JOURNALOFTHE AMERICANMATHEMATICALSOCIETY Volume15,Number2,Pages295{366 S0894-0347(02)00387-9 ArticleelectronicallypublishedonJanuary18,2002 NOETHERIAN HEREDITARY ABELIAN CATEGORIES SATISFYING SERRE DUALITY I. REITEN AND M. VAN DEN BERGH Contents Notations and conventions 296 Introduction 296 I. Serre duality and almost split sequences 300 I.1. Preliminaries on Serre duality 300 I.2. Connection between Serre duality and Auslander{Reiten triangles 304 I.3. Serre functors on hereditary abelian categories 307 II. Hereditary noetherian abelian categories with non-zero projective objects 309 II.1. Hereditary abelian categories constructed from quivers 310 II.2. Hereditary abelian categories generated by preprojectives 314 II.3. Derived equivalences 318 II.4. The classi(cid:12)cation 324 III. Sources of hereditary abelian categories with no projectives or injectives 329 III.1. Hereditary abelian categorieswith Serre functor and all objects of (cid:12)nite length 329 III.2. Sheaves of hereditary orders and graded rings 331 III.3. Hereditary abelian categories associated to in(cid:12)nite Dynkin and tame quivers 333 IV. Hereditary noetherian abelian categories with no projectives or injectives 345 IV.1. Preliminaries 345 IV.2. Completion 349 IV.3. Description in terms of a pullback diagram 353 IV.4. The (cid:12)nite orbit case 356 IV.5. The in(cid:12)nite orbit case 358 V. Applications 360 V.1. Saturatedness 361 ReceivedbytheeditorsDecember6,2000. 2000 Mathematics Subject Classi(cid:12)cation. Primary 18E10, 18G20, 16G10, 16G20, 16G30, 16G70. Key words and phrases. Noetherian hereditary abelian categories, Serre duality, saturation property. ThesecondauthorisaseniorresearcherattheFundforScienti(cid:12)cResearch. Thesecondauthor also wishes to thank the Clay Mathematics Institute for material support during the period in whichthispaperwaswritten. (cid:13)c2002 American Mathematical Society 295 296 I. REITEN AND M. VAN DEN BERGH V.2. Graded rings 362 Appendix A. Some results on abelian categories 363 References 365 Notations and conventions Most notations will be introduced locally. The few global ones are given below. Unless otherwise speci(cid:12)ed k will be an algebraically closed (cid:12)eld and all rings and categories in this paper will be k-linear. If A is a ring, then mod(A) will be the category of (cid:12)nitely generated right A- modules. Similarly if R is a Z-graded ring, then gr(R) will be the category of (cid:12)nitely generated graded right modules with degree zero morphisms, and Gr(R) the category of all graded right R-modules. If R is noetherian, then following [3] tors(R) will be the full subcategory of gr(R) consisting of graded modules with right bounded grading. Also following [3] we put qgr(R)=gr(R)=tors(R). For an abelian category C we denote by Db(C) the bounded derived category of C. Introduction One of the goals of non-commutative algebraic geometry is to obtain an under- standingofk-linearabeliancategoriesC,fora(cid:12)eldk,whichhavepropertiescloseto thoseofthecategoryofcoherentsheavesoveranon-singularproperscheme. Hence some obvious properties one may impose on C in this context are the following: (cid:15) C is Ext-(cid:12)nite, i.e. dim Exti(A;B)<1 k for all A;B 2C and for all i. (cid:15) C has homological dimension n<1, i.e. Exti(A;B)=0 for A;B 2C and i>n, and n is minimal with this property. Throughout this paper k will denote a (cid:12)eld, and even though it is not always necessarywewillforsimplicityassumethatk isalgebraicallyclosed. Allcategories will be k-linear. When