Maltsiniotiss first conjecture for K_1

MAL TSINIOTIS’S FIRST CONJECTURE F OR K 1 FERNANDO MURO Abstract. W e sho w that K 1 ( E ) of an exa ct category E agrees with K 1 ( D E ) of the associated triangulated deriv ator D E . More generally w e show tha t K 1 ( W ) of a W aldhausen category W with cylinders and a saturated class of weak equiv alenc es agrees with K 1 ( D W ) of the asso ciated ri gh t pointed deriv ator D W . Introduction F or a lo ng time there w as a n int erest in defining a nice K -theory for tria ngulated categorie s such that Quillen’s K -theory o f an exact category E agrees with the K - theory of its b ounded derived categor y D b ( E ). Schlic h ting [Sch02 ] showed that such a K -theo ry for tr iangulated categories ca nno t exist. It w as then natura l to ask ab out the definition of a nice K -theory for alg ebraic structures in terp olating betw een E and D b ( E ). The b est kno wn intermediate structure is C b ( E ), the W aldhausen ca tegory of bo unded complexes in E , with qua si-isomorphisms as weak equiv a lences and cofi- brations g iv en by c hain morphisms which are levelwise admissible mono morphisms. The derived catego r y D b ( E ) is the lo ca lization of C b ( E ) with res p ect to weak equiv- alences. The Gillet-W aldhausen theorem 1 , relating Quillen’s K -theory to W ald- hausen’s K -theory , states that the homomorphisms τ n : K n ( E ) − → K n ( C b ( E )) , n ≥ 0 , induced b y the inclusion E ⊂ C b ( E ) of co mplexes co ncent rated in degr e e 0, a r e isomorphisms. The category C b ( E ) is considered to b e too close to E so one would still lik e to find an alg e braic stucture with a nice K -theory in terp olating betw een C b ( E ) and D b ( E ). The notion o f a tria ngulated deriv ator [Gro 90, Mal07] seems to be a str ong candidate. Maltsiniotis [Mal07] defined a K -theory for tria ng ulated deriv ator s together with natural homomorphisms ρ n : K n ( E ) − → K n ( D E ) , n ≥ 0 , 1991 M athematics Subje ct Classific ation. 18E1 0, 18E30, 18F25, 19B99, 55S 45. Key wor ds and phr ases. K -theory , exact category , triangulated deriv ator, Postnik o v in v arian t, stable quadratic mo dule. The author was partially s upp orted b y the Spanish M i nistry of Education and Science under MEC-FEDER grants MTM2004-03629 and MTM 2007-63277 , and a Juan de la Cierv a r esearc h con tract. 1 The proof due to Thomason-T robaugh [ TT90, Theorem 1.11.7] corr ects Gillet’s [ Gil81, 6.2] and uses an extra hypothesis on E . This h yp othesis is not strictly necessary , since the general case follows then f rom cofinality arguments, see [Cis02]. 1 2 FERNANDO MURO where D E is the triangula ted deriv ator asso c iated to an exact category E , con- structed by K eller in the app endix of [Ma l07]. Cisinski and Neeman pr ov ed the additivity of triangula ted deriv ator K -theory [CN05]. Maltsiniotis also conjectured that ρ n is an isomorphism for all n . He succe e ded in pro ving the conjecture for n = 0. The following theor em is the main result of this pap er. Theorem A. L et E b e an exact c ate gory. The natu r al homomorph ism ρ 1 : K 1 ( E ) ∼ = − → K 1 ( D E ) is an isomorphism. In order to obtain Theorem A we use tech niques introduced in [MT07]. Ther e we give a presentation of an ab elian 2-group D ∗ W which enco des K 0 ( W ) and K 1 ( W ) of a W aldhausen category W , and moreov er the 1-t yp e of the K -theory spectrum K ( W ) whose homotopy gro ups a r e the K -theory groups of W . This presentation is a higher dimensional analogue of the c lassical presentation of K 0 ( W ). Here we similarly define an a belia n 2-gro up D der ∗ W which mo dels the 1-type of the K -theory sp ectrum K ( D W ) o f the right 2 po in ted der iv ator D W asso ciated to a W aldhausen category W with cylinders and a s aturated cla ss of weak equiv alences, such as W = C b ( E ). The K -theor y for this kind o f deriv a tors, more genera l than triang ulated deriv a tors, was defined b y Garkusha [Gar06] extending the work of Maltsiniotis [Mal07]. The r e are defined compariso n homo morphisms µ n : K n ( W ) − → K n ( D W ) , n ≥ 0 . These homomorphisms ca nnot b e isomorphisms in general, as sho wn in [TV04]. Nevertheless we here pro ve the following result. Theorem B. L et W b e a Wald hausen c ate gory with cyli nders and a satur ate d class of we ak e quivalenc es. Th e natur al homomorphism µ 0 : K 0 ( W ) ∼ = − → K 0 ( D W ) , µ 1 : K 1 ( W ) ∼ = − → K 1 ( D W ) , ar e isomorphi sms. In Remark 5.3 we c o mmen t on the case wher e the hypothesis on the satur ation of weak equiv alences is replaced b y the 2 o ut of 3 axio m, which is a weaker assumption. Theorem A is a ctually a corollar y o f the Gillet-W aldhausen theorem and Theo- rem B, since D C b ( E ) = D E and the natur al homomorphisms ρ n factor as ρ n : K n ( E ) τ n − → K n ( C b ( E )) µ n − → K n ( D E ) , n ≥ 0 . W e assume the rea der certa in familiarity with exact, W a ldha usen and derived categorie s, with simplicia l constructions and with homotopy theor y . W e refer to [W ei, GM03, GJ9 9] fo r the basics. Ac kno wledgements. I am very grateful to Grigory Garkusha for suggesting the po ssibility of using [MT07] in order to tackle Maltsiniotis’s first co njecture in di- mension 1. I also feel indebted to Denis- Cha rles Cisinski, who kindly indica ted how to extend the results of a preliminary v ersion of this pap er to a broader generality . 