An introduction to algebraic models for rational G-spectra
aa r X i v : . [ m a t h . A T ] A p r AN INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA DAVID BARNES AND MAGDALENA KĘDZIOREK
Abstract.
The project of Greenlees et al. on understanding rational G –spectra in terms ofalgebraic categories has had many successes, classifying rational G –spectra for finite groups, SO (2) , O (2) , SO (3) , free and cofree G –spectra as well as rational toral G –spectra for arbitrarycompact Lie groups.This paper provides an introduction to the subject in two parts. The first discusses rational G –Mackey functors, the action of the Burnside ring and change of group functors. It gives acomplete proof of the well-known classification of rational Mackey functors for finite G . Thesecond part discusses the methods and tools from equivariant stable homotopy theory neededto obtain algebraic models for rational G –spectra. It gives a summary of the key steps in theclassification of rational G –spectra in terms of a symmetric monoidal algebraic category.Having these two parts in the same place allows one to clearly see the analogy between thealgebraic and topological classifications. Contents
1. Introduction 2
Part 1. The structure of rational Mackey functors
32. An introduction to rational Mackey functors 33. Change of group functors 74. The classification of rational Mackey functors 105. The diagonal decomposition 146. Comparison to equivariant spectra 167. Monoidal properties 17
Part 2. The structure of rational G -spectra G -spectra 199. Idempotents of the rational Burnside ring 2010. Left and right Bousfield localisations and splittings 2511. Change of groups and localisations 2912. An algebraic model for rational G -spectra - overview of some cases 34References 36
1N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 2 Introduction
The project of Greenlees et al. on understanding rational G –spectra in terms of algebraic cate-gories has had many successes, classifying rational G –spectra for finite groups, SO (2) , O (2) , SO (3) ,free and cofree G –spectra as well as rational toral G –spectra for arbitrary compact Lie groups. Theproject has expanded to consider (commutative) ring spectra in terms of these algebraic mod-els. This paper provides an introduction to this body of work, whose papers often assume a deepfamiliarity with rational equivariant homotopy theory.Starting from the definition of rational G -Mackey functors, we explain how the rational Burnsidering acts on this category and how change of groups functors behave. Combining these functors, wegive an accessible account of the structure and classification of rational G -Mackey functors in termsof group rings and a comparison of the monoidal structures. We explain how this classification isthe template for the classifications of rational G –spectra for varying G .The second half of the paper considers rational G –spectra for G a compact Lie group. Here therational Burnside ring appears as the ring of self maps of the sphere spectrum. We describe thestructure of this ring and its idempotents. Following the template, we show how the same approach(Burnside ring actions, restriction to subgroups and fixed points) is used in the various classificationsof rational G –spectra. We also discuss the additional complexities (isotropy separation, localisationsand cellularisations) that are needed for spectra.The conjecture by Greenlees states that for any compact Lie group G there is a nice gradedabelian category A ( G ) , such that the category d A ( G ) of differential objects in A ( G ) with a certainmodel structure is Quillen equivalent to the category of rational G –spectra G – Sp Q ≃ Q d A ( G ) . Nice here means that the category A ( G ) is of injective dimension equal to the rank of G and of aform that is easy to use in calculations. If we find such A ( G ) and d A ( G ) equipped with a modelstructure Quillen equivalent to G – Sp Q , we say that A ( G ) is an abelian model and d A ( G ) is an algebraic model for rational G –spectra. The conjecture is known for quite a number of groups insome form. Particularly useful examples are the case of O (2) as given in [Bar17] and [Gre98b]; and SO (3) as given in [Kęd17b] and [Gre01]. We refer to [GS17] for a more complete summary of theknown cases.Since [GS17] was published there was significant development in the field. This includes extendingthe existence of algebraic models to profinite groups (see [BS20] and [Sug19]) as well as takingvarious complexities with monoidal structure into account (see [BGK18a], [BGK18b] and [PW19]).We refer the reader to [BG19] for a related result stating that a nice stable, monoidal model categoryhas a model built from categories of modules over completed rings in an adelic fashion.The aim of this paper is to give a new introduction and explanation to some of these existingresults while demonstrating the analogy between the algebraic and topological sides. By doing so,we intend to give an overview of the methods and tools used in obtaining algebraic models forrational G –spectra and provide a step-by-step guide, at least in some cases. Acknowledgements.
The second author is grateful for support from the Dutch Research Council(NWO) under Veni grant 639.031.757.
N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 3 Part The structure of rational Mackey functors An introduction to rational Mackey functors
For G a finite group, the category of Mackey functors is an abelian category that is important togroup theorists and algebraic topologists working equivariantly. Working over the rationals greatlysimplifies the category, rationally it splits into a direct product of modules over group rings of theWeyl groups of subgroups of G (counted up to conjugacy). We use the rationals for definiteness,but it can be see than any ring such that | G | is invertible will give a splitting result.This result is stated formally as Theorem 4.7. It was proven independently by two sources,Greenlees and May [GM95, Appendix A] and Thévenaz and Webb [TW95]. The former took anapproach from equivariant stable homotopy theory, the latter from algebra. We find the formerapproach simpler, so we follow it, expanding substantially on the proofs. General references for theresults on Mackey functors are Greenlees [Gre92], Greenlees and May [GM92] and Webb [Web00].For a discussion on Mackey functors for compact Lie groups see [Lew98].From the many equivalent definitions of a Mackey functor, we choose one in terms of inductionand restriction maps. Definition 2.1. A rational G -Mackey functor M is: • a collection of Q -modules M ( G/H ) for each subgroup H G , • for subgroups K, H G with K H and any g ∈ G we have a restriction map, an induction map and a conjugation map R HK : M ( G/H ) → M ( G/K ) , I HK : M ( G/K ) → M ( G/H ) and C g : M ( G/H ) → M ( G/gHg − ) . These maps satisfy the following conditions.(1) For all subgroups H of G and all h ∈ HR HH = Id M ( G/H ) = I HH and C h = Id M ( G/H ) . (2) For L K H subgroups of G and g, h ∈ G , there are composition rules I HL = I HK ◦ I KL , R HL = R KL ◦ R HK , and C gh = C g ◦ C h . The first two are transitivity of induction and restriction. The last is associativity of con-jugation.(3) For g ∈ G and K H subgroups of G , there are composition rules R gHg − gKg − ◦ C g = C g ◦ R HK and I gHg − gKg − ◦ C g = C g ◦ I HK . This is the equivariance of restriction and induction.(4) For subgroups
K, L H of GR HK ◦ I HL = X x ∈ [ K (cid:31) H (cid:30) L ] I KK ∩ xLx − ◦ C x ◦ R LL ∩ x − Kx . This condition is known as the
Mackey axiom .We denote the category of rational Mackey functors by Mackey ( G ) .To save space, many texts shorten the input and write M ( H ) := M ( G/H ) . This notation alignsbetter with the terms induction and restriction, but precludes the following remark. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 4 Remark 2.2.
Since every finite G –set is (up to non-canonical isomorphism) a disjoint union oforbits G/H , we can (by choosing such an isomorphism) extend any Mackey functor to take inputfrom the category of finite G –sets and G –maps by sending disjoint union to direct sums. We willrepeatedly use this extension (without further notice) in the adjunctions on Mackey functors thatwe define later.Lindner [Lin76] uses this extension to give an equivalent definition of Mackey functors in terms ofa pair of covariant and contravariant functors from finite G –sets to Q -modules. These functors agreeon objects, send disjoint unions to direct sums and satisfy a pullback condition (that is equivalentto the Mackey axiom). The equivalence is proven via the decomposition G/K × G/H = a x ∈ [ K (cid:31) H (cid:30) L ] G/ ( H ∩ xKx − ) . A further definition in terms of spans of G –sets (the Burnside category) is also given in thatreference.We illustrate how the structure works for two small groups. Example 2.3.
Let G = C = { , σ } . A rational Mackey functor is a pair of Q -modules M ( C /C ) and M ( C / { } ) . The conjugation maps imply that both Q -modules have an action of C , but it istrivial on the first module. There is a restriction map, which commutes with the C -actions M ( C /C ) −→ M ( C / C ֒ → M ( C / { } ) . Similarly there is an induction map, which commutes with the C -actions M ( C / { } ) −→ M ( C / { } ) /C −→ M ( C /C ) . The Mackey axiom (for H = C , K = L = { } ) says that R C { } I C { } = X x ∈ [ { } (cid:31) C (cid:30) { } ] I { }{ } ◦ C { } ◦ R { }{ } = X x ∈ C C x = Id + C σ Example 2.4.
Let G = C . A rational Mackey functor consists of four Q -modules with mapsbetween them. We draw this as a diagram below. The looped arrows indicate the group that actson each module. M ( C /C ) { } (cid:7) (cid:7) R C C (cid:24) (cid:24) R C C (cid:6) (cid:6) M ( C /C ) C /C (cid:15) (cid:15) R C C , , I C C F F M ( C /C ) C /C (cid:15) (cid:15) R C C r r I C C X X M ( C /C ) C G G I C C I C C l l N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 5 The Mackey axiom also implies that R C C ◦ I C C = I C C ◦ R C C and R C C ◦ I C C = I C C ◦ R C C . There are several general constructions that give examples of Mackey functors.
Example 2.5.
The constant Mackey functor at a Q -module A takes value A at each G/H , theconjugation and restriction maps are the identity map of A , induction from G/K to G/H is multi-plication by index of K inside H . Given that the restriction maps are identities, the Mackey axiomprevents the induction maps from being identity maps.We may also define the co-constant Mackey functor at a Q -module A takes value A at each G/H , the conjugation and induction maps are the identity of A and restriction from G/H to G/K is multiplication by index of K inside H .The similarity between the constant and co-constant Mackey functors is an example of dualityof Mackey functors. Lemma 2.6.
Given a Mackey functor M , there is a dual Mackey functor DM , which at G/H takes value DM ( G/H ) = Hom( M ( G/H ) , Q ) . The conjugation maps for M induce conjugation maps for DM , though the contravariance of D ( − ) requires us to use C g − for M to define C g for DM . The induction maps of DM are induced fromthe restriction maps of M and the restriction maps are induced from the induction maps of M . Many well-known structures arising from group theory can be assembled into Mackey functors.
Example 2.7.
Let R ( G ) denote the ring of complex representations of the finite group G . Wedefine a rational Mackey functor M R by M R ( G/H ) = R ( H ) ⊗ Q , with induction and restrictioninduced by induction and restriction of representations.The ring structure on R ( G ) gives more structure to this Mackey functor, it is in fact a Tambarafunctor . See Strickland [Str12] for a survey of such functors and related notions like Green functors.
Example 2.8.
The equivariant stable homotopy groups of a G –spectrum are a Mackey functor.For X an orthogonal G –spectrum over a complete G -universe, let [ − , X ] G ⊗ Q denote the functorwhich sends G/H to [Σ ∞ G/H + , X ] G ⊗ Q ∼ = [Σ ∞ S , X ] H ⊗ Q ∼ = π H ( X ) ⊗ Q . We leave the induction, restriction and conjugation maps to the standard references of May [May96]and Elmendorf et al. [EKMM97].We also note that G –equivariant cohomology theories use Mackey functors as their coefficients,rather than abelian groups. Example 2.9.
Given a Q [ G ] -module V , we may define a rational Mackey functor Mack G ( V ) astaking value V H at G/H . The restriction maps are inclusion of fixed points and the induction mapsare given by coset orbits.We could also define a Mackey functor by taking value
V /H at G/H . The two functors arerelated via duality, and in the rational case they are isomorphic, as we now explain. Since G isfinite, there is a diagram V H inclusion / / V quotient / / av H w w V /H av ′ H y y N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 6 where av H ( x ) = 1 | H | X h ∈ H hx and av ′ H ([ x ]) = 1 | H | X h ∈ H hx. The composite of inclusion and quotient V H ∼ = V /H is an isomorphism with inverse given by thecomposite av H ◦ av ′ H .When V = Q with trivial G -action, Mack G ( Q ) is an instance of the constant Mackey functor,see Example 2.5. Example 2.10.
The rational Burnside rings for subgroups of G assemble into a Mackey functor, A Q ( G/H ) = A Q ( H ) , the rational Grothendieck ring of finite H –sets. The structure maps are theusual restriction and induction of sets with group actions. Moreover, the restriction maps are mapsof rings.As is well-known, the rational Burnside ring splits. Lemma 2.11.
For G a finite group, there is an isomorphism of rings A Q ( G ) −→ C (Sub( G ) /G, Q ) = C (Sub( G ) , Q ) G = Y ( H ) G Q where Sub( G ) /G is the set of conjugacy classes of subgroups of G and C (Sub( G ) /G, Q ) is the setof continuous maps between the two spaces (both equipped with the discrete topology). We define C (Sub( G ) , Q ) to have a G -action by conjugation on the domain.We define e GH ∈ A Q ( G ) to be the element of the Burnside ring corresponding to the characteristicmap of ( H ) in C (Sub( G ) /G, Q ) .Proof. The isomorphism is defined by sending a G –set T to the map ( H )
7→ | T H | . Since the domainand codomain have the same dimension, the result follows from proving the map is surjective, whichfollows from the formulas of the following lemma. (cid:3) We can compare idempotents with the additive basis by a formula from Gluck [Glu81].
Lemma 2.12.
