Auslander Algebras of Self-Injective Nakayama Algebras

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1 Pure Mathematical Sciences, Vol. 2, 2013, no. 2, HIKARI Ltd, Auslander Algebras of Self-Injective Nakayama Algebras Ronghua Tan Department of Mathematics Hubei University for Nationalities Enshi Hubei, P.R. China ronghua Copyright c 2013 Ronghua Tan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For the Auslander algebras E of self-injective Nakayama algebras, the Δ-filtrations of the submodules of indecomposable projective E- modules are determined, a class of Δ-filtered E-modules without selfextensions are constructed, and the Ringel dual of E is described. Mathematics Subject Classifications: 16G10 Keywords: Auslander algebra, Δ-filtered modules, Tilting modules, Nakayama algebra 1 Introduction For a quasi-hereditary algebra A, we can obtain a new quasi-hereditary endomorphism algebra by forming its Ringel dual. Starting from a representationfinite algebra A, we also can get a new quasi-hereditary endomorphism algebra E of global dimension 2 which is called the Auslander algebra of A. Recall that Auslander algebras can be constructed in the following way: let A be a representation-finite finite dimensional algebra, M is a finite dimensional A-module such that every indecomposable A-module is isomorphic to a direct summand of M. Then the endomorphism algebra E := End A (M) is the Auslander algebra of A.

2 90 Ronghua Tan For quasi-hereditary Nakayama algebras, the structure of their Ringel duals is determined in [6]. On the other hand, for Nakayama algebras which are not quasi-hereditary, we shall concentrate on their Auslander algebras. For a special Nakayama algebra k[x]/ <x n > with admissible sequence (n) (here, k is a field, x a variable and n a natural number) which is not quasi-hereditary, its Auslander algebra A n has been studied in connection with the actions of parabolic groups in general linear groups in [2]. The Δ-filtered modules without self-extensions over A n have been determined. Moreover, this class of A n -modules can be described purely combinatorially. In this paper we shall consider the category F(Δ) of Δ-filtered modules over the Auslander algebras E of Nakayama algebras A, where the Nakayama algebras A are self-injective. More precisely, we shall study the Δ-filtrations of the submodules of indecomposable projective E-modules and the structure of the Ringel dual algebra of E, and construct a class of Δ-filtered E-modules without self-extensions. The main result is as follows: Theorem 1.1. (1) All submodules of projective E-modules belong to F(Δ). (2) Suppose that X is an indecomposable E-module with Δ-support I = {(j 1 i 1 ),, (j r i r )} for r 1 and the multiplicities [X :Δ(j t i t )] = 1 for 1 t r. Then (i) Ext 1 E (X, Δ(j r+1i r+1 )) is one-dimensional if and only if (j r+1 i r+1 ) is of the form (j r + s, i r + s +1) with s 0 such that i r+1 > i r. Otherwise, Ext 1 E (X, Δ(j r+1i r+1 )) = 0. (ii) The extension module Y of X by Δ(j r+1 i r+1 ) is a submodule of P (j r+1 i r+1 + n, n r). (3) Let X and Y be submodules of indecomposable projective E-modules with Δ-support I and J, respectively. If (ji) J there exists (j i ) I with i = i, then Ext 1 E (Y,X) = 0 = Ext1 E (X, Y ). (4) The Ringel dual R of E is isomorphic to E. As a consequence of parts (2) and (3) in Theorem 1.1, we can construct a class of Δ-filtered E-modules Δ(D) with Δ-dimension vector D such that Ext 1 E (Δ(D), Δ(D))=0. 2 Preliminaries For a finite dimensional algebra A over a field k, ana-module M is called uniserial if the set of its submodules is totally ordered by inclusion. The algebra A is said to be a Nakayama algebra if all the indecomposable projective and indecomposable injective modules are uniserial. Further, an algebra A is

3 Auslander algebras of self-injective Nakayama algebras 91 called a left (right) serial algebra if each indecomposable projective (injective) module is uniserial. Obviously, a Nakayama algebra is both left serial and right serial. Sometimes Nakayama algebras are called serial algebras. It is well-known (see [1]) that the quiver Γ of an indecomposable Nakayama algebra A is either a chain or a cycle. That is, Γ is either of the form n n if A has a simple projective module, or of the form 1 n 2 n 1 n 2 if A has no simple projective modules. An indecomposable Nakayama algebra A with a simple projective module is not self-injective. On the other hand, if an indecomposable Nakayama algebra A with no simple projective module is both self-injective and quasi-hereditary, then A is semisimple. In this paper, we only study the Auslander algebras E of self-injective Nakayama algebras A, where the self-injective Nakayama algebras A are not quasi-hereditary. Throughout this paper, we always work with finite dimensional algebras A over an algebraically closed field k. Denote by P j, 1 j m, the indecomposable projective A-modules, I j, 1 j m, the indecomposable injective A-modules, and A-mod the category of finitely generated left A-modules. An admissible sequence is a sequence of positive integers (a 1,,a m ) satisfying the following conditions: a j a j for 1 j m 1 and a m a 1 1. If P j is a projective cover of radp j+1 for 1 j m 1, and P m is a projective cover of radp 1 or radp 1 = 0, then (l(p 1 ),,l(p m )) is called the admissible sequence of A. Given two homomorphisms f : L M and g : M N for L, M, N A-mod, the composition of f and g is a homomorphism from L to N and is denoted by fg. 3 The Auslander algebras of Nakayama algebras In this section, we shall study the category F(Δ) of the Δ-filtered modules over the Auslander algebras E of self-injective Nakayama algebras A, where A

