Equivariant Kasparov theory of finite groups via Mackey functors

EQUIV ARIANT KASP AR O V THEOR Y OF FINITE GR OUPS VIA MA CKEY FUNCTORS IV O DELL’AMBROGIO Abstract. Let G be any ﬁnite group. In this pap er, we systematically ex- ploit general homological metho ds i n or der to reduce the computation of G - equiv arian t K K -theory to topological equiv ariant K -theory . The k ey o bserv a- tion is that the f unctor on KK G that assigns to a G -C ∗ -algebra A the collection of its K - theory groups { K H ∗ ( A ) : H 6 G } admits a lifting to the abelian cat- egory of Z / 2-graded Mack ey mo dules o ve r the represen tation Green functor for G ; mor eo v er, this lifting is the univ ersal exact homologica l functor for the resulting relativ e homological algebra i n KK G . It follows that there is a spec- tral seque nce abutting to KK G ∗ ( A, B ), who se second page displays Ext groups computed in the category of Mack ey m o dules. Thanks to the nice prop er- ties of Mac ke y funct ors, w e obt ain a simil ar K ¨ unneth sp ectral seque nce whic h computes the equiv arian t K -theory groups of a tensor pro duct A ⊗ B . Both spectral sequences b eha v e nicely if A belongs to the localizing sub categ ory of KK G generated b y the algebras C ( G/H ) for all subgroups H 6 G . Contents 1. Int ro duction 2 2. G -cell alg ebras 4 2.1. Restriction, induction and conjuga tion 4 2.2. The catego ry of G -cell alg e bras 5 3. Recollections on Mack ey and Green functors 8 3.1. The subgro up picture 8 3.2. The G -set picture 10 3.3. The functorial picture and the Bur ns ide-Bouc c a tegory B R 10 3.4. The tensor a belia n ca tegory of R -Mack ey mo dules 11 3.5. Induction a nd r e striction of Mack ey functor s 12 4. Equiv ariant K-theory as a Ma c key mo dule 13 4.1. The repres en tation Gr een functor 13 4.2. Equiv ariant K-theory 13 4.3. The R G -Mack ey mo dule k G ( A ) 14 4.4. The extension to the K asparov categor y 17 4.5. The Burnside-B ouc categ ory as equiv ariant KK-theo ry 18 5. Relative homolo gical a lgebra and G -cell algebra s 20 5.1. Recollections and notation 21 5.2. The gra ded r estricted Y oneda functor 22 5.3. The universal co eﬃcients sp ectral seq uenc e 25 5.4. The K ¨ unneth sp ectral seque nce 26 References 27 2000 Mathematics Subject Classiﬁc ation. 46L80, 46M18, 19A22. 1 2 IVO DELL’ AMBR OGIO 1. Introduction The theory of Mack ey f unctors ([ 5 , 9 , 16 , 28 , 30 ]. . . ) has proved itself to b e a powerful conceptual a nd computational to ol in many branches of mathematics : group cohomology , equiv ar ian t stable homotop y , a lgebraic K-theory of group rings, algebraic num b er theory , etc.; in short, any theo ry where one has ﬁnite gr oup actions and induction/ transfer maps. W e refer to the survey article [ 30 ]. Equiv ar iant Kas parov theor y KK G ([ 12 , 21 , 2 3 , 25 ]. . . ), a lthough typically more preo ccupied with top ologic a l groups o r inﬁnite discrete groups, is alr eady quite int eresting when G is a ﬁnite group – see for instance the work of C. H. Phillips [ 25 ] on the freeness of G -actio ns on C ∗ -algebra s. T he r e e x ist induction maps for K K G - theory , s o it is natur a l to ask whether the theory o f Ma ckey functors has a n ything useful to say in this context. As we will s hortly see, the ans w er is deﬁnitely “ Y es”. Recall that for a ﬁnite (or, more genera lly , co mpact) group G and every G - C ∗ -algebra A , we hav e the natural iden tiﬁcation KK G ∗ ( C , A ) ∼ = K G ∗ ( A ) with G - equiv ar ia n t top ological K- theory K G ∗ , which genera lizes Atiy ah and Seg al’s clas- sical G - e quiv ariant vector bundle K-co homology of spaces and has s imilar pro per - ties. Consequent ly , equiv ariant K-theory is often easier to compute tha n gene r al equiv ar ia n t KK -theory . In view of all this, it is natural to ask: Question. T o what extent , and how, is it p ossible t o r e duc e t he c omputation of e quivariant KK-the ory gr oups to that of e quivariant K-the ory gr oups? In order to answer this questio n pr ecisely , the following tw o obser v ations will be crucial. First, for any ﬁx e d G -C ∗ -algebra A the collection of all its equiv aria n t top ological K -theory gr oups K H ∗ (Res G H A ) ∼ = KK G ∗ ( C ( G/H ) , A ) ( H 6 G ) a nd o f the asso ciated restriction, induction and co njugation maps, forms a graded Mackey functor for G , that we denote k G ∗ ( A ). In fac t, k G ∗ ( A ) carr ies the structure of a Mack ey mo dule ov er the repres en tation Green functor R G , and if we denote by R G - Mac Z / 2 the category of Z / 2- graded mo dules ov er R G – whic h is a p erfectly nice Grothendieck tensor a belian c ategory – we obtain in this wa y a lifting o f K -theory to a homologica l functor k G ∗ : K K G → R G - Mac Z / 2 on the tr iangulated K asparov category of sepa rable G -C ∗ -algebra s. The seco nd basic observ ation is that k G ∗ is the “b est” s uc h lifting o f K-theory to some ab elian ca tegory approximating KK G : in the technical jarg o n of [ 22 , 24 ], the functor k G ∗ (or more precis e ly: its res triction to countable mo dules) is the univ ersal stable homological functor on KK G for the relative homolo gical a lgebra deﬁned b y the K-theor y functors { K H ∗ ◦ Res G H } H 6 G . Another, but equiv alent, w ay to formulate this seco nd obser v ation is the following: the categ ory R G - Mac of R G -Mack ey modules is equiv a len t to the c a tegory of addi- tive contra v ar iant functors p erm G → Ab , where p erm G denotes the f ull subc a tegory in K K G of p ermut ation algebr as , i.e. , those of the form C ( X ) fo r X a ﬁnite G -set. (See § 4.5 .) Once all of this is prov ed, it is a straig h tforward matter to apply the g e neral techn iques of rela tiv e homolo gical algebr a in tr ia ngulated categ ories ([ 1 , 6 , 20 , 22 ]) in o rder to obtain a universal co eﬃcient sp ectral sequence whic h will provide our answer to the ab ov e q uestion. By further exploiting the nicely-b ehav ed tensor pro duct o f Mack ey mo dules, we simila rly obtain a K ¨ unneth sp ectral sequence for equiv ar ia n t K- theory , which moreov er has b e tter convergence prop erties. The natural domain of conv ergence of these spe c tral sequences consists of G -c el l algebr as , namely , those a lg ebras contained in the lo calizing triangulated sub cate- gory of KK G generated b y perm G (equiv alently: generated b y the alg ebras C ( G/H ) for all subgroups H 6 G ). W e note that the categor y , Cell G , of G -ce ll algebr as is rather lar ge. F or instance it co n tains a ll ab elian separa ble G -C ∗ -algebra s and is closed under a ll the cla s sical “ bo otstra p” op eratio ns (see Remar k 2.4 ). EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 3 W e can now for mulate our results pre cisely . Theorem (Thm. 5.16 ) . L et G b e a ﬁ nite gr oup. F or every A and B in K K G , and dep ending functorial ly on them, t her e exists a c ohomolo gic al ly indexe d right half-plane sp e ctr al se quenc e of the form E p,q 2 = Ext p R G ( k G ∗ A, k G ∗ B ) − q n = p + q = ⇒ KK G n ( A, B ) . The sp e ctr al se quenc e c onver ges c onditionally whenever A is a G -c el l algebr a, and c onver ges str ongly if mor e over A is such that K K G ( A, f ) = 0 for every morphism f which c an b e written, for e ach n > 1 , as a c omp osition of n maps e ach of which vanishes under k G ∗ . If A is a G -c el l algebr a and the R G -mo dule k G ∗ A has a pr oje ctive r esolution of ﬁnite lengt h m > 1 , then t he sp e ctr al se quenc e is c onﬁne d in the r e gion 0 6 p 6 m + 1 and thus c ol lapses at t he p age E ∗ , ∗ m +1 = E ∗ , ∗ ∞ . The gr oups display e d in the second page E ∗ , ∗ 2 are the homogeneo us components of the gr aded Ext functors o f R G - Mac Z / 2 . Concr etely , for M , N ∈ R G - Mac Z / 2 Ext n R G ( M , N ) ℓ = M i + j = ℓ Ext n R G ( M i , M j ) ( ℓ ∈ Z / 2 , n > 0) where the right-hand-side E xt n R G ( , M j ) deno tes the n - th rig h t derived functor of the Hom functor R G - Mac ( , M j ) o n the a b elian catego r y of R G -Mack ey mo dules. The latter categor y , as well as their ob jects k G ∗ ( A ), are explained in grea t detail in Sections 3 and 4 . Theorem (Thm. 5.17 ) . L et G b e a ﬁnite gr oup. F or al l sep ar able G - C ∗ -algebr as A and B , and dep ending functorial ly in them, ther e is a homolo gic al ly indexe d right half-plane sp e ctr al se quenc e of the form E 2 p,q = T or R G p ( k G ∗ A, k G ∗ B ) q n = p + q = ⇒ K G n ( A ⊗ B ) which str ongly c onver ges if A is a G -c el l algebr a. If mor e over A is su ch that k G ∗ A has a pr oje ctive r esolution of ﬁn ite length m > 1 , then the sp e ctr al se quenc e is c onﬁne d in the r e gion 0 6 p 6 m and thus c ol lapses at the p age E m +1 ∗ , ∗ = E ∞ ∗ , ∗ . Now the sec o nd page E 2 ∗ , ∗ contains the left der iv ed functors of the tensor pr od- uct  R G of Z / 2-gr aded R G -Mack ey mo dules, which is explained in § 3.4 . F rom these sp ectral sequences there fo llo w the us ual consequences and sp ecial cases. Her e w e o nly furnish, a s a s imple illustration, the following v anis hing result. Theorem 1.1. L et A and B b e two G - C ∗ -algebr as for a ﬁnite gr oup G , and assu me that either A or B is a G -c el l algebr a. If K E ∗ (Res G E A ) = 0 for al l elementary sub gr oups E of G , t hen K G ∗ ( A ⊗ B ) = 0 . Pr o of. The h yp othesis on K-theory implies that k G ∗ ( A ) = 0 (see Lemma 2.10 b e- low) and therefore the second pa ge of the K ¨ unneth sp ectral sequence is zero. By symmetry of the tensor pr o duct we may ass ume that A ∈ Cell G , a nd we conclude that K G ∗ ( A ⊗ B ) = 0 by the stro ng convergence.  