Let H be a subgroup of G , then e GH ∈ A Q ( G ) is given by the formula e GH = X K H | K || N G H | µ ( K, H ) G/K where µ ( K, H ) = Σ i ( − i c i for c i the number of strictly increasing chains of subgroups from K to H of length i . The length of a chain is one less than the number of subgroups involved and µ ( H, H ) = 1 for all H G .Let H and K be subgroups of G , then G/H = X K H | N G K || H | e K . Example 2.13.
Let G = C , then A Q ( G ) is generated by the one-point space C /C which isthe monoidal unit, and C / { } . The only non-evident multiplication is C / { } × C / { } = 2 C / { } . It follows that e = (1 / C is an idempotent, as is e C = 1 − e . Looking at the fixed points ofthese sets show that the idempotents are correctly named and we recover the isomorphism A Q ( C ) ∼ = Q h e i × Q h e C i . N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 7 Remark 2.14.
The restriction map A Q ( H ) → A Q ( K ) in terms of C (Sub( H ) /H, Q ) → C (Sub( K ) /K, Q ) corresponds to precomposing with the map including subgroups Sub( K ) → Sub( H ) and takingsuitable orbits. We can use this description to see how the restriction map interacts with idempo-tents. Let A and H be subgroups of G . Then the restriction of the idempotent e GH to A is still anidempotent, but it is not always e AH . Instead, R GA ( e GH ) = X K A AK ∈ ( H ) G e AK where the sum runs over A -conjugacy classes of subgroups K of A , such that K is G -conjugate to H .We see that if H is not G –subconjugate to A , this will be zero. Contrastingly, if H is G -conjugateto A , then the only term in the summand will be K = A and R GA ( e GH ) = e AA .Given a G -Mackey functor M , we can define an action of the Burnside ring A Q ( H ) on the abeliangroup M ( G/H ) by [ H/K ] := I HK R HK : M ( G/H ) −→ M ( G/H ) and extending linearly from the additive basis for A Q ( H ) given by H/K for subgroups K of H .The Mackey axiom implies that this action is compatible with the multiplication of A Q ( H ) , so that M ( G/H ) is a module over A Q ( H ) . Moreover, the following square commutes. A Q ( G/H ) ⊗ M ( G/H ) / / R HK ⊗ R HK (cid:15) (cid:15) M ( G/H ) R HK (cid:15) (cid:15) A Q ( G/K ) ⊗ M ( G/K ) / / M ( G/K ) The action of Burnside rings is compatible with induction in the sense of the
Frobenius reciprocity relations. Let α ∈ A Q ( G/H ) , β ∈ A Q ( G/K ) , m ∈ M ( G/H ) and n ∈ N ( G/H ) α · I HK ( m ) = I HK ( R HK ( α ) · m ) I HK ( β ) · n = I HK ( β · R HK ( n )) See [Yos80, Definition 2.3 and Example 2.11].
Lemma 2.15.
Given an idempotent e ∈ A Q ( G ) and a G -Mackey functor M we can define a newMackey functor eM by ( eM )( G/H ) = R GH ( e ) M ( G/H ) . Proof.
The conjugation and restriction maps are as for M , since these actions are compatible withrestriction.By Frobenius reciprocity, the induction map for K H gives a map R GK ( e ) M ( G/K ) I HK −−→ R GH ( e ) M ( G/H ) . (cid:3) Change of group functors
As one should expect, we have adjunctions coming from inclusions of subgroups and projectionsonto quotients.
N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 8 Definition 3.1.
Given an inclusion of a subgroup i : H → G , there are functors i : Mackey ( G ) −→ Mackey ( H ) and i : Mackey ( H ) −→ Mackey ( G ) . Using the extension of Mackey functors to finite G –sets, we may define the functor i as pre-composition with the forgetful functor on sets with group actions. The functor i is defined bypre-composition with extension of groups. Thus for M ∈ Mackey ( G ) , N ∈ Mackey ( H ) , A a G –setand B a H –set, ( i M )( B ) = M ( G × H B ) ( i N )( A ) = N ( i ∗ A ) . Similar definitions hold for the induction, restriction and conjugation maps; and for morphisms ofMackey functors.
Lemma 3.2.
Given an inclusion of a subgroup i : H → G , there is an adjunction i : Mackey ( G ) / / Mackey ( H ) : i o o with each functor both left and right adjoint to each other.Proof. To see that this is an adjunction with i as the left adjoint, we take a map f : M → i N and construct a map ¯ f : i M → N . Consider an H –set B , then ¯ f ( B ) is given by the composite M ( G × H B ) f ( B ) −−−→ N ( i ∗ ( G × H B )) N ( η B ) −−−−→ N ( B ) where the second map is induced (by using restriction maps) from the canonical map of H –sets η B : B −→ i ∗ ( G × H B ) . Conversely, given g : i M → N we construct ˆ g : M → i N in a similarway. Given a G –set A , ˆ g ( A ) is the composite M ( A ) M ( ε A ) −−−−→ M ( G × H i ∗ A ) g ( i ∗ A ) −−−−→ N ( i ∗ A ) where the first map is induced (by using restriction maps) from ε A : G × H i ∗ A −→ A .Now we take a map f : M → i N and show that it is equal to ˆ¯ f : M → i N (the other case of ¯ˆ g = g is similar). The map ˆ¯ f is defined by taking the lower path in the following diagram. M ( A ) f ( A ) / / M ( ε A ) (cid:15) (cid:15) N ( i ∗ A ) M ( G × H i ∗ A ) f ( G × H i ∗ A ) / / N ( i ∗ G × H i ∗ A ) N ( η i ∗ A ) O O That we have an adjunction follows as N ( η i ∗ A ) ◦ f ( G × H i ∗ A ) ◦ M ( ε A ) = N ( η i ∗ A ) ◦ N ( i ∗ ε A ) ◦ f ( A ) = f ( A ) by the triangle identity for sets with group actions.The proof that ( i , i ) is an adjunction is very similar to the previous case. The primarydifference is that one uses induction maps rather than restriction maps. (cid:3) We want to reproduce this construction for a quotient ε : G → G/N . To make an adjunction,we need to restrict the category of G -Mackey functors somewhat. We take a strong restriction, sothat the two functors we produce will be both left and right adjoint to each other. Definition 3.3.
For N a normal subgroup of G , the category Mackey ( G ) /N is the full subcategoryof Mackey ( G ) which are trivial on those G/K where K does not contain N . N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 9 Definition 3.4.
Given an quotient map ε : G → G/N for N a normal subgroup of G , there arefunctors ε : Mackey ( G ) /N −→ Mackey ( G/N ) and ε : Mackey ( G/N ) −→ Mackey ( G ) /N. Thus for M ∈ Mackey ( G ) /N , M ′ ∈ Mackey ( G/N ) , K a subgroup of G containing N and B a G/N –set, we define ε M ( B ) = M ( ε ∗ B ) and ε M ′ ( G/K ) = M ′ (( G/N ) / ( K/N )) . If K does not contain N we set ε M ′ ( G/K ) = 0 .The structure maps of M and M ′ are defined in terms of these formulae, as are maps of Mackeyfunctors. Lemma 3.5.
Given N a normal subgroup of G , there is an adjunction ε : Mackey ( G ) /N / / Mackey ( G/N ) : ε o o with each functor both left and right adjoint to each other.Proof. Both cases are similar and use the fact that ε ∗ ( G/N ) / ( K/N ) =
G/K.
We give one part of the proof as an illustration.Take f : M −→ ε M ′ a map of G -Mackey functors which are trivial on those G/K where K does not contain N . We want to construct ¯ f : ε M −→ M ′ . Take a subgroup K/N of G/N , thenwe define ¯ f (( G/N ) / ( K/N )) = f ( G/K ) : M ( G/K ) → M ′ (( G/N ) / ( K/N )) . (cid:3) We give one more adjunction, between the category of rational G -Mackey functors and Q -moduleswith an action of G . Lemma 3.6.
There is an adjunction ( − )( G/e ) :
Mackey ( G ) / / Q [ G ]– mod : Mack G o o with each functor both left and right adjoint to each other.The functor ( − )( G/e ) sends a G -Mackey functor to the value M ( G/e ) . Its adjoint Mack G isdefined in Example 2.9 as the fixed points of V (or equally, the orbits of V ).Proof. Take a map f : M → Mack G ( V ) . Evaluating at G/e gives a map ¯ f : M ( G/e ) → V . In theother direction, one starts with a map g : M ( G/e ) → V of Q [ G ] -modules. The restriction map R He : M ( G/H ) → M ( G/e ) takes values in M ( G/e ) H as conjugation by elements of H is trivial in M ( G/e ) . We define ˆ g as f ( G/e ) H ◦ R He .For the adjunction in the other direction, we use the equivalent description of Mack G ( V ) in termsof orbits and follow a similar pattern, using the induction maps of M to define the adjoint of a map V → M ( G/e ) . (cid:3) N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 10 The classification of rational Mackey functors
Let e GH ∈ A Q ( G ) = Q ( H ) G Q be the idempotent which is 1 on factor H and zero elsewhere. Asdescribed above, we can form a full subcategory of Mackey ( G ) consisting of those Mackey functorsof the form e GH M . Applying e GH defines a functor Mackey ( G ) −→ e GH Mackey ( G ) . It follows that wehave a splitting Mackey ( G ) ∼ = Y H G G e GH Mackey ( G ) . To classify rational Mackey functors, it therefore suffices to classify the categories e GH Mackey ( G ) .The key step is the following theorem giving a sequence of adjunctions. The proof of the theoremoccupies the rest of this section. Theorem 4.1.
For H G , there is a sequence of adjunctions of exact functors. e GH Mackey ( G ) i / / R GN G H ( e GH ) Mackey ( N G H ) i o o ε / / Mackey ( N G H/H ) ε o o ( − )( W G H/e ) / / Q [ W G H ]– mod Mack
WGH o o with each pair both left and right adjoint to each other. Lemma 4.2.
The adjunction ( i , i ) restricts to an adjunction i : e GH Mackey ( G ) / / R GN G H ( e GH ) Mackey ( N G H ) : i o o . The functors are exact and are both left and right adjoint to each other.Proof.
Take M ∈ e GH Mackey ( G ) and K N G H . Since M = e GH M , we have the first equality below ( i M )( N G H/K ) = R GK ( e GH ) M ( G/K )= R N G HK R GN G H ( e GH ) M ( G/K )= (cid:0) R GN G H ( e GH )( i M ) (cid:1) ( N G H/K ) . Thus ( i M ) ∈ R GN G H ( e GH ) Mackey ( N G H ) .Conversely, let M ′ ∈ R GN G H ( e GH ) Mackey ( N G H ) and K G . Then ( i M ′ )( G/K ) = M ′ ( i ∗ G/K ) = ⊕ λ ∈ Λ M ′ ( N G H/L λ ) where i ∗ G/K decomposes as ` λ ∈ Λ N G H/L λ . Since M ′ ( N G H/L λ ) = R N G HL λ R GN G H ( e GH ) M ′ ( N G H/L λ ) = R GL λ ( e GH ) M ′ ( N G H/L λ ) , it follows that i M ′ = e GH i M ′ .The functors are additive and left and right adjoint to each other. Hence they are exact. (cid:3) Lemma 4.3.
For any Q [ W G H ] -module V , there is a canonical isomorphism of N G H -Mackeyfunctors R GN G H ( e GH ) ε Mack W G H ( V ) ∼ = ε Mack W G H ( V ) . It follows that we have an adjunction R GN G H ( e GH ) Mackey ( N G H ) / / Q [ W G H ]– mod o o with the functors both left and right adjoint to each other. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 11 Proof.
The inclusion of an idempotent summand gives the map. To see that this inclusion is anisomorphism, we evaluate both sides a subgroup A N G H that contains H . Both domain andcodomain take value zero on subgroups which do not contain H .We first decompose R GA ( e GH ) into idempotents of the rational Burnside ring of AR GA ( e GH ) = X K A AK ∈ ( H ) G e AK . Secondly, we use Lemma 2.12, ( e AK ) is a sum of | W A K | − [ A/K ] and rational multiples of basis ele-ments [ A/K ′ ] for K ′ a proper subgroup of K . The element [ A/L ] acts on ε Mack W G H ( V )( N G H/A ) through I AL R AL . This is zero unless L contains H . Hence each [ A/K ′ ] acts as zero, and [ A/K ] onlyacts non-trivially when K contains H . Since K is also G -conjugate to H , we see that K = H .Hence, (cid:16) X K A AK ∈ ( H ) G e AK (cid:17) ε Mack W G H ( V )( N G H/A ) = e AH ε Mack W G H ( V )( N G H/A ) = e AH V A/H with e AH acting through | W A H | − I AH R AH .Thirdly, I AH R AH is the composite V A −→ V H −→ V A with the first map the inclusion and the second map taking the sum over A/H -coset representatives.Hence this map is multiplication by | A/H | . Since H is normal in A , it follows that | W A H | − I AH R AH acts through the identity, giving the first statement.For the second statement, the inclusion of the full subcategory R GN G H ( e GH ) Mackey ( N G H ) −→ Mackey ( N G H ) /H has an adjoint, which is applying the idempotent R GN G H ( e GH ) . This adjoint is both left and rightadjoint to the inclusion.Composing this with the adjunctions ( ε , ε ) and (( − )( W G H/e ) , Mack W G H ) gives the result.The functors in each adjunction are additive, and are left and right adjoint to each other. Hencethey are exact. (cid:3) Definition 4.4.
