A Brown representability theorem via coherent functors

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1 Topology 41 (2002) A Brown representablty theorem va coherent functors Hennng Krause Fakultat fur Mathematk, Unverstat Belefeld, Postfach , Belefeld, Germany Receved 20 October 2000; accepted 23 January 2001 Abstract In ths paper we dscuss the Brown Representablty Theorem for trangulated categores havng arbtrary coproducts. Ths theorem s an extremely useful tool and varous versons appear n the lterature. All of them requre a set of objects whch generate the category n some approprate sense. Dependng onthe proof, there are essentally two types: the rst type s based onthe analogue of terated attachng of cells whch s used nthe topologcal case; the second type s based onsoluton sets and apples a varant of Freyd s Adjont Functor Theorem. Motvated by recent work of Neeman (A. Neeman, Trangulated Categores, Annals of Mathematcs Studes, 148, Prnceton Unversty Press, Prnceton, NJ, 2001) and Franke (On the Brown representablty theorem for trangulated categores, Topology, to appear), we prove a new theorem of the rst type (Theorem A) and add, as an applcaton, a Brown Representablty Theorem for covarant functors (Theorem B). The nal Theorem C establshes a ltraton of a trangulated category whch clares the relatonbetweenresults of the rst and the second type.? 2002 Elsever Scence Ltd. All rghts reserved. MSC: 18E30 (55P42, 55P65, 55U35) Keywords: Brown representablty; Trangulated category; Coherent functor 1. The man theorem Let T be a trangulated category and suppose that T has arbtrary coproducts. Denton 1. A set of objects S 0 perfectly generates T f the followng holds: (G1) anobject X T s zero provded that (S; X ) = 0 for all S S 0, Tel.: ; fax: E-mal address: hennng@mathematk.un-belefeld.de (H. Krause) /02/$ - see front matter? 2002 Elsever Scence Ltd. All rghts reserved. PII: S (01)

2 854 H. Krause / Topology 41 (2002) (G2) for every countable set of maps X Y n T the nduced map ( S; ) ( X S; ) Y s surjectve for all S S 0 provded that (S; X ) (S; Y ) s surjectve for all and S S 0. Here, (X; Y ) denotes the maps from X to Y. For example, (G2) holds f every S S 0 s small, that s, the functor (S; ) preserves arbtrary coproducts. Therefore the followng result generalzes the classcal Brown Representablty Theorem for trangulated categores havng a set of small generators [5]. Theorem A. Let T be a trangulated category wth arbtrary coproducts, and suppose that T s perfectly generated by a set of objects. Then a functor F : T op Ab s representable f and only f F s cohomologcal and sends coproducts n T to products. In[7], Neemanproves a BrownRepresentablty Theorem for trangulated categores whch are well generated. A well generated trangulated category has a set of perfect generators so that one gets a quck proof for Neeman s result. Examples of perfectly generated categores arse very naturally from trangulated categores havng a set of small generators. Take for nstance the stable homotopy category of CW-spectra or the unbounded derved category of modules over an assocatve rng. Let T 0 be a set of objects and let T be the smallest full trangulated subcategory whch s closed under coproducts and contans T 0. Then T s perfectly generated by some set of objects. The proof of Theorem A s based ona reformulatonof condton(g2) whch s gven nlemma 3. Let us start wth some preparatons. We x anaddtve category T. Followng Auslander [1], a functor F : T op Ab nto the category of abelan groups s called coherent f there exsts anexact sequence ( ;X) ( ;Y) F 0: The natural transformatons between two coherent functors form a set, and the coherent functors T op Ab form an addtve category wth cokernels whch we denote by ˆT (see also [3] for ths concept). A basc tool s the Yoneda functor T ˆT; X ( ;X): Gvenanaddtve functor f : S T, we denote by f : Ŝ ˆT the rght exact functor whch sends ( ;X)to( ;fx). Lemma 1. Let T be an addtve category. (1) If T has weak kernels, then ˆT s an abelan category. (2) If T has arbtrary coproducts, then ˆT has arbtrary coproducts and the Yoneda functor preserves all coproducts.

