Cluster Tilting for Representation-Directed Algebras

Transcription

1 U.U.D.M. Report 8: Cluster Tilting for Representation-Directed Algebras Laertis Vaso Filosofie licentiatavhandling i matematik som framläggs för offentlig granskning den 9 november 8, kl., sal, Ångströmlaboratoriet, Uppsala Department of Mathematics Uppsala University

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3 CLUSTER TILTING FOR REPRESENTATION-DIRECTED ALGEBRAS This licentiate dissertation consists of the following two papers, referred to by their roman numerals. I. L. Vaso, n-cluster tilting subcategories of representation-directed algebras, J. Pure Appl. Algebra 8), Elsevier. Reproduced with permission. All rights reserved. II. Vaso, L. Gluing of n-cluster tilting subcategories for representation-directed algebras, preprint. Introduction An associative) algebra over a field K is a unital ring which is also a K-vector space such that the ring multiplication is compatible with multiplication with scalars. From the point of view of representation theory, one way of studying a finite-dimensional algebra Λ is to describe the category mod Λ of finitely generated Λ-modules. Although describing mod Λ can be very hard in general, there are some cases which are easier to work with. One such case is when Λ is representationfinite, that is when there exist finitely many indecomposable Λ-modules up to isomorphism. In this case, classical Auslander-Reiten theory, developed by Auslander and Reiten in the late 97 s, gives a combinatorial approach to describiling all Λ-modules and all morphisms between them. Another case where we can obtain some results is when Λ is hereditary, that is when every submodule of a projective module is projective. Expressed in another language, the hereditary algebras are those which have global dimension at most. In particular, the module categories of algebras which are both representation-finite and hereditary are very well understood. More recently, Iyama introduced a higher-dimensional analogue of Auslander- Reiten theory for algebras of arbitrary global dimension, based on replacing the focus from the whole module category mod Λ to a subcategory C mod Λ with suitable homological properties called an n-cluster tilting subcategory [Iya], [Iya7], [Iya8]). For n > it is unknown if there exists an n-cluster tilting subcategory C containing infinitely many nonisomorphic indecomposable modules and so all known examples can be thought of as higher dimensional versions of representation-finite algebras. When an n-cluster tilting subcategory C exists, one can use techniques analogous to classical Auslander-Reiten to study C. If gl. dim Λ = d, then we have that n > d implies that Λ is semisimple, and so we can restrict to the case n d. If in particular there exists a d-cluster tilting subcategory C, then C can be thought of as a generalization of the module category of a hereditary algebra. Note that the existence of an n-cluster tilting subcategory, for any n d, is far from guaranteed. Moreover, there are many known examples in the literature concerning the case n = d, while there are very few examples known when n < d. This is the main motivation for the work in this dissertation. The main aim of this dissertation is to find examples of n-cluster tilting subcategories for algebras of global dimension d, for any possible pair n, d) with n d. The first step to managing this is to find new examples of n-cluster tilting subcategories in general. Since the problem of the existence of an n-cluster tilting subcategory

4 CLUSTER TILTING FOR REPRESENTATION-DIRECTED ALGEBRAS for an arbitrary algebra Λ is hopeless, we restrict ourselves to a particular class of algebras called representation-directed algebras. These are algebras for which there are no cycles of nonzero nonisomorphisms between indecomposable Λ-modules. In particular these are representation-finite and hence they are quite well understood. The second step is to extrapolate from these new examples a way that allows us to find even more examples but where the global dimension can be easily computed. In paper I we give a characterization of n-cluster tilting subcategories for representation-directed algebras based on the so-called n-auslander-reiten translations. This characterization allows us to reduce the problem of existence of n-cluster tilting subcategories for representation-directed algebras to classical Auslander-Reiten theory. We apply this characterization to classify acyclic Nakayama algebras with homogeneous relations which admit an n-cluster tilting subcategory in terms of a simple numerical equation. Furthermore, using this classification we also find all Nakayama algebas of global dimension d < which admit a d-cluster tilting subcategory. In paper II we introduce a method to iteratively construct n-cluster tilting subcategories for representation-directed algebras. The first step in doing this is defining the process of gluing of representation-directed algebras. This process takes us inputs two representation-directed algebras A and B, a uniserial projective A- module P and a uniserial injective B-module I satisfying certain conditions, and it outputs a new representation-directed algebra Λ := B P I A, with the advantage that the representation theory of Λ is completely described by the representation theories of A and B. The second step is introducing the concept of n-fractured subcategories. These are defined by generalizing the characterization of n-cluster tilting subcategories for representation-directed algebras which we obtained in paper I. Under reasonable compatibility conditions, we show that if A and B admit n-fractured subcategories, then the glued algebra Λ = B P I A also admits an n- fractured subcategory. Under some conditions, repeatingly gluing algebras results in an algebra with an honest n-cluster tilting subcategory. One such case is when the n-fractured subcategories of A and B are actual n-cluster tilting subcategories and the modules P and I are simple. In this case, computing the global dimension of Λ is easy. As a result, we show that if n is odd and d n then there exists an algebra admitting an n-cluster tilting subcategory and having global dimension d. We show the same result if n is even and d is odd or d n. It should be noted that except for providing the aforementioned examples where the interest lies in the pair n, d), the methods developed in this licentiate dissertation can be used to provide examples of algebras with many other interesting properties that also admit n-cluster tilting subcategories. One such example is the existence of a bound quiver algebra Λ = KQ/I such that Q has an arbitrary number of sinks and sources and Λ admits a -cluster tilting subcategory. Acknowledgements I am grateful to my supervisor Martin Herschend for the constant help, suggestions and support during the preparation of this work. References [Iya] Osamu Iyama. Auslander correspondence. Adv. Math. 7), no., 8, May, math/v. [Iya7] O. Iyama. Higher-dimensional Auslander Reiten theory on maximal orthogonal subcategories. Advances in Mathematics, ):, 7. [Iya8] O. Iyama. Auslander-Reiten theory revisited. In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pages Eur. Math. Soc., Zürich, 8.