For H G , define F H : Q [ W G H ]– mod −→ e GH Mackey ( G ) to be the composite ofthe lower level functors from diagram in Theorem 4.1. Define U H : e GH Mackey ( G ) −→ Q [ W G H ]– modto be the composite of the upper level functors from diagram in Theorem 4.1.We see immediately that the additive functors F H and U H are both left and right adjoint to eachother and that U H M = M ( G/H ) . Proposition 4.5.
For K G , F H ( V )( G/K ) = ( Q [( G/K ) H ] ⊗ V ) W G H . Moreover, the Mackey functor F H ( V ) is both projective and injective, and e GH F H ( V ) = F H ( V ) .Proof. From the definitions, the composite is given by
G/K M λ ∈ Λ V L λ /H N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 12 where G/K decomposes as ` λ ∈ Λ N G H/L λ and K contains H . If K does not contain H , thecomposite takes value zero. Each factor in this decomposition corresponds to an N G H -orbit in theset of N G H -maps N G H/H → i ∗ G/K . The N G H -action is by right multiplication by the inverse on N G H/H . Such a map corresponds to a G -map G/H → G/K , which is simply an element α of theset ( G/K ) H . By thinking of ( G/K ) H as a W G H –set, we can sum over all α to obtain the formula G/K (cid:16) M α ∈ ( G/K ) H V α (cid:17) W G H with W G H permuting the summands (it acts by right multiplication by the inverse on ( G/K ) H ).Replacing summands by a tensor product gives the formula F H ( V )( G/K ) = ( Q [( G/K ) H ] ⊗ V ) W G H . Every Q [ W G H ] -module is both injective and projective. Hence, F H ( V ) is both projective andinjective as the functors F H and U H are exact.Lemmas 4.2 and 4.3 give the statement about idempotents. (cid:3) It will be useful later to have a clear description of the induction and restriction maps of F H ( V ) . Lemma 4.6.
For L K G , the induction map ( Q [( G/L ) H ] ⊗ V ) W G H = F H ( V )( G/L ) −→ F H ( V )( G/K ) = ( Q [( G/K ) H ] ⊗ V ) W G H is induced by the projection α : G/L → G/K .The restriction map F H ( V )( G/K ) −→ F H ( V )( G/L ) is induced by the map Q [( G/K ) H ] → Q [( G/L ) H ] , which sends gK to the sum of the elements in itspreimage under the projection α : G/L → G/K .Proof.
Write ( Q [( G/L ) H ] ⊗ V ) W G H as (cid:0) M σ : G/H → G/L V σ (cid:1) W G H , we use the subscript σ on V to keep track of the factors. The W G H -action is given by both actingon V and permuting the summands. That is, for w ∈ W G H and v ∈ F H ( V )( G/L ) , we define wv tohave component in summand σ given by g ( v σ ◦ r ( w − ) ) where r ( w − ) : G/H −→ G/H is right multiplication by w − .Chasing through the definitions, it follows that the restriction map is given by ( R KL y ) σ = y α ◦ σ . The induction map is given by ( I KL y ) τ = X α ◦ σ = τ y σ the sum over those summands σ which map to τ by α . (cid:3) N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 13 Theorem 4.7.
For each H G there is an equivalence of categories U H : e H Mackey ( G ) / / Q [ W G H ]– mod : F H o o Hence there is an equivalence of categories
Mackey ( G ) ∼ = Y ( H ) G Q [ W G H ]– mod where the product runs over G -conjugacy classes of subgroups of G .Proof. We have already seen that U H and F H are both left and right adjoint to each other. Theunit is an isomorphism: V −→ U H F H V = ( Q [( G/H ) H ] ⊗ V ) W G H ∼ = V. It follows that the counit is an isomorphism of Mackey functors of the form F H V .The rest of the proof shows that any Mackey functor is a finite direct sum of Mackey functorsof the form F H V for varying H and V .We partition the set of subgroups of G into sets, which we may think of as their height in thesubgroup lattice. We start with S = { e } , then we define S j as those groups not in S j − but allof whose subgroups are in S i for i < j . Each S j is closed under conjugation, with n j conjugacyclasses. Choose a H j,k in each conjugacy class, k n j . We say that a Mackey functor M is oftype ( j, k ) if M ( G/H j,k ) is non-zero, but M ( G/H j ′ ,k ′ ) = 0 for j ′ < j and for j ′ = j and k ′ < k. We argue via descending induction. Starting at the top, if M ( G/H ) = 0 for all proper subgroups H , then M = F G M ( G/G ) . Fix ( j, k ) inductively and assume that all Mackey functors of type ( j ′ , k ′ ) for j ′ > j and for j ′ = j and k ′ > k are finite direct sums Mackey functors of the form F J V J where V J is a W G J -module and J ∈ { H j ′′ ,k ′′ | j ′′ > j ′ or j ′′ = j ′ and k ′′ > k ′ } . Let M be a Mackey functor of type ( j, k ) . For H = H j,k , there is a map of Mackey functors κ : M −→ F H M ( G/H ) = e GH F H M ( G/H ) which is the identity on G/H . The kernel and cokernel of κ are, by inductive assumption, of the form F J V J . It follows that e GH applied to the kernel and cokernel are zero. Thus e GH κ is an isomorphismand so κ , which is equal to the epimorphism M → e GH M followed by e GH κ , is an epimorphism.Since F H M ( G/H ) is projective, the epimorphism splits and M is a direct sum of F H M ( G/H ) and Mackey functors of the form F J V J . (cid:3) Corollary 4.8.
Every rational Mackey functor is both projective and injective.
By Remark 2.14, for H G we have R GH ( e GH ) = e HH . This gives the following corollary ofTheorem 4.7 Corollary 4.9. A G -Mackey functor M is uniquely determined by the collection { e HH M ( G/H ) ∈ Q [ W G H ]– mod | ( H ) G } where we index over G -conjugacy classes of subgroups of G . The remaining question is how to conveniently find the values M ( G/H ) of the Mackey functor M from such a collection. The next section gives a formula which provides a satisfying answer. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 14 The diagonal decomposition
Rational Mackey functors for compact Lie groups are considered in Greenlees [Gre98a]. ExamplesC i) and Corollary 5.3 of that reference give the following decomposition formula for rational G -Mackey functors for finite G . The reference proves the result using equivariant stable homotopytheory, a direct algebraic proof is given by Sugrue [Sug19, Lemma 6.1.9]. We use the structureresults to prove it via a calculation on Mackey functors of the form F A V . Theorem 5.1.
Let M be a G -Mackey functor and let K H be subgroups of G . Then e HK M ( G/H ) ∼ = (cid:0) e KK M ( G/K ) (cid:1) W H K . Proof.
By Theorem 4.7, we may assume M is of the form F A V , for V a Q [ W G A ] -module. Lemma5.2 implies that we only need to consider the case A = K .By Remark 2.14 and Proposition 4.5, we may remove the idempotent e KK from the formula. Thuswe must prove e HK ( Q [( G/H ) K ] ⊗ V ) W G K = e HK F K ( G/H ) ∼ = (cid:0) F K ( G/K ) (cid:1) W H K = (cid:0) ( Q [( G/K ) K ] ⊗ V ) W G K (cid:1) W H K . By Lemma 2.12, ( e HK ) is a sum of | W H K | − [ H/K ] and rational multiples of basis elements [ H/L ] for L a proper subgroup of K . An element [ H/L ] acts on F K ( G/H ) through I HL R HL . As F K ( V )( G/L ) = 0 unless L contains K (up to H -conjugacy), ( e HK ) acts as | W H K | − [ H/K ] = I HK R HK .Using Lemma 4.6, we can identify the action of induction and restriction by looking at the Q -modules Q [( G/H ) K ] −→ Q [( G/K ) K ] −→ Q [( G/H ) K ] . Take an element gK which is fixed by left multiplication by elements of K (that is, an elementof N G K ), then gH is also K –fixed. Take gK and g ′ K which are K –fixed with gH = g ′ H . Then g ′ = gh for some h ∈ H , and K = ( g ′ ) − Kg ′ = ( gh ) − Kgh = h − Kh so h ∈ N H K . It follows that Q [( G/K ) K ] /W H K −→ Q [( G/H ) K ] is injective.Take gH in the image of ( G/K ) K → ( G/H ) K , the composite sends this to the sum of those aH such that aK = gK . By the previous argument, this sum is | W H K | gH . Now take gH which is notin the image of ( G/K ) K → ( G/H ) K , then the first map sends this to zero.It follows that the composite ( Q [( G/H ) K ] ⊗ V ) W G K −→ ( Q [( G/K ) K ] ⊗ V ) W G K −→ ( Q [( G/H ) K ] ⊗ V ) W G K induces an isomorphism of (cid:16) ( Q [( G/K ) K ] ⊗ V ) W G K (cid:17) W H K onto the image of I HK R HK in ( Q [( G/H ) K ] ⊗ V ) W G K . (cid:3) Lemma 5.2.
Let A , B and C be subgroups of G , with C B and let V be a Q [ W G A ] -module.Then e BC (cid:0) F A ( V )( G/B ) (cid:1) = 0 unless A and C are G -conjugate. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 15 Proof.
By Proposition 4.5, e GA F A ( V ) = F A ( V ) . Hence, e BC (cid:0) F A ( V )( G/B ) (cid:1) = e BC R GB ( e GA ) (cid:0) F A ( V )( G/B ) (cid:1) . The composite e BC R GB ( e GA ) is zero unless A is G -conjugate to a subgroup A ′ of G and that subgroup A ′ is B -conjugate to C . We also require that A is G –subconjugate to B , as otherwise Q [( G/B ) A ] = 0 .This is equivalent to requiring that A is G -conjugate to C . (cid:3) We illustrate this decomposition with two examples.
Example 5.3.
Let M be a rational Mackey functor for C p . Define Q -modules V = M ( C p /C ) , V = e C p M ( C p /C p ) , V = e C p M ( C p /C p ) , V = e C p M ( C p /C p ) . where e C pi ∈ A Q ( C p i ) is the idempotent with support C p i , i ∈ { , , , } . Note that for i = 0 thisis ∈ A Q ( C ) = Q . The Q -module V i has an action of Q [ W C p C p i ] = Q [ C p /C p i ] .The classification theorem implies that M ∼ = F e V ⊕ F C p V ⊕ F C p V ⊕ F C p V . Writing out the values of M at varying subgroups gives the diagram V ⊕ V C p /C p (cid:15) (cid:15) ⊕ V C p /C p (cid:15) (cid:15) ⊕ V C p /C (cid:15) (cid:15) = M ( C p /C p ) (cid:15) (cid:15) V O O ⊕ V C p /C p (cid:15) (cid:15) O O ⊕ V C p /C (cid:15) (cid:15) O O = M ( C p /C p ) (cid:15) (cid:15) O O V O O ⊕ V C p /C (cid:15) (cid:15) O O = M ( C p /C p ) (cid:15) (cid:15) O O V O O = M ( C p /C ) O O With vertical maps indicating induction and restriction.
Remark 5.4.
Given a G -Mackey functor M and H G , we can construct M ( G/H ) , the quotientof M ( G/H ) by the images of the induction maps from proper subgroups of H . This example showshow M ( G/H ) = e HH M ( G/H ) , so that the classification result is based around stripping out theimages of the induction functors. Example 5.5.
