math.CT 2010-04-13 0

Hereditary Categories with Serre Duality which are generated by Preprojectives

We show that every k-linear abelian Ext-finite hereditary category with Serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver.

Authors: Carl Fredrik Berg, Adam-Christiaan van Roosmalen

HEREDIT AR Y CA TEGORIES WITH SERR E DUALITY WHICH ARE GENERA TED BY PREPROJECTIVES CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Abstract. W e sho w that every k - linear abelian Ext-finite hereditary category with Serre du- ality which is generated by prepro jectiv e ob jects is derived equiv alen t to the categ ory of repre- sen tations of a strongly lo cally finite thread quiver. Contents 1. Int ro duction 1 2. Conv en tions and Pr eliminaries 4 2.1. Conv en tions 4 2.2. Ab e lian her editary catego ries 4 2.3. Serre duality and a lmost split maps 4 2.4. Thread q uivers a nd dualizing k -v arieties 5 2.5. Sketc hing categories 6 3. Round T rip Distance and Light Cone Dista nce 6 3.1. Light cone distance 6 3.2. Connection with directing o b jects 8 3.3. Round trip distance 8 4. Hereditary s ections 9 4.1. Split t -s tructures 9 4.2. Definition a nd characteriza tio n of hereditary sections 10 4.3. (Co)reflective sub categorie s o f heredita r y sections 11 4.4. Criterium for b eing a dualizing k -v ar ie t y 12 4.5. Light cone 12 5. Hereditary s ections Z -equiv alen t to dua lizing k -v arieties 13 5.1. The condition (*) 13 5.2. Finding a dualizing k -v ariety Z -e q uiv alent to Q 15 6. Nonth read o b jects and threa ds in hereditar y sections 19 6.1. d • -in-b etw een and threa ds 19 6.2. Nonthread ob jects 22 6.3. Rays a nd co rays 25 7. Categorie s g enerated by Z Q 28 7.1. Marks and comar ks 28 7.2. Enlarg ing hereditary sectio ns 29 References 31 1. Introduction Throughout, let k b e an alg ebraically closed field. In [12], Reiten a nd V a n den Bergh classify k - linear ab elian he r editary Ext-finite noether ia n catego ries with Serre duality . One r esult in there is that every such catego ry is a direct sum of a categor y without nonze r o pro jectives, and a category generated by pr e pro jective o b jects. The la tter c a tegories were o f sp ecific interest since there was 2010 Mathematics Subje ct Classific ation. 18E10, 18E30,16E35. 1 2 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN no known w ay to relate them –throug h equiv ale nces or derived equiv a lences– to known abelian categorie s. Reiten and V a n den Bergh gav e a construction by formally inv erting the (right) Serre functor, and in [15] Ringel g av e a construction using ray quivers. (In [5] it was shown that these categorie s were derived equiv a le n t to representations of str ongly lo cally finite quivers, i.e. quivers whose indecomp os able pro jective a nd injective representations hav e finite length.) Reiten and V an den Be r gh asked whether every hereditary categories with Serre dua lit y is derived equiv alent to a noetheria n o ne, a nd thus fit –up to der ived equiv alence– in to their classifi- cation. In [1 4] how ever, Ringel gav e a class o f counterexamples. Reiten then a sks in [1 3] whether it is feas ible to have a classifica tion of hereditar y categor ie s with Serre duality whic h a r e gener ated by pr epro jectives, but not nece s sarily no etheria n. This pa p e r is the third pap er of the author s to answer this question (the o ther two being [5, 6]); we pro vide an a nswer to this question up to der ived equiv a lence in terms o f representations of thread quivers (see b elow): Theorem 1.1. L et A b e a k - line ar ab elian her e ditary Ext-finite c ate gory with Serr e duality which is gener ate d by pr epr oje ct ive obje ct s. Then D b A ∼ = D b rep k Q wher e Q i s a str ongly lo c al ly finite thr e ad quiver. The undefined concepts in this theo rem will be intro duced b e low. Ro ughly sp eaking a t hread quiver is a (p ossibly infinite) quiver where some of the ar r ows ha ve b een replaced b y loca lly discre te (=without a ccumu lation p oints) linearly or dered set. Strong lo ca l finiteness is an additiona l fin teness prop erty ensuring that the category of finitely presented representations has Ser re dua lity . The pro of of this theo rem consists out of tw o steps. In the firs t step (up to a nd including § 5 ) we prov e a version of T he o rem 1.1 under an additiona l assumption, namely condition (* ) explained below. The r est of this pap er will b e devoted to removing this condition. The first part of this pa per ( § 3, § 4, and § 5 ) follows the pro o f of [5 , Theorem 4 .4] clos ely . Although we r eintroduce all relev ant concepts, some familiarity with the pro of of [5 , Theorem 4.4] mig ht be helpful to the r eader to b etter understand our a rguments b elow. W e will start our overview o f the paper with § 4, where w e discuss so-called spl it t - s t ructur es (for de finitio n, we refer to § 4 .1). Our main res ult is the following theor em (compa r e with [16, Theorem 1 ]), which describ es the heart of a b ounded split t - s tructure. Theorem 1.2. L et A b e an ab elian c ate gory and let H b e a ful l sub c ate gory of D b A such that D b A is the additive c losur e of S t ∈ Z H [ t ] and Hom( H [ s ] , H [ t ]) = 0 for t < s , then H is an ab elian her e ditary c ate gory derive d e quivalent with A . Let A b e a n a b elia n her editary E xt-finite category with Serre duality . W e a r e thus interested in finding a split t -struc tur e s uch that the hear t is of the form rep Q for a strongly lo cally finite threa d quiver Q . In particular this mea ns that the categ ory of pro jectives Q of H is a semi-her editary dualizing k -v ariety , i.e. a Hom-finite Karoubian category Q such that mo d Q is ab e lian, hereditar y , and has Ser re duality . T o help find such t -structures, w e introduce hereditar y sections: a full additive sub category o f D b A is a hereditar y section if there is a split t -structure on D b A and the categor y of pro jectives of its hereditary hea rt coincides with Q (s e e Theo r em 4.15 ). Given a her editary section Q in D b A , the full replete (=close d under isomo r phisms) additive sub c ategory genera ted b y all indecompo sables of the form τ n X , X ∈ ind Q and n ∈ Z will be denoted by Z Q . This co incides with the full a dditive sub catego ry of D b A generated by all indecomp osables lying in the Auslander - Reiten comp onents of D b A intersecting with Q . In order to find hereditary sections, we first introduce the (rig ht ) light cone dista nce d • ( − , − ) and the round trip distance d ( − , − ) w orking on ind D b A in § 3 as follows: for a ll X , Y ∈ ind D b A d • ( X, Y ) = inf { n ∈ Z | there is a path from X to τ − n Y } and d ( X, Y ) = d • ( X, Y ) + d • ( Y , X ) . W e then hav e following characterization o f a her e dita ry section (Prop osition 4.14): a full additive sub c ategory Q of D b A is a her e ditary section if and only if HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 3 (1) d • ( X, Y ) ≥ 0, for all X , Y ∈ ind Q , a nd (2) if X ∈ ind Q and d ( X, Y ) < ∞ for a Y ∈ ind D b A , then Y ∈ Z Q . F or a set T ⊆ ind Z Q , we define d • ( T , X ) = inf T ∈T d • ( T , X ), d • ( X, T ) = inf T ∈T d • ( T , X ), and d ( T , X ) = d • ( T , X ) + d • ( X, T ) . F ollowing the pr o of o f [5, Theorem 4.4], we find a set T ∈ ind Q such that d ( T , X ) < ∞ for all X ∈ ind Q and we choose a heredita r y section Q T such tha t d • ( T , X ) =  d ( T , X ) 2  for all X ∈ ind Q T where ⌊·⌋ is the flo or function. If T is chosen to satisfy some ex tra pro per ties (as given in Lemma 5.4, but in particular T has to be co untable), then Theo rem 5.10 yields that the Q T is indeed a semi-hereditary dualizing k -v ar iety . Thus if Z Q gener ates D b A as thick triangula ted category , then D b A ∼ = D b rep Q for a strongly lo cally finite thre a d quiver Q . That T can indeed b e chosen to satisfy the extra needed assumptions, is exactly the condition (*) mentioned earlier . Condition (*) can eas ily b e sta ted as (see § 5.1): (*) : there is a c ountable subse t T ⊆ ind Z Q such that d ( T , X ) < ∞ , for a ll X ∈ ind Z Q . Hereditary sections not satisfying co ndition (*) seem to b e rather artificia l yet they do o ccur, even when the co rresp onding heart is, for exa mple, g e ne r ated by prepr o jectives (see Ex a mple 5 .3)! W e now come to the second part of the a r ticle ( § 6 and 7) wher e we will remove the c o ndition (*) fro m the as sumptions. The fir st s tep to unders ta nding c o ndition (* ) b etter is to make a distinction b etw een thread ob jects and nonthread ob jects in Q , whose definitions we now give. As an easy c o nsequence of Serre duality on D b A , it will turn out that Q has left and r ight almost split maps, th us for every A ∈ ind Q , there are nonsplit maps f : A → M and g : N → A in Q suc h that every nons plit map A → X or Y → A facto rs though f or g , resp ectively . W e will say A is a thread ob ject if b oth M and N a re indecompos able. An indecomp o sable ob ject which is not a thread ob ject will b e called a nonthread ob ject. One ma jor s tep in understanding co ndition (*) will be showing that there are only countably many non