4 92 Ronghua Tan has m non-isomorphic simple A-modules and the admissible sequence of A is (n, n,,n). 3.1 The structure of the submodules of indecomposable projective E-modules Let A be an indecomposable Nakayama algebra with admissible sequence (n,, n). All indecomposable projective modules have the same length n, and [P j : L j ] 1 for 1 j m. In particular, these indecomposable projective modules are also injective. A is a representation finite algebra with precisely nm isomorphism classes of indecomposable representations M ji, 1 j m, 1 i n, and it is convenient to choose the indexing such that M ji is of length n i + 1 for all j. As the submodules of P j = M j1, the uniserial modules M ji can embed into each other as follows M jn M j,n 1 M j2 M j1. Let E = End A (M) be the Auslander algebra of A, where M = i,j M ji. Note that E is a quasi-hereditary algebra (see [4]). In fact, Dlab and Ringel proved the quasi-heredity of Auslander algebras for several different partial orders (see [4]). For example, if an indecomposable k-algebra B is given by quiver 2 1, then the quiver of its Auslander algebra C is of the form: a b c. It is clear that the Auslander algebra C is quasi-hereditary with respect to the orders a>b>c, a<b<cand c>a>b, respectively. Write P (ji) = Hom A (M,M ji ) be the indecomposable projective left E- module, E(ji) = top(p (ji)) be the simple E-module corresponding to the index (ji). Thus, for all j, the indecomposable projective E-modules P (ji) have the following inclusion P (jn) P (j, n 1) P (j2) P (j1). From now on, for self-injective Nakayama algebras A, we will concentrate on their Auslander algebras E which are quasi-hereditary with respect to the fixed order: (ji) > (j i )ifi>i. Lemma 3.1. The standard E-module Δ(ji) is the uniserial module of length i with socle E(j i +1, 1). Proof. Since Δ(ji) = P (ji)/trace (j i )>(ji) P (j i )(P (ji)), we have Δ(ji) = P (ji)/p (j, i+1) from the structure of the indecomposable projective E-modules. It follows that the length of Δ(ji) is just the number of the uniserial modules M st such that M ji is a quotient of M st, which is equal to n length(m ji )+1.

5 Auslander algebras of self-injective Nakayama algebras 93 Since the length of M ji is n i + 1, we get the length of Δ(ji) isi. Since M ji is a submodule of M j1 = P j, we have topm ji = L j i+1. Thus, socle of Δ(ji) is E(j i +1, 1). E(ji) E(j 1,i 1) More precisely, the standard module Δ(ji) = 1 j m, we have. E(j i +1, 1). Hence, for Δ(jn) Δ(j 1,n 1) Δ(j 2,n 2) Δ(j n +1, 1). Remark 1. We should notice that if there are two numbers j i + 1 and a with j i +1>m,1 a m such that (j i +1) a = tm for some integer t, then the notation we have chosen implies L j i+1 = La. Remark 2. Since Δ(jn) =P (jn) for all j, the standard E-module Δ(ji) is a submodule of a projective E-module. As a Δ-filtered E-module, the indecomposable projective module P (ji) has the Δ- filtration with factors Δ(ji), Δ(j, i +1),, Δ(jn). Lemma 3.2. All submodules of the projective E-modules belong to F(Δ). Proof. Since F(Δ) is closed under extensions, it is sufficient to consider the submodules of the indecomposable projective E-module P (ji). For i = n, P (jn)=δ(jn), we have P (jn) Δ(j 1,n 1) Δ(j 2,n 2) Δ(j n +1, 1) with Δ(j s, n s)/δ(j s 1,n s 1) = E(j s, n s) for 0 s n 2. Thus, the submodules of P (jn) belong to F(Δ). Suppose that for i +1 n, the claim is true, that is, the submodules of P (j, i + 1) belong to F(Δ). Now we consider the submodules of P (ji). Let V P (ji), we have the following exact sequence 0 P (j, i +1) V V V/(P (j, i +1) V ) 0, where V/(P (j, i +1) V ) = (V + P (j, i + 1))/P (j, i +1) P (ji)/p (j, i +1)= Δ(ji) Δ(j i + n, n) =P (j i + n, n) and P (j, i +1) V P (j, i + 1). By induction, V/(P (j, i +1) V ) F(Δ), and P (j, i +1) V F(Δ). Hence, V F(Δ) since F(Δ) is closed under extensions. Dually, we have Lemma 3.3. (i) The costandard E-module (ji) = Hom k (Δ (ji),k) is the uniserial module of length i with top E(j1). (ii) All factor modules of the injective E-modules belong to F( ).

