we write the ambient group as a subscript, so that R G denotes the Q (G)-diagram of rings. The fixed point functor is is defined on extended R G -modules. Definition 6.1. The functor Φ M : e- R G -mod −→ e- R G/M -mod is defined by restrictinga module X from the diagram Q (G) to the subdiagram Q (G / M). In other words, it isdefined by the formula (Φ M X )( G/H ) G/K = X ( G/H ) G/K where M ⊆ K ⊆ H ⊆ G and where bars denote the image in G/M .The following property is built in to the definition of qce R G -modules. Lemma 6.2. The functor Φ M takes quasi-coherent modules to quasi-coherent modules. (cid:3) Unfortunately, Φ M is not a right adjoint, but we will see that there is a convenient sub-stitute.6.B. Inflation. Inflation involves filling in the diagram with extensions of scalars. Definition 6.3. The inflation functor inf = inf GG/M : e- R G/M -mod −→ e- R G -mod is definedby extension of scalars. To describe this in more detail, we do not suppose that M is containedin any of the other subgroups as we would usually do, but we will assume L ⊆ K ⊆ H ⊆ G .For an R G/M -module Y the R G -module inf Y is defined on the leading diagonal via(inf Y )( G/H ) G/H = Y ( G/H ) G/H where H denotes the image of H in G = G/M . The remaining entries are given by extendingscalars along the vertical maps. The horizontal h HK : inf Y ( G/K ) G/M −→ inf Y ( G/H ) G/M is obtained from h HK : Y ( G/K ) G/K −→ Y ( G/H ) G/K by extending scalars along the inflation G/K −→ G/M .From the definition, good behaviour on quasi-coherent modules is clear. Lemma 6.4. The functor inf takes quasi-coherent modules to quasi-coherent modules. (cid:3) A substitute for a left adjoint. There is no left adjoint to the geometric fixed pointfunctor, but we have the following description which fulfils a similar purpose. For spectra,there is a familiar connection between representations and geometric fixed points extendingthe one for spaces. For this, we write S ∞ V ( M ) = [ V M =0 S V . he statement for spectra is that[ A, Φ M X ] G/M = [inf A, X ∧ S ∞ V ( M ) ] G or that Lewis-May and geometric fixed points are related byΦ M X ≃ ( X ∧ S ∞ V ( M ) ) M . The counterpart to this in algebra is built in to the category of modules at the level of theunderlying abelian category. Proposition 6.5. If X and A are quasi-coherent, there is an isomorphism Hom R G/M ( A, Φ M X ) = Hom R G (inf A, lim → V M =0 Σ V X ) . If in addition A is small (for example if it is a sphere), this is lim → V M =0 Hom R G (Σ − V inf A, X ) . Proof : For brevity, we write X ⊗ S ∞ V ( M ) = lim → V M =0 Σ V X . First note that Φ M ( X ⊗ S ∞ V ( M ) ) = Φ M X ; indeed, for subgroups H containing M , we have (Σ V X )( H ) = Σ V H ( X ( H )) = X ( H ), and all the maps in the direct system are the identity. Accordingly, taking geometricfixed points gives a map θ : Hom R G (inf A, X ⊗ S ∞ V ( M ) ) −→ Hom R G/M ( A, Φ M X ) . We consider applying θ to a map f : inf A −→ X ⊗ S ∞ V ( M ) . If θf = Φ M f = 0 we can see f = 0. Indeed, if we evaluate at H containing M we see f ( H ) = 0 since f ( H ) = (Φ M f )( H ).If H does not contain M then the basing map( X ⊗ S ∞ V ( M ) )( H ) β MHH ∼ = / / ( X ⊗ S ∞ V ( M ) )( M H ) = X ( M H ) , gives an isomorphism, and the commutative square(inf A )( M H ) f ( MH )=(Φ M f )( MH ) −→ ( X ⊗ S ∞ V ( M ) )( M H ) ↑ ↑∼ =(inf A )( H ) f ( H ) −→ ( X ⊗ S ∞ V ( M ) )( H )shows f ( H ) = 0.To see θ is an epimorphism, we note that Φ M is the identity on subgroups H containing M , so we may view the problem as that of extending a map g : A −→ Φ M X to a map f : inf A −→ X ⊗ S ∞ V ( M ) . Now if H is a subgroup not containing M the above commutativesquare shows that we must take f ( H ) to be the composite(inf A )( H ) −→ (inf A )( M H ) g −→ X ( M H ) = ( X ⊗ S ∞ V ( M ) )( M H ) . The required compatibility of basing maps associated to an inclusion K ⊆ H comes fromthat of g for M K ⊆ M H . (cid:3) . The associated extended functor The purpose