we say that C is Ext-(cid:12)nite, it will be understood that this is with respect to the (cid:12)eld k. In most of this paper we will assume that C is an Ext-(cid:12)nite abelian category of homologicaldimension at most 1, in which case we say that C is hereditary. A slightly more subtle property of non-singular proper schemes is Serre duality. Let X be a non-singular proper scheme over k of dimension n, and let coh(X) denote the category of coherent O -modules. Then the classical Serre duality X theorem asserts that for F 2coh(X) there are natural isomorphisms Hi(X;F)(cid:24)=Extn(cid:0)i(F;! )(cid:3) X where ((cid:0))(cid:3) =Hom ((cid:0);k). k AveryelegantreformulationofSerredualitywasgivenbyBondalandKapranov in [8]. It says that for any E;F 2Db(coh(X)) there exist natural isomorphisms Hom (E;F)(cid:24)=Hom (F;E (cid:10)! [n])(cid:3): Db(coh(X)) Db(coh(X)) X NOETHERIAN HEREDITARY ABELIAN CATEGORIES 297 StatedinthiswaytheconceptofSerredualitycanbegeneralizedtocertainabelian categories. (cid:15) C satis(cid:12)es Serre duality if it has a so-called Serre functor. The latter is by de(cid:12)nition an autoequivalence F : Db(C) ! Db(C) such that there are isomorphisms (cid:24) (cid:3) Hom(A;B)=Hom(B;FA) which are natural in A;B. On the other hand hereditary abelian Ext-(cid:12)nite categories C with the additional propertyofhavingatiltingobjecthavebeenimportantfortherepresentationtheory of(cid:12)nitedimensionalalgebras. RecallthatT isatiltingobjectinCifExt1(T;T)=0, and if Hom(T;X) = 0 = Ext1(T;X) implies that X is 0. These categories C are importantin the study of quasitilted algebras,which by de(cid:12)nition arethe algebras of the form EndC(T) for a tilting object T [16], and which contain the important classesoftiltedandcanonicalalgebras. Aprominentpropertyintherepresentation theory of (cid:12)nite dimensional algebras is having almost split sequences, and also the Ext-(cid:12)nite hereditary abelian categories with tilting object have this property [16]. In view of the above it is interesting, and useful, to investigate the relationship between Serre duality and almost split sequences. In fact, this relationship is very close in the hereditary case. The more general connections are on the level of triangulated categories, replacing almost split sequences with Auslander{Reiten triangles. In fact one of our (cid:12)rst results in this paper is the following (see xI for more complete results): Theorem A. 1. C has a Serre functor if and only if Db(C) has Auslander{ Reiten triangles (as de(cid:12)ned in [14]). 2. If C is hereditary, then C has a Serre functor if and only if C has almost split sequences and there is a one-one correspondence between the indecomposable projective objects P and the indecomposable injective objects I, such that the simple top of P is isomorphic to the socle of I. Hence Ext-(cid:12)nite hereditary abelian categories with Serre duality are of interest both for non-commutative algebraic geometry and for the representationtheory of (cid:12)nite dimensional algebras. The main result of this paper is the classi(cid:12)cation of the noetherian ones. To be able to state our result we (cid:12)rst give a list of hereditary abelian categories satisfying Serre duality. (a) If C consists of the (cid:12)nite dimensional nilpotent representations of the quiver A~ or of the quiver A1, with all arrows oriented in the same direction, then n 1 it is classical that C has almost split sequences, and hence Serre duality. (b) LetX beanon-singularprojectiveconnectedcurveoverk