Homological algebra in bivariant K-theory and other triangulated categories. II

HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y AND OTHER TRIANGULA T ED CA TEGORIES. I I RALF MEYER Abstrac t. W e use homological ideals i n triangulate d categories to get a suﬃ- cien t criterion for a pair of subcategories in a triangulated category to be c om- plemen tary . W e apply this cri teri on to construct the Baum–Connes assem bly map for lo cally compac t groups and torsion-f r ee d iscrete quantu m groups. Our methods ar e related to th e abstract version of the A da ms sp ect ral sequence b y Brinkmann and Christensen. 1. Introduction The framework of t ri angulate d c a te go ries is ideal to extend ba sic cons t ructions from homotopy theory to catego ries of C ∗ -algebra s. It pr o vides a uniform setting for v ar ious problems in non-commutativ e top ology , including homotop y co limit s and May er–Vietoris sequences, univ ersal co eﬃcient theorems, and generalisations of the Baum–Connes as sem bly map (see [16–20]). More sp eciﬁcally , the Baum–Connes assembly map for co actions of cer tain compact Lie gro ups, whic h is studied in [17], is always an iso morphism and it is closely rela ted to a universal co eﬃcien t theorem for eq ui v a rian t Kaspar o v t heory by Jonathan Rosenberg a nd Claude Schochet ([22]). Univ ersal co eﬃcien t theorems for Kirch be rg’s biv ariant K-theory for C ∗ -algebra s ov er certa in ﬁnite top ological spaces are derived in [19, 20]. This article contin ues [18], which deals w it h a fr amew ork for car rying ov er famil- iar notions fro m homolog ical algebra to g eneral triangulated ca tegories. Before we explain what this article is ab out, w e outline some imp ortant ideas from [1 8 ]. The lo calisation (or tota l derived functor) of an a ddit ive functor b et w een Ab elian categories is a functor b et w een their derived categories. Mapping chain complexes to chain complexes , it be longs to the w orld of triangulated categories b y deﬁni- tion. Although the more classical der iv ed functors orig inally live in the under lying Abelian categories, they can b e carried over to triangulated categor ies as w ell. Both lo calisations and derived functors requir e additional s t ructure on a trian- gulated categor y to be deﬁned. F or the lo calisation of a functor, we specify the sub c ate gory to localise at, consisting of all obj e cts o n whic h the lo calisation v a n- ishes. F or its derive d functors , we sp ecify an ide al , consis t ing of all morph isms on which the derived functors v anish. The idea to use ideals in triang ul ated categories go es back to Daniel Chris- tensen [7]. Some imp ortan t related concepts are due to Apostolos Beligiannis [3], who uses a slightly diﬀerent but equiv alent setup, whic h is inspired by the notion of an exa ct catego ry in homological algebra . A homolo gic al ide al in a tria ngulated catego ry T is , by deﬁnition, the kernel of a stable homo logical functor (see [1 8 ]). Such an ideal I allows us to ca rry over v arious notions of homological algebra to T . The ultimate explanation for this is that a homologica l ideal generates a canonical homological funct or to a certain Abelian category , namely , the universal I -exact stable homological functor H I : T → A I T . All homolog ical notions in T deﬁned using the ideal I re ﬂ ect familiar notions in 2000 Mathematics Subje c t Classiﬁc ation. 18E30, 19K35, 46L80, 55U35. 1 2 RALF MEYER this Ab elian categor y . The homologica l alg ebra in the target Ab elian categ ory A I T provides a r ough A b elian appr oximation to the category T . An interesting and typical example is the G -equiv a rian t Kasparov ca tegory KK G for a countable discrete group G . Let I b e the ideal deﬁned by the K-theory functor, that is, an elemen t of KK G ( A, B ) b elongs to the ideal if it induces the zero map K ∗ ( A ) → K ∗ ( B ) . The resulting Abelian approximation A I ( KK G ) is the categor y of all Z / 2 -graded count able modules ov er the group ring Z [ G ] , and the universal functor maps a C ∗ -algebra A with a n action of G to its K-theory , equipped with the induced actio n of G (this is a sp ecial case of a r esult in [1 8 ]). Notice that the passag e to the universal functor adds the gr oup action o n K ∗ ( A ) . F orgetting this group action do es no t change the idea l deﬁned by the functor, but it kills most int eresting ho mological algebra. (In Section 7, we will a ctually consider a smaller ideal in KK G that is mo re clo sely r elated to the Baum–Connes conjecture, but leads to a more complicated Ab elian approximation.) The Abelian catego ry A I T is usually no t a lo calisation o f T : we must mo dify bo th mor ph isms a nd ob jects to get an Abelian category . Instead, it is descr ibed in [3] as a lo calisation o f the Abelian category containing T constructed by P eter F reyd. The main inno v ation in [18] is a concrete criterion for a stable homo logical functor to b e universal, which in v olves its par tially deﬁned left adjoin t. Using this criterion, we ca n often ﬁnd rather concrete mo dels for the universal functor – as in the example men tioned ab o v e – and then compute derived functors ass ociated to the ideal. What do the derived functors of a homological functor on T tell us abo ut the original functor ? In genera l, these derived functors ar e alw ays related to the original functor by a s pectral seq u ence, whose conv ergence we will discuss b elow. This result is mainly of theor etical impor t ance b ecause sp ectral sequence computations are almost impossible without additional simplifying assumptions. But giv en how m uch information is lo st by passing to an Ab elian categ ory , w e cannot hope for m uch more than a spec t ral sequence. The sp ectral sequence tha t links a functor to its derived functors w as already discov ered in the 1960 s b efore tr iangulated c ategories b ecame p opular. First F rank A dams treated an imp ortant sp ecial case in stable homotopy theory – the A dams sp ectral s equence [1]. This was re f ormulated in an abstrac t setting by Hans-Ber nd t Brinkmann [6]. Daniel Chris t ensen [7] form ulated the A dams sp ectral sequence in the setting of triangulated categories, apparently unaware o f Br inkmann’s work. Given the so urces of the sp ectral sequence, we call it the ABC sp e ctr al se quenc e here. W e describe its construction and its higher pages in gr eater detail than previous authors and weaken the assumptions needed to g uaran tee its c on vergence. I was drawn to wards this theory b ecause similar idea s pro vide a n eﬀectiv e metho d to prov e that pairs of sub categories ar e complement ary; this is the most diﬃcult tech nical aspe ct o f the construction of the Baum– Connes a ssem bly map in [1 6 ]. In Section 7, we ﬁrst a pp ly o ur new criterion to the group case already treated in [16] and then deﬁne an analog ue of the Baum–Connes a ssem bly map for all “torsio n- free” discrete q uan tum groups. More precisely , we co nstruct an assemb ly map for all discrete quantu m g roups, but since this map do es not take into account torsio n, it is not the rig h t analo gue o f the Ba u m–Connes asse m bly map unless the quantum group in question is torsion-free. A built-in feature of our new as sem bly map is that its domain is computed by a sp ectral sequence – the ABC spectra l s equence – whos e second page is quite accessible. The sp ectral sequence computation is very diﬃcult, but an opera tor algebraist might consider it to b e a top ological problem, that is, Someone Else’s Problem. His own problem is to ﬁnd out when the a ssem bly map is an iso morphism. HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 3 Given our exp erience with the group ca se, this should happ en often but not always. So far – b esides classica l groups – o nly the duals of certain compact Lie gr oups and quan tum SU (2) have been treated in [17] and in [24], resp ectiv ely . F or the alternative approach by Aderemi Kuku and Debashish Go sw ami in [11], it is unclear whether the domain of the assembly map is computable b y to pological metho ds. Our criterio n for complementarit y of tw o sub categories is also useful in situations that hav e nothing to do with biv aria n t K-theory . The improve ment upo n similar criteria in [3] is that we can co ver categories that are not compactly generated: what w e need is an ideal with enough pro jective ob jects that is compatible with countable direct sums. This assumption is still sa t is ﬁed for the ideals that appear in connection with the B aum–Connes as sem bly map, although the catego ries in question are pro bably not co mpactly gener ated. More pr ecisely , the criterion is the follo w in g. Let I be a homo l ogical ideal in a triangulated ca tegory T . W e assume tha t T has countable direct sums and that the ideal I is compatible with co un table direct sums in a suitable sense. F urthermore , we assume t hat I has enough pro jective ob jects. Let P I ⊆ T be the class of I -pro jective o b jects in T and let h P I i b e the localis in g sub category generated by it, t hat is, the smallest triang ulated sub category that is clo sed under countable direct sums and con tains P I . Finally , let N I be the subcatego ry o f I -contractible ob jects. Under the assumptions abov e, the pair of s ubcategories ( h P I i , N I ) is complement ary , that is, T ∗ ( P, N ) = 0 whenever P ∈∈ h P I i a nd N ∈∈ N I , a nd any ob ject A ∈∈ T is pa rt of an exa ct triangle P → A → N → P [1] with P ∈∈ h P I i and N ∈∈ N I . Equiv alently , the sub category N I is r eﬂ e ctive , that is, the embedding N I → T has a right a d joint functor. Our pro of also provides the following structural information on the category h P I i . First, we ge t a n incr easing chain ( P n I ) n ∈ N of s ubcategories, consisting o f the pro jective o b jects for the idea ls I n ; these can also be g enerated iteratively from P I using exact triangles. W e show that any ob ject of h P I i is a homotopy colimit of an inductive s ystem P n with P n ∈∈ P n I . Notation 1.1. W e write f ∈ C for a morphism and A ∈∈ C for a n o b ject of a category C . W e denote the categor y of Abelian groups by A b . W e usua lly write T for t riangulated, A for Ab elian, and C for additive categor ies. The translation automorphism in a triangulated c ategory is deno ted by A 7→ A [1] . 2. Homological ideals, po wers, and fil tra tions The con vergence of a spectral sequence a lw ays inv olves a ﬁltration on the limit group. Hence we exp ect a homological ideal I in T to genera te ﬁltrations on the cat- egory T itself and on homological and cohomologica l functors on T . After reca lling some basic notions , we intro duce these ﬁltrations here. W e will use the results and the no tation of [18]. In particular, a stable categor y is a category with a tr anslation automo rphism, denoted A 7→ A [1] , and a stable functor is a functor F together with natural isomo rphisms F ( A [1]) ∼ = ( F A )[1] for all ob jects A . Let F : T → A be a stable homolog ical functor fro m a tr iangulated c ategory T to a s t a ble Ab elian category A . W e deﬁne an ideal ker F in T by ker F ( A, B ) : = { ϕ ∈ T ( A, B ) | F ( ϕ ) = 0 } . Ideals of this form a re ca ll ed homolo gic al ide als . A ho mological ideal is used in [1 8 ] to carry over v arious notions from Ab elian to triangula ted c at e gories. This includes I -epimorphisms, I -e xact chain complexes, I -exact functors, I -pro jectiv e ob jects, and I -pro jective res olutions. The ﬁrst three of these can be tested using the func- tor F ; for instance, a chain complex with en tries in T is I -exa ct if and o nly if its 4 RALF MEYER F -image is an exact c hain complex in the Ab elian category A . Pro jectiv e ob jects can only b e describ ed in ter ms of F if F is the un iversal I -exact stable homolog ical functor, see [1 8 ]. W e a lso call a mo rphism an I -phantom map if it b elongs to I . Most of our constr uct ions require T to contain enough I -pr oje ctive obje cts – that is, any ob ject should b e the rang e of an I - epimorphism with I -pro jective domain. This is equiv alent to the existence of I -pro jective resolutions for all o b jects. R emark 2 .1 . Daniel Christensen uses a somewhat diﬀerent ter mi nology in [7]. His pr oje ctive classes ( I , P ) turn out to be the same as a homolo gical ideal I with enough pro jective o b jects together with its class P = P I of pro jective ob jects. The ideal I in a pro jective clas s is homological bec ause, in the presence of enough pro jective ob jects, the univ ersa l homologica l functor with kernel I is w ell-deﬁned. There are tw o wa ys to constr uct this universal functor, which involv e a lo calisation of categories in o ne step. Ap ostolos Beligiannis [3] ﬁrst em b eds the category T into an Abelian category and then lo calises the latter at a Serre subca tegory . The au- thors use the heart of a t-structure on a suitable derived catego ry of chain c omplexes ov er T in [18, §3.2.1 ]. In b oth ca ses, the morphisms in the r elev ant loca lisations can b e computed using pro jective resolutions, so that the localisa tion is again a category with mor phism sets instead o f mor phism classes . 