Products, multiplicative Chern characters, and finite coefficients via Non-commutative motives

PR ODUCTS, MUL TIPLICA TIVE CHERN CHARA CTERS, AND FINITE COEFFICIENTS VIA NON-COMMUT A TIVE MOTIVES GONC ¸ ALO T ABUADA Abstract. Products, multiplicativ e Chern c haracters, and finite coefficients, are unarguably among the most important t o ols in algebraic K - theory . Al- though they admit nu merous different constructions, they are not y et f ully understoo d at the concept ual l ev el. In this article, making use of the theory of non-commuta tive motiv es, we c hange this state of affairs b y cha racterizing these construct ions in terms of simple, elegant , and precise universal prop er- ties. W e illustrate the potent ial of our results by dev eloping t wo of its man yf old consequenc es: (1) the multiplicativit y of the negative Chern characte rs follo ws directly f rom a si mple factorization of the mixed complex construction; (2) Kassel’s biv arian t Chern c haracter admits an adequat e extension, from the Grothendiec k group lev el, to all higher algebraic K-theory . 1. Intr oduction This article is pa rt of a long -term pro ject, whose ob jective is the study of (higher) algebraic K -theor y via non-co mmu tative motives. Previous contributions can b e found in [ 2 , 5 , 6 , 30 , 31 , 32 , 33 , 3 4 ]. Here, we focus on pr o ducts, multiplicativ e Chern characters, and finite co efficients. Since the ea r ly days, it has b een exp ected that alg ebraic K -theor y w ould come equipp e d with so me kind of pr o ducts . Over the last decades such pro ducts were constructed by Lo day , May , McCarthy , Milnor, Quillen, W aldhausen, a nd others in different settings a nd by making use of differe nt to ols [ 22 , 2 4 , 25 , 2 6 , 29 , 37 , 38 ]. Among o ther impo rtant applica tio ns, they y ielded new elements in alge br aic K - theory and simplified pro ofs of Riemann-Ro ch theorems; see W eib el’s survey [ 39 ]. Algebraic K -theory is a very p ow erful and subtle inv a riant whose ca lc ula tion is often out of reach. In or der to ca ptur e some of its (multiplicative) informa tion Connes-Ka r oubi, Dennis, Goo dwillie, Ho o d-Jo ne s , Ka ssel, McCarthy , and others constructed (multiplic ative) Chern char acters tow ards simpler theories by mak ing use of a v ar iety of highly inv olved techniques [ 8 , 9 , 11 , 14 , 19 , 25 ]. In order to attack the Lich ten baum-Quillen conjectures, Browder [ 4 ] introduced the algebr aic K -t he ory with finite c o efficients . Sin ce then, this flex ible theor y be- came one of the most imp ortant (working) to ols in the field. Although the imp ortance of the aforementioned co nstructions extends well be- yond alge br aic K -theory , they remained rather ad ho c and somewhat m ysterious. Therefore, a solution to the following q uestion is of ma jor impor tance: Question: Is it p ossible t o char acterize a l l the afor ementione d c onst ructions by simple, ele gant, and pr e cise un iversal pr op erties ? Date : October 26, 2018. 2000 Mathematics Subje ct Classific ati on. 19D45, 19D55, 18G55. Key wor ds and phr ases. Algebraic K -theory , Chern characters, Non-commutativ e motives. The author was partially supp orted by the Clay Mathematics Institute, the M idwest T opol ogy Net work, and the Calouste Gulb enkian F oundation. 1 2 GONC ¸ ALO T ABUADA In this article we a ffirmatively a nswer this q ue s tion. Mo reov er, the theory of non-c ommutative motives plays a cen tral role is such characteriza tions. Non-commutativ e m otiv es. A differ ent ial gr ade d (= dg) c ate gory , ov er a fixed base commut ative ring k , is a ca tegory enriched o ver co chain complexe s of k - mo dules; se e § 5 for deta ils. All the clas sical inv ariants such as cyclic ho mology (and its v ariants), alg ebraic K -theor y , a nd even top olo gical cy clic ho mology , e x tend naturally from k -a lgebras to dg categor ies. In or der to study all these inv ar ia nts simult aneously the notio n of additive invariant w as introduced in [ 30 ]. This no- tion, that w e now r ecall, makes use of the languag e of Gro thendieck deriv ators, a formalism which a llows us to state and prov e precise univ ersal proper ties; see § 3.2 . The category dgcat of dg categories carries a Quillen model structure, whose weak equiv ale nc e s are the deriv e d Morita e quivalenc es (see § 5 .2 ), and so it gives rise to a deriv a tor HO ( dgcat ). Let D b e a tria ngulated deriv a tor and E : HO ( dgcat ) → D a morphism of der iv a to rs. W e say that E is an additive invariant if it preserves filtered homotopy colimits and sends split exact sequences ( i.e. sequences o f dg ca tegories which b ecome split exact after pas sage to the asso cia ted derived c a tegories ; see [ 30 , Definition 13.1]) to direct sums. By the additivity res ults o f Blumberg-Mandell, Keller, W a ldhausen, a nd others [ 3 , 17 , 35 , 37 ] all the mentioned inv aria nts g ive rise to additive inv ariants. In [ 30 ] the universal additive invariant was constructed U A : HO ( dgcat ) − → Mot A . Given any triangulated deriv a tor D we hav e a n induced equiv alenc e of categorie s (1.1) ( U A ) ∗ : Hom ! (Mot A , D ) ∼ − → Hom A ( HO ( dgcat ) , D ) , where the left-hand side denotes the categor y o f homotopy colimit preserving mor- phisms of deriv ators and the right-hand side the categ ory of additive inv ar iants. Because of this universal pro pe r ty , which is r eminiscent o f the theor y of motives, the deriv a tor Mot A is called the additive motivator , a nd its base categor y Mo t A ( e ) the triangulate d c ate gory of non-c ommutative motives . The tenso r pro duct extends na turally from k -a lg ebras to dg c a tegories , giving rise to a (derived) sy mmetr ic mono idal structure − ⊗ L − on HO ( dgcat ) whose unit is the dg category k with one ob ject and with k as the dg algebra o f endomorphisms . In [ 6 ] this monoidal str ucture was extended to Mot A in a universal way 1 : given any triangulated der iv a to r D , endow ed with a homotopy colimit preser ving monoidal structure, the ab ov e equiv alence ( 1.1 ) admits a symmetric monoida l sharp ening (1.2) ( U A ) ∗ : Hom ⊗ ! (Mot A , D ) ∼ − → H om ⊗ A ( HO ( dgcat ) , D ) . As any triangula ted der iv ator, Mo t A is naturally enr iched over sp ectra; let us denote by R Hom ( − , − ) this enrichmen t. In [ 30 ] it w as proved that algebraic K -theor y not only descends uniquely to Mot A (since it is an additive inv aria nt ) but, moreov er, it bec omes co-repre s entable by the unit ob ject, i.e. for every dg catego ry A we have a natural equiv a lence of spec tr a (1.3) R Hom ( U A ( k ) , U A ( A )) ≃ K ( A ) . In the triangulated categor y Mot A ( e ) we have ab elian gro up is o morphisms (1.4) Hom ( U A ( k ) , U A ( A )[ − n ]) ≃ K n ( A ) n ≥ 0 . 