On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

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1 Cent. Eur. J. Math. 12(1) DOI: /s Central European Journal of Mathematics On the number of terms in the middle of almost split sequences over cycle-finite artin algebras Research Article Piotr Malicki 1, José Antonio de la Peña 2, Andrzej Skowroński 1 1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, Toruń, Poland 2 Centro de Investigación en Mathemáticas (CIMAT), Guanajuato, México Received 20 February 2013; accepted 26 April 2013 Abstract: We prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5. MSC: 16G10, 16G70, 16G60 Keywords: Auslander Reiten quiver Almost split sequence Cycle-finite algebra Versita Sp. z o.o. 1. Introduction and the main result Throughout this paper, by an algebra is meant an artin algebra over a fixed commutative artin ring K, which we moreover assume (without loss of generality) to be basic and indecomposable. For an algebra A, we denote by mod A the category of finitely generated right A-modules, by ind A the full subcategory of mod A formed by the indecomposable modules, by Γ A the Auslander Reiten quiver of A, and by τ A and τa 1 the Auslander Reiten translations DTr and TrD, respectively. We do not distinguish between a module in ind A and the vertex of Γ A corresponding to it. The Jacobson radical rad A of mod A is the ideal generated by all nonisomorphisms between modules in ind A, and the infinite radical rad A of mod A is the intersection of all powers rad i A, i 1, of rad A. By a theorem of Auslander [4], rad A = 0 if and only if A is of finite representation type, that is, ind A admits only a finite number of pairwise nonisomorphic modules (see [22] for an alternative proof of this result). On the other hand, if A is of infinite representation type then (rad A ) 2 0, by a theorem proved in [12]. pmalicki@mat.uni.torun.pl jap@cimat.mx skowron@mat.uni.torun.pl 39

2 On the number of terms in the middle of almost split sequences over cycle-finite artin algebras A prominent role in the representation theory of algebras is played by almost split sequences introduced by Auslander and Reiten in [5] (see [7] for general theory and applications). For an algebra A and a nonprojective module X in ind A, there is an almost split sequence 0 τ A X Y X 0, (1) with τ A X a noninjective module in ind A called the Auslander Reiten translation of X. So we may associate to X the numerical invariant α(x) being the number of summands in a decomposition Y = Y 1 Y r of Y into a direct sum of modules in ind A. Then α(x) measures the complication of homomorphisms in mod A with domain τ A X and codomain X. Therefore, it is interesting to study the relation between an algebra A and the values α(x) for all modules X in ind A (we refer to [6, 8, 11, 24, 28, 29, 32, 46, 47] for some results in this direction). In particular, it has been proved by Bautista and Brenner in [8] that, if A is of finite representation type and X a nonprojective module in ind A, then α(x) 4, and if α(x) = 4 then the middle term Y of an almost split sequence in mod A with the right term X admits an indecomposable projective-injective direct summand P, and hence X = P/soc P. In [24] Liu generalized this result by showing that the same holds for any nonprojective module X in ind A over an algebra A provided τ A X has a projective predecessor and X has an injective successor in Γ A, as well as for X lying on an oriented cycle in Γ A. It has been conjectured by Brenner and Butler in [10, Conjecture 1] that α(x) 5 for any nonprojective module X in ind A for an arbitrary tame finite dimensional algebra A over an algebraically closed field K. In fact, it is expected that this also holds for nonprojective indecomposable modules over arbitrary generically tame (in the sense of [13, 14]) artin algebras. The main aim of this paper is to prove the following theorem which gives the affirmative answer for the above conjecture in the case of cycle-finite artin algebras. Main Theorem. Let A be a cycle-finite artin algebra and X be a nonprojective module in ind A, and (1) be the associated almost split sequence in mod A. The following statements hold. (i) α(x) 5. (ii) If α(x) = 5 then Y admits an indecomposable projective-injective direct summand P, and hence X P/ soc(p). We