6 94 Ronghua Tan Lemma 3.4. (i) F( ) is closed under factor modules. (ii) F(Δ) is closed under submodules. Proof. (i) Since Δ(jn) =P (jn), 1 j m, we have proj.dim E Δ(jn) =0. For 1 i<n, a projective resolution of Δ(ji) is given by 0 P (j, i +1) P (ji) Δ(ji) 0. Thus, proj.dim E Δ(ji) 1 for all i, j. Hence, F( ) is closed under factor modules by Lemma 4.1 in [3]. (ii) A dual consideration shows (ii). Note that the left injective E-module I(ji) = Hom k (P (ji),k) has a - filtration with factors (ji), (j +1,i+1),, (j i + n, n). For any Δ-filtered module X, we denote by (dim Δ X) ji =[X :Δ(ji)] the multiplicity of Δ(ji) in a Δ-filtration of X (these multiplicities are uniquely determined, since we deal with the standard modules of a quasi-hereditary algebra). The m n matrix dim Δ X with entries (dim Δ X) ji is said to be the Δ-dimension vector of X. The Δ-support of X is the set of indices (ji) such that (dim Δ X) ji 0. We denote by T (ji) the unique indecomposable module corresponding to index (ji), which has both a Δ-filtration and a -filtration. Note that T (jn)= P (j1) for 1 j m, since P (j1) = I(j n +1, 1) is both projective and injective. Lemma 3.5. (i) A module X is isomorphic to a nonzero submodule of P (j1), 1 j m if and only if soc(x) =E(j n +1, 1). Such a module X is indecomposable and X F(Δ). (ii) A module X is isomorphic to a nonzero factor module of P (j1), 1 j m if and only if top(x) =E(j1). Such a module X is indecomposable and X F( ). Proof. The equivalence stated in (i) is due to the fact that P (j1) = I(j n + 1, 1) is both projective and injective. This also implies that any nonzero submodule X of P (j1) is indecomposable. Such a module X belonging to F(Δ) is from Lemma 3.2. A dual consideration shows (ii). Proposition 3.6. T (ji) = P (j1)/p (j, i +1) for 1 i, j n. Proof. Since P (j, i + 1) has a Δ-filtration with factors Δ(j, i +1), Δ(j, i + 2),, Δ(jn); P (j1) has a Δ-filtration with factors Δ(j1), Δ(j2),, Δ(jn). It follows that the factor module P (j1)/p (j, i + 1) has a Δ-filtration with factors Δ(j1), Δ(j2),, Δ(ji). On the other hand, P (j1) is injective and

7 Auslander algebras of self-injective Nakayama algebras 95 F( ) is closed under factor modules, we have P (j1)/p (j, i +1) F( ). Thus, P (j1)/p (j, i +1) = T (ji) since P (j1)/p (j, i + 1) is indecomposable. More precisely, T (ji) F(Δ(j1), Δ(j2),, Δ(ji)) F( (ji), (j 1,i 1),, (j i +1, 1)). Proposition 3.7. The modules in F(Δ) are just the E-modules with projective dimension at most one. Proof. In order to bound the projective dimension of the modules in F(Δ), it is sufficient to consider the standard modules by Horseshoe Lemma. From the proof of Lemma 3.4, we know that proj.dim E Δ(ji) 1. It follows that the projective dimension of any module in F(Δ) is at most 1. Conversely, assume that X E-mod with projective dimension 1. It is sufficient to show that Ext t E (X, T(ji)) = 0 for all t 1. Since Extt E (X, T(ji)) = 0 for all t>1 from the assumption proj.dim E X 1, it remains to consider Ext 1 E (X, T(ji)). From the short exact sequence we get the following exact sequence 0 P (j, i +1) P (j1) T (ji) 0, 0 Hom E (X, P(j, i + 1)) Hom E (X, P(j1)) Hom E (X, T(ji)) Ext 1 E (X, P(j, i + 1)) Ext1 E (X, P(j1)) Ext1 E (X, T(ji)) 0, which implies that Ext 1 E (X, T(ji)) = 0 since Ext1 E (X, P(j1)) = 0 by the injectivity of P (j1). This completes the proof. In the following we will aim at determining the Δ-filtrations of the submodules of indecomposable projective modules. Proposition 3.8. Ext 1 E (Δ(ji), Δ(j i )) is one-dimensional if and only if (j i ) is of the form (j + s, i + s +1) with s 0 such that i > i. Otherwise, Ext 1 E (Δ(ji), Δ(j i )) = 0. Proof. Apply Hom E (, Δ(j i )) to the projective resolution of Δ(ji): we have the following exact sequence 0 P (j, i +1) P (ji) Δ(ji) 0, 0 Hom E (Δ(ji), Δ(j i )) Hom E (P (ji), Δ(j i )) Hom E (P (j, i +1), Δ(j i )) Ext 1 E (Δ(ji), Δ(j i )) 0. Observe that dim k Hom E (P (ji), Δ(j i )) = [Δ(j i ):E(ji)] 1. If [Δ(j i ): E(ji)] = 0, then dim k Hom E (Δ(ji), Δ(j i )) = 0. If [Δ(j i ):E(ji)] = 1, then

8 96 Ronghua Tan Δ(ji) Δ(j i ) from the structure of the standard modules. By the above discussion, we have dim k Hom E (Δ(ji), Δ(j i )) = dim k Hom E (P (ji), Δ(j i )). Since dim k Hom E (P (j, i +1), Δ(j i )) = [Δ(j i ):E(j, i + 1)], we have that Ext 1 E (Δ(ji), Δ(j i )) is one-dimensional if and only if [Δ(j i ):E(j, i + 1)] = 1. Since the composition factors of Δ(j i ) are E(j i ),E(j 1,i 1),,E(j i +1, 1), we get that E(j, i + 1) must be some E(j s, i s) with s 0. Therefore, (j i ) is of the form (j + s, i + s + 1) with s 0 such that i >i. Suppose that Ext 1 E (Δ(ji), Δ(j i )) is one-dimensional. We want to describe the structure of the extension X of the standard module Δ(ji) byδ(j i ). Proposition 3.9. If Ext 1 E (Δ(ji), Δ(j i )) is one-dimensional, then the extension module X is a submodule of P (j i + n, n 1). Proof. Since (j i ) is of the form (j + s, i + s + 1) from Proposition 3.8, we know that Soc(Δ(ji)) = Soc(Δ(j i + n, n 1)) = E(j i +1, 1), which implies Δ(ji) Δ(j i + n, n 1). From the exact sequence 0 P (j i + n, n) P (j i + n, n 1) Δ(j i + n, n 1) 0 and the injective map ι :Δ(ji) Δ(j i + n, n 1), we get the following exact commutative pullback diagram 0 P (j i + n, n) X Δ(ji) 0 g ι 0 P (j i + n, n) P (j i + n, n 1) Δ(j i + n, n 1) 0, which implies g is injective. Similarly, from the exact sequence 0 Δ(j i ) X Δ(ji) 0 and the injective map ι :Δ(j i ) Δ(j i +n, n), we get the following exact commutative pushout diagram 0 Δ(j i ) X Δ(ji) 0 ι f 0 Δ(j i + n, n) X Δ(ji) 0, which implies f is injective. Since (j i +n, n) is of the form (j+s, i+s+1), we have that Ext 1 E (Δ(ji), Δ(j i + n, n)) is one-dimensional from Proposition 3.8. It follows that the two short exact sequences 0 Δ(j i + n, n) X Δ(ji) 0 and

