Localisation and colocalisation of KK-theory at sets of primes

LOCALISA TION AND COLOCALI SA TION OF KK-THEOR Y A T SETS OF PRIMES HVEDRI INASSARIDZE, T AMAZ KANDELAKI, AND RALF MEYER Abstract. Give n a set of prime nu mbers S , we localise equiv ariant biv ariant Kasparo v theory a t S and compare this lo cali sation with Kasparov theory b y an exact sequenc e. More precisely , w e define the localisation at S to be KK G ( A, B ) ⊗ Z [ S − 1 ]. W e study the prop erties of the resulting v ariants of Kasparo v theory . 1. Introduction Lo calisation at a prime n umber is a standard tec hnique fo r computations in stable homotopy theory . W e c o nsider a more genera l situation here. Let R b e a co mm utative ring a nd le t T be an R - linear triangulated catego ry , that is, the morphism spaces in T are enriched to R -mo dules a nd the comp ositio n in T is R -bilinear . Let S ⊆ R T \ { 0 } b e a mult iplicatively closed subset. Le t S − 1 R be the loca lis ation of the ring R at S . W e define the lo calisation of T at S to b e the category T [ S − 1 ] with morphism spaces T [ S − 1 ]( A, B ) : = T ( A, B ) ⊗ R S − 1 R. This yields ag ain a triangulated catego ry be c a use S − 1 R is a flat mo dule ov er R . Equiv alently , we may describ e T [ S − 1 ] as a loc a lisation o f T at a certain thick sub c ategory . V ery similar results are due to P aul Balmer [2] (see [2 , Theorem 3 .6] and [6 , Theorem 2 .3 2]). F o r instance, let R : = Rep( G ) b e the repre s entation ring o f a co mpact Lie group G and let T b e the equiv a riant Kaspa rov category KK G . The r ing Rep( G ) is alwa ys a commutativ e No etherean ring, its prime ideals ar e des crib ed in [1 5]. Let ˆ S be a s et of prime ideals of Rep( G ) and let S : = Rep( G )  [ p / ∈ ˆ S p be the as s o ciated m ultiplicatively closed subset of elements that are invertible out- side S . The resulting lo calisa tio n of KK G at S is used, fo r instance, in [6]. A particula rly simple e x ample is ra tional KK G -theory . Her e we take R = Z and S = Z \ { 0 } , so tha t Z [ S − 1 ] = Q . The lo calisa tion of KK G at S has morphism spaces KK G ( A, B ; Q ) : = KK G ( A, B ) ⊗ Q . 2000 Mathematics Subje c t Classific ation. 19K99, 19K35, 19D55. This researc h was supp orted b y t he V olkswagen F oundation (Georgian–German non- comm utativ e partnership). The third author was supported by the German Researc h F oundation (Deutsc he F orsch ungsgemeinsc haft (DFG )) through the Institutiona l Strategy of the Unive rsity of G¨ ottingen. 1 2 HVEDRI INASSARIDZE, T AMAZ KANDE LAKI, AND RALF ME YER This differs from the definition of Kaspa rov theory with co efficie nts in [3, Exe r cise 23.15.6 ]. Our definition has several a dv antages. Most imp ortantly , the ca tegory defined b y the gro ups KK G ( A, B ; Q ) is aga in triangulated. F o r trivial G and A = C , we get K-theory with co efficients in S − 1 Z , K ∗ ( A ; S − 1 Z ) : = K ∗ ( A ) ⊗ S − 1 Z ; here our definition is equiv alent to the one in [3, Exercise 23.15 .6]. K-theory with co efficients in Q is pa rticularly imp or tant beca use the Cher n character iden tifies K ∗ ( X ) ⊗ Q for a compact spac e X with the ratio na l cohomolo gy of X . W e r eturn to the gener al case of a loca lisation of a n R -linear triangulated cat- egory T at a mu ltiplicatively closed subset S of R . The embedding R → S − 1 R induces a cano nical map from T to T [ S − 1 ]. Since T [ S − 1 ] is a lo c a lisation of T , the construction in [8] pro vides a natural long exact sequence (1.1) · · · → T 0 ( A, B ; S − 1 R/R ) → T 0 ( A, B ) → T 0 ( A, B ; S − 1 R ) → T − 1 ( A, B ; S − 1 R/R ) → · · · . The groups T ( A, B ; S − 1 R/R ) behave like the mo rphism spaces in a triang ulated category , except that they lac k unit morphisms. F o r instance, as sume T = KK , R = Z , S = Z \ { 0 } , and A = C . T hen the torsion theory KK ∗ ( C , B ; S − 1 Z / Z ) ag rees with the K-theo r y of B with S − 1 Z / Z -co efficients, and the long ex act seq uence (1.1) already appears in [5, Section 8.1]. The definition in [8] is not useful to actually compute T ( A, B ; S − 1 R/R ). T o address this problem, recall that S − 1 R/R ∼ = lim − → x − 1 R/R = lim − → R/ ( x ) , where ( x ) = x · R ∼ = R is the principal idea l gener ated by x . Hence we exp ect that T ( A, B ; S − 1 R/R ) is a colimit of theories T ( A, B ; s ) “with finite co efficien ts.” W e ma ke this precise be low. Finite co e fficie n t theories on C ∗ -algebra s are a lready considered in [14]. The ex amples we consider in this article only inv o lve the simple sp ecia l ca se R = Z . W e w ork in the mor e general situation described ab ov e b ecause the definitions and pr o ofs are all literally the sa me as in the special case R = Z . The exact seque nc e (1.1) allows us to split a pro blem concer ning T into tw o problems concer ning T [ S − 1 ] and T ( A, B ; S − 1 R/R ). The firs t of these tw o may be consider ably simpler, for suitable choice o f S ; the latter inv olves only torsio n R -mo dules with the possible torsion controlled by S . As an illustratio n, we consider the Ba um– Connes assembly map with c o efficien ts. Another example is the compa rison b etw een r eal and complex K a sparov theo ry . Let A and B b e real C ∗ -algebra s and let A C : = A ⊗ R C , B C : = B ⊗ R C , then the results in [13] yield a na tural is omorphism KK G n ( A C , B C ; Z [1 / 2]) ∼ = KKO G n ( A, B ; Z [1 / 2]) ⊕ KKO G n − 2 ( A, B ; Z [1 / 2]) . W e plan in a forthco ming article to study the situation wher e G is a finite group and S is the set of prime n um be r s dividing the order of G . It se ems plau- sible that there should be a tractable Universal Co efficient Theorem computing KK G ∗ ( A, B ; S − 1 Z ) for this choice o f S for a larg e class of gr oups. Then the long exact sequence (1.1) reduces the computatio n of KK G ∗ ( A, B ) to th at of the t orsion theory KK G ∗ ( A, B ; S − 1 Z / Z ). LOCALISA TION AND COLOC ALISA TION 3 Finally , we consider the exponent of KK G ∗ ( A, B ; Z /q ) for q ∈ N ≥ 2 . F or complex C ∗ -algebra s, it is ea sy to see that q · KK G ∗ ( A, B ; Z /q ) = { 0 } for all q . In the re al case, we s how q · KKO G ∗ ( A, B ; Z /q ) = { 0 } for odd q and 2 q · KKO G ∗ ( A, B ; Z /q ) = { 0 } for even q . W e remark without pr o of that the latter easy res ult is not optimal: if 4 | q , then q · K KO G ∗ ( A, B ; Z /q ) = { 0 } as well, by an analo g ue of a Theorem by Browder, Karoubi and L a mbre for a lgebraic K-theo ry ( see [4 , 10]). 2. Central localisa tion and colo cal isa tion Cent ral lo calisa tio n in the setting of tensor tria ngulated ca tegories is already studied b y Paul Balmer [2] and Ivo dell’Am broglio [6]. Here we define it using a tensor pro duct, and we check that tw o natur al definitions ar e equiv alen t. One of them s hows that the lo ca lisation is again a triangulated category . F o llowing [8], we then in tr o duce central c o lo ca lis ations and describe them more explicitly . W e also co nsider homo lo gy with finite co efficients, ass uming the natur a l- it y of certain cones. Let R b e a commutativ e unital ring and let S be a multiplicativ ely closed s ubset of R . Let S − 1 R denote the lo calisation of R at S (see [1]). This is a unital ring equipp e d with a natural unit al r ing ho momorphism i S : R → S − 1 R . T he following constructions o nly dep end on t his ring extension and not on the c ho ice o f S . Let T b e a n R -linea r triangulated category , t hat is, each morphism space in T is an R -mo dule and comp osition of morphisms is R - linear. 2.1. The lo calisatio n. Definition 2.1. The lo c alisa tion o f T at S is the S − 1 R -linear additive cate- gory S − 1 T with morphism spaces S − 1 T ( A, B ) : = T ( A, B ) ⊗ R S − 1 R and the obvious comp os itio n. The na tural map i S : R → S − 1 R induces an R - linear functor T → S − 1 T . Example 2.2 . Any tria ng ulated category is Z -linea r. Le t S 0 be a set of prime num- ber s and let S b e the set of all natural nu mbers whose prime factor decompositio n contains only primes in S 0 . This yields a lo calisation at the set S 0 of prime num b ers. In par ticular, w e ma y loca lise a t a single prime n um ber . F o r instance, if S 0 is the set of all primes , then S − 1 Z = Q and S − 1 T is a ratio nal version of T . Example 2.3 . Recall that the centre Z C of an additive category C is the set of natural tra nsformations id C ⇒ id C . That is, an element o f the c e n tre is a family o f maps φ x : x → x for all o b jects x that is central in the sense that f ◦ φ x = φ y ◦ f for all morphisms f : x → y . The cen tr e is a comm uta tive unital ring in a natural wa y , and C is Z C -linear. An R -linea r structure o n C is equiv alent to a unital ring homomorphism fr o m R to Z C . Example 2.4 . Let G be a compact metr isable group a nd let R = Rep( G ) be its repre- sentation ring. Let K K G denote the G -equiv ariant Kas parov categ ory . Reca ll that an ob ject of K K G is a separ able C ∗ -algebra s with a strong ly contin uous action of G ; the mor phism spaces in KK G are the G -equiv aria nt biv aria nt K-groups K K G ( A, B ) defined by Gennadi Kaspar ov [1 1]. The ring Rep( G ) is isomo rphic to KK G ( C , C ) and a c ts on KK G ( A, B ) by exterior product. Thus KK G is Rep( G ) -linear. 4 HVEDRI INASSARIDZE, T AMAZ KANDE LAKI, AND RALF ME YER It is conc e iv able that the centre o f KK G is is omorphic to Rep( G ), but this seems difficult to prov e, even for the trivia l group G , b ecause we know v ery little a b o ut the algebraic pr op erties of KK b eyond the bo otstrap category . Example 2.5 . Let T be a tenso r triangulated categor y with tensor unit 1 . Then R : = T ( 1 , 1 ) is a commutativ e ring that a c ts on T ( A, B ) for a ll A, B by exterio r pro duct, so that T bec o mes R -linear . This situation is co ns idered in [2 , 6] and contains Example 2 .4 as a sp ecial ca s e. The following results gener a lise [2, Theorem 3.6] and [6, Theo r em 2 .3 2] for this sp ecial case. Our next goal is to r ealise S − 1 T as a lo c a lisation of T . This will also show that S − 1 T inher its from T a tr ia ngulated catego r y structur e. As a pr eparation, we describ e S − 1 R as a filter ed colimit. Definition 2.6. Let C S be the categ ory whose o b ject set is S and whose mo rphism space from s to t is the s et of all u ∈ S with su = t . The comp ositio n of morphis ms in C S is the m ultiplication in S . Lemma 2. 