Naive-commutative structure on rational equivariant K -theory for abelian groups
Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, Magdalena Kędziorek, Clover May
aa r X i v : . [ m a t h . A T ] F e b NAIVE-COMMUTATIVE STRUCTURE ON RATIONALEQUIVARIANT K -THEORY FOR ABELIAN GROUPS ANNA MARIE BOHMANN, CHRISTY HAZEL, JOCELYNE ISHAK,MAGDALENA KĘDZIOREK, AND CLOVER MAY
Abstract.
In this paper, we calculate the image of the connective and peri-odic rational equivariant complex K -theory spectrum in the algebraic modelfor naive-commutative ring G -spectra given by Barnes, Greenlees and Kędziorekfor finite abelian G . Our calculations show that these spectra are unique asnaive-commutative ring spectra in the sense that they are determined up toweak equivalence by their homotopy groups. We further deduce a structuretheorem for module spectra over rational equivariant complex K -theory. Introduction
Modeling rational spectra via algebraic data has a long and fruitful history inhomotopy theory. Serre’s original calculations of stable homotopy groups of spheres[18] imply that the rational homotopy category Ho ( S p Q ) is equivalent to the cate-gory of graded rational vector spaces. An analogous equivalence was later obtainedat the level of derived categories by Robinson [15] and at the level of model cate-gories by Shipley [20] as a zig-zag of symmetric monoidal Quillen equivalences S p Q ≃ Q Ch ( Q -mod ) . Work of Richter and Shipley [14] further shows that there is a zig-zag of Quillenequivalences between rational commutative ring spectra and commutative differen-tial graded algebras over Q . Hence rational CDGAs are an algebraic model for therational commutative ring spectra.It is something of a truism in algebraic topology that “algebra is easy,” in thesense that once one can reduce a topological question to a matter of algebra, the re-maining algebraic computations should be straightforward. Like most truisms, thisone is mostly false: algebraic computations come equipped with a plethora of sub-tleties. Moreover, the abstract knowledge that one can reduce a problem to algebrais often quite separate from the explicit reduction in a given case. In particular,for a concrete rational commutative ring spectrum X , it may be nontrivial to findthe explicit rational CDGA corresponding to X under the Richter–Shipley zig-zagof Quillen equivalences. Nevertheless, algebraic models of homotopy theory—andmore general algebraicizations of topological questions—are of great utility in bothstructural and computational understanding of homotopy theory.In this paper, we focus on specific, concrete computations in algebraic modelsfor rational G -equivariant spectra over a finite group G . That is, our main goal isto find explicit models for rational G -spectra in the algebraic categories modelingthese spectra. The main spectra of interest are commutative ring spectra.For any finite group G , there is a model for the homotopy category of rational G -spectra given by work of Greenlees and May [10]. What we call an algebraic model in this paper is not the model for the homotopy category of rational G -spectra, butan algebraic model category that is Quillen equivalent to the category of spectra inquestion. In the case of G -spectra, [12] uses Greenlees and May’s result to producean algebraic model category A ( G ) Q that is Quillen equivalent to the stable modelcategory of rational orthogonal G -spectra.In the nonequivariant case, Richter and Shipley’s result says that commutativealgebra objects in the algebraic model for rational spectra are a model for rationalcommutative algebra spectra. In the equivariant case, the story is more intricate.There is a hierarchy of types of “equivariant commutativity” [5], and commutativealgebra objects in A ( G ) Q only model the lowest level of this commutativity, which issometimes referred to as “naive commutative” [3]. We denote this algebraic modelfor rational naive-commutative ring G -spectra by Comm A ( G ) Q . We provide adetailed description of the algebraic models A ( G ) Q and Comm A ( G ) Q in Section 2.Our main theorem is as follows. It appears later as Theorem 5.8. Theorem.
Let G be a finite abelian group. The image of KU G Q in the algebraicmodel Comm A ( G ) Q is given by ( V H ) ( H ) ≤ G where • V H = 0 if H is not cyclic and • when H is cyclic of order n , V H ∼ = Q ( ζ n )[ β ± ] where Q ( ζ n ) is the fieldextension of Q by a primitive n -th root of unity ζ n and β is in degree . Finding this image of KU G Q in the algebraic model is not simply a matter oftracing through the various functors in zig-zag of Quillen equivalences betweenrational naive-commutative ring G -spectra and Comm A ( G ) Q . This zig-zag includesfunctors for which we do not have explicit computational control. What the zig-zag retains is control over the homology of the image in the algebraic model ofa given spectrum X ; in general this does not suffice to determine the algebraicobject itself. Hence the strategy of proof is to compute the homotopy groups ofthe geometric fixed points of KU G Q . These homotopy groups encode the homologyof the algebraic model of KU G Q as a naive commutative ring G -spectrum. We thenshow any commutative differential graded algebra with this homology is formal.The formality result finally determines the image of KU G Q in the algebraic model.Our main result has several consequences. Firstly, it shows that all modules over KU G Q are free over the idempotent pieces of KU G Q . This result is stated as Corollary4.6. In fact, our calculations show this holds for both abelian and nonabeliangroups. Theorem.
Let G be a finite group and X be a module spectrum over KU G Q . Then X ≃ M ( H ) ( e ( H ) KU G Q ) ⊕ i H ⊕ (Σ e ( H ) KU G Q ) ⊕ j H , where H ≤ G , and i H and j H are nonnegative integers. Our formality result also shows that KU G Q admits a unique naive-commutative E ∞ structure. This result is stated as Corollary 5.10. Theorem.
Let G be a finite abelian group. Then KU G Q and ku G Q admit uniquestructures as naive commutative G -ring spectra, i.e., as naive E ∞ -algebras in G -spectra. That is, if X is a rational naive-commutative G -ring spectrum whose gradedGreen functor of homotopy groups is isomorphic to that of KU G Q or ku G Q , then there AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 3 is a weak equivalence of rational naive-commutative G -ring spectra between X andKU G Q or ku G Q , respectively. The computations in this paper also set the stage for an analysis for KU G Q asa genuine commutative ring spectrum. This analysis, which uses recent work ofWimmer [24], is the subject of forthcoming work by the authors.1.1. Notation.
Throughout the paper we assume that G is a finite group. We usethe notation A ( G ) Q for the algebraic model of rational G -spectra and Comm A ( G ) Q for the algebraic model of rational naive-commutative ring G -spectra; see Defi-nition 2.4 and Definition 2.6, respectively. We use the notation ≃ Q to denotea zig-zag of Quillen equivalences between model categories. If X is a rationalnaive-commutative ring G -spectrum then we denote by θ ( X ) its derived image inComm A ( G ) Q .1.2. Acknowledgments.