Related w ork. T o o ur knowledge, [ 17 ] is the only published work wher e s pectra l sequences are systematically co mputed in ab elian categories of Mackey modules; this is done in the context of equiv a riant stable homotopy . There may b e some ov erlap b etw een their res ults and our s; sp eciﬁcally , it should b e p ossible to use lo c. cit. to reprov e our results in the sp ecial case of c ommut ative C ∗ -algebra s. W e also ment ion that [ 29 ] per forms explicit computations of Ext functors in the ca te- gory of Mack ey mo dules ov er R G for some sma ll gro ups G . F or G a co nnected L ie gr oup with tors ion-free fundamen tal gro up, and for suﬃ- ciently nice G -C ∗ -algebra , there are the K ¨ unneth and universal co eﬃcient sp ectral 4 IVO DELL’ AMBR OGIO sequences of [ 27 ], which are computed in the ordinar y mo dule c a tegory ov er the complex r e pr esentation r ing of G . It seems plausible that a uniﬁed trea tmen t of their and our results might b e b oth obtainable and desir a ble, p ossibly in terms of Mack ey functors for compact Lie gro ups ( cf . Remark 4.13 ). Quite recently , a universal co eﬃcient short exact sequence was constructed in [ 15 ] for K K G when G is a cyclic group of pr ime order . The in v ar iant used in lo c. cit. is a slight ly more complica ted lifting o f K- theory than our Mack ey mo dule k G ∗ , and contains more information. The r ange of applicability is the same though: the ﬁrs t algebra must b elong to Cell G . Con v en tions. F or simplicity , we will work only with co mplex C ∗ -algebra s and complex group r epresentations, a lthough the alert reader will see without any tro u- ble how to adapt all r esults to the r eal ca s e. Our notatio n Res G H for the restr ic tion functor fro m G to H is at odds with e.g. [ 23 ], wher e Res H G is used instead, but is compatible with the co mmon indexing conv en tions in the context o f Mack ey func- tors. W e alw ays wr ite C ( X , Y ) for the set o f morphisms from the o b ject X to the ob ject Y in a categor y C . W e use the shor t-hand notations g H := g H g − 1 and H g := g − 1 H g for the conjugates of a subgr oup H 6 G . If H , L 6 G are subgr oups, the no tation [ H \ G/L ] denotes a full set of representativ es of the double cosets H g L ⊆ G . Ac knowledgements. Our warm thanks go to Ser g e Bouc for several illuminating discussions on the virtues and vices of Mack ey functor s . 2. G -cell algebras After some rec o llections on the equiv a riant Ka sparov catego ry , we in tro duce the sub c ategory of G -cell algebra s and der iv e its ﬁrst prop erties. 2.1. Restriction, induction and conjugation. Let KK G be the Ka s parov ca te- gory o f separable G -C ∗ -algebra s, fo r a second countable lo ca lly compact g roup G . W e r efer to the ar ticles [ 21 , 2 3 ] for an acco un t of KK- theo ry conside r ed from the categoric al p oint of view; ther e in the reader will ﬁnd pro ofs or re fer ences for the facts r ecalled in this subsection. F or each G , the catego ry KK G is a dditiv e a nd has ar bitrary countable copro ducts, given b y the C ∗ -algebra ic dir e ct sums L i A i on which G ac ts co ordinatewise. Mor e over, it is equipp ed with the str ucture of a tria ngulated categor y (s e e [ 23 ], esp. App endix A); in par ticular every morphism f ∈ KK G ( A, B ) ﬁts into a distinguishe d triangle A → B → C → A [1], and the collection of distinguished triangles satisﬁes a set o f axioms tha t capture the ho mo- logical b ehaviour of KK-theor y . Here the shift (o r susp ension, tr a nslation) functor A 7→ A [1] is the endo equiv a le nce of K K G given by A [1] = C 0 ( R ) ⊗ A . By the B o tt isomorphism C 0 ( R ) ⊗ C 0 ( R ) ∼ = C 0 ( R ), this functor is its own qua si-inv e r se. Using a standar d trick, it is alwa ys p ossible to “ correct” the shift functor making it a (strict) automor phism (see [ 23 , § 2 .1] and [ 1 3 , § 2]). Therefor e, in o rder to s implify notation, we shall pretend tha t ( )[1] : KK G → KK G is str ic tly inv ertible, with [2] def. = [1 ] ◦ [1] = id KK G . The tr iangulated categor y K K G is a lso endow ed with a compatible symmetric monoidal structure KK G × KK G → KK G , which is induced by the spa tia l tensor pro duct A ⊗ B of C ∗ -algebra s on which G acts diag o nally (in fact, we have already used this to deﬁne the shift functor). The unit ob ject 1 G (or simply 1 if no confusion arises) is the algebr a C o f complex num ber s with the tr ivial G -action. The tenso r pro duct is not the only construction at the C ∗ -algebra ic lev el that extends to a triangula ted functor on the K asparov categor ies. F or instance, there EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 5 is an e v iden t restriction functor Res G H : KK G → KK H for every subg roup H 6 G , which co mm utes with co pr o ducts a nd is also (strict) symmetric monoidal: Res G H ( A ⊗ B ) = Re s G H ( A ) ⊗ Res G H ( B ) and Res G H ( 1 G ) = 1 H . If H is clo sed in G ther e is also a copro duct-preser ving induction functor Ind G H : KK H → KK G which on ea c h H -C ∗ -algebra A ∈ KK H is given by the function C ∗ -algebra Ind G H ( A ) = { G ϕ → A | hϕ ( xh ) = ϕ ( x ) ∀ x ∈ G, h ∈ H ; ( xH 7→ k ϕ ( x ) k ) ∈ C 0 ( G/H ) } equipp e d with the G -action ( g · ϕ )( x ) := ϕ ( g − 1 x ) ( g , x ∈ G ). If G/H is discrete, then induction is left adjoint to r estriction, i.e. , there is a natural isomorphism KK G (Ind G H A, B ) ∼ = KK H ( A, Res G H B ) for all A ∈ KK G and B ∈ KK H . In ter estingly , if instea d G/H is compact then induction is right adjoint to restriction. There is also a F r ob enius isomorphism Ind G H ( A ) ⊗ B ∼ = Ind G H ( A ⊗ Res G H ( B )) (2.1) natural in A ∈ KK H and B ∈ KK G . The induction a nd r estriction functors w ill be us ed co nstant ly in this a r ticle. F or every s ubgoup H 6 G a nd every element g ∈ G , we w ill also consider the c onjugation funct or g ( ) : KK H → KK g H which sends the H -C ∗ -algebra A to the g H -C ∗ -algebra g A who se underlying C ∗ - algebra is just A , equipp ed with the g H -a ction g h g − 1 a := ha ( h ∈ H , a ∈ A ). Like restriction – and for the sa me re a sons – each co njugation functor preser v es copro ducts, t riangle s and tenso r pro ducts. More over, it is a n isomorphism of tensor triangulated ca tegories with inv e rse g − 1 ( ). 2.2. The category of G -cell algebras. F o r every closed s ubgroup H 6 G , we hav e the “standard or bit” G -C ∗ -algebra C 0 ( G/H ). The idea is that G -cell alge bras are those (separ able) G -C ∗ -algebra s that ca n b e pro duced o ut of these by applying all the sta ndard op erations of tr iangulated c a tegories. Although we will b e mos tly concerned with ﬁnite groups, in this subsection we brieﬂy study G -cell algebr a s in greater gener ality , for future reference. Deﬁnition 2.2. W e deﬁne the Kasp ar ov c ate gory of G -c el l algebr as to b e the lo calizing triang ulated sub category of KK G generated by all C 0 ( G/H ), in sy m bo ls: Cell G :=  C 0 ( G/H ) | H 6 G  lo c ⊆ KK G . This means tha t Cell G is the smallest triang ulated sub category of KK G that co ntains all C 0 ( G/H ) and is closed under the formation o f inﬁnite direct sums . R emark 2.3 . The same notion of G -c e ll algebr a is cons ide r ed in [ 15 ], and is prop osed as a KK -analogue of G -CW-co mplex es. An even b etter analo gy would be “cellula r ob jects” in a (mo del) category o f equiv ariant spa ces, where the order o f attac hment of the cells is completely free, like here. In order to obtain a mor e rigid notio n of noncommutativ e G -CW-co mplexes – which would serve similar purp oses a s in the commutativ e case – one s ho uld rather extend to the equiv ariant se tting the deﬁnition of nonco mm utative CW-complexes of [ 10 ]. 6 IVO DELL’ AMBR OGIO R emark 2.4 . The c la ss Cell G contains many int eresting G -C ∗ -algebra s, althoug h this may not b e appar en t from the deﬁnition. F or insta nce if G is a compact (non necessarily connected) L ie g roup, by [ 15 , Thm. 9.5 ] the c lass Cell G contains all separa ble commutativ e G -C ∗ -algebra s and is closed under the usual bo otstr ap op erations, in the sense that it enjoys the following clos ure prop erties: (1) F or every extensio n J ֌ A ։ B of nuclear separable G -C ∗ -algebra s, if tw o out of { J, A, B } are in Cell G then so is the thir d. (2) Cell G is closed under the fo r mation (in the category of G -C ∗ -algebra s a nd G -equiv ar ian t ∗ -homomorphisms) of colimits of countable inductive system of nuclear se parable G -C ∗ -algebra s. (3) Cell G is closed under exterio r equiv alence of G -a ctions. (4) Cell G is closed under G -stable iso morphisms. (5) Cell G is closed under the formation of cross ed pr oducts with r e spect to Z - and R -actions that commute with the given G -action. Next, we sho w that m uc h of the functoriality of eq uiv ariant K K-theory descends to G -cell algebr a s. Lemma 2.5. L et T b e a triangulate d c ate gory e quipp e d with a symmetric tensor pr o duct which pr eserves c opr o ducts ( whatever ar e available in T ) and triangles. Then hE i lo c ⊗ hF i lo c ⊆ hE ⊗ F i lo c for any two su b classes E , F ⊆ T . Pr o of. First, we cla im that (2.6) hE i lo c ⊗ F ⊆ hE ⊗ F i lo c . F or ev ery ob ject B ∈ T , the functor ⊗ B commutes with copro ducts and tr ia ngles by h yp othesis. Thu s S B := { A ∈ T | A ⊗ B ⊆ hE ⊗ B i lo c } is a lo calizing tria ngulated sub c ategory of T , which mor eov er co n tains E ; hence hE i lo c ⊆ S B . Therefore for every B ∈ F we hav e hE i lo c ⊗ B ⊆ hE ⊗ B i lo c ⊆ hE ⊗ F i lo c , from whic h ( 2.6 ) follo ws. Similarly , for every A ∈ T w e see that U A := { B ∈ T | A ⊗ B ⊆ hE ⊗ F i lo c } is lo calizing, and therefore also U := { B ∈ T | hE i lo c ⊗ B ⊆ hE ⊗ F i lo c } = T A ∈hE i loc U A . By ( 2.6 ), U contains F , so it must co n tain hF i lo c . This was precisely the claim.  Prop osition 2.7. Assu me that G is a discr ete gr oup or a c omp act Lie gr oup. Then Cell G is a tensor-triangulate d sub c ate gory of KK G . Mor e over, al l r estriction, induction and c onjugation fun ctors Res G H , Ind G H and g ( − ) , desc end to the appr opri- ate Kasp ar ov su b c ate gories of c el l algebr as. Pr o of. The tensor unit 1 G = C 0 ( G/G ) b elongs to Cell G . Moreover, for all c losed subgroups H, L 6 G , the algebr a C 0 ( G/H ) ⊗ C 0 ( G/L ) ∼ = C 0 ( G/H × G/L ) b elongs again to Cell G . Indeed, if G is a compact Lie group this follows fro m Remark 2.4 bec ause the algebra is commut ative; if G is discr e te, then it follows simply b y applying the co pr o ducts - preserving functor C 0 to the orbit decomp osition o f G -sets G/H × G/L ∼ = a x ∈ [ H \ G /L ] G/ ( H ∩ x L ) . W e conclude by Lemma 2.5 , with E = F := { C 0 ( G/H ) | H 6 G clo s ed } , that Cell G ⊗ Cell G = hE i lo c ⊗ hE i lo c ⊆ hE ⊗ E i lo c ⊆ hE i lo c = Cell G . This proves that Cell G is a tens or sub catego ry o f KK G . The induction functors satisfy Ind G H ◦ Ind H L ∼ = Ind G L for all a ll L 6 H 6 G , and each Ind H G commutes with triangles and copr o ducts . Thus Ind G H ( C 0 ( H/ L )) = Ind G H ◦ Ind H L ( C ) = C 0 ( G/L ), and we conclude that Ind G H ( Cell H ) ⊆ Cell G . Similar ly , the identiﬁcations g C 0 ( G/H ) ∼ = C 0 ( g G/ g H ) in KK g G for all H 6 G show that EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 7 g ( Cell G ) ⊆ Cell g G (and thus Cell G ∼ = Cell g G ). Finally , for G discr e te the isomorphism of H - C ∗ -algebra s Res G H C 0 ( G/L ) = C 0 (Res G H G/L ) ∼ = M x ∈ [ H \ G /L ] C 0 ( H/ H ∩ x L ) shows that Res G H C ( G/L ) ∈ Cel l H for a ll L 6 G a nd there fo re Res G H ( Cell G ) ⊆ Cell H . If G is a compact Lie g roup, we notice that Res G H C ( G/L ) is commutativ e and app eal again to Remark 2.4 .  