3 H. Krause / Topology 41 (2002) Proof. Recall that a map X Y s a weak kernel for Y Z f the nduced sequence ( ;X) ( ;Y) ( ;Z) s exact. The proof of (1) and (2) s straghtforward. Note that for every famly of functors F havng a presentaton ( ;X ) ( ;) ( ;Y ) F 0 the coproduct F = F has a presentaton ( ; X ) ( ; ) ( ; Y ) F 0: Gvena class S of objects nanaddtve category T, we denote by Add S the closure of S n T under all coproducts and drect factors. Lemma 2. Let T be an addtve category wth arbtrary coproducts and weak kernels. Let S 0 be a set of objects n T and denote by f : S T the ncluson for S = AddS 0. (1) S has weak kernels and Ŝ s an abelan category. (2) The assgnment F F S nduces an exact functor f : ˆT Ŝ. (3) The functor f : Ŝ ˆT s a left adjont for f. (4) f f = d and f nduces an equvalence ˆT=Ker f Ŝ. Proof. Frst observe that for every X T, there exsts an approxmaton X X such that X S and (S; X ) (S; X ) s surjectve for all S S. Ths follows from Yoneda s lemma f we take X = S S 0 X S where X S = (S;X ) S. (1) To prove that Ŝ s abelant s sucent to show that every map ns has a weak kernel. To obtan a weak kernel of a map Y Z n S, take the composte of a weak kernel X Y n T and an approxmaton X X. (2) We need to check that for F ˆT the restrcton F S belongs to Ŝ. It s sucent to prove ths for F =( ;Y). To obtan a presentaton, let X Y be a weak kernel of an approxmaton Y Y. The composte X Y wth anapproxmatonx X gves anexact sequence ( ;X ) S ( ;Y ) S F S 0: Clearly, F F S s exact. (3) Let F Ŝ and G ˆT. Suppose rst that F =( ;X). Then (f F; G)=(( ;fx);g) = G(fX ) = (F; f G): Ths mples the adjontness somorphsm for an arbtrary F snce f s rght exact. (4) We have (f f )( ;X)=( ;X) for all X S, andf f = d follows snce f f s rght exact. For the rest we refer to PropostonIII.5 n[4].

4 856 H. Krause / Topology 41 (2002) Lemma 3. Let T be a trangulated category wth arbtrary coproducts. Let S 0 be a set of objects n T and let S = AddS 0. Then the functor h : T Ŝ; X ( ;X) S s cohomologcal. It preserves countable coproducts f and only f (G2) holds for S 0. Proof. We apply Lemma 2. To ths end wrte h as the composton h : T ˆT Ŝ: f The Yoneda functor s cohomologcal and f s exact. Therefore h s cohomologcal. It s clear that h preserves coproducts f and only f f preserves coproducts. We know that f : ˆT Ŝ nduces an equvalence ˆT=Ker f Ŝ, and t s not hard to see that f preserves coproducts f and only f Kerf s closed under takng coproducts. We x a coproduct F = F n ˆT and for each F a presentaton ( ;X ) ( ;) ( ;Y ) F 0: Now suppose that F Ker f for all. Thus (S; ) s surjectve for all S S and all. We have F Ker f f and only f the nduced map ( S; ) ( X S; ) Y s surjectve for all S S. Clearly, t s sucent to have ths for all S S 0, and we conclude that h preserves countable coproducts f and only f (G2) holds for S 0. Proof of the Theorem A. We x a perfectly generatng set S 0 of objects n T and put S = AddS 0. Replacng S 0 by { n S n Z; S S 0 }, we may assume that (S 0 )=S 0. Let F : T op Ab be a cohomologcal functor whch sends coproducts n T to products. We construct nductvely a sequence X 0 X1 X2 of maps n T and a set of maps :( ;X ) F for 0 as follows. Let U = S S 0 FS. Each x U corresponds to an element n FS x and we put X 0 = x U S x. We get anelement n FS x = FX0 ; x U and usng Yoneda s lemma, ths gves a map 0 :( ;X 0 ) F. Suppose we have already constructed :( ;X ) F for some 0. Let K = Ker and let U = S S 0 K S. We dene T = x U S x and apply agan Yoneda s lemma to obtan a map T X. We complete ths to a trangle T X X+1 T

