A guided tour through the garden of noncommutative motives

A GUIDED TOUR THROUG H THE GARDEN OF NONCOMM UT A TIVE MOTIVES GONC ¸ ALO T ABUADA Abstract. These are the extended notes of a surv ey talk on noncomm uta tiv e motiv es give n at the 3 era Escuela de Inverno Luis Santal´ o-CIMP A Rese ar c h Scho ol: T op ics in Nonc ommutative Ge ometry , Buenos Aires, July 26 to Au- gust 6, 2010 . Contents 1. Higher alg e braic K -theory 2 2. Noncommutativ e a lg ebraic geometry 2 3. Derived Mor ita equiv alences 4 4. Noncommutativ e pure motives 5 5. Noncommutativ e mixed motiv es 9 6. Co-repre sentabilit y 12 7. Symmetric monoidal structure 12 8. Higher Chern c haracters 14 Appendix A. Grothendieck deriv a tors 15 References 15 Ac kno wle dgments: I would like to start by tha nk ing all the o rganizer s o f this wint er schoo l and sp ecia lly profess or Guillermo Corti ˜ nas for kindly giving me the opp ortunity to present my work. I am also very gr ateful to the Clay Mathematics Institute for financially s uppo rting my participation. A sp e cial thanks go es a lso to Eugenia Ellis who made me discover the mag ic of the Ar gentine T ango a s well as the fascinating city of Buenos Air e s. Finally , I would lik e to thank the Depar tment of Mathematics o f MIT for a warm reception. ⋆ ⋆ ⋆ In order to make these no tes access ible to a br oad audience, I have decided to em- phasize the conceptual ideas b ehind the theory of noncommutativ e motives r ather than its technical a sp ects. I will start by stating tw o foundational questions. O ne concerning higher alge br aic K -the ory ( Question A ) a nd ano ther one concerning nonc ommu tative algebr aic ge ometry ( Questi on B ). One o f the main go als of this guided tour will b e not only to provide precise answers to these distinct questions but moreover to explain wha t is the relation b etw een the corresp onding answers. Date : Nov em ber 27, 2024. 2000 Mathematics Subje ct Classific ation. 14C15, 14F40, 18D20, 18G55, 19D23 , 19D35, 19D55. Key wor ds and phr ases. Algebraic K -theory , noncommut ativ e algebraic geometry , pure and mixed motiv es, no ncomm utativ e motiv es, Chern c haracte rs, Grothendiec k deriv ato rs. 1 2 GONC ¸ ALO T ABUADA 1. Higher algebraic K -theor y Algebraic K-theory go es back to Gro thendieck’s w ork [ 20 ] on the Riemann-Ro ch theorem. Giv en a comm utative ring R (or mor e generally an algebr aic v ariet y), he int ro duced the now adays called Gr othendie ck gr oup K 0 ( R ) of R . La tter, in the six- ties, B a ss [ 2 ] defined K 1 ( R ) as the abelia nization of the general linear g roup GL( R ). These t wo ab elian groups, whos e applications r ange from arithmetic to surgery of manifolds, are very well understo o d from a conceptual and computationa l p oint of view; see W eibel’s survey [ 57 ]. After B ass’ work, it b ecame clear that these gro ups should b e part of a whole family of higher algebr a ic K -theory groups. After several attempts made by se veral mathematicians , it was Quillen who devised a n elegant top olo gic al construction; see [ 39 ]. He introduced, the now adays ca lled Quillen’s plus c onst ruction ( − ) + , by whic h we simplify the fundamental group of a spa c e without changing its (co-)homology g roups. By applying this co nstruction to the classifying space BGL( R ) (where simplification in this case mea ns ab elianiza tio n), he defined the higher alg ebraic K -theory gr oups as K n ( R ) := π n (BGL( R ) + × K 0 ( R )) n ≥ 0 . Since Quillen’s founda tional work, hig he r alge braic K -theor y has found extra ordi- nary a pplications in a wide range of research fields; consult [ 18 ]. How ev er, Quillen’s mechanism for man ufacturing thes e hig her alg e br aic K -theor y g roups r emained rather m ysterious until to day . Hence, the follo wing question is of ma jo r impo r- tance: Question A: How t o c onc eptual ly char acterize higher algebr aic K -t he ory ? 2. Nonco m mu t a tive algebraic geometr y Noncommutativ e alg ebraic g eometry go es ba ck to Bondal-K apranov’s work [ 7 , 8 ] on exceptional collections of coher ent sheav es. Since then, Drinfeld, Ka le din, Kontsevic h, Orlov, V an den Bergh, a nd others, have ma de impor tant adv ances; see [ 9 , 1 0 , 16 , 17 , 2 5 , 3 0 , 31 , 32 , 3 3 ]. Let X b e an algebra ic v ariety . In order to study it, we c a n pro ceed in tw o distinct directions. In one dir e c tion, we can asso ciate to X several (functorial) inv ariants like the Grothendieck gro up ( K 0 ), the hig her K -theo r y gr oups ( K ∗ ), the neg ative K -theory groups ( I K ∗ ), the cyclic ho mology groups ( H C ∗ ) and all its v ar iants (Ho chsc hild, per io dic, neg ative, . . . ), the to p o logical cyclic ho mo logy groups ( T C ∗ ), e tc. Each one of these in v aria nts e nco des a pa rticular ar ithmetic/geometric feature of the algebraic v a riety X . In the o ther dir ection, we can asso cia te to X its der ived ca teg ory D p erf ( X ) o f per fect complexes of O X -mo dules. The imp or tance o f this tr iangulated ca teg ory relies on the fact that any corresp o ndance b etw een X and X ′ which induces an equiv ale nc e b etw een the der ived catego r ies D p erf ( X ) and D p erf ( X ′ ), induces also an is o morphism on all the ab ov e inv aria nt s. Hence, it is natural to ask if the ab ov e inv ariants of X c a n b e r ecov ered dir e ctly out of D p erf ( X ). This can b e done in very pa rticular ca ses ( e.g. the Grothendieck group) but not in full generality . The reason being is that when we pa s s from X to D p erf ( X ) w e loos e too muc h information co ncerning X . W e should therefore “stop somewhere in the middle”. In or der to for malize this insig ht, Bondal a nd Kapra nov introduced the following notion. A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 3 Definition 2.1 . (Bondal-Kapr a nov [ 7 , 8 ]) A differ ential gr ade d (= dg) c ate gory A , ov er a (fixed) bas e commutativ e ring k , is a catego ry enriched over complexes of k -mo dules (morphism sets A ( x, y ) are complexes) in such a wa y that comp osition fulfills the Leibniz rule: d ( f ◦ g ) = d ( f ) ◦ g + ( − 1) deg( f ) f ◦ d ( g ). A d iffer ential gr a de d (=dg) fun ctor is a functor which pre s erves the different ial graded structure; consult Keller’s ICM adres s [ 2 8 ] for further details . The category of (small) dg catego ries (ov er k ) is denoted b y dgcat . Asso ciated to the algebraic v ariety X there is a natur al dg catego ry D dg p erf ( X ) which enhances 1 the derived categor y D p erf ( X ), i.e. the latter category is o btained from the former o ne b y a pplying the 0 th -cohomolo gy group funct or at each complex of mor phisms. By considering D dg p erf ( X ) instead of D p erf ( X ) we solve many of the (tec hnical) problems inher ent to triangula ted ca tegories like the non-functoriality of the cone. More imp ortantly , we are a ble to recover all the ab ov e in v ariants of X directly out of D dg p erf ( X ). This circle o f ideas is depicted in the following dia gram: X z & & g # #  Inv ariants / / K 0 ( X ) , K ∗ ( X ) , I K ∗ ( X ) , H C ∗ ( X ) , . . . , T C ∗ ( X ) , . . . D dg p erf ( X ) _ H 0   ! 