COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS THAT CONTAIN NON-PERIODIC HELLER LATTICES OF STRING MODULES: THE CASE OF THE KRONECKER ALGEBRA O[X,Y]/(X2,Y 2) OVER A COMPLETE D.V.R. 6 1 0 KENGO MIYAMOTO 2 n Abstract. We consider the Kronecker algebra A = O[X,Y]/(X2,Y2), where O is a a completediscretevaluationring. Then,sinceA⊗κisasymmetricspecialbiserialalgebra, J where κ is the residue field of O, we may give a complete list of Heller lattices. In this 1 paper,wedetermineacomponentofthestableAuslander–Reitenquiverforthecategory 3 of A-lattices that contains non-periodic Heller lattices of the string modules over A⊗κ. ] T R . h Contents t a m Acknowledgment 1 [ Introduction 1 3 1. Preliminaries 3 v 1.1. Almost split sequences 3 6 1.2. Stable Auslander–Reiten quivers 5 5 2 2. The Kronecker algebra and almost split sequences 7 6 2.1. Heller lattices of κ[X,Y]/(X2,Y2) 7 0 1. 2.2. Almost split sequence ending at Zn 13 2.3. Almost split sequence ending at E 20 0 1 6 3. Candidates for a component of the stable AR quiver that contains Z 24 n 1 3.1. Valencies of vertices in C 25 : v 3.2. Candidates for C 34 i X 4. The main result 40 r References 43 a Acknowledgment My heartfelt appreciation goes to Professor Susumu Ariki (Osaka Univ.) who provided helpful comments and suggestions. I would also like to thank Professor Shigeto Kawata (Osaka City Univ.), Professor Michihisa Wakui (Kansai Univ.), Dr. Ryoichi Kase (Nara Women’s Univ.) and Dr. Liron Speyer (Osaka Univ.) whose meticulous comments were an enormous help to me. Introduction One of issues of the Auslander–Reiten theory is to give classification of modules. In the theory, we visualize the categories of modules by using Auslander–Reiten quivers, which encapsulate much information on indecomposable modules and irreducible morphisms. 1 2 KENGO MIYAMOTO Therefore describing Auslander–Reiten quivers for various algebras is one of the classical problems in the representation theory of algebras. However, in the case of algebras over a complete discrete valuation ring, there are few examples, since it is difficult to compute almost split sequences of such algebras. Let O be a complete discrete valuation ring, ε a uniformizer, K its fraction field and κ its residue field. In this paper, tensor products are taken over O. Let A be a symmetric O-order and mod-A the category of finitely generated right A-modules. An additive full subcategory of mod-A consisting of A-lattices is denoted by latt-A. According to [AR], there exists an almost split sequence ending at M if and only if M is not a projective A-lattice and M ⊗ K is projective as an A ⊗ K-module, and dually, there exists an almost split sequence starting at M if and only if M is not an injective A-lattice and M ⊗K is injective as an A⊗K-module. Since A is symmetric, an almost split sequence ending at M exists if and only if an almost split sequence starting at M exists. Here, we denote the property of A-lattices by (♮) (♮) M ⊗K is projective as an A⊗K-module. Heller lattices, which are direct summands of the kernel of a projective cover of an indecomposable A⊗κ-module viewed as an A-module, have the property (♮). It provides the possibility that we may determine components of stable Auslander–Reiten quivers that contain Heller lattices. In [AKM], we presented the technique of constructing almost split sequences for A-lattices with (♮), and determined such components for the truncated polynomial ringover O. IfA⊗κisaspecial biserial algebra, thatis, itisMorita equivalent to an algebra κQ/I, where Q is a quiver, such that: (i) the number of incoming and outgoing arrows in Q is at most two for each vertex, (ii) for each arrow α, there is at most one arrow β with αβ ∈/ I, (iii) for each arrow α, there is at most one arrow γ with γα ∈/ I, (iv) I