SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE - AN ALGEBRAIC APPROACH HENNING KRAUSE Abstract.We prove a modified version of Ravenel's telescope conjecture. * *It is shown that every smashing subcategory of the stable homotopy category is gener* *ated by a set of maps between finite spectra. This result is based on a new charac* *terization of smashing subcategories, which leads in addition to a classification of t* *hese subcate- gories in terms of the category of finite spectra. The approach presente* *d here is purely algebraic; it is based on an analysis of pure-injective objects in a com* *pactly gener- ated triangulated category, and covers therefore also situations arising* * in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from* * lo- calization functors l :S ! S which induce for every spectrum X a natural isomor* *phism l(X) ' X ^ l(S) between the localization of X and the smash product of X with t* *he localization of the sphere spectrum S. In fact, a localization functor has this* * property if and only if it preserves arbitrary coproducts in S. Therefore one calls a full * *subcategory R of S smashing if R = {X 2 S | l(X) = 0} for some localization functor l :S ! S which preserves coproducts. In this paper we study smashing subcategories from* * an algebraic point of view. The main result is a new characterization of smashing * *subcat- egories which leads to a classification in terms of certain ideals in the categ* *ory of finite spectra. One motivation for this work is the telescope conjecture of Ravenel an* *d Bous- field which states that every smashing subcategory is generated by finite spect* *ra. The approach presented here is purely algebraic and covers therefore also situation* *s arising in algebraic geometry and representation theory where one studies certain trian* *gulated categories having a number of formal properties in common with the stable homot* *opy category. Let C be a compactly generated triangulated category, for example the stable * *homo- topy category. Thus C is a triangulated category with arbitrary coproducts, an* *d C is generated by a set of compact objects (an object X in C is compact if the repre* *sentable functor Hom (X; -) preserves coproducts). Recall that a full triangulated subca* *tegory B of C is localizing if B is closed under taking coproducts. We say that a loc* *alizing subcategory B is strictly localizing if the inclusion functor B ! C has a right* * adjoint, and B is called smashing if there exists a right adjoint for the inclusion B ! * *C which preserves coproducts. Note that a full subcategory B is strictly localizing if * *and only if there exists a localization functor l :C ! C such that B = {X 2 C | l(X) = 0}, * *and B is smashing if and only if the corresponding localization functor preserves coprod* *ucts. Theorem A. Let B be a localizing subcategory of C, and denote by I the ideal* * of maps between compact objects in C which factor through some object in B. Then the fo* *llowing conditions are equivalent: (1) B is smashing; 1 2 HENNING KRAUSE (2) an object X in C belongs to B if and only if every map C ! X from a compa* *ct object C factors through a map C ! D in I; (3) an object X in C satisfies Hom (B; X) = 0 if and only if Hom (I; X) = 0. Let us mention an immediate consequence: The smashing subcategories of C form* * a set of cardinality at most 2 where denotes the cardinality of the set of isomorph* *ism classes of maps between compact objects in C. For example, the stable homotopy category has precisely 2@0 smashing subcategories because, in this case = @0, and arith* *metic localization gives rise to a smashing subcategory for every set of primes. Given any class I of maps in C, we say that a localizing subcategory B is gen* *erated by I if B is the smallest localizing subcategory of C such that every map in I * *factors through some object in B. For example, B is generated by a class I = {idXi | i * *2 I} of identity maps if and only if B is the smallest localizing subcategory containin* *g Xi for all i 2 I. Corollary. Every smashing subcategory is generated by a set of maps between com* *pact objects. The statement of the corollary is a modified version of the following "telesc* *ope con- jecture" which is based on conjectures of Ravenel [23, 1.33] and Bousfield [6, * *3.4] for the stable homotopy category: Every smashing subcategory is generated by a set of identity maps between compa* *ct objects. In this generality, the conjecture is known to be false. In fact, Keller give* *s an example of a smashing subcategory which contains no non-zero compact object [14]. Despi* *te some efforts of Ravenel [24], the conjecture remains open for the stable homotopy ca* *tegory. The characterization of smashing subcategories leads to a classification in t* *erms of certain ideals which we now explain. We denote by C0 the full triangulated subc* *ategory of compact objects in C and call an ideal I of maps in C0 exact if there exists* * an exact functor f :C0 ! D into a triangulated category D such that I = {OE 2 C0 | f(OE)* * = 0}. Theorem B. Let C be a compactly generated triangulated category and suppose t* *hat every cohomological functor Cop0! Ab is isomorphic to Hom (-; X)|C0 for some ob* *ject X in C. Then the maps B 7! {OE 2 C0 | OE factors through an object in}B and I 7! {X 2 C | every map C ! X, C 2 C0, factors through a map C ! D in I} induce mutually inverse bijections between the set of smashing subcategories of* * C and the set of exact ideals in C0. Note that the additional assumption on C in the preceding theorem is automati* *cally satisfied if there are at most countably many isomorphism classes of maps betwe* *en compact objects in C; in particular the stable homotopy category has this prope* *rty [21]. The classification of smashing subcategories has the following consequence. Corollary. A localizing subcategory B of C is smashing if and only if B is gene* *rated by a class of maps between compact objects in C. Moreover, given any class I of* * maps between compact objects in C, there exists a localizing subcategory of C which * *is generated by I. SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 3 The preceding corollary amounts to a classical result of Bousfield and Ravene* *l if I is a class of identity maps. In fact, they showed for the stable homotopy category t* *hat every class of compact objects generates a localizing subcategory which is smashing [* *6, 23]. However, if I is a class of arbitrary maps in C, it is not clear that there exi* *sts a localizing subcategory which is generated by I. Our analysis of smashing subcategories is based on the concept of purity for * *compactly generated triangulated categories. Let us call a map X ! Y in C a pure monomorp* *hism if the induced map Hom (C; X) ! Hom (C; Y ) is a monomorphism for all compact o* *b- jects C. An object X is called pure-injective if every pure monomorphism X ! Y * *splits. These definitions are motivated by analogous concepts for the category of modul* *es over a ring [7]. In this context one frequently studies the indecomposable pure-inje* *ctive mod- ules; they form the Ziegler spectrum of the ring [28]. We shall see that the is* *omorphism classes of indecomposable pure-injective objects in C form a set which we denot* *e by Sp C. Theorem C. Let B be a smashing subcategory of C, and let U be the set of obj* *ects Y in Sp C such that Hom (B; Y ) = 0. Then the following holds for any object X in* * C: (1) X 2 B if and only if Hom (X; U) = 0; Q (2) Hom (B; X) = 0 if and only if there is a pure monomorphism X ! i2IYi wi* *th Yi2 U for all i. We obtain the following consequence if we put B = 0. Q Corollary. Every object X in C admits a pure monomorphism X ! i2IYi with Yi2 Sp C for all i. In particular, Hom (X; Y ) = 0 for all Y 2 SpC implies X = 0. The concept of purity is closely related to the occurence of phantom maps. R* *ecall that a map X ! Y is a phantom map if the induced map Hom (C; X) ! Hom (C; Y ) is zero for all compact objects C. From the existence of pure-injective envelopes * *in C we derive for every object X the existence of a universal phantom map ending in X * *and a universal pure monomorphism starting in X. Theorem D. For every object X in C there exists, up to isomorphism, a unique* * triangle X0 -ff!X -fi!X00-fl!X0[1] having the following properties: (A1) a map OE: Y ! X is a phantom map if and only if OE factors through ff; (A2) every endomorphism OE of X0 satisfying ff = ff OOE is an isomorphism. The same triangle is characterized, up to isomorphism, by the following propert* *ies: (B1) a map OE: X ! Y is a pure monomorphism if and only if fi factors through * *OE; (B2) every endomorphism OE of X00satisfying fi = OE Ofi is an isomorphism. Our main tool in this paper is a functor h: C ! M into a module category M wh* *ich has the following universal property: (1) h: C ! M is a cohomological functor into an abelian AB 5 category which p* *re- serves coproducts; (2) any functor h0:C ! M0 as in (1) has a unique factorization h0= f Oh such * *that f :M ! M0 is exact and preserves coproducts. In Section 1 of this paper we exploit the fact that h induces an equivalence * *between the full subcategory of pure-injective objects in C and the full subcategory of* * injective objects in M. We continue in Section 2 with the problem of extending cohomologi* *cal 4 HENNING KRAUSE functors. For instance, we prove the following result where C0 denotes the full* * triangu- lated subcategory which is formed by the compact objects in C. Theorem E. Every cohomological functor f :C0 ! A into an abelian AB 5 category A extends, up to isomorphism, uniquely to a cohomological functor f0: C ! A whi* *ch preserves coproducts. Moreover, if A is the category of abelian groups, then f0* * preserves products if and only if f0 ' Hom (X; -) for some compact object X in C. In Section 3 we derive from the universal property of h: C ! M a strong relat* *ion between localizing subcategories in C and localizing subcategories in M. This i* *nterplay between triangulated and module categories is crucial for our characterization * *of smash- ing subcategories. The final Section 4 is devoted to the proofs for the main re* *sults of this paper. Acknowledgement. I would like to thank Dan Christensen and Bernhard Keller for a number of helpful comments concerning the material of this paper. Thanks als* *o to Amnon Neeman for pointing out a mistake in a preliminary version of this paper.