5 H. Krause / Topology 41 (2002) and get an exact sequence F(T ) FX F F F +1 FX FT snce F s cohomologcal. The constructon mples (F ) =0 and ths gves an element +1 FX +1 such that (F ) +1 =. Thus we have a factorzaton :( ;X ) ( ;) ( ;X +1 ) +1 F: For each 0 the map nduces an epmorphsm ( ;T ) S K S and we get therefore an exact sequence ( ; ) S S ( ;T ) S ( ;X ) S F S 0: We obtannŝ for each 0 the followng commutatve dagram wth exact rows: S 0 K S ( ;X ) S F S 0 0 d +1 S 0 K +1 S ( ;X +1 ) S F S 0 where =( ; ) S. Each has a factorzaton S :( ;X ) S F S ( ;X+1 ) S and therefore +1 S = d. Ths gves the followng commutatve dagram: ( ;X 1 ) S ( ;X 2 ) S ( ;X 3 ) S F S K1 S d 0 F S K2 S d 0 F S K3 S d 0 and takng colmts n ( ) 0 Now consder the trangle X d Ŝ, we get a coproduct of two splt exact sequences d ( S ) ( ;X ) S ( ;X ) S F S 0: X X ( X )

6 858 H. Krause / Topology 41 (2002) and observe that ( ) FX = F( X ) nduces a map :( ;X) F. We apply the functor T Ŝ; X ( ;X) S ; whch s cohomologcal and preserves countable coproducts by Lemma 3. Ths gves an exact sequence d ( ;X ) S ( ;X ) S ( ;X) S d ( ; X ) S ( ; X ) S : We compare ths sequence wth ( ). The map d s a monomorphsm, snce (S)=S, and t follows that S :( ;X) S F S s ansomorphsm. Moreover, the subcategory of all Y T such that Y s an somorphsm s trangulated, contans S 0, and s closed under arbtrary coproducts. Now let T be the localzng subcategory of T whch s generated by S 0. Thus T s the smallest trangulated subcategory of T whch contans S 0 and s closed under coproducts. We clam that T = T. To see ths let Y T and apply the constructon n the rst part of ths proof to F =( ;Y). The correspondng map :( ;X) ( ;Y) s nduced by a map X Y snce X T. We complete ths map to a trangle W X Y W and use (G1) to obtan W =0=W snce (S; X ) (S; Y ) s ansomorphsm for all S S 0, and (S 0 )=S 0 by our assumpton. Thus T = T and we obtan ( ;X) = F nthe rst part of the proof. Corollary. Let T be a trangulated category wth arbtrary coproducts whch s perfectly generated by a set of objects S 0. Suppose that T s a full trangulated subcategory whch s closed under countable coproducts and contans all coproducts of objects n S 0. Then T =T. Thus perfect generators are strong generators n the sense of [2]. Note that for every cardnal ℵ o,a-perfect generatng set n the sense of [7] s automatcally perfect as n Denton Brown representablty for the dual Let T be a trangulated category wth arbtrary coproducts. Inths sectonwe prove a Brown Representablty Theorem for T op. The rst result of ths type s due to Neeman[6] and requres T to be generated by a set of small objects. Ths has been generalzed n [7]. The concept whch s used here stresses the symmetry between T and T op.

7 H. Krause / Topology 41 (2002) Denton 2. A set S 0 of objects s a set of symmetrc generators for T f the followng holds: (G1) anobject X T s zero provded that (S; X ) = 0 for all S S 0 ; (G3) there exsts a set T 0 of objects n T such that for every map X Y n T the nduced map (S; X ) (S; Y ) s surjectve for all S S 0 f and only f (Y; T) (X; T ) s njectve for all T T 0. It s clear that (G3) mples (G2), and that T has a set of symmetrc generators f and only f T op has a set of symmetrc generators. Therefore the followng Brown Representablty Theorem for T op s an mmedate consequence of Theorem A. Theorem B. Let T be a trangulated category wth arbtrary coproducts, and suppose that T has a set of symmetrc generators. Then T has arbtrary products, and a functor F : T Ab s representable f and only f F s cohomologcal and preserves products. Proof. We have a set of perfect generators for T and therefore arbtrary products n T. Infact, Theorem A mples that for every famly (X ) I of objects the functor ( ;X ) s represented by anobject whch s X. The set T 0 whch arses n(g3) s a set of perfect generators for T op, and t follows from Theorem A that a functor F : T Ab s representable f and only f F s cohomologcal and preserves products. An example for a set of symmetrc generators s any set S 0 of small objects satsfyng (G1). To see ths, take for T 0 the set of objects representng the functors T op Ab; X ((S; X ); Q=Z); where S S 0. Ths shows that the stable homotopy category of CW-spectra or the unbounded derved category of modules over an assocatve rng have sets of symmetrc generators. Remark. Let S 0 be a set of perfect generators and let S = AddS 0. Each njectve object I Ŝ gves rse to anobject nt representng T op Ab; X (( ;X) S ;I): Therefore (G3) holds for S 0 f and only f Ŝ has an njectve cogenerator. Neeman s Brown Representablty Theorem for the dual n [7] nvolves the exstence of an njectve cogenerator for a category whch s equvalent to some Ŝ; t s therefore a consequence of Theorem B. 3. A ltraton Let T be a trangulated category wth arbtrary coproducts. Inths sectonwe study a ltraton T = S whch s dened n terms of a set S 0 of approprate generators.