2 The ref erences [Gro90, Cis03] and [Gar06, RB07 ] foll ow a different conv en tion with respect to sides. Here w e foll o w the conv en tion in [Gro90, Cis03], so what we call a ‘r igh t p oin ted deriv ator’ is the same as a ‘l eft p oin ted deriv ator’ in [Gar06 ]. MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 3 1. The bounded derived ca tegor y of an exact ca tegor y In this sectio n we o utline the t wo-step co nstruction o f the derived ca tegory D b ( E ) of an ex act category E . This constructio n is a sp ecial case of the homotopy categor y Ho W of a W a ldhausen catego ry W with cylinders satisfying the 2 o ut of 3 axiom, D b ( E ) = Ho C b ( E ). Definition 1.1. A Waldhausen c ate gory is a category W with a distinguished zero ob ject 0 and tw o distinguis hed sub categories w W and c W , whos e morphisms are called c ofibr ations and we ak e quivalenc es , resp ectively . A morphism which is both a weak equiv alence a nd a cofibration is said to b e a trivial c ofibr ation . The arrow ֌ stands for a cofibration and ∼ → for a weak equiv alence. • All morphisms 0 → A are cofibrations. All iso morphisms are cofibrations and weak equiv alences. • The push-out of a morphism along a cofibra tion is alw ays defined A / / / /   push B   X / / / / X ∪ A B and the low er map is also a cofibration. • Giv en a commutativ e diagram X ∼   A ∼   o o / / / / B ∼   X ′ A ′ o o / / / / B ′ the induced map X ∪ A B ∼ → X ′ ∪ A ′ B ′ is a weak equiv alence. Notice that copro ducts A ∨ B = A ∪ 0 B are defined in W . A functor W → W ′ betw een W aldhausen ca tegories is exact if it pr eserves cofi- brations, w eak eq uiv alences, push-outs along cofibr ations and the distinguished zero ob ject. Example 1.2 . Recall that an exact c ate gory E is a full sub categor y of an ab elian category A such that E co n tains a zero ob ject of A and E is clo sed under extensions in A . A short exact se quenc e in E is a short exact sequence in A b etw een o b j ects in E . A mor phism in E is a n admissible monomorphism if it is the initial morphism of s ome s hort exa ct seque nce . The category E is a W aldhausen categor y with admissible monomo rphisms as co fibrations and is omorphisms as weak equiv alences. In order to complete the structure we fix a zero o b j ect 0 in E . W e denote by C b ( E ) the category of b ounded complexes in E , · · · → A n − 1 d − → A n d − → A n +1 → · · · , d 2 = 0 , A n = 0 for | n | ≫ 0 . A c hain morphism f : A → B in C b ( E ) is a quasi-isomo rphism if it induces an iso- morphism in ho mology computed in the ambien t a belia n ca tegory A . The categ ory C b ( E ) is a W aldha us en categ ory . W ea k equiv alences are quasi-isomor phisms and cofibrations are levelwise a dmissible monomorphisms. There is a full exact inclusion of W aldhausen catego ries E ⊂ C b ( E ) sending an ob ject X in E to the complex · · · → 0 → X → 0 → · · · , 4 FERNANDO MURO with X in degree 0. Definition 1.3. The homotopy c ate gory Ho W of a W aldhausen categ o ry is a cat- egory equipp ed with a functor ζ : W − → Ho W sending weak equiv alences to isomorphisms. Mor eov er, ζ is initial amo ng a ll func- tors W → C sending weak equiv ale nc e s to isomor phisms, so Ho W is well de fned up to ca nonical iso morphism ov er W . This category ca n b e cons tr uctued as a categor y of fractions, in the sense of [GZ67], by for mally in verting w eak equiv alences in W . The class of weak equiv alences is satu r ate d if any morphism f : A → B in W such that ζ ( f ) is a n isomorphism in Ho W is indeed a weak equiv a lence f : A ∼ → B . Example 1.4 . W eak equiv alence s in C b ( E ), i.e. quas i-isomorphisms, are s a turated since they are detected by a functor H ∗ : C b ( E ) → A Z , the cohomology functor from b ounded complexes in E to Z -graded ob jects in A , see [CF00 , Prop osition 1.1]. The homo to p y category alwa ys exists up to set theoretical difficulties which do not a rise if W is a sma ll categ ory , for instance. This is not a harmful assump- tion if one is in terested in K -theory since smallness may also b e required in order to ha ve well defined K -theor y gr oups. The homotopy catego ry can howev er b e constructed in a more straighforward w ay if the W aldha usen category W sa tisfies further prop erties. Definition 1. 5 . A W aldhausen catego ry W satisifies the 2 out of 3 axiom provided given a co mm utative diagr a m in W C A @ @        / / B ^ ^ = = = = = = = if t wo arrows are w eak equiv alences then the third one is also a weak equiv alence. Given an o b ject A in W a cylinder I A is an ob ject together with a factorization of the folding map (1 , 1 ) : A ∨ A → A as a cofibration follow ed by a weak eq uiv alence, A ∨ A ֌ i I A ∼ − → p A. W e say that W has cylinders if all ob j ects hav e a cylinder. Example 1.6 . The W aldhausen category C b ( E ) has cylinders. The cylinder of a bo unded complex A can be funct oria lly c hosen as ( I A ) n = A n ⊕ A n +1 ⊕ A n , d =   d − 1 0 0 − d 0 0 1 d   : ( I A ) n − → ( I A ) n +1 . R emark 1.7 . The 2 out of 3 axiom is often called the saturation axiom. W e do no t use this terminology in this pap er in order to av oid confusion with Definition 1.3. Usually o ne considers more structure d cylinders in W aldhausen ca tegories, com- pare [W ei, Definition IV.6.8 ]. F or the purp oses of this pap er it is enoug h to consider cylinders as defined ab ov e. MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 5 R emark 1.8 . As one can easily chec k, a W aldha usen categor y with a s a turated class of w eak equiv alences satisfies the 2 out of 3 axiom. This applies to C b ( E ). A W aldhausen category with cylinder s W sa tisfying the 2 out of 3 axio m is an example of a right derivable c ate gory , in the sense of [Cis03], also called pr e c ofi- br ation c ate gory in [RB07], see [Cis03, E xample 2.2 3] or [RB07, Prop osition 2.4.2]. In particular any mo rphism in W can b e fac to red as a cofibration followed b y a weak equiv a le nce which is left in verse to a trivia l cofibr ation, see [RB0 7, Pr op osi- tion 1.3 .1]. Moreov er, one can define a homo top y rela tion in W and co nstruct the homotopy category Ho W by a homotopy calculus of left fra ctions as w e indicate below, see [Cis03, Section 1] or [RB07, Section 5.4 ]. Let W b e a W aldhausen category with cylinders satisfying the 2 out of 3 axio m. As usual we say that tw o mo rphisms f , g : A → B in W are strictly homotopic if there is a morphism H : I A → B with H i = ( f , g ). The maps f , g are homotopi c f ≃ g if there exists a weak equiv alence h : B ∼ → B ′ such that hf and hg are strictly homotopic. ‘B eing homotopic’ is a natura l equiv alence relation a nd the quo tien t category is denoted by π W . The homotop y category Ho W is obtained b y calculus of left frac tions in π W . Ob jects in Ho W ar e the same as in W . A morphism A → B in Ho W is re pr esented by a diagram in W , A − → α 1 X ∼ ← − α 2 B . Another diagram A − → α ′ 1 Y ∼ ← − α ′ 2 B represents the s a me morphism if there is a diagram in W X > > α 1 } } } } } } } ` ` ∼ α 2 A A A A A A A A Z / / o o ∼   O O B Y α ′ 1 A A A A A A A ~ ~ ∼ α ′ 2 } } } } } } } whose pro jection to π W is commutativ e. Notice that, by the 2 out of 3 axio m, the vertical arr ows in this diagram a re also w eak eq uiv alences. The compo site of t wo morphisms A → α B → β C in Ho W represent ed b y A − → α 1 X ∼ ← − α 2 B − → β 1 Y ∼ ← − β 2 C is defined as follows. If β 1 is a cofibration then the push-out B push / / β 1 / / ∼ α 2   Y ∼ ¯ α 2   X / / ¯ β 1 / / X ∪ B Y is defined, ¯ α 2 is a weak equiv alence, and β α : A → C is repr esented by A − → ¯ β 1 α 1 X ∪ B Y ∼ ← − ¯ α 2 β 2 C. 6 FERNANDO MURO In general we can factor β 1 as cofibration follow ed by a weak e quiv alence β 1 : B ֌ β ′ 1 Z ∼ − → r Y such that there is a mor phism s : Y ∼ ֌ Z with rs = 1 Y . The dia gram Y > > β 1 ~ ~ ~ ~ ~ ~ ~ ` ` ∼ β 2 @ @ @ @ @ @ @ B Z / / β ′ 1 / / o o ∼ sβ 2 r O O C commutes in W , s o β : B → C is also repr esented by B ֌ β ′ 1 Z ∼ ← − sβ 2 C, where the first ar row is a cofibratio n, and we can use this repres e ntative to define the comp osite β α : A → C . The functor ζ : W − → Ho W is the iden tit y on ob jects a nd sends a mo r phism f : A → B to the morphism ζ ( f ) : A → B represented b y A − → f B ∼ ← − 1 B B . If f : A ∼ → B is a w eak equiv a lence then ζ ( f ) is an isomorphism and ζ ( f ) − 1 is represented by B − → 1 B B ∼ ← − f A, hence a morphism α : A → B in Ho W repr e sent ed by A − → α 1 X ∼ ← − α 2 B coincides with ζ ( α 2 ) − 1 ζ ( α 1 ) = α . R emark 1.9 . If α ab ov e is an isomorphism in Ho W then ζ ( α 1 ) = ζ ( α 2 ) α is als o an isomorphism. In particula r if W has a saturated class of weak equiv a lences then α 1 : A ∼ → X is neces sarily a w eak equiv alence. R emark 1.10 . F or W = C b ( E ) the categor y π W = H b ( E ) is usually termed the b oun de d homotopy c ate gory , while Ho W = D b ( E ) is called the b ounde d derive d c ate gory of E . 2. On W aldhausen and derived K -theor y Recall that a c ofib er se quenc e in a W aldhausen catego ry W A ֌ B ։ B / A is a push-out diagra m A / / / /   push B     0 / / / / B / A Therefore the quo tient B / A is o nly defined up to canonical isomo rphism ov er B , although the notation B / A is standar d in the literature. MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 7 The K -theories w e deal with in this paper a re constructed b y using the W a ld- hausen categorie s S n W that we now recall. Definition 2.1. An ob ject A •• in the category S n W , n ≥ 0, is a comm utative diagram in W (2.2) A nn . . . O O A 22 / / · · · / / A 2 n O O A 11 / / A 12 / / O O · · · / / A 1 n O O A 00 / / A 01 / / O O A 02 / / O O · · · / / A 0 n O O such that A ii = 0 and A ij ֌ A ik ։ A j k is a cofib er sequence for all 0 ≤ i ≤ j ≤ k ≤ n . Notice that these conditio ns imply that the whole diagram is determined, up to canonical isomorphism, by the s e quence of n − 1 comp osable cofibr a tions (2.3) A 01 ֌ A 02 ֌ · · · ֌ A 0 n . A mo rphism A •• → B •• in S n W is a na tural transfor mation b et ween diagra ms given by morphisms A ij → B ij in W . The catego ry S n W is a W a ldhausen catego ry . A mo rphism A •• ∼ → B •• is a weak equiv alenc e if all morphisms A ij ∼ → B ij are weak equiv alenc e s in W . A cofibration A •• ֌ B •• is a mor phism such that A ij ֌ B ij and B ij ∪ A ij A ik ֌ B ik are cofibra tio ns, 0 ≤ i ≤ j ≤ k ≤ n . The distinguished zero ob ject is the dia g ram with 0 in all en tries. The catego ries S n W assemble to a simplicial category S. W . The face functor d i : S n W → S n − 1 W is defined by removing the i th row and the i th column, and the degeneracy functor s i : S n W → S n +1 W is defined b y duplicating the i th row and the i th column, 0 ≤ i ≤ n . F aces and degener acies are exact functors. F or the definition of the simplicia l str ucture it is crucial to consider the whole diagram (2.2) instead of just (2.3). One c an obtain a p ointed space out of the simplicial categor y S. W as follo ws. W e restr ict to the sub categor ies of weak equiv alences w S. W , then we take lev elwise the nerv e in or de r to get a bis implicial set Ner w S. W , w e consider the diago nal simplicial set Diag Ner w S. W , and its g eometric realization | Diag Ner wS. W | . This p ointed space, actually a reduced C W -complex , is the 1-sta ge of the Wald- hausen K -the ory spectr um K ( W ) [W al85], which is an Ω-sp ectrum, hence the K - theory groups of W are the homotopy g roups K n ( W ) = π n +1 | Diag Ner w S. W | , n ≥ 0 . 8 FERNANDO MURO W e now assume that W has cylinders and satisifies the 2 out of 3 axiom, so that the ass o ciated r ig h t pointed deriv ator D W is defined, see [Cis03, Corollar y 2.24 and the duals of Lemmas 4 .2 a nd 4.3]. Then the W a ldha usen categories S n W also hav e cylinder s and satisfy the 2 out of 3 axiom. W e will neither recall the notion o f deriv ator nor the de finitio n of the der iv ator D W but just the K -theory o f D W , we refer the interested reader to [Gro 9 0, Mal07, Gar0 6, RB07]. F or this w e consider the homotopy catego ries Ho S n W and the subgroup oids of isomorphisms i Ho S n W . These group oids for m a simplicia l g roup oid i Ho S. W and we can co nsider the po in ted space | Diag Ner i Ho S. W | , which is the 1 -stage of Garkusha’s derive d K -the ory Ω-sp ectrum D K ( W ). Garkusha [Gar0 5] considers derived K -theor y for W = C b ( E ), and mo re g enerally for W a nice complicial biW aldhause n categor y , although the definition immediately extends to W aldha usen categorie s with cy linders sa tisfying the 2 out o f 3 axiom, a s indicated her e. Moreover, Ga rkusha shows that there is a natural weak equiv a le nce D K ( W ) ∼ → K ( D W ) b et ween the derived K -theory s pectr um of a nic e co mplicial biW aldhausen category W and the K -theory sp ectrum of the asso ciated deriv ator D W , compare [Gar05, Corollar y 4.3]. Nevertheless [Gar 05, Co rollary 4.3] o nly uses the fact that a ll morphisms in W factor a s a cofibr ation follow ed by a weak equiv alenc e , compare also [Gar05, Lemmas 4.1 a nd 4.2], so w e also ha ve a natural weak equiv alence D K ( W ) ∼ → K ( D W ) for W a W aldha us en ca tegory with cylinder s satisfying the 2 out of 3 axiom, and therefore K n ( D W ) ∼ = π n +1 | Diag Ner i Ho S. W | , n ≥ 0 . The functors ζ : S n W → Ho S n W res tr ict to w S n W → i Ho S n W . These functors give rise to a map | Diag Ner w S. W | − → | Diag Ner i Ho S. W | which induces the co mparison homomorphisms in homotopy gr oups, µ n : K n ( W ) − → K n ( D W ) , n ≥ 0 . This map is actually the 1-stag e of a compar is on map o f s pectr a (2.4) K ( W ) − → K ( D W ) . In the r e st o f this pap er w e will b e mainly concerned with the structure of the bisimplicial sets X = Ner w S. W and Y = Ner i Ho S. W in lo w dimensions, that w e now re v iew mo re thoroughly . A bisimplicia l set Z co ns ists of sets Z m,n , m, n ≥ 0, together with horizontal and v ertical face and degenera cy maps d h i : Z m,n − → Z m − 1 ,n , s h i : Z m,n − → Z m +1 ,n , 0 ≤ i ≤ m, d v j : Z m,n − → Z m,n − 1 , s v j : Z m,n − → Z m,n +1 , 0 ≤ j ≤ n, satisfying some relations that w e will not recall here, compar e [GJ9 9]. An element z m,n ∈ Z m,n is a bisimple x of bide gr e e ( m, n ) and total d e gr e e m + n . A g eneric bisimplex z m,n of bidegree ( m, n ) can be depicted as the pro duct o f t wo geometric simplices of dimensions m and n with vertices labelled b y the pro duct set { 0 , . . . , m } × { 0 , . . . , n } , see Figs. 1 and 2. The horizontal i th face d h i ( z m,n ) is the face obtained by removing MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 9 (0 , 0) (1 , 0) z 1 , 0 (0 , 1) (1 , 1) (0 , 0) (1 , 0) z 1 , 1 (0 , 0) (1 , 0) (2 , 0)                    z 2 , 0 Figure 1. Bisimplices of total degree 1 and 2. the interior, the v ertices ( i, j ), for all j , and the inciden t fa ces of the b oundar y . Similarly the vertical j th face d v j ( z m,n ) is obtained by removing the in terior , the vertices ( i, j ), for all i , and the incident face s of the bo undary . (0 , 2) (1 , 2) ? ? ? ? ? ? ? ? (0 , 1) (0 , 0) (1 , 0) (1 , 1)         z 1 , 2 z 2 , 1 (0 , 1) (2 , 1) ? ? ? ? ? ? ? ? ? ? (1 , 1) o o o o o o o o o o o o o o o (0 , 0) ? ? ? ? ? ? ? ? ? ? (1 , 0) (2 , 0) o o o o o o o o o o o o o o o z 3 , 0 (0 , 0) T T T T T T T T T T T T T T T T T T T T T T t t t t t t t t t t t t t t t t t t t t t t t t t (1 , 0) o o o o o o o o o o o o o o o o (2 , 0) (3 , 0) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Figure 2. Bisimplices of total degr ee 3 . The bisimplicial sets X and Y ar e horizontally r educed, i.e. X 0 ,n = Y 0 ,n are singletons for all n ≥ 0, X 1 , 0 = Y 1 , 0 is the set o f ob jects in W , X 1 , 1 is the set of weak equiv alences in W , Y 1 , 1 is the set of isomorphisms in Ho W , and X 2 , 0 = Y 2 , 0 is the set of cofib er sequences, see Fig. 3. The set X 1 , 2 consists of pairs o f c o mpo sable w eak equiv alences in W , Y 1 , 2 is the set of comp osa ble isomorphisms in Ho W , X 2 , 1 is the set of weak equiv alences betw een co fiber sequences i.e. weak equiv alence s in S 2 W which are co mm utative diagrams in W (2.5) A ′ / / / / B ′ / / / / B ′ / A ′ A / / / / ∼ O O B / / / / ∼ O O B / A ∼ O O 10 FERNANDO MURO A x 1 , 0 = y 1 , 0 A ′ A ∼ O O x 1 , 1 A ′ A ∼ = O O y 1 , 1 A B                 B / A O O O O / / / / x 2 , 0 = y 2 , 0 Figure 3. Bisimplices of total degree 1 and 2 in X and Y . Y 2 , 1 is the set of isomo rphisms in Ho S 2 W , and X 3 , 0 = Y 3 , 0 is the set of fo ur cofib er sequences assicated to pairs of comp osable cofibratio ns (2.6) C /B B / A / / / / C / A O O O O A / / / / B / / / / O O O O C O O O O see Fig. 4. Suppo se that W has a saturated class of w eak equiv a lences. T he n the categ o ries S n W inherit this prop erty . Therefor e the isomorphism y 2 , 1 in Ho S 2 W is repr esented by a commutativ e diagram in W (2.7) A ′ / / / / B ′ / / / / B ′ / A ′ X / / / /   ∼ α 2 O O ∼ α 1 Y / / / /   ∼ β 2 O O ∼ β 1 Y /X   ∼ γ 2 O O ∼ γ 1 A / / / / B / / / / B / A where the hor izontal line s are cofib er seq