2.1. P ow e r s and in tersections of ideals. At ﬁrst, w e do not care whether the ideals w e are dealing with ar e homologica l. Let C b e a n additive categor y . If ( I α ) α ∈ S is a set of ideals , then the intersection T I α is aga in an ideal. If I 1 , I 2 ⊆ C are ideals, deﬁne I 1 ◦ I 2 ( A, B ) : = { f 1 ◦ f 2 | f 1 ∈ I 1 ( X, B ) , f 2 ∈ I 2 ( A, X ) for some X ∈∈ C } . This is a s ub group of C ( A, B ) b ecause we can deco mpose f 1 ◦ f 2 + f ′ 1 ◦ f ′ 2 as A  f 2 f ′ 2  / / X ⊕ X ′ ( f 1 f ′ 1 ) / / B . Thu s I 1 ◦ I 2 is an idea l in C . W e have I 1 ◦ I 2 ⊆ I 1 ∩ I 2 . Now the p owers of an ideal I ⊆ C are deﬁned recur siv ely: we let I 0 : = C consist of all mor phisms and deﬁne I n : = I n − 1 ◦ I for 1 ≤ n < ∞ . The sequence of ideals ( I n ) n ∈ N is decreasing, a nd we hav e I m ◦ I n = I m + n for a ll m, n ∈ N . If I n = I n +1 for some n ∈ N , then I n = I N for all N ≥ n . W e also let I ∞ : = T n ∈ N I n and I n ∞ : = ( I ∞ ) n . In g eneral, the ideals I ◦ I ∞ and I ∞ ◦ I ma y diﬀer from I ∞ (see the r emark after Prop osition 4.7). Theo rem 3.2 7 shows that I n ∞ = I 2 ∞ for all n ≥ 2 if I is compatible with countable direct sums. Now we r eplace the additive catego ry C by a tr iangulated catego ry T and r estrict atten tion to ho mological ideals. It is no t obvious whether the p o wers of a homolog- ical ideal are aga in homolo gical. If I = k er F , then a functor with kernel I 2 ⊂ I contains mo re information than F because it has a smaller k er nel. Therefore, we cannot hop e to construct such a functor out of F . Nevertheless, I exp ect that products and int e rsections of homolo gical ideals are again homologica l, at least if the ca tegories in question a re small to rule out set theoretic diﬃculties with loca lisation o f catego ries. A pro of co uld use Belig iannis’ axiomatic characterisa tion of homological ideals. Since we only need the m uch easier case where there a re enough pro jective ob jects, I ha ve not completed the argument. Prop osition 2.5 and Theorem 3 .1 in [3] s h ow that our “homo logical ideals” a re exactly the “satura t e d Σ -stable ideals” in Beligia nnis’ notatio n . Clear ly , pro ducts and in ters ecti ons o f Σ -stable ideals remain Σ -stable, and intersections of saturated ideals remain saturated. It is less clear whether pro ducts of saturated ideals remain sa t urated; the pro of should inv olve the octahedral a xiom. HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 5 Here w e only consider the easy c ase of ideals with enough pro jective ob jects, where we ca n describ e which ob jects are pro jective for pro ducts and intersections: Prop osition 2. 2 ([7, Pro position 3 .3]) . L et I 1 and I 2 b e homolo gic al ide als in T with enough pr oje ctive obje cts. Then I 1 ◦ I 2 is a homolo gic al ide al with enough pr oje ctive obje cts. An obje ct A of T is I 1 ◦ I 2 -pr oje ctive if and only if ther e ar e I j -pr oje ctive obj e cts P j and an exact triangle P 2 → P → P 1 → P 2 [1] , such t h at A is a dir e ct summand of P . Prop osition 2.3 (see [7, Prop osition 3.1]) . L et ( I α ) α ∈ S b e a set of homolo gic al ide als in T with enough pr oje ctive obje cts. Supp ose that T has di re ct sums of c ar- dinality | S | . Then I S : = T α ∈ S I α is a homolo gic al ide al with enough pr oje ctive obje cts. A n obje ct A of T is I S -pr oje ctive if and only if ther e ar e I α -pr oje ctive obje cts P α such that A is a dir e ct sum m and of L α ∈ S P α . W e ma y use Christensen’s results be cause o f Remar k 2.1. Deﬁnition 2. 4. Let I be a homo logical ideal in a triangula t e d category T with enough pr o jectiv e ob jects. W e write P I for the cla ss of I -pro jectiv e ob jects, and P n I for the clas s of I n -pro jective o b jects for n ∈ N ∪ {∞} . The class P I is a lw ays closed under dir ect summands, susp ensions, and direct sums that exist in T . Prop ositions 2.2 and 2.3 show tha t the p o wers I n for n ∈ N ∪ {∞} hav e enough pro jective ob jects. Mo reo ver, P n I for n ∈ N consists of all direct summands of ob jects A n ∈∈ T for which there is an exact tria ngle A n − 1 → A n → A 1 → A n − 1 [1] with A n − 1 ∈∈ P n − 1 I , A 1 ∈∈ P I ; and P ∞ I consists of all retracts of ob jects o f the form L n ∈ N A n with A n ∈∈ P n I . The phan tom castle introduced in Deﬁnit io n 3.15 explicitly decomp oses o b jects of P n I in to ob jects o f P I ; es sen tially , its co nstruction is the pro of of Pr oposition 2.2. Example 2.5 . Let I b e a homologica l ideal with enough pro jective ob jects. If I = I 2 , then Pr oposition 2.2 implies that P I is c losed under extensions. Since this subc ategory is alw ays closed under direct summands, susp ensions, iso morphism, and direct sums, P I is a lo calising sub category o f T . Conv ersely , if P I is a triangulated subca tegory , then I a nd I 2 hav e the same pro jective ob jects. Si nce an ideal with enough pro jectiv e ob jects is determined by its class of pro jective ob jects, this implies I = I 2 . Homological idea ls with I = I 2 , but p ossibly without enough pro jective o b jects, play an imp ortant role in [14] as a substitute for loca lising sub categories. W e usually know very little ab out the Ab elian approximations g enerated by I n for n ≥ 2 , even if the situation for I itself is rather s im ple. Deriv ed functors for I and I 2 do not se em closely related. This is particularly o b vious in case s where I 6 = 0 and I 2 = 0 . F or instance, this ha ppens if I is the kernel o f the homo logy functor on the derived category of the categor y of Abe li an gr oups. Here the universal I -exact functor is the ho mology functor to Ab Z ; the univ er sal I 2 -exact functor is the F reyd embedding of the der iv ed catego ry into an Ab elian catego ry . 2.2. The phan tom ﬁltrations. Let I be a n ideal in an additive catego ry C . Since I α ⊆ I β for α ≥ β , w e g et a decr easing ﬁltration C ( A, B ) = I 0 ( A, B ) ⊇ I 1 ( A, B ) ⊇ I 2 ( A, B ) ⊇ · · · ⊇ I ∞ ( A, B ) ⊇ { 0 } , called the phantom ﬁltr ation [3]. W e shall also need re lated ﬁltratio n s on contrav a ri- ant and co v ar ian t functors on C . 6 RALF MEYER Let G : C op → Ab b e a c ontra variant functor and A ∈∈ C . W e deﬁne a decreasing ﬁltration G ( A ) = I 0 G ( A ) ⊇ I 1 G ( A ) ⊇ I 2 G ( A ) ⊇ · · · ⊇ I ∞ G ( A ) ⊇ { 0 } on G ( A ) by I α G ( A ) : = { f ∗ ( ξ ) | f ∈ I α ( A, B ) , ξ ∈ G ( B ) for s ome B ∈∈ C } . If w e apply this construction to the repres en table functor C ( , B ) w e get back the ﬁltration I α ( A, B ) o n C ( A, B ) . If G is compatible w it h direct sums, then (5.3) asserts that T n ∈ N I n G ( A ) = I ∞ G ( A ) . The functoriality of G restricts to maps I β ( A, B ) ⊗ I α G ( B ) → I α + β G ( A ) , f ⊗ x 7→ f ∗ ( x ) , for all α, β . In particular, I α G is a contra v aria n t functor o n C . The ideal I β acts trivially on the sub quotien ts I α G ( A ) / I α + β G ( A ) , which therefore descend to functors on the quo t ient ca t e gory C / I β . W e may also view G a s a right module ov e r the ca tegory C and C / I α as a C -bimo dule. The quotient G/ I α G corr esponds to the right C -mo dul e G ⊗ C C / I α . F or a cov a rian t functor F : C → Ab , w e deﬁne a n increa sing ﬁltration { 0 } = F : I 0 ( A ) ⊆ F : I 1 ( A ) ⊆ F : I 2 ( A ) ⊆ · · · ⊆ F : I ∞ ( A ) ⊆ F ( A ) for any A ∈∈ C b y F : I α ( A ) : = { x ∈ F ( A ) | f ∗ ( x ) = 0 for all f ∈ I α ( A, B ) , B ∈∈ C } . If I and F are compatible with direct sums, then F : I ∞ ( A ) = S n ∈ N F : I n ( A ) (see Theor em 5 .1), but this need not be the case in genera l. The functoriality of F restricts to maps I β ( A, B ) ⊗ F : I α + β ( B ) → F : I α ( A ) , f ⊗ x 7→ f ∗ ( x ) , for all α, β . In particular, F : I α is a cov ar ian t functor on C . The ideal I β acts trivially o n the subquotients F : I α + β ( A )  F : I α ( A ) , whic h ther ef ore descend to functors on C / I β . W e ma y also v iew F as a left mo dule over the ca tegory C a nd C / I α as a C -bimo dule. Then F : I α corresp onds to the left C -mo dule Hom C ( C / I α , F ) . The ﬁltration F : I α ( A ) is closely r elated to pro jective r esolutions o f A . In contrast, the ﬁltra t ion I α F ( A ) : = { f ∗ ( ξ ) ∈ F ( A ) | f ∈ I α ( B , A ) , ξ ∈ F ( B ) for some B ∈∈ C } is related to inje ctive (co)resolutions. There is also an increasing ﬁltration G : I n for a con tr a v ariant functor G . The ﬁltrations I α F a nd G : I α will not be used in this article. 3. Fr o m pr ojective resolutions to complement ar y p airs First we r eﬁne a pro jective reso luti on b y adjoining certain pha n tom ma ps. This yields the phantom tower ov er an ob ject (see also [3]). W e show that the pro jec- tiv e resolution determines this tow er uniquely up to non-canonica l isomorphism. There is a n other tower over an ob ject, the c el lular appr oximation t o wer . These tw o tow ers are related by v ario us commuting diag rams and exact triangles ; we call the collection of all these ex act or comm uting triang les the phantom c astle . The goal of this section is to sho w that the categories h P I i a nd N I are comple- men tary if I is compatible with direct sums. Before w e co me to that, we recall the notion of complementary pair o f s ubcategories and deﬁne what it means for an ideal to b e compatible with countable direct sums. The main ing redien ts in the pr oof are the homo t opy co lim its of the phantom tower a n d the cellular approximation tow er. HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 7 Finally , we describe a metho d for chec king that a given lo calising s u bca tegory is reﬂective, that is, pa rt of a complemen tar y pair. All results inv olving inﬁnite direct sums require that the category T has co un table direct sums. T r iangulated categories inv olv in g biv aria n t K-theory ha ve no more than countable direct sums b ecause o f built-in separability a ssumptions that make the a nalysis be h ind the scenes work. This is why we only us e c ount able direct sums. Of co urse, everything remains true if we drop the word “countable” or re p lace it b y another car dinalit y constraint. A triangula t ed sub category of T is called lo c alising (more precisely , ℵ 0 -lo c alising ) if it is closed under co un table direct sums. Lo calising subcateg ories are automati- cally thick, that is, closed under dire ct summands (se e [2 1 ]). Let I b e a homo logical ideal in a triang ulated category T . Recall that an I -pr oje ctive r esolution of an o b ject A of T is a c hain co m plex · · · δ n +1 − − − → P n δ n − → P n − 1 δ n − 1 − − − → · · · δ 2 − → P 1 δ 1 − → P 0 of I -pr o jectiv e ob jects P n , augmented b y a map π 0 : P 0 → A , such that the a ug- men ted c hain complex is I -exact. If I = ker F for a stable ho m ological functor F to some Ab elian category A , then I - exactness means that the chain c omplex · · · F ( δ n +1 ) − − − − − → F ( P n ) F ( δ n ) − − − − → F ( P n − 1 ) F ( δ n − 1 ) − − − − − → · · · F ( δ 1 ) − − − → F ( P 0 ) F ( π 0 ) − − − − → F ( A ) is e xact in A . W e say that I has enough pr oje ctive obje cts if each A ∈∈ T has such an I -pro jectiv e r esolution. 3.1. The phantom to wer. Deﬁnition 3. 1. A pha ntom tower ov er an ob ject A of T is a diagram (3.2) A N 0 ι 1 0 / / N 1 ι 2 1 / / ◦    ǫ 0      N 2 ι 3 2 / / ◦    ǫ 1      N 3 / / ◦    ǫ 2      · · · ◦         P 0 π 0 [ [ 6 6 6 6 6 6 P 1 π 1 [ [ 6 6 6 6 6 6 ◦ δ 1 o o P 2 π 2 [ [ 6 6 6 6 6 6 ◦ δ 2 o o P 3 π 3 [ [ 6 6 6 6 6 6 ◦ δ 3 o o · · · [ [ 6 6 6 6 6 6 6 ◦ o o with I -phan tom maps ι n +1 n and I -pr o jectiv e ob jects P n for n ∈ N , suc h that the triangles P n π n − − → N n ι n +1 n − − − → N n +1 ǫ n − → P n [1] in (3.2) are exact for a ll n ∈ N and the other triangles in (3.2) co mm ute, that is, δ n +1 = ǫ n ◦ π n +1 for all n ∈ N . Notice our conv ent io n that cir cled arrows ar e maps of degree 1 . Since the maps δ n in the phan to m tow er hav e degree 1 , w e sligh tly mo dify o ur notion of pro jective reso lu tion, letting the b oundary maps hav e degree 1 . Lemma 3.3. The maps δ n for n ∈ N ≥ 1 and π 0 in a phantom tower over A form an I -pr oje ctive r esolution of A . Conversely , any I -pr oje ctive re s olut i on c an b e emb e dde d in a phantom tower, whi ch is unique up to non-c anonic al isomorphism. A morphism f : A → A ′ lifts ( non-c anonic al ly ) to a morphi sm b etwe en two given phantom towers over A and A ′ . A chain map b etwe en pr oje ctive r esolutions of A and A ′ extends to the phantom towers that c ontain these re solutions. Pr o of. Let P n , π 0 , and δ n be part of a pha n tom tow er ov er A . The ob jects P n are I -pro jective b y deﬁnition, a nd δ n ◦ δ n +1 = 0 for all n ∈ N and π 0 ◦ δ 1 = 0 beca u se these pro ducts inv o lv e t wo consecutive ar ro ws in an exact tria n gle. Hence the maps δ n and π 0 form a chain complex. W e c laim that it is I -exa ct . 