A TRIANGULATED CATEGORY WITHOUT MODELS FERNANDO MURO Abstract.We exhibit a triangulated category which is neither the stable category of a Frobenius category nor a full triangulated subcategory of * *the homotopy category of a stable model category. 1.Introduction Triangulated categories are fundamental tools in both algebra and topology. In algebra they use to arise as the stable category of a Frobenius category ([Hel6* *8, 4.4], [GM03 , IV.3 Exercise 8]). In topology they usually appear as a full triangulat* *ed subcategory of the homotopy category of a Quillen stable model category ([Hov99* * , 7.1]). The triangulated categories which belong, up to exact equivalence, to on* *e of these two families will be termed algebraic and topological, respectively. We b* *orrow this terminology from [Kel06, 3.6] and [Sch06]. Algebraic triangulated categori* *es are generally also topological, but there are many well-known examples of topologic* *al triangulated categories which are not algebraic. In the present paper we produce the first example of a triangulated category which is neither algebraic nor topological. Let P(R) be the category of finitely generated projective modules over a com- mutative ring R. Theorem 1.1. The pair given by the category P(Z=4) together with the identity functor can be endowed with a unique triangulated category structure. This theorem is proved in Section 2. There we also show that (1.2) Z=4 -2!Z=4 -2!Z=4 -2!Z=4, is an exact triangle in P(Z=4), therefore P(Z=4) cannot be algebraic. Indeed gi* *ven an object X in an algebraic triangulated category T and an exact triangle X 2.1X-!X -! Y -! X, the equation 2 . 1Y = 0 holds, compare [Kel06, 3.6] and [Sch06]. This needs not happen in a topological triangulated category. This fact is illustrated by the * *clas- sical example of X = S the sphere spectrum in the Spanier-Whitehead category. Nevertheless we prove as a main theorem that the triangulated category P(Z=4) is not topological. ____________ 1991 Mathematics Subject Classification. 18E30, 55P42, 16E40. Key words and phrases. Triangulated category, stable model category, cohomol* *ogy of cate- gories, Toda bracket, Mac Lane cohomology. The author was partially supported by the Spanish Ministry of Education and * *Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship E* *X2004- 0616, and a Juan de la Cierva research contract. 1 2 FERNANDO MURO Theorem 1.3. The triangulated category P(Z=4) with the identity translation fun* *c- tor is not exact equivalent to a full triangulated subcategory of the homotopy * *category Ho M of a stable model category M. The proof of this theorem is completed at the very end of the paper. Unlike in the algebraic case, one cannot prove that P(Z=4) is not topological by using lo* *cal properties. In order to prove Theorem 1.3 we use the existence of the universal Toda bracket in the cohomology of the homotopy category 2 H3(Ho M, Hom HoM( , -)), as defined in [BD89 ]. This universal Toda bracket determines all Toda brackets* * in Ho M, and Toda brackets determine the exact triangles, see [Hel68, Theorem 13.2* *]. We will explicitly compute H3(P(Z=4), Hom Z=4) and show that no element in this group would yield (1.2) as an exact triangle. The cohomological theory of triangulated categories is studied in [BM05a , BM* *05b ]. There, given an additive category A, a self equivalence : A ~!A, and a cohomol- ogy class r 2 H3(A, Hom A( , -)), we describe conditions on r, or rather on a lift of r to the translation cohomo* *l- ogy group H3(A, ), under which the family of exact triangles determined by r yields a triangulated structure on A with translation