thread ob jects (Pro po sition 6.19); this will b e the main result in § 6.2 . Thu s without enlarging the set T ⊆ Q ab ove to m uch, we may assume it contains ev ery nonthread o b ject in ind Q . If Z Q do es no t satisfy conditio n (*), then there a re ob jects X which lie “ to o far from nonthread ob jects”, th us d ( A, X ) = ∞ for ev er y nonthread ob ject A . Suc h ob jects X w ill b e divided into tw o classes: ra y ob jects and coray ob jects. If there is a nonthread ob ject A such that d • ( A, X ) < ∞ , then the thread ob ject X will b e called a ray o b ject; if ther e is a nonthread ob ject A such that d • ( X, A ) < ∞ , then X will is called a coray ob ject. If Q has nonthread ob jects (and we may alwa ys reduce to this c a se), connectedness implies one of these conditions is satisfied. On the r ay ob jects, we define an equiv alence r elation (see § 6.3) given by X ∼ Y if and only if d • ( X, Y ) < ∞ or d • ( Y , X ) < ∞ . A full additiv e ca tegory generated b y an equiv alence cla ss of ray ob jects will be called a ray a nd it is shown in 6.29 that there may only be a countable num ber or rays. In o rder to enlarge Q , we will add an ob ject M for every ray R , calle d the mark of R . This should b e seen a s a no nthread ob ject “lying on the far side of R ”. As shown in Example 7.3 w e cannot expect to find a hereditar y section Q ′ such that Q and all marks of a ll rays lie in Z Q ′ . Ho w ever, if D b A is ge ne r ated by Z Q , then this will always be the case. Thu s in § 7 we will c onstruct a hereditary section Q ′ such that Z Q ′ satisfies condition (*) and Z Q ⊆ Z Q ′ . Theorem 1.1 will follow fro m this. Ac knowledgmen ts The authors like to thank Idun Reiten, Sverre Smalø , J an ˇ S ˇ tov ´ ıˇ cek, a nd Michel V a n den B ergh for many useful discussions and helpfu l idea s. The second author a lso 4 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN gratefully ackno wledges the hos pitality of the Max-Pla nck-Institut f ¨ ur Mathema tik in Bonn and the Nor wegian Universit y of Science and T echnology . 2. Conventions and Preliminaries 2.1. Conv entions. Throughout, let k b e an algebr a ically clo sed field. All categor ies will be assumed to be k -linea r. W e will fix a universe U and assume that (unless explicitly noted) all our catego ries a re U - categorie s, thus Hom C ( X, Y ) ∈ U for any catego r y C and all ob jects X , Y ∈ Ob C . A category C is called U -sm al l (or just small) if Ob C ∈ U . Let C b e a K rull-Schmidt c ategory . By ind C we will deno te a set o f chosen repres ent atives of isomorphism classes of indecomp osa ble ob jects of C . If C ′ is a Krull- Sch midt sub categ ory o f C , w e will as sume ind C ′ ⊆ ind C . If C is a triangulated catego ry with Serre duality (se e below) and Q is a full Krull-Schmidt sub- category , then w e will denote by Z Q the unique full additive replete (= closed under isomorphisms) sub c ategory of C with ind Z Q = { τ n X | X ∈ ind Q , n ∈ Z } . If Q 1 and Q 2 are K r ull-Schmidt sub- categorie s o f C such that Z Q 1 ∼ = Z Q 2 as sub categor ies o f C , then we will say Q 1 and Q 2 are Z -e quivalent . An (or der e d) p ath b etw een indecomp osa bles X and Y in a K rull-Schmidt category C is a sequence X = X 0 , X 1 , . . . , X n = Y of indecomp osables s uch that Hom( X i , X i +1 ) 6 = 0 for all 0 ≤ i ≤ n − 1. A n ontrivial p ath is a path where ther e are i, j ∈ { 0 , 1 , . . . , n } such that ra d( X i , X j ) 6 = 0 . If there is no nontrivial path from X to X , then we will say X is dir e cting . W e will say a K rull-Schmidt categ o ry C is c onne cte d if for all indecompo sables X , Y , there is a sequence X = X 0 , X 1 , . . . , X n = Y of indecomp osa bles such that there is either a path from X i to X i +1 or from X i +1 to X i , for a ll 0 ≤ i ≤ n − 1. If C is a Krull-Schmidt category and A, B ∈ ind C then we will denote by [ A, B ] the full replete a dditiv e categ ory containing every indecomp os a ble C ′ ∈ ind C with Hom( A, C ′ ) 6 = 0 and Hom( C ′ , B ) 6 = 0. W e define ] A, B ] similarly , but with the extr a c o ndition that C ′ 6 ∼ = A . The sub c ategories [ A, B [ and ] A, B [ are defined in an obvious w ay . 2.2. Ab e lian hereditary categories. An ab elian ca tegory A is sa id to be Ext-finite if dim Ext i ( X, Y ) < ∞ for all i ∈ N and X , Y ∈ Ob A . If E xt i ( X, Y ) = 0 for all i ≥ 2, then A is ca lled her e ditary . If Ext i ( X, Y ) = 0 for all i ≥ 1, we will say A is semi-simple . F or a n ab elian categor y A , we will denote by D b A its b o unded der ived categ ory . There is a fully faithful functor i : A → D b A mapping every X ∈ A to the complex whic h is X in degr ee 0 and 0 in all other degrees . W e will o ften suppress this embedding and write X ∈ Ob D b A instead of iX ∈ O b D b A . When A is hereditary , the b ounded de r ived categor y D b A has the following well-known de- scription ([9, 10, 17]): ev ery ob ject X ∈ D b A is is omorphic to the direc t sums o f its ho mologies. 2.3. Serre dual it y and almost s plit maps. Let C be a k -line a r Hom-finite triangulated ca te- gory . A S err e functor [7] is a k -linear additive equiv alence S : C → C s uch that for any tw o ob jects A, B ∈ O b C , ther e is an isomorphis m Hom( A, B ) ∼ = Hom( B , S A ) ∗ of k -v ector spaces, natura l in A and B . Here, ( − ) ∗ denotes the vector space dual. A Serre functor will alwa ys b e an exact equiv alence. If A is an Ext-finite ab elia n catego r y , then we will say that A has Serr e duality if and only if D b A has a Serre functor. It has been shown in [1 2] that an Ex t-finite hereditary category has Serre dualit y if a nd only if A has Auslander-Reiten sequences and there is a 1-1 -corr e s po ndence b e tw een the indecomp osable pro jective ob jects and the indeco mp os able injective ob jects via their simple top and simple so cle, resp ectively . It has a lso b een shown in [12] that S ∼ = τ [1] where τ : D b A → D b A is the Ausla nder-Reiten translate. In particular, an Ext-finite tr iangulated ca tegory has Serre duality if and only if it has Auslander-Reiten tria ngles. HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 5 A ma p f : A → B is said to b e left (or right) almost split if every non-s plit map A → X (or X → B ) factors throug h f . 2.4. Thread quiv ers and dual i zing k - v arie ti es. W e recall s ome definitions from [1 , 2]. A Hom- finite additive categ o ry a where idemp otents split will b e ca lled a finite k - variety . The functors a ( − , A ) a nd a ( A, − ) ∗ from a to mod k will b e called standar d pr oje ctive r epr esentations and stan- dar d inje ctive re pr esentations , resp ectively . W e will write mo d a for the category of contra v a riant functors a → mo d k which are finitely presentable b y standard pro jectives. F ollowing [6, Prop o sition 4.1] we will say a finite k - v ariety a is dualizing [2 ] if a nd only if a has pseudokernels and pseudo cokernels (thus mod a a nd mo d a ◦ are ab elian, where a ◦ is the dual category o f a ), every standard pro jectiv e ob ject is cofinitely generated by standa rd injectives, and every s tandard injective ob ject is finitely g enerated by standard pro jectives. A finite k -v ariety a is called semi-her e ditary if and only if the categor y mo d a is a belia n and hereditary . It has b een shown ([18, Prop osition 4.2], see also [3, Theor e m 1.6 ]) that a is semi- hereditary if and o nly if every full (prea dditive) sub categ ory with finitely many elements is semi- hereditary . Let a b e a finite k -v ariety . It has b een shown in [6] that mo d a is an ab elian and hereditary category with Ser re duality if and only if a is a semi-hereditary dualizing (finite) k -v ar ieties. Thread quiver w ere then intro duced in order to classify these semi-heredita ry dualizing k -v a rieties. A thr e ad quiver consis ts of the following information: • A quiver Q = ( Q 0 , Q 1 ) where Q 0 is the set of vertices and Q 1 is the set of arr ows. • A decomp osition Q 1 = Q s ` Q t . Arr ows in Q s will be ca lled s t andar d arr ows , while arr ows in Q t will be refer r ed to as thr e ad arr ows . Threa d a r rows will b e drawn by dotted ar rows. • With e very thr e a d a rrow α , there is an a sso ciated linea rly o r dered set T α , p oss ibly empty . When not empty , we will write this p oset as a lab el for the thread arrow. A finite linear ly ordered p oset will just b e deno ted by its num b er of elements. When Q is a thr ead quiver, we will deno te by Q r the underlying quiver, thus forgetting lab els and the difference b etw een arrows and threa d a r rows. W e will say Q is str ongly lo c al ly fin ite when Q r is strongly lo cally finite, i.e . all indecomp osa ble pro jective and injective representations hav e finite dimension as k -v ector spaces. Let Q be a strongly lo cally finite thread quiver. With ev er y thread t ∈ Q t , we denote by f t : k ( · → · ) − → k Q r the functor asso cia ted with the obvious embedding ( · → · ) − → Q r . W e define the functor f : M t ∈ Q t k ( · → · ) − → k Q r . With ev ery thread t , there is a n a sso ciated linea rly o r dered set T t . W e will write L t = N · ( T t → × Z ) · − N . Denote by g t : k ( · → · ) − → k L t a chosen fully faithful functor given by mapping the extremal p oints o f · → · to the minimal and maximal ob jects o f L , r esp ectively . W e will write g : M t ∈ Q t k ( · → · ) − → M t ∈ Q t k L t . The catego r y k Q is defined to b e a 2-push-out of the following diag ram. L t ∈ Q t k ( · → · ) f / / g   k Q r i      L t ∈ Q t k L t j / / _ _ _ _ k Q W e ha ve the following result whic h classifies the semi- her editary dualizing k -v arieties in function of str ongly lo cally finite thread quivers. 