is an admissible ideal, then indecomposable A⊗κ modules are classified in string modules and band modules, and one can calculate a complete set of isoclasses of indecomposable modules [BR],[Erd] and the Heller lattices. We denote by Γ (A) the stable Auslander–Reiten quiver for latt-A, whose arrows are s determined by almost split sequences for latt-A. Since A is symmetric, Γ (A) is stable s under the Auslander–Reiten translation. If a component, say C, of Γ (A) does not have s loops, then one can apply the Riedtmann structure theorem (Theorem 1.9) to C, and we have an isomorphism C ≃ ZT/G, where T is a directed tree and G is an admissible group (see Subsection 1.2). The underlying undirected tree T is called the tree class of C. Thus, in order to know the shape of C, we determine the tree class and the admissible group. A technique to determine the shape a valued quiver is to construct a subadditive function [HPR]. If C admits a subadditive function d, then T is one of Dynkin diagrams or Euclidean diagrams. On the other hand, according to [Z], if there exists a non-zero Z subadditive function with values in on a non-periodic connected stable translation ≥0 quiver Q, which admits loops, then Q is either smooth or of the shape of Z∆ for some valued quiver ∆. Moreover, If Q has loops then Q is smooth. Therefore, the existence of loops is a point to determine C. In this paper, we consider the case of the Kronecker algebra A = O[X,Y]/(X2,Y2), and calculate Heller lattices and almost split sequences. The aim of this paper is to deter- mine the unique component, say C, of the stable Auslander–Reiten quiver that contains COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS 3 non-periodic Heller lattices of the string A⊗κ-modules. The main idea is to construct a function d′, which is defined by the following formula: d′ : C ∋ X 7−→ ♯{non-projective indecomposable direct summands of X ⊗κ} ∈ Z , ≥0 and the function d′ allows us to prove that C has no loops. 1. Preliminaries 1.1. Almost split sequences. InordertointroducethestableAuslander–Reitenquivers for Gorenstein O-orders, we recall some notions of irreducible, minimal, and almost split morphisms. Let A be an abelian category with enough projectives, C an additive full subcategory that is closed under extensions and direct summands. We call a morphism in C a section when it admits a left inverse, and we call a morphism in C a retraction when it admits a right inverse. A morphism f : L → M in C is called left minimal if every h ∈ EndC(M) satisfying the following diagram is an isomorphism: f L // M ●●●●f●●●(cid:9)●●●●## (cid:15)(cid:15) h M and left almost split if it is not a section and for every h ∈ HomC(L,W) which is not a section, there exists u in C that makes the following diagram commutative: f L // M (cid:9) h ∃ u (cid:15)(cid:15) {{ W Dually, a morphism g : M → N in C is called right minimal if every h ∈ EndC(M) satisfying the following diagram is an isomorphism: g M // N h (cid:15)(cid:15) ✇✇✇(cid:9)✇✇✇g✇✇✇✇✇;; M and right almost split if for every h ∈ HomC(W,N) which is not a retraction, there exists v in C that makes the following diagram commutative: g M // N cc OO (cid:9) h ∃ v W A morphism f is said to be left minimal almost split in C if f is both left minimal and left almost split, and dually f is said to be right minimal almost split in C if f is both right minimal and right almost split. Then the following proposition holds [A, Proposition 4.4]. 