Let G = S , K = h (12) i , H = h (12) , (34) i . Then N G K = H ( G/K ) K = W G K = H/K = { K, (34) K } , ( G/H ) K = { H, (14)(23) H } and the W G A –fixed points of Q [( G/A ) A ] is isomorphic to Q . Now we consider the maps Q [ { H, (14)(23) H } ] −→ Q [ { K, (34) K } ] −→ Q [ { H, (14)(23) H } ] . The first sends H to K + (34) K and (14)(23) H to zero. The second map sends K and (34) K to H .If we take V = Q [ W G K ] , and consider F K ( V ) , we see that e HK Q [( G/H ) K ] ∼ = ( Q [( G/K ) K ] W H K ∼ = Q . N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 16 Comparison to equivariant spectra
For G a finite group, we have a classification of rational G –spectra in terms of an algebraic model,see Theorem 12.1. The algebraic model is built from chain complexes of Q [ W G H ] -modules for H running over conjugacy classes of subgroups of G .The most modern approach to the classification takes several steps • idempotent splitting • restriction to normalisers • passing to Weyl groups (by taking fixed points) • algebraicisationwhich we see are analogous to our classification of rational Mackey functors. At the level of homo-topy categories, one takes a spectrum X , and then splits it into e H X for varying H . Then one forgetsto N G H –spectra and takes H –fixed points. Taking homology of the algebraicisation of a spectrum e H X gives the homotopy groups of the spectrum. This results in a graded Q [ W G H ] -module π ∗ ( (cid:0) i ∗ ( e H X ) (cid:1) H ) = e H π H ∗ ( X ) = e H ([ − , X ] G ∗ ⊗ Q )( G/H ) . This is exactly the functor U H applied to the Mackey functor [ − , X ] G ∗ ⊗ Q .A major difference between the method we use for Mackey functors and the approach for rational G –spectra is that in the latter one proves that the various model categories are Quillen equivalentat each stage, rather than arguing via the composite functor. This is partly due to adjunctionsin the topological setting not being both left and right adjoint and partly due to the difficulty ofworking with complex composite functors in model categories. For Mackey functors, we see thatmost adjunctions in e GH Mackey ( G ) i / / R GN G H ( e GH ) Mackey ( N G H ) i o o ε / / Mackey ( N G H/H ) ε o o ( − )( W G H/e ) / / Q [ W G H ]– mod Mack
WGH o o are not equivalences. To resolve this, we can restrict Mackey ( N G H/H ) and R GN G H ( e GH ) Mackey ( N G H ) to the full subcategories in the image of the functors from Q [ W G H ]– mod. We write Mackey ( N G H/H ) and R GN G H ( e GH ) Mackey ( N G H ) for these categories. Our calculation of the counit of the ( F H , U H ) adjunction shows that Id → ( Mack W G H V )( W G H/e ) and Id → ( ε ε Mack W G H V )( W G H/e ) are isomorphisms. Hence the functors ε and Mack W G H are full and faithful. It follows that onthese full subcategories, we have equivalences of categories. e GH Mackey ( G ) i / / R GN G H ( e GH ) Mackey ( N G H ) i o o ε / / Mackey ( N G H/H ) ε o o ( − )( W G H/e ) / / Q [ W G H ]– mod Mack
WGH o o Passing to a full subcategory is the algebraic equivalent of localisation at an idempotent as used inthe classification of rational G –spectra for finite G .We will discuss the topological analogues of these results in more detail in Part 2. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 17 Monoidal properties
We end this part with a discussion of the monoidal structure on Mackey functors. Details canbe found in Green [Gre71] and Luca [Luc96]. Given G -Mackey functors M and N , we define T ( H ) = M K H M ( G/K ) ⊗ Q N ( G/K ) ( M (cid:3) N )( G/H ) = T ( H ) /I ( H ) where I ( H ) is the Q –submodule of T ( H ) generated by R KL ( x ) ⊗ y ′ − x ⊗ I KL ( y ′ ) for x ∈ M ( G/K ) , y ′ ∈ N ( G/L ) , L K Hx ′ ⊗ R KL ( y ) − I KL ( x ′ ) ⊗ y for x ′ ∈ M ( G/L ) , y ∈ N ( G/K ) , L K HC h ( x ) ⊗ y − x ⊗ C − h ( y ) for x ∈ M ( G/K ) , y ∈ N ( G/hKh − ) , L K H. Theorem 7.1.
For M and N G -Mackey functors, the construction ( M (cid:3) N )( G/H ) = T ( H ) /I ( H ) defines a Mackey functor when equipped with the conjugation, restriction and induction maps de-scribed below. We call this Mackey functor the box product of M and N . Conjugation is given by the diagonal action C h ( x ⊗ y ) = C h ( x ) ⊗ C h ( y ) . Induction from H to H ′ is given by the inclusion (cid:16) M K H M ( G/K ) ⊗ Q N ( G/K ) (cid:17) −→ (cid:16) M K H ′ M ( G/K ) ⊗ Q N ( G/K ) (cid:17) followed by taking quotients with respect to I ( H ) and I ( H ′ ) . Restriction from H ′ to H is inducedby the map T ( H ′ ) → T ( H ) /I ( H ) given by x ⊗ y X l ∈ [ H (cid:31) H ′ (cid:30) K ] R lKl − H ∩ lKl − C l ( x ) ⊗ R lKl − H ∩ lKl − C l ( y ) for x ∈ M ( K ) , y ∈ N ( K ) and K H .One can also define the box product via a convolution product (a left Kan extension over theproduct of G –sets), using the definition of Mackey functors in terms of spans of G –sets (the Burnsidecategory). The unit for the box product is the Burnside ring Mackey functor.While not immediately obvious, one can check that a (commutative) monoid for this box prod-uct is a rational Mackey functor M , such that each M ( G/H ) is a (commutative) Q -algebra, theconjugation and restriction maps are maps of algebras and for K H , the Frobenius relations hold: x · I HK ( y ) = I HK ( R HK ( x ) · y ) I HK ( y ) · x = I HK ( y · R HK ( x )) for x ∈ M ( G/H ) and y ∈ M ( G/K ) . We call such a (commutative) monoid Mackey functor a(commutative) Green functor.The category of Q [ W G H ] -modules has a monoidal product, given by tensoring two modulesover Q and equipping the result with the diagonal Q [ W G H ] -action. We then see that U H sends(commutative) Green functors to (commutative) monoids in Q [ W G H ] -modules. In fact, we showthat U H is a symmetric monoidal functor. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 18 Lemma 7.2.
Let M and N be Mackey functors. Then e GH ( M (cid:3) N ) ∼ = ( e GH M (cid:3) N ) ∼ = ( M (cid:3) e GH N ) ∼ = ( e GH M (cid:3) e GH N ) . Hence (cid:0) e GH M (cid:3) e GH N (cid:1) ( G/H ) = e GH M ( G/H ) ⊗ Q e GH N ( G/H ) . Proof.
The first statement is a calculation of the action of the Burnside ring on the box product.For the second, the Mackey functor e GH M is trivial on proper subgroups of H , from which itfollows that T ( H ) = e GH M ( G/H ) ⊗ Q e GH N ( G/H ) and I ( H ) = 0 . (cid:3) The first statement of Lemma 7.2 implies that the category e GH Mackey ( G ) is monoidal withrespect to (cid:3) with the unit e GH A Q . Corollary 7.3.
For each H G the equivalence of categories U H : e GH Mackey ( G ) / / Q [ W G H ]– mod : F H o o is strong symmetric monoidal.Moreover, the splitting result Mackey ( G ) ∼ = Y H G G e GH Mackey ( G ) is strong symmetric monoidal. The topological equivalent of this result is Barnes, Greenlees and Kędziorek [BGK18a]. Thisgives a description of E ∞ -algebras in rational G –spectra in terms of differential graded algebras in Y ( H ) G Q [ W G H ]– mod . The more complicated case of commutative ring G –spectra (or N ∞ -algebras) is considered in workof Wimmer [Wim19]. The extra data here comes from multiplicative norm maps, which are relatedto Tambara functors (commutative Green functors with additional structure), see Strickland [Str12],Mazur [Maz13] and Hill and Mazur [HM19]. The idempotent splitting result we use destroys theadditional structure of a Tambara functor, leaving only a commutative Green functor. Hence,there is no immediate extension of the above results to Tamabara functors. The question of whichidempotents and splittings persevere norms in the Burnside ring is answered fully in work of Böhme[Böh19]. Part The structure of rational G -spectra For G a compact Lie group, it is natural to study the homotopy theory of G –spectra as Brownrepresentability holds equivariantly, see [May96, Section XIII.3]. That is, G –equivariant cohomol-ogy theories are represented by G –spectra, so the category of G –equivariant cohomology theoriesand stable natural transformations between them, is equivalent to the homotopy category of G –spectra. Due to the complexity of the non-equivariant case, one cannot expect a complete analysisof either G –equivariant cohomology theories or G –spectra integrally. However, if we restrict our-selves to G –equivariant cohomology theories with values in rational vector spaces, the situation isgreatly simplified, whilst valuable geometric and group theoretic structures remain. For this reason,the programme of understanding G –equivariant cohomology theories begun by Greenlees restrictsattention to rational G –equivariant cohomology theories and rational G –spectra. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 19 In this part, we discuss the methods and tools used to obtain algebraic models for rationalequivariant spectra. Recall from the introduction that an algebraic model for rational G –spectrais a model category d A ( G ) , that is Quillen equivalent to G –spectra. This category must consist ofdifferential objects (and morphisms) in a graded abelian category A . To start our journey we beginby recalling some useful facts about G –spectra.8. Preliminaries on G -spectra Let G be a compact Lie group. We work with orthogonal G –spectra, see Mandell and May[MM02] for more details. Unless otherwise stated, our categories of G –spectra will be indexed on acomplete G -universe U .For H a closed subgroup of G , one can define homotopy groups of an orthogonal G –spectrum X with structure map σ as π H ( X ) = colim V [ S V , X ( V )] H where the maps in the colimit send a map α : S V −→ X ( V ) to the composite S W ∼ = S V ∧ S V ⊥ α ∧ Id −→ X ( V ) ∧ S V ⊥ σ −→ X ( V ⊕ V ⊥ ) ∼ = X ( W ) . Here V runs through the G representations in the universe U . More generally, the integer gradedhomotopy groups of a G –spectrum X are defined using shift and loop functors on spectra and theformula above. A map f of G –spectra is a weak equivalence, also called a stable equivalence , inorthogonal G –spectra if and only if π Hp ( f ) is an isomorphism for all closed subgroups H of G andall integers p . The class of stable equivalences is part of a stable model structure on G –spectra, G –Sp O .Orthogonal G –spectra with the stable model structure is a convenient model category for G –equivariant homotopy theory. In particular, the homotopy category is a symmetric monoidal trian-gulated category with unit the sphere spectrum S , see Hovey, [Hov99, Section 7]. Furthermore, thestable equivalences can be detected by objects in the category in the following sense. For a closedsubgroup H in G , an orthogonal spectrum X and integers p ≥ and q > (8.0.1) [Σ p S ∧ G/H + , X ] G ∼ = π Hp ( X ) [ F q S ∧ G/H + , X ] G ∼ = π H − q ( X ) where [ − , − ] G denotes morphisms in the homotopy category of G –Sp O and F q ( − ) is the left adjointto the evaluation functor at R q : Ev R q ( X ) = X ( R q ) . In particular, F q ( S ) models S − q , the q –folddesuspension of the sphere spectrum. We can put this relation between the shifts of G/H + and theweak equivalences into the formalism of [SS03, Section 2]. Definition 8.1.
Let C be a triangulated category with infinite coproducts. A full triangulatedsubcategory of C (with shift and triangles induced from C ) is called localising if it is closed undercoproducts in C . A set P of objects of C is called a set of generators if the only localising subcategoryof C containing objects of P is the whole of C . An object of a stable model category is called a generator if it is so when considered as an object of the homotopy category.An object X in C is homotopically compact . if for any family of objects { A i } i ∈ I the canonicalmap M i ∈ I [ X, A i ] C −→ [ X, a i ∈ I A i ] C is an isomorphism in the homotopy category of C . There are different names used in the literature - compact, small. We chose to use the name homotopicallycompact here.
N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 20 The set of suspensions and desuspensions of
G/H + , where H varies through all closed subgroupsof G , is a set of homotopically compact generators in the stable model category G –Sp O . Thoseobjects are compact since homotopy groups commute with coproducts and it is clear from [SS03,Lemma 2.2.1] and Equation (8.0.1) that this is a set of generators for G –Sp O .There is an easy-to-check condition for a Quillen adjunction between stable model categorieswith sets of homotopically compact generators to be a Quillen equivalence. It is used often in thesetting of algebraic models. Also notice that the derived functors of Quillen equivalences preservehomotopically compact objects. Lemma 8.2.
Suppose F : C ⇄ D : U is a Quillen pair between stable model categories with setsof homotopically compact generators, such that the right derived functor RU preserves coproducts(or equivalently, such that the left derived functor sends homotopically compact generators to ho-motopically compact objects).If the derived unit and counit are weak equivalences for the respective sets of generators, then ( F, U ) is a Quillen equivalence.Proof. The result depends upon the fact that the homotopy category of a stable model category isa triangulated category. First notice that since the derived functor RU preserves coproducts, thederived unit and counit are triangulated natural transformations. If the derived unit condition isan isomorphism for a set of objects K then they are also satisfied for every object in the localisingsubcategory for K . Since we assume that K consists of homotopically compact generators, thelocalising subcategory for K is the whole category and the derived unit is an isomorphism. Thesame argument applies to the counit and the result follows. (cid:3) To construct a model category of rational G –spectra will we need to introduce the language ofBousfield localisations, see Section 10. Since we will often localise the model category of rational G –spectra at idempotents of the rational Burnside ring, we first look at this ring.9. Idempotents of the rational Burnside ring
For G a compact Lie group, the Burnside ring A ( G ) was defined by tom Dieck in [tD75] in terms of G -manifolds. For a survey on the subject see, for example, Fausk [Fau08]. When working rationally,several descriptions of this ring exist. We give these descriptions and use them to understand theidempotents of the rational Burnside ring. These idempotents are fundamental to the constructionof the algebraic model and the calculations therein.9.1. Two ways of understanding rational Burnside ring.