6 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Figure 1. Sketc hes of Z A ∞ -, Z A ∞ ∞ -, and Z D ∞ -comp onents Theorem 2 .1. [6 ] Every semi-her e ditary dualizing k -variety is e quivalent t o a c ate gory of t he form k Q wher e Q is a str ongly lo c al ly finite thr e ad quiver. W e will also use the following result ([6 , Cor ollary 6.4]). Prop ositio n 2 .2. A semi-her e ditary dualizing k - variety has only c ount ably many sinks and sour c es. 2.5. Sketc hing categories. Throughout this paper, sketc hes of categories (or more precisely , the Auslander-Reiten quiver) will b e provided for the b enefit of the rea der. All examples will b e directed categ o ries, and we will use the conv e n tions used in [18] (see also [14, 15]). W e will consider only thr ee shap es o f Auslander- Reiten comp onents: those o f the form Z A ∞ , Z A ∞ ∞ , and Z D ∞ , which will b e represented by squares , triangles , and tria ngles with a do ubled side, resp ectively (see Fig ure 1). These c omp o nents will b e ordered s uch that the maps go fr om left to right. Whenever a triangulated ca tegory comes eq uiped with a t -structure, this will be suitably indi- cated on the co rresp onding sketc h. 3. Round Trip Dist ance and Light Cone Dist ance In [5 ], the r ound trip dis tance and lig ht cone distance were introduced for sta ble tra nslation quivers o f the form Z Q . These dis tances pr ov e d v aluable to discuss sections of Z Q . Our goal o f describing the ca tegory of pro jectives Q is similar and we wish to employ simila r techniques to this case . W e will have to generalize the tec hniques o f [5] somewha t since the category Z Q do es not hav e to b e gener alized standar d in o ur pres ent setting. The definitions co incide in case this connecting comp onent is g eneralized standar d. In this section, let C = D b A where A is an abelia n E xt-finite ca tegory with Serre duality . Although A is not req uired to be hereditary , it follows from Corolla ry 3.9 tha t our definitions are only nontrivial if C has directing ob jects, which implies that A is der ived equiv alen t to a hereditar y category (see Theorem 4 .4). 3.1. Light cone di stance. F o r all X , Y ∈ ind C , we define the (right) light c one distanc e as d • ( X, Y ) = inf { n ∈ Z | ther e is a path from X to τ − n Y } . In par ticular, d • ( X, Y ) ∈ Z ∪ {±∞} . R emark 3 .1 . E ven when X and Y lie in the same Auslander-Reiten compo ne nt, the right light cone distance do e s not nee d to coincide with the o ne g iven in [5 ], as the following example illustrates. The difference is that the definition ab ov e takes all maps into acco unt when determining paths, while the definition in [5] o nly consider s irreducible mor phisms. Example 3.2. Let a b e the se mi-hereditary dualizing k -v a riety whose thread quiver is • / / / / • The Ausla nder -Reiten quiv er of D b mo d a containing the sta ndard pro jectives of mod a via the standard embedding is of the form Z A ∞ ∞ . O n the left hand side of Figure 2 we have lab eled the vertices w ith the rig ht light cone distanc e d • ( X, − ) as a s table tr anslation q uiver (as in [5]), while on the right ha nd side we hav e used the definition of right light cone distance given in this a rticle. F or the benefit of the r eader, the arrows b etw een indecomp osable pro jective ob jects have be en drawn in black. HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 7 Figure 2. The Auslander-Reiten quiver of the category Z Q in Example 3.2 whe r e every vertex has been lab eled with d • ( X, − ). F or this, the Auslander -Reiten quiver on the left has b een in terpreted as a sta ble tr a nslation quiver, while on the right w e hav e us e d the category Z Q to determine the right light cone dis atnce. The following lemma is stated for easy r e ference. Lemma 3.3 . F or al l X , Y ∈ ind C , we have d • ( X, τ n Y ) = d • ( τ − n X , Y ) = d • ( X, Y ) + n . Note that the function d • is no t symmetric. It do es how ever s a tisfy the tr iangle inequa lit y . Prop ositio n 3.4 . F or al l X , Y , Z ∈ ind C , we have d • ( X, Z ) ≤ d • ( X, Y ) + d • ( Y , Z ) , whenever this su m is define d. Pr o of. The pr o of follows directly fro m the definition.  F or a subsets T 1 , T 2 ⊆ ind C , we define the right light co ne distance in a n obvious way: d • ( T 1 , T 2 ) = inf T 1 ∈ T 1 T 2 ∈ T 2 d • ( T 1 , T 2 ) . The following result follows fro m the tria ngle inequality . Corollary 3.5. L et X ∈ ind C and T 1 , T 2 ⊆ ind C , we have d • ( T 1 , T 2 ) ≤ d • ( T 1 , X ) + d • ( X, T 2 ) , whenever this su m is define d. W e now contin ue to define a right and left light c one distanc e spher e by S • ( X, n ) = { Y ∈ ind D b A | d • ( X, Y ) = n } and S • ( X, n ) = { Y ∈ ind D b A | d • ( Y , X ) = n } , resp ectively , for any n ∈ Z and X ∈ O b C . Finally , we will denote S • Q ( X, n ) = S • ( X, n ) ∩ ind Q a nd S Q • ( X, n ) = S • ( X, n ) ∩ ind Q . 8 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN 3.2. Conne ctio n with directing ob jects. Although the left and rig ht light cone distances betw een any tw o indecomp osables a r e defined, we can only exp ect nontrivial r esults in the case where b oth are dir ecting. W e sta rt by recalling following result. Prop ositio n 3.6. [16, Lemma 3 ] L et X u / / Y v / / Z w / / X [1] b e a triangle wher e X , Y ar e inde c omp osable and u is nonzer o and n oninvertible. L et Z 1 b e a dir e ct summand of Z . The maps v 1 : Y → Z 1 and w 1 : Z 1 → X [1] induc e d by v and w , r esp e ctively, ar e nonzero and noninvertible. Prop ositio n 3.7. L et X , Y , Z ∈ ind C such that d • ( X, Z ) = 0 . F or al l non-zer o f ∈ Hom( X, Y ) and g ∈ Hom( Y , Z ) we have that g f is non-zer o. In p articular, d • ( X, Z ) = 0 implies Hom( X, Z ) 6 = 0 . Pr o of. Without loss o f generality , w e may ass ume g is not an isomo rphism, and hence C = cone( g : Y → Z ) is nonzer o. It follows from Pro po sition 3.6 that Hom( Z, C i ) 6 = 0 fo r every direct summand C i of C . Using Serre duality we find Hom( C i [ − 1] , τ Z ) 6 = 0, a nd therefore d • ( C i [ − 1] , Z ) ≤ − 1. The tria ngle inequalit y then gives d • ( X, C i [ − 1]) ≥ d • ( X, Z ) − d • ( C i [ − 1] , Z ) ≥ 1 a nd hence Hom( X , C [ − 1]) = 0. W e deduce that f : X → Y do es not fa ctor through C [ − 1] a nd hence g f is non-zero.  Prop ositio n 3. 8. An obje ct X ∈ ind C is dir e cting if and only if d • ( X, X ) = 0 , or e quivalently, X is non-dir e cting if and only if d • ( X, X ) = −∞ . Pr o of. It is clear that directing implies d • ( X, X ) = 0 . T o prove the o ther implication, ass ume there is a nontrivial path X = X 0 f 0 → X 1 f 1 → · · · f n − 1 → X n f n → X . Since d • ( X, X ) = 0, the triang le inequa lit y yields d • ( X i , X j ) = 0 fo r all i, j ∈ { 0 , . . . , n } . Pr op o- sition 3.7 now giv es that f = f n . . . f 1 f 0 is non-zer o. Since X is indecomp osable, End X is a finite dimensiona l lo c al algebra and th us every ele ment is either nilpo ten t or in vertible. Propo sition 3.7 yields f is not nilp otent, hence it is inv ertible, a contradiction.  Corollary 3.9. L et X , Y ∈ ind C such t hat d • ( X, Y ) ∈ Z , t hen b oth X and Y ar e dir e cting. Pr o of. Using triangle inequality , we hav e d • ( X, Y ) ≤ d • ( X, X ) + d • ( X, Y ), and hence 0 ≤ d • ( X, X ). W e alw ays hav e d • ( X, X ) ≤ 0, so we get d • ( X, X ) = 0. P rop osition 3.8 shows X is directing. Sho wing Y is directing is similar.  Corollary 3.10. L et X ∈ ind C . If X is dir e cting, then so is every ind e c omp osable Y in the Auslander-R eiten c omp onen t of X . Pr o of. Since Y lies in the same Auslander-Reiten compo nen t as X , we know d • ( X, Y ) < ∞ . Then by Pr op osition 3.8 a nd tria ngle inequality , 0 = d • ( X, X ) ≤ d • ( X, Y ) + d • ( Y , X ), and hence d • ( Y , X ) > −∞ . Inv oking Coro llary 3.9 completes the pro o f.  3.3. Ro un d trip dis tance. F or X , Y ∈ ind C , we define the r oun d trip distanc e d ( X, Y ) a s the symmetrization o f the r ight lig h t cone distance, thus d ( X, Y ) = d • ( X, Y ) + d • ( Y , X ) , whenever this is well-defined. It is easy to see that d ( X, Y ) dep ends only on the τ -o rbit of X and Y , thus d ( X , Y ) = d ( τ n X , τ m Y ) fo r all m, n ∈ Z (compare with Lemma 3.3 ). When we restr ict ourselves to indecomp os ables o f Z Q , where Q is the catego ry of pro jectives of a hereditary categ ory A with Serre dualit y , then we know that both d • ( X, Y ) and d • ( Y , X ) will be in Z ∪ {∞} , hence d ( X , Y ) will b e well-defined. F ollowing propo sition shows d defines a pseudo metr ic. Prop ositio n 3.1 1 . L et Z Q as ab ove. F or al l X , Y , Z ∈ ind Z Q we have (1) d ( X , Y ) ≥ 0 (2) d ( X , X ) = 0 HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 9 (3) d ( X , Y ) = d ( Y , X ) (4) d ( X , Z ) ≤ d ( X , Y ) + d ( Y , Z ) Pr o of. The claims (2), (3), and (4 ) follow from Pr o p osition 3.8, the definition, and Pr op osition 3.4, r esp ectively . Since then 0 = d ( X , X ) ≤ d ( X , Y ) + d ( Y , X ) = 2 d ( X, Y ), the first c laim holds as well.  