functor . Not only algeb* *raic and topological triangulated categories arise in this way. More generally, the * *tri- angulated homotopy category of a stable S-category S, in the sense of [TV04 , 7* *], is associated to the classifying cohomology class ([BJ02 ]) of the groupoid-enr* *iched category obtained by taking fundamental groupoid on the morphism simplicial sets of S. This cohomology class can be regarded as the first Postnikov invariant of S, compare [DKS86 ]. The theory of stable 1-categories developed in [Lur06] is equivalent to the theory of stable S-categories, compare [Joy], so the homotopy category of a stable 1-category is also cohomologically triangulated in the sen* *se of [BM05a , BM05b ]. The results of this paper show that P(Z=4) is not cohomo- logically triangulated so it cannot be obtained by any of the procedures descri* *bed above. Acknowledgements. Stefan Schwede's talk on algebraic vs. topological triangu- lated categories at the ICM 2006 Satellite Workshop on Triangulated Categories, held in Leeds, stimulated my interest on this subject. I am grateful to Bernhard Keller for conversations on the results of this paper. I feel indebted to Amnon Neeman, who suggested the possibility of constructing a triangulated structure * *on P(Z=4) by using Heller's theory. 2.The triangulated category Recall that a triangulated category is an additive category T together with a self-equivalence : T ~!T, called translation functor, and a collection of diag* *rams (2.1) A -f!B -i! C -q! A, called distinguished or exact triangles, which satisfy Puppe's axioms ([Pup62 ]* *) and Verdier's octahedral axiom ([Ver77]), see [Nee01] for a unified reference. If w* *e do not require the octahedral axiom then we speak of a pretriangulated category, as considered for example in [Hel68]. An exact functor between (pre)triangulated A TRIANGULATED CATEGORY WITHOUT MODELS 3 categories is an additive functor commuting with the translation functors up to natural isomorphism and preserving exact triangles. Heller developed in [Hel68] a method to "count" pretriangulated structures on* * an additive category T which is the category of injective objects in a Frobenius c* *ategory B, with a given translation functor . The following proposition is essentiall* *y a restatement of [Hel68, Theorem 16.4] for B the category of finitely generated Z* *=4- modules. Proposition 2.2. There is a bijective correspondence between the pretriangulated category structures on P(Z=4) with identity translation functor and the automor- phisms of the identity functor in P(Z=2). Proof.The category B of finitely generated Z=4-modules is a Frobenius category, see for instance [Rot79, Theorems 4.35 and 4.37]. Any object in B is isomorphic* * to a finite direct sum of copies of Z=2 and Z=4. Moreover, Z=4 is an injective hul* *l of Z=2, therefore the stable category B is equivalent to P(Z=2) through the funct* *or P(Z=2) B i B . The suspension functor S in B is determined by the choice of short exact sequences in B j r X ,! CX i SX, with CX injective, for each X in B. For our convenience we choose CX to be an injective hull of X, hence SX is always a Z=2-vector space. For X injective we can take CX = X and SX = 0, and for X a Z=2-vector space SX = X and jr = 2: CX ! CX. Notice that S in B restricts to the identity functor in P(Z=2). Notice also that any automorphism OE of the identity functor in P(Z=2) satisfies OE + OE = 0. Now the proposition follows from [Hel68, Theorem 16.4]. The category P(Z=4) has at least the pretriangulated structure associated to * *the identity automorphism of the identity functor in P(Z=2), for which we will prov* *e