THE LEFT PART AND THE AUSLANDER-REITEN COMPONENTS OF AN ARTIN ALGEBRA

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1 REVIST DE L UNIÓN MTEMÁTIC RGENTIN Volumen 46, Número 2, 2005, Páginas THE LEFT PRT ND THE USLNDER-REITEN COMPONENTS OF N RTIN LGEBR IBRHIM SSEM, JUN ÁNGEL CPP, MRÍ INÉS PLTZECK, ND SONI TREPODE Dedicated to the memory of Ángel Rafael Larotonda bstract. The left part L of the module category of an artin algebra consists of all indecomposables whose predecessors have projective dimension at most one. In this paper, we study the uslander-reiten components of (and of its left support λ ) which intersect L and also the class E of the indecomposable Ext-injectives in the addditive subcategory addl generated by L. Introduction Let be an artin algebra and mod denote the category of finitely generated right modules. The class L, called the left part of mod, isthefullsubcategory of mod having as objects all indecomposable modules whose predecessors have projective dimension at most one. This class, introduced in [15], was heavily investigated and applied (see, for instance, the survey [4]). Our objective in this paper is to study the uslander-reiten components of an artin algebra which intersect the left part. Some information on these components was already obtained in [2, 3]. Here we are interested in the components which intersect the class E of the indecomposable Ext-injectives in the full additive subcategory add L having as objects the direct sums of modules in L.Westartby proving the following theorem. Theorem (). Let be an artin algebra, and Γ be a component of the uslander- Reiten quiver of. IfΓ E, then: (a) Each τ -orbit of Γ L intersects E exactly once. (b) The number of τ -orbits of Γ L equals the number of modules in Γ E. (c) Γ L contains no module lying on a cycle between modules in Γ Mathematics Subject Classification. 16G70, 16G20, 16E10. Key words and phrases. artin algebras, uslander-reiten quivers, sections, left and right supported algebras. This paper was completed during a visit of the first author to the Universidad Nacional del Sur in Bahía Blanca (rgentina). He would like to thank María Inés Platzeck and María Julia Redondo, as well as all members of the argentinian group, for their invitation and warm hospitality. He also acknowledges partial support from NSERC of Canada. The other three authors gratefully acknowledge partial support from Universidad Nacional del Sur and CONICET of rgentina, and the fourth from NPCyT of rgentina. The second author has a fellowship from CONICET, and the third and the fourth are researchers from CONICET. 19

2 20 I. SSEM, J.. CPP, M. I. PLTZECK ND S. TREPODE If, on the other hand, Γ E =, then either Γ L or else Γ L =. We recall that, by [3] (3.3), the class E contains only finitely many non-isomorphic modules (hence only finitely many uslander-reiten components intersect E). s a consequence, we give a complete description of the uslander-reiten components lying entirely inside the left part. We then try to describe the intersection of E with a component Γ of the uslander-reiten quiver Γ(mod). We find that, in general, Γ E is not a section in Γ (in the sense of [20, 23]) but is very nearly one. This leads us to our second theorem, for which we recall that a component Γ of Γ(mod) is called generalised standard if rad (X, Y ) = 0 for all X, Y Γ, see [23]. Theorem (B). Let be an artin algebra and Γ be a component of Γ(mod) such that all projectives in Γ belong to L.If Γ E, then: (a) Γ E is a section in Γ. (b) Γ is generalised standard. (c) /nn(γ E) is a tilted algebra having Γ as a connecting component and Γ E as a complete slice. In particular, such a component Γ has only finitely many τ -orbits. The situation is better if we look instead at the intersection of E with the uslander-reiten components of the left support λ of. We recall from [3, 24] that the left support λ of is the endomorphism algebra of the direct sum of the indecomposable projective -modules lying in L. It is shown in [3, 24] that every connected component of λ is a quasi-tilted algebra (in the sense of [15]). We prove the following theorem. theorem (C). Let be an artin algebra and Γ be a component of the uslander- Reiten quiver of the left support λ of. If Γ E, then: (a) Γ E is a section in Γ. (b) Γ is directed, and generalised standard. (c) λ /nn(γ E) is a tilted algebra having Γ as a connecting component and Γ E as a complete slice. We then apply our results to the study of left supported algebras. We recall from [3] that an artin algebra is left supported provided addl is contravariantly finite in mod. Several classes of algebras are left supported, such as all representationfinite algebras, and all laura algebras which are not quasi-tilted (see [3, 4]). It is shown in [1] that an artin algebra is left supported if and only if L consists of all the predecessors of the modules in E. Wegivehereaproofofthisfactwhich, in contrast to the homological nature of the proof in [1], uses our theorem and the full power of the uslander-reiten theory of quasi-tilted algebras. Our proof also yields a new characterisation: an algebra is left supported if and only if every projective -module which belongs to L is a predecessor of E. Weendthepaper with a short proof of the theorem of D. Smith [25] (3.8) which characterises the left supported quasi-tilted algebras. Rev. Un. Mat. rgentina, Vol 46-2