A r ound t rip distanc e spher e is defined in an obvious wa y . 4. Heredit ar y sections Let A b e an ab elian hereditary Ext-finite category with Serr e duality . In what follows, we shall discuss the categ ory of pro jectiv es of her editary catego ries H derived equiv a lent to A . Thes e pro jectives will for m hereditary s ections in D b A and, likewise, a hereditar y section in D b A will give a hereditary ca teg ory H derived equiv alent to A . W e sta rt with a some r esults concer ning split t -str uctures. 4.1. Spli t t -s tructures. The concept o f a t -structure was intro duce d by Be ˘ ılinso n, Bernstein and Deligne in [4]. Sp ecifically , we will b e interested in so-ca lled split t -s tructures of which the heart will b e a heredita ry categor y ([16]). Definition 4.1 . A t -structur e on a triangulated category C is a pair ( D ≥ 0 , D ≤ 0 ) of non-zero full sub categor ies of C satisfying the following conditions, where we denote D ≤ n = D ≤ 0 [ − n ] and D ≥ n = D ≥ 0 [ − n ] (1) D ≤ 0 ⊆ D ≤ 1 and D ≥ 1 ⊆ D ≥ 0 (2) Hom( D ≤ 0 , D ≥ 1 ) = 0 (3) ∀ Y ∈ C , there exists a triang le X → Y → Z → X [1] with X ∈ D ≤ 0 and Z ∈ D ≥ 1 . Let D [ n,m ] = D ≥ n ∩ D ≤ m . W e will say the t -structure ( D ≥ 0 , D ≤ 0 ) is b ounde d if and o nly if every ob ject of C is c ontained in some D [ n,m ] . W e call ( D ≥ 0 , D ≤ 0 ) split if e very triangle o ccurr ing in (3) is split. It is shown in [4] that the he art H = D ≤ 0 ∩ D ≥ 0 is an ab elian catego r y . Unfortunately , if A is an ab elian category , then no t every t -structure on D b A defines a heart H whic h is derived equiv alent to A . F ollowing pro p osition shows that in our setting w e may exp ect derived equiv alence betw een A and H . Prop ositio n 4.2. L et A b e an ab elian c ate gory and let ( D ≥ 0 , D ≤ 0 ) b e a b ounde d t -structur e on D b A . If al l the triangles X → Y → Z → X [1] with X ∈ D ≤ 0 and Z ∈ D ≥ 1 split, then D ≤ 0 ∩ D ≥ 0 = H is her e ditary and D b A ∼ = D b H as triangulate d c ate gories. Pr o of. It is w ell known tha t the ca tegory Ind A of left ex a ct co nt rav a riant functors from A to Mo d k is a k -linear Gr othendieck catego ry and that the Y one da embedding of A into Ind A is a full a nd ex act embedding. By [11, P rop osition 2.14 ], this embedding extends to a full and ex a ct embedding D b A → D b Ind A . Since all triang les X → Y → Z → X [1] with X ∈ D ≤ 0 and Z ∈ D ≥ 1 split, we may use [12, Lemma I.3.5] to see that H is hereditar y . It is now an easy conseque nc e o f [4, Pro po sition 3.1.16] that D b A ∼ = D b H as triangulated categories .  R emark 4.3 . Since the ab ove categor y Ind A is not a U -catego ry , we tacitly assume an enlarge ment of the universe. W e will say a subcatego ry D o f D b A is close d un der s uc c essors if it satisfies t he follo wing prop erty: if X ∈ D and Y ∈ ind D b A suc h that Hom( X , Y ) 6 = 0 or Y ∼ = X [1], then Y ∈ D . As the following theorem shows, this is a useful prop erty to find s plit t -str uc tur es. Theorem 4 .4. L et A b e a c onne cte d ab elian c ate gory and let D b e a nontrivial ful l sub c ate gory of D b A close d under suc c essors, then ( D ≥ 0 , D ≤ 0 ) is a b ounde d and split t -structu r e on D b A wher e D ≤ 0 = D and D ≥ 1 = D ⊥ . 10 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Pr o of. It is straig htf orward to check ( D ≥ 0 , D ≤ 0 ) defines a split t -structure. It follows from [16, Lemma 7] that it is als o bo unded.  Combining the previous theorem with [16, Theorem 1], w e get the following attractive descrip- tion o f a hereditar y heart in a der ived category . Corollary 4.5. L et A b e an ab elian c ate gory and let H b e a ful l su b c ate gory of D b A such t hat D b A is the additive closur e of S t ∈ Z H [ t ] and Hom( H [ s ] , H [ t ]) = 0 for t < s , then H is an ab elian her e ditary c ate gory derive d e quivalent with A . R emark 4 .6 . In [16, The o rem 1] one starts with a full sub categor y H of a tria ngulated catego r y T (not necess arily a de r ived categor y) and o btains that T is e quiv alent as additive c ate gory to D b H , for a her editary ca tegory H . In Theo rem 4.4 and Co rollar y 4.5 we restr ic t ourselves to the case where T is a derived categ ory (with the induced triang ula ted str uc tur e) and find T ∼ = D b H as triangulate d c ate gories . 4.2. Defini ti on and c haracterization of hereditary section s . Before defining a hereditary section, we need a pr eliminary co ncept. Throughout, let A b e an abe lia n Ext-finite c a tegory with Serre dua lity and write C = D b A . Definition 4 .7. Le t Q b e a full sub catego ry of C . W e will say Q is c onvex if every path in C starting a nd ending in Q lies en tirely in Q . A sub categor y Q of C is called τ -c onvex if Z Q is conv ex. Example 4. 8 . Any ob ject X ∈ ind C spans a convex sub categor y Q of C if and only if X is directing in C . R emark 4.9 . Since there is alwa ys a trivial pa th b etw een isomorphic o b jects, a convex sub categ ory will always b e r eplete. In what follows Q will consists only o f directing ob jects. In this case , we may give a n alternative formulation of τ -co nv ex: Q will b e τ -conv ex if a nd only if for every X ∈ ind C , the condition d ( Q , X ) 6 = ∞ implies that Q meets the τ -or bit of X . Definition 4 . 10. A her e ditary se ction is a nontrivial (= having at least one no nz e r o ob ject), full, conv ex, and τ -conv ex additive subcateg o ry Q of C s uch that Q meets e very τ -orbit at mos t once. R emark 4 .11 . The notion of a hereditary section is self-dual. If Q is a hereditar y section in C , then Q ◦ is a her editary sec tion in C ◦ . R emark 4.12 . If A is s emi-simple, then S ∼ = id D b A such that τ ∼ = [ − 1]. Since a her editary section Q o f D b A may meet every τ -o rbit at most once, we hav e that X ∈ Ob Q implies that X [ n ] 6∈ O b Q for all n 6 = 0 . Example 4. 13. If A is a hereditary a b elian Ext- finite categor y with Serre dua lity with Q A as category of pro jectives, then Q A is a hereditar y s e c tion in D b A . In Theorem 4.15 the co n verse of this statement will b e shown. Prop ositio n 4.14. The sub c ate gory Q is a her e ditary s e ction if and only if it is a ful l and τ -c onvex additive sub c ate gory Q of C such that d • ( X, Y ) ≥ 0 for al l X , Y ∈ ind Q . Pr o of. W e ma y as s ume D b A is connected. F urthermo re, the statement is triv ial if A is semi-simple, th us a s sume the g lobal dimension o f A is at le ast one. Assume Q is a hereditary section in D b A . If d • ( X, Y ) < 0, then there is a path from X to τ Y . Since A is not semi-s imple, there is als o a pa th from τ Y to Y and thus, using that Q is convex, we s e e that τ Y ∈ Q , a co nt radiction. This pr oves o ne direction. Assume Q is a full and τ -conv ex additive sub categor y of ind C such that d • ( X, Y ) ≥ 0 for all X , Y ∈ Q . Since d • ( X, τ − n X ) < 0 for all n > 0, Q con tains a t mos t one ob ject from ea ch τ -orbit. Assume X , Y ∈ Q with paths from X to Z a nd fro m Z to Y , th us d • ( X, Z ) ≤ 0 and d • ( Z, Y ) ≤ 0. Since Q is τ -co nv ex , Q contains a n o b ject o f the τ -o r bit of Z . Using the triangle inequality , we find d • ( X, Y ) ≤ d • ( X, Z ) + d • ( Z, Y ) ≤ 0. Since w e hav e assumed d • ( X, Y ) ≥ 0, we se e d • ( X, Z ) = 0 and d • ( Z, Y ) = 0. Th us Lemma 3.3 s hows that the ob ject Q con tains from the τ -or bit of Z must b e Z its e lf. Hence Q is co nv ex.  HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 11 W e now come to the main result ab o ut hereditary s ections, characterizing them to b e ca tegories of pr o jectives of a heredita ry hear t. Theorem 4.15. L et A b e a c onne cte d Ext-finite ab elian c ate gory with Serr e duality and let Q b e a her e ditary se ction of D b A , then ther e exists an Ext -finite ab elian her e ditary c ate gory H with Serr e duality, such that A is derive d e quivalent to H and the c ate gory of pr oje ct ives of H is given by Q . Pr o of. If A is semi-simple, the categor y H is j ust Q itself. Th us assume now that A is not semi-simple. Let D b e the full r eplete additive sub categ o ry of D b A spanned by all indecomp osa ble ob jects X with d • ( X, Q ) ≥ 0 and d • ( Q , X ) < ∞ . W e chec k that D satisfies the conditions of Theorem 4.4. Let X ∈ ind Q . Since d • ( X, Q ) = d • ( Q , X ) = 0 w e know tha t X ∈ Ob D , and L emma 3 .3 shows that τ X 6∈ Ob D such that D is indeed a nontrivial sub categ o ry of D b A . Let X ∈ ind D and Y ∈ ind D b A suc h that Hom( X , Y ) 6 = 0 , o r th us in particular d • ( X, Y ) ≤ 0. The triangle inequality implies that Y ∈ ind D . F urthermor e, the conditions on A imply there is an Auslander-Reiten triangle X → M X → τ − X → X [1 ], such that Prop osition 3.6 yie lds that d • ( X, X [1]) ≤ 0. As above the triangle inequa lit y will implies that X [1] ∈ D . W e conclude tha t the co nditions of Theorem 4 .4 are indeed sa tisfied such tha t there is a split t -structure on D b A with D ≤ 0 = D and D ≥ 1 = D ⊥ . Denote the her editary heart by H . W e only need to show that the ca tgeory of pr o jectives Q H of H coincides with Q in D b A . Note that Q ⊆ D and Q [ − 1] ⊆ D ⊥ , so that Q ⊆ H . Let X ∈ ind Q H , thus X ∈ ind D but τ X 6∈ ind D . In this ca se, we have d • ( X, Q ) = 0 such that τ -co nv exity implies that X ∈ ind Q . If X ∈ ind Q , then τ X 6∈ ind D but X ∈ H such that X is a pr o jectiv e o b ject in H and hence X ∈ ind Q H . W e conclude that Q ∼ = Q H as subcateg ories of D b A .  