in this section the octahedral axiom. By Proposition 3.3 this is the only automorp* *hism of the identity functor in P(Z=2). This will complete the proof of Theorem 1.1. We now recall some useful definitions from [Nee01]. A candidate triangle in a pretriangulated category is a diagram as (2.1) where if, qi, and ( f)q are zero morphisms. A morphism of candidate triangles is a commutative diagram (2.3) A __f__//_B_i__//_C_q__// A |k0| |k1| |k2| ||k0 fflffl|ffflffl|0fflffl|i|fflffl0q0 A0 _____//B0___//_C0___//_ A0 This morphism k = (k0, k1, k2) is homotopic to another morphism k0= (k00, k01, * *k02) between the same candidate triangles if there are morphisms B -ff1!A0, C -ff2!B0, A -ff0!C0, such that -k1 + k01 = f0ff1 + ff2i, -k2 + k02 = i0ff2 + ff0q, - k0 + k00 = q0ff0 + (ff1f). 4 FERNANDO MURO A candidate triangle is contractible if the identity morphism is homotopic to t* *he zero morphism. We also recall that the mapping cone of a morphism like (2.3) is the candidate triangle ` -i 0 ' ` -q 0' ` - f 0 ' k1 f0 0 k2 i0 0 k0 q0 0 B A0 - ! C B - ! A C -! B A . The mapping cones of two homotopic morphisms are isomorphic. Proposition 2.4. Consider the pretriangulated category structure on P(Z=4) with identity translation functor associated to the identity automorphism of the ide* *ntity functor in P(Z=2). The exact triangles of this triangulated category are the c* *an- didate triangles which are isomorphic to the direct sum of a contractible candi* *date triangle with a candidate triangle of the form X -2! X -2! X -2! X. Proof.Contractible candidate triangles are always exact, see [Nee01, Proposition 1.3.8]. Any object X in P(Z=4) is, up to isomorphism, of the form CY for some Z=2-vector space Y . The canonical resolution of Y given by the choice of cones* * and suspensions in the proof of Proposition 2.2 takes the form (a) CY -2! CY -2! CY -2! CY ! . .,. therefore the description of exact triangles in P(Z=4) given in [Hel68] shows t* *hat the triangle of the statement is exact. Let us now prove that all exact triangles can be decomposed as indicated in t* *he statement. Let (b) A -f!B -i! C -q! A be an exact triangle. The kernel of f in the category of Z=4-modules decomposes* * as a direct sum Kerf ~=X Y with X injective and Y a Z=2-vector space. Since (a) is a minimal injective resolution of Y , combining elementary homological algeb* *ra with axiom (Tr3) for pretriangulated categories, see [Nee01], we obtain a morph* *ism k of candidate triangles given by monomorphisms j" 2 2 2 Y"O`O```//`CY____//"C`Y___//"C`Y___//"C`Y" ` O |k0 |k1 k2| k0| fflfflOO fflffl||"fflffl||ffflffl||ifflffl||q Ker f` ` `//A______//B______//C_____//_A such that the square with dashed arrows on the left (which is not part of the morphism k) commutes. Since CY is injective the exact triangle (b) decomposes as the direct sum of the following two candidate triangles (c) CY -2! CY -2! CY -2! CY, ~f ~i ~q (d) Cokerk0 -! Coker k1 -! Coker k2 -! Coker k0. The axioms of a pretriangulated category show that if we extend (b) to a 3-peri* *odic cochain complex then the resulting cochain complex is acyclic, as it happens wi* *th A TRIANGULATED CATEGORY WITHOUT MODELS 5 (c), therefore the long exact sequence in cohomology yields the same property f* *or (d). In particular we can decompose (d) as a diagram p0 i1 p1 i2 p2 i0 X0 i0,!Cokerk0 i X1 ,!Coker k1 i X2 ,!Coker k2 i X0 ,!Coker k0. where each subdiagram o ,! o i o is a short exact sequence. By construction X0 * *~= X, the injective direct summand of Kerf. Since proyectives and injectives coinc* *ide we derive that all Xi are injective, as the Cokerki, so we can take morphisms r0 l0 r1 l1 