Brown representability often fails for homotopy categories of complexes

BR O WN REPRESENT ABILITY OFTEN F AILS F OR HOMOTOPY CA TEGORIES OF COMPLEXES GEORGE CIPRIAN MODOI AND JAN ˇ S ˇ TO V ´ I ˇ CEK Abstra ct. W e sho w that for the homotopy category K (Ab) of com- plexes of ab elian groups, b oth Brown represen tability and Bro wn repre- senta bility for the du al fai l. W e also pro vide an example of a localizing sub category of K (Ab) for which the inclusion into K (Ab) d oes not have a righ t adjoin t. Introduction Inspired b y a result of Casacub erta and Neeman [5], we pro vide in this note a class of naturally o ccurr ing triangulated catego ries with copro du cts and pr o ducts, for w hic h b oth Brown representa b ility and Bro wn repre- sen tabilit y for the dual (in the sense of [14, Definition 8.2.1]) fail. I n p artic- ular, the homotop y catego ry of complexes of ab elian group s b elongs to this class. Let u s s hortly exp lain the motiv ation. Giv en a triangulated category T with copro ducts and an ob ject Y ∈ T , the con tra v arian t f u nctor T ( − , Y ) : T − → Ab is cohomolog ical and s ends copro d ucts in T to pro ducts in Ab. By saying that T satisfies Br own r epr esentability , we mean that the con v erse h olds for T . That is, eac h cohomological f u nctor T → Ab wh ic h sends copro du cts in T to pro ducts is naturally equiv alen t to T ( − , Y ) for some Y ∈ T . Dually , giv en a catego ry T w ith pr o ducts, we ma y b e interested in whether co v ariant Hom–functors T ( Y , − ) are c haracterized b y the p r op ert y that they are h omologi cal and send pro du cts in T to pro du cts of ab elian groups. In suc h a case we sa y that T satisfies Br own r epr esentability for the dual . As noted in [5], there ha v e recentl y b een several results confirm ing Brown represent abilit y for v arious classes of triangulated catego ries. A lot of atten- tion has been given to wel l gener ate d triangulated categ ories; w e r efer to [14, p. 274] or [12, § 6] for a p recise defin ition. This is a wide class of triangulated catego ries and we hav e the follo wing im p lications (cf. [14, Chapter 8]): Date : July 7, 2018. 2010 Mathematics Subje ct Classific ation. 18G35 (primary), 16D90, 55U35 (secondary). Key wor ds and phr ases. Brown represen tability , Adjoint functor, T riangulated cate- gory , Homotop y category of complexes. Second named author supp orted by GA ˇ CR P201/10/ P084, researc h pro ject MSM 00216 20839 and the DFG Sc hw erpunk t SPP 1388. 1 2 GEORG E CIPRIAN M ODOI AND JAN ˇ S ˇ TOV ´ I ˇ CEK T is a w ell generated triangulated category . ⇓ T satisfies Bro wn representa bility . ⇓ A triangulated fun ctor G : T → U whic h send s coprod ucts in T to copro ducts in U has a r ight adjoin t. Ho w ev er, it turned out that man y natur ally o ccurr ing algebraic triangu- lated categories are not well generate d . In particular, the homotop y category K (Mo d - R ) is w ell generated if and only if R is a righ t p ure semisimple rin g, [20, Prop osition 2.6]. Here w e sho w that for this type of catego ries, the sec- ond link in the implication c hain ab o ve also fails for a v ery formal reason. P ostp oning the explanation of the terminology to the next section, we can state: Theorem 1. L et T b e a lo c al ly wel l gener ate d triangulate d c ate gory. Then T satisfies Br own r epr esentability if and only if T is wel l gener ate d. In p articular, if R is a ring which is not right pur e semisimple, for