We would like to thank Hausdorff Research Institutefor Mathematics in Bonn for their hospitality in hosting the Women in TopologyIII workshop, where much of this research was carried out. We also extend ourthanks to the organizers of the workshop, Julie Bergner, Angélica Osorno, and SarahWhitehouse, for making the event both possible and productive. Many thanks aredue to Brooke Shipley, who helped shape the initial stages of this research. Warmthanks also to Dan Dugger and Mike Hill for many helpful conversations.In addition to funding for the workshop from the Hausdorff Institute, we aregrateful to Foundation Compositio Mathematica and to the National Science Foun-dation of the United States, both of which provided funding for the workshop.NSF support was via the grants NSF DMS 1901795 and NSF HRD 1500481: AWMADVANCE. Additionally, the fourth author was supported by a NWO Veni grant639.031.757. The first author was partially supported by NSF Grant DMS-1710534.2.
Review of Rational Models
In this section, we recall the construction of algebraic models in the world ofrational equivariant stable homotopy theory. We begin by reviewing the story atthe level of homotopy groups and homotopy categories, followed by a discussion ofthe more structured story at the level of algebraic models via Quillen equivalences.Given any G -spectrum X and any integer n , the collection of homotopy groups { π Hn ( X ) | H ≤ G } forms what is called a Mackey functor . The description ofMackey functors we follow is due to Dress [6]. For an introduction to the theory ofMackey functors we refer the reader to [23], [21], or [11, §3.1]. When X is a rational G -spectrum, its Mackey functor of homotopy groups π n ( X ) is a rational Mackeyfunctor, meaning for every subgroup H of G , π Hn ( X ) is a rational vector space. Forexample, if X is the rational equivariant sphere spectrum S Q , then the homotopygroups Mackey functor π ( S Q ) is the rational Burnside ring Mackey functor A Q ,which is defined by A Q ( G/H ) := A ( H ) ⊗ Q , where A ( H ) is the Burnside ring of H , i.e. the Grothendieck ring of finite H -sets.All higher homotopy groups of S Q vanish. Remark . The Burnside ring Mackey functor A Q has more structure than simplythat of a Mackey functor. It is a commutative Green functor, which reflects the BOHMANN, HAZEL, ISHAK, KĘDZIOREK, AND MAY fact that S Q is a (naive) commutative G -spectrum. In fact, A Q has the even richerstructure of a Tambara functor, although we will not make use of it in this paper.In this rational setting, Greenlees and May [10] show that the algebraic structureof Mackey functor homotopy groups determine the homotopy category of spectrain the following sense: they produce an equivalence of categoriesHo ( G - S p Q ) → gr ( M ack ( G ) Q ) from the homotopy category of rational G spectra to the category of graded rationalMackey functors that is given by taking homotopy groups. We note here that thisfunctor is not induced by a Quillen equivalence of model categories. Once in thealgebraic setting of Mackey functors, idempotents in the Burnside ring allow afurther splitting of rational Mackey functors into families of modules over grouprings Q [ W G H ] , where W G H is the Weyl group of a subgroup H of G . That is,Greenlees and May prove the following theorem. Theorem 2.2 (Greenlees–May [10]) . Idempotent splitting produces an equivalenceof categories gr ( M ack Q G ) → Y ( H ) ≤ G gr ( Q [ W G H ] -mod ) . where W G H = N G H/H is the Weyl group of H as a subgroup of G . There is thusan equivalence of categoriesHo ( G - S p Q ) → Y ( H ) ≤ G gr ( Q [ W G H ] -mod ) . At each level of the grading, the functor from M ack Q ( G ) to Q Q [ W G H ] -mod isgiven by sending a Mackey functor M to the W G H -module e H M ( G/H ) where e H is an idempotent in the rational Burnside ring for G associated to the subgroup H .We discuss these idempotents in more detail below in Section 3.We are interested in incorporating additional structure that is not present inthe homotopy category of rational G -spectra. First, we wish to work at the modelcategorical level. Theorem 2.2’s splitting of the homotopy category of rational G -spectra for finite G is mimicked to give a zig-zag of symmetric monoidal Quillenequivalences of monoidal model categories in [12](2.3) G - S p Q ≃ Q Y ( H ) ≤ G Ch ( Q [ W G H ] -mod ) , where the product is over conjugacy classes of subgroups of G . The model category Q ( H ) ≤ G Ch ( Q [ W G H ] -mod ) has the objectwise projective model structure. Definition 2.4.
The model category Y ( H ) ≤ G Ch ( Q [ W G H ] -mod ) is called the algebraic model for rational G -spectra and is denoted A ( G ) Q .Note that the monoidal structure on A ( G ) Q is given by tensor product over Q in every product factor. One of the consequences of this result is that the derivedimage of the unit is the unit. That is, the sphere spectrum is sent to the constantsequence Q concentrated in degree with trivial Weyl group actions. AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 5 Next we consider commutative ring structures on spectra. This considerationis more subtle than in the non-equivariant case. G -spectra have a hierarchy oflevels of “equivariant commutativity” [5]. Ring G -spectra with the lowest level ofcommutativity are called naive -commutative. Naive-commutative ring G -spectraare algebras for a G -operad equipped with a trivial G -action which is underlying E ∞ when one forgets the G -action. An example of such a G -operad is the linearisometries operad on a trivial G -universe.Barnes, Greenlees and Kędziorek [3] showed that commutative algebras in thealgebraic model A ( G ) Q for rational G -spectra model these naive-commutative ring G -spectra. That is, there is a zig-zag of Quillen equivalences(2.5) Comm Naive ( G S p Q ) ≃ Q Comm ( Y ( H ) ≤ G Ch ( Q [ W G H ] -mod )) . Definition 2.6.