8 8 D p erf ( X ) > > i j l m n o q r t u w x z . F r om the p oint of view o f the inv aria n ts, there is absolutely no difference b etw een the algebraic v ariety X and the dg category D dg p erf ( X ). This is the main idea behind noncommutativ e algebr aic ge o metry: g iven a dg ca tegory , we should consider it as being the dg derived categor y of p erfect complexes over a hypo thetica l noncommu- tative space a nd try to do “algebra ic geometry” dir ectly on it. Citing Drinfeld [ 17 ], noncommutativ e algebr aic geometry can b e defined as: “the study of dg c ate gories and their homolo gic al invariants ” . Example 2.2 . (Beilinson [ 3 ]) Suppo s e that X is the n th -dimensional pro jective s pace P n . The n, there is an eq uiv alence of dg ca tegories D dg p erf ( P n ) ≃ D dg p erf ( B ) , where B is the algebra E nd( O (0 ) ⊕ O (1) ⊕ . . . ⊕ O ( n )) op . Note that the a b elian cate- gory of quasi-coherent sheav es o n P n is far from being the categor y of mo dules ov er an algebr a. Beilins o n’s remar k able res ult show us that this situation changes ra di- cally when we pass to the derived setting. Intuitiv ely sp eaking , the n th -dimensional pro jective space is a n “a ffine o b ject” in noncommutativ e alg ebraic g eometry since it is des c rib ed by a single (noncomm utative) algebr a. In the commutativ e world, Gr othendieck envisioned a theory of motives as a gatewa y b etw e e n alge braic geometry and the assor tment of the cla ssical W e il co- homology theories (de Rham, Betti, l - adic, c rystalline, and others); consult the monogra ph [ 23 ]. 1 Consult Lun ts-Orlov [ 35 ] for the uniqueness of this enhan cemen t. 4 GONC ¸ ALO T ABUADA In the noncommutativ e w orld w e can envision a similia r picture. The role o f the algebraic v arieties and of the classical W eil cohomolog ie s is play ed, resp ectively , b y the dg c a tegories and the numerous (functoria l) inv ariants 2 (2.3) dgcat K ∗ , I K ∗ , H C ∗ ,...,T C ∗ ,... / / Ab The Gr othendieckian idea of motives consists then o n co mb ing this skein of in v ar i- ants in or der to is o late the truly fundamental one: Ab Ab dgcat U / / K ∗ - - I K ∗ - - H C ∗ 3 3 T C ∗ 3 3 Mot 5 5 2 2 / / , , Ab Ab The gatew ay categor y Mot , throug h which all inv ariants factor uniquely , should then b e ca lled the catego ry of nonc ommutative motives and the functor U the universal invariant . Note that in this y oga, the different inv ariants are simply different r epresentations of the motivic catego r y Mot . In particular, any result which ho lds in Mot , ho lds everywhere. This b eautiful circle of idea s le ads us to the following down-to-earth question: Question B: Is ther e a wel l-define d c ate gory of nonc ommutative motives ? 3. Derived Morit a equiv alences Note first that all the classical constructio ns whic h c an b e performed with k - algebras can also be per formed with dg categories; consult [ 28 ]. A dg functor F : A → B is called a derive d Morita e quivalenc e if the induced restrictio n of scalars functor D ( B ) ∼ → D ( A ) is an equiv alence of (triangulated) categories. Thanks to the work o f B lum ber g-Mandell, Ke lle r, Schlic h ting, and Thoma son-T robaugh, all the inv a riants ( 2.3 ) in vert derived Morita equiv ale nces; se e [ 6 , 29 , 40 , 53 ]. Intu itively sp eaking, although de fined at the “dg le vel”, these inv aria nts only dep end o n the underlying derived categ o ry . Hence, it is cr ucial to understand dg ca tegories up to derived Morita equiv alence. The following res ult is central in this direc tion. Theorem 3.1. ( [ 42 , 48 ] ) The c ate gory dgcat c arries a (c ofibr antly gener ate d) Quil len mo del stru ctur e 3 whose we a k e quivalenc es ar e the derive d Morita e quivalenc es. The homo topy ca teg ory obtained is deno ted by Hmo . Theo r em 3.1 allow us to study the pur ely alge br aic setting of dg categories using idea s, techniques, and insights of to po logical nature. Here are some examples: Bondal-Kaprano v’s pre-triangulated env el op e. Using “one-sided twisted com- plexes”, Bondal a nd Kapra nov constructed in [ 7 ] a pre- triangulated en v elope A pre - tr of every dg category A . In tuitiv ely sp eaking , their co nstruction c onsists o n for- mally adding to A (de-)susp ensions, co nes, cones o f morphisms b etw een cones, etc. 2 In order to simplif y the (graphical) exposition, w e hav e decided to forget the k -li near structure of the cyc lic homology groups H C ∗ . 3 An analogous model structure in the set ting of spectral categories w as dev elop ed in [ 45 ]. A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 5 Thanks to Theor em 3.1 , this inv olv ed co ntribution can be co nceptually character - ized as b eing simply a functorial fibra nt resolutio n functor ; see [ 42 ]. Drinfeld’s DG quotient. The most us eful op eration which can b e p erformed on triangulated ca tegories is the pas s age to a V erdier quotient. Recen tly , thr ough a very elegant construction (reminiscent from Dwy er-Kan lo ca lization), Drinfeld [ 16 ] lifted this op er ation to the w orld of dg categories. Although v ery elegant, this construction didn’t seem to satisfy any obvious universal prop erty . Theo rem 3.1 allow ed us to co mplete this asp ect of Drinfeld’s work b y character izing the DG quotient as a homotopy c ofib er constr uction; see [ 44 ]. Kon tsevic h’s saturated dg categories. Kon tsevich understo o d prec isely how to express smo o th and pr op erness in the noncommutativ e world. Definition 3.2 . (Kontsevic h [ 30 , 31 ]) A dg ca teg ory A is called: • smo oth if it is p erfect as a bimo dule ov er itself; • pr op er if its complexes of k -mo dules A ( x, y ) a re p erfect; • satur ate d if it is smo oth and prop er. Definition 3.2 is justified b y the following fact: given a quasi- compact and qua si- separated scheme X , the dg category D dg p erf ( X ) is smo oth a nd prop er if and only if X is smo o th and prop er in the sense o f clas s ical alg e braic geo metry . Other exa mples of satur a ted dg catego r ies a ppe ar in study of Deligne- Mumford stacks, quan tum pro jective v arieties, Landau-Ginzburg mo dels , etc. Now, note that the tensor pro duct of k -a lgebras extends naturally to dg cat- egories. By deriving it (with resp ect to der ived Mo rita equiv alences), we obtain then a symmetric monoidal structure on Hmo . Making use of it, the saturated dg ca tegories c an b e conceptually characterized a s b eing pr ecisely the dualizable (or rigid) ob jects in the symmetric monoidal category Hmo ; s ee [ 12 ]. As in any symmetric monoidal category , w e can define the Euler c haracteristic of a dualizable ob ject. In to po logy , for instance, the Euler characteris tic of a finite C W -co mplex is the alternating s um of the num ber of cells. In Hmo , we ha ve the following result. Prop ositi on 3.3. (Cisinski & T ab. [ 12 ] ) L et A b e a satur ate d dg c ate gory. Then, its Euler char acteristic χ ( A ) in Hmo is the Ho chsch ild homolo gy 4 c omplex H H ( A ) of A . Prop ositio n 3.3 illustrates the Gr o thendieckian idea o f com bining the skein of inv a riants ( 2 .3 ) “as far as p o ssible” in order to understand, directly on Mot , their conceptual nature. By simply inv erting the class of derived Morita equiv alences, Ho chsc hild homolo gy can b e conceptually understo o d as the Euler characteristic. 