instanc e R = Z , then K (Mo d- R ) do es not satisfy Br own r epr esentability. When considerin g the Bro wn representabilit y for the dual, the situation is more delicate. First, it is still an op en p roblem wh ether eve ry well gener- ated triangulated category h as this prop erty . Ho we ver, [14, Th eorem 8.4.4] dualizes smo othly and w e ha ve the implication: T satisfies Br own representabilit y for th e dual. ⇓ A triangulated fu nctor G : T → U which sends pro du cts in T to pro ducts in U h as a left adjoint. Unfortunately , w e do not hav e a th eorem c haracterizing the categories whic h satisfy Brown rep r esen tabilit y for the d u al and co vering such a wide class of categories as Theorem 1. Nevertheless, we give a necessary con- dition for the repr esen tabilit y showing that man y homotop y catego ries of complexes, including K (Ab), cannot satisfy Bro wn representa bility for the dual. Namely , let B b e an additiv e catego ry with pro d ucts. F or a sub categ ory U ⊆ B , we denote b y Pro d( U ) the closure of U in B und er p ro ducts and direct summands . If U = { X } is a singleton, w e write just Pro d( X ). F urther, w e sa y that B has a pr o duct gener ator if there exists X ∈ B such that B = Pro d( X ). W e then obtain the follo wing resu lt: Theorem 2. L et B b e an additive c ate gory with pr o ducts. If K ( B ) satisfies Br own r epr ese ntability f or the dual, then B has a pr o duct gener ator. In p articular K (Ab) do es not satisfy Br own r epr esentability for the dual. Finally , w e touc h the m ost delicate question, w hen a triangulated functor G : T → U h as an adjoint. As a matter of the fact, th e existence of ad j oin t functors can sometimes b e p ro v ed ev en if Bro wn representabilit y fails. This w as done by Neeman [16], and other attempts h av e follo wed in [18, 11, 4]. Using recen t results from [8, 19, 3] (and p ostp oning the explanation of th e terminology again), we giv e, how ever, a statemen t sho wing th at the existence of adjoin ts do es not come for fr ee either: BRO WN REPRESENT A B ILITY OFTEN F A ILS 3 Theorem 3. L e t R b e a c ountable ring and let D b e the class of al l right flat Mittag–L effler R –mo dules. Then K ( D ) is always close d under c opr o ducts in K (Mo d- R ) , but the inclusion functor K ( D ) → K (Mod- R ) has a right adjoint if and only if R is a right p erfe ct ring. In p articular, a right adjoint do es not exist for R = Z . Ac kno wledgments. Th e fi rst named author would lik e to thank Simion Breaz for indicating him the argumen t sh o wing that the category of ab elian groups do es not h av e a pro d uct generator, w hic h is u sed in the pro of of Theorem 2. 1. Re p resent ability for locall y well genera ted triangula ted ca tegories Let T b e a triangulate d category and den ote b y R the (prop er) class of all infinite r egular cardinal n umb ers. In what f ollo w s we often need an increasing c hain S ℵ 0 ⊆ S ℵ 1 ⊆ S ℵ 2 ⊆ · · · ⊆ S κ ⊆ . . . of ske letally small triangulated sub categories of T indexed b y R suc h that the union S κ ∈ R S κ is the wh ole of T . R emark 4 . S trictly sp eaking, it is not clear ho w to obtain suc h a c hain in general using the axioms of ZFC alone. But there are tw o w ork arounds. First, if we w ork with a more concrete triangulated category , it may b e p os- sible to construct the chain directly . F or example, if T = K (Mo d- R ), th en S κ can b e defi n ed as the sub categ ory of all complexes