Recall that for H a subgroup of G , N G H = { g ∈ G | gH = Hg } is the normaliser of H in G . We write W = W G H = N G H/H for theWeyl group of H in G .Let F ( G ) be the set of closed subgroups of G with finite index in their normalizer. That is, allclosed H G such that N G H/H is finite. We give this set the topology induced by the Hausdorffmetric, see [LMSM86, Section V.2].By work of tom Dieck [tD79, Propositions 5.6.4 and 5.9.13], there is an isomorphism of rings A ( G ) ⊗ Q ∼ = C ( F ( G ) /G, Q ) , where C ( F ( G ) /G, Q ) denotes the ring of continuous functions on the orbit space F ( G ) /G withvalues in discrete space Q . From now on, we will use notation A Q ( G ) for A ( G ) ⊗ Q . This isomorphismgeneralises that of Lemma 2.11. Notice that if G is a finite group, then Sub( G ) = F ( G ) , where Sub( G ) is the set of all subgroups of G . N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 21 From the ring isomorphism above, it is clear that idempotents of the rational Burnside ringof G correspond to the characteristic functions of open and closed subspaces of the orbit space F ( G ) /G (or equivalently, to open and closed G -invariant subspaces of F ( G ) ). We write e V , for theidempotent corresponding to an open and closed subset V of F ( G ) /G .Every inclusion i : H −→ G induces a ring homomorphism i ∗ : A Q ( G ) −→ A Q ( H ) . In general, itis difficult to explicitly describe the image of a given idempotent in terms of open and closed setsunder i ∗ : C ( F ( G ) /G, Q ) −→ C ( F ( H ) /H, Q ) . Even before taking conjugacy classes into account, notice that a subgroup K H with finite indexin the normaliser N H K does not have to have a finite index in the normaliser N G K . Thus the map i ∗ : C ( F ( G ) /G, Q ) −→ C ( F ( H ) /H, Q ) is not always induced by a map from F ( H ) to F ( G ) . Theexception is of course, when G, H are finite groups, as we discussed in Remark 2.14.A better approach to investigate the action of i ∗ on idempotents is to view idempotents ascorresponding to certain subspaces of the space of all closed subgroups of G as follows. We put atopology on the set of all closed subgroups of G , Sub( G ) . This topology is called the f -topology inGreenlees [Gre98a, Section 8].For a closed subgroup H G and ε > we define a ball O ( H, ε ) = { K ∈ F ( H ) | d ( H, K ) < ε } in Sub( G ) , where the distance above is measured with respect to the Hausdorff metric. Thus,subgroups close to H which have infinite Weyl groups are ignored, for example if H = SO (2) is atorus then O ( SO (2) , ε ) is a singleton. Given also a neighbourhood A of the identity in G consider O ( H, ε, A ) = ∪ a ∈ A O ( H, ε ) a , where O ( H, ε ) a is the set of a –conjugates of elements of O ( H, ε ) . Definition 9.1.
For G a compact Lie group, the f–topology on Sub( G ) is generated by the sets O ( H, ε, A ) as H, ε, A vary. We write
Sub f ( G ) for this topological space.We say that subgroups K H of G are cotoral if H/K is a torus. We write K ∼ H for theequivalence relation generated by the cotoral pairs. An idempotent in a rational Burnside ring A Q ( G ) corresponds to an open and closed, G –invariant subspace of Sub f ( G ) which is a union of ∼ –equivalence classes.Let V be an open and closed G –invariant set in Sub f ( G ) which is a union of ∼ –equivalenceclasses. Let i ∗ V be the preimage of V under i ∗ : Sub f ( H ) −→ Sub f ( G ) . We then let i ∗ V be thesmallest G -invariant open and closed set of Sub f ( H ) , which is the union of ∼ –equivalence classescontaining i ∗ V . Using the techniques of Greenlees [Gre98a, Section 8] one can show the following. Lemma 9.2.
Let i : H → G be an inclusion of a closed subgroup. Let e V an idempotent of A Q ( G ) corresponding to V , an open and closed G –invariant set in Sub f ( G ) which is a union of ∼ –equivalence classes. Then i ∗ ( e V ) = e i ∗ V . Remark 9.3.
As it is more common in the literature, an idempotent e V will come from a openand close subset V of F ( G ) /G unless otherwise stated.9.2. Special idempotents.
There are two situations of particular interest to us. In these caseswe have an idempotent in the rational Burnside ring and we can provide an algebraic model for thepiece of homotopy theory of rational G –spectra that this idempotent governs. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 22 The first situation is where there is an idempotent which remembers only one, special subgroup.The second author called such a subgroup exceptional in [Kęd17a]. The second situation is wherethe idempotent corresponds to the maximal torus T in G and all its subgroups. This is called the toral part of rational G –spectra in [BGK19].When we look at idempotents defined by subsets of F ( G ) /G the above two cases look identicalat first glance: both idempotents are indexed by one subgroup. However in the case of a torus,there are subgroups of the torus which are “hidden” in the torus idempotent. This is visible whenone uses the space Sub f ( G ) to describe the idempotent. The subgroups which are cotoral in T areresponsible for making the algebraic model for that part substantially more difficult than in thecase of an exceptional subgroup.We will start our analysis with the case of an exceptional subgroup of G . Definition 9.4.
Suppose G is a compact Lie group. We say that a closed subgroup H G is exceptional in G if W G H is finite, there exist an idempotent e GH in the rational Burnside ring of G corresponding to the conjugacy class of H in G (via tom Dieck’s isomorphism) and H has nocotoral subgroups.If H is exceptional in G , then { K | K ∈ ( H ) G } is an open and closed G –invariant subspace of Sub f ( G ) , which already is a union of ∼ –equivalence classes, since H does not contain any cotoralsubgroup and W G H is finite. The other implication also holds; if there is an idempotent corre-sponding to { K | K ∈ ( H ) G } in Sub f ( G ) , then H is an exceptional subgroup of G . Thus we couldrephrase the definition in terms of the space Sub f ( G ) , but we decided to use the more familiar F ( G ) /G with the topology given by the Hausdorff metric.Any subgroup of a finite group G is exceptional. In O (2) only finite dihedral subgroups areexceptional; in particular none of the finite cyclic subgroups are exceptional (since finite cyclicsubgroups do not have idempotents in the rational Burnside ring of O (2) ). The maximal torus SO (2) in O (2) has an idempotent in the rational Burnside ring of O (2) , however it is not anexceptional subgroup, since it contains cotoral subgroups, for example the trivial one. In SO (3) all finite dihedral subgroups are exceptional except for D , which is conjugate to C and thereforeis a cotoral subgroup of a torus. There are four more conjugacy classes of exceptional subgroups: A , Σ , A and SO (3) , where A denotes rotations of a tetrahedron, Σ denotes rotations of a cubeand A denotes rotations of a dodecahedron, see [Kęd17b].If a trivial subgroup is exceptional in G , then G has to be finite. This holds as the normaliserof a trivial subgroup is the whole G , W G { } = G and the condition that the Weyl group is finiteimplies that G is a finite group.Given an exceptional subgroup H , we may use the corresponding idempotent in the rationalBurnside ring to split (see Section 10) the category of rational G –spectra into the part over anexceptional subgroup H and its complement. [Kęd17a] presents the model for rational G –spectraover an exceptional subgroup H .The exceptional subgroups of a group G can be divided into two sets, according to how theiridempotent behaves once restricted to the normaliser of the exceptional subgroup. We closely follow[Kęd17a] in analysis of these different behaviours. Definition 9.5.
Suppose H A are closed subgroups of G such that H is exceptional in G .Suppose further that i : A −→ G is an inclusion. We say that H is A – good in G if i ∗ ( e GH ) = e AH and A – bad in G if it is not A –good, i.e. i ∗ ( e GH ) = e AH . the name was motivated by the exceptional behaviour of the algebraic model over such a subgroup. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 23 Notice that the above definition is all about subgroups conjugate to H in A and in G and theirrelation to each other. If L A is such that L is conjugate to H in A , then it is also true that L is conjugate to H in G . Thus if H is A –bad in G it just means that there exists L ′ A such that ( L ′ ) G = ( H ) G and ( L ′ ) A = ( L ) A . An exceptional subgroup H in a compact Lie group G is always H –good in G . Lemma 9.6. [Kęd17a]
For the exceptional subgroups in G = SO (3) , we have the following relationbetween H and its normaliser N G H :(1) A is A –good in SO (3) .(2) Σ is Σ –good in SO (3) .(3) A is Σ –good in SO (3) .(4) D is Σ –bad in SO (3) .Proof. We only need to prove Part (3) and (4), since any exceptional subgroup H in a compact Liegroup G is H –good in G . Part (3) follows from the fact that there is one conjugacy class of A in Σ , as there is just one subgroup of index 2 in Σ . Part (4) follows from the observation that thereare two subgroups of order 4 in D (so also in Σ ) and they are conjugate by an element g ∈ D ,which is the generating rotation by 45 degrees (thus g / ∈ D and thus g / ∈ Σ ). (cid:3) Remark 9.7.
Notice that we can generalise Definition 9.5 to non-exceptional subgroups using theequivalent description in terms of conjugacy classes of H in A and in G . In that case, if G = SO (3) , A = O (2) and H = C A , then H is A –bad in G , which follows from the fact that D A is G -conjugate to H , but not A -conjugate. This bad behaviour of C in SO (3) is visible in theadjunctions used to obtain the algebraic model for toral part of rational SO (3) –spectra in [Kęd17b]which we recall in Proposition 11.7.Finishing the discussion about idempotents of rational Burnside ring, we note that there is alwaysan idempotent corresponding to the maximal torus T in G and all its subgroups. This fact wasused in [BGK19] to obtain an algebraic model for rational toral G –spectra, thus the ones that havegeometric isotropy contained in the set of subgroups of the maximal torus.9.3. Examples.
Closed subgroups of SO (2) . Recall that SO (2) is the group of rotations of R . The closedsubgroups of SO (2) are the finite cyclic groups C n . Each C n is cotoral in SO (2) , that is, it is normalin SO (2) and SO (2) /C n ∼ = SO (2) . The only subgroup of SO (2) with finite index in its normaliseris SO (2) itself. Hence, the space F ( SO (2)) /SO (2) is a single point and the rational Burnside ringof SO (2) is Q . Similar arguments show that A Q ( T ) = Q for T a torus of any rank.9.3.2. Closed subgroups of O (2) . Recall that O (2) is the group of rotations and reflections of R .The closed subgroups are the finite cyclic groups, T = SO (2) , O (2) and finite dihedral groups. Forfixed n , the finite dihedral groups of order n are all conjugate. We Write D n for this conjugacyclass. The space F ( O (2)) /O (2) consists of two parts, which we call the toral part and the dihedralpart. The toral part e T , is just one point T corresponding to the maximal torus and all its subgroups.The dihedral part e D , is the set of all dihedral subgroups together with their limit point O (2) . Thus,we have idempotents e e T and e e D in the rational Burnside ring of O (2) which sum to the identity.The toral idempotents for O (2) and SO (3) will behave very differently when we discuss theinteractions between localisations and change of group functors in Section 11. To help the notationfor this comparison, we use a tilde to denote the dihedral and toral parts of F ( O (2)) /O (2) and notilde for SO (3) . N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 24 Space F ( O (2)) /O (2)Part (Subspace) e T T e D D D D D D ... O (2) Closed subgroups of SO (3) . Recall that SO (3) is a group of rotations of R . We choose amaximal torus T in SO (3) with rotation axis the z -axis. We divide the closed subgroups of G intothree types: toral T , dihedral D and exceptional E . This division is motivated by our preferredsplitting of the category of rational SO (3) –spectra. The toral part consist of all tori in SO (3) andall cyclic subgroups of these tori. Note that for any natural number n there is one conjugacy classof subgroups from the toral part of order n in SO (3) .The dihedral part consists of all dihedral subgroups D n (dihedral subgroups of order n ) of SO (3) where n is greater than 2, together with all subgroups isomorphic to O (2) . Note that O (2) is the normaliser for itself in SO (3) . Moreover, there is only one conjugacy class of a dihedralsubgroup D n for each n greater than . The normaliser of D n in SO (3) is D n for n > .We deliberately exclude the conjugacy classes of D and D from the dihedral part. Conjugatesof D are excluded from the dihedral part, as D is conjugate to C in SO (3) and that subgroupis already taken into account in the toral part. Conjugates of D are excluded from the dihedralpart since its normaliser in SO (3) is Σ (symmetries of a cube), thus its Weyl group Σ /D is oforder , whereas all other finite dihedral subgroups D n , n > have Weyl groups of order . Forsimplicity we decided to treat D separately.There are five conjugacy classes of subgroups which we call exceptional, namely SO (3) itself,the rotation group of a cube Σ , the rotation group of a tetrahedron A , the rotation group of adodecahedron A and D , the dihedral group of order 4. Normalisers of these exceptional subgroupsare as follows: Σ is equal to its normaliser, A is equal to its normaliser and the normaliser of A is Σ , as is the normaliser of D .Consider the space F ( SO (3)) /SO (3) of conjugacy classes of subgroups of SO (3) with finite indexin their normalisers. Recall that the topology on this space is induced by the Hausdorff metric.The division into these parts is an indication of idempotents of the rational Burnside ring for SO (3) that are chosen to obtain an algebraic model for rational SO (3) –spectra. Space F ( SO (3)) /SO (3)Part (Subspace) E SO(3) Σ A A D T T D D D D ... O (2) N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 25 The topology on E is discrete, T consists of one point T and D forms a sequence of pointsconverging to O (2) .Note the difference between the dihedral parts for O (2) and SO (3) : the conjugacy class of D and D . At a first glance, the toral part for SO (3) looks the same as the toral part for O (2) . However,for SO (3) it contains information about D O (2) (since D is conjugate to C in SO (3) ), whereasfor O (2) it does not. These differences will become significant when we look at the interactionsbetween localisations at idempotents and change of groups functors in Section 11.We use the following idempotents in the rational Burnside ring of SO (3) : e T corresponding tothe characteristic function of the toral part T , e D corresponding to the characteristic function ofthe dihedral part D and e E corresponding to the characteristic function of the exceptional part E . Since E is a disjoint union of five points, it is in fact a sum of five idempotents, one for every(conjugacy class of a) subgroup in the exceptional part: e SO (3) , e Σ , e A , e A and e D . We use asimplified notation e H to mean e SO (3) H here. Remark 9.8.
