Maltsiniotis's first conjecture for K1

arXiv:0707.1892v2 [math.KT] 19 Oct 2007 MALTSINIOTIS’S FIRST CONJECTURE FOR K1 FERNANDO MURO Abstract. We show that K1(E) of an exact category Eagrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category Wwith cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW. Introduction For a long time there was an interest in defining a nice K-theory for triangulated categories such that Quillen’s K-theory of an exact category Eagrees with the Ktheory of its bounded derived category Db(E). Schlichting [Sch02] showed that such a K-theory for triangulated categories cannot exist. It was then natural to ask about the definition of a nice K-theory for algebraic structures interpolating between Eand Db(E). The best known intermediate structure is Cb(E), the Waldhausen category of bounded complexes in E, with quasi-isomorphisms as weak equivalences and cofibrations given by chain morphisms which are levelwise admissible monomorphisms. The derived category Db(E) is the localization of Cb(E) with respect to weak equivalences. The Gillet-Waldhausen theorem1, relating Quillen’s K-theory to Waldhausen’s K-theory, states that the homomorphisms τn:Kn(E)−→ Kn(Cb(E)), n ≥0, induced by the inclusion E⊂Cb(E) of complexes concentrated in degree 0, are isomorphisms. The category Cb(E) is considered to be too close to Eso one would still like to find an algebraic stucture with a nice K-theory interpolating between Cb(E) and Db(E). The notion of a triangulated derivator [Gro90, Mal07] seems to be a strong candidate. Maltsiniotis [Mal07] defined a K-theory for triangulated derivators together with natural homomorphisms ρn:Kn(E)−→ Kn(DE), n ≥0, 1991 Mathematics Subject Classification. 18E10, 18E30, 18F25, 19B99, 55S45. Key words and phrases. K-theory, exact category, triangulated derivator, Postnikov invariant, stable quadratic module. The author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-03629 and MTM2007-63277, and a Juan de la Cierva research contract. 1The proof due to Thomason-Trobaugh [TT90, Theorem 1.11.7] corrects Gillet’s [Gil81, 6.2] and uses an extra hypothesis on E. This hypothesis is not strictly necessary, since the general case follows then from cofinality arguments, see [Cis02]. 1 2 FERNANDO MURO where DEis the triangulated derivator associated to an exact category E, constructed by Keller in the appendix of [Mal07]. Cisinski and Neeman proved the additivity of triangulated derivator K-theory [CN05]. Maltsiniotis also conjectured that ρnis an isomorphism for all n. He succeeded in proving the conjecture for n= 0. The following theorem is the main result of this paper. Theorem A. Let Ebe an exact category. The natural homomorphism ρ1:K1(E)∼ = −→ K1(DE) is an isomorphism. In order to obtain Theorem A we use techniques introduced in [MT07]. There we give a presentation of an abelian 2-group D∗Wwhich encodes K0(W) and K1(W) of a Waldhausen category W, and moreover the 1-type of the K-theory spectrum K(W) whose homotopy groups are the K-theory groups of W. This presentation is a higher dimensional analogue of the classical presentation of K0(W). Here we similarly define an abelian 2-group Dder ∗Wwhich models the 1-type of the K-theory spectrum K(DW) of the right2pointed derivator DWassociated to a Waldhausen category Wwith cylinders and a saturated class of weak equivalences, such as W= Cb(E). The K-theory for this kind of derivators, more general than triangulated derivators, was defined by Garkusha [Gar06] extending the work of Maltsiniotis [Mal07]. There are defined comparison homomorphisms µn:Kn(W)−→ Kn(DW), n ≥0. These homomorphisms cannot be isomorphisms in general, as shown in [TV04]. Nevertheless we here prove the following result. Theorem B. Let Wbe a Waldhausen category with cylinders and a saturated class of weak equivalences. The natural homomorphism µ0:K0(W)∼ = −→ K0(DW), µ1:K1(W)∼ = −→ K1(DW), are isomorphisms. In Remark 5.3 we comment on the case where the hypothesis on the saturation of weak equivalences is replaced by the 2 out of 3 axiom, which