4. Nonco m mu t a tive pure motives In or der to answer Question B we need to start b y identifying the prop erties common to all the in v ar iants ( 2.3 ). In the pre v ious section w e hav e alr e a dy observed that they are derive d Morita invariant , i.e. they send de r ived Morita equiv alences 4 More generally , the trace of an endomorphisms is give n by Ho c hsc hild homology with coefficients. 6 GONC ¸ ALO T ABUADA to iso morphisms. In this sec tion, w e identif y another common prop erty . An upp er triangular matrix M is given by M :=  A X 0 B  , where A and B ar e dg categor ies and X is a A - B -bimo dule. The tota lization | M | of M is the dg category whose set of ob jects is the disjoint union o f the sets of ob jects of A and B , and whose morphisms are given b y: A ( x, y ) if x, y ∈ A ; B ( x, y ) if x, y ∈ B ; X ( x, y ) if x ∈ A and y ∈ B ; 0 if x ∈ B and y ∈ A . Co mp os ition is induced by the compositio n op eration on A a nd B , a nd by the A - B -bimo dule structure of X . Note that we hav e tw o natural inclusion dg functor s ι A : A → | M | and ι B : B → | M | . Definition 4.1 . Let E : dgcat → A be a functor with v alues in an additive category . W e say that E is an additive invariant of dg c ate gories if it is der ived Morita inv a riant and s a tisfies the following c ondition: for every upp er triang ula r matrix M , the inclusio n dg functors ι A and ι B induce an isomorphism E ( A ) ⊕ E ( B ) ∼ − → E ( | M | ) . It follo ws from the work of Blumber g-Mandell, Keller , Sc hlic h ting, and Thomaso n- T r obaugh, that a ll the inv ariants ( 2.3 ) satisfy additivity , a nd hence are additive inv a riant of dg categ ories; see [ 6 , 2 9 , 40 , 5 3 ]. The universal additive inv ariant of dg catego ries was constructed in [ 42 ]. It can b e describ ed 5 as follows: let Hmo 0 be the categ ory w ho se ob jects are the dg categ ories a nd who se mo rphisms are g iven by Hom Hmo 0 ( A , B ) := K 0 rep ( A , B ), whe r e rep ( A , B ) ⊂ D ( A op ⊗ L B ) the full trian- gulated sub catego ry of those A - B -bimo dules X such that X ( a, − ) ∈ D p erf ( B ) fo r every ob ject a ∈ A . Co mpo sition is induced by the tensor pro duct of bimo dules. Note that we hav e a natural functor U A : dgcat − → Hmo 0 which is the identit y on o b jects and which maps a dg functor to the clas s (in the Grothendieck gr oup) of the naturally asso c ia ted bimodule. The ca tegory Hmo 0 is additive and the functor U A is a dditive in the s ense of Definition 4.1 . Mor eov er, it is characterized by the following universal prop erty . Theorem 4.2. ( [ 42 ] ) Given an addi tive c ate gory A , we have an i nduc e d e quivalenc e of c ate gories ( U A ) ∗ : F un add ( Hmo 0 , A ) ∼ − → F un additivity ( dgcat , A ) , wher e the left hand-side denotes the c ate gory of additive fun ctors and the right hand-side the c ate gory of additive invariants in the sen se of Definition 4.1 . The additive categ o ry Hmo 0 (and U A ) is our fir st answ er to Questi on B . A second answer will be describ ed in Se c tio n 5 . Note that by Theorem 4.2 , a ll the in- v ar iants ( 2.3 ) factor uniquely through H mo 0 . This motivic category enabled several (tangential) applica tions. Here are tw o examples: Example 4.3 . (Chern characters) The Chern character maps ar e o ne of the most impo rtant working to ols in mathematics. Although they admit nu merous differ e nt constructions, they were no t fully under s to o d at the c o nceptual level. Making 5 A similar co nstruction in the setting of sp ectral categories was deve loped in [ 43 ]. A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 7 use of the a dditiv e categ ory Hmo 0 and of Theo rem 4.2 we hav e br idged this gap by c haracteriz ing the Chern character maps, from the Grothendieck gro up to the (negative) cyclic homology groups, in terms o f simple universal proper ties ; see [ 47 ]. Example 4.4 . (F undamental theorem) The fundamental theorems in homotopy al- gebraic K - theory and p erio dic cyclic homolog y , prov ed resp ectively b y W e ib el [ 56 ] and K assel [ 26 ], a re of ma jor imp ortance . Their pro ofs are not o nly very different but also quite inv olved. Mak ing us e of the a dditiv e categor y Hmo 0 and of Theo- rem 4.2 , we ha v e giv en a simple, unified and conceptual pro o f of these fundamen tal theorems; see [ 46 ]. Noncommuta tiv e Chow motives. By r estricting himself to saturated dg cate- gories, whic h mor ally are the “noncommutativ e smo oth pro jectiv e v arieties”, Kont- sevich intro duced the following categor y . Definition 4.5 . (Kontsevic h [ 30 , 33 ]; [ 52 ]) Let F b e a field of co efficients. The category NCho w F of nonc ommu tative Chow motives (ov er the base ring k and with co efficients in F ) is defined a s follows: first consider the F - line a rization Hmo 0; F of the additive ca tegory Hmo 0 . Then, pass to its idempo ten t completion Hmo ♮ 0; F . Finally , take the ide mp otent complete full s ub ca tegory of Hmo ♮ 0; F generated by th e saturated dg categories . The precise relation b etw een the cla ssical category of C how motiv es a nd the cat- egory of noncommutativ e Chow motives is the following: recall that the catego ry Chow Q of Chow motives (with rationa l co efficient s) is Q -linear , additive and s y m- metric mo noidal. More over, it is endow ed with an impo rtant ⊗ -inv ertible ob ject, namely the T ate motive Q (1). The functor − ⊗ Q (1) is an a utomorphism of Chow Q and so we can consider the asso ciated orbit category Chow( k ) Q / −⊗ Q (1) ; c o nsult [ 52 ] for details . Informally spea k ing, Chow motives which differ from a T a te twist bec ome isomor phic in the o rbit ca tegory . Theorem 4.6. (Kontsevich [ 30 , 33 ] ; [ 52 ] ) Ther e ex ists a ful ly-faithful, Q -line ar, additive, and symmetric monoidal functor R making the diagr am (4.7) SmProj op D dg perf ( − ) / / M   dgcat U A   Chow Q π   Hmo 0 ( − ) ♮ Q   Chow Q / −⊗ Q (1) R / / NChow Q ⊂ Hmo ♮ 0; Q c ommu te (up to a natur al isomorphism). Int uitiv ely spe aking, Theorem 4.6 formalizes the co nceptual idea tha t the com- m utative world can b e em bedded into the noncommutativ e w orld after factor izing out b y the action of the T ate motiv e. The above diagram ( 4.7 ) opens new horizonts and opp or tunities of resea rch b y enabling the in terc hange of results, techniques, ideas, and insights betw een the commutativ e and the nonco mmutative w orld. This yoga was developed in [ 52 ] in what rega rds Sch ur and K imura finiteness, motivic measures, and motivic zeta functions. 