8 RALF MEYER Let F b e a stable homologica l functor with ker F = I . Recall that a c hain complex is I -ex act if a nd only if its F -image is exact in the usua l sense by [1 8 , Lemma 3.9]. The exact tria ngles in the phan to m tower yield short exa ct s equences F ∗ +1 ( N n +1 ) ֌ F ∗ ( P n ) ։ F ∗ ( N n ) for all n ∈ N b ecause ι n +1 n ∈ I ; here F ∗ ( A ) : = F ( A [ − n ]) . Splicing these e xten sions as in the deﬁnition of the Y oneda pro duct, w e get a n exact c hain complex. Since this chain c omplex is · · · → F ∗ +2 ( P 2 ) → F ∗ +1 ( P 1 ) → F ∗ ( P 0 ) → F ∗ ( A ) → 0 , we hav e g ot an I - exact chain complex and hence a n I -pro jective resolution. Now let π 0 : P 0 → A and δ n : P n → P n − 1 [1] fo r n ∈ N ≥ 1 form a n I -pro jective resolution. W e recursively construct the triangles that comprise the phantom tow er. T o begin w it h, w e embed π 0 in an exact triangle P 0 π 0 − → A ι 1 0 − → N 1 ǫ 0 − → P 0 [1] . Since π 0 is I -epic, ι 1 0 is an I -phantom map and ǫ 0 is I -monic. Thus our exact triangle yields a s hort exa ct sequence T ∗ +1 ( P, N 1 ) ֌ T ∗ ( P, P 0 ) ։ T ∗ ( P, A ) for an y I -pro jective ob ject P . In particular, this applies to P = P 1 and sho ws that δ 1 factors uniquely as δ 1 = ǫ 0 ◦ π 1 with π 1 ∈ T 0 ( P 1 , N 1 ) . W e claim that π 1 is I -epic. Let F b e a deﬁning functor for I a s above. Then F ( P 1 ) → F ( P 0 ) → F ( A ) is exa ct at F ( P 0 ) , a nd F ( N 1 ) ֌ F ( P 0 ) ։ F ( A ) is a shor t exact sequence. Hence the ra nge of F ( δ 1 ) is isomor phic to F ( N 1 ) . This implies that F ( π 1 ) is an epimor phi sm, that is, π 1 is I -epic. Thu s the ma p s π 1 and δ n for n ∈ N ≥ 2 form an I -pr o jectiv e resolution of N 1 . W e may now repea t the above pro cess and recursively co nstruct the phan tom tower. Thu s any I -pro jective resolution embeds in a phan tom to wer. F urthermore, since the exac t tria ngle con ta inin g a giv en mo rphism is unique up to isomo rphism and the liftings π 1 ab o ve a re unique, there is, up to iso morphism, only one phant om tow er that contains a g iv en I -pr o jectiv e resolution. O f co urse, diﬀerent res olutions yield diﬀerent phantom tow ers. Finally , it remains to lift a morphism f : A → A ′ to a transformation b et ween t wo given phantom tow ers. First we can lift f to a ch ain map b et ween the I - pro jectiv e resolutions contained in these tow ers (see [18]); let P n ( f ) : P n → P ′ n for n ∈ N be this c ha in map. It r emains to co nst ruct maps N n ( f ) : N n → N ′ n that together with t he maps P n ( f ) in ter t wine the v arious maps in the phan tom to wers. W e already have the map N 0 ( f ) = f . The triangula t ed catego ry axio m s provide a map N 1 ( f ) : N 1 → N ′ 1 making the diag ram P 0 P 0 ( f )   π 0 / / A f   ι 1 0 / / N 1 N 1 ( f )   ǫ 0 / / P 0 [1] P 0 ( f )[1]   P ′ 0 π ′ 0 / / A ′ ι ′ 1 0 / / N ′ 1 ǫ ′ 0 / / P ′ 0 [1] commut e. W e claim that N 1 ( f ) ◦ π 1 = π ′ 1 ◦ P 1 ( f ) . As abov e, we get short exact sequences T ∗ +1 ( P 1 , N ′ 1 ) ֌ T ∗ ( P 1 , P ′ 0 ) ։ T ∗ ( P 1 , A ′ ) . Hence it suﬃces to chec k ǫ ′ 0 ◦ N 1 ( f ) ◦ π 1 = ǫ ′ 0 ◦ π ′ 1 ◦ P 1 ( f ) = δ ′ 1 ◦ P 1 ( f ) . But this is true b ecause ǫ ′ 0 ◦ N 1 ( f ) = P 0 ( f ) ◦ ǫ 0 and the maps P n ( f ) form a chain map. Thu s the map N 1 ( f ) has all required pr operties. Iterating this co nstruction, we get the HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 9 maps N n ( f ) for all n ∈ N . B y the w ay , they need not be unique even if the ma ps P n ( f ) are ﬁxed.  The following deﬁnition forma lises a n imp ortan t prop ert y of the maps ι n +1 n in a phant om tow er. Deﬁnition 3.4 . Let I ⊆ T b e a n ideal. Let A, B ∈∈ T . W e call f ∈ I ( A, B ) I -versal if, for any C ∈∈ T , any g ∈ I ( A, C ) factor s as g = h ◦ f for some h ∈ T ( B, C ) : A f / / g   @ @ @ @ @ @ @ B h ∃   C W e do not require this factorisation to b e unique. Since I is an idea l, any map of the for m h ◦ f be longs to I . Lemma 3.5. The maps ι n +1 n in a phantom t ower ar e I -versal for al l n ∈ N . Pr o of. Let f ∈ I ( N n , B ) . Since P n is I -pro jectiv e , I ∗ ( P n , B ) = 0 . Th us f ◦ π n = 0 . This forces f to facto r throug h ι n +1 n beca use T ∗ ( , B ) is cohomologic al.  Lemma 3 . 6. L et I 1 and I 2 b e ide als in a triangulate d c ate gory. If f 1 ∈ I 1 ( B , C ) and f 2 ∈ I 2 ( A, B ) ar e I 1 - and I 2 -versal maps, r esp e ctively, then f 1 ◦ f 2 : A → C is I 1 ◦ I 2 -versal. Pr o of. Let h ∈ I 1 ◦ I 2 ( A, D ) , write h = h 1 ◦ h 2 with h 1 ∈ I 1 and h 2 ∈ I 2 . Using versalit y of f 1 and f 2 , we ﬁnd the maps h ′ 2 and h ′ in the following diagra m: A f 2 / / h 2   @ @ @ @ @ @ @ @ B f 1 / / h ′ 2 ∃   C h ′ ∃   • h 1 / / D . Thu s h factor s through f 1 ◦ f 2 as required.  As a consequence, the maps ι n + k n : = ι n + k n + k − 1 ◦ · · · ◦ ι n +1 n : N n → N n + k in a phantom to wer are I k -versal for all n , k ∈ N . Lemma 3.7. A map f : A → B is I k -versal if and only if I k ( A, C ) = r ange  f ∗ : T ( B , C ) → T ( A , C )  for al l C ∈∈ T . L et f : A → B b e I k -versal. If F : T → Ab is homolo gic al, then F : I k ( A ) = ker  f ∗ : F ( A ) → F ( B )  ; if G : T op → Ab is c ohomolo gic al, then I k G ( A ) = ra nge  f ∗ : G ( B ) → G ( A )  . Pr o of. This follo ws immediately from the deﬁnitions.  As a consequence, w e can compute the ﬁlt rations F : I k ( A ) and I k G ( A ) o f Section 2.2 from the phantom tow er. 10 RALF MEYER 3.2. The phan tom castle. Now we extend the phantom tow er to the phan tom castle, which con tains a mong other things the cellular approximation tower. W e start with a pha n tom tow er over some ob ject A ∈∈ T . Let ι n : = ι n n − 1 ◦ · · · ◦ ι 0 1 : A = N 0 → N n and embed ι n in an exa ct triangle (3.8) ˜ A n α n − − → A ι n − → N n γ n − → ˜ A n [1] . The o ctahedral axio m relates the mapping cones ˜ A n +1 , P n , and ˜ A n of the maps ι n +1 , ι n +1 n , a nd ι n beca use ι n +1 = ι n +1 n ◦ ι n (see [21, P roposition I.4 .6] o r [1 6 , P roposition A.1]). More precisely , the o ctahedral axiom allows us to choose maps (3.9) ˜ A n α n +1 n − − − → ˜ A n +1 σ n − − → P n κ n − − → ˜ A n [1] , such that this triangle is exact and the following diagra m co mm utes: (3.10) N n ι n +1 n / / ◦ γ n   N n +1 ◦ G G G G ǫ n # # G G G G ◦ γ n +1   ˜ A n α n +1 n / / α n " " E E E E E E E E E ˜ A n +1 α n +1   σ n / / P n π n   ◦ κ n / / ˜ A n A ι n / / ι n +1 # # H H H H H H H H H N n ◦ y y y y γ n < < y y y y ι n +1 n   N n +1 In addition, we can achiev e that the triangle (3.11) N n [ − 1] → ˜ A n +1  α n +1 σ n  − − − − − → A ⊕ P n ( ι n ,π n ) − − − − − → N n is exact, that is, the square in the middle o f (3.1 0 ) is homotop y Cartesian and the diagonal of the top r igh t squar e provides its diﬀere n tial. Lemma 3.12 . The obje ct ˜ A n is I n -pr oje ctive for e ach n ∈ N . Pr o of by induction on n . The case n = 0 is clear. Sin ce P n ∈∈ P I for all n ∈ N , the exact triang les (3.9) a nd Pro position 2.2 provide the inducti on step.  F urthermo re, the map α n : ˜ A n → A is I n -epic b ecause ι n ∈ I n , so that it is the ﬁrst step of an I n -pro jective res olution of A . This provides ano t her expla n ation why the map ι n is I -versal (co mp are Lemma 3.5). R emark 3.13 . The cone of the map ι m + k m is I k -pro jective for all m, k ∈ N by a similar arg umen t. Hence A = N 0 ι k 0 − → N k ι 2 k k − − → N 2 k ι 3 k 2 k − − → N 3 k → · · · together with the exact tria ngles that contain the maps ι j k + k j k is an I k -phantom tow er and hence yields an I k -pro jective r esolution by Lemma 3.3. As a result, a n I -phantom to wer determines I k -phantom tow ers for all k ∈ N . Deﬁnition 3.14. The seq uence o f exa ct triang les (3.9) is called the c el lular ap- pr oximation tower ov e r A . HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 11 The motiv ation for our terminology is the following. If A has a pro jective res- olution o f ﬁnite length, then w e can choose a phant o m to wer with P n = 0 for n ≫ 0 . Suppose, in addition, that A b elongs to the thick tria n gulated subca t egory generated by P I . Then the pro of of Pro position 4.10 yields N n = 0 for n ≫ 0 . The exa ct triangles (3.8) mean that the maps α n : ˜ A n → A b ecome in vertible for n ≫ 0 , that is, ˜ A n ∼ = A . Therefore, we think of the ob jects N n as “obstructions” that should get smaller for n → ∞ , and of the ob jects ˜ A n as b etter and b etter approximations to A . They are called “ P I -cellular” b ecause they are constructed out of I -pro jective ob jects – the cells – by iterated exac t triangles. Deﬁnition 3.15. A phantom c astle ov er A is a sequence o f ob jects N n , P n , ˜ A n with maps ι n +1 n , π n , ǫ n , ι n , α n , γ n , σ n , κ n such that the triangles (4 .1 ) , (3.8), (3.9), and (3.11) ar e exact a nd the diag ram (3.10) co mm utes. W e will use most of the information enco ded in this deﬁnition to identify the sp ectral sequences genera t e d by the phantom tower a nd the cellular approximation tow er; only the commutativit y of the sq uare in the middle of (3.10) a nd the exact sequence (3.11) seem ir relev ant in the followin g. 3.3. Complem en tary pairs of s ub categories and lo calisation. W e call t wo thic k subcatego ries L and N of T c omplementary if T ∗ ( L, N ) = 0 for all L ∈∈ L , N ∈∈ N and, for any A ∈∈ T , there is an ex act triangle L → A → N → L [1] with L ∈∈ L and N ∈∈ N (see [16, Deﬁnition 2 .8]). Similar situa t io ns have b een studied by v arious authors, under v ario us names, such as loca lisation pair s, stable t-structures, tors ion pairs; a complemen tary pair is equiv a len t to a lo calisation functor L on T , where L is the cla ss o f L - local ob jects and N is the clas s of L -ac yclic ob jects. The following assertions are contained in [16, Prop osition 2.9]. Let ( L , N ) b e complement ary . Then the exact triangle L → A → N → L [1] with L ∈∈ L and N ∈∈ N is unique and functoria l, and the resulting functors L : T → L and N : T → N ma ppi ng A to L and N , r espectively , are left adjoint to the em b edding functor L → T and righ t adjoint to the em b edding functor N → T , resp ectiv ely . That is, the sub category N is reﬂective and L is co reﬂectiv e. The co m po site functors L → T → T / N a n d N → T → T / L are eq ui v a lences of categ ories. Conv ersely , let N ⊆ T be a reﬂective sub category and let N : T → N b e the left adjoint of the em b edding functor N → T . Let L = { A ∈∈ T | N ( A ) = 0 } be the left ortho gonal c omplement of N . Then ( L , N ) is a complementary pair of subcatego ries, a nd L is the only p ossible partner for N . Thus co mp lemen tary pairs are essen tially the sa me as reﬂective sub categories. Dually , a sub category L is c oreﬂectiv e if and only if it is par t of a complemen ta ry pair ( L , N ) , and the only candidate for N is the righ t o rthogonal co mplem ent of L . If F : T → C is a cov aria n t functor, then its lo c alisation L F with r espect to N is deﬁned by L F : = F ◦ L , where L : T → L is the rig h t adjoint of the embedding L → T . The natural maps L ( A ) → A provide a na tu ral transforma t ion L F ⇒ F . If G : T op → C is a contrav a rian t functor, then the lo calisation G ◦ L is denoted b y R G . It comes tog et her with a na t ur al tra nsformation G ⇒ R G . This lo calisation pro cess is a n imp ortant to ol to construct functors . Sp ecial cases are derived functors in homologica l algebra and the doma in of the Ba um –Connes assembly map (se e [16]). Although the deﬁnition of a co mp lemen tary pair is sy mm e t ric, the sub categories L and N hav e a ra th er diﬀeren t nature in most examples. Usually , one of them – here it is alwa ys N – is deﬁned directly and the other o ne is only des cribed b y 12 RALF MEYER generator s. This mak es it hard to tell which ob jects it co n tains and to ﬁnd the exact triangles needed fo r complementarit y . Here homo logical ideals help. Let I be a homological ideal with enough pro jec- tiv e ob jects in a triang ulated category T . L et h P I i b e the lo calising s ubcategory generated by P I , that is, the smallest lo calising sub category of T that contains P I . Since the name “pro jectiv e ” is alr eady taken, we call ob jects of h P I i P I -c el lular . W e ha ve P n I ⊆ h P I i for all n ∈ N ∪ { ∞ , 2 ∞ , . . . } by Prop ositions 2.2 and 2.3. Deﬁnition 3.16. Let N I be the full sub category of I -contractible ob jects, that is, ob jects N with id N ∈ I ( N , N ) . An ob ject N is I -contractible if and only if 0 → N is an I -pro jective resolution. Thu s all I -der iv ed functors v a nish on N I . Now the following question arises: is the pa ir o f sub categories ( h P I i , N I ) com- plemen ta ry? It is eviden t that T ( P, N ) = 0 if P ∈ P I and N ∈ N I . This extends to P ∈ h P I i b ecause the left orthogona l complement of N I is loca lising. This is the easy half of the deﬁnition of a complementary pair . The other, non-trivial half requires an additional condition on the ideal I . 