All finite dihedral subgroups in SO (3) are exceptional, hence each has an idempotentcorresponding to it. However, as there are countably many conjugacy classes of dihedral subgroups,we cannot write e D as the sum of all these idempotents. Similarly, the characteristic function ofthe point O (2) is not a continuous map to Q , hence it does not correspond to an idempotent.10. Left and right Bousfield localisations and splittings
There are two well-understood ways of making a homotopy category of a given model category smaller . Both ways boil down to adding weak equivalences in a tractable way. The first one keepsthe cofibrations the same and is called a left Bousfield localisation (the particular version we use isalso called a homological localisation ). The second one keeps the fibrations the same and is calledthe right Bousfield localisation (or cellularisation).10.1.
Left Bousfield localisation.
The general theory of left Bousfield localisations is given inHirschhorn [Hir03]. For homological localisation we use the following result, which is [MM02,Chapter IV, Theorem 6.3].
Theorem 10.1.
Suppose E is a cofibrant object in G –Sp O or a cofibrant based G –space. Then thereexists a new model structure called the E -local model structure on G –Sp O , denoted L E ( G –Sp O ) ,defined as follows. A map f : X −→ Y is • a weak equivalence if it is an E –equivalence, that is, Id E ∧ f : E ∧ X −→ E ∧ Y is a stableequivalence, • a cofibration if it is a cofibration with respect to the stable model structure, • a fibration if it has the right lifting property with respect to all trivial cofibrations.The E –fibrant objects Z are the fibrant G –spectra which are E -local, that is, the map [ f, Z ] G : [ Y, Z ] G −→ [ X, Z ] G is an isomorphism for all E –equivalences f . For X a G –spectrum, E –fibrant approximation givesBousfield localisation λ : X −→ L E X of X at E . We will refer to the above model structure as the left Bousfield localisation of the category of G –spectra at E . This model category is proper, stable, symmetric monoidal and cofibrantly generated.An E –equivalence between E –local objects is a weak equivalence by [Hir03, Theorems 3.2.13 and3.2.14]. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 26 As previously mentioned, the first simplification of the category of G –spectra is rationalisation.This means localisation at the Moore spectrum for Q , S Q . For details see [Bar09b, Definition 5.1].This spectrum has the property that π ∗ ( X ∧ S Q ) = π ∗ ( X ) ⊗ Q . We refer to this model category asthe model category of rational G –spectra.The self-maps of the rational sphere spectrum in the homotopy category of G –spectra are givenby the rational Burnside ring A Q ( G ) ∼ = [ S , S ] G –Sp O ⊗ Q ∼ = [ S , S ] L SQ G –Sp O ∼ = [ S Q , S Q ] G –Sp O . It follows that e ∈ A Q ( G ) can be represented by a map e : S Q −→ S Q . We define e S Q to be thehomotopy colimit (a mapping telescope) of the diagram S Q e / / S Q e / / S Q e / / ... . We ask for this spectrum to be cofibrant either by choosing a good construction of homotopy colimit,or by cofibrantly replacing the result in the stable model structure for G –spectra. We thus havemodel structures L e S Q ( G –Sp) and L (1 − e ) S Q ( G –Sp) . Fibrant replacement in L e S Q ( G –Sp O ) is given bytaking the fibrant replacement of X ∧ e S Q . Since this commutes with taking infinite coproducts, thelocalisation is smashing in the sense of Ravenel [Rav84] and Hovey et al. [HPS97]). In particular,this localisation preserves homotopically compact generators.We know from Section 9 that e corresponds to an open and closed, G –invariant subspace of Sub f ( G ) which is a union of ∼ –equivalence classes, call it V e . By considering the geometric fixedpoint functors Φ H , for all H G (see [MM02, Section V.4]), we can see that the homotopy categoryof L e S Q ( G –Sp O ) is the homotopy category of rational G –spectra X with geometric isotropy GI ( X ) = { H G | Φ H ( X )
6≃ ∗} , concentrated over the subgroups H which are in V e .10.2. Splitting.
A common step in the classification of rational G –spectra is to split the categoryusing idempotents of the rational Burnside ring. Work of the first author [Bar09b] allows us toperform a compatible splitting at the level of model categories. Theorem 10.2. [Bar09b, Theorem 4.4]
Let e be an idempotent in the rational Burnside ring A Q ( G ) .There is a strong symmetric monoidal Quillen equivalence: △ : L S Q ( G –Sp O ) / / L e S Q ( G –Sp O ) × L (1 − e ) S Q ( G –Sp O ) : Π o o The left adjoint is a diagonal functor, the right adjoint is a product and the product category onthe right is considered with the objectwise model structure (a map ( f , f ) is a weak equivalence, afibration or a cofibration if both factors f i are so). One can also look at splittings non-rationally, as in Böhme [Böh19].10.3.
Cellularisation.
A cellularisation of a model category is a right Bousfield localisation at aset of objects. Such a localisation exists by [Hir03, Theorem 5.1.1] whenever the model categoryis right proper and cellular. When we are in a stable context, the results of [BR13] can be used,which allows us to relax the cellularity condition.The most common use of cellularisation in the context of algebraic models is the CellularisationPrinciple, which we recall in Theorem 10.5.
N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 27 Definition 10.3.
Let C be a stable model category and K a stable set of objects of C , i.e. a setsuch that a class of K –cellular objects of C is closed under desuspension (Note that this class isalways closed under suspension). We call K a set of cells . We say that a map f : A −→ B of C is a K –cellular equivalence if the induced map [ k, f ] C∗ : [ k, A ] C∗ −→ [ k, B ] C∗ is an isomorphism of graded abelian groups for each k ∈ K . An object Z ∈ C is said to be K –cellular if [ Z, f ] C∗ : [ Z, A ] C∗ −→ [ Z, B ] C∗ is an isomorphism of graded abelian groups for any K –cellular equivalence f .The following is Hirschhorn [Hir03, Theorem 5.1.1]. Theorem 10.4.
For K a set of objects in a right proper, cellular model category C , the rightBousfield localisation or cellularisation of C with respect to K is the (right proper) model structure K – cell – C on C defined as follows. • The weak equivalences are K –cellular equivalences, • the fibrations of K – cell – C are the fibrations of C , • the cofibrations of K – cell – C are defined via left lifting property.The cofibrant objects of K – cell – C are called K –cofibrant and are precisely the K –cellular and cofi-brant objects of C . When C is stable and K is a stable set of cofibrant objects, then the cellularisation of a proper,cellular stable model category is proper, cellular and stable by Barnes and Roitzheim [BR13, The-orem 5.9].We can further ask the cells K to be homotopically compact objects. By [BR13, Section 9] thehomotopy category K – cell – C is the full triangulated subcategory of the homotopy category of C generated by K . In particular, K is a set of homotopically compact generators for K – cell – C . Theseideas lead to the following theorem. For examples of its use, see Section 10.4, Theorem 11.9 orTheorem 12.2. Theorem 10.5 (The Cellularisation Principle) . Let M and N be right proper, stable, cellular modelcategories with ( F, U ) a Quillen adjunction between M and N . Let Q be a cofibrant replacementfunctor in M and R a fibrant replacement functor in N . • Let K be a set of objects in M with F QK its image in N . Then F and U induce a Quillenadjunction F : K – cell – M / / F QK – cell – N : U o o between the K -cellularisation of M and the F QK -cellularisation of N . • If K is a stable set of homotopically compact objects in M such that for each A in K theobject F Q A is homotopically compact in N and the derived unit Q A → U R F Q A is a weakequivalence in M , then F and U induce a Quillen equivalence between the cellularisations: K – cell – M ≃ F QK – cell – N. • If L is a stable set of homotopically compact objects in N such that for each B in L the object U R B is homotopically compact in M and the derived counit F Q U R B → R B is a weakequivalence in N , then F and U induce a Quillen equivalence between the cellularisations: U RL – cell – M ≃ L – cell – N. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 28 Alternatives to splitting.
In the case of SO (2) , the rational Burnside ring is Q , so thereare no idempotents to give a splitting. Instead, one must look for replacements for the idempotentsor other methods of simplifying the category of rational SO (2) –spectra. One approach comes frominducing idempotents from the smaller subgroups. Suppose H is a subgroup of SO (2) such that A Q ( H ) has an idempotent e . Then SO (2) + ∧ H e S is a retract of SO (2) /H + that does not come froman idempotent of A Q ( SO (2)) . The set of these spectra as H and e vary give a better behaved set ofhomotopically compact generators for rational SO (2) –spectra. We can think of this constructionas applying an induced idempotent to SO (2) /H + . While they are not used directly in constructingthe algebraic model for rational SO (2) –spectra, they are highly useful in understanding it.Generalising the situation above, the rational Burnside ring of any torus T has no idempotents.Greenlees and Shipley [GS17] provided a new method of obtaining an algebraic model in this case.Suppose F is the family of all proper subgroups of T , we define the universal space E F + as a T -CW-complex with the following universal property ( E F + ) H ≃ ( S iff H ∈ F∗ otherwise.The universal space E F + is part of a cofiber sequence called the isotropy separation sequence E F + / / S / / e E F which can be turned into a homotopy pullback diagram in T –spectra . In the case of T = SO (2) ,this is also called the Hasse square : S / / (cid:15) (cid:15) e E F (cid:15) (cid:15) DE F + / / DE F + ∧ e E F . The diagram with S removed is called the punctured cube and is denoted by S y . Using [GS14b],we may construct a model category of modules over S y in rational SO (2) –spectra, which we call S y -mod (slightly abusing notation and not mentioning the ambient category). Any SO (2) –spectrum X defines a module over the diagram by smashing with the ring spectra e E F , DE F + and DE F + ∧ e E F . This functor has a right adjoint that is a type of pullback, giving an adjunction between S y -mod and rational SO (2) –spectra. The Cellularisation Principle, Theorem 10.5, can be used toconstruct a Quillen equivalence from this adjunction, see either [GS17] or [BGKS17] for details.In case of a torus of rank r , repeatedly using the isotropy separation sequence one can obtain a r + 1 -dimensional cube diagram. The terms of this cube are all genuine-commutative equivariantring T –spectra by Greenlees [Gre20]. We again use the notation S y for the punctured cube of thesering T –spectra and obtain the following theorem. Theorem 10.6. [GS17]
There is a strong symmetric monoidal Quillen equivalence L S Q ( T –Sp O ) ≃ QE K –cell– S y -modwhere K is the image in S y -mod of the set of compact generators for L S Q ( T )–Sp O ) . We slightly abuse the notation and whenever we write a T –space we actually mean its suspension spectrum. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 29 When G is a finite group, we let F be the family of all proper subgroups of G . The homotopypullback diagram obtained by using the isotropy separation sequence gives exactly the idempotentsplitting, since E F + ≃ Y ( H ) , H ∈F e GH S ≃ DE F + , e E F ≃ e GG S and DE F + ∧ e E F ≃ ∗ .However, the spectra e GH S are not genuine-commutative equivariant ring spectra (they are onlynaïve-commutative). Hence, it is easier to use the splitting approach for finite G . See Böhme[Böh19] for a complete explanation of the relation between genuine-commutative equivariant ringspectra and localisation at idempotents.An interesting case when there are some, but not enough idempotents, is the case of the dihedralpart of O (2) –spectra, see [Bar17]. In that case, there is no idempotent whose support is exactly O (2) . The abelian (resp. algebraic) model for the dihedral part of rational O (2) –spectra is given interms of sheaves of Q [ W ] -modules (resp. differential Q [ W ] -modules) over the space e D , where thestalk over the point O (2) has a trivial W -action. The stalk over O (2) can be described in terms ofa virtual idempotent – a colimit of idempotents, see [Bar17, Section 5].A similar approach occurs for profinite groups in work of Barnes and Sugrue [BS20] and Sugrue[Sug19]. 11. Change of groups and localisations
Once we split the category of rational G –spectra using idempotents, our main aim is to get ridof the remaining equivariance in each piece separately by applying certain fixed points functors.Assume we are working with the category L e S Q ( G –Sp O ) and we want to take H fixed points. Firstwe must move to the category N –Sp O where N is the normaliser of H in G , appropriately localised.We need N , since we want to have a residual Weyl group ( W = N G H/H ) action. At the same timewe need to localise N –Sp O at some idempotent of the rational Burnside ring of N correspondingto e , since we want to obtain a Quillen equivalence with L e S Q ( G –Sp O ) .In work of the second author [Kęd17a] and [Kęd17b], there was a precise analysis of two adjunc-tions: the induction–restriction and restriction–coinduction adjunctions in relation to localisationsof categories of equivariant spectra at idempotents. Below we summarise how these results allowus to make the restriction–coinduction adjunction into Quillen equivalence in suitable situations.Our examples are based on finite groups, O (2) and SO (3) .11.1. Restriction–coinduction adjunction and localisations.
