A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS HENNING KRAUSE Abstract.We discuss the Brown Representability Theorem for triangulated * *cate- gories having arbitrary coproducts. In this paper we discuss the Brown Representability Theorem for triangulated * *cate- gories having arbitrary coproducts. This theorem is an extremely useful tool an* *d various versions appear in the literature. All of them require a set of objects which g* *enerate the category in some appropriate sense. Depending on the proof, there are essential* *ly two types: the first type is based on the analogue of iterated attaching of cells w* *hich is used in the topological case; the second type is based on solution sets and applies * *a variant of Freyd's Adjoint Functor Theorem. Motivated by recent work of Neeman [7] and Franke [2], we prove a new theorem* * of the first type (Theorem A) and add, as an application, a Brown Representability The* *orem for covariant functors (Theorem B). The final Theorem C establishes a filtratio* *n of a triangulated category which clarifies the relation between results of the first* * and the second type. The main theorem Let T be a triangulated category and suppose that T has arbitrary coproducts. Definition 1. A set of objects S0 perfectly generates T if the following holds: (G1) an object X 2 T is zero provided that (S; X) = 0 for all S 2 S0; (G2) for every countable set of maps Xi! Yi in T the induced map a a (S; Xi) -! (S; Yi) i i is surjective for all S 2 S0 provided that (S; Xi) ! (S; Yi) is surjectiv* *e for all i and S 2 S0. Here, (X; Y ) denotes the maps from X to Y . For example, (G2) holds if every* * S 2 S0 is small, that is, the functor (S; -) preserves arbitrary coproducts. Therefor* *e the fol- lowing result generalizes the classical Brown Representability Theorem for tria* *ngulated categories having a set of small generators [5]. Theorem A. Let T be a triangulated category with arbitrary coproducts, and * *suppose that T is perfectly generated by a set of objects. Then a functor F :T op ! A* *b is representable if and only if F is cohomological and sends coproducts in T to p* *roducts. In [7], Neeman proves a Brown Representability Theorem for triangulated categ* *ories which are well generated. A well generated triangulated category has a set of * *perfect generators so that one gets a quick proof for Neeman's result. Examples of perfectly generated categories arise very naturally from triangul* *ated cat- egories having a set of small generators. Take for instance the stable homotopy* * category 1 2 HENNING KRAUSE of CW-spectra or the unbounded derived category of modules over an associative * *ring. Let T0 be a set of objects and let T be the smallest full triangulated subcateg* *ory which is closed under coproducts and contains T0. Then T is perfectly generated by a* * set of objects [7]. The proof of Theorem A is based on a reformulation of condition (G2) which is* * given in Lemma 3. Let us start with some preparations. We fix an additive category * *T . Following Auslander [1], a functor F :T op! Ab into the category of abelian gro* *ups is called coherent if there exists an exact sequence (-; X) -! (-; Y ) -! F -! 0: The natural transformations between two coherent functors form a set, and the c* *oherent functors T op! Ab form an additive category with cokernels which we denote by b* *T(see also [3] for this concept). A basic tool is the Yoneda functor T - ! bT; X 7! (-; X): Given an additive functor f :S ! T , we denote by f* :bS! bTthe right exact fun* *ctor which sends (-; X) to (-; fX). Lemma 1. Let T be an additive category. (1) If T has weak kernels, then bTis an abelian category. (2) If T has arbitrary coproducts, then bT has arbitrary coproducts and the * *Yoneda functor preserves all coproducts. Proof.Recall that a map X ! Y is a weak kernel for Y ! Z if the induced sequence (-; X) -! (-; Y ) -! (-; Z) is exact. The proof of (1) and (2) is straightforward. Note that for every fa* *mily of functors Fi having a presentation (-;OEi) (-; Xi) - ! (-; Yi) -! Fi- ! 0 ` the coproduct F = iFi has a presentation a (-;` OEi) a (-; Xi) -! (-; Yi) -! F -! 