3.4. Compatibili t y with direct sums. Deﬁnition 3.17. An ideal I is called c omp atible with c ountable di re ct sums if, fo r any countable family ( A i ) i ∈ I of ob jects of T , the canonical iso morphism T  M i ∈ I A i , B  ∼ = Y i ∈ I T ( A i , B ) restricts to an iso morphism I  L i ∈ I A i , B  ∼ = Q i ∈ I I ( A i , B ) . An ideal I is compatible with co un table direct sums if and only if the following holds: given countable families of ob jects ( A i ) , ( B i ) and maps f i ∈ I ( A i , B i ) for i ∈ I , we ha ve L f i ∈ I  L A i , L B i  . Recall that dir ect sums of exact triangles a re aga in exact (see [21]) and that the ideal determines and is det ermined b y the cla sses of I -epimorphisms or of I -monomorphisms. Ther ef ore, I is compatible with direct sums if a nd only if direct sums of I -monomorphisms are aga in I -monomo rphisms, if and only if direct sums of I -epimorphisms ar e ag ain I -epimorphisms. Moreov er , if I is compatible with coun table direct sums, then a dir ect sum of I -equiv a lences is aga in an I -equiv a lence, and N I is a lo calising sub category of T . Moreov er , a dir ect sum o f phantom castles ov er A i is a phantom castle ov er L A i . Example 3.18 . Let F be a stable homologic al functor or an exact functor to another triangulated ca t e gory , a nd s uppose that F commutes with countable direct sums. Then ker F is a homo logical ideal a nd compatible with coun ta ble dir ect sums. This example is, in fact, already the mos t genera l case: Prop osition 3.19. L et T b e a t ri angulate d c ate gory with c ountable dir e ct sums and let I b e a homolo gic al ide al in T . L et F : T → A I T b e a universal I -exact stable homolo gic al functor. The ide al I is c omp atible with c ount a ble dir e ct su ms if and only if the A b elian c ate gory A I T has exact c ountable dir e ct sums and the functor F : T → A I T c ommutes with c oun t a ble dir e ct sums. Pr o of. One direction is just the asser t ion in Exa m ple 3.18. The o th er direction requires so m e description of the universal functor F . W e us e the description in [18], which star t s with the homotopy categor y Ho( T ) of chain complexes with entries in T . Since T has countable dir ect sums, so has Ho( T ) . The I -exact chain co mpl exes form a thic k sub category E of Ho( T ) ; it is c losed under count able direct sums b ecause I HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 13 is c ompatible with count able direct sums. Hence the lo calisation Ho( T ) / E still has countable direct sums. The Abelian appro ximation A I T is equiv alent to the heart of a canonical trun- cation structure on Ho( T ) / E des cribed in [18] and consis t s of chain co mp lexes that are exact in degrees not equal to 0 . The univ ers al functor F is the obvious o ne, viewing an ob ject of T a s a chain complex suppo rted in degree 0 . It is evident tha t the sub category A I T ⊆ Ho ( T ) / E is closed under countable dir ect sums. Count able direct sums of extensions in A I T remain extensions b ecause the ana logous asse rtion holds for direct sums of exact triangle s in a n y triangula t e d ca t e gory (see [21]) and extensions in the heart ar e related to exact triang les in the ambien t tria ngulated category . Clearly , the functor T → A I T preserves countable direct sums.  Example 3 .20 . The ideal of ﬁnite rank op erators on the category of vector spaces is an idea l that is no t co mpatible with co u nt a ble direct sums. 3.5. Complem en tarity and structur e of cellular ob jects. The r esults in this section generalise results o f Ap ostolos Beligiannis (see [3, Theorem 6.5 ], [3, Corol- lary 5.12]) in the case where P I is generated by a single compact o b ject. Theorem 3.21. L et T b e a t riangulate d c ate gory with c ountable dir e ct sums, and let I b e a homolo gic al ide al in T with enough pr oje ctive obje cts. Supp ose that I is c omp atible with c ount able dir e ct sums. Then the p air of lo c alising sub c ate gories ( h P I i , N I ) in T is c omplementary. W e will prese n t t wo indep enden t pro ofs, one using phantom tow ers , the other cellular approximation tow e rs. Both requir e homotopy co li mits: Deﬁnition 3. 22. L et ( D n , ϕ n +1 n ) b e an inductiv e system in T . Deﬁne the shif t S : M D n → M D n , S | D n : D n ϕ n +1 n − − − → D n +1 ⊆ M D n . The homotopy c olimit ho- lim − → ( D n , ϕ n +1 n ) is the third leg in the exa ct triangle L D n id − S / / L D n / / ho- lim − → ( D n , ϕ n +1 n ) / / L D n [1] . Recall that id − S determines this triangle uniquely up to isomorphism. Pr o of of The or em 3.21 . Since the class of A ∈∈ T with T ∗ ( A, B ) = 0 for all B ∈∈ N I is localising , we ha ve T ∗ ( A, B ) = 0 if A ∈∈ h P I i a nd B ∈∈ N I . It remains to construct, for each A ∈∈ T , a n exact tria ngle ˜ A → A → N → ˜ A [1] with N ∈∈ N I and ˜ A ∈∈ h P I i . Construct a phantom castle ov er A and le t N : = ho - lim − → ( N n , ι n +1 n ) be the ho- motopy colimit of the phan tom tow er . W e also use the homo t opy colimit of the constant inductive sy stem ( A, id A ) . This is just A b ecause of the split exact tria ngle (3.23) M A id − S − − − → M A ∇ − → A 0 − → M A [1] , 14 RALF MEYER where ∇ is the co diagonal map. B y [2, Pr oposition 1.1.11] (and a r otation), we can ﬁnd ˜ A and the do t ted arr o ws in the following diagram (3.24) L ˜ A n / / L α n   L ˜ A n / / L α n   ˜ A ◦ / /   L ˜ A n L α n   L A id − S / / L ι n   L A L ι n   ∇ / / A ◦ 0 / /   L A L ι n   L N n id − S / / ◦ L γ n   L N n ◦ L γ n   / / N ◦ / / ◦   − L N n ◦ − L γ n   L ˜ A n / / L ˜ A n / / ˜ A [1] ◦ / / L ˜ A n so that the rows a nd columns are exact tria ngles and the squares commute except for the one ma rk ed with a min us sign, whic h anti-comm utes. Lemma 3.12 yields ˜ A n ∈∈ P n I for all n ∈ N . Hence L ˜ A n ∈∈ P ∞ I ⊆ h P I i by Prop osition 2.3. The exactness of the ﬁrst r o w in (3.2 4 ) implies ˜ A ∈∈ h P I i . W e claim that N ∈∈ N I . Hence the third column in (3.24) is the kind o f ex act triangle we need fo r h P I i and N I to b e c omplemen tary . Let F b e a stable homolog ical functor with ker F = I . W e must show F ( N ) = 0 . The map S factors through L ι n +1 n ; this ma p b elongs to I = k er F beca use I is compatible with direct sums. Hence F ( id − S ) = F ( id ) is invertible. By a lo ng exact sequence, this implies F ( N ) = 0 , that is , N ∈∈ N I .  Suppos e from now on that we are in the situation of Theorem 3.21. Since h P I i and N I are complemen ta ry , there is a unique exact triangle ˜ A → A → N → ˜ A [1] with I -co n tractible N and P I -cellular ˜ A ; w e call ˜ A the P I -c el lular appr oximation of A . E v en mo re, ˜ A a nd N dep end functorially on A , so that we get t wo functors L : T → h P I i and N : T → N I . The pro of of Theore m 3 .21 ab o ve provides an explicit mo del for N ( A ) : it is the homotopy colimit o f the phan tom tow e r of A . Prop osition 3 .25. L et A ∈∈ T and c onst ruct a phantom c astle over A . Then L ( A ) = ˜ A is t h e homotop y c olimit of the c el lular appr oximation tower ( ˜ A n ) n ∈ N . This do es not yet follow from (3.24) because we cannot cont rol the dotted maps. Pr o of. Let ˜ A : = ho- lim − → ( ˜ A n , α n +1 n ) . W e c ompare the exact triangle that deﬁnes the homo top y colimit ˜ A with the tria ngle (3.23). The triangulated categor y ax ioms provide f ∈ T ( ˜ A, A ) that makes the follo wing dia gram commute: (3.26) L ˜ A n id − S / / L α n   L ˜ A n / / L α n   ˜ A f   ◦ / / L ˜ A n L α n   L A id − S / / L A ∇ / / A ◦ 0 / / L A. W e claim that f is an I - equiv alence. Eq uiv a len tly , the cone of f is I -contractible, so that the mapping cone tr iangle for f has entries in h P I i and N I ; this implies that L ( A ) = ˜ A . Let F b e a stable homolog ical functor with ker F = I . W e chec k that F ( f ) is in vertible. The direct sum of the triangles (3.8) for n ∈ N is ag ain a n exact triangle. HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 15 On the long e xact homolo gy sequence · · · − → F m +1  M N n  − → F m  M ˜ A n  − → F m  M A  − → F m  M N n  − → · · · for this ex act triangle, co nsider the o perator induced b y id − S on ea c h entry . Since I is compatible with direct sums, the shift ma p S on L N n is a phan tom map, so that F ( id − S ) acts identically on F  L N n  . On F  L A  , the map id − S induces a split mono morphism with cokernel F ( ∇ ) : F  L A  → F ( A ) . No w a diagram chase shows that the map o n F  L ˜ A n  induced by id − S is injective and has the cokernel F  M ˜ A n  F  L α n  − − − − − − → F  M A  F ( ∇ ) − − − → F ( A ) . Comparing the lo ng exact homology sequences for the t wo r o ws in (3 .26) , we co n- clude that F ( f ) is indeed in vertible.  W e have proved P roposition 3.25 by constructing an I -equiv alence b et ween an arbitrary o b ject o f T and the homotopy co limit o f its cellular appr o ximation tow er. This provides ano t her, indep enden t pr oof of Theo rem 3.21. Since the exact triangle ˜ A → A → N → ˜ A [1] with ˜ A ∈∈ h P I i a nd N ∈∈ N I is unique up to isomo rphism, bo th pro ofs c onstruct the same exact tr iangle. The ﬁrst pr oof shows that N is the homotop y colimit of the phantom tow er , the second one shows that ˜ A is the homotopy colimit of the cellula r approximation tow er. W e conclude, therefo re, that the ob ject ˜ A in (3.24) is the homotopy colimit of the pha n tom tow er and that the map ˜ A → A in (3.24) ag rees with the map f from (3.26). Theorem 3.27. L et T b e a t riangulate d c ate gory with c ountable dir e ct sums, and let I b e a homolo gic al ide al in T that is c omp atible with c ount ab le di re ct sums and has enough pr oje ctive obje cts. L et A ∈∈ T and c onstruct a pha n tom c astle over A . The fol lo wing ar e e quivalent: (1) A is P I -c el lular, that is, A ∈∈ h P I i ; (2) A is isomorphic t o the homotopy c olimit of its c el lular appr oximation t o wer; (3) A is isomorp hic to t h e homotopy c olimit of an inductive system ( P n , ϕ n ) with P n ∈∈ S k ∈ N P k I for al l n ∈ N ; (4) A is I 2 ∞ -pr oje ctive. A s a c onse qu enc e, I 2 ∞ = I n ∞ for al l n ≥ 2 . Pr o of. Since L ( A ) ∼ = A if and only if A ∈∈ h P I i , P roposition 3 .25 yields the equiv a- lence of (1) and (2). The implication (2 ) = ⇒ (3) is trivial: the cellular approximation tow er provides an inductiv e system of the required kind. W e chec k that (3) implies (4). Let ( P n , ϕ n ) b e an inductiv e system as in (3). First, Propo sition 2.3 shows that L P n is I ∞ -pro jective. Then Prop osition 2 .2 shows that the homotop y colimit is I 2 ∞ -pro jective. Prop ositions 2.3 and 2.2 show recursively that all I α -pro jective ob jects b elong to h P I i for n = 0 , 1 , 2 , 3 , . . . , ∞ , 2 · ∞ . Hence (4) implies (1), so t hat all four conditions are equiv alent. Finally , since P 2 ∞ I = h P I i , it follows from Prop osition 2.2 that the powers I n ∞ for n ≥ 2 have the same pr o jectiv e ob jects. Therefor e, they ar e all equa l.  Prop osition 3.28. The P I -c el lular appr oximation functor L : T → h P I i maps a phantom c astle over A ∈∈ T to a phantom c astle over L ( A ) . A morphism f ∈ T ( A, B ) b elongs to I α for some α if and only if L ( f ) ∈ T  L ( A ) , L ( B )  do es. 16 RALF MEYER Pr o of. Since L is an exact functor, it preserves the commuting diag rams and exact triangles required for a phantom ca stle. It also maps P I to itself b ecause L ( B ) ∼ = B for all B ∈∈ h P I i . It remains to check that L ( f ) ∈ I α  L ( B ) , L ( B ′ )  if and only if f ∈ I α ( B , B ′ ) . Le t F b e a sta b le homo logical functor with I α = ker F . Then F ( N ) = 0 for all N ∈∈ N I . Therefore, F desc ends to the lo calisation T / N I ; equiv a len tly , the natural transforma t ion F ◦ L ⇒ F is an iso m orphism. In pa rticular, F ( f ) = 0 if and only if F  L ( f )  = 0 .  As a result, it makes almost no diﬀerence whether we work in T or T / N I . W e work in T most of the time and allow N I to b e non-trivial in order to formulate Theorem 3.21. The direct s um s L ˜ A n in (3.26) are I ∞ -pro jective by Pro position 2 .3 . The map ∇ ◦ L α n : L ˜ A n → A is I ∞ -epic b ecause it is I n -epic for a ll n ∈ N . W e ma y replace A b y ˜ A in this statement b y Prop osition 3 .28 . Th us the top ro w in (3.26) is an I ∞ -exact triangle. This means that the c ha in complex · · · → 0 → M ˜ A n id − S − − − → M ˜ A n → ˜ A is I ∞ -exact a nd hence a n I ∞ -pro jective resolution of ˜ A . Once again, Pro po- sition 3 .28 allo ws us to replace ˜ A by A in this statement , that is, we get an I ∞ -pro jective r esolution o f length 1 (3.29) · · · → 0 → M ˜ A n id − S − − − → M ˜ A n → A. This will allow us to ana lyse the convergence of the ABC spectr al sequence. 