Observ ation 4.16. Since every hereditar y section is the image o f the catego r y of pro jectives of a her editary ca tegory in its derived categ ory , we see that every her editary section Q of C is semi-hereditar y , a par tia l tilting set, has left and right almost split ma ps , and cons ists o f only directing ob jects. R emark 4.17 . Theorem 4.15 shows that, given a hereditar y section Q , there is a t -s tructure on D b A such that Q is the catego r y of pro jectives of the heart H . How ever, the t - structure is no t uniquely deter mined by Q as the next example illustra tes. Example 4.18. Let Q b e the thread quiver x 1 / / y , and let Q be the standard heredita ry section in D b rep Q . Denote by P x ∈ Q the indecomp o sable ob ject asso cia ted with x . The category D b rep Q is sketc hed in Fig ure 3. W e find a smaller her editary section Q ′ spanned by all ob jects A ∈ Q with d ( P x , A ) < ∞ , thus Q ′ contains exactly those indecompo s ables of Q which lie in the same Ausla nder -Reiten comp onent of D b rep Q a s P x . There are at least tw o different hea rts in D b rep Q such that Q ′ is the categor y of pr o jectiv es, as shown in Figure 3 . The middle picture co rresp onds to the t - structure g iven in the pr o of of Theorem 4 .1 5. The following statement is a sp e c ial case of P r op osition 4.14 . Corollary 4.19 . L et A b e an ab elian her e ditary Ext - finite k -line ar c ate gory satisfying Serr e duality and let Q b e the c ate gory of pr oje ct ives of A . L et Q ′ b e a ful l pr e addi tive sub c ate gory of D b A such that Z Q = Z Q ′ and d • ( X, Y ) ≥ 0 for al l X , Y ∈ ind Q ′ , then Q ′ is a her e ditary se ction in D b A . 4.3. (Co)reflective sub categories of hereditary sections. W e prove an ana lo gue of P rop o- sition [6 , P rop osition 5 .2] for hereditar y sections. W e intro duce the fo llowing nota tion. Let A be an Ext-finite ab elian ca tegory and Q a heredita r y s ection in D b A . F or any ob ject Z ⊆ Ob D b A , we define Q ⊥ Z as the full subca tegory of Q left-orthogo nal on Z , thus A ∈ O b Q ⊥ Z ⇔ ∀ Z ∈ Z : RHom( A, Z ) = 0 . 12 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Figure 3. Illustration of Ex a mple 4.18 Prop ositio n 4.2 0 . L et Q b e a her e ditary se ct ion in D b A . (1) L et Z ⊂ Ob D b A with P Z ∈Z dim Hom( A, Z ) < ∞ for al l A ∈ Q and Hom( Z 1 , Z 2 [ n ]) = 0 for al l Z 1 , Z 2 ∈ Ob D b A and n ∈ Z \ { 0 } . Then the emb e dding Q ⊥ Z → Q has a left and a right adjoint. (2) L et X , Y ∈ Q . The emb e ddi ng [ X , Y ] → Q has a left and a right adjoint. Pr o of. Theorem 4.15 yields there is a hereditary category H ⊂ D b A with Q as its categor y of pro jectives. The pr o of is obtained by rep eating the pro of o f [6, P rop osition 5.2 ] in H .  4.4. Criteri u m for b eing a dualizing k -v ariety. W e will b e interested in her editary sections which are dualizing k - v arieties. The following criterion will b e useful. Prop ositio n 4.21. A her e ditary se ction Q is dualizing if and only if for every A ∈ ind Q ther e ar e C 1 , C 2 ∈ Ob Q such that for every B ∈ ind Q (1) Hom( B , C 1 ) 6 = 0 when d • ( A, B ) = 0 , and (2) Hom( C 2 , B ) 6 = 0 when d • ( B , A ) = 0 . Pr o of. Let H b e a hereditary catego r y of which Q is the ca tegory of pro jectives (Theorem 4.15). The fir st statement is equiv alent to sa ying there is an epimor phism Q ( − , C 1 ) → Q ( A, − ) ∗ and the second s tatement is equiv alent to s aying there is a mono mo rphism Q ( − , A ) → Q ( C 2 , − ) ∗ . Since the cokernel of the first ma p and the kernel of the second map ar e finitely g enerated pro jectives, we k now that Q ( − , A ) is cofinitely presented and Q ( A, − ) is finitely pres e nted. By Obser v ation 4.16 Q is semi-heredita ry and thu s C o rollar y 2.1 yields the requir ed result.  4.5. Light c one. Let A b e an a belia n ca tegory with Serre duality and X ∈ D b A b e an inde- comp osable directing o b ject. W e define the lig ht cone centered o n X to b e full reple te a dditive category Q X with ind Q X = S • ( X, 0), th us Q X is generated by those indecomp osa ble ob jects Y such that X admits a path to Y , but no path to τ Y . Using Pr o p osition 4.14 o ne ea sily chec ks that Q X is a hereditary section. If A is connected then Theorem 4.1 5 shows that Q X defines a t -structure with hear t a hereditary category H X . W e will refer to H X as the light cone tilt centered o n X . A s imilar construction has b een used by Ringel in [16]. Dually we define the co-light cone and the co-light cone tilt cent ered on X . HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 13 Lemma 4.22. In the light c one tilt c enter e d on X , we have Hom( X , P ) 6 = 0 , for al l pr oje ctives P . Pr o of. The r esult follows directly fro m Prop os ition 3.7.  Lemma 4.23. In the light c one tilt c ent er e d on X , al l pr oje ctives obje ct s have an inje ctive re so- lution. Pr o of. Let P b e a pro jective and consider the ca nonical map P → S X ⊗ Hom( P , S X ) ∗ with kernel K . Since P is pr o jectiv e, the kernel needs to b e pr o jectiv e as well. It is straightforw ard to chec k that Hom( X , K ) = 0, hence K = 0 a nd the canonical ma p is a monomo r phism. An injective resolution is then giv en b y 0 → P f → S X ⊗ Hom( P, S X ) ∗ → coker f → 0 .  Prop ositio n 4.24. In a light c one t ilt, al l pr epr oje ctives have pr oje ct ive and inje ctive r esolutions. Pr o of. It suffices to show this for all indecomp osa ble prepro jective ob jects. Every such ob ject is of the form τ − n Y for an indecomp osa ble pro jective ob ject Y . W e will prove the statement by induction on n . If n = 0 then the statement is Lemma 4 .23. Assume that τ − n Y has a pro jective and an injective r esolution. If 0 → τ − n Y → I → J → 0 is an injective resolution of τ − n Y then 0 → S − 1 I → S − 1 J → τ − n − 1 Y → 0 is a pro jective resolution of τ − n − 1 Y . Since the pro jectives S − 1 I and S − 1 J hav e injective resolutions, the same holds for τ − n − 1 Y .  5. Heredit ar y sections Z -equiv alent to dual izing k -v arieties 5.1. The condi tion (*). Let A b e a co nnected ab elia n hereditary Ext-finite categ ory satisfying Serre dua lity and denote the c ategory of pro jectives by Q . W e will as sume Z Q is connected. If Q is a dualizing k -v a r iety , then Q ( − , A ) is cofinitely presented. This means that at least one source S ma ps no n-zero to A , hence d • ( S, A ) = 0 . Dually we find that A maps non-zer o to at least one sink T , such that d • ( A, T ) = 0. Prop ositio n 2.2 yields there are only a count able amount of sinks and so ur ces, hence Q satisfies the following proper ty: there is a co untable subset T ⊆ ind Q such that d ( T , X ) = 0, f or all X ∈ ind Q . W e will w eaken this prop er t y to : (*) : ther e is a countable subset T ⊆ ind Z Q s uch that d ( T , X ) < ∞ , for all X ∈ ind Z Q . It is thus clear (*) needs to b e satisfied when Q is a dualizing k - v ariety . Mo r eov er if ther e is a her e ditary s ection Q ′ in D b A with Z Q = Z Q ′ where Q ′ is a dualizing k - v ariety , then Z Q a lso needs to satisfy conditio n (*). Before giving an example where co ndition (*) is not satisfied, we recall following definitions. Definition 5.1. Let P be a p oset. The s ubs et T ⊆ P is s aid to be c ofinal if for every X ∈ P there is a Y ∈ T such that X ≤ Y . The least c ardinality of the cofinal subsets of P is calle d the c ofinality of P and is denote by cofin P . Dually , one defines a c oinitial subset of P and the c oinitiality of P is denoted by coinit P . Next exa mple shows (* ) is no t always sa tisfied. Example 5.2. Let L b e a linearly order ed and lo cally discrete set such that cofin L > ℵ 0 . F o r example, if T is a linear ly orde r ed set with cofin T > ℵ 0 we may define the p oset L = T → × Z . Let P b e the p oset N · ( T → × Z ) · − N , th us k P is the s emi-hereditary dualizing k -v ariety given by the thread q uiver · T / / · . W e may sk etch the categor y as the upper par t of Figure 4. In mo d k P , we consider a new hereditary categ ory H b y choo sing a here dita ry section Q in mo d k P generated by all s tandard pro jectives of the form P ( − , A ) where A ∈ N or A ∈ L . The category H is marked with gray in Figur e 4. The new category H has ca tegory of pr o jectiv es Q a nd Z Q do es not sa tisfy (*). 