4 KENGO MIYAMOTO 1.1. Proposition. Let L,E and M be objects of C. Then the following statements are equivalent for a short exact sequence g f 0 −→ L −−−−→ E −−−−→ M −→ 0. (1) f is right almost split in C, and g is almost split in C. (2) f is minimal right almost split in C. (3) f is right almost split and EndC L is local. (4) g is minimal left almost split in C. (5) g is left almost split in C and EndC M is local. We now define a particular type of short exact sequence. Recall that the radical radHom (M,N) is defined by the following formula: A radHom (M,N) = {f ∈ Hom (M,N) | 1−gf is invertible for any g ∈ Hom (N,M)}, A A A where M,N ∈ latt-A. 1.2. Definition. Let A be an O-order and M,E,L objects of latt-A. A short exact sequence p 0 −→ L −→ E −−−−→ M −→ 0 is called an almost split sequence ending at M if (i) the epimorphism p is not a retraction, (ii) L and M are indecomposable, (iii) the epimorphism p induces the epimorphism Hom (X,p) : Hom (X,E) −→ radHom (X,M) A A A for every indecomposable A-lattice X. It follows from Proposition 1.1 that if there exists an almost split sequence ending at M with starting term L, then such sequences are uniquely determined up to isomorphism by M, and similarly, such sequences are uniquely determined up to isomorphism by L. 1.3. Definition. Let M and N be objects of latt-A. A morphism f ∈ Hom (M,N) is A said to be an irreducible morphism provided that (i) the morphism f is neither a section nor a retraction, (ii) ifthereexists thefollowing commutative diagraminlatt-A, theneither f isasection 1 or f is a retraction. 2 f M // N ❄❄ (cid:9) ⑧?? ❄ ⑧ ❄❄ ⑧⑧ f1 (cid:31)(cid:31) ⑧ f2 W 1.4. Lemma ([ARS]). Let A be an O-order and M an A-lattice. Suppose that M satisfies the property (♮). Then the short exact sequence ι p 0 −→ L −−−−→ E −−−−→ M −→ 0 is almost split if and only if ι and p are irreducible morphisms. From now on, we denote an almost split sequence ending at M by E(M), and it is computed as follows [AKM, Proposition 1.14]. COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS 5 1.5. Proposition. Let A be a Gorenstein O-order, M an indecomposable non-projective A-lattice with the property (♮) and p : P → M its projective cover, ν the Nakayama functor on latt-A. For each homomorphism ϕ : M → ν(M) in latt-A, we obtain the commutative diagram 0 // L // E // M // 0 ϕ (cid:15)(cid:15) (cid:15)(cid:15) 0 // L // ν(P) // ν(M) // 0 ν(p) with exact rows, where E is the pullback along ϕ, and L = D(Coker(Hom (p,A))). Then A the following conditions are equivalent. (1) The upper short exact sequence is almost split. (2) The following three conditions hold. (i) the morphism ϕ does not factor through ν(p), (ii) L is an indecomposable A-lattice, (iii) the composition ϕ◦f factors through ν(p), for all f ∈ radEnd (M). A Note that, since A = O[X,Y]/(X2,Y2) is symmetric, the lower short exact sequence in the Proposition 1.5 becomes p 0 −→ Ker(p) −→ P −−→ M −→ 0. Thus the Auslander–Reiten translation is the syzygy functor on the stable category mod- A. 1.2. Stable Auslander–Reiten quivers. In this subsection we introduce stable Auslander–Reiten quivers for latt-A [AKM]. We recall some notions about translation quivers. A main reference for the notions is [Z]. Let Q = (Q ,Q ) be a quiver, where Q and Q are the set of vertices and arrows, 0 1 0 1 respectively. If a map v : Q → Z × Z is given, the pair (Q,v) is called a valued 1 ≥0 ≥0 quiver, and the values of the map v are called valuations. For an arrow x → y ∈ Q , 1 we write v(x → y) = (d ,d ), and if there is no arrow between x and y, we understand xy yx that d = d = 0. If v(x → y) = (1,1) for all arrows x → y in Q, then v is said to be xy yx trivial. For each vertex x ∈ Q , two sets x+ and x− are defined by 0 x+ = {y ∈ Q | x → y ∈ Q }, x− = {y ∈ Q | y → x ∈ Q }. 0 1 0 1 A quiver Q is localy finite if the number of vertices of x+ ∪x− is finite for any x ∈ Q . 0 In other words, Q is locally finite if there are only finitely many incoming and outgoing arrows at each vertex. A stable translation quiver (Q,τ) is given by a quiver which is locallyfinitewithnomultiplearrows, andaquiver automorphismτ satisfying x− = (τx)+. 