formed by κ –presented mo dules. Second, if w e insist on general T , we can adopt some suitable ax- iomatiza tion of set theory which allo ws u s to w ell-order the unive rs e of all sets (eg. the vo n Neumann–Berna ys–G¨ odel set theory). Then we can easily construct the chai n using the in d uced we ll-ordering of ob jects of T . The same applies to the p r o of of Prop osition 5 b elo w, where w e strictly sp eaking use the Axiom of Choice for p rop er classes. Giv en an arbitrary full su b category S ⊆ T , we denote ⊥ S = { X ∈ T | T ( X, S ) = 0 for all S ∈ S } S ⊥ = { X ∈ T | T ( S, X ) = 0 for all S ∈ S } , Note th at if S is a triangulate d sub category of T , so are ⊥ S and S ⊥ . No w we can formulate a simp le but imp ortan t obstruction to Bro wn represen tabilit y . Prop osition 5. L et T b e a triangulate d c ate gory with c opr o ducts. Supp ose that T p ossesses an incr e asing chain ( S κ | κ ∈ R ) of skeletal ly smal l tri- angulate d sub c ate gories such that T = S κ ∈ R S κ and S ⊥ κ 6 = 0 for al l κ ∈ R . Then T do e s not satisfy Br own r epr esentability. Dual ly, supp ose T is triangulate d, has pr o ducts and has an incr e asing chain ( S κ | κ ∈ R ) of skeletal ly smal l triangulate d sub c ate gories such that T = S κ ∈ R S κ and ⊥ S κ 6 = 0 for al l κ ∈ R . Then T do es not satisfy Br own r epr e sentability for the dual. 4 GEORG E CIPRIAN M ODOI AND JAN ˇ S ˇ TOV ´ I ˇ CEK Pr o of. W e prov e only the fir s t p art, the other is dual. Cho ose for eac h κ ∈ R an ob ject 0 6 = Y κ ∈ S ⊥ κ . W e consid er the fun ctor F = Y κ ∈ R T ( − , Y κ ) : T − → Ab . Note th at F is a w ell-defined fun ctor. Indeed, recall that an y X ∈ T is con tained in S κ for some κ ∈ R , so T ( X , Y λ ) = 0 for all λ ≥ κ and the pro du ct defining F X is essen tially set-indexed. Moreo v er, F is homolog ical and sends copro du cts to pro du cts. No w we are essen tially done, since if F were repr esen ted by some ob ject in T , it would h a v e to b e the pro duct of ( Y κ | κ ∈ R ) in T , whic h cannot exist. T o giv e a formal argumen t, assume for the momen t that there is some Y ∈ T and a natural equiv alence η : T ( − , Y ) − → F. F or eac h κ ∈ R we then h a v e an idemp oten t natural tr an s formation ǫ κ : F → F give n as th e comp osition F − → T ( − , Y κ ) − → F whic h, b y the Y oneda lemma, induces an idemp oten t morphism e κ : Y → Y in T . Since ( ǫ κ | κ ∈ R ) is a p rop er class of pairwise orthogonal n on -zero idemp oten t end otransformations of F , the collec tion ( e κ | κ ∈ R ) would ha v e to b e a prop er class of endomorp h isms of Y with the same prop erties. This is absu rd since T ( Y , Y ) is a set.  Let us next exp lain the terminology n ecessary for Th eorem 1. Definition 6. L et T b e a triangulated category with copro ducts. A full triangulated sub category L of T is called lo c alizing if it is also closed u nder taking copro ducts in T . Giv en a class of ob jects S ⊆ T , we d enote b y Loc S the smallest lo calizing s ub category of T con taining S . The category T is called lo c al ly wel l gener ate d (in the sense of [20, Def- inition 3.1]) if for any set S (not a p r op er class!) of ob jects of T , Lo c S is w ell generated. No w we are ready to giv e a pr o of of Theorem 1. F or a more concrete construction of a non-representable fu n ctor K (Ab) → Ab, see Example 12 b elo w. Pr o of of The or em 1. If T is w ell generated, or equiv alently T = Lo c S for some set S , then Bro wn