8 GONC ¸ ALO T ABUADA Noncommuta tiv e numerical moti v es. In order to formalize and s olve “ c ount- ing pr oblems”, such as coun ting the n um ber of common p oints to t w o pla nar curv es in gener al po sition, the classical ca tegory of Chow motives is not appr opriate as it makes use of a very refined no tion of equiv a lence. Motiv ated by these “co unt ing problems”, Grothendieck developed in the sixties the catego r y Num F of numerical motives; see [ 23 ]. Its noncommutativ e ana logue ca n b e describ ed a s fo llows: let A and B b e tw o saturated dg categ ories and X = [ P i a i X i ] ∈ Hom NChow F ( A , B ) and Y = [ P j b j Y j ] ∈ H om NChow F ( B , A ) tw o nonc ommutative c orr esp ondanc es . Their interse ctio n numb er is given by the formula (4.8) h X · Y i := X i,j,n ( − 1) n a i · b j · rk H H n ( A ; X i ⊗ L B Y j ) ∈ F , where rk H H n ( A ; X i ⊗ B Y j ) denotes the rank of the n th -dimensional Ho chsc hild homology gr oup of A with co efficients in the A - A -bimodule X i ⊗ L B Y j . A noncom- m utative corr esp ondance X is numeric al ly e quivalent to zer o if for every no nc o m- m utative corr esp ondence Y the intersection num ber h X · Y i is zero. As prov ed in [ 36 ], these c o rresp ondence s fo r m a ⊗ - ideal of NChow( k ) F , which we deno te b y N . Definition 4.9 . ( Mar c ol li & T ab. [ 36 ]) The categor y NNum F of nonc ommutative numeric al motives (o ver the base ring k and with co efficients in F ) is the idempo tent completion of the quotient c a tegory NChow F / N . The relatio n be t ween Chow motiv es and no ncommutativ e motives describ ed in diagram ( 4.7 ) admits the following numerical ana logue. Theorem 4.10. (Mar c ol li & T ab. [ 36 ] ) Ther e exists a ful ly-faithful, Q -line ar, ad- ditive, and symmetric monoidal functor R N making the diagr am Chow Q π   u u k k k k k k k k k k k k Num Q π   Chow Q / −⊗ Q (1) u u k k k k k k k k k R / / NChow Q w w o o o o o o o o Num Q / −⊗ Q (1) R N / / NNum Q c ommu te (up to natur al isomorphism). Int uitiv ely speaking , Theorem 4.10 for malizes the conceptual idea that Ho chschild homology is the cor rect wa y to expr ess “counting” in the noncommutativ e w orld. In the commutative w orld, Grothendieck conjectured that the catego ry of n umerical motives Num F was ab elian semi-simple. J annsen [ 24 ], thirty years latter, pr ov ed this conjecture without the us e of any o f the standard c onjectures. Recently , we gav e a further step forward by proving that Grothendieck’s conjecture holds more broadly in the noncommutativ e world. Theorem 4.11. (Mar c ol li & T ab. [ 36 ] ) A ssume one of t he fol lowing two c onditions: (i) The b ase ring k is lo c al (or mor e gener al ly t hat K 0 ( k ) = Z ) and F is a k - algebr a; a lar ge class of examples is given by taking k = Z and F an arbitr ary field. (ii) The b ase ring k is a field extension of F ; a lar ge class of examples is given by taking F = Q and k a field of char acteristic zer o. A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 9 Then, the c ate gory NNum F is ab elia n semi-simple. Mor e over, if J is a ⊗ -ide al in NChow F for which the idemp otent c omp letion of the quotient c ate gory NChow F / J is ab elian semi-simple, then J agr e es with N . Roughly sp ea king, Theorem 4.1 1 shows that the unique wa y to obtain an a belia n semi-simple ca tegory out of NChow F is through the use of the ab ov e “counting formula” ( 4.8 ), defined in terms of Ho chsc hild homo logy . Among other applications, Theorem 4.11 allow ed us to obtain an alternative pro of of J annsen’s result; see [ 36 ]. Kon tsevic h’s noncommuta tiv e n umerical moti v es. Making use of a well- behaved bilinear form on the Grothendieck of satur a ted dg ca teg ories, K ontsevic h int ro duced in [ 30 ] a category NCNum F of noncommutativ e numerical motiv es. Via duality ar g ument s, the authors prov ed the following agreement result. Theorem 4 .12. (M ar c ol li & T ab. [ 37 ] ) The c ate gories NCNum F and NNum F ar e e quivalent. By com bining Theor em 4.12 with Theor em 4.1 1 , we then conclude that NCNum F is a be lian semi-simple. Kontsevic h conjectur e d this la tter re s ult in the particular case where F = Q and k is o f character is tic zer o. W e obs e r ve that Kontsevic h’s bea utiful insight not only holds m uc h more generally , but mo r eov er it do es not require the a ssumption of any (p olar ization) conjecture. 5. Nonco m mu t a tive mixed motives Up to now, we hav e b een considering in v ariants with v a lues in additive categ ories. F r om now on we will co nsider “richer in v ariants”, taking v a lues not in additive categorie s but in “highly structured” triangulated categ ories. In order to make this precise we will use the lang uage of Gr othendie ck derivators , a fo rmalism which allow us to s tate and prove precise universal prop erties ; the reader who is unfamiliar with this language is in vited to co nsult Appendix A at this point. Recall from Drinfeld [ 16 ] that a sequence of dg functors A I → B P → C is called exact if the induced se q uence o f derived categorie s D ( A ) → D ( B ) → D ( C ) is ex a ct in the sense of V erdier. F or example, if X is quasi-c o mpact and quasi-sepa rated scheme, U ⊂ X a q uasi-compac t op en s ubs cheme and Z := X \ U the closed complementary , then the sequence of dg functor s D dg p erf ( X ) Z − → D dg p erf ( X ) − → D dg p erf ( U ) is exact; see Thomaso n-T robaugh [ 53 ]. An exact sequence o f dg functors is called split-exact if there exist dg functors R : B → A and S : C → B , right adjoints to I and P , res pe ctively , such that R ◦ I ≃ Id and P ◦ S ≃ Id via the adjunction morphisms; co ns ult [ 41 ] for details. Definition 5.1 . Let E : HO ( dgcat ) → D be a filtered homotopy colimit preserving morphism of deriv a tors, from the deriv ator asso ciated to the Quillen mo del struc- ture of Theorem 3.1 , tow ards a strong triangula ted deriv ato r. W e say that E is a lo c alizing invariant if it sends exa ct sequences to distinguished tr iangles A − → B − → C 7→ E ( A ) − → E ( B ) − → E ( C ) − → E ( A )[1] in the base categ ory D ( e ) of D . W e say tha t E is a n additiv e invariant if it s ends split-exact seq uences to direc t sums A / / B / / v v C v v 7→ E ( A ) ⊕ E ( C ) ∼ → E ( B ) . 