9 Auslander algebras of self-injective Nakayama algebras 97 0 Δ(j i + n, n) X Δ(ji) 0 are equivalent, that is, X = X. Thus, we have the following commutative diagram with exact rows 0 Δ(j i ) X Δ(ji) 0 ι f 0 Δ(j i + n, n) X Δ(ji) 0 g ι 0 P (j i + n, n) P (j i + n, n 1) Δ(j i + n, n 1) 0, which implies that X P (j i +n, n 1) since the composition fg is injective. Such a module X is indecomposable with simple socle E(j i +1, 1). For such a module X obtained in this way, we consider further the extension module Y of X by some standard module Δ(st). Theorem Suppose that X is an indecomposable E-module with Δ-support I = {(j 1 i 1 ),, (j r i r )} for r 1 and the multiplicities [X :Δ(j t i t )] = 1 for 1 t r. Then (i) Ext 1 E(X, Δ(j r+1 i r+1 )) is one-dimensional if and only if (j r+1 i r+1 ) is of the form (j r + s, i r + s +1) with s 0 such that i r+1 > i r. Otherwise, Ext 1 E (X, Δ(j r+1i r+1 )) = 0. (ii) The extension module Y of X by Δ(j r+1 i r+1 ) is a submodule of P (j r+1 i r+1 + n, n r). Proof. (i) For r = 1, the claim holds from Proposition 3.8. Suppose that r 1 and the claim holds for r. We show that the claim is true for r +1. Note that if there exists no index (j r+2 i r+2 ) with (j r+2 i r+2 ) > (j r+1 i r+1 ), then the proof is finished. So we assume that there exists such index (j r+2 i r+2 ). Firstly, we look at the following short exact sequence 0 Δ(j r+2 i r+2 ) P (j r+2 i r+2 + n, 1) C 0. Let C = T (j r+2 i r+2 )/Δ(j r+2 i r+2 )=T (j r+2,i r+2 1) be the submodule of C. Thus, C/C = T (j r+2 i r+2 + n, n)/t (j r+2 i r+2 ) has a -filtration with factors (j r+2 i r+2 + n, n), (j r+2 i r+2 + n 1,n 1),, (j r+2 +1,i r+2 + 1). Observe that Hom E (X, (st)) = 0 for all (st) / I with I the Δ-support of X, in particular for all (st) > (j r+1 i r+1 ), thus Hom E (X, C) = Hom E (X, C ). Note that X has a Δ-filtration with factors Δ(j 1 i 1 ),, Δ(j r+1 i r+1 ), and C has a -filtration with factors (j r+2,i r+2 1), (j r+2 1,i r+2 2),, (j r+2 i r+2 +2, 1). So we have Hom E (X, C ) 0 if and only if there exist (j r i r ) I of the form (j r+2 s, i r+2 s 1).

