Universal suspension via Non-commutative motives

UNIVERSAL SUSPENSION VIA NON-CO MMUT A TIVE MOTIVES GONC ¸ ALO T A BUADA Abstract. In this article we further the study of non-comm utativ e motives, initiated i n [ 5 , 6 , 26 ]. Our main result is the construction of a simple mo del, give n in terms of infin ite matrices, for the suspension in the triangulate d cate- gory of non-commut ativ e motives. As a consequenc e, this simple model holds in all the classical inv ariant s suc h as Ho c hsch ild homology , c yclic homology and its v ariants (perio dic, negativ e, . . . ), algebraic K -theory , top ological Ho chsc hild homology , topological cyclic homology , . . . 1. Introduction Non-commutativ e motives. A differ ential gr ade d (= dg) c ate gory , ov er a co mm u- tative base ring k , is a category enriched ov er complexes of k -mo dules (morphisms sets are such complexes) in such a wa y that comp osition fulfills the Leibniz rule : d ( f ◦ g ) = ( d f ) ◦ g + ( − 1) deg ( f ) f ◦ ( dg ). Dg categ ories enha nce and s olve many of the techn ical proble ms inherent to triang ulated ca teg ories; se e Keller’s IC M adress [ 13 ]. In non-c ommut ative algebr aic ge ometry in the sense of Bondal, Drinfeld, Kaledin, Kapranov, Kontsevic h, T o¨ en, V an den Ber gh, . . . [ 2 , 3 , 7 , 10 , 17 , 1 8 , 31 ], they are considered as dg-enhance ments of der ived catego ries of (qua si-)coherent sheaves o n a hypothetic non-co mm utative space. All the classica l (functorial) inv ar iants, such as Ho chsc hild ho mology H H , cyclic homology H C , (non-connective) alg ebraic K -theor y I K , top ologica l Ho chsc hild homology T H H , and top olo gical cyclic homolo gy T C , e xtend naturally from k - algebras to dg categor ie s. In order to study al l these c lassical inv ariants s im ultane- ously the a uthor introduced in [ 26 ] the no tion of lo c alizing invariant . This notion, that we now recall, makes use o f the la nguage of Gr othendieck deriv ators [ 9 ], a formalism which allows us to state a nd pr ov e precis e universal prop erties. Let L : HO ( dgc at ) → D b e a morphism o f deriv a tors, from the deriv ator asso cia ted to the derived Morita mo del str ucture on dg catego ries (see § 2.2 ), to a tria ngu- lated der iv ator . W e say that L is a lo c alizing invariant if it preserves filtered ho- motopy colimits as well a s the terminal ob ject, a nd sends exa ct sequences of dg categorie s (see § 2.3 ) A − → B − → C 7→ L ( A ) − → L ( B ) − → L ( C ) − → L ( A )[1] Date : August 28, 2018. 2000 Mathematics Subje ct Classific ation. 18D20, 19D35, 19D55. Key wor ds and phr ases. Non-comm utativ e motives, Infinite matrix algebras, Algebraic K - theory , (T opol ogical) Hochsc hild and cyclic homology . The author w as partially supp orted by the Estimulo ` a In v estiga¸ c˜ ao Aw ard 2008 - Calouste Gulbenkian F oundation. 1 2 GONC ¸ ALO T ABUADA to distinguished triangles in the base categ ory D ( e ) o f D . Thanks to the work of Keller [ 14 , 1 5 ], Thomason-T robaug h [ 30 ], Schlich ting [ 22 ], and Blumber g -Mandell [ 1 ] (see also [ 29 ]), all the mentioned inv a r iants satis fy lo calizatio n 1 , and so g ive rise to lo calizing inv a riants. In [ 26 ], the author constructed the universal lo ca lizing inv ar ia nt U lo c dg : HO ( dgcat ) − → Mot lo c dg , i.e. given any tr iangulated der iv ator D , we hav e a n induced eq uiv alence o f catego ries (1.1) ( U lo c dg ) ∗ : Hom ! (Mot lo c dg , D ) ∼ − → Hom lo c ( HO ( dgcat ) , D ) , where the left-hand s ide denotes the catego r y of homoto py colimit preserving mo r - phisms of deriv ators, and the r ight-hand side denotes the categor y of lo caliz ing inv ar ia nts. B e cause of this universality prop er ty , which is a reminiscence o f mo- tives, Mot lo c dg is ca lled the lo c alizing motivator , and its base categ ory Mot lo c dg ( e ) the c ate