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6 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) Journal of Pure and Applied Algebra ) Contents lists available at ScienceDirect Journal of Pure and Applied Algebra n-cluster tilting subcategories of representation-directed algebras Laertis Vaso Department of Mathematics, Uppsala University, P.O. Box 8, 7 Uppsala, Sweden a r t i c l e i n f o a b s t r a c t Article history: Received 8 August 7 Received in revised form February 8 Available online xxxx Communicated by S. Koenig We give a characterization of n-cluster tilting subcategories of representationdirected algebras based on the n-auslander Reiten translations. As an application we classify acyclic Nakayama algebras with homogeneous relations which admit an n-cluster tilting subcategory. Finally, we classify Nakayama algebras of global dimension d < which admit a d-cluster tilting subcategory. 8 Elsevier B.V. All rights reserved.. Introduction In representation theory of finite dimensional algebras, one aims to understand the modules over an algebra and the homomorphisms between them. In the case of a representation-finite algebras, classical Auslander Reiten theory gives a complete picture of the module category, see for example []. In Osamu Iyama s higher-dimensional Auslander Reiten theory, introduced in [] and [7], one replaces the module category with a subcategory with suitable homological properties called an n-cluster tilting subcategory, where n is a positive integer. If an n-cluster tilting subcategory exists, it behaves similarly to the module category from the perspective of Auslander Reiten theory. In particular, it contains all the projective and injective modules and there are many higher-dimensional analogues of classical notions. For instance, n-almost split sequences and the n-auslander Reiten translations τ n and τn become almost split sequences and the Auslander Reiten translations τ and τ when n =. If an n-cluster tilting subcategory admits an additive generator M, then M is called an n-cluster tilting module and we say that the algebra is weakly n-representation-finite. If moreover n is equal to the global dimension d of the algebra, the d-cluster tilting subcategory is unique and we say that the algebra is d-representation-finite. In Theorem. of [] it is shown that d-representation-finite algebras play the role of hereditary representation-finite algebras in higher-dimensional Auslander Reiten theory. address: laertis.vaso@math.uu.se. -9/ 8 Elsevier B.V. All rights reserved.

7 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Since the existence of an n-cluster tilting subcategory is far from guaranteed, it is natural to ask under which conditions an n-cluster tilting subcategory exists. We study this question in the case of representationdirected algebras and give the following characterization. Theorem. Assume Λ is a representation-directed algebra and let C be a full subcategory of modλ, closed under direct sums and summands. Denote by C P and C I the sets of isomorphism classes of indecomposable nonprojective respectively noninjective Λ-modules in C. Then C is an n-cluster tilting subcategory if and only if the following conditions hold: ) Λ C, ) τ n and τ n induce mutually inverse bijections τ n C P C I, τ n ) Ω i M is indecomposable for all M C P and < i < n, ) Ω i N is indecomposable for all N C I and < i < n. Remark. Let us make three remarks about Theorem : a) ) and ) are known to be necessary for any finite dimensional algebra [7], Theorem.8). Moreover, ) and ) are also necessary for any finite dimensional algebra by Corollary.. b) Let C be an n-cluster tilting subcategory of modλ where Λis representation-directed, and M Cbe indecomposable. By representation-directedness, ) ) imply that τn i M =and τnm j =for i and j large enough. Then ) implies that M = τn N P for some projective indecomposable module P and ) r some N. Using ) and ) we conclude that C =add r τn Λ). c) We do not know of any examples where conditions ) and ) hold and either of conditions ) and ) fails. As an application, we characterize the acyclic Nakayama algebras with homogeneous relations which admit an n-cluster tilting subcategory. Theorem. Let Q m be the quiver a m a m a m a a Q m : m m m. Then KQ m /rad KQ m ) l admits an n-cluster tilting subcategory if and only if l =and m = nk + for some k or n is even and m = n l + + knl l +) for some k. Cyclic Nakayama algebras with homogeneous relations which admit n-cluster tilting subcategories are classified by Darpö and Iyama in []. The case l =in Theorem was first considered by Jasso in [8], Proposition.. Moreover Iyama and Opperman completely classify -representation finite acyclic Nakayama algebras in [], Theorem.. It turns out that d-representation-finite Nakayama algebras arise only as acyclic Nakayama algebras with homogeneous relations. Therefore, we also give a complete classification of d-representation-finite Nakayama algebras.

8 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Theorem. Let Λ be a Nakyama algebra of global dimension d <. The following are equivalent. i) Λ is d-representation-finite. ii) Λ = KQ m /rad KQ m ) l and d is even or l =. iii) Λ = KQ m /rad KQ m ) l and l m or l =. Then d = m l. Acknowledgments. The author wishes to thank his advisor Martin Herschend for the constant support and help during the preparation of this article. The author would also like to thank an anonymous referee for the helpful suggestions, in particular for suggesting the use of Theorem. in [], as well as Steffen Oppermann for suggesting the proof of Proposition.. The author was funded by Uppsala University.. Preliminaries Throughout the paper, K will be a field and Λa finite dimensional unital associative K-algebra. We denote by modλ the category of finitely generated right Λ-modules and in the following we say module instead of right Λ-module. We will denote by d the global dimension of Λ and by D the duality Hom Λ, K). Recall that if M is an indecomposable nonprojective module, then there exists an almost split sequence τm E M in modλ and, similarly, if N is an indecomposable noninjective module, then there exists an almost split sequence N F τ N where τ and τ are the Auslander Reiten translations. In particular, we have the Auslander Reiten formulas Ext ΛM,N) = DHom Λ τ N,M) = DHom Λ N,τM). For further details we refer to chapter IV in []. Let X modλ. We will denote by ΩX the syzygy of X, that is the kernel of P X, where P is the projective cover of X and by Ω X the cosyzygy of X, that is the cokernel of X I where I is the injective hull of X. Note that ΩX and Ω X are unique up to isomorphism. Following [7], we denote by τ n and τ n the n-auslander Reiten translations defined by τ n X = τω n X)and τ n X = τ Ω n ) X). In this paper, all subcategories considered will be full and closed under direct sums and summands. Let C be a subcategory of modλ. A morphism f : M X with X Cis called a left C-approximation if _ f :Hom Λ X, X ) Hom Λ M, X )is surjective for any X C; if moreover for any M modλ there exists a left C-approximation, we say that C is covariantly finite. Dually we define a right C-approximation and a contravariantly finite subcategory. If C is both covariantly and contravariantly finite, we say that C is functorially finite. Functorially finite subcategories were first introduced in []. A morphism f : M N in modλ will be called left minimal if whenever f is isomorphic to M N N, we have N =; if f is also a left C-approximation, we will say that f is a minimal left approximation. Dually we define right minimal morphisms and minimal right approximations. It is well-known that minimal approximations are unique up to isomorphism. For the rest of the paper n will be a positive integer. The following definition is due to Iyama [7], []). f )