of this section is to give a construction of a functor Γ v replacing an R -moduleby an extended R -module, so that its vertical structure maps become extensions of scalars. Theorem 7.1. There is a right adjoint Γ v = k ! to the inclusione- R -mod k −→ R -mod . We will give an explicit construction of the functor k ! . The main complication is notational,so we begin with two examples.7.A. Examples. When G is the circle group, the ring is O F /Gv (cid:15) (cid:15) O F h / / E − G O F /G , An arbitrary module takes the form V α (cid:15) (cid:15) P β / / Q. This induces a diagram E − O F ⊗ V ˜ α (cid:15) (cid:15) P β / / Q, and the associated extended module is the pullback β ′ : P ′ −→ E − O F ⊗ V. Turning to a group of rank 2, we know that for every circle subgroup K , the constructionof k ! on the G/K part of the diagram must do as above. Using the ˜ α maps we start with adiagram E − G O F ⊗ O F /G M ( G/G ) G/G (cid:15) (cid:15) E − K O F ⊗ O F /K M ( G/K ) G/K / / (cid:15) (cid:15) E − G O F ⊗ O F /K M ( G/G ) G/K (cid:15) (cid:15) M ( G/ G/ / / M ( G/K ) G/ / / M ( G/G ) G/ ow apply the rank 1 construction on the top two rows. We may then omit M ( G/G ) G/G inthe top row since the ( G/G ) G/K entry is extended from it. This gives us the α -diagram M ( G/K ) ′ G/K / / (cid:15) (cid:15) E − G/K O F /K ⊗ O F /G M ( G/G ) G/G (cid:15) (cid:15) M ( G/ G/ / / M ( G/K ) G/ / / M ( G/G ) G/ . Extending scalars, we obtain the ˜ α -diagram E − K O F ⊗ O F /K M ( G/K ) ′ G/K / / (cid:15) (cid:15) E − G O F ⊗ O F /G M ( G/G ) G/G (cid:15) (cid:15) M ( G/ G/ / / M ( G/K ) G/ / / M ( G/G ) G/ . Remembering that there are in fact infinitely many circle subgroups K , we take the pull-back of the resulting diagram to give M ( G/ ′ G/ , and the resulting extended module is M ( G/ ′ G/ / / E − K O F ⊗ O F /K M ( G/K ) ′ G/K / / E − G O F ⊗ O F /G M ( G/G ) G/G . The construction. We are ready to give the construction in arbitrary rank. Definition 7.2. Given a diagram M with G of rank r , the construction of the associatedextended module k ! M proceeds in r steps with M = k !0 M λ ←− k !1 M λ ←− · · · λ r ←− k ! r M = k ! M. Assuming k ! n M has been defined in such a way that the top n vertical maps in each columnare extensions of scalars, we define k ! n +1 M to agree with k ! n M except in the rank n + 1 row.To fill in the rank n + 1 row, we work from the rank 0 column back towards the rank n + 1column.We start in the rank 0 column with k ! n +1 M ( G/G ) G/L := E − G/L O F /L ⊗ M ( G/G ) G/G . To fill in the ( G/K ) G/L entry, where rank( G/L ) = n + 1, we suppose that the ( G/H ) G/L entries with K ( H are already filled in. We then obtain a diagram ∆ n +1 M , with two rows(row 0 and row 1). In each row there are entries ( G/H ) G/L for H ⊇ L . Row 1 is the ( n + 1)strow of k ! n M : ∆ n +1 M ( G/H, G/L = k ! n M ( G/H ) G/L . Row 0 is obtained from the rank n row of k ! n M . For each H ) L we take∆ n +1 M ( G/H, G/L = E − H/L O F /L ⊗ O F /H k ! n M ( G/H ) G/H , noting that, since the ˜ α -structure maps of k ! n M above the rank n + 1 row are alreadyisomorphisms, this is also the extension of k ! n M ( G/H ) G/K for any K between L and H .Finally, we take k ! n +1 M ( G/K ) G/L = lim ← ∆ n +1 M. emma 7.3. The maps λ n +1 : k ! n +1 M −→ k ! n M induce isomorphisms ( λ n +1 ) ∗ : Hom( L, k ! n +1 M ) −→ Hom( L, k ! n M ) for any extended R -module L . In particular k ! is right adjoint to the inclusion i : e- R -mod −→ R -mod . Proof: To see ( λ n +1 ) ∗ is an epimorphism, suppose f : L −→ k ! n M is a map, and we attemptto lift it to a map f ′ : L −→ k ! n +1 M is a map. Of course we take f to agree with f ′ exceptin the rank n + 1 row, and we use f ( G/G ) G/G to give the map on the rank 0 column. Wethen work along the rank n + 1 row; when we come to define f ′ ( G/K ) G/L we suppose that f ′ ( G/H ) G/L has already been defined when K ( H . Since k ! n +1 M ( G/K ) G/L is defined as aninverse limit, we