10 98 Ronghua Tan For studying the dimension of Ext 1 E (X, Δ(j r+2i r+2 )), we consider the following exact sequence 0 Hom E (X, Δ(j r+2 i r+2 )) Hom E (X, P(j r+2 i r+2 + n, 1)) Hom E (X, C) Ext 1 E (X, Δ(j r+2i r+2 )) 0. Observe that dim k Hom E (X, P(j r+2 i r+2 + n, 1)) = [X : E(j r+2 i r+2 +1, 1)] by the injectivity of P (j r+2 i r+2 + n, 1). Suppose that Hom E (X, C ) 0. If(j r1 i r1 ), (j r2 i r2 ),, (j rt i rt ) are of the form (j r+2 s, i r+2 s 1) with i r1 <i r2 < <i rt, then j r+2 i r+2 +1= j r1 i r1 = j r1 +1 i r1 +1+1,j r+2 i r+2 +1 = j r2 i r2 = j r2 +1 i r2 +1+1,,j r+2 i r+2 +1=j rt i rt = j rt+1 i rt by induction. We are going to consider two different cases. In the case of (j r+1 i r+1 ) not of the form (j r+2 s, i r+2 s 1), we have dim k Hom E (X, P(j r+2 i r+2 + n, 1)) = dim k Hom E (X, C )=t and Hom E (X, Δ(j r+2 i r+2 )) = 0. Thus, Ext 1 E (X, Δ(j r+2i r+2 )) = 0 in this situation. In the case of (j r+1 i r+1 )=(j rt i rt ) of the form (j r+2 s, i r+2 s 1), we have dim k Hom E (X, P (j r+2 i r+2 +n, 1)) = t 1 = dim k Hom E (X, C ) 1, but Hom E (X, Δ(j r+2 i r+2 )) = 0. Therefore, Ext 1 E (X, Δ(j r+2i r+2 )) is one-dimensional in this situation. If Hom E (X, C ) = 0, that is, all (j r i r ) I are not of the form (j r+2 s, i r+2 s 1), then dim k Hom E (X, Δ(j r+2 i r+2 )) = dim k Hom E (X, P(j r+2 i r+2 +n, 1)) = 1 when Δ(j 1 i 1 ) Δ(j r+2 i r+2 ), otherwise dim k Hom E (X, Δ(j r+2 i r+2 )) = dim k Hom E (X, P(j r+2 i r+2 + n, 1)) = 0. Thus, Ext 1 E (X, Δ(j r+2i r+2 )) is onedimensional if and only if (j r+2 i r+2 ) is of the form (j r+1 + s, i r+1 + s + 1) with s 0 such that i r+2 >i r+1. This completes the statement (i). (ii) We do induction on r. Forr = 1, the statement is true from Proposition 3.9. Suppose that r 1 and the statement is true for r. In the following we prove that the statement is true for r +1. Since the indecomposable projective E-module P (j r+2 i r+2 + n, n r 1) is filtered by Δ(j r+2 i r+2 +n, n r 1), Δ(j r+2 i r+2 +n, n r),, Δ(j r+2 i r+2 + n, n), we have the following exact sequence 0 Δ(j r+2 i r+2 + n, n) P (j r+2 i r+2 + n, n r 1) Z 0, where Z = P (j r+2 i r+2 + n, n r 1)/Δ(j r+2 i r+2 + n, n) is filtered by Δ(j r+2 i r+2 +n, n r 1), Δ(j r+2 i r+2 +n, n r),, Δ(j r+2 i r+2 +n, n 1). We get that Z is a submodule of P (j r+2 i r+2 + n +1,n r) =P (j r+1 i r+1 + n, n r) by induction. From the form of (j r+1 i r+1 ) and (j r+2 i r+2 ), we know i r+1 is at most n 1. It follows that Δ(j r+1 i r+1 ) Δ(j r+1 i r+1 + n 1,n 1) = Δ(j r+2 i r+2 + n, n 1), similar discussion tells us Δ(j 1 i 1 )

11 Auslander algebras of self-injective Nakayama algebras 99 Δ(j r+2 i r+2 + n, n r 1), Δ(j 2 i 2 ) Δ(j r+2 i r+2 + n, n r),, Δ(j r i r ) Δ(j r+2 i r+2 + n, n 2). It follows that X is a submodule of Z. Thus, we get the following exact commutative pullback diagram 0 Δ(j r+2 i r+2 + n, n) Y X 0 g ι 0 Δ(j r+2 i r+2 + n, n) P (j r+2 i r+2 + n, n r 1) Z 0 with g injective. Similarly, we have the following exact commutative pushout diagram 0 Δ(j r+2 i r+2 ) Y X 0 ι f 0 Δ(j r+2 i r+2 + n, n) Y X 0 with f injective. Since (j r+2 i r+2 + n, n) is of the form (j r+1 + s, i r+1 + s + 1), we have that Ext 1 E (X, Δ(j r+2 i r+2 + n, n)) is one-dimensional from the statement (i). It follows that the two short exact sequences 0 Δ(j r+2 i r+2 + n, n) Y X 0 and 0 Δ(j r+2 i r+2 + n, n) Y X 0 are equivalent, that is, Y = Y. Thus, we get Y P (j r+2 i r+2 + n, n r 1) since fg is injective. Such a module Y is indecomposable with simple socle E(j r+2 i r+2 +1, 1). The above Theorem 3.10 describes the Δ-filtrations of the submodules of indecomposable projective E-modules. 3.2 The Δ-filtered E-modules without self-extensions In this subsection, we are going to construct a class of Δ-filtered modules without self-extensions. Theorem Let X and Y be submodules of indecomposable projective E- modules with Δ-support Iand J, respectively. If (ji) J there exists (j i ) I with i = i, then Ext 1 E(Y,X) = 0 = Ext 1 E(X, Y ). Proof. We only need to consider the case where Y =Δ(ji) with (ji) J since if the result is true for this special case, then by using dimension shifting the theorem can be proved. We first consider the situation where j j. If i = n, then Y =Δ(jn) is projective, thus Ext 1 E (Y, ) =0. For1 i<n, take the projective resolution of Δ(ji) 0 P (j, i +1) P (ji) Δ(ji) 0.