1.6. Remark. In standard textbooks, loops are not allowed when we define a stable translation quiver. However, we note that the definition of a stable translation quiver in [Z] allows loops, and we adopt this definition of a stable translation quiver. (cid:3) A valued stable translation quiver consists of (Q,v,τ), where, (Q,v) is a valued quiver, (Q,τ) is a stable translation quiver, and if v(x → y) = (a,b), then v(τy → x) = (b,a). Riedtmann introduced the valued stable translation quiver Z∆, where (∆,v) is a valued quiver, as follows [Ri]: • arrows are (n,x) → (n,y) and (n−1,y) → (n,x) for x → y in ∆ and n ∈ Z, 6 KENGO MIYAMOTO • valuations are defined by v((n,x) → (n,y)) = (a,b), v((n−1,y) → (n,x)) = (b,a) if v(x → y) = (a,b) in ∆, • translations are defined by τ((n,x)) = (n−1,x). Z The valued stable translation quiver ∆ has no loops whenever ∆ has no loops. Let (Q,v,τ) be a connected valued stable translation quiver and x a vertex of Q. The vertex x is called τ-periodic if x = τkx for some k > 0. If there are τ-periodic vertices in Q, then all vertices in Q are τ-periodic. In rhis case (Q,v,τ) is called τ-periodic. [HPR], [B]. (Q,v,τ) is said to be smooth if its valuation is trivial and ♯x+ = 2 for each vertex x ∈ Q . A subadditive function on (Q,v,τ) is a function d from Q to the positive 0 0 rational numbers satisfying d(x) + d(τX) ≥ d d(y) for each vertex x ∈ Q . A y6=x yx 0 subadditive function d is called additive if d(x)+d(τx) = d d(y) for any vertices P y6=x yx x ∈ Q . In this paper, we require that d(x) = d(τx). Then, the following theorem holds 0 P [Z, p.653, 669]. 1.7. Theorem. Let (Q,v,τ) be a τ-non-periodic connected valued stable translation quiverwhichadmitsanon-zerosubadditivefunctiondsuchthatitsvaluesarenon-negative integer. Then, either (i) (Q,v,τ) is smooth and d is both additive and bounded, or (ii) (Q,v,τ) is of the form Z∆ for some valued quiver ∆. Moreover if Q has cyclic paths, then Q is smooth and d is additive. 1.8. Definition. Let (Q,τ) be a stable translation quiver. A full subquiver, say C, of Q is called a component of (Q,τ) provided that a proper subquiver of C satisfying the following two conditions does not exist. (i) C is stable under the quiver automorphism τ, (ii) C is a disjoint union of connected components of the underlying undirected graph. A group, say G, of valued translation quiver automorphisms of (Q,τ) is called admis- sible if each G-orbit Gx = {g(x) | g ∈ G} of x ∈ Q intersects y− ∪{y} and y+ ∪{y} in 0 at most one vertex respectively, for each vertex y ∈ Q . For an admissible group G, we 0 may define a valued stable translation quiver Q/G = (Q/G,τ ) as follows: Q/G • the vertices are the G-orbits in Q , 0 • the arrows are the G-orbits in Q , 1 • the translation τ is induced by τ. Q/G 1.9. The Riedtmann structure theorem. Let (Q,τ) be a stable translation quiver without loops and C a component of (Q,τ). Then there is a directed tree T and an admissible group G ⊆ Aut(ZT) such that C ≃ ZT/G as a stable translation quiver. Moreover, (1) the underlying undirected graph T of T is uniquely determined by C, (2) the admissible group G is unique up to conjugation in Aut(ZT). In Theorem 1.9, the underlying undirected tree T is called the tree class of the com- ponent C. The following lemma is useful to determine the shape of a valued quiver [B, Theorem 4.5.8]. COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS 7 1.10. Theorem. Let (∆,v) be a valued quiver without loops and multiple arrows, and we assume that the underlying undirected graph ∆ is connected. Then the following hold. (1) Suppose that (∆,v) admits a subadditive function. Then the underlying undirected graph ∆ is either a Dynkin diagram or an Euclidean diagram. (2) If (∆,v) admits a subadditive function which is not additive, then ∆ is either a finite Dynkin diagram or A . ∞ (3) (∆,v) admits an unbounded subadditive function if and only if ∆ is A . ∞ We define stable Auslander–Reiten quivers for symmetric O-orders as follows [AKM]. 