3 THE LEFT PRT 21 Clearly, the dual statements about the right part of the module category, also hold true. Here, we only concern ourselves with the left part, leaving the primaldual translation to the reader. We now describe the contents of the paper. fter a brief preliminary section 1, the sections 2, 3 and 4 are respectively devoted to the proofs of our theorems (), (B) and (C). In our final section 5, we consider the applications to left supported algebras. 1. Preliminaries Notation. For a basic and connected artin algebra, let mod denote its category of finitely generated right modules and ind a full subcategory consisting of exactly one representative from each isomorphism class of indecomposable modules. We sometimes consider as a category, with objects a complete set {e 1,,e n } of primitive orthogonal idempotents, and where e i e j is the set of morphisms from e i to e j. n algebra B is a full subcategory of if there is an idempotent e, which is a sum of some of the distinguished idempotents e i, such that B = ee. Itisconvex in if, for any sequence e i = e i0,e i1,,e it = e j of objects of such that e il+1 e il 0(with0 l<t)ande i, e j objects of B, all e il are in B. Given a full subcategory C of mod, wewritem Cto indicate that M is an object in C, and we denote by addc the full subcategory with objects the direct sums of summands of modules in C. Given a module M, weletpdm stand for its projective dimension. We also denote by Γ(mod) the uslander-reiten quiver of and by τ =DTr,τ 1 = TrD the uslander-reiten translations. For further notions or facts needed on mod, we refer to [7, 22] Paths. Let be an artin algebra and M,N ind. path M N is a sequence f 1 f 2 f t ( ) M = M 0 M1 Mt 1 Mt = N where the f i are non-zero morphisms and the M i lie in ind. We call M a predecessor of N and N a successor of M. path from M to M involving at least one non-isomorphism is a cycle. n indecomposable module M lying on no cycle is called directed. path ( ) is called sectional if each f i is irreducible and τ M i+1 M i 1 for all i. refinement of ( ) isapath M = M 0 f 1 M 1 f 2 M s 1 f s M s = N such that there exists an order-preserving injection σ : {1,,t 1} {1,,s 1} with M i = M σ(i) for all i. full subcategory C of ind is convex if, for any path ( ) withm, N C,alltheM i lie in C. 2. Ext-injectives in the left part Let be an artin algebra. The left part L of mod is the full subcategory of ind defined by L = {M ind pd L 1 for any predecessor L of M}. Rev. Un. Mat. rgentina, Vol 46-2

4 22 I. SSEM, J.. CPP, M. I. PLTZECK ND S. TREPODE n indecomposable module M L is called Ext-projective (or Ext-injective)in addl if Ext 1 (M, ) L =0(orExt 1 (,M) L = 0, respectively), see [9]. While the Ext-projectives in addl are the projective modules lying in L (see [3] (3.1)), the Ext-injectives are more interesting. Before stating their characterisations we recall that, by [9] (3.7), M L is Ext-injective in addl if and only if τ 1 M/ L. Lemma [5] (3.2), [3] (3.1). Let M L. (a) The following are equivalent : (i) There exists an indecomposable injective module I such that Hom (I,M) 0. (ii) There exist an indecomposable injective module I and a path I M. (iii) There exist an indecomposable injective module I and a sectional path I M. (b) The following conditions are equivalent for M L which does not satisfy conditions (a): (i) There exists an indecomposable projective module P / L such that Hom (P, τ 1 M) 0. (ii) There exist an indecomposable projective module P / L and a path P τ 1 M. (iii) There exist an indecomposable projective module P / L and a sectional path P τ 1 M. Letting E 1 (or E 2 ) denote the set of all M L verifying (a) (or (b), respectively), and setting E = E 1 E 2,thenM is Ext-injective in addl if and only if M E The following lemma will also be useful. Lemma [3] (3.2) (3.4). (a) ny path of irreducible morphisms in E is sectional. (b) Let M E and M N with N L. Then this path can be refined to a sectional path and N E.Inparticular,E is convex in ind The following immediate corollary will be useful in the proof of our theorem (). Corollary ll modules in E are directed. Proof. ssume M = M 0 M 1 M s = M is a cycle in ind, with M E. By (2.2) above, such a cycle can be refined to a sectional cycle with all indecomposables lying in E. Now compose two copies of this cycle to form a larger cycle in E of irreducible morphisms. By (2.2), this cycle is also sectional, in contradiction to [11, 12] Theorem (). Let be an artin algebra, and Γ be a component of the uslander-reiten quiver of. If Γ E, then: (a) Each τ -orbit of Γ L intersects E exactly once. Rev. Un. Mat. rgentina, Vol 46-2