r2 l2 r0 X0 j Cokerk0 - X1 j Cokerk1 - X2 j Cokerk2 - X0 j Cokerk0 such that rsis = 1Xs, psls = 1Xs+1, and isrs + lsps = 1Cokerksfor all s 2 Z=3 = {0, 1,.2}Now one can easily check that the morphisms ffs = rsls-1 yield a con- tracting homotopy for the candidate triangle (d). Now we use this explict description of the exact triangles in P(Z=4) to prove* * the octahedral axiom. Proposition 2.5. The pretriangulated category structure on P(Z=4) with identity translation functor associated to the identity automorphism of the identity fun* *ctor in P(Z=2) is indeed triangulated. Proof.We are going to show that P(Z=4) satisfies axiom (Tr4') in [Nee01, Defini* *tion 1.3.13], which is equivalent to Verdier's octahedral axiom. We have to show that for any morphism of exact triangles k :T ! T 0we can modify k2 so that the mapping cone of k is again an exact triangle. The mapping cone of a morphism only depends on its homotopy class, and any morphism with contractible source or target is homotopic to the zero morphism, therefore by Proposition 2.4 we on* *ly need to consider the case where T and T 0are defined by two objects X and X0 in P(Z=4) and the multiplication by 2, as in the statement of Proposition 2.4. The exact triangle given by an object Y and multiplication by 2 will be denoted by Y (2), so T = X(2) and T 0= X0(2). We can suppose without loss of generality that X = X1 X2 X3, X0 = X10 X2 X4,1 1 0 0 k0 = @ 0 2 0 A . 0 0 0 Let 0 1 0 0 0 k02 = k1 + @ 0 2 0 A . 0 0 0 Obviously k0= (k0, k1, k02): T ! T 0is also a morphism of candidate triangles. * *We are going to show that the mapping cone of k0 is exact. Since k is a morphism 2k0 = 2k1, therefore there exists g :X ! X0 such that k1 = k0 + 2g. Then k0 is homotopic to k00= (k0, k0, k002), where 0 1 1 0 0 k002= @ 0 0 0 A , 0 0 0 6 FERNANDO MURO so it will suffice to show that the mapping cone of k00is exact. The mapping co* *ne of k00is isomorphic to the direct sum of four candidate triangles, namely X3(2), X4(2), the mapping cone of the identity morphism in X1(2) (which is contractibl* *e), and the mapping cone of (2, 2, 0): X2(2) ! X2(2), which we denote by T2. The candidate triangle T2 is also exact since we have an isomorphism `` ' ' 1 0 ~= 1 1 , 1, 1:T2 -! (X2 X2)(2). Now this proposition follows from Proposition 2.4. Remark 2.6. One can actually check that given a prime p and a possitive integer n the category P(Z=pn) with the identity translation functor has a pretriangula* *ted structure if and only if n = 1 or p = n = 2. In all cases the pretriangulated structure is unique and triangulated. For n = 1, P(Z=p) is the stable category * *of the Frobenius abelian category of finitely generated Z=p2-modules, so P(Z=p) is algebraic. This illustrates the singularity of the triangulation of P(Z=4). 3. Cohomology of categories and Mac Lane cohomology Let us recall from [BW85 ] the definition of the Baues-Wirsching cohomology * *of categories. Definition 3.1. Let C be a category. A C-bimodule L is a functor L: Copx C ! Ab , where Ab is the category of abelian groups. The Baues-Wirsching complex F *(C, L) is given by the following products indexed by all sequences of morphi* *sms of length n in C Y F n(C, L)= L(Xn, X0), n 0. X0oe1...oenXn In this formula we assume that a sequence of length 0 is an object X0 in C which we also identify with the identity morphism 1X0. The coordinate of c 2 F n(C, L) in X0 oe1. .o.enXn will be denoted by c(oe1, . .,.oen). The value of the differ* *ential d over an n-cochain c, n 1, is defined as d(c)(oe1, . .,.oen+1)=L(Xn+1, oe1)c(oe2, . .,.oen+1) nX + (-1)ic(oe1, . .,.oeioei+1, . .,.oen+1) i=1 +(-1)n+1L(oen+1, X0)c(oe1, . .,.oen). For