0: i i __ |_* *_| Given a class S of objects in an additive category T , we denote by Add S the* * closure of S in T under all coproducts and direct factors. Lemma 2. Let T be an additive category with arbitrary coproducts and weak ke* *rnels. Let S0 be a set of objects in T and denote by f :S ! T the inclusion for S = * *Add S0. (1) S has weak kernels and bSis an abelian category. (2) The assignment F 7! F |S induces an exact functor f*: bT-! bS. (3) The functor f* :bS-! bTis a left adjoint for f*. (4) f*O f* ~=id and f* induces an equivalence bT=Ker f*! bS. Proof.First observe that for every X 2 T , there exists an approximation X0 ! X* * such that X0 2 S and (S; X0)`! (S; X) is surjective`for all S 2 S. This follows from* * Yoneda's lemma if we take X0 = S2S0XS where XS = (S;X)S. A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS 3 (1) To prove that bSis abelian it is sufficient to show that every map in S h* *as a weak kernel. To obtain a weak kernel of a map Y ! Z in S, take the composite of a we* *ak kernel X ! Y in T and an approximation X0 ! X. (2) We need to check that for F 2 bTthe restriction F |S belongs to bS. It is* * sufficient to prove this for F = (-; Y ). To obtain a presentation, let X ! Y 0be a weak k* *ernel of an approximation Y 0! Y . The composite X0 ! Y 0with an approximation X0 ! X gives an exact sequence (-; X0)|S -! (-; Y 0)|S -! F |S -! 0: Clearly, F 7! F |S is exact. (3) Let F 2 bSand G 2 bT. Suppose first that F = (-; X). Then (f*F; G) = ((-; fX); G) ~=G(fX) ~=(F; f*G): This implies the adjointness isomorphism for an arbitrary F since f* is right e* *xact. (4) We have (f*O f*)(-; X) = (-; X) for all X 2 S, and f*O f* ~= idfollows si* *nce __ f*O f* is right exact. For the rest we refer to Proposition III.5 in [4]. * * |__| Lemma 3. Let T be a triangulated category with arbitrary coproducts. Let S0 * *be a set of objects in T and let S = Add S0. Then the functor h: T - ! bS; X 7! (-; X)|S; is cohomological. It preserves countable coproducts if and only if (G2) holds f* *or S0. Proof.We apply Lemma 2. To this end write h as composite h: T - ! bT-f*!bS: The Yoneda functor is cohomological and f* is exact. Therefore h is cohomologic* *al. It is clear that h preserves coproducts if and only if f* preserves coproducts. W* *e know that f*: bT! bSinduces an equivalence bT=Ker f*! bS, and it is not hard to see * *that f* preserves coproducts`if and only if Ker f* is closed under taking coproducts. W* *e fix a coproduct F = iFi in bTand for each Fi a presentation (-;OEi) (-; Xi) - ! (-; Yi) -! Fi- ! 0: Now suppose that Fi2 Kerf* for all i. Thus (S; OEi) is surjective for all S 2 S* * and all i. We have F 2 Kerf* if and only if the induced map a a (S; Xi) -! (S; Yi) i i is surjective for all S 2 S. Clearly, it is sufficient to have this for all S * *2 S0, and_we conclude that h preserves countable coproducts if and only if (G2) holds for S0* *. |__| Proof of the Theorem A.We fix a perfectly generating set S0 of objects in T an* *d put S = Add S0. Replacing S0 by {nS | n 2 Z; S 2 S0}, we may assume that (S0) = S0. Let F :T op! Ab be a cohomological functor which sends coproducts in T to produ* *cts. We construct inductively a sequence X0 -OE0!X1 -OE1!X2 -OE2!. . . S of maps in T and a set of maps ssi:(-; Xi) ! F for i 0 as follows.`Let U = S* *2S0F S. Each x 2 U corresponds to an element in F Sx and we put X0 = x2USx. We get an 4 HENNING KRAUSE element in Y F Sx ~=F X0; x2U and using Yoneda's lemma, this gives a map ss0: (-; X0) ! F . Suppose weShave a* *lready constructed ssi:(-;`Xi) ! F for some i 0. Let Ki= Ker ssi and let Ui= S2S0Ki* *S. We define Ti = x2UiSx and apply again Yoneda's lemma to obtain a map Ti ! Xi. We complete this to a triangle Ti-AEi!Xi-OEi!Xi+1-Oi!Ti and get an exact sequence F (Ti) FOi-!F Xi+1-FOEi!F Xi-FAEi!F Ti since F is cohomological. The construction implies (F AEi)ssi= 0 and this gives* * an element ssi+12 F Xi+1 such that (F OEi)ssi+1= ssi. Thus we have a factorization (-;OEi) ssi+1 ssi:(-; Xi) - ! (-; Xi+1) -! F: For each i 0 the map AEiinduces an epimorphism (-; Ti)|S ! Ki|S and we get the* *refore an exact sequence (-;AEi)|S ssi|S (-; Ti)|S - ! (-; Xi)|S -! F |S -! 0: We obtain in bSfor each i 0 the following commutative diagram with exact rows ssi|S 0 - ! Ki|S? -! (-;?Xi)|S -! F?|S -! 0 ?y0 ?y ? i y id ssi+1|S 0 - ! Ki+1|S -! (-; Xi+1)|S -! F |S -! 0 where i= (-; OEi)|S. Each i has a factorization ssi|S i i:(-; Xi)|S -! F |S -! (-; Xi+1)|S and therefore ssi+1|S Oi= id. This gives the following commutative diagram (-; X1)|S? -!1 (-; X2)|S? - 2! (-;?X3)|S -!3 . . . ?yo ?yo ?yo ` id`0 ` id`0 ` id`0 F |S K1|S -! F |S K2|S - ! F |S K3|S -! . . . and taking colimits in bS, we get an exact sequence a id- ia (ssi|S) (*) 0 -! (-; Xi)|S - ! (-; Xi)|S - ! F |S -! 