5 THE LEFT PRT 23 (b) The number of τ -orbits of Γ L equals the number of modules in Γ E. (c) Γ L contains no module lying on a cycle between modules in Γ. If, on the other hand, Γ E =, then either Γ L or else Γ L =. Proof. ssume first that Γ E, that is, the component Γ contains an Extinjective in addl. (a) If Γ contains an injective module, then the statement follows from [3] (3.5). We may thus assume that Γ contains no injective. But then Γ E 1 =, and therefore Γ E 2 =Γ E. Thus, by (2.1), there exist an indecomposable projective P in Γ such that P / L, a module M Γ E 2 and a sectional path P τ 1 M. Now let X Γ L. Since Γ contains no injective, there exists s>0 such that τ s s X is a successor of P. Hence τ X / L. Since X itself lies in L, there exists j 0 such that τ j X L but τ j 1 X/ L, so that τ j X is Ext-injective in add L. This shows that every τ orbit of Γ L intersects E at least once. Furthermore, it intersects it only once: if Y and τ t Y (with t>0) both belong to Γ E then, by (2.2), all the modules on the path Y τ 1 Y τ t Y belong to L. In particular, τ 1 Y L and this contradicts the Ext-injectivity of Y. This completes the proof of (a). (b) It follows from (a) that the number of τ -orbits in Γ L does not exceed the cardinality of Γ E (note that by [3] (3.3), the cardinality of E is finite and does not exceed the rank of the Grothendieck group K 0 () of). Since clearly, any element of Γ E belongs to exactly one τ -orbit in L, this establishes (b). f 1 f 2 f t (c) Let ( ) M 0 M1... Mt = M 0 be a cycle with M 0 Γ L and all M i in Γ. Clearly, all M i belong to Γ L. By (2.2) and (2.3), none of the M i belongs to E and none of the f i factors through an injective module. Indeed, if f i factors through the injective I, then some indecomposable summand of I would belong to L and thus M i would lie in E, contradicting (2.3). Then the cycle ( ) induces a cycle τ 1 M 0 τ 1 M 1 τ 1 M t = τ 1 M 0, and every module in this cycle belongs to Γ L. We can iterate this procedure and deduce that, for any m>0, the module τ m M 0 lies on a cycle in Γ L. However, as shown in (a), there exists s>0 such that τ s M 0 does not belong to L, and this contradiction proves (c). Now assume that the component Γ contains no Ext-injective, that is, Γ E =. If Γ contains both a module in L and a module which is not in L, then there exists an irreducible morphism X Y with X Γ L and Y Γ \L. Since Γ E =, then τ 1 X L. But this is a contradiction, because Y / L and Hom (Y, τ 1 X) 0. This shows that either Γ L = or Γ L, as required. We observe that part (c) of the theorem was already proven in [3] (1.5) under the additional hypothesis that Γ contains an injective module. Rev. Un. Mat. rgentina, Vol 46-2

6 24 I. SSEM, J.. CPP, M. I. PLTZECK ND S. TREPODE 2.5. Corollary [3] (1.6). Let be a representation-finite artin algebra. Then L is directed We have a good description of the uslander-reiten components which completely lie in L. We need to recall a definition. The endomorphism algebra λ of the direct sum of all the projective modules lying in L is called the left support of, see [3, 24]. Clearly, λ is (isomorphic to) a full convex subcategory of, closed under successors, and any -module lying in L has a natural λ -module structure. It is shown in [3] (2.3), [24] (3.1) that λ is a product of connected quasi-tilted algebras, and that L L λ. The following corollary generalises [3] (5.5). Corollary. Let be a representation-infinite not hereditary artin algebra, and Γ be a component of Γ(mod) lying entirely in L.ThenΓ is one of the following: a postprojective component, a regular component (directed, stable tube or of type Z ), a semiregular tube without injectives, or a ray extension of Z. Proof. Indeed, the component Γ lies entirely in mod λ andthusisacomponent of Γ(mod λ ). Since L L λ, then Γ is a component of Γ(mod λ )lyinginthe left part L λ. The statement then follows from the well-known description of the uslander-reiten components of quasi-tilted algebras, as in [13, 18]. 3. Ext-injectives as sections in Γ(mod) We recall the following notion from [20, 23]. Let be an artin algebra and Γ be a component of Γ(mod). full connected subquiver Σ of Γ is called a section if it satisfies the following conditions: (S 1 ) Σ contains no oriented cycle. (S 2 ) Σ intersects each τ -orbit of Γ exactly once. (S 3 ) Σ is convex in Γ. (S 4 ) If X Y is an arrow in Γ with X Σ, then Y Σ or τ Y Σ. (S 5 ) If X Y is an arrow in Γ with Y Σ, then X Σorτ 1 X Σ. s we show next, the intersection of E with a component of Γ(mod) satisfies several of these conditions (but generally not all). Proposition. ssume Γ is a component of Γ(mod) which intersects E. Then Γ E satisfies (S 1 ), (S 3 ), (S 5 ) above, and the following conditions (S 2 ) Γ E intersects each τ -orbit of Γ at most once. (S 4) If X Y is an arrow in Γ with X Γ E and Y non-projective, then Y E or τ Y E. Proof. (S 1 ) follows from Theorem () (c). (S 2) follows from Theorem () (a). (S 3 ) follows from (2.2). (S 4 )IfY L,thenX Eand (2.2) imply Y E. Otherwise, since Y is non-projective, there exists an arrow τ Y X. Since X E,thenτ Y L. Since Y = τ 1 (τ Y ) / L,wegetτ Y E. Rev. Un. Mat. rgentina, Vol 46-2