Suppose we have an inclusion i : N ֒ → G of a subgroup N in a group G . This gives a pair of adjoint functors at the levelof orthogonal spectra (see for example [MM02, Section V.2 ]), namely induction, restriction andcoinduction as below (the left adjoint is above the corresponding right adjoint). We note here, thatfor the induction functor to be a left Quillen functor we must take care over the universes involved. G –Sp O i ∗ / / N –Sp O F N ( G + , − ) k k G + ∧ N − s s We assume that G –spectra are indexed over a complete G –universe U and N –spectra are indexedover one of two universes. In the case where we want to use the restriction functor as a right adjoint,we use the restriction of U to an N –universe. If we consider restriction as a left Quillen functor we N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 30 use a complete N –universe. With these conventions, the two pairs of adjoint functors are Quillenpairs with respect to stable model structures by [MM02, Chapter V, Proposition 2.3 and 2.4]. Giventhis, we slightly abuse the notation by not mentioning universes or the change of universe functorsof [MM02, Section V.2].The restriction functor as a right adjoint is often used when we want to take (both categorical andgeometric) H –fixed points of G –spectra, where H is not a normal subgroup of G . The procedureis to restrict to N G H –spectra and then to take H –fixed points to land in W G H –spectra. This isusually done in one go, since the restriction functor and the H –fixed points functor are both rightQuillen functors.It is natural to ask when the pair of adjunctions above passes to the localised categories, in ourcase localised at e GH S Q and e NH S Q respectively. The answer is related to H being a good or badsubgroup in G . The induction–restriction adjunction does not always induce a Quillen adjunction onthe localised categories, unless H is N -good in G . However, the restriction–coinduction adjunctioninduces a Quillen adjunction on these localised categories, for all exceptional subgroups H . Beforewe discuss this particular adjunction we state a general result. Lemma 11.1.
Suppose that F : C ⇄ D : R is a Quillen adjunction of model categories where theleft adjoint is strong (symmetric) monoidal. Suppose further that E is a cofibrant object in C andthat both L E C and L F ( E ) D exist. Then F : L E C / / L F ( E ) D : U o o is a strong (symmetric) monoidal Quillen adjunction. Furthermore, if the original adjunction wasa Quillen equivalence, then the induced adjunction on localised categories is as well.Proof. Since the localisation did not change the cofibrations, the left adjoint F still preservesthem. To show that it also preserves acyclic cofibrations, take an acyclic cofibration f : X −→ Y in L E C . By definition, f ∧ Id E is an acyclic cofibration in C . Since F was a left Quillen functorbefore localisation, F ( f ∧ Id E ) is an acyclic cofibration in D . As F was strong monoidal, we have F ( f ∧ Id E ) ∼ = F ( f ) ∧ Id F ( E ) , so F ( f ) is an acyclic cofibration in L F ( E ) D which finishes the proofof the first part.To prove the second part of the statement we use Part (2) from [Hov99, Corollary 1.3.16].Since F is strong monoidal, and the original adjunction was a Quillen equivalence, F reflects F ( E ) –equivalences between cofibrant objects. It remains to check that the derived counit is an F ( E ) –equivalence. An F ( E ) –fibrant object is fibrant in D and the cofibrant replacement functorremains unchanged by localisation. Thus the claim follows from the fact that ( F, U ) was a Quillenequivalence before localisations. (cid:3) We will use this result in several cases. We start with the restriction–coinduction adjunction.
Corollary 11.2.
Let i : N −→ G denote the inclusion of a subgroup and let E be a cofibrant objectin G –Sp O . Then i ∗ : L E ( G –Sp O ) / / L i ∗ ( E ) ( N –Sp O ) : F N ( G + , − ) o o is a strong symmetric monoidal Quillen pair. Notice that if E = e S Q for some idempotent e ∈ A Q ( G ) then we get the following N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 31 Corollary 11.3.
Suppose G is any compact Lie group, i : N −→ G is an inclusion of a subgroupand e is an idempotent in A Q ( G ) . Then the adjunction i ∗ : L e S Q ( G –Sp O ) / / L i ∗ ( e ) S Q ( N –Sp O ) : F N ( G + , − ) o o is a Quillen pair. Exceptional part of rational G -spectra. We will repeatedly use the above result, mainlyin situations where after further localisation of the right hand side we will get a Quillen equivalence.
Corollary 11.4. [Kęd17a]
Suppose G is a compact Lie group and H is an exceptional subgroup of G . Then i ∗ : L e GH S Q ( G –Sp O ) / / L e NH S Q ( N –Sp O ) : F N ( G + , − ) o o is a Quillen pair.Proof. For N -good H , the result follows from the fact that the idempotent on the right hand sidesatisfies e NH = i ∗ ( e GH ) . For N -bad H , it is true since the left hand side is a further localisation of L i ∗ ( e GH ) S Q ( N − Sp O ) at the idempotent e NH : L e GH S Q ( G –Sp O ) i ∗ / / L i ∗ ( e GH ) S Q ( N –Sp O ) F N ( G + , − ) o o Id / / L e NH S Q ( N − Sp O ) Id o o Note that since H is bad, e NH = i ∗ ( e GH ) and e NH i ∗ ( e GH ) = e NH . (cid:3) Theorem 11.5. [Kęd17a]
Suppose H is an exceptional subgroup of G . Then the adjunction i ∗ N : L e GH S Q ( G –Sp O ) / / L e NH S Q ( N –Sp O ) : F N ( G + , − ) o o is a strong symmetric monoidal Quillen equivalence. Part of the difficulty in providing an algebraic model for a piece of homotopy category of rational G –spectra governed by an idempotent e comes from balancing two things. On the one hand, onewants to simplify the ambient category as much as one can. On the other one must preserve all therelevant homotopical information. This balancing act requires deep understanding of the homotopycategory of L e S Q G –Sp . In the case of an exceptional subgroup H of G , this is achieved by passingto L e NH S Q N –Sp using restriction as a left Quillen functor, as we described above.There was a reason why we considered restriction to be a left Quillen functor and it is relatedto the good and bad exceptional subgroups in G . Proposition 11.6. [Kęd17a]
Suppose H is an exceptional subgroup of G which is N -bad in G .Then i ∗ : L e GH S Q ( G –Sp O ) / / L e NH S Q ( N –Sp O ) : G + ∧ N − o o is not a Quillen adjunction. If H is an N -good subgroup in G , then the above adjunction is aQuillen pair. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 32 Toral part of rational SO (3) -spectra. As it was shown in [Kęd17b], i ∗ is not always aright Quillen functor, when considered between categories localised at the toral idempotents either.One can argue that this is because the toral idempotents do not always correspond with each other.One example is when G = SO (3) , T = SO (2) and N = O (2) . In that case the proof is based onthe fact that D is conjugate to C in SO (3) and thus i ∗ ( e T ) = e e T . Proposition 11.7. [Kęd17b]
Suppose e T is the toral idempotent of SO (3) and e e T is the toralidempotent of O (2) .That is, e T is the idempotent in A Q ( SO (3)) corresponding to the characteristicfunction of the toral part T (i.e. all subconjugates of the maximal torus of SO (3) ) and e e T is theidempotent in A Q ( O (2)) corresponding to the characteristic function of the toral part e T , i.e. allsubconjugates of the maximal torus of O (2) (see Sections 9.3.2 and 9.3.3). Then i ∗ : L e T S Q ( SO (3)–Sp) / / L e e T S Q ( O (2)–Sp) : SO (3) + ∧ O (2) − o o is not a Quillen adjunction. The restriction–coinduction adjunction is often better behaved with respect to localisation atidempotents.
Proposition 11.8.
Let i : O (2) −→ SO (3) be the inclusion. Then the following adjunction i ∗ : L e T S Q ( SO (3)–Sp) / / L e e T S Q ( O (2)–Sp) : F O (2) ( SO (3) + , − ) o o is a strong symmetric monoidal Quillen adjunction. The proof follows the same argument as Corollary 11.4 above, in the sense that the adjunctionis a composite of the restriction–coinduction adjunction localised at an idempotent e T (and itsrestriction i ∗ ( e T ) ) followed by a further localisation of O (2) –spectra (which excludes subgroup D ).This adjunction of restriction and coinduction is not quite a Quillen equivalence. Howevercellularising the right hand side at the derived images of the homotopically compact generators K for rational toral SO (3) –spectra and using the Cellularisation Principle (see Theorem 10.5) gives aQuillen equivalence. Theorem 11.9. [Kęd17b]
The following adjunction i ∗ : L e T S Q ( SO (3)–Sp) / / i ∗ ( K )–cell– L e e T S Q ( O (2)–Sp) : F O (2) ( SO (3) + , − ) o o is a Quillen equivalence, where K denotes the set of homotopically compact generators for L e T S Q ( SO (3)–Sp) . Dihedral part of rational SO (3) -spectra. In other cases of idempotents it is not alwaysclear to which category one should restrict. For the dihedral idempotent in rational SO (3) –spectra,restricting to certain part of the rational dihedral O (2) –spectra is the correct choice, but in generalthere is no good recipe for obtaining an algebraic model.In the dihedral part of SO (3) we can use restriction as a right or left Quillen functor, we chosethe following one, which also follows from Lemma 11.1. Corollary 11.10.
Let D denote the dihedral part of SO (3) and e D the corresponding idempotent.Let i : O (2) −→ SO (3) be the inclusion. Then i ∗ : L e D S Q ( SO (3)–Sp) / / L i ∗ ( e D ) S Q ( O (2)–Sp) : F O (2) ( SO (3) + , − ) o o is a Quillen adjunction. Remark 11.11.
The idempotent on the right hand side i ∗ ( e D ) corresponds to the dihedral partof O (2) excluding all subgroups D and D . Thus, i ∗ ( e D ) = i ∗ ( e D ) e e D . N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 33 Inflation and fixed point adjunction.
Suppose H is a normal subgroup of N and considerthe natural projection ε : N −→ N/H = W . Then there is a pair of adjoint functors ε ∗ : W –Sp O / / N –Sp O : ( − ) H o o where the right adjoint is the H fixed points functor and the left adjoint is called inflation. Fordetails see [MM02, Section V.3].We would like to understand the interaction between the localisation at idempotents and theabove adjunction. Notice that since inflation is strong symmetric monoidal, the result below followsfrom Lemma 11.1. Corollary 11.12.
Let ε : N −→ W denote the projection of groups, where H is normal in N and W = N/H . Let E be a cofibrant object in W –Sp O . Then ε ∗ : L E ( W –Sp O ) / / L ε ∗ ( E ) ( N –Sp O ) : ( − ) H o o is a strong symmetric monoidal Quillen pair. Lemma 11.13. [Kęd17a]
Suppose H is an exceptional subgroup of N , then the adjunction ε ∗ : L e W S Q ( W –Sp O ) / / L e NH S Q ( N –Sp O ) : ( − ) H o o is a Quillen equivalence. Here e W denotes an idempotent for the trivial subgroup { } W . In case of a torus T , we define ( S y ) T to be the diagram of commutative ring spectra obtained bytaking objectwise T –fixed points of S y (from Section 10.4). We illustrate this in the case T = SO (2) . S y = e E F (cid:15) (cid:15) DE F + / / DE F + ∧ e E F ( S y ) T = e E F T (cid:15) (cid:15) DE F T + / / ( DE F + ∧ e E F ) T The inflation–fixed point adjunction lifts to the level of module categories over the diagrams ofrings S y and ( S y ) T by [GS14a]. This adjunction is a Quillen equivalence and by the CellularisationPrinciple, Theorem 10.5, it induces a Quillen equivalence on the cellularised categories as follows.We refer the reader to [GS17] for more details. Theorem 11.14. [GS17]
Let T be a torus. The fixed point functor induces strong symmetricmonoidal Quillen equivalences S y -mod ≃ QE ( S y ) T -mod K –cell– S y -mod ≃ QE K T –cell–( S y ) T -modwhere K is the image in S y -mod of the set of compact generators for L S Q ( T –Sp O ) and K T its imagein ( S y ) T -mod. The advantage of this last theorem is that it gives a model for rational T –spectra in terms ofnon-equivariant spectra.The base idea for the toral part of rational N –spectra (where T is normal in N ) is to use thesame steps, but in a context where after taking T –fixed points we land in a category of spectra withan action of W = N/ T . This requires some very detailed constructions to make precise, which weleave to [BGK19]. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 34 An algebraic model for rational G -spectra - overview of some cases In this section we provide a summary of the necessary steps to obtain an algebraic model for a(part of) rational G –spectra in two cases. The first case is when G is a finite group and we followthe steps presented in the algebraic case in Part 1. The second case is when we are interested inthe toral part of rational G –spectra, for any compact Lie group G . We discuss briefly the series ofsimplifications required for the classification result in this case.12.1. An algebraic model for rational G -spectra for finite G . Building on the results ofSections 11.2 and 11.5 we can sketch the passage to the algebraic model for rational G –spectrawhen G is a finite group.Theorem 10.2 allows us to split the category of rational G –spectra into a finite product Y ( H ) G L e GH S Q ( G –Sp O ) . The next step uses restriction–coinduction Quillen equivalence L e GH S Q ( G –Sp O ) ≃ QE L e NH S Q ( N –Sp O ) for each factor of the product seperately. We then follow with the inflation-fixed point Quillenequivalence L e NH S Q ( N –Sp O ) ≃ QE L e W S Q ( W –Sp O ) of the previous section.The model category L e W S Q ( W –Sp O ) obtained after taking H –fixed points of L e NH S Q ( N –Sp O ) can be described in a much easier way. It is Quillen equivalent to the model category Sp O [ W ] oforthogonal spectra with the W action, where the model structure is created from the one on Sp O bythe forgetful functor U : Sp O [ W ] −→ Sp O . This allows us to remove the equivariance from inside of the complicated category W –Sp O (where it appeared in the indexing spaces for the spectrum)to the outside of much simpler Sp O [ W ] .Shipley [Shi07] gives a (zig-zag of weak) symmetric monoidal Quillen equivalence between rationalspectra and chain complexes of Q -modules (with the projective model structure). This is oftenreferred to in the literature as a algebraicisation . This result readily extends to a Quillen equivalencebetween rational spectra with a finite group action and rational chain complexes with a finite groupaction. Hence, we obtain an algebraic model for L e NH S Q ( N –Sp O ) in terms of chain complexes of Q [ W G H ] -modules.Combining all the steps mentioned in this section we obtain the following result. Theorem 12.1.