7. The c ate gory C S is filter e d. Map s ∈ S t o the r ank-1 fr e e R -mo du le R and the morphism u : s → t in C S to the map R → R , r 7→ u · r . This defines an inductive s yst em of R -mo dules with c olimit S − 1 R . L et M b e an R -mo dule. If we map s ∈ S t o M and the m orphism u : s → t t o t he map M → M , f 7→ u · f , then we get an inductive system with c olimit S − 1 M = S − 1 R ⊗ R M . Pr o of. If s and t are ob jects of C S , then the ob ject st domina tes b oth beca use of the m orphisms t : s → st and s : t → st . Tw o mor phisms u , u ′ : s ⇒ t are equalised by s : t → st because su = t = su ′ if u , u ′ bo th m ap s to t . Thus C S is directed. It is clear tha t our prescriptions ab ov e yield inductive systems indexed by C S . Elements of the colimit of this sys tem are equiv alence classes of pairs ( a, s ) with a ∈ R (or a ∈ M ), s ∈ S , and ( a, s ) and ( b, t ) r epresent the same element in the colimit if there exis t s ′ , t ′ ∈ S with ss ′ = tt ′ and as ′ = bt ′ . This equiv a lence rela tion on M × S for an R -mo dule M des crib es t he localisa tio n S − 1 M (see [1]).  Definition 2.8. A morphism f in T ( A, B ) is called an S -e quivalenc e if there are g , h ∈ T ( B , A ) and s, t ∈ S such that g ◦ f = s · id A and f ◦ h = t · id B . Lemma 2. 9. A morphism f in T ( A, B ) is an S -equiv a lence if and only if t her e ar e g ∈ T ( B , A ) and s ∈ S with g ◦ f = s · id A and f ◦ g = s · id B . Pr o of. If g, h, s, t a r e a s in Definition 2 .8, then tg = g ◦ f ◦ h = sh, ( tg ) ◦ f = st · id A , f ◦ ( sh ) = s t · id B . Thu s tg = sh ∈ T ( B , A ) and st ∈ S will do .  Definition 2. 10. An ob ject A o f T is called S -fin ite if s · id A = 0 fo r some s ∈ S . Prop ositio n 2.1 1. L et A f − → B → C → A [1] b e an ex act t riangle in T . Then the fol lowing ar e e quivalent: (1) the morphism f is an S -e quivalenc e; (2) the morphism f b e c omes invertible in S − 1 T ; (3) the obje ct C is S -finite; (4) the obje ct C b e c omes a zer o obje ct in S − 1 T . LOCALISA TION AND COLOC ALISA TION 5 Pr o of. (1) ⇐ ⇒ (2): Since the morphism spaces in S − 1 T ar e S − 1 R -mo dules and elements of S b eco me inv ertible in S − 1 R , (1) implies that f be c omes invertible in S − 1 T . Conv er sely , suppo s e that f b ecomes in vertible in S − 1 T . Lemma 2.7 shows that its inv er se is o f the form s − 1 g for some g ∈ T ( B , A ), s ∈ S . The relation f · ( s − 1 g ) = id means that f · ( u g ) = us for so me u ∈ S . Similarly , we g et ( v g ) · f = v s for so me v ∈ S . Hence f is an S -equiv alence, so that (2) implies (1). (3) ⇐ ⇒ (4): The ob ject C b e comes zero in S − 1 T if and only if the zero map on C b ecomes in v ertible in S − 1 T . Since there is only one map on t he zero o b ject, the inv er se ma p in S − 1 T m us t be the imag e of id C . This means that s · id C = s · 0 for s ome s ∈ S . Thus (3 ) ⇐ ⇒ (4). (2) ⇐ ⇒ (4): Since S − 1 R is a fla t R -mo dule, the functor S − 1 T ( D, ) is ho molog- ical for any o b ject D of T . By the Y oneda Lemma, f b ecomes inv er tible in S − 1 T if and only if S − 1 T ( D, f ) is inv ertible for all D . By the long exact se q uence fo r the exact triangle A → B → C → A [1] a nd the homolog ical functor S − 1 T ( D , ), this is equiv alent to S − 1 T ( D, C ) = 0 f or all D . And this is equiv alen t to C ∼ = 0 in S − 1 T by a nother application of the Y oneda Lemma.  Corollary 2.12. The class N S of S -finite obje cts is a thick sub c ate gory of T . Pr o of. It is clear that N S is clo sed under isomorphism and susp ensio n. It is c lo sed under dir ect summands because the functor T → S − 1 T is additive. If B and C in an exact triangle A → B → C → A [1] a r e S -finite, then B ∼ = 0 in S − 1 T and the map A → B b ecomes an isomorphism in S − 1 T by Pr op osition 2.11. Hence A ∼ = 0 in S − 1 T , s o that A is S -finite b y Prop osition 2.1 1.  Theorem 2.13. The c ate gory S − 1 T to gether with the functor T → S − 1 T is the lo c alisation of T at the thick sub c ate gory N S . Pr o of. It follo ws fr om the results above and the s ta ndard description of the locali- sation that the lo calisation of T at N S is the category theoretic localisa tion a t the class of S -equiv alences ( see [7]). Let W b e the set of all mo r phisms in T of the form s · f with s ∈ S and an inv ertible morphism f in T . It is easy to see that W has b oth left a nd rig ht ca lculi of fra c tions with lo c alisation S − 1 T . It remains to show that the categor y theor etic loc a lisations of T at W and at the class of a ll S - equiv alences are equal. That is, a functor defined on T maps all elements o f W to isomorphisms if and only if it maps all S -equiv alences to isomor- phisms. One direction is clear becaus e W consists of S -equiv alences. Conv ersely , if f is an S -equiv alence , then there are morphisms g and h suc h tha t f g and hf belo ng to W . Hence a functor that maps W to isomorphisms maps f g and hf to isomorphisms. The n it maps f to an iso morphism as well.  Definition 2. 14. L et F : T → Ab be a homolo gical functor. Its lo c alisation at S is the functor S − 1 F : T → A b , S − 1 F ( A ) : = F ( A ) ⊗ R S − 1 R. Since S − 1 R is a flat R -mo dule, the functor S − 1 F is again homo lo gical. By definition, the loca lisation of the homolo gical functor T ( A, ) for an ob ject A of T agrees with S − 1 T ( A, ). Prop ositio n 2.15. The functor S − 1 F is the lo c alisation of F with r esp e ct to the thick sub c ate gory of S -finite obj e cts. 6 HVEDRI INASSARIDZE, T AMAZ KANDE LAKI, AND RALF ME YER Pr o of. The lo calisation of F at the S - finite ob jects maps an ob ject A of T to the colimit of F ( B ) f , where f runs throug h the directed set of S -equiv alences f : A → B (see also [8]). As in the pro o f of Theorem 2.13, we may re place S -equiv alences by s · id A for s ∈ S , where we us e the filtered categ ory C S int ro duced in Definition 2.6. The colimit ov er C S agrees with S − 1 F ( A ) by Lemma 2.7.  