gory of non-c ommutative motives . W e invite the reader to consult [ 5 , 6 , 26 ] for several applications o f this theory of non-commutativ e motives. Univ ersal susp e nsion. The purp o s e of this a rticle is to construc t a simple mo del for the susp ension in the triangulated category of non-co mm utative motives. Consider the k -alg ebra Γ of N × N -matrice s A which sa tisfy the following tw o conditions : (1) the set { A i,j | i, j ∈ N } is finite; (2) there exists a natural n umber n A such that each row and ea ch column has at most n A non-zero e n tries; se e Definition 3 .5 . Let Σ be the quotient o f Γ by the tw o-sided ide a l consisting of those matrices with finitely many non- zero entries; s ee Definition 3.1 . Alternatively , take the (left) loca lization of Γ with resp ect to the matrices I n , n ≥ 0, with en tries ( I n ) i,j = 1 for i = j > n a nd 0 otherwise; se e Prop ositio n 3.11 . The algebra Σ go es back to the work of Karoubi and Villamay or [ 12 ] on negative K -theory . Recen tly , it w as used by Cor ti ˜ nas and Thom [ 4 ] in the constructio n of a biv ariant algebr aic K -theory . Given a dg ca tegory A , we denote by Σ( A ) the tensor pr o duct of A with Σ; see § 2.1 . The main result of this article is the following. Theorem 1.2. F or every dg c ate gory A we have a c anonic al isomorphi sm U lo c dg (Σ( A )) ∼ − → U lo c dg ( A )[1] . The pro of of The o rem 1.2 is bas ed on several pro p er ties of the categ ory o f non- commutativ e motiv es (see Section 6 ), o n an exa ct seq uence r e lating A and Σ( A ) (see Section 4 ), and on the flasqueness of Γ (see Sectio n 5 ). Let us now describ e some applications of Theor em 1.2 . Applications. A r e aliza tion of the category of non-commut ative motiv es is a tri- angulated functor R : Mo t lo c dg ( e ) → T . An imp ortant aspect of a realization is the fact that every result which holds on Mot lo c dg ( e ) also holds o n T . In pa r ticular, given a dg categor y A , Theorem 1.2 furnish us a cano nical isomorphism ( R ◦ U lo c dg )(Σ( A )) ∼ − → ( R ◦ U lo c dg )( A )[1] . 1 In the case of algebraic K -theory we consider its non-connectiv e version. UNIVERSAL SUSPENSION VIA NON-COMMUT A TIVE MOTIVES 3 Thanks to the ab ove equiv alence ( 1.1 ) every lo caliz ing inv ariant g ives ris e to a realization. Ther e fore, w e obtain the cano nical isomorphisms : H H (Σ( A )) ≃ H H ( A )[1] H H ∗ +1 (Σ( A )) ≃ H H ∗ ( A ) (1.3) H C (Σ( A )) ≃ H C ( A )[1] H C ∗ +1 (Σ( A )) ≃ H C ∗ ( A ) (1.4) I K (Σ( A )) ≃ I K ( A )[1] I K ∗ +1 (Σ( A )) ≃ I K ∗ ( A ) (1.5) T H H (Σ( A )) ≃ T H H ( A )[1] T H H ∗ +1 (Σ( A )) ≃ T H H ∗ ( A ) (1.6) T C (Σ( A )) ≃ T C ( A )[1] T C ∗ +1 (Σ( A )) ≃ T C ∗ ( A ) . (1.7) Negative cyclic homology H C − and p erio dic cyclic homo logy H P are not exa mples of lo calizing inv a riants since they do n ot preserve filtered (homo topy) colimits. Nevertheless, as expla ined in [ 6 , Examples 8.1 0 a nd 8 .11], they factor thro ugh Mot lo c dg th us giv ing ris e to r ealizations. W e obtain then the canonica l isomorphis ms : H C − (Σ( A )) ≃ H C − ( A )[1] H C − ∗ +1 (Σ( A )) ≃ H C − ∗ ( A ) (1.8) H P (Σ( A )) ≃ H P ( A )[1] H P ∗ +1 (Σ( A )) ≃ H P ∗ ( A ) . (1.9) Note that since H P is 2-p erio dic, the homologies o f Σ( A ) and A can b e obtained from each other by simply switching the degrees. T o the b est o f the author’s knowledge the is omorphisms ( 1.3 )-( 1.9 ) are new. They show us that Σ( A ) is a simple mo de l for the susp ension in all these classical in v aria nt s. 2 W e would like to mention that Kassel constr ucted an isomorphism related to ( 1.4 ) but for or dinary algebr a s ov er a field and with cyclic ho mology replac ed by biv aria nt cyclic cohomology; see [ 11 , Theo rem 3.1 ]. Instea d of Γ, he considered the larger algebra of infinite matrices which have finitely many non-zero entries in ea ch line