The model category Comm ( Q ( H ) ≤ G Ch ( Q [ W G H ] -mod )) is de-noted by Comm A ( G ) Q and it has weak equivalences and fibrations created in A ( G ) Q . It is the algebraic model for rational naive-commutative ring G -spectra.Remark . Note that the product of these commutative differential graded alge-bras is equivalent to a diagram categoryComm A ( G ) Q = Y ( H ) ≤ G Q [ W G H ] -CDGA ∼ = Orb × G -CDGA Q , Here the category Orb G is the orbit category spanned by transitive G -sets G/H for H ≤ G , and the morphisms are given by the set of G -equivariant maps. Thecategory Orb × G is the full subcategory of Orb G consisting of isomorphisms.The image of a (naive-commutative ring) G -spectrum in the algebraic model isnot very explicit, as the Quillen equivalences of (2.3) and (2.5) used in establishingthe algebraic model use Shipley’s result [20] (and Richter–Shipley’s result [14],respectively), which is not computationally trackable.Let θ ( X ) denote the image of a naive-commutative rational G -spectrum X in thealgebraic model Comm A ( G ) Q . The algebraic splitting of the category of gradedMackey functors (or commutative Green functors) using idempotents of the ratio-nal Burnside ring A ( G ) ⊗ Q is compatible with splitting rational G -spectra usingthe idempotents by [10, Appendix A]. Hence, by [3] we know that the homotopygroups of the geometric fixed points of a naive-commutative ring G -spectrum X areisomorphic to the homology of θ ( X ) . That is, for each conjugacy class of subgroups ( H ) , the homology of the chain complex θ ( X ) ( H ) is given by the homotopy groupsof the H -geometric fixed points Φ H ( X ) :(2.8) H ∗ ( θ ( X ) ( H ) ) = π ∗ (Φ H ( X )) . In fact, using this observation we can calculate the homology of the image of X inthe algebraic model using the splitting of rational Mackey functors, since π ∗ (Φ H ( X )) ∼ = e H π ∗ ( X )( H ) where e H is an idempotent element in the Burnside ring A ( G ) ⊗ Q . This compati-bility is shown in [10].A key to identifying the image of a spectrum X in the rational model is thereforeto calculate the idempotent pieces of the Mackey functors π ∗ ( X ) . In the next sectionwe concentrate on understanding the action of the Burnside ring Green functor ona given Mackey functor and the behavior of the algebraic idempotent splitting. BOHMANN, HAZEL, ISHAK, KĘDZIOREK, AND MAY Splitting Mackey functors via idempotents in the Burnside ring
In this section, we give an overview of the idempotent splitting of rational Mackeyfunctors for a finite group with the goal of providing the context necessary for thecalculations in Section 4. As mentioned, these results originate in [10, AppendixA]. For more details on the action of the Burnside ring Green functor on a Mackeyfunctor X and modern account of the idempotent splitting see [4]. We review theconstruction of the idempotent splitting in enough detail to suggest the essentialcalculational result, Lemma 3.4. This result is proved in [16].Let G be a finite group. For the remainder of the paper, we suppress the notationfor rationalization and let A ( G ) denote the rational Burnside ring for G . Recallthat if X is a finite G -set, then X decomposes into orbits G · x , . . . , G · x n and foreach x i the orbit G · x i is isomorphic to G/ Stab ( x i ) . For subgroups H and K in G ,the orbits G/H and
G/K are isomorphic as G -sets if and only if H is conjugate to K . Thus a basis for A ( G ) is given by { [ G/H ] | ( H ) ≤ G } , where ( H ) ≤ G is used to denote a conjugacy class of subgroups in G . We willabuse notation by writing ( H ) for both the set of subgroups conjugate to H andfor a single representative of this conjugacy class. Note if K is another subgroupof G , the notation ( H ) ≤ K indicates H is subconjugate to K by an element of G .By tom Dieck’s result [22, 5.6.4, 5.9.13], the ring map Φ : A ( G ) → Y ( H ) ≤ G Q defined by [ X ] ( | X H | ) ( H ) . is an isomorphism. Thus it can be used to find idempotents in the ring A ( G ) . Define e J to be the pre-image of the projection onto the ( J ) -th factor in the product. Thatis, e J = Φ − (( δ J ( H )) ( H ) ) , where δ J ( H ) = ( , if ( J ) = ( H )0 , otherwise.Let M be a rational Mackey functor on a finite group G . We can define an actionof the Burnside ring A ( G ) on M as follows. Let X and Y be finite G -sets, and let π : Y × X → Y denote the projection. The action of [ X ] ∈ A ( G ) on M ( Y ) is given by the composite(3.1) M ( Y ) π ∗ −→ M ( Y × X ) π ∗ −→ M ( Y ) , as is shown, for example, in [9] or [21]. One can check this action is through ringmaps, and so using the description of the idempotents in terms of the additive basis,we can decompose the Mackey functor M as(3.2) M ∼ = M ( H ) ≤ G e H M. This is a decomposition as Mackey functors. To deduce Theorem 2.2, Greenleesand May make a further essential reduction by showing that for any H , the Mackey AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 7 functor e H M is freely generated by the W G H -module e H M ( G/H ) . Indeed, theungraded case of Theorem 2.2 is an equivalence M ack ( G ) Q → Y ( H ) ≤ G Q [ W G H ] -modgiven by sending a Mackey functor M to the sequence of modules ( V H ) defined by V H = e H M ( G/H ) . The Weyl group action on V H is the inherent W G H -action on the value of theMackey functor e H M at G/H .In order to understand the idempotent pieces of a Mackey functor more con-cretely, we would like an explicit description of the elements e H ∈ A ( G ) . Theformula for the idempotents in terms of the additive basis was first introduced byGluck in [8]. Lemma 3.3 ([8]) . Let H be a subgroup of G , then e H ∈ A ( G ) is given by theformula e H = 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 sub-groups from K to H of length i . The length of a chain is one less than the numberof subgroups involved and µ ( H, H ) = 1 for all H G . Thus the action of the idempotent element e H on a Mackey functor M can becalculated from the action of orbits G/K on the groups M ( G/J ) for subgroups K ≤ H . From the description of the action of the Burnside ring (3.1), we knowthat G/K acts on M ( G/J ) by a sum of composites tr J ? ◦ res J ? between J and variousgroups that are subconjugate to both J and K , but the coefficients of this sum arenot transparent. Thus, in general, it is not that easy to calculate e H M ( G/J ) , where ( H ) ≤ J . However, in this paper we only need to compute V H = e H M ( G/H ) asa Q [ W G H ] -module. In [16], Schwede gives an inductive argument on the lattice ofsubgroups to obtain the following elegant description. Lemma 3.4. [16, Theorem 3.4.22]
For H ≤ G , V H ∼ = M ( G/H ) /t H M, where t H M is the subgroup of M ( G/H ) generated by transfers from proper sub-groups of H . Note that the Weyl group action on the Mackey functor descends toa W G H -action on the quotient V H because of the compatibility axiom c g,K tr KH = tr gKgH c g,H , where H ≤ K , g ∈ G and c g,H is the conjugation map c g,H : M ( G/H ) → M ( G/gHg − ) . Remark . Observe that Maschke’s theorem [7, Proposition 1.5] applies to showthat since t H M is a Q [ W G H ] -module of M ( G/H ) , it has a complement and thusthe quotient M ( G/H ) /t H M is in fact a direct summand of M ( G/H ) . This is ofcourse necessary if M ( G/H ) /t H M is to be the direct summand e H M ( G/H ) . BOHMANN, HAZEL, ISHAK, KĘDZIOREK, AND MAY
Lemma 3.4 provides a tool at the heart of our strategy for computing the image θ ( X ) of a rational G -spectrum in the algebraic model. For reference, we describethis strategy explicitly. This is the procedure we employ in the next two sectionsto calculate the image of KU G Q . Strategy 3.6.