9 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Definition.. We call a subcategory C of modλ an n-cluster tilting subcategory if it is functorially finite and where C n C = C n = n C, := {X modλ Ext i ΛC,X) = for all <i<n}, n C := {X modλ Ext i ΛX, C) = for all <i<n}. Our main result is inspired by the following necessary condition for n-cluster tilting subcategories due to Iyama. Proposition.. [7], Theorem.8) Let C be an n-cluster tilting subcategory of modλ. Then τ n and τ n induce mutually inverse bijections τ n C P C I. τ n For M modλ we denote by addm the subcategory of modλ containing all modules isomorphic to direct summands of finite direct sums of M. Note that addm is always functorially finite. Hence addm is an n-cluster tilting subcategory if and only if addm =addm n = n addm. In that case we will call M an n-cluster tilting module. Observe that if Λis representation-finite, then any additive subcategory of modλ is of the form addm for some M modλ. Moreover it is clear from the definition that any n-cluster tilting subcategory contains Λand DΛ. If there exists an n-cluster tilting subcategory with n > d, then Ext i ΛDΛ, Λ) = for all i d <, so Λ is semisimple. Therefore, when Λis not semisimple, we have n d. Observe also that modλ is the unique -cluster tilting subcategory of modλ so in the following we assume n d. To keep track of the Ext Λ -vanishing conditions in Definition., it is useful to consider the following sets of modules. Definition.. Let X modλ. Define the left Ext n Λ-)support of X, denoted LS n X), to be LS n X) ={Y modλ <i<n:ext i ΛX, Y ) }. Similarly, define the right Ext n Λ-)support of X, denoted RS n X), to be RS n X) ={Y modλ <i<n:ext i ΛY,X) }. A path from M to M t in modλ is a sequence of nonzero nonisomorphisms f k : M k M k between indecomposable modules M, M,, M t for t. We define the relation M N on indecomposable modules M and N as the transitive hull of Hom Λ M, N). Then M N if and only if M = N or there is a path from M to N. Λ is called representation-directed if there is no path from M to N in modλ with M = N. Note that representation-directed algebras are representation-finite; for a proof and more details on paths and representation-directed algebras we refer to []. Note also that in this case M N and N M implies M = N. Therefore, we will write M <Nif M N and M N. In the following lemma we collect some basic results that will be used throughout.

10 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Lemma.. Let M, N modλ be indecomposable and n. Then, i) M is projective if and only if LS n M) =, i ) N is injective if and only if RS n N) =, ii) N LS n M) if and only if M RS n N), iii) if < k<nand Ω k M is nonprojective, then Ω k M LS n M), iii ) if < k<nand Ω k ) N is noninjective, then Ω k N RS n N), iv) if X is an indecomposable summand of ΩM, then X M, iv ) if Y is an indecomposable summand of Ω N, then N Y. If in addition Λ is representation-directed, v) M N and N M imply M = N, vi) if τm, then τm < M, vi ) if τ N, then N <τ N, vii) if X is an indecomposable summand of ΩM, then X τm, vii ) if Y is an indecomposable summand of Ω N, then τ N Y. Proof. Statements i), i ) and ii) follow immediately from the definitions. Statement iii) follows by noticing Ext k ΛM, Ω k M) = Ext ΛΩ k M, Ω k M). Statement iv) follows because if P is the projective cover of M then there exists some indecomposable summand P of P with X P and since P M, it follows that X M. Statement v) follows since otherwise there are paths from M to N and from N to M. Statement vi) follows since there is a path from τm to M and τm M for representation-directed algebras. Statement vii) follows because if P is the projective cover of M and ΩM ι P p M is a short exact sequence, then we have DHom Λ X, τm) = Ext ΛM,X) = Hom Λ ΩM,X)/Im_ ι), where the last inequality follows since the canonical projection ΩM X does not factor through ι, as it is a radical morphism. Statements iii ), iv ), vi ) and vii ) follow similarly to iii), iv), vi) and vii) respectively.. n-cluster tilting subcategories of representation-directed algebras The aim of this section is to give a proof of Theorem. Subsection. collects some preliminary results while subsection. uses those results to provide the proof. Before we proceed, let us give an outline of the strategy of the proof. Theorem asserts that C is an n-cluster tilting subcategory if and only if conditions ) ) in that theorem hold. As we mentioned in Remark, conditions ) and ) are known to be necessary without assuming Λto be representation-directed. Proposition., and its Corollaries. and. connect conditions ) and ) to Ext Λ -vanishing. In particular, Corollary. shows that conditions ) and ) are also necessary without assuming representation-directedness. This establishes one direction of the equivalence in Theorem. For the other direction, first notice that as we mentioned in Remark, if Λis a representation-directed ) r algebra, the only candidate for an n-cluster tilting subcategory is C =add r τn Λ). By applying n-auslander Reiten duality Theorem. in []) and in particular Proposition. we show that if the conditions ) ) hold, then C C n = n C. To show C n Cwe take X C n and consider the