and column. Now, let X a quasi-compact and quasi-separ ated s cheme. It is well-known that the ca teg ory of p er fect complex e s in the (unbounded) derived category of quasi- coherent sheaves on X admits a dg- enhancement p erf dg ( X ); see for instance [ 2 , 21 ] or [ 6 , Exa mple 4.5]. Thanks to [ 1 , Theorem 1.3], [ 15 , § 5.2 ] and [ 22 , § 8 Theo rem 5] the alg ebraic K -theory and the (top olog ic al) cyclic homolog y 3 of the scheme X can b e o btained from the dg categor y p erf dg ( X ) by applying the corresp o nding inv ar ia nt. Therefor e, when A = p erf dg ( X ), the ab ov e isomorphisms ( 1.3 )-( 1.9 ) suggest us that the dg ca tegory Σ( p erf dg ( X )) sho uld b e consider ed as the “non- commutativ e susp ension” o r “non-commutative delo oping” of the scheme X . This will be the sub j ect of future research. Ac kno wl edgments : Theore m 1.2 a nswers affirmatively a ques tio n raise d by Maxim Kontsevic h in my P h.D. thesis defense [ 25 ]. I dee ply thank him for his insight. I am also gra teful to Ber nhard Keller, Marco Schlic h ting and Bertra nd T o¨ en for use ful conv ersations and/o r references. Convention 1 .1 0 . Throughout the article k will denote a c ommut ative base ring with unit 1 . Given a dg algebr a H we will denote by H the dg categor y with a single ob ject ∗ a nd with H as the dg algebra of endomorphisms . 2 Recall that all these inv ariant s tak e v alues in arbitr ar y degrees. 3 In fact we can consider an y v ariant of (topological) cyclic homology . 4 GONC ¸ ALO T ABUADA 2. Backg round on dg ca tegories In this section we collect s ome no tions and results on dg categor ies which will be used throughout the article. Let C ( k ) b e the categ ory of (unbounded) complexes of k -mo dules; we use co- homologica l no ta tion. A differ ential gr ade d (=dg) c ate gory is a categor y enr iched ov er C ( k ) a nd a dg fun ctor is a functor enriched over C ( k ); consult Keller’s ICM adress [ 13 ] for a survey on dg catego ries. The catego r y of dg ca tegories will b e denoted b y dgca t . Notation 2.1 . Let A be a dg c a tegory . The catego ry Z 0 ( A ) has the s ame ob jects as A and morphisms given by Z 0 ( A )( x, y ) := Z 0 ( A ( x, y )). The catego ry H 0 ( A ) has the same o b jects as A and morphisms given b y H 0 ( A )( x, y ) := H 0 ( A ( x, y )). The opp osite dg categor y A op of A has the s a me ob jects a s A and complexes of morphisms given by A op ( x, y ) := A ( y , x ). 2.1. (Bi)mo du l es. Let A b e a dg categor y . A right A -mo dule M is a dg functor M : A op → C dg ( k ) with v alues in the dg ca tegory C dg ( k ) of complexes o f k - mo dules. W e will denote by C ( A ) the categor y of rig ht A -mo dules ; see [ 13 , § 2.3]. As explained in [ 13 , § 3.1] the differential graded structure o f C dg ( k ) makes C ( A ) naturally into a dg category C dg ( A ). Recall from [ 13 , Theor em 3.2] that C ( A ) carries a s tandard pro jective C ( k )-mo del structur e. The derive d c ate gory D ( A ) of A is the lo calizatio n of C ( A ) with resp ect to the class of ob jectwise qua s i-isomor phis ms . Notation 2.2 . W e denote by perf ( A ), resp. by p erf dg ( A ), the full sub categor y o f C ( A ), resp. full dg subcateg ory of C dg ( A ), who se ob jects a re the cofibra nt r ight A - mo dules tha t ar e compac t [ 20 , Definition 4.2.7] in the tria ng ulated categ ory D ( A ). Given dg categories A a nd B their t ensor pr o duct A ⊗ B is defined as follows: the set of ob jects is the ca r tesian pro duct and given ob jects ( x, z ) and ( y , w ) in A ⊗ B , w e set ( A ⊗ B )(( x, z ) , ( y , w )) := A ( x, y ) ⊗ B ( z , w ). A A - B -bimo dule X is a dg functor X : A op ⊗ B → C dg ( k ), i.e. a r ight A op ⊗ B -mo dule. 2.2. Deriv ed Mo rita e quiv alences. A dg functor F : A → B is a called a derive d Morita e quivalenc e if its deriv ed e x tension of scalars functor L F ! : D ( A ) ∼ → D ( B ) (see [ 31 , § 3]) is a n equiv alence of triangulated ca tegories. Thanks to [ 27 , The- orem 5.3] (and [ 28 ]) the catego ry dgc at ca rries a (cofibra nt ly generated) Q uillen mo del structure whose weak equiv alences are the derived Mor ita equiv alences. W e denote by Hmo the ho mo topy categ ory he nc e obtained. The tensor pro duct of dg catego ries ca n b e derived int o a bifunctor − ⊗ L − on Hmo . Moreover, thanks to [ 31 , Theorem. 