3.6. Complem en tarity via partially de ﬁne d adjoin ts. Suppose that we are given a thick sub category N of a tria ngulated categ ory T and that w e want to us e Theorem 3.2 1 to sho w that it is reﬂective, that is, there is another thick subcate- gory L such that ( L , N ) is complementary . T o hav e a c hance of doing so, T must ha ve countable di rect sums, a nd the subc ategory N must be lo calising, that is, closed under count a ble direct sums: this happ ens whenev er Theorem 3.21 applies. By [16, Prop osition 2 .9], the only candidate for L is the left orthog onal c omplemen t L : = { A ∈∈ T | T ( A, N ) = 0 for all N ∈∈ N } of N , which is another lo calising subca tegory . The starting p oin t of o ur metho d is a s table additive functor F : T → C with N = N F : = { A ∈∈ T | F ( A ) = 0 } . This functor yields a stable ideal I F : = ker F . In applications, F is either a stable homologica l functor to a stable Ab elian categor y or a n exa ct functor to another triangulated c ategory; in either ca se, the ideal ker F is homological a nd N F is the class o f all I F -contractible ob jects. In addition, w e assume F to c ommute with c ountable dir e ct sums , so that I F is compatible with coun table direc t sums. In order to apply Theorem 3.21, it re m ains to prov e that there are enough I F -pro jective ob jects in T . Then the pair of subcategor ies ( h P I F i , N ) is comple- men tary . F or a go o d choice of F , this may be muc h easier than pro ving directly that ( L , N ) is complementary . The c hoice in the following example never helps. But often, there is another ch oice for F that do es. Example 3.30 . The loca lisation functor T → T / N is a p ossible choice for F – that is, it has the r igh t kernel o n ob jects – but it should b e a ba d one b ecause it tells us nothing new ab out N . In fact, the ker F -pro jective o b jects a re precisely the ob jects of L , s o that we hav e not g ained anything. HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 17 W e now discuss a suﬃcien t condition for enough pr o jectiv e ob jects from [18]. The left adj oint of F : T → C is deﬁned o n an o b ject A ∈∈ C if the functor B 7→ C  A, F ( B )  on T is representable, that is, t her e is an o b ject F ⊢ ( A ) of T and a natural isomorphism T ( F ⊢ ( A ) , B ) ∼ = C  A, F ( B )  for all B ∈∈ T . W e say that F ⊢ is deﬁne d on enough obje ct s if, for an y ob ject B of C there is an epimor p hism B ′ → B suc h that F ⊢ is deﬁned o n B ′ . The following theorem as serts that I F has enough pro jective ob jects if F ⊢ is deﬁned on enough ob jects. The statemen t is somewhat more inv olved because it is often useful to shr ink the domain of deﬁnition of F ⊢ to a s uﬃ cient ly big subc ategory PC . Theorem 3.31. L et T , C , and F b e as ab ove, that is, T is a t ri angulate d c ate gory with c ount a ble dir e ct su ms, C is either a stable A b elian c ate gory or a t riangulate d c ate gory, and F : T → C is a stable functor c ommut i ng with c ountable dir e ct sums and either homolo gic al ( if C is A b elian ) or exact ( if C is triangulate d ) . L et I F : = ker F and let N F b e the class of F -c ontra ctible obje cts as ab ove. L et PC ⊆ C b e a sub c ate gory with two pr op erties: ﬁrst, for any A ∈∈ T , ther e exists an epimorphism P → F ( A ) with P ∈∈ PC ; se c ond ly, the left ad joint func- tor F ⊢ of F is deﬁne d on PC , that is, for e ach P ∈ PC , t h er e is an obje ct F ⊢ ( P ) in T with T ( F ⊢ ( P ) , B ) ∼ = T  P, F ( B )  natur al ly for al l B ∈∈ T . Then I F has enough pr oje ctive obje cts, the sub c ate gory N F is r eﬂe ctive, and the p air of lo c alising sub c ate gories  h F ⊢ ( PC ) i , N F  is c omplementary. Pr o of. [18, P ropositio n 3.37 ] shows that I F has enough pro jective ob jects and that any pro jective ob ject is a direct summ a nd o f F ⊢ ( P ) for some P ∈ PC . No w Theorem 3.21 yields the assertions.  Theorem 3.31 is non-trivial even if N F contains o nly zero ob jects, that is, if F ( A ) = 0 implies A = 0 . Then it asserts h F ⊢ ( PC ) i = T . In the situation of Theor em 3.31, w e also understand how ob jects of h F ⊢ ( PC ) i are to b e constructed from the building blocks in F ⊢ ( PC ) . Let C 1 ⋆ C 2 for subcateg ories C 1 , C 2 ⊆ T be the sub category o f all o b jects A for which there is an exact sequence A 1 → A → A 2 → A 1 [1] with A 1 ∈∈ C 1 and A 2 ∈∈ C 2 . W e a bbreviate P F : = F ⊢ ( PC ) and recursively deﬁne P ⋆n F for n ∈ N b y P ⋆ 0 F : = { 0 } and P ⋆n F : = P ⋆n − 1 F ⋆ P F for n ≥ 1 . Theorem 3.3 2 . In t h e situation of The or em 3.3 1 , any obje ct of h F ⊢ ( PC ) i is a homotopy c olimit of an inductive system ( A n ) n ∈ N with A n ∈ P ⋆n F . Pr o of. This follows from Theo rem 3.27. But an extra observ ation is needed here beca use we do no t adjoin direct summands of ob jects in the deﬁnition o f the sub- categories P ⋆n F , so that they do not neces sarily contain all I n F -pro jective o b jects. There is an I F -pro jective resolution with entries in P F , which w e embed in a phant om ca stle. The resulting ce ll ular approximation tow er satisﬁes ˜ A n ∈∈ P ⋆n F , so that Theo rem 3.27 yields an inductive system o f the r equired form.  W e hav e considered tw o cases abov e: homolog ical and e xact functors. F or ho- mological functors with v alues in the catego ry o f Abelian groups, our results were obtained previously by Ap ostolos Beligiannis [3]. Let F : T → Ab Z be a s t able homologica l functor that c omm utes with dir ect sums. Suppose that F is deﬁned on suﬃcient ly many ob jects. Then there m ust b e a s urjectiv e map A ։ Z for which F ⊢ ( A ) is deﬁned. Since Z is pro jectiv e, Z is a retract o f A . Since we assume T to hav e direct sums, idempotent mor phisms in T hav e range ob jects. Thus F ⊢ ( Z ) is deﬁned as w ell. By deﬁnition, F ⊢ ( Z ) is a representin g ob ject for F . Co n versely , if F is r epresen table, then F ⊢ can b e deﬁned o n all free Ab elian groups. Hence the 18 RALF MEYER adjoint F ⊢ is deﬁned o n suﬃcien tly many ob jects if a nd o nly if F is r epresen table. F urthermo re, we ca n take P F to b e the set of all dir ect sums of transla ted copies of the representing o b ject F ⊢ ( Z ) . The a ssumption that F commute with direct sums means that F ⊢ ( Z ) is a compact ob ject. Summing up, if F is a stable homolo gical functor to Ab Z , then our metho ds a pply if and only if F ( A ) ∼ = T ∗ ( D , A ) for a compa ct generator D of T . This situation is considered already in [3 ]. 4. The ABC spectral sequence When we apply a homologica l or c ohomological functor to the phantom tow er, we get ﬁrst an ex act couple and then a sp ect ral sequence. W e call it the ABC sp ectral sequence after A dams, Brinkmann, and Christensen. Its sec ond page only inv olves derived functors. The higher pages can b e describ ed in terms o f the phantom tower, but are more c omplicated. It is r emarkable that the ABC sp ectral sequence is well- deﬁned and functorial on the level o f triangulated categories, that is, all the higher bo und ary ma p s a re uniquely determined and functorial without in tro ducing ﬁner structure like, say , mo del ca tegories. Several results in this section ar e already know to the experts or can b e ex- tracted from [3, 4, 7]. W e hav e included them, nevertheless, to give a r easonably self-contained ac coun t. 4.1. A sp ectral se q uence from the phan tom tow er. W e are going to construct exact co up les out of the pha n tom tow er , extending r esult s of Daniel Christensen [7]. W e ﬁx A ∈∈ T and a phantom tower (3.2) over A . In addition, we let P n : = 0 , N n : = A, and ι n +1 n : = id A for n < 0 . Thus the tr iangles (4.1) P n π n − − → N n ι n +1 n − − − → N n +1 ǫ n − → P n [1] are exact for all n ∈ Z . Of co urse, ι n +1 n rarely b elongs to I for n < 0 . Let F : T → Ab b e a homologica l functor. Deﬁne bigraded Ab elian groups D : = X p,q ∈ Z D pq , D pq : = F p + q +1 ( N p +1 ) , E : = X p,q ∈ Z E pq , E pq : = F p + q ( P p ) , and homogeneous gr oup homomo rphisms D i / / D j         E k Z Z 4 4 4 4 4 4 i pq : = ( ι p +2 p +1 ) ∗ : D p,q → D p +1 ,q − 1 , deg i = (1 , − 1) , j pq : = ( ǫ p ) ∗ : D p,q → E p,q , deg j = (0 , 0) , k pq : = ( π p ) ∗ : E p,q → D p − 1 ,q , deg k = ( − 1 , 0) . Since F is homologica l a nd the tria ngles (4.1) are exact, the chain complexes · · · / / F m ( P n ) π n ∗ / / F m ( N n ) ι n +1 n ∗ / / F m ( N n +1 ) ǫ n ∗ / / F m − 1 ( P n ) / / · · · are ex act for all m ∈ Z . Hence the data ( D , E , i, j, k ) ab o ve is an exact couple (see [15, Section XI.5]). W e brieﬂy recall how an exact couple yields a sp ectral s equence, see also [15, page 336–3 37] or [4]. The ﬁrst s t e p is to form derive d exact c ouples . Let D r : = i r − 1 ( D ) ⊆ D , E r : = k − 1 ( D r )  j (ker i r − 1 ) , HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 19 for a ll r ≥ 1 . Let i ( r ) : D r → D r be the restriction of i ; let k ( r ) : E r → D r be induced b y k : E → D ; and let j ( r ) : D r → E r be induced by the rela t io n j ◦ i 1 − r . It is s h own in [15] that E r +1 ∼ = H ( E r , d ( r ) ) for all r ∈ N , where d ( r ) = j ( r ) k ( r ) ; the map d ( r ) has bidegree ( − r, r − 1) . W e call this spectral sequence the ABC sp e ctr al se quenc e for F and A . W e ar e going to descr ibe the derived ex act couples explicitly . First, we claim that (4.2) D r +1 p − 1 ,q ∼ =      F p + q ( A ) for p ≤ 0 , F p + q ( A )  F p + q : I p ( A ) for 0 ≤ p ≤ r , F p + q ( N p − r )  F p + q : I r ( N p − r ) for r ≤ p . By deﬁnition, D r +1 p − 1 ,q for r ∈ N is the range of the map i r : D p − 1 − r,q + r → D p − 1 ,q . This is the iden tity map on F p + q ( A ) for p ≤ 0 , the map ι p ∗ : F p + q ( A ) → F p + q ( N p ) for 0 ≤ p ≤ r , and the ma p ( ι p p − r ) ∗ : F p + q ( N p − r ) → F p + q ( N p ) for r ≤ p . No w r ecall that the maps ι n m are I n − m -versal for all n ≥ m ≥ 0 a nd use Lemma 3 .7 . Prop osition 4.3. L et 1 ≤ r < ∞ . Then we have E r +1 pq ∼ = 0 for p ≤ − 1 ; for 0 ≤ p ≤ r , ther e is an exact se quenc e 0 → F p + q +1 ( N p ) F p + q +1 : I r +1 ( N p ) ( ι p +1 p ) ∗ − − − − − → F p + q +1 ( N p +1 ) F p + q +1 : I r ( N p +1 ) → E r +1 pq → F p + q : I p +1 ( A ) F p + q : I p ( A ) → 0; for p ≥ r , ther e is an exact se quenc e 0 → F p + q +1 ( N p ) F p + q +1 : I r +1 ( N p ) ( ι p +1 p ) ∗ − − − − − → F p + q +1 ( N p +1 ) F p + q +1 : I r ( N p +1 ) → E r +1 pq → F p + q : I r +1 ( N p − r ) F p + q : I r ( N p − r ) → 0 . Final ly, for r = ∞ , let Bad pq : = F q ( N p ) . [ r ∈ N F q : I r ( N p ); then we get E ∞ pq ∼ = 0 for p ≤ − 1 , and exact se quenc es 0 → Bad p,p + q +1 → Bad p +1 ,p + q +1 → E ∞ pq → F p + q : I p +1 ( A ) F p + q : I p ( A ) → 0 . Ther efor e, if S r ∈ N F q : I r ( N p ) = F q ( N p ) for al l p ∈ N , then E ∞ pq ∼ = F p + q : I p +1 ( A ) F p + q : I p ( A ) , that is, the ABC sp e ctr al se quenc e c onver ges towar ds F m ( A ) , and the induc e d in- cr e asing ﬁltr ation on the limi t is  F m : I r ( A )  r ∈ N . Pr o of. W e hav e E r +1 pq ∼ = 0 for p ≤ − 1 b ecause already E pq = E 1 pq = 0 . Let p ≥ 0 . W e use (4.2) and the exactness o f the derived exact couple ( D r +1 , E r +1 ) to com- pute E r +1 b y an extensio n inv olving the k ernel a nd co k ernel of the restriction of i to D r +1 . Since ι m +1 m is I -versal, x ∈ F ( N m ) satisﬁes ( ι m +1 m ) ∗ ( x ) ∈ F : I r ( N m +1 ) if and only if x ∈ F : I r +1 ( N m ) . Plugging this into the ex t ension that descr ibes E r +1 , we get the assertion, at least for ﬁnite r . The case r = ∞ is similar. Now (4.4) E ∞ : = \ r ∈ N k − 1 ( i r D ) . [ r ∈ N j (ker i r ) . 20 RALF MEYER The injecti vity of the map Bad p,p + q +1 → Bad p +1 ,p + q +1 follows from the exa ctness of c olimit s of Abelia n gro ups. Using i ( D ) = ker j , ker i = k ( E ) , and (4.4), we get a short exact seq u ence (4.5) 0 → D pq .  i ( D p − 1 ,q +1 ) + [ ker i r  → E ∞ pq → D p − 1 ,q ∩ ker i ∩ \ i r ( D ) → 0; the ﬁrst map is induced b y j , the second one by k . The in tersection T i r ( D ) is describ ed by (4.2) for r = ∞ , so that the third case in (4.2) is missing. Hence the quotient in (4.5) is ker i ∩ \ i r ( D ) ∼ = F p + q : I p +1 ( A )  F p + q : I p ( A ) . The versalit y of the maps ι n m yields S ker i r ∩ D p − 1 ,q = S r F p + q : I r ( N p ) . Hence the kernel in (4.5) is Bad p +1 ,p + q +1 / Bad p,p + q +1 . If Bad = 0 , then the g roups E ∞ pq for p + q = m are the subquo t ient s o f the ﬁltration F m : I p ( A ) on F m ( A ) . Mo reo ver, since Bad 0 m = F m ( A ) . S p ∈ N F m : I p ( A ) , our hypothesis includes the statemen t that F m ( A ) = S p ∈ N F m : I p ( A ) .  Dual cons t r uctions apply to a co homological functor G : T op → Ab . Equa- tion (4.1) yields a sequence of exact chain complexes · · · / / G m − 1 ( P n ) ǫ ∗ n / / G m ( N n +1 ) ( ι n +1 n ) ∗ / / G m ( N n ) π ∗ n / / G m ( P n ) / / · · · . Therefore, the following deﬁnes an exa ct couple: ˜ D pq : = G p + q +1 ( N p +1 ) , ˜ E pq : = G p + q ( P p ) , ˜ D i / / ˜ D j         ˜ E k Y Y 3 3 3 3 3 3 i pq : = ( ι p +1 p ) ∗ : ˜ D p,q → ˜ D p − 1 ,q +1 , deg i = ( − 1 , 1) , j pq : = π ∗ p +1 : ˜ D p,q → ˜ E p +1 ,q , deg j = (1 , 0) , k pq : = ǫ ∗ p : ˜ E p,q → ˜ D p,q , deg k = (0 , 0) . Again we form derived exact couples ( ˜ D r , ˜ E r , i r , j r , k r ) , a nd ( ˜ E r , d r ) with d r : = j r k r is a sp ectral sequence. The ma p d r has bidegree ( r, 1 − r ) . W e call this sp ectral sequence the ABC sp e ctr al se quenc e for G a nd A . W e can describ e the der iv ed exact couples a s ab ov e. T o begin with, (4.6) ˜ D p − 1 ,q r +1 =      G p + q ( A ) for p ≤ 0 , I p G p + q ( A ) for 0 ≤ p ≤ r , I r G p + q ( N p − r ) for r ≤ p . Prop osition 4.7. L et 1 ≤ r < ∞ . Then ˜ E pq r +1 ∼ = 0 for p ≤ − 1 , and ther e ar e exact se qu e nc es 0 → I p G p + q ( A ) I p +1 G p + q ( A ) → ˜ E pq r +1 → I r G p + q +1 ( N p +1 ) ( ι p +1 p ) ∗ − − − − − → I r +1 G p + q +1 ( N p ) → 0 for 0 ≤ p ≤ r and 0 → I r G p + q ( N p − r ) I r +1 G p + q ( N p − r ) → ˜ E pq r +1 → I r G p + q +1 ( N p +1 ) ( ι p +1 p ) ∗ − − − − − → I r +1 G p + q +1 ( N p ) → 0 for p ≥ r . F or r = ∞ , let g Bad pq : = \ r ∈ N I r G q ( N p ) , HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 21 then we get E ∞ pq ∼ = 0 for p ≤ − 1 , and exact se quenc es 0 → I p G p + q ( A ) I p +1 G p + q ( A ) → ˜ E pq ∞ → g Bad p +1 ,p + q +1 → g Bad p,p + q +1 . Ther efor e, if T r ∈ N I r G q ( N p ) = 0 for al l p, q , then ˜ E pq ∞ ∼ = I p G p + q ( A ) I p +1 G p + q ( A ) , that is, the ABC sp e ctr al se quen c e c onver ges towar ds G m ( A ) , and the induc e d de- cr e asing ﬁltr ation on the limi t is  I r G m ( A )  r ∈ N . Pr o of. This is pr o ved exactly a s in the homological case.  