The fundamental isomorphism conjecture via non-commutative motives

THE FUND AMENT AL ISOMORPHISM CO NJECTURE VIA NON-COM MUT A TIV E MOTIVES P AUL BALMER AND GONC ¸ ALO T ABUADA Abstract. Giv en a group, we con struct a fundamenta l additiv e functor on its orbit cate gory . W e prov e that an y isomorphism co njecture v alid for th is fundamen tal additive functor holds for all additive f unctors, like K -theory , cyclic homology , top ological Ho ch schild homology , etc. Finally , w e reduce this fundamen tal isomor phi s m conjecture to K -theoretic ones. 1. Introduction and st a tement of resul ts 1.1. Isomorphism conjectures. The F a r rell-Jo nes isomorph ism c onje ctur es are impo rtant driv ing forces in current mathema tical resea rch and imply well-kno wn conjectures due to Bass, Borel, Ka plansky , Novik ov; se e a survey in L¨ uc k [ 19 ]. Given a group G , the F arrell-J ones conjectures predict the v alue of alge br aic K - and L - theory of the group r ing R G in terms of their v alues o n the virtua lly cyclic subgroups of G ; here R is a fixed ba se commutativ e ring. In [ 8 ], Davis and L ¨ uc k prop osed the following unified setting for these isomor phism conjectur e s ; se e § 2 . Let F b e a family of s ubg roups of G and E : Or( G ) → Spt a functor from the orbit category of G to spe ctra. The ( E , F , G ) -assembly map is the induced map (1.1.1) ho colim Or( G, F ) E − → ho co lim Or( G ) E = E ( G ) , where Or( G, F ) ⊂ Or( G ) is the o rbit catego ry restricted on F . W e say that the functor E has the F -assembly pr o p erty for G when the map ( 1.1.1 ) is a stable w eak equiv alence, i.e. when it induces an is omorphism on stable homotopy gro ups. When we sp eak of the ( E , F , G ) -isomorphi sm c onje ctur e , we refer to the expr essed hop e that this prop erty ho lds for a par ticular choice of E , F and G . Da vis and L ¨ uc k prov ed (see also [ 11 ] for deta ils on the pro of ) that the F ar r ell-Jones conjecture in K -theory for G is equiv alen t to the ( K , V C, G )-isomor phis m conjecture, where K is non-connective K -theory (see § 4.2 ) and V C the family of vir tually cyclic subgr oups of G ; and similarly for L -theor y . The first s tep in their approa ch is the construction of a functor to R -linear categories (1.1.2) Or( G ) ? − → Grp R [ − ] − → R - cat , comp osed of the tr a nsp ort gr o up oid functor and the R -linear iz ation functor ; see § 2 . In addition, the literature contains man y v ariations on the ab ove theme, re- placing t he K - and L -theory functors by o ther functor s E , and t he category of Date : Nov ember 2, 2018. 2000 M athematics Subject Classific ation. 18D20 , 19D50, 19D55. Key wor ds and phr ases. F arr el l -Jones conjectures, assem bly map, non-commut ative motiv es, algebraic K -theory , (topol ogical) Hochsc hi ld homology , dg categories, Grothendiec k deriv ators. The first author was supp orted by N SF grant DMS-0969644 and the second author b y the Clay Mathematics Institute and by the FCT-P ortugal gr an t PTDC/MAT/09831 7/2008 . 1 2 P AUL BALME R AND GONC ¸ ALO T ABUADA sp ectra b y o ther mo del categories M . See for instance the isomor phis m conjecture for ho motopy K -theory ( K H ) [ 2 , § 7], for Ho chschild homology ( H H ) and cyclic homology ( H C ) [ 20 , § 1], or for topo logical Ho chschild homology ( T H H ) [ 18 , § 6]. This simple idea of letting the functor E and the category M float freely generates a profusion o f po tent ial isomorphism conjectures : (1.1.3) Spt Spt Or( G ) K 2 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ K H 1 1 ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ H H / / H C - - ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ T H H , , ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ E . . . * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Ch( R ) Ch( R ) Spt M , where Ch( R ) sta nds for t he categ ory o f complexes of R -mo dules . Each of thes e isomorphism conjectures ha s already b een proved for lar ge c la sses of g r oups us ing a v ar iety of differen t metho ds. Our go a l in this article is no t to prove a ny of these conjectures for any class of gr oups. W e ar e rather in terested in the g eneral organi- zation and deep er pr op erties b ehind this s omewhat exub e rant her d of conjectures. Our guiding q ue s tions are the following : Question A : Is ther e a fundamental isomorphism c onje ctur e implyi ng al l others ? Question B : If this is the c ase, c an this fun damental isomorphism c onje ct u r e b e describ e d solely in terms of classic al invariants ? W e provide p ositive answers to those questions. As w e s hall see, the isomorphism conjecture for K -theory will play a central role, probably confirming the secret feelings of some ex per ts. The pre c ise formulation of our a nswers uses the theo r y of non-c ommutative motives initiated in [ 31 ]. Connecting those tec hniques to the world of isomor phism conjectures is the technical bulk o f the a r ticle. W e b elieve that this bridge will encourag e the adoption of these new techniques by r esearchers working on isomorphism conjectures. 1.2. N on-commuta tive motives. A differ en t ial gr ade d (= dg) c ate gory , ov er o ur fixed base commutativ e ring R , is a categor y enriched over co chain complexes of R -mo dules (mor phisms sets are complex e s) in suc h a w ay that comp osition fulfills the Leibniz rule : d ( f ◦ g ) = d ( f ) ◦ g + ( − 1) deg( f ) f ◦ ( dg ); see Keller [ 16 ] and § 3 for further explana tions ab out dg ca teg ories. There is a Quillen mo del structure on dgcat , the categor y of small dg categ ories, with weak equiv alenc e s being der ived Morita equiv alence s (see § 3.2 ). Many of the cla ssical inv ariants such as Ho chsc hild and cy c lic homo logy , con- nective, non- c onnective, and homoto p y K -theor y , and even to po logical Ho chsc hild homology , extend naturally from R -algebr as to dg categorie s ; see § 4 . In order to study a ll these inv ariants s im ultaneous ly the notion of additive invaria nt w as int ro duced in [ 29 , § 15 ]. This theory makes use of the la nguage o f Gro thendieck deriv ator s, a formalism which allows us to state and prov e pr e cise univ ersa l pro p- erties ; see App endix B . Let E : HO ( dg cat ) → D be a mor phism o f der iv ator s, fro m the deriv ator assoc ia ted to dg cat , to a strong triang ulated deriv a to r D . W e sa y THE FUND AMENT AL ISOMORPHISM CONJECTURE 3 that E is an additive invariant if it pre s erves filtered homotopy colimits and the terminal ob ject, and if it sends split exact sequence s to direct sums A I / / B T o o P / / C S o o 7− → [ E ( I ) E ( S )] : E ( A ) ⊕ E ( C ) ∼ − → E ( B ) . By the additivit y results of Keller [ 17 ], W aldhausen [ 34 ], Schlic hting [ 2 7 ], W e ib el [ 3 5 ], and Blum b erg-Ma ndell [ 3 ] (see also [ 28 ]), all the ab ov e classical theorie s ar e addi- tive inv a riants (see § 4 ). In [ 2 9 , Def. 15.1] the universal additive invariant was constructed U add dg : HO ( dgc at ) − → Mot add dg . It is univ ersal in the fo llowing sense . Giv en a ny strong tr ia ngulated de r iv ator D we hav e an equiv alence of categories (1.2.1) ( U add dg ) ∗ : Hom !  Mot add dg , D  ∼ − → Ho m add  HO ( dgcat ) , D  , where the left-hand side denotes the categor y of homo topy co limit pr eserving mo r- phisms of deriv ato rs and the rig ht -hand side denotes the category of additive in- v aria n ts ; see [ 29 , Thm. 15.4]. In words, this means that every a dditiv e inv ar iant on dg catego ries (an ob ject in the right-hand c a tegory) fac to rs e s sentially uniquely via U add dg , that is, via the universal additive deriv ator Mo t add dg . F urthermor e , this universal a dditive der iv ator Mot add dg admits an explicit Quillen mo del M ot add dg ; see § 5 . Because of its universal prop er ty , which is reminiscent of the theory o f motives, the deriv ator Mot add dg is calle d the additive m otivator , and its base categ ory Mot add dg ( e ), which is the homoto py catego ry Ho ( M ot add dg ), is ca lled the triangulate d c ate gory of non-c ommutative motives . 