withfunction(cid:12)eld K, and let O be a sheaf of hereditary O -orders in M (K) (see [21]). Then X n one proves exactly as in the commutative case that coh(O) satis(cid:12)es Serre duality. (c) Let Q be either A11 or D1 with zig-zag orientation. It is shown in xIII.3 that there exists a noetherian hereditary abelian categoryC which is derived equivalent to the category rep(Q) of (cid:12)nitely presented representations of Q, and which has no non-zero projectives or injectives. Depending on Q we call this category the ZA11 or the ZD1 category. Since Serre duality is de(cid:12)ned in terms of the derived category, it follows that C satis(cid:12)es Serre duality. 298 I. REITEN AND M. VAN DEN BERGH If Q=A1,then C is nothing but the categorygr (k[x;y])=((cid:12)nite length) 1 Z2 considered in [30]. If Q=D1 and chark 6=2, then C is a skew versionof the ZA11 category (see xIII.3). The ZA11 category and the ZD1 category have also been considered by Lenzing. (d) Wenowcometomoresubtleexamples(seexII). LetQbeaconnectedquiver. Thenforavertexx2QwehaveacorrespondingprojectiverepresentationP x andaninjectiverepresentationI . IfQislocally(cid:12)niteandthereisnoin(cid:12)nite x pathendingatanyvertex,thefunctorP 7!I maybederivedtoyieldafully x x faithful endofunctor F : Db(rep(Q)) ! Db(rep(Q)). Then F behaves like a Serrefunctor,exceptthatitisnotingeneralanautoequivalence. Wecallsuch F a rightSerre functor (see xI.1). Luckily givena rightSerre functor there is a formal procedure to invertit so as to obtain a true Serre functor (Theorem II.1.3). This yields a hereditary abeliancategoryrfep(Q) whichsatis(cid:12)es Serre duality. Under the additional hypotheses that Q consists of a subquiver Q o with no path of in(cid:12)nite length, with rays attached to vertices of Q , then o rfep(Q) turns out to be noetherian (see Theorem II.4.3). Here we mean by a ray an A1 quiver with no vertex which is a sink. An interesting feature of the noetherian categories rfep(Q), exhibiting a new type of behavior, is that they are generated by the preprojective objects, but not necessarily by the projective objects. Now we can state our main result. Recall that an abelian category C is connected if it cannot be non-trivially written as a direct sum C (cid:8)C . 1 2 Theorem B. LetC beaconnectednoetherianExt-(cid:12)nitehereditaryabeliancategory satisfying Serre duality. Then C is one of the categories described in (a){(d) above. The cases (a), (b), (c) are those where there are no non-zero projective objects. Those in (a) are exactly the C where all objects have (cid:12)nite length. For the C having some objects of in(cid:12)nite length, then either all indecomposable objects of (cid:12)nite length have (cid:12)nite (cid:28)-period (case (b)) or all have in(cid:12)nite (cid:28)-period (case (c)). Here the object (cid:28)C is de(cid:12)ned by the almost split sequence 0!(cid:28)C !B !C !0 for C indecomposable in C, and we have (cid:28)C =F(C)[(cid:0)1]. UndertheadditionalassumptionthatC hasatiltingobject,suchaclassi(cid:12)cation was given in [19]. The only cases are the categories of (cid:12)nitely generated modules modA for a (cid:12)nite dimensional indecomposable hereditary k-algebra A and the categoriescohX of coherentsheavesona weightedprojective line X in the sense of [12]. Fromthepointofviewoftheabovelistofexamplesthe (cid:12)rstcasecorresponds to the (cid:12)nite quivers Q in (d), in which case rfep(Q) = rep(Q) is equivalent to mod(kQ), where kQ is the path algebra of Q over k. The categories cohX are a special case of (b), corresponding to the case where the projective curve is P1 (see [23]). OurproofofTheoremBisratherinvolvedandcoversthe(cid:12)rstfoursections. The main steps are as follows: 1. In the (cid:12)rst two subsections of xII we construct the categories rfep(Q), and we show that they are characterized by the property of having noetherian injectives and being generated by preprojectives. 