uences and the vertical arr ows are w eak equiv alenc e s. The face d v 1 ( y 2 , 1 ) is a cofib er sequence in W which is the source of the isomor phism in Ho S 2 W , a nd the face d v 0 ( y 2 , 1 ) is the target. The faces d h 2 ( y 2 , 1 ), d h 1 ( y 2 , 1 ), d h 0 ( y 2 , 1 ) co rresp ond, in this or de r , to the isomorphisms α , β , γ in Ho S 2 W represented by the v ertical lines in the previo us diag ram. Notice that the representativ e of y 2 , 1 corres p onds to the pas ting of tw o bisim- plices of bidegree (2 , 1) in X through a common face, see Fig. 5. The degenerate bisimplices of total degree 1 and 2 in X and Y ar e depicted in Fig. 6 . MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 11 C ? ? ? ? ? ? ? ? A         B ∼ ? ? ∼ _ _ ∼ O O x 1 , 2 C ? ? ? ? ? ? ? ? A         B ∼ = ? ? ∼ = _ _ ∼ = O O y 1 , 2 x 2 , 1 B ′ A ′ ? ? ? ? ? ? ? ? B ′ / A ′ o o o o o o o o o o B A ? ? ? ? ? ? ? ? B / A o o o o o o o o o o 7 7 7 7 o o o o o 7 7 7 7     ? ?     ∼ O O ∼ O O ∼ O O y 2 , 1 B ′ A ′ ? ? ? ? ? ? ? ? B ′ / A ′ o o o o o o o o o o B A ? ? ? ? ? ? ? ? B / A o o o o o o o o o o 7 7 7 7 o o o o o 7 7 7 7     ? ?     in Ho( S 2 W ) ∼ = O O x 3 , 0 = y 3 , 0 A T T T T T T T T T T T T T T T T T T T T C t t t t t t t t t t t t t t t t t t t t t t B / A o o o o o o o o o o o o C / A C /B ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? B * * * * / / / / 7 7 7 7 O O O O _ _ _ _ _ _ _ _ ? ? ? ? ? ? ? * * * * T T T T T T T T 7 7 7 7 o o o Figure 4. Bisimplices of total degree 3 in X and Y . The choice of binary co pro ducts A ∨ B in W gives rise to a biexac t functor ∨ : W × W → W which induces maps o f bisimplical sets [W al8 5, Gar05] X × X ∨ − → X , Y × Y ∨ − → Y , in the ob vious wa y . These maps induce co- H -multiplications in | Diag X | and | Diag Y | , whic h come from the fact that they a re infinite lo op spaces. 3. Abelian 2 -gr oups In this section we re c all the definition of sta ble quadr a tic mo dules, introduced in [Bau91, Definition IV.C.1]. Related structures are stable crossed mo dules [Con84] and symetric categorical groups [BCC9 3, CMM04]. All these algebraic structures yield equiv alent 2-dimensio nal extensions of the theory of ab elian groups. Among 12 FERNANDO MURO B ′ A ′ ? ? ? ? ? ? ? ? B ′ / A ′ o o o o o o o o o o Y X ? ? ? ? ? ? ? ? Y /X o o o o o o o o o o 7 7 7 7     ? ?       ∼   ∼   ∼ B A ? ? ? ? ? ? ? ? B / A o o o o o o o o o o 7 7 7 7 o o o o o 7 7 7 7     ∼ O O ∼ O O ∼ O O Figure 5. A representativ e o f y 2 , 1 given by the pasting of t wo x 2 , 1 ’s. 0 s h 0 (0) A A s v 0 ( A ) A A               0 / / / / s h 1 ( A ) 0 A               A O O O O s h 0 ( A ) Figure 6. Degenerate bisimplices of total degree 1 and 2 in X a nd Y . them stable quadratic mo dules a re sp ecially con venien t since they are a s small and strict as po ssible. Definition 3.1. A stable quad r atic m o dule C ∗ is a diagr am of group homomor- phisms C ab 0 ⊗ C ab 0 h· , ·i − → C 1 ∂ − → C 0 such that given c i , d i ∈ C i , i = 0 , 1, (1) ∂ h c 0 , d 0 i = [ d 0 , c 0 ], (2) h ∂ ( c 1 ) , ∂ ( d 1 ) i = [ d 1 , c 1 ], (3) h c 0 , d 0 i + h d 0 , c 0 i = 0. Here [ x, y ] = − x − y + x + y is the co mm utator of t w o elements x, y ∈ K in a group K , and K ab is the ab e lianization of K . A m orphism f : C ∗ → D ∗ of stable quadra tic mo dules is giv en by gr oup homo- morphisms f i : C i → D i , i = 0 , 1, compatible w ith the structure homomo rphisms of C ∗ and D ∗ , i.e. f 0 ∂ = ∂ f 1 and f 1 h· , ·i = h f 0 , f 0 i . MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 13 R emark 3.2 . It follows from Definition 3 .1 that the image of h· , ·i and Ker ∂ are central in C 1 , the groups C 0 and C 1 hav e nilpo tency class 2 , and ∂ ( C 1 ) is a normal subgroup of C 0 . There is a natural right action of C 0 on C 1 defined by c c 0 1 = c 1 + h c 0 , ∂ ( c 1 ) i . The axio ms of a s table quadr a tic mo dule imply tha t commutators in C 0 act trivia lly on C 1 , and that C 0 acts trivially on the image of h· , ·i and on Ker ∂ . The action gives ∂ : C 1 → C 0 the s tr ucture of a crossed mo dule. Indeed a stable quadratic mo dule is the s ame as a comm utative mo noid in the category of crossed mo dules such that the monoid pro duct of t w o elemen ts in C 0 v anishes when one of them is a comm utator, see [MT07, Lemma 4.18 ]. R emark 3.3 . The forgetful functor from stable quadratic mo dules to pairs of sets squad − → Se t × Set : C ∗ 7→ ( C 0 , C 1 ) has a left adjoint. This makes p ossible to define a free stable q ua dratic mo dule with generating set E 0 in dimension 0 a nd E 1 in dimension 1. One can more generally define a stable quadra tic module b y a presentation with gener ators and re la tions in degr ees 0 and 1. The explicit c o nstruction o f a stable qua dratic mo dule with a given presentation c a n b e found in the app endix o f [MT07]. F or the pur pos es of this pap er it will be enough to assume the existence of this construction, satisfying the obvious universal prop erty as in the case of gr oups. W e no w reca ll the connection of stable quadratic mo dules with stable homotop y theory . Definition 3.4. The homotopy gr oups of a stable quadr a tic mo dule C ∗ are π 0 C ∗ = C 0 /∂ ( C 1 ) , π 1 C ∗ = Ker[ ∂ : C 1 → C 0 ] . Notice that thes e groups are a belia n. Homotopy groups ar e ob viously functors in the categor y squad of stable quadratic modules . A mor phism in squad is a we ak e quivalenc e if it induces iso morphisms in π 0 and π 1 . The k -inva riant of C ∗ is the natural homomorphism k : π 0 C ∗ ⊗ Z / 2 − → π 1 C ∗ defined as k ( x ⊗ 1 ) = h x, x i . W eak equiv alences in the Bousfield-F riedlander categ ory Spe c 0 of connective sp ectra of simplicia l sets [BF78] a re also morphis ms inducing isomorphisms in ho - motopy groups. Extending Definition 1.3, if C is a category endowed with a clas s of weak equiv ale nc e s we denote b y Ho C the loca lization of C with resp ect to weak equiv alenc e s in the sense of [GZ67]. Lemma 3.5. [MT07, Lemma 4.22] Th er e is define d a functor λ 0 : Ho Sp ec 0 − → Ho squad to gether with natu r al isomo rphisms π 0 λ 0 X ∼ = π 0 X , π 1 λ 0 X ∼ = π 1 X . 