1.3. F undamental is omorphism conjecture. The ab ov e notion of additivity combined with functor ( 1 .1 .2 ) yields the notion o f an addi tive functor on the orbit c ate gory ; s e e Definition 6.0.2 . In particula r, all functors mentioned in diagr am ( 1.1.3 ) ar e additive on Or( G ). 1.3.1. Rema rk ( Limitations) . W e would like to mention that inv aria nt s inv olving enriche d structures, like top olog ical K -theory or L -theory are not a dditiv e inv ar i- ants in this first, elementary sens e. Including the Baum-Connes conjecture in our treatment w ould require the definition o f to po logical K -theo ry of the reduced C ∗ - algebra via dg catego ries and this do es not ex is t at the mo ment. Similarly , including L -theory would require to carr y over dualities througho ut the game and this is an- other story . An application o f universality ( 1.2 .1 ) is the following : 1.3.2. Theorem (see Thm. 6.0.5 ) . L et G b e a gr oup and let R b e a c ommutative ring. Then t her e exists a fundamental additive functor on its orbit c ate gory E fund : Or( G ) ? − → Grp R [ − ] − → R - cat ⊂ dgc at U add dg − → M ot add dg thr ough which al l additive functors on Or( G ) factor. 4 P AUL BALME R AND GONC ¸ ALO T ABUADA Int uitively Theo rem 1.3.2 allows us to com b the skein ( 1.1.3 ) from the left to isolate a fundamental additive functor Spt Spt Or( G ) E fund / / M ot add dg K 5 5 ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ K H 2 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ H H / / H C , , ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ T H H ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ E . . . ' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ Ch( R ) Ch( R ) Spt M A key p oint is that the right-hand functors E pr eserve ho mo topy colimits (no t o nly filtered o nes). Hence they will preserve any assembly prop erty that E fund might enjoy . W e then obtain the following answer to our Question A : 1.3.3. Corollary (s e e Cor . 6.0.6 ) . L et G b e a gr oup and F a family of sub gr oups. If t he fundamental addi tive functor E fund has the F -assembly pr op erty, so do al l additive fu n ctors on Or( G ) . Let us make clear that the F -assembly pr op erty fo r E fund has essentially no chance to hold for r andom choices of G , F a nd R . F o r insta nce , if F = V C this prop erty w ould imply the ( K , V C , G )-isomorphism conjectur e for R = Z and for K b eing c onne ctive K - theory (see § 4.1 ). And this is known to fail be c ause of the Bass-Heller -Swan decomp osition; see [ 21 , Rem. 1 5]. Howev e r , if R is a r e gu lar r ing ( i.e. no etheria n and of finite pro jective dimension) in which the or ders of all finite subgroups of G are in vertible, then the ab ov e obstruction v anishes b ecause the ( K, V C, G )-isomor phism conjecture follows from the F arrell- Jones co njectur e ; see Prop ositio n 2.5.2 . This might sugg est the following “mother” of many iso morphism conjectures : Mamma Conjecture. Gi ven a gr oup G , t he fundamental additive invariant E fund has the V C -assembly pr op erty when the b ase ring R is r e gular and the o r ders of al l finite s u b gr oups of G ar e invertible in R . A lar ge class of examples is given by taking R = Q or C and G arbitr ary. Another lar ge class of examples is given by taking R = Z and G torsion-fr e e. Corollar y 1.3.3 says that the Mamma conjecture implies al l additive c o njectures on the ma rket, for that bas e ring R and that g roup G , with resp ect to virtually cyclic subgr oups. Note that our choice of the family o f virtually cy clic gro ups is merely b orr ow e d from F arrell- J ones and ano ther family F might b e preferable. In any case, the main result is that once this is achieved for some family F , then a ll additive functors will automatically inherit the same F -assembly pr op erty . Stated differently , instead o f m ultiplying the ar ticles o n v aria tions on the iso - morphisms conjectures for this and that additive inv ariant, mathematicians can now try to attack one single conjecture and deduce all other ones. It is of cours e impo rtant to know how wild this conjecture is, as asked in o ur Question B. W e discuss this pr oblem now. W e w ould like to r educe the F - assembly prop erty for E fund , whose impor tance should now be clea r , to the F -assembly pr op erty for mo re down-to-earth functor s. THE FUND AMENT AL ISOMORPHISM CONJECTURE 5 T o do this, we consider functors whic h are co oked up via K -theory and dg catego ries as follows. Given a small dg category B , co nsider the functor K ( − ; B ) : O r( G ) → Spt defined for e very G/H ∈ Or( G ) by K ( G/H ; B ) := K  rep dg  B , R [ G/H ]  . Some explana tions ar e in order . F or any sma ll dg categ ory A , we denote by rep dg ( B , A ) the internal Hom-functor, b etw een B a nd A , in the derived Morita homotopy categor y; see § 3.3 . If B is the dg ca tegory R with one ob ject and with R as dg algebra of endomorphisms, then the functor K ( − ; B ) r educes to the usual connective K -theor y functor K . Hence, when B is a gener a l small dg ca tegory , the functor K ( − ; B ) can b e thought o f as a “co e fficient s v ariant” of K ; se e Exa m- ple 6.0.3 . The functor K ( − ; B ) is not additive in gener al, mainly becaus e B mig ht be to o la rge. Therefore, w e restrict to dg c ategories B whic h ar e homotopic al ly finitely pr esente d ; see Definition A.0.1 . Heuristically , this condition is the homo - topical version o f the classical notion of finite pres entation. In pa rticular the above example B = R is homotopically finitely presented. Our answer to Questio n B is : 1.3.4. Theorem. L et G b e a gr oup and F b e a family of su b gr oups. Then t he fol lowing c onditions ar e e quivalent : (1) The fundamental additive fun ctor E fund has the F -assembly pr op ert y for G . (2) The additive fu n ctors K ( − ; B ) have t he F - assembly pr op erty for G , for al l homotopic al ly finitely pr esente d dg c ate gories B . (3) The additive functors K ( − ; B ) have the F - assembly pr op erty for G for al l strictly finite dg c el ls B (se e D efinition 3.1.1 ). The proof occ upies § 7 a nd is based on the co -representabilit y theor em 7.0.7 . The strictly finite dg cells of (3) form a set o f homotopically finitely presented dg categorie s which are espe cially s mall. Roughly speak ing, they a r e the dg category analogues of finite CW-complexes, na mely they ar e built by a ttaching finitely many basic cells, chosen a mong the dg analogues S ( n − 1) → D ( n ) of the top olog ical inclusion S n − 1 ֒ → D n ; see Definition 3.1.1 . Via Theor em 1.3.4 , the Ma mma conjecture now bo ils down to K -theor y : Mamma Conjecture (revisited). Given a gr oup G , the fun ctors K ( − ; B ) have the V C -assembly pr op erty for al l strictly finite dg c el ls B , when the b ase ring R is r e gular and the or ders of al l fi nite sub gr oups of G ar e invertible in R . When R is a r egular r ing where the orders of all finite subgro ups of G are inv er tible and B is the dg categor y B = R , the a b ov e conjecture ba sically is the F arre ll- Jones co njecture; see Remar k 2.5.3 . Hence, the Mamma conjecture a mo unt s to a co efficients v ariant of the classic a l F arr el-Jones conjecture, with strictly finite dg co efficients B . Its imp or tance (a nd that of Theorem 1.3.4 ) r elies on the fact that it simultaneously implies all additive isomorphism conjectures on the market and yet is desc r ib ed solely in terms of K -theor y . O ne can therefore exp ect that future resear ch will adapt ex isting pro ofs of the F arrell- Jones co njecture for sp ecific cla sses of groups to prov e the Mamma conjecture, with the benefits explained a b ove. A t some sta ge, and at lea s t b efor e § 4.6 , the reader who is not familiar with the langua ge of Grothendieck deriv ators should pr o ceed to Appendix B , where we also prov e that the op erations of stabilization and of left Bo usfield lo calizatio n of deriv ator s commut e (Theor e m B.4.1 ). The latter result is of indep endent interest. 