represen tabilit y holds by [14, Prop osition 8.4.2]. Let us, therefore, assume that T is not w ell generated. As discussed ab o ve, we hav e an increasing chain ( S κ | κ ∈ R ) of sk eletally small triangulated su b categories of T such that T = S κ ∈ R S κ . Let us p ut L κ = Lo c S κ ; b y defin ition eac h L κ is w ell generated and our assump tion ensures L κ $ T . It follo ws from [14, 9.1.13 and 9.1 .19] that eac h X ∈ T admits a triangle Γ κ X − → X − → L κ X − → Γ κ X [1] ( ∗ ) with Γ κ X ∈ L κ and L κ X ∈ S ⊥ κ . So giv en arbitrary X ∈ T \ L κ it follo ws 0 6 = L κ X ∈ S ⊥ κ . Now w e j ust apply Prop ositio n 5. BRO WN REPRESENT A B ILITY OFTEN F A ILS 5 Finally , th e second p art of the theorem follo ws fr om [20, 2.6 and 3.5]: If R is a ring which is not righ t p ure semisimple, K (Mo d- R ) is lo cally w ell generated but n ot w ell generated.  2. Re p resent ability for the dual In this section we discuss Bro wn representa bility for the du al and pro v e Theorem 2 . Based on [13 , Defin ition 2.24], we introdu ce the follo wing con- cept: Definition 7. Let B b e an additiv e categ ory and X a full sub category . Giv en M ∈ B , by an augmente d pr op e r right X –r esolution w e und er s tand a complex of th e f orm X M : . . . − → 0 − → 0 − → M − → X 0 − → X 1 − → X 2 − → · · · , suc h that X i ∈ X for all i ≥ 0 and Hom K ( B ) ( X M , X ′ [ n ]) = 0 for all X ′ ∈ X and n ∈ Z . A fa v orable fact is that su c h resolutions often do exist. Lemma 8. L et B b e an additive c ate gory with pr o ducts and splitting idem- p otents, let X ∈ B and put X = Pro d( X ) . Then any M ∈ B admits an augmente d pr op er right X –r e solution X M ∈ K ( B ) . Mor e over, X M = 0 in K ( B ) if and only if M ∈ X . R emark 9 . The lemma is true also without B h a ving splitting idemp otents, but we keep the assump tion f or the sak e of simp licit y . Pr o of. W e will construct the terms X i of an augmen ted resolution X M : . . . − → 0 − → 0 − → M d − 1 − → X 0 d 0 − → X 1 d 1 − → X 2 d 2 − → · · · b y induction on i . W e pu t X 0 = X Hom B ( M ,X ) and tak e for d − 1 the obvio us morphism. Ha ving constructed X i for i ≥ 0, w e set Z i = { f ∈ Hom B ( X i , X ) | f ◦ d i − 1 = 0 } , Then we can tak e X i +1 = X Z i and construct d i : X i → X i +1 in the obvio u s w a y . F or the s econd part, assume that X M = 0 in K ( B ), so it is a cont ractible complex. In particular, d − 1 : M → X 0 splits, so M ∈ Pr o d( X ) = X . The other implication is easy .  No w we show a consequence of non-existence of a pro d uct generator for B , whic h is imp ortant in connection with Pr op osition 5. Prop osition 10. L et B b e an additive c ate gory with pr o ducts and splitting idemp otents. If B do es not have a pr o duct g ener ator, then ⊥ S 6 = 0 in K ( B ) for every set (not a pr op er class!) S ⊆ K ( B ) . Pr o of. Supp ose B has no pro d uct generator and S ⊆ K ( B ) is a set of com- plexes. Let U ⊆ B b e the set of all ob jects o ccurr ing in the comp on ents of complexes in S , and let X = Prod( U ). Then clearly S ⊆ K ( X ), so it suffices to sho w that ⊥ K ( X ) 6 = 0 in K ( B ). T o this end, we ha v e X $ B since B has no p r o duct generator. Thus, w e can tak e an ob ject M ∈ B \ X and construct, using Lemma 8, an augmen ted 6 GEORG E CIPRIAN M ODOI AND JAN ˇ S ˇ TOV ´ I ˇ CEK prop er righ t X –resolution X M of M such that X M 6 = 0 in K ( B ). W e would lik e to see that X M ∈ ⊥ K ( X ), but this h as b een pr o v ed by Murfet in [13, Prop osition 2.27] (using cru cially the fact that X M is a complex w hic h is b ound ed b elo w).  