11 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) sequence τnx. k We show that by directedness we have that τn l X is projective for some l and, using Lemma., we show that X = τn l ) τn l X C. We proceed to show the technical results needed to justify the steps outlined in the above strategy... Preparation In this section we present the results that will be used in the proof of Theorem. In the first part of this section we only assume that Λ is a finite dimensional algebra. In fact we show that the conditions ) ) in Theorem are necessary without the use of the assumption of representation-directedness. In the second part of this section, we additionally assume that Λ is representation-directed to prove some technical results which are used in the reverse implication. We begin by giving a necessary condition for the existence of an n-cluster tilting subcategory. We thank Steffen Oppermann for suggesting the proof of the following result. Proposition.. Let Λ be a finite dimensional algebra. a) Let M modλ be indecomposable and nonprojective and let P be the projective cover of M. If ΩM is decomposable, then Ext ΛM, P ). b) Let N modλ be indecomposable and noninjective and let I be the injective hull of N. If Ω N is decomposable, then Ext ΛI, N). Proof. We only prove a); b) is proved similarly. Assume towards a contradiction that ΩM = X X with X and X in particular, M is not projective) and Ext ΛM, P ) =. Consider the short exact sequence ΩM ι P p M ; by applying Hom Λ, P )we get the long exact sequence Hom Λ M,P) _ p Hom Λ P, P) _ ι Hom Λ ΩM,P) Ext ΛM,P). By our assumption, Ext ΛM, P ) =so that _ ι is surjective. Hence ι is a left addp )-approximation. Moreover, it is minimal left for if P P is a direct sum decomposition of P such that ι is isomorphic to ι ) ΩM P P, then P is a direct summand of M, and since M is not projective and indecomposable, it follows that P =. Now let f : X P and f : X P be minimal left addp )-approximations. Then f f is a minimal left addp )-approximation of X X, and therefore it is isomorphic to ι as a map. As P is the projective cover of M, we have that f and f are both monomorphisms but not isomorphisms. Hence coker f and coker f. But then M =cokerf coker f contradicts M being indecomposable. We have two immediate corollaries. Corollary.. Let Λ be a finite dimensional algebra. Then a) If M modλ is indecomposable nonprojective such that LS n M) = RS n τ n M), then Ω i M and Ω i τ n M are indecomposable for all < i < n. b) If N modλ is indecomposable noninjective such that RS n N) = LS n τ n N), then Ω i N and Ω i τ n N are indecomposable for all < i < n. Proof. We only prove a); b) is proved similarly. Since M is nonprojective, LS n M) by Lemma.i). Then, by assumption, we have that RS n τ n M) and so τ n M is nonzero and even noninjective Lemma.i )). In particular, Ω i M for < i < n.

12 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P.7 -) L. Vaso / Journal of Pure and Applied Algebra ) 7 Let us now prove that Ω i τ n M for < i < n. Assume towards a contradiction that there is < i < n, such that Ω i τ n M =, while Ω i ) τ n M. In particular <i, since τ n M is noninjective. Then Ω i ) τ n M is injective and nonzero, so that Ω i ) τ n M is noninjective. By Lemma.iii ), Ω i ) τ n M RS n τ n M) = LS n M), which contradicts Ω i ) τ n M being injective. Next, assume towards a contradiction that Ω i M is decomposable for some <i <nminimal. Then Ω i M is indecomposable nonprojective, since Ω i M. Let P be the projective cover of Ω i M. By Proposition.a), Ext ΛΩ i M, P ) and so Ext i ΛM, P ). But then P LS n M) = RS n τ n M), which contradicts P being projective. Hence Ω i M is indecomposable for < i < n. Similarly, using Proposition.b) we prove that Ω i τ n M is indecomposable for < i < n. Corollary.. Let Λ be a finite dimensional algebra and C be an n-cluster tilting subcategory of modλ. Then a) Ω i M is indecomposable for all M C P and < i < n, b) Ω i N is indecomposable for all N C I and < i < n. Proof. This follows from Theorem. in [] and Corollary.. More precisely, in the case of a) we have from the aforementioned theorem that LS n M) = RS n τ n M)and so Corollary. implies then that Ω i M is indecomposable. The case b) is similar. Corollary. gives a necessary condition for a subcategory C to be n-cluster tilting: the syzygy and a cosyzygy of an indecomposable module in C must be either indecomposable or. In particular, we have now proved that if C is an n-cluster tilting subcategory then ) ) in Theorem hold, since ) is immediate by the definition, ) follows from Proposition. and ) and ) from Corollary.. More generally, we have shown that conditions ) ) being necessary is true for any finite dimensional algebra, since we have not used representation-directedness yet. In the rest of this section we develop the tools needed for the reverse implication. In the next proposition we apply the n-auslander Reiten duality to gain control of the left and right support of an indecomposable module. Proposition.. Let Λ be a finite dimensional algebra. a) Let N modλ be indecomposable. Then if RS n N) LS n τ n N), there exists an indecomposable injective module I such that I RS n N). b) Let M modλ be indecomposable. Then if LS n M) RS n τ n M), there exists an indecomposable projective module P such that P LS n M). Proof. This follows from Theorem. in []. More precisely, in the case a) we have from the aforementioned theorem that if we assume DΛ / RS n N), then Ext n i Λ Z, N) = DExt i Λτn N, Z) for any Z modλ, which in turn implies RS n N) = LS n τn N). The case b) is similar. From now on we will additionally assume that Λ is representation-directed. We begin with the following easy lemma. Lemma.. Let Λ be a representation-directed algebra. a) Let N, X modλ be indecomposable such that X RS n N) and τ N X. Then τ N < X and N<X. b) Let M, Y modλ be indecomposable such that Y LS n M) and Y τm. Then Y <τmand Y <M.