Lemma 2.8. F or ﬁn ite gr oups H 6 G , ther e is an isomorphism KK G ( A ⊗ C ( G/H ) , B ) ∼ = KK G ( A, C ( G/H ) ⊗ B ) natur al in A, B ∈ KK G . Pr o of. Since G/H is ﬁnite, Ind G H and Res G H are adjoint to ea c h other on both sides. W e obtain the following comp osition o f na tural isomorphis ms : KK G ( A ⊗ C ( G/H ) , B ) = KK G  A ⊗ Ind G H ( 1 H ) , B  ∼ = KK G  Ind G H (Res G H ( A ) ⊗ 1 H ) , B  ∼ = KK H  Res G H ( A ) ⊗ 1 H , Res G H ( B )  ∼ = KK H (Res G H ( A ) , 1 H ⊗ Res G H ( B )) ∼ = KK G ( A, Ind G H ( 1 H ⊗ Res G H ( B ))) ∼ = KK G ( A, Ind G H ( 1 H ) ⊗ B )) = KK G ( A, C ( G/H ) ⊗ B )) . where we have also used F ro benius ( 2.1 ) in the second a nd sixth lines.  The next prop osition s ays that, at least when G is ﬁnite, G - c e ll a lgebras for m a rather nice tenso r triangulated c a tegory . Prop osition 2.9 . F or ev ery ﬁn ite gr oup G , the ten s or triangulate d c ate gory Cell G is gener ate d by the ( ﬁn ite ) set { C ( G/H ) , C ( G/H )[1] | H 6 G } of ri gid and c omp act ℵ 1 obje cts, in the sen se of [ 8 ] . In p articular, Ce ll G is c omp actly ℵ 1 gener ate d and its sub c ate gory Cell G c of c omp act ℵ 1 obje cts c oincides with that of its rigid obje cts, and is ther efor e a t ensor t r iangulate d su b c ate gory. Pr o of. T o prove the ﬁrst part, cons ider the natural iso morphism KK G ( C ( G/H ) , A ) ∼ = KK H ( 1 , Res G H A ) = K H 0 (Res G H A ) ∼ = K 0 ( H ⋉ Res G H A ) provided by the Ind G H - Res G H adjunction a nd th e Gr een-Julg theor em [ 25 , § 2.6]. If A is separable, then so is the cross-pr o duct H ⋉ Res G H A , from which it follows that the ordina ry K-theor y gr o up on the right-hand side is countable; moreover, w e see that KK G ( C ( G/H ) , ) sends a copro duct in K K G to a copr o duct of ab elian groups. These tw o facts toge ther state precisely tha t C ( G/H ) is a compact ℵ 1 ob ject of KK G . The same follows immediately for the susp ensions C ( G/H )[1]. The second claim follows for mally , whenever the set o f co mpact ℵ 1 generator s consists of r igid ob jets and contains the tensor unit. The latter is obvious, since 1 G = C ( G/G ), and it follows immediately fro m Lemma 2.8 that each genera- tor C ( G/H ) (and thus a lso each C ( G/H )[1]) is r igid – in fact, self-dual.  Recall that a ﬁnite g roup is elementary if it has the form P × C , wher e C is cyclic and P is a p - group for some pr ime p not dividing the o r der o f C . Lemma 2.10. L et A b e a G - C ∗ -algebr a for a ﬁ nite gr ou p G . Then: (1) If K E ∗ (Res G E A ) = 0 for al l elementary s u b gr oups E 6 G , then K G ∗ ( A ) = 0 . 8 IVO DELL’ AMBR OGIO (2) If K C ∗ (Res G E A ) ⊗ Z Q = 0 for al l cyclic C 6 G , then K G ∗ ( A ) ⊗ Z Q = 0 . Pr o of. (1) Denote by E ( G ) the set of all ele men tary subgroups of G . B rauer’s clas- sical inductio n theor em ([ 2 , Thm. 5 .6 .4 and p. 18 8]) says that the homomo rphism P E ind G E : L E ∈E ( G ) R ( E ) → R ( G ) is surjectiv e, where R ( H ) deno tes th e represen- tation ring of a ﬁnite gr oup H . In par ticular, there exist ﬁnitely man y E i ∈ E ( G ) and x i ∈ R ( E i ), such that 1 = P i ind G E i ( x i ) in R ( G ). Now consider an A ∈ KK G such that K H ∗ (Res G E A ) = 0 for a ll E ∈ E ( G ). Since the equiv ar iant K-theo ry of A is a Mackey mo dule ov er the representation ring (s e e Section 4 ), we compute for every x ∈ K G ∗ ( A ) x = 1 R ( G ) · x = X i ind G E i ( x i ) · x = X i ind G E i ( x i Res G E i ( x ) | {z } = 0 ) = 0 by apply ing the v anis hing h ypo thesis. The pro of of (2) is similar , but now we must use Artin’s inductio n theorem instea d.  F or ﬁnite G , denote by Cel l G Q the ratio nalization of the category Cell G which is co mpa tible with countable copro ducts, i.e. , the o ne obtained by applying [ 8 , Thm. 2.33] to T := Cell G and S := ( Z r { 0 } ) · 1 1 . Thus Cell G Q is again a compactly ℵ 1 generated tenso r triangulated categ ory with the same ob jects, and it has the pr op- erty that Cell G Q ( A, B ) ∼ = KK G ( A, B ) ⊗ Z Q for all compact ℵ 1 algebras A ∈ Cell G c (th us in particula r Cell G Q ( C ( G/H )[ i ] , B ) ∼ = K H i (Res G H B ) ⊗ Z Q for every H 6 G ). By mes hing familiar tricks from the theor y of Mack ey functor s and fro m the theory of triangulated categor ies, we obta in the following generatio n result for G - cell alg ebras and rationa l G -cell alg ebras. Prop osition 2.11. L et G b e a ﬁn ite gr oup. Then: (1) Cell G = h C ( G/H ) | H is an element ary su b gr oup of G i lo c . (2) Cell G Q = h C ( G/H ) | H is a cyclic sub gr oup of G i lo c . Pr o of. If T is a triangula ted category with countable copro ducts and if E 1 , E 2 ⊆ T c are tw o co untable sets o f compact ob jects which are clo sed under sus pensio ns and desusp ensions, then hE 1 i lo c = hE 2 i lo c whenever E 1 and E 2 hav e the sa me rig h t orthogo nal in T , i.e. , if E ⊥ 1 := { B ∈ T | T ( A, B ) = 0 ∀ A ∈ E 1 } equa ls E ⊥ 2 := { B ∈ T | T ( A , B ) = 0 ∀ A ∈ E 2 } (see [ 8 , § 2.1 ] for e x planations). Thus par t (1) fo llo ws immediately , using T = KK G or T = Cell G , by combining Prop osition 2.9 with Lemma 2.10 (1), while part (2) uses Lemma 2.10 (2) instead (and T = Ce ll G Q ).  3. Recollections o n Mackey and G reen functors Throughout this s e c tion, we ﬁx a ﬁnite gro up G . Mack ey functors, and the rela ted notions o f Green functors and mo dules ov er them, can b e deﬁned from v arious diﬀerent p oint of views . The three most imp or- tant (all of which are treated in deta il in [ 5 ]) are the deﬁnition in terms o f s ubg roups of G , that in terms of G -sets, a nd that in terms of functor catego ries. Since we ar e going to need a ll three of them, let us pro ceed without further dela y . 3.1. The subgroup picture. This is the most concrete of the three points of view. A Mackey functor M ( for G ) consists of a family of ab elian gr oups M [ H ], one for each subgro up H 6 G , together with a restriction homomophism res H L : M [ H ] → M [ L ] and an induction homomor phism ind H L : M [ L ] → M [ H ] for a ll L 6 H 6 G , and a conjugation homomor phis m con g,H : M [ H ] → M [ g H ] for all g ∈ G and all EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 9 H 6 G . These three families of maps must sa tisfy the following six families of relations: res H L res G H = r es G L , ind G H ind H L = ind G L ( L 6 H 6 G ) con f , g H con g,H = co n f g ,H ( f , g ∈ G, H 6 G ) con g,H ind H L = ind g H g L con g,L ( g ∈ G, L 6 H 6 G ) con g,L res H L = res g H g L con g,H ( g ∈ G, L 6 H 6 G ) ind H H = res H H = con h,H = id M [ H ] ( h ∈ H , H 6 G ) res H L ind H K = X x ∈ [ L \ G/K ] ind L ∩ x K L con x,L x ∩ K res K L x ∩ K ( L, K 6 H 6 G ) The last r elation is the Mackey formula . A morphism ϕ : M → N of Mack ey functors is a family of k -linear maps ϕ [ H ] : M [ H ] → N [ H ] which c omm ute with restriction, induction and conjuga tion maps in the evident wa y . A ( c ommu tative ) Gr e en functor is a Mack ey functor R such that each R [ H ] carries the s tructure o f a (commutativ e) asso ciative unital r ing, the r estriction and conjugation maps are unital ring homomorphis ms , and the follo wing F r ob enius formulas hold: ind H L (res H L ( y ) · x ) = y · ind H L ( x ) , ind H L ( x · res H L ( y )) = ind H L ( x ) · y for all L 6 H 6 G , x ∈ R [ L ] and y ∈ R [ H ]. Similarly , a ( left ) Mackey mo dule over R (or simply R -mo dule ) is a Mack ey functor M where each M [ H ] carr ies the structure of a (left) R [ H ]-module, in such a wa y that: res H L ( r · m ) = res H L ( r ) · res H L ( m ) ( L 6 H 6 G, r ∈ R [ H ] , m ∈ M [ H ]) con g,H ( r · m ) = con g,H ( x ) · con g,H ( m ) ( g ∈ G, H 6 G, r ∈ R [ H ] , m ∈ M [ H ]) r · ind H L ( m ) = ind H L (res H L ( r ) · m ) ( L 6 H 6 G, r ∈ R [ H ] , m ∈ M [ L ]) ind H L ( r ) · m = ind H L ( r · res H L ( m )) ( L 6 H 6 G, r ∈ R [ L ] , m ∈ M [ H ]) A mo rphism of R -Mackey mo dules, ϕ : M → M ′ , is a morphism of the underly ing Mack ey fu nctors s uc h that each co mponent ϕ [ H ] is R [ H ]-linear. W e will deno te b y R - Mac the ca tegory of R -Mack ey mo dules. W e will see that it is a Grothendiec k ab elian category with a pro jective generator, and that it has a nice tenso r pro duct when R is commutativ e. Example 3.1 . The Burnside ring Gr een functor, R = B u r , is deﬁned b y setting B ur [ H ] := K 0 ( H - set ), the Gro thendiec k ring o f the ca tegory of ﬁnit e H -sets with ⊔ and × yielding sum and multiplication, and with the structure ma ps induced by the usual restriction, inductio n and conjugation op erations for H -sets. It turns out that B ur acts unique ly on all Mack e y functor s, so that B ur - Mac is just Mac , the category of Ma c key functors. (This is analog ous to Z - M od = Ab ). R emarks 3.2 . Instead of using a belia n groups for the base catego ry , it is o ften useful in applications to a llow mo re general ab elian categorie s , s uch as mo dules ov er some base comm utative r ing k , p ossibly graded. It is s traightforw ard to adapt the deﬁni- tions. F or our applicatio ns, it will so metimes b e useful to let our Mackey functors take v alues in the catego r y of Z / 2 -graded a belia n g roups and deg r ee pre s erving homomorphisms. (A simila r rema rk holds for the tw o other pictures.) 10 IVO DELL’ AMBR OGIO 3.2. The G -set pi cture. The s econd pictur e is in terms of “bifunctors” on the category of ﬁnite G -sets. Now a Mack ey functor is deﬁned to b e a pair of functors M = ( M ⋆ , M ⋆ ) fr om G -sets to ab elian gr oups, with M ⋆ contra v aria n t and M ⋆ cov a riant, having the same v alues on ob jects: M ⋆ ( X ) = M ⋆ ( X ) =: M ( X ) for all X ∈ G - set . Mor e o ver, t wo axio ms have to b e satisﬁed: (1) M sends every copro duct X → X ⊔ Y ← Y to a direct-s um diagram in Ab . (2) M ⋆ ( g ) M ⋆ ( f ) = M ⋆ ( f ′ ) M ⋆ ( g ′ ) for every pull-bac k square · g ′   f ′ / / · g   · f / / · in G - set . Morphisms are na tural transformations ϕ = { ϕ ( X ) } X , where naturalit y is req uired with resp ect to b oth functoria lities. Every Mack ey functor in this new s ense deter- mines a unique Mack ey functor in the previous sense, by setting M [ H ] := M ( G/H ) and re s H L := M ⋆ ( G/L ։ G/H ), ind H L := M ⋆ ( G/L ։ G/H ) and con g,H := M ⋆ ( g H ∼ = H ) = M ⋆ ( H ∼ = g H ). Conv ersely , by decomp osing e a c h G -set into orbits we see how a Ma ckey functor in the old sense determines an (up to isomorphism, unique) Mack ey functor in the new s e nse. 3.3. The functorial picture and the Burnsi d e -Bouc category B R . Since Lindner [ 19 ], it is k nown that one can “push” the t wo functor ialities of Mack ey functors into the domain categor y , so that Ma ckey functors a re – as their name would sugg est – just or dinary (additive) functors on a suitable ca tegory . It was prov ed b y Serge Bouc that a s imilar trick c a n be p erformed also for Mack ey modules , as follows (see 1 [ 5 , § 3.2]). F or any Mack ey functor M a nd any ﬁnite G -set X , let M X be the Mack ey functor which, in the G -set picture , is given by M X ( Y ) := M ( Y × X ) ( Y ∈ G - set ) . Let R b e a Green functor . If M is a n R -mo dule, then M X inherits a natura l structure of R -mo dule, and the ass ignmen t M 7→ M X extends to an endofunctor on R - M ac w hich is its own right and left adjoint ([ 5 , Lemma 3.1.1]). By [ 5 , Pro p. 3 .1 .3], there is an is omorphism α X,M : R - Mac ( R X , M ) ∼ = M ( X ) (3.3) natural in X ∈ G - set and M ∈ R - Mac . This lo oks s uspiciously like the Y oneda lemma. In fac t, it can b e turn e d into the Y o neda lemma ! It suﬃces to deﬁne an (essentially small Z -linear) category B R as follows. Its ob jects are the ﬁnite G -sets, and its morphism g roups a re deﬁned by B R ( X, Y ) := R ( X × Y ). The comp osition of morphis ms in B R is induced b y that o f the ca tegory of R -Mack ey mo dules, via the natura l bijection α X,M . The resulting embedding B R → R - Mac , X 7→ R X , extends along the (additive) Y oneda em bedding B R → Ab ( B R ) op , X 7→ B R ( , X ), to an e q uiv alence of categ ories ([ 5 , Theo rem 3.3 .5]) Ab ( B R ) op ≃ R - Mac . Thu s the functor B R → R - Mac se nding X to R X is identiﬁed with the Y oneda embedding, and ( 3.3 ) with the Y oneda lemma. In particula r, the catego ry of Mac key mo dules over R is a n ab elian functor catego ry , a nd we see that the repre sen tables R G/H ( H 6 G ) furnish a ﬁnite set of pro jective gener ators. 