0: i i Now consider the triangle a id-OEia a Xi -! Xi- ! X -! ( Xi) i i i and observe that Y a (ssi) 2 F Xi~= F ( Xi) i i induces a map ss :(-; X) ! F . We apply the functor T - ! bS; X 7! (-; X)|S; A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS 5 which is cohomological and preserves countable coproducts by Lemma 3. This give* *s an exact sequence a id- ia a id- i a (-; Xi)|S - ! (-; Xi)|S -! (-; X)|S -! (-; Xi)|S -! (-; Xi)|S: i i i i We compare this sequence with (*). The map id - i is a monomorphism, since (S) = S, and it follows that ss|S :(-; X)|S ! F |S is an isomorphism. Moreover,* * the subcategory of all Y 2 T such that ssY is an isomorphism is triangulated, cont* *ains S0, and is closed under arbitrary coproducts. Now let T 0be the localizing subcategory of T which is generated by S0. Thu* *s T 0 is the smallest triangulated subcategory of T which contains S0 and is closed * *under coproducts. We claim that T 0= T . To see this let Y 2 T and apply the construc* *tion in the first part of this proof to F = (-; Y ). The corresponding map ss :(-; X) !* * (-; Y ) is induced by a map X ! Y since X 2 T 0. We complete this map to a triangle W -! X -! Y - ! W and use (G1) to obtain W = 0 = W since (S; X) ! (S; Y ) is an isomorphism for a* *ll S 2 S0, and (S0) = S0 by our assumption. Thus T 0= T and we obtain (-; X) ~=F i* *n __ the first part of the proof. * * |__| Corollary. Let T be a triangulated category with arbitrary coproducts which is * *perfectly generated by a set of objects S0. Suppose that T 0is a full triangulated subcat* *egory which is closed under countable coproducts and contains all coproducts of objects in * *S0. Then T 0= T . Thus perfect generators are strong generators in the sense of [2]. Note that * *for every cardinal fi > @0 a fi-perfect generating set in the sense of [7] is automatical* *ly perfect as in Definition 1. Remark. If T has a set S0 of perfect generators, then the construction of eac* *h X 2 T implies that the functor X 7! (-; X)|S preserves arbitrary coproducts. Therefo* *re the following stronger condition holds for S0: (G20) for every set of maps Xi! Yi in T the induced map a a (S; Xi) -! (S; Yi) i i is surjective for all S 2 S0 provided that (S; Xi) ! (S; Yi) is surjectiv* *e for all i and S 2 S0. Brown representability for the dual Let T be a triangulated category with arbitrary coproducts. In this section w* *e prove a Brown Representability Theorem for T op. The first result of this type is due t* *o Neeman [6] and requires T to be generated by a set of small objects. This has been gen* *eralized in [7]. The concept which is used here stresses the symmetry between T and T op. Definition 2. A set S0 of objects is a set of symmetric generators for T if the* * following holds: (G1) an object X 2 T is zero provided that (S; X) = 0 for all S 2 S0; (G3) there exists a set T0 of objects in T such that for every map X ! Y in* * T the induced map (S; X) ! (S; Y ) is surjective for all S 2 S0 if and only* * if (Y; T ) ! (X; T ) is injective for all T 2 T0. 6 HENNING KRAUSE It is clear that (G3) implies (G2), and that T has a set of symmetric genera* *tors if and only if T ophas a set of symmetric generators. Therefore the following * *Brown Representability Theorem for T opis an immediate consequence of Theorem A. Theorem B. Let T be a triangulated category with arbitrary coproducts, and s* *uppose that T has a set of symmetric generators. Then T has arbitrary products, and a * *functor F :T ! Ab is representable if and only if F is cohomological and preserves prod* *ucts. Proof.We have a set of perfect generators for T and therefore arbitrary product* *sQin T . In fact, Theorem A implies that for everyQfamily (Xi)i2Iof objects the functor * * i(-; Xi) is represented by an object which is iXi. The set T0 which arises in (G3) is * *a set of perfect generators for T op, and it follows from Theorem A that a functor F :T * *! Ab __ is representable if and only if F is cohomological and preserves products. * * |__| An example for a set of symmetric generators is any set S0 of small objects s* *atisfying (G1). To see this, take for T0 the set of objects representing the functors T op-! Ab ; X 7! ((S; X); Q=Z) where S 2 S0. This shows that the stable homotopy category of CW-spectra or the unbounded derived category of modules over an associative ring have sets of sym* *metric generators. Remark. Let S0 be a set of perfect generators and let S = Add S0. Each injecti* *ve object I 2 bSgives rise to an object in T representing T op-! Ab ; X 7! ((-; X)|S; I): Therefore (G3) holds for S0 if and only if bShas an injective cogenerator. Neeman's Brown Representability Theorem for the dual in [7] involves the exis* *tence of an injective cogenerator for a category which is equivalent to some bS; it i* *s therefore a consequence of Theorem B. A filtration Let T be a triangulatedScategory with arbitrary coproducts. In this section w* *e study a filtration T = S which is defined in terms of a set S0 of appropriate gen* *erators.