2. In xII.4 we give necessary and su(cid:14)cient conditions for rfep(Q) to be noether- ian, andfurthermorewe provea decompositiontheoremwhich statesthat an Ext-(cid:12)nite noetherian hereditary abelian category with Serre functor can be NOETHERIAN HEREDITARY ABELIAN CATEGORIES 299 decomposed as a direct sum of a hereditary abelian category which is gener- ated by preprojectives and a hereditary abelian category which doesn’t have non-zero projectives or injectives. 3. We are now reduced to the case where there are no non-zero projectives or injectives. The case where all objects have (cid:12)nite length is treated in xIII.1. 4. Thecasewheretherearenonon-zeroprojectivesorinjectivesandatleastone objectofin(cid:12)nitelengthiscoveredinxIV. Itturnsoutthatthiscasenaturally falls into two subcases: ((cid:11)) Thesimpleobjectsare(cid:28)-periodic. Inthatcase,usingtheresultsin[3],we show that C is of the form qgr(R) for R a two-dimensional commutative graded ring, where qgr(R) is the quotient category grR/(cid:12)nite length. Using [1] it then follows that C is of the form (b). ((cid:12)) The simple objects are not (cid:28)-periodic. We show that if such C exists, then it is characterized by the fact that it has either one or two (cid:28)-orbits of simple objects. Since the ZA11 and ZD1 categoryhavethis property, we are done. Our methods for constructing the new hereditary abelian categories rfep(Q) are somewhat indirect although we believe they are interesting. After learning about our results Claus Ringel has recently found a more direct construction for these categories [24]. AllthehypothesesforTheoremBarenecessary. Forexamplethenon-commuta- tive curvesconsideredin [30] arenoetherian hereditaryabeliancategoriesofKrull- dimension one which in general do not satisfy Serre duality (except for the special case listed in (c)). If C is the opposite category to one of the categories (b), (c), (d), then it is not noetherian, but it satis(cid:12)es the other hypotheses. NeverthelessitistemptingtoaskwhetheraresultsimilartoTheoremBremains valid without the noetherian hypothesis if we work up to derived equivalence. In particular,is any such categoryderived equivalentto a noetherian one? Under the additional assumption that C has a tilting object, this has been proved by Happel in [15], and it has recently been shown by Ringel that this is not true in general [25]. Inthe(cid:12)nalsectionweuseTheoremBtodrawsomeconclusionsonthestructure of certain hereditary abelian categories. To start with we discuss the \saturation" property. This is a subtle property of certain abelian categories which was discovered by Bondal and Kapranov [8]. Recall that a cohomological functor H : Db(C) ! mod(k) is of (cid:12)nite type if for every A 2 Db(C) only a (cid:12)nite number of H(A[n]) are non-zero. We have already de(cid:12)ned what it means for C to have homological dimension n < 1. It will be convenientto saymore generallythat C has (cid:12)nite homologicaldimension if for any A;B in C there is at most a (cid:12)nite number of i with Exti(A;B)6=0. In particular, the analogue of this de(cid:12)nition makes sense for triangulated categories. (cid:15) LetC