would like to mention that, for finite dimensional cycle-finite algebras A over an algebraically closed field K, the theorem was proved by de la Peña and Takane [29, Theorem 3], by application of spectral properties of Coxeter transformations of algebras and results established in [24]. Let A be an algebra. Recall that a cycle in ind A is a sequence f X 1 f 0 X1... X r r 1 Xr = X 0 of nonzero nonisomorphisms in ind A [36], and such a cycle is said to be finite if the homomorphisms f 1,..., f r do not belong to rad A. Then, following [3, 41], an algebra A is said to be cycle-finite if all cycles in ind A are finite. The class of cycle-finite algebras contains the following distinguished classes of algebras: the algebras of finite representation type, the hereditary algebras of Euclidean type [15, 16], the tame tilted algebras [18, 20, 36], the tame double tilted algebras [33], the tame generalized double tilted algebras [34], the tubular algebras [36], the iterated tubular algebras [30], the tame quasi-tilted algebras [23, 44], the tame generalized multicoil algebras [27], the algebras with cycle-finite derived categories [2], and the strongly simply connected algebras of polynomial growth [42]. On the other hand, frequently an algebra A admits a Galois covering R R/G = A, where R is a cycle-finite locally bounded category and G is an admissible group of automorphisms of R, which allows to reduce the representation theory of A to the representation theory of cycle-finite algebras being finite convex subcategories of R (see [17, 28, 43] for some general results). For example, every finite dimensional selfinjective algebra of polynomial growth over an algebraically closed field admits a canonical standard form A (geometric socle deformation of A) such that A has a Galois covering R R/G = A, where R is a cycle-finite selfinjective locally bounded category and G is an admissible infinite cyclic group of automorphisms of R, the Auslander Reiten quiver Γ A of A is the orbit quiver Γ R /G of Γ R, and the stable Auslander Reiten quivers of A and A are isomorphic (see [37, 45]). Recall also that, a module X in ind A which does not lie on a cycle in ind A is called directing, and its support algebra is a tilted algebra, by a result of Ringel [36]. Moreover, it has been proved independently by Peng and Xiao [31] and Skowroński [39] that the Auslander Reiten quiver Γ A of an algebra A admits at most finitely many τ A -orbits containing directing modules. 40

3 P. Malicki, J.A. de la Peña, A. Skowroński 2. Preliminary results Let H be an indecomposable hereditary algebra and Q H the valued quiver of H. Recall that the vertices of Q H are the numbers 1, 2,..., n corresponding to a complete set S 1, S 2,..., S n of pairwise nonisomorphic simple modules in mod H and there is an arrow from i to j in Q H if Ext 1 H(S i, S j ) 0, and then to this arrow is assigned the valuation ( dimendh (S j ) Ext 1 H(S i, S j ), dim EndH (S i ) Ext 1 H(S i, S j ) ). Recall also that the Auslander Reiten quiver Γ H of H has a disjoint union decomposition of the form Γ H = P(H) R(H) Q(H), where P(H) is the preprojective component containing all indecomposable projective H-modules, Q(H) is the preinjective component containing all indecomposable injective H-modules, and R(H) is the family of all regular components of Γ H. More precisely, we have: if Q H is a Dynkin quiver, then R(H) is empty and P(H) = Q(H); if Q H is a Euclidean quiver, then P(H) = ( N)Q op H, Q(H) = NQ op H and R(H) is a strongly separating infinite family of stable tubes; if Q H is a wild quiver, then P(H) = ( N)Q op H, Q(H) = NQ op H and R(H) is an infinite family of components of type ZA. Let T be a tilting module in mod H and B = End H (T ) the associated tilted algebra. Then the tilting H-module T determines the torsion pair (F(T ), T(T )) in mod H, with the torsion-free part F(T ) = {X mod H : Hom H (T, X) = 0} and the torsion part T(T ) = { X mod H : Ext 1 H(T, X) = 0 }, and the splitting torsion pair (Y(T ), X(T )) in mod B, with the torsion-free part Y(T ) = { Y mod B : Tor B 1 (Y, T ) = 0 } and the torsion part X(T ) = {Y mod B : Y B T = 0}. Then, by the Brenner Butler theorem, the functor Hom H (T, ): mod H mod B induces an equivalence of T(T ) with Y(T ), and the functor Ext 1 H(T, ): mod H mod B induces an