1.11. Definition. Let A be a symmetric O-order. A stable Auslander–Reiten quiver of A is a valued quiver such that • vertices areisoclasses ofnon-projective A-latticesM such thatM⊗K isprojective, (a,b) • valued arrows M −−→ N for irreducible morphisms M → N where the value (a,b) of the arrow is given as follows. (i) For a minimal right almost split morphism f : E → N, M appears a times in E as a direct summand. (ii) For a minimal left almost split morphism g : M → E, N appears b times in E as a direct summand. A component of a stable Auslander–Reiten quiver is defined in the similar way as stable translation quivers. Let C be a component of the stable Auslander–Reiten quiver of a symmetric O-order A. By the definition, C can not have multiple arrows, and if M is a vertex of C, then there exists an almost split sequence 0 −→ τM −→ E −→ M → 0. Thus M− = (τM)+ holds and C is a valued stable translation quiver. 2. The Kronecker algebra and almost split sequences 2.1. Heller lattices of κ[X,Y]/(X2,Y2). LetA = O[X,Y]/(X2,Y2). Inthissubsection, we give a complete list of the Heller lattices of the indecomposable A ⊗ κ-modules and examine its properties. Let k be a positive integer, and {e } the canonical O-basis l l=1,2,...,k of O⊕k. We set {e ,Xe ,Ye ,XYe } as an O-basis of the direct sum of k copies of l l l l l=1,2,...,k A and call the O-basis the canonical basis. Note that A⊗κ is a special biserial algebra since there is an isomorphism κ[X,Y]/(X2,Y2) ≃ κ α 66 e1 hh β (α2,β2,αβ −βα). (cid:18) (cid:19)(cid:30) 2.1. Remark. Let k be an algebraically closed field. Then, the path algebra B = kQ where Q is the Kronecker quiver ◦ // ◦ // 2 1 is also called the Kronecker algebra. For example, see [ASS]. The algebra B is not isomorphic to A = k[X,Y]/(X2,Y2) since B is hereditary but A is not, and they are related by an isomorphism k 0 B ≃ . rad(A/soc(A)) k (cid:18) (cid:19) 8 KENGO MIYAMOTO It follows that mod-B ≃ mod-A/soc(A). In particular, stable Auslander–Reiten quivers of B and A/soc(A) are isomorphic as valued quivers [ARS]. (cid:3) First we recall some basic facts on string modules over special biserial algebras [BR], [Erd]. LetA = kQ/(ρ)beaspecialbiserialalgebra,wherekisanalgebraicallyclosedfield, Q = (Q ,Q ,s,t) is a quiver and ρ is a set of relations for Q. Without loss of generality, 0 1 we assume that ρ consists of zero-relations and commutativity relations. If r = w −w 1 2 is a commutative relation, then we say that the paths w and w are contained in r, and 1 2 define a set ρ as follows: ρ = {paths in ρ}⊔{paths contained in commutative relations in ρ}. Given an arrow α ∈ Q , the formal inverse α−1 is defined by the following formula: 1 s(α−1) = t(α), t(α−1) = s(α). Then we understand (α−1)−1 = α. A string path with length n ≥ 1 in (Q,ρ) is a sequence c c ···c satisfying the following properties: 1 2 n (i) for 1 ≤ i ≤ n, c are of the form α or α−1 for some α ∈ Q , i i i i 1 (ii) for 1 ≤ i ≤ n−1, t(c ) = s(c ), i i+1 (iii) for 1 ≤ i ≤ n−1, c 6= c−1, i+1 i (iv) for 1 ≤ i < j ≤ n, neither c c ···c nor c−1c−1 ···c−1 belong to ρ. i i+1 j j j−1 i Let C = c ···c be a string pathwith length n ≥ 1. The source s(C), the target t(C) and 1 n the formal inverse C−1 are defined by s(c ), t(c ) and c−1···c−1 respectively. In addition, 1 n n 1 for each vertex v ∈ Q , two string paths with length 0, say 1 and 1 , are defined 0 (v,1) (v,−1) by s(1 ) = t(1 ) = v for t = 1,−1. Then we define 1−1 = 1 for t = 1,−1. (v,t) (v,t) (v,t) (v,−t) For each string path C, a representation M(C) of Q is constructed as follows: if C is of the form 1 for some v ∈ Q and t ∈ {1,−1}, then M(C) is the simple representation (v,t) 0 S(v) of Q associated with v ∈ Q . Assume that C = c ···c with length n ≥ 1. Let 0 1 n u : {1,2,...,n+1} → Q be a map defined as 0 s(c ) if k = 1, u(k) = 1 t(c ) if 2 ≤ k ≤ n+1, k−1 (cid:26) and let M(C) be a k-vector space with basis vectors z (i ∈ u−1({v})) for v ∈ Q . If v i 0 α : e → e′ is an arrow, then the coresponding k-linear map ϕα : M(C)e −→ M(C)e′ is defined by the following: z if α = c , i+1 i ϕ : z 7−→ z if α−1 = c , α i  i−1 i−1 0 otherwise.  