2.2. The colo calisation. F ollowing [8], we embed the functor T → S − 1 T into an exact sequence (2.1) · · · → T 1 ( A, B ) → S − 1 T 1 ( A, B ) → T 1 ( A, B ; S − 1 R/R ) → T 0 ( A, B ) → S − 1 T 0 ( A, B ) → T 0 ( A, B ; S − 1 R/R ) → T − 1 ( A, B ) → · · · . In the notation of [8], S − 1 T ( A, B ) : = T / N S ( A, B ) and T n +1 ( A, B ; S − 1 R/R ) : = ( T / N S ) ⊥ n ( A, B ), where N S denotes the c la ss o f S - finite o b jects. The de g ree shift is natural if we think of T ∗ ( A, B ) and S − 1 T ∗ ( A, B ) as T with co efficients in R and S − 1 R . More gener a lly , if F : T → A b is a homo logical functor, then the map F → S − 1 F embeds in an exa ct sequence (2.2) · · · → F 1 ( A ) → S − 1 F 1 ( A ) → F 1 ( A ; S − 1 R/R ) → F 0 ( A ) → S − 1 F 0 ( A ) → F 0 ( A ; S − 1 R/R ) → F − 1 ( A ) → S − 1 F − 1 ( A ) → · · · with F n +1 ( A ; S − 1 R/R ) : = R ⊥ n F ( A ) in the notation of [8 ]. Prop ositio n 2.16. A natur al tr ansformation Φ : F ⇒ G is invertible if and only if b oth its c olo c alisation Φ( ; S − 1 R/R ) : F ( ; S − 1 R/R ) ⇒ G ( ; S − 1 R/R ) and its lo c alisation S − 1 Φ : S − 1 F ⇒ S − 1 G ar e invertible. Pr o of. This is [8, Corollary 4.4].  The last r esult hints a t an a pplica tion of colo ca lisation and lo calisa tion: they break up computations in T into tw o hop efully simpler co mputatio ns. W e will return t o this po int in some examples later. Prop ositio n 2.17. F or an obje ct A of T and a homolo gic al fun ct or F : T → Ab , we have T o r R n ( F ∗ ( A ); S − 1 R/R ) = 0 for al l n ≥ 2 , and ther e is a n atur al gr oup extension T o r R 0 ( F 0 ( A ); S − 1 R/R ) ֌ F 0 ( A ; S − 1 R/R ) ։ T or R 1 ( F − 1 ( A ); S − 1 R/R ) . Pr o of. Since R and S − 1 R are flat R -mo dules, the defining exa ct seque nc e · · · → 0 → R → S − 1 R → S − 1 R/R is a flat resolution of S − 1 R/R . W e use this resolution to compute T or R n ( ; S − 1 R/R ). This yields the v anishing for n ≥ 2 a nd (2.3) T o r R 1 ( M ; S − 1 R/R ) = ker( M → S − 1 M ) , T o r R 0 ( M ; S − 1 R/R ) = coker( M → S − 1 M ) . The iso morphisms in (2.3) and the ex act se q uence (2.2 ) fin ish the pro of.  Prop ositio n 2.17 justifies our notation F ( A ; S − 1 R/R ): we expect ex actly such an extens io n for a homolo gy theory with co efficients S − 1 R/R . The same re mark applies to T ( A, B ; S − 1 R/R ) b ecause it is a sp ecia l cas e of F ( B ; S − 1 R/R ). LOCALISA TION AND COLOC ALISA TION 7 It f ollows directly from the definition of S − 1 M that (2.4) ker( M → S − 1 M ) = [ s ∈ S ker( M s − → M ) , coker ( M → S − 1 M ) = lim − → s ∈∈C S coker ( M s − → M ) , where s also deno tes the endomor phism m 7→ s m on M . These endo morphisms form a n inductive sy stem indexed b y C S bec ause o f t he comm uting diagrams M t / / M M s / / M u O O for arrows u : s → t in C S . This explains the colimit o f coker ( M s − → M ) for s ∈ C S . Of co ur se, co ker ( s : M → M ) = M /sM . 2.3. Homo l ogical functors with co e fficien ts. Can we describ e F ( ; S − 1 R/R ) as a filtered co limit, in a nalogy to (2.4) ? More precisely , given s ∈ S , is ther e a homologica l functor A 7→ F ( A ; s ) tha t fits in to an extensio n (2.5) coker  s : F 0 ( A ) → F 0 ( A )  ֌ F 0 ( A ; s ) ։ ker  s : F − 1 ( A ) → F − 1 ( A )  , and is F ( A ; S − 1 R/R ) a filter ed colimit of F ( A ; s ) in a natural way? If s is not a zero divisor in R , then (2.5) is the exp ected b ehaviour for F with co efficients in the R -mo dule R /sR ; if s is a zer o divisor, then we should view F ( A ; s ) as a theory with c o efficients in the c hain complex R s − → R concentrated in degrees 1 and 0. Let A s be the cone of the map s · id A : A → A . Then the long exact homology sequence f or F a pplied to the defining exact triangle (2.6) A s − → A → A s → A [1] yields an extension as in (2.5). This sug g ests to define F 0 ( A ; s ) as F ( A s ). How ev er, without further assumptions it is no t clear whether the construction of A s is nat- ural, that is, whether F ∗ ( A s ) is functoria l in A . T o pro ceed, we therefore assume that (2.6) is the obje ct-p art of a functor fr om T to t he c ate gory of exact triangles in T , such that the r esulting functor A 7→ A s is triangulate d. Then F ( A ; s ) : = F ( A s ) defines a homologica l functor on T that fits in to na tural exact sequences as in (2 .5). R emark 2.18 . If T is a tensor triangula ted categ ory and R ⊆ T ( 1 , 1 ), then the ab ov e as sumption is a utomatic b eca use we get (2.6) b y tensoring the triang le 1 s − → 1 → 1 s → 1 [1 ] by A . Lemma 2.19. L et F ′ ∗ ( A ) : = F ∗ ( A ; S − 1 R/R ) . Then the n atur al tra nsformation F ′ ∗ +1 ( A ) → F ∗ ( A ) induc es natur al isomorph isms F ′ ∗ +1 ( A ; s ) ∼ = F ∗ ( A ; s ) for s ∈ S . Pr o of. Since A s is the cone of the S -equiv a lence s : A → A , it is S -finite. Hence F ∗ ( A s ; S − 1 R ) = 0. B y the lo ca lisation–colo calisation exact sequence, this implies an isomor phism F ∗ +1 ( A s ; S − 1 R/R ) ∼ = F ∗ ( A s ). This is equiv ale nt to the as s ertion.  