1 Bew are that we prefer to use the opp osite category , thinking of preshea ve s, so that Bouc’s original notation C A denotes the same category as our ( B R ) op (his A b eing the Green f unctor R ). This is r ather i mmaterial t hough, in view of the isomorphism B R ∼ = ( B R ) op EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 11 Example 3.4 . F or the Burnside ring R = B ur , the c a tegory B B ur is just the Burn- side c ate gory B , whic h has ﬁnite G -sets for ob jects, Hom s e ts B ( X, Y ) = K 0 (G- set ↓ X × Y ), and comp osition induced by the pull-back of G -s e ts. W e recover this wa y Lindner’s pictur e M ac ≃ Ab B op of Mackey functor s. The pr o duct X × Y of G -sets clearly pr ovides a tensor pro duct ( i.e. , a symmetr ic monoida l structure) on B with unit ob ject G/ G . By the theory o f Kan extensions ( i.e. , “Day conv olution” [ 7 ]), there is, up to ca nonical iso morphism, a unique closed sy mmetric mo no idal struc- ture on the pr esheaf category Ab B op which mak es the Y oneda embedding B ֒ → Ab B op a symmetric mono idal functor. This is usually called the b ox pr o duct o f Mack ey functors and is denoted b y  . It turns out that a Green functor is quite simply a monoid (= ring ob ject) in the tenso r categor y ( Mac ,  , B ur ), and it follows tha t one can study the whole sub ject of Gre en functors and Mack ey mo dules from the categoric al p o in t of view; it is the fr uitful a pproach taken by L. G. Lewis [ 16 ]. 3.4. The tens or ab elian category o f R -Mac k ey mo dul es. If we consider a commutativ e Green functor R to b e a commutativ e monoid in ( Mac ,  , B ur ), as in Example 3.4 , then the tensor product M  R N of t w o R -mo dules M and N with structure maps ρ M : R  M → M and ρ N : R  N → N , resp ectively , is deﬁned by the following co equaliz er in Mac M  R  N M  ρ N / / ( ρ M ◦ γ )  N / / M  N / / M  R N , where γ denotes the symmetry iso mo rphism o f the b ox pro duct. Co ncretely , the v alue of M  R N a t a G -set X is the quotient ( M  R N )( X ) = M α : Y → X M ( Y ) ⊗ Z M ( Y ) ! / J , where the sum is ov er all G -maps into X , a nd where J is the subgro up g e nerated by the elements M ⋆ ( f )( m ) ⊗ n ′ − m ⊗ N ⋆ ( f )( n ′ ) , M ⋆ ( f )( m ′ ) ⊗ n − m ′ ⊗ N ⋆ ( f )( n ) , m · r ⊗ n − m ⊗ r · n for all r ∈ R ( Y ), m ∈ M ( Y ), m ′ ∈ M ( Y ′ ), n ∈ N ( Y ), n ′ ∈ N ( Y ′ ) and all morphisms f : ( Y , α ) → ( Y ′ , α ′ ) in the slice categor y G - set ↓ X , i.e. , all G -maps f : Y → Y ′ such that α ′ ◦ f = α (see [ 5 , § 6.6]). As usual, this extends to deﬁne a clos e d sy mmetric monoida l structure on R - Mac with unit ob ject R . The in ternal Hom functor Hom R ( , ) : ( R - Mac ) op × R - Mac → R - Mac , which of co urse is character ized by the natural iso morphism R - Mac ( M  R N , L ) ∼ = R - Mac ( M , Hom R ( N , L )) , (3.5) has also the following mor e c o ncrete, and rather useful, descr iption: Hom R ( M , N )( X ) = R - Mac ( M , N X ) (3.6) for every G -set X (see [ 5 , P rop. 6 .5 .4]). Finally , the tensor pro duct extends to gr ade d R -Mackey mo dules M a nd N by the familiar formula ( M  R N ) ℓ := M i + j = ℓ M i  R N j . W e will consider g rading b y a n inﬁnite or ﬁnite cyclic gr o up Z / π ( π ∈ N ), cf. § 5 . R emark 3.7 . It follows fro m the natural isomo rphism R X  R R Y ∼ = R X × Y (see [ 16 , Prop. 2.5]) that the tensor pro duct r estricts to repr esent able mo dules in the functorial picture R - Mac ≃ Ab ( B R ) op , inducing a tensor pro duct on B R which is 12 IVO DELL’ AMBR OGIO simply X × Y o n ob jects. Therefore , w e ma y r ecov e r  R as the Day conv olution pro duct extending the tensor s tr ucture of B R back to all R - modules. 3.5. Induction and restriction of M ack ey functors. Jus t for a moment, let us see what happ ens if we allow the group G to v ary . Given a Ma c key functor M for G and a subgro up G ′ 6 G , there is an ev iden t restricted Mack ey functor for G ′ , written Res G G ′ ( M ), which is simply Res G G ′ ( M )[ H ] := M [ H ] at each H 6 G ′ . W e obtain this wa y a functor Res G G ′ from the c ategory of Mack ey functors for G to the catego ry of Mackey functor s for G ′ . In particula r, we see that if R is a Green functor for G then Res G G ′ ( R ) is a Green functor for G ′ , and tha t restrictio n may b e considered as a functor Res G G ′ : R - Mac → Res G G ′ ( R )- Mac . Int erestingly , there is also a n induction functor Ind G G ′ going the opp osite way which is b oth left and rig h t adjoint to Res G G ′ (see [ 5 , § 8.7 ]). It c a n b e construc ted as follows: given a Res G G ′ ( R )-mo dule M and a n H 6 G , set (3.8) Ind G G ′ ( M )[ H ] := M a ∈ [ G ′ \ G/H ] M [ G ′ ∩ a H ] . Each summand is made into an R [ H ]-mo dule in the evident w ay , that is , v ia the comp osite r ing homomor phism con a,G ′ a ∩ H res H G ′ a ∩ H : R [ H ] → R [ G ′ ∩ a H ] . In the subgroup picture of Mackey functors, we have the following s imple formulas: Ind G G ′ ( M )( X ) = M (Res G G ′ X ) , Re s G G ′ ( N )( Y ) = N (Ind G G ′ Y ) for all G -sets M and G ′ -sets N . These res triction and induction functors for Mack ey modules sa tisfy “higher versions” of the exp ected r elations. F or instance, there is a Mackey formula iso- morphism ([ 28 , Prop. 5.3]), as well a s the following F r ob e nius isomorphism (see also [ 5 , § 10.1 ] for more genera l r esults of this type). Prop osition 3.9. Ther e is a natur al isomorphism of R -Mackey mo dules Ind G G ′ ( M )  R N ∼ = Ind G G ′  M  Res G G ′ ( R ) Res G G ′ ( N )  for al l N ∈ R - Mac and M ∈ Res G G ′ ( R ) - Mac . Pr o of. W e will use for this pro of the G -set picture o f Mack ey functors. Since there is no ambiguit y , we will drop the decor ations o n all induction and restriction functors in o rder to av oid clutter. Let us star t – inno cently enough – w ith a m uch more evident F ro benius isomor phism, namely , the natural iso mo rphism of G -sets Ind ( X ) × Y ∼ = Ind ( X × Res Y ) that exis ts for all G ′ -sets X and all G -s ets Y . It follows from this that, for an arbitrar y L ∈ R - Mac , we may identify (3.10) Res ( L Y ) ∼ = Res ( L ) Res Y bec ause of the computation Res ( L Y )( X ) = L Y (Ind X ) = L ((Ind X ) × Y ) ∼ = L (Ind ( X × Res Y )) = Res L ( X × Res Y ) = (Res L ) Res Y ( X ) . Next, we cla im the existence of a na tural iso morphism (3.11) Ind Hom Res ( R ) ( M , Res L ) ∼ = Hom R (Ind M , L ) EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 13 for all M and L . Indeed, ev a luating at every Y ∈ G - set we ﬁnd Ind Hom Res ( R ) ( M , Res L )( Y ) = Hom Res ( R ) ( M , Res L )(Res Y ) = Res ( R )- Mac ( M , Res ( L ) Res Y ) ∼ = Res ( R )- Mac ( M , Res ( L Y )) ∼ = R - Mac (Ind M , L Y ) = Hom R (Ind M , L )( Y ) by using ( 3 .6 ) in the seco nd a nd in the last lines, ( 3.10 ) in the third line, and the (Ind , Res )-a djunction in the fourth. Finally , there is a na tur al isomo rphism R - Mac  Ind ( M  Res ( R ) Res N ) , L  ∼ = Res ( R )- Mac  M  Res ( R ) Res N , Res L  ∼ = Res ( R )- Mac  Res N , Hom Res ( R ) ( M , Res L )  ∼ = R - Mac  N , Ind Hom Res ( R ) ( M , Res L )  ∼ = R - Mac  N , Hom R (Ind M , L )  ∼ = R - Mac  (Ind M )  R N , L  by consecutive application o f the (Ind , Res )-adjunction, the (  , Hom )-adjunction ( 3.5 ), the (Ind , Res )-adjunction once aga in, the isomor phis m ( 3.11 ), a nd the other (  , Ho m)-adjunction. Since this isomo rphism is natural in L and sinc e L is an arbitrar y R -module, we conclude b y Y o neda the existence of a natural isomor phism Ind ( M )  R N ∼ = Ind  M  Res ( R ) Res ( N )  of R -mo dules.  4. Equiv ariant K-theor y as a Mackey module 4.1. The representa tion Green functor. Le t us describ e the commutativ e Gr e e n functor that will co ncern us here, the r epr esentation Gr e en functor , that we de- note R G . It is als o one of the most cla s sical ex a mples. By deﬁnition, the v alue R G [ H ] at the subgro up H 6 G is the c o mplex representa- tion ring R ( H ) := K 0 ( C H - mo d ). Addition is induced b y the direct sum of mo dules and multiplication b y their tensor pro duct ov er the base ﬁeld C , equipp ed with the diagonal G a ction. F or L 6 H 6 G , the restr iction maps res H L : R ( H ) → R ( L ) are deﬁned b y restricting the action of a C H - module to C L via the inclusion C L → C H , a nd the induction maps ind H L : R ( L ) → R ( H ) are deﬁned by the usual induction of mo dules, M 7→ C H ⊗ C L M ( M ∈ C L - mo d ). The conjugatio n maps co nj g,H : R ( H ) → R ( g H ), similarly to the restriction maps, a re induced by precomp osition w ith the isomorphisms C g H → C H , x 7→ g − 1 xg . The v eriﬁcation that R G satisﬁes the axioms of a comm utative Green functor is an easy exercise, and follows immediately fro m general text- b o ok pro per ties o f mo dules over group rings ( e.g. [ 2 , § 3.3]). 