6.1] the bifunctor − ⊗ L − admits an int ernal Hom-functor rep ( − , − ). 4 Given dg catego r ies A a nd B , rep ( A , B ) is the full dg sub catego ry of C dg ( A op ⊗ L B ) spa nned by the co fibrant A - B -bimo dules X such that, for every ob ject x in A , the right B -mo dule X ( x, − ) is co mpact in D ( B ). The set o f mor phisms in Hmo from A to B is given by the set of isomorphism classes of the triangula ted categor y H 0 ( rep ( A , B )). 2.3. Exact se quences. A sequence o f triangulated categor ies 0 − → R I − → S P − → T − → 0 4 Denoted b y R Hom ( − , − ) in lo c . cit. UNIVERSAL SUSPENSION VIA NON-COMMUT A TIVE MOTIVES 5 is called exact if the c o mpo sition is zero, the functor I is fully-faithful and the induced functor from the V erdier quotient S / R to T is c ofinal , i.e. it is fully- faithful and every ob ject in T is a direc t summand o f an ob ject of S / R ; see [ 20 , § 2]. A seq ue nc e in Hmo 0 − → A X − → B Y − → C − → 0 is called exact if the induced sequence of triangula ted categor ies 0 − → D ( A ) −⊗ L A X − → D ( B ) −⊗ L B Y − → D ( C ) − → 0 is exact; see [ 13 , § 4.6]. 3. Infinite ma trix a l gebras In this section w e in tro duce the matr ix algebras used in the construction of the universal susp ensio n. Definition 3.1. Giv en n ∈ N , w e denote b y M n the k - a lgebra of n × n -matric e s with co efficients in k . Let M ∞ := ∞ [ n =1 M n be the k -algebr a of finite matric es , where M n ⊂ M n +1 via the map A 7→  A 0 0 0  . Note that M ∞ do es not hav e a unit ob ject. Moreover, tra nsp osition o f matrices gives rise to an isomorphism of k -alge bras (3.2) ( − ) T : ( M ∞ ) op ∼ − → M ∞ . Notation 3.3 . Given k , l ∈ N , w e denote by E kl ∈ M ∞ the matrix ( E kl ) i,j :=  1 if i = k and j = l 0 otherwise Note that given k , l , m, n ∈ N , the pro duct E kl · E nm equals E km if l = n and is zero otherwise. Given a non-negative integer n ≥ 0, we denote by I n ∈ M ∞ the matrix ( I n ) i,j :=  1 if i = j ≤ n 0 otherwise In particular, I 0 stands for the zero ma trix. Lemma 3. 4. The k -algebr a M ∞ has idemp o tent lo ca l units , i.e. f or e ach fin ite family A s , s ∈ S , of elements in M ∞ ther e exists an idemp oten t E ∈ M ∞ such that E · A s = A s · E = A s for al l s ∈ S . Pr o of. Since the matr ic e s A s , s ∈ S , hav e only a finite num ber of no n-zero en tries there exist natural n umbers m s , s ∈ S , such that ( A s ) i,j = 0 when i or j is greater than m s . Let m := max { m s | s ∈ S } . The n, if E is the idemp otent matrix I m we observe that I m · A s = A s · I m = A s for all s ∈ S . √ Definition 3.5. Let Γ b e the k -algebra o f N × N -ma trices A with co efficients in k and satisfying the following tw o conditions : (1) the set { A i,j | i , j ∈ N } is finite; 6 GONC ¸ ALO T ABUADA (2) there exists a natural n umber n A (whic h depe nds o n A ) such that ea ch row and each column has at most n A non-zero entries. The k -mo dule structure is defined entrywise and the multiplication is given by the ordinary ma tr ix multiplication law; note that if A, B ∈ Γ w e ca n take n A × n B as the natural num b er n A · B . In c ontrast with M ∞ , Γ do es have a unit ob ject I i,j :=  1 if i = j 0 otherwise Moreov er, tra nsp osition of matrices induces an iso mo rphism of k -algebr as ( − ) T : Γ op ∼ − → Γ which extends is o morphism ( 3.2 ). Now, let us fix a bijection θ : N ∼ − → N × N n 7→ ( θ 1 ( n ) , θ 2 ( n )) ; take for instance the inv er se of Cantor’s classical pairing function. As in [ 23 , Lemma 19], we define a k - algebra homomorphism φ : Γ − → Γ A 7→ φ ( A ) as follows φ ( A ) i,j :=  A θ 1 ( i ) ,θ 1 ( j ) if θ 2 ( i ) = θ 2 ( j ) 0 otherwise Note that the non-zero elements in line i , r esp. in column j , of the matrix φ ( A ) are precisely the non-zero elements in line i , res p. in column j , o f the matrix A . Definition 3 .6. Let W b e the Γ-Γ-bimo dule, whic h is Γ as a left Γ-mo dule, and whose right Γ-actio n is given by Γ × Γ − → Γ ( B , A ) 7→ B · φ ( A ) . Lemma 3.7. Ther e exists a natur al Γ - Γ -bimo dule isomorph ism Γ ⊕ W ∼ → W . Pr o of. Consider the elements α i,j :=  1 if θ ( j ) = ( i, 0) 0 otherwise and β i,j :=  1 if θ ( j ) = θ ( i ) + (0 , 1) 0 otherwise in Γ. Using α and β , we define maps : Γ ⊕ W − → W ( A, B ) 7→ A · α + B · β (3.8) W − → Γ ⊕ W B 7→ ( B · α T , B · β T ) . (3.9) The map ( 3.8 ) is a left Γ-mo dule homomor phism. The fact that it is a lso a right Γ-mo dule homomorphism follows fr om the following equalities : β · α T = 0 α · α T = β · β T = I α T · α + β T · β = I . Moreov er, since for ev ery A ∈ Γ w e hav e A · α = α · φ ( A ) and φ ( A ) · β = β · φ ( A ) we conclude that the maps ( 3.8 ) and ( 3.9 ) ar e inv er se of each other. √ UNIVERSAL SUSPENSION VIA NON-COMMUT A TIVE MOTIVES 7 Notation 3.10 . Clearly the k - a lgebra M ∞ forms a tw o-sided idea l in Γ. W e denote by Σ the a sso ciated quotient k -a lgebra Γ / M ∞ . Alternatively , w e can describ e the q uotient k -a lg ebra Σ as follows. Prop ositi o n 3.11. The m atric es I n := I − I n (see Notation 3.3 ) form a left denominator set S in Γ [ 19 , § 4] , i.e. I ∈ S , S · S ⊂ S and (i) given I n ∈ S and E ∈ Γ , ther e ar e I m ∈ S and E ′ ∈ Γ such that E ′ · I n = I m · E ; (ii) if I n ∈ S and E ∈ Γ satisfy E · I n = 0 , ther e is I m ∈ S such that I m · E = 0 . Mor e over, t he lo c alize d k -algebr a Γ[ S − 1 ] 5 is natur al ly isomorph ic t o Σ . Pr o of. In or de r to simplify the pro of we cons ide r the following blo ck-matrix g raph- ical notation E = k l  E a E b E c E d  ∈ Γ , where k , l ∈ N , E a is a k × l -matr ix, E b is a k × N -matr ix, E c is a N × l -matr ix, and E d is a N × N -matrix. Under this nota tio n w e hav e, for n ∈ N , the equalities (3.12) I n · n n  E a E b E c E d  = n n  0 0 E c E d  and (3.13) n n  E a E b E c E d  · I n = n n  0 E b 0 E d  . By definition I = I 0 ∈ S . Equalities ( 3.1 2 ) and ( 3.1 3 ), a nd the fact that I 0 = I , imply that I n · I m = I max { n,m } n ≥ 0 . This shows that S · S ⊂ S . (i) Note first that whe n n = 0 the cla im is trivia l. Since E b elongs to Γ, there exist natural n umbers m j , 1 ≤ j ≤ n , such that E i,j = 0 for i ≥ m j and 1 ≤ j ≤ n . T ake m = ma x { n, m j | 1 ≤ j ≤ n } . Then, w e ha ve the following equality (3.14) I m · m n  E a E b E c E d  = m n  0 0 0 E d  . Since m ≥ n , the ab ove equality ( 3.13 ) s hows us that we can tak e for E ′ the above matrix ( 3.14 ). T his pr ov es the cla im. (ii) When n = 0 the claim is trivial. If E · I n = 0 the ab ov e equality ( 3 .13 ) s hows us that (3.15) E = n n  E a 0 E c 0  . 5 Since S is a left denominator set this k -algebra is given b y lef t fr actions, i. e. equiv alence classes of pairs ( I n , E ) mo dulo the relation whic h iden tifies ( I n , E ) with ( I m , E ′ ) if there are B , B ′ ∈ Γ suc h that B · I n = B ′ · I m belongs to S and B · E = B ′ · E ′ . 