12 100 Ronghua Tan By applying the functor Hom E (,X) to the above sequence, we get the following exact sequence 0 Hom E (Δ(ji),X) Hom E (P (ji),x) Hom E (P (j, i+1),x) Ext 1 E (Δ(ji),X) 0. Recall that X is filtered by Δ(j 1 i 1),, Δ(j i)=δ(j s i s),, Δ(j r i r) with r s 1. In order to determine the dimension of Hom E (P (ji),x), we only need to consider the multiplicity [M : E(ji)], where M X with Δ-support {(j s+1i s+1 ),, (j ri r )}. If [M : E(ji)] 0, then there exist (j r 1 i r1 ),, (j r t i rt ) with (j r t i rt ) > > (j r 1 i r1 ) > (j i) and t 1 such that [Δ(j r 1 i r1 ):E(ji)] = 1,, [Δ(j r t i rt ): E(ji)] = 1. Hence, [Δ(j r 1 +1i r1 +1) : E(j, i + 1)] = 1,, [Δ(j r t+1i rt+1) : E(j, i + 1)] = 1, and [M : E(ji)] = t. Now we will deal with two different situations. When (j r t i rt )=(j ri r ), we have [M : E(j, i + 1)] = [X : E(j, i + 1)] = t 1 and dim k Hom E (Δ(ji),X) = 1 since Δ(ji) Δ(j r t i rt ) = Δ(j r i r) X. Thus, dim k Hom E (P (ji),x) = dim k Hom E (Δ(ji),X)+ dim k Hom E (P (j, i+ 1),X), which implies that Ext 1 E (Δ(ji),X) = 0. When (j r t i rt ) < (j ri r ), then dim k Hom E (P (ji),x) = dim k Hom E (P (j, i+1),x)=tand Hom E (Δ(ji),X)= 0, which tells us that Ext 1 E (Δ(ji),X)=0. If [M : E(ji)] = [X : E(ji)] = 0, then Hom E (Δ(ji),X) = 0. And Hom E (P (j, i +1),X) = 0 since if not, then [X : E(ji)] 1, a contradiction. This shows that Ext 1 E (Δ(ji),X)=0. In the situation j = j, we also can get that Ext 1 E (Δ(ji),X)=0. On the other hand, we want to see that Ext 1 E (X, Δ(ji)) = 0. In fact, we can show it by using a method similar to that of Proposition 1 in [2] and the information we have obtained in Theorem Now we will focus on a class of Δ-filtered modules which are suitable direct sums of submodules of indecomposable projective E-modules. For any m n matrix D with integral entries d ji such that d 1i = d 2i = = d mi for all 1 i n, we will construct a Δ-filtered module Δ(D) with Δ-dimension vector D, that is, [Δ(D) : Δ(1i)] = [Δ(D) : Δ(2i)] = = [Δ(D) : Δ(mi)] for all i. It is convenient to use the following visualization for Δ(D). Let D = ((d j1 ), (d j2 ),, (d jn )) with (d ji )= d 1i. d mi the column vector and d i = d 1i = = d mi for all i, write m = d 1 + d d n. We start with an arrangement of m vertices in n columns, such that the ith column contains precisely d i

13 Auslander algebras of self-injective Nakayama algebras 101 vertices. For example, set m =3,n = 4 and D = with the following arrangement: , we work Next, we connect these vertices by horizontal edges, whenever possible; in this manner, the rows give rise to the connected components of a graph: Note that the rows represent the various direct summands of Δ(D); a row with vertices in the columns i 1,,i s,s 1, corresponds to the direct sum j M Ij, where M Ij is the submodule of some indecomposable projective module with Δ-support I j = {(ji 1 ), (l2i j 2 ),, (ls ji s)}, here l j s +1 i s +1 = l j s i s 1 for 1 <s <sand l j 2 i 2 = j i 1 1. In our example, we see that Δ(D) = M {11,12,13,14} M {21,22,23,24} M {31,32,33,34} M {11,12,13} M {21,22,23} M {31,32,33} M {11,23} M {21,33} M {31,13} M {11} M {21} M {31}, where M I denotes the submodule of some indecomposable projective module with Δ-support I. Here 3 j=1 M {(j1),(j2),(j3),(j4)} corresponds to the first row, 3 j=1 M {(j1),(j2),(j3)} corresponds to the second row, 3 j=1 M {(j1),(j+1,3)} corresponds to the third row, 3 j=1m {(j1)} corresponds to the bottom row. (In case some of the d i s are zero, it may be necessary to add the labelling of the columns, in order to be able to recover D from the diagram.) Therefore, we have the following result which shows that we do construct a class of Δ-filtered modules without self-extensions. Corollary Suppose that X F(Δ) with Δ-dimension vector D and Δ(D) is defined as before. If X = Δ(D), then Ext 1 E (X, X) =0. Proof. From the construction of Δ(D), we know that any two indecomposable direct summands of Δ(D) satisfy the assumption of Theorem Thus, Ext 1 E (X, X) =0.

14 102 Ronghua Tan 3.3 The Ringel dual of the Auslander algebra E In this subsection we are going to study the structure of the Ringel dual of E. Recall that the Auslander algebra E is quasi-hereditary with respect to the order: (j i ) > (ji) ifi >i, and the characteristic tilting module is T = i,j T (ji). Thus, the Ringel dual R = End E (T ) is quasi-hereditary with respect to the opposite order: (j i ) > (ji) ifi <i. For the Ringel dual R, we denote by P R (ji) = Hom E (T,T(ji)) the indecomposable left projective R-module. By S(ji) we denote the top of P R (ji). Thus S(ji) is simple. From the -filtration of the tilting E-modules ( T (ji) is filtered by (ji), (j 1,i 1),, (j i +1, 1).) and the short exact sequence 0 V (ji) T (ji) (ji) 0 with V (ji) filtered by (j i ) s with (j i ) < (ji), we get that T (j 1,i 1) is a submodule of T (ji). More precisely, T (jn) T (j 1,n 1) T (j 2,n 2) T (j n +1, 1). Thus, we have P R (jn) P R (j 1,n 1) P R (j 2,n 2) P R (j n +1, 1). The standard R-module Δ R (ji) = Hom E (T, (ji)) is the uniserial module of length n i + 1, which is equal to the number of the elements in the set {T (st) :T (st) filtered by Δ(ji)}. In fact, the elements in the set are just S(ji) T (ji),t(j, i+1),,t(jn), thus Δ R (ji) =. and soc(δ R (ji)) = S(jn). S(jn) On the other hand, we know that Δ R (ji) =P R (ji)/p R (j 1,i 1), it follows that P R (ji) is filtered by Δ R (ji), Δ R (j 1,i 1),, Δ R (j i +1, 1). The dimension of P R (ji) as a vector space over the algebraically field k is dim k P R (ji) = dim k Δ R (ji)+dim k Δ R (j 1,i 1) + +dim k Δ R (j i +1, 1) = (n i +1)+(n i + 2)+ + n. For indecomposable projective E-module P (j, n i + 1), dim k P (j, n i + 1) = dim k Δ(j, n i + 1)+dim k Δ(j, n i + 2) + +dim k Δ(jn) =(n i +1)+(n i +2)+ + n. Thus, dim k R = i,j dim k P R (ji)= i,j dim kp (ji) = dim k E. Now we are ready to compare the structure of E with that of R. For the following Theorem 3.13, we first recall the definition of the rational completion of a module M. Let I(M) be an injective envelope of M, the rational completion R(M) ofm is the intersection of the kernels of all endomorphisms φ of I(M) which vanish on M; in particular, we always have M R(M) I(M).