13 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P.8 -) 8 L. Vaso / Journal of Pure and Applied Algebra ) Proof. We only prove a); b) is proved similarly. Since X RS n N), there exists < j<nsuch that Ext j Λ X, N). Using dimension shift and the Auslander Reiten formula we have DHom Λ τ Ω j ) N,X) = Ext ΛX, Ω j ) N) = Ext j Λ X, N). If j =, we have N <τ N<Xsince τ N X and the result is proved. Now that assume j. Then for some indecomposable summand Y of τ Ω j ) N we have Y X. Then τy is an indecomposable summand of Ω j ) N. Then, by Lemma.iv ), there exists an indecomposable summand Z of Ω N such that Z τy. By Lemma.vii ), we have τ N Z. All together, we have N<τ N Z τy < Y X, as required. Next we give sufficient conditions for τ n τ n N = N as well as τ n τ n M = M in the case of representationdirected algebras, to be used in the proof of Theorem. Lemma.. Let Λ be a representation-directed algebra. a) Let N modλ be indecomposable noninjective. Then RS n N) = LS n τ n N) implies τ n τ n N = N. b) Let M modλ be indecomposable nonprojective. Then LS n M) = RS n τ n M) implies τ n τ n M = M. Proof. We only prove a); b) is proved similarly. As N is noninjective, we have that RS n N) and since LS n τ n N), it follows that τ n N. Let <i <n. First note that Ω i N and Ω i τ n N are indecomposable by Corollary.. In particular, τ n N is indecomposable as well. Therefore, it is enough to show that Ω n τ n N = τ N. We will show this by showing τ N Ω n τ n N and Ω n τ n N τ N. Since Ω n τ n N, we have Ω n τ n N LS n τ n N) = RS n N) and so by Lemma. we have τ N Ω n τ n N. Now, since τ N RS n N) = LS n τ n N) there exists some j such that Ext j Λ τ n N, τ N). In particular Hom Λ Ω j τ n N, τ N) so Ω n τ n N Ω j τ n N τ N, which finishes the proof... Proof of Theorem With the preparation from the previous section, we can give a proof of the following more general form of Theorem. Theorem. Let Λ be a representation-directed algebra and C be a subcategory of modλ. Then the conditions a), b) and c) are equivalent. a) a) Λ C, a) τ n and τ n induce mutually inverse bijections

14 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P.9 -) L. Vaso / Journal of Pure and Applied Algebra ) 9 τ n C P C I, τ n a) Ω i M is indecomposable for all M C P and < i < n, a) Ω i N is indecomposable for all N C I and < i < n. b) b) Λ C, b) for all M C P, we have τ n M C and LS n M) = RS n τ n M), b) for all N C I, we have τ n N C and RS n N) = LS n τ n N). c) C is an n-cluster tilting subcategory. Proof. First note that as we mentioned before we have already proved c) implies a) by Corollary. and Proposition.. Next note that b) implies a) and b) implies a) by Corollary.. Moreover b) and b) imply a) by Lemma.. This shows that b) implies a). Next we will prove a) implies b) and finally a) and b) imply c). a) implies b): We only prove a) implies b); a) implies b) is similar and a) implies b) is clear. Let N C I. Then τn N C P by a), so it remains to show that RS n N) = LS n τn N). Assume instead that RS n N) LS n τn N) and we will reach a contradiction. Our strategy will be to construct an infinite sequence X k ) k Cof indecomposable nonisomorphic modules, contradicting the fact that Λis representation-directed. The sequence X k will be defined using τ n and injective modules, which are all in C as a) and a) imply DΛ Csince Λis representation-directed and so representation-finite. By Proposition.a) there exists an injective indecomposable module I RS n N). More generally, there is a sequence X k, k, satisfying: X = τ n N, X = I, X k = τ n X k if RS n τ n X k ) = LS n X k ), X k RS n τ n X k )andx k indecomposable injective if RS n τ n X k ) LS n X k ). In particular, X k Cfor all k. We claim that X k RS n τ n X k )for all k. We will prove this by induction. For k =we have X = I RS n N) a) = RS n τ n τ n N) =RS n τ n X ). For the induction step, assume that X k RS n τ n X k ). We want to prove X k+ RS n τ n X k ). If RS n τ n X k ) LS n X k ), then X k+ is an indecomposable injective module in RS n τ n X k ) by construction. Otherwise, RS n τ n X k ) = LS n X k )and X k+ = τ n X k. By induction assumption we have X k RS n τ n X k )and so X k+ = τ n X k LS n X k )=RS n τ n X k ) as required. In particular, X k C P for all k. Next we use X k RS n τ n X k )to show that X k <X k. Since X k RS n τ n X k ), there exists some < i < nwith Ext i ΛX k, τ n X k ). In particular, Hom Λ X k, Ω i τ n X k ). Since τ n X k C I, we have that Ω i τ n X k is indecomposable by a) and so X k Ω i τ n X k. Since Ω j τ n X k is indecomposable for i < j<n, X k Ω i τ n X k Ω n ) τ n X k <τ Ω n ) τ n X k.

15 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Since by a) we have τ n τ n X k = X k, we get X k <X k. So, the sequence X k is an infinite sequence of indecomposable modules such that <X k <X k < <X <X which contradicts the fact that Λis representation-directed and representation-finite. a) and b) imply c): We first show C n = n C. We have C n = {X modλ Ext i ΛC,X) = for all <i<n} = {X modλ Ext i ΛM,X) = for all <i<nand M C} = {X modλ Ext i ΛM,X) = for all <i<nand M C P } = {X modλ X/ LS n M) for all M C P } = {X modλ X LS n M) c for all M C P } = M C P LS n M) c. Similarly, Hence n C = RS n N) c. N C I C n = LS n M) c b) = RS n τ n M) c a) = RS n N) c = n C. M C P M C P N C I It remains to show that C = n C. Let us first show that n C C. Let X n C and without loss of generality we can assume that X is indecomposable otherwise use additivity of Ext Λ ). Moreover, if X is projective then X Cby a) so we further assume that X is nonprojective. Consider the sequence τnx k for k. We consider two possible cases. In the first case we show that X C, while in the second we reach a contradiction. Case : LS n τn k X) = RS n τnx) k for all k. Then, since Λis representation-directed, there exists some minimal l such that τnx l =. Since = RS n ) = RS n τnx) l =LS n τn l X), τn l X is projective, and so τn l X C. Since l was minimal, we have τn l X. Consider the modules Ω i τnx k where i <nand k l. Since τn l X, they are all nonprojective. Using Corollary. and induction on k we find that they are all indecomposable. Hence τn l X = τω n τn l X is also indecomposable. Since RS n τn l X) = LS n τn l X) because τn l X is nonprojective), it follows that τn l X is noninjective and so τn l X C I. On the other hand, Lemma. implies τn τnx k = τn k X for all k l, and so τn l ) τn l X = X. Since τn l X C I, it follows that X C. Case : There exists some m such that LS n τn m X) RS n τn m X). In particular, τn m X. Pick l minimal such that LS n τn l X) RS n τnx). l Then we have LS n τn k X) = RS n τnx) k for all < k < l, and as in Case, τn k X is indecomposable nonprojective. Moreover, Lemma. implies τn τnx k = τn k X and so τnx k is noninjective for < k<l. In particular, by Proposition.b), there exists an indecomposable projective module P such that P LS n τn l X). Then τn l X RS n P ) = LS n τn P ) by b). Equivalently, τn P RS n τn l X) = LS n τn l X). Repeating this argument, we get τn l ) P