8 GONC ¸ ALO T ABUADA Since E belo ng s to Γ, there exis t natural num b er s m j , 1 ≤ j ≤ n , s uch that E i,j = 0 for i ≥ m j and 1 ≤ j ≤ n . T a ke m = max { m j | 1 ≤ j ≤ n } . Then, the a b ov e description ( 3.15 ) combined with equality ( 3.1 3 ) show us that I m · E = 0. This prov es the claim. W e now show that the loca lized k -algebr a Γ[ S − 1 ] is naturally isomo rphic to Σ. Since the matrices I n = I − I n n ≥ 0 belo ng to M ∞ , we conclude that all the elements of the set S b eco me the iden tit y ob ject in Σ. Therefor e, by the universal prop er ty o f Γ[ S − 1 ] we obta in a k - algebra map (3.16) Γ[ S − 1 ] − → Σ . On the o ther hand, the k ernel of the lo calization map Γ → Γ [ S − 1 ] consists of those matrices E ∈ Γ for whic h I n · E = 0 for some n ≥ 0 . Thanks to equalit y ( 3.12 ) w e observe tha t the elements of M ∞ satisfy this condition. Therefore, by the univ ersal prop erty of Σ := Γ / M ∞ we o btain a k -algebr a map (3.17) Σ − → Γ[ S − 1 ] . The maps ( 3.16 ) a nd ( 3.17 ) are clear ly inv erse of each other a nd so the pro o f is finished. √ Lemma 3.18 . The algebr as M ∞ , Γ and Σ ar e flat as k -mo dules. Pr o of. W e sta rt by proving this prop o sition in the particular case where the base ring k is Z . In this case the underlying Z -modules of ( M ∞ ) Z and Γ Z are torsio nfree and so by [ 33 , Corollar y 3.1.5], they ar e flat. Tha nks to Prop os itio n 3.11 , Σ Z ident ifies with the (left) lo c alization of Γ Z with r esp ect to the set S , a nd so a standard argument shows us that the right Γ Z -mo dule Σ Z is flat. Since Γ Z is flat as a Z -mo dule, we conclude that Σ Z is also flat as a Z -mo dule. Let us now co nsider the gener al cas e . Clearly w e have a natural iso mo rphism of k -mo dules ( M ∞ ) Z ⊗ Z k ∼ − → ( M ∞ ) k . Thanks to [ 4 , Lemma 4.7 .1 ], we hav e a lso natural isomorphisms of k - mo dules Γ Z ⊗ Z k ∼ − → Γ k and Σ Z ⊗ Z k ∼ − → Σ k . Therefore, since flat mo dules are stable under extensio n of s c a lars, the pro of is achiev ed. √ 4. An exact sequence Let H b e a k -algebra and J ⊂ H a tw o-sided ideal. Definition 4. 1. The c ate gory J of idemp otents of J is defined as follows : its ob jects are the sym bo ls u , where u is an idempotent of J ; the k - mo dule J ( u , u ′ ) of morphisms fro m u to u ′ is uJ u ′ ; co mpo sition is given b y multiplication in J and the unit o f each ob ject u is the idemp otent u . Asso cia ted to H and J there is also a J - H -bimo dule X such that X ( u , ∗ ) := uJ , w ith left and right actions given b y m ultiplication. UNIVERSAL SUSPENSION VIA NON-COMMUT A TIVE MOTIVES 9 Recall fr om [ 16 , Example 3.3 (b)] that if H a nd J are flat as k -mo dules a nd J has idemp otent lo c al u n its ( i.e. for each finite family a s , s ∈ S , of elements in J there exists an idemp otent u ∈ J such that ua s = a s u = a s for all s ∈ S ) w e have a exact sequence in Hmo 0 − → J X − → H − → H/ J − → 0 . Thanks to Lemmas 3.4 and 3.18 , if we take H = Γ and J = M ∞ , we then obtain then the following exa ct sequence in Hmo (4.2) 0 − → M ∞ X − → Γ − → Σ − → 0 . Prop ositi o n 4.3. The dg funct or k − → M ∞ ∗ 7→ E 11 ( se e N otation 3.3 ) (4.4) is a derive d Morita e quivalenc e. Pr o of. W e will prove a stronger statement, namely that the ab ove functor ( 4.4 ) is a Mor ita equiv alence; see [ 24 , § 2 ]. The category M ∞ is by definition enriched ov er k -mo dules and the classica l theory of Morita holds in this setting. Let Mo d- M ∞ be the ab e lia n category of rig ht M ∞ -mo dules ( i.e. co nt rav ariant k - linear functor s from M ∞ to k -mo dules) and d ( − ) : M ∞ − → Mo d- M ∞ E 7→ M ∞ ( − , E ) =: b E the (enriched) Y oneda functor . F ollowing [ 24 , Theorems 2.2 a nd 2.5 ], we need to show that d E 11 is a small pro jective gener ator of Mo d- M ∞ and that its ring of endomorphisms is isomo rphic to k . Note that w e hav e natural iso morphisms Hom Mo d- M ∞ ( d E 11 , d E 11 ) ≃ M ∞ ( E 11 , E 11 ) = E 11 · M ∞ · E 11 ≃ k . Moreov er, d E 11 is s mall a nd pro jective by definition. Therefor e, it only rema ins to show that d E 11 is a generator , i.e. , that every right M ∞ -mo dule P is an epimo rphic image of a sum of (pos sibly infinitely many) copies of d E 11 . Given an ob ject E in M ∞ we have, by the (enr iched) Y oneda lemma , an isomorphism Hom Mo d- M ∞ ( b E , P ) ≃ P ( E ) and so we obtain a natural epimorphism L E ∈M ∞ L P ( E ) b E / / / / P . This shows that it suffices to treat the case where P is of shap e b E . W e consider first the cases E = E nn , n ∈ N . The following morphisms in M ∞ E 11 E 11 · E 1 n · E nn / / E nn and E nn E nn · E n 1 · E 11 / / E 11 show us that E 11 and E nn are isomor phic a nd so the claim follows. W e cons ider now the ca ses E = I m , m ∈ N . The natura l morphisms in M ∞ E nn E nn · E nn · I m / / I m 1 ≤ n ≤ m give rise to a map in Mod- M ∞ (4.5) m M n ≥ 1 d E nn − → c I m . 10 GONC ¸ ALO T ABUADA In order to show that the map ( 4.5 ) is sur jective, we need to show tha t its ev aluation (4.6) m M n ≥ 1 M ∞ ( B , E nn ) − → M ∞ ( B , I m ) at each ob ject B o f M ∞ is s ur jective. W e have I m = P m n ≥ 1 E nn , and s o ( 4.6 ) ident ifies with the natura l map m M n ≥ 1 ( B · M ∞ · E nn ) − → B · M ∞ ·   m X n ≥ 1 E nn   , which is easily seen to b e s ur jective. Since E 11 is iso morphic to E nn , n ∈ N , the claim is prov ed. Finally , we consider the case of a general ob ject E in M ∞ . Since E has only a finite num ber of non-zero entries there exists a natura l num b er m such that E i,j = 0 when i or j is greater than m . W e have then the equality E · I m = I m · E = E . This implies that the co mpo sition E E · I m · I m / / I m I m · I m · E / / E equals the ident ity ma p of the ob ject E . Since the a belia n categor y Mo d- M ∞ is idempo ten t co mplete, we conclude that the right M ∞ -mo dule b E is a direct fac to r of c I m . This achieves the pro o f. √ By co m bining the exact sequence ( 4.2 ) with the derived Mor ita equiv alence ( 4.4 ) we obtain an exac t sequence in Hmo (4.7) 0 − → k − → Γ − → Σ − → 0 . Notation 4 .8 . Given a dg ca teg ory A , we denote by Γ( A ) the dg catego ry Γ ⊗ A and b y Σ( A ) the dg ca tegory Σ ⊗ A ; see § 2.1 . Prop ositi o n 4.9. F or every dg c ate gory A we have an exact se quenc e in Hmo 0 − → A − → Γ( A ) − → Σ( A ) − → 0 . Pr o of. The exact seq ue nc e ( 4.7 ) and [ 7 , Pro po sition 1.6.3] lea d to the exact sequence in Hmo 0 − → k ⊗ L A − → Γ ⊗ L A − → Σ ⊗ L A − → 0 . By Lemma 3.18 the algebras Γ and Σ are flat as k -mo dules and so the derived tensor pro ducts are identified in Hmo with the ordinar y ones. Mor eov er, we hav e a natural isomorphism k ⊗ L A ≃ A . √ 5. Flasqueness of Γ Definition 5 . 1. Let A b e a dg categ ory with sums ( i.e. the diagona l dg functor ∆ : A → A × A admits a left a djo int ⊕ : A × A → A ) such that Z 0 ( A ) is equiv alent to pe rf ( A ); see Notations 2.1 and 2.2 . Under these hypothes is , we say that A is flasque if there exists a dg functor τ : A → A and a natura l iso morphism Id ⊕ τ ≃ τ . Prop ositi o n 5.2. The dg c ate gory p erf dg (Γ) is flasque. UNIVERSAL SUSPENSION VIA NON-COMMUT A TIVE MOTIVES 11 Pr o of. Notice first that since we hav e a n isomorphism of k -alg ebras ( − ) T : Γ op ∼ − → Γ it is equiv alent to show that the dg category p erf dg (Γ op ) is flasque. By definition, p erf dg (Γ op ) has sums and Z 0 ( pe rf dg (Γ op )) is eq uiv alent to p erf (Γ op ). Now, recall from Definition 3.6 the constructio n o f the Γ-Γ-bimodule W . As explained in [ 1 3 , § 3.8] the bimo dule W gives rise to a Q uillen adjunction C (Γ op )   C (Γ op ) , W ⊗ Γ − O O which is more over c ompatible with the C ( k )-enrichmen t. Since the Γ-Γ-bimodule W is Γ as a left Γ-mo dule, the left Quillen dg functor W ⊗ Γ − : C dg (Γ op ) − → C dg (Γ op ) restricts to a dg functor τ : p erf dg (Γ op ) − → p erf dg (Γ op ) . Moreov er, g iven an ob ject P in perf dg (Γ op ) we have a functorial iso morphism P ⊕ τ ( P ) = P ⊕ ( W ⊗ Γ P ) ≃ (Γ ⊕ W ) ⊗ Γ P ψ ∼ / / W ⊗ Γ P = τ ( P ) , where ψ is obtained by tensoring the Γ-Γ-bimo dule isomo r phism Γ ⊕ W ∼ → W of Lemma 3.7 with P . T his a chiev es the pro of. √ Lemma 5. 