is a weaker assumption. Theorem A is actually a corollary of the Gillet-Waldhausen theorem and Theorem B, since DCb(E) = DEand the natural homomorphisms ρnfactor as ρn:Kn(E)τn −→ Kn(Cb(E)) µn −→ Kn(DE), n ≥0. We assume the reader certain familiarity with exact, Waldhausen and derived categories, with simplicial constructions and with homotopy theory. We refer to [Wei, GM03, GJ99] for the basics. Acknowledgements. I am very grateful to Grigory Garkusha for suggesting the possibility of using [MT07] in order to tackle Maltsiniotis’s first conjecture in dimension 1. I also feel indebted to Denis-Charles Cisinski, who kindly indicated how to extend the results of a preliminary version of this paper to a broader generality. 2The references [Gro90, Cis03] and [Gar06, RB07] follow a different convention with respect to sides. Here we follow the convention in [Gro90, Cis03], so what we call a ‘right pointed derivator’ is the same as a ‘left pointed derivator’ in [Gar06]. MALTSINIOTIS’S FIRST CONJECTURE FOR K13 1. The bounded derived category of an exact category In this section we outline the two-step construction of the derived category Db(E) of an exact category E. This construction is a special case of the homotopy category Ho Wof a Waldhausen category Wwith cylinders satisfying the 2 out of 3 axiom, Db(E) = Ho Cb(E). Definition 1.1. AWaldhausen category is a category Wwith a distinguished zero object 0 and two distinguished subcategories wWand cW, whose morphisms are called cofibrations and weak equivalences, respectively. A morphism which is both a weak equivalence and a cofibration is said to be a trivial cofibration. The arrow ֌stands for a cofibration and ∼ →for a weak equivalence. •All morphisms 0 →Aare cofibrations. All isomorphisms are cofibrations and weak equivalences. •The push-out of a morphism along a cofibration is always defined A////  push B  X////X∪AB and the lower map is also a cofibration. •Given a commutative diagram X ∼  A ∼  oo////B ∼  X′A′ oo////B′ the induced map X∪AB∼ →X′∪A′B′is a weak equivalence. Notice that coproducts A∨B=A∪0Bare defined in W. A functor W→W′between Waldhausen categories is exact if it preserves cofibrations, weak equivalences, push-outs along cofibrations and the distinguished zero object. Example 1.2.Recall that an exact category Eis a full subcategory of an abelian category Asuch that Econtains a zero object of Aand Eis closed under extensions in A. A short exact sequence in Eis a short exact sequence in Abetween objects in E. A morphism in Eis an admissible monomorphism if it is the initial morphism of some short exact sequence. The category Eis a Waldhausen category with admissible monomorphisms as cofibrations and isomorphisms as weak equivalences. In order to complete the structure we fix a zero object 0 in E. We denote by Cb(E) the category of bounded complexes in E, · · · → An−1d −→ And −→ An+1 → · · · , d2= 0, An= 0 for |n| ≫ 0. A chain morphism f:A→Bin Cb(E) is a quasi-isomorphism if it induces an isomorphism in homology computed in the ambient abelian category A. The category Cb(E) is a Waldhausen category. Weak equivalences are quasi-isomorphisms and cofibrations are levelwise admissible monomorphisms. There is a full exact inclusion of Waldhausen categories E⊂Cb(E) sending an object Xin Eto the complex · · · → 0→X→0→ · · · , 4 FERNANDO MURO with Xin degree 0. Definition 1.3. The homotopy category Ho Wof a Waldhausen category is a category equipped with a functor ζ:W−→ Ho W sending weak equivalences to isomorphisms. Moreover, ζis initial among all functors W→Csending weak equivalences to isomorphisms, so Ho Wis well defned up to canonical isomorphism over W. This category can be constructued as a category of fractions, in the sense of [GZ67], by formally inverting weak equivalences in W. The class of weak equivalences is saturated if any morphism f:A→Bin W such that ζ(f) is an isomorphism in Ho Wis indeed a weak equivalence f:A∼ →B. Example 1.4.Weak equivalences in Cb(E), i.e. quasi-isomorphisms, are saturated since they are detected by