equivalence of F(T ) with X(T ) (see [9, 18]). Further, the images Hom H (T, I) of the indecomposable injective modules I in mod H via the functor Hom H (T, ) belong to one component C T of Γ B, called the connecting component of Γ B determined by T, and form a faithful section T of C T, with T the opposite valued quiver Q op H of Q H. Recall that a full connected valued subquiver Σ of a component C of Γ B is called a section (see [1, (VIII.1)]) if Σ has no oriented cycles, is convex in C, and intersects each τ B -orbit of C exactly once. Moreover, the section Σ is faithful provided the direct sum of all modules lying on Σ is a faithful B-module. The section T of the connecting component C T of Γ B has the distinguished property: it connects the torsion-free part Y(T ) with the torsion part X(T ), because every predecessor in ind B of a module Hom H (T, I) from T lies in Y(T ) and every successor of τb Hom H(T, I) in ind B lies in X(T ). We note that, by a result proved in [26, 38], an algebra A is a tilted algebra if and only if Γ A admits a component C with a faithful section such that Hom A (X, τ A Y ) = 0 for all modules X and Y from. We refer also to [19] for another characterization of tilted algebras involving short chains of modules. The following proposition follows from the structure of the Auslander Reiten quiver of a hereditary algebra of Euclidean type and the shape of Euclidean graphs (see [15]). Proposition 2.1. Let H be a hereditary algebra of Euclidean type. Then, for any nonprojective indecomposable module X in mod H, we have α(x) 4. An essential role in the proof of the main theorem will be played by the following theorem. Theorem 2.2. Let A be a cycle-finite algebra, C a component of Γ A, and D be an acyclic left stable full translation subquiver of C which is closed under predecessors. Then there exists a hereditary algebra H of Euclidean type and a tilting module T in mod H without nonzero preinjective direct summands such that for the associated tilted algebra B = End H (T ) the following statements hold. 41

4 On the number of terms in the middle of almost split sequences over cycle-finite artin algebras (i) B is a quotient algebra of A. (ii) The torsion-free part Y(T ) C T of the connecting component C T of Γ B determined by T is a full translation subquiver of D which is closed under predecessors in C. (iii) For any indecomposable module N in D, we have α(n) 4. Proof. Since A is a cycle-finite algebra, every acyclic module X in Γ A is a directing module in ind A. Hence D consists entirely of directing modules. Moreover, it follows from [31, Theorem 2.7] and [39, Corollary 2], that D has only finitely many τ A -orbits. Then, applying [25, Theorem 3.4], we conclude that there is a finite acyclic valued quiver such that D contains a full translation subquiver Γ which is closed under predecessors in C and is isomorphic to the translation quiver N. Therefore, we may choose in Γ a finite acyclic convex subquiver such that Γ consists of the modules τa mx with m 0 and X indecomposable modules lying on. Let M be the direct sum of all indecomposable modules in C lying on the chosen quiver. Let I be the annihilator ann A (M) = {a A : Ma = 0} of M in A, and B = A/I the associated quotient algebra. Then I = ann A (Γ) (see [38, Lemma 3]) and consequently Γ consists of indecomposable B-modules. Clearly, B is a cycle-finite algebra, as a quotient algebra of A. Now, using the fact that Γ N and consists of directing B-modules, we conclude that rad B (M, M) = 0 and Hom B (M, τ B M) = 0. Then, applying [40, Lemma 3.4], we conclude that H = End B (M) is a hereditary algebra and the quiver Q H of H is the dual valued quiver op of. Further, since M is a faithful B-module with Hom B (M, τ B M) = 0, we conclude that pd B M 1 and Ext 1 B(M, M) = D Hom B (M, τ B M) = 0 (see [1, Lemma VIII.5.1 and Theorem IV.2.13]). Moreover, it follows from the definition of M that, for any module Z ( in ind B with Hom B (M, Z) 0 and not on, we have Hom B τ 1 B M, Z) 0. Since M is a faithful module in mod B ( there is a monomorphism B M s for some positive integer s. Then rad B (M, M) = 0 implies Hom B τ 1 B M, B) = 0, and consequently id B M 1. Applying now [35, Lemma 1.6] we conclude that M is a tilting B-module. Further, applying the Brenner Butler theorem (see [1, Theorem