6 P AUL BALME R AND GONC ¸ ALO T ABUADA 2. The Da vis and L ¨ uck appro ach In this section, we recall Da vis a nd L ¨ uc k’s refor mulation [ 8 ] of the F ar rell-Jones conjecture in K -theory . This will b e the stepping stone for the c o nstruction of the fundamen tal additive functor in § 6 . Let G b e a (fixe d) group. 2.1. The orbit category . The orbit c ate gory Or( G ) of G has as ob jects the ho - mogeneous G -spaces G/H , considered as left G - s ets, and as morphisms the G - equiv aria nt maps. A family F of sub gr oups of G is a no n-empty set of subgr oups of G which is closed under conjugation and finite intersection. Examples of families of subgroups a re given b y the family F in of finite subgroups, b y the family of cyclic subgroups (finite and infinite), and b y the family V C of virtually cyc lic subgroups ; recall that H is v irtually cyclic if it contains a cyclic s ubg roup of finite index. The o rbit c ate gory Or( G, F ) r estricte d on F is the full s ubca tegory of O r ( G ) consisting of thos e ob jects G/H fo r which H b elo ngs to F . 2.2. F -assembly prop e rt y. The F - assembly pr op erty can be generalized from sp ectra ( § 1.1 ) to any target model catego ry M . Let F be a family o f subgro ups of G and let E : Or( G ) → M b e a functor. The ( E , F , G ) -assembly map is the map (2.2.1) ho colim Or( G, F ) E − → ho colim Or( G ) E = E ( G ) in M . W e say that E has the F - assembly pr op erty (for G ) when that map is an isomorphism in Ho ( M ). A typical approach in the Davis and L¨ uck philosophy (mostly with M = Spt ) is the following : Given G a nd E , find as small a family F as p ossible for which E has the F -as sembly pr op erty . F or instance, for the F ar rell-Jones isomorphism con- jectures in K - and L -theor y , one exp ects F to r educe to virtually cyclic subgroups. Conceptually , the F -assembly prop erty for a functor E : Or( G ) → M essentially means that it is induced from its res tr iction to Or( G, F ), up to ho motopy , i.e. it belo ngs to the image of the functor on homotopy categorie s L Ind : Ho  F un (Or( G, F ) , M )  − → Ho  F un (Or( G ) , M )  left adjoint to the obvious functor in the other direction, defined b y restriction from Or( G ) to Or( G, F ). This is explained in [ 1 ], where we say that the functor E satisfies Or( G, F ) -c o desc ent if E b elongs to the image of L Ind up to isomor phism in Ho  F un (Or( G ) , M )  . This is e quiv alent to the F -a s sembly pro p erty for G a nd for all its subgroups. How ever, w e shall not use the langua ge of [ 1 ] here. 2.3. T ransp ort g roup oid. Let S b e a left G -s e t. The tr ansp ort gr oup oid S asso- ciated to S has S as the set of o b jects and the following morphisms Hom S ( s, t ) := { g ∈ G | g s = t } for s, t ∈ S . Compositio n is given b y gro up mu ltiplication. This defines a functor ? : Or( G ) − → Grp from the or bit categor y to the catego r y o f gro up o ids. No te that for every subgr oup H of G , the group oid G/H is connected. Hence it is equiv alent to the full sub cat- egory on any of its o b jects, for instance the ca nonical ob ject eH ∈ G/H , whose group of automor phisms is H . So , if we think of the group H as a one-ob ject cate- gory , denoted H , we ha ve an equiv alence o f gro upo ids H ∼ → G/H . In other words, the gro upo id G/H is a natural s e veral-o b ject repla cement of the group H . THE FUND AMENT AL ISOMORPHISM CONJECTURE 7 2.4. R -line arization. W e now recall the pas sage fro m gro up o ids to R -categ ories, i.e. additiv e categor ies enr iched over the symmetric monoidal categor y of R -mo dules. Let C b e a gr oup oid. The asso ciate d R -c ate gory R [ C ] is the idemp otent c omple- tion of the R -catego r y R [ C ] ⊕ whose o b jects a re the formal finite direct sums of ob jects o f C and whose morphisms are the ob v ious ma tr ices with entries in the free R -mo dules R [ C ( X, Y )] generated by the sets C ( X , Y ). Co mpo sition in R [ C ] ⊕ is induced from compo sition in C a nd matrix multiplication. Idemp o tent completion is the usual formal creation o f images and kernels for idempotent endomorphis ms . The construction C 7→ R [ C ] yields a w ell-defined functor R [ − ] : Grp − → R - cat with v alues in the categor y of (idemp otent co mplete) small R - categor ies. F or in- stance, for a one-ob ject gr oup oid H , the catego ry R [ H ] ⊕ is equiv a lent to that of free RH -mo dules of finite rank and its idemp otent completion R [ H ] is equiv alent to the categ ory of finitely g enerated pro jective RH -mo dules. 2.5. K -the o ry. Rec all from [ 25 ] that w e can asso cia te to every R -catego ry C its non-connective K -theory s p ectr um K ( C ), defining a functor K : R - cat → Spt . Putting all these constructions together, we obtain the following co mpo sed functor (2.5.1) Or( G ) ? − → Grp R [ − ] − → R - cat K − → Spt . As usual, one obtains the K -theory gr oups K ∗ by taking (sta ble) homotopy groups. Thanks to the arguments in § 2.3 - 2.4 we hav e the fo llowing identifications K ∗ ( RH ) = π ∗ K ( RH ) ∼ = π ∗ K ( R [ H ]) ∼ = π ∗ K ( R [ G/H ]) , which expla in w hy the K -theory functor ( 2.5.1 ) defined on O r( G ) is indeed the exp ected one. This allow ed Davis a nd L ¨ uc k to pr ov e in [ 8 ] the e q uiv alence betw een the F arrell-Jo nes co njectur e in K -theor y for G and the ( K , V C, G )-isomor phism conjecture, i.e . the statement that the functor ( 2.5.1 ) has the V C -as sembly prop erty . Of cour se, there is a lso a cla ssical connective K -theory functor, here s imply denoted by K : R - cat → Spt . W e now discuss a connection b e tw een K and K . 2.5.2. Prop osition (L¨ uck-Reic h [ 21 , Pr o p. 70 ]) . L et R b e a re gular ring in which the or ders of al l fi nite sub gr oups of G ar e invertible. The ( K , V C , G ) -isomorphism c onje ct ur e ( i.e. the F arr el l-Jones c onje ctur e) implies the ( K , F in, G ) -isomorphism c onje ct ur e, and a fortiori the ( K, V C, G ) -isomorphism c onje ctu r e. Pr o of. W e ha ve a co mm utative diagram of na tur al maps ho colim Or( G, F in ) K γ ≃   α / / K ( R G ) β   ho colim Or( G, F in ) K δ ≃ / / ho colim Or( G, V C ) K ǫ / / K ( RG ) . Under the stated assumptions on R , [ 21 , Prop. 70] implies that δ is a stable weak equiv alence. Mo reov er, as sho wn in the pro of of [ 21 , Pro p. 70 ], the group rings RH , with H < G finite, are regular rings. This implies tha t γ is a stable weak equiv alence. Since β induces a monomorphism on stable homoto py gr oups, if ǫ is a stable weak equiv alence then so is α .  8 P AUL BALME R AND GONC ¸ ALO T ABUADA 2.5.3. R emark. Conv ers e ly , under the ab ove assumptions a b o ut R and G , one ex- pec ts the sp ectrum K ( RG ) to b e connective; s e e [ 21 , § 2.4.1]. If this is the c a se, the ab ov e pro of also gives the conv erse to the statemen t of Prop osition 2.5.2 . 