8 HVEDRI INASSARIDZE, T AMAZ KANDE LAKI, AND RALF ME YER Lemma 2.19 and (2 .5) yield a sho rt ex act se q uence (2.7) coker  s : F ∗ +1 ( A ; S − 1 R/R ) → F ∗ +1 ( A ; S − 1 R/R )  ֌ F ∗ ( A ; s ) ։ ker  s : F ∗ ( A ; S − 1 R/R ) → F ∗ ( A ; S − 1 R/R )  , Next we w ant to wr ite F ( A ; S − 1 R/R ) as a filtered colimit of the finite co efficient theories F ( A ; s ). Given s, t ∈ S and u ∈ C S ( s, t ), that is, t = u · s , we ma y find a dotted arrow ϕ that mak es the following diagr am commute: (2.8) A s / / A / / u   A s / / ϕ   A [1] A t / / A / / A t / / A [1] W e let C S ( A ; s, t ) be the set of all suc h pair s ( u, ϕ ). Lemma 2.20. The c ate gory with obje ct set S and morphism s p ac es C S ( A ; s, t ) is filter e d. Pr o of. Any tw o ob jects in C S ( A ; s, t ) are dominated b y another ob ject because C S is filtered and an arrow ϕ as ab ov e exists for a ny u . Let ( u 1 , ϕ 1 ) and ( u 2 , ϕ 2 ) b e t wo parallel a rrows in C S ( A ; s, t ). In the co nstruction o f the lo calis ation–colo c aliation exact sequence in [8], it is shown that w e may equalise ( u 1 , ϕ 1 ) and ( u 2 , ϕ 2 ) by a morphism of ex act tria ngles A t / / A / / g   A t / / ψ   A [1] A tg / / A / / A tg / / A [1] where g is an S -equiv alence. Hence there is g ′ such that g ′ g = v ∈ R . W e may embed g ′ in a mo rphism o f exact triang le s ( g ′ , ψ ′ ). Then the composite morphism ( v , ψ ′ ◦ ψ ) be longs to C S ( A ; t, tv ) and equalises ( u 1 , ϕ 1 ) and ( u 2 , ϕ 2 ). Thus the category C S ( A ) is filtered.  Prop ositio n 2.21. The c olimit of the inductive syst em of extensions (2.5) is the extension in Pr op osition 2.17 , T o r R 0 ( F 0 ( A ); S − 1 R/R ) ֌ F 0 ( A ; S − 1 R/R ) ։ T or R 1 ( F − 1 ( A ); S − 1 R/R ) . Pr o of. Recall that the lo calis ation–colo ca lisation exact sequence in [8] is constructed as a colimit over the filtered categor y of all triangles N → A → B → N [1] where the map A → B is an S -equiv ale nce . W e g et the same colimit if we use the filt ered category with ob ject s et S and morphisms C S ( A ; s, t ) b ecause the arr ows s : A → A with s ∈ S are cofinal in the filtered category of S -eq uiv alences A → B . This o bser- v ation is alre a dy implicit in the pro of of Lemma 2.20. Ther efore, the colimit of the long exact s e quences from the ab ove filtered sub ca tegory yields the lo calisa tion– colo calisa tion long exact sequenc e . Splitting this into extensions then yields the assertion of the prop osition.  R emark 2.22 . The filtered category C S ( A ) with morphism spa ces C S ( A ; s, t ) used ab ov e may b e rather large b ecause w e a llow all pair s ( u, ϕ ) making (2 .8) co mmute. In nice cases, we exp ect a canonical c hoice ϕ u for each u , coming fro m a natural LOCALISA TION AND COLOC ALISA TION 9 cone constructio n in some ca tegory whose ho motopy categ ory is T . Even more, this natural choice should define a functor C S → C S ( A ), u 7→ ( u, ϕ u ). If thes e nice things happ en, then w e may r eplace t he filtered c a tegory C S ( A ) abov e by the filtered sub ca tegory C S bec ause c olimits over C S and C S ( A ) ag ree. Lemma 2. 23. Multiplic ation by s 2 vanishes on F ∗ ( A ; s ) . Pr o of. It is clear that multiplication by s v anis he s on the cokernel and k ernel of multiplication by s on F ∗ ( A ). Hence the ass ertion follows from the ex act se- quence (2.5) .  R emark 2.24 . Sometimes the extension (2.5 ) splits unnaturally (see, for instance, [14]). In those ca ses, already mult iplication by s v anishes on F ∗ ( A ; s ). Lemma 2. 25. L et s, t ∈ S . Then ther e is a long exact se quenc e · · · → F 0 ( A ; s ) → F 0 ( A ; st ) → F 0 ( A ; t ) → F − 1 ( A ; s ) → F − 1 ( A ; st ) → F − 1 ( A ; t ) → · · · , which is natur al in F with r esp e ct t o natur al tr ansformations. Pr o of. The o ctahedra l axio m for the maps s, t, st : A → A yields a commuting diagram w ith exact r ows a nd columns A s / / A t   / / A s / /   A [1] A st / /   A / /   A st   / / A [1]   0   / / A t   A t / /   0   A [1] s [1] / / A [1] / / A s [1] / / A [2] . Here we us e the uniqueness of cones to iden tify the o b jects A s , A t , and A st in the diagram. Apply the functor F to the third r ow to get the asser ted long exact sequence f or homology with coefficients.  Corollary 2.26. L et Φ : F ⇒ G b e a natur al tr ansformation b etwe en t wo homolo g- ic al functors on T . L et S 0 ⊆ S b e a gener ating subset, that is, any ele ment of S is a pr o duct of elements in S 0 . Then Φ induc es an invertible map F ( A ; S − 1 R/R ) → G ( A ; S − 1 R/R ) for al l A if and only if Φ induc es invertible maps F ( A ; s ) → G ( A ; s ) for al l s ∈ S 0 and al l obje cts A of T . Pr o of. Assume first that the maps Φ A,s : F ( A ; s ) → G ( A ; s ) are inv ertible for a ll s ∈ S 0 . Let S 1 be the set o f a ll s ∈ S fo r whic h Φ A,s is in vertible. Lemma 2.25 and the Five Lemma s how that S 1 is closed under multiplication. Hence Φ A,s is inv ertible for all s ∈ S . The n Prop osition 2.21 sho w s that Φ induces an in vertible map F ( A ; S − 1 R/R ) → G ( A ; S − 1 R/R ). 10 HVEDRI INASSARIDZE, T AMAZ KANDE LAKI, AND RALF ME YER F o r the c o nv er se implication, L e mma 2 .19 a llows us to replace Φ by the induced natural tra nsformation Φ ′ betw een the functors F ′ ∗ ( A ) : = F ∗ +1 ( A ; S − 1 R/R ) , G ′ ∗ ( A ) : = G ∗ +1 ( A ; S − 1 R/R ) bec ause Φ and Φ ′ induce equiv alent maps b etw een finite co efficie n t theories. If Φ ′ is an isomorphism fo r a ll A , it is one for A s , so that Φ A,s is in vertible.  