* * In addition, I am grateful to an anonymous referee for numerous suggestions. 1. Purity 1.1. Pure-exactness. Let C be a triangulated category [26, 27] and suppose that* * arbi- trary coproducts exist in`C. An object X in C is`called compact if for every fa* *mily (Yi)i2I in C the canonical map iHom (X; Yi) ! Hom (X; iYi) is an isomorphism. We de* *note by C0 the full subcategory of compact objects in C and observe that C0 is a tri* *angulated subcategory of C. Following [20], the category C is called compactly generated * *provided that the isomorphism classes of objects in C0 form a set, and Hom (C; X) = 0 fo* *r all C in C0 implies X = 0 for every object X in C. Examples of compactly generated tr* *iangu- lated categories arise in stable homotopy theory, algebraic geometry, and repre* *sentation theory. Definition 1.1. Let C be a compactly generated triangulated category. (1) A map X ! Y in C is said to be a pure monomorphism if the induced map Hom (C; X) ! Hom (C; Y ) is a monomorphism for all compact objects C in C. (2) An object X in C is called pure-injective if every pure monomorphism OE: * *X ! Y splits, i.e. there exist a map OE0:Y ! X such that OE0OOE = idX. (3) A triangle X ! Y ! Z ! X[1] is called pure-exact if the induced sequence 0 ! Hom (C; X) ! Hom (C; Y ) ! Hom (C; Z) ! 0 is exact for all compact objects C in C. The preceding definition is motivated by analogous definitions for the catego* *ry of mod- ules over a ring [7]. However, contrary to the concept for modules, a pure mono* *morphism in C is usually not a monomorphism in the categorical sense. For the sake of co* *mplete- ness we include the following definition. Definition 1.2. Let C be a compactly generated triangulated category. (1) A map Y ! Z in C is said to be a pure epimorphism if the induced map Hom (C; Y ) ! Hom (C; Z) is an epimorphism for all compact objects C in C. (2) An object Z in C is called pure-projective if every pure epimorphism :Y* * ! Z splits, i.e. there exist a map 0:Z ! Y such that O 0= idZ. The concept of purity is closely related to the occurence of phantom maps. R* *ecall that a map X ! Y is a phantom map provided that the induced map Hom (C; X) ! Hom (C; Y ) is zero for all compact objects C in C. SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 5 Lemma 1.3. For a triangle X !OEY ! Z O! X[1] the following are equivalent: (1) OE is a phantom map; (2) is a pure monomorphism; (3) O is a pure epimorphism; (4) the shifted triangle Y ! Z ! X[1] ! Y [1] is pure-exact. Proof.Clear, since the induced sequence Hom (C; X) ! Hom (C; Y ) ! Hom (C; Z) !* * __ Hom (C; X[1]) is exact for every C 2 C0. * * |__| Lemma 1.4. The following conditions are equivalent for an object X in C: (1) X is pure-injective; (2) if OE: Y ! X is a phantom map, then OE = 0; (3) if OE: V ! W is a pure monomorphism, then every map V ! X factors through OE. Proof.(1) , (2) follows immediately from the preceding lemma, and the direction* * (3) ) (1) is also clear. To prove (1) ) (3), let OE: V ! W be a pure monomorphism a* *nd let :V ! X be a map with X pure-injective. We obtain a commutative diagram Ufl- ! V? -OE! W? -! U[1]fl flfl ?y ?y flfl U - ! X -! Y -! U[1] such that both rows are triangles. The map U ! V is a phantom map since OE is* * a pure monomorphism, and it follows that U ! X is a phantom map. Therefore the map X ! Y is a pure monomorphism which splits since X is pure-injective. It follows* * that_ factors through OE. * * |__| 1.2. Modules. Let C be any additive category. A C-module is by definition an ad* *ditive functor Cop ! Ab into the category Ab of abelian groups, and we denote for C-mo* *dules M and N by Hom (M; N) the class of natural transformations M ! N. A sequence L ! M ! N of maps between C-modules is exact if the sequence L(X) ! M(X) ! N(X) is exact for all X in C. A C-module M is finitely generated if there exists an exa* *ct sequence Hom (-; X) ! M ! 0 for some X in C, and M is finitely presented if there exists* * an exact sequence Hom (-; X) ! Hom (-; Y ) ! M ! 0 with X and Y in C. Note that Hom (M; N) is a set for every finitely generated C-module M by Yoneda's lemma. * *The finitely presented C-modules form an additive category with cokernels which we * *denote by mod C. It is well-known that mod C is abelian if and only if every map Y ! Z* * in C has a weak kernel X ! Y , i.e. the sequence Hom (-; X) ! Hom (-; Y ) ! Hom (-; * *Z) is eaxct. In particular, mod C is abelian if C is triangulated. Suppose now that C is skeletally small. Then the C-modules form together with the natural transformations an abelian category which we denote by Mod C. Note * *that ModQC has arbitraryQproducts and coproducts which are defined pointwise. For ex* *ample, ( iMi)(X) = iMi(X) for a family (Mi)i2Iin Mod C and X in C. We denote for ev* *ery X in C by HX = Hom (-; X) the corresponding representable functor and recall t* *hat Hom (HX ; M) ' M(X) for every module M by Yoneda's lemma. It follows that HX is a projective object in Mod C. We shall also need to use the fact that Mod * * C is a Grothendieck category, which as far as we are concerned means that it has injec* *tive envelopes [9]. 6 HENNING KRAUSE Our main tool for studying a compactly generated triangulated category C is t* *he restricted Yoneda functor hC: C -! Mod C0; X 7! HX = Hom (-; X)|C0: 1.3. Brown representability. Recall that a (covariant) functor f :C ! A from a * *tri- angulated category C into an abelian category A is cohomological if for every t* *riangle X ! Y ! Z ! X[1] in C the sequence f(X) ! f(Y ) ! f(Z) ! f(X[1]) is exact. Examples of cohomological functors are the representable functors Hom (X; -): C* * ! Ab and Hom (-; X): Cop ! Ab for any X in C. The Brown representability theorem cha* *r- acterizes the representable cohomological functors Cop ! Ab for a compactly gen* *erated triangulated category C. Theorem (Brown).` LetQf :Cop ! Ab be a cohomological functor such that the ca* *non- ical map f( iXi) ! if(Xi) is an isomorphism for every family (Xi)i2Iof objec* *ts in C. Then f ' Hom (-; X) for some object X in C. * * __ Proof.See Theorem 3.1 in [20]. * *|__| The existence of arbitrary products in C is a well-known consequence of the B* *rown representability theorem. Lemma 1.5. The category C has arbitrary products. Q Proof.Let (Xi)i2Ibe a family of objects in C and let f = iHom (-; Xi). Clearl* *y, f is a cohomological functor which sends coproducts to products. Thus f ' HomQ(-; X)* * by __ the Brown representability theorem, and it is easily checked that X = iXi in * *C. |__| 1.4. Pure-injectives. Our analysis of pure-injective objects in a compactly gen* *erated triangulated category C is based on some properties of the restricted Yoneda fu* *nctor hC: C ! Mod C0. We need two lemmas. Recall that a module M is fp-injective if Ext1(N; M) = 0 for every finitely presented module N. Lemma 1.6. The C0-module HX is fp-injective for every X in C. Proof.A finitely presented C0-module N has a projective presentation HA -! HB -! HC -! N -! 0 coming from a triangle A ! B ! C ! A[1] in C with objects in C0. Thus one can compute Ext1 as the cohomology of the complex Hom (HC ; HX ) -! Hom (HB ; HX ) -! Hom (HA ; HX ): This is, however, isomorphic to Hom (C; X) ! Hom (B; X) ! Hom (A; X), so it is * *exact._ Therefore Ext1(N; HX ) = 0 and HX is fp-injective. * * |__| Lemma 1.7. Let M be an injective C0-module. Then there exists, up to isomorph* *ism, a unique object X in C such that M ' HX . Moreover, hC induces an isomorphism Hom (Y; X) ! Hom (HY ; HX ) for all Y in C. Proof.Let f = Hom (-; M) OhC. Then f is a cohomological functor since hC is coh* *omo- logical and Hom (-; M) is exact.` Moreover,QhC preserves coproducts and Hom (-;* * M) induces an isomorphism Hom ( iNi; M) ' iHom (Ni; M) for every family (Ni)i2I* *of C0-modules. Therefore f ' Hom (-; X) for some object X in C by the Brown repre- sentability theorem. The induced map Hom (X; X) ' f(X) = Hom (HX ; M) sends idX to a map OE: HX ! M which is an isomorphism since HX (C) = Hom (C; X) ' Hom (HC ; M) ' M(C) SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 7 for every compact object C by Yoneda's lemma. The inverse OE-1 :M ! HX induces * *an isomorphism Hom (Y; X) ' Hom (HY ; M) ' Hom (HY ; HX ) * * __ which is precisely the map induced by hC. This finishes the proof. * * |__| The following theorem collects a number of characterizing properties of pure-* *injective objects. We denote for every object X and every set I by XI the product and by * *X(I) the coproduct of cardI copies of X. Theorem 1.8. The following conditions are equivalent for an object X in C: (1) X is pure-injective; (2) HX = Hom (-; X)|C0 is an injective C0-module; (3) the map Hom (Y; X) ! Hom (HY ; HX ), OE 7! Hom (-; OE)|C0, is an isomorph* *ism for all Y in C; (4) if OE: Y ! X is a phantom map, then OE = 0; (5) for every set I the summation map X(I)! X factors through the canonical m* *ap X(I)! XI. Proof.