3. Dg ca tegories W e review some asp ects of the theory of dg categorie s and introduce the notio n of strictly finite dg cell. F or a survey article, we invite the reader to c onsult Keller [ 16 ]. Let A b e a small dg catego r y ( § 1.2 ). The opp osite dg c ate gory A op of A ha s the same ob jects as A and complexes of mor phisms given by A op ( x, y ) := A ( y , x ). The category Z 0 ( A ) has the same ob jects as A a nd mo rphisms given by Z 0 ( A )( x, y ) := Z 0 ( A ( x, y )), the 0-co cycles in the co chain complex A ( x, y ). The homotopy c ate gory H 0 ( A ) of A has the sa me ob jects as A and mor phisms given by H 0 ( A )( x, y ) := H 0 ( A ( x, y )). Recall from [ 16 , § 3.1 ] that a right dg A -mo dule (or simply a n A - mo dule) is a dg functor A op → C dg ( R ), with v alues in the dg ca tegory C dg ( R ) o f complexes of R -mo dules. W e denote by C ( A ) (resp. b y C dg ( A )) the category (resp. dg catego ry) o f A -mo dules. Recall fr o m [ 16 , Thm. 3.2] that C ( A ) car ries a standard pro jective mo del s tructure. The derive d c ate gory D ( A ) of A is the lo calizatio n of C ( A ) with resp ect to quasi-iso morphisms. Finally , let p erf dg ( A ) b e the dg category of p erfe ct A -mo dules, i.e. the full dg sub categ ory of C dg ( A ) spanned by the cofibr ant A -mo dules that b ecome compact [ 24 , Def. 4.2.7 ] in the triangula ted c a tegory D ( A ). 3.1. Stri ctly fi ni te dg cells . Let R be the small dg ca tegory w ith one o b ject ∗ and such that R ( ∗ , ∗ ) := R (in degr ee zer o), wher e R is the ba se ring. F or n ∈ Z , let S n be the complex R [ n ] (with R concentrated in degre e n ) and let D n be the mapping cone o n the identit y of S n − 1 . W e denote by S ( n ) the dg ca tegory with tw o ob jects 1 and 2 such that S ( n )(1 , 1) = R , S ( n )(2 , 2) = R , S ( n )(2 , 1) = 0 , S ( n )(1 , 2) = S n and comp os ition given by multip lication. W e denote by D ( n ) the dg categor y with t wo ob jects 3 and 4 suc h that D ( n )(3 , 3) = R , D ( n )(4 , 4) = R , D ( n )(4 , 3) = 0 , D ( n )(3 , 4) = D n and with comp osition giv en by multiplication. Finally , let ι ( n ) : S ( n − 1) → D ( n ) b e the dg functor that sends 1 to 3, 2 to 4 a nd S n − 1 int o D n via the map incl : S n − 1 → D n which is the identit y on R in degree n − 1 : S ( n − 1) ι ( n ) / / D ( n ) 1 R   S n − 1   ✤ / / 3 R   D n   incl / / 2 R D D ✤ / / 4 R D D where S n − 1 incl / / D n 0 / /   0   0 / /   R id   R id / /   R   (degree n − 1) 0 / / 0 W e denote by I the set consis ting of the dg functor s { ι ( n ) } n ∈ Z and the dg functor ∅ → R (where the empty dg categ o ry ∅ is the initia l one). 3.1.1. Defin ition. A small dg category A is a strictly finite dg c el l (compare with Hirschhorn [ 14 , Def. 10.5.8 ]) if it is o btained from ∅ b y a finite n umber of pusho uts THE FUND AMENT AL ISOMORPHISM CONJECTURE 9 along the dg functors of the set I . W e denote by dgcat sf the full sub c a tegory of dgcat consisting of strictly finite dg cells. 3.2. Q uillen mo de l structure. Recall from [ 30 , Thm. 5 .3] that the catego ry dgcat is endow ed with a (cofibrantly ge ne r ated) derive d M orita mo del structur e, whose weak equiv alences ar e the derive d Morita dg fun ctors , i.e. the dg functors F : A → B which induce an equiv alence o n the derived categor ie s D ( B ) ∼ → D ( A ). W e denote by Ho ( dgcat ) the homotopy ca tegory hence obtained. 3.3. Internal Hom-functor. Given dg categor ies B and A their t ensor pr o duct B ⊗ A is defined as follows. The set of ob jects is the cartes ian pro duct a nd, giv en ob jects ( z , x ) and ( w , y ) in B ⊗ A , we set ( B ⊗ A )(( z , x ) , ( w , y )) := B ( z , w ) ⊗ A ( x, y ). This tensor pro duct can b e naturally derived into a bifunctor (3.3.1) − ⊗ L − : Ho ( dgcat ) × Ho ( dgcat ) − → Ho ( dgcat ) , which gives rise t o a symmetric mono idal str ucture on Ho ( dgcat ). By T o¨ en [ 32 , Thm. 6.1] the bifunctor ( 3.3.1 ) admits an internal Hom-functor rep dg ( − , − ). 1 Given small dg categor ies B and A , rep dg ( B , A ) is the full dg sub catego ry of C dg ( B op ⊗ L A ) spanned by the cofibra nt B - A -bimodules X such that, for every ob ject z in B , the A -mo dule X ( z , − ) belong s to p erf dg ( A ); by a B - A -bimo dule we mean a dg functor B op ⊗ A → C dg ( R ), i.e. a B op ⊗ A -mo dule. Equiv alently , rep dg ( B , A ) is formed by the cofibra nt B - A -bimo dules X such that the induced functor − ⊗ L B X : D ( B ) − → D ( A ) takes the representable B -mo dules to p er fect A -mo dules. Suc h a bimodule yields a functor H 0 ( B ) → H 0 ( pe rf dg ( A )), which suggests that rep dg ( B , A ) ca n be thought o f as the dg category of r epresentations “up to homotopy” of B in p erfect A -modules. Note that rep dg ( R , B ) is derived Morita equiv alent to perf dg ( B ) (see [ 16 , § 4]). 4. Additive inv ariants of dg ca tegories Recall fro m § 1.2 the notion of additive in v ar iant of dg categories ; consult [ 29 , § 15] for further details. In this section we collect several examples of additive inv aria n ts and introduce a “co efficients v ariant”. 4.1. C o nnectiv e K -theory. Given a sma ll dg category A the R -linear catego r y Z 0 ( pe rf dg ( A )) is a category with cofibr ations and weak equiv alences in the sense of W aldhausen [ 34 ]. The cofibra tions are the morphisms of A -mo dules which admit retractions as morphisms of gr aded A -mo dules and the weak e q uiv alences are the quasi-isomo rphisms; se e [ 16 , § 5.2]. The c onn e ctive K -the ory sp e ctrum K ( A ) of A is obtained by applying W a ldhausen’s c onstruction [ 34 , § 1.3] to Z 0 ( pe rf dg ( A )). Thanks to [ 29 , Ex ample 15.6] this gives rise to an a dditive in v ar iant of dg c a tegories K : HO ( dgcat ) − → HO ( Spt ) . 4.2. N on-connectiv e K -theory. Given a small dg categor y A , its non-c onne ctive K - the ory sp e ctrum K ( A ) is obtained b y applying Schlic hting’s construction to the F rob enius pair natur a lly a sso ciated to Z 0 ( pe rf dg ( A )) ; see [ 27 , § 6 .4]. Thanks to [ 29 , Thm. 10.9 ] this gives ris e to a n a dditive inv ariant of dg categor ie s K : HO ( dgcat ) − → HO ( Spt ) . 1 Denoted by R Hom ( − , − ) in lo c. cit. 10 P AUL BALME R AND GONC ¸ ALO T ABUADA 4.3. H omotopy K -theo ry. Recall from W eib el [ 35 , § 1 ] the s implicial R -algebr a ∆ • (viewed a s a simplicial o b ject in dg cat ), wher e ∆ n := R [ t 0 , . . . , t n ] / P n i =0 t i − 1 . Given a s ma ll dg catego ry A , its homotopy K -the ory sp e ctrum K H ( A ) is given by ho colim n K ( A ⊗ ∆ n ). Then we obtain a w ell-defined mor phism of deriv ators K H : HO ( dgcat ) − → HO ( Spt ) . By constructio n it pr eserves filtered colimits a nd the terminal o b ject. Since t he functors − ⊗ ∆ n send split exact sequences to split e x act seq uences, we conclude that K H is also an exa mple of an a dditiv e inv a riant. 4.4. H o c hschild and cyc lic homolo g y. Let A b e a small dg catego ry . Recall from [ 16 , § 5.3] the construction of the Ho chsc hild and c y clic homo logy complexes H H ( A ) and H C ( A ), a nd of the mixed complex C ( A ). Thanks to [ 29 , Thm. 10 .7], this gives rise to additive inv aria nts of dg categories C : HO ( dgcat ) − → HO (Λ- Mo d) H H , H C : HO ( dgcat ) − → HO (Ch( R )) , where Λ := R [ B ] / ( B 2 ), with B of degree − 1 and dB = 0 , and Ch( R ) denotes the category of co mplexes of R - mo dules endow ed with its pro jective mo del structure. 4.5. T op ological Ho c hs c hild homolog y. Let A be a small dg c ategory . Reca ll from [ 3 , § 3] or [ 28 , § 8.1] the top olo gic al Ho chschild homolo gy sp e ctrum T H H ( A ). Thanks to [ 28 , Pr o p. 8.9] this gives rise to an additive inv a riant o f dg categ ories T H H : HO ( dgcat ) − → HO ( Spt ) . 4.5.1. R emark. Any R -a lg ebra A can b e seen a s a small dg categ o ry A with one ob ject and with A as the dg algebr a of endomo rphisms conce ntrated in degree zero. Note that the above inv ariants § 4.1 - 4.5 verify the “ag reement pro p er ty”, i.e. when we apply them to A we recover the cla s sical inv aria nt s asso cia ted to A . 4.6. C o efficients v arian t. Given a small dg category B , the functor rep dg ( B , − ) (see § 3.3 ) naturally g ives rise to a morphism o f deriv ator s (4.6.1) rep dg ( B , − ) : HO ( dgcat ) − → HO ( dgcat ) . 