10 GONC ¸ ALO T ABUADA Clearly , every lo c alizing inv ariant is additive. Here are some clas sical examples. Example 5.2 . (Connective K -theory) As explained in [ 41 ], co nnec tive K -theory gives rise to an a dditiv e inv ariant K : HO ( dgcat ) − → HO ( Spt ) with v alues in the triangula ted deriv a tor as so ciated to the (stable) Quillen mo del category of spectra . Quillen’s higher K -theory groups K ∗ can then be obtained from this sp ectr um b y taking stable homotopy groups. This inv ariant, a lthough additive, is n ot lo ca lizing. The following exa mple corre c ts this default. Example 5.3 . (Nonconnective K -theory) As explained in [ 41 ], nonco nnective K - theory gives rise to a lo calizing inv aria nt I K : HO ( dgcat ) − → HO ( Spt ) . As in the previous example, Bass’ negative algebraic K -theory gro ups I K ∗ can b e obtained from this sp ectrum b y taking (negative) stable homo to py gro ups. Example 5.4 . (Mixed c o mplex) F o llowing Kass el [ 26 ], let Λ be the dg algebra k [ ǫ ] /ǫ 2 where ǫ is of deg ree − 1 and d ( ǫ ) = 0. Under this notation, a mixe d c omplex is simply a r ight dg Λ-mo dule. As explained in [ 41 ], the mixed complex construction gives rise to a lo calizing inv aria nt C : HO ( dgcat ) − → HO (Λ-Mo d) with v alues in the triangula ted deriv a tor as so ciated to the (stable) Quillen mo del category of right dg Λ -mo dules. Cyclic homolog y and all its v ar iants (Hochschild, per io dic, nega tive, . . . ) ca n b e obtaine d from this mixed complex construction by a simple pro cedure s ; see [ 26 ]. Example 5.5 . (T op ologica l cyclic homology) As expla ined b y Blumberg and Mandell in [ 6 ] (see also [ 50 ]), top olo gical cyclic homolog y g ives rise to a lo ca lizing in v ar iant T C : HO ( dgcat ) − → HO ( Spt ) . The top o lo gical cyclic homolog y gr oups T C ∗ can be obtained from this s pe ctrum by taking stable homotopy groups. In order to simultaneously s tudy all the a bove classical examples, the univ ersal additive and lo calizing inv ariants U add dg : HO ( dgcat ) − → Mot add dg U lo c dg : HO ( dgcat ) − → Mo t lo c dg were constructed 6 in [ 41 ]. They a re c haracterized (in the 2-catego ry of Grothendieck deriv a tors) by the following universal prop erty . Theorem 5.6. ( [ 41 ] ) Given a stro ng triangulate d derivator D , we have induc e d e quivalenc es of c ate gories ( U add dg ) ∗ : Hom ! (Mot add dg , D ) ∼ − → Hom add ( HO ( dgcat ) , D ) ( U lo c dg ) ∗ : Hom ! (Mot lo c dg , D ) ∼ − → Hom lo c ( HO ( dgcat ) , D ) , wher e the right hand-sides denote, r esp e ctively, t he c ate gories of additive and lo c al- izing invariants. 6 A simil ar approac h in the setting of ∞ -categories was developed b y Bl umberg, Gepner and the aut hor in [ 4 ]. Besides algebraic and geomet ric examples, the authors stu died also topological examples lik e A -theo ry . A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 11 R emark 5.7 . (Quillen mo del) The additive and the lo calizing mo tiv ator admit nat- ural Quillen mo dels given in ter ms of a Bousfield localiza tion of presheav es of (sym- metric) sp ectra; c o nsult [ 41 ] for details. Because of these universal prop erties , Mot add dg is called the additive motivator , Mot lo c dg the lo c alizing motivator , U add dg the universal additive invariant , U lo c dg the uni- versal lo c alizing invariant , Mot add dg ( e ) the triangulate d c ate gory of nonc ommutative additive motives , and Mot lo c dg ( e ) the triangulate d c ate gory of nonc ommutative lo c al- izing motives . Note that since loca lization implies additivit y , we ha ve a well-defined (homotopy colimit preser ving) morphism of deriv ators Mot add dg → Mot lo c dg . The tri- angulated categor y Mo t add dg ( e ) (and Mot lo c dg ( e )) is our second a nswer to Q uestion B . Note that by Theor em 5.6 , all the inv ariants o f Examples 5.2 - 5.5 facto r uniquely through Mot add dg ( e ). Since the c omp osed functor dgcat − → Hmo U add dg ( e ) − → Mot add dg ( e ) is an additive in v ar iant o f dg categor ies in the sense of Definition 4.1 , we obtain by Theorem 4.2 an induced additive functor Hmo 0 → Mot add dg ( e ), which turns out to b e fully-faithful. In tuitiv ely sp eaking, our second a nswer to Questio n B contains the first one. In other words, the w orld of nonco mmutative pur e motives is co nt ained in the world of noncommutativ e mixed motives. As we will see in the next section, the latter w orld is muc h richer than the former one. In Exa mple 2.2 , we o bserved that the dg categor y D dg p erf ( P n ) is derived Morita equiv ale nt to the alg ebra End( O (0) ⊕ O (1 ) ⊕ . . . ⊕ O ( n )) op . By pas sing to the triangulated categor y of noncommutativ e additiv e mo tives, w e obtain the fo llowing splitting: U add dg ( D dg p erf ( P n )) ≃ U add dg ( k ) ⊕ · · · ⊕ U add dg ( k ) | {z } ( n +1)-copies . The reason b ehind this pheno menon is a s e mi-orthogo nal decomp os itio n of the triangulated categor y D p erf ( X ). Intuitiv ely spe a king, the nonco mmutative additive motive of the n th -dimensional pr o jective space c onsists simply o f n + 1 “ p o ints”. The motivic categor y Mo t add dg ( e ) enable d several (tangential) applica tions, Here is one illustra tive exa mple: Example 5.8 . (F arre ll- Jones isomorphism conjectures ) The F a r rell-Jo nes iso mor- phism conjectures ar e important driv ing forces in current mathema tica l res earch and imply well-kno w conjectures due to Bass, Borel, Kaplansk y , No vik ov; see L¨ uck- Reich’s s urvey in [ 18 ]. Given a gro up G , they predict the v a lue of algebr aic K - and L -theory of the gro up ring k [ G ] in ter ms o f its v a lue s on the virtually cyclic sub- groups of G . In addition, the liter ature contains man y v ariations of this theme, o b- tained b y replac ing the K - and L - theory functors by other functors like Hochsc hild homology , to p o logical cyclic homolo gy , etc. Dur ing the last deca des eac h o ne of these isomor phism co njectures has b een prov ed for large classes of groups using a v ar iety of different methods. Ma king use of Theo rem 5.6 , Balmer a nd the a uthor organize d this exuber ant herd of conjectures by explicitly describing the funda- men tal isomorphism co njecture; see [ 1 ]. I t turns out that this fundamen tal con- jecture, which implies a ll the ex is ting isomor phism co njectures on the mar ket, can be describ ed s olely in terms o f algebraic K -theor y . More pr ecisely , it is a simple “co efficient v aria nt ” of the classic a l F a rrell-Jo nes conjecture in algebr aic K -theory . 