a functor H∗:Cb(E)→AZ, the cohomology functor from bounded complexes in Eto Z-graded objects in A, see [CF00, Proposition 1.1]. The homotopy category always exists up to set theoretical difficulties which do not arise if Wis a small category, for instance. This is not a harmful assumption if one is interested in K-theory since smallness may also be required in order to have well defined K-theory groups. The homotopy category can however be constructed in a more straighforward way if the Waldhausen category Wsatisfies further properties. Definition 1.5. A Waldhausen category Wsatisifies the 2 out of 3 axiom provided given a commutative diagram in W C A @@       //B ^^= = = = = = = if two arrows are weak equivalences then the third one is also a weak equivalence. Given an object Ain Wacylinder IA is an object together with a factorization of the folding map (1,1): A∨A→Aas a cofibration followed by a weak equivalence, A∨A֌ iIA ∼ −→ pA. We say that Whas cylinders if all objects have a cylinder. Example 1.6.The Waldhausen category Cb(E) has cylinders. The cylinder of a bounded complex Acan be functorially chosen as (IA)n=An⊕An+1 ⊕An, d =  d−1 0 0−d0 0 1 d  : (IA)n−→ (IA)n+1. Remark 1.7.The 2 out of 3 axiom is often called the saturation axiom. We do not use this terminology in this paper in order to avoid confusion with Definition 1.3. Usually one considers more structured cylinders in Waldhausen categories, compare [Wei, Definition IV.6.8]. For the purposes of this paper it is enough to consider cylinders as defined above. MALTSINIOTIS’S FIRST CONJECTURE FOR K15 Remark 1.8.As one can easily check, a Waldhausen category with a saturated class of weak equivalences satisfies the 2 out of 3 axiom. This applies to Cb(E). A Waldhausen category with cylinders Wsatisfying the 2 out of 3 axiom is an example of a right derivable category, in the sense of [Cis03], also called precofibration category in [RB07], see [Cis03, Example 2.23] or [RB07, Proposition 2.4.2]. In particular any morphism in Wcan be factored as a cofibration followed by a weak equivalence which is left inverse to a trivial cofibration, see [RB07, Proposition 1.3.1]. Moreover, one can define a homotopy relation in Wand construct the homotopy category Ho Wby a homotopy calculus of left fractions as we indicate below, see [Cis03, Section 1] or [RB07, Section 5.4]. Let Wbe a Waldhausen category with cylinders satisfying the 2 out of 3 axiom. As usual we say that two morphisms f, g:A→Bin Ware strictly homotopic if there is a morphism H:IA →Bwith Hi = (f, g). The maps f, g are homotopic f≃gif there exists a weak equivalence h:B∼ →B′such that hf and hg are strictly homotopic. ‘Being homotopic’ is a natural equivalence relation and the quotient category is denoted by πW. The homotopy category Ho Wis obtained by calculus of left fractions in πW. Objects in Ho Ware the same as in W. A morphism A→B in Ho Wis represented by a diagram in W, A−→ α1 X∼ ←− α2 B. Another diagram A−→ α′ 1 Y∼ ←− α′ 2 B represents the same morphism if there is a diagram in W X >> α1 } } } } } } }`` ∼ α2 A A A A A A A A Z //oo∼  OOB Y α′ 1 A A A A A A A~~ ∼ α′ 2 } } } } } } } whose projection to πWis commutative. Notice that, by the 2 out of 3 axiom, the vertical arrows in this diagram are also weak equivalences. The composite of two morphisms A→ αB→ βCin Ho Wrepresented by A−→ α1 X∼ ←− α2 B−→ β1 Y∼ ←− β2 C is defined as follows. If β1is a cofibration then the push-out B push //β1 // ∼ α2  Y ∼¯α2  X//¯ β1 //X∪BY is defined, ¯α2is a weak equivalence, and βα:A→Cis represented by A−→ ¯ β1α1 X∪BY∼ ←− ¯α2β2 C. 6 FERNANDO MURO In general we can factor β1as cofibration followed by a weak equivalence β1:B֌ β′ 1 Z∼ −→ rY such that there is a morphism s:Y∼ ֌Zwith rs = 1Y. The diagram Y >> β1 ~ ~ ~ ~ ~ ~ ~`` ∼ β2 @ @ @ @ @ @ @ B Z // β′ 1 //oo∼ sβ2 r OO C commutes in W, so β:B→Cis also represented by B֌ β′ 1 Z∼ ←− sβ2 C, where the first arrow is a cofibration, and we can use this representative to define the composite βα:A→C. The functor ζ:W−→ Ho W is the identity on objects and sends a