(1) ) (2) Let HX ! M be an injective envelope in Mod C0. It follows fr* *om Lemma 1.7 that M ' HY for some object Y in C, and the map HX ! M ' HY is of the form HOEfor some OE: X ! Y . Clearly, OE is a pure monomorphism, and OE spl* *its since X is pure-injective. Thus HX is a direct summand of M and therefore injective. (2) ) (3) Use Lemma 1.7. (3) ) (4) If OE: Y ! X is a phantom map, then Hom (-; OE)|C0 = 0. Thus it fol* *lows from (3) that OE = 0. (4) ) (1) Use Lemma 1.4. (2) ) (5) Suppose that M = HX is an injective C0-module. It follows that t* *he summation map M(I)! M factors through the canonical monomorphism M(I)! MI. The corresponding map HXI ! HX is of the form HOEfor some map OE: XI ! X by Lemma 1.7, and it follows that the composition of OE with X(I)! XI is the summa* *tion map. (5) ) (2) M = HX is an fp-injective C0-module by Lemma 1.6 which is injective* * if the summation map M(I)! M factors through the canonical monomorphism M(I)! MI * * __ for every set I by [17, Theorem 2.6]. * * |__| We discuss a number of consequences. Corollary 1.9. The restricted Yoneda functor C ! Mod C0 induces an equivalence * *be- tween the full subcategory of pure-injective objects in C and the full subcateg* *ory of injec- tive objects in Mod C0. Proof.The restricted Yoneda functor sends pure-injectives to injectives by Theo* *rem 1.8,_ and it is fully faithful and dense by Lemma 1.7. * * |__| Recall that an object X in`any additive category is indecomposable if X 6= 0 * *and every decomposition X = X1 X2 implies X1 = 0 or X2 = 0. The isomorphism class* *es of indecomposable injective objects in Mod C0 form a set since every indecompo* *sable injective C0-module arises as an injective envelope of a finitely generated C0-* *module. It follows that the indecomposable pure-injective objects in C form a set which we* * denote by Sp C. Q Corollary 1.10. Every object X in C admits a pure monomorphism X ! i2IYi with Yi2 SpC for all i. In particular, Hom (X; Y ) = 0 for all Y 2 SpC implies X = 0. 8 HENNING KRAUSE Proof.We observe first that the indecomposable injective C0-modules cogenerate * *Mod C0. In fact, one could take the injective envelopes of all simple modules. To see t* *his, observe that every non-zero module M has a finitely generated non-zero submodule U whic* *h has a maximal submodule V by Zorn's lemma. This gives a non-zero map from M to the injective envelope of U=V . Now letQX be an object in C and choose a monomorphi* *sm HX ! M in Mod C0 such that M = iMi is a product of indecomposable injective C0-modules.QIt follows from Lemma 1.7 that this map comes from a pure monomorph* *ism X ! i2IYi with Mi ' HYi for all i, and each Yi is indecomposable pure-injecti* *ve_by Corollary 1.9. * *|__| Remark 1.11. The set SpC carries two natural topologies. A subset U of SpC is Z* *iegler- closed if and only if U = {X 2 Sp C | Hom (OE; X) = 0 for allOE 2 I} for some c* *lass I of maps in C0; see [15, Lemma 4.1]. A subset U of Sp C is Zariski-open if and * *only if there exists some class I of maps in C0 such that U = {X 2 Sp C | Hom (OE; X* *) = 0 for someOE 2 I}; see [9, Chap. VI]. We refer to [18] for a detailed discussio* *n of both topologies in the context of modules over a ring. A map OE: X ! Y in C is said to be a pure-injective envelope of X if Y is pur* *e-injective and a composition OOE with a map :Y ! Z is a pure-monomorphism if and only * *if is a pure monomorphism. Lemma 1.12. The following are equivalent for a pure monomorphism OE: X ! Y in* * C: (1) OE is a pure-injective envelope of X; (2) Y is pure-injective and every endomorphism of Y satisfying OOE = OE* * is an isomorphism; (3) HOE:HX ! HY is an injective envelope in Mod C0. * *__ Proof.Straightforward. |* *__| Corollary 1.13. Every object X in C admits a pure-injective envelope OE: X ! Y . If OE0:X ! Y 0is another pure-injective envelope, then there exists an isomorph* *ism :Y ! Y 0such that OE0= OOE. Proof.The assertion is a consequence of Theorem 1.8 and the existence of inject* *ive_ envelopes in Mod C0. |* *__| We are now in a position to prove Theorem D. In fact, the existence of a univ* *ersal phantom map X0 ! X ending in a fixed object X follows from the existence of a p* *ure- injective envelope X ! X00. We recall Theorem D for the convenience of the read* *er. Theorem 1.14. For every object X in C there exists, up to isomorphism, a uniq* *ue triangle X0 -ff!X -fi!X00-fl!X0[1] having the following properties: (A1) a map OE: Y ! X is a phantom map if and only if OE factors through ff; (A2) every endomorphism OE of X0 satisfying ff = ff OOE is an isomorphism. The same triangle is characterized, up to isomorphism, by the following propert* *ies: (B1) a map OE: X ! Y is a pure monomorphism if and only if fi factors through * *OE; (B2) every endomorphism OE of X00satisfying fi = OE Ofi is an isomorphism. SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 9 Proof.Let X be an object in C and complete the pure-injective envelope fi :X ! * *X00 to a triangle X0 -ff!X -fi!X00-fl!X0[1]: The map ff is a phantom map by Lemma 1.3 since fi is a pure monomorphism, and the property (4) in Theorem 1.8 implies that ff is a universal phantom map endi* *ng in X since X00is pure-injective. On the other hand, fi is a universal pure monomor* *phism starting in X by Lemma 1.4 since X00is pure-injective. This establishes (A1) an* *d (B1). Condition (B2) is an immediate consequence of Lemma 1.12, and (A2) then follows* * from (B2). It is easily checked that each pair of conditions characterizes the above* * triangle,_ and therefore the proof is complete. * * |__| It is interesting to observe that the full subcategory of pure-injective obje* *cts in C is completely determined by the full subcategory C0 of compact objects in C. Corollary 1.15. Let C and D be compactly generated triangulated categories, and* * sup- pose that there exists an equivalence f :C0 ! D0 between the full subcategories* * of com- pact objects in C and D. Then f induces an equivalence between the full subcate* *gories of pure-injective objects in C and D. Proof.The functor hC: C ! Mod C0 induces an equivalence between the full subcat* *egory of pure-injectives in C and the full subcategory of injective C0-modules by Cor* *ollary 1.9. The assertion now follows since an equivalence f :C0 ! D0 induces an equivalenc* *e_ Mod C0 ! Mod D0. |__| 1.5. Pure-injective modules. The concept of purity has been studied extensively by algebraists. Pure-exactness and pure-injectivity for modules over a ring ha* *ve been introduced by Cohn [7], and we refer to [13] for a modern treatment of this sub* *ject. Let us recall briefly the relevant definitions. Let be an associative ring w* *ith identity. We consider the category Mod of (right) -modules. A sequence 0 ! X ! Y ! Z ! 0 of maps in Mod is pure-exact if the induced sequence 0 ! Hom (C; X) ! Hom (C; Y ) ! Hom (C; Z) ! 0 is exact for all finitely presented -modules C. T* *he map X ! Y in such a sequence is called a pure monomorphism. Note that any pure- exact sequence is automatically an exact sequence in the usual sense. A module * *X is pure-injective if every pure monomorphism X ! Y splits. Suppose now that is a quasi-Frobenius ring, i.e. projective and injective -m* *odules coincide. In this case the stable category Mod__ is triangulated; e.g. see [1* *2]. Recall that the objects in Mod__ are those of Mod , and for two -modules X; Y one defi* *nes Hom__(X; Y ) to be Hom (X; Y ) modulo the subgroup of maps which factor through* * a pro- jective -module. Note that the projection functor Mod ! Mod__ preserves produc* *ts and coproducts. Thus Mod__ has arbitrary coproducts, and it is not difficult to* * check that an object X in Mod__ is compact if and only if X ' Y in Mod__ for some fin* *itely presented -module Y . Therefore Mod__ is compactly generated. Proposition 1.16. A -module X is pure-injective if and only if X is a pure-inj* *ective object in Mod__. Proof.We use the following characterization of pure-injectivity for -modules wh* *ich is due to Jensen and Lenzing [13, Proposition 7.32]: A -module X is pure-injective* * if and only if for every set I the summation map oeI: X(I)! X factors through the cano* *nical map I: X(I)! XI. We now combine this characterization with the characterization of pure-injectivity in Mod__ from Theorem 1.8. Thus any pure-injective -module * *is a 10 HENNING KRAUSE pure-injective object in Mod__. To prove the converse, let X be a pure-injectiv* *e object in Mod__ and fix a set I. Thus there exists a map OE: XI ! X in Mod such that oeI- OE OI factors through a projective -module P , i.e. oeI- OE OI = fi Off fo* *r some map ff: X(I)! P . The map ff factors through the monomorphism I since P is injecti* *ve, i.e. ff = ff0OI for some map ff0, and therefore oeI = fi Off + OE OI = (fi Off0+ OE) OI: * * __ Thus oeI factors through I, and this finishes the proof. * * |__| For some further discussion of the relation between pure-injectives in Mod * *and Mod__ we refer to [16, 5]. 2. Cohomological and exact functors 2.1. Extending functors. Let C be any triangulated category. We recall the foll* *owing well-known property of the Yoneda functor h: C ! mod C, X 7! Hom (-; X). Lemma 2.1. Every additive functor f :C ! A into an abelian category A extends* *, up to isomorphism, uniquely to a right exact functor f0: mod C ! A such that f = f* *0Oh. The functor f0 is exact if and only if f is a cohomological functor. Proof.Any finitely presented C-module M has a projective presentation Hom(-;OE) Hom(-; ) Hom (-; X) -! Hom (-; Y ) -! Hom (-; Z) -! M -! 0 coming from a triangle X -OE!Y - ! Z -O! X[1] in C. We obtain a right exact functor f0: mod C ! A if we define f0(M) = Cokerf* *( ). Clearly, f = f0Oh holds by construction. Exactness of f0implies that f is cohom* *ological, since h is cohomological. Suppose now that f is cohomological. Taking project* *ive presentations of the modules in an exact sequence 0 ! M1 ! M2 ! M3 ! 0 in mod C as above, one obtains the following commutative diagram: 0? 0? 0? 0? ?y ?y ?y ?y f(X1)? -! f(Y1)? - ! f(Z1)? -! f0(M1)? -! 0 ?y ?y ?y ?y ` ` ` f(X1 ? X3) -! f(Y1? Y3) - ! f(Z1? Z3) -! f0(M2)? -! 0 ?y ?y ?y ?y f(X3)? -! f(Y3)? - ! f(Z3)? -! f0(M3)? -! 0 ?y ?y ?y ?y 0 0 0 0 The rows are exact since f is cohomological, and therefore the exactness of the* * first three columns implies the exactness of the sequence 0 ! f0(M1) ! f0(M2) ! f0(M3) ! 0.