1 In [ 6 ] we ha v e considered the lo calizing analogue of Mot A . Ho wev er, the arguments in the additiv e case are completely similar. PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 3 2. St a tement of resul ts Pro ducts. Let A and B b e dg ca tegories. On o ne ha nd, we can consider the asso ciated categor ies b A , b B and \ A ⊗ B o f p erfect mo dules (see Notation 5.1 ) and the bi-exact functor (see § 4.2 ) b A × b B − → \ A ⊗ B ( M , N ) 7→ M ⊗ k N . (2.1) F ollowing W aldhausen (see § 5.3 ) we obtain then a pairing in algebraic K -theor y (2.2) K ( A ) ∧ K ( B ) − → K ( A ⊗ B ) . On the other hand, if A (or B ) is k -flat ( i.e. for every pair ( x, y ) of ob jects in A the functor A ( x, y ) ⊗ − preserves q ua si-isomor phisms; see [ 16 , § 4.2 ]) A ⊗ L B ≃ A ⊗ B and so the symmetric monoida l str ucture on Mot A ( e ) combined with ( 1.3 ) g ives rise to another pairing in algebr aic K -theory (2.3) K ( A ) ∧ K ( B ) − → K ( A ⊗ L B ) ≃ K ( A ⊗ B ) . Theorem 2.4. The p airings ( 2.2 ) and ( 2.3 ) agr e e up to homotopy. In p articular, the ab elian gro up homomorphisms K i ( A ) ⊗ Z K j ( B ) → K i + j ( A ⊗ B ) induc e d by the ab ove p airing ( 2.2 ) agr e e with the ab elian gro up homomorph isms obtaine d by c ombining the symmetric monoidal stru ctur e on Mot A ( e ) with ( 1 .4 ) . Example 2.5 . (Commutativ e alg ebras) Let A = B = A , with A a k -flat commutativ e algebra. Since A is commutativ e, its multiplication is a mor phism of k -alg ebras a nd hence a dg functor. By comp osing ( 2.3 ) with the induced map K ( A ⊗ A ) ≃ R Hom ( U A ( k ) , U A ( A ⊗ A )) − → R H om ( U A ( k ) , U A ( A )) ≃ K ( A ) we recov er the algebraic K -theory pairing on A constructed by W aldhausen in [ 38 ]. In particular , we r ecov er the (g raded-co mmutative) multiplicativ e structure on K ∗ ( A ) co nstructed originally by Lo day in [ 22 ]; see [ 3 9 , § 4] for the agreement betw een the approa ch of W aldhausen and the approach of Lo day . Example 2.6 . (Schemes) Let X b e a quasi-compa c t and quasi-separ ated scheme ov er a base field k . Recall from [ 23 ] (or from [ 6 , Example 4.5(i)]) that the de- rived categ ory of p erfect complexes of O X -mo dules admits a na tur al differential graded (= dg) enha nc e ment D dg p erf ( X ). Under this enhancement, the bi-exact func- tor ( E · , F · ) 7→ E · ⊗ L O X F · lifts to a dg functor D dg p erf ( X ) ⊗ D dg p erf ( X ) → D dg p erf ( X ). Therefore, s ince the a lgebraic K -theory of X c a n b e r ecov ered from D dg p erf ( X ) (see [ 16 , § 5.2]), the comp osition of ( 2.3 ) with the induced ma p R Hom ( U A ( k ) , U A ( D dg p erf ( X ) ⊗ D dg p erf ( X ))) − → R H om ( U A ( k ) , U A ( D dg p erf ( X ))) gives rise to the alg ebraic K -theor y pairing on X constr ucted origina lly b y Thoma son- T ro baugh in [ 36 , § 3.15]. Note that Theorem 2.4 (and Exa mples 2.5 - 2 .6 ) offer s a n elegant conceptual char- acterization of the algebraic K - theory pro ducts. Informally sp eaking, algebra ic K -theory is the additive inv ariant co-r epresented by the unit of Mot A ( e ), and the algebraic K -theor y pro ducts are the o pe r ations naturally induced by the symmet- ric mo no idal structure of Mo t A ( e ). W or k in pro gress with Blumberg a nd Ge pner suggests that a stronger result, in the s etting of (stable) infinit y catego ries, ca n be prov ed: algebra ic K -theory carries a unique E ∞ -mult iplicative structure. 4 GONC ¸ ALO T ABUADA Multipli cativ e Chern c haracters. Let E : HO ( dgcat ) → D be a symmetric monoidal a dditiv e inv a r iant. Thanks to equiv alence ( 1.2 ) there is a unique sym- metric monoidal morphism of deriv ators E ma king the diagra m HO ( dgcat ) E / / U A   D Mot A E : : u u u u u u u u u u commute. Let us write 1 := E ( k ) for the unit of D ( e ). Then, for ev ery dg category A the morphism E co mbined with eq uiv alence ( 1.3 ) gives rise to a canonical map ch : K ( A ) ≃ R Hom ( U A ( k ) , U A ( A )) − → R H om D ( e ) ( 1 , E ( A ) ) . Now, reca ll from [ 6 , Examples 7.9 -7.10 ] that the mixed co mplex ( C ) and the Ho chsc hild homolog y ( H H ) constructions give ris e to symmetric monoidal addi- tive inv ariants C : HO ( dgcat ) − → HO ( C (Λ)) H H : HO ( dgcat ) − → HO ( C ( k )) . Here, Λ is the dg algebra k [ ǫ ] /ǫ 2 with ǫ of degree − 1 and d ( ǫ ) = 0. Moreov er, there is a symmetric mo noidal for getful morphism H O ( C (Λ)) → HO ( C ( k )) whose pre-comp ositio n w ith C equals H H . By the above considera tions we o bta in then a canonical commutativ e diagram of s pe c tra (2.7) R Hom ( k , C ( A ))   K ( A ) ch Λ 7 7 n n n n n n n n n n n n ch k / / R Hom ( k , H H ( A )) . As shown in [ 6 , Examples 8 .9-8.10 ] we hav e na tural isomo r phisms of a b elian g roups Hom D (Λ) ( k , C ( A )[ − n ]) ≃ H C − n ( A ) Hom D ( k ) ( k , H H ( A )[ − n ]) ≃ H H n ( A ) , where H C − n ( A ) denotes the n th negative cyc lic homolog y gro up o f A . T he r efore, by passing to the homotopy gr oups in the ab ov e diagram ( 2.7 ) (or equiv a lently by considering the mor phisms in the triangula ted categor ies Mot A ( e ), D (Λ) and D ( k )) we obtain canonica l c ommut ative dia grams of ab elian gr oups H C − n ( A )   K n ( A ) ch Λ n 9 9 r r r r r r r r r r ch k n / / H H n ( A ) . Theorem 2.8. When A = A , with A a k - algebr a, the ab ove ab elian gr oup homo- morphisms ch Λ n and ch k n agr e e, r esp e ctively, with the n th ne gative Chern char acter indep endently c onst ructe d by Ho o d-Jones [ 14 ] and by Go o dwil lie [ 11 ] , and with the n th Dennis tr ac e map original ly c onst r u cte d by Dennis [ 9 ] . Theorem 2.8 provides a s imple conceptual characterization of the highly inv olved work of Dennis, Go o dwillie, and Ho o d-Jones. Intuitiv ely it tell us that the nega tive Chern character, r esp. the Dennis trace map, is the unique symmetric mo noidal factorization o f the mixed complex construction, r e s p. of the Ho chschild homolo gy construction, through the universal additive inv ariant. PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 5 When the k -alg ebra A is co mmutative, the mu ltiplicative structures o n H C − ∗ ( A ) and H H ∗ ( A ) can b e rec overed, r esp ectively , from the symmetric monoidal struc- tures o f D (Λ) and D ( k ); se e [ 15 , § 5]. Therefore, by combining Theorem 2.8 with Example 2 .5 , we obta in for fr e e the following res ult (proved originally by Ho o d- Jones [ 14 , § 5] using a highly elab or ate construction), which was one of the main driving forces b e hind the development of negative cyclic homology . Corollary 2. 9. L et A b e a c ommutative k -algebr a. Then, the ne gative Chern char acters and the Denn is tr ac e maps ar e m u ltiplic ative. Biv arian t picture. Jones and K assel, by dr awing inspiration fro m Ka sparov’s K K -theory , introduced in [ 15 ] the bivariant cyclic c ohomolo gy the ory of k -alg ebras. Immediately afterwards, Ka ssel [ 19 ] in tro duced the bivariant algebr aic K -the ory and constructed a (m ultiplicative) biv ar iant Chern character from it to biv a r iant cyclic c o homology . These inv olved c o nstructions can b e compac tly expr essed in terms of a symmetric monoidal functor ch B : BKK − → D (Λ) . Roughly , BKK is the additive catego ry 2 whose ob jects are the k -algebr as and whose (ab elian gr o up) morphisms Hom BKK ( B , A ) are the Gro thendieck groups of the exa ct categorie s o f those B - A -bimodules which are pro jective and of finite t yp e as A - mo dules; for details see the pro o f of Theorem 2.10 . Kassel’s biv ar iant Chern character was quite an impor tant contribution. F or example, it yielded a simple pro of of the inv ar iance of cyclic homolog y under Mor ita equiv ale nc e s . How ever, it has a serious drawback: it is only defined at the level of the Grothendieck groups. Ka s sel claimed in [ 19 , page s 368- 369] that an extension to the highe r a lg ebraic K -theor y gr o ups should exist but, to the b est of the author ’s knowledge, this imp orta nt pro blem re ma ined wide op en during the last decades. The following res ult c hanges this state of affairs . Theorem 2. 10. Ther e is a natur al symmetric m onoidal additive functor N making the fol lowing diagr am BKK N   ch B / / D (Λ ) Mot A ( e ) C ( e ) : : u u u u u u u u u c ommu te. Mor e over, given k -algebr as A and B , with B homotopica lly finitely pr e - sented (t he homotopic al version of t he classic al notion of finite pr esent ation; se e [ 16 , § 4.7] ), we have an induc e d isomorphi sm (2.11) Hom BKK ( B , A ) ∼ − → Hom Mot A ( e ) ( N ( B ) , N ( A )) . As proved in [ 30 ] (see ( 8.3 )-( 8.4 )), equiv alence ( 1.3 ) and isomorphisms ( 1.4 ) admit na tur al biv ariant extens io ns. Ther efore, T he o rem 2.1 0 (combined with The- orem 2.8 ) shows us that Mo t A ( e ) and C ( e ) should b e co ns idered a s the corr e c t “higher-dimensio nal ex tensions” of the additive category BKK and of the biv ari- ant Cher n character ch B , re s pe ctively . Moreover, in contrast with Ka ssel’s ad ho c construction, the triangula ted categ o ry Mot A ( e ) a nd the (sy mmetric monoidal) tr i- angulated functor C ( e ) a re characterized by simple and pr ecise universal prop erties. 2 The ini tials BKK stand for biv ari an t, K -theory , and Kassel resp ective ly . 6 GONC ¸ ALO T ABUADA Finite co efficients. In this s ubsection we assume for simplicity that k = Z . Given an ob ject X in a triangulated categor y T and an integer l ≥ 2, we write · l for the l - fold multiple of the ident ity morphism in the a belia n g roup Hom T ( X, X ). F ollowing the top olo gical con ven tion, we define the mo d- l Mo or e obje ct X/l of X as the cone of · l , i.e. a s the ob ject which is part of a distinguished tria ngle X · l − → X − → X/l − → X [1 ] . Prop ositi on 2.12 . L et l ≥ 2 b e an int e ger. Then, for every dg c ate gory A we have an e quivalenc e of sp e ctr a R Hom ( U A ( Z ) /l , U A ( A )) ≃ K ( A ; Z /l )[ − 1] , wher e K ( A ; Z /l ) denotes Br owder’ s mo d- l algebr aic K -the ory sp e ctru m of A (se e § 5.3 ). Mor e over, if l = pq with p and q c oprime inte gers, we have an isomorphi sm U A ( Z ) /l ≃ U A ( Z ) /p ⊕ U A ( Z ) /q in Mot A ( e ) . Making use of the ab ov e triangle (with X = U A ( Z )) and of iso morphisms ( 1.4 ) we recov er then Br owder’s lo ng exact sequence r e lating K -theory with mo d- l K -theor y · · · → K n ( A ) · l → K n ( A ) → K n ( A ; Z /l ) → K n − 1 ( A ) · l → K n − 1 ( A ) → · · · . Prop os itio n 2.12 gives a precise conceptual characteriza tion of Browder’s construc- tion. Roughly spea king, mo d- l algebr aic K -theory is the additive inv a riant co- represented by the mo d- l Mo or e ob ject o f the unit U A ( k ) of Mot A ( e ). Moreov er, Prop os itio n 2.12 shows us that we can alw ays as s ume that l is a prime p ower. Example 2.1 3 . (Rings and Schemes) Let l b e a prime n umber and l ν a p ower of it. When A = A , with A a ring, the sp ectr um K ( A ; Z /l ν ) agrees with the one K ( A ; Z / l ν ) constructed orig inally by Br owder in [ 4 ]. When A = D dg p erf ( X ), with X a qua si-compact and qua si-separ a ted scheme (see Exa mple 2.6 ), the spe c- trum K ( D dg p erf ( X ); Z /l ν ) ag rees with the one K ( X ; Z /l ν ) constr ucted or iginally by Thomason-T roba ugh in [ 36 , § 9.3]. Ac kno wledgment s : The author is very gra teful to Paul Ba lmer, Alexa nder Beilinson, Andrew Blumber g, Guillermo Corti ˜ nas, David Gepner , Christian Haes e- meyer, B ernhard Keller, a nd Randy McCarthy for stimulating co nv ersations. He would like also to thank the depa rtments of mathematics of UCLA and UIC fo r its hospitality a nd the Clay Ma thematics Institute, the Midwest T op ology Netw ork, and the Calouste Gulb enkian F oundation for financial supp ort. 3. P reliminaries 3.1. Notations . W e will work ov er a fix ed commut ative ba s e ring k . The catego ry of finite order ed se ts and non-decr easing monotone maps will be denoted by ∆. Given an in teger r ≥ 1, w e will write ∆ ( r ) for the pro duct of ∆ with itself r times. The geo metric rea lization of a (m ulti-)simplicial set and of a (m ulti-simplicial) category (obtained by firs t passing ot the nerv e N . ) will be denoted simply b y | − | . W e will write Spt for the classical categ ory of sp ectra [ 1 ]. Given a mo del category M in the sense of Q uillen [ 28 ] we w ill write Ho ( M ) for its homoto py categ ory and Map ( − , − ) for its homoto py function co mplex; see [ 1 3 , Definition 17 .4.1]. Given categorie s C a nd D , we will denote by F un ( C , D ) the catego r y o f functors with natural transformations as morphisms. Finally , the adjunctions will b e displayed vertically with the left (re sp. right) adjoint o n the left- (resp. right-) ha nd side. PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 7 3.2. Grothendie c k deri v ators. W e will use the ba sic languag e of Grothendieck deriv a tors [ 12 ] which can b e easily a c q uired by skimming thro ugh [ 7 , § 1] or [ 5 , 6 , Appendix A]. An arbitr a ry deriv ato r will b e denoted by D . The essential example to keep in mind is the (triangulated) der iv a to r D = HO ( M ) asso ciated to a (stable) Quillen mo del categor y M and defined for every small category I by HO ( M )( I ) := Ho (F un ( I op , M )) . W e will denote by e the 1-p oint categor y with one ob ject and one identit y mor phism. Heuristically , the catego ry D ( e ) is the ba sic “der ived” ca tegory under consider ation in the deriv a tor D . F or instance, if D = HO ( M ) then D ( e ) is the homotopy categ ory Ho ( M ). As shown in [ 5 , § A.3], every tr iangulated der iv ato r D is canonically en- riched ov er sp ectra . W e will denote by R Hom D ( e ) ( X, Y ) the sp ectr um o f maps from X to Y in D ( e ). When there is no a mbiguit y , we will write simply R Hom ( X, Y ). 4. W aldhausen’s constructions In this section we recall s ome o f W aldha us en’s [ 37 ] foundationa l constructions. This will give us the occa sion to fix imp ortant notations which will a llow us to greatly simplify the pro ofs of Theor e ms 2.4 , 2.8 , 2.10 and of Pr o p osition 2.12 . 4.1. K -theo ry sp ect rum. Given a non-negative in teger q we will denote by [ q ] the ordered set { 0 < 1 < · · · < q } co nsidered as a ca tegory . W e will write Ar[ q ] for the categor y o f arrows in [ q ] ( i.e. the functor categor y F un([1] , [ q ])) and ( i/ j ) fo r the arrow from i to j with i ≤ j . Let C b e a categ ory with co fibrations a nd weak equiv ale nc e s in the s ense of W aldhausen [ 37 , § 1.2]. W e will denote by S q C the full sub c ategory of F un (Ar[ q ] , C ) whose ob jects a re the functors A : ( i/ j ) 7→ A i,j such that A j,j = ∗ , and for every triple i ≤ j ≤ k the following diagr am A i,j / / / /   y A i,k     ∗ ≃ A j,j / / A j,k is a co-ca rtesian. As mentioned by W aldhause n in [ 37 , page 3 28] an ob ject A ∈ S q C corres p o nds to a sequence of cofibrations ( ∗ = A 0 , 0 ֌ A 0 , 1 ֌ · · · ֌ A 0 ,q ) in C together with a c hoice of sub quo cients A i,j = A 0 ,j / A 0 ,i . Moreov er, S q C ca n b e considered as a ca tegory with cofibrations and weak e quiv alence s in a natur al wa y; see [ 37 , pag e 329 ]. By letting q v ar y we obtain a s implicia l ca tegory with co fibrations and weak eq uiv alences S. C and by iterating the S. -constr uction we obtain mult i- simplicial ca tegories with co fibr ations and w eak equiv alences S. ( n ) C := S. · · · S.C | {z } n tim e s . Given non-negative integers q 1 , . . . , q n , the categ ory S q 1 · · · S q n C will b e denoted by S ( n ) q 1 ,...,q n C . By passing to the geometr ic rea lization of the multi-simplicial ca tegories wS. ( n ) C of weak e quiv alence s w e obtain finally the algebr aic K - the ory sp e ctru m : (4.1) K ( C ) : Ω | wS. C | | w S. C | | w S. (2) C | · · · | wS. ( n ) C | · · · . As shown in [ 37 , pag e 3 30] this is a connec tive Ω- sp ectrum with structure maps S 1 ∧ | w S. ( n ) C | → | w S. ( n +1) C | induced b y the natural identification C ≃ S 1 C . 8 GONC ¸ ALO T ABUADA 4.2. Pa irings. Let A , B and C b e catego r ies with cofibrations a nd weak eq uiv- alences and T : A × B → C a bi-exact funct or in the sense of W aldhausen [ 37 , page 342], i.e. a functor satisfying the following t wo conditions: for every pair of ob ject A ∈ A and B ∈ B , the induced functors T ( A, − ) and T ( − , B ) are exa ct; for every pair of cofibratio ns A ֌ A ′ and B ֌ B ′ in A and B , r esp ectively , the induced map T ( A ′ , B ) ∐ T ( A,B ) T ( A, B ′ ) ֌ T ( A ′ , B ′ ) is a cofibr ation in C . Given non-negative integers q and p , we obtain then an induced bi-exact functor S q A × S p B → S (2) q,p C which sends the pa ir (( ∗ = A 0 , 0 ֌ A 0 , 1 ֌ · · · ֌ A 0 ,q ) , ( ∗ = B 0 , 0 ֌ B 0 , 1 ֌ · · · ֌ B 0 ,p )) ∈ S q A × S p B to the element o f S (2) q,p C represented by T ( A 0 , 0 , B 0 , 0 )     / / / / T ( A 0 , 0 , B 0 , 1 ) / / / /     · · · / / / / T ( A 0 , 0 , B 0 ,p )     T ( A 0 , 1 , B 0 , 0 )     / / / / T ( A 0 , 1 , B 0 , 1 ) / / / /     · · · / / / / T ( A 0 , 1 , B 0 ,p )     . . .     . . .     . . . . . .     T ( A 0 ,q , B 0 , 0 ) / / / / T ( A 0 ,q , B 0 , 1 ) / / / / · · · / / / / T ( A 0 ,q , B 0 ,p ) with asso ciated subq uotients T ( A 0 ,j , B 0 ,l ) /T ( A 0 ,i , B 0 ,k ) := T ( A j /i , B l/k ). The it- eration of this constructio n furnish us bi-exact functors S ( n ) q 1 ,...,q n A × S ( m ) p 1 ,...,p m B − → S ( n + m ) q 1 ,...,q n p 1 ,...,p m C , where q 1 , . . . , q n and p 1 , . . . , p m are non-negative integers. By passing to the geo- metric realization of the m ulti-simplicial categor ies of w eak equiv alences we obtain induced maps | wS. ( n ) A| × | wS. ( m ) B | − → | wS. ( n + m ) C | which factor through | wS. ( n ) A| ∧ | wS. ( m ) B | . Finally , due to the Ω-s tructure of the algebraic K -theory sp ectrum, these latter ma ps assemble themselves in a well- defined algebra ic K -theo r y pairing K ( A ) ∧ K ( B ) − → K ( C ) . 5. Differential graded ca tegories In this section we co lle ct the notions and res ults co ncerning dg categories whic h will be us e d in the pro o fs of Theorems 2.4 , 2.8 , 2.10 and of Pr o p osition 2 .12 . W e will deno te by C ( k ) b e the categ ory of (unbounded) co mplexes o f k -modules. W e will use co -homologic al nota tion, i.e. the differential incr eases the degree. A differ ential gra de d (=dg) c ate gory is a catego ry enriched ov er C ( k ) (morphisms sets are complexes) in such a wa y that compo sition fulfills the Leibniz rule : d ( f ◦ g ) = ( d f ) ◦ g + ( − 1) deg( f ) f ◦ ( dg ). Given a k -alg e bra A , we will write A for the dg ca tegory with one o b ject and with A as the dg a lgebra of endomo rphisms (concentrated in degree zer o). F or a sur vey ar ticle, we invite the reader to consult K eller’s ICM adress [ 16 ]. PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 9 5.1. Dg (bi -)mo dul es. Let A b e a dg ca tegory . The opp osite dg c ate gory A op has the same ob jects as A and co mplexes of mo rphisms given by A op ( x, y ) := A ( y , x ). Recall from [ 16 , § 3.1 ] that a right dg A -mo dule (or simply a A -mo dule) is a dg functor M : A op → C dg ( k ) with v a lues in the dg category C dg ( k ) of complexes of k - mo dules. W e w ill denote by C ( A ) (resp. by C dg ( A )) the categor y (resp. dg catego ry) of A -mo dules . Reca ll fro m [ 16 , Theor em 3.2] that C ( A ) ca rries a standar d pr o jectiv e mo del s tructure whose weak equiv alences a r e the quasi-is o morphisms. The derive d c ate gory D ( A ) of A is the lo calizatio n o f C ( A ) with resp ect to the class of quasi- isomorphisms. Notation 5.1 . In o rder to simplify the exp o sition, we will deno te by b A (resp. by b A dg ) b e the full sub catego ry of C ( A ) (resp. of C dg ( A )) formed b y the A -mo dules which ar e co fibrant and that b ecome compact in D ( A ). As shown in [ 10 , § 3] the category b A , endo wed with the cofibrations a nd weak equiv alences of the pro jective mo del structure, is a category with co fibrations and w eak equiv a lences in the sense of W a ldhausen [ 37 ]. The ob jects of b A (and o f b A dg ) will b e called the p erfe ct A - mo dules. Recall from [ 16 , § 2.3] that the tens or pr o duct A ⊗ B of t wo dg catego ries is defined as follows: the set of ob jects is the cartes ia n pro duct of the set o f o b jects o f A and B and ( A ⊗ B )(( x, z ) , ( y , w )) := A ( x, y ) ⊗ A ( z , w ). By a right dg A - B -bimo dule (or simply a A - B -bimo dule) we mean a dg functor A op ⊗ B → C dg ( k ). 5.2. Derived M o rita e quiv alences. A dg functor F : A → B is called a derive d Morita e quivalenc e if it induces an equiv alence D ( B ) ∼ → D ( A ) o f triangulated cat- egories; see [ 16 , § 4.6]. Recall from [ 16 , Theorem 4.10] that dg cat c arries a Quillen mo del s tructure, whose weak equiv a lences ar e the derive d Morita e quivalenc es . W e will denote by Hmo the homo topy catego r y hence o btained. The tensor pro duct of dg categ o ries ca n b e natur ally derived − ⊗ L − , thus giving rise to a sy mmetr ic monoidal structure on Hmo . Given dg categories A and B , let rep ( B , A ) b e the full triangulated subc ategory of D ( B op ⊗ L A ) spanned b y the cofibrant B - A -bimo dules X such that for every o b ject x ∈ B the a sso ciated A -mo dule X ( x, − ) b eco mes compact in D ( A ). Recall from [ 16 , § 4.2 a nd § 4.6 ] that there is a natura l bijection betw een Hom Hmo ( B , A ) and the isomorphism cla sses o f ob jects in rep ( B , A ). More- ov er, comp osition in Hmo corresp o nds to the (derived) tensor pro duct of bimo dules. Let R ( B , A ) be the subcateg ory of C ( B op ⊗ L A ) with the s ame ob jects as rep ( B , A ) and whos e mor phis ms ar e the quasi-isomo r phisms. As explained in [ 16 , § 4] there is a ca nonical weak equiv a lence o f simplicial sets b etw een M ap Hmo ( B , A ) and the nerve of the categ ory R ( B , A ). Finally , r ecall fro m [ 16 , § 4.3 ] that the (derived) symmetric mono idal structure on Hmo is closed. Given dg categories A and B , the int ernal Ho m-functor rep dg ( B , A ) is the full dg sub ca tegory of C dg ( B op ⊗ L A ) with the same ob jects as rep ( B , A ). 5.3. Alge b raic K- the ories. L e t A b e a dg ca tegory . Rec a ll from [ 16 , § 5.2 ] that the algebr aic K - the ory sp e ctrum K ( A ) of A is the sp e ctrum ( 4.1 ) asso ciated to the categ ory with cofibra tions and weak equiv alences b A . Now, assume that k = Z and consider the disting uis hed tria ng le S · l → S → S /l → S [1] in Ho ( Spt ), wher e l ≥ 2 is an integer, S is the spher e s pe c trum, and · l is the l -fo ld multiple of the ident ity morphism. F ollowing Browder [ 4 ], the mo d- l algebr aic K - the ory sp e ctrum K ( A ; Z / l ) of A is the de r ived smas h pro duct spectr um S /l ∧ L K ( A ). 10 GONC ¸ ALO T ABUADA 6. P roof of Theorem 2.4 Pr o of. The pro of will consist on showing that the maps (6.1) K ( A ) n × K ( B ) m − → K ( A ⊗ B ) n + m n, m ≥ 0 asso ciated to the pairings ( 2.2 ) and ( 2 .3 ) a gree up to ho motopy; note that ( 2.2 ) and ( 2.3 ) ar e na turally defined on K ( A ) × K ( B ). W e star t by observing that it suffices to study the cases n, m ≥ 1 . Given a n arbitrary dg categ o ry C and an int eger r ≥ 0 , let us write K ( C )[ r ] for the sp ectr um K ( C )[ r ] n := K ( C ) n + r with the naturally induced structure maps. As explained in § 4.2 , ( 2.2 ) is obtained by lo oping t wice the pair ing K ( A )[1] ∧ K ( B )[1] − → K ( A ⊗ B )[2] induced by the bi-e x act functor ( 2.1 ). In what c o ncerns ( 2.3 ) the same phenomenon holds: the category Mot A ( e ) is tria ngulated and its mono idal structure is ho motopy colimit pres erving, which implies that ( 2.3 ) is obtained by lo oping twice the pair ing R Hom ( U A ( k ) , U A ( A )[1]) ∧ R Ho m ( U A ( k ) , U A ( B )[1]) − → R H om ( U A ( k ) , U A ( A ⊗ B )[2]) . Therefore, we c a n assume that n a nd m are (fixed) integers ≥ 1. Now, r ecall from § 5.3 that we hav e the equalities: K ( A ) n = | w S. ( n ) b A| K ( B ) m = | w S. ( m ) b B | K ( A ⊗ B ) n + m = | w S. ( n + m ) \ ( A ⊗ B ) | . F ollowing § 4.2 , the maps ( 6.1 ) asso ciated to the pair ing ( 2.2 ) cor resp ond to the ones | wS. ( n ) b A| × | w S. ( m ) b B | − → | w S. ( n + m ) \ ( A ⊗ B ) | induced by the bi-e x act functor ( 2.1 ). By cons truction, these latter maps can b e furthermore expressed as the following homotopy colimit (6.2) ho colim q 1 ,...,q n p 1 ,...,p m  | wS ( n ) q 1 ,...,q n b A| × | wS ( m ) p 1 ,...,p m b B | − → | wS ( n + m ) q 1 ,...,q n p 1 ,...,p m \ ( A ⊗ B ) |  induced by the bi-ex act functors (6.3) S ( n ) q 1 ,...,q n b A × S ( m ) p 1 ,...,p m b B − → S ( n + m ) q 1 ,...,q n p 1 ,...,p m \ ( A ⊗ B ) , where q 1 , . . . , q n and p 1 , . . . , p n are non- ne g ative integers. The remainder o f the pro of consists of s howing that the maps ( 6.1 ) ass o ciated to the pair ing ( 2.3 ) can also b e expressed as the ab ov e ho motopy colimit ( 6.2 ). W e start by obser ving that the bi- exact functors ( 6.3 ) admit a natural dg enrichmen t. Giv en an abstract dg category C , the dg enrich ment C dg ( k ) o f C ( k ) gives r ise to a dg enrichmen t b C dg of b C ; see No tation 5.1 . By construction, S q b C and all the maps used in the S. -constr uction inherit a natur al dg enrichmen t from b C dg . In sum, we obtain well-defined multi- simplicial dg ca tegories S. ( n ) b C dg . Under this dg enrichmen t the ab ove bi-exact functors ( 6.3 ) b ecome k -bilinear in the differential graded sense. Therefore, they give rise to well-defined dg functors (6.4) S ( n ) q 1 ,...,q n b A dg ⊗ S ( m ) p 1 ,...,p m b B dg − → S ( n + m ) q 1 ,...,q n p 1 ,...,p m \ ( A ⊗ B ) dg which assemble themselv es in a morphism (6.5) Φ ( n + m ) : S. ( n ) b A dg ⊗ S. ( m ) b B dg − → S. ( n + m ) \ ( A ⊗ B ) dg PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 11 in HO ( dgcat )(∆ ( n + m ) ). Now, recall from [ 30 , § 1 5 ] that the universal additive in- v ar iant is defined as the following compos ition U A : HO ( dgcat ) U u A − → Mot u A ϕ − → Mot A . The deriv ator Mot u A is the unst able analogue of Mot A and the morphism ϕ cor re- sp onds to the stabiliza tion pro cedure. Let us write Σ fo r the susp ension functor in the p ointed ba se ca tegory Mot u A ( e ). Recall from [ 7 , § 1 ] that since Mot u A is a deriv a tor, w e hav e the following adjunction Mot u A (∆ ( r ) ) π !   Mot u A ( e ) π ∗ O O asso ciated to the pro jection functor π : ∆ ( r ) → e . Lemma 6.6. L et C b e a dg c ate gory. Then, for every inte ger r ≥ 1 we have a natur al isomorphism π ! U u A ( S. ( r ) b C dg ) ≃ Σ ( r ) U u A ( b C dg ) . Pr o of. The pro of go es by induction on r . The ca se r = 1 was prov ed in [ 30 , Prop os itio n 1 4.11]. Now, suppos e w e hav e a natural isomorphism (6.7) π ! U u A ( S. ( r ) b C dg ) ≃ Σ ( r ) U u A ( b C dg ) . F ollowing McCarthy [ 25 , § 3.3], we consider the s equence of morphisms S. ( r ) b C dg − → P S. ( r +1) b C dg − → S. ( r +1) b C dg in HO ( dgcat )(∆ ( r +1) ), where S. ( r ) b C dg is constant in one simplicial direc tio n and P S . ( r +1) b C dg denotes the simplicial path o b ject of S. ( r +1) b C dg . As in the pro of of [ 30 , Pr op osition 14 .1 1] we observe that the a b ove sequence of morphisms is split exact at each comp o nent and that P S. ( r +1) b C dg is simplicia lly contractible. Hence, we obtain the following natural isomorphism (6.8) π ! U u A ( S. ( r +1) b C dg ) ≃ Σ  π ! U u A ( S. ( r ) b C dg )  . By combining ( 6.8 ) with ( 6.7 ) w e conclude that π ! U u A ( S. ( r +1) b C dg ) is natura lly is o- morphic to Σ ( r +1) U u A ( b C dg ), which achiev es the pro of.  Lemma 6.9. The induc e d morphism π ! U u A (Φ ( n + m ) ) (se e ( 6.5 ) ) b e c omes invertible in Mot u A ( e ) . Pr o of. The deriv ator Mot u A carries a ho mo topy colimit preserving sy mmetric monoida l structure and b oth mor phisms U u A and ϕ ar e sy mmetric monoidal. There fore, the pro of is a consequence of the following natural isomorphisms: π ! U u A ( S. ( n ) b A dg ⊗ S. ( m ) b B dg ) ≃ π ! U u A ( S. ( n ) b A dg ⊗ L S. ( m ) b B dg ) (6.10) ≃ π ! U u A ( S. ( n ) b A dg ) ⊗ π ! U u A ( S. ( m ) b B dg ) (6.11) ≃ Σ ( n ) U u A ( b A dg ) ⊗ Σ ( m ) U u A ( b B dg ) (6.12) ≃ Σ ( n + m ) U u A ( b A dg ⊗ L b B dg ) (6.13) ≃ Σ ( n + m ) U u A ( \ ( A ⊗ B ) dg ) (6.14) ≃ π ! U u A ( S. ( n + m ) \ ( A ⊗ B ) dg ) . (6.15) 12 GONC ¸ ALO T ABUADA Isomorphism ( 6.10 ) follows from the a ssumption that A or B (and hence S. ( n ) b A dg or S. ( m ) b B dg ) is k - flat. Iso morphisms ( 6.11 ) a nd ( 6.13 ) follow from the fact that the morphism U u A is symmetric monoida l and that the symmetr ic mono idal structure on Mo t u A ( e ) is ho motopy co limit pr eserving; note that π ! and Σ are tw o (differ- ent ) examples of homoto py co limits. Isomo rphisms ( 6.12 ) and ( 6.15 ) follow fr om Lemma 6.6 . Finally , is omorphism ( 6.14 ) follows fro m the natural der ived Mo rita equiv ale nc e s b A dg ⊗ L b B dg ≃ b A dg ⊗ b B dg ≃ A ⊗ B ≃ \ ( A ⊗ B ) dg .  Now, let us co nsider the following (co mp o s ed) maps (6.16) | Map ( U u A ( k ) , π ! U u A ( S. ( n ) b A dg )) | × | Map ( U u A ( k ) , π ! U u A ( S. ( m ) b B dg )) |   | Map ( U u A ( k ) , π ! U u A ( S. ( n ) b A dg ⊗ S. ( m ) b B dg )) | ≃   | Map ( U u A ( k ) , π ! U u A ( S. ( n + m ) \ ( A ⊗ B ) dg )) | The “upp er” map is the o ne induced b y the s ymmetric mono idal structure o n Mot u A and b y the natural isomo r phism π ! U u A ( S. ( n ) b A dg ) ⊗ π ! U u A ( S. ( m ) b B dg ) ≃ π ! U u A ( S. ( n ) b A dg ⊗ S. ( m ) b B dg ) . The “low er” map is the isomor phism induced by π ! U u A (Φ ( n + m ) ); see Lemma 6.9 . A careful analysis of the pro o f of [ 30 , Theorem 15.9] show us tha t given a n arbitra r y dg category C and an integer r ≥ 1, we have natur al w eak equiv a le nc e s K ( C ) r ≃ R Ho m ( U A ( k ) , U A ( C )) r ≃ | Map ( U u A ( k ) , π ! U u A ( S. ( r ) b C dg )) | . Therefore, since the mor phism ϕ : Mo t u A → Mot A is symmetric monoida l, we con- clude that the ma ps ( 6.1 ) asso ciated to the pairing ( 2.3 ) agree with the ab ov e (comp osed) maps ( 6.16 ). Throug h a simple iteration of [ 3 0 , Pro p osition 14.12 ] w e obtain moreov er a na tural weak equiv alence | Map ( U u A ( k ) , π ! U u A ( S. ( r ) b C dg )) | ≃ ho colim q 1 ,...,q r | Map ( k , S ( r ) q 1 ,...,q r b C dg ) | . This description, combined with the fact that the functor U u A ( e ) : HO ( dgcat )( e ) = Hmo − → Mot u A ( e ) is sy mmetric monoidal, allows us to c o nclude that the ab ov e (comp osed) maps ( 6.16 ) can b e expres sed as the follo wing homotopy co limits ho colim q 1 ,...,q n p 1 ,...,p m       | Map ( k , S ( n ) q 1 ,...,q n b A dg ) | × | Map ( k , S ( m ) p 1 ,...,p m b B dg ) | ↓ | Map ( k , S ( n ) q 1 ,...,q n b A dg ⊗ S ( m ) p 1 ,...,p m b B dg ) | ↓ | Map ( k , S ( n + m ) q 1 ,...,q n p 1 ,...,p m \ ( A ⊗ B ) dg ) |       . The “upp er” map is the one induced by the symmetric monoidal structure on Hmo . The “low er” ma p is the one obtained by a pplying the functor | Map ( k , − ) | to the dg functor ( 6.4 ). No w, in order to co nclude the pr o of it re mains to show that this homoto py co limit agr ees with the one des c rib ed in ( 6.2 ). Note that given an arbitrar y dg ca tegory C , the category R ( k , C ) (see § 5.2 ) identifies naturally with b C which implies that | Map ( k , C ) | ≃ | w b C | . In the particular case of the dg categ o ries PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 13 S ( r ) q 1 ,...,q r b C dg , the asso cia ted categ ories of p erfect mo dules ar e naturally identified with S ( r ) q 1 ,...,q r b C . Therefor e | Map ( k , S ( r ) q 1 ,...,q r b C dg ) | ≃ | w S ( r ) q 1 ,...,q r b C | and so the maps in the ab ov e homotopy colimit are obtained by applying | w − | to the c o mp o sed functors S ( n ) q 1 ,...,q n b A × S ( m ) p 1 ,...,p m b B − → \ S ( n ) q 1 ,...,q n b A dg ⊗ S ( m ) p 1 ,...,p m b B dg − → S ( n + m ) q 1 ,...,q n p 1 ,...,p m \ ( A ⊗ B ) . The left-hand side functor is the one induced b y the symmetric monoida l structure on H mo . The right-hand side functor is the clas sical extensio n of scalars asso c i- ated to the dg functor ( 6.4 ). Finally , a direct ins p ec tio n show us that the ab ov e comp osed functors a g ree with the ones describ ed in ( 6.3 ). T his implies tha t the ab ov e homotopy colimit agr e es with the one describ ed in ( 6.2 ) and so the pro of is finished.  7. P roof of Theorem 2.8 Pr o of. W e start by s howing that ch k n agrees with the n th Dennis trace map. Thanks to K eller-McCar th y’s delo o ping theorem (s e e [ 17 , § 1.13] and [ 25 , Cor ollary 3.6 .3]) the sp ectrum R Hom D ( k ) ( k , H H ( A )) ca n be des crib ed as follows: (7.1) Ω | H H DK ( S. b A dg ) | | H H DK ( S. b A dg ) | · · · | H H DK ( S ( n ) . b A dg ) | · · · . Some explanatio ns are in o rder. As in the pro of o f Theorem 2.4 , S. ( n ) b A dg can be considered as a multi-simplicial dg categor y in a natural wa y . By taking at each degree the s implicial k -mo dule ( H H DK ), which (via the Dold-Ka n equiv a- lence) corresp onds to the Ho chschild homology complex ( H H ), we obtain a m ulti- simplicial k - mo dule H H DK ( S. ( n ) b A dg ) whose geometric re a lization we denote by | H H DK ( S. ( n ) b A dg ) | . The structure maps o f ( 7.1 ) ar e induced by the fibr ation se- quences | H H DK ( S. ( n ) b A dg ) | − → | H H DK ( P S. ( n ) b A dg ) | − → | H H DK ( S. ( n +1) b A dg ) | , where the middle term is co ntractible; see [ 25 , Theorem 3.3.3]. No w, s ince K ( A ) (see § 5.3 ) and ( 7.1 ) are connective Ω-sp ectra, it suffices to study the restr iction of the map of sp ectra ch k (see ( 2.7 )) to its first comp onent. Concretly , we need to study the induced map (7.2) K ( A ) 1 ≃ R Ho m ( U A ( k ) , U A ( A )) 1 − → | H H DK ( S. b A dg ) | . As explained in the pro of of Theor em 2.4 , we have natural weak e q uiv alences R Hom ( U A ( k ) , U A ( A )) 1 ≃ | N .wS. b A | ≃ | Map ( k , S. b A dg ) | . On the other hand we have also the following natural w eak equiv alence | H H DK ( S. b A dg ) | ≃ | Map D ( k ) ( k , H H ( S. b A dg )) | . Therefore, ( 7.2 ) is obtained by pa ssing to the geometric re a lization of the map (7.3) Map ( k , S. b A dg ) − → Map D ( k ) ( k , H H ( S. b A dg )) induced by the Ho chschild homolo gy functor . Now, r ecall from McCa rthy [ 25 ] that asso ciated with the k -algebr a A we hav e not only the ca tegory b A of per fect A - mo dules but a lso the exa ct category P A of finitely g enerated pro jectiv e A -mo dules. As with a ny exact categor y , we can co nsider P A as a categor y with cofibrations a nd weak equiv a lences, wher e the weak eq uiv alence s are the is omorphisms. Mo r eov er, 14 GONC ¸ ALO T ABUADA since P A is k -linear w e can also view it a s a dg catego ry . W e hav e then an inclusion functor P A ֒ → b A and a n inclusio n dg functor P A ֒ → b A dg . These functors allow us to construct the following (solid) dia gram of bisimplicial sets: ob j ( S. P A ) M / / _ _ _ _ _ α ≃   H H DK ( S. P A ) ≃ γ   N .iS. P A β ≃   N .wS. b A ( 7.3 ) / / H H DK ( S. b A dg ) . The bisimplicial set ob j( S. P A ) is constant in one simplicial direction and the mor- phism α cor resp onds to the inclus io n o f ob jects. As proved by W aldha usen [ 37 ], α is a weak equiv alence since we are co nsidering the simplicia l ca tegory iS. P A of iso mor- phisms. The morphis m β is the one induced by the inclusion functor P A ֒ → b A . As prov ed b y Thomaso n-T r obaugh [ 36 , Theor em 1.11 .7], β is also a weak equiv alence since b A identifies naturally with the ca tegory of b ounded co chain complex e s in P A . The morphis m γ is the one induced by the inclusion dg functor P A ֒ → b A dg . Thanks to K eller-McCar thy’s ag reement prop erty (see [ 17 , § 1.5 ] a nd [ 2 5 , Pro po sition 2.4 .3]), γ is also a weak equiv alence. Fina lly , M is the induced mo rphism. Now, let q b e a non-negative integer and x an element o f the set ob j ( S q P A ). Under the weak equiv alences α and β , this element corres p o nds to an ob ject x of the catego ry S q b A . Using the natural dg enrichmen t S q b A dg of S q b A , w e ca n repr esent x by a dg functor x : k → S q b A dg which maps the unique ob ject of k to the o b ject x . This dg functor is one of the 0-simplices of the simplicial set Map ( k , S q b A dg ) (see § 5.2 ) and so it is mapped by the ab ove map ( 7.3 ) to the following morphism o f complexes H H ( x ) : k ≃ H H ( k ) − → H H ( S q b A dg ) . A direct insp ection shows that this morphism cor resp onds to the zero cycle of H H ( S q b A dg ) given b y the iden tity o f x , or equiv alently to the 0-simplice of the sim- plicial k - mo dule H H DK ( S q b A dg ) given by the iden tit y of x . Therefore, if we denote by H H DK ( S. P A ) 0 the 0-s implices of H H DK ( S. P A ), the mor phis m M a dmits the following factoriza tion ob j ( S. P A ) id − → H H DK ( S. P A ) 0 inc ֒ → H H DK ( S. P A ) as in [ 25 , § 4.4]. McCar thy has shown in [ 2 5 , § 4.5] that this latter construction agrees with the Dennis trace map. Therefor e , since the ab ov e mor phisms α , β and γ are w eak equiv a lences, the pro of concer ning the Dennis trace map is finished. W e now s how tha t ch Λ n agrees with the n th negative Chern character. Via the Dold-Ka n cor resp ondance b etw een complexes and sp ectra , R Hom D (Λ) ( k , C ( A )) ident ifies w ith the (des us pe ns ion o f the) total negative complex asso cia ted to the simplicial mixed co mplex C ( S. b A dg ); see [ 25 , § 3]. F ollowing the sa me arguments a s those of the Dennis trace map, we ne e d to study the induced mor phism of mixed complexes C ( x ) : k ≃ C ( k ) − → C ( S q b A dg ) . PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 15 As shown in [ 15 , § 2] this morphism of mixed complexes corr esp onds to the following morphism of complexes (7.4) B − C ( x ) : B − ( k ) − → B − ( S q b A dg ) , where B − denotes Connes’ B − -construction. No w, if we write 1 for the unit of k , a simple computation sho w us that B − ( k ) is canonically endow ed with the canonical zero cycle ∞ Y t =0 ( − 1) t (2 t )! t ! (1 ⊗ · · · ⊗ 1) . Therefore, since 1 is mapp ed to the identit y of x , the a b ove morphism ( 7.4 ) corre- sp onds to the following ze r o cycle (7.5) ∞ Y t =0 ( − 1) t (2 t )! t ! (id x ⊗ · · · ⊗ id x ) ∈ Z 0 B − ( S. b A dg ) . W e o bta in then the same factorization ob j( S. P A ) ( 7.5 ) − → Z 0 B − ( S. b A dg ) inc ֒ → B − ( S. b A dg ) of the morphism M as in [ 25 , § 4.4 ]. McCar th y has shown in [ 25 , § 4.5 ] that this latter constr uction agr ees with the nega tive Chern character. There fo re, since the ab ov e mor phisms α , β and γ are weak equiv alences, the pro of is finished.  8. P roof of Theorem 2.10 Pr o of. The construction of the symmetric mo noidal additive functor N makes use of an auxiliar categor y Hmo 0 and is divided in tw o steps: N : BKK N 1 − → Hmo 0 N 2 − → Mot A ( e ) . W e start by describing the functor N 2 . Recall from [ 31 ] that there is a symmetric monoidal functor U a : dgcat − → Hmo − → H mo 0 , with v alues in an additive catego ry which, heur is tically , is the “zero- dimensional” analogue of the universal additive inv ariant U A . T he ob jects of Hmo 0 are the dg categorie s a nd the (ab elian g roup) morphisms Hom Hmo 0 ( B , A ) are the Grothendieck group o f the triang ulated catego ry rep ( B , A ); see § 5.2 . Comp osition is induced by the (derived) tensor pr o duct of bimo dules. The symmetric monoida l structur e is induced b y the (derived) tensor pro duct of dg categor ie s. Note that w e hav e a natural symmetric monoida l functor Hmo → Hmo 0 which sends an element X of rep ( B , A ) to the corres po nding cla ss [ X ] in the Grothendieck group K 0 rep ( B , A ). A simple symmetr ic monoidal s ha rp ening of [ 31 , Theor e m 6.3] provides us the following universal characterizatio n o f U a : given a ny additive categor y A , endow ed with a s ymmetric mo no idal str ucture, we hav e an induced equiv alence of categorie s (8.1) ( U a ) ∗ : F un ⊗ add ( Hmo 0 , A ) ∼ − → F un ⊗ a ( dgcat , A ) , here the left-hand side denotes the ca tegory of sy mmetric monoidal additive func- tors fro m Hmo 0 to A and the rig ht -hand side the categ ory of symmetric mono idal functors from dg cat to A , which inv ert derived Morita equiv alences and s end split exact sequences to dir e c t sums. Therefore, s inc e the catego ry Mot A ( e ) is additive and the symmetric monoidal functor U A ( e ) : Hmo = HO ( dgcat )( e ) − → Mot A ( e ) 16 GONC ¸ ALO T ABUADA sends split exact sequenc e s to direct sums, equiv a lence ( 8.1 ) furnishes us a unique symmetric monoidal additive functor N 2 making the following diag ram commute (8.2) Hmo   U A ( e ) % % K K K K K K K K K K Hmo 0 N 2 / / Mot A ( e ) . Now, recall from [ 16 , § 4.7] that a dg ca tegory B is called homotopic al ly fin it ely pr esent e d if for every filtered direct sys tem of dg ca tegories {C j } j ∈ J , the ca nonical map ho colim j ∈ J Map ( B , C j ) − → Map ( B , ho colim j ∈ J C j ) is an isomor phism of s implicial sets. An imp ortant clas s of examples is pr ovided by the sm o oth and pr op er dg categories in the sense of Kontsevich; see [ 20 , 21 ]. As prov ed in [ 30 , Theorem 15.1 0], equiv alence ( 1.3 ) and isomorphisms ( 1.4 ) can be gr eatly generalized: g iven dg ca tegories A a nd B , with B homo topically finitely presented, we ha ve a natural equiv alence of sp ectra (8.3) R Hom ( U A ( B ) , U A ( A )) ≃ K rep dg ( B , A ) and isomorphisms of ab elian gro ups (8.4) Hom ( U A ( B ) , U A ( A )[ − n ]) ≃ K n rep dg ( B , A ) n ≥ 0 . Equiv a lence ( 1.3 ), resp. isomo r phisms ( 1.4 ), can b e recov ered fro m ( 8.3 ), resp. from ( 8.4 ), by tak ing B = k . Moreov er, the categ ory of per fect rep dg ( B , A )-mo dules is equiv ale nt to rep ( B , A ) which implies that K 0 rep dg ( B , A ) is na turally is omorphic to the Gro thendieck group of the triangulated c a tegory rep ( B , A ). This allow us to conclude that the functor N 2 induces the following is o morphisms (8.5) Hom Hmo 0 ( B , A ) ∼ − → Hom Mot A ( e ) ( N 2 ( B ) , N 2 ( A )) . Now, r ecall that the mixed complex construc tio n A 7→ C ( A ) inv erts derived Morita equiv ale nc e s , sends split e x act sequences to direct sums, a nd is moreover sy mmetr ic monoidal. Hence, the ab ov e equiv alence ( 8.1 ) furnishes us a unique symmetric monoidal additive functor e C : Hmo 0 → D (Λ) such that e C ◦ U a = C . Lemma 8. 6. The fol lo wing diagr am c ommutes Hmo 0 N 2   e C / / D (Λ ) Mot A ( e ) C ( e ) : : u u u u u u u u u . Pr o of. Both functor s e C and C ( e ) ◦ N 2 are sy mmetr ic mono idal and additive. There- fore, by pre-comp os ing them with U a we obtain functors from dgcat to D (Λ) which inv ert der ived Mo rita equiv alences a nd send split e x act seq uences to direct sums. Thanks to the ab ov e commutativ e diagr am ( 8.2 ), we observe tha t these latter func- tors ar e in fact the same, namely they ar e b oth the mixe d c omplex co nstruction. Making use of equiv alence ( 8.1 ) we then co nclude tha t e C = C ( e ) ◦ N 2 , which achieves the pro of.  PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 17 W e now describ e the functor N 1 : BKK → Hmo 0 . Let us start with the additive category BK K. T he ob jects of BKK are the unital k -a lgebras and the (ab elian group) mor phisms Hom BKK ( B , A ) ar e the Gro thendieck gro ups of the exa ct ca t- egories Rep ( B , A ) of those B - A -bimo dules which are pro jective and of finite type as A -modules; see [ 19 , § II Definition 1.1]. Compo sition is induced by the (derived) tensor pro duct of bimo dules; se e [ 19 , § I I.2]. The symmetric monoidal structure is the one induced b y the tensor pro duct of k -a lgebras ; see [ 19 , § I I.3]. Giv en unital k -alge br as A a nd B , we consider the natural functor Rep ( B , A ) − → rep ( B , A ) P 7→ P , where P is co ncentrated in deg ree zero. This functor is fully-faithful and sends every short ex a ct s equence of the exact ca tegory Re p ( A, B ) to a distinguished triangle o f the tria ngulated category rep ( B , A ). Therefor e, it g ives rise to an abelia n g roup homomorphism K 0 Rep ( B , A ) − → K 0 rep ( B , A ) [ P ] 7→ [ P ] . (8.7) Lemma 8. 8. The ab elian gr oup homomorphism ( 8.7 ) is invertible. Pr o of. Let X b e a n ob ject of rep ( B , A ). Since this bimo dule is co mpact as an ob ject in D ( A ), we can assume that it is given by a bounded complex of B - A - bimo dules ( X n ) n ∈ Z which are pro jective a nd o f finite type as A -mo dules. Hence, the assignment [ X ] 7− → X n ( − 1) n [ X n ] ∈ K 0 Rep ( B , A ) extends b y linearity to K 0 rep ( B , A ) and gives ris e to the in verse of ( 8.7 ).  The a b ove a belia n g roup iso morphisms ( 8.7 ) ar e compatible with the co mpo sition op erations as well as with the symmetric monoidal structures of the ca tegories BKK and Hmo 0 . Hence, they ass emble themselves in a well-defined fully-faithful symmetric monoida l additive functor N 1 : BKK − → Hmo 0 . Mor eov er, the biv ar iant Chern character co nstructed by Ka ssel in [ 19 , § I I.4] corr esp onds, under the ab ove description of BKK, to a symmetric mono idal functor ch B : BKK → D (Λ). Lemma 8. 9. The fol lo wing diagr am c ommutes BKK N 1   ch B / / D (Λ ) Hmo 0 e C ; ; v v v v v v v v v . Pr o of. Let us start b y recalling from [ 19 , § II.4] Kassel’s constructio n of the biv ar i- ant Chern character. Let A and B b e tw o unital k -alg ebras and P an ob ject o f Rep ( B , A ). Since P is pro jective and of finite t yp e as a A -mo dule, ther e is a A - mo dule Q a nd an isomorphism α : P ⊕ Q ∼ → A n . W e ca n then consider the following morphism of k -algebr as i : B γ − → End A ( P ) ֒ → End A ( P ⊕ Q ) Ad( α ) − → ∼ M n ( A ) . The morphism γ is the one induced by the B - A -bimo dule structur e of P and Ad( α ) denotes the conjugation by α . Note that although each one of the k -algebras in the ab ov e morphism is unital, the inclusion mo rphism End A ( P ) ֒ → End A ( P ⊕ Q ) do es 18 GONC ¸ ALO T ABUADA not pr e s erve the unit. In particular , i is no t unit preser ving. Nevertheless, there is a well-defined morphism be t ween the as so ciated mixed complexes. Kassel’s biv ariant Chern character ch B ([ P ]) of P is by definition the co mpo sed mo r phism T r ◦ C ( i ) in D (Λ), where T r is the g eneralized trac e map. In par ticular, this construction is independ of the choices o f Q , n , and α . Now, recall fro m [ 18 , § 2.4] that given unital k -alg e br as A a nd B and a (not necessarily unit preserving ) homo mo rphism ϕ : B → A , we can cons tr uct the B - A -bimo dule ϕ ( 1 B ) A A , whose B - A -action is g iven by b · ϕ (1) a · a ′ := ϕ ( b ) aa ′ . This B - A -bimodule gives rise to a n ob ject ϕ ( 1 B ) A A of rep ( B , A ) and the v a lue o f the mixed co mplex cons truction on it agree s with the v a lue of the mixed complex construction on ϕ . Consider then the following dia gram B γ − → End A ( P ) ֒ → E nd A ( P ⊕ Q ) Ad( α ) − → ∼ M n ( A ) e 11 ← − A in Hmo , where e 11 : A → M n ( A ) is the k -alg ebra homomor phism which maps A to the p osition (1 , 1 ). Since e 11 induces a n equiv alence b etw een A -mo dules a nd M n ( A )-modules , the mor phism e 11 bec omes inv ertible in Hmo . Moreov er, the B - A - bimo dule P can be expr essed as the c o mp o sition e 11 − 1 ◦ i . Hence, by the a bove consideratio ns, w e conclude that the v alue of e C on [ P ] is the c omp osed mor phism C ( e 11 ) − 1 ◦ C ( i ) in D (Λ). Finally , s ince C ( e 11 ) and the genera lized tra ce map T r are inv erse of ea ch other (see [ 19 , § 3]) we conclude that ch B ([ P ]) = e C ([ P ]). B y linearity , this implies that ch B = e C ◦ N 1 and so the pro of is finished.  W e are now ready to conclude the pro of o f Theor e m 2.10 . Let N b e the com- po sed functor N 2 ◦ N 1 . Then, the diagra m of Theor em 2.10 can b e o btained by concatenating the commutativ e diagrams of L emmas 8.6 and 8.9 . In par ticular, it is commut ative. Recall from [ 16 , § 4.7] that a k -algebra B is ho motopically finitely presented a s a dg k - algebra if and o nly if the na tur ally as so ciated dg category B is homotopically finitely presented as a dg categor y . Ther efore, isomorphis ms ( 2.11 ) follow from ( 8.5 ) and Lemma 8 .8 .  9. P roof of Proposition 2.12 Pr o of. By construction, the a dditive motiv ator Mot A is triangulated. Ther efore, as shown in [ 5 , § A.3], it co mes equipped with a cano nical action HO ( Spt ) × Mot A − → Mot A ( E , X ) 7→ E ⊗ X of the deriv ator asso cia ted to sp ectra. Giv en any sp ectr um E and ob jects X and Y in Mo t A ( e ), the following natur a l w eak equiv alence of sp ectra holds (9.1) R Hom Mot A ( e ) ( E ⊗ X , Y ) ≃ R Hom Ho ( Spt ) ( E , R Hom Mot A ( e ) ( X, Y )) . Let us now conside r the distinguished triangle of sp ectra (9.2) S · l − → S − → S /l − → S [1] , where S is the sphere sp ectr um a nd · l the l -fo ld m ultiple of the identit y mo rphism. By applying the functor − ⊗ U A ( Z ) to it we obtain a distinguished triang le in the category of non-commutativ e motiv es U A ( Z ) · l − → U A ( Z ) − → S / l ⊗ U A ( Z ) − → U A ( Z )[1] , PRODUCT S, MUL TIPLICA TIVE CHERN CHARACTERS, AND FINITE COEFFICIENTS 19 which allow us to conclude that S /l ⊗ U A ( Z ) is the mo d- l Mo or e ob j ect U A ( Z ) /l of U A ( Z ). Therefore, b y combining ( 9.1 ) with ( 1.3 ) we obtain the w eak equiv alence R Hom ( U A ( Z ) /l , U A ( A )) ≃ R Ho m Ho ( Spt ) ( S /l , K ( A )) . Note that b y construction S / l is a dualizable o b ject in the symmetric monoidal category Ho ( Spt ). Its dual ( S /l ) ∨ is given by R Hom Ho ( Spt ) ( S /l , S ). By applying the duality functor R Hom Ho ( Spt ) ( − , S ) to ( 9.2 ) we o btain the distinguished triangle S [ − 1] − → ( S /l ) ∨ − → S · l − → S , and so by “rotating it” twice (see [ 27 , § 1.1]) we co nclude that ( S /l ) ∨ ident ifies with ( S /l )[ − 1]. There fo re, we hav e the following weak equiv alences R Hom Ho ( Spt ) ( S /l , K ( A )) ≃ ( S /l ) ∨ ∧ L K ( A ) ≃ ( S /l ∧ L K ( A ))[ − 1] . Finally , since b y definition K ( A ; Z /l ) = S /l ∧ L K ( A ) we have R Hom ( U A ( Z ) /l , U A ( A )) ≃ K ( A ; Z /l )[ − 1] . Now, let us a ssume that l = pq with p and q coprime integers. In this ca se, Z /l ≃ Z / p × Z /q and so the natural maps S /l → S / p and S / l → S /q induce an isomorphism S / l ≃ S /p ∨ S /q in Ho ( Spt ). Ther e fo re, since we hav e the natural ident ifications U A ( Z ) /l ≃ S /l ⊗ U A ( Z ) U A ( Z ) /p ≃ S / p ⊗ U A ( Z ) U A ( Z ) /q ≃ S /q ⊗ U A ( Z ) , and the functor Ho ( Spt ) − → Mot A ( e ) E 7→ E ⊗ U A ( Z ) preserves sums, we o btain the isomor phism U A ( Z ) /l ≃ U A ( Z ) /p ⊕ U A ( Z ) /q in Mot A ( e ).  References [1] J. F. Adams, Stable homotopy and gener alize d homotopy . U ni v ersity of C hi cago press (1974). [2] A. Blumberg, D. Gepner, and G. T abuada, A universal char acterization of algebr aic K - the ory . Av ailable at [3] A. Blumberg and M. Mandell, Lo c alization the or ems in top olo gic al Ho chschild homolo gy and top olo gic al cy clic homolo gy . Av ail able at arXiv:0802 . 3938. [4] W. Browde r, Al gebr aic K -the ory with c o efficients Z /p . Lecture notes in Mathematics 657 (1978), 40–84. [5] D.-C. Ci sinski and G. T abuada, Non-co nne ctive K -the ory via universal invariants . Av ailable at arXi v:0903.3717v2. T o appear i n Comp ositio Mathematica. [6] , Symmetri c monoidal structur e on Non-co mmutative motives . Av ailable at [7] D.-C. Cisinski and A. Neeman, A dditivity for derivator K -the ory . Adv. Math. 217 (2008) , no. 4, 1381–1475. [8] A. Connes and M. Karoubi, Car act` er e multiplic atif d’un mo dule de F r e dholm . K -theory 2 (1988), 431–463. [9] K. Dennis, In se ar ch of a new homol o gy the ory . Unpublished man uscript (1976). [10] D. Dugger and B. Shipley , K -the ory and derive d equivalenc es . Duk e M ath. J. 124 (2004), no. 3, 587–617. [11] J. Go o dwillie, Rela tive algebr aic K -the ory and cyclic homolo gy . Ann. Math. 124 (1986), 347–402. [12] A. Grothe ndiec k, L es D´ eri v ateurs . Av ailable at http://p eople.math. jussieu.fr/maltsin/groth/Derivateurs.html . [13] P . Hir sc hhorn, Mo del c ate gories and their lo ca lizations . Mathematical Surveys and Mono- graphs 9 9 . American Mathematical So ciety (2003). [14] C. Ho o d and J.D.S. Jones, Some algebr aic pr op ertie s of cyclic homolo gy gr oups . K -Theory 1 (1987), no. 4, 361–384. 20 GONC ¸ ALO T ABUADA [15] J.D.S. Jones and C. Kassel, Bivariant cyclic the ory . K -theory 3 (1989), 339–365. [16] B. Keller, On diffe r e ntial g ra de d ca te gories , Internat ional Congress of Mathematicians (Madrid), V ol. I I, 151–190. Eur. Math. So c., Z ¨ urich (2006). [17] , O n the cyc lic homolo gy of exact ca te gories . J. Pure Appl. Algebra 1 36 (1999), 1–56. [18] , Invarianc e and lo c alization for cyclic homolo gy of DG algebr as . J. Pure Appl. Alge- bra 123 (1998), 223–273. [19] C. Kassel, Car act` er e de Chern bivariant . K -theory 3 (1989), 367–400. [20] M. Kontse vich, Non-c ommutative motives . T alk at the Institute for Adv anced Study on the o ccasion of the 61st birthday of Pierr e Deligne, October 2005. Video a v ailable at http://v ideo.ias.ed u/Geometry- a nd- Ari thmetic . [21] , Mixe d non-com mutative motives . T alk at the W orkshop on Ho- mological Mir ror Symmetry . University of Miami. 2010. Notes av ail able at www- math.mit.ed u/auroux/frg/miami10- notes . [22] J.-L. Lo da y , K -th´ eorie alg´ ebrique et r epr´ esent ati ons des gro ups . Ann. Sci. ´ Ecole Norm. Sup. 4 (1976), 305–377. [23] V. Lunts and D. Orl o v, Uniqueness of enhanc ement for triangulate d c ate gories . J. Amer. Math. So c. 23 (2010), 853–908. [24] P . M ay , Pairings of c ate gories and sp e ctr a . J. Pur e Appl. Algebra 1 9 (1980), 299–346. [25] R. McCarth y , The cyclic homolo gy of an exact c ate gory . J. Pure Appl. Al gebra 93 (199 4), no 3, 251–296. [26] J. M ilnor, Intr o duction to algebr aic K -the ory . Annals of M athematics Studies 72 . Princeton Unive rsity Press (1974). [27] A. Neeman, T riangulate d c atego ries . Annals of Mathematics Studies 14 8 . Princeton Univer- sity Press (2001). [28] D. Quillen, Homotopic al algebra . Lecture notes i n Mathematics 43 , Spr i nger-V erlag (1967). [29] , Higher algebr aic K -the ory I . Lecture notes in M athematics 341 (1973), 85–147. [30] G. T abuada, Higher K -the ory via universal invariants . Duke M ath. J. 145 (2008), 121–206. [31] , A dditive invariants of dg ca te gories . In t. Math. Res. Not. 53 (2005), 3309–3339. [32] , Unive rsal susp ension via non-c ommu tative motives . A v ailable at ar X iv:1003.4425. [33] , T r ansfer maps and pr oje ction formul as . Av ailable at arXi v:1006.2742. [34] , Bivariant cyclic c ohom olo gy and Connes’ b iline ar p airings in Non-c ommutative mo- tives . Av ailable at arXi v:1005.2336v2. [35] , Gener alize d sp e ctr al ca te gories, top olo gic al Ho c hschild homolo gy, and tra c e maps . Algebr. Geom. T op ol. 10 (2010), 137–213. [36] R. Thomason and T. T r obaugh, Higher algebr aic K -the ory of schemes and of derive d ca te- gories . The Grothendiec k F estschrift, vol. II I, Progress in M athematics 88 , 247–435. [37] F. W aldhausen, Algebr aic K-the ory of sp ac e s . Algebraic and geometric topology (New Brunswick, N. J., 1983), 318–419, Lecture notes in Mathematics 112 6 . Springer (1985) . [38] , Algebr aic K -the ory of gene r alize d fr ee pr o ducts . Ann. of Math. 108 (1978), 135–256. [39] C. W eibel, A survey of pr o ducts in algeb r aic K -t he ory . Lecture notes in Mathematics 854 (1981), 494–517. Gonc ¸ alo T ab uada, Dep ar t amento de Ma tema tica, FCT-UNL , Quint a da Torre, 2829- 516 Cap arica, Por tug al E-mail addr e ss : tabuada @fct.unl.p t