14 FERNANDO MURO The k - invariant of λ 0 X c orr esp onds to the action of the stable Hopf map in the stable homotopy gr oups of spher es 0 6 = η ∈ π s 1 ∼ = Z / 2 , π 0 X ⊗ Z / 2 − → π 1 X : x ⊗ 1 7→ x · η . Mor e over, λ 0 r est ricts to an e quivalenc e of c ate gories on the ful l su b c ate gory of sp e ctr a with homotopy gr oups c onc ent r ate d in dimensions 0 and 1 . W e interpret this lemma as follows. Chain complexes of ab elian gr oups · · · → 0 → B 1 ∂ − → B 0 → 0 → · · · do not model all spectra with homotopy groups concentrated in dimensions 0 and 1 since these co mplexes neg lect the stable Ho pf map. How ever these sp ectra are mo delled by stable quadra tic mo dules, which can be regarded a s non-a belia n chain complexes · · · → 0 → C 1 ∂ − → C 0 → 0 → · · · endow ed with a n extra map C ab 0 ⊗ C ab 0 h· , ·i − → C 1 which keeps track of the behaviour o f comm utators in C 1 and C 0 . The homology of this non-ab elian chain complex a re the homotopy groups of the corre spo nding sp ectrum. Moreover, squa ring the brack et h· , ·i w e recover the a ction of the s table Hopf map. In Sec tio n 2 we recalled that K -theory sp ectra a re sp ectra o f top ologica l spaces . In this section we have stated Lemma 3.5 for sp ectra of simplicial sets. The geomet- ric rea liz ation functor from simplicial se ts to spaces induces an eq uiv alence be t ween the the homotopy categ ories of spectr a of simplicial sets and spec tra of top ologica l spaces. Therefor e in the nex t s e ction we feel free to apply the functor λ 0 in Lemma 3.5 to K -theory spectr a. 4. Algebraic models f or lower K -theor y In [MT07] w e define a stable qua dratic module D ∗ W for an y W aldhausen cate- gory W w hich is na tur ally iso morphic to λ 0 K ( W ) in the homotopy ca tegory o f sta ble quadratic mo dules, therefore D ∗ W is a mo del for the 1-type of the W aldha usen K - theory of W . The stable quadratic mo dule D ∗ W is defined by a pres en tation with as few generators as p ossible. W e no w recall this presentation. Definition 4.1. W e define D ∗ W as the stable quadratic module genera ted in di- mension zero b y the symbols (G1) [ A ] for any ob ject in W , and in dimension one by (G2) [ A ∼ → A ′ ] for an y weak equiv alence, (G3) [ A ֌ B ։ B / A ] for any cofiber seq uence, such that the following relations hold. (R1) ∂ [ A ∼ → A ′ ] = − [ A ′ ] + [ A ]. (R2) ∂ [ A ֌ B ։ B / A ] = − [ B ] + [ B / A ] + [ A ]. (R3) [0] = 0. (R4) [ A 1 → A ] = 0. (R5) [ A 1 ֌ A ։ 0] = 0, [0 ֌ A 1 ։ A ] = 0. MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 15 (R6) F or any pa ir of comp osable w eak equiv alences A ∼ → B ∼ → C , [ A ∼ → C ] = [ B ∼ → C ] + [ A ∼ → B ] . (R7) F or an y weak equiv ale nc e betw een cofiber sequences in W , given by a com- m utative diagram (2.5), w e hav e [ A ∼ → A ′ ] + [ B / A ∼ = → B ′ / A ′ ] [ A ] = − [ A ′ ֌ B ′ ։ B ′ / A ′ ] + [ B ∼ → B ′ ] + [ A ֌ B ։ B / A ] . (R8) F or any commutativ e diagram consisting of four cofib er sequences in W asso ciated to a pair of comp osable cofibrations (2.6) we ha ve [ B ֌ C ։ C /B ] + [ A ֌ B ։ B / A ] = [ A ֌ C ։ C / A ] + [ B / A ֌ C / A ։ C /B ] [ A ] . (R9) F or any pa ir of ob jects A , B in W h [ A ] , [ B ] i = − [ B i 2 ֌ A ∨ B p 1 ։ A ] + [ A i 1 ֌ A ∨ B p 2 ։ B ] . Here A i 1 / / A ∨ B p 1 o o p 2 / / B i 2 o o are the inclusions and pro jections of a copro duct in W . R emark 4.2 . The pres en tation of the stable qua dr atic mo dule D ∗ W is completely determined by the bisimplicial s tructure of X = Ner w S. W and the map ∨ : X × X → X in total degree ≤ 3 , see Section 2. More prec is ely , D ∗ W is g enerated in degr ee 0 by the bisimplices of total degr e e 1 and in degr e e 1 by the bisimplices o f tota l degree 2, see Fig . 3. Relations (R1) and (R2) identify the imag e by ∂ o f a deg ree 1 gener a tor with the summation, in an a ppropriate order , of the faces o f the boundar y of the corresp onding bisimplex of total degree 2, see again Fig. 3. Relations (R3)–(R5) say that degenerate bisim- plices of tota l degree 1 or 2 ar e trivial in D ∗ W , see Fig. 6. Rela tions (R6)–(R8) tell us tha t the summation, in an a ppropriate order, of the faces of the b oundar y of a bisimplex of total deg ree 3 is zero in D ∗ W , see Fig 4. Finally (R9) sa ys that the brack et h [ A ] , [ B ] i c oincides with − [ s h 0 ( A ) ∨ s h 1 ( B )] + [ s h 1 ( A ) ∨ s h 0 ( B )] , i.e. it is obtained as fo llows. W e fir st take the t wo po ssible degenera te bisimplices of bidegree (2 , 0) asso ciated to A and B in the following o rder. 0 A               A O O O O B B               0 / / / / A A               0 / / / / 0 B               B O O O O 16 FERNANDO MURO W e then take the copr o duct of the first and the s econd pair of degener ate bisim- plices. B A ∨ B               A O O O O / / / / A A ∨ B               B O O O O / / / / Finally w e take the diff erence betw een the corresp onding generator s in D 1 W −         B A ∨ B               A O O O O / / / /         +         A A ∨ B               B O O O O / / / /         . There is a non-ab elian E ilen b erg-Zilb er theorem b ehind this formula, compar e [MT07, Theorem 