use its universal property.To see ( λ n +1 ) ∗ is a monomorphism, suppose the two maps f , f : L −→ k ! n +1 M give thesame map to k ! n M . Evidently f and f agree except perhaps in the rank n + 1 row, andwe must check they agree there. To start with they agree at ( G/G ) G/L , and we then workalong the rank n + 1 row. When we come to ( G/K ) G/L we observe that f and f alreadyagree at ( G/H ) G/L when K ( H . Since k ! n +1 M ( G/K ) G/L is defined as a pullback, the factthat f and f agree there follows. (cid:3) Torsion functors Various obvious constructions on qce modules (i.e., objects of A ( G )) give objects whichare extended but not quasi-coherent (i.e., objects of e- R -mod = ˆ A ( G )). It is thereforeconvenient to have a right adjoint Γ h to j : A ( G ) = qce- R -mod −→ e- R -mod = ˆ A ( G ) . Thus Γ h replaces an extended module by a qce module, which is one in which the horizontalstructure maps are extensions of scalars. We refer to as a torsion functor since its restrictionto objects concentrated at 1 is the E -torsion functor. Theorem 8.1. There is a right adjoint Γ h : ˆ A ( G ) −→ A ( G ) to the inclusion j : A ( G ) −→ ˆ A ( G ) . The construction is a natural extension of that given in Chapters 17-20 of [6], but wewill spend less time here describing alternative approaches. The construction also works forsimilar inclusions, so for brevity we write j : A −→ ˆ A , and allow ourselves to give examplesin rather simpler categories.8.A. Motivation. For the purposes of discussion suppose Y is an object of ˆ A , and considerthe properties Γ h Y is forced to have. Note first that the inclusion j is full and faithful, sowe may write Hom to denote maps both in ˆ A and in A without ambiguity. Thus if T is anobject of A ( G ) then Hom( T, Y ) = Hom( T, Γ h Y ).We need only use spheres S − W as test objects; as before we write Y ( W ) = Hom( S − W , Y ) . ccording to Lemma 2.1, this is the set of elements of (Σ − W Y )( U (1)) with the same footprintas the characteristic element of S − W . Thus we have(Γ h Y )( W ) = Y ( W )for all representations W .For example, as in Proposition 6.5, this allows us to deduce φ G (Γ h Y ) by the calculation φ G (Γ h Y ) = Hom( S , (Γ h Y ) ⊗ S ∞ V ( G ) )= lim → V G =0 Hom( S − V , Γ h Y )= lim → V G =0 Hom( S − V , Y )= lim → V G =0 Y ( V )In fact, the main difference between Y and Γ h Y is that in Γ h Y the elements with differentfootprints must be more separate. Example 8.2. We consider the case of semifree objects for the circle group, so that ˆ A consists of all maps β : N −→ Q [ c, c − ] ⊗ V , and A is the subcategory in which the map β becomes an isomorphism when c is inverted (see [6, Chapter 18] for more details).The case Y = ( Q [ c ] −→ 0) is instructive (see [6, 18.3.2]). In this case, the relevant spheresare S − kz = (Σ − k Q [ c ] c k −→ Q [ c, c − ] ⊗ Q ) . Here β (1) = c k ⊗ 1, so that the footprint is c k . Thus Y ( kz ) = Y ( c k ) = { y ∈ Y ( U (1)) | β ( y ) = c k ⊗ y ′ for some y ′ } . Thus we see Y ( c k ) = Q [ c ] for all k . On the other hand φ G (Γ h Y ) = Q [ c, c − ] so that, assubspaces of (Γ h Y )( U (1)), the subspaces (Γ h Y )( c k ) = Y ( c k ) = Q [ c ] for different k onlyintersect in 0. Indeed, we findΓ h Y = ( N −→ Q [ c, c − ] ⊗ Q [ c, c − ])where N = ker( µ : Q [ c, c − ] ⊗ Q [ c, c − ] −→ Q [ c, c − ]) . Torsion and fixed points. In describing the torsion functor it is convenient to beginby observing some of the properties it will need to have. Accordingly we start by assumingthe right adjoint exists (we could easily establish this by checking the formal properties ofthe inclusion, but since we construct the functor explicitly, this is unnecessary).The following is immediate from Proposition 6.5. Lemma 8.3. If A is quasi-coherent then for any extended module X we have Hom R G/M ( A, Φ M Γ h X ) = lim → V M =0 Hom R G (inf A ⊗ S − V , X ) . (cid:3) Taking A = S and M = G we find the vertex of Γ h X . Corollary 8.4. (Γ h X )( G/G ) G/G = lim → V G =0 Hom R G ( S − V , X ) . .C. Internal and external Euler classes. If W, W ′ are complex representations with W ⊆ W ′ , then the inclusion S W −→ S W ′ is associated to Euler classes in various ways.Firstly, the map Σ W Y −→ Σ W ′ Y induces multiplication by the Euler class e ( W ′ − W ) on evaluation at the trivial subgroup.On the other hand, we can form another version of this by using internal Euler classes.Thus, the inclusion S W −→ S W ′ of spheres induces a map Y ( W ) = Hom( S − W , Y ) −→ Hom( S − W ′ , Y ) = Y ( W ′ )Note that there are maps Y ( W ) −→ (Σ W Y )( U (1)), and that the square Y ( W ) / / (cid:15) (cid:15) (Σ W Y )( U (1)) (cid:15) (cid:15) Y ( W ′ ) / / (Σ W Y )( U (1))commutes. There are therefore two compatible ways to pass to limits over diagrams ofrepresentations.By definition, for qce modules we have E − K Y ( U (1)) = E − K O F ⊗ O F /K φ K Y . The followinglemma records the fact that therefore the internal and external limits agree for qce modules. Lemma 8.5. If Y is qce, and W is a representation of G/K then lim → V K =0 Hom( S − ( W ⊕ V ) , Y ) = ( φ K Y )( W ) . (cid:3) In view of this, it is convenient to write Y ( W ⊕ ∞ V ( K )) := lim → V K =0 Y ( W ⊕ V ) . The construction. The idea in building Γ h Y is to build up the geometric fixed points φ L Γ h Y in order of increasing codimension of L , referring to a localization diagram to spreadout the parts with each footprint.To start with, as suggested by the discussion in Subsection 8.A, we must take φ G (Γ h Y ) = (Γ h Y )( ∞ V ( G )) := Y ( ∞ V ( G )) . Now, suppose c ≥ 1, that φ K (Γ h Y ) has been constructed for all K of codimension ≤ c − L is of codimension c .We form φ L (Γ h Y ) by modifying φ L Y so as to be compatible with the values φ K (Γ h Y ) for K of codimension ≤ c − 1. In fact we use a pullback square of the following form φ L (Γ h Y ) / / (cid:15) (cid:15) φ L Y (cid:15) (cid:15) CH ( L ; Φ L Γ h Y ) / / CH ( L ; lφ L Y ) , where the lower horizontal needs to be explained. For the present, the important thing isthat it is defined using only values on subgroups of codimension ≥ c + 1. This defines anextended sheaf Γ h Y of O -modules with a natural transformation Γ h Y −→ Y . o explain the lower horizontal, we begin with the ˇCech functor CH ( L ; F ). It is definedfor a functor F on the non-trivial connected subgroups of G/L . First we choose subgroups S , S , . . . , S c so that we have a direct product decomposition G/L = S × S × · · · × S c .Now form the associated ˇCech-type complexˇ C ( L ; F ) = "Y i F ( S i ) −→ Y i Independence of choices. For proofs it is helpful to note the analogy with well-known constructions from commutative algebra. If we have a commutative ring R and anelement x we may form the stable Koszul complex K •∞ ( x ) = ( R −→ R [1 /x ]) , and given elements x , . . . , x c , and a module M , we may form K •∞ ( x , . . . , x c ; M ) = K •∞ ( x ) ⊗ · · · ⊗ K •∞ ( x c ) ⊗ M. The ˇCech complex is obtained by deleting the M in degree 0 and shifting degree, so thatthere is a fibre sequence K •∞ ( x , . . . , x c ; M ) −→ M −→ ˇ C ( x , . . . , x c ; M ) . Indeed, if we replace elements by multiplicative sequences, we findˇ C ( L ; lφ L Y ) = ˇ C ( E S , . . . , E S c ; φ L Y ) . We have the analogue of the well know fact that ˇCech cohomology is geometric. Lemma 8.6. For the two functors F = φ L Γ h Y and F = lφY , the cohomology CH ( L ; F ) isindependent of the choice of subgroups S , . . . , S c . roof: This is the analogue of the fact that the ˇCech complex for a sequence of elementsdepends only on the ideal they generate. An elementary proof begins by observing that if x ∈ ( x , . . . , x c ) then the natural mapˇ C ( x , . . . , x c , x ; M ) −→ ˇ C ( x , . . . , x c ; M )is a chain equivalence. Indeed, providing we use the right proof, it applies in our situation.The point is to recognize the fibre of this map as K •∞ ( x , . . . , x c ; M [1 /x ]) . Now apply this to our context, assuming that S c +1 is another circle intersecting each of S , . . . , S c only in the identity. From the commutative algebra, we see that the fibre ofˇ C ( L ; F ) S ,...,S c −→ ˇ C ( L ; F ) S ,...,S c +1 consists of a stable Koszul complex formed from values of F on which it is already defined.Furthermore, the multiplicative set E S c +1 is inverted on all of those values, and every elementof this multiplicative set is in the ideal generated by the multiplicative sets E S , . . . , E S c . Thefibre is therefore acyclic.To complete the proof, we can move between any two sets of circles giving G as a productby adding and removing circles in general position. (cid:3) Properties. It remains to check that the construction has the desired properties. Lemma 8.7. If Y is quasi-coherent then Γ h Y = Y . Proof: Again we prove this at L by induction on the codimension of L . When L = G wehave φ G Y = Y ( ∞ V ( G )) as required.Now suppose L is of codimension c ≥ φ K Γ h Y −→ φ K Y is an isomorphism for all K of codimension ≥ c + 1. It follows that the lower horizontal in the defining pullback squareis an isomorphism, and hence the upper is an isomorphism as required. (cid:3) Lemma 8.8. For any Y , the module Γ h Y is quasicoherent. Proof: We need to show that E − K/L φ L Γ h Y = E − K/L O F /L ⊗ O F /K φ K (Γ h Y ) when L is of codi-mension 1 in K . For notational simplicity we treat the case L = 1.Choose S = K in the decomposition of G . Consider the diagram obtained from thedefining pullback square by inverting E K . First note that E − K Y ( U (1)) = E − K CH ( L, lY ( U (1)));this is most easily seen by noting that the augmented complex Y ( U (1)) −→ ˇ C ( L ; lY ) is thestable Koszul complex K •∞ ( E K , E S , . . . , E S c ; Y ( U (1))) . It is therefore a tensor product of stable Koszul complexes, the first of which, Y ( U (1)) −→E − K Y ( U (1)), becomes acyclic when we invert E K .From the properties of pullbacks, the left hand vertical becomes an isomorphism when E K is inverted. We therefore need to show that E − K CH ( L ; φ L Γ h Y ) = E − L O F ⊗ O F /K φ K Γ h Y s an isomorphism. We want to apply the same argument, but we need to avoid reference tothe value of Γ h Y at 1, which has not yet been defined. For example we may pick off each ofthe quotient complexes E − K ( E − S j O F ⊗ O F /Sj φ S j Γ h Y ) −→ E − K × S j O F ⊗ O F /K × Sj φ K × S j Γ h Y for j = 1, which are acyclic by induction, until we are left with just E − K ( E − K O F ⊗ O F /K φ K Γ h Y ) = E − K O F ⊗ O F /K φ K Γ h Y in degree 0. (cid:3) We thus have a unit Γ h jX −→ X which is an isomorphism and counit j Γ h Y −→ Y , whichis an isomorphism on A . It follows that the functor Γ h is right adjoint to inclusion. References [1] W.G.Dwyer and J.P.C.Greenlees “The equivalence of categories of torsion and complete modules.”American Journal of Mathematics (2002) 199-220.[2] W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar “Duality in algebra and topology.” Advances in Maths (2006) 357-402[3] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and Algebras in Stable HomotopyTheory , Volume 47 of Amer. Math. Soc. Surveys and Monographs . 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(2001) 99-125[8] J.P.C.Greenlees “Rational torus-equivariant cohomology theories I: calculating groups of stable maps.”JPAA (2008) 72-98[9] J.P.C.Greenlees “Rational S -equivariant elliptic cohomology.” Topology (2005) 1213-1279[10] J.P.C.Greenlees “Algebraic groups and equivariant cohomology theories.” Proceedings of 2002 NewtonInstitute workshop ‘Elliptic cohomology and chromatic phenomena’, Cambridge University Press, (2007)89-110pp[11] J.P.C.Greenlees and B.E.Shipley “An algebraic model for rational torus-equivariant spectra.” Preprint(2011) 75pp, arXiv:1101.2511 Department of Pure Mathematics, The Hicks Building, Sheffield S3 7RH. UK. E-mail address : [email protected]@sheffield.ac.uk