4.2. Equiv arian t K- the ory. F or every sepa rable G -C ∗ -algebra A ∈ C ∗ sep G , we wan t to deﬁne a Z / 2-graded R G -Mack ey mo dule k G ∗ ( A ) := { K H ǫ (Res G H ( A )) } H 6 G ǫ ∈ Z / 2 by collecting all its top ological K -theor y groups. In or der to describe the structure maps of this R G -mo dule as concr etely as possible, w e now brieﬂy reca ll from [ 25 , § 2] the deﬁnition of equiv ar iant K -theor y in terms of (Ba nach) mo dules. Assume ﬁrst that A is unital. A ( G, A ) -mo dule E consists of a r igh t mo dule E over the ring A , together with a r epresentation G → L ( E ) of G by co n tinu ous linear op erators on E , such that g ( e a ) = ( g e )( g a ) for a ll g ∈ G, e ∈ E , a ∈ A . Of course , for L ( E ) to make sense, E m ust b e endow ed with a top ology; we do 14 IVO DELL’ AMBR OGIO not b elab o r this p oint, beca use w e will b e exclusively concerned with mo dules that are pro jective and ﬁnitely g e nerated ov er A , and which therefore inher it a Banach space structure (and a uniq ue top ology) from that of A . The direct sum of tw o ( G, A )-mo dules is deﬁned in the evident way with the diagonal G -action, and a morphism of ( E , A )-mo dules is a contin uous A -mo dule map ϕ : E → E ′ commuting with the G - a ction: ϕ ( g e ) = g ϕ ( e ) for all g ∈ G, e ∈ E . Let ˜ K G 0 ( A ) b e the Grothendieck g roup o f isomo rphism cla sses of ﬁnitely gen- erated A -pr o jectiv e ( G, A )-mo dules, with addition induced by the direct sum. If V is a ﬁnite dimensional C G -mo dule and E a ( G, A )-mo dule, we may equip the tensor pro duct V ⊗ C E with the diago nal G -a ction g ( v ⊗ e ) := g v ⊗ g e and the right A -action ( v ⊗ e ) a := v ⊗ ea ( g ∈ G, v ∈ V , e ∈ E , a ∈ A ), thereby inducing a le ft R ( G )-action on the ab elian g roup ˜ K G 0 ( A ). The ass ignmen t A 7→ ˜ K G 0 be- comes a c o v ar iant functor from unital G -C ∗ -algebra s to R ( G )-mo dules b y extension of scalars ; indeed, given a unital G -e q uiv ariant *-homomo rphism f : A → B and a ( G, A )-mo dule E , we equip the ﬁnitely generated pro jectiv e B -mo dule E ⊗ A B with the G -a ction g ( e ⊗ b ) := g e ⊗ gb . If A is a g eneral, p ossibly no n unital, G -C ∗ -algebra , then b y the usual trick we set K G 0 ( A ) := ker( ˜ K G 0 ( π A : A + → C )), where π A is the na tural augmen tation o n the functorial unitization A + of A . Then K G ( A ) = ˜ K G ( A ) for unital A , and K G 0 yields a functor C ∗ alg G → R ( G )- Mod on G -C ∗ -algebra s. 4.3. The R G -Mac k ey mo dul e k G ( A ) . W e now deﬁne our K -theory Mack ey mo d- ule. F or all A ∈ C ∗ alg G and H 6 G , s et k G ( A )[ H ] := K H 0 (Res G H ( A )) . F or the deﬁnition of the structur e ma ps, assume a t ﬁrs t that A is unital. The r e striction ma ps res H L are simply induced by restricting the H -action on ( H, A )-mo dules to L , as in the ca se o f R G . The c o njugation maps are a s follows. Given an ( H, Res G H A )-mo dule E , let Con g ( E ) denote E equippe d with the H - and A -actions ( h g · e ) := g − 1 hg e , ( e · g a ) := e ( g − 1 a ) ( e ∈ E , h ∈ H, g ∈ G ) . Lemma 4.1. The ab ove formulas pr oide a wel l-deﬁne d ( g H, Res G g H ( A )) -mo dule Con g E , and t he assignment E 7→ Con g ( E ) induc es a wel l-deﬁne d R ( H ) -line ar ho- momorphism con g,H : K H 0 (Res G H A ) → K g H 0 (Res G g H A ) . Mor e over, con f , g H con g,H = co n f g ,H for al l f , g ∈ G, H 6 G , and con g,H = id K H 0 ( A ) whenever g ∈ H . Pr o of. The t wo ac tions are certainly w ell- deﬁned (to see this for the A -action, rec a ll that G acts on A by alg e bra homomorphisms, which must b e unital if A is unital), and they a re compatible by the computation h g · ( e · g a ) = ( g − 1 hg )( e · g − 1 a ) = ( g − 1 hg e )( g − 1 hg g − 1 a ) = ( h g · e ) · g ( h · a ) ( h ∈ H , a ∈ A, e ∈ E ). L e t E b e ﬁnitely genera ted pro jectiv e ov er A . But then Con g E is a ls o ﬁnitely generated pro jective, b ecause (ignoring the g roup actio ns) the map E → Con g E , e 7→ g − 1 e , is a n A -linea r isomorphis m: g − 1 ( ea ) = ( g − 1 e )( g − 1 a ) = ( g − 1 e ) · g a ( e ∈ E , a ∈ A ) . The res t is s imilarly straig h tforward.  R emark 4.2 . Perhaps a more natural w ay to understand the conjugation maps is to note that every ( H , A )-mo dule E can be considered as an ( g H, g A )-mo dule, say g E , where g A is the g H -C ∗ -algebra with underlying C ∗ -algebra A and with the g H -a ction EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 15 g hg − 1 g · a = ha , as in § 2.1 . This is just as for the res triction maps: b o th gr oup actions, that on E and that on A , are precomp osed with a gr oup homomor phism, in this cas e the conjuga tion isomor phis ms g H → H , h 7→ g − 1 hg (for restriction, the inclusion of a subgroup). Similar ly , we le t g A act o n g E simply by e · a = e a , just as A a cted on E , and the compatibilit y condition fo r g E is trivially satisﬁed bec ause it is for E . No w note that, if the H -action on A comes fr om a n ac tion o f the whole g roup G , then the C ∗ -algebra iso morphism g − 1 : A ∼ → g A pr ovided b y the action is G -equiv ariant, since g − 1 ( ha ) = ( g − 1 hg )( g − 1 a ) = h g · ( g − 1 a ) for all h ∈ G and a ∈ A . Clear ly , the re s triction of g E a lo ng g − 1 is precisely the ( g H, A )-mo dule Con g E deﬁned a b ov e (o r, with extension of sca la rs: ( g − 1 ) ∗ (Con g E ) ∼ = g E ). W e now deﬁne the induction maps, following [ 25 , § 5.1]. Let L 6 H 6 G . If E is an ( L, A )-mo dule, we deﬁne an ( H , A )-mo dule Ind H L ( E ) := { ϕ : H → E | ϕ ( x ℓ ) = ℓ − 1 ϕ ( x ) ∀ ℓ ∈ L , x ∈ H } with the following A - and H -actions: ( ϕ · a )( x ) := ϕ ( x )( x − 1 a ) , ( h · ϕ )( x ) := ϕ ( h − 1 x ) for all ϕ ∈ Ind H L ( E ), a ∈ A , and x, h ∈ H . B y [ 25 , Pr op. 5.1 .3], the r e sulting functor E 7→ Ind H L ( E ) from ( L, A )-mo dules to ( H , A )-mo dules preserves ﬁnitely genera ted pro jectives, and the induced homo morphism ind H L : K L 0 (Res G L A ) → K H 0 (Res G H A ) is R ( H )-linea r. (Here as always, we turn K L 0 (Res G L A ) into an R ( H )-mo dule v ia the ring ho momorphism res H L : R ( H ) → R ( L ).) R emark 4.3 . Note that, when A = C is the tr ivial G -C ∗ -algebra , there are evi- dent canonical is omorphisms K H 0 ( C ) ∼ = R ( H ) ( H 6 G ) that identify the r espec- tive induction, restriction and co njugation maps. In other w ords, w e can identify k G ( C ) = R G as Mack ey functors. As usual with C ∗ -algebra s, it is e a sy to use the functorial unitisation to extend the deﬁnitions of res H L , ind H L and con g,H to general, p ossibly nonunital, algebras A . F or instance, ind H L is the map induced on kernels in the following morphism of short exact sequences : K H 0 (Res G H A ) / / / / K H 0 (Res G H A + ) / / / / K H 0 (Res G H C ) = R ( H ) K H 0 (Res G L A ) / / / / ind H L O O K H 0 (Res G L A + ) / / / / ind H L O O K H 0 (Res G L C ) = R ( L ) ind H L O O and similarly for r es H L and con g,H . Because of the naturality o f the deﬁnition, it will suﬃce to verify equalities betw een restriction, conjugation and induction maps for the ca se of unital a lgebras. Lemma 4.4. Ther e is an isomorph ism of ( H , A ) -mo dules Res G H Ind G L ( E ) ∼ = M z ∈ [ H \ G/L ] Ind H H ∩ z L Con z Res L H z ∩ L ( E ) for every ( L, A ) -mo dule E and al l sub gr oups H , L 6 G . Mor e over, onc e the set of r epr esentatives [ H \ G/L ] is ﬁxe d, t he isomorphism is natur al in E . Pr o of. Every choice o f the set [ H \ G/L ] yields a basic deco mp osition H G L ∼ = a z ∈ [ H \ G/L ] H z L 16 IVO DELL’ AMBR OGIO of ( H , L )-bise ts . There follows a decomp osition of ( H , A )-mo dules Res G H Ind G L ( E ) =    ϕ : a z ∈ [ H \ G/L ] H z L → E | ℓϕ ( xℓ ) = ϕ ( x ) ∀ x ∈ G, ℓ ∈ L    = M z ∈ [ H \ G/L ] { ϕ : H z L → E | ℓϕ ( xℓ ) = ϕ ( x ) ∀ x ∈ H z L, ℓ ∈ L } | {z } =: V z . Of co ur se the H -action on each summand V z is still given b y ( h · ϕ )( x ) = ϕ ( h − 1 x ), and the A -ac tion by ( ϕ · a )( x ) = ϕ ( x )( x − 1 a ) (for a ll x ∈ H z L, h ∈ H, a ∈ A ). F or every z , let W z := Ind H H ∩ z L Con z Res L H z ∩ L ( E ) deno te the corresp onding sum- mand of the r ight hand side of the Mack ey for m ula. Here Con z Res L H z ∩ L ( E ) is E equipp e d with the conjugated H ∩ z L -action h z · e = ( z − 1 hz ) e (for h ∈ H ∩ z L and e ∈ E ), so that W z =  ψ : H → E | ( z − 1 hz ) ψ ( y h ) = ψ ( y ) ( y ∈ H, h ∈ H ∩ z L )  =  ψ : H → E | ℓψ ( y z ℓ z − 1 ) = ψ ( y ) ( y ∈ H, ℓ ∈ H z ∩ L )  . On W z to o the H -action is ag ain ( h · ψ )( y ) = ψ ( h − 1 y ), but now, bec ause of co nju- gation, the A -action lo oks as follows: ( ψ · z a )( y ) = ψ ( y )( z − 1 y − 1 a ) ( ψ ∈ W z , y ∈ H, a ∈ A ) . W e cla im that V z ∼ = W z via the function ϕ 7→ ˜ ϕ given by ˜ ϕ ( y ) := ϕ ( y z ) for all y ∈ H . The function is well-deﬁned, b ecause y z ∈ H z L for all y ∈ H and ℓ · ˜ ϕ ( yz ℓz − 1 ) = ℓ · ϕ ( y z ℓ ) = ℓℓ − 1 ϕ ( y z ) = ˜ ϕ ( y ) for all ℓ ∈ L . It is also ev iden tly H - linear, and it is A -linear by the co mputation ( g ϕ · a )( y ) = ( ϕ · a )( y z ) = ϕ ( y z )( z − 1 y − 1 a ) = ( ˜ ϕ · z a )( y ) ( ϕ ∈ V z , a ∈ A, y ∈ H ). Finally , w e claim that the inv erse map ψ 7→ ˆ ψ , W z → V z , is given by the formula ˆ ψ ( x ) := ℓ − 1 ψ ( h ) for each x = hz ℓ ∈ H z L . The ma p ˆ ψ is well-deﬁned: if x = hz ℓ = h 1 z ℓ 1 ∈ H z L and ψ ∈ W z , then ℓ − 1 ψ ( h ) = ℓ − 1 ψ ( h 1 z ℓ 1 ℓ − 1 z − 1 ) = ℓ − 1 ψ ( h 1 z ℓ 1 z − 1 ( z ℓ − 1 z − 1 )) = ψ ( h 1 z ℓ 1 z − 1 ) ( ψ ∈ W z ) = ℓ − 1 1 ψ ( h 1 ) ( ψ ∈ W z ) Moreov er, the computation (with x = hz ℓ ∈ H z L, ℓ ′ ∈ L ) ℓ ′ ˆ ψ ( xℓ ′ ) = ℓ ′ ( ℓℓ ′ ) − 1 ψ ( h ) = ℓ − 1 ψ ( h ) = ˆ ψ ( x ) shows that indeed ˆ ψ ∈ V z for all ψ ∈ W z . The veriﬁcation that ( ˆ ψ ) ∼ = ψ and ( ˜ ϕ ) ˆ = ϕ is equa lly immediate: ( ˜ ϕ ) ˆ = ℓ − 1 ˜ ϕ ( h ) = ℓ − 1 ϕ ( hz ) = ϕ ( hz ℓ ) = ϕ ( x ) ( ϕ ∈ V z , x = hz ℓ ∈ H z L ) , ( ˆ ψ ) ∼ ( y ) = ˆ ψ ( y z ) = ψ ( y ) ( ψ ∈ W z , y ∈ H ) . Hence we obtain the claimed iso morphism V z ∼ = W z of ( H , A )-mo dules. There- fore we hav e an isomorphism as claimed in the lemma, and it is evident from its construction that it is na tural in the ( L, A )-mo dule E .  Prop osition 4. 5. The mo dules K H 0 ( A ) and the maps re s H L , ind H L and con g,H de- scrib e d ab ove deﬁne an R G -Mackey mo dule k G ( A ) , and the f unctorialities of al l K H 0 assemble to yield a functor k G : C ∗ alg G → R G - Mac , with k G ( 1 ) = R G . EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 17 Pr o of. First of a ll, let us ﬁx a G -C ∗ -algebra A , and let us verify that k G ( A ) is a Mack ey functor. The Ma c key for m ula ho lds by Lemma 4.4 . The ﬁr st and ﬁfth relations (see § 3.1 ) are either con tained in Lemma 4.1 or in [ 25 , Prop. 5.1.3]. There remain the co mpatibilities o f the conjuga tio n maps with r estrictions a nd induct ions, which ar e straightforward a nd are left to the reader. Thus k G ( A ) is a Mack ey functor. Next, w e must verify that the co llected actions of R ( H ) o n K G 0 (Res G H A ) turn k G ( A ) into an R G -mo dule. The third and fourth relations for Mackey mo dules are prov ed in [ 25 , Prop. 5 .1.3] (the third o ne under the guise of the R ( H )-linearity of ind H L ). The R G -linearity of restriction and conjugation maps (ﬁrst and se c ond relations), ar e ea s ier and ar e left to the reader . Finally , the functoriality o f A 7→ k G ( A ) for equiv ariant ∗ -homomorphisms follows immediately from that of e ach H - equiv ar ia n t K -theory , and we hav e already seen (Remark 4.3 ) that k G ( 1 ) = R G .  