Notice that w e do not claim that t he maps g Bad p +1 ,p + q +1 → g Bad p,p + q +1 are surjective. If G is representable, that is, G ( A ) ∼ = T ( A, B ) for some B ∈∈ T , then the question whether the maps g Bad p +1 ,p + q +1 → g Bad p,p + q +1 are surjective is r elated to the question whether I ∞ · I = I ∞ . In this ca se, g Bad p, ∗ = I ∞ ( N p [ ∗ ] , B ) . Since the maps ι n +1 n are I -versal, range  g Bad p +1 , ∗ → g Bad p, ∗  = I ∞ ◦ I ( N p [ ∗ ] , B ) ⊆ I ∞ ( N p [ ∗ ] , B ) . Theorem 4.8. St arti ng with the se c ond p age, the ABC sp e ctr al se quenc es for ho- molo gic al and c ohomolo gic al functors ar e indep en d en t of auxiliary choic es and func- torial in A . Their se c ond p ages c ontain the derive d fun c tors: E 2 pq ∼ = L p F q ( A ) , ˜ E pq 2 ∼ = R p G q ( A ) . Pr o of. W e only formulate the pro of for homologic al functors; the cohomological case is s im ilar. The map d : = j k : E → E is induced by the maps δ p : P p +1 → P p [1] in the phantom tow er . B y Lemma 3.3, these ma ps form a P - pro jectiv e resolution of A . This together with coun ting o f susp ensions yields the descr ipt ion of E 2 pq . Let f : A → A ′ be a mo rphism in T . By Lemma 3.3, it lifts to a mor phism betw e en the phantom tow ers over A and A ′ . This induces a morphism o f exact couples and hence a morphism o f sp ectral sequences. The maps P n → P ′ n form a chain map b et ween the I -pro jective reso lu tions em b edded in the phan tom tow ers . This chain map lifti ng o f f is unique up to c hain homotopy (see [18, Prop osition 3.36]). Hence the induced map on E 2 is unique and functoria l. W e get E r for r ∈ N ≥ 2 ∪ {∞} as subq uotien ts of E 2 . Since our map o n E 2 is part of a mor ph ism of exact couples, it descends to these subquotients in a unique and functorial way . Thu s E r is functorial for all r ≥ 2 .  The naturality of the ABC spectra l sequence do es n ot mean that the exact sequences in Prop ositions 4 .3 and 4 .7 are natural. They use the exa ct couple under- lying the ABC sp ectral sequence, a nd this exact couple is not na t ural. It is easy to chec k that the maps E r +1 pq → F p + q : I p +1 ( A )  F p + q : I p ( A ) in Prop osition 4.3 ar e canonical for 0 ≤ p ≤ r ≤ ∞ . But F p + q : I r +1 ( N p − r )  F p + q : I r ( N p − r ) depends on auxiliary choices. Our results so far only formulate the conv er gence problem for the ABC sp ectral sequence. It remains to c heck whether the relev ant o bstructions v anish. The easy sp ecial ca se wher e the pr o jectiv e resolutions have ﬁnite length is alrea dy dealt with in [7]. Recall that P n I denotes the class of I n -pro jective o b jects. Lemma 4.9. L et k ∈ N and A ∈∈ P k I . Then ι m + k m = 0 and N m ∈∈ P k I for al l m ∈ N . Pr o of. Since ι m + k m is I k -versal, we hav e ι m + k m = 0 if a nd only if N m ∈∈ P k I . W e prov e ι m + k m = 0 by induction on m . The case m = 0 is clea r beca use N 0 = A . 22 RALF MEYER If ι m + k m = 0 , then ι m + k m +1 factors through the map ǫ m : N m +1 → P m [1] by the lo ng exact homology s equence for the triangles (4.1). If w e compo se the resulting map P m [1] → N m + k with ι m + k +1 m + k ∈ I , we ge t zero becaus e P m [1] ∈∈ P . Th us ι m + k +1 m +1 v anis h es as well.  Prop osition 4.10. L et F : T → Ab b e a homolo gic al functor and let m ∈ N . If A ∈∈ P m +1 I , then the ABC sp e ctr al se qu enc e for F and A c ol lapses at E m +2 and c onver ges towar ds F ∗ ( A ) , and its limiting p age E ∞ = E m +2 is supp orte d in the r e gion 0 ≤ p ≤ m . If, inste ad, A has a P -pr oje ctive r esolution of length m , then the ABC sp e ct r al se quenc e for F and A is supp orte d in the r e gion 0 ≤ p ≤ m fr om the se c ond p age onwar d, so that it c ol lapses at E m +1 . If, in add ition, A b elongs to the lo c alising sub c ate gory of T gener ate d by P I , then t h e sp e ctra l se quenc e c onver ges towar ds F ∗ ( A ) . Similar assertions hold in the c ohomolo gic al c ase. Pr o of. W e only wr it e down the argument in the homological ca se. If A ∈∈ P m +1 I , then N p ∈∈ P m +1 I for all p ∈ N b y Lemma 4.9. Therefore, F : I r ( N p ) = F ( N p ) for all r ≥ m + 1 . P lu gging this in to P roposition 4.3, w e get E r pq = 0 for r ≥ m + 2 and p ≥ m + 1 . This forces the b oundary ma ps d r to v a nish for r ≥ m + 2 , so that E ∞ pq = E m +2 pq . Suppos e now that A has a pro jective resolution of length m . Embed such a resolution in a phan tom tow er by L emm a 3.3, so that P p = 0 and N p ∼ = N p +1 for p > m . Then E r is supp orted in the region 0 ≤ p ≤ m for all r ≥ 1 . F or r ≥ 2 , this holds for any choice of phan tom to wer b y Theorem 4.8. As a consequence, d r = 0 for r ≥ m + 1 and hence E ∞ = E m +1 . Suppos e, in addition, that A b elongs to the lo calising subca tegory of T genera t ed b y P I . W e cla im that N p ∼ = 0 for p > m . This implies that the ABC sp ectral sequence conv er ges towards F ∗ ( A ) . If D ∈ P I , then there a re ex act sequences T ∗ +1 ( D , N p ) ֌ T ∗ ( D , P p − 1 ) ։ T ∗ ( D , N p − 1 ) for all p ∈ N ≥ 1 (see the pr oof o f Lemma 3.3). Therefore, T ∗ ( D , N p ) = 0 for p > m if D ∈∈ P I . The cla ss of ob jects D with this prop ert y is localising, tha t is, closed under susp ensions, direct sums, direct summands, and exact triangles. Hence it contains the loca lising subca t egory generated by P I . This includes A = N 0 b y assumption. Since it contains a ll P n as well, it contains N n for all n ∈ N b ecause o f the exact triangles in the phantom tow er, so that T ∗ ( N n , N p ) v anishes fo r a ll n ∈ N . Thu s T ∗ ( N p , N p ) = 0 , for cing N p = 0 .  If A b elongs to the loca lising sub category generated b y the I -pro jective ob- jects, then the existence o f a pro jective resolution o f length n implies that A is I n -pro jective. This fails without an additional h yp othesis b ecause I -con tr actible ob jects hav e pro jective res olutions of length 0 . The conv erse assertion is usually far fro m true (see [7]). Prop osition 2 .2 shows that an ob ject A of T is I 2 -pro jective if and only if ther e is a n exact triangle P 2 → P 1 → A → P 2 [1] with I -pro jective ob jects P 1 and P 2 . The resulting chain complex 0 → P 2 → P 1 → A is an I - pro jectiv e resolution if and only if the map A → P 2 [1] is an I -phan to m map, if and only if the map P 2 → P 1 is I -monic. But this need no t b e the case in general. Recall that th e derived functor s of the contra v aria n t functor A 7→ T ∗ ( A, B ) are the ex tensio n gr oups E xt ∗ T , I ( A, B ) . These ag ree with extension gro up s in the Abelian approximation, that is, the targ et category of the univ ers al I -exa ct stable homologica l functor. In particular, Ext 0 T , I ( A, B ) is the space of morphisms b et ween HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 23 the imag es of A and B in the Abelia n approximation (see [18]). Theorem 4 .8 and Prop osition 4 .7 yield exa ct sequences (4.11) 0 → T ( A, B ) I ( A, B ) → Ext 0 T , I ( A, B ) → I ( N 1 [ − 1] , B ) ( ι 1 0 ) ∗ − − − → I 2 ( A [ − 1] , B ) → 0 and (4.12) 0 → I ( A, B ) I 2 ( A, B ) → Ext 1 T , I ( A, B ) → I ( N 2 [ − 1] , B ) ( ι 2 1 ) ∗ − − − → I 2 ( N 1 [ − 1] , B ) → 0 . In particular , we get injectiv e maps T / I ( A, B ) → Ext 0 T , I ( A, B ) , I / I 2 ( A, B ) → Ext 1 T , I ( A, B ) . But these maps need not be surjective. What they do is easy to understand. The ﬁrst map is simply the functor from T to the Ab elian a pp roximation. The second map em b eds a mor ph ism in I in an exact triangle. This triang le is I -exact b e- cause it inv olves a phantom map, and hence provides a n extension in the Abelian approximation. The higher quotien ts I n / I n +1 ( A, B ) are a lso related to Ext n T , I ( A, B ) , but this is merely a rela t io n in the formal sense, that is, it is no lo nger a map. T o construct this re lation, we use the I n -versalit y of the map ι n : A → N n . Thus any map f ∈ I n ( A, B ) factors through a map ˆ f : N n → B , and we can choo se ˆ f ∈ I ( N n , B ) if f ∈ I n +1 ( A, B ) . Now we use the map T / I ( N n , B ) → E x t n T , I ( A, B ) provided by Theorem 4 .8 and Prop osition 4 .7 . Since f does not determine the c lass of ˆ f in T / I ( N n , B ) uniquely , we only get a relation. The ambiguit y in this rela t io n disapp ears on the n th page of the ABC spectra l sequence b y P roposition 4.7. Once we hav e chosen ˆ f as a bov e, w e can also extend it to a morphism b et ween the phantom tow er s of A and B that shifts degrees down by n – the extension to the left of N n is induced by ˆ f ◦ ι n m : N m → B for m < n and v anishes o n P m for m < n . Thu s w e get a mor phism b et ween the ABC spectr al seq u ences for A and B for any homologica l o r cohomolo gical functor – s h ifting degrees down b y n , of course. 4.2. An equiv alent exact couple. The cellular approximation tow er pro duces a sp ectral seq uence in the same wa y as the phantom tow er. W e extend the phantom tow er to n < 0 b y ˜ A n = 0 and P n = 0 for a ll n < 0 . The tria ngles (3.9) are exact for all n ∈ Z . A homological functor F ma ps these exact triangles to exact c ha in complexes · · · / / F m ( ˜ A n ) α n +1 n ∗ / / F m ( ˜ A n +1 ) σ n ∗ / / F m ( P n ) κ n ∗ / / F m − 1 ( ˜ A n ) / / · · · . As ab o ve, these amount to a n exact co upl e D ′ i ′ / / D ′ j ′         E ′ k ′ [ [ 6 6 6 6 6 6 D ′ pq : = F p + q ( ˜ A p +1 ) , E ′ pq : = F p + q ( P p ) , i ′ pq : = ( α p +2 p +1 ) ∗ : D ′ p,q → D ′ p +1 ,q − 1 , j ′ pq : = ( σ p ) ∗ : D ′ p,q → E ′ p,q , k ′ pq : = ( κ p ) ∗ : E ′ p,q → D ′ p − 1 ,q . Part of the commuti ng dia gram (3.10) as serts that the identi ty maps E → E ′ and the ma ps γ p +1 , ∗ : D pq → D ′ pq form a mo rphism of exact co uples betw een the exact couples from the phantom tow er and the cellular approximation tow er . This induces a morphism betw e en the resulting sp ectral s equences. Since this morphism acts ident ically on E 1 , the induced mor phisms on E r m ust be in vertible for all 24 RALF MEYER r ∈ N ≥ 1 ∪ {∞} . Hence our new sp ectral sequence is isomor p hic to the ABC sp ectral sequence. Although the sp ectral seq uen ces are isomorphic, the underlying exa ct co up les are diﬀerent a nd thus provide isomo rphic but diﬀerent descriptions of E ∞ . An imp ortan t diﬀerence b et ween the tw o exa ct couples is that D ′ pq = 0 for p ≤ 0 . Hence a n y element of D ′ pq is annihilated by a suﬃciently high p o wer of i . Therefore, the kernel in (4.5) v anishes and E ∞ pq ∼ = D ′ p − 1 ,q ∩ ker i ′ ∩ \ r ∈ N ( i ′ ) r ( D ′ ) . Let L pq : = lim − → range  α r p ∗ : F q ( ˜ A p ) → F q ( ˜ A r )  ; these spaces deﬁne an increasing ﬁltration ( F pq ) p ∈ N on lim − → F q ( ˜ A r ) – we for m the limit with the maps α n m . Using (4.5) a nd the exactness of co lim its of Abelia n gr oups, we get isomorphisms E ∞ pq ∼ = L p +1 ,p + q L p,p + q . Hence E ∞ pq conv erge s tow ards lim − → F q ( ˜ A r ) a nd induces the ﬁltration ( L p,p + q ) o n its limit – without a n y assumption on the ideal or the homological functor. In the cohomologica l case, the exact triang les (3.9) yield an ex act couple as well, and the morphism be t ween the tw o exact co uples from the cellular approxima- tion tower a nd the phantom tow er induces an isomorphism betw een the asso ciated sp ectral seq uences. Again, we g et a new description of ˜ E ∞ . But the result is not as simple a s in the ho m ological ca se b ecause the pro jective limit functor for Abelian g roups is not ex act. Let ˜ L pq : = \ r ≥ p range  α r ∗ p : G q ( ˜ A r ) → G q ( ˜ A p )  . Then ˜ E pq ∞ ∼ = ker  ˜ L p +1 ,p + q → ˜ L p,p + q  . In genera l, we ca nnot say muc h more than this. If ˜ L pq is the range of the map lim ← − r G q ( ˜ A r ) → G q ( ˜ A p ) for a ll p and q , then the ABC sp ectral sequence co n verges tow ards lim ← − r G q ( ˜ A r ) and induces on this limit the decreasing ﬁltration by the sub- spaces ker  lim ← − r G q ( ˜ A r ) → G q ( ˜ A p )  for p ∈ N . 