7 THE LEFT PRT 25 (S 5 )IfX is injective then, since it lies in L (because it precedes Y ), it belongs to E. So assume it is not and consider the arrow Y τ 1 1 X.Ifτ X/ L then, again, X E while, if τ 1 X L,thenY E and (2.2) imply τ 1 X E Example. Let k be a field and be the radical square zero k-algebra given by the quiver Here, is representation finite and E consists of the two indecomposable projectives P 1 and P 2 corresponding to the points 1 and 2, respectively. Clearly, E = {P 1,P 2 } is not a section in Γ(mod) : indeed, there is an arrow P 1 P 3 with P 3 / E and, moreover, E does not intersect each τ -orbit of Γ(mod) We are now in a position to prove our second main theorem. Theorem (B). Let be an artin algebra and Γ be a component of Γ(mod) such that all projectives in Γ belong to L.If Γ E, then: (a) Γ E is a section in Γ. (b) Γ is generalised standard. (c) /nn(γ E) is a tilted algebra having Γ as a connecting component and Γ E as a complete slice. Proof. (a) We start by observing that, if X P is an arrow in Γ, with X E and P projective then, by hypothesis, P L. Thus, (2.2) implies P E. This shows that (S 4 ) is satisfied. In view of the lemma, it suffices to show that Γ E cuts each τ orbit of Γ. We claim that if M Eand N Γ lie in two neighbouring orbits, then E intersects the τ -orbit of N. This claim and induction imply the statement. We assume that E does not intersect the orbit of N and try to reach a contradiction. There exist n Z andanarrowτ nm X or X τ nm,withx in the τ -orbit of N, where we may suppose, without loss of generality, that n is minimal. Suppose first that n<0. If there exists an arrow X τ n M then there exists an arrow τ n+1 M X, a contradiction to the minimality of n. If, on the other hand, there exists an arrow τ n M X, thenthereisapathinγoftheform M τ 1 1 M X. Since M Ethen τ M / L. Hence X / L. In particular, X is not projective, so there exists an arrow τ n+1 M τ X, contrary to the minimality of n. Suppose now that n>0. If there exists an arrow τ n M X, then there exists an arrow X τ n 1 M, a contradiction to the minimality of n. If, on the other hand, there exists an arrow X τ nm,thenthereisapathinγoftheform X τ nm M. Hence X L. In particular, X is not injective (otherwise, X E, a contradiction). Hence there exists an arrow τ 1 X τ n 1 M, contrary to the minimality of n. Rev. Un. Mat. rgentina, Vol 46-2