R emark 4.6 . As usual we set K G 1 ( A ) := K G 0 ( A [1]), so that we get a functor K G ∗ : C ∗ alg G → R ( G )- Mo d Z / 2 , K G ∗ ( A ) := { K G ǫ ( A ) } ε ∈ Z / 2 to g r aded mo dules and degree pr eserving morphisms. W e similarly obtain a functor k G ∗ : C ∗ alg G → R ( G )- Mac Z / 2 , k G ∗ ( A ) := { K H ǫ ( A ) } H 6 G ε ∈ Z / 2 int o the categor y of Z / 2-graded Mack ey mo dules o v er R G . Alternatively , the targ et category o f k G ∗ may b e under sto od as the categor y of Mack ey mo dules ov er R G based in the catego ry of Z / 2 -graded a belia n gro ups. 4.4. The extension to the Kasparo v category. Let us restrict our attention to the the sub category of se pa rable G -C ∗ -algebra s, C ∗ sep G . Next, we extend our functor k G to the G -equiv aria n t Kasparov catego ry and study the proper ties of the extension. Lemma 4 . 7. The funct or k G has a unique lifting, that we also denote by k G , to the Kasp ar ov c ate gory KK G along the c anonic al functor C ∗ sep G → KK G . Pr o of. By the universal prop erty of the ca no nical functor C ∗ sep G → KK G , as proved in [ 20 ], the exis tence and uniqueness of such a lif ting is equiv alent to the functor k G being homotopy inv aria n t, C ∗ -stable and split exa ct (in the G -equiv ariant sense). This follows immediately fro m the basic fact that eac h K - theo ry functor K H 0 ◦ Res G H do es enjoy the thr ee prop erties.  Note that, fo r a ll H 6 G , the Green functor R H is just the restriction o f R G at H , in the sense of § 3.5 : R H = Res G H ( R G ). Therefor e, a s explained there, the evident restriction functor Res G H : R G - Mac → R H - Mac has a left-and-right adjoint Ind G H . Lemma 4.8. F or al l H 6 G , the diagr ams KK G k G / / Res G H   R G - Mac Res G H   KK H k H / / R H - Mac KK G k G / / R G - Mac KK H k H / / Ind G H O O R H - Mac Ind G H O O c ommut e up to isomorphism of functors. Pr o of. The claim in volving the restriction functor s is evident from the de ﬁnitio ns; in this case, the squar e even commutes strictly . Now we prove the claim for induction. Let A ∈ KK H and L 6 G . In the cas e of the r ank-one free mo dule, the Ma c key formula of L e mma 4.4 ca n b e easily rewritten as the following isomor phism of L - C ∗ -algebra s: Res G L Ind G H ( A ) ∼ = M x ∈ [ L \ G/H ] Ind L L ∩ x H Res x H L ∩ x H ( x A ) . 18 IVO DELL’ AMBR OGIO Once we have ﬁxed the set of r e presentativ es for L \ G/H , the iso morphism b ecomes natural in A . Therefore we get a natura l is omorphism ( k G ◦ Ind G H ( A ))[ L ] = K L 0  Res G L Ind G H ( A )  ∼ = M x ∈ [ L \ G/H ] K L 0  Ind L L ∩ x H Res x H L ∩ x H ( x A )  ∼ = M x ∈ [ L \ G/H ] K L ∩ x H 0  Res x H L ∩ x H ( x A )  ∼ = M x ∈ [ L \ G/H ] k x H ( x A )[ L ∩ x H ] ∼ = M x ∈ [ L \ G/H ] k H ( A )[ H ∩ L x ] = (Ind G H ◦ k H ( A ))[ L ] . In the third line we have used the (Ind , Res )-adjunction for K asparov theor y , and in the ﬁft h w e ha ve us ed the H , x H -eq uiv ariant isomor phism A ∼ = x A of Remar k 4.2 and the iso morphism K H 0 (Res H L x ∩ H A ) ∼ = K x H 0 (Res x H L ∩ x H x A ) it induces; the last line is ( 3.8 ) with a = x − 1 . This prov es the claim.  The next theor em is the ma in r esult o f this ar ticle. Theorem 4.9. The r estriction of k G : KK G → R G - Mac to the ful l sub c ate gory { C ( G/H ) : H 6 G } of KK G is a ful ly faithful fun ct or. Pr o of. Iden tifying k G ◦ Ind = Ind ◦ k H and k H ◦ Res = Res ◦ k G as in Lemma 4.8 , for all H , L 6 G we hav e the following commutativ e diagr a m: KK G ( C ( G/H ) , C ( G/L )) k G / / R G - Mac ( k G C ( G/H ) , k G C ( G/L )) KK G (Ind G H 1 , Ind G L 1 ) k G / / ∼ =   R G - Mac (Ind G H k H ( 1 ) , Ind G L k L ( 1 )) ∼ =   KK H ( 1 , Res G H Ind G L 1 ) k G / / ∼ =   R H - Mac ( R H , k H (Res G H Ind G L 1 )) can ∼ =   K H 0 (Res G H Ind G L 1 ) can ∼ = / / R ( H )- Mo d ( R ( H ) , K H 0 (Res G H Ind G L 1 )) Therefore the upp er map la beled k G is bijective.  4.5. The Burnside- B o uc category as equi v arian t KK-theory. T o complete the pictur e , we can no w descr ibe the Burns ide - Bouc categor y a sso ciated with the representation ring R G in terms of G -equiv ariant Kaspar o v theory . The r e lation is a very simple and sa tisfying one. Deﬁnition 4.10. In analo gy with p ermutation modules, we call a G -C ∗ -algebra of the form C 0 ( X ), for some G -set X , a p ermu tation algebr a . Let Per m G resp. p erm G be the full sub c ategory of KK G of separ able pe rm utation a lgebras, res pectively of ﬁnite dimensio nal pe r m utation a lgebras. Note that they are precisely those of the form A ∼ = L i ∈ I C ( G/H i ) for some countable, r espec tively ﬁnite, index set I . Note EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 19 also that, by vir tue of the natural isomor phisms C ( X ) ⊕ C ( Y ) ∼ = C ( X ⊔ Y ) and C ( X ) ⊗ C ( Y ) ∼ = C ( X × Y ), bo th Perm G and p erm G are a dditive tenso r sub categories of KK G . Theorem 4.11. F or every ﬁnite gr oup G , the functor k G : KK G → R G - Mac of § 4.4 r estricts to a tensor e qu ivalenc e of p erm G with the ful l su b c ate gory of r epr esentable R G -mo dules, i.e. , with the Burn side-Bouc c ate gory B R . Pr o of. Iden tifying R G - Mac = Ab ( B R G ) op as in § 3.3 , w e obta in the follo wing diagr am of functors, which we cla im is commutativ e (up to iso morphism). ( G - set ) op can / / C   C % % K K K K K K K K K K B X 7→ R G X   can / / B R G Y oneda                    C ∗ sep G can y y s s s s s s s s s s k G % % K K K K K K K K K K β ∼ = KK G k G / / R G - Mac Indeed, the left, b ottom and right triangles (strictly) comm ute by deﬁnition. W e m ust show that ther e is a natura l isomorphism β X : k G ◦ C ( X ) ∼ = R G X , making the central triangle c o mm ute. F or X = G/H and Y = G/L ∈ G - set , we obtain the following isomor phis ms β G/H ( G/L ): k G ( C ( G/H ))( G/L ) = K L 0 (Res G L C ( G/H )) = K L 0 ( C (Res G L G/H )) ∼ = K L 0 C   a x ∈ [ L \ G/H ] L/ ( L ∩ x H )   ∼ = M x ∈ [ L \ G/H ] K L 0 C ( L/ ( L ∩ x H )) ∼ = M x ∈ [ L \ G/H ] R ( L ∩ x H ) ∼ = R G   M x ∈ [ L \ G/H ] G/ ( L ∩ x H )   = R G ( G/L × G/H ) = R G G/H ( G/L ) W e leave to the rea der the veriﬁcation that, by letting L 6 G v ary , these deﬁne an isomorphism β G/H : k G C ( G/H ) ∼ = R G G/H , and that the la tter can b e extended to a natural isomo rphism β X as requir ed. The statement of the theor em follows now from the fact that the botto m hor- izontal k G is fully faithful on the ima ge o f C , by Theo rem 4.9 ; the “tenso r” par t follows from the identiﬁcation k G ( C ) = R G and fro m the natural iso morphism φ X,Y : k G ( C ( X ))  R G k G ( C ( Y )) ∼ = R G X  R G R G Y ∼ = R G X × Y ∼ = k G ( C ( X × Y )) ∼ = k G ( C ( X ) ⊗ C ( Y )) 20 IVO DELL’ AMBR OGIO for a ll X , Y ∈ G - s et , o btained by combining β with the sy mmetr ic mo noidal struc- tures of the functor C and of the Y one da embedding X 7→ R X . Clearly the squar e k G C ( X )  R G k G C ( Y ) φ X,Y   k G (switch) / / k G C ( Y )  R G k G C ( X ) φ Y ,X   k G ( C ( X ) ⊗ C ( Y )) switch / / k G ( C ( Y ) ⊗ C ( X )) is comm utativ e, showing that φ turns k G int o a s ymmetric monoidal functor on the image of C .  Corollary 4 . 12. The c ate gory of additive functors ( p erm G ) op → Ab is e quivalent to the c ate gory of Mackey mo dules over the r epr esent ation Gr e en functor R G . I f we e quip functor the c ate gory with t he Day c onvolution pr o duct, we have a symmetric monoidal e qu ivalenc e. The same hold s for the c ate gory of c opr o duct-pr eserving additive functors ( Perm G ) op → Ab , Pr o of. W e know from Theo rem 4.11 that k G ∗ : p erm G ≃ B G R G as tenso r categorie s, so this is just Bo uc’s functorial picture for R G -Mack ey modules ( § 3.3 ). Day con vo- lution provides the cor r ect tensor structure by co nstruction, cf . Remar k 3.7 .  R emark 4 .13 . Cor ollary 4.12 should be compar ed w ith the following result, see [ 18 , Prop osition V.9.6]: the Bur nside categ ory B = B B ur is equiv a len t to the f ull sub cat- egory in the sta ble homotopy category of G -equiv a r iant sp ectra, SH G , with ob jects all susp ension sp ectra Σ ∞ X + for X ∈ G - set . The a utho r s of lo c. cit. deﬁne Mack ey functors for a compact Lie group G precisely so that the ana logous statement re- mains true in this case. It would b e interesting to kno w whether the same deﬁnition prov es useful for the study of K K G when G is a compact Lie gro up. R emark 4.1 4 . In principle, it must b e p ossible to prov e Theorem 4.11 directly , without app ealing to Theorem 4.9 . Firs t notice that KK G ( C ( G/H ) , C ( G/L )) ∼ = KK H ( 1 , Res G H C ( G/L )) ∼ = M x ∈ [ H \ G /L ] R ( H ∩ x L ) = R G ( G/H × G/L ) def. = B R G ( G/H, G/L ) for all H , L 6 G , by the (Ind , Res )-a djunction in KK-theory . Then it remains “only” to prove that this identiﬁcation takes the co mpos ition o f KK G to the c o mpos ition of B R G . But this seems like a lot of work: the K a sparov pro duct is famously diﬃcult to co mpute ex plicitly (althoug h, admittedly , we ar e dea ling here with an easy s pecia l ca s e), and the explicit formula for the comp osition in the Burnside- Bouc catego ry is a lso rather in volv ed (see [ 5 , § 3 .2]). In o rder to do this, one could per haps use the corresp ondences of [ 11 ] a nd their geometric picture of the Kaspar o v pro duct. Anyw ay , o nce Theorem 4.11 is proved it is then pos sible to use abstract consideratio ns to derive from it T he o rem 4.9 , ra ther than the other wa y round. 5. Rela tive homol ogical algebra and G -cell algebras W e beg in by reca lling fro m [ 24 ] and [ 22 ] a few deﬁnitions and results o f r elative homolo gic al algebr a in triangulate d c ate gories . This will allow us to e s tablish some notation that will b e used thr oughout the re st o f the ar ticle. EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 21 In the fo llo wing, let T b e a triangulated categor y endo wed with a rbitrary co- pro ducts; for simplicity , w e still assume that the s hift functor T → T , A 7→ A [1], is a s trict automorphism, r a ther than just a s elf-equiv alence. Deﬁnition 5. 1. It will b e co n venien t to deﬁne the p erio dici ty , written π , of the shift functor [1] : T → T to be the smallest p o sitive integer π s uch that there exists an isomor phism [ n ] ∼ = id T , if such an integer exists; if it do es not, we set π := 0. 