5. Convergence o f the ABC spectral sequence There is a n o b vious obstruction to the c on vergence of the ABC sp ectral sequence: the sub category N I of I -co n tractible ob jects. Since I -derived functors v a n ish on N I , the s pectral sequence ca nnot co n verge tow a rds the original functor unless it v anishes on N I as well. A t b est, the ABC sp ectral sequence may co n verge to the lo c alisation of the g iv en functor at N I . W e sho w that this is indeed the case for homolog ical functors that comm ute with direct sums, provided the ideal I is compatible with direct sums. The situation for cohomo logical functors is less satisfactory b ecause the pro jective limit functor for Ab elian groups is not exa ct . W e contin ue to ass um e throughout this section that the category T has co un table direct sums. V ario us notions of conv ergence of spectra l sequences are discussed in [4]. The following results deal only with strong conv erg e nce. Theorem 5.1. L et I b e a homolo gic al ide al c omp atible with dir e ct sums in a tri- angulate d c ate gory T ; let F : T → Ab b e a homolo gic al functor t h at c ommut es with HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 25 c ountable dir e ct sums, and let A ∈∈ T ; let L F b e the lo c alisation of F at N I . Then F : I 2 ∞ ( A ) = F : I ∞ ( A ) = [ r ∈ N F : I r ( A ) = rang e  L F ( A ) → F ( A )  , and the ABC sp e ctr al se quenc e for F and A c onver ges towar ds L F ( A ) with the ﬁltr ation  L F : I k ( A )  k ∈ N . W e have S k ∈ N L F : I k ( A ) = L F ( A ) . Pr o of. Lemma 3.7 implies that F : I r ( A ) is the range of α r ∗ : F ( ˜ A r ) → F ( A ) for all r ∈ N and that F : I ∞ ( A ) is the range of F  L ˜ A r  → F ( A ) beca u se (3.29) is an I ∞ -pro jective r esolution. Now F  L ˜ A r  = L F ( ˜ A r ) shows that F : I ∞ ( A ) is the union o f F : I r ( A ) for r ∈ N . Let ˜ A b e as in (3.26), so that ˜ A = L ( A ) and L F ( A ) = F ( ˜ A ) . Since the inductive limit functor for Abelia n gr oups is exact, the map id − S on L F ( ˜ A r ) is injectiv e and ha s cokernel lim − → F ( ˜ A r ) . Since the top row in (3.26) is an exa ct triangle, the long exa ct s equence yields F ( ˜ A ) ∼ = lim − → F ( ˜ A r ) . A s a consequence, the rang e of f ∗ : F ( ˜ A ) → F ( A ) is equal to F : I ∞ ( A ) . Since ˜ A is I 2 ∞ -pro jective by Theorem 3.27 and f : ˜ A → A is an I -equiv alence, this map is an I 2 ∞ -pro jective r esolution o f A . Hence the r ange of f ∗ also agrees with F : I 2 ∞ ( A ) by Lemma 3.7. Esp ecially , F : I ∞ ( A ) = F ( A ) if A ∈∈ h P I i . F or such A , all o b jects that o ccur in the phantom castle b elong to h P I i as well, so that F ( N p ) = S r ∈ N F : I r ( N p ) for all p ∈ N . Hence Pro position 4.3 yields the conv er gence of the ABC sp ectral sequence to F ( A ) a s asser t ed. Since the ABC spectra l sequences fo r A and ˜ A are isomorphic b y Prop osition 3.28, w e get co n vergence tow ards L F ( A ) for general A .  The conv erg ence of the ABC sp ectral sequences is more pr oblematic for a c oho- mological functor G : T op → Ab b ecause pro jective limits of Abelia n gro ups ar e not exact. In the following, w e assume that G maps dir e ct sums to dir e ct pr o ducts – this is the co rrect compatibility with direct sums for cont r a v ariant functors. The exactness of the ﬁrst row in (3.2 6 ) yields an exa ct s equence (5.2) lim ← − 1 G ∗− 1 ( ˜ A n ) ֌ G ∗ ( ˜ A ) ։ lim ← − G ∗ ( ˜ A n ) for an y A (this also follows from [18, Theorem 4.4] applied to the ideal I ∞ ). F ur- thermore, G ∗ ( ˜ A ) = R G ∗ ( A ) . Since (3.26) is an I ∞ -pro jective r esolution, we have lim ← − 1 G − 1 ( ˜ A n ) ∼ = I ∞ G ( ˜ A ) . The same a rgumen t as in the homological case yields (5.3) I ∞ G ( A ) = \ n ∈ N I n G ( A ) . Using compatibility of G with dire ct sums, we can als o rewrite the obstructions to the conv er gence o f the ABC s pectral sequence in Prop osition 4.7: g Bad pq ∼ = I ∞ G q ( ˜ N p ) , where ˜ N p is the p th ob ject in a phan tom tow er o ver ˜ A instead of A . The sp ectral sequence conv er ges towards R G ( A ) if these obstructions all v anish. Prop osition 5 .4. L et I b e a homolo gic al ide al with enough pr oje ctives that is c om- p atible with dir e ct sums, and let G : T op → Ab b e a c ohomolo gic al functor that maps dir e ct sums to dir e ct pr o ducts. L et A ∈∈ T and let L ( A ) ∈∈ h P I i b e its P I -c el lular appr oximation. If L ( A ) is I ∞ -pr oje ctive, then t h e ABC s p e ct r al se qu enc e for A and G c onver ges towar ds R G ( A ) = G ◦ L ( A ) . 26 RALF MEYER Pr o of. Pr oposition 3.2 8 implies that A and L ( A ) have canonically is omorphic ABC sp ectral sequences. Hence we may replace A b y L ( A ) and assume that A itself is I ∞ -pro jective. By Prop osition 2.3, A is a direct summand of L n ∈ N A n with I n -pro jective ob jects A n . The ABC spectra l se quence for eac h A n conv erge s b y Prop osition 4 .10 . Since I is compatible with co un table direct sums, a direct sum of phan tom castles o ver A n is a phant o m castle over L A n . Th us the ABC sp ectral sequence for L n ∈ N A n is the direct pro du ct o f the ABC sp ectral sequences for A n ; here we use that G maps dir ect sums to direct pro ducts. Hence the ABC sp ectral sequence for L n ∈ N A n conv erge s tow ards Q G ( A n ) = G  L A n  . Since the ABC sp ectral sequence is an a ddit ive functor on T , this implies that the ABC sp ectral seq u ence for a n y direct summand of L A n conv erge s. This y ields the co n vergence of the ABC sp ectral se quence for L ( A ) , as desir ed.  6. A classical spec ial case Before we a pply our r esults to equiv aria n t biv ariant K-theory , we brieﬂy discuss a more classical applica t io n in homological algebra, where we recover results by Marcel Bökstedt a nd Amnon Neeman [5] a nd where the ABC sectral sequence sp ecialises to a spectra l sequence due to Alexander Gro t hendieck. Let A b e an Abelia n categ ory with eno u gh pr o jectiv e ob jects and exact countable direct sums. Let Ho( A ) b e the homotopy ca t egory of chain complexes over A . W e require no ﬁniteness conditions, so that Ho( A ) is a triangulated catego ry with countable direct sums. W e are interested in the derived category of A and therefore wan t to lo calise a t the full sub category N ⊆ Ho( A ) of exa ct chain complexes. This subc ategory is lo calising b ecause co un table direct sums of exact chain complexes are ag ain exact by a ssumpti on. The obvious functor deﬁning this subca tegory N is the functor H : Ho( A ) → A Z that maps a c hain complex to its homolog y . The functor H is a stable ho m ological functor that co mm utes with direct s um s. Hence its kernel I H is a homo logical ideal that is compatible with direct sums. Let PA Z ⊆ A Z be the full sub category of pro jective ob jects. Since w e assume A to hav e enough pro jective ob jects, any ob ject of A Z admits an epimorphism from an ob ject in PA Z . It is easy to see that the left a djoin t of the homolo gy functor is deﬁned o n PA Z and maps a sequence ( P n ) of pro jective ob jects to the chain complex ( P n ) with v anishing b oundary map. Since this functor is clearly fully faithful, w e use it to view PA Z as a full sub category of Ho( A ) , o mitt ing the functor H ⊢ from our notation. Using the criterion of [18], it is ea sy to ch eck that the functor H ab o ve is the univ er sal I H -exact homolo gical functor . Theorems 3.31 a nd 3.32 apply here. The ﬁrst one shows that ( h PA Z i , N ) is a complemen ta ry pair of sub categories. Thus h PA Z i is equiv alent to the derived category of A . F urthermor e, any ob ject o f the derived category is a homotopy colimit of a diag ram with en tries in ( PA Z ) ⋆n for n ∈ N . Let F : A → Ab b e an additive cov a rian t functor that commutes with direct sums. W e extend F to an exact functor Ho( F ) : Ho( A ) → Ho( Ab ) . Let ¯ F q = H q ◦ Ho( F ) : Ho( A ) → Ab be the functor that maps a chain complex C • to the q th homology of Ho( F )( C • ) . This is a homolog ical functor. Its derived functors with r espect to I H are computed in [18]: for a chain complex C • , we have L p ¯ F q ( C • ) = ( L p F )  H q ( C • )  , HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 27 that is, we apply the usual deriv ed functor s of F to the homolog y of C • . Thus the ABC spectral sequence computes the homolo gy of the total derived functor of F applied to C • in terms of the derived functors of F , applied to H ∗ ( C • ) . Such a sp ectral seq uence was already constructed by Alexander Grothendieck. 7. Constr uction of the Ba um–Connes assembl y map Finally , we apply o ur g eneral machinery to construct the Baum–Connes asse m bly map with co eﬃcients ﬁrst for lo cally co m pact groups and then for certain discrete quantum gro up s. In the gro up ca se, we get a simpler a rgumen t than in [1 6 ]. 7.1. The ass em bly map for l o cally compact groups. Let G b e a second co un t- able lo cally compact gro u p and let KK G be the G -equiv a rian t Kaspa ro v category; its ob jects are the separa ble C ∗ -algebra s with a strongly cont inuous action of G , its morphism spa ce A → B is KK G 0 ( A, B ) . It is shown in [16] that this categor y is triangulated (we must exclude Z / 2 -g raded C ∗ -algebra s for this). The ca tegory KK G has coun table direct sums – they a re just direct sums of C ∗ -algebra s. But uncoun table direct sums usually do no t exist b ecause of the sepa- rability assumption in the deﬁnition of KK G , which is needed to mak e the ana lysis work. Alternative deﬁnitions of biv aria n t K-theory b y Joachim Cuntz [8] still work for non-separ able C ∗ -algebra s, but it is not clear whether dir ect sums of C ∗ -algebra s remain dire ct sums in this category beca use the deﬁnition of the Ka sparo v gro ups for insepa rable C ∗ -algebra s inv olves colimits, which do not commute with the direct pro ducts that appear in the deﬁnition of the dir ect s um . With enough eﬀort, it should be possible t o extend K K G to a category with uncount able direct sums. But it seems easier to av oid this by imp osing c ardinalit y restrictions on dir ect sums. F or an y closed subg roup H ⊆ G , w e hav e induction and res t riction functors Ind G H : KK H → KK G , Res H G : KK G → KK H ; the la t ter functor is quite trivial a nd simply forgets pa rt of the gro u p action. These functors give r ise to tw o subcatego ries of KK G , which play a cr ucial role in [16]. Deﬁnition 7. 1. Let F b e the set of a ll co mpact subgr oups of G . C C : = { A ∈∈ KK G | Res H G ( A ) = 0 for all H ∈ F } , C I : = { Ind G H ( A ) | A ∈∈ KK H and H ∈ F } . Whereas the sub category C C is lo calising by deﬁnition, C I is not. Therefore, the lo calising subcateg ory it gener ates, hC I i , plays a n impo rtan t r ole a s w ell. Since G acts prop erly on ob jects of C I , they satisfy the Baum–Connes conjecture, that is, the Baum–Connes assembly map is an isomorphism for coeﬃcients in C I . Since domain and tar get of the assembly map are e xact functors on KK G , this extends to the catego ry hC I i . On o b jects of C C the domain of the Ba um–Connes as sem bly ma p is known to v anish, so that the Baum–Co nn es conjecture predicts K ∗ ( G ⋉ r A ) = 0 for A ∈∈ C C . On a tec hnical level, the main to ol in [16] is that the pair of sub categories ( hC I i , C C ) is complementary . Hence the Baum–Connes assembly map is determined b y what it do es on these tw o sub categories. This implies that its domain is the lo calisation of the functor A 7→ K ∗ ( G ⋉ r A ) a t C C and that the assembly map is the natural tra nsformation from this localis ation to the original functor . Put diﬀerently , the Baum–Co nnes assembly map is the o nly natura l transfor ma- tion from an e xact functor on KK G to the functor K ∗ ( G ⋉ r ) that is an is omorphism on C I and whose domain v anishes on C C (w e give s ome more details ab out this ar- gument in the related quant um gro up ca se b elo w). 28 RALF MEYER In order to prov e that ( hC I i , C C ) is co m plemen ta ry , we in tro duce the follo wing ideal: Deﬁnition 7. 2. Let I = T H ∈F ker Res H G . This ideal consists of the mo rphisms that vanish for c omp act sub gr oups in the notation of [16]. Clea rly , an ob ject b elongs to C C if and only if its iden tity ma p belo ngs to I , that is, N I = C C . Moreov er, [16, Prop osition 4 .4] implies that ob jects of C I a re I -pro jective; even mor e, f ∈ I ( A, B ) if and only if f induces the zero map KK G ∗ ( D , A ) → KK G ∗ ( D , B ) for all D ∈∈ C I . W e can also describ e I as the k er nel of a s ingle functor: F = (Res H G ) H ∈F : KK G → Y H ∈F KK H . The functor F commutes with direct sums be cause each functor Res H G clearly does so. Hence I is compatible w it h coun ta ble direct sums. The following theorem contains the main asser t ion in [16, Theorem 4.7]. W e will provide a simpler pro of here than in [1 6]. Theorem 7.3. The pr oje ctive obje ct s for I ar e the r et r acts of dir e ct su ms of obje cts in C I , and the ide al I has enough pr oje ct i ve obje cts. Henc e the p air of sub c ate gories ( hC I i , C C ) is c omplementary. Pr o of. As in Theo rem 3.3 1 , we study the partially deﬁned left a dj o in t of the func- tor F above o r, equiv a len tly , of the functors Res H G for H ∈ F . The discrete case is particularly simple b ecause then all H ∈ F a re op en sub- groups. If H ⊆ G is o pen, then Ind G H is left adjoin t to Res H G . Thus we ma y ta k e PC = Q H ∈F KK H in Theorem 3.31 and get F ⊢  ( A H ) H ∈F  = L H ∈F Ind G H ( A H ) . Notice that the set F is countable if G is discrete, so that this deﬁnition is legiti- mate. It follows that I has enough pro jective ob jects and that they are all direct summands of L H ∈F Ind G H ( A H ) for suitable families ( A H ) , as a sserted. F or lo cally compact G , the a rgumen t g et s more complicated b ecause the functor Res H G do es not alw ays hav e a left adjoin t, and if it has, it need not be simply Ind G H . But there a re still e nough co mp act subgroups H for which the left adjoin t is deﬁned on enough H -C ∗ -algebra s and close eno ugh to the induction functor for the arg um ent a bov e to go through. A go od wa y to understand this is the duality theory developed in [9, 10]. This is r elev a n t b ecause the induction functor provides an equiv alence of categories KK H ≃ KK G ⋉ G/H , where w e use the g roupoid G ⋉ G/H , that is, w e consider G -equiv a rian t bundles of C ∗ -algebra s ov er G/H . This equiv alence o f ca tegories reﬂects the equiv alence b et ween the g roupoids H and G ⋉ G/H . Iden tifying KK H ≃ KK G ⋉ G/H , the re striction functor Res H G beco mes the functor p ∗ G/H : KK G → KK G ⋉ G/H that pulls b ack a G -C ∗ -algebra to a trivial