beanExt-(cid:12)niteabeliancategoryof(cid:12)nitehomologicaldimension. Then C is saturated if every cohomological functor H : Db(C) ! mod(k) of (cid:12)nite type is of the form Hom(A;(cid:0)) (i.e. H is representable). It is easy to show that a saturated category satis(cid:12)es Serre duality. It was shown in [8] that coh(X) for X a non-singular projective scheme is saturated and that saturationalsoholds forcategoriesof the formmod((cid:3)) with(cid:3) a (cid:12)nite dimensional algebra. Inspired by these results we prove the following result in xV.1. 300 I. REITEN AND M. VAN DEN BERGH Theorem C. Assume that C is a saturated connected noetherian Ext-(cid:12)nite hered- itary abelian category. Then C has one of the following forms: 1. mod((cid:3)) where (cid:3) is a connected (cid:12)nite dimensional hereditary k-algebra. 2. coh(O) where O is a sheaf of hereditary O -orders (see (b) above) over a X non-singular connected projective curve X. Itiseasytoseethatthehereditaryabeliancategorieslistedintheabovetheorem are of the form qgr(R). We refer the reader to [9] (see also xV.2) where it is shown in reasonable generality that abelian categories of the form qgr(R) are saturated. Therearealsoapplicationstotherelationshipbetweenexistenceoftiltingobjects and the Grothendieck group being (cid:12)nitely generated. This was one of the original motivations for this work, and is dealt with in another paper [23]. We would like to thank Claus Ringel for helpful comments on the presentation of this paper. I. Serre duality and almost split sequences It has been known for some time that there is a connection between classical Serre duality and existence of almost split sequences. There is a strong analogy between the Serre duality formula for curves and the formula D(Ext1(C;A)) ’ (cid:3) Hom(A;DTrC)forartinalgebras(whereD =Hom ((cid:0);k)),onwhichtheexistence k of almost split sequences is based (see [5]). Actually, existence of almost split sequences in some sheaf categories for curves can be proved either by using an analogousformulaforgradedmaximalCohen{MacaulaymodulesorbyusingSerre duality [4, 27]. The notion of almost split sequences was extended to the notion of Auslander{Reitentrianglesintriangulatedcategories[14],andexistenceofsuchwas provedfor Db(mod(cid:3))when (cid:3) is a k-algebraof(cid:12)nite globaldimension [14]. In this case the corresponding translate is given by an equivalence of categories. On the otherhandanelegantformulationofSerredualityintheboundedderivedcategory, together with a corresponding Serre functor, was given in [8]. These developments provide the basis for further connections, which turn out to be most complete in the setting of triangulated categories. For abelian categories we obtain strong connectionsinthehereditarycase. Infact,weshowthatwhenC ishereditary,then C hasSerredualityifandonlyifithasalmostsplitsequencesandthereisaone-one correspondence between indecomposable projective objects P and indecomposable injectiveobjectsI,suchthatP moduloitsuniquemaximalsubobjectisisomorphic to the socle of I. I.1. Preliminaries on Serre duality. Let A be a k-linear Hom-(cid:12)nite additive category. A right Serre functor is an additive functor F : A ! A together with isomorphisms (I.1.1) (cid:17) :Hom(A;B)!Hom(B;FA)(cid:3) A;B for any A;B 2 A which are natural in A and B. A left Serre functor is a functor G:A!A together with isomorphisms (I.1.2) (cid:16) :Hom(A;B)!Hom(GB;A)(cid:3) A;B foranyA;B 2AwhicharenaturalinAandB. Belowwestateandproveanumber of properties of right Serre