4.10 and Example 4.13]. The main result of [MT07] is the following theorem. Theorem 4.3. [MT0 7, Theorem 1.7] L et W b e a Waldhausen c ate gory. Ther e is a n atur al isomorp hism in Ho squad D ∗ W ∼ = − → λ 0 K ( W ) . This result is meaningful since λ 0 K ( W ) is huge compa red with D ∗ W , while D ∗ W is directly defined in terms o f the basic structure o f the W aldhausen category W . As a consequence we hav e an exact sequence of gro ups K 1 ( W ) ֒ → D 1 W ∂ − → D 0 W ։ K 0 ( W ) . W e now extend Theorem 4.3 to derived K -theo r y . Definition 4.4. W e define D der ∗ W as the stable q uadratic module generated in dimension zero by the sy m b ols (DG1) [ A ] for an y ob ject in W , i.e. the same as (G1), and in dimension one by (DG2) [ A ∼ = → A ′ ] for any is omorphism in Ho W , (DG3) [ A ֌ B ։ B / A ] for any co fiber se q uence in W , i.e. the same as (G3), such that the following relations hold. (DR1) ∂ [ A ∼ = → A ′ ] = − [ A ′ ] + [ A ]. (DR2) = (R2). (DR3) = (R3). (DR4) [ A 1 → A ] = 0 . (DR5) = (R5). (DR6) F or any pair of comp o sable isomorphisms A ∼ = → B ∼ = → C in Ho W , [ A ∼ = → C ] = [ B ∼ = → C ] + [ A ∼ = → B ] . MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 17 (DR7) F or any comm utative dia gram in W as (2.7) w e hav e [ α : A ∼ = → A ′ ] + [ γ : B / A ∼ = → B ′ / A ′ ] [ A ] = − [ A ′ ֌ B ′ ։ B ′ / A ′ ] + [ β : B ∼ = → B ′ ] + [ A ֌ B ։ B / A ] . Here α = ζ ( α 2 ) − 1 ζ ( α 1 ), β = ζ ( β 2 ) − 1 ζ ( β 1 ) and γ = ζ ( γ 2 ) − 1 ζ ( γ 1 ). (DR8) = (R8). (DR9) = (R9). If W is a W aldhausen ca tegory with cylinder s and a satura ted c la ss o f weak equiv- alences then the presen tation of the stable qua dratic module D der ∗ W is determined by the bisimplicial str ucture of Y = Ner i Ho S. W and the map ∨ : Y × Y → Y in total degree ≤ 3, se e Section 2, exa ctly in the same way as D ∗ W is determined b y X = Ner wS . W and ∨ : X × X → X , see Remark 4.2. There fore replacing X b y Y in the pro of of [MT07, Theorem 1.7] w e obtain the following result. Theorem 4.5. L et W b e a Waldhausen c ate gory with cylinders and a satu r ate d class of we ak e quivalenc es. Ther e is a natur al iso morphism in Ho squad D der ∗ W ∼ = − → λ 0 K ( D W ) . As a consequence we have an exa ct sequence of groups K 1 ( D W ) ֒ → D der 1 W ∂ − → D der 0 W ։ K 0 ( D W ) . R emark 4.6 . As one can ea sily c heck, taking λ 0 in the compa rison map of sp ectra (2.4) which induces µ n : K n ( W ) → K n ( D W ) in homotopy gr oups co rresp onds to the natural morphism in squad , ¯ µ : D ∗ W − → D der ∗ W , [ A ] 7→ [ A ] , [ f : A ∼ → A ′ ] 7→ [ ζ ( f ) : A ∼ = → A ′ ] , [ A ֌ B ։ B / A ] 7→ [ A ֌ B ։ B / A ] , under the na tural isomor phis ms of Theorems 4.3 and 4 .5. In particula r taking π 0 and π 1 in this mor phism of stable quadratic mo dules we obtain µ 0 and µ 1 , resp ectively . This fact will b e used b elow in the proo f of Theorem B. 5. Proof of Theorem B Theorem B is a consequence of the following result. Theorem 5.1. L et W b e a Waldhausen c ate gory with cylinders and a satu r ate d class of we ak e quivalenc es. The natur al morphi sm in squad ¯ µ : D ∗ W − → D der ∗ W , define d in Remark 4.6, is an iso morphism. The key fo r the pro of of Theorem 5.1 is the follo wing lemma. Lemma 5.2. L et W b e a Waldhausen c ate gory with cyli nders satisfying the 2 out of 3 axiom. Two we ak e quivalenc es f , g : A ∼ → A ′ which ar e homotopic f ≃ g r epr esen t the same element in D 1 W , [ f : A ∼ → A ′ ] = [ g : A ∼ → A ′ ] . 18 FERNANDO MURO Pr o of. Let I A be a cylinder of A and A ∨ A ֌ i I A ∼ − → p A a factor iz a tion of the folding map, i.e. if i = ( i 0 , i 1 ) then pi 0 = pi 1 = 1 A . Since bo th p and 1 A are weak equiv alences we deduce from the 2 out of 3 axio m that i 0 and i 1 are also weak equiv alences. Moreov er, for j = 0 , 1, 0 (R4) = [ A 1 A → A ] = [ pi j : A ∼ → A ] (R6) = [ p : I A ∼ → A ] + [ i j : A ∼ → I A ] , therefore [ i 0 : A ∼ → I A ] = − [ p : I A ∼ → A ] = [ i 1 : A ∼ → I A ] . F urthermo r e, f ≃ g , so there is a weak equiv alence h : A ′ ∼ → A ′′ and a mo r phism H : I A → A ′′ such that H i 0 = hf a nd H i 1 = hg . Again by the 2 out of 3 axiom H is a w eak equiv alence, and [ h : A ′ ∼ → A ′′ ] + [ f : A ∼ → A ′ ] (R6) = [ hf = H i 0 : A ∼ → A ′′ ] (R6) = [ H : I A ∼ → A ′′ ] + [ i 0 : A ∼ → I A ] = [ H : I A ∼ → A ′′ ] + [ i 1 : A ∼ → I A ] (R6) = [ hg = H i 1 : A ∼ → A ′′ ] (R6) = [ h : A ′ ∼ → A ′′ ] + [ g : A ∼ → A ′ ] , hence w e are done.  W e ar e no w ready to prov e Theorem 5 .1. Pr o of of The or em 5.1. W e are going to define the inv erse of ¯ µ , ¯ ν : D der ∗ W − → D ∗ W . W e first sho w that ¯ ν 0 [ A ] = [ A ] , ¯ ν 1 [ α : A ∼ = → A ′ ] = − [ α 2 : A ′ ∼ → X ] + [ α 1 : A ∼ → X ] , ¯ ν 1 [ A ֌ B ։ B / A ] = [ A ֌ B ։ B / A ] , defines a stable quadratic mo dule morphism ¯ ν . Here A ∼ − → α 1 X ∼ ← − α 2 A ′ is a r e pr esentativ e of the isomorphism α . F or this w e are going to prove that the image of [ α : A ∼ = → A ′ ] do es not dep end on the choice of a r epresentativ e. Suppo se that A ∼ − → α ′ 1 Y ∼ ← − α ′ 2 A ′ MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 19 also represents α . Then there is a diagra m in W X > > α 1 ~ ~ ~ ~ ~ ~ ~ ~ ` ` α 2 A A A A A A A A A Z / / f 1 o o f 2   g O O g ′ A ′ Y α ′ 1 @ @ @ @ @ @ @ @ ~ ~ α ′ 2 } } } } } } } } where all a rrows ar e weak equiv a lences and the fo ur triangles comm ute up to ho- motopy , so − [ α 2 : A ′ ∼ → X ] + [ α 1 : A ∼ → X ] = − [ α 2 : A ′ ∼ → X ] − [ g : X ∼ → Z ] +[ g : X ∼ → Z ] + [ α 1 : A ∼ → X ] (R6) = − [ g α 2 : A ′ ∼ → Z ] + [ g α 1 : A ∼ → Z ] Lemma 5.2 = − [ f 2 : A ′ ∼ → Z ] + [ f 1 : A ∼ → Z ] Lemma 