15 Auslander algebras of self-injective Nakayama algebras 103 In case I(M) is of finite length, this submodule R(M) of I(M) is characterized as follows: it contains M, no composition factor of R(M)/M embeds into M, whereas every composition factor of the socle of I(M)/R(M) does embed into M. Theorem The Ringel dual R is isomorphic to E. Proof. By Theorem 3.10, T (ji) is a submodule of P (j i+n, n i+1) since T (ji) is filtered by Δ(j1), Δ(j2),, Δ(ji). Thus, the canonical embedding T = i,j T (ji) i,j P (j i + n, n i +1)= i,j P (ji) =E is the rational completion. In fact, soct (ji)=e(j i +1, 1) is not a subfactor of P (j i + n, n i+1)/t (ji) since [P (j i+n, n i+1) : E(j i+1, 1)] = [T (ji):e(j i+ 1, 1)]; on the other hand, P (j i+n, 1)/P (j i+n, n i+1) = T (j i+n, n i) with socle E(j +1, 1) for i<n, and P (j i + n, 1)/P (j i + n, n i +1)=0 for i = n. This rational completion shows that the endomorphism algebra R of T embeds into the endomorphism algebra of the regular module E E,thus into the opposite algebra of E. Since E is isomorphic to its opposite algebra E op, R can be regarded as a subalgebra of E. But dim k R = dim k E. Therefore, R = E. 4 Examples and application For an indecomposable Nakayama algebra, we know the structure of the indecomposable modules and the almost split sequences (see [1]), so we can construct the Auslander-Reiten-quiver which depends only on their admissible sequences. In particular, we can get the quiver of the Auslander algebra of a Nakayama algebra by using another result in [1]: let A be an artin algebra of finite representation type and let B be the associated Auslander algebra. Then the ordinary quiver of B is the opposite of the transpose of the Auslander- Reiten-quiver of A. (Let Γ be a valued quiver. The transposed valued quiver Γ tr has the same underlying quiver as Γ, and if an arrow α has valuation (a, b) in Γ, it has valuation (b, a) inγ tr.) Let (Q, ρ) be the quiver of the Auslander algebra E, its set of vertices is Q 0 = {(ji) :1 j m, 1 i n}, the set Q 1 consists of the arrows α ji : (ji) (j 1,i 1) for 1 j m, 1 < i n and β ji : (ji) (j, i + 1) for 1 j m, 1 i < n. The set ρ consists of the relations α jn α j n+i,i β j n+i 1,i 1 = 0 for 1 j m, i 1, and α j+1,i+1 β ji β j,i+1 β j,n 1 = 0 for 1 j m, i n.

16 104 Ronghua Tan 4.1 Examples The following Example 4.1 will illustrate our results. Example 4.1. Let the indecomposable Nakayama algebra A with admissible sequence (4, 4, 4, 4) be given by quiver and relations as follows: β 3 2 α γ 4 δ 1, αβγδ =0,βγδα=0, γδαβ =0,δαβγ =0. Let M ji be the submodule of M j1 = P j of length 4 i+1 for 1 i 4. The Auslander algebra E = End A ( i,j M ji )ofa is quasi-hereditary with respect to the order: ji > j i if i>i. The quiver of the Auslander algebra E is as follows: where the same vertices are identified. The standard E-modules Δ(ji) are the uniserial modules of length i with socle E(j i +1, 1) of the form: Δ(44) = , Δ(34) = , Δ(24) = , Δ(14) = Δ(33) = , Δ(23) = , Δ(13) = , Δ(43) = Δ(22) = 22 11, Δ(12) = 12 41, Δ(42) = 42 31, Δ(32) = Δ(11) = 11, Δ(41) = 41, Δ(31) = 31, Δ(21) = 21.

17 Auslander algebras of self-injective Nakayama algebras 105 The indecomposable projective-injective E-modules are : P (41) = , P(31) = , P (21) = , P(11) = We can check that submodules X of indecomposable projective modules have Δ-support I with property mentioned in Theorem All submodules of the projective modules belong to F(Δ). The tilting modules T (ji) are just the factor modules P (j1)/p (j, i + 1) for i =1, 2, 3 and T (j4) = P (j1). The quiver of the Ringel dual R = End E ( i,j T (ji)) is given by where the same vertices are identified, ji is the index corresponding to that of T (ji). By S(ji) we denote the top of the indecomposable R-module P R (ji) = Hom E (T,T(ji)). Note that R is quasi-hereditary with respect to the opposite order: ji > j i if i<i. Here, the standard R-modules Δ R (ji) is uniserial of length 4 i + 1 with socle S(j4). The standard R-modules are as follows: Δ R (11) = , Δ R (21) = , Δ R (31) = , Δ R (41) =