No w we are ready to p ro v e Theorem 2: Pr o of of The or em 2. First of all, we may w ithout loss of generalit y assume that B has splitting idemp oten ts. If not, we replace B by its idemp oten t completion ˜ B (see e.g. [2, § 1]). Since K ( B ) has splitting idemp oten ts by [14, Prop osition 1.6.8 and Remark 1.6 .9], it follo w s that the inclusion K ( B ) ⊆ K ( ˜ B ) is a triangle equiv alence. Next we supp ose th at B h as n o pro du ct generator and prov e the existence of a n on-represen table homological p ro duct-preserving fu nctor F : K ( B ) → Ab. Namely , we c ho ose an increasing chain S ℵ 0 ⊆ S ℵ 1 ⊆ S ℵ 2 ⊆ · · · ⊆ S κ ⊆ . . . of skel etally small triangulated sub cat egories of K ( B ) indexed by R such that the u nion S κ ∈ R S κ is the wh ole of K ( B ) (cf. Remark 4). Th en, ho we ver, Prop osition 10 ensures that ⊥ S κ 6 = 0 in K ( B ) for eac h κ ∈ R , and so w e are in the situation of Prop ositio n 5, wh ic h asserts th e existence of suc h a functor. Finally , w e m ust pro v e that Ab h as no p ro duct generator. F or this pur- p ose, let u s fix a p rime n u m b er p ∈ N . Using the n otatio n from [9, § XI.65], w e d efine inductiv ely for eve ry ab elian group G and every ordinal σ : p σ G =        G, if σ = 0 p ( p σ − 1 G ) , if σ is n on limit . T ρ<σ p ρ G, if σ is limit . The length l ( G ) of the group G is th en by definition the minimum ordinal λ suc h th at p λ +1 G = p λ G . Note that for any family ( G i | i ∈ I ) of ab elia n groups, we ha v e the formula l  Y G i  = sup  l ( G i ) | i ∈ I  . Th u s, to prov e that Ab h as n o pro duct generator, it suffi ces to construct ab elian groups of arbitrary length. Ho w ev er, such families of group s are kno wn. F or ins tance W alk er’s group s P β [21] (wh ose construction can also b e found in [14, Construction C.2.1]) or generalized P r ¨ ufer groups [9, pp. 85–86 ].  R emark 11 . T he non-existence of a pro duct generator for Mo d- R seems to b e a muc h more widespr ead ph enomenon. If X ∈ Mo d- R is a pro duct generator, then Ext 1 R ( M , X ) = 0 implies that M is pro jectiv e for eac h M ∈ Mo d- R . That is, X is a pr oje ctive test mo dule in the sens e of [7 , p. 408]. If R is n ot right p erfect it is, h o w eve r, consistent with ZFC + GCH that there are no pr o j ectiv e test mo d ules and in particular no pro d uct generators. W e are grateful to the anon ymous referee f or making us a ware of th is fact. W e conclude the section with more co ncr ete examples of non-represen table (co)homolo gical functors K (Ab) → Ab. BRO WN REPRESENT A B ILITY OFTEN F A ILS 7 Example 12 . Let us for eac h κ ∈ R denote by A κ the full sub ca tegory of Ab formed by all groups of cardinalit y smaller than κ , and p ut T = K (Ab). If we take for a giv en κ a group P κ of length κ + 1 (e.g. W alke r’s group P κ from [21]), then clearly P κ 6∈ P r o d( A κ ), since the length of any group from Pro d A κ is at most κ . Thus, recalling the argumen ts ab o v e, w e s ee th at the augmen ted p rop er righ t Pr o d( A κ )–resolution of P κ , wh ic h w e denote b y Y κ , is nonzero in K (Ab) and b elongs to ⊥ K (Pro d( A κ )). In particular, the functor F = Y κ ∈ R T ( Y κ , − ) : T − → Ab , is a we ll-defined homologic al functor wh ich sends pro ducts in T to p ro ducts of ab elian groups, but it is n ot representable b y an ob ject of T . Let u s also