8 26 I. SSEM, J.. CPP, M. I. PLTZECK ND S. TREPODE We have thus shown that necessarily n = 0, that is, there is an arrow M X or X M. IfM X then, by (S 4 ), X Eor τ X E, in any case a contradiction. If X M, then (3.1) yields X Eor τ 1 X E, again a contradiction in any case. This completes the proof of (a). (b) By [23], Theorem 2, it suffices to show that for any X, Y Γ E,wehave Hom (X, τ Y )=0. ButY E implies pdy 1. Therefore the Ext-injectivity of X in addl implies that Hom (X, τ Y ) DExt 1 (Y,X)=0. (c) This follows directly from [20] (2.2) Example. Let k be a field and be the radical square zero algebra given by the quiver Let Γ be the component containing the injective I 1 corresponding to the point 1. Clearly, I 1 E,sothatΓ E. On the other hand, the only projective lying in Γ is P 3, and it belongs to L. Thus, the hypotheses of the theorem apply here. Note that /nn(γ E) is equal to the left support λ of, thatis,the full convex subcategory with objects {1, 2, 3}. 4. Ext-injectives and the left support 4.1. In this section we study the intersection of E with the components of the uslander-reiten quiver of the left support λ of the artin algebra. We observe first that if Y is an λ -module and τ Y L then τ Y = τ λ Y. In particular, Y is not projective in mod λ. Indeed, since mod λ is closed under extensions in mod, then the inclusion L ind λ implies that the almost split sequence in mod ending at Y is entirely contained in mod λ (See also [7], p. 187). Similarly, if τ 1 λ -module. Y L,thenτ 1 Y = τ 1 λ Y,andY is not an injective Lemma. If an indecomposable injective λ -module I is a predecessor of E, then I E. Proof. This is clear if I is an indecomposable injective -module. So assume it is not. Since I precedes E, theni L. By the above observation we obtain that τ 1 I/ L, because I is λ -injective. This proves that I E, as desired The following is an easy consequence of (3.1) and the results in [3]. Lemma. Let E = X E X. Then E is a convex partial tilting λ -module. In particular, E rk K 0 ( λ ). Rev. Un. Mat. rgentina, Vol 46-2

9 THE LEFT PRT 27 Proof. Indeed, since L L λ (see [3], (2.1)), E L implies pd λ E 1. Since Ext 1 (E,E) =0and λ is a full convex subcategory of, wealsohave Ext 1 λ (E,E) = 0. Finally, the convexity of E in ind λ follows from its convexity in ind (see (2.2)) theorem C. Let be an artin algebra and Γ be a component of the uslander- Reiten quiver of the left support λ of. If Γ E, then: (a) Γ E is a section in Γ. (b) Γ is directed, and generalised standard. (c) λ /nn(γ E) is a tilted algebra having Γ as a connecting component and Γ E as a complete slice. Proof. (a) In order to show that Γ E is a section in Γ, we just have to check the conditions of the definition in (3.1). Clearly, (S 1 ) follows from (2.3) and the observation that any cycle in ind λ induces one in ind. lso,(s 3 ) follows from (4.2). We start by proving (S 4 )and(s 5 ). (S 4 ) ssume X Y is an arrow in Γ, with X E. If Y L, then (2.2) implies Y E. ssume Y / L. Then, in particular, Y is not a projective λ -module. Since Y is an λ -module, it is not a projective -module either, so there is an irreducible morphism τ Y X in mod. Thenτ Y precedes X E and therefore lies in L. Thus, as we observed in (4.1), τ Y = τ λ Y. Since (τ λ Y )=Y / L, we conclude that τ λ Y E, as required. (S 5 ) ssume X Y is an arrow in Γ, with Y E.IfX/ E,thenτ 1 X L and, again by the observation in 4.1, we know that X is not an injective λ - module. Hence τ 1 λ X = τ 1 X L. Since there is an arrow Y τ 1 λ X,we conclude that τ 1 λ X E, as required. There remains to prove (S 2 ), that is, that E intersects each orbit of Γ exactly once. We use the same technique as in the proof of Theorem (B). Clearly, the situation is different and so the arguments vary slightly. We start by proving that E intersects each orbit of Γ at least once. We claim that if M Eand N Γ lie in two neighbouring orbits, then E intersects the τ λ -orbit of N. This claim and induction imply the statement. We assume that E does not intersect the orbit of N and try to reach a contradiction. There exist n Z and an arrow τ n λ M X or X τ n λ M,withX in the τ λ -orbit of N, where we may suppose, without loss of generality, that n is minimal. τ 1 Suppose first that n < 0. If there exists an arrow X τ n λ M then there exists an arrow τ n+1 λ M X, a contradiction to the minimality of n. If, on the other hand, there exists an arrow τ n λ M X, then there is a path in Γ of the form M τ 1 λ M X. Now, M Eimplies τ 1 M / L. By [7] p. 186, there exists an epimorphism τ 1 M τ 1 λ M. Hence τ 1 λ M / L and so X/ L.Inparticular,Xis not a projective λ -module, so there exists an arrow τ n+1 λ M τ λ X, contrary to the minimality of n. Suppose now that n>0. If there exists an arrow τ n λ M X, then there exists an arrow X τ n 1 λ M, a contradiction to the minimality of n. If, on the other hand, there exists an arrow X τ n λ M, then there is a path in Γ of the form Rev. Un. Mat. rgentina, Vol 46-2