* * __ Thus f0 is exact and this finishes the proof. * * |__| Recall that an abelian category A satisfies Grothendieck's AB 5 condition if * *A has arbitrary coproducts and taking filtered colimits preserves exactness. For exa* *mple, any module category is an AB 5 category. Suppose now that C is a skeletally sm* *all triangulated category and consider the Yoneda functor h: C ! Mod C. SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 11 Lemma 2.2. Every additive functor f :C ! A into an abelian AB 5 category A ex- tends, up to isomorphism, uniquely to a right exact functor f0: Mod C ! A which preserves coproducts and satisfies f = f0Oh. The functor f0 is exact if and onl* *y if f is a cohomological functor. Proof.Any C-module M has a projective presentation a (Hom (-;OEij))a Hom (-; Xi) -! Hom (-; Yj) -! M -! 0 i j which is given by a family of maps OEij:Xi! Yj in C. We obtain`a functor`f0: Mo* *d C ! A if we define f0(M) as the cokernel of the map (f(OEij)): if(Xi) ! jf(Yj) * *in A. It is easily checked that f0 preserves colimits, and that f = f0Oh. The restrictio* *n f0|mod C is exact if and only if f is cohomological by the preceding lemma. Now observe* * that any exact sequence 0 ! L OE!M ! N ! 0 in Mod C can be written as a filtered co* *limit of exact sequences 0 ! Li!OEiMi! iNi! 0 in mod C. To see this, write OE as a fi* *ltered colimit of maps OE0i:L0i! Mi in mod C. Denoting by i:Mi ! Ni the cokernel of e* *ach OE0i, we obtain a filtered system of exact sequences 0 ! Li OEi!Mi !iNi ! 0 in * *mod C with colimit 0 ! L !OEM ! N ! 0. It follows that f0 is exact if and only if f* * is_ cohomological since A is an AB 5 category. * * |__| We are now in a position to prove the first part of Theorem E. To this end su* *ppose that C is compactly generated and consider the restricted Yoneda functor hC: C ! Mod* * C0. Proposition 2.3. Let C be a compactly generated triangulated category. Then e* *very cohomological functor f :C0 ! A into an abelian AB 5 category A extends, up to * *iso- morphism, uniquely to a cohomological functor f0:C ! A which preserves coproduc* *ts. Proof.We denote by f* :Mod C0 ! A the exact colimit preserving functor which e* *x- tends f, and define f0 = f* OhC. Clearly, f0 is cohomological, preserves copro* *ducts, and f0|C0 = f. Suppose there is another functor f00:C ! A with these properties* *. We construct a natural transformation j :f0 ! f00as follows. If X is coproduct of * *compact objects in C, then we obtain a unique isomorphism jX :f0(X) ! f00(X) since f0 a* *nd f00preserve coproducts. Now let X = X0 be an arbitrary object in C. We can choo* *se pure-exact triangles Xi+1 ! Pi! Xi! Xi+1[1] with Pi being a coproduct of compact objects for i = 0; 1, and we obtain a sequence of maps P1 ! P0 ! X in C such th* *at HP1 ! HP0 ! HX ! 0 is exact. This gives a commutative diagram f0(P1)? -! f0(P0)? -! f0(X) -! 0 ?yj ? P1 yjP0 f00(P1) -! f00(P0) -! f00(X) where the upper row is exact since f* is exact. Thus there is a unique map jX :* *f0(X) ! f00(X) since the composition P1 ! P0 ! X is zero. Now let B be the full subcate* *gory formed by the objects X in C such that jX is an isomorphism. Clearly, B contain* *s C0, and it is triangulated since f0 and f00are cohomological. Furthermore, B is clo* *sed under taking coproducts since f0 and f00preserve coproducts. Thus B = C by [20, Lemma* * 3.2],_ and therefore j :f0 ! f00is an isomorphism. * * |__| The following consequence generalizes a result from [8]. 12 HENNING KRAUSE Corollary 2.4. Let C be a compactly generated triangulated category and let f :* *C ! A be a cohomological functor into an abelian AB 5 category A. Suppose also th* *at f preserves coproducts. Then there exists, up to isomorphism, a unique exact fun* *ctor f0: Mod C0 ! A which preserves coproducts and satisfies f = f0OhC. Proof.Let f0: Mod C0 ! A be the colimit preserving functor extending f|C0 which exists by Lemma 2.2. We have f ' f0OhC by the preceding theorem since both func* *tors __ are cohomological and preserve coproducts. This gives the uniqeness of f0. * * |__| Corollary 2.5. The following are equivalent for a map OE: X ! Y in a compactly generated triangulated category C: (1) OE is a phantom map; (2) f(OE) = 0 for every cohomological functor f :C ! A into an abelian AB 5 c* *ategory A which preserves coproducts; (3) the induced map Hom (Y; Q) ! Hom (X; Q) is zero for every (indecomposab* *le) pure-injective object Q in C. Proof.The equivalence (1) , (2) is an immediate consequence of Corollary 2.4. * *The equivalence (1) , (3) follows from the fact that OE is a phantom map if and onl* *y if the map in a triangle X !OEY ! Z O! X[1] isQa pure monomorphism. In addition, on* *e uses the existence of a pure monomorphism Y ! i2IZi into a product of indecomposab* *le __ pure-injectives which has been established in Corollary 1.10. * * |__| 2.2. Adjoint functors. We study pairs of adjoint functors between compactly gen* *er- ated triangulated categories. This is based on properties of adjoint functors * *between module categories. We start with some notation. Let f :C ! D be an additive fun* *ctor between skeletally small additive categories. Then we denote by f*: Mod D ! Mod* * C, X 7! X Of the corresponding restriction functor, and f* :Mod C ! Mod D denotes the unique functor which preserves colimits and sends Hom (-; X) to Hom (-; f(X* *)) for every X in C. Applying Yoneda's lemma, we get for every X in C and every D-modu* *le M a functorial isomorphism Hom (f*(Hom (-; X)); M) ' M(f(X)) = f*(M)(X) ' Hom (Hom (-; X); f*(M)) which shows that f* is a left adjoint for f*. Proposition 2.6. Let f :C ! D be an exact functor between compactly generated * *tri- angulated categories. Suppose also that f preserves coproducts, and that the ri* *ght adjoint g :D ! C of f preserves coproducts. (1) f induces a functor f0: C0 ! D0 which makes the following diagrams commut* *a- tive: C? -f! D? D? -g! C? ?yh ? ? ? C yhD yhD yhC (f0)* (f0)* Mod C0 -! Mod D0 Mod D0 -! Mod C0 (2) The functors (f0)* and (f0)* are both exact. (3) The functor g sends pure-exact triangles to pure-exact triangles, and pur* *e-injectives to pure-injectives. Proof.The existence of the right adjoint g :D ! C is an immediate consequence o* *f the Brown representability theorem, since for every object X in D there exists a un* *ique object Y = g(X) in C such that Hom (-; X) Of ' Hom (-; Y ). SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 13 (1) Given a compact object X in C, it is well-known that f(X) is compact sinc* *e g pre- serves coproducts. This follows from the following sequence of canonical isomor* *phisms for every family (Yi)i2Iof objects in D: a a a Hom (f(X); Yi)' Hom (X; g(Yi)) ' Hom (X; g(Yi)) i i a i a ' Hom (X; g( Yi)) ' Hom (f(X); Yi): i i Therefore f induces a functor f0: C0 ! D0. The composition hD Of is a cohomo- logical functor which preserves coproducts. Thus there exists a unique exact f* *unctor f0: Mod C0 ! Mod D0 commuting with coproducts and satisfying hD Of = f0OhC by Corollary 2.4. We claim that (f0)* = f0. In fact, both functors are right exa* *ct, pre- serve coproducts, and coincide on the full subcategory of finitely generated pr* *ojective objects in Mod C0. The assertion follows since every object M in Mod C0 has a p* *rojective presentation a (OEij)a HXi -! HYj -! M ! 0 i j with Xi and Yj in C0 for all i and j. To prove hC Og = (f0)*O hD , observe that for every C in C0 and X in D we have (hC Og)(X)(C) = Hom (C; g(X)) ' Hom (f(C); X) = HX (f(C)) = ((f0)*O hD )(X)(C): (2) The exactness of (f0)* has already been noticed, and the restriction (f0)* ** is au- tomatically exact. (3) The first assertion follows directly from the adjointness formula and the* * fact that f preserves compactness. The second assertion follows from the characterizatio* *n of pure-injectivity in part (5) of Theorem 1.8, and the fact that g preserves prod* *ucts_and coproducts. |* *__| 2.3. Flat modules. Let C be a skeletally small additive category. Recall that * *there exists a tensor product Mod C x Mod Cop -! Ab ; (M; N) ! M C N where for any C-module M, the tensor functor M C - is determined by the fact th* *at it preserves colimits and M C Hom (X; -) ' M(X) for all X in C. Observe that the existence of such a tensor product is an immediate consequence of Lemma 2.2. A* * C- module M is flat if the tensor functor M C - exact, and we denote by FlatC the full subcategory of flat C-modules. Recall that a C-module M is flat if and onl* *y if M is a filtered colimit of representable functors [22, Theorem 3.2]. Therefore F* *latC is equivalent to the category of ind-objects over C in the sense of Grothendieck a* *nd Verdier [11]. In particular, FlatC is a category with filtered colimits, and every func* *tor f :C ! D into a category D with filtered colimits extends uniquely to a functor f0: Flat* *C ! D preserving filtered colimits and satisfying f0(Hom (-; X)) = f(X) for all X in* * C. Suppose now that C is triangulated. Then we have the following characterizati* *on of flat C-modules which has been observed independently by Beligiannis [3]. Lemma 2.7. The following are equivalent for an additive functor M :Cop ! Ab : (1) M is a flat C-module; (2) M is a cohomological functor; (3) M is a fp-injective C-module. 14 HENNING KRAUSE Proof.