` Let ff be a cardinal.`Recall that an object S is ff-small if every map S ! * *i2IXi factors through i2JXi for some J I with cardJ < ff. Theorem C. Let T be a triangulated category with arbitrary coproducts, and s* *uppose that T is perfectly generated by a set S0 of ff-small objects. Let be the succ* *essor of ff for some cardinal a sup{card(S; Si) | S; Si2 S0 and cardI < ff} + cardS0: i2I Then the objects X 2 T satisfying card(S; X) < for all S 2 S0 form a subcateg* *ory S having the following properties: (1) S is a triangulated subcategory of T which contains S0; (2) S is closed under taking coproducts of less than objects; (3) the isomorphism classes of objects in S form a set; (4) every subcategory T 0of T which satisfies (1) and (2) contains S ; (5) every object in S is -small. A BROWN REPRESENTABILITY THEOREM VIA COHERENT FUNCTORS 7 A triangulated category which is well generated in the sense of Neeman [7] sa* *tisfies the assumption of the preceding theorem. The conclusion of this theorem implies* * the condition on a triangulated category which Franke assumes in [2] for his proof * *of the Brown Representability Theorem. Note that the proof in [2] is based on a varia* *nt of Freyd's Adjoint Functor Theorem. Thus Theorem C provides a link between results having completely different proofs. Proof of the Theorem C.(1) is clear. To prove (2), let S 2 S0 and (Xi)i2Ibe a`f* *amily of less than objects`in S . Suppose first that Xi2 S0 for all i. Every map`S ! * *i2IXi factors through i2JXifor some J I with cardJ < ff. We have card(S; i2JXi) * * , and I has at most (ff)ff= ffsubsets of cardinality less than ff. Therefore a card(S; Xi) ff. = ff< : i Now let each Xi2 S be arbitrary. We have for each i 2 I a map Ti! Xi such that* * Ti is a coproduct of less than objects from S0 and the induced map (S; Ti) ! (S; * *Xi) is surjective for all S 2 S0. Using (G20), it follows that the induced map a a (S; Ti) -! (S; Xi) i i ` ` is surjective for all S 2 S0. Thus iXi belongs to S since iTi2 S by the f* *irst part of this proof. (3) and (4) follow from the proof of Theorem A where it is shown that each ob* *ject in S can be constructed in countably many steps from objects in S0 by taking copr* *oducts of less than factors and cofibers. Note that in each step there is only a set * *of possible choices. (5) follows from (4) since the -small objects form a triangulated subcategory* * which_ satisfies (1) and (2). * * |__| As an example take the stable homotopy category S of CW-spectra. The set S0 = {nS | n 2 Z} of suspensions of the sphere spectrum S = S0 is a set of perfect g* *enerators where ff = @0. For every regular cardinal > @0 the subcategory S = {X 2 S | cardss*(X) < } has the properties (1) - (5) of the preceding theorem. Acknowledgement. I would like to thank Dan Christensen and Nora Ganter for a number of helpful conversations about the topic of this paper. Thanks in addit* *ion to Amnon Neeman for various comments on (t)his work. References [1]M. Auslander, Coherent functors, in: Proceedings of the conference on categ* *orical algebra (La Jolla, 1965), Springer-Verlag (1966), 189-231. [2]J. Franke, On the Brown representability theorem for triangulated categorie* *s, Topology, to appear. [3]P. Freyd, Stable homotopy, in: Proceedings of the conference on categorical* * algebra (La Jolla, 1965), Springer-Verlag (1966), 121-172. [4]P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 3* *23-448. [5]A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and * *Brown representabil- ity, J. Amer. Math. Soc. 9 (1996), 205-236. [6]A. Neeman, Brown representability for the dual, Invent. Math. 133 (1998), 9* *7-105. [7]A. Neeman, Triangulated categories, Princeton Univ. Press, to appear. 8 HENNING KRAUSE Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33501 Bielefeld, Germany E-mail address: henning@mathematik.uni-bielefeld.de