Let X be a naive-commutative rational G -spectrum. A generalstrategy for attempting to calculate θ ( X ) is to do the following for a representative H of each conjugacy class of subgroups of G :(1) Use Lemma 3.4 to calculate V H = e H π ∗ ( X )( G/H ) , together with its gradedalgebra structure.(2) Show that V H is formal as a commutative differential graded Q [ W G H ] -algebra.(3) Use (2.8) to conclude that the ( H ) -coordinate of θ ( X ) is weakly equivalentto V H .These steps imply that each component of θ ( X ) is weakly equivalent to V H . Sincethe model category Comm A ( G ) Q is a product, we obtain a weak equivalence θ ( X ) ≃ ( V H ) . We begin by illustrating this strategy on two simple examples, the Eilenberg–MacLane spectrum for the constant commutative Green functor Q and the Eilenberg–MacLane spectrum for the rational Burnside Green functor A Q . In both cases, theformality of Step 2 is immediate and the focus is on calculating the idempotentpieces of the Green functors using Lemma 3.4. Example . Let G be a finite group. Suppose Q is the constant Green functorwith value Q , i.e. the value at any orbit G/H is given by Q ( G/H ) = Q , where the action of the Weyl group W G H is trivial. For K ≤ H ≤ G , all restrictionand conjugation maps are the identities and the transfer maps are given bytr HK : Q → Q x X γ ∈ W H K γ · x = X γ ∈ W H K x = | W H K | x. Hence, for any subgroup H , the image of the transfer from the trivial subgroup e is ℑ ( tr He ) = Q . Thus the homology of the image of the equivariant rational Eilenberg–MacLanespectrum H Q in the algebraic model is V H = Q /t H Q = 0 , for all subgroups H ≤ G except for the trivial subgroup, where V e = Q ( G/e ) = Q . Notice that V e = Q is concentrated in degree and is formal as a CDGA. Thusthe image of H Q in the algebraic model is weakly equivalent to the sequence ofCDGAs with value Q at the trivial group and zero at other groups. AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 9 Example . Let G be a finite group. The rational Burnside Mackey functor A Q is the representable functor B opG ( − , G/G ) ⊗ Q where B G is the Burnside category.One can check that this is isomorphic to the Mackey functor whose value on theorbit G/H is given by A ( H ) , the rational Burnside ring of H . For H ≤ K ≤ G ,the restriction maps are given byres KH ([ Y ]) = [ i ∗ H ( Y )] , where Y is a K -set, and i ∗ H : S et K → S et H is the forgetful functor from the category S et K of finite K -sets to the category offinite H -sets. The transfer maps are given by induction, i.e.tr KH ([ X ]) = [ K × H X ] , where K × H X is the quotient of K × X given by ( kh, x ) ∼ ( k, hx ) for all h ∈ H .Note that K acts on the left coordinate of the set K × H X .Recall that the rational equivariant sphere spectrum S Q is the Eilenberg–MacLanespectrum HA Q for the Burnside ring Mackey functor. Using Lemma 3.4 we cancalculate the image of S Q in the algebraic model Comm A ( G ) Q by calculating theimage of the transfers.For each subgroup H , A Q ( G/H ) = A ( H ) has an orbit basis given by { [ H/J ] | ( J ) H ≤ H } . Here it is important that we consider conjugation by H instead of conjugation by G ,and in general the conjugacy class ( J ) H may contain strictly fewer subgroups than ( J ) G . Let J be a proper subgroup of H . Observe that the image of the element [ J/e ] ∈ A ( J ) under the transfer map tr HJ : A ( J ) → A ( H ) is [ H × J J/e ] = [
H/J ] . Hence all basis elements of the form [ H/J ] for proper J ≤ H are in the image of thetransfer. Moreover, explicit calculation shows that no fixed H -set is in the imageof a transfer map tr HJ : A ( J ) → A ( H ) . Therefore, for each conjugacy class ( H ) , wehave an isomorphism V H = A ( H ) /t H A Q ∼ = Q { [ H/H ] } , concentrated in degree zero. The W G H -action on A ( H ) is via conjugation, and ishence trivial on the basis element [ H/H ] ; thus V H has a trivial W G H -action.This determines the homology of θ ( S Q ) in Comm A ( G ) Q . Since for each ( H ) , Q concentrated in degree zero is formal as an object of Q [ W G H ] -CDGA, we find that θ ( S Q ) is weakly equivalent to the constant sequence of Q ’s with trivial Weyl groupactions. Remark . In fact, the image θ ( S Q ) can be deduced from the construction of thezig-zag of Quillen equivalences in [12]. This is a zig-zag of (symmetric) monoidalQuillen equivalences and thus, as mentioned in Section 2, it sends the unit S Q inrational G -ring spectra to the unit in A ( G ) Q . Since the zig-zag of Quillen equiva-lences for naive-commutative rational ring G -spectra from [3] is a lift of the zig-zagof [12], the statement follows. The calculation in Example 3.8 is presented as anillustration of the computational techniques on a familiar example. Rational representation rings and Bott periodicity: homologylevel calculations
Our main goal is to calculate the image of the ring spectrum KU G Q in the algebraicmodel Comm A ( G ) Q by implementing Strategy 3.6. Thus the first step towardsunderstanding the image θ ( KU G Q ) is to calculate the homotopy Mackey functors π ∗ ( KU G Q ) and their idempotent splittings via the techniques of Section 3. As inDisplay (2.8), the result of these calculations is the homology of θ ( KU G Q ) , which isrecorded as Lemma 4.5.Recall that π KU G Q ∼ = RU G Q where RU G Q is the rationalized representation ringMackey functor. That is, the value of RU G Q at an orbit G/H is the rationalization ofthe Grothendieck ring of complex H -representations RU ( H ) , the restriction mapsare given by the restriction of representations, and the transfer maps are givenby the induction of representations. The action of W G H on RU ( H ) is given by g · [ V ] = [ V g ] where V g is the H -representation such that h · v = ( ghg − ) v . Webegin by studying the Eilenberg–MacLane spectrum for this Mackey functor. Inorder to describe the action of the Weyl group on the homology, we first define thefollowing function. Definition 4.1.