n = 0 and oe :X ! Y , d(c)(oe) = L(1, oe)c(X) - L(oe, 1)c(Y ). The cohomology of C with coefficients in L is the cohomology of the complex F *(C, L). It is denoted by H*(C, L). For the functorial behaviour of cohomology of categories see [Mur06 ]. If the bimodule L takes values in the category of k-modules for k a commutati* *ve ring then H*(C, L) is a graded k-module. It follows from the very definition that given an additive category A the coh* *omol- ogy group H0(A, Hom A) is isomorphic to the endomorphism group of the identity functor in A, see [JP91, Proposition 3.2]. Given an R-module M the following co- homology groups of P(R) coincide with Mac Lane cohomology by [JP91, Theorem A TRIANGULATED CATEGORY WITHOUT MODELS 7 A and Corollary 3.11]. (3.2) H*(P(R), Hom R(-, M R -)) ~= HML*(R, M). For R = M = Z=2 the 0-dimensional group was computed in [FLS94 ], H0(Z=2, Z=2) * *~= Z=2. From this isomorphism we derive the following proposition. Proposition 3.3. The identity functor in P(Z=2) has only two endomorphisms, the identity and the zero endomorphisms. In particular it has only one automorphism, the identity morphism. Let us recall now the Pirashvili-Waldhausen homology of categories defined in [PW92 ], which is defined in a dual way to Baues-Wirsching cohomology. Definition 3.4. Let C be a category and let L be a C-bimodule. The Pirashvili- Waldhausen complex F*(C, L) is given by the direct sums M Fn(C, L) = L(X0, Xn), n 0. X0oe1...oenXn An element c 2 L(X0, Xn) regarded as an n-chain in the direct factor correspodi* *ng to X0 oe1. .o.enXn will be denoted by c . (oe1, . .,.oen). The differential is * *defined by d(c . (oe1, . .,.oen))=L(oe1, Xn)(c) . (oe2, . .,.oen) n-1X + (-1)ic . (oe1, . .,.oeioei+1, . .,.oen) i=1 +(-1)nL(X0, oen)(c) . (oe1, . .,.oen-1), for n 2, and for n = 1 and oe :X ! Y we have d(c . oe) = L(oe, X)(c) . X - L(Y, oe)(c) . Y . This differential differs from the one defined in [PW92 ] by * *a sign. Of course this does not affect the homology. The homology of C with coefficients in L is the homology of the complex F*(C,* * L). It is denoted by H*(C, L). If L takes values in the category of k-modules for k a commutative ring then H*(C, L) is a graded k-module. By [PW92 ] the homology of P(R) is naturally isomorphic to the Mac Lane homol- ogy of the ring R and to the topological Hochschild homology of the Eilenberg-M* *ac Lane ring spectrum HR in the following case, (3.5) H*(P(R), Hom R) ~= HML*(R) ~= T HH*(HR). In the following theorem we construct a universal coefficients spectral seque* *nce for the Mac Lane cohomology of commutative rings which is crucial for the com- putations in this paper. Theorem 3.6. For any commutative ring R and any R-module M there is a spec- tral sequence Ep,q2= ExtpR(HMLq(R), M) =) HMLp+q(R, M). Here M is regarded as a symmetric R-bimodule for the definition of the Mac Lane cohomology. 8 FERNANDO MURO Proof.Given two finitely generated projective R-modules P, Q there are natural isomorphisms Hom R(Hom R (Q, P ), M) -ffM R Hom R(P, Q) -fi!HomR (P, M R Q), defined by ff(m f)(g)= m . trace(fg), g 2 Hom R(Q, P ), fi(m f)(x)= m f(x), x 2 P. Indeed the homomorphisms ff and fi are clearly natural and biadditive, so it is enough to check that they are isomorphisms for P = Q = R, and this case is trivial. The isomorphisms ff and fi determine a cochain isomorphism Hom R(F*(P(R), Hom R), M) ~= F *(P(R), Hom R(-, M R -)), compare Definitions 3.1 and 3.4. Now if E* is an injective resolution of M then the spectral sequence of the statement is the spectral sequence of the bicomplex Hom R (F*(P(R), Hom R), E*). The following proposition is a simple application of the previous theorem. Proposition 3.7. For any Z=4-module M there is a natural isomorphism HML3(Z=4, M) ~= Hom Z=4(Z=2, M). Proof.The topological Hochschild homology of Z=4 is computed in [Bru00]. The lower homology groups are 8 >> 0, n = 1, >: Z=4, n = 2, Z=2, n = 3. This implies that the E2-term of the universal coefficients spectral sequence s* *atisfies Ep,q2= 0 in case p > 0 and q < 3. Therefore the isomorphism of the statement follows. 