(1) , (2) M is flat if and only if the restriction M C -|mod Copis exact * *since every exact sequence in Mod Cop can be written as a filtered colimit of exact s* *equences in mod Cop. Thus M is flat if and only if M C -|Cop is a cohomological functor* * by Lemma 2.1. The assertion now follows since M ' M C -|Cop. * * __ (2) , (3) Use the argument from the proof of Lemma 1.6. * *|__| We combine the preceding lemma with our results about cohomological functors * *on compactly generated triangulated categories. Note that the following theorem ge* *neral- izes a result of Christensen and Strickland in [8]. Theorem 2.8. Let C be a compactly generated triangulated category. Then the f* *ollowing categories are pairwise equivalent: (1) the category of cohomological functors C ! Ab which preserve coproducts; (2) the category of cohomological functors C0 ! Ab ; (3) the category of ind-objects over (C0)op. __ Proof.Combine Proposition 2.3 and Lemma 2.7. |_* *_| The finitely presented modules over a ring are characterized by the fact tha* *t the corresponding tensor functor M - preserves arbitrary products of op-modules. * *In fact, it is sufficient to assume that M - preserves products of finitely gene* *rated projective modules; e.g. see [25, Lemma I.13.2]. This result generalizes to r* *ings with several objects and leads to a characterization of cohomological functors C ! A* *b which preserve products; it is the second part of Theorem E. Proposition 2.9. Let C be a compactly generated triangulated category. Then th* *e fol- lowing are equivalent for a cohomological functor f :C ! Ab which preserves cop* *roducts: Q Q (1) f(Q iXi) ' Q if(Xi) for every family (Xi)i2Iof objects in C; (2) f( iXi) ' if(Xi) for every family (Xi)i2Iof compact objects in C; (3) f ' Hom (C; -) for some compact object C in C. Proof.The directions (1) ) (2) and (3) ) (1) are clear. Therefore suppose that* * f preserves products of compact objects. The functor f extends uniquely to a col* *imit preserving functor f0: Mod C0 ! Ab by Corollary 2.4. We have f0 ' - C0 M for M = f|C0 and M is flat by Lemma 2.7. Moreover, M is finitely presented since f0* * pre- serves products of finitely generated projective C0-modules. Any flat module is* * finitely presented if and only if it is finitely generated projective (e.g. see [25, Cor* *ollary I.11.5]), and therefore M ' Hom (C; -) for some C in C0. We obtain f(Y ) = f0(HY ) ' HY C0Hom (C; -) ' HY (C) = Hom (C; Y ) * * __ for every Y in C, and therefore f ' Hom (C; -). * * |__| 2.4. Pure-semisimplicity. A compactly generated triangulated category C is pure- semisimple if every pure monomorphism in C splits; equivalently if every object* * in C is pure-injective. Our aim is a characterization of pure-semisimplicity, using the* * fact that this property is equivalent to a number of familiar properties of the module ca* *tegory Mod C0. For instance, Bass has characterized the rings for which every flat mo* *dule is projective. This can be generalized to rings with several objects and then desc* *ribes when every flat C0-module is a projective C0-module, see [13, Theorem B.12]. On the* * other hand, noetherian rings can be characterized by the fact that every fp-injective* * module is injective. Moreover, Matlis showed that a ring is noetherian if and only if* * every injective module is a coproduct of indecomposable modules. These results gener* *alize SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 15 to rings with several objects as well, see [13, Theorem B.17]. We obtain theref* *ore the following characterization of pure-semisimplicity, since the restricted Yoneda * *functor C ! Mod C0 identifies every object in C with a C0-module which is flat and fp-i* *njective by Lemma 1.6 and Lemma 2.7. Theorem 2.10. The following are equivalent for a compactly generated triangul* *ated category C: (1) C is pure-semisimple; (2) every object in C is a coproduct of indecomposable objects with local end* *omorphism rings; (3) every compact object is a finite coproduct of indecomposable objects with* * local endomorphism rings, and, given a sequence X1 -OE1!X2 -OE2!X3 -OE3!: : : of non-isomorphisms between indecomposable compact objects, the compositi* *on OEn O: :O:OE2O OE1 is zero for n sufficiently large; (4) the restricted Yoneda functor hC: C ! Mod C0, X 7! Hom (-; X)|C0, is * *fully faithful; (5) C has filtered colimits. This characterization, and indeed a host of other equivalent statements have * *been obtained independently by Beligiannis in [4]. 3. Localization 3.1. Cohomological ideals. Let C be an additive category. An ideal I in C consi* *sts of subgroups I(X; Y ) in Hom (X; Y ) for every pair of objects X; Y in C such t* *hat for all OE in I(X; Y ) and all maps ff: X0 ! X and fi :Y ! Y 0in C the composition fi O* *OE Off belongs to I(X0; Y 0). Definition 3.1. An ideal I in a triangulated category C is called cohomological* * if there exists a cohomological functor f :C ! A into an abelian category A such that I * *= {OE 2 C | f(OE) = 0}. Given an ideal I in C, we denote by SI the full subcategory of objects M in m* *od C such that M ' Im HOEfor some OE in I. If C is skeletally small, then TI denotes* * the full subcategory of filtered colimits lim-!Mi in Mod C such that Mi belongs to SI f* *or all i. Recall that a full subcategory S of an abelian category A is a Serre subcategor* *y provided that for every exact sequence 0 ! X0 ! X ! X00! 0 in A the object X belongs to S if and only if X0 and X00belong to S. Lemma 3.2. Let I be a cohomological ideal in a triangulated category C. (1) SI is a Serre subcategory of mod C. (2) If C is skeletally small, then TI is a Serre subcategory of Mod C. Proof.(1) Let f :C ! A be a cohomological functor such that I = {OE 2 C | f(OE)* * = 0}, and denote by f0: mod C ! A the exact functor extending f which exists by Lemma* * 2.1. The full subcategory S = {M 2 mod C | f0(M) = 0} is a Serre subcategory of mod C since f0 is exact. Now observe that every finitely presented C-module M with pr* *ojective presentation HX ! HY ! M ! 0 is isomorphic to Im HOEwhere OE is the map occuring in the triangle X ! Y !OEZ ! X[1]. Given an arbitrary map OE in C, we have f(OE* *) = 0 16 HENNING KRAUSE if and only if f(Im HOE) = 0, and therefore SI = S. Thus SI is a Serre subcateg* *ory of mod C. * * __ (2) See [15, Theorem 2.8]. * *|__| Let f :C ! D be an additive functor between additive categories. We denote by f* :mod C ! mod D the unique right exact functor which sends Hom (-; X) to Hom (-; f(X)) for all X in C. If C and D are skeletally small, then f* extends * *uniquely to a colimit preserving functor Mod C ! Mod D which we also denote by f*. Lemma 3.3. Let f :C ! D be an exact functor between triangulated categories. * *Then I = {OE 2 C | f(OE) = 0} is a cohomological ideal in C. Moreover, the following* * holds: (1) SI = {M 2 mod C | f*(M) = 0}. (2) If C and D are skeletally small, then TI = {M 2 Mod C | f*(M) = 0}. Proof.Let f0:C ! mod D be the composition of f with the Yoneda functor D ! mod * *D. This functor is cohomological, and f(OE) = 0 if and only if f0(OE) = 0 for ever* *y map OE 2 C since the Yoneda functor is faithful. Thus I is a cohomological ideal. (1) The functor f* :mod C ! mod D is the unique exact functor extending f0. T* *here- fore SI = {M 2 mod C | f*(M) = 0} by the argument given in the proof of Lemma 3* *.2. (2) We denote by T the full subcategory of C-modules M such that f*(M) = 0. * *It follows from (1) that TI T since f* preserves filtered colimits. To prove th* *e other inclusion, we use the right adjoint f*: Mod D ! Mod C, M 7! M Of for f*. We den* *ote by t: Mod C ! Mod C the functor which is obtained from the functorial exact se* *quence 0 -! t(M) -! M M-! (f*O f*)(M): Note that t induces a right adjoint for the inclusion T ! Mod C since f*(M ) * *is an isomorphism for all M. Moreover, t preserves filtered colimits since f* and f** * have this property. Now let M 2 mod C, and write t(M) = lim-!Mi as a filtered colim* *it of finitely generated submodules. For all i, we have Mi2 T since T is closed under* * taking submodules, and Mi 2 mod C since C has weak kernels and therefore finitely gene* *rated submodules of finitely presented modules are again finitely presented. It foll* *ows that t(M) is a filtered colimit of modules in S = T \ mod C. Given any module M in * *T , we can write M = lim-!Mi as a filtered colimit of finitely presented modules. * * Thus M = t(lim-!Mi) ' lim-!t(Mi) is a filtered colimit of modules in S, and T TI f* *ollows_ since S = SI by (1). * *|__| 3.2. Localization for triangulated categories. Let C be a compactly generated t* *ri- angulated category. Recall that a full triangulated subcategory B of C is loca* *lizing if B is closed under taking coproducts. The quotient category C=B is, by definiti* *on, the category of fractions C[-1] (in the sense of [10]) with respect to the class o* *f maps Y ! Z which admit a triangle X ! Y ! Z ! X[1] with X in B. Thus the correspond- ing quotient functor C ! C[-1] is the universal functor which inverts every map* * in . Note that C[-1] is a large category which means that the maps between fixed obj* *ects are not assumed to form a set. Let us mention a few basic facts about the forma* *tion of the quotient category C=B which we shall use frequently without further referen* *ce. Lemma 3.4. The quotient functor f :C ! C=B has the following properties: (1) The triangulation of C induces a triangulation for C=B and f is an exact * *functor. (2) Let X be an object in C. Then f(X) = 0 if and only if X 2 B. (3) Let OE be a map in C. Then f(OE) = 0 if and only if OE factors through so* *me object in B. SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 17 * * __ Proof.See [27, Corollaire 2.2.11]. * * |__| The following lemma characterizes the existence of a right adjoint for the qu* *otient functor C ! C=B. Lemma 3.5. Let B be a localizing subcategory of a compactly generated triangu* *lated category C. Then the following are equivalent: (1) the maps between fixed objects in C=B form a set; (2) the quotient functor f :C ! C=B has a right adjoint g :C=B ! C; (3) the inclusion functor B ! C has a right adjoint e: C ! B. Moreover, in this case there is for every object X in C a triangle (g Of)(X)[-1] ffX-!e(X) fiX-!X -flX!(g Of)(X) which is functorial in X. A localizing subcategory B which satisfies the equivalent conditions of the p* *receding lemma admits a localization functor C ! C which is, by definition, the composit* *ion of the quotient functor C ! C=B with a right adjoint C=B ! C. To prove Lemma 3.5 we shall need the following lemma about C[-1]. Lemma 3.6. Let C be any category with coproducts.`Suppose that is a class of* * maps in C which admits a calculus of left fractions. If ioei2 for every family (o* *ei)i2Iin , then the quotient category C[-1] has coproducts and the quotient functor C ! C[* *-1] preserves coproducts. Proof.Recall from [10] that the objects in C[-1] are those of C, and that the m* *aps X ! Y in C[-1] are equivalence classes of left fractions X !OEZ oeY with`oe 2 * *. Now let (Xi)i2Ibe a family of objects in C[-1]. We claim that the coproduct iXi i* *n C is also a coproduct`in C[-1].QThus we need to show that for every object Y , the c* *anonical map ff: Hom ( iXi; Y ) ! iHom (Xi; Y ) between Hom-sets in C[-1] is bijectiv* *e. To check surjectivity, let (Xi OEi!Zi oeiY )i2Ibe a family of left fractions.