Let H be a cyclic subgroup of G of order n with a chosen generator g . Let m H denote the function m H : W G H → Z /n where m H ( a ) ∈ { , . . . , n } issuch that a − ga = g m H ( a ) .In fact, m H : W G H → Z /n is a homomorphism into the (multiplicative) groupof units ( Z /n ) × ⊂ Z /n . Lemma 4.2.
The map m H : W G H → Z /n is a group homomorphism W G H → ( Z /n ) × .Proof. It is straightforward to check that m H ( ab ) = m H ( a ) m H ( b ) and if e ∈ W G H is the identity, m H ( e ) = 1 ; moreover m H ( a − ) is clearly the inverse to m H ( a ) sothat the image of m H is contained in the units. (cid:3) We can now state the homology of the image of H RU G Q in the algebraic model. Lemma 4.3.
The homology of θ ( H RU G Q ) , the image of H RU G Q in the algebraicmodel, is given by ( V H ) ( H ) ≤ G where V H = 0 when H is not cyclic and when H is cyclic of order n , V H ∼ = Q ( ζ n ) , where Q ( ζ n ) is the field extension of Q by aprimitive n -th root of unity ζ n . The action of a ∈ W G H on Q ( ζ n ) is given by a · ζ n = ζ m H ( a ) n . That is, the action of W G H is given via the homomorphism m H : W G H → ( Z /n ) × ∼ = Gal( Q ( ζ n ) / Q ) .Proof. Let H be a subgroup of G and consider the map M C ≤ H ind HC : M C ≤ H RU ( C ) ⊗ Q → RU ( H ) ⊗ Q where C runs over all cyclic subgroups of H . By a theorem of Artin, this map issurjective (see [19, 9.2.17], for example). By Lemma 3.4, the module V H is foundby quotienting the image of all transfers of proper subgroups, so we immediatelysee V H = 0 if H is not cyclic.Now suppose H is a cyclic subgroup of order n . We first show V H ∼ = Q ( ζ n ) as a Q -algebra. Fix a generator g for H . For each divisor d of n , there is one subgroup AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 11 of H of order d , and the fixed generator for this subgroup will be g n/d . Denotethis subgroup by H d . Note that each subgroup H d gives rise to a unique conjugacyclass of subgroups in G .In what follows, fix a primitive n -th root of unity ζ n and let ζ d be the primitive d -th root of unity ζ n/dn . The complex representation ring for the cyclic group oforder d is isomorphic to Q [ x d ] / ( x dd − where x d corresponds to the one-dimensionalirreducible representation such that the generator g n/d acts via multiplication by ζ d . Here we use the subscript d to keep track of which subgroup we are considering.The choices made in the previous paragraph show the restriction maps are givenby res H d H d ( x d ) = x d for two divisors of n such that d | d .The polynomial x dd − factors as a product of cyclotomic polynomials Φ j where j | d . By the Chinese remainder theorem, the map Q [ x d ] / ( x dd − → Y j | d Q [ x d ] / (Φ j ( x d )) f ( x ) ( f ( x ) , . . . , f ( x )) is an isomorphism. We will use this interpretation throughout our computation.We need to compute the image of the various transfer maps in RU ( H ) ⊗ Q ∼ = Y j | n Q [ x n ] / (Φ j ( x n )) ∼ = Y j | n Q ( ζ j ) . For a divisor d , let’s start by finding tr HH d (1) . The unit is given by the one-dimensional trivial representation, and inducing the trivial H d -representation givesthe H -representation C [ H/H d ] . Using character theory, one can check [ C [ H/H d ]] = 1 + x dn + x dn + · · · + x ( n / d − dn . We can factor this as a product of cyclotomic polynomials by observing x nn − x dn − x dn + x dn + · · · + x ( n / d − dn ) , and sotr HH d (1) = 1 + x dn + x dn + · · · + x ( n / d − dn = Y j ∤ d Φ j ( x n ) . Thus the transfer of is zero in all factors indexed by divisors of n that do notdivide d . For the divisors of d , the above shows the j -th factor is given by ( tr HH d (1)) j = ( n / d − d + 1 = n − d + 1 because x dn = ( x jn ) d / j = 1 in this factor. To summarize, let δ j,d be defined by δ j,d = 1 if j | d and δ j,d = 0 if j ∤ d . We have shown ( tr HH d (1)) j = δ j,d · ( n − d + 1) . By Frobenius reciprocity, tr HH d ( x ℓd ) = tr HH d ( res HH d ( x ℓn ) ·
1) = tr HH d (1) · x ℓn , and so ( tr HH d ( x ℓd )) j = δ j,d ( n − d + 1) · ζ ℓj . We conclude the image of the transfer is given by
Im( tr HH d ) = Span { ( δ j,d ζ ℓj ) j | n | ℓ = 0 , . . . , d − } . To find V H , we need to quotient by Im( tr HH d ) for all divisors d of n such that d = n . We can do this inductively beginning with d = 1 , and the above shows thateverything will be killed except for the factor Q [ x n ] / (Φ n ( x n )) = Q ( ζ n ) . Thus V H ∼ = Q ( ζ n ) .Next we determine the action of the Weyl group. Observe the action of a ∈ W G H on H is given by a · g = a − ga = g m H ( a ) . The action on x n ∈ RU ( H ) is determinedas follows. The class x n is represented by the representation V that is a one-dimensional complex vector space such that g acts via multiplication by ζ n . Thetwisted representation a · V = V a has the same underlying vector space as V , butthe action of g is given by first conjugating by a . Thus g acts in V a as a − ga acts in V . Hence the action of g on V a is given by multiplication by ζ m H ( a ) n and a · x n = x m H ( a ) n . From the proof above, we see this corresponds in the quotient V H to a · ζ n = ζ m H ( a ) n . (cid:3) Remark . If G is abelian, then the conjugation action of the Weyl group usingthe function m H of Definition 4.1 is trivial. Thus the above analysis implies thatthe action of W G H on Q ( ζ n ) is trivial.Using equivariant Bott periodicity (see for example [17] or [13, XIV.3]), we extendLemma 4.3 to a result about KU G Q . Lemma 4.5.