8 860 H. Krause / Topology 41 (2002) Let be a cardnal. Recall that an object S s -small f every map S I X factors through J X for some J I wth card J. We shall use a set S 0 of perfect generators satsfyng the followng stronger condton: (G2 ) for every set of maps X Y n T the nduced map (S; X ) (S; Y ) s surjectve for all S S 0 provded that (S; X ) (S; Y ) s surjectve for all and S S 0. Theorem C. Let T be a trangulated category wth arbtrary coproducts, and suppose that T has a set S 0 of -small objects satsfyng (G1) and (G2 ). Let be the successor of for some cardnal { ( sup card S; ) } S S; S S 0 and card I + card S 0 : I Then the objects X T satsfyng card(s; X ) for all S S 0 form a subcategory S havng the followng propertes: (1) S s a trangulated subcategory of T whch contans S 0. (2) S s closed under takng coproducts of less than objects. (3) The somorphsm classes of objects n S form a set. (4) Every subcategory T of T whch satses (1) and (2) contans S. (5) Every object n S s -small. A trangulated category, whch s well generated n the sense of Neeman [7], satses the assumptonof the precedng theorem. The conclusonof ths theorem mples the condtonona trangulated category whch Franke assumes n [2] for hs proof of the Brown Representablty Theorem. Note that the proof n [2] s based on a varant of Freyd s Adjont Functor Theorem. Thus Theorem C provdes a lnk between results havng completely derent proofs. Proof of the Theorem C. (1) s clear. To prove (2), let S S 0 and (X ) I be a famly of less than objects n S. Suppose rst that X S 0 for all. Every map S I X factors through J X for some J I wth card J. We have card(s; J X ) 6, andi has at most ( ) = subsets of cardnalty less than. Therefore ( card S; ) X 6 = : Now let each X S be arbtrary. We have for each I a map T X such that T s a coproduct of less than objects from S 0 and the nduced map (S; T ) (S; X ) s surjectve for all S S 0. Usng (G2 ), t follows that the nduced map ( S; ) ( T S; ) X s surjectve for all S S 0. Thus X belongs to S snce T S by the rst part of ths proof.

9 H. Krause / Topology 41 (2002) (3) and (4) follow from the proof of Theorem A, where t s shown that each object n S can be constructed n countably many steps from objects n S 0 by takng coproducts of less than factors and cobers. Note that n each step there s only a set of possble choces. (5) follows from (4) snce the -small objects form a trangulated subcategory whch satses (1) and (2). As anexample take the stable homotopy category S of CW-spectra. The set S 0 ={ n S n Z} of suspensons of the sphere spectrum S = S 0 s a set of perfect generators where = ℵ 0. For every regular cardnal ℵ 0 the subcategory S = {X S card (X ) } has the propertes (1) (5) of the precedng theorem. Acknowledgements I would lke to thank Dan Chrstensen and Nora Ganter for a number of helpful conversatons about the topc of ths paper. Thanks naddtonto AmnonNeemanfor varous comments on (t)hs work. References [1] M. Auslander, Coherent functors, n: Proceedngs of the Conference on Categorcal Algebra, La Jolla, 1965, Sprnger, Berln, 1966, pp [2] J. Franke, On the Brown representablty theorem for trangulated categores, Topology, to appear. [3] P. Freyd, Stable homotopy, n: Proceedngs of the Conference on Categorcal Algebra La Jolla, 1965, Sprnger, Berln, 1966, pp [4] P. Gabrel, Des categores abelennes, Bull. Soc. Math. France 90 (1962) [5] A. Neeman, The Grothendeck dualty theorem va Bouseld s technques and Brown representablty, J. Amer. Math. Soc. 9 (1996) [6] A. Neeman, Brown representablty for the dual, Invent. Math. 133 (1998) [7] A. Neeman, Trangulated Categores. Annals of Mathematcs Studes, 148, Prnceton Unversty Press, Prnceton, NJ, 2001.