morphism f:A→Bto the morphism ζ(f): A→Brepresented by A−→ fB∼ ←− 1B B. If f:A∼ →Bis a weak equivalence then ζ(f) is an isomorphism and ζ(f)−1is represented by B−→ 1B B∼ ←− fA, hence a morphism α:A→Bin Ho Wrepresented by A−→ α1 X∼ ←− α2 B coincides with ζ(α2)−1ζ(α1) = α. Remark 1.9.If αabove is an isomorphism in Ho Wthen ζ(α1) = ζ(α2)αis also an isomorphism. In particular if Whas a saturated class of weak equivalences then α1:A∼ →Xis necessarily a weak equivalence. Remark 1.10.For W=Cb(E) the category πW=Hb(E) is usually termed the bounded homotopy category, while Ho W=Db(E) is called the bounded derived category of E. 2. On Waldhausen and derived K-theory Recall that a cofiber sequence in a Waldhausen category W A֌B։B/A is a push-out diagram A////  push B  0////B/A Therefore the quotient B/A is only defined up to canonical isomorphism over B, although the notation B/A is standard in the literature. MALTSINIOTIS’S FIRST CONJECTURE FOR K17 The K-theories we deal with in this paper are constructed by using the Waldhausen categories SnWthat we now recall. Definition 2.1. An object A•• in the category SnW,n≥0, is a commutative diagram in W (2.2) Ann . . . OO A22 //··· //A2n OO A11 //A12 // OO ··· //A1n OO A00 //A01 // OO A02 // OO ··· //A0n OO such that Aii = 0 and Aij ֌Aik ։Ajk is a cofiber sequence for all 0 ≤i≤j≤ k≤n. Notice that these conditions imply that the whole diagram is determined, up to canonical isomorphism, by the sequence of n−1 composable cofibrations (2.3) A01 ֌A02 ֌···֌A0n. A morphism A•• →B•• in SnWis a natural transformation between diagrams given by morphisms Aij →Bij in W. The category SnWis a Waldhausen category. A morphism A•• ∼ →B•• is a weak equivalence if all morphisms Aij ∼ →Bij are weak equivalences in W. A cofibration A•• ֌B•• is a morphism such that Aij ֌Bij and Bij ∪Aij Aik ֌Bik are cofibrations, 0 ≤i≤j≤k≤n. The distinguished zero object is the diagram with 0 in all entries. The categories SnWassemble to a simplicial category S.W. The face functor di:SnW→Sn−1Wis defined by removing the ith row and the ith column, and the degeneracy functor si:SnW→Sn+1Wis defined by duplicating the ith row and the ith column, 0 ≤i≤n. Faces and degeneracies are exact functors. For the definition of the simplicial structure it is crucial to consider the whole diagram (2.2) instead of just (2.3). One can obtain a pointed space out of the simplicial category S.Was follows. We restrict to the subcategories of weak equivalences wS.W, then we take levelwise the nerve in order to get a bisimplicial set Ner wS.W, we consider the diagonal simplicial set Diag Ner wS.W, and its geometric realization |Diag Ner wS.W|. This pointed space, actually a reduced CW-complex, is the 1-stage of the Waldhausen K-theory spectrum K(W) [Wal85], which is an Ω-spectrum, hence the Ktheory groups of Ware the homotopy groups Kn(W) = πn+1|Diag Ner wS.W|, n ≥0. 8 FERNANDO MURO We now assume that Whas cylinders and satisifies the 2 out of 3 axiom, so that the associated right pointed derivator DWis defined, see [Cis03, Corollary 2.24 and the duals of Lemmas 4.2 and 4.3]. Then the Waldhausen categories SnW also have cylinders and satisfy the 2 out of 3 axiom. We will neither recall the notion of derivator nor the definition of the derivator DWbut just the K-theory of DW, we refer the interested reader to [Gro90, Mal07, Gar06, RB07]. For this we consider the homotopy categories Ho SnWand the subgroupoids of isomorphisms iHo SnW. These groupoids form a simplicial groupoid iHo S.Wand we can consider the pointed space |Diag Ner iHo S.W|, which is the 1-stage of Garkusha’s derived K-theory Ω-spectrum DK(W). Garkusha [Gar05] considers derived K-theory for W=Cb(E), and more generally for Wa nice complicial biWaldhausen category, although the definition immediately extends to Waldhausen categories with cylinders satisfying the 2 out of 3 axiom, as indicated here. Moreover, Garkusha shows that there is a natural weak equivalence DK(W)∼ →K(DW) between the derived K-theory spectrum of a nice complicial biWaldhausen category Wand