18 106 Ronghua Tan For j =1, 2, 3, 4, we have Δ R (j4) Δ R (j3) Δ R (j2) Δ R (j1). In fact, R is isomorphic to E by choosing a bijection between the vertices. 4.2 Application In the following we will define a category of collections of flags of vector spaces being acted upon by certain linear maps. The category to be defined will be shown to be equivalent to the category F(Δ) over the Auslander algebra E. For the fixed m and n, define a category F(m, n) as follows. The objects are of the form ((F j ), (f j )), where F j for 1 j m is a flag {0} = V j,n+1 V jn V j1 = V j of length n of subspaces of some finite dimensional k-vector space V j, and f j is a collection {f ji : V ji V j 1,i+1 } 1 i<n of k-linear maps such that θ jn θ ji f j,i 1 =0,θ j,i+1 f ji f j 1,i+1 f j n+i+1,n 1 = 0 with θ ji : V ji V j,i 1 injective for 1 <i n and such that the following diagram V ji θ ji V j,i 1 f ji V j 1,i+1 f j,i 1 V j 1,i θ j 1,i+1 commutes. Let ((F j ), (f j )) and ((F j), (f j)) be two objects in F(m, n). A morphism ϕ :((F j ), (f j )) ((F j ), (f j )) is a collection {ϕ ji : V ji V ji } of k-linear maps such that the following diagrams θ V ji ji f V j,i 1 V ji ji V j 1,i+1 ϕ ji ϕ j,i 1 ϕ ji ϕ j 1,i+1 V ji θ ji V j,i 1 V ji f ji V j 1,i+1 commute. For ((F j ), (f j )) F(m, n), set d ji := dim k V ji dim k V j,i+1 for 1 j m, 1 i n, the m n matrix D =(d ji ) with entries d ji is said to be the dimension vector of ((F j ), (f j )). If ϕ :((F j ), (f j )) ((F j), (f j)) is an isomorphism in F(m, n), then D = D and dim k V j = dim k V j = for all j. After identifying (F j) with (F j ), we see that f ji = ϕ 1 ji f jiϕ j 1,i+1. On the other hand, for the quiver (Q, ρ) of the Auslander algebra E, denote by Rep(Q, ρ) the category of finite dimensional representations of (Q, ρ) over

19 Auslander algebras of self-injective Nakayama algebras 107 k. We will consider the full subcategory M(m, n) of Rep(Q, ρ) whose objects M satisfy the condition that for each arrow α ji :(ji) (j 1,i 1) the corresponding linear map M(α ji ) is injective. Let ((F j ), (f j )) be in F(m, n), we define a representation M((F j ), (f j )) of the quiver Q by V m1 V m 1,1 V 21 V 11 V m1 V Vm2 m,n 1 V Vm 1,2 m 1,n 1 V V22 2,n 1 V V12 1,n 1 V Vm2 m,n 1 V mn V m 1,n V 2n V 1n V mn where the same k-vector spaces are identified, the maps from the bottom up (that is, the maps corresponding to arrows α ji ) are simply inclusions θ ji and the maps in the opposite direction are f ji. From the definition of ((F j ), (f j )), we have M((F j ), (f j )) M(m, n). Conversely, give a representation M of M(m, n), the injectivity of the maps M(α ji ) provides the flag structure on each space V j1 for 1 j m. Since the maps M(α ji ),M(β ji ) satisfy the relations ρ, we get that the maps f ji are just M(β ji ). Hence, we have Proposition 4.2. The categories F(m, n) and M(m, n) are equivalent for any m, n N. In particular, let m =1,F(1,n) is the category F(n) defined in [5] and M(1,n) is the category M(n) defined in [5]. Moreover, M(n) is precisely the category F(Δ) of Δ-filtered modules over the Auslander algebra of the representation-finite Nakayama algebra k[x]/ <x n > (see [3]). More generally, we have that the category M(m, n) is the category F(Δ) of Δ-filtered modules over the Auslander algebra E of the representation-finite Nakayama algebra A with admissible sequence (l(p 1 )=n, l(p 2 )=n,,l(p m )=n). ACKNOWLEDGMENT. This work is supported by the NNSF of China (No ) and the Youth Foundation of Educational Department of Hubei Province of China (Q ). The author would like to express her gratitude to Prof. Changchang Xi and Prof. Steffen Koenig for their valuable suggestions. References [1] Auslander, M., Reiten, I., Smalø, S. (1995). Representation theory of Artin algebras. Cambridge studies in advanced mathematics 36, Cambridge University Press.

20 108 Ronghua Tan [2] Brüstle, T., Hille, L., Ringel, C. M., Röhrle, G. (1999). The Δ-filtered modules without self-extensions for the Auslander algebra of k[t ]/ <T n >. Algebra and Represent. Theory 2: [3] Dlab, V., Ringel, C. M. (1992). The module theoretical approach to quasihereditary algebras. In: Tachikawa, H., Brenner, S. eds. Representation of algebras and Related topics, London Math. Soc. LNS 168. Cambridge University Press, pp [4] Dlab, V., Ringel, C. M. (1989). Auslander algebras as quasi-hereditary algebra. J. London Math. Soc. 39(2): [5] Hille, L., Röhrle, G. (1999). A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical. Transformation Groups 4: [6] Tan, R., Koenig, S. (2005). Tilting modules over quasi-hereditary Nakayama algebras. Comm. Algebra 33: Received: February 4, 2013