16 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) LS n X). Set τn l ) P = N; then N C I and X RS n N), contradicting X n C = N C I RS n N) c and so Case is impossible. Finally, let us show that C C n. Assume towards a contradiction that Y Cis indecomposable but there exists some M C P such that Y LS n M) = RS n τ n M). By representation-directedness and because of a), there exists some minimal l such that τny l or τnm l is indecomposable projective. Since τ n M LS n Y ), Y is nonprojective, and so τ n M LS n Y ) = RS n τ n Y )byb), or τ n Y LS n τ n M) = RS n τnm). Repeating this argument we get τny l RS n τn l+ M), which is a contradiction since either τny l is projective or τn l+ M =. Example.7. Let us give an example of a -cluster tilting subcategory using Theorem. Let Λ be the path algebra of the quiver with relations Note that Λis representation-directed, special biserial and that indecomposable modules are determined uniquely by their dimension vectors. The Auslander Reiten quiver of Λ is. Let M be the direct sum of all encircled modules. Note that their syzygies and cosyzygies are indecomposable or zero and computing τ and τ applied to them we get τ τ τ τ τ τ, τ τ τ, τ. If we let C =addm, conditions a) of Theorem are satisfied for C and so C is a -cluster tilting subcategory. A simple computation shows that gl.dim.λ = ; as far as we know this is the first example of an algebra with global dimension that admits a -cluster tilting subcategory.. n-cluster tilting subcategories of acyclic Nakayama algebras with homogeneous relations.. Motivation In this section we aim to use Theorem to produce examples of n-cluster tilting subcategories for representation-directed algebras. We begin with a necessary condition.

17 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Proposition.. Let Q be a connected quiver with m vertices, Λ = KQ/I where I is an admissible ideal and n. Let k be a vertex in Q, which is a sink or a source such that the full subquiver of Q with vertex set Q \{k} is disconnected. Then Λ = KQ/I admits no n-cluster tilting subcategory. Proof. Let us prove the proposition when k is a sink; the other case is similar. Let Q be the full subquiver of Q with vertex set Q \{k} and write Q as the disjoint union of two nonempty quivers Q A and Q B. Without loss of generality, assume Q A ) = {,..., k } and Q B ) = {k +,..., m}. Consider the indecomposable projective module P k) corresponding to the vertex k. Its dimension vector is.. dim P k) =.. Moreover it is noninjective and its injective hull, Ik), has Ik) k = K since k is a sink. Furthermore, in dim Ik), there is at least one nonzero entry in a position i < ksince there exists an arrow from a vertex in Q A to k. Similarly, there is at least one nonzero entry in a position j >k. Therefore we have dim Ik) = a.. a k b k+. b m, dim Ω P k) = where a,, a k ),, ) and b k+,, b m ),, ). Let Ω P k) = M i, φ α ) i Q,α Q. Let f =f i ) i Q where f i : M i M i is identity if i < kand zero otherwise. Note that f and f Id. We will prove that f is an endomorphism of M i, φ α ). Let α : a b be an arrow in Q. Note that we cannot have a < k<bor b < k<asince Q is disconnected and we cannot have a = k since k is a sink. We need to show that a.. a k b k+. b m, φ α f a = f b φ α..) If a, b < k, f a = f b =Idand.) becomes φ α = φ α. If k <a, b then f a = f b =and.) becomes =. If b = k, then since M k =, we have that φ α =and.) becomes = again. Hence f EndΩ P k)) with f = f but f and f Id, and so EndΩ P k)) is not local, which implies that Ω P k) is not indecomposable. Since any n-cluster tilting subcategory must contain the projective modules, Λdoesn t admit an n-cluster tilting subcategory by Corollary.. Example.. Let Q be a quiver with underlying graph the Dynkin diagram A m for m, with nonlinear orientation. Pick any source or sink k with degree. Then Proposition. implies that there exists no n-cluster tilting KQ/I-module for I an admissible ideal of KQ. Example. suggests that perhaps the simplest class of representation-directed algebras for which one should try to find n-cluster tilting subcategories is quotients of the path algebra of the quiver

18 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) a m a m a m a a Q m : m m m by an admissible ideal. Such algebras are called acyclic Nakayama and for more details on them we refer to []... Computations In this section we will consider acyclic Nakayama algebras with homogeneous relations. That is, for m and l, we will denote Λ m,l = KQ m /rad KQ m ) l. As we will see later, it turns out that this is a necessary condition for a Nakayama algebra to be d-representation finite. Since our main tool will be Theorem, we will need to compute syzygies, cosyzygies and n-auslander Reiten translations for Λ m,l -modules. Recall that the isomorphism classes of the indecomposable modules for any acyclic Nakayama algebra can be described by the representations Mi, j) of the form K K m i+j- i with Mi, j)i =[], Gabriel s Theorem). We will use the convention that Mi, j) = if the coordinates i, j) do not define a module. In particular, for Λ m,l -modules we have Mi, j) if and only if i m, j l and i + j m +. For a vertex k Q, we will denote by P k) respectively Ik) the corresponding indecomposable projective respectively injective Λ m,l -module. In the rest of this section, all modules will be Λ m,l -modules. Lemma.. Let Mi, j). Then { M,k) if k l, a) P k) = M + k l, l) if l k m. { Mk, l) if k m l +, b) Ik) = Mk, m + k) if m l + k m. c) Mi, j) is both projective and injective if and only if j = l and i m l +. Proof. c) follows immediately by a) and b). We only prove a); b) is proved similarly. Note that for k l, P k) as a representation is isomorphic to K K m k which is precisely M, k). Similarly, when l k m, P k) is isomorphic to K K m k k-l+ which is precisely M + k l, l). Next we wish to compute syzygies and cosyzygies of the indecomposable Λ m,l -modules.