4.Toda brackets in triangulated and homotopy categories We recall from [Hel68, 13] the definition of Toda bracket in a triangulated c* *ate- gory. Definition 4.1. Let T be a triangulated category with translation functor . Gi* *ven three composable morphisms in T W -f! X -g! Y -h! Z, such that gf = 0 and hg = 0, the Toda bracket 2 _________T(_W,_Z)__________h . T( W, Y ) + T( X, Z* *) . ( f) is defined in the following way. If W -f! X -i! Cf -q! W A TRIANGULATED CATEGORY WITHOUT MODELS 9 is an exact triangle in T the axioms of a triangulated category imply the exist* *ence of a commutative diagram f i q W ____//_X____//Cf____// W || || | | || || a| |b || || fflffl| fflffl| W _f__//_X_g__//_Y__h__//_Z The morphism b is a representative of the Toda bracket . The Toda brac* *ket does not depend on the choices made for its definition since we divide out the indeterminacy. Moreover, all representatives of can be obtained in t* *his way. Example 4.2. We derive from the very definition that the Toda bracket of the morphisms of an exact triangle like (2.1) contains the identity morphism 1 A 2 . In fact exact triangles are characterized among candidate triangles * *by this property and a further homological condition, see [Hel68, Therorem 13.2]. This and Proposition 2.4 imply that, in the triangulated category P(Z=4) of Theorem 1.1, the sequence of morphisms Z=4 -2!Z=4 -2!Z=4 -2!Z=4 has Toda bracket 0 6= <2, 2, 2> 2 __________P(Z=4)(Z=4,_Z=4)___________2~.=P(Z=4)(Z=4,ZZ=4* *)=+2P(Z=4)(Z=4,.Z=4) . 2 Toda brackets in the homotopy category of a stable model category are deter- mined by a class in Baues-Wirsching cohomology of categories. A 3-dimensional class in the Baues-Wirsching cohomology of a category with zero object defines Toda brackets as we now indicate. The Toda category Toda introduced in [BD89 ] has five objects *, 1, 2, 3, 4. The object * is a zero ob* *ject, and there are only three morphisms which are neither identities nor zero morphisms, namely jn :n ! n + 1 for n = 1, 2, 3. The Toda category can be represented by t* *he following commutative diagram _0___________________________________________* *___________________________________ _______________________________________________* *___________________________________________ ________________________________________________ Toda = 1 __j1_//2j2_//_____>>__________3j3_//_4. __________________________________________* *_________________________ ________________________________________* *_____________________________________________________________________________* *_________________ 0 If M is a Toda -bimodule with M(*, -) = 0 and M(-, *) = 0 then H3(Toda , M) ~= ____________M(1,_4)___________M(1,.j 3)M(1, 3) + M(j* *1, 4)M(2, 4) The isomorphism follows from the fact that, with this kind of coefficients, the* * Baues- Wirsching cohomology can be computed by using cochains which are normalized with respect to identity and zero morphisms, see [BD89 ]. This isomorphism sends a cohomology class represented by such a normalized 3-cocycle c to the class of c(j3, j2, j1). 