functors. We leave the proofs of the corresponding properties for left Serre functors to the reader. NOETHERIAN HEREDITARY ABELIAN CATEGORIES 301 Let (cid:17) : Hom(A;FA) ! k be given by (cid:17) (id ), and let f 2 Hom(A;B). A A;A A Lookingat the commutative diagram(which follows fromthe naturality of(cid:17) in A;B B) Hom(A;A) (cid:0)(cid:0)(cid:17)(cid:0)A(cid:0);A! Hom(A;FA)(cid:3) ? ? ? ? Hom(A;f)y Hom(f;FA)(cid:3)y Hom(A;B) (cid:0)(cid:0)(cid:17)(cid:0)A;(cid:0)B! Hom(B;FA)(cid:3) we (cid:12)nd for g 2Hom(B;FA) that (I.1.3) (cid:17) (f)(g)=(cid:17) (gf): A;B A Similarly by the naturality of (cid:17) in A we obtain the following commutative dia- A;B gram: Hom(B;B) (cid:0)(cid:0)(cid:17)(cid:0)B(cid:0);B! Hom(B;FB)(cid:3) ? ? ? ? Hom(f;B)y Hom(B;Ff)(cid:3)y Hom(A;B) (cid:0)(cid:0)(cid:17)(cid:0)A;(cid:0)B! Hom(B;FA)(cid:3) This yields for g 2Hom(B;FA) the formula (I.1.4) (cid:17) (f)(g)=(cid:17) (F(f)g) A;B B and we get the following description of the functor F. Lemma I.1.1. The following composition coincides with F: Hom(A;B)(cid:0)(cid:17)(cid:0)A(cid:0);!B Hom(B;FA)(cid:3) (cid:0)((cid:0)(cid:17)(cid:0)B(cid:3)(cid:0);F(cid:0)A(cid:0))(cid:0)(cid:0)!1 Hom(FA;FB): Proof. Toprovethisweneedtoshowthatforf 2Hom(A;B)andg 2Hom(B;FA) one has (cid:17) (f)(g)=(cid:17)(cid:3) (Ff)(g). Thanks to the formulas(I.1.3), (I.1.4) we ob- A;B B;FA tain (cid:17) (f)(g)=(cid:17) (F(f)g) and also (cid:17)(cid:3) (Ff)(g)=(cid:17) (g)(Ff)=(cid:17) (F(f)g). A;B B B;FA B;FA B Thus we obtain indeed the correct result. We have the following immediate consequence. Corollary I.1.2. If F is a right Serre functor, then F is fully faithful. Also note the following basic properties. Lemma I.1.3. 1. If F and F0 are right Serre functors, then they are naturally isomorphic. 2. A has a right Serre functor if and only if Hom(A;(cid:0))(cid:3) is representable for all A2A. Fromtheabovediscussionitfollowsthatthereisalotofredundancyinthedata (F;((cid:17) ) ). In fact we have the following. A;B A;B Proposition I.1.4. In order to give (F;((cid:17) ) ) it is necessary and su(cid:14)cient A;B A;B to give the action of F on objects, as well as k-linear maps (cid:17) :Hom(A;FA)!k A such that the composition (I.1.5) Hom(A;B)(cid:2)Hom(B;FA)!Hom(A;FA)(cid:0)(cid:17)!A k yields a non-degenerate pairing for all A;B 2A. If we are given (cid:17) , then (cid:17) is A A;B obtained from the pairing (I.1.5). Furthermore the action F on maps F :Hom(A;B)!Hom(FA;FB) 302 I. REITEN AND M. VAN DEN BERGH is de(cid:12)ned by the property that for f 2Hom(A;B) we have (cid:17) (gf)=(cid:17) (F(f)g) for A B all g 2Hom(B;FA). Proof. It is clear from the previous discussion that the data (F;((cid:17) ) ) gives A;B A;B rise to ((cid:17) ) with the requiredproperties. So converselyassume that we are given A A ((cid:17) ) and the action of F on objects. We de(cid:12)ne ((cid:17) ) and the action of F on A A A;B A;B maps as in the statement of the proposition. We (cid:12)rstshowthatF is afunctor. IndeedletA;B;C 2Aandassumethatthere are maps g : A ! B and h : B ! C. Then for all f 2 Hom(C;FA) we have (cid:17) (fhg)=(cid:17) (F(hg)f), but also (cid:17) (fhg)= (cid:17) (F(g)fh) =(cid:17) (F(h)F(g)f). Thus A C A B C by non-degeneracy we have F(hg)=F(h)F(g). It is easy to see that the pairing (I.1.5) de(cid:12)nes an isomorphism (cid:17) :Hom(A;B)!Hom(B;FA)(cid:3) :f 7!