VI.3.8]), we conclude that M is a tilting module in mod H op and B = End H op(m). Since H is a hereditary algebra, T = D(M) is a tilting module in mod H with B = End H (T ), and consequently B is a tilted algebra of type Q H = op. Moreover, the translation quiver Γ is the torsion-free part Y(T ) C T of the connecting component C T of Γ B determined by the tilting H-module T (see [1, Theorem VIII.5.6]). Observe that then Y(T ) C T is the image Hom H (T, Q(H)) of the preinjective component Q(H) of Γ H via the functor Hom H (T, ): mod H mod B. In particular, we conclude that H is of infinite representation type (Q H is not a Dynkin quiver) and C T does not contain a projective module, and hence T is without nonzero preinjective direct summands (see [1, Proposition VIII.4.1]). Finally, we prove that Q H = op is a Euclidean quiver. Suppose that Q H is a wild quiver. Since T has no nonzero preinjective direct summands, it follows from [21] that Γ B admits an acyclic component Σ with infinitely many τ B -orbits, with the stable part ZA, contained entirely in the torsion-free part Y(T ) of mod B. Since B is a cycle-finite algebra, Σ consists of directing B-modules, and hence Γ B contains infinitely many τ B -orbits containing directing modules, a contradiction. Therefore, Q H is a Euclidean quiver and B is a tilted algebra of Euclidean type Q H = op. This finishes proof of the statements (i) and (ii). In order to prove (iii), consider a module N in D and an almost split sequence 0 τ A N E N 0 in mod A with the right term N. Since D is left stable and closed under predecessors in C, we have in mod A almost split sequences 0 τ m+1 A N τ m A E τ m A N 0 for all nonnegative integers m. In particular, there exists a positive integer n such that 0 τ n+1 A N τ n A E τn A N 0 is an exact sequence in the additive category add(y(t ) C T ) = add(γ). Since Y(T ) C T = Hom H (T, Q(H)), this exact sequence is the image via the functor Hom H (T, ): mod H mod B of an almost split sequence 0 τ H U V U 0 with all terms in the additive category add(q(h)) of Q(H). Then, applying Proposition 2.1, we conclude that α(n) = α ( τ n A N) = α ( τ n B N) = α(u) 4. 42

5 P. Malicki, J.A. de la Peña, A. Skowroński 3. Proof of Main Theorem We will use the following results proved by Liu in [24, Theorem 7, Proposition 8, Lemma 6 and its dual]. Theorem 3.1. Let A be an algebra, and let 0 τ A X r Y i X 0 (2) be an almost split sequence in mod A with Y 1,..., Y r from ind A. Assume that one of the following conditions holds. (i) τ A X has a projective predecessor and X has an injective successor in Γ A. (ii) X lies on an oriented cycle in Γ A. Then r 4, and r = 4 implies that one of the modules Y i is projective-injective, whereas the others are neither projective nor injective. Proposition 3.2. Let A be an algebra, and let (2) be an almost split sequence in mod A with r 5 and Y 1,..., Y r from ind A. Then the following statements hold. (i) If there is a sectional path from τ A X to an injective module in Γ A, then τ A X has no projective predecessor in Γ A. (ii) If there is a sectional path from a projective module in Γ A to X, then X has no injective successor in Γ A. We are now in position to prove the main result of the paper. Let A be a cycle-finite algebra, and let (2) be an almost split sequence in mod A with Y 1,..., Y r from ind A, and let C be the component of Γ A containing X. Assume r 5. We claim that then r = 5, one of the modules Y i is projective-injective, whereas the others are neither projective nor injective. Since r 5, it follows from Theorem 3.1 that τ A X has no projective predecessor or X has no injective successor in Γ A. Assume that τ A X has no projective predecessor in Γ A. We claim that then one of the modules Y i is projective. Suppose it is not the case. Then for any nonnegative integer m we have in mod A an almost split sequence 0 τa m+1 X r τa m Y i τa m X 0 with r 5 and τ m A Y 1,..., τ m A Y r from ind A, because τ A X has no projective predecessor in Γ A. Moreover, it follows from Theorem 3.1, that τ m A X, m 0, are acyclic modules in Γ A. Then it follows from [25, Theorem 3.4] that the modules τ m A X, m 0, belong to an acyclic left stable full translation subquiver D of C which is closed under predecessors. But then the assumption r 5 contradicts Theorem 2.2 (iii). Therefore, one of the modules Y i, say Y r is projective. Observe now that the remaining modules Y 1,..., Y r 1 are noninjective. Indeed, since Y r is projective, we have l(τ A X) < l(y r ) and consequently r 1 l(y i) < l(x). Further, Y r is a projective predecessor of X in Γ A, and hence, applying Proposition 3.2 (ii), we conclude that X has no injective successors in Γ A. We claim that Y r is injective. Indeed, if it is not the case, we have in mod A almost split sequences 0 τa m+1 X r τa m Y i τa m X 0 for all nonnegative integers m. Then, applying the dual of Theorem 2.2, we obtain a contradiction with r 5. Thus Y r is projective-injective. Observe that then the modules Y 1,..., Y r 1 are nonprojective, because Y r injective forces the inequalities l(x) < l(y r ) and r 1 l(y i) < l(τ A X). 43

6 On the number of terms in the middle of almost split sequences over cycle-finite artin algebras Finally, since τ A X has no projective predecessor in Γ A, we have in mod A almost split sequences 0 τa m+1 X r 1 τa m Y i τa m X 0 for all positive integers m. Applying Proposition 3.2 again, we conclude (as in the first part of the proof) that r 1 4, and hence r 5. Therefore, α(x) = r = 5, one of the modules Y i is projective-injective, whereas the others are neither projective nor injective. Moreover, if Y i is a projective-injective module, then X = Y i /soc Y i. Acknowledgements This work was completed with the support of the research grant DEC-2011/02/A/ST1/00216 of the Polish National Science Center and the CIMAT Guanajuato, México. References [1] Assem I., Simson D., Skowroński A., Elements of the Representation Theory of Associative Algebras, 1, London Math. Soc. Stud. Texts, 65, Cambridge University Press, Cambridge, 2006 [2] Assem I., Skowroński A., Algebras with cycle-finite derived categories, Math. Ann., 1988, 280(3), [3] Assem I., Skowroński A., Minimal representation-infinite coil algebras, Manuscripta Math., 1990, 67(3), [4] Auslander M., Representation theory of artin algebras II, Comm. Algebra, 1974, 1(4), [5] Auslander M., Reiten I., Representation theory of artin algebras III. Almost split sequences, Comm. Algebra, 1975, 3(3), [6] Auslander M., Reiten I., Uniserial functors, In: Representation Theory, 2, Ottawa, August 13 25, 1979, Lecture Notes in Math., 832, Springer, Berlin, 1980, 1 47 [7] Auslander M., Reiten I., Smalø S.O., Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., 36, Cambridge University Press, Cambridge, 1995 [8] Bautista R., Brenner S., On the number of terms in the middle of an almost split sequence, In: Representations of Algebras, Puebla, August 4 8, 1980, Lecture Notes in Math., 903, Springer, Berlin, 1981, 1 8 [9] Brenner S., Butler M.C.R., Generalizations of the Bernstein Gel fand Ponomarev reflection functors, In: Representation Theory, II, Ottawa, August 13 25, 1979, Lecture Notes in Math., 832, Springer, Berlin, 1980, [10] Brenner S., Butler M.C.R., Wild subquivers of the Auslander Reiten quiver of a tame algebra, In: Trends in the Representation Theory of Finite-Dimensional Algebras, Seattle, July 20 24, 1997, Contemp. Math., 229, American Mathematical Society, Providence, 1998, [11] Butler M.C.R., Ringel C.M., Auslander Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra, 1987, 15(1-2), [12] Coelho F.U., Marcos E.N., Merklen H.A., Skowroński A., Module categories with infinite radical square zero are of finite type, Comm. Algebra, 1994, 22(11), [13] Crawley-Boevey W., Tame algebras and generic modules, Proc. London Math. Soc., 1991, 63(2), [14] Crawley-Boevey W., Modules of finite length over their endomorphism rings, In: Representations of Algebras and Related Topics, Tsukuba, 1990, London Math. Soc. Lecture Note Series, 168, Cambridge University Press, Cambridge, 1992, [15] Dlab V., Ringel C.M., Indecomposable Representations of Graphs and Algebras, Mem. Amer. Math. Soc., 173, American Mathematical Society, Providence, 1976 [16] Dlab V., Ringel C.M., The representations of tame hereditary algebras, In: Representation Theory of Algebras, Philadelphia, May 24 28, 1976, Lecture Notes in Pure Appl. Math., 37, Marcel Dekker, New York, 1978, [17] Dowbor P., Skowroński A., Galois coverings of representation-infinite algebras, Comment. Math. Helv., 1987, 62(2),

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