The homology of θ ( KU G Q ) , the image of KU G Q in the algebraic model,is given by V H = 0 when H is not cyclic and V H = Q ( ζ n )[ β ± ] with | β | = 2 when H is a cyclic group of order n . The action of a ∈ W G H on Q ( ζ n )[ β ± ] is given by a · ζ n = ζ m H ( a ) n and a · β = β .Proof. We begin by reviewing the Mackey functor structure of π ∗ KU G Q . Recall π ( KU G Q ) ∼ = RU G Q . In fact, π ( KU G ) ∼ = RU G before rationalizing. For any com-plex representation V and any finite pointed G - CW complex X , Bott periodicityprovides a natural isomorphism [ X, KU G ] G ∼ = [ S V ∧ X, KU G ] G which is given by multiplication by the Bott class β V [13, Theorem XIV.3.2]. Let β denote β C where C is the one-dimensional trivial representation.If X = G/H , then the Bott periodicity isomorphism shows KU jG ( G/H ) = [
G/H + , Σ j KU G ] G ∼ = [ S n ∧ G/H + , Σ j KU G ] G = KU − n + jG ( G/H ) where the isomorphism is given by multiplying by β n . For any G -spectrum E , thecoefficient Mackey functor E −∗ ( pt ) is isomorphic to the homotopy group Mackeyfunctor π ∗ ( E ) . Thus on the homotopy group level, π j ( KU G )( G/H ) ∼ = π j +2 n ( KU G )( G/H ) . When j is odd, note π ( KU G )( G/H ) = 0 from the comments in [13, Section XIV.3],and thus by periodicity, all homotopy groups in odd degrees are zero. We have de-termined that as a graded ring, π ∗ ( KU G )( G/H ) ∼ = RU ( H )[ β ± ] where | β | = 2 . Forclarity of the proof, we will decorate the polynomial generator β ∈ π ∗ ( KU G )( G/H ) with a subscript H to keep track of which level of the Mackey functor it lives in. AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 13 We next determine the restriction, transfer, and Weyl group action on the el-ements β H . For H ≤ K , the restriction map is induced by the quotient map G/H → G/K , and so the naturality of the Bott class implies the following diagramcommutes KU G ( G/K ) · β K / / res KH (cid:15) (cid:15) KU G ( G/K ) res KH (cid:15) (cid:15) KU G ( G/H ) · β H / / KU nG ( G/H ) Consider the image of ∈ KU G ( G/K ) . Going around the diagram the two differentways will show res KH ( β K ) = β H in π ∗ ( KU G ) . The transfer map is also induced bya stable map of orbits and tr KH (1) = | W K H | , so we have tr KH ( β H ) = | W K H | β K .To find the action of the Weyl group, note the action by a ∈ W G H is inducedby the map a : G/H → G/H , eH aH . By taking H = K in the diagram aboveand replacing restriction by the map induced by a , we can consider the image ofthe unit and use that a · to see a · β H = β H .We now return to the algebraic model. The value of V H as a Q -algebra followsreadily from Lemma 4.3 and the periodicity shown above. The action of W G H on Q ( ζ n ) is the same as that of Lemma 4.3, and the action on β is trivial since it wastrivial in the original Mackey functor. (cid:3) Since the homotopy groups of idempotent pieces of KU G Q are a graded field, weobtain the following result. Corollary 4.6.
Let G be a finite group and X be a module spectrum over KU G Q .Then X ≃ M ( H ) ( e ( H ) KU G Q ) ⊕ i H ⊕ ( e ( H ) KU G Q ) ⊕ j H , where H ≤ G , and i H and j H are nonnegative integers.Proof. Let X be a module spectrum over KU G Q . Then X is determined by its imagein Comm A ( G ) Q , which is determined by the idempotent pieces e ( H ) ( X ) . Since X is a module over KU G Q , Barnes’s splitting result [2] implies that for each conjugacyclass of subgroups ( H ) , e ( H ) X is a module over e ( H ) KU G Q . The homotopy groups of e ( H ) KU G Q are a graded field, so any module spectrum over e ( H ) KU G Q is free, and isthus a wedge of suspensions of e ( H ) KU G Q . Since e ( H ) KU G Q is -periodic, it’s enoughto consider the suspensions e ( H ) KU G Q and Σ e ( H ) KU G Q . (cid:3) Let ku G Q denote the connective cover of KU G Q . Bott periodicity also provides thehomology of the image of ku G Q in the algebraic model. Lemma 4.7.
The homology of θ ( ku G Q ) , the image of ku G Q in the algebraic model, isgiven by V H = 0 if H is not cyclic and V H = Q ( ζ n )[ β ] with | β | = 2 if H is a cyclicgroup of order n . The action of a ∈ W G H on Q ( ζ n )[ β ± ] is given by a · ζ n = ζ m H ( a ) n and a · β = β . Formality: The image of KU G Q in the algebraic model Our main goal is to find the image of KU G Q in the algebraic model Comm A ( G ) Q when G is a finite abelian group. Lemma 4.5 calculates the homology of the image θ ( KU G Q ) , but in general, the homology of a rational CDGA is not enough to deter-mine its isomorphism class in the homotopy category. In the abelian case, followingStrategy 3.6, we will show that θ ( KU G Q ) is formal. That is, if ( A • ) ( H ) is an objectof Comm A ( G ) Q such that ( H ( A • )) ( H ) is isomorphic to π ( φ K KU G Q ) in the category Q gr Q [ W G H ] -alg, then there exists a zig-zag of quasi-isomorphisms of CDGAs from ( A • ) ( H ) to ( H ( A • )) ( H ) where ( H ( A • )) ( H ) is the tuple of chain complexes with zerodifferentials given by the homologies of the complexes ( A • ) ( H ) . This will imply ( A • ) ( H ) ∼ = ( H ( A • )) ( H ) in the homotopy category Ho ( Comm A ( G ) Q ) .In an effort to simplify the exposition, we prove the main formality result wewant in several lemmas, which show formality of increasingly complicated Q [ W G H ] -CDGAs. In the case of interest, while we know that the action of W G H is trivialon homology, we cannot assume that the action on the underlying chain complexitself is trivial. We begin by looking at chain complexes in degree zero and add afree generator.Our essential technique is to construct a single zig-zag of quasi-isomorphismsbetween a chain complex A • and its homology H ( A • ) of the form A • D • H ( A • ) . ≃≃ where D • is an appropriately “free” commutative differential graded algebra D • ∈ Q [ W G H ] -CDGA, given by a polynomial algebra tensored with an exterior algebra.It is free in the sense that an algebra map out of D • is determined by defining achain map on the generators. In the case where H ( A • ) has trivial action, we canchoose D • to have trivial action.As a warm-up and illustration of the construction, we consider the algebraicmodel for KU Q nonequivariantly. This is the special case where G is the trivialgroup. Lemma 5.1.