10 28 I. SSEM, J.. CPP, M. I. PLTZECK ND S. TREPODE X τ n λ M M, hence X is a predecessor of E. Since X / E, by hypothesis, then we know by (4.1) that X is not injective in mod λ. Hence there exists an arrow τ 1 λ X τ n 1 λ M, contrary to the minimality of n. This shows that necessarily n = 0, that is, there is an arrow M X or X M. If M X, then (S 4 ) yields X Eor τ λ X E, in any case a contradiction. If X M, then(s 5 ) yields X E or τ 1 λ X E, again a contradiction in any case. We proved that E intersects each τ λ -orbit of Γ. Suppose now that M E and τ t λ M E with t>0. Then the epimorphism τ 1 M τ 1 λ M yields a path τ 1 M τ 1 λ M τ t λ M,sothatτ 1 M L. This is a contradiction because M E.Thus(a)isproven. (b) Since, by [13], directed components of quasi-tilted algebras are postprojective, preinjective or connecting, thus always generalised standard (see [20, 23]), it suffices to show that Γ is directed. If this is not the case then, by [18] (4.3), Γ is a stable tube, of type Z or obtained from one of these by finitely many ray or coray insertions. We first notice that by (2.3), any E 0 Γ E is directed in ind, hence in ind λ. In particular, Γ is neither a stable tube, nor of type Z. Therefore Γ is obtained from one of these by ray or coray insertions. ssume first that Γ is an inserted tube or component of type Z,andlet E 0 Γ E. We claim that E 0 E 2. Indeed, if this is not the case, then there exists an injective -module I such that Hom (I,E 0 ) 0, by (2.1). However, I L implies that I is an λ -module, so that I is an injective λ -module. But this is impossible because no injective λ -module precedes an inserted tube or component of type Z. This establishes our claim. Thus, there exists an indecomposable projective module P / L such that Hom (P, τ 1 E 0) 0, by (2.1). On the other hand, τ 1 λ E 0 Γ, therefore there exist a non-directed projective P Γanda path τ 1 λ E 0 P in Γ. This is clear if Γ is an inserted tube, and follows from [10, 17] if Γ is an inserted component of type Z. Hence there exists a path P τ 1 E 0 τ 1 λ E 0 P in ind. Since P / L,thenP / L. However, P Γ, hence P is a projective -module lying in L, a contradiction. ssume next that Γ is a co-inserted tube or component of type Z, and let E 0 Γ E. Then, among the predecessors of E 0 lies a cycle M 0 M 1 M t = M 0, with all M i Γ. Since all M i precede E 0 and, by hypothesis, E 0 E L L λ, then this cycle lies in L λ. This contradicts Theorem () (c) (also [3] (1.5) (b)). (c) This follows directly from [20] (2.2) Example. It is important to underline that, while the components of Γ(mod λ )whichcute are directed, and even generalised standard, the same does not hold for the components of Γ(mod). Indeed, let k be a field and be given by the quiver Rev. Un. Mat. rgentina, Vol 46-2

11 THE LEFT PRT 29 3 β 4 α 5 ε γ δ bound by γδ = 0, γɛ =0andαβ = 0. Letting, as usual, P i and S i denote respectively the indecomposable projective and the simple modules corresponding to the point i, we have an almost split sequence 0 P 3 P 4 S 4 0. Moreover, radp 5 = S 4 S 2,whereS 2 lies in a regular component of type Z in the uslander-reiten quiver of the wild hereditary algebra H which is the full subcategory of with objects the points 1, 2 and 6. Now, the projective P 4 is also injective and lies in L (because its unique proper predecessor is P 3 ), hence in E. Therefore, the component of Γ(mod) containing it is neither directed, nor generalised standard Lemma. Let Γ be a component of Γ(mod λ ). (a) If Γ is a non-connecting postprojective component, then Γ E =. (b) If Γ is a non-connecting preinjective component, then Γ E =. (c) If Γ intersects E, thenγ is connecting. (d) If a connected component B of λ is not tilted, then modb E =. Proof. (a) ssume that Γ is a non-connecting postprojective component of Γ(mod λ ) such that Γ E. Let B be the (unique) connected component of λ such that Γ consists of B-modules. We claim that Γ does not contain every indecomposable projective B module. Indeed, if this is not the case, then the number of τ B orbits in Γ coincides with rk K 0 (B). By Theorem (C) (a), E intersects each τ B orbit of Γ exactly once. Hence Γ E has rk K 0 (B) elements. From this and (4.2) we deduce that E 0 = X Γ E X is a convex tilting B-module. By [6], (2.1), Γ E is acompletesliceinmodb. But this is a contradiction, because Γ was assumed to be non-connecting. This establishes our claim. Now, let Q / Γ be an indecomposable projective B module. Since B is a connected algebra, there exists a walk of projective B modules P = P 0 P 1... P s = Q, with P Γ. Thus there exists i such that P i ΓandP i+1 / Γ. Since Γ does not receive morphisms from other components of Γ(modB), then Hom B (P i,p i+1 ) 0. By [21] (2.1) there exists, for each s>0, a path f 1 f 2 P i = M 0 M1 M2 f s Ms = L f P i+1 with f i irreducible. Since s is as large as we want, and E intersects each τ B orbit of Γ, we may choose s so that L is a proper successor of Γ E. On the other hand, P i+1 is a projective B module, hence a projective module lying in L. Thus L L. Since L is a successor of E, by (2.2), L E, a contradiction which proves (a). Rev. Un. Mat. rgentina, Vol 46-2