14 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN { { P S f r a g r e p la c e m e n t s L L Figure 4. Illustration of Example 5 .2 { P S f r a g r e p la c e m e n t s L Figure 5 . Sketc h o f a ca teg ory gener ated by prepro jective ob jects, but which do es not satisfy co nditio n (*). Example 5.3. Let H ′ be the dual category o f the category H defined in Example 5.2 (see Figure 5). This catego r y is generated by pre pro jective ob jects. Denote by Q ′ the categor y o f pro jectives of H ′ . It is clear tha t Z Q ′ do es not satisfy co nditio n (*). The following lemma says that, under the condition (*), we can choose the set T to satisfy some additiona l prop erties. Lemma 5.4 . L et Q b e a her e ditary se ct ion such that Z Q satisfy c ondition (*). Ther e is a c ountable subset T = { T i } i ∈ I ⊆ ind Z Q , with I ⊆ N , satisfying the fol lowing pr op erties. (1) d ( T , X ) < ∞ for al l X ∈ ind Z Q , (2) d ( T j , T k ) = ∞ for al l j < k and wher e T j = { T i } i ≤ j , (3) d • ( T i , T j ) ≥ max { i , j } for al l i 6 = j . Pr o of. The fir st co ndition is exactly condition (*), so w e may assume there is a co un table subset T = { T i } i ∈ I ⊆ ind Z Q sa tisfying the first prop erty . F or the second pr op erty , co nsider T ′ = { T k ∈ T | ∀ j < k : d ( T j , T k ) = ∞} instead of T . It is clear that T ′ ⊆ ind Z Q satis fies the second co ndition. It follows from the tr ia ngle inequality that d ( T ′ , X ) < ∞ for a ll X ∈ Z Q . HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 15 F or the last prop e rty , assume T = { T i } i ∈ I ⊆ ind Z Q is a count able set s atisfying the fir st t wo prop erties. T o ea se notations, assume I = { 0 , 1 , . . . , n } or I = N . W e will define sets S i recursively . Firstly let S 0 = { T 0 } . F or every i > 0 , choose an ob ject S i on the τ -o rbit of T i such that d • ( S i − 1 , S i ) ≥ i and d • ( S i , S i − 1 ) ≥ i (see Lemma 3 .3). This is p ossible since, by the second condition, one of these will be infinite. The set S = ∪ i ∈ I S i satisfies the require d prop erties.  5.2. Findi ng a dualizing k -v ariet y Z -equiv alen t to Q . L e t A be a connected Ext-finite ab elian category with Ser re dua lit y and let Q b e a hereditary section. W e have r emarked a bove that Q (or Z Q ) needs to satisfy condition (*) for ther e to b e a her e dita ry section Q ′ which is a dualizing k -v ar iety and Z -equiv alent to Q . The main result of this section will be to show the co ndition (*) is a lso sufficient, namely if Q is a her e ditary section in D b A such that there is a c o unt able set T ⊆ ind Z Q with d ( T , X ) < ∞ for a ll X ∈ ind Z Q , then Q is Z -equiv a lent to a semi-he r editary dualizing k - v ariety Q T . W e start by cho osing s uch a set T and constructing a n asso ciated hereditary section Q T . W e will then show that Q T is a dualizing k -v a riety . Construction 5. 5. W e s tart by choosing a set T with the prop erties of Lemma 5.4. Asso ciated to this set T , w e will consider the full subca tegory Q T of D b A as follows: for every X ∈ ind Z Q , fix a τ -shift of X such that d • ( T , X ) =  d ( T , X ) 2  . Example 5.6. Let a be the dualizing k -v ariety given by the thread quiver · 2 / / · thus a is equiv alent to k ( N · Z · Z · − N ). The Auslander-Reiten quiver of D b mo d a is as sketc hed in the upper part of Figure 6. W e will consider the hereditary section Q spanned b y all ob jects of a ⊂ D b mo d a lying in a Z A ∞ ∞ -comp onent. The corres po nding heredita ry ca tegory A is as g iven by the middle pa rt o f Figure 6 . W e choos e a set T = { T 0 , T 1 } as in Fig ure 7, satisfying the conditions d ( T 0 , T 1 ) = ∞ a nd d • ( T 0 , T 1 ) ≥ 1 fro m Lemma 5.4. In Figure 7, the lig ht cones S • ( T , 0 ) and S • ( T , 0 ) hav e be e n marked by bla ck arr ows, and the corres po nding full s ubca tegory Q T of D b A ha s b een indicated by ’ • ’. W e fir st verify that Q H defined ab ove is indeed a hereditary sec tion. Prop ositio n 5.7 . The sub c ate gory Q define d in Construction 5.5 is a her e ditary se ction. Pr o of. According to Cor ollary 4.1 9 we only need to ch eck that d • ( Y , Z ) ≥ 0 for all Y , Z ∈ ind Q H . Using the triangle inequa lit y , we find d • ( Y , Z ) ≥ d • ( T , Z ) − d • ( T , Y ) =  d ( T , Z ) 2  −  d ( T , Y ) 2  ≥ 0 if d ( T , Y ) ≤ d ( T , Z ), and d • ( Y , Z ) ≥ d • ( Y , T ) − d • ( Z, T ) =  d ( T , Y ) 2  −  d ( T , Z ) 2  ≥ 0 . if d ( T , Z ) ≤ d ( T , Y ).  Lemma 5.8 . L et A, B ∈ ind Q T with Hom( A, B ) 6 = 0 , then (1) d ( T , A ) − 1 ≤ d ( T , B ) ≤ d ( T , A ) + 1 , (2) (a) d • ( T , A ) − 1 ≤ d • ( T , B ) ≤ d • ( T , A ) (b) d • ( A, T ) ≤ d • ( B , T ) ≤ d • ( A, T ) + 1 16 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Figure 6. Illustrations for mo d a , A , and H of Ex a mple 5.6 Pr o of. Since Hom( A, B ) 6 = 0, one finds 0 = d • ( A, B ) ≥ d • ( T , B ) − d • ( T , A ) =  d ( T , B ) 2  −  d ( T , A ) 2  . Hence d • ( T , B ) ≤ d • ( T , A ) and d ( T , B ) ≤ d ( T , A ) + 1. Likewise, one finds 0 = d • ( A, B ) ≥ d • ( A, T ) − d • ( B , T ) =  d ( T , A ) 2  −  d ( T , B ) 2  so that d • ( A, T ) ≤ d • ( B , T ) and d ( T , A ) − 1 ≤ d ( T , B ). The r equired inequalities follow re a dily .  Lemma 5.9 . F or any A ∈ ind Q T , ther e is a fi n ite subset T A • ⊆ T with the fol lowing pr op erty: ∀ B ∈ ind Q T : d • ( A, B ) = 0 ⇒ d • ( B , T ) = d • ( B , T A • ) . Pr o of. Fix a T i ∈ T such that d • ( T , A ) = d • ( T i , A ). Let T j ∈ T such that d • ( B , T ) = d • ( B , T j ) for some B ∈ S • Q ( A, 0). If i 6 = j , then using the triangle inequality we find d • ( T , A ) + d • ( B , T ) = d • ( T i , A ) + d • ( B , T j ) = d • ( T i , A ) + d • ( A, B ) + d • ( B , T j ) ≥ d • ( T i , T j ) ≥ max { i , j } . HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 17 P S f r a g r e p la c e m e n t s T 0 T 1 • • • • • • • • • Figure 7. The lig ht co nes and chosen hereditar y section o f Example 5 .6 By Lemma 5.8 we know tha t d • ( B , T ) ≤ d • ( A, T ) + 1 so that d • ( T , A ) + d • ( A, T ) + 1 ≥ max { i , j } . This shows that j is b ounded and hence that T A • is finite.  Theorem 5.10. L et A b e a c onne cte d a b elian her e ditary c ate gory satisfying Serr e d uality with c ate gory of pr oje ctives Q A . Assu me that Z Q A satisfies (*). Then ther e is a her e ditary s e ction Q T in Z Q A which is a dualizing k - variety, and Z Q T = Z Q . Pr o of. Let Q T be a hereditar y sectio n as descr ibe d in Cons truction 5 .5 . W e need to chec k that the t wo conditions of P rop osition 4.21 are satisfied. W e will o nly pr ove the fir st par t, the seco nd part is shown dua lly . 18 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Let A ∈ ind Q T and divide the set of indeco mpo sables B ∈ ind Q T with d • ( A, B ) = 0 into subsets S T ,i = { B ∈ ind Q | d • ( A, B ) = 0 , d • ( B , T ) = d • ( B , T ) = i } where i ∈ Z , T ∈ T . It follows from Lemmas 5.8 and 5 .9 that only finitely many o f these subsets are none mpty . F or each of these nonempty subsets S T ,i we will co nstruct, in tw o s teps, an ob ject C T ,i ∈ Q T such tha t Hom( B , C T ,i ) 6 = 0 when B ∈ S T ,i . The ob ject C = M S T ,i 6 = ∅ C T ,i is then the r equired ob ject fro m the first condition o f Prop os ition 4.21. Let Q A be the light cone centered o n A and let H A be an a sso ciated her editary category in the sense of Theorem 4.15, thus H A is the hereditary heart o f a t -s tructure on D b A such that the catego r y of pro jectives of H A corres p o nd to Q A . In particular any B ∈ S T ,i corres p o nds to a pro jective ob ject in H A and b ecause Ho m( B , τ − i T ) 6 = 0 (due to Prop osition 3.7 ) we kno w that τ − i T ∈ Ob H A [0]. Moreov er, since d • ( A, τ − i T ) 6 = −∞ , we know that τ − i T is even a prepr o jective ob ject in H A . Pr op osition 4.24 shows ther e is a pro jective cov er X → τ − i T in H A . Note that Hom( B , X ) 6 = 0 for all B ∈ S T ,i . Let Y b e a maximal direct summand of X s uch that for every indecomp osable direct summand Y ′ of Y there is a B ∈ S T ,i with Hom( B , Y ′ ) 6 = 0 , thus d • ( B , Y ′ ) = 0 . Using the tria ngle inequality we find d • ( Y ′ , T ) = d • ( Y ′ , T ) = i , and d • ( T , Y ′ ) ≥ d • ( T , T ) − d • ( Y ′ , T ) = − i . In general the ob ject Y do es not hav e to lie on Q T . In the second s tep of this construction, we will use the ob ject Y to construct the req uir ed ob ject C T ,i . Let j ∈ Z b e the smalles t integer such that ⌊ i + j 2 ⌋ = j , th us an ob ject Z ′ ∈ ind Z Q T with j ≤ d • ( T , Z ′ ) ≤ d • ( T , B ) and d • ( Z ′ , T ) = i w ould lie in the sub categor y Q T , if B ∈ ind S T ,i . Note that j ≤ d • ( T , B ) for a ll B ∈ S T ,i . Let T f ⊆ T b e the subset consisting of all ob jects T k ∈ T such that d • ( T k , T ) < i + j . Since T satisfies the conditio ns of Lemma 5.4 , this is necessarily a finite set. Note tha t the triangle inequality implies that a ny T ′ ∈ T w ith d • ( T ′ , Y ′ ) < j lies in T f . W e now apply Lemma 5 .11 below to the hereditar y sectio n Q A with i 1 = − i and i 2 = j + 1. W e obtain a full sub ca tegory Q ′ A of Q A and a right adjoint r : Q A → Q ′ A to the embedding. W e will write Z fo r the maximal direct summand of r ( Y ) s uch that for every indecomp osable direct summand Z ′ of Z there is a B ∈ S T ,i with Hom( B , Z ′ ) 6 = 0 . Note that Hom( B , Z ) 6 = 0 for all B ∈ S T ,i . W e claim that every direct summand Z ′ of Z lies in Q T . Note that Hom( Z ′ , r ( Y )) 6 = 0 a nd thus Hom( Z ′ , Y ) 6 = 0. This implies that d • ( Z ′ , Y ′ ) 6 = 0 , fo r a dir e ct s ummand Y ′ of Y , and thus the triangle inequa lity gives d • ( Z ′ , T ) ≤ d • ( Y ′ , T ) = i . There is also a B ∈ S T ,i with d • ( B , Z ′ ) = 0, and we use the triangle ine q uality to shows that d • ( Z ′ , T ) ≥ i . W e conclude that d • ( Z ′ , T ) = i . Next, Lemma 5.11 yields that j ≤ d • ( T f , Z ′ ). Since T f ⊆ T we kno w that d • ( T f , Z ′ ) ≤ d • ( T , Z ′ ). T o pro of that j ≤ d • ( T , Z ′ ), let T ′ ∈ T such that d • ( T ′ , Z ′ ) < j . Then d • ( T ′ , T ) ≤ d • ( T ′ , Z ′ ) + d • ( Z ′ , T ) < j + i a nd thus by definition we hav e T ′ ∈ T f . This s hows that indeed j ≤ d • ( T , Z ′ ). Since there is a B ∈ S T ,i with d • ( B , Z ′ ) = 0 we know that d • ( T , Z ′ ) ≤ d • ( T , B ). W e conclude that Z ′ ∈ Q T . This shows that Z is the req uir ed ob ject C T ,i ∈ Q .  