Suppose there exists A • ∈ CDGA Q such that H ( A • ) ∼ = Q [ β ± ] . Then A • is formal, i.e. there is a zig-zag of quasi-isomorphism algebra maps A • D • Q [ β ± ] . ≃≃ Proof.
Let α ∈ A be an element representing β so that [ α ] = β . It is not necessarilythe case that α is invertible in A • , so we may not be able to define a map directlyfrom Q [ β ± ] to A • . Instead, we will replace Q [ β ± ] with a quasi-isomorphic freecommutative differential graded algebra D • . Let ¯ α ∈ A − be a class that represents [ β − ] . Although it is possible that α ¯ α = 1 , because α is invertible in homology thereexists σ ∈ A such that d ( σ ) = 1 − α ¯ α .Define the replacement differential graded algebra D • to be the polynomial al-gebra on classes γ and ¯ γ tensored the exterior algebra on a class y with | γ | = 2 , | ¯ γ | = − , and | y | = 1 , where d ( γ ) = d (¯ γ ) = 0 and d ( y ) = 1 − γ ¯ γ . That is, D • = Q [ γ, ¯ γ ] ⊗ E ( y ) . To define ϕ : Q [ γ, ¯ γ ] ⊗ E ( y ) → A • we only need to spec-ify ϕ ( γ ) , ϕ (¯ γ ) , and ϕ ( y ) such that ϕ is a chain map. We can define ϕ ( γ ) = α , ϕ (¯ γ ) = ¯ α , and ϕ ( y ) = σ . Notice that ϕ is a quasi-isomorphism by construction.Finally we can define a map ψ : Q [ γ, ¯ γ ] ⊗ E ( y ) → Q [ β ± ] by ψ ( γ ) = β , ψ (¯ γ ) = β − and ψ ( y ) = 0 . This map induces an isomorphism on homology and so we havea zig-zag of quasi-isomorphism algebra maps A • Q [ γ, ¯ γ ] ⊗ E ( y ) Q [ β ± ] ≃ ≃ AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 15 and hence A • is formal. (cid:3) As an immediate corollary, this shows that as an E ∞ ring nonequivariant KU Q is unique. The uniqueness of KU Q , and indeed of KU , as an E ∞ ring was shownpreviously by Baker and Richter in [1] using obstruction theory.Before turning to the formality arguments in the general context of Q [ W G H ] -CDGAs, we make the following useful observation. Lemma 5.2 (Averaging) . Let A • be a Q [ W G H ] -CDGA, and suppose a homologyclass x ∈ H ( A • ) is fixed under the W G H -action. Then x has a representative a ∈ A • that is fixed under the W G H -action. Similarly if y ∈ A • such that d ( y ) is fixedunder the W G H -action, then there exists a fixed element b such that d ( b ) = d ( y ) .Proof. Choose an arbitrary cycle a in A • representing x . Then the element a = av W G H ( a ) := 1 | W G H | X g ∈ W G H g · a is a cycle, it also represents x in homology and is W G H -fixed.For the class y , define b = av W G H ( y ) . Since d ( y ) is fixed and the differential isan equivariant map, d ( b ) = d ( y ) and b is fixed by construction. (cid:3) In Lemmas 5.3, 5.5 and 5.6 we often apply the averaging trick of Lemma 5.2 tochose fixed representatives for homology classes without further mention.
Lemma 5.3.
Let A • be a Q [ W G H ] -CDGA such that its homology H ( A • ) is iso-morphic to Q ( ζ n ) concentrated in degree zero and has trivial W G H action. Then A • is a formal as a Q [ W G H ] -CDGA.Proof. We use a standard Koszul resolution. Let D • ∈ Q [ W G H ] -CDGA be the freecommutative differential graded algebra with trivial W G H action Q [ t ] ⊗ E ( z ) where | t | = 0 and | z | = 1 . Let d ( t ) = 0 and d ( z ) = Φ n ( t ) . Notice that z is in an odddegree, so graded commutativity implies z = 0 . Thus the chain complex D • is → Q [ t ] { z } → Q [ t ] → with D = Q [ t ] , D = Q [ t ] { z } , and all other D k = 0 . By construction D • hashomology Q ( ζ n ) concentrated in degree zero with trivial action.To define a map ϕ : D • → A • , choose a class a ∈ A that represents ζ n so [ a ] = ζ n .Now ζ n is a root of Φ n ( x ) but it is possible that a is not a root in A . However Φ n ( a ) must be a boundary, so there exists a class ρ such that d ( ρ ) = Φ n ( a ) . ByLemma 5.2, we may assume a is fixed under W G H . Since a is fixed, the polynomial Φ n ( a ) is fixed, and so we can also assume ρ is fixed. Thus we may define ϕ ( t ) = a and ϕ ( z ) = ρ . Notice the map ϕ is now a quasi-isomorphism.We may easily define a quasi-isomorphism ψ : D • → Q ( ζ n ) by ψ ( a ) = ζ n and ψ ( z ) = 0 . Thus we have a zig-zag of quasi-isomorphisms A • D • Q ( ζ n ) , ≃ ≃ which completes the proof that A • is formal. (cid:3) Lemma 5.3 applies to the homology of the representation ring Mackey func-tor, which shows that θ ( H RU G Q ) is weakly equivalent to the homology specified inLemma 4.3. Corollary 5.4.
For G abelian, the image of H RU G Q is unique in Comm A ( G ) Q . Proof.
By Lemma 5.6 we have formality of the algebraic model θ ( H RU G Q ) at each ( H ) . The values of the algebraic model at each conjugacy class of subgroup areindependent so the image of H RU G Q is unique up to equivariant quasi-isomorphismin Comm A ( G ) Q . (cid:3) More generally, if R is the quotient of a polynomial Q -algebra by a regularsequence, viewed as a Q [ W G H ] -algebra with trivial action, then R [ β ± ] is formal.We prove this in two steps. The first generalizes Lemma 5.3. Lemma 5.5.