* * We obtain a commutative diagram ` ` iOEi` `ioei ` iXi -! iZi? - ?iY ?y ?yss Y Z -oe Y ` where ssY : iY ! Y is the summation map and oe 2 . It is easily checked that (Xi! Z oeY ) ~ (Xi!OEiZi oeiY ) ` oe OEi oei for all i 2 I, and therefore ff sends iXi! Z Y to the family (Xi! Zi Yi* *)i2I. ` OE0 oe0 ` OE00 oe00 To check injectivity, let iXi! Z0 Y and iXi! Z00 Y be left fraction* * such that OE0i0oe0 OE00i00oe00 (Xi! Z Y ) ~ (Xi! Z Y ) for all i. We may assume that Z0= Z = Z00and oe0= oe = oe00since we can choose * *maps o0:Z0! Z and o00:Z00! Z with o0Ooe0= o00Ooe002 . Thus there are maps i:Z ! Zi * * __ with iOOE0i= iOOE00iand iOoe 2 for all i. Each i belongs to the saturation* * of which is the class of all_maps_in C which become an isomorphism in C[-1]. Note * *that a map ff in C belongs to if and only if there are maps ff0and ff00such that f* *f Off0and 18 HENNING KRAUSE __ * * __ ff00Off belong to . Therefore is also closed under taking coproducts. Moreo* *ver, admits a calculus of left fractions, and we obtain therefore a commutative diag* *ram ` ` ssZ oe iXi -! iZ? -! Z? - Y ?y` ? ` i i y o * iZi -! Z __ __ with o 2 . Thus o Ooe 2 , and we have a OE0 oe a OE00 oe ( Xi! Z Y ) ~ ( Xi! Z Y ) ` i ` i since ssZ O iOE0i= OE0 and ssZ O iOE00i= OE0. Therefore ff is also injective,* * and_this com- pletes the proof. * *|__| Proof of Lemma 3.5. (1) ) (2) The quotient functor preserves coproducts by Lemm* *a 3.6, since is closed under taking coproducts. Given an object X in C=B, the composi* *tion Hom (-; X) Of is a cohomological functor which sends coproducts to products. T* *hus there exists Y in C with Hom (-; X) Of ' Hom (-; Y ) by the Brown representabi* *lity theorem. We put g(X) = Y , and it is easily checked that this gives a right ad* *joint g :C=B ! C for f. (2) ) (1) Let X = f(X0) and Y be objects in C=B. Then Hom (X; Y ) ' Hom (X0; * *g(Y )) since g is a right adjoint of f. Thus the maps between objects in C=B form a se* *t. (2) ) (3) Suppose that f has a right adjoint g. Completing the canonical map flX :X ! (g Of)(X) to a triangle (g Of)(X)[-1] -! Y - ! X -flX!(g Of)(X) for every X in C gives a functor e: C ! B if we put e(X) = Y . In fact, f(flX * *) is an isomorphism and therefore f(Y ) = 0 which implies Y 2 B. Given Y 02 B, one appl* *ies Hom (Y 0; -) to the above triangle and gets an isomorphism Hom (Y 0; Y ) ! Hom * *(Y 0; X). Thus e is a right adjoint for the inclusion B ! C. (3) ) (2) Suppose that the inclusion B ! C has a right adjoint e, and let X =* * f(X0) be an object in C=B. Completing the canonical map fiX0 :e(X0) ! X0 to a triangle fiX0 0 Y [-1] -! e(X0) -! X -! Y gives a functor g :C=B ! C if we put g(X) = Y . It is not hard to check that th* *is defines a right adjoint for the quotient functor C ! C=B, but we leave the details to t* *he reader. The last assertion is an immediate consequence of the construction given in (* *2)_) (3). |* *__| We continue with a series of lemmas which collect some basic properties of th* *e quotient functor and its right adjoint, assuming that it exists. The notation of Lemma 3* *.5 remains fixed. Lemma 3.7. The natural transformation idC! g Of induces a functorial isomorph* *ism Hom ((g Of)(X); Y ) ! Hom (X; Y ) for all X and Y such that Hom (B; Y ) = 0. * * __ Proof.Apply Hom (-; Y ) to the triangle in Lemma 3.5. * * |__| Given any class B of objects in C, we say that an object Y in C is B-local if* * Hom (X; Y ) = 0 for all X in B. The full subcategory of B-local objects is denoted by B? . Th* *e definition of I-local objects for a class I of maps in C is analogous. Lemma 3.8. The functor g induces an equivalence between C=B and B? . SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 19 Proof.An inverse is the composition of the inclusion B? ! C with the quotient f* *unctor_ C ! C=B; use Lemma 3.5. |_* *_| 3.3. Cohomological ideals and localization. Let C be a compactly generated tria* *n- gulated category. Given an ideal I in C0, we define a full subcategory CI as fo* *llows: CI = {X 2 C | every map C ! X, C 2 C0, factors through a map C ! D in I}: Given a full additive subcategory B of C, we define an ideal IB as follows: IB = {OE 2 C0 | OE factors through an object in}B: We are interested in properties of the category CI and collect them in two tech* *nical lemmas. Lemma 3.9. Let I be an ideal in C0 and X 2 C. Then X 2 CI if and only if HX 2* * TI. Proof.Suppose first that HX 2 TI. Thus HX = lim-!ImHOEiwith OEi2 I for all i. N* *ow let OE: C ! X be any map with C 2 C0. We have Hom (C; X) = HX (C) = lim-!ImHOEi(C) and obtain therefore a factorization C ! Ci!OEiDi! X of OE for some i. The comp* *osition C ! Ci!OEiDi belongs to I since OEi2 I, and this implies X 2 CI. To prove the converse, suppose that X 2 CI. Every module is a filtered colim* *it of finitely presented ones. More precisely, HX = lim-!i2IMi where I denotes the * *filtered category of maps i:Mi! HX with Mi2 mod C0. We claim that the full subcategory J of maps i:Mi! XX with Mi2 SI is cofinal, i.e., for every i 2 I there exists a* * map :Mi! Mj for some j 2 J such that i= jO . To prove this claim, let OEi:Ci! Di be a map in C0 with Mi ' Im HOEiwhich exists by the argument given in the proof* * of Lemma 3.2. We get a factorization Mi ! HDi ! HX of i since HX is fp-injective by Lemma 1.6, and the corresponding map Di ! X has a factorization Di ! E ! X for some :Di ! E in I since X 2 CI. Thus i factors through the map Im H ! HX with Im H 2 SI. Therefore J is cofinal in I, and the inclusion J ! I induces * *an_ isomorphism lim-!i2JMi' lim-!i2IMi' HX which proves HX 2 TI. |* *__| Lemma 3.10. Let I be a cohomological ideal in C0 such that OE[n] 2 I for all * *OE 2 I and n 2 Z. (1) CI is a localizing subcategory of C. (2) If I = I(CI), then the inclusion CI ! C has a right adjoint and (CI)? = I* *? . Proof.(1) Clearly, CI is closed under the shift in C since I is closed under th* *e shift. Now let X !OEY ! Z !O X[1] be a triangle in C with X; Y 2 CI. We need to show that Z 2 CI. We apply the description of CI given in Lemma 3.9. The triangle in* *duces an exact sequence 0 ! Im H ! HZ ! Im HO ! 0 in Mod C0. The category TI is a Serre subcategory of Mod C0 by Lemma 3.2 since I is cohomological, and therefor* *e HZ belongs to TI. Thus Z 2 CI. Furthermore, CI is closed under taking coproducts b* *ecause TI has this property, and we conclude that CI is localizing. (2) In order to show that the inclusion CI ! C has a right adjoint, it is by * *Lemma 3.5 sufficient to show that for two objects X and Y in C the maps X ! Y in C=CI for* *m a set. In fact, it is sufficient to check this for all X 2 C0 and Y 2 C since C0* * generates C. To prove this claim, we consider the exact quotient functor q :Mod C0 ! Mod* * C0=TI with respect to the Serre subcategory TI and observe that the maps in Mod C0=TI form a set [9, Proposition III.8]. The composition of q with the Yoneda functor* * h: C ! 20 HENNING KRAUSE Mod C0 annihilates CI by Lemma 3.9, and therefore q Oh induces a cohomological * *functor h0:C=CI ! Mod C0=TI making the following diagram of functors commutative: C? -! C=CI? ?yh ?yh0 Mod C0 -q! Mod C0=TI We claim that h0 induces an injective map Hom (X; Y ) ! Hom (h0(X); h0(Y )) for* * all X 2 C0 and Y 2 C. To this end choose a map ff: X ! Y in C=CI which is by defini* *tion a left fraction X !OEZ oeY , and assume that h0(ff) = 0. It follows that q(HO* *E) = 0 since q(Hoe) is an isomorphism, and therefore Im HOE2 TI since q is exact. Thus* * Im HOE is a filtered colimit of objects in SI. An argument similar to that given in t* *he proof of Lemma 3.9 shows that the map OE: X ! Z factors through a map :X ! X0 in I since X 2 C0. Thus OE factors through an object in CI by our assumption on I,* * and therefore ff = 0. We conclude that the maps between fixed objects in C=CI form * *a set, and therefore the inclusion CI ! C has a right adjoint. It remains to show that (CI)? = I? . Clearly, (CI)? I? since every map in I * *factors through an object in CI. To prove the other inclusion, let B = CI and consider * *for any object X in C the triangle (g Of)(X)[-1] ffX-!e(X) fiX-!X -flX!(g Of)(X) as in Lemma 3.5. Now suppose that Hom (I; X) = 0. Every map OE: C ! e(X) with 0 OE00 C in C0 has a factorization C !OED ! e(X) with OE0 2 I, and therefore fiX OOE * *= 0. Thus OE factors through ffX . The same argument shows that OE00factors through* * ffX , and therefore OE = 0 since Hom (OE0; (g Of)(X)[-1]) = 0 by Lemma 3.7. We conclu* *de_that e(X) = 0 and therefore Hom (CI; X) = 0. * *|__| 3.4. Approximations. We need to recall the following definition from [2]. Let Y* * be a class of objects in a category C. Then a map X ! Y is a left Y-approximation of* * X if Y belongs to Y and if the induced map Hom (Y; Y 0) ! Hom (X; Y 0) is surjective f* *or every Y 0in Y. For example, if we view Y as a full subcategory of C and assume the ex* *istence of a left adjoint f :C ! Y for the inclusion Y ! C, then the canonical map X ! * *f(X) is a left Y-approximation for every X in C. In general, a left Y-approximation* * is far from being unique. Suppose now that I is a class of maps in a triangulated category C such that * *their isomorphism classes form a set. Recall that I? denotes the full subcategory of * *objects X in C satisfying Hom (OE; X) = 0 for all OE 2 I. We construct for any object * *X in C a left I? -approximation flX;I?: X ! XI?. To this end we define inductively * *maps ffn :Xn ! Xn+1 for every n 0. By definition, set X0 = X. Let n be a representa* *tive set of non-zero maps :C ! Xn which factor through`some map C ! D in I. We obtain ffn if we complete the canonical map 2n C ! Xn to a triangle a ffn a C -! Xn -! Xn+1 -! ( C)[1]: 2n 2n We denote by XI? the homotopy colimit hocolimXn of the sequence X = X0 ff0-!X1 ff1-!X2 ff2-!: ::: SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 21 More precisely, XI? is obtained from the triangle a id-ffa a Xn -! Xn -! hocolimXn -! ( Xn)[1]: n n n We denote by flX;I?: X ! XI? the canonical map from X0 into hocolimXn, but this map is only unique up to a non-unique isomorphism since the construction involv* *es the completion of various maps to triangles. Given any map :X ! Y in C, we obtain* * a sequence of commuting diagrams X? -id! X0? -ff0! X1? -ff1!X2? ff2-!: : : ?y ?y ? ? 