the K-theory spectrum of the associated derivator DW, compare [Gar05, Corollary 4.3]. Nevertheless [Gar05, Corollary 4.3] only uses the fact that all morphisms in Wfactor as a cofibration followed by a weak equivalence, compare also [Gar05, Lemmas 4.1 and 4.2], so we also have a natural weak equivalence DK(W)∼ →K(DW) for Wa Waldhausen category with cylinders satisfying the 2 out of 3 axiom, and therefore Kn(DW)∼ =πn+1|Diag Ner iHo S.W|, n ≥0. The functors ζ:SnW→Ho SnWrestrict to wSnW→iHo SnW. These functors give rise to a map |Diag Ner wS.W| −→ | Diag Ner iHo S.W| which induces the comparison homomorphisms in homotopy groups, µn:Kn(W)−→ Kn(DW), n ≥0. This map is actually the 1-stage of a comparison map of spectra (2.4) K(W)−→ K(DW). In the rest of this paper we will be mainly concerned with the structure of the bisimplicial sets X= Ner wS.Wand Y= Ner iHo S.Win low dimensions, that we now review more thoroughly. Abisimplicial set Zconsists of sets Zm,n,m, n ≥0, together with horizontal and vertical face and degeneracy maps dh i:Zm,n −→ Zm−1,n, sh i:Zm,n −→ Zm+1,n,0≤i≤m, dv j:Zm,n −→ Zm,n−1, sv j:Zm,n −→ Zm,n+1,0≤j≤n, satisfying some relations that we will not recall here, compare [GJ99]. An element zm,n ∈Zm,n is a bisimplex of bidegree (m, n) and total degree m+n. A generic bisimplex zm,n of bidegree (m, n) can be depicted as the product of two geometric simplices of dimensions mand nwith vertices labelled by the product set {0,...,m} × {0,...,n}, see Figs. 1 and 2. The horizontal ith face dh i(zm,n) is the face obtained by removing MALTSINIOTIS’S FIRST CONJECTURE FOR K19 (0,0) (1,0) z1,0 (0,1) (1,1) (0,0) (1,0) z1,1 (0,0) (1,0) (2,0)                    z2,0 Figure 1. Bisimplices of total degree 1 and 2. the interior, the vertices (i, j), for all j, and the incident faces of the boundary. Similarly the vertical jth face dv j(zm,n) is obtained by removing the interior, the vertices (i, j), for all i, and the incident faces of the boundary. (0,2) (1,2) ? ? ? ? ? ? ? ? (0,1) (0,0) (1,0) (1,1)         z1,2z2,1 (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 z3,0 (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 degree 3. The bisimplicial sets Xand Yare horizontally reduced, i.e. X0,n =Y0,n are singletons for all n≥0, X1,0=Y1,0is the set of objects in W,X1,1is the set of weak equivalences in W,Y1,1is the set of isomorphisms in Ho W, and X2,0=Y2,0 is the set of cofiber sequences, see Fig. 3. The set X1,2consists of pairs of composable weak equivalences in W,Y1,2is the set of composable isomorphisms in Ho W,X2,1is the set of weak equivalences between cofiber sequences i.e. weak equivalences in S2Wwhich are commutative diagrams in W (2.5) A′////B′////B′/A′ A//// ∼ OO B//// ∼ OO B/A ∼ OO 16 FERNANDO MURO We then take the coproduct of the first and the second pair of degenerate bisimplices. B A∨B              A OO OO//// A A∨B              B OO OO//// Finally we take the difference between the corresponding generators in D1W −        B A∨B              A OO OO////         +        A A∨B              B OO OO////         . There is a non-abelian Eilenberg-Zilber theorem behind this formula, compare [MT07, Theorem 4.10 and Example 4.13]. The main result of [MT07] is the following theorem. Theorem 4.3. [MT07, Theorem 1.7] Let Wbe a Waldhausen category. There is a natural isomorphism in Ho squad D∗W∼ = −→ λ0K(W). This result is meaningful since λ0K(W) is huge compared with D∗W, while D∗W is directly defined in terms of the basic structure of the Waldhausen category W. As a consequence we have an exact sequence of groups K1(W)֒→D1W∂ −→ D0W։K0(W). We now extend Theorem 4.3 to derived K-theory. Definition 4.4. We define Dder ∗Was the stable quadratic module generated in dimension zero by the symbols (DG1) [A] for any object in W, i.e. the same as (G1), and in dimension one by (DG2) [A∼ = →A′] for any isomorphism in Ho W, (DG3) [A֌B։B/A] for any cofiber sequence in W, i.e. the same as (G3), such that the following relations hold. (DR1) ∂[A∼ = →A′] = −[A′] + [A]. (DR2) = (R2). (DR3) = (R3). (DR4) [A1 →A] = 0. (DR5) = (R5). (DR6) For any pair of composable isomorphisms A∼ = →B∼ = →Cin Ho W, [A∼ = →C] = [B∼ = →C] + [A∼ = →B]. MALTSINIOTIS’S FIRST CONJECTURE FOR K117 (DR7) For any commutative diagram in Was (2.7) we have [α:A∼ = →A′] + [γ:B/A ∼ = →B′/A′][A]=−[A′֌B′։B′/A′] + [β:B∼ = →B′] + [A֌B։B/A]. Here α=ζ(α2)−1ζ(α1), β=ζ(β2)−1ζ(β1) and γ=ζ(γ2)−1ζ(γ1). (DR8) = (R8). (DR9) = (R9). If Wis a Waldhausen category with cylinders and a saturated class of weak equivalences then the presentation of the stable quadratic module Dder ∗Wis determined by the bisimplicial structure of Y= Ner iHo S.Wand the map ∨:Y×Y→Yin total degree ≤3, see Section 2, exactly in the same way as D∗Wis determined by X= Ner wS.Wand ∨:X×X→X, see Remark 4.2. Therefore replacing Xby Y in the proof of [MT07, Theorem 1.7] we obtain the following result. Theorem 4.5. Let Wbe a Waldhausen category with cylinders and a saturated class of weak equivalences. There is a natural isomorphism in Ho squad Dder ∗W∼ = −→ λ0K(DW). As a consequence we have an exact sequence of groups K1(DW)֒→Dder 1W∂ −→ Dder 0W։K0(DW). Remark 4.6.As one can easily check, taking λ0in the comparison map of spectra (2.4) which induces µn:Kn(W)→Kn(DW) in homotopy groups corresponds to the natural morphism in squad, ¯µ:D∗W−→ Dder ∗W, [A]7→ [A], [f:A∼ →A′]7→ [ζ(f): A∼ = →A′], [A֌B։B/A]7→ [A֌B։B/A], under the natural isomorphisms of Theorems 4.3 and 4.5. In particular taking π0and π1in this morphism of stable quadratic modules we obtain µ0and µ1, respectively. This fact will be used below in the proof of Theorem B. 5. Proof of Theorem B Theorem B is a consequence of the following result. Theorem 5.1. Let Wbe a Waldhausen category with cylinders and a saturated class of weak equivalences. The natural morphism in squad ¯µ:D∗W−→ Dder ∗W, defined in Remark 4.6, is an isomorphism. The key for the proof of Theorem 5.1 is the following lemma. Lemma 5.2. Let Wbe a Waldhausen category with cylinders satisfying the 2 out of 3 axiom. Two weak equivalences f, g:A∼ →A′which are homotopic f≃grepresent the same element in D1W, [f:A∼ →A′] = [g:A∼ →A′]. 18 FERNANDO MURO Proof. Let IA be a cylinder of Aand A∨A֌ iIA ∼ −→ pA a factorization of the folding map, i.e. if i= (i0, i1) then pi0=pi1= 1A. Since both pand 1Aare weak equivalences we deduce from the 2 out of 3 axiom that i0 and i1are also weak equivalences. Moreover, for j= 0,1, 0(R4) = [A1A →A] = [pij:A∼ →A] (R6) = [p:IA ∼ →A] + [ij:A∼ →IA], therefore [i0:A∼ →IA] = −[p:IA ∼ →A] = [i1:A∼ →IA]. Furthermore, f≃g, so there is a weak equivalence h:A′∼ →A′′ and a morphism H:IA →A′′ such that Hi0=hf and Hi1=hg. Again by the 2 out of 3 axiom H is a weak equivalence, and [h:A′∼ →A′′] + [f:A∼ →A′](R6) = [hf =Hi0:A∼ →A′′] (R6) = [H:IA ∼ →A′′] + [i0:A∼ →IA] = [H:IA ∼ →A′′] + [i1:A∼ →IA] (R6) = [hg =Hi1:A∼ →A′′] (R6) = [h:A′∼ →A′′] + [g:A∼ →A′], hence we are done.  We are now ready to prove Theorem 5.1. Proof of Theorem 5.1. We are going to define the inverse of ¯µ, ¯ν:Dder ∗W−→ D∗W. We first show that ¯ν0[A] = [A], ¯ν1[α:A∼ = →A′] = −[α2:A′∼ →X] + [α1:A∼ →X], ¯ν1[A֌B։B/A] = [A֌B։B/A], defines a stable quadratic module morphism ¯ν. Here A∼ −→ α1 X∼ ←− α2 A′ is a representative of the isomorphism α. For this we are going to prove that the image of [α:A∼ = →A′] does not depend on the choice of a representative. Suppose that A∼ −→ α′ 1 Y∼ ←− α′ 2 A′ MALTSINIOTIS’S FIRST CONJECTURE FOR K119 also represents α. Then there is a diagram in W X >> α1 ~ ~ ~ ~ ~ ~ ~ ~``α2 A A A A A A A A A Z // f1oof2  g OO g′ A′ Y α′ 1 @ @ @ @ @ @ @ @~~α′ 2 } } } } } } } } where all arrows are weak equivalences and the four triangles commute up to homotopy, so −[α2:A′∼ →X] + [α1:A∼ →X] = −[α2:A′∼ →X]−[g:X∼ →Z] +[g:X∼ →Z] + [α1:A∼ →X] (R6) =−[gα2:A′∼ →Z] + [gα1:A∼ →Z] Lemma 5.2 =−[f2:A′∼ →Z] + [f1: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 check that the definition of ¯νon generators is compatible with the defining relations. The only non-trivial part concerns relations (DR1), (DR6) and (DR7). Compatibility with (DR1) follows from ¯ν0∂[α:A∼ = →A′] = ∂¯ν1[α:A∼ = →A′] =−∂[α2:A′∼ →X] + ∂[α1:A∼ →X] (R1) =−(−[X] + [A′]) + (−[X] + [A]) =−[A′] + [A] =−¯ν0[A′] + ¯ν0[A]. In order to check compatibility with (DR6) we consider two composable isomorphisms in Ho W A∼ = −→ αB∼ = −→ βC and we take representatives of α,βand βα as in the following commutative diagram of weak equivalences in W X∪BY aa¯α2 C C C C C C C C== ¯ β1 =={ { { { { { { { Xaa α2D D D D D D D DBB α1       push Y\\ β2 8 8 8 8 8 8 8 A B ==β1 == { { { { { { { {C 20 FERNANDO MURO Then ¯ν1[βα:A∼ = →C] = −[¯α2β2:C∼ →X∪BY] + [¯ β1α1:A∼ →X∪BY] =−[¯α2β2:C∼ →X∪BY] + [¯α2β1:B∼ →X∪BY] −[¯ β1α2:B∼ →X∪BY] + [¯ β1α1:A∼ →X∪BY] (R6) =−([¯α2:Y∼ →X∪BY] + [β2:C∼ →Y]) +[¯α2:Y∼ →X∪BY] + [β1:B∼ →Y] −([¯ β1:X∼ →X∪BY] + [α2:B∼ →X]) +[¯ β1:X∼ →X∪BY] + [α1:A∼ →X] =−[β2:C∼ →Y] + [β1:B∼ →Y] −[α2:B∼ →X] + [α1:A∼ →X] = ¯ν1[β:B∼ = →C] + ¯ν1[α:A∼ = →B]. Let us now check compatibility with (DR7). −¯ν1[A′֌B′։B′/A′] +¯ν1[β:B∼ = →B′] +¯ν1[A֌B։B/A] = −[A′֌B′։B′/A′]−[β2:B′∼ →Y] +[X֌Y։Y/X]−[X֌Y։Y/X] +[β1:B∼ →Y] + [A֌B։B/A] (R7) =−([α2:A′∼ →X] + [γ2:B′/A′∼ →Y/X][A′]) +[α1:A∼ →X] + [γ1:B/A ∼ →Y/X][A] Rem. 3.2 =−[γ2:B′/A′∼ →Y/X]−[α2:A′∼ →X] +[α1:A∼ →X] + [γ1:B/A ∼ →Y/X] −h[A′], ∂[γ2]i+h[A], ∂[γ1]i Defn. 3.1 (2) and Rem. 3.2 =−[α2:A′∼ →X] + [α1:A∼ →X] −[γ2:B′/A′∼ →Y/X] + [γ1:B/A ∼ →Y/X] +h−∂[α2] + ∂[α1],−∂[γ2]i − h[A′], ∂[γ2]i+h[A], ∂[γ1]i (R1) =−[α2:A′∼ →X] + [α1:A∼ →X] −[γ2:B′/A′∼ →Y/X] + [γ1:B/A ∼ →Y/X] +h−(−[X] + [A′]) + (−[X] + [A]),−∂[γ2]i +h[A′],−∂[γ2]i+h[A], ∂[γ1]i =−[α2:A′∼ →X] + [α1:A∼ →X] −[γ2:B′/A′∼ →Y/X] + [γ1:B/A ∼ →Y/X] +h[A],−∂[γ2]i+h[A], ∂[γ1]i = +¯ν1[α:A∼ = →A′] + ¯ν1[γ:B/A ∼ = →B′/A′] +h¯ν0[A], ∂¯ν1[γ:B/A ∼ = →B′/A′]i Rem. 3.2 = ¯ν1[α:A∼ = →A′] + ¯ν1[γ:B/A ∼ = →B′/A′]¯ν0[A]. This establishes that ¯νis a well defined morphism of stable quadratic modules. MALTSINIOTIS’S FIRST CONJECTURE FOR K121 Let us now check that ¯µ¯ν= 1Dder ∗Wand ¯ν¯µ= 1D∗W. Both equations are obvious on generators (G1) = (DG1) and (G3) = (DG3). For (G2) ¯ν1¯µ1[f:A∼ →A′] = ¯ν1[ζ(f): A∼ = →A′] =−[1A′:A′∼ →A′] + [f:A∼ →A′] (R4) = [f:A∼ →A′]. If α:A∼ = →A′is an isomorphism in Ho Wwe have the following equation in Dder 1W, 0(DR4) = [A1A →A] = [α−1α:A∼ = →A] (DR6) = [α−1:A′∼ = →A] + [α:A∼ = →A′], so [α−1:A′∼ = →A] = −[α:A∼ = →A′]. Now for (DG2) ¯µ1¯ν1[α:A∼ = →A′] = −¯µ1[α2:A′∼ →X] + ¯µ1[α1:A∼ →X] =−[ζ(α2): A′∼ = →X] + [ζ(α1): A∼ = →X] = [ζ(α2)−1:X∼ = →A′] + [ζ(α1): A∼ = →X] (DR6) = [α=ζ(α2)−1ζ(α1): A∼ = →A′]. The proof of Theorem 5.1 is now finished.  Remark 5.3.Let Wbe a Waldhausen category with cylinders satisfying the 2 out of 3 axiom. We do not assume that Whas a saturated class of weak equivalences. However we can endow the underlying category with a new Waldhausen category structure which does have a saturated class of weak equivalences. We consider the Waldhausen category Wwith the same underlying category as W. Cofibrations in Ware also de same as in W. Weak equivalences in Ware the morphisms in Wwhich are mapped to isomorphisms in Ho Wby the canonical functor ζ:W→Ho W. Therefore weak equivalences in Ware also weak equivalences in Wbut the converse need not hold. This indeed defines a Waldhausen category W with cylinders and a saturated class of weak equivalences, and the obvious exact functor W→Winduces an isomorphism on the associated derivators DW∼ =DW, compare [Cis03, dual of Proposition 3.16] and [RB07, Theorem 6.2.2]. Hence we have a commutative diagram for n= 0,1, Kn(W)µn//  Kn(DW) ∼ =  Kn(W)µn ∼ =//Kn(DW) Here the lower arrow is an isomorphism by Theorem B. Now we can use the ‘fibration theorem’, [Wal85, 1.6.7] and [Sch06, Theorem 11], to embed the morphisms µn:Kn(W)→K(DW), n= 0,1, in an exact sequence. More precisely, let W0 be the full subcategory of Wspanned by the objects which are isomorphic to 0 in Ho W. The category W0is a Waldhausen category where a morphism is a cofibration, resp. a weak equivalence, if and only if it is a cofibration, resp. a weak 22 FERNANDO MURO equivalence, in W. There is an exact sequence K1(W0)−→ K1(W)µ1 −→ K1(DW)δ −→ K0(W0)−→ K0(W)µ0 −→ K0(DW)→0. The group K0(W0) has also been considered by Weiss in [Wei99]. Weiss