4.6.2. Lemma. If the dg c ate gory B is homotopic al ly finitely pr esente d (Def. A.0.1 ), then the morphism ( 4.6 .1 ) pr eserves filter e d homotop y c olimits, the terminal obje ct, and split exact se quenc es. Pr o of. The morphism ( 4.6.1 ) clearly preser ves the terminal o b ject as well as split exact sequences. Since B is homotopica lly finitely pr e sented, [ 4 , Thm. 3.3(3)] (where rep dg was denoted by rep ) implies that rep dg ( B , − ) als o preser ves filtered homotopy colimits.  Let E : HO ( dgcat ) → D b e an a dditive in v aria nt of dg categor ies and B a homo- topically finitely pres e n ted dg ca tegory . Thanks to Lemma 4.6.2 w e can co nstruct a new additive inv ar ia nt E ( − ; B ) : HO ( dgcat ) → D as follows A 7→ E ( A ; B ) := E ( rep dg ( B , A )) . If B = R , then the dg c a tegory rep dg ( R, A ) is derived Morita equiv alent to A and so E ( − ; R ) reduces to E . Hence, when B is a gener al homotopica lly finitely pre sented dg categor y , E ( − ; B ) can b e thought of as a “co efficients v ariant” of E . THE FUND AMENT AL ISOMORPHISM CONJECTURE 11 5. Reordering the model of th e additive motiv a tor W e mo dify the Quillen mo del for the a dditiv e motiv ator Mot add dg of dg c ategories . This will b e the main technical to ol in the pro of of The o rem 1.3.4 ; see § 7 . 5.1. The original m o del. In [ 29 , § 5] the second author introduced the s mall category dgcat f of finite I -c el ls as b eing the smalles t full sub categ o ry of dgcat which contains the strictly finite dg cells (see § 3.1 ) and which is stable under the co-simplicial and fibrant reso lution functors of [ 29 , Def. 5.3]. Then, he co nsidered the pro jective mo del s tr ucture on the categor y Fun ( dgcat op f , sSet • ) of presheav es of po int ed simplicial sets and to o k its left Bousfield lo calizatio n (5.1.1) L g E s un , p, Σ F un ( dgcat op f , sSet • ) with resp ect to sets of morphisms g E s un , p and Σ; see [ 29 , § 14] for details. Heuris- tically , inv erting Σ is resp onsible for inv erting Morita equiv a lences, inv erting p is resp onsible for preserving the terminal ob ject and, inv e rting g E s un is resp onsible for mapping split exact sequences of dg c a tegories to split triangles in the homo topy category . 5.1.2. R emark. In F un ( dgcat op f , sSet • ), sequential homotopy colimits co mm ute with finite pro ducts a nd homo topy pullbacks a nd so by Remark B.1.3 , the ass o ciated deriv ator is r egular (Def. B.1.2 ). Since the do mains and c o domains o f the sets of morphisms g E s un , p and Σ are homotopically finitely presented (Def. A.0.1 ), Re- mark B.3.2 implies that the deriv a tor asso ciated to the left B o usfield lo ca lization ( 5.1.1 ) is als o regular. In [ 29 , Def. 15 .1] the second author defined the a dditive motiv ator Mo t add dg as the triangulated deriv ator asso ciated (as in B.1 ) to the s table mo del categ ory of sp ectra of ob jects in ( 5.1.1 ), i.e. (5.1.3) Mot add dg := H O  Spt  L g E s un , p, Σ F un ( dgcat op f , sSet • )   . 5.2. A ne w Quillen mo del. Recall fro m App endix A that since dgcat f is a small category , the categor y Fun ( dgcat op f , Spt ) carries naturally a simplicial pro jective mo del structure. Moreov er, w e hav e a natural (Quille n) iden tification (5.2.1) Spt ( F un ( dgcat op f , sSet • )) ≃ Fun ( dgcat op f , Spt ) . Now, cons ider the Y oneda functor h : dgcat f − → F un ( dgcat op f , Spt ) B 7→ Σ ∞ dgcat f ( − , B ) , where every set dgca t f (? , B ) is considered as a simplicia lly -constant simplicial set and Σ ∞ ( − ) denotes the infinite susp ension sp ectr um. If F is a fibrant o b ject in F un ( dgcat op f , Spt ), w e hav e the following weak equiv alences : Map ( h ( B ) , F ) ≃ F ( B ) 0 Map ( h ( B ) , F ) ≃ F ( B ) ; consult Remark A.0.2 for the definition o f M ap and Map . W e a ls o hav e a homo- topic al Y oneda functor h : dgc at − → Fun ( dgcat op f , Spt ) A 7→ Σ ∞ Map ( − , A ) , where Map ( − , − ) deno tes the homotopy function complex (see App. A ) of the de- rived Morita mo del structure on dgcat (see § 3.2 ). By c o nstruction, homo topy (co)limits in Fun ( dgcat op f , Spt ) are calculated ob jectwise. This implies tha t the s hift 12 P AUL BALME R AND GONC ¸ ALO T ABUADA mo dels in Spt for the susp ensio n and lo o p spac e functors in Ho ( Spt ) (see Jar dine [ 15 , § 1]) induce ob ject wise s hift mo dels in Fun ( dgcat op f , Spt ) for the susp ensio n and lo op space functors in the triangula ted category Ho ( F un ( dgca t op f , Spt )). 5.2.2. Prop osi tion. The additive motivator ( 5.1.3 ) admits another Quil len mo del M ot add dg := L Ω( g E s un ) , Ω( p ) , Ω(Σ) F un ( dgcat op f , Spt ) , wher e Ω( g E s un ) , Ω( p ) and Ω(Σ) ar e obtaine d by stabilizing the sets g E s un , p and Σ in F un ( dgcat op f , Spt ) un der the obje ctwise lo op sp ac e funct or. Pr o of. The pro of follows from the combination of Theor e m B.4.1 , [ 29 , Thms. 4.4 and 8.7], Remar k 5.1.2 and the above identification ( 5.2.1 ).  Our new construction can be s ummed up as follo ws ; compar e with [ 29 , Rem. 15 .2]. HO ( dgcat f ) / / HO ( h )   HO ( dgcat ) R h t t U add dg o o HO  L Ω(Σ) F un ( dgcat op f , Spt )    Mot add dg Here, HO ( dgcat f ) is the prederiv ator asso cia ted with the full sub categor y dgcat f of dgcat (see B.1.1 and [ 2 9 , § 5] for details) and U add dg is the compo s ition of the functor R h induced by Y oneda a nd the lo calization morphism HO  L Ω(Σ) F un ( dgcat op f , Spt )  − → HO  M ot add dg  = Mot add dg . 5.2.3. Prop o sition. An obje ct F ∈ M ot add dg is fibr ant if and only if the fol lowing four c onditions ar e verifie d : (1) F ( B ) ∈ Spt is stably fibr ant, for al l B ∈ dg cat f . (2) F or every derive d Morita e qu ivalenc e B → B ′ in dgcat f , the induc e d mor- phism F ( B ′ ) → F ( B ) is a stable we ak e quivalenc e in Spt . (3) F ( ∅ ) ∈ Spt is c ont r actible. (4) Every (left-hand) split exact se quenc e in dgcat f (se e [ 29 , Def. 13.1] ) gives rise to a (right-hand) homotopy fib er se quenc e in H o ( Spt ) B ′ I / / B T o o P / / B ′′ S o o 7→ F ( B ′′ ) F ( P ) − → F ( B ) F ( I ) − → F ( B ′ ) . Pr o of. Condition (1) cor resp onds to the fa c t that F is fibr a nt in Fun ( dgcat op f , Spt ) since we use the pro jective mo del. Thanks to the shift mo dels in Fun ( dgcat op f , Spt ) for the susp e nsion and lo op space functors in Ho ( F un ( dgcat op f , Spt )), the construc- tion of the lo ca lized mo del structur e yields : An o b ject F is Ω(Σ)-lo ca l if and only if for every derived Morita equiv alence B → B ′ in dgcat f , the mor phism F ( B ′ ) → F ( B ) is a levelwise weak equiv alence in sSet • . Since F ( B ′ ) and F ( B ) are stably fibrant this is equiv alent to condition (2). An ob ject F is Ω( p )-lo ca l if and only if F ( ∅ ) n is contractible for every n ≥ 0. Since F ( ∅ ) is stably fibrant this is equiv alent to condition (3). W e now discuss condition (4). The construction of the set Ω( g E s un ) (see [ 29 , Not. 14.5] and Prop os itio n 5 .2.2 ) and the fact that the functor Map (? , F ) : Ho ( F un ( dgcat op f , Spt )) op − → Ho ( sSet • ) THE FUND AMENT AL ISOMORPHISM CONJECTURE 13 sends homotopy cofib er s e quences into homotopy fiber sequences, implies that an ob ject F is Ω( g E s un )-lo cal if and only if every split exact sequence in dgcat f induces a homotopy fib er sequence in Ho ( sSet • ) for every n ≥ 0; see [ 29 , Pr op. 14.8]. B ′ I / / B T o o P / / B ′′ S o o 7→ F ( B ′′ ) n F ( P ) n − → F ( B ) n F ( I ) n − → F ( B ′ ) n Once aga in, since F ( B ′ ), F ( B ) and F ( B ′′ ) are stably fibrant, this is equiv alent to condition (4). The pro of is then co ncluded, thanks to general Bo usfield lo ca lization theory; see [ 14 , Pr o p. 3.4.1].  Using the description of the fibrant o b jects of Pro p o sition 5.2.3 , we now prov e a key technical result. 