5.2 = − [ g ′ α ′ 2 : A ′ ∼ → Z ] + [ g ′ α ′ 1 : A ∼ → Z ] (R6) = − [ α ′ 2 : A ′ ∼ → Y ] − [ g ′ : Y ∼ → Z ] +[ g ′ : Y ∼ → Z ] + [ α ′ 1 : A ∼ → Y ] = − [ α ′ 2 : A ′ ∼ → X ] + [ α ′ 1 : A ∼ → X ] . Now we chec k that the definition of ¯ ν on genera tors is compatible with the defining r e lations. The only non-triv ial part conc e rns relations (DR1), (DR6) a nd (DR7). Compatibilit y with (DR1) follows from ¯ ν 0 ∂ [ α : A ∼ = → A ′ ] = ∂ ¯ ν 1 [ α : A ∼ = → A ′ ] = − ∂ [ α 2 : A ′ ∼ → X ] + ∂ [ α 1 : A ∼ → X ] (R1) = − ( − [ X ] + [ A ′ ]) + ( − [ X ] + [ A ]) = − [ A ′ ] + [ A ] = − ¯ ν 0 [ A ′ ] + ¯ ν 0 [ A ] . In order to c heck compatibility with (DR6) we c o nsider t wo comp osable isomor- phisms in Ho W A ∼ = − → α B ∼ = − → β C and we take representatives of α , β a nd β α as in the following co mm utative diagram of w eak equiv alences in W X ∪ B Y a a ¯ α 2 C C C C C C C C = = ¯ β 1 = = { { { { { { { { X a a α 2 D D D D D D D D B B α 1        push Y \ \ β 2 8 8 8 8 8 8 8 A B = = β 1 = = { { { { { { { { C 20 FERNANDO MURO Then ¯ ν 1 [ β α : A ∼ = → C ] = − [ ¯ α 2 β 2 : C ∼ → X ∪ B Y ] + [ ¯ β 1 α 1 : A ∼ → X ∪ B Y ] = − [ ¯ α 2 β 2 : C ∼ → X ∪ B Y ] + [ ¯ α 2 β 1 : B ∼ → X ∪ B Y ] − [ ¯ β 1 α 2 : B ∼ → X ∪ B Y ] + [ ¯ β 1 α 1 : A ∼ → X ∪ B Y ] (R6) = − ([ ¯ α 2 : Y ∼ → X ∪ B Y ] + [ β 2 : C ∼ → Y ]) +[ ¯ α 2 : Y ∼ → X ∪ B Y ] + [ β 1 : B ∼ → Y ] − ([ ¯ β 1 : X ∼ → X ∪ B Y ] + [ α 2 : B ∼ → X ]) +[ ¯ β 1 : X ∼ → X ∪ B Y ] + [ α 1 : A ∼ → X ] = − [ β 2 : C ∼ → Y ] + [ β 1 : B ∼ → Y ] − [ α 2 : B ∼ → X ] + [ α 1 : A ∼ → X ] = ¯ ν 1 [ β : B ∼ = → C ] + ¯ ν 1 [ α : A ∼ = → B ] . Let us now chec k compatibility with (DR7). − ¯ ν 1 [ A ′ ֌ B ′ ։ B ′ / A ′ ] + ¯ ν 1 [ β : B ∼ = → B ′ ] + ¯ ν 1 [ A ֌ B ։ B / A ] = − [ A ′ ֌ B ′ ։ B ′ / A ′ ] − [ β 2 : B ′ ∼ → Y ] +[ X ֌ Y ։ Y / X ] − [ X ֌ Y ։ Y / X ] +[ β 1 : B ∼ → Y ] + [ A ֌ B ։ B / A ] (R7) = − ([ α 2 : A ′ ∼ → X ] + [ γ 2 : B ′ / A ′ ∼ → Y /X ] [ A ′ ] ) +[ α 1 : A ∼ → X ] + [ γ 1 : B / A ∼ → Y /X ] [ A ] Rem. 3.2 = − [ γ 2 : B ′ / A ′ ∼ → Y /X ] − [ α 2 : A ′ ∼ → X ] +[ α 1 : A ∼ → X ] + [ γ 1 : B / A ∼ → Y /X ] −h [ A ′ ] , ∂ [ γ 2 ] i + h [ A ] , ∂ [ γ 1 ] i Defn. 3.1 (2) and Rem . 3.2 = − [ α 2 : A ′ ∼ → X ] + [ α 1 : A ∼ → X ] − [ γ 2 : B ′ / A ′ ∼ → Y /X ] + [ γ 1 : B / A ∼ → Y /X ] + h− ∂ [ α 2 ] + ∂ [ α 1 ] , − ∂ [ γ 2 ] i − h [ A ′ ] , ∂ [ γ 2 ] i + h [ A ] , ∂ [ γ 1 ] i (R1) = − [ α 2 : A ′ ∼ → X ] + [ α 1 : A ∼ → X ] − [ γ 2 : B ′ / A ′ ∼ → Y /X ] + [ γ 1 : B / A ∼ → Y /X ] + h− ( − [ X ] + [ A ′ ]) + ( − [ X ] + [ A ]) , − ∂ [ γ 2 ] i + h [ A ′ ] , − ∂ [ γ 2 ] i + h [ A ] , ∂ [ γ 1 ] i = − [ α 2 : A ′ ∼ → X ] + [ α 1 : A ∼ → X ] − [ γ 2 : B ′ / A ′ ∼ → Y /X ] + [ γ 1 : B / A ∼ → Y /X ] + h [ A ] , − ∂ [ γ 2 ] i + h [ A ] , ∂ [ γ 1 ] i = + ¯ ν 1 [ α : A ∼ = → A ′ ] + ¯ ν 1 [ γ : B / A ∼ = → B ′ / A ′ ] + h ¯ ν 0 [ A ] , ∂ ¯ ν 1 [ γ : B / A ∼ = → B ′ / A ′ ] i Rem. 3.2 = ¯ ν 1 [ α : A ∼ = → A ′ ] + ¯ ν 1 [ γ : B / A ∼ = → B ′ / A ′ ] ¯ ν 0 [ A ] . This establishes that ¯ ν is a w ell defined morphism of stable quadr a tic mo dules. MAL TSINIOTIS’S FIRST CONJECTURE FOR K 1 21 Let us now c heck that ¯ µ ¯ ν = 1 D der ∗ W and ¯ ν ¯ µ = 1 D ∗ W . Both equatio ns a re obvious on generator s (G1) = (DG1 ) and (G3) = (DG3). F or (G2) ¯ ν 1 ¯ µ 1 [ f : A ∼ → A ′ ] = ¯ ν 1 [ ζ ( f ) : A ∼ = → A ′ ] = − [1 A ′ : A ′ ∼ → A ′ ] + [ f : A ∼ → A ′ ] (R4) = [ f : A ∼ → A ′ ] . If α : A ∼ = → A ′ is an isomorphism in Ho W w e hav e the following equation in D der 1 W , 0 (DR4) = [ A 1 A → A ] = [ α − 1 α : A ∼ = → A ] (DR6) = [ α − 1 : A ′ ∼ = → A ] + [ α : A ∼ = → A ′ ] , so [ α − 1 : A ′ ∼ = → A ] = − [ α : A ∼ = → A ′ ]. No w for (DG2) ¯ µ 1 ¯ ν 1 [ α : A ∼ = → A ′ ] = − ¯ µ 1 [ α 2 : A ′ ∼ → X ] + ¯ µ 1 [ α 1 : A ∼ → X ] = − [ ζ ( α 2 ) : A ′ ∼ = → X ] + [ ζ ( α 1 ) : A ∼ = → X ] = [ ζ ( α 2 ) − 1 : X ∼ = → A ′ ] + [ ζ ( α 1 ) : A ∼ = → X ] (DR6) = [ α = ζ ( α 2 ) − 1 ζ ( α 1 ) : A ∼ = → A ′ ] . The pro of of Theorem 5.1 is now finished.  R emark 5.3 . Let W b e a W aldhausen categ ory with cylinders satisfying the 2 out of 3 axio m. W e do not assume that W has a saturated cla ss of weak equiv alences. How ev er we can endow the underly ing category with a new W aldhausen category structure which do es hav e a saturated class of weak equiv alences. W e consider the W aldhaus en category W with the same underlying categor y as W . Cofibra tions in W are also de same as in W . W eak equiv alence s in W are the morphisms in W which are mapp ed to isomorphisms in Ho W by the canonical func- tor ζ : W → Ho W . Therefore weak equiv a lences in W ar e also weak equiv a lences in W but the conv erse need not hold. This indeed defines a W a ldhausen catego ry W with cylinder s and a saturated c lass of weak equiv alences, and the obvious exact functor W → W induces an iso morphism on the ass o cia ted deriv ators D W ∼ = D W , compare [C is 03, dual of Prop osition 3.16] and [RB07, Theorem 6.2.2]. Hence we hav e a co mm utative dia g ram for n = 0 , 1, K n ( W ) µ n / /   K n ( D W ) ∼ =   K n ( W ) µ n ∼ = / / K n ( D W ) Here the lo wer arrow is an isomor phism by Theorem B. No w we can use the ‘fibra- tion theor em’, [W al8 5, 1.6.7 ] and [Sc h06, Theorem 11], to embed the morphisms µ n : K n ( W ) → K ( D W ), n = 0 , 1, in an exact sequence. More pr ecisely , let W 0 be the full subcatego ry of W spanned b y the ob j ects whic h are iso morphic to 0 in Ho W . The category W 0 is a W aldhausen ca tegory where a morphism is a cofi- bration, resp. a w eak equiv alence, if and only if it is a co fibration, resp. a w eak 22 FERNANDO MURO equiv alenc e , in W . 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