explicitly construct a co ntra v ariant non-represen table functor. In fact, w e will use the formally dual statemen t to Theorem 2 and its p ro of for this rather th an Theorem 1 . The k ey p oin t is [6, Theorem 3.1] b y Chase, whic h implies that for an y un coun table κ ∈ R , we ha v e Z κ 6∈ Add A κ . Here, Add A κ denotes as u sual the closure of A κ in Ab under taking d irect sums and summand s. Therefore, denoting by Y ′ κ the augmen ted prop er left Add( A κ )–resolution of Z κ (with the obvious meaning), w e can infer exactly as b efore that the functor F ′ = Y κ ∈ R T ( − , Y ′ κ ) : T − → Ab , is a we ll-defin ed co h omological functor wh ic h sends copr o ducts in T to pro d- ucts of ab elian groups, but it is not repr esen table by an ob ject of T . 3. The non-ex istence of right adj oint The final section is fo cused on Theorem 3 , wh ic h is in fact an easy con- sequence of recen t results from [8, 19, 3]. Let us explain th e terminology . Giv en a r ing R , a r igh t R –mod ule M is called M i ttag–L effler if the canonica l map of groups M ⊗ R Y i ∈ I N i ! − → Y i ∈ I ( M ⊗ R N i ) is injectiv e for eac h family of left R -mo d ules ( N i | i ∈ I ). T h is concept comes from [17]. Let D b e th e class of all flat Mittag–Le ffler R –mod ules. Th er e are sev eral c haracterizations of m o d ules in D already in work of Ra ynaud and Gru - son [17], but the late st one is due to Herb era and T rlifa j, [10, Theorem 2.9]: Flat Mittag-Leffle r mo d u les coincide with so called ℵ 1 -pro jectiv e mo dules. F or R = Z , th is simply means that G ∈ D if and only if eac h coun table subgroup of G is free, whic h is a sp ecial case of [1, Prop osition 7] pro v ed by Azuma y a and F acc hini. Let us now pro v e th e last theorem. Pr o of of The or em 3. It is rather easy to s ee that D is closed und er direct sums and cont ains all pro jectiv e mo d ules. Assume first that R is r igh t p erf ect. Th en D coincides w ith the class of pro jectiv e mo d u les. In particular, K ( D ) is a w ell-generated triangulated 8 GEORG E CIPRIAN M ODOI AND JAN ˇ S ˇ TOV ´ I ˇ CEK catego ry b y [15, Theorem 1.1], so the inclusion K ( D ) → K (Mo d- R ) has a righ t ad j oin t by [14, Theorem 8.4.4]. On the other hand, assume that K ( D ) → K (Mod - R ) has a righ t adjoin t. Giv en an y G ∈ Mo d - R and considering it as a stalk complex in degree 0, w e hav e the counit of adjunction ε G : D → G . Let us tak e the R –mo dule homomorphism f = ε 0 G : D 0 → G in degree 0. Clearly D 0 ∈ D and it is easy to see that an y R –mo dule homomorphism f ′ : D ′ → G with D ′ ∈ D factors through f . That is, D is what is usually called a pr eco v ering class in Mo d- R . Ho w ev er, according to [3, Th eorem 6], this implies for a coun table ring R th at it is right p erfect.  Referen ces [1] G. Azumay a and A. F acchini. Rings of p u re global d imension zero an d Mittag-Leffler mod u les. J. 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I n Symp osia Mathematic a, Vol. XIII (Conve gno di Gruppi Ab eliani, IN DAM, Rome, 1974) , p ages 245–255. Academic Press, Lon d on, 1974. Babes ¸ –Bol y ai Univ ersity, F a cul ty of Ma thema tics and Computer Sci ence, 1, Mih ail Kog ˘ alniceanu, 400084 Cluj–Napoca, Ro mani a E-mail addr ess : cmodoi@math. ubbcluj.ro Charles University in Pra gue, F acul ty of Ma them a tics and Physics, De- p ar tment of Algebra, Sok olo vska 83, 186 75 Praha 8, Cz e ch Rep ublic E-mail addr ess : stovicek@kar lin.mff.cuni.cz