12 30 I. SSEM, J.. CPP, M. I. PLTZECK ND S. TREPODE (b) ssume that Γ is a non-connecting preinjective component of Γ(mod λ ) such that Γ E. Using the same reasoning as in (a), there exist M Γ, which is a proper predecessor of Γ E, and an indecomposable injective λ module I/ Γ such that Hom λ (I,M) 0. Since I precedes E then, by (4.1), I E. The convexity of E yields the contradiction M E. This establishes (b). (c)itisshownin[3,24]thatevery connected component of λ is quasi-tilted. By Theorem (C), E intersects only directed components of Γ(mod λ ). Furthermore, directed components of quasi-tilted algebras are necessarily postprojective, preinjective or connecting. Now the result follows from (a) and (b). (d) Is a consequence of (c) Proposition Let B be a connected component of the left support λ, such that modb E. Then B is a tilted algebra and modb E is a complete slice in modb. Proof. Let Γ be a component of Γ(mod λ ) such that Γ E. By (c) of the previous lemma, we know that Γ is a connecting component. Since, on the other hand, E intersects each τ B -orbit of Γ exactly once (by Theorem (C) (a)), we have Γ E =rkk 0 (B). But by (4.2), Γ(modB) E rkk 0 (B). Hence Γ E =Γ(modB) E and the direct sum of the modules in Γ(modB) E is a convex tilting B-module. The result then follows from [6](2.1). Observe that if Γ is a component of Γ(modB) such that Γ E, thenit follows from the proof of (4.6) that Γ E is a complete slice in modb. Therefore, by [23], B = λ /nn(γ E) Example. It is possible to have λ = B B,and E modb, while E modb =. Indeed, let be given by the quiver 1 γ 2 β γ 2 3 β 2 6 α 2 γ β α α 3 8 λ 12 µ 11 σ 10 ρ 9 bound by α 1 β 1 γ 1 + α 2 β 2 γ 2 + α 3 β 3 γ 3 =0,λα 1 =0,λα 2 =0,,λα 3 =0,µσ =0, σρ = 0. Let B denote the (tilted) full subcategory of having as objects 9, 10 and 11, and B denote the (tubular) full subcategory of having as objects 1, 2, 3, 4, 5, 6, 7, 8. Then λ = B B, E modb (it consists of the indecomposable modules P 10, P 11 and S 10 ) while E modb = (because B is a tubular algebra). 5. Left supported algebras n artin algebra is left supported if addl is contravariantly finite in mod, in the sense of [8]. It is shown in [3] (5.1) that an artin algebra is left supported if and only if each connected component of λ is tilted and the restriction of E to this component is a complete slice. Several other characterisations of Rev. Un. Mat. rgentina, Vol 46-2