0 y 1 y 2 Y -id! Y0 -fi0! Y1 -fi1!Y2 -fi2!: : : and we denote by I? :XI? ! YI? a map which makes the following diagram commu- tative ` id-ff` ` nXn? -! ?Xn -! hocolim?Xn - ! ( n?Xn)[1] ?y` ?` ? ? ` n y n y I? y( n)[1] ` id-fi ` ` nYn -! Yn -! hocolimYn - ! ( n Yn)[1] Note that the map I? is not unique. Proposition 3.11. Let C be a triangulated category and suppose that I is a cla* *ss of maps between compact objects in C such that their isomorphism classes form a se* *t. Then the map flX;I?: X ! XI? is a left I? -approximation. Proof.We need to show that XI? is I-local. To this end let OE: C ! D be a map i* *n I. The canonical maps n :Xn ! hocolimXn induce an isomorphism lim-!Hom(D; Xn) -! Hom (D; hocolimXn) since D is compact; e.g. see [19, Lemma 1.5]. Therefore any map :D ! XI? has a factorization = n O 0 for some n 2 N. The construction of ffn implies ffn O 0* *OOE = 0, and therefore OOE = 0. Thus XI? is I-local. We have YI? = Y for every I-lo* *cal object Y , and therefore every map :X ! Y with Y 2 I? factors through flX;I? * *via * * __ I? :XI? ! YI? = Y . Thus flX;I? is a left I? -approximation. * * |__| We include the following lemma for later reference. Lemma 3.12. Let f :C ! D be an exact functor between triangulated categories * *which preserves coproducts. Suppose that I is a class of maps in C such that their is* *omorphism classes form a set. If f(I) = 0, then the map f(flX;I?): f(X) ! f(XI?) is a sp* *lit monomorphism. Proof.The construction of each ffn implies that f(ffn) is a split monomorphism * *for_every_ n. It follows that idf(X)factors through f(hocolim Xn) ' hocolimf(Xn). * * |__| 4. Smashing subcategories 4.1. A characterization of smashing subcategories. Let C be a compactly gen- erated triangulated category and suppose that B is a localizing subcategory of * *C. We denote by D = C=B the quotient category and f :C ! C=B denotes the corresponding quotient functor. A right adjoint of f is denoted by g :C=B ! C, provided it e* *xists. Recall that B is smashing if the inclusion B ! C has a right adjoint which pres* *erves 22 HENNING KRAUSE coproducts. Note that this is equivalent to the fact that the quotient functor * *has a right adjoint which preserves coproducts. Lemma 4.1. Let B be a smashing subcategory of a compactly generated triangula* *ted category C. Then C=B is a compactly generated triangulated category. Proof.C=B has coproducts by Lemma 3.6, and the argument in the proof of Proposi- tion 2.6 shows that f(C0) D0. Suppose now that Hom (D; X) = 0 for all D in D0 * *and some X in D. Then Hom (C; g(X)) ' Hom (f(C); X) = 0 for all C in C0, and theref* *ore X = 0, since C is compactly generated and g is faithful by Lemma 3.8. Thus C=B* *_is_ compactly generated. |* *__| We are now in a position to prove the characterization of smashing subcategor* *ies which is stated in Theorem A. We reformulate this theorem as follows. Theorem 4.2. Let B be a localizing subcategory of a compactly generated trian* *gulated category C, and denote by I the ideal of maps between compact objects in C whic* *h factor through some object in B. Then the following conditions are equivalent: (1) B is smashing; (2) B = CI; (2')B CI; (3) I? = B? ; (3')I? B? . Proof.(1) ) (2) We know from the preceding lemma that D = C=B is a compactly ge* *n- erated triangulated category. We have therefore by Proposition 2.6 an induced f* *unctor (f0)*: Mod C0 ! Mod D0 such that (f0)*O hC = hD Of. An object X in C belongs to* * B if and only if (f0)*(HX ) = 0 since hD (Y ) = 0 if and only if Y = 0. Therefore* * B = CI by Lemma 3.3 and Lemma 3.9. (2') ) (3) It is clear that I? B? . The condition (2') implies that I = I(CI* *), and therefore I? = (CI)? by Lemma 3.10. Using again (2'), we have (CI)? B? and this implies I? B? . (3') ) (2') Let X be an object in C and suppose X 62 CI. It follows from Lemm* *a 3.9 that HX 62 TI, and we find a maximal subobject T HX with T 2 TI since TI is a * *Serre subcategory of Mod C0 which is closed under taking coproducts by Lemma 3.2. Cho* *osing an injective envelope HX =T ! M, we have Hom (HX ; M) 6= 0 and Hom (TI; M) = 0 * *by construction. Applying Lemma 1.7, we find an object Y in C such that HY ' M and Hom (X; Y ) ' Hom (HX ; M) 6= 0. Moreover, Hom (I; Y ) = 0 since Hom (TI; M) =* * 0. Assuming (3'), it follows that X 62 B. Thus (2') holds. (2') ) (1) The condition (2') implies that I = I(CI), and therefore I? = (CI* *)? by Lemma 3.10. We claim that B = CI. To this end let X 2 CI and consider the I? -approximation flX;I?: X ! XI? from Proposition 3.11. We have flX;I? = 0 si* *nce I? = (CI)? , and the quotient functor f :C ! C=B sends flX;I? to a split monomo* *rphism by Lemma 3.12. Thus f(X) = 0 and therefore X belongs to B. We conclude from Lemma 3.5 that the inclusion B ! C has a right adjoint, and it remains to show * *that this right adjoint preserves coproducts. To this end consider the right adjoint g :C* *=B ! C of the quotient functor which identifies C=B with B? by Lemma 3.8. Now let (Xi)* *i2I be a family`of objects in B? . Using (3), we have`Hom (I; Xi) = 0 for all i, an* *d therefore Hom (I; iXi) = 0 since I belongs to C0. Thus iXi belongs to B? . It follows* * that_g preserves coproducts and therefore B is smashing. * * |__| SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 23 Remark 4.3. Localizations in C and Mod C0 are closely related. In fact, if B is* * a smash- ing subcategory of C, then one can use Proposition 2.6 to show that f Og ' idD * *im- plies (f0)*O(f0)* ' idMod D0 for D = C=B. Therefore (f0)* induces an equivalen* *ce Mod C0=TI ! Mod (C=B)0 where Mod C0=TI denotes the quotient category with respe* *ct to the localizing subcategory TI = {M 2 Mod C0 | (f0)*(M) = 0}; e.g. see [9, Pr* *oposi- tion III.5]. This leads to the following commutative diagram: B? -! C? -f! C=B? ?y ?yh ? C y TI -! ModfC0l -! Mod C0=TI? flfl ?yo (f0)* Mod C0 -! Mod (C=B)0 Note that the composition C=B ! Mod C0=TI ! Mod (C=B)0 is just the restricted Yoneda functor hC=B. We proceed with the proof of Theorem C which we recall for the convenience of* * the reader. Theorem 4.4. Let B be a smashing subcategory of a compactly generated triangu* *lated category C, and let U be the set of objects Y in Sp C such that Hom (B; Y ) = * *0. Then the following holds for any object X in C: (1) X 2 B if and only if Hom (X; U) = 0; Q (2) Hom (B; X) = 0 if and only if there is a pure monomorphism X ! i2IYi wi* *th Yi2 U for all i. Proof.We identify D = C=B via g with the full subcategory of objects X in C suc* *h that Hom (B; X) = 0. This is possible by Lemma 3.8. In particular, this identifies S* *p D with U = {X 2 SpC | Hom (B; X) = 0} since g preserves pure-injectivity by Propositio* *n 2.6. (1) Clearly, X 2 B implies Hom (X; U) = 0. Conversely, Hom (X; U) = 0 implies Hom ((g Of)(X); U) = 0 by Lemma 3.7, and this implies (g Of)(X) = 0 since U cog* *en- erates D by Corollary 1.10. Thus f(X) = 0 since g is faithful, and therefore X * *belongs to B. (2) Suppose firstQthat Hom (B; X) = 0. We apply Corollary 1.10 and get a pure monomorphism X ! iYi in D with Yi 2 U for all i. The inclusion D ! C preserves pure monomorphisms by Proposition 2.6, and this proves one direction. Now suppo* *se that we have a pure monomorphism X ! Y in C with Hom (B; Y ) = 0. It follows fr* *om part (3) in Theorem 4.2 that Hom (B; X) = 0, and therefore the proof of Theorem* *_4.4 is complete. |_* *_| Let B be a localizing subcategory of a triangulated category C. Recall that * *a map X ! Y in C is a B-localization of X if Y is B-local and the induced map Hom (Y;* * Y 0) ! Hom (X; Y 0) is bijective for every B-local object Y 0. Let us describe an expl* *icit construc- tion of the B-localization provided that B is smashing. We use the left I? -app* *roximation flX;I?: X ! XI? with respect to an ideal I in C0 which has been constructed in * *Propo- sition 3.11. Recall from [2] that a map OE: X ! Y is left minimal if every endo* *morphism of Y such that OE = OOE is an isomorphism. 24 HENNING KRAUSE Theorem 4.5. Let B be a smashing subcategory of a compactly generated triangu- lated category C and let I = IB be the corresponding ideal in C0. Then the lef* *t I? - approximation flX;I?: X ! XI? of an object X in C has a decomposition a (fl0; fl00): X -! XI? = Y 0 Y 00 such that fl0 is left minimal and fl00= 0. In this case, the map fl0:X ! Y 0is* * a B- localization of X. Proof.The I? -approximation flX;I?: X ! XI? is also a B? -approximation since I* *? = B? by Theorem 4.2. There exists a B-localization flX :X ! Y of X by Lemma 3.5. We obtain therefore maps ff: Y ! XI? and fi :XI? ! Y such that flX;I? = ff OflX* * and flX = fi OflX;I?. We have fi Off = idY since flX is a B-localization of X, and* * this gives a decomposition flX;I? = (fl0; fl00) such that fl0 is isomorphic to flX and fl* *00=_0. This finishes the proof. * * |__| There are examples where the left I? -approximation flX;I?: X ! XI? is differ* *ent from the B-localization of X. Take for instance a smashing subcategory B 6= 0 * *with B \ C0 = 0. 