3. Applica tion to Kasp aro v theor y Now we apply the g eneral theory developed ab ove to eq uiv ariant Kaspar ov theory , viewed as a triang ulated category (see [12]). W e only consider central lo calis ations where R is the r ing Z of integers, as in Example 2.2. Finer information may b e obtained b y considering the larger ring Rep( G ) instead (see Example 2.4) , but w e leav e this to f uture in vestigation. Let us fir st consider the r ational KK G -the ory . Here S = Z \ { 0 } and S − 1 Z = Q . F o llowing the definitions a b ove, we define (3.1) KK G n ( A, B ; Q ) : = KK G n ( A, B ) ⊗ Q , where A and B are G -C ∗ -algebra s. This differs from the de finitio n in [3, Exercis e 2 3.15.6], wher e ratio nal KK-theory for complex C ∗ -algebra s is defined to b e K K n ( A, B ⊗ D Q ) for a C ∗ -algebra D Q in the bo otstrap cla ss with K 0 ( D Q ) = Q and K 1 ( D Q ) = 0 . The example KK 0 ( D Q , C ⊗ D Q ) ∼ = Q and KK 0 ( D Q ; C ) ⊗ Q = 0 shows that the tw o theories are different. In KK( , ; Q ), the C ∗ -algebra s C a nd D Q are no t isomor phic b eca us e ther e is no mo rphism from D Q to C . The map C → D Q corres p o nding to the embedding Z → Q is not a n S -equiv alence . Its co ne is the susp ension of a C ∗ -algebra D Q / Z with K 0 ( D Q / Z ) = Q / Z and K 1 ( D Q / Z ) = 0 . This C ∗ -algebra is not S -finite, so that KK 0 ( D Q / Z , D Q / Z ; Q ) 6 = 0 . But D Q / Z ⊗ D Q ∼ = 0 in KK bec a use this tens or pr o duct b elongs to the b o otstra p ca tegory and its K-theo ry K ∗ ( D Q / Z ⊗ D Q ) ∼ = Q / Z ⊗ Q v anishes. The definition of K K( A, B ; Q ) ab ov e yields aga in a triang ulated category . This is cr ucial to apply m etho ds fro m stable homotopy theo ry a nd homological algebra. Do e s the definition in [3] also yield a triangulated categ o ry? This seems unclear if we co ns ider all of KK ; but w e get a positive answ er in th e b o otstrap class. F or A in the bo otstr ap cla ss, the Univ ersal Co e fficient Theorem yields KK 0 ( A, B ⊗ D Q ) ∼ = Hom(K ∗ ( A ) , K ∗ ( B ) ⊗ Q ) ∼ = Hom Q (K ∗ ( A ) ⊗ Q , K ∗ ( B ) ⊗ Q ) bec ause Ab elian groups of the form K ∗ ( B ) ⊗ Q ar e injectiv e. Hence the b o o tstrap class with these mor phisms is equiv ale nt to the catego ry of countable Q -vector spaces. This catego r y is tria ng ulated and A be lia n at the same time. And we may also view it as the lo calis a tion of KK at the class o f C ∗ -algebra s with v anishing rational K -theory K ∗ ( ) ⊗ Q . But this observ a tio n dep ends on a n explicit co m- putation of the categor y . O nce w e go beyond the bo o ts tr ap category or consider equiv aria n t situations , it is no longer clear whe ther the definition in [3] y ie lds a triangulated ca tegory . More ge ne r ally , if S is a ny multiplicativ ely clos e d subset of Z , then we may replace Q b y S − 1 Z a nd define KK G ∗ ( A, B ; S − 1 Z ) = S − 1 KK G ∗ ( A, B ) : = KK G ∗ ( A, B ) ⊗ S − 1 Z . LOCALISA TION AND COLOC ALISA TION 11 By o ur general theor y , these groups form the mo rphism spaces of a n S − 1 Z -linear triangulated category . It is the lo ca lisation of KK G at the cla ss o f S -finite G -C ∗ - algebras . Here A is S -finite if a nd only if there is s ∈ S with s · id A = 0. The colo calisa tion also pro duces an S -torsion the ory KK G ∗ ( A, B ; S − 1 Z / Z ) that fits into a natural lo ng ex act seq uence (3.2) · · · → KK G n +1 ( A, B ) → K K G n +1 ( A, B ; S − 1 Z ) → K K G n +1 ( A, B ; S − 1 Z / Z ) → K K G n ( A, B ) → K K G n ( A, B ; S − 1 Z ) → K K G n ( A, B ; S − 1 Z / Z ) → · · · . In par ticular, t his includes a torsion the ory KK G ∗ ( A, B ; Q / Z ). The S -ra tional and S -torsion theories inherit basic proper ties like homoto py in- v ariance , C ∗ -stability , excision and Bo tt p erio dicity from KK G . All this is co ntained in the statement that they are bifunctors on KK G that ar e ho mological in the first and cohomological in the s econd v ariable. F ur thermore, the maps in (3.2) ar e nat- ural transformations. Since the S -rational theory is a gain a triangulated category , we g et an asso ciative pr o duct KK G n ( A, B ; S − 1 Z ) ⊗ S − 1 Z KK G m ( B , C ; S − 1 Z ) → K K G n + m ( A, C ; S − 1 Z ) . It is the pro duct of the natural isomor phis m (KK G n ( A, B ) ⊗ S − 1 Z ) ⊗ S − 1 Z (KK G n ( B , C ) ⊗ S − 1 Z ) ∼ =  KK G n ( A, B ) ⊗ KK G n ( B , C )  ⊗ S − 1 Z and the homomorphism KK G n ( A, B ) ⊗ K K G n ( B , C ) ⊗ S − 1 Z → KK G n ( A, C ) ⊗ S − 1 Z induced b y the Kaspa rov pro duct. As for t he rational theory , the torsion KK-theory K K( A, B ; Q / Z ) is, in general, not iso morphic t o K K ∗ ( A, B ⊗ D Q / Z ). As in Section 2.3, we now rea lis e the S -torsion theory as a filtered colimit of finite c o efficient Kaspar ov theorie s for s ∈ S . These functors hav e alrea dy b een studied by Claude Schochet [14]. This is a nice case in the sense of Remark 2 .22, that is, w e get canonical na tural tr ansformatio ns A s ⇒ A t for s , t ∈ S with s | t . Let S 1 be the unit circle in C with base point ∗ = 1 ∈ S 1 . The ma p ˜ q : S 1 → S 1 , x 7→ x q , for q ∈ N ≥ 2 is called standar d q t h p ower map . It pr eserves the base p o int. Let C 0 ( S 1 ) ∼ = C 0 ( R ) b e the C ∗ -algebra o f co ntin uous functions on S 1 v anishing at ∗ . The map ˜ q induces a ∗ -ho momorphism ˆ q : C 0 ( S 1 ) → C 0 ( S 1 ) , f ( s ) 7→ f ( s q ) . Let C q be the mapping cone C ∗ -algebra o f ˆ q : C q : = { ( x, f ) ∈ C 0 ( S 1 ) ⊕ C 0 ( S 1 ) ⊗ C 0 [0 , 1) | ˆ q ( x ) = f (0 ) } . Up to a desusp ensio n, w e ma y define the finite coefficient theories b y A