5.1. Recollectio ns and notation. A stable ab elian categ ory is an abelia n cate- gory equipped with an automorphis m M 7→ M [1], ca lled sh ift . A stable homolo gic al functor is an additive functor F : T → A to a stable ab elian categor y A , which commutes with the shift and which sends distinguished triang les to exact s equences in A . In particula r, a s table homologica l functor is homolo g ical in the usua l s e ns e. Conv ersely , if F : T → A is a ho mological functor to some abelia n catego r y A , then we can construct a stable homolo gical functor F ∗ : T → A Z /π as follows (reca ll that we allow π = 0 , in which case we hav e Z /π = Z ). Here A Z /π denotes the stable ab elian catego r y of Z / π - graded ob jects in A with degr ee preserv- ing morphisms. As a categor y , it is simply the pr oduct A Z /π = Q i ∈ Z /π A ; we write M i for the i -th comp onent of a n ob ject M , and similarly fo r mor phis ms. The shift functor is given by ( M [1]) i := M i − 1 , and ditto for morphisms. Then we deﬁne F ∗ by F ( A ) i = F i ( A ) := F ( A [ − i ]). R emark 5.2 . This choice of degree follows the usua l ( ho mological) indexing con- ven tio ns , a ccording to which a distinguished triangle A → B → C → A [1] giv es r ise to a long exact sequence of the form . . . → F i A → F i B → F i C → F i − 1 A → . . . . Note that, if instead F : T op → A is a contrav a riant homological functor, then the usual conv en tion requires us to write indices up , F i ( A ) := F ( A [ − i ]), to indicate that diﬀ erentials no w incr ease deg ree: . . . F i − 1 ( A ) → F i ( C ) → F i ( B ) → F i ( A ) . . . . If one must insist in using homo logical notation (as we will do later with graded Y oneda a nd gra ded Ext gr oups), then one uses the conversion rule F i = F − i . A homolo gic al ide al I in T is the collection of mor phis ms of T v anishing under some stable homo logical functor H : I = ker H := { f ∈ Mo r( T ) | F ( f ) = 0 } . Thu s in particular I is a categor ical ideal which is closed under shifts of ma ps. Note that diﬀerent s table homolo gical functors H can deﬁne the same homologic a l ideal I , but it is the latter datum that is of primary interest and will determine all “relative” ho mo logico-alg ebraic notions. 2 A homological functor F : T → A is I -exact if I ⊆ ker F . An ob ject P ∈ T is I -pr oje ctive if T ( P, ) : T → Ab is I -exact. An I -pr oje ctive r esolution of a n o b ject A ∈ T is a diagr am . . . P n → P n − 1 → . . . → P 1 → P 0 → A → 0 in T such that each P n is I - pr o jectiv e a nd such that the sequence is I - ex act in a suitable sense (se e [ 24 , § 3.2 ]). Let F : T → A be an additiv e (usually homological) functor to an ab elian cat- egory , a nd let n > 0 b e a nonnegative integer. The n -th I - r elative left derive d functor of F , written L I n F , is the functor T → A obtained by taking an ob ject A ∈ T , cho osing a pro jective res olution P • for it, applying F to the complex P • and taking the n -th ho mology of the r e sulting complex in A — in the usual w ay . In the ca s e of a contrav a riant functor, F : T op → A , we ca n still use I -pro jective resolutions in the same wa y in T to deﬁne the I - relative right derived functors R n I F : T op → A . 2 There i s a n elega nt ax iomatic approac h due to Beligiannis [ 1 ] th at does justice to this obser- v ation, but we w i ll not use it here. 22 IVO DELL’ AMBR OGIO R emark 5.3 . One of co urse has to prove that the re c ip es fo r L I n F and R n I F yield well- deﬁned functors. This is a lw ays the case – as in our examples – as so on as there ar e enough I -pro jective ob jects, in the precise sense that for ev e ry A ∈ T there exists a morphism P → A ﬁtting in to a distinguished triangle B → P → A → B [1] where P is I -pro jective and ( A → B [1]) ∈ I . All our ex amples hav e enough I -pro jectives but po ssibly not enough I -injectives, which causes the a bove asy mmetrical de ﬁnitio n of derived functors . R emark 5.4 . It is immediate from the deﬁnitions that one may stabilize either be fo re or after ta king derived functors , namely : ( L I n F ) ∗ = L I n ( F ∗ ) and ( R n I F ) ∗ = R n I ( F ∗ ). 5.2. The graded restricted Y oneda functor. Assume no w that we ar e given an (essentially) small set G ⊆ T of c omp act o b jects; that is, the f unctor T ( X , ) : T → Ab commutes with arbitra ry co pr o ducts for ea c h X ∈ G . Our goa l is to understand the homologica l algebra in T relative to G , that is , relative to the homolo g ical ideal I := \ X ∈G ker T ( X , ) ∗ = { f ∈ Mor( T ) | T ( X [ i ] , f ) = 0 ∀ i ∈ Z /π , X ∈ G } . The re a son we b other with this gener alit y is that, already at this level, Ralf Ma y er’s ABC sp ectral seque nc e [ 22 ] sp ecializes to a pleasa n t-lo oking universal co eﬃcient sp ectral sequence (see Theor e m 5.15 below). Let T ( A, B ) ∗ = {T ( A [ i ] , B ) } i ∈ Z /π denote the g raded Hom in T induced by the shift automo phism, a nd let T ∗ denote the Z /π -gr aded categor y with the sa me ob jects as T and comp osition g iv en by T ( B [ j ] , C ) × T ( A [ i ] , B ) → T ( A [ i + j ] , B ) ( g , f ) 7→ g f := g ◦ f [ j ] . Similarly , denote by G ∗ the full gr aded sub categor y of T ∗ containing the ob jects of G . Let GrMod - G ∗ be the ca tegory of gr ade d right G ∗ -mo dules . Its o b jects are the deg ree-preser ving functors M : ( G ∗ ) op → ( Ab Z /π ) ∗ int o the gr ade d catego ry of graded ab elian gro ups, and its morphisms are grading pr eserving natural transfo r - mations ϕ : M → M ′ , i.e. , families ϕ i,X : M i ( X ) → M ′ i ( X ) ( i ∈ Z /π , X ∈ G ) o f homomorphisms commuting with ma ps M ( f ) of all degrees. Note that GrMo d - G ∗ is a stable a belia n c a tegory with shift functor giv en b y ( M [ k ]) i := M i − k and ( f [ k ]) i := f i − k . Every A ∈ T deﬁnes a gra ded G ∗ -mo dule h ∗ ( A ) := { T (( )[ i ] , A ) ⇂ G ∗ } i ∈ Z/π in a natural wa y , so tha t we get a ( r estricte d ) Y one da functor h ∗ : T − → Gr Mo d - G ∗ which is stable homo logical a nd moreov er preserves copr o ducts , since the ob jects of G are c ompact. Lemma 5.5. Ther e is a natur al isomorphism of Z / π - gr ade d ab elian gr oups (5.6) GrMo d - G ∗ ( h ∗ ( X ) , M ) ∗ ∼ = M ( X ) for al l X ∈ G and al l M ∈ GrMo d - G ∗ , which sends the natura l tr ansformation ϕ : h ∗ ( X )[ i ] → M to the element ϕ i,X (1 X ) ∈ M i ( X ) . Pr o of. This follows from the Y oneda lemma for Z /π -g raded Z -linear catego ries, i.e. , for categories enriched over the clo s ed s ymmetric monoidal ca tegory Ab Z /π (see [ 14 ]). It ca n a lso b e e asily prov ed by hand.  W e see in par ticular that the co llection { h ∗ ( X )[ i ] | i ∈ Z /π , X ∈ G } for ms a set of pro jective gener ators for GrMo d - G ∗ . EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 23 W e also note that h ∗ detects the v anishing of any ob ject in the lo ca lizing trian- gulated sub categor y Cell ( T , G ) := hG i lo c ⊆ T generated by G . Example 5.7 . Let T = KK G be the equiv ariant Ka sparov categor y of a ﬁnite group G , a nd let G := { C ( G/H ) | H 6 G } . By Bott p erio dicity , we m ust grade ov er Z / 2. Then Cell ( T , G ) = Cell G , and the sta ble ab elian categor y GrMo d - G ∗ is just R G - Mac Z / 2 , the categ ory of Ma c key mo dules over the repr esent ation Green functor for G . This is b ecause in this ca se G ( X [1] , Y ) = 0 for a ll X, Y ∈ G , so that GrMo d - G ∗ is quite simply the categ o ry of Z / 2- g raded ob jects in the usual un- graded mo dule categ ory Mod - G , and w e kno w from Corollary 4 .12 that the latter is equiv alent to R G - Mac . If G = 1 is the trivia l gr oup, then Cell ( T , G ) is the classica l Bo otstrap categ ory of separ able C ∗ -algebra s [ 26 ], and GrMo d - G ∗ is the category of graded ab elian g roups, Ab Z / 2 . Prop osition 5.8. The r est ricte d Y one da functor h ∗ : T → Gr Mo d - G ∗ is the univer- sal I -exact stabl e homolo gic al f unctor on T . In p articular, h ∗ induc es an e quivalenc e b etwe en the c ate gory of I - pr oje ctive obje ct s in T and that of pr oje ctive gr ade d right G - mo dules: h ∗ : Proj ( T , I ) ≃ Pro j ( GrMod - G ∗ ) , and for every A ∈ T it induc es a bije ction b etwe en I -pr oje ctive r esolutions of A in T and pr oje ctive r esolutions of the gr ade d G -mo dule h ∗ ( A ) . Pr o of. Since the stable homo logical functor h ∗ is I -exact by deﬁnition, and since idempo ten t morphis ms in T split (b ecause of the existence of coun table copro ducts, [ 4 , Pro p. 3.2]), we may use the criter ion o f [ 2 4 , Theor em 57 ]. Since the ab elian category GrMo d - G ∗ has enough pro jectives, it rema ins to show the ex is tence of a partial right-in verse and partial left adjoint to h ∗ deﬁned on pro jective modules, i. e. , the existence o f a functor ℓ : Proj ( GrMo d - G ∗ ) → T and tw o natural iso morphisms h ∗ ◦ ℓ ( P ) ∼ = P and T ( ℓP, B ) ∼ = GrMo d - G ∗ ( P, h ∗ B ) ( P ∈ Proj - G ∗ , B ∈ T ) . Since the pro jective ob jects in GrMo d - G ∗ are additively genera ted by the re pr e- sentables h ∗ ( X [ i ]) = h ∗ ( X )[ i ], for X ∈ G a nd i ∈ Z /π , by [ 24 , Remark 58] we need only deﬁne the functor ℓ and the natura l isomorphisms on the full sub cat- egory { h ∗ ( X )[ i ] | X ∈ G , i ∈ Z } ⊂ GrMo d - G ∗ . F or ea c h s uc h ob ject, we set ℓ ( h ∗ ( X )[ i ]) := X [ i ], so that indeed h ∗ ℓ ( h ∗ ( X )[ i ]) = h ∗ ( X )[ i ], and also T ( ℓ ( h ∗ X [ i ]) , B ) = h 0 ( B )( X [ i ]) ∼ = GrMo d - G ∗ ( h ∗ X [ i ] , h ∗ B ) by the Y oneda Lemma 5.5 . It is now evident how to extend ℓ on mo rphisms.  R emark 5.9 . If T do e s no t have all set-indexed copro ducts, the same argument still works if instea d the following tw o hypotheses are satisﬁed: (1) There exists an unco un table r egular cardinal ℵ , such that T has all small ℵ copro ducts, i .e. , those indexed b y sets of ca rdinality strictly s maller than ℵ . (In particula r T has at least a ll countable copr oducts.) (2) Every ob ject X ∈ G in o ur g enerating s e t is smal l ℵ , that is , the functor T ( X , ) commutes with small ℵ copro ducts a nd sends e v ery ob ject A ∈ T to a small ℵ set: |T ( X , A ) | < ℵ . Then Prop ositio n 5.8 rema ins true, with precis e ly the same pro of, if in its s tatemen t we substitute GrMo d - G ∗ with the ca tegory of smal l ℵ graded G ∗ -mo dules, that is, those M ∈ GrMod - G ∗ such that each M ( X ) i is small ℵ . F or our application to KK- theory , we will have to use this ℵ -rela tiv e version of the statement with ℵ = ℵ 1 . 