12 GONC ¸ ALO T ABUADA 6. Co-represent ability As in any tria ngulated deriv a tor, the a dditiv e a nd lo calizing motiv ators ar e canonically enriched ov er spectr a. Let us denote b y R Hom ( − , − ) their sp ectra of mor phisms; see App endix A . Connective algebraic K - theory is an example o f an additive inv ariant while nonc o nnective algebraic K -theory is an exa mple of a lo calizing inv a riant. Therefore , b y Theorem 5.6 , they descend to the additive and lo calizing motiv ator, res pe c tively . The following result s how us that they b ecome co-repr e sentable b y the nonco mmutative motive a sso ciated to the base ring . Theorem 6.1. ( [ 41 ] ; Cisinski & T ab. [ 11 ] ) Given a dg c ate gory A , we have n atu r al e quivalenc es of sp e ctr a R Hom ( U add dg ( k ) , U add dg ( A )) ≃ K ( A ) R Ho m ( U lo c dg ( k ) , U lo c dg ( A )) ≃ I K ( A ) . In the t riangulate d c ate gories of nonc ommutative motives, we have natu r al isomor- phisms of ab elian gr oups Hom ( U add dg ( k ) , U add dg ( A )[ − n ]) ≃ K n ( A ) n ≥ 0 Hom ( U lo c dg ( k ) , U lo c dg ( A )[ − n ]) ≃ I K n ( A ) n ∈ Z . Example 6.2 . (Sch emes) By taking A = D dg p erf ( X ) in Theorem 6.1 , with X a quasi- compact and quasi-s e pa rated s cheme, w e recover the connectiv e K ( X ) a nd no ncon- nective I K ( X ) K -theory sp ectr um of X . R emark 6 .3 . (Biv aria nt K - theo ry) Theor em 6 .1 is in fact richer. In w ha t concerns the additiv e motiv ator, the base ring k can b e replaced by any homotopically finitely presented dg ca tegory B (the homoto pical version of the clas s ical notion of finite presentation) and K ( A ) b y the biv ariant K -theory of B - A -bimo dules. In what concerns the lo calizing motiv ator, the base ring k can b e repla ced b y any satur ated dg catego ry B and I K ( A ) by the spec tr um I K ( B op ⊗ A ); consult [ 11 , 12 , 41 ]. R emark 6.4 . (Biv ariant cyclic homolog y) Classical theories like biv ariant cyclic cohomolog y (a nd the asso cia ted Connes’ bilinea r pairings ) can also be expr essed as morphisms sets in the c a tegory of no ncommutativ e motives; s ee [ 51 ]. Theorem 6.1 is our answer to Question A . Note that while the r ight-hand s ide s are, res p ectively , co nnective a nd nonconnec tive algebr aic K -theory , the left-ha nd sides ar e defined so lely in terms o f pre c ise universal prop erties: a lgebraic K -theor y is never used (o r even mentioned) in their c onstruction. Hence, the equiv alences of Theorem 6.1 provide us with a conceptual characterization o f higher alg ebraic K - theory . T o the bes t of the author’s knowledge, this is the first conceptual char- acterization of algebraic K -theor y s ince Q uillen’s foundational work. W e can e ven take these e quiv alence s as the v ery definition o f higher alg e braic K - theory . The precise relatio n b etw een the answers to Questions A and B is by no w clear. Int uitiv ely speaking , connective (resp. nonconnective) a lg ebraic K -theor y is the additive (resp. lo ca lizing) inv aria nt co-repres e n ted by the noncommut ative motive asso ciated to the ba s e ring , which as explained in the next section is simply the ⊗ -unit ob ject. 7. Symmetric monoidal structure The tensor pro duct of k -algebr as extends naturally to dg catego ries, giv ing rise to a symmetric mo noidal structure on HO ( dg cat ). The ⊗ -unit is the base ring k A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 13 (considered as a dg category). Making use of a derived version of Day’s conv olution pro duct, the a uthors pr oved the following result. Theorem 7.1. (C isinski & T ab. [ 12 ] ) The additive and lo c alizing motivators c arry a c anonic al s ymmet ric m onoidal structu re making the universal additive and lo c al- izing invariants symmetric monoidal. Mor e ov er, these symmetric monoidal struc- tur es pr eserve homotopy c oli mits in e ach variable and ar e char acterize d by t he fol- lowing universal pr op erty: given any st r ong triangulate d derivator D , endowe d with a symmetric monoidal stru ctur e, we have induc e d e qu ivalenc e of c ate gories: ( U add dg ) ∗ : Hom ⊗ ! (Mot add dg , D ) ∼ − → Hom ⊗ add ( HO ( dgcat ) , D ) ( U lo c dg ) ∗ : Hom ⊗ ! (Mot lo c dg , D ) ∼ − → Hom ⊗ lo c ( HO ( dgcat ) , D ) . Kon tsevic h’s noncomm utativ e m ixed moti v es. In [ 30 , 3 3 ], Kontsevic h int ro- duced a categor y K MM of nonc ommu tative mixe d motives (over the base ring k ). Roughly sp eaking, KMM is obtained by taking a for mal ide mp otent completion of the triangulated en velope of the category of sa turated dg categories (with biv ariant algebraic K -theory sp ectra as mo rphism sets). Making use Theorem 7.1 , the ca te- gory KMM can b e “realiz e d” inside the triangula ted categor y o f noncommut ative motives. Prop ositi on 7.2. (Cisinski & T ab. [ 12 ] ) Ther e is a n atu r al ful ly-faithful emb e d- ding (enriche d over sp e ctr a) of Kontsevich’s c ate gory K MM of nonc ommutative mixe d motives into the triangulate d c ate gory Mot lo c dg ( e ) of nonc ommutative lo c aliz- ing motives. The essential image is the thick triangulate d sub c ate go ry sp anne d by the nonc ommutative motives of satu r a te d dg c ate gories. R emark 7.3 . (Relation with V o evodsky’s motives) In the same vein as Theorem 4.6 , V o evo dsky’s tr iangulated ca tegory DM of motives [ 54 ] relates to (a A 1 -homotopy v ar iant of ) K ontsevic h’s categ o ry KMM o f noncommutativ e mixed motives. The author and Cisinski are nowada ys in the pro cess of writing down this res ult. Pro ducts in algebraic K -theo ry. Let A a nd B b e tw o dg categ ories. On one hand, following W a ldha usen [ 55 ], we hav e a classical algebr aic K - theory pairing K ( A ) ∧ K ( B ) − → K ( A ⊗ B ) . (7.4) On the other hand, by com bining the co-repr e sentabilit y Theor em 6.1 with Theo- rem 7.1 , w e obtain another w ell-defined algebr aic K -theory pairing K ( A ) ∧ K ( B ) − → K ( A ⊗ B ) . (7.5) Theorem 7.6. ( [ 49 ] ) The p airi ngs ( 7.4 ) and ( 7.5 ) agr e e up to homotopy; a similar r esu lt holds for nonc onn e ct ive K -t he ory. Example 7.7 . (Commutativ e alg ebras) Le t A = B = A , with A is a c ommutative k -alge br a. Then, b y comp osing the pairing ( 7.5 ) with the multiplication map K ( A ⊗ A ) ≃ R Hom ( U add dg ( k ) , U add dg ( A ⊗ A )) − → R Hom ( U add dg ( k ) , U add dg ( A )) ≃ K ( A ) we r ecov er inside Mot add dg the algebr a ic K -theory pairing on K ( A ) co nstructed orig- inally by W aldha usen. In particula r, we recover the (gr aded-commutativ e) m ulti- plicative struc tur e on K ∗ ( A ) constructed o riginally by Lo day [ 34 ]. 14 GONC ¸ ALO T ABUADA Example 7.8 . (Sc hemes) When A = B = D dg p erf ( X ), w ith X a quasi-compa c t and quasi-sepa r ated k -scheme, a n ar gument similar to the one of the above exa mple allow us to recover inside Mot add dg the alg ebraic K -theo r y pair ing on X c onstructed originally by Thomason-T roba ugh [ 53 ]. Theorem 7.6 (and Exa mples 7 .7 - 7.8 ) o ffers an elegant conceptual c haracteriza tio n of the alg ebraic K -theory pro ducts. Int uitively speak ing , while Theorem 6.1 shows us that connective a lgebraic K -theory is the additive inv ariant co-r epresented by the ⊗ -unit of Mot add dg , Theo r em 7.6 shows us that the classical algebr aic K -theo ry pro ducts ar e simply the o per ations naturally induced by the symmetric monoidal structure on Mot add dg . 