3. L et A and B b e two dg c ate gories, with A flasque. Then the dg c ate gory rep ( B , A ) (se e § 2.2 ) is also flasque. Pr o of. By construction, the dg catego ry rep ( B , A ) has sums and Z 0 ( rep ( A , B )) is equiv alent to p erf ( rep ( A , B )). Moreov e r , s ince A is flasque and rep ( B , − ) is a 2- functor which pr eserves (derived) pro ducts, we obtain a dg functor rep ( B , τ ) : rep ( B , A ) − → rep ( B , A ) and a natural isomor phism Id ⊕ rep ( B , τ ) ≃ rep ( B , τ ). √ Let us now recall the definition of the a lg ebraic K -theo ry of dg ca tegories . Given a dg category A we denote by p erf W ( A ) the W aldhausen ca tegory p erf ( A ), whose weak equiv alences and c o fibrations a re those of the Q uillen mo del structur e on C ( A ); se e [ 8 , § 3]. The algebr aic K - the ory sp e ctru m K ( A ) of A is the W aldhausen’s K - theory sp ectr um [ 32 ] of p erf W ( A ). Given a dg functor F : A → B , the ex- tension of s c a lars left Quillen functor F ! : C ( A ) → C ( B ) preser ves weak e q uiv- alences, cofibr a tions, and pushouts. Therefo r e, it restricts to a n exac t functor F ! : p erf W ( A ) → p erf W ( B ) betw een W aldhausen categorie s a nd s o it gives r ise to a morphism of sp ectra K ( F ) : K ( A ) → K ( B ). Lemma 5. 4. L et A b e a flasque dg c ate gory. Then, its algebr aic K -t he ory sp e ctru m K ( A ) is c ontr actible. 12 GONC ¸ ALO T ABUADA Pr o of. By applying the functor Z 0 ( − ) to A and τ we obta in an exact functor Z 0 ( τ ) : p erf W ( A ) → p erf W ( A ) and a natural isomo rphism Id ⊕ Z 0 ( τ ) ≃ Z 0 ( τ ). Since W aldhausen’s K -theory satisfies additivity [ 3 2 , Prop os ition 1 .3.2(4)] w e hav e the following equality K (Id) + K ( Z 0 ( τ )) = K ( Z 0 ( τ )) in the ho motopy catego ry of sp ectra . Therefore , we conclude that Id K ( A ) ≃ K (Id) is the trivial map. This shows that the algebraic K -theory sp ectr um K ( A ) is contractible. √ 6. Proof of Theorem 1.2 W e star t b y showing that Γ( A ) (see Notation 4.8 ) b eco mes the zero ob ject in the triangulated category Mot lo c dg ( e ) after a pplication of U lo c dg . Since Γ( A ) ≃ Γ ⊗ L A and U lo c dg is symmetric mono ida l with resp ect to a homotopy co limit pr eserving symmetric monoidal structure on Mot lo c dg (see [ 6 , Theor em 7.5]), it suffices to show that Γ bec omes the zero o b ject in Mot lo c dg ( e ). Recall from [ 26 , § 17] that the universal lo calizing inv a riant a dmits the follo wing factorization (6.1) U lo c dg : HO ( dgcat ) U add dg − → Mot add dg γ − → Mot lo c dg , where Mot add dg is the additive motiv ator 6 and γ is a lo calizing mor phism be tw een tri- angulated deriv ators. Moreov er, thanks to [ 5 , P rop ositio n 3 .7] the o b jects U add dg ( B )[ n ], with B a dg cell and n ∈ Z , for m a se t of (compa ct) generato rs of the tria ngu- lated category Mot add dg ( e ). Ther e fore, U add dg (Γ ) is the zero ob ject in Mot add dg ( e ) if and only the sp ectra of morphisms R Hom ( U add dg ( B ) , U add dg (Γ)), with B a dg cell, is (ho- motopically) trivial; see [ 5 , § A.3]. By [ 26 , Theo rem 1 5.10] w e hav e the following equiv alence s R Hom ( U add dg ( B ) , U add dg (Γ)) ≃ K rep ( B , Γ) ≃ K rep ( B , pe rf dg (Γ)) . Therefore, Prop os ition 5.2 and Lemmas 5.3 and 5.4 imply that U add dg (Γ) is the zer o ob ject in Mot add dg ( e ). Thanks to the ab ov e factoriza tion ( 6.1 ) we co nclude that U lo c dg (Γ ) (and so U lo c dg (Γ( A ))) is the zero o b ject in Mot lo c dg ( e ). Now, reca ll from Prop ositio n 4.9 that we ha ve an exact seque nc e 0 − → A − → Γ( A ) − → Σ( A ) − → 0 . By applying the universal loca lizing in v ariant to the preceding exact sequenc e we obtain a distinguished tria ngle U lo c dg ( A ) − → U lo c dg (Γ( A )) − → U lo c dg (Σ( A )) − → U lo c dg ( A )[1] in Mot lo c dg ( e ). 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