19 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Lemma.. Let Mi, j). Then a) If Mi, j) is nonprojective, ΩMi, j) = b) If Mi, j) is noninjective, { M,i ) if i + j l, Mi + j l, l j) if l + i + j. Ω Mi, j) = { Mi + j, l j) if i m l +, Mm + i l + j, l j) if m l + i m. Proof. We only prove a); b) is proved similarly. Assume first that l + i + j and consider the following commutative diagram i+j l+l ) Mi + j l, l j) : K... K... u Mi + j l, l) :... K... K K... K... s Mi, j) :... K... K i+j j i l i+j l+l j ) l j i+j l where the arrows K K are the identity and all other arrows are the zero map. Then Mi + j l, l j) u Mi + j l, l) v Mi, j) is a short exact sequence. Since Mi + j l, l) = P i + j ) by Lemma., we have ΩMi, j) = Mi + j l, l j). If i +j l, similarly we have the short exact sequence M, i ) M, i +j ) Mi, j) with M, i + j ) = P i + j ) and so ΩMi, j) = M, i ). Corollary.. Let Mi, j) with j <land k. Then a) Denote Ω k Mi, j) = Mi k, j k ) and assume that l + i k + j k. Then { M i + j k+ Ω k Mi, j) = l, l j) if k is odd, M i k l, j) if k is even. b) Denote Ω k Mi, j) = Mi k, j k ) and assume that i k m l +. Then { M i + j + k Ω k Mi, j) = l, l j) if k is odd, M i + k l, j) if k is even.

20 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) Proof. Immediate by using Lemma. and induction on k. Proposition.. Let Mi, j) and Mi, j +). Then the sequence [ r ] p Mi, j) Mi, j +) Mi +,j ) [ t q] Mi +,j) is almost split, where r, t are the natural inclusions, p, q the natural projections, and by convention Mi, ) =. Proof. This follows from Theorem V.. in [] by noting that rad t Mi, j ) = Mi, j t) for any t and any indecomposable Λ m,l -module Mi, j ). Lemma.7. Let Mi, j). Then a) If Mi, j) is nonprojective, τmi, j)) = Mi, j). b) If Mi, j) is noninjective, τ Mi, j)) = Mi +, j). Proof. Immediate by Proposition. and by uniqueness, up to isomorphism, of almost split sequences see [], Chapter IV.). Lemma.8. Let Mi, j). Then a) If Mi, j) is nonprojective, we have τ n Mi, j) = b) If Mi, j) is noninjective, we have τ n Mi, j) = { M i + j n l,l j) if n is even, M i n l,j) if n is odd. { M i + j + n l +,l j) if n is even, M i + n l +,j) if n is odd. Proof. Immediate by Corollary. and Lemma.7. Recall that by convention Mi, j) if and only if i m, j l and i + j m +... Proof of Theorem With our basic computations done, we are ready to prove Theorem. Theorem. Λ m,l admits an n-cluster tilting subcategory if and only if one of the following holds: i) l =and m = nk + for some k, or ii) n is even and m = n l + + knl l +) for some k. Proof. For the case l =we refer to Proposition. in [8]. Assume now that l. Set C =add r= τ r n Λ m,l ).

21 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) By Remark b) it is enough to prove that C satisfies condition a) of Theorem if and only if n is even and m = n l + + knl l +)for some k. Assume first that C satisfies condition a) and n is odd and we will reach a contradiction. Using Lemma.8 and an easy induction we can show that if n is odd and j <l, we have ) ) n τn k M,j)) = M +k l +,j. Since l, M, ) and M, ) are indecomposable projective noninjective by Lemma.. Therefore, by condition a) of Theorem there exist integers k, k > such that τ k n M, ) and τ k n M, ) are indecomposable injective. Computing and using Lemma. we find that τ k n M, ) = M τ k n M, )) = M M M are the only possibilities. In particular, n +k l + ) n +k l + ) n +k l + n +k n +k l + ) ) ) l +,, ) ), = Mm, ), ),, ), = Mm, ) ) n = m, +k l + = m which imply k k ) n l +) =, contradicting n >. Hence, n must be even; an easy induction here shows that for j <lwe have τ k n M,j)) = { M +j + k l + k n l +),l j ) if k is odd, M + k l + k n l +),j ) if k is even. As before, τ k n M, ) and τ k n M, ) must be indecomposable injective for some integers k, k >. If we assume that k and k have different parities or are both even, we reach a contradiction as in the case of n being odd. Therefore k must be odd and we have τ k n M, ) = M + k ) ) n l + k l +,l = Mm + l, l ) as the only possibility by Lemma.. This implies + k l + k n l +) = m + l or equivalently m = n l + + k nl l +)so we get the result for k = k. Now, assume that n is even and that m = n l + + knl l +)and we will show that condition a) of Theorem holds for C. a) holds by construction. Note that by Lemma.8, τn k M, j) is indecomposable or zero. For s = k +we have τn s M,j)=M +j + s ) ) n l + s l +,l j = Mm ++j l, l j),

22 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P.7 -) L. Vaso / Journal of Pure and Applied Algebra ) 7 which is injective by Lemma.. Therefore τn k M, j) is nonzero for k s, it is projective for k = and injective for k = s. Then a) holds since by Lemma.8, we have that τ n τn Mi, j) = Mi, j) if τn Mi, j) and τn τ n Mi, j) = Mi, j) if τ n Mi, j). Finally a) and a) hold by Lemma. and the proof is complete. Example.9. For m = 9, l =, n = and k =the Auslander Reiten quiver of Λ 9, = KQ 9 /rad KQ 9 ) is, ), ), ), ), ), ) 7, ), ), ), ), ), ), ) 7, ) 8, ), ), ), ), ), ), ) 7, ) 8, ) 9, ) where we write i, j) instead of Mi, j). The circled modules are the indecomposable summands of the -cluster tilting module of Λ 9, and they satisfy τ τ τ, ), ), ) 8, ) τ τ τ, τ τ τ, ), ), ) 9, ) τ τ τ.. d-representation-finite Nakayama algebras In this section we classify the Nakayama algebras admitting a d-cluster tilting subcategory, where d = gl.dim.λ. Even though cyclic Nakayama algebras are not representation-directed, we include a proof that shows that no cyclic Nakayama algebra is d-representation-finite to present the full classification. Note that the following proposition shows that the homogeneous case of the previous chapter plays a special role. Proposition.. Let Λ be a Nakayama algebra and assume that Λ admits a d-cluster tilting subcategory. Then Λ = Λ m,l, for some m, l. Proof. Let us first assume that Λ = KQ m /I is an acyclic Nakayama algebra that admits a d-cluster tilting subcategory C. Assume to a contradiction that I radkq m ) l. In particular, I {}. Then we claim that there exist some x and y such that at least one of the two following cases is true: a) Mx +, y) and Mx +, y +)are projective and x, b) Mx, y +)and Mx, y) are injective and x + y<m +. To see this, notice that there exists a path α i α i k I such that either α i+ α i k+ / I or α i α i k / I. In the first case, the coordinates i k, k +) do not correspond to an indecomposable module and it follows that Mi k+, k+) = P i +) and Mi k+, k+) = P i +) satisfy the assumptions of case a). Similarly, in the second case Mi k, k +) = Ii k) and Mi k, k +) = Ii k ) satisfy the assumptions of case b). Let us prove that case a) leads to a contradiction; the case b) is similar. Since the ideal I is admissible, we have y. In this case, the relevant part of the Auslander Reiten quiver looks like