8. Higher Chern characters Higher algebraic K -theo r y is a very p ow erful and subtle inv ariant whose ca lcula- tion is often out o f r each. In order to capture some of its information, Connes- Karoubi, Dennis, Go o dwillie, Ho o d- J ones, Kassel, McCarthy , and others, con- structed hig her C he r n c haracters tow ards simpler theories by making use of a v arie t y of highly in v olved techniques; se e [ 14 , 15 , 19 , 22 , 27 , 38 ]. Making use of the theory of noncommutativ e motives, these higher Chern char- acters can b e constructed, and conceptually characterized, in a simple a nd elegant wa y; see [ 11 , 12 , 41 , 49 , 50 ]. Let us now illus tr ate this in a particular case: choo se your fa v orite additive in v ariant E with v alues in the deriv a to r asso cia ted to sp ectr a . A cla ssical ex a mple is g iven by co nnective algebraic K - theory . Tha nks to T heo - rem 5.6 , we obtain then (ho mo topy colimit preserving ) mor phisms of deriv ators K , E : Mot add dg − → HO ( Spt ) such that K ◦ U add dg = K and E ◦ U add dg = E . Recall fr o m Theo rem 6.1 that the functor K is co-repr esented by the no ncommutativ e additive motive U add dg ( k ). Hence, the enr iched Y oneda lemma furnishes us a natur al eq uiv alence of sp ectra R Nat ( K , E ) ≃ E ( k ), wher e R Nat denotes the sp ectrum of natural transfor mations. Using Theorem 5.6 aga in, we obtain a natural equiv alenc e R N at ( K, E ) ≃ E ( k ). By passing to the 0 th -homotopy gr o up, we conclude that there is a natur a l bijection betw een the natural transfor mation (up to homoto py) from K to E and π 0 E ( k ). In sum, the theor y of noncommutativ e motives allow us to fully classify in s imple and elegant terms a ll pos sible natural transformation from connective K -theo ry tow ards any a dditiv e inv ariant; a similar result ho lds for nonconnective K -theory . Example 8 .1 . (Chern c haracter) Let E be the cyclic ho mology H C additive funct or (promoted to an inv aria nt taking v a lue s in s pe c tr a). Then, w e hav e the following ident ifications: Nat ( K, H C ) ∼ → k ≃ H C 0 ( k ) { Chern character } 7→ 1 . Example 8.1 provides a conceptual character ization of the Che r n character as being pr ecisely the unit among all p o ssible natural transfor mations. A similar characterization of the cy c lo tomic trace ma p, in the setting of ∞ -categ o ries, was recently developed by Blumberg, Gepner and the author in [ 5 ]. A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 15 Appendix A. Grothendieck deriv a tors The orig inal reference for the theory of deriv ators is Grothendieck’s o r iginal man- uscript [ 21 ]. See als o a short account b y Cisinski and Neeman in [ 13 ]. Deriv ators originate in the problem of higher homo topies in derived categor ies. F or a triangu- lated catego ry T a nd for X a small ca tegory , it esse ntially never happ ens that the diagram catego ry F un( X , T ) = T X remains triang ulated; it alrea dy fails for the category o f ar rows in T , that is, for X = ( • → • ). Now, very often, our triangu- lated catego ry T app ear s as the homotopy categor y T = H o ( M ) of so me Quillen mo del M . In this ca se, we can consider the categ ory F un( X , M ) of diagra ms in M , whose homo topy categ ory Ho (F un( X, M )) is often triang ulated and provides a reasona ble appr oximation for F un( X, T ). Mo r e impo rtantly , one ca n let X mov e. This nebula of categ o ries H o (F un( X , M )), indexed by small categories X , a nd the v ar ious functors and natur a l transfor mations betw een them is what Grothendieck formalized into the conce pt of derivator . A deriv ator D consists of a strict contrav a riant 2-functor from the 2- categor y of small catego ries to the 2 -catego r y of a ll categor ies D : Cat op − → CA T , sub ject to certain conditions; consult [ 13 ] for details. The ess ent ial example to keep in mind is the deriv ator D = HO ( M ) asso ciated to a (cofibra ntly genera ted) Quillen mo del category M a nd defined for every s mall catego ry X by HO ( M )( X ) = Ho (F un( X op , M )) . W e denote by e the 1-p oint category with o ne ob ject a nd one identit y morphis m. Heuristically , the category D ( e ) is the basic “derived” ca tegory under consideration in the der iv ator D . F or instance, if D = HO ( M ) then D ( e ) = Ho ( M ). Let us now recall tw o slig htly technical prop erties o f deriv ators. - A deriv a tor D is called str ong if for ev ery finite free categ ory X and every small ca tegory Y , the natural functor D ( X × Y ) − → F un( X op , D ( Y )) is full and essentially surjective. - A deriv ator D is called triangulate d (or s table ) if it is p ointed and if every global commutativ e square in D is cartesia n exa ctly when it is co cartesia n. A source of ex a mples is provided b y the deriv ators HO ( M ) asso cia ted to stable Quillen mo del categories M . Recall from [ 13 ] that given any triangulated deriv ator D and sma ll category X , the category D ( X ) has a canonica l triangulated structure. In par ticula r, the catego ry D ( e ) is triangulated. Recall also from [ 11 ] that any triangula ted deriv ator D is canonically enriched o v er spectra , i.e. we hav e a w ell-defined morphism of deriv ators R Hom ( − , − ) : D op × D − → HO ( Spt ) . Finally , given deriv a tors D a nd D ′ , we denote by Hom ( D , D ′ ) the category of all morphisms of deriv a tors and b y Hom ! ( D , D ′ ) the categor y of morphisms of deriv a- tors which preserve ar bitrary homotopy c olimits. References [1] P . Balmer and G. T abuada, The fundamental isomorphism isomorphism c onje ctur e via non- c om mutative motiv e s . Av ailable at [2] H. Bass, K - t he ory and stable algebr a . Publications Math´ ematiques de l ’IH ´ ES 22 (1964), 1–60. 16 GONC ¸ ALO T ABUADA [3] A. Beilinson, The derived c ate gory of c oher ent she aves on P n . Select a Math. So viet. 3 (198 3), 233–237. [4] A. Bl umberg, D. Gepner, and G. T abuada, A universal c har acterization of higher algebr aic K -the ory . Av ailable at arxiv:1001.2282 v3. [5] , Uniqueness of t he multiplic ative cyc lotomic tr ac e . Av ailable at arxi v:1103.3923. [6] A. Blum berg and M. Mande ll, L o c alization t he or ems in top olo gic al Ho chschild homolo gy and top olo gic al c yclic homolo gy . Av ailable at arX iv:0802 . 3938. [7] A. Bondal and M. Kapranov, F r ame d triangulate d c ate gories . (Russian) M at. Sb. 1 81 (1990 ), no. 5, 669 –683; translation in M ath. USSR-Sb. 70 (1991), no. 1, 93–107 . [8] , R epr esentable f unctors, Serr e functors, and mutations . Izv. Ak ad. Nauk SSSR Ser. Mat., 53 (19 89), no. 6, 11 83–1205. [9] A. Bondal and D. Orlov, Derive d ca te gories of c oh er e nt she aves . Inte rnational Congress of Mathematicians, V ol . II, Beijing, 2002, 47–56 . [10] A. Bondal and M. V an den Bergh, Gener ators and r epr esentabilit y of functors in c ommutative and nonc omm utative ge ometry . Mosc. Mat h. J. 3 (2003), no. 1, 1–36. [11] D.- C. Cisinski and G. T abuada, Non-co nne c tive K -the ory via universal invariants . Comp o- sitio Mathemat ica, V ol. 147 (20 11), 1281–1320. [12] , Symmetric monoidal struct ur e on Non-co mmutative motives . Av ailable at arXiv:1001.0228v2. T o app ear in Journal of K - theory . [13] D. Cis i nski and A. Neeman, A dd itivity for deriv ator K -the ory , Adv. Math. 217 (2008), no. 4, 1381 –1475. [14] A. Connes and M. Karoubi, Car act ` er e multiplic atif d’un mo du le de F r e dholm . K -theory 2 (1988), 431–4 63. [15] K. Dennis, In se ar ch of a new homolo gy the ory . Unpub lished man uscript (1976 ). [16] V. Drinfeld, DG quotients of DG c ate gories . J. Algebra 272 (20 04), 643–691. [17] , DG c ate g ories . University of Chicago Geometric Langlands Seminar 2002. Notes a v ail able at www.math .utexas. edu/users/benzvi/GRASP/lectures/Langlands.html . [18] E. F riedlander and D. Grayson, Handb o ok of K -the ory . V ol. 1 and 2. Edited by Springer- V erlag, Berlin, 20 05. [19] J. Goo dwill ie, R elative algebr aic K -the ory and cy clic homolo gy . Ann. Math. 12 4 (1986), 347–402. [20] A. Grothendiec k, Th´ eorie des interse ctions et th´ e or ` eme de Riemann-R o ch . Springer-V erlag, Berline, 197 1, S ´ emi naire de G ´ eometrie Alg´ ebrique du Bois-Marie 1996 -1967 (SGA 6), Dirig ´ e par P . Berthelot, A. Grothendiec k et L. Ill usie. Avec la collaboration de D. F err and, J. P . Jouanolou, O. Jussi l a, S. Kleiman, M. Ra ynaud et J. P . Serre, Lecture notes in M athe- matics, V ol. 225 . [21] , L es D´ erivateurs , a v ail able at http://peo ple.math .jussieu.fr/ ∼ maltsin/groth/ Derivate urs.html . [22] C. Ho o d a nd J.D.S. Jones, Some al gebr aic pr op ertie s of cyclic hom olo gy gr oups . K -Theory 1 (1987), no. 4, 361–384. [23] U. Jannsen, S. Kleiman and J.-P . Serr e, Motives . Pro ceedings of the AMS- IMS-SIAM Join t Summer Research Conference held at the U nive rsity of W ashington, Seattle, W ashington, July 20–August 2, 1991. Pro ceedings of Symposi a i n Pure Mathematics 55 , part 1 and 2. American Mathemat ical So ci ety , Providence , RI, 1994. [24] U. Jannsen, Motive s, numeric al e quivalenc e, and semi-simplicity . In v en t. Math. 107 (1992), no. 3, 447 –452. [25] D. Kaledin, Motivic structur es in nonc ommutative ge ometry . Av ailable at arX iv:1003 . 3210. T o appear in the Proceedings of the ICM 20 10. [26] C. Kassel, Cyclic homolo gy , Como dules, and mixe d c omplexes . Journal of Algebra 107 (1987), 195–216. [27] , Car act` e r e de Chern bivariant . K -theory 3 (1989 ), 367–400. [28] B. Kell er, On d iffer ential gr ade d c ate gories . In ternational Congress of M athematicians (Madrid), V ol . I I, 151–190. Eur. Math. Soc., Z¨ urich (2006). [29] , On the c yclic ho molo gy of exact c ategories . J. Pure Appl. Algebra 136 (1999), 1–56 . [30] M. Kontsevic h, Nonc om mutative motives . T alk at the Institute for Adv anc ed Study on the o ccasion of the 61 st birthda y of Pi erre Deligne, October 2005. Video av ailable at http://v ideo.ias .edu/Geometry- and- Arithmetic . A GUIDED TOUR THR OUGH THE GARDEN OF NONCOMMUT A TIVE MOTIVES 17 [31] , T riangulate d c ate gories and ge om etry . Course at the ´ Ecole Norm al e Sup´ eri eure, Pa ris, 1998. N otes a v ailable at www.math .uchicag o.edu/mitya/langlands.html [32] , Mixe d no nc ommutative motives . T alk at the W orkshop on Ho- mological Mi rror Symmetry . Univ ersity of Mi ami . 2010. N otes av ailable at www- math.mit.ed u/auroux /frg/miami10- notes . [33] , Notes on motives in finite char acte ri stic . Algebra, ari thmetic, and geometry: in honor of Y u. I. Manin. V ol. II, 213–247, Progr. Math., 2 70 , Birkhuser Bost on, MA, 2009. [34] J.-L. Loday , K - th´ eorie alg´ ebrique et r epr ´ esentations des gr oup s . Ann. Sci. ´ Ecole Norm. Sup. 4 (197 6), 305–377. [35] V. Lun ts and D. Orl o v, Uniqueness of e nhanc ement for tri angulate d c ate gories . J. Amer. Math. Soc. 23 (2010 ), 853–908. [36] M. Marcolli and G. T abuada, Nonc ommutative motives, numeric al e quivalenc e, and se mi- simplicity . Av ail able at [37] , Kontsevich’s nonc ommutative numeric al motives . Av ailable at arXi v:0302013. [38] R. McCarthy , The cyclic homolo gy of an exact c ate gory . J. Pure Appl. Algebra 93 (1994 ), no 3, 251–296. [39] D. Quillen, Higher algebr aic K -the ory I . Lecture notes in Mathematics 341 (1 973), 85–147. [40] M. Sc hlic h ting, Ne gativ e K-the ory of derive d c ate gories . Math. Z. 253 (2006 ), no. 1, 97–134. [41] G. T abuada, Higher K -t he ory via universal invariants . Duke Math. J. 1 45 (2008), no. 1, 121–206. [42] , A dditive invariants of dg c ategories . In t. Math. Res. Not. 53 (2005 ), 3309–3339. [43] , M atrix invariants of sp e ctr al c ate gories . In t. Math. Res. Not. 13 (2010), 2459–2 511. [44] , On Drinfeld’s D G quotient . Jo urnal of Algebra, 323 (2010), 1226–1240. [45] , Homotop y t he ory of sp e ctra l c ate gories . Adv. in Math., 221 (2009), no. 4, 1122–114 3. [46] , The fundamental the or em v i a derive d Morita invarianc e , lo c alization, and A 1 - homotopy invarianc e . Av ailable at arXiv:1103.5936. T o appear in Journal of K -theory . [47] , A universal char acterization of the Chern char acter maps . Proc. Amer. Math. Soc. 139 (20 11), 1263–127 1. [48] , A Quil len mo del structur e on the c ate gory of dg c ate gories . CRAS Pa ris, 340 (20 05), 15–19. [49] , Pr o ducts, multiplic ative Chern char act ers, and finit e c o efficients via non- c om mutative motiv e s . Av ailable at [50] , Gener alize d sp ectr al c ate gories, t op olo gic al Ho chschild homolo gy, and tr ac e maps . Algebraic and Geometric T op ology , 10 (2010) , 137–213. [51] , Bivariant cy clic c oh omolo gy and Connes’ biline ar p airings in Non-c ommutative mo- tives . Av ailable at [52] , Chow motives ve rsus nonc ommutative motives . Av ailable at arXi v:1103.0200. T o appear in Journal of Noncomm utat ive Ge ometry . [53] R. Thomason and T. T robaugh, Higher algebr aic K -the ory of schemes and of derived c ate- gories . The Groth endiec k F estsc hrift, v ol. III, Progress in Mathe matics 88 , 247–4 35. [54] V. V oevodsky , T riangulate d c ate gories of motives over a field . Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 , Princeton Universit y Pr ess, Princeton, NJ, 20 00, 188–238 . [55] F. W aldhausen, Algebr aic K-t he ory of sp ac es . Al gebraic and geometric topology ( New Brunswick, N. J., 1983), 318 –419, Lecture notes in M athematics 1126 . Springer (1985). [56] C. W eibel, Homotopy algebr aic K -theo ry . Con temporary Mathematics 83 (198 9). [57] , The development of algebr a ic K -the ory b ef or e 1980. Av ailable at http://w ww.math. rutgers.edu/wei bel Gonc ¸ alo T ab uada, Dep ar tment of Ma thema tics, MIT, Cam bridge, MA 021 39 E-mail addr e ss : tabuada@math.mi t.edu