W e have used the following lemma. Lemma 5 .11. L et Q b e a her e ditary se ction in D b A , and let T f ⊂ ind Z Q b e a finite set. L et i 1 , i 2 ∈ Z with i 1 ≤ i 2 . Ther e is a ful l sub c ate gory Q ′ ⊆ Q satisfying the fol lowing pr op ert ies: (1) the emb e dding Q ′ → Q has a left and a right adjoint, (2) if A ∈ ind Q with i 2 < d • ( T , A ) , t hen A ∈ ind Q ′ , (3) if A ∈ ind Q with i 1 ≤ d • ( T , A ) ≤ i 2 , then A 6∈ ind Q ′ . Pr o of. Let H b e a hereditary hea rt corres p o nding to the hereditary s e ction Q a s in the dual of Theorem 4.15, thus s uch that Q corr esp onds to the image of the categ ory of injectives of H into D b A . HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 19 W rite T f = { T 0 , T 1 , . . . , T k } a nd co nsider the set Z = { τ j T i | T i ∈ T f , i 1 ≤ j ≤ i 2 } . By po ssibly removing some elements from Z , w e may assume every elemen t Z ∈ Z lies in H ⊂ D b A . F urthermore , every element of Z is directed s o w e ca n write Z = { Z 0 , Z 1 , . . . , Z l } suc h tha t Ext( Z b , Z a ) = 0 whenever a ≤ b . W e define a full replete sub categ ory Q ′ of Q as follows: A ∈ Ob Q ′ ⇔ ∀ Z ∈ Z : RHom( Z , A ) = 0 . W e prove that the categ ory Q ′ is the categor y from the sta temen t of the lemma . Note that Lemma 3.3 implies that RHom( Z , A ) = 0 for all Z ∈ Z when i 2 < d • ( T f , A ), and that P rop osition 3.7 implies that Hom( Z , A ) 6 = 0 for so me Z ∈ Z when i 1 ≤ d • ( T f , A ) ≤ i 2 . Set Z (0) = Z 0 . F o r 0 < a ≤ l we define Z ( a ) = Z ( a − 1) ⊕ Z a if Ext( Z a , Z ( a − 1) ) = 0, and by the universal extension 0 → Z ( a − 1) → Z ( a ) → Z a ⊗ k Ext( Z a , Z ( a − 1) ) → 0 if Ext( Z a , Z ( a − 1) ) 6 = 0. It is straig ht forward to verify that Ext( Z ( l ) , Z ( l ) ) = 0. Since A is an injectiv e o b ject in H , we no w hav e A ∈ O b Q ′ ⇔ Hom( Z ( l ) , A ) = 0 . The r e quired result now fo llows from (the dual o f ) Pr op osition 4.20 .  6. Nonthread objects and threads in h eredit ar y sections 6.1. d • -in-b etw een and threads. As with dualizing k - v arieties, the co ncepts of threads will be paramount in our discussion of here dita ry sec tio ns. How ev er, a ma jor difference b etw ee n dualiz ing k -v ar ieties and hereditary sectio ns is that in the la tter one can encoun ter so- called broken threads and a sort of ha lf-o p en thre a ds, called rays or cor ays. T o de s crib e these cases, we s ta rt w ith a definition. Let Q b e a hereditar y sectio n in D b A where A is a n ab elian categ ory with Serr e duality and let X, Y ∈ ind Q with d • ( X, Y ) < ∞ . W e will say Z ∈ ind Q is d • -in-b etwe en X and Y if d • ( X, Y ) = d • ( X, Z ) + d • ( Z, Y ). W e denote the full replete additive sub category of Q generated by a ll indeco mp os ables d • -in-b etw een X and Y by [ X , Y ] • Q , thus ind[ X , Y ] • Q = { Z ∈ ind Q | d • ( X, Z ) + d • ( Z, Y ) = d • ( X, Y ) } . If there is no confusion, we will often write [ X , Y ] • instead of [ X , Y ] • Q . W e will define ] X , Y ] • Q to b e the full replete additive sub catego ry of [ X , Y ] • Q spanned by the ob jects not s uppo r ted on X . Likewise one defines [ X , Y [ • Q and ] X , Y ] • Q R emark 6.1 . If d • ( X, Y ) = 0, then [ X , Y ] • Q = [ X , Y ]. Example 6.2. Let Q b e the quiver given by b     > > > > > > > > a ? ?         / /   ? ? ? ? ? ? ? ? c e d O O ? ?        Denote by P i the pro jective ob ject in r ep Q assoc ia ted with the v ertex i of Q . Let Q b e the standard her editary section in D b rep Q . W e have ind[ P a , P e ] • Q = { P a , P b , P d , P e } ind[ P b , P d ] • Q = { P a , P b , P c , P d , P e } ind[ P a , P d ] • Q = { P a , P d } ind[ P d , P a ] • Q = { P a , P d , P c } Note that ind[ P b , P d ] • 6⊆ ind[ P a , P e ] • (even though P b , P d ∈ ind[ P a , P e ] • ) and that [ P a , P d ] • 6 ∼ = [ P d , P a ] • . 20 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Figure 8. Illustration of Example 6 .8 R emark 6.3 . As the previous exa mple indicates, the s ubca tegories [ X , Y ] • are the replac ement of d • -geo desics on a q uiver. Prop ositio n 6.4. L et X , Y ∈ Q with n = d • ( X, Y ) < ∞ . T he s et s ind[ X , Y ] • Q and ind[ X , τ − n Y ] interse ct t he same τ - orbits. Pr o of. This follows immediately from Le mma 3.3 .  Example 6.5. Let Q be the quiver from Example 6.2. The light cones centered on P a , P b and P d are g iven b y P b   @ @ @ @ @ @ @ @ P b   " " D D D D D D D D { { w w w w w w w w w τ − P b P a > > ~ ~ ~ ~ ~ ~ ~ ~ / / @ @ @ @ @ @ @ @ P c P e τ − P a # # G G G G G G G G G P c   o o P e | | z z z z z z z z τ − P a ; ; w w w w w w w w w P c O O o o P e a a D D D D D D D D P d O O > > ~ ~ ~ ~ ~ ~ ~ ~ τ − P d P d c c G G G G G G G G G O O = = z z z z z z z z Corollary 6.6. Le t X , Y , Z ∈ ind Q with d • ( X, Y ) < ∞ and Z ∈ [ X , Y ] • . In this c ase [ X , Z ] • ⊆ [ X , Y ] • and [ Z, Y ] • ⊆ [ X , Y ] • . Definition 6.7. Le t X ∈ ind Q . It follows from Observ a tion 4.16 that Q ha s right and left a lmost split maps. Let X → M and N → X be a left and r ight almos t split map, res pe c tiv ely . If M and N a re indecomp osable, w e will say X is a thr e ad obje ct . W e will denote M and N by X + and X − , resp ectively . An ob ject which is not a thread o b ject is calle d a nonthr e ad obje ct . If [ X , Y ] • consists of only thread ob jects in Q , then we c a ll [ X , Y ] • a thr e ad . If furthermo re d • ( X, Y ) > 0 or d • ( X, Y ) = 0, then we ca ll [ X, Y ] • a br oken thr e ad or an unbr oken thr e ad , resp ectively . Example 6.8. Let a = k Q where Q is the thread q uiver · / / · Thus a = k ( N · − N ) and the indecomp osable pr o jectiv es of mo d a ar e given b y a ( − , n ) and a ( − , − n ) for n ∈ N . The Auslander-Reiten quiv er of D b mo d a ma y b e sk etched as in the upp er part of Figure 8 where the triangles repres ent Z A ∞ -comp onents and where the categor y mo d a has b een marked with gr ay . W e will denote by Q the here dita ry sectio n in D b mo d a corr e sp o nding to the pro jectives of mo d a . The interv a l [ a ( − , 1) , a ( − , − 1)] = [ a ( − , 1) , a ( − , − 1)] • ⊂ ind Q is an (unbroken) thread. HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 21 Consider the hereditar y section Q ′ ⊆ D b mo d a spanned by a ll ob jects of the for m a ( − , n ) a nd τ a ( − , − n ) whe r e n ∈ N as in the low er part of Fig ure 8. Now [ a ( − , 1) , τ a ( − , − 1)] • ⊂ ind Q ′ is a broken thr e ad. A reaso n to introduce thread o b jects is g iven by the following obser v ation: let X , Y ∈ ind Q and consider the left adjoint l to the embedding i : [ X, Y ] → Q (see P rop osition 4 .20). Let A b e any indecompo sable ob ject in Q . If A do es not lie in [ X , Y ], then the only thread ob ject which can o ccur as a direct summand of l ( A ) is X . Indeed, let Z b e a thr ead ob ject which is a direct summand o f l ( A ). If Z 6 ∼ = X then Z − ∈ Ob[ X, Y ]. Since Q is semi-hereditar y , we know that dim Hom( l ( A ) , Z ) > dim Hom( l ( A ) , Z − ). Ho wev er , since no map A → Z is a split map, w e hav e dim Hom( A, iZ ) = dim Hom( A, iZ − ). A co ntradiction. This proves the following lemma. Lemma 6 .9. L et X , Y ∈ ind Q and let l b e a left adjoint to the emb e dding i : [ X , Y ] → Q . A thr e ad obje ct in ] X , Y ] c annot b e a dir e ct sum m and of l ( A ) , for any A ∈ ind Q \ ind[ X, Y ] . Example 6.10. Le t Q b e the q uiver f             = = = = = = = = a / / b / / c / / d / / e Denote b y P i ∈ ind rep Q a n indec o mpo sable pro jective a s so ciated with the vertex i of Q , let Q be the standard her e ditary section in D b rep Q , and let l : Q → [ P b , P d ] be the left a djoint to the embedding [ P b , P d ] → Q . W e hav e that l ( P f ) ∼ = P b ⊕ P c . Let A, B ∈ ind Q . While the subca tegory [ A, B ] will only be nontrivial if Hom( A, B ) 6 = 0, a similar statement is not true for [ A, B ] • . In fact, as Remark 6.1 indica tes we will mostly b e int erested in cas e s whe r e d • ( A, B ) 6 = 0. This means ho w ever, as the following ex ample shows, tha t we ca n encounter situations wher e we consider [ A, B ] • where d • ( B , A ) = 0 . Example 6.11. Le t Q b e the q uiver A 5 with linear orientation, th us Q is given b y a / / b / / c / / d / / e Denote by P i ∈ ind rep Q