defines the Whitehead group of Was Wh(W) = K0(W0). Moreover, for any morphism f:A→A′which becomes an isomorphism in Ho Whe defines the Whitehead torsion τ(f)∈Wh(W), which is the obstruction for fto be a weak equivalence in W. If f:A→Ais an endomorphism which maps to an automorphism in Ho W then one can check that δ[ζ(f): A∼ = →A] = −τ(f), therefore an automorphism in Ho Wcomes from a weak equivalence in Wif and only if its class in derivator K1comes from Waldhausen K1. References [Bau91] H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, Walter de Gruyter, Berlin, 1991. [BCC93] M. Bullejos, P. Carrasco, and A. M. Cegarra, Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg-Mac Lane’s classification theorem, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 1, 163–189. [BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. [CF00] C. Casacuberta and A. Frei, On saturated classes of morphisms, Theory Appl. Categ. 7(2000), no. 4, 43–46. [Cis02] D.-C. Cisinski, Th´eor`emes de cofinalit´e en K-th´eorie (d’apr`es Thomason), Seminar notes, http://www.math.univ-paris13.fr/~cisinski/cofdev.dvi, 2002. [Cis03] ,Cat´egories d´erivables, Preprint, http://www.math.univ-paris13.fr/~cisinski/kthwt.dvi, 2003. [CMM04] P. Carrasco and J. Mart´ınez-Moreno, Simplicial cohomology with coefficients in symmetric categorical groups, Appl. Categ. Structures 12 (2004), no. 3, 257–285. [CN05] D.-C. Cisinski and A. Neeman, Additivity for derivator K-theory, Preprint, http://www.math.univ-paris13.fr/~cisinski/kthder.dpf, 2005. [Con84] D. Conduch´e, Modules crois´es g´en´eralis´es de longueur 2, J. Pure Appl. Algebra 34 (1984), no. 2-3, 155–178. [Gar05] G. Garkusha, Systems of diagram categories and K-theory II, Math. Z. 249 (2005), no. 3, 641–682. [Gar06] ,Systems of diagram categories and K-theory I, Algebra i Analiz 18 (2006), no. 6, 131–186, (Russian). [Gil81] H. Gillet, Riemann-Roch theorems for higher algebraic K-theory, Adv. Math. 40 (1981), 203–289. [GJ99] P. J. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics, no. 174, Birkh¨auser Verlag, Basel, 1999. [GM03] S. I. Gelfand and Y. I. Manin, Methods of homological algebra, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. [Gro90] A. Grothendieck, Les d´erivateurs, Manuscript, http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html, 1990. [GZ67] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, SpringerVerlag, Berlin-Heidelberg-New York, 1967. [Mal07] G. Maltsiniotis, La K-th´eorie d’un d´erivateur triangul´e, Categories in algebra, geometry and mathematical physics (A. Davydov et al., ed.), Contemp. Math., no. 431, American Mathematical Society, 2007, with an appendix “Le d´erivateur triangul´e associ´e `a une cat´egorie exacte” by B. Keller, pp. 341–368. [MT07] F. Muro and A. Tonks, The 1-type of a Waldhausen K-theory spectrum, Advances in Math. 216 (2007), no. 1, 178–211. MALTSINIOTIS’S FIRST CONJECTURE FOR K123 [RB07] A. Radulescu-Banu, Cofibrations in homotopy theory, Preprint, arXiv:math/0610009v2, 2007. [Sch02] M. Schlichting, A note on K-theory and triangulated categories, Invent. Math. 150 (2002), no. 1, 111–116. [Sch06] ,Negative K-theory of derived categories, Math. Z. 253 (2006), 97–134. [TT90] R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and derived categories, The Grothendieck festschrift, a collection of articles written in honor pf the 60th birthday of Alexander Grothendieck, Progress in Math. 88, vol. III, Brikh¨auser, Boston, 1990, pp. 247–435. [TV04] B. To¨en and G. Vezzosi, A remark on K-theory and S-categories, Topology 43 (2004), no. 4, 765–791. [Wal85] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. [Wei] C. Weibel, An introduction to algebraic K-theory, Book in progress available at http://math.rutgers.edu/~weibel/Kbook.html. [Wei99] M. Weiss, Hammock localization in Waldhausen categories, J. Pure Appl. Algebra 138 (1999), no. 2, 185–195. Universitat de Barcelona, Departament d’ ` Algebra i Geometria, Gran via de les corts catalanes 585, 08007 Barcelona, Spain E-mail address:[email protected]