q : = A ⊗ C q [ − 2] , F ∗ ( A ; q ) : = F ∗− 2 ( A ⊗ C q ) . In par ticular, w e define KK G n ( A, B ; Z /q ) : = KK G n − 2 ( A, B ⊗ C q ) , 12 HVEDRI INASSARIDZE, T AMAZ KANDE LAKI, AND RALF ME YER In the co mplex case, the P upp e exact se q uence and Bott p er io dicity imply K 0 ( C q ) ∼ = Z /q and K 1 ( C q ) ∼ = 0. F ur thermore, C q belo ngs to the b o o tstrap class and is the unique o b ject o f the bo otstra p cla s s with this K-theor y . By definition, C q fits in to a pull-ba ck diagr a m C q / / q 0   C 0 ( S 1 ) ⊗ C 0 [0 , 1) e 0   C 0 ( S 1 ) ˆ q / / C 0 ( S 1 ) . If q divides p , then w e g et a natural co mmuting diagr am C q / / Θ p,q   C 0 ( S 1 ) ˆ q / / C 0 ( S 1 ) d p/q   C p / / C 0 ( S 1 ) ˆ p / / C 0 ( S 1 ) W e get Θ p,r Θ r,q = Θ p,q if q | r | p by [14 , Pr op osition 2.1 (3)]. This provides a functor C S → K K G , s 7→ A s , p/q 7→ Θ p,q . W e get a natural isomor phism KK G n ( A, B ; S − 1 Z / Z ) ∼ = lim − → C S KK G n ( A, B ; Z /s ) , that is, we may replace the filtered ca tegory C S ( A ) that is used in Section 2.3 by the m uc h smaller categor y C S . F urthermore, it can b e chec k ed that the maps F ∗ ( A ; s ) → F ∗ ( A ; st ) in Lemma 2.25 may b e chosen to be the maps induced by Θ s,st . The finite co efficient theory is related to KK G by a natural ex act se quence · · · → KK G n ( A, B ) q − → K K G n ( A, B ) → K K G n ( A, B ; Z /q ) → K K G n − 1 ( A, B ) q − → KK G n − 1 ( A, B ) → K K G n − 1 ( A, B ; Z /q ) → · · · for q ∈ S b y (2.5); here q denotes the ma p of mult iplication by q . Simila rly , (2.7) yields a natural exact sequence · · · → KK G n ( A, B ; Z /q ) → KK G n ( A, B ; S − 1 Z / Z ) q − → K K G n ( A, B ; S − 1 Z / Z ) → KK G n − 1 ( A, B ; Z /q ) → KK G n − 1 ( A, B ; S − 1 Z / Z ) q − → K K G n − 1 ( A, B ; S − 1 Z / Z ) → · · · . Lemma 2 .25 provides natura l exac t sequences · · · → KK G n ( A, B ; Z /p ) → KK G n ( A, B ; Z /pq ) → KK G n ( A, B ; Z /q ) → K K G n − 1 ( A, B ; Z /p ) → K K G n − 1 ( A, B ; Z /pq ) → KK G n − 1 ( A, B ; Z /q ) → · · · for all p, q ∈ Z , see also [14, Prop osition 2.6.(1)]. The maps in this sequence come from a natural ex act tria ngle C p → C pq → C q → C p [1] The exp onent of a group G is the smallest a ∈ N ≥ 1 with a · g = 0 for all g ∈ G . At first s ight, it ma y seem pla us ible that the expo nen t of KK G ∗ ( A, B ; Z /q ) should divide q . Ther e is, howev er , an obstruction to this for rea l C ∗ -algebra s. The issue is discussed in [1 4, Prop osition 2.4]. Complex KK-theory is goo d in the sense of [14] b ecaus e the KK-class o f the H opf map S 3 → S 2 belo ngs to the group KK 0  C 0 ( R 2 ) , C 0 ( R 3 )  = 0 . As a conseq uence, the exact sequence (2.5) for com- plex KK-theory splits unnaturally a nd, in particular , multiplication b y q v anishes LOCALISA TION AND COLOC ALISA TION 13 on KK G ∗ ( A, B ; Z /q ) for a ll q ∈ Z . This a lso follows easily fr o m the isomorphism KK 0 ( C q , C q ) ∼ = Z /q , whic h follows from the Universal Co efficient Theor e m. Now c onsider real C ∗ -algebra s, denote their KK-theo r y by K KO . The extension in (2.5) for K KO G ∗ ( A, B ) is equiv alen t to an extension (3.3) KKO G ∗ ( A, B ) ⊗ Z /q ֌ KKO G ∗ ( A, B ; Z /q ) ։ T or(KKO G ∗− 1 ( A, B ) , Z / q ) bec ause H ⊗ Z /q ∼ = coker( q : H → H ) , T or( H, Z /q ) ∼ = ker( q : H → H ) . The ab ov e extension splits unnaturally for o dd q by [14, P rop osition 2.4], s o that KKO G ∗ ( A, B ; Z /q ) has exp o ne nt q . But since the class of the Hopf map is the generator of KKO 0  C 0 ( R 2 ) , C 0 ( R 3 )  ∼ = Z / 2, the ho mology theory KKO G ∗ ( A, ) is not go o d. Hence the ab ov e extens ion need not split for even q . Instea d, general arguments from homoto p y theory show that the exp onent o f K KO G ∗ ( A, B ; Z /q ) divides 2 q for all even q , and it divides q if 4 | q . W e o nly e x plain why 2 q a nnihilates KKO G ∗ ( A, B ; Z /q ) for even q ; the str onger a ssertion for 4 | q is more difficult. W e assume that q is even. Since KKO 0 ( C q , C q ) = K KO 0 ( C q , R ; Z /q ), Equa- tion (3.3 ) y ields exa ct seq uences 0 → K KO 0 ( C q , R ) ⊗ Z /q → KKO 0 ( C q , C q ) → T o r(KKO − 1 ( C q , R ) , Z /q ) → 0 for all q > 1. Since KKO 1 ( R , R ) = Z / 2, KKO 0 ( R , R ) = Z and KKO − 1 ( R , R ) = 0, we g et a long exac t sequence · · · → Z / 2 q − → Z / 2 → KKO 0 ( C q , R ) → Z q − → Z → KKO − 1 ( C q , R ) → 0 → · · · Since q : Z → Z is a monomor phism, we hav e exact sequences 0 → Z q − → Z → KKO − 1 ( C q , R ) → 0 and Z / 2 q − → Z / 2 → KKO 0 ( C q , R ) → 0 Thu s KKO − 1 ( C q , R ) = Z /q and KKO 0 ( C q , R ) = Z 2 bec ause q is even. P lugging this into the UCT ex act se quence yie lds 0 → Z / 2 ⊗ Z / q → K KO 0 ( C q , C q ) → T o r( Z /q , Z /q ) → 0 . W e have T or( Z /q , Z /q ) = Z /q and Z / 2 ⊗ Z /q ∼ = Z / 2 because q is even. Therefore , KKO 0 ( C q , C q ) is annihilated b y 2 q . Since KKO G ( A, B ; Z /q ) is a mo dule over KKO 0 ( C q , C q ), it is annihilated by q for o dd q and b y 2 q for o dd q . 