24 IVO DELL’ AMBR OGIO F or the following nex t g e ne r al statements, we may e ither assume that T ha s arbitrar y c opro ducts and the ob jects of G are compact, or that T and G sa tisfy the hypothesis of Remark 5.9 . Notation 5.10 . Let Ext n G ∗ ( M , N ) ∗ be the graded Ext functor in Gr Mo d - G ∗ . In other words, Ext n G ∗ ( , N ) ∗ denotes the right derived functors of the gr ade d Hom func- tor GrMo d - G ∗ ( , N ) ∗ : GrMo d - G ∗ → Ab Z /π ; as usua l, it is co mputed by pro jective resolutions of gra de d G -mo dules. If, as in Example 5.7 , the categ ory G ∗ = G has only ma ps in degr e e zero, then GrMo d - G ∗ = ( Mo d - G ) Z /π , and we may compute the graded Ext in terms o f the ungraded E xt functor s, according to the for m ula Ext n G ∗ ( M , N ) ℓ = M i + j = ℓ Ext n G ( M i , M j ) ( n ∈ N , ℓ ∈ Z /π ) . Prop osition 5.11. If F is a homolo gic al functor F : T op → Ab sending c opr o ducts in T to pr o ducts of ab elian gr oups, then ther e ar e natur al isomorphisms R n I F ∗ ∼ = Ext n G ∗ ( h ∗ ( ) , F ⇂ G ∗ ) −∗ ( n ∈ N ) c omputing its right I -r elative derive d fu n ctors. Her e F ⇂ G ∗ : G ∗ → Ab Z / 2 denotes the gr ade d G ∗ -mo dule obtaine d by c onsidering the r estriction of F to the ful l sub c ate gory { X [ i ] | X ∈ G , i ∈ Z /π } ⊆ T . Pr o of. By P rop osition 5.8 , we kno w that h ∗ : T → GrMo d - G ∗ is the universal I - exact functor. By [ 2 4 , Theore m 5 9], there exists (up to canonical isomo r phism) a unique left exa ct functor F : ( GrMod - G ∗ ) op → Ab suc h tha t F ◦ h ∗ ( P ) = F ( P ) for every I -pro jective ob ject P of T . Since h ∗ induces a bijection betw een I - pro jective resolutions of A ∈ T and pr o jectiv e resolutions of h ∗ ( A ) ∈ GrMo d - G ∗ , there follows easily the existence of isomorphisms (5.12) R n I F ∼ = R n F ◦ h ∗ ( n ∈ N ) . By Lemma 5.5 for the mo dule M = F ⇂ G ∗ , there a re natural iso morphisms F ( X [ i ]) = F − i ( X ) = ( F ⇂ G ∗ ) i ( X ) ∼ = GrMo d - G ∗ ( h ∗ X , F ⇂ G ∗ ) i = GrMo d - G ∗ ( h ∗ ( X [ i ]) , F ⇂ G ∗ ) for all X ∈ G and i ∈ Z /π . Since every I - pro jective ob ject is a direct s umma nd of a copro duct of such X [ i ], we may extend this additively to a na tur al isomorphism F ( P ) ∼ = GrMo d - G ∗ ( h ∗ ( P ) , F ⇂ G ∗ ) (5.13) for all P ∈ Proj ( T , I ). Moreov er, the Ho m functor GrMod - G ∗ ( , F ⇂ G ∗ ) is left exact. Hence by ( 5.13 ) we can identif y GrMo d - G ∗ ( , F ⇂ G ∗ ) with F , b ecause o f the uniqueness prop erty of the latter. By injecting this knowledge int o ( 5.12 ) we g et the requir ed is omorphisms.  As an importa n t sp ecial ca se, we can no w compute the I -r elative Ext functors ( cf. [ 22 , p. 195]). Corollary 5.1 4. Ther e ar e isomorphisms R n I ( T ( , B )) ∼ = Ext n G ∗ ( h ∗ ( ) , h ∗ B ) 0 ( n ∈ N ) of functors T op → Ab for al l B ∈ T . Pr o of. F or every B ∈ T , the functor F := T ( , B ) : T op → Ab satisﬁes the h y- po thesis of Pr o po s ition 5.11 , and in this case we have F ⇂ G ∗ = h ∗ ( B ) by de ﬁnitio n. Now we apply the pro po s ition and lo ok a t degree zero .  EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 25 5.3. The universal co e ﬃ cien ts sp ectral sequence. W e ar e ready to prov e the following gener al for m of the universal coeﬃcie nt theorem. W e do no t claim any originality for this result, a s it is already essentially in [ 6 ] and [ 22 ]. Theorem 5.15 . L et T b e a triangulate d c ate gory with arbitr ary c opr o ducts and let G b e a smal l set of its c omp act obje cts — or assume the ℵ -r elative versions of this hyp othesis, as in R emark 5.9 . F or al l obje ct s A, B ∈ T , ther e is a c ohomolo gic al ly indexe d right half-plane sp e ctr al se quenc e of the form E p,q 2 = E xt p G ∗ ( h ∗ A, h ∗ B ) − q n = p + q = ⇒ T ( A [ n ] , B ) dep ending fun ctorial ly on A and B . The sp e ctr al se quenc e c onver ges c onditional ly ([ 3 ]) if A ∈ Cell ( T , G ) def. = hG i lo c ⊆ T . We have str ong c onver genc e if A is mor e over I ∞ -pr oje ct ive, that is, if T ( A , f ) = 0 for every morphism f which c an b e written, for every n > 1 , as a c omp osition of n morphisms e ach of which vanishes under h ∗ . If A b elongs to Cell ( T , G ) and mor e over has an I -pr oje ctive r esolution of ﬁn ite length m > 1 (e quivalently: if A ∈ Cell ( T , G ) and h ∗ A has a pr oje ctive r esolution in GrMo d - G ∗ of length m ), then the sp e ctr al se quenc e is c onﬁne d in the r e gion 0 6 p 6 m and ther efor e c ol lapses at t he ( m + 1 ) - st p age E ∗ , ∗ m +1 = E ∗ , ∗ ∞ . Pr o of. W e de ﬁne our sp ectral sequenc e to b e the ABC sp ectral sequence of [ 22 ] asso ciated to the triangulated category T , its homo logical ideal I = ker h ∗ , the contra v aria n t homolo g ical functor F = T ( , B ) into ab elian gro ups, and the ob- ject A ∈ T ; the h ypo thes es that T has coun table copr oducts , that I is closed under them, and that F sends them to pro ducts, ar e all satisﬁed. By [ 22 , Theorem 4.3], the ABC sp ectral s equence is (fro m the sec ond page onw ards) functorial in A , a nd ours is clear ly also functorial in B by construction. Mo reov er, its seco nd page con- tains the gro ups E p,q 2 = R p I F q ( A ), which take the requir ed form by Co rollary 5.14 . The criterio n for s trong conv ergence is proved in [ 22 , Pr opo sition 5.2] (where , in the no tation o f lo c. cit. , A = LA and R F = F b ecause A ∈ Cell ( T , G )), and the criterion for collapse is part of [ 22 , Pr opo sition 4.5 ]. Conditional conv ergence is prov ed as in [ 6 , P rop osition 4.4]. The h yp othesis of lo c. cit. is that G -pro jective o b jects gener ate, i.e. , that Cell ( T , G ) = T . How ever, for ﬁxed A and B , the ar gument only uses that A ∈ Cell ( T , G ), not B : this s till implies that X k ∈ Cell ( T , G ) for all the stages of the Adams resolution [ 6 , (4.1)], i.e. , of the phan tom to wer [ 22 , (3.1)], and the conclusion fo llo ws exactly with the same pro of.  Spec ia lizing Theor em 5.15 to E xample 5.7 , we obtain the ﬁrst of the r esults promised in the Introductio n. Theorem 5. 16. L et G b e a ﬁnite gr oup. F or every A, B ∈ KK G such that A is G -c el l algebr a, and dep ending funct orially on t hem, ther e exists a c ohomolo gic al ly indexe d, right half plane, c onditional ly c onver gent sp e ctr al se quenc e E p,q 2 = Ext p R G ( k G ∗ A, k G ∗ B ) − q n = p + q = ⇒ KK G n ( A, B ) which c onver ges st r ongly or c ol lapses u nder the same hyp othesis of The or em 5.15 . Pr o of. Since K K G only has countable copr oducts, we ado pt the hypothes es o f Re- mark 5.9 with ℵ := ℵ 1 . Note that the gener a tors G = { C ( G/H ) | H 6 G } are compact ℵ 1 by Propo sition 2.9 . The universal G -exact stable homo lo gical functor of Prop osition 5.8 is just our k G ∗ : KK G → R G - Mac ℵ 1 Z / 2 (where the “ ℵ 1 ” indicates that we must r estrict attention to countable mo dules), a nd the rest follows.  26 IVO DELL’ AMBR OGIO 5.4. The K ¨ unneth sp ectral sequence. W e have a fairly g oo d idea of what should b e the most a ppropriate level of abstra ction for proving a nice gener al K ¨ unneth sp ectral sequence, s imila r to the g eneral universal co eﬃcient sp ectra l sequence of § 5.2 . But this would inv olve inﬂicting on the reader more abstract nonsense than might b e decently included in this a r ticle, a nd we ther efore reserve such thoughts for a diﬀere nt pla ce and a future time. F or Kaspar o v theory , in any c a se, we hav e the following. Theorem 5. 17. L et G b e a ﬁnite gr oup. F or al l sep ar able G - C ∗ -algebr as A and B , ther e is a homolo gic al ly indexe d right half-plane sp e ctr al se quenc e of t he form E 2 p,q = T or G ∗ p ( k G ∗ A, k G ∗ B ) q n = p + q = ⇒ K G n ( A ⊗ B ) which is str ongly c onver gent whenever A ∈ Cell G . The key is to s how that the genera tors in G are suﬃciently nice with r espe ct to the universal I -exact functor, s o that we may corr e c tly identify the second pag e. Lemma 5.18. F or every H 6 G , t her e is an isomorphism of gr ade d R G -mo dules ϕ H : k G ∗ ( C ( G/H ) ⊗ B ) ∼ = k G ∗ ( C ( G/H ))  R G k G ∗ ( B ) natur al in B ∈ C ∗ alg G . Pr o of. W e deﬁne ϕ H by the following commutativ e diag ram. k G ∗ ( C ( G/H ) ⊗ B ) ϕ H / / k G ∗ ( C ( G/H ))  R G k G ∗ ( B ) ∼ = Le mma 4.8   k G ∗ (Ind G H ( 1 H ) ⊗ B ) F rob enius ( 2.1 ) ∼ =   Ind H G ( R H )  R G k G ∗ ( B ) ∼ = F robe nius 3.9   k G ∗ (Ind G H ( 1 H ⊗ Res G H ( B ))) ∼ =   Ind G H ( R H  R H Res G H ( k G ∗ ( B ))) ∼ =   k G ∗ (Ind G H Res G H (( B ))) ∼ = Lemma 4.8 / / Ind G H Res G H ( k G ∗ ( B )) Because each iso morphism is natural in B , their comp osition is to o.  Prop osition 5.19. F or the stable homolo gic al functor F ∗ := k G ∗ ( ⊗ B ) : K K G → Ab Z / 2 , ther e ar e c anonic al isomorp hisms L I n F ∗ ∼ = T or R G n ( k G ∗ ( ) , k G ∗ ( B )) ∗ ( A ∈ KK G , n ∈ Z ) of functors KK G → Ab Z / 2 b etwe en its I - r elative left derive d functors and the left de- rive d functors of the tensor pr o duct  R G of gr ade d R G -Mackey mo dules. Mor e over, if we e quip b oth sides with t he natur al ly induc e d R G -action, these isomorphisms b e c ome isomorphisms of functors KK G → R G - Mac Z / 2 . Pr o of. The pro of is quite simila r to that of Pro pos ition 5.11 , but let us go a gain through the motions. By [ 24 , Theorem 59], there ex is ts (up to c a nonical isomor- phism) a unique r ight exact functor F ∗ : R G - Mac Z / 2 → Ab such that F ∗ ◦ k G ∗ ( P ) = F ∗ ( P ) for all I -pr o jective ob jects P ∈ K K G . Moreov er, it follows that there are isomorphisms L I n F ∗ ∼ = L n F ∗ ◦ k G ∗ ( n > 0) . (5.20) Because of the isomorphisms ϕ H of Lemma 5.18 , and the consequent isomorphisms ϕ H [1] : k G ∗ ( C ( G/H )[1] ⊗ B ) ∼ = k G ∗ ( C ( G/H ))[1]  R G k G ∗ ( B ) , EQUIV ARIANT KASP ARO V THEOR Y OF FINITE GROUPS VIA MACKEY FUNCTORS 27 we see that k G ∗ ( )  R G k G ∗ ( B ) agrees with F ∗ on I -pr o jectiv e ob jects. There fore, since ( )  R G k G ∗ ( B ) is right exa c t, we can further identify F ∗ = ( )  R G k G ∗ ( B ). Hence by ta king left derived functors w e deduce that the identiﬁcation ( 5.20 ) takes the form o f the claimed is omorphisms. The las t cla im of the pro po s ition is clear from the na tur alit y of the isomorphisms.  Pr o of of The or em 5.17 . W e deﬁne the K ¨ unneth spectra l seq ue nc e to b e the ABC sp ectral sequence of [ 22 ] a sso ciated with the tria ngulated categor y T = KK G , the homologica l ideal I = T H 6 G ker( K H ∗ ◦ Res G H ), the o b ject A , and the cov ariant ho- mological functor F = k G ( ⊗ B ) : KK G → Ab ; the hypotheses that T has countable copro ducts, that I is closed under them and that F prese rves them ar e all sa tis- ﬁed. The desc r iption of the s econd page follows from [ 22 , Theor em 4.3] and the computation in Pro pos ition 5.19 . The strong conv ergence to F n ( A ) = K G n ( A ⊗ B ) follows from [ 22 , Theor em 5.1] together with the hypothesis that A b elongs to Cell G , namely , to the lo calizing sub category g enerated b y the I -pr o jective ob jects, which implies that F ( A [ n ]) = L F ( A [ n ]) in the nota tio n of lo c. cit . 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