(cid:17) ((cid:0)f) A;B A which is natural in A and B. The proof is now complete. A Serre functor is by de(cid:12)nition a right Serre functor which is essentially surjec- tive. The following is easy to see. Lemma I.1.5. A has a Serre functor if and only it has both a right and a left Serre functor. From this we deduce the following [8]. Lemma I.1.6. A has a Serre functor if and only if the functors Hom(A;(cid:0))(cid:3) and Hom((cid:0);A)(cid:3) are representable for all A2A. Remark I.1.7. In the sequel A will always be a Krull{Schmidt category (in the sense that indecomposable objects havelocal endomorphismrings). In that case it isclearlysu(cid:14)cienttospecify(cid:17) ;(cid:17) ;F,etc.onthefullsubcategoryofAconsisting A;B A of indecomposable objects. If one is given a right Serre functor, then it is possible to invert it formally in suchawaythattheresultingadditivecategoryhasaSerrefunctor. Thenextresult is stated in somewhat greater generality. Proposition I.1.8. Let A be an additive category as above, and let U :A!A be a fully faithful additive endofunctor. Then there exists an additive category B with the following properties: 1. There is a fully faithful functor i:A!B. 2. There is an autoequivalence U(cid:22) : B ! B together with a natural isomorphism (cid:23) :U(cid:22)i!iU. 3. For every object B 2 B there is some b 2 N such that U(cid:22)bB is isomorphic to i(A) with A2A. Furthermore a quadruple (B;i;U(cid:22);(cid:23)) with these properties is unique (in the appro- priate sense). Proof. Let us sketch the construction of B. The uniqueness will be clear. The objects in B are formally written as U(cid:0)aA with A2A and a2Z. A mor- phismU(cid:0)aA!U(cid:0)bBisformallywrittenasU(cid:0)cf withf 2HomA(Uc(cid:0)aA;Uc(cid:0)bB) where c2Z is such that c(cid:21)a, c(cid:21)b. We identify U(cid:0)cf with U(cid:0)c(cid:0)1(Uf). Thefunctoriisde(cid:12)nedbyi(A)=U0AandthefunctorU(cid:22) isde(cid:12)nedbyU(cid:22)(U(cid:0)a)(A) =U(cid:0)a+1(A). It is clear that these have the required properties. NOETHERIAN HEREDITARY ABELIAN CATEGORIES 303 The followinglemma providesa complementto this propositioninthe casethat A is triangulated. Lemma I.1.9. Assume that in addition to the usual hypotheses one has that A is triangulated. Let (B;i;U(cid:22);(cid:23)) be as in the previous proposition. Then there is a unique way to make B into a triangulated category such that i and U(cid:22) are exact. Proof. If we require exactness of i and U(cid:22), then there is only one way to make B intoatriangulatedcategory. Firstwemustde(cid:12)netheshiftfunctorby(U(cid:0)aA)[1]= U(cid:0)a(A[1]) and then the trianglesin B mustbe those diagramsthat are isomorphic to U(cid:0)cA(cid:0)U(cid:0)(cid:0)(cid:0)c!f U(cid:0)cB (cid:0)U(cid:0)(cid:0)(cid:0)c!g U(cid:0)cC (cid:0)U(cid:0)(cid:0)(cid:0)c!h U(cid:0)cA[1] where A(cid:0)!f B (cid:0)!g C (cid:0)!h A[1] is a triangle in A (note that the exactness of U(cid:22) is equivalent to that of U(cid:22)(cid:0)1). To show that this yields indeed a triangulated category one must check the axioms in [34]. These all involve the existence of certain objects/maps/triangles. By applying a su(cid:14)ciently high power of U(cid:22) we can translate such problems into ones involving only objects in A. Then we use the triangulatedstructure of A and afterwards we go back to the original problem by applying a negative power of U(cid:22). In the sequel we will denote by U(cid:0)1A the category B which was constructed in Proposition I.1.8. Furthermore we will consider A as a subcategory of U(cid:0)1A through the functor i. Finally we will usually write U for the extended functor U(cid:22). Belowwewillonlybe interestedinthe specialcasewhereU =F isa rightSerre functor on A. In that case we have