For G a finite group, there is a zig-zag of symmetric monoidal Quillen equivalencesbetween L e GH S Q ( G –Sp O ) and Ch( Q [ W G H ]) .The algebraic model for rational G –spectra is therefore Y ( H ) G Ch( Q [ W G H ]) . Moreover, if X is a rational G –spectrum with corresponding object ( A H ) ( H ) G in the algebraicmodel, then π ∗ ( (cid:0) i ∗ ( e GH X ) (cid:1) H ) ∼ = π ∗ (Φ H X ) ∼ = H ∗ ( A H ) . Here i ∗ and ( − ) H denote derived functors of restriction and fixed points discussed in Sections 11.2and 11.5, respectively. N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 35 Morita equivalences.
A different approach to obtaining an algebraic model for rational G –spectra for a finite group G is presented in [Bar09a] and uses Morita equivalences developed inthe spectral setting by Schwede and Shipley [SS03].The idea is to present L e GH S Q ( G –Sp O ) as a category of modules over the endomorphism ringspectrum of the compact generator e GH G/H + .Let Hom( − , =) denote the enrichment of G –spectra in non-equivariant spectra, then E H = Hom( e GH G/H + , e GH G/H + ) (with fibrant replacements omitted from the notation) is a ring spectrum under composition. Fur-thermore, the model category of modules over E H (in non-equivariant spectra) is Quillen equivalentto L e GH S Q ( G –Sp O ) . One can then use algebraicisation (the results of Shipley [Shi07])to obtain analgebraic model for this part of rational G –spectra.However, E H it is not (in general) a commutative ring spectrum in orthogonal spectra. Theproblem is fundamental and can be seen by looking at homotopy groups. The homotopy groups of E H are non-trivial only in degree , where they take value π (Hom( e GH G/H + , e GH G/H + )) = Q [ W G H ] . While this has a cocommutative Hopf algebra structure, it does not have a commutative ringstructure in general.This makes it much harder to obtain a comparison that takes into account the monoidal struc-tures. In particular, we would need to check that the algebraicisation of the ring spectrum E H alsohas a cocommutative Hopf algebra structure. As we only have control over the homology of thealgebraicised object, we would also need a formality argument that preserves the cocommutativeHopf algebra structure.12.3. An algebraic model for the toral part of rational G -spectra. We give a brief overviewof the remaining steps needed to classify rational toral G –spectra. Details are left to the references.While reading the summary, the reader may like to keep in mind the case T = SO (2) , G = SO (3) and N = O (2) . These are the easiest cases of interest and have been discussed in previous sections.Greenlees and Shipley [GS17] give an algebraic model for rational T –spectra, where T is a torus.See also [Gre99] for a full explanation of the algebraic model and [BGKS17] for the classificationof rational SO (2) –spectra. The first two steps of the classification are to apply Theorems 10.6 and11.14. The next step is to algebraicise using work of Shipley [Shi07]. This gives an algebraic modelfor rational T –spectra in terms of (a cellularisation of) a category of modules over a diagram ofcommutative differential graded algebras. Formality of these commutative dgas allows us to simplifythe rings in the diagram. An additional simplification of the algebra removes the cellularisationand gives the algebraic model for rational T –spectra.Work of the authors and Greenlees gives an algebraic model for the toral part of rational G –spectra for any compact Lie group G , see [BGK19]. Given G , we let T be a maximal torus and N its normaliser in G . We can lift the classification for rational T –spectra to a classification ofrational toral N –spectra. We then use the following result to reduce problem of classifying rationaltoral G –spectra to understanding a cellularisation of rational toral N -spectra. Theorem 12.2. [BGK19]
The following adjunction i ∗ : L e G T S Q ( G –Sp) / / i ∗ ( L )–cell– L e N T S Q ( N –Sp) : F N ( G + , − ) o o N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 36 is a Quillen equivalence, where the idempotent on both sides corresponds to the families of allsubgroups of maximal torus T ≤ N ≤ G and L denotes the set of homotopically compact generatorsfor L e G T S Q ( G –Sp) . By the Cellularisation Principle, Theorem 10.5, we can cellularise each term of the classificationof rational toral N -spectra at the derived images of the cells L . This gives a classification of rationaltoral G -spectra in terms of a cellularisation of the algebraic model for rational toral N –spectra. Thefinal simplification is to remove this cellularisation, which is based on another formality argument. References [Bar09a] D. Barnes. Classifying rational G -spectra for finite G . Homology, Homotopy Appl. , 11(1):141–170, 2009.[Bar09b] D. Barnes. Splitting monoidal stable model categories.
J. Pure Appl. Algebra , 213(5):846–856, 2009.[Bar17] D. Barnes. Rational O (2) -equivariant spectra. Homology Homotopy Appl. , 19(1):225–252, 2017.[BG19] S. Balchin and J.P.C. Greenlees. Adelic models of tensor-triangulated categories. arXiv:1903.02669, 2019.[BGK18a] D. Barnes, J. P. C. Greenlees, and M. Kędziorek. An algebraic model for rational naïve-commutative G –equivariant ring spectra for finite G . Homology, Homotopy and Applications , (21(1)):73–93, 2018.[BGK18b] D. Barnes, J. P. C. Greenlees, and M. Kędziorek. An algebraic model for rational naïve-commutativering SO(2)-spectra and equivariant elliptic cohomology. arXiv:1810.03632, 2018.[BGK19] D. Barnes, J. P. C. Greenlees, and M. Kędziorek. An algebraic model for rational toral G–spectra.
Algebr.Geom. Topol. , 19(7):3541–3599, 2019.[BGKS17] D. Barnes, J. P. C. Greenlees, M. Kędziorek, and B. Shipley. Rational
SO(2) -equivariant spectra.
Algebr.Geom. Topol. , 17(2):983–1020, 2017.[Böh19] B. Böhme. Multiplicativity of the idempotent splittings of the Burnside ring and the G -sphere spectrum. Adv. Math. , 347:904–939, 2019.[BR13] D. Barnes and C. Roitzheim. Stable left and right Bousfield localisations.
Glasgow Mathematical Journal ,FirstView:1–30, 2 2013.[BS20] D. Barnes and D. Sugrue. The equivalence between rational G -sheaves and rational G -mackey functorsfor profinite G . In preparation, 2020.[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and algebras in stable ho-motopy theory , volume 47 of
Mathematical Surveys and Monographs . American Mathematical Society,Providence, RI, 1997. With an appendix by M. Cole.[Fau08] H. Fausk. Survey on the Burnside ring of compact Lie groups.
J. Lie Theory , 18(2):351–368, 2008.[Glu81] D. Gluck. Idempotent formula for the Burnside algebra with applications to the p -subgroup simplicialcomplex. Illinois J. Math. , 25(1):63–67, 1981.[GM92] J. P. C. Greenlees and J. P. May. Some remarks on the structure of Mackey functors.
Proc. Amer. Math.Soc. , 115(1):237–243, 1992.[GM95] J. P. C. Greenlees and J. P. May. Generalized Tate cohomology.
Mem. Amer. Math. Soc. ,113(543):viii+178, 1995.[Gre71] J. A. Green. Axiomatic representation theory for finite groups.
J. Pure Appl. Algebra , 1(1):41–77, 1971.[Gre92] J. P. C. Greenlees. Some remarks on projective Mackey functors.
J. Pure Appl. Algebra , 81(1):17–38,1992.[Gre98a] J. P. C. Greenlees. Rational Mackey functors for compact Lie groups. I.
Proc. London Math. Soc. (3) ,76(3):549–578, 1998.[Gre98b] J. P. C. Greenlees. Rational O(2)-equivariant cohomology theories. In
Stable and unstable homotopy(Toronto, ON, 1996) , volume 19 of
Fields Inst. Commun. , pages 103–110. Amer. Math. Soc., Providence,RI, 1998.[Gre99] J. P. C. Greenlees. Rational S -equivariant stable homotopy theory. Mem. Amer. Math. Soc. ,138(661):xii+289, 1999.[Gre01] J. P. C. Greenlees. Rational
SO(3) -equivariant cohomology theories. In
Homotopy methods in algebraictopology (Boulder, CO, 1999) , volume 271 of
Contemp. Math. , pages 99–125. Amer. Math. Soc., Provi-dence, RI, 2001.[Gre20] J. P. C. Greenlees. Couniversal spaces which are equivariantly commutative ring spectra.
HomologyHomotopy Appl. , 22(1):69–75, 2020.
N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 37 [GS14a] J. P. C. Greenlees and B. Shipley. Fixed point adjunctions for equivariant module spectra. Algebr. Geom.Topol. , 14(3):1779–1799, 2014.[GS14b] J. P. C. Greenlees and B. Shipley. Homotopy theory of modules over diagrams of rings.
Proc. Amer.Math. Soc. Ser. B , 1:89–104, 2014.[GS17] J. P. C. Greenlees and B. Shipley. An algebraic model for rational torus-equivariant spectra. Acceptedto J. Topol., arXiv:1101.2511v5, 2017.[Hir03] P. S. Hirschhorn.
Model categories and their localizations , volume 99 of
Mathematical Surveys andMonographs . American Mathematical Society, Providence, RI, 2003.[HM19] M. A. Hill and K. Mazur. An equivariant tensor product on Mackey functors.
J. Pure Appl. Algebra ,223(12):5310–5345, 2019.[Hov99] M. Hovey.
Model categories , volume 63 of
Mathematical Surveys and Monographs . American Mathemat-ical Society, Providence, RI, 1999.[HPS97] M. Hovey, J. H. Palmieri, and N. P. Strickland. Axiomatic stable homotopy theory.
Mem. Amer. Math.Soc. , 128(610):x+114, 1997.[Kęd17a] M. Kędziorek. An algebraic model for rational G -spectra over an exceptional subgroup. Homology Ho-motopy Appl. , 19(2):289–312, 2017.[Kęd17b] M. Kędziorek. An algebraic model for rational
SO(3) -spectra.
Algebr. Geom. Topol. , 17(5):3095–3136,2017.[Lew98] L. Gaunce Lewis, Jr. The category of Mackey functors for a compact Lie group. In
Group representations:cohomology, group actions and topology (Seattle, WA, 1996) , volume 63 of
Proc. Sympos. Pure Math. ,pages 301–354. Amer. Math. Soc., Providence, RI, 1998.[Lin76] H. Lindner. A remark on Mackey-functors.
Manuscripta Math. , 18(3):273–278, 1976.[LMSM86] L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure.
Equivariant stable homotopy theory ,volume 1213 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1986. With contributions by J.E. McClure.[Luc96] F. Luca.
The algebra of Green and Mackey functors . ProQuest LLC, Ann Arbor, MI, 1996. Thesis(Ph.D.)–University of Alaska Fairbanks.[May96] J. P. May.
Equivariant homotopy and cohomology theory , volume 91 of
CBMS Regional ConferenceSeries in Mathematics . Published for the Conference Board of the Mathematical Sciences, Washington,DC, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C.Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner.[Maz13] K. Mazur.
On the Structure of Mackey Functors and Tambara Functors . ProQuest LLC, Ann Arbor,MI, 2013. Thesis (Ph.D.)–University of Virginia.[MM02] M. A. Mandell and J. P. May. Equivariant orthogonal spectra and S -modules. Mem. Amer. Math. Soc. ,159(755):x+108, 2002.[PW19] L. Pol and J. Williamson. The left localization principle, completions, and cofree G -spectra.arXiv:1910.01410, 2019.[Rav84] D. C. Ravenel. Localization with respect to certain periodic homology theories. Amer. J. Math. ,106(2):351–414, 1984.[Shi07] B. Shipley. H Z -algebra spectra are differential graded algebras. Amer. J. Math. , 129(2):351–379, 2007.[SS03] S. Schwede and B. Shipley. Stable model categories are categories of modules.
Topology , 42(1):103–153,2003.[Str12] N. Strickland. Tambara functors. arXiv: 1205.25161 , 2012.[Sug19] D. Sugrue. Rational G -spectra for profinite G . arXiv Math. Ann. , 215:235–250, 1975.[tD79] T. tom Dieck.
Transformation groups and representation theory , volume 766 of
Lecture Notes in Math-ematics . Springer, Berlin, 1979.[TW95] J. Thévenaz and P. Webb. The structure of Mackey functors.
Trans. Amer. Math. Soc. , 347(6):1865–1961, 1995.[Web00] P. Webb. A guide to Mackey functors. In
Handbook of algebra, Vol. 2 , volume 2 of
Handb. Algebr. , pages805–836. Elsevier/North-Holland, Amsterdam, 2000.[Wim19] C. Wimmer. A model for genuine equivariant commutative ring spectra away from the group order. 2019.arXiv.org:1905.12420 [math.AT].[Yos80] T. Yoshida. On G -functors. I. Transfer theorems for cohomological G -functors. Hokkaido Math. J. ,9(2):222–257, 1980.
N INTRODUCTION TO ALGEBRAIC MODELS FOR RATIONAL G -SPECTRA 38 (Barnes) Mathematical Sciences Research Centre, Queen’s University Belfast (Kędziorek)(Kędziorek)