23 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P.8 -) 8 L. Vaso / Journal of Pure and Applied Algebra ) x +,y+) x +,y) where the cross indicates the absence of the indecomposable module Mx, y +), while the dotted lines indicate the presence of some indecomposable modules along them. In other words, since x, we have that x +, y) is not in the same diagonal as, ). Since Mx +, y +), we have that Mx +, y) is noninjective. Then, by Proposition., τ d Mx +, y)) = N is an indecomposable nonprojective module and moreover, by the same proposition, τ d N) = Mx +, y). By applying τ on this we get Mx +,y)=τ Mx +,y)) = τ τω d N))), so that Mx +,y)=ω d N). We have pd Ω d N ), since otherwise we would have pdn >d. Moreover, Ω d N is not projective since τ d N), so pd Ω d N ) =. Therefore pd Mx +,y)) =. But since Mx +, y +) is projective, the short exact sequence Mx +, ) Mx +,y+) Mx +,y) implies that ΩMx +, y) = Mx +, ). But Mx +, ) is nonprojective, since it is a simple module different than M, ) which contradicts pdmx +, y) =. To complete the proof, it remains to show that a cyclic Nakayama algebra Λ with gl.dim. Λ = d < admits no d-cluster tilting subcategory. This case is very similar to the previous one so we omit most of the details. Let Λ =K Q m /I be a cyclic Nakayama algebra where Q m is the quiver a m m a a m Q m : m a a m. Then I rad Q m ) l since otherwise Λ is self-injective and thus of infinite global dimension. Then there exists an indecomposable projective noninjective module P and we must have pdτ P ) = as in the previous case. Similarly to the previous case, it is not difficult to see that Ωτ P is simple, which is a contradiction since there exists no simple projective Λ-module.

24 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P.9 -) L. Vaso / Journal of Pure and Applied Algebra ) 9.. Global dimension of Λ m,l Since given m and l we know by Theorem when Λ m,l admits an n-cluster tilting subcategory, it is enough to see what the global dimension of Λ m,l is and then check under what conditions on m and l we have n = d. Proposition.. Let Λ = Λ m,l. Then a) Let Mx, y) and assume x = or y = l. Then pdmx, y) =. b) Let Mx, y) and assume x > and y <l. Write x = ql + r with r<l. We have pdmx, y) = { q + if y<l r, q + if y l r..) c) Let m = q l + r, r l. Then d) Proof. pdmm + j, j) = { m l m l m gl.dim.λ = + l + m l if r =or j r, + m.) l otherwise. m a) Follows immediately from Lemma. since Mx, y) is projective. b) Throughout, we use l. pdmx, y) =pdωmx, y)) +..) We first prove.) for x + y l. In that case x < x + y l so that q =and r = x. Then by Lemma. ΩMx, y) = M, x ) which is projective by Lemma.. Therefore, pdmx, y) = = q+ as required, since y <l x ). Now we use induction on x + y l. The base case was just proved. Assume that.) holds when l x + y k. Let Mx, y) be such that x + y = k. Since x + y l +, Lemma. implies ΩMx, y) = Mx + y l, l y). In particular,.) holds for ΩMx, y) by induction assumption. Let x = ql + r and assume first that y <l r. We calculate x + y l =ql + r ++y l =q )l + r + y, where r + y<r+ l y = l, so x + y l = q l + r with q = q, r = r + y. To apply.) to ΩMx, y), we need to compare l y with l r so from r we get l y l r y = l r + y) =l r, and thus by.) we have pdmx + y l, l y) = q +. Then, we have as required. pdmx, y) =pdmx + y l, l y)+=q ++ =q )++=q +,

25 JID:JPAA AID:9 /FLA [ml; v.; Prn:/7/8; :] P. -) L. Vaso / Journal of Pure and Applied Algebra ) For the last case, let x = ql + r and y l r. Now we have x + y l =ql + r ++y l =q )l + l +r + y l) =ql +r + y l). Since l r y<l, we get r + y l<r l. So x + y l = ql + r with r = r + y l. We compare l y with l r =l r y, so from l >r we get or l r y>l y l r >l y. So pdmx + y l, l y) = q +by.) and.) now gives pdmx, y) =pdmx + y l, l y)+=q ++=q + as required. c) We will prove c) using b). Let m + j = ql + r so that m = ql + j + r for r l. Assume first that j <l r so that q = q and r = j + r<l. Then r and j j + r = r, so that m l + m ql + j + r = l l j + r =q + l + + ql + j + r l j + r as required. Assume now that j l r so that j + r l and l <j+r<l = q ++=q +.) = pdmm + j, j), m =ql + r + j = ql + l + r + j l =q +)l +r + j l) =q l + r. Note that j r gives l r, a contradiction, so j >r. If r =we have m l + m = l as required. Finally, if r we have m l + m l = which completes the proof of c). q l + l q l + r + l r + j l =q + l q l l =q =q +)=q +.) = pdmm + j, j), q l + r l +.) = pdmm + j, j), r + j l l <r+j l<l = q +++ =q +