13 THE LEFT PRT 31 left supported algebras are given in [1, 3]. In particular, it is shown in [1] that is left supported if and only if L =PredE, whereprede denotes the full subcategory of ind having as objects all the M ind such that there exists E 0 E and a path M E 0. Our objective in this section is to give another proof of this theorem, using the results above. Our proof also yields a new characterisation of left supported algebras. Theorem. Let be an artin algebra. Then the following conditions are equivalent: (a) is left supported. (b) L =PredE. (c) Every projective module which belongs to L is a predecessor of E. Proof. (a) implies (b). ssume that is left supported. By [3](4.2), L is cogenerated by the direct sum of the modules in E. In particular, L Pred E. Since the reverse inclusion is obvious, this completes the proof of (a) implies (b). Clearly (b) implies (c). To prove that (c) implies (a) we assume that every projective module which belongs to L is a predecessor of E. Let B be a connected component of λ and P be an indecomposable projective B-module. Since P L,thereexistE 0 Eand a path P E 0 in L, hence in modb. Therefore, modb E. By (4.6), B is a tilted algebra and modb E is a complete slice in modb. Hence is left supported We end this paper with a short proof of a result by D. Smith. Theorem.([25] (3.8)) Let be a quasi-tilted algebra. Then is left supported if and only if is tilted having a complete slice containing an injective module. Proof. Since is quasi-tilted, then = λ. ssume that is left supported. Then L =PredE. By (5.1), is tilted and E is a complete slice in Γ(mod). Furthermore, since is quasi-tilted, then all projective -modules lie in L,so that E 2 = and E = E 1.ThusE must contain an injective module. Conversely, if has a complete slice containing an injective, then there exists a complete slice Σ having all its sources injective. By (2.1), Σ E. Since Σ = rkk 0 (), it follows from [3] (3.3) that is left supported. References 1. I. ssem, J.. Cappa, M. I. Platzeck and S. Trepode, Some characterisations of supported algebras, J.Pureppl.lgebra.InPress. 2. I. ssem, F. U. Coelho, Two-sided gluings of tilted algebras, J. lgebra269 (2003) I. ssem, F. U. Coelho, S. Trepode, Theleftandtherightpartsofamodulecategory,J. lgebra 281 (2004) I.ssem,F.U.Coelho,M.Lanzilotta,D.Smith,S.Trepode,lgebras determined by their left and right parts, Proc. XV Coloquio Latinoamericano de Álgebra, Contemp. Math. 376 (2005) 13-47, mer. Math. Soc. Rev. Un. Mat. rgentina, Vol 46-2

14 32 I. SSEM, J.. CPP, M. I. PLTZECK ND S. TREPODE 5. I.ssem,M.Lanzilotta,M.J.Redondo,Laura skew group algebras, toappear. 6. I. ssem, M. I. Platzeck, S. Trepode, On the representation dimension of tilted and laura algebras, J. oflgebra. 283 (2005) M.uslander,I.Reiten,S.O.Smalø,Representation Theory of rtin lgebras, Cambridge Studies in dvanced Mathematics 36, Cambridge Univ. Press, (1995). 8. M. uslander, S. O. Smalø, Preprojective modules over artin algebras, J. lgebra66 (1980) M. uslander, S. O. Smalø, lmost split sequences in subcategories, J. lgebra69 (1981) D. Baer, Wild hereditary artin algebras and linear methods, Manuscripta Math., 55 (1986) R. Bautista, S. O. Smalø, Non-existent cycles, Comm. lgebra, 11 (16) (1983) K. Bongartz, On a result of Bautista and Smalø on cycles, Comm. lgebra, 11 (18) (1983) F. U. Coelho, Directing components for quasi-tilted algebras, Coll. Math. 82 (1999), D. Happel, characterization of hereditary categories with tilting object, Invent. Math. 144 (2001) D.Happel,I.Reiten,S.O.Smalø,Tilting in belian Categories and Quasitilted lgebras, Memoirs MS 575 (1996). 16. O. Kerner, Tilting wild algebras, J. London Math. Soc. (2) 39 (1989), O. Kerner, Representation of wild quivers, Proc. Workshop UNM, México 1994 CMS Conf. Proc. 19 (1996), H.Lenzing,.Skowroński, Quasi-tilted algebras of canonical type, Coll. Math. 71 (2) (1996), S. Liu, The connected components of the uslander-reiten quiver of a tilted algebra, J. lgebra 161 (2) (1993) S. Liu, Tilted algebras and generalized standard uslander-reiten components, rch.math. 61 (1993) C. M. Ringel, Report on the Brauer-Thrall conjectures, Proc. ICR II (Ottawa, 1979), Lecture Notes in Math. 831 (1980) , Springer-Verlag. 22. C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math (1984), Springer-Verlag Skowroński, Generalized standard uslander-reiten components without oriented cycles, Osaka J. Math. 30 (1993) Rev. Un. Mat. rgentina, Vol 46-2

15 THE LEFT PRT Skowroński, On artin algebras with almost all indecomposable modules of projective or injective dimension at most one, Central European J. Math. 1 (2003) D. Smith, On generalized standard uslander-reiten components having only finitely many non-directing modules, J. lgebra279 (2004) Ibrahim ssem Département de Mathématiques, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1 ibrahim.assem@usherbrooke.ca Juan Ángel Cappa Instituto de Matemática, Universidad Nacional del Sur, 8000 Bahía Blanca, rgentina jacappa@yahoo.com.ar María Inés Platzeck Instituto de Matemática, Universidad Nacional del Sur, 8000 Bahía Blanca, rgentina platzeck@uns.edu.ar Sonia Trepode Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Funes 3350, Universidad Nacional de Mar del Plata, 7600 Mar del Plata, rgentina strepode@mdp.edu.ar Recibido: 26 de enero de 2006 ceptado: 7 de agosto de 2006 Rev. Un. Mat. rgentina, Vol 46-2