bundle of G -C ∗ -algebra s ov er G/H . F ollowing [12], it is sho wn in [9] that the left adjoint of p ∗ G/H is deﬁned on all trivial bundles if G/ H is a smo oth manifold. W e will see that this is e n ough for our purposes . As in [1 6 ], we call a compact subgroup lar ge if it is a maximal compac t subgro up in an op en, almost connected subg roup o f G . Let H b e larg e. Then G/H is a smoo t h manifold and a n y compact subgroup is contained in a large one by [16, Lemma 3.1]. F urthermo re, since G is second countable there is a sequence ( U n ) n ∈ N of almost connected op en subgro ups of G such that an y other o ne is con tained in U n for some n ∈ N . Pick a maximal compact subgroup H n ⊆ U n for each n ∈ N . Then an y compact subgroup of G is HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 29 subc onju gate to H n for so me n ∈ N . Therefore, w e already have I = T n ∈ N Res H n G beca use Res K G factors through Res H G if K is sub conjugate to H . F or a compact subgr oup H ⊆ G , let RKK G ( G/H ) ⊆ KK G ⋉ G/H be the full subc ategory o f tr ivial bundles over G/H or, equiv alently , the esse n tial ra nge of the functor p ∗ G/H . W e do no t ca re whether this ca tegory is triangula t ed, it is certainly additive. W e replace the functors Res H G b y p ∗ G/H : KK G → RKK G ( G/H ) for H ∈ F . F or the large compact subgroups H n selected ab o ve, the results in [10] show that the left adjoin t of p ∗ G/H is deﬁned on all of RKK G ( G/H ) and maps the trivial bundle with ﬁbre A to C 0 ( T G/H ) ⊗ A with the diago nal a ct ion of G ; here T G/H denotes the tang en t space o f G/H (w e are not allo wed to use the Cliﬀord algebra dual considered in [9] b ecause it inv olves Z / 2 -gra ded C ∗ -algebra s, which do not belo ng to our c ategory). Thu s we hav e veriﬁed the h yp otheses of Theor em 3.31 and can conclude that I has enough pro jectiv e ob jects and that C C is reﬂectiv e. It remains to observe that the pro jective ob jects a re precisely the direct summands of co un table di- rect sums of ob jects of C I . W e have alre ady obser v ed that o b jects of C I are I -pr o jectiv e. Conversely , Theorem 3.31 shows that the pro jective ob jects are re- tracts of L n ∈ N C 0 ( T G/H n ) ⊗ A n for suitable G -C ∗ -algebra s A n . The summands are isomorphic to Ind G H n ( C 0 ( T 1 G/H n ) ⊗ A n ) where T 1 G/H n denotes the tangen t space of G/H n at the ba se point 1 · H n . Hence all pro jective o b jects are of the required form.  Since the stable homolog ical functor F ∗ ( A ) : = K ∗ ( G ⋉ r A ) commutes with direct sums, Theo rem 5.1 applies to it and shows that the ABC sp ectral sequence for the ideal I con verges tow a rds the do m ain of the Baum–Connes assembly map – which is the lo calisation of F ∗ at C C b y [16, Theo rem 5 .2]. It turns out that for a totally disco nnected gro up G the ABC spectral sequence agrees with a known spectr al sequence that we g et from the older deﬁnition of the Baum–Connes asse m bly map and the skeletal ﬁltration o f a G -CW-mo del for the univ ersal pro per G -spac e E G (see [13]). W e omit the pro of of this statement, which requires some work. 7.2. An ass e m bl y map for torsi o n-fr ee discrete quant um groups. Before we turn to the a ssem bly map, we mu st disc u ss some op en pro b lems that lead us to restrict attention to the torsion-free case. The ﬁrst issue is the cor rect deﬁnition of “to rsion” for loca ll y compact quantu m groups. The tors ion in a loca lly compact group is the family of co mp act sub- groups. Quan tum groups exhibit some torsion phenomena that do not app ear for groups, and it is conceiv able that we hav e not yet found all of them. First, compact quantum s ub groups are not enough: they should b e replaced by prope r quan tum homogeneous spaces, so that open subgroups provide torsio n in C ∗ ( G ) whenever G is disconnected. Secondly , pro jective r epresen tations of co m pact gr oups with a no n - trivial co cycle also provide tor sion (in their discrete dual); for instance, C ∗  SO (3)  is not tor sion-free b ecause of its pro jective repr esen tation on C 2 . If we considered C ∗  SO (3)  to b e tor sion-free, then the Ba um –Connes asse m bly map for it (which we describ e b elo w) would fail to b e an isomorphism. The co rrect formulation of the Baum–Connes conjecture for C ∗  SO (3)  turns out to b e equiv- alent to the Ba um –Connes conjecture for C ∗  SU (2)  – which is torsion-free – so that there is no need to discuss it in its own r igh t in [17]. I pro pose to approa c h torsion in discr ete quantum groups b y studying actions of its compact dual quant um group on ﬁnite-dimensional C ∗ -algebra s. A dis crete 30 RALF MEYER quantum g roup is torsion-fr e e if a n y such a ct ion is a dir ect s um o f a cti ons that are Morita equiv alent to the trivial action on C . The ab o ve deﬁnition of tor sion gives the exp ected results in simple cases. First, C 0 ( G ) fo r a discr ete g roup G is torsion-free if and only if G contains no ﬁnite subgroups. Secondly , C ∗ ( G ) for a compact group is torsion-free if and only if G is connected and has torsion-free fundamental group; this is exa ctly the generality in which Universal Co eﬃcien t Theorems for equiv ar ian t Kaspar o v theor y work (see [17, 2 2 ]). Christian V oig t shows in [24] that the quant um deformations of simply connected Lie gr oups such a s SU q ( n ) are to rsion-free. Another issue is to ﬁnd analogues of the re striction and induction functors for the non-classica l torsion that ma y app ear, and to prov e analogues o f the adjointness relations used in the pro of of Theorem 7.3. F o r honest quan tum subgro u ps, the restriction functor is evident, and Stefaan V aes has constructed induction functors for a ct ions of quan tum group C ∗ -algebra s in [23]. I expec t restriction to be left adjoint to induction for op en qua n tum subgroups a nd , in particular, for quan tum subgroups of discr ete quantum g roups. F or the time being, w e av oid these pro blem s and limit our atten tio n to the torsion-free case. Mo re precise ly , w e consider ar bit r ary discrete quant um groups, but disregard torsio n. The resulting assem bly map sho uld not b e an iso morphism for quantum groups with torsio n . The discrete qua n tum gro ups a re precisely the dua ls of co mpact quantum groups; we use reduced duals here be cause these app ear also in the Baum–Connes conjecture. It is useful to reformu late results a bout a discrete quantum group in terms of its compact dual as in [18, Remark 2.9]. Let G be a compact quantum gr oup and let b G b e its discrete dua l. Since we pr et end that b G is torsion-free, there is only one “restriction functor” to consider : the forgetful functor KK b G → KK that forg ets the action of b G altogether. The catego ry KK b G is e quiv a len t to KK G b y Baa j–Skandalis dualit y . Under this e quiv a lence, the for getfu l functor KK b G → KK corr esponds to the cros sed pro duct functor G ⋉ : KK G → KK , A 7→ G ⋉ A. The induction functor from the trivial subgroup to b G corresp onds under Baa j– Skandalis dualit y to the functor τ : KK → KK G that equips a C ∗ -algebra with the trivial action of G . This functor is left adjoint to the crosse d pro duct functor. Hence the relev ant sub categories C I , C C and the ideal I corresp ond to C I = { τ ( A ) | A ∈∈ KK } , C C = { A ∈∈ KK G | G ⋉ A ≃ 0 } , where ≃ means K K-equiv alence, that is, isomorphism in KK , and I = { f ∈ KK G | G ⋉ f = 0 } . The ideal I is a lready studied in [18, §5]. It is shown there that I has enough pro jective ob jects, and the univ er sal homological functor for it is describ ed. The target category inv o lv es actions o f the representation ring Rep( G ) of the c ompact quantum g roup G o n o b jects of KK ; s uc h an actio n on A is, by deﬁnition, a r ing ho - momorphism Rep( G ) → KK 0 ( A, A ) . The catego ry KK [Rep( G )] of Rep( G ) -mo dules in KK is not yet Abelian b ecause KK is not Abelian. T o remedy this, we must re- place KK by its F r eyd category of coher en t functors KK → Ab . But this co mplet ion do es not aﬀect homolog ical a lgebra muc h bec ause KK [Rep( G )] is an exact sub cat- egory that con tains all pro jective ob jects; hence we can compute derived functors without leaving the subcatego ry KK [Rep( G )] . W e could modify the ideal I and consider all f for which G ⋉ f induces the zero map on K-theory . This leads to a simpler Abelian appr o ximation, na m ely , HOMOLOGICAL ALGEBRA IN BIV ARIANT K-THEOR Y 31 the ca t e gory of all countable Z / 2 -gr aded Rep( G ) -mo dules. But this larg er ideal no longer leads to the subca tegories C C and C I above. Theorem 7.4 . L et G b e any c omp act quantum gr oup. Then the ide al I is c om- p atible with c ountable dir e ct su ms and has enough pr oje ctive obje cts. The p air of sub c ate gories ( hC I i , C C ) is c omplementary. Pr o of. The ideal I ha s enoug h I -pro jective ob jects by [16, Lemma 5.2], which also shows that the I -pro jective ob jects are prec isely the direct summands of ob jects in C I . The ideal I is compatible with direct sums b ecause the cros sed pr oduct functor commut e s with direct sums. Now Theor em 3.21 shows that the pair of subc ategories ( hC I i , C C ) is complemen ta ry .  Deﬁnition 7.5 . Let F : KK G → A be some homo logical functor . The assembly map for F with co eﬃcien ts in A is the map L F ( A ) → F ( A ) , where the lo calisation L F is formed with resp ect to the sub category C C . T o get an analogue of the Baum–Connes a ssem bly map, we should consider the functor F ( A ) : = K ∗ ( A ) because it corresp onds to the functor B 7→ K ∗ ( G ⋉ r B ) under Baa j–Ska ndalis duality . A torsion-free discrete quan tum group has the Baum–Connes pr op erty with co eﬃcien ts if the assembly map L F ( A ) → F ( A ) is a n isomorphism for all A for this functor. Prop osition 7.6. L et F : KK G → A b e a homolo gic al functor that c ommutes with dir e ct sums. The assembly map L F ⇒ F is the unique natur al t r ansformation fr om a functor ˜ F to F with the fol lowing pr op erties: • ˜ F is homolo gic al and c ommutes with dir e ct sum s ; • t h e natura l tr ansformation is an isomorphism for obje cts in C I ; • ˜ F vanishes on C C . Pr o of. Let ˜ F ⇒ F b e a natura l transfor mation with the r equired pr operties. Since bo th functor s in volv e d are homolog ical, the Five Lemma implies that the class o f ob jects for which the natural transformation ˜ F ⇒ F is an is omorphism is tria n- gulated. It is also closed under direct sums b ecause b oth functors commute with direct sums. Hence the natural transfor mation ˜ F ⇒ F is a n isomor p hism for all ob jects in hC I i b ecause this holds for o b jects in C I . Since ˜ F v anishes o n C C and is homolog ical, the universal pro perty of the lo cal- isation shows that the natural tr ansformation ˜ F ⇒ F fa ct ors uniquely thro ugh the assem bly map: ˜ F ⇒ L F ⇒ F . B oth ˜ F a nd L F descend to the category KK G / C C , which is equiv a len t to hC I i . Since b oth natura l transforma t ions ˜ F ⇒ F and L F ⇒ F are in vertible on ob jects of hC I i , we get the desired natura l isomor- phism ˜ F ∼ = L F .  The critical prop ert y in Prop osition 7.6 is the v anishing on C C . This cannot b e exp ected if b G ha s torsion. The Ba um –Connes ass em bly map is an isomorphism for F if and only if F v anishes on C C : one direction is trivia l, and the other follows b y taking ˜ F = F in Pr oposition 7.6. While this refor m ulation of the Baum–Connes conjecture came to o late to be used in v er if ying the co nject ure for g roups, it is quite helpful for duals of compact gro ups (see [17]) and probably also for t heir deformations. 8. Conclusion The idea of lo calisation – cent r al bo t h in homologica l alg eb r a a nd in homotopy theory – is beco min g more impor t ant in non-commutativ e top ology as well. When reﬁned using homolog ical ideals, it uniﬁes v ar ious new and old universal co eﬃcien t theorems, the Ba um –Connes conjecture, and its extensions to quantum gro u ps. 32 RALF MEYER Homological idea ls provide s ome bas ic top ological to ols in the general setting of triangulated catego ries. This includes • imp ortant notions from ho mological a lgebra like pro jectiv e resolutions and derived functors (these w er e already dealt with in [18]); • a n eﬃcien t method to c heck that pairs of subca t e gories in a triang u lated category are complementary; • so me control on how ob jects of the categ ory are constr u cted from genera tors, that is, from the pro jective ob jects for the ideal; • a natural sp ectral sequence that computes the lo calisation of a homological functor from its v alues o n g enerators. Since the a ssumpt ions on the underlying category are quite w eak, all this applies to equiv a rian t biv ariant K-theory . W e hav e applied this general ma c hinery to construct the Baum–Connes assembly map fo r torsion-fr ee quantum g roups, whose domain is o f top ological nature in the sense that it ca n be computed by top ological techniques such as sp ectral s equences. But m uch remains t o be done here. 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Homological algebra in bivariant K-theory and other triangulated categories. II

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