Let R be the quotient of a finitely generated polynomial algebra over Q by a finite regular sequence and let A • ∈ Q [ W G H ] -CDGA have homology H ( A • ) ∼ = R concentrated in degree zero with trivial W G H action. Then A • is formal.Proof. By assumption, R ∼ = Q [ x , . . . , x n ] /I where I is an ideal generated by aregular sequence of finitely many polynomials I = ( f , . . . , f m ) . Let a , . . . , a n ∈ A be W G H -fixed elements representing the homology classes x , . . . , x n and choose b , . . . , b m ∈ A also W G H -fixed such that d ( b i ) = f i ( a , . . . , a n ) . Define D • ∈ Q [ W G H ] -CDGA to be the free CDGA with trivial action given by D • = Q [ t , . . . , t n ] ⊗ E ( z , . . . , z m ) where | t i | = 0 , | z i | = 1 , and d ( z i ) = f i ( t , . . . , t n ) . That is, D • is the Koszulcomplex for the regular sequence ( f , . . . , f m ) . By the regularity of the sequence ( f , . . . , f m ) , the homology H ( D • ) is isomorphic to R concentrated in degree zero.Now define ϕ : D • → A • by ϕ ( t i ) = a i and ϕ ( z i ) = b i ; again, this map isequivariant by our choices of W G H -fixed representatives a i and b i . The map ϕ is aquasi-isomorphism. Define ψ : D • → R by ψ ( t i ) = x i and ψ ( z i ) = 0 . This is also aquasi-isomorphism. Thus we have constructed a zig-zag of quasi-isomorphisms A • D • R ≃ ≃ and hence A • is formal. (cid:3) Now we generalize the technique of Lemma 5.1 to incorporate the invertible class β . Lemma 5.6.
Let A • ∈ Q [ W G H ] -CDGA with H ( A • ) ∼ = R [ β ± ] where R is a thequotient of a finitely generated polynomial algebra over Q by a regular sequence, | β | = 2 and where R and β have trivial W G H -action. Then A • is formal.Proof. As in the proof of Lemma 5.5, let D • ∼ = Q [ t , . . . , t n ] ⊗ E ( z , . . . , z m ) be theKoszul complex so that H ( D • ) ∼ = R . To adjoin the invertible class β ± , we tensor D • with the chain complex Q [ γ, ¯ γ ] ⊗ E ( y ) constructed in Lemma 5.1. Since we areworking over a field, the Künneth theorem for chain complexes implies that thereare isomorphisms on homology H ( D • ⊗ Q [ γ, ¯ γ ] ⊗ E ( y )) ∼ = H ( D • ) ⊗ H ( Q [ γ, ¯ γ ] ⊗ E ( y )) ∼ = R ⊗ Q [ β ± ] . Let α ∈ A be a W G H -fixed representative of β and ¯ α ∈ A − be a W G H -fixedrepresentative of β − . It is possible that α ¯ α = 1 in A • but since [ α ][¯ α ] = 1 inhomology, there exists a W G H -fixed element c ∈ A such that d ( c ) = 1 − α ¯ α . Weextend ϕ from the previous lemma to ¯ ϕ : D • ⊗ Q [ γ, ¯ γ ] ⊗ E ( y ) → A • via ¯ ϕ ( γ ) = α , ¯ ϕ (¯ γ ) = ¯ α , and ¯ ϕ ( y ) = c . We also extend the map ψ from Lemma 5.5 to ¯ ψ : D • ⊗ Q [ γ, ¯ γ ] ⊗ E ( y ) → R [ β ± ] via ¯ ψ ( γ ) = β , ¯ ψ (¯ γ ) = β − and ¯ ψ ( y ) = 0 . Now ¯ ϕ and ¯ ψ define a zig-zag of quasi-isomorphisms AIVE-COMMUTATIVE STRUCTURE ON RATIONAL K -THEORY 17 A • D • ⊗ Q [ γ, ¯ γ ] ⊗ E ( y ) R [ β ± ] ≃ ≃ and hence A • is formal. (cid:3) Notice this last lemma would also hold with | β | = 2 n , adjusting the degrees of γ and ¯ γ appropriately.As discussed in Remark 4.4, when G is an abelian group, the actions on thehomology of θ ( KU G Q ) are trivial. Hence Lemma 5.6 applies to show that θ ( KU G Q ) isformal for an abelian group G . Lemma 5.7.
Let G be a finite abelian group. Then θ ( KU G Q ) is formal in Comm A ( G ) Q .Proof. In Lemma 5.6 we have shown formality of the algebraic model θ ( KU G Q ) ateach conjugacy class of subgroups ( H ) . As above, the values of the algebraic modelat each conjugacy class of subgroup are independent so θ ( KU G Q ) is formal. (cid:3) Theorem 5.8.
Let G be a finite abelian group. The image of periodic K -theoryKU G Q in the algebraic model is given by Q ( ζ n )[ β ± ] with | β | = 2 with trivial actionof the Weyl group for each cyclic subgroup C n ≤ G and is zero for non-cyclicsubgroups.Proof. This follows from Lemma 4.5 and Lemma 5.7. (cid:3)
A similar formality argument as above together with calculations from Section4 show the following result.
Theorem 5.9.
When G is a finite abelian group, the image of connective K -theoryku G Q in the algebraic model is given by Q ( ζ n )[ β ] with | β | = 2 for each C n ≤ G andzero otherwise. The action of the Weyl group W G ( C n ) on Q ( ζ n ) is trivial. Corollary 5.10. KU G Q and ku G Q admit unique structures as naive commutative G -ring spectra, i.e., as naive E ∞ -algebras in G -spectra. That is, if X is a rationalnaive-commutative G -ring spectra whose graded Green functor of homotopy groupsis isomorphic to that of KU G Q or ku G Q , then there is a weak equivalence of rationalnaive-commutative G -ring spectra between X and KU G Q or ku G Q (respectively).Proof. For concreteness, suppose X is a naive-commutative rational G -spectrumand the graded commutative homotopy group Green functor of X is isomorphic tothat of KU G Q . Let θ ( X ) be the image of X in Comm A ( G ) Q . Then the homologyof θ ( X ) is isomorphic to the homology of θ ( KU G Q ) ; since we know the latter to beformal, there is a zig-zag of quasi-isomorphisms θ ( X ) ∼ θ ( KU G Q ) in Comm A ( G ) Q .The zigzag of Quillen equivalencesComm Naive G - S p Q ≃ Q Comm A ( G ) Q implies that X and KU G Q are thus weakly equivalent in Comm Naive G - S p Q . Theproof for ku G Q —or indeed, for any spectrum whose image we know to be formal inComm A ( G ) Q —is the same. (cid:3) References [1] Andrew Baker and Birgit Richter. On the Γ -cohomology of rings of numerical polynomialsand E ∞ structures on K -theory. Comment. Math. Helv. , 80(4):691–723, 2005.[2] D. Barnes. Splitting monoidal stable model categories.
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