an indecomp osable pro jective a sso ciated with the vertex i of Q , a nd let Q b e the standar d hereditary section in D b rep Q . W e hav e that [ P d , P b ] • = [ P b , P d ]. In some sense the in terv al [ P d , P b ] • from the previous example do es not hav e the “natura l” orientation. The following lemma a nd Pro po sition 6.1 4 b elow indicate that we ca n lo o k a t the neighbors of P b and P d to so mewhat comp ensate for this lack of orientation. Lemma 6.1 2. L et Q b e a her e ditary se ction, and let A, B ∈ ind Q with d • ( A, B ) < ∞ . (1) If A 6 = B , then A and B ha ve a le ast one neighb or lying in [ A, B ] • . An X ∈ ind[ A, B ] • with A 6 = X 6 = B has at le ast two (n on- isomorphi c) dir e ct neighb ors in [ A, B ] • . (2) Ass u me [ A, B ] • is a thr e ad (with A 6 = B ) in Q . If B − 6∈ [ A, B ] • , then A − , B + ∈ ind[ A, B ] • and d • ( B , A ) < ∞ . Pr o of. The first r esult follows immediately from Prop ositio n 6 .4. F o r the seco nd result, let B = B 0 → B 1 → B 2 → · · · be a (pos sibly finite) sequence o f direct successo rs. Since B − do es not lie in [ A, B ] • , and B is a thread ob ject, we know B 1 lies in [ A, B ] • . If B 1 6 = A , then it is a thread ob ject and we know B 2 also lies in [ A, B ] • . Iteration s hows either the entire se q uence lie s in [ A, B ] • , o r some B i = A . Since d • ( B j +1 , B j ) = 1, we find that d • ( B j , B ) = j . This shows that B i = A where i = d • ( A, B ) and that b oth B + = B 1 and A − = B i − 1 lie in [ A, B ] • .  Example 6.13. Le t Q b e the q uiver given b y a / / b / / c / / d / / e 22 CARL FREDRIK BERG AND ADAM-CHRISTIAAN V AN ROOSMALEN Denote by P i ∈ ind rep Q an indecomp osable pro jective a sso ciated with the vertex i of Q , a nd let Q b e the sta ndard hereditary section in D b rep Q . Since w e have ind[ P b , P d ] • = { P b , P c , P d } = ind[ P d , P b ] • , b oth [ P b , P d ] • and [ P d , P b ] • are threads. W e eas ily s ee that the results o f the previous lemma a re v alid in this case. If we replace the q uiver Q by a / / b / / c d o o e o o then, with the same notations as ab ov e, ind[ P d , P b ] • = { P b , P c , P d } would not b e a thread. No te that P − b 6∈ [ P d , P b ] • , but also P − d 6∈ [ P d , P b ] • . The following prop osition res em bles Pro p osition [6, Pr op osition 6.2]. Prop ositio n 6.14. L et [ X , Y ] • b e a thr e ad. If [ X, Y ] • and [ X, Y ′ ] • shar e an inde c omp osable ap art fr om X , t hen [ X, Y ] • ⊆ [ X , Y ′ ] • or [ X , Y ′ ] • ⊆ [ X , Y ] • . Pr o of. W e will work in the ligh t cone Q X centered on X . W rite n = d • ( X, Y ) a nd n ′ = d • ( X, Y ′ ). The as sumption in the sta tement shows there is a Z ∈ ind] X, τ − n Y ] ∩ ind] X, τ − n ′ Y ′ ]. Note that ] X , τ − n Y ] is a threa d. Prop ositio n 4.20 yields that the embedding [ X, τ − n Y ] → Q X has a right adjo int r : Q → [ X , τ − n Y ]. Since Z ∈ ind] X , τ − n Y ] and Hom( iZ, τ − n ′ Y ′ ) 6 = 0, w e find that r ( τ − n ′ Y ′ ) has a nonzero direct summand lying in ] X , τ − n Y ]. Lemma 6.9 yields that either τ − n ′ Y ′ ∈ Ob] X, τ − n Y [ and t hus [ X , τ − n ′ Y ′ ] ⊆ [ X , τ − n Y ] by Corollar y 6.6, or that Ho m( τ − n Y , r ( τ − n ′ Y ′ )) 6 = 0 a nd th us τ − n Y ∈ Ob] X, τ − n ′ Y ′ ] so that [ X , τ − n Y ] ⊆ [ X , τ − n ′ Y ′ ]. Applying Pro po sition 6.4 shows the required prop er ty .  Example 6.15. Le t Q b e the q uiver A 5 with linear orientation, th us Q is given b y a / / b / / c / / d / / e Denote by P i ∈ ind rep Q an indecomp osable pro jective a sso ciated with the vertex i of Q , a nd let Q b e the s ta ndard heredita ry section in D b rep Q . The threads [ P c , P b ] • and [ P c , P d ] • do have P c in common, but no other indecomp osable. Neither thread is a sub catego r y of the other such that the res ult from Prop osition 6.14 do es not hold. 6.2. Nonthread ob je cts. In this subsection, we will g ive a short discussio n of the non thread ob jects of Q . Our ma in result will be that, if Z Q is co nnected, Q has only co un tably ma ny nonthread ob jects. Lemma 6.1 6. L et Q b e a her e ditary se ction in D b A and let X ∈ ind Q . (1) F or every Y ∈ ind Q with d • ( X, Y ) = 0 , ther e ar e only finitely many nonthr e ad obje cts in [ X , Y ] . (2) F or every Y ∈ ind Q with d • ( X, Y ) ∈ Z , ther e ar e only finitely many nont hr e ad obje ct s in [ X , Y ] • . (3) Ass u me X is a nonthr e ad obje ct. F or every Y ∈ ind Q with d • ( X, Y ) ∈ Z , ther e i s a nonthr e ad obje ct Z ∈ [ X , Y ] • such that ] Z , Y [ • has no nonthr e ad obje cts. Pr o of. (1) Let A b e a nonthread ob ject in [ X , Y ]. If A is not isomor phic to X or Y , then Lemma 6.12 implies there are (nonzero) almos t split ma ps N A → A and A → M A in Q . Since A is a nonthread o b ject, either M A or N A is not indeco mpo sable. Seeking a contradiction, assume there ar e infinitely many nonthread ob jects A such that N A is no t indecomp osable. Let A be a hea rt of a t -structur e asso c iated with Q as in Theorem 4 .15. W e de no te Z = im( Y → S X ⊗ Hom( Y , S X ) ∗ ) and K = ker( Y → S X ⊗ Hom( Y , S X ) ∗ ). Since Q is semi- hereditary (Obse r v ation 4.1 6 ) and dim Ho m( X, Y ) < ∞ , there ca n o nly be finitely ma n y o b jects A ∈ ind[ X , Y ] such that dim Hom( X, N A ) > Hom( X , A ). There are hence infinitely many ob jects A ∈ ind[ X , Y ] such that dim Hom( X , N A ) = Hom( X , A ). An y direct summa nd B of N A not lying in [ X , Y ] is nece s sarily a direct summand o f K , HEREDIT AR Y CA TEGORIES WITH SERRE DUALITY WHICH ARE GENERA TE D BY PREPROJECTIVES 23 but K is a finitely g enerated pr o jectiv e ob ject. W e c o nclude that ther e a re infinitely man y nonthread o b jects A ∈ [ X , Y ] such that N A is not indecomp osable. Likewise, one shows there are only finitely many nonthread ob jects A ∈ [ X , Y ] such that M A is not indecomp osable. (2) Seeking a contradiction, assume [ X , Y ] • Q has infinitely many no nt hread ob jects in Q . Denote n = d • ( X, Y ) and let Q X be the light cone cen tered on X . It follows from the pre vious part tha t [ X , τ − n Y ] has only finitely ma ny nonthread ob jects in Q X , thus infinitely many no n thread ob jects in [ X , Y ] • Q are either a sink or a so urce with ex actly t wo direct neig hbors. Denote by { A i } i ∈ I ⊆ ind[ X , Y ] • Q such an infinite se t of sinks a nd sources, and denote b y A ′ i the ob ject in ind[ X, τ − n Y ] lying in the same τ -or bit as A i (see Prop ositio n 6.4). W e define a par tial order ing on I b y i ≤ j ⇔ Hom( A ′ i , A ′ j ) 6 = 0 . Since Q X is semi-hereditary (Observ a tio n 4.16 ) and dim Hom( X , τ − n Y ) < ∞ , we know that there is an infinite linearly ordered subp oset J o f I . F urthermore either infinitely many elemen ts of { A j } j ∈ J are sinks o r infinitely many are source s. If A j is a sink, then d • ( A j , A k ) > 0 when j, k ∈ J with j < k . Also note that, since A ′ j ∈ [ X , A ′ k ], Pro po sition 6.4 shows that A j ∈ [ X , A k ] • and th us d • ( X, A k ) = d • ( X, A j ) + d • ( A j , A k ). W e infer that d • ( X, A j ) > d • ( X, A k ) for a ny k > j and hence { A j } j ∈ J cannot hav e infinitely many sinks. Likewise one shows that { A j } j ∈ J cannot have infinitely ma n y sources. A contradiction. (3) T a ke a no nt hread ob ject Z ∈ ind[ X , Y [ • such that [ Z, Y [ • has a minimal num ber of nonthread o b jects. Using Corollary 6.6 it is ea sy to see t hat Z is the only non thread ob ject in [ Z , Y [ • .  Lemma 6.1 7. L et Q b e a her e ditary se ction in D b A and let X ∈ ind Q . (1) Ther e ar e only finitely many nont hr e ad obje cts Y such that ] X , Y [ is nonempty and has no nonthr e ad obje cts. (2) Ther e ar e only c ountably many nonthr e ad obje cts Y ∈ ind Q with d • ( X, Y ) = 0 . (3) Ther e ar e only c ountably many nonthr e ad obje cts Y ∈ ind Q with d • ( X, Y ) < ∞ . Pr o of. (1) Let X → M b e a left almost split map in Q . Let M ′ be an indeco mpo sable sum- mand of M a nd let Y 1 , Y 2 be tw o no nisomorphic nonthread ob jects with Hom( M ′ , Y 1 ) 6 = 0 and Hom( M ′ , Y 2 ) 6 = 0 such that ] X , Y 1 [ and ] X , Y 2 [ are nonempt y and hav e no nonthread ob jects. Note that this implies that Y 1 6 ∼ = M ′ 6 ∼ = Y 2 . In par ticula r we know that M ′ is a thread ob ject in Q . The embedding i : [ M ′ , Y 1 ] → Q has a r ight adjoin t i R . Since Hom( Y 1 , Y 2 ) = 0 we know that the ob ject i ◦ i R ( Y 2 ) lies in [ M ′ , Y 1 [. Ho wev er , e very indecomp osa ble dir ect summand of i ◦ i R ( Y 2 ) is a threa d ob ject in Q , contradicting Lemma 6.9. (2) Denote b y N X i the set of all nonthread o b jects Y ind Q such that d • ( X, Y ) = 0 and ] X , Y [ has exactly i no nthread ob jects. Lemma 6.1 6 yields that it is sufficient to pr ov e that the set ∪ i ∈ N N X i is countable. It was shown ab ove tha t N X 0 is finite for all X ∈ ind Q ; we will pro ceed by induction. Assume therefor e tha t N Z j is finite for every j < i and every Z ∈ ind Q . W e will pr ov e that N X i is finite. Let Y ∈ N X i and let Z b e a nonthread ob ject in ] X , Y [, th us Z ∈ N X j for some j < i . Cor ollary 6.6 y ields that Y ∈ N Z k for some k < i so that N X i ⊆ [ j,k

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