3.1. Lo ok at the Baum–Connes C o njecture. The lo calisatio n– colo calisa tion exact se q uence allows to reduce computations in KK G to computations in the S -rational a nd S -torsio n v aria nts of KK G . The computatio n for S -torsion inv ari- ants, in turn, reduces to computations for KK G with finite c o efficients. Here we explain what this means for the Ba um–Connes conjecture. How ever, we know of no concrete applica tion where this r eduction of the problem would help to settle this conjecture . The formulation of the Baum– C o nnes Conjecture inv olv es the groups K top n ( G, A ), called the top olo gic al K -the ory of G with c o efficients in A , and a natural transfor- mation µ A : K top ∗ ( G, A ) → K ∗ ( G ⋉ r A ) , 14 HVEDRI INASSARIDZE, T AMAZ KANDE LAKI, AND RALF ME YER called the Baum–Connes assembly map . The Baum–Connes Conje ct ur e for G with co efficients in A asser ts that µ A is an isomorphis m. Note that K top ∗ ( G, ) and K ∗ ( G ⋉ r ) ar e homology theor ies of the kind studied in Section 2 .3 (see also [12]). Therefore, the constructions ab ov e yield S -rational, finite, and S -torsion versions of the Baum–Connes as sembly map: µ S − 1 Z A : K top ∗ ( G, A ; S − 1 Z ) → K ∗ ( G ⋉ r A ; S − 1 Z ) , µ Z /q A : K top ∗ ( G, A ; Z /q ) → K ∗ ( G ⋉ r A ; Z /q ) , µ S − 1 Z / Z A : K top ∗ ( G, A ; S − 1 Z / Z ) → K ∗ ( G ⋉ r A ; S − 1 Z / Z ) . Theorem 3.1. The fol lowing c onje ct ur es ar e e quivalent: (1) the Baum–Connes Conje ctu r e with c o efficients; (2) the S -r ational and S - torsion Baum–Co nnes Conje ctu r es with c o efficients; (3) the S -r ational and q -finite Baum–Connes Conje ctu r es with c o efficients for al l primes q ∈ S . Pr o of. This fo llows immediately from Prop o sition 2.16 and Coro llary 2.26.  3.2. Real v ers u s compl ex Kasparov theory . T o illustra te the usefulness of lo- calisation, we reform ulate some w ell-k nown r esults a b o ut th e relationship b etw e en real a nd co mplex K asparov theory and K-theory . Roughly sp eaking, these tw o the- ories beco me equiv alent when we lo calise at 2, tha t is, work with Z [ 1 / 2 ]-co efficients. The re s ults in this se c tion ar e due to Max Karoubi [9] and Thoma s Schic k [13]. In [1 3], Tho mas Schic k related the KK-theor ies of t w o rea l C ∗ -algebra s A and B and their complexifications A C and B C by an exact sequence (3.4) · · · → KKO Γ n − 1 ( A, B ) χ − → K KO Γ n ( A, B ) c − → KK Γ n ( A C , B C ) δ − → KKO Γ n − 2 ( A, B ) χ − → K KO Γ n − 1 ( A, B ) c − → KK Γ n − 1 ( A C , B C ) → · · · , extending previous r e s ults for real and complex K-theory , KO a nd K. In [13], Γ is assumed to b e a discr ete g roup, but the sa me arguments work if Γ is replaced by a lo cally co mpact g roup o r even gr oup oid; A and B are separa ble r eal Γ-C ∗ -algebra s; χ is giv en by Kaspa rov pr o duct with the g e ne r ator of KK O Γ 1 ( R , R ) = Z / 2; c is the complexification functor; and δ is the comp osition of the inverse of the co mplex Bott p erio dicit y iso morphism with “forgetting the complex struc tur e.” The pro of of [13, Corolla r y 2.4] shows that, a fter in verting 2, the long exact sequence abov e bec o mes a natur al ly split shor t exact sequence KKO Γ n ( A, B ) ⊗ Z [ 1 / 2 ] c ֌ KK Γ n ( A C , B C ) ⊗ Z [ 1 / 2 ] δ ։ K KO Γ n − 2 ( A, B ) ⊗ Z [ 1 / 2 ] . In our notation, this yields a natural isomorphism KK Γ n ( A C , B C ; Z [ 1 / 2 ]) ∼ = KKO Γ n ( A, B ; Z [ 1 / 2 ]) ⊕ KKO Γ n − 2 ( A, B ; Z [ 1 / 2 ]) . Besides the exactness o f (3.4), the pro o f uses tw o observ ations. First, 2 χ = 0, s o that (3.4) s plits into short exact sequences after tensoring with Z [ 1 / 2 ]. Seco ndly , the m ap KKO Γ n − 2 ( A, B ) c − → KK Γ n − 2 ( A C , B C ) ∼ = KK Γ n ( A C , B C ) δ − → KKO Γ n − 2 ( A, B ) , where the middle isomo r phism is Bott p erio dicity , is multiplication by 2 ([13, Lemma 3.9]). Hence c provides a na tural sectio n for the r esulting extensions, up to inv erting 2. More generally , th e same argument yields: LOCALISA TION AND COLOC ALISA TION 15 Theorem 3. 2. L et G b e a se c ond c oun table lo c al ly c omp act gr oup, let A and B b e sep ar able r e al G - C ∗ -algebr as. Ther e is a natur al i somorphism KK Γ n ( A C , B C ; H ) ∼ = KKO Γ n ( A, B ; H ) ⊕ K KO Γ n − 2 ( A, B ; H ) for the fol lowing c o efficients: (1) H = S − 1 Z with 2 ∈ S ( lo c alisation ) ; (2) H = Z /s Z with o dd s ( finite c o efficients ) ; (3) H = S − 1 Z / Z if S c ontains only o dd numb ers ( c olo c alisation ) . Pr o of. T ensoring (3.4) with S − 1 Z , we get an a nalogous long exa ct sequence for the S - rational theories. By assumption on S , 2 is in vertible on t he S -rational the- ory . Hence exa ctly the sa me ar g ument as for Z [ 1 / 2 ] works. Studying KK-theories with c o efficien ts in Z /s amo un ts to r eplacing B by B s : = B ⊗ R s , an o p eration that commutes with complexificatio n and the v ar ious other constructions needed for the exact sequence (3.4). Hence there is an ana lo gous exac t sequence with co ef- ficient s Z /s . Since s 2 annihilates the theory with co efficients Z /s b y Lemma 2.23, the element s 2 + 1 acts as the identit y on Z /s , and 2 is inv ertible in Z /s for o dd s . This proves the seco nd case . Finally , we write the S -torsion theory as a filtere d colimit of finite co efficient theories a s in P rop osition 2.21. Only co efficients Z /s with o dd s appea r here b y assumption. 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Alexidze Street 1, 380093 Tbilisi, Georgia E-mail addr ess : inassari@ gmail.co m E-mail addr ess : tam.kande l@gmail. com Ralf Meyer: M a them atis ches Institut and, Couran t Centre “Higher order struc- tures”, Georg-August Universit ¨ at G ¨ ottingen, Bunsenstraße 3–5, 370 73 G ¨ ottingen, Ger- many E-mail addr ess : rameyer@u ni-math. gwdg.de