5.2.4. Prop osi tion. F or every smal l dg c ate gory C ∈ dgcat f , the functor Map ( h ( C ) , − ) : Ho ( M ot add dg ) − → Ho ( Spt ) pr eserves homotopy c olimits. Pr o of. W e star t b y observing that by cons truction, the res ult holds in the stable mo del categor y F un ( dgcat op f , Spt ). Tha nks to Rema rk A.0.3 it suffices to prove the following : If { F j } j ∈ J is a diagr am of fibrant ob jects in the lo caliz ed cate- gory M ot add dg , then its ho motopy colimit sa tis fie s conditions (2)-(4) of Prop os i- tion 5.2.3 . Since homotopy colimits in Fun ( dgcat op f , Spt ) are calculated o b jectwise, conditions (2)-(3) are clear ly verified. In what co ncerns condition (4), notice that Ho ( Spt ) is a triangulated catego ry and so the homoto py fib er sequences F j ( B ′′ ) − → F j ( B ) − → F j ( B ′ ) are also homo topy cofib er seq uences. This implies that ho colim j ∈ J F j ( B ′′ ) − → ho colim j ∈ J F j ( B ) − → ho colim j ∈ J F j ( B ′ ) is a homotopy cofib er sequence and so als o a homotopy fib e r sequence. This shows condition (4) a nd so the pr o of is finis hed.  W e finish this subsection by describing an explicit set of generator s. 5.2.5. Prop osition. The set of strictly finite dg c el ls { h ( B ) | B ∈ dgcat sf } (D ef. 3.1.1 ) form a set of homotopic gener ators (Def. A.0.4 ) in M ot add dg . Pr o of. Notice that the o b jects { h ( B ) | B ∈ dgcat f } are ho mo topic genera tors in the mo del category Fun ( dgcat op f , Spt ) by the very definition of weak equiv alences. Recall from § 5.1 that dgcat f is the s mallest full s ubca tegory of dg cat which con- tains the strictly finite dg cells and which is stable under the co-simplicial and fibrant reso lutio n functors o f [ 29 , Def. 5.3]. Ther efore, every ob ject in dgc at f is de- rived Mo rita equiv a lent to an o b ject in dgcat sf . This implies b y Lemma A.0.6 a nd Prop ositio n 5.2.3 , that the ob jects { h ( B ) | B ∈ dgca t sf } a re homotopic generator s in L Ω(Σ) F un ( dgcat op f , Spt ). Once again by Lemma A.0 .6 we can lo calize further with resp ect to the s ets Ω( g E s un ) and Ω( p ), which completes the pro o f.  14 P AUL BALME R AND GONC ¸ ALO T ABUADA 6. Fundament al additive functor W e in tro duce the no tion of additive functor on the o rbit ca teg ory , give several examples, and co nstruct the fundamental functor which satisfies additivity . Note that every R -catego ry (see § 2.4 ) can b e naturally consider ed as a dg cate- gory (with complexes of morphisms co ncentrated in deg ree zer o ). Given a gro up G , we thus obtain a co mp os ed functor (6.0.1) Or( G ) ? − → Grp R [ − ] − → R - cat ⊂ dg cat . This functor is the bas ic piece. W e now consider all functor s obtained from com- po sing it with an additive inv ariant of dg categ ories. 6.0.2. Defin ition. Let M b e a stable mo del categ ory (see Rem. B.1.3 ) a nd E : Or( G ) → M a f unctor. W e say tha t E is additive if it factor s through ( 6.0 .1 ) follow ed by a fu nctor E : dgcat → M whos e asso cia ted morphism of deriv ato rs E : HO ( dgcat ) → HO ( M ) is an additive in v ar iant of dg categories (see § 1.2 ). The factorization of Definition 6.0 .2 should not b e confused with the one we wan t to establis h in Theor e m 6.0.5 (that is, via the fundamen tal additive functor E fund ). W e rather restr ict attention to functor s on the orbit ca tegory that only dep end on the asso cia ted dg catego ry . This is a mild res tr iction since many of the classical functors hav e bee n extended to dg categ o ries, as explained in § 4 . 6.0.3. Examples. Recall fr om § 4 several examples of functors E : dgcat → M defined on the category of dg categor ies ( e.g. connective, non-connective, and ho- motopy K - theory , Hochschild a nd cy clic homology , and topolo gical Ho chsch ild ho- mology), whos e asso ciated morphisms of deriv ator s E : HO ( dgcat ) → HO ( M ) are additive inv ariant o f dg catego ries. By pre-c o mpo sing them with the functor ( 6.0.1 ) we obtain several ex amples of additive functor s E : Or( G ) → M in the se ns e of Definition 6.0.2 . Mor eov er, if B is a homotopically finitely presented dg category B , we obtain a “co efficients v aria nt ” E ( − ; B ) (see § 4.6 ) defined as follows Or( G ) ∋ G/H 7→ E ( G/H ; B ) := E ( rep dg ( B , R [ G/H ])) . Note that if B = R , the additive functor E ( − ; B ) reduces to the compo sition E : Or( G ) ? − → Grp R [ − ] − → R - cat − → dgcat E − → M . 6.0.4. Definition. The fundamental additiv e functor E fund is the compo sition Or( G ) ? − → Grp R [ − ] − → R - cat ⊂ dg cat U add dg − → M ot add dg . The universalit y theorem [ 29 , Thm. 1 5.4] (see eq uiv alence ( 1.2 .1 )) yields : 6.0.5. Theorem. L et G b e a gr oup and E : Or( G ) → M an additive funct or. T hen ther e exists a homotopy c olimit pr eserving morphism of derivators E : Mot add dg → HO ( M ) , which makes the fol lowing diagr am c ommute (up to isomorphism) Or( G ) E fund / / E $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ M ot add dg / / Mot add dg ( e ) E ( e )   M / / Ho ( M ) . THE FUND AMENT AL ISOMORPHISM CONJECTURE 15 Pr o of. By Definition 6.0.2 , E factors through a functor E : dgcat → M whose asso ciated morphism o f deriv ato r s E : H O ( dgcat ) → HO ( M ) is an additive inv aria nt of dg c ategories . By [ 29 , Thm. 1 5 .4], see ( 1.2.1 ), this E descends to a homotopy colimit preserving mo rphism o f deriv ators E : Mot add dg → HO ( M ), whos e v a lue at the base ca tegory makes the ab ov e diagram commute (up to isomor phism).  Using the general notion of assembly prop erty of § 2.2 , we get : 6.0.6. Corollary . L et G b e a gr oup and let F b e a family of su b gr oups. If the fundamental additive funct or E fund has the F -assembly pr op erty, then so do al l additive fu n ctors. Pr o of. Simply a pply the mo rphism E to the ( E fund , F , G )-as sembly map and use the fact that E preserves ar bitrary homotopy c olimits.  7. Reduction to strictl y finite dg cells Pr o of of The or em 1.3.4 . The main ing redient of the pro of is the following result: 7.0.7. Theorem . (se e 2 [ 29 , T hm. 1 5.10] ) Given dg c ate gories A and B , with B ho- motopic al ly finitely pr esente d, ther e is a c anonic al s t able we ak e quivalenc e of sp e ctr a Map Mot add dg  U add dg ( B ) , U add dg ( A )[1]  ∼ = K ( rep dg ( B , A )) . Thanks to Corolla r y 6.0.6 , condition (1 ) of Thm. 1.3 .4 implies condition (2). Thanks to [ 29 , Prop. 5.2 and E x. 5.1] a dg categ ory is homotopica lly finitely pre- sented (Def. A.0.1 ) if and only if it is derived Morita equiv a lent to a retra ct in Ho ( dgcat ) (see § 3.2 ) of a strictly finite dg cell (Def. 3 .1.1 ). Therefor e, every strictly finite dg cell is ho motopically finitely presented, and so condition (2) implies con- dition (3). W e now sho w that co ndition (3) implies condition (1). Recall the construction of the fundamen tal additiv e functor E fund : Or( G ) ? − → Grp R [ − ] − → R - cat ⊂ dg cat U add dg − → M ot add dg . Assuming condition (3), w e need to show that the induced map ho colim Or( G, F ) U add dg ( R [ G/H ]) − → U add dg ( R [ G/G ]) is a n isomo rphism in Mot add dg ( e ). Since the catego ry Mot add dg ( e ) is triangula ted, it suffices to show that the s uspe nsion map ho colim Or( G, F )  U add dg ( R [ G/H ])[1]  ≃  ho colim Or( G, F ) U add dg ( R [ G/H ])  [1] − → U add dg ( R [ G/G ])[1] is an isomor phism. By Prop o s ition 5.2.5 , the set of ob jects { h ( B ) | B ∈ dg cat sf } form a set of ho mo topic generator s in M ot add dg and so it is enough to pr ov e that, for every B ∈ dgcat sf , the induced ma p of sp ectra Map  h ( B ) , ho colim Or( G, F ) U add dg ( R [ G/H ])[1]  − → Map  h ( B ) , U add dg ( R [ G/G ])[1]  is a s table w eak equiv a le nce. By Pr op osition 5.2.4 , the functor Map ( h ( B ) , − ) pre- serves ho mo topy colimits and so we have Map  h ( B ) , ho colim Or( G, F ) U add dg ( R [ G/H ])[1]  ≃ ho colim Or( G, F ) Map  h ( B ) , U add dg ( R [ G/H ])[1]  . 2 In lo c. c it. Map wa s denoted by Hom Sp N , K by K c , and rep dg b y rep mor . 16 P AUL BALME R AND GONC ¸ ALO T ABUADA By co nstruction of Mot add dg (see [ 29 , § 15]), we o bs erve that s ince B ∈ dgcat sf the ob ject h ( B ) iden tifies with U add dg ( B ). Hence, the ab ove co-r epresentabilit y theo- rem 7.0.7 provides stable weak eq uiv alences Map ( h ( B ) , U add dg ( R [ G/H ])[1]) ∼ = K ( rep dg ( B , R [ G/H ])) , for every B ∈ dgcat sf and H ∈ Or( G, F ). In conclusion, we ar e r e duced to show that for every strictly finite dg cell B , the map ho colim Or( G, F ) K ( rep dg ( B , R [ G/H ])) − → K ( rep dg ( B , R [ G/G ])) is a stable weak equiv alence. But now, this is prec is ely our hypothesis, na mely that the additive functors K ( − ; B ) hav e the F -a ssembly prop er t y for G .  Appendix A. Mod el ca tegor y tools In this app endix we re c a ll some material from the theo r y of Quillen mo del struc- tures [ 26 ] and prove a technical lemma conc e rning homotopic gener ators. Let sSet (resp. sSet • ) b e the model category of ( p ointed) simplicial s e ts ; see Go erss-J ardine [ 9 , § I]. Given a Quillen model catego ry M , we denote by Map ( − , − ) : M op × M → H o ( sSet ) its homotopy function co mplex; see [ 14 , Def. 17 .4.1]. Recall that if M is a simplicial mo del category [ 9 , § I I.3], its homotopy function complex is g iven, for X, Y ∈ M , b y the simplical set Ma p ( X, Y ) n := M ( X c ⊗ ∆[ n ] , Y f ), where X c is a cofibra nt reso lution of X and Y f is a fibrant resolution of Y . More- ov er, if Ho ( M ) denotes the homotop y category of M , we hav e an iso morphism π 0 Map ( X , Y ) ≃ Ho ( M )( X , Y ). A.0.1. Definition. An ob ject X in M is homotopic al ly finitely pr esente d if for an y diagram Y : J → M in M (for any shap e, i.e. small categ ory , J ), the induce d map ho colim j ∈ J Map ( X , Y j ) − → Map ( X , ho c o lim j ∈ J Y j ) is an isomor phism in Ho ( sSet ). Let Spt be the (mo del) category of sp ectra [ 9 , § X.4]. If X is a spectr um, we denote by X [ n ] , n ≥ 0 its n th susp ension , i.e. the sp ectrum defined as X [ n ] m := X n + m , m ≥ 0. If X and Y are tw o sp ectra, we define its homotopy function sp e ctrum Map ( X, Y ) b y Ma p ( X , Y ) n := Map ( X , Y [ n ]), wher e the bonding maps are the natura l ones. A.0.2. R emark. L e t I be a small c a tegory . B y [ 15 , Thm. 3 .3 ], the categor y of presheav es of spectra 3 F un ( I op , Spt ) = Spt I op carries the pr oje ctive mo del struc- ture, with w ea k-equiv a lences a nd fibrations defined ob jectwise. If w e denote by Map ( − , − ) its homotop y function complex, the homotop y funct ion sp e ct rum be- t ween t wo presheav es F a nd G is given (as in the cas e of sp ectra ) by Map ( F, G ) n := Map ( F, G [ n ]), w her e G [ n ] is the n th obje ctwise su sp ension of G . A.0.3. Rema rk. Let S b e a set of morphisms in F un ( I op , Spt ) and L S ( F un ( I op , Spt )) its left B o usfield lo ca lization with respect to S ; see [ 14 , Thm. 4.1 .1]. Since the categorie s F un ( I op , Spt ) and L S ( F un ( I op , Spt )) hav e the same cofibr ations, and hence the same trivial fibra tions, the simplicial cofibrant replacement functor Γ ∗ (see [ 14 , § 16 ]) is the same in b oth cas es. Hence, the homotopy function sp ectrum 3 In [ 15 ] the category Fu n ( I op , Spt ) of preshea ve s of spectra i s denoted by Spt ( I ). W e do not use that notation since it already app ears in ( 5.1.3 ) and ( 5.2.1 ) wi th a different meaning. THE FUND AMENT AL ISOMORPHISM CONJECTURE 17 of L S ( F un ( I op , Spt )) can be computed as M ap ( − , Q ( − )), where Q ( − ) is a fibrant resolution functor in L S ( F un ( I op , Spt )). Let M b e a left Bo usfield lo calizatio n o f Fun ( I op , Spt ). A.0.4. Definition. A set of homotopic gener ators is a set of ob jects { G j } j ∈ J in M such that a morphism f : F → F ′ is a weak equiv a lence in M if (and only if ) for every ob ject G j the induced ma p of sp ectra (A.0.5) f ∗ : Map ( G j , F ) − → Map ( G j , F ′ ) is a stable w eak equiv a lence. A.0.6. Lemma. L et S b e a set of m orphisms in M . If the { G j } j ∈ J ar e homotopic gener ators in M then they ar e homotopic gener ators in L S ( M ) as wel l. Pr o of. W e use the homotopy function s pe ctrum Map ( − , Q ( − )) in L S ( M ) a s in Remark A.0.3 . Let f : F → F ′ be a morphism in M which induces a stable equiv alences under Map ( G j , Q ( − )) for all j ∈ J . Consider the commutativ e square F f   ∼ / / Q ( F ) Q ( f )   F ′ ∼ / / Q ( F ′ ) . Since b y hypothesis the { G j } j ∈ J are homotopic generators in M , the map Q ( f ) is a w eak equiv alence in M and so a weak equiv alence in L S ( M ). By the tw o -out-of- three prop erty , we co nclude that f is a weak equiv alence in L S ( M ).  Appendix B. Grothendieck Deriv a tors: st abiliza tion and lo caliza tion In this app endix w e giv e a brief in tro ductio n to deriv ators , recall so me basic facts, and then prov e that the op er ations of sta bilization (see [ 29 , § 8 ]) and left Bousfield lo caliza tion (see [ 29 , § 4]) co mm ute. B.1. Deriv ators. The original re fer ence is Grothendieck’s manuscript [ 10 ]. See also Maltsiniotis [ 23 ] or a shor t account in Cis inski-Neeman [ 7 , § 1]. Deriv ator s originate in the problem of higher homotopies in derived ca tegories . F or a non- z e ro triang ulated categor y D a nd for X a s mall ca teg ory , it esse ntially never ha ppe ns that the dia gram catego ry Fun ( X, D ) = D X remains tria ngulated (it alrea dy fails for the category of ar rows in D , that is, for X = [1] = ( • → • )). Now, very often, our tr ia ngulated category D a ppea rs as t he homotopy cate- gory D = Ho ( M ) of so me mo del M . In this cas e, we ca n consider the categor y F un ( X , M ) of diag rams in M , whose homotopy catego ry Ho ( F un ( X , M )) is often triangulated and provides a reasona ble approximation for F un ( X , D ). More imp or - tantly , one can let X mov e. This nebula of categories Ho ( Fun ( X, M )), index ed by small ca tegories X , and the v arious functors and natural tr a nsformations betw een them is what Grothendieck for malized in to the concept of deriva tor . A deriv ator D consists o f a strict contrav ar ia nt 2-functor fro m the 2-catego ry of small catego ries to the 2- c ategory of all catego ries (a. k. a. a prederiv ato r) D : Cat op − → CA T , sub ject to certain co nditions. W e shall no t list them her e for it would b e too long but we refer to [ 7 , § 1]. The essential exa mple to keep in mind is the deriv ator 18 P AUL BALME R AND GONC ¸ ALO T ABUADA D = HO ( M ) asso ciated to a cofibrantly gener a ted Quillen mode l categ o ry M and defined for every small catego ry X by (B.1.1) HO ( M ) ( X ) = Ho  F un ( X op , M )  . W e denote by e the 1- p o int categ o ry with one ob ject and o ne ident ity morphism. Heuristically , the catego ry D ( e ) is the basic “derived” catego ry under consider ation in the deriv ator D . F or instance, if D = HO ( M ) then D ( e ) = Ho ( M ). B.1.2. Definitions. W e now reca ll three slightly tec hnical pr o p erties of deriv ators. (1) A deriv a tor D is str ong if for every finite free catego r y X and every sma ll category Y , the natural functor D ( X × Y ) − → Fun ( X op , D ( Y )) (see [ 7 , § 1 .1 0]) is full a nd essentially surjective. (2) A deriv a tor D is p ointe d if for a ny clo sed immersion i : Z → X in Cat the cohomological direct image functor i ∗ : D ( Z ) − → D ( X ) has a r ight adjoint, a nd if moreover and dually , for any op en immersion j : U → X the homologica l direct imag e functor j ! : D ( U ) − → D ( X ) has a left adjoint; see details in [ 7 , Def. 1.1 3 ]. (3) A deriv ator D is t riangulate d o r stable if it is p ointed and if every glo bal commutativ e square in D is cartesian exa ctly when it is co cartesia n; see details in [ 7 , Def. 1.1 5 ]. B.1.3. R emark. If M is a cofibrantly g enerated Quillen mo del category , [ 6 , Pro p. 2.15] applied to all the homotopy categor ies Ho ( F un ( X op , M )) allows us to conclude that the der iv ator HO ( M ) is strong . If M is p ointed then so is HO ( M ). Finally , if M is a s table mo del category , then its a sso ciated deriv ator HO ( M ) is triangulated. In short, the reader who wishes to restrict atten tion to deriv a tors of the form HO ( M ) can as well consider pro per ties (1)-(3) of Definition B.1.2 as mild o nes. B.1.4. Theorem (Maltsiniotis [ 22 ]) . F or any t r iangulate d derivator D a nd s mal l c ate gory X the c ate gory D ( X ) has a c anonic al triangulate d structur e. (An explicit des cription of the triangulated str ucture is also giv en [ 7 , § 7.9 ].) B.1.5. Notation. Le t D and D ′ be der iv ators . W e denote by Hom ( D , D ′ ) the categor y of all mor phisms of deriv ators and by Hom ! ( D , D ′ ) the categ ory of mor phisms of deriv ator s whic h prese r ve homotopy c olimits ; see details in Cisinski [ 5 , § 3.25]. B.2. Stabilization. In [ 12 , 13 ] Heller developed the no tion of a smal l homotopy the ory or in other words the notion of a r e gular deriv ator, i.e. a deriv ator D w her e se - quential homotopy colimits co mmute with finite pro ducts and homotopy pullbacks; see [ 13 , IV, Section 5 ]. F or instance, if M is a cofibrantly g enerated mo del catego ry where s e quential homotopy colimits commut e with finite pro ducts and homotopy pullbacks, then the a sso ciated deriv ator HO ( M ) is r e gular. By adopting the la tter notation, Heller’s work [ 12 ] can de des crib ed as follows: let D b e a r egular po int ed strong deriv ator . Heller constructed the universal morphism stab : D → St ( D ) tow a rds a triang ulated strong deriv a to r, which pres erves homo topy co limits, and satisfies the following universal pr op erty . B.2.1. Theorem (Heller [ 12 ]) . L et T b e a triangulate d str ong derivator. Then the morphism stab : D → St ( D ) induc es an e quivalenc e of c ate gories ( stab ) ∗ : Ho m ! ( St ( D ) , T ) ∼ − → Hom ! ( D , T ) . THE FUND AMENT AL ISOMORPHISM CONJECTURE 19 B.3. Left Bousfield lo cali zation. Let D b e a deriv ato r a nd S a cla ss of m or - phisms in the base categor y D ( e ). B.3.1. Definition. The deriv ato r D admits a left Bousfield lo c alization with resp ect to S if there exists a morphism of der iv ator s γ : D → L S D , whic h preserves ho mo- topy co limits, sends the elements of S to isomorphisms in L S D ( e ), a nd satisfies the following univ ersa l prop erty : F or every deriv ator D ′ the morphism γ induces a n equiv alence o f categorie s γ ∗ : Ho m ! ( L S D , D ′ ) ∼ − → Hom ! ,S ( D , D ′ ) , where Hom ! ,S ( D , D ′ ) denotes the categor y of morphisms o f deriv ato rs which pre- serve homo to py colimits a nd send the elemen ts of S to isomorphisms in D ′ ( e ). B.3.2. R emark. L e t M be a left prop er, cellular mo del categor y a nd L S M its left Bousfield lo calization (see [ 14 , T hm. 4 .1.1]) with resp ect to a set of morphisms S . Then, t he induced mor phism o f deriv ators HO ( M ) → HO ( L S M ) is a left Bous - field lo calization of deriv ators with r e sp e ct to the image of S in Ho ( M ) ; see [ 29 , Thm. 4.4]. Moreover, if the domains and co domains of the set S are homotopi- cally finitely presented ob jects (Def. A.0.1 ), the functor H o ( L S M ) → Ho ( M ), right adjoint to t he localizatio n functor, preserves filtered homotop y colimits ; s ee the pro of of [ 29 , Lem. 7.1]. Under these hypo thes es, if HO ( M ) is regular then so is HO ( L S M ). B.3.3. R emark. By [ 29 , Lem. 4 .3], the Bo usfield lo ca liz ation L S D of a triangulate d deriv ator D remains triangulated as long as S is sta ble under the lo o p spa c e functor . F or mor e general S , to remain in the world of triangulate d deriv a tors, one has to lo calize with r esp ect to the s et Ω( S ) generated b y S a nd lo ops, as follows. B.3.4. Prop os ition. L et D b e a triangulate d derivator and S a class of morphisms in D ( e ) . L et us denote by Ω( S ) the smal lest class of morphisms in D ( e ) which c ontains S and is stable under the lo op sp ac e functor Ω : D ( e ) → D ( e ) . Then, for any tr ia ngulated derivator T , we have an e quivalenc e of c ate gories (B.3.5) Hom ! , Ω( S ) ( D , T ) ≃ Hom ! ,S ( D , T ) . Heuristic al ly, L Ω( S ) D is the triang ula ted lo c alization of D with r esp e ct to S . Pr o of. F or F an element of Hom ! ( D , T ), the functor F ( e ) : D ( e ) → T ( e ) pre s erves homotopy colimits, hence it co mm utes in pa rticular with the susp ensio n functor. Since both D and T ar e tria ngulated, susp ensio n and lo op space functors are inv erse to each other. Hence F ( e ) also comm utes with Ω. It is then o bvious that F ( e ) sends S to isomorphisms if and only if it does so with Ω( S ).  B.4. Commuting stabil ization with l o calization. Let D be a p ointed, str ong and regular der iv ator a nd S a class of mor phisms in D ( e ). Assume that D admits a left Bousfield lo c alization L S D with resp ect to S . W e then obtain a der iv ator L S D which is still po int ed and strong . If it is also regular (see Remark B.3.2 ), we can consider its sta bilization St ( L S D ) as in § B.2 . On the other ha nd, we can first co nsider the triangulated der iv ator St ( D ). W e still denote by S the image o f the c la ss S under the morphism of der iv ator s stab : D → St ( D ). Supp ose that the left Bousfield lo ca lization L Ω( S ) St ( D ) by Ω( S ) a lso 20 P AUL BALME R AND GONC ¸ ALO T ABUADA exists. W e then ha ve t wo constr uctio ns D γ t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ stab + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ L S D stab $ $ ■ ■ ■ ■ ■ St ( D ) γ x x q q q q q St ( L S D ) L Ω( S ) St ( D ) and we cla im that they agree, namely : B.4.1. Theorem. With the ab ove notations and hyp otheses, the derivators L Ω( S ) St ( D ) and St ( L S D ) ar e c anonic al ly e qu ivalent, under D . Pr o of. Both deriv ators are triangulated (for L Ω( S ) St ( D ), see Remark B.3.3 ) and strong. So, it suffices to show that for any triangulated strong deriv ator T , w e hav e the following equiv alences of categories: Hom ! ,S ( D , T ) Hom ! ( L S D , T ) ≃ γ ∗ 2 2 ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ Hom ! ,S ( St ( D ) , T ) ( B.3.5 ) ≃ stab ∗ i i ❚ ❚ ❚ ❚ ❚ ❚ ❚ Hom ! , Ω( S ) ( St ( D ) , T ) Hom ! ( St ( L S D ) , T ) ≃ stab ∗ h h ◗ ◗ ◗ ◗ ◗ ◗ Hom ! ( L Ω( S ) St ( D ) , T ) . ≃ γ ∗ 4 4 ✐ ✐ ✐ ✐ ✐ ✐ ✐ The tw o e quiv alences o n the left-hand side as well as the lower-right one all follow from Theorem B.2.1 or Definition B.3.1 . Equiv a lence stab ∗ : Hom ! ,S ( St ( D ) , T ) ∼ − → Hom ! ,S ( D , T ) requires a comment : By Theore m B.2.1 we have an equiv alence stab ∗ : Hom ! ( St ( D ) , T ) ∼ − → Ho m ! ( D , T ) a nd it is straightforward to chec k that it preser ves the ab ov e sub categor ies.  Ac knowledgmen ts: The authors ar e grateful to Ber nhard Keller and Bertra nd T o¨ e n for useful dis cussions, Denis-Char le s Cisinsk i for p ointing out a missing hy- po thesis in Theor em B.4.1 , Arthur Bartels and W olfga ng L ¨ uck for pre c ious com- men ts on a previous draft, and Holger Reich for k indly providing P rop osition 2.5.2 . The second author would like to thank the UCLA Mathematics Depar tmen t for hospitality a nd excellent working conditions, when this work w as initiated. References [1] P . Balmer and M . 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P aul Balmer, Dep ar tment of Ma thema tics, UCLA, Los An geles, CA 90095 -1555, USA E-mail addr ess : balmer@math .ucla.edu URL : http://w ww.math.ucla. edu/ ∼ balmer Gonc ¸ alo T abuada, Dep ar tment of Ma them a tics, MIT, Cambridge, MA 02139, USA E-mail addr ess : tabuada@mat h.mit.edu URL : http://w ww.math.mit.e du/ ∼ tabuada