10 FERNANDO MURO Definition 4.3. Given a category C with zero object *, a C-bimodule L with L(*, -) = 0 and L(-, *) = 0, and a cohomology class i 2 H3(C, L), then the i-Toda bracket of a diagram in C W -f! X -g! Y -h! Z, with gf = 0 and hg = 0, is defined as follows. The diagram corresponds to a functor ': Toda ! C with '(*) = *, '(j1) = f, '(j2) = g, and '(j2) = h, and the i-Toda bracket is the pull-back along ' of the cohomology class i i = '*i 2 H3(Toda , L(', ')) ~= ___________L(W,_Z)___________L(W,.h)L* *(W, Y ) + L(f, Z)L(X, Z) If c is a cocycle representing i which is normalized with respect to identity a* *nd zero morphisms then i is represented by c(h, g, f). If a triangualted category T = Ho M is the homotopy category of a stable model category M then Toda brackets in T are determined by the universal Toda bracket 2 H3(T, Hom T( , -)), i.e. Toda brackets in the triangulated category Ho M are -Toda brackets in t* *he sense of the previous definition. This universal Toda bracket is defined in [BD* *89 ] on the homotopy category of fibrant-cofibrant objects, but it can be placed in the cohomology of the full homotopy category by using the fact that equivalences of categories induce isomorphisms in Baues-Wirsching cohomology. By [Hel68, Theorem 13.2] the universal Toda bracket also determines the exact triangles in Ho M. Proposition 4.4. For any i 2 H3(P(Z=4), Hom Z=4) ~=Z=2 the i-Toda bracket of Z=4 -2!Z=4 -2!Z=4 -2!Z=4 vanishes. Proof.The isomorphism H3(P(Z=4), Hom Z=4) ~=Z=2 follows from the isomorphism (3.2) and Proposition 3.7. Actually the isomorphism class of this group is not important for the proof. We just want to remark that if it were trivial there w* *ould be nothing to prove, but this is not the case, so the argument below is needed. Let ': Toda ! P(Z=4) be the functor corresponding to Z=4 -2!Z=4 -2!Z=4 -2!Z=4, and let i: Z=2 ,! Z=4 be the inclusion. The functorial behaviour of Baues- Wirsching cohomology implies that the following diagram commutes (a) Hom Z=4(-,i -)* H3(P(Z=4), Hom Z=4(-, Z=2 Z=4-))____________//_H3(P(Z=4), Hom Z=4) '* || |'*| fflffl| fflffl| H3(Toda , Hom Z=4(', Z=2 Z=4'))Hom_______//_H3(Toda , Hom Z=4(', ')) Z=4(',i ')* Here an upper * indicates the pull-back along a functor and the lower * indicat* *es a push-forward along a bimodule morphism. A TRIANGULATED CATEGORY WITHOUT MODELS 11 The isomorphism in (3.2) together with Proposition 3.7 prove that we have a commutative diagram Hom Z=4(-,i -)* H3(P(Z=4), Hom Z=4(-, Z=2 Z=4-))__________//H3(P(Z=4), Hom Z=4) ~=|| |~=| fflffl| HomZ=4(Z=2,i) fflffl| Hom Z=4(Z=2, Z=2)___________________//_HomZ=4(Z=2, Z=4) ~=|| |~=| fflffl| fflffl| Z=2 __________________________________Z=2 Hence the upper horizontal homomorphism in (a) is an isomorphism. On the other hand we have another commutative diagram HomZ=4(',i ')* H3(Toda , Hom Z=4(', Z=2 Z=4'))_____________//H3(Toda , Hom Z=4(', ')) ~=|| |~=| fflffl| fflffl| ________HomZ=4(Z=4,Z=2)_____ _HomZ=4(Z=4,i)//_____HomZ=4(Z=4,Z=4)_____ 2.HomZ=4(Z=4,Z=2)+HomZ=4(Z=4,Z=2).2 2.HomZ=4(Z=4,Z=4)+HomZ=4(Z=4,Z=4).2 ~=|| |~=| fflffl| HomZ=4(Z=4,i) fflffl| Hom Z=4(Z=4, Z=2)____________________//HomZ=4(Z=4, Z=4)=(2 . 1Z=4) ~=|| |~=| fflffl| fflffl| Z=2__________________0__________________//Z=2 Therefore the lower horizontal homomorphism in diagram (a) is 0, so the vertical homomorphism in the right of diagram (a) must also be zero. This finishes the proof. Now we are ready to prove Theorem 1.3. Proof of Theorem 1.3.Suppose by the contrary that there is a fully faithful exa* *ct functor _ :P(Z=4) ! Ho M. Such a functor preserves Toda brackets, therefore Toda brackets in P(Z=4) are _*-Toda brackets, hence by Proposition 4.4 the Toda bracket of Z=4 -2!Z=4 -2!Z=4 -2!Z=4 vanishes. 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