the following. Proposition I.1.10. The canonical extension of F to F(cid:0)1A is a Serre functor. Proof. By construction F is an automorphism on F(cid:0)1A. To prove that F is a Serre functor we have to construct suitable maps (cid:17)F(cid:0)aA;F(cid:0)bB. Pick c (cid:21) a, c (cid:21) b. Then we have HomF(cid:0)1A(F(cid:0)aA;F(cid:0)bB)=HomA(Fc(cid:0)aA;Fc(cid:0)bB) (cid:17)Fc(cid:0)aA(cid:24)=;Fc(cid:0)bB HomA(Fc(cid:0)bB;Fc(cid:0)a+1A)(cid:3) =HomF(cid:0)1A(F(cid:0)bB;F(cid:0)a+1A)(cid:3): We de(cid:12)ne (cid:17)F(cid:0)aA;F(cid:0)bB as the composition of these maps. It follows easily from Lemma I.1.1 that the constructed map is independent of c, and it is clear that (cid:17) has the required properties. F(cid:0)aA;F(cid:0)bB We shall also need the following easily veri(cid:12)ed fact. Lemma I.1.11. If A = A (cid:8) A is a direct sum of additive categories, then a 1 2 (right) Serre functor on A restricts to (right) Serre functors on A and A . 1 2 304 I. REITEN AND M. VAN DEN BERGH I.2. Connection between Serre duality and Auslander{Reiten triangles. In this section we prove that existence of a right Serre functor is equivalent to the existence of right Auslander{Reiten triangles, in triangulated Hom-(cid:12)nite Krull{ Schmidt k-categories. Hence the existence of a Serre functor is equivalent to the existence of Auslander{Reiten triangles. In the sequel A is a Hom-(cid:12)nite k-linear Krull-Schmidt triangulated category. Following [14] a triangle A (cid:0)!f B (cid:0)!g C (cid:0)!h A[1] in A is called an Auslander{Reiten triangle if the following conditions are satis(cid:12)ed: (AR1) A and C are indecomposable. (AR2) h6=0. (AR3) IfD isindecomposable,thenforeverynon-isomorphismt:D !C wehave ht=0. It is shown in [14] that, assuming (AR1) and (AR2), then (AR3) is equivalent to (AR3)0 If D0 is indecomposable, then for every non-isomorphism s : A ! D0 we have sh[(cid:0)1]=0. We say that right Auslander{Reiten triangles exist in A if for all indecompos- ables C 2 A there is a triangle satisfying the conditions above. Existence of left Auslander{Reiten triangles is de(cid:12)ned in a similar way, and we say that A has Auslander{ReitentrianglesifithasbothrightandleftAuslander{Reitentriangles. (Note that in [14] one says that A has Auslander{Reiten triangles if it has right Auslander{Reiten triangles in our terminology.) Itisshownin[14,x4.3]thatgivenCthecorrespondingAuslander{Reitentriangle A!B !C !A[1]isuniqueuptoisomorphismoftriangles. (Bydualityasimilar resultholdsifAisgiven.) ForagivenindecomposableC welet(cid:28)~C be anarbitrary objectinA, isomorphicto Ain the Auslander{Reitentrianglecorrespondingto C. The followingcharacterizationofAuslander{Reitentrianglesisanalogoustothe corresponding result on almost split sequences (see [5]). Proposition I.2.1. Assume that A has right Auslander{Reiten triangles, and as- sume that we have a triangle in A (I.2.1) A(cid:0)!f B (cid:0)!g C (cid:0)!h A[1] with A and C indecomposable and h6=0. Then the following are equivalent: 1. The triangle (I.2.1) is an Auslander{Reiten triangle. 2. The map h is in the socle of Hom(C;A[1]) as a right End(C)-module and (cid:24) A=(cid:28)~C. (cid:24) 3. The map h is in the socle of Hom(C;A[1]) as a left End(A)-module and A= (cid:28)~C. Proof. We will show that 1. and 2. are equivalent. The equivalence of 1. and 3. is similar. 1:)2: By de(cid:12)nition we have A (cid:24)= (cid:28)~C. Assume that t 2 End(C) is a non- automorphism. Then by (AR3) we have ht=0. 2:)1: Let (I.2.2) A(cid:0)f!0 B0 (cid:0)g!0 C (cid:0)h!0 A[1]