4.2. The modified telescope conjecture. We are now in a position to prove the corollary of Theorem A; it will be an immediate consequence of the following pr* *oposition. Proposition 4.6. Let B be a smashing subcategory of a compactly generated tria* *ngu- lated category C and let IB be the corresponding ideal in C0. Suppose that f :* *C ! D is an exact functor into a triangulated category D which preserves coproducts. * * Then f(IB) = 0 if and only if f(B) = 0. Proof.Let I = IB. Clearly, f(B) = 0 implies f(I) = 0. Suppose now that f(I) =* * 0. Let X be an object in B and let flX;I?: X ! XI? be the left I? -approximation f* *rom Proposition 3.11. Theorem 4.2 implies B? = I? and therefore flX;I? = 0 since * *XI? belongs to I? . On the other hand, f(flX;I?) is a split monomorphism by Lemma 3* *.12._ Thus f(X) = 0. |_* *_| Corollary 4.7. Let B be a smashing subcategory of a compactly generated triangu* *lated category C. Then B is generated by the corresponding ideal I = IB in C0. More p* *recisely, (1) B is a localizing subcategory of C and every map in I factors through som* *e object in B; (2) if B0 is any localizing subcategory of C such that every map in I factors* * through some object in B0, then B B0. Proof.(1) follows immediately from the definitions of B and I. To prove (2), le* *t B0be a localizing subcategory of C such that every map in I factors through some objec* *t in B0, and denote by f :C ! C=B0 the corresponding quotient functor. Note that f prese* *rves coproducts by Lemma 3.6. Clearly, f(I) = 0 and therefore f(B) = 0 by the preced* *ing_ proposition. Thus B B0. |* *__| 4.3. A classification of smashing subcategories. In this section we consider a * *com- pactly generated triangulated category C such that the following additional pro* *perty holds: (B) Every cohomological functor Cop0! Ab of the form Hom (f(-); f(C)) (where f :C0 ! D is any exact functor into a triangulated category D and C is any object in C0) is isomorphic to Hom (-; X)|C0 for some object X in C. SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 25 This condition is a weak form of Brown representability. For example, (B) hold* *s for the stable homotopy category [1]. More generally, (B) holds if the category C0* * has a countable skeleton [21]. Our aim in this section is a classification of the sma* *shing sub- categories of C. To this end we introduce the following class of ideals for a t* *riangulated category. Definition 4.8. An ideal I in a triangulated category C is called exact if ther* *e exists an exact functor f :C ! D into a triangulated category D such that I = {OE 2 C | f* *(OE) = 0}. The following result gives a classification of smashing subcategories. Theorem 4.9. Let C be a compactly generated triangulated category and suppose* * that (B) holds. Then the maps B 7! {OE 2 C0 | OE factors through an object in}B and I 7! {X 2 C | every map C ! X, C 2 C0, factors through a map C ! D in I} induce mutually inverse bijections between the set of smashing subcategories of* * C and the set of exact ideals in C0. The proof of this theorem is based on the following lemma. Lemma 4.10. Let C be a compactly generated triangulated category and suppose * *that (B) holds. If I is an exact ideal in C0, then I = I(CI). Proof.Let f :C0 ! D0 be an exact functor such that I = {OE 2 C0 | f(OE) = 0}. We may assume that D0 is a skeletally small triangulated category by taking the* * full subcategory formed by the objects in the image of f. Adding successively new ob* *jects arising from the completion of maps to triangles gives a full triangulated subc* *ategory which needs to be skeletally small since C0 is skeletally small. Now observe t* *hat the inclusion I(CI) I is obvious from the definitions. To prove the other inclusion* *, we use the pair of adjoint functors f* :Mod C0 ! Mod D0 and f*: Mod D0 ! Mod C0 which have already been introduced. Let OE: C ! D be a map in I and consider the cano* *nical map : HD ! (f*O f*)(HD ). Using our assumption on C, there exists an object X in C such that (f*O f*)(HD ) ' HX since (f*O f*)(HD ) ' Hom (f(-); f(D)). We obta* *in a map :D ! X with = H and consider the corresponding triangle X[-1] -oe!V -o! D -! X in C which induces an exact sequence [-1] HD[-1]- ! HX[-1]- ! HV -! HD - ! HX in Mod C0. The maps f*() and f*([-1]) are isomorphisms, and therefore f*(HV ) =* * 0 since f* is exact. Thus V 2 CI by Lemma 3.3 and Lemma 3.9. Now observe that H OHOE= OHOE= 0 since the following diagram is commutative HC? -! (f*O f*)(HC?) ?yH ? * OE y(f*Of )(HOE) HD -! (f*O f*)(HD ) and (f*O f*)(HOE) = f*(Hf(OE)) = 0 by our assumption on OE. Thus OOE = 0 sinc* *e C is compact, and therefore OE factors through V which is an object in CI. We conclu* *de_that_ I I(CI)and this finishes the proof. * * |__| 26 HENNING KRAUSE We are now in a position to give the proof of the theorem which states the cl* *assification of the smashing subcategories of a compactly generated triangulated category C. Proof of Theorem 4.9.Let B be a smashing subcategory of C and denote by f :C ! * *C=B the corresponding quotient functor. It is clear that IB is an exact ideal in C* *0 since IB = {OE 2 C0 | f(OE) = 0}. Suppose now that I is an exact ideal in C0. We ha* *ve I = I(CI)by the preceding lemma, and a combination of Lemma 3.10 and Theorem 4.2 then shows that CI is a smashing subcategory of C. Given a smashing subcategory* * B, we have C(IB)= B by Theorem 4.2. Conversely, I(CI)= I holds for every exact id* *eal in C0 by Lemma 4.10. Thus the maps B 7! IB and I 7! CI are mutually inverse, an* *d __ therefore the proof is complete. * * |__| We continue with a number of applications of the above theorem. In fact, we * *are interested in the interplay between ideals in C0 and localizing subcategories o* *f C. The following lemma will be useful. T Lemma 4.11. Let (Ii)i2Ibe a family of exact ideals. Then i2IIi is exact. Proof.SupposeTthat each Iiis given by some exact functor fi:C0 ! Di. The inters* *ection I = i2IIi isQagain exact since I = {OE 2 C0 | f(OE) = 0} where f denotes the * *exact_ functor C0 ! iDi, X 7! (fi(X))i. * *|__| Recall that a lattice is complete if every subset has a least upper bound and* * a greatest lower bound. Corollary 4.12. Let C be a compactly generated triangulated category and suppos* *e that (B) holds. Then the smashing subcategories of C form a partially ordered set wh* *ich is a complete lattice. Proof.Theorem 4.9 translates the assertion of this corollary into a statement a* *bout the lattice of ideals in C0. Clearly, the cardinality of this lattice is bounded by* * 2 where denotes the cardinality of the set of isomorphism classes of maps in C0. The ex* *act ideals_ in C0 form a complete lattice by the preceding lemma, and this finishes the pro* *of. |__| Given a localizing subcategory B of C, it is not clear that the maps between * *fixed objects in the quotient category C=B form a set. Therefore one calls a category* * large to point out that the maps between fixed objects are not assumed to form a set. Lemma 4.13. Let B be a skeletally small subcategory of a large triangulated c* *ategory C. Then there exists a skeletally small triangulated subcategory of C which con* *tains B. Proof.We construct inductively a chain C1 C2 C3 : :o:f classes of maps in C * *and a chain B = B0 B1 B2 : :o:f skeletally small subcategories of C as follows: * *Let n 1 and assume that Bn-1 is already defined. Let Cn be a class of maps in C sa* *tisfying the following conditions: o if OE 2 Bn-1 and r 2 Z, then OE[r] 2 Cn; o if X !OEY ! Z O! X[1] is a triangle in C with OE 2 Bn-1, then ; O 2 Cn; o if there is a commutative diagram X? -! Y? - ! Z -! X[1]? ?yff ?yfi ?yff[1] X0 -! Y 0 - ! Z0 -! X0[1] in Bn-1 such that the rows are triangles in C, then there is a map Z ! Z0 in Cn* * making the diagram commutative; SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE 27 o if there is a set of maps in Bn-1 satisfying the assumptions of the octahed* *ral axiom, then there are maps in Cn such that the octahedral axiom holds. We may assume that the isomorphism classes of maps in Cn form a set since Bn-1 * *is skeletally small. Now define Bn to beSthe smallest additive subcategory of C co* *ntaining Cn. It is easily checked that B1 = n2NBn is a skeletally small triangulated * *subcategory_ of C which contains B. * *|__| Corollary 4.14. Let C be a compactly generated triangulated category and suppos* *e that (B) holds. Suppose also that B is a localizing subcategory and denote by IB the* * ideal of maps between compact objects in C which factor through some object in B. Then t* *here exists a unique smashing subcategory B0 of C such that IB0 = IB. Moreover, B0 B. Proof.Let I = IB and let f :C ! C=B be the quotient functor corresponding to B. Clearly, I = {OE 2 C0 | f(OE) = 0} and we claim that I is an exact ideal in C0.* * By Lemma 4.13, there exists a skeletally small triangulated subcategory D of C=B c* *ontaining the image of f, and we obtain therefore an exact functor f0:C0 ! D, X 7! f(X) w* *ith I = {OE 2 C0 | f0(OE) = 0}. Thus I is an exact ideal, and there exists a unique* * smashing subcategory B0 = CI such that IB0 = I by Theorem 4.9. Finally f(I) = 0 implies* * __ f(B0) = 0 by Proposition 4.6, and therefore B0 B. * * |__| The preceding corollary suggests the following definition. Definition 4.15. A localizing subcategory B of a triangulated category C is sai* *d to be generated by a class I of maps in C if the following holds: (1) every map in I factors through some object in B; (2) if B0 is a localizing subcategory of C such that every map in I factors t* *hrough some object in B0, then B B0. For example, B is generated by a class I = {idXi | i 2 I} of identity maps if* * and only if B is the smallest localizing subcategory containing Xi for all i 2 I. A* * classical result of Bousfield and Ravenel for the stable homotopy category says that ever* *y class of identity maps of compact objects generates a localizing subcategory which is sm* *ashing [6, 23]. This can be generalized as follows. Corollary 4.16. Let C be a compactly generated triangulated category and suppos* *e that (B) holds. Then a localizing subcategory B of C is smashing if and only if B is* * generated by a class of maps between compact objects in C. 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Verdier, Des categories derivees des categories abeliennes, Asterisque* * 239 (1996). [28]M. Ziegler, Model theory of modules, Ann. of Pure and Appl. Logic 26 (1984)* *, 149-213. Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33501 Bielefeld, Germany E-mail address: henning@mathematik.uni-bielefeld.de