Obviously, M(C) = (M(C) ,ϕ ) is a representation of Q which satisfies the v α v∈Q0,α∈Q1 relations in ρ, and M(C) is called a string module over A. Note that string A-modules are indecomposable, and two string modules M(C ) and M(C ) are isomorphic if and 1 2 only if two string paths C and C satisfy either C = C or C = C−1. 1 2 1 2 1 2 We return to the Kronecker algebra A = O[X,Y]/(X2,Y2). An A⊗κ-module is given byapairofsquarematricesX andY whichcommuteandhavesquarezero. Forsimplicity, we visualize an A⊗κ-module as follows: • points represent basis vectors of the underlying κ-vector spaces, COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS 9 • arrows of the form −→ represent the action of X, and 99K represent the action of Y. For example, the indecomposable projective module A⊗κ is described by A⊗κ = κ1⊕κX ⊕κY ⊕κXY = 1 ♥P♥♥XPY♥♥P♥♥P66((XY ♥P♥♥P♥XY♥P♥((66XY . Let m ≥ 0,n ≥ 1 and λ ∈ P1(κ) = κ⊔{∞}. Then there are non-projective indecom- posable A⊗κ-modules as follows: (i) The indecomposable A⊗κ-module M(m) is given by the formula: v u1 ❣❴❣❣❴❣❣❣❴❣❣❴❣//33 v10 m m . . . . M(m) = κui ⊕ κvj = ... ... ! ! . . Mi=1 Mj=0 um ❣❴❣❣❴❣❣❴❣❣❴33v// vmm−1 (ii) The indecomposable A⊗κ-module M(−n) is given by the formula: u1 ❣ ❣ ❣ ❣ ❣//33v01 n+1 n . . . . M(−n) = κui ⊕ κvj = ... ... ! ! . . Mi=1 Mj=1 uvnn+1❣ ❣ ❣ ❣//33//v0n (iii) The indecomposable A⊗κ-module M(λ) is given by the formula: n n n ui // vi M(λ)n = κui ⊕ κvj = u1 ❴ ❴ ❴ ❴// λv1 Mi=1 ! Mj=1 ! ui ❴ ❴ ❴//λvi +vi−1 and for λ = ∞: u1 ❣❴❣❣❴❣❣❣❴❣❣❴❣//33v01 n n . . . . M(∞)n = κui ⊕ κvj = ... ... ! ! . . Mi=1 Mj=1 un ❣❴❣❣❴❣❣❣❴❣❣❴33v// vn−n1 The above indecomposable A ⊗ κ modules form a complete set of isoclasses of non- projective indecomposable A ⊗ κ-modules, and M(0) is the unique simple module, and A ⊗ κ is the unique projective module, which also is the unique injective module. We understand that M(λ) is zero for λ ∈ P1(κ). Then M(m), M(−n), M(0) and M(∞) 0 n n are string A⊗κ-modules. 10 KENGO MIYAMOTO 2.2. Remark. Through the stable equivalence in Remark 2.1 between mod-A⊗κ/soc(A⊗ κ) and mod-κQ, where Q is the Kronecker quiver, non-projective indecomposable A⊗κ- modules correspond to indecomposable κQ-modules as follows: t(1m 0) (1n 0) M(m) = κm //// κm+1 , M(−n) = κn+1 //// κn , t(0 1m) (0 1n) 1n J(n,0) M(λ) = κn // κn , M(∞) = κn // κn , n // n // J(n,λ) 1n where m ≥ 2, n ≥ 1, λ ∈ κ and J(n,λ) is the Jordan normal form of size n for the eigenvalue λ. (cid:3) 2.3. Remark. Almost split sequences of A⊗ κ = κ[X,Y]/(X2,Y2) are known to be as follows: 0 −→ M(−1) −→ (A⊗κ)⊕M(0)⊕M(0) −→ M(1) −→ 0, 0 −→ M(n−1) −→ M(n)⊕M(n) −→ M(n+1) −→ 0 if n 6= 0, 0 −→ M(λ) −→ M(λ) ⊕M(λ) −→ M(λ) −→ 0 n ≥ 1,λ ∈ P1(κ). n n−1 n+1 n (cid:3) In Remark 2.2, the sizes of the representing matrices of the actions of X and Y on M(m) and M(−n) are vertically long or horizontally long. Hence, in this paper, we call M(m) and M(−n) horizontal and vertical, respectively. Let M be a non-projective A⊗κ-module. Then the projective cover of M is given by π : A⊕♯{ui} −→ M, e 7→ u . M i i We denote it by π+, π−, πλ or π∞ according to the type of the module M: p p p p π+ if M = M(p), p  π− if M = M(−p),  p π =  M   πλ if M = M(λ) ,  p p  π∞ if M = M(∞) .  p p      Then Heller lattices are direct summands of the following A-lattices: