The Ringel dual of the Auslander–Dlab–Ringel algebra

arXiv:1708.05766v1 [math.RT] 18 Aug 2017 THE RINGEL DUAL OF THE AUSLANDER-DLAB-RINGEL ALGEBRA TERESA CONDE AND KARIN ERDMANN Abstract. The ADR algebra RA of a finite-dimensional algebra A is a quasihereditary algebra. In this paper we study the Ringel dual R(RA ) of RA . We prove that R(RA ) can be identified with (RAop ) op , under certain ‘minimal’ regularity conditions for A. We also give necessary and sufficient conditions for the ADR algebra to be Ringel selfdual. Contents 1. Introduction 2. Background 2.1. Quasihereditary algebras 2.2. The ADR algebra as an ultra strongly quasihereditary algebra 3. Theorem A 3.1. Motivation for Theorem A 3.2. Towards the proof of Theorem A 3.3. Proof of Theorem A 4. Cartan matrices and multiplicities 4.1. The Cartan matrix of RA 4.2. The Cartan matrix of R (RA ) 4.3. The Cartan matrix of SA 4.4. Comparing C(R(RA )) with C(SA ) 5. Ringel selfdual ADR algebras References 1 3 4 5 6 7 8 14 18 18 19 20 20 23 24 1. Introduction Quasihereditary algebras were introduced by Cline, Parshall and Scott ([4]) to investigate highest weight categories arising in algebraic Lie theory, and they were extensively studied from the perspective of finite-dimensional algebras by Dlab and Ringel ([11, 10]). Since then, quasihereditary algebras have been discovered in many other contexts. In particular, to every finite-dimensional algebra A (or more generally, to every Artin algebra A) there is a canonical quasihereditary algebra R̃A which contains A Date: September 25, 2018. 2010 Mathematics Subject Classification. Primary 16S50, 16W70. Secondary 16G10, 16G20. Key words and phrases. Quasihereditary algebra, Ringel dual, ADR algebra. The first named author was supported by the grant SFRH/BD/84060/2012 of Fundação para a Ciência e a Tecnologia while part of this work was carried out. 1 2 TERESA CONDE AND KARIN ERDMANN as an idempotent subalgebra. That is, there is an idempotent ξ in R̃A such that A = ξ R̃A ξ. The algebra R̃A was introduced by Auslander in [1], to show that any algebra occurs as idempotent subalgebra of an algebra of finite global dimension. Subsequently, Dlab and Ringel showed in [8] that R̃A is in fact quasihereditary. To define R̃A , take the direct sum of all radical powers of A, that is G̃ = L M A/ Radi A, i=1 where L denotes the Loewy length of A. Then R̃A := EndA (G̃)op . To study its representation theory, one considers the basic version of R̃A instead. We denote such basic algebra by RA and call it the Auslander–Dlab–Ringel algebra (ADR algebra) associated to A. In specific cases, an algebra A may occur in many different ways as an idempotent subalgebra of some quasihereditary algebras. A famous example is the Schur algebra S (n, r). For n ≥ r, it has the group algebra of a symmetric group of degree r as an idempotent subalgebra. One might think of ADR algebras as analogues of Schur algebras, which exist in general. In [11], Ringel proved that quasihereditary algebras come in pairs. For every quasihereditary algebra (B, Φ, ⊑), there is another quasihereditary algebra (R (B) , Φ, ⊑ op ), unique up to isomorphism, such that R (R (B)) is again Morita equivalent to B. The algebra R (B) is defined as the endomorphism algebra EndB (T ) op of a special B-module T , called the characteristic tilting module. We call R (B) a Ringel dual of B. The aim of this paper is to study the Ringel dual R (RA ) of an ADR algebra RA . Quasihereditary algebras arising in Lie theory are often isomorphic to their own Ringel dual. For example, this is the case for category O ([19]), or for Schur algebras S(n, r) for n ≥ r, and q-analogues ([12]). However, Ringel selfduality appears to be less prevalent in general. Nevertheless, there is a natural description of the Ringel dual R (RA ) of an ADR algebra RA when the radical structure of A satisfies certain symmetry conditions. Our main result is as follows. Theorem A (Main Theorem). Let A be a finite-dimensional algebra with Loewy length L and assume that all projective and injective indecomposable A-modules are rigid with Loewy length L. The Ringel dual of the quasihereditary algebra RA is isomorphic to the opposite of the ADR algebra of Aop . That is, R (RA ) ∼ = (RAop ) op . Here a module is said to be rigid if its radical series coincides with its socle series. Connected selfinjective algebras with radical cube zero but radical square nonzero trivially satisfy the conditions of Theorem A. Such class of algebras was studied in [13], and contains blocks of symmetric group algebras of weight 1 (see [6, Section 7] for an example). According to [17], blocks of symmetric group algebras of weight 2 also satisfy the assumptions of Theorem A when the underlying field has characteristic p > 2. Along the way, we prove preliminary results of independent interest. For instance, in Theorem 3.12, a complete description of the ∆-filtrations of the tilting RA -modules is given in terms of the socle series of the injective indecomposable THE RINGEL DUAL OF THE ADR ALGEBRA 3 A-modules, when A is an algebra whose projectives are rigid modules with Loewy length L (here L denotes the Loewy length of A). Philosophically speaking, the technicalities encountered towards the proof of Theorem A are due to the fact that we are seeking to identify two algebras in a “noncanonical way”. Furthermore, we give some details to explain why the assumptions on the Loewy length and on the rigidity are needed for Theorem A to hold. First, it is necessary that the projective cover Pi and the injective hull Qi of a simple module Li of A must have the same Loewy length; otherwise there cannot be a canonical correspondence between the labelling sets for weights between R (RA ) and (RAop ) op . In fact, we show that the conditions in the statement of Theorem A are somehow ‘minimal’. Theorem B. Let A be a finite-dimensional connected algebra with Loewy length L. Suppose that dim EndA (Li ) = 1 for every simple module Li , and assume that the projective cover Pi and the injective hull Qi of Li have both the same Loewy length li . If the Cartan matrix of R(RA ) coincides with the Cartan matrix of (RAop )op (up to a ‘natural’ permutation of rows and colums), then Pi and Qi are rigid modules, and li = L for every i. Our final result shows that the algebra RA is not usually Ringel selfdual. Theorem C. The ADR algebra RA is Ringel selfdual if and only if A is a selfinjective Nakayama algebra. It was already proved in [20] that selfinjective Nakayama algebras are Ringel selfdual, but in our setting this comes out as a special case. The layout of the paper is the following. Section 2 contains background on quasihereditary algebras and on the ADR algebra. Section 3 is dedicated to the proof of our main result: Theorem A. We start out this section by providing some evidence that supports the statement of Theorem A. For the proof of this theorem, we collect a number of auxiliary results which relate the ∆- and the ∇-filtrations of the tilting modules over the ADR algebra, the highlight being Theorem 3.12. In Section 4, we show that the regularity conditions for the algebra A assumed in the statement of Theorem A are, in a certain sense, minimal. This is attained by comparing several multiplicities for the algebras R (RA ) and (RAop ) op , and culminates with the proof of Theorem B. In Section 5, we discuss Ringel selfduality for the ADR algebra and prove Theorem C. 2. Background In this section we give some background on quasihereditary algebras and on the ADR algebra. Throughout this paper the letters B and A shall denote arbitrary Artin algebras over some underlying commutative artinian ring K. All the modules will be finitely generated left modules. The notation mod B will be used for the category of (finitely generated) B-modules. The case when K is a field is perhaps the most significant one. In this situation, A is just a finite-dimensional K-algebra, and the modules are finite-dimensional. For the general case, the technology works the same, and details may be found in the text book [2], Chapter II. 4 TERESA CONDE AND KARIN ERDMANN 2.1. Quasihereditary algebras. Given an Artin algebra B, we may label the isomorphism classes of simple B-modules by the elements of a finite poset (Φ, ⊑). Denote the simple B-modules by Li , i ∈ Φ, and use the notation Pi (resp. Qi ) for the projective cover (resp. injective hull) of Li . Let ∆ (i) be the largest quotient of Pi whose composition factors are all of the form Lj , with j ⊑ i, and call ∆ (i) the standard module with label i ∈ Φ. Dually, denote the costandard module with label i by ∇ (i), i.e. let ∇ (i) be the largest submodule of Qi with all composition factors of the form Lj , with j ⊒ i. Denote by ∆ (resp. ∇) the set of all standard modules (resp. costandard modules). Given a class of modules Θ, let F (Θ) be the category of all B-modules which have a Θ-filtration, that is, a filtration whose factors are isomorphic to modules in Θ. The notation [M : L] will be used for the multiplicity of a simple module L in the composition series of M . In a similar manner, (M : ∆(i)) shall denote the multiplicity of ∆(i) in a ∆-filtration of a module M in F (∆). Define (M : ∇(i)), M ∈ F (∇), in the same way. Definition 2.1. We say that (B, Φ, ⊑) is quasihereditary if the following hold for every i ∈ Φ: (1) [∆(i) : Li ] = 1; (2) Pi ∈ F (∆); (3) (Pi : ∆ (i)) = 1, and (Pi : ∆ (j)) 6= 0 ⇒ j ⊒ i. An algebra (B, Φ, ⊑) is quasihereditary if and only if (B op , Φ, ⊑) is quasihereditary. The standard modules and the costandard modules have striking homological properties. The following is well known: (1) if Ext1B (∆(i), ∆(j)) 6= 0, then i ❁ j; (2) Ext1B (∆(i), ∇(j)) = 0 for all i, j ∈ Φ. 2.1.1. Ringel duality. In [15] Ringel introduced the concept of characteristic tilting module over a quasihereditary algebra. This is a multiplicity free B-module T , satisfying F (∆) ∩ F (∇) = add T . Here for a module M , we denote by add M the full subcategory of mod B consisting of all modules isomorphic to a direct summand of a finite direct sum of copies of M . It is common to refer to a module in F (∆) ∩ F (∇) as a tilting module. The indecomposable tilting modules are in bijection with the elements of Φ. We write L T (i) – the indecomposable summands T (i) are characterised by the T = i∈Φ following result. Lemma 2.2 ([15]). Let (B, Φ, ⊑) be an arbitrary quasihereditary algebra. For every i in Φ there is a short exact sequence 0 ∆ (i) φ T (i) X (i) 0 , with φ a left minimal F (∇)-approximation of ∆ (i) and with X (i) a module lying in F ({∆ (j) : j ❁ i}). The endomorphism algebra EndB (T )op is known as the Ringel dual of B, and shall be denoted by R (B). Let Pi′ be the projective indecomposable R (B)-module HomB (T, T (i)) and let L′i be its top. According to [15], the algebra R (B) is quasihereditary with respect to the poset (Φ, ⊑ op ), and moreover R (R (B)) is THE RINGEL DUAL OF THE ADR ALGEBRA 5 Morita equivalent to B. Denote the standard and the costandard R (B)-modules by ∆′ (i) and ∇′ (i). 2.2. The ADR algebra as an ultra strongly quasihereditary algebra. Fix an Artin algebra A. Given a module M in mod A, we shall denote its Loewy length by LL(M ). Let A have Loewy length L (as a left module). We want to study the LL basic version of the endomorphism algebra of j=1 A/ Radj A. For this, let {P1 , . . . , Pn } be a complete set of projective indecomposable Amodules and let li be the Loewy length of Pi . Define G = GA := li n M M Pi / Radj Pi . i=1 j=1 The Auslander–Dlab–Ringel algebra of A (ADR algebra of A) is defined as RA := EndA (G) op . Observe that the functor HomA (G, −) : mod A −→ mod RA is fully faithful as G is a generator of mod A. The projective indecomposable RA -modules are given by
Pi,j := HomA G, Pi / Radj Pi , for 1 ≤ i ≤ n, 1 ≤ j ≤ li . Denote the simple quotient of Pi,j by Li,j and define (1) Λ := {(i, j) : 1 ≤ i ≤ n, 1 ≤ j ≤ li }, so that Λ labels the simple RA -modules. Define a partial order, ✂, on Λ by (2) (i, j) ✁ (k, l) ⇔ j > l. It turns out that the ADR algebra RA is a quasihereditary algebra with respect to the poset (Λ, ✂), and its quasihereditary structure is specially neat. Let us clarify the previous assertion. Following Ringel ([16]), a quasihereditary algebra (B, Φ, ⊑) is said to be right strongly quasihereditary if Rad ∆ (i) ∈ F (∆) for all i ∈ Φ. This property holds if and only if the category F (∆) is closed under submodules (see [9], [11, Lemma 4.1*] and [16, Appendix]). Let (B, Φ, ⊑) be an arbitrary quasihereditary algebra, as before. Additionally, suppose that B satisfies the following two conditions: (A1): Rad ∆ (i) ∈ F (∆) for all i ∈ Φ (that is, B is right strongly quasihereditary); (A2): Qi ∈ F (∆) for all i ∈ Φ such that Rad ∆ (i) = 0. We call these algebras right ultra strongly quasihereditary algebras (RUSQ algebras, for short). The algebra (B, Φ, ⊑) is said to be a left ultra strongly quasihereditary algebra (LUSQ) if the quasihereditary algebra (B op , Φ, ⊑) is RUSQ. Remark 2.3. It was proved in [5, §2.5.1] that the definition of RUSQ algebra given in [6] is equivalent to the one above. According to [6, §4], (RA , Λ, ✂) is a RUSQ algebra. The ADR algebra is the prototype of a RUSQ algebra. 6 TERESA CONDE AND KARIN ERDMANN Theorem 2.4 ([6, §4]). The algebra (RA , Λ, ✂) is a RUSQ algebra. If Qi is the injective A-module with simple socle isomorphic to Top Pi , then the following identity holds T (i, 1) = Qi,li = HomA (G, Qi ) . 2.2.1. Properties of RUSQ algebras. Our results about the ADR algebra rely on its properties as a RUSQ algebra. Next, we outline the main features of RUSQ algebras. Let (B, Φ, ⊑) be a RUSQ algebra. It is always possible to label the elements in Φ as Φ = {(i, j) : 1 ≤ i ≤ n, 1 ≤ j ≤ li }, for certain n, li ∈ Z>0 , so that [∆ (k, l) : Li,j ] 6= 0 implies that k = i and j ≥ l (see [6, §5]). We shall always assume that the elements in Φ are labelled in such a way. The labelling poset (Λ, ✂) for the ADR algebra defined in (1) and (2) is compatible with this. The following theorem summarises the main properties of the RUSQ algebras. Theorem 2.5 ([6, §5]). Let (B, Φ, ⊑) be a RUSQ algebra. The following hold: (1) F (∆) is closed under submodules; (2) Rad ∆ (i, j) = ∆ (i, j + 1) for j < li , and ∆ (i, li ) = Li,li ; (3) each ∆ (i, j) is uniserial and has composition factors Li,j , . . . , Li,li , ordered from the top to the socle; (4) Qi,li ∼ = T (i, 1) and T (i, j + 1) ⊆ T (i, j) for j < li ; (5) T (i, li ) ∼ = ∇ (i, j) = ∇ (i, li ), and for j < li we have T (i, j) /T (i, j + 1) ∼ and Qi,j ∼ = T (i, 1) /T (i, j + 1); (6) for M ∈ F (∆), the total number of standard modules appearing in a ∆P filtration of M is given by ni=1 [M : Li,li ]; (7) a module M belongs to F (∆) if and only if Soc M is a (finite) direct sum of modules of type Li,li . RUSQ algebras are well behaved with respect to Ringel duality. Theorem 2.6 ([6, §6]). Let (B, Φ, ⊑) be a RUSQ algebra. Then (R (B) , Φ, ⊑ op ) is a LUSQ algebra. The costandard module ∇′ (i, 1) is isomorphic to L′i,1 , and for j > 1 we have ∼ ∇′ (i, j) /L′ . ∇′ (i, j − 1) = i,j ′ Each ∇ (i, j) is uniserial and has composition factors L′i,1 , . . . , L′i,j , ordered from ′ ′ ′ ∼ for j < li . This ⊆ Pi,j the top to the socle. Moreover, Pi,1 = T ′ (i, li ) and Pi,j+1 ′ ′ ′ ′ ∼ ′ ∆ (i, l ) and P /P gives rise to a filtration with factors Pi,li ∼ = i i,j i,j+1 = ∆ (i, j) for j < li . 3. Theorem A The goal of this section is to prove Theorem A stated in the Introduction. In order to attain this, we investigate in detail the ∆-filtrations of the tilting modules over the ADR algebra RA . THE RINGEL DUAL OF THE ADR ALGEBRA 7 3.1. Motivation for Theorem A. Given an Artin algebra A, define C = CA := n LL(Q Mi ) M i=1 Socj Qi . j=1 This is a cogenerator of mod A. Set SA := EndA (C) op . It turns out that the algebras SA and R (RA ) have a very similar structure. In fact, the statement of Theorem A can be loosely rephrased as: the algebras SA and R (RA ) are isomorphic provided that A is “nice enough”. Before delving into the technical results necessary to prove Theorem A, we will try to illustrate (informally) why the algebras SA and R (RA ) should be related. Ln Lli T (i, j). In this setting, R (RA ) is equal to EndRA (T ) op , where T = i=1 j=1 This algebra is quasihereditary with respect to (Λ, ✂op ), where (Λ, ✂) is the poset associated with the ADR algebra described in (1) and (2). According to Theorem 2.6, (R (RA ) , Λ, ✂op ) is a LUSQ algebra. Turning the attention to the algebra SA , we have that   n LL(Q Mi ) M Socj Qi  op SA = EndA  i=1   = EndA D  j=1 n LL(Q Mi ) M i=1 j=1 op op  PiA / Radj PiA  op ∼ = EndAop (GAop ) = (RAop )op , op where D is the standard duality and PiA denotes the projective indecomposable Aop -module D (Qi ). To avoid ambiguity, denote the poset corresponding to the ADR algebra RAop of Aop by (ΛAop , EAop ) and represent its elements by [i, j]. Note that SA is quasihereditary, as RAop is. To be precise, (SA , ΛAop , EAop ) is a LUSQ algebra since (RAop , ΛAop , EAop ) is RUSQ. So both (R (RA ) , Λ, ✂op ) and (SA , ΛAop , EAop ) are LUSQ algebras. We take this analogy further by comparing the posets (Λ, E op ) , Λ = {(i, j) : 1 ≤ i ≤ n, 1 ≤ j ≤ li = LL(Pi )}, op (ΛAop , EAop ) , ΛAop = {[i, j] : 1 ≤ i ≤ n, 1 ≤ j ≤ LL(PiA ) = LL(Qi )}. For the algebras R (RA ) and SA to be isomorphic they must have the same number of simple modules, i.e. the sets Λ and ΛAop must have the same cardinality. It seems then reasonable to require that LL(Pi ) = LL(Qi ), for all 1 ≤ i ≤ n. Ideally, an isomorphism between R (RA ) and SA would somehow preserve the orders E op and EAop of Λ and ΛAop , respectively. As R (RA ) and SA are LUSQ 8 TERESA CONDE AND KARIN ERDMANN algebras, they both have uniserial costandard modules. By Theorem 2.6, the costandard R (RA )-module with label (i, j), ∇′ (i, j), has the following structure (i, 1) (i, 2) . .. . (i, j) The costandard SA -module with label [i, j], ∇SA [i, j], is isomorphic to the module D(∆RAop [i, j]), where ∆RAop [i, j] is the standard RAop -module with label [i, j]. Therefore, the submodule lattice of ∇SA [i, j] is ‘dual’ to the submodule lattice of ∆RAop [i, j]. Using part 3 of Theorem 2.5, we deduce that ∇SA [i, j] has the following structure [i, LL(Qi )] [i, LL(Qi ) − 1] . .. . [i, j] If we suppose that LL(Pi ) = LL(Qi ) = li for all i, then the modules ∇′ (i, j) and ∇SA [i, li − j + 1] have the same length for every 1 ≤ i ≤ n, 1 ≤ j ≤ li . The stronger assumption that LL(Pi ) = LL(Qi ) = L for all i, actually implies that the bijection (i, j) 7−→ [i, L − j + 1] preserves the partial orders. In this case, we have (i, j) ✁ op (k, l) ⇔ (i, j) ✄ (k, l) ⇔j<l ⇔ L − j + 1 > L − l + 1 ⇔ [i, L − j + 1] ✁Aop [k, L − l + 1]. These observations support the assumptions and the claim of Theorem A. Theorem A. Suppose that A satisfies LL(Pi ) = LL(Qi ) = L for all i, 1 ≤ 1 ≤ n. Moreover, suppose that all projectives Pi and all injectives Qi are rigid. Then R (RA ) ∼ = (RAop )op . = SA ∼ Recall that a module is rigid if its radical series coincides with its socle series. The assumptions in the statement of Theorem A will be further discussed in Section 4. 3.2. Towards the proof of Theorem A. Roughly speaking, the quasihereditary structure of the algebra SA depends on the socle series of the injective indecomposable A-modules, whereas the structure of the algebra R (RA ) depends on the filtrations 0 ⊂ T (i, li ) ⊂ · · · ⊂ T (i, j) ⊂ · · · ⊂ T (i, 1) = Qi,li mentioned in Theorem 2.5. The results in this subsection explore the connections between these two filtrations. Furthermore, we determine the ∆-filtration of the tilting modules T (i, j) completely, when A satisfies the relevant conditions for Theorem A. THE RINGEL DUAL OF THE ADR ALGEBRA 9 3.2.1. ∆-semisimple filtrations for the ADR algebra. Recall that the ADR algebra is a RUSQ algebra. Our proof of Theorem A uses the special properties of ∆filtrations of modules over RUSQ algebras. We recall the definition P of trace of a module. The trace of Θ in a B-module M is given by Tr(Θ, M ) := f : f ∈HomB (U,M), U∈Θ Im f . This is the largest submodule of M generated by Θ. A module M is said to be ∆-semisimple if it isomorphic to a direct sum of standard modules. When the underlying algebra is a RUSQ algebra, the ∆-semisimple modules are particularly well behaved. Proposition 3.1 ([7, Corollary 3.3, Proposition 3.8]). Let (B, Φ, ⊑) be a RUSQ algebra. The following hold: (1) every submodule of a ∆-semisimple B-module is still ∆-semisimple; (2) every module M ∈ F (∆) has a unique ∆-semisimple submodule which is maximal among the class of all ∆-semisimple submodules of M ; (3) the largest ∆-semisimple submodule of M ∈ F (∆) is given by δ (M ) := Tr(∆, M ), and moreover M/δ (M ) lies in F (∆). Write δ := Tr(∆, −) and let δ0 be the zero functor in mod B. For i ≥ 1 and M in mod B, define δi+1 (M ) as the module satisfying the identity δi+1 (M ) /δi (M ) = δ (M/δi (M )). Note that δ1 = δ. Let (B, Φ, ⊑) be a RUSQ algebra. Using Proposition 3.1, we deduce that every module M in F (∆) has a special filtration with proper inclusions 0 ⊂ δ (M ) ⊂ · · · ⊂ δm (M ) = M whose factors are ∆-semisimple modules. This is the ∆-semisimple filtration of M . The integer m is called the ∆-semisimple length of M . We write ∆. ssl M = m. The operators δi can be regarded as subfunctors of the identity functor 1mod B . In fact, the functors δi are left exact subfunctors of 1mod B , or in other words, they satisfy δi (N ) = N ∩ δi (M ) for every N and M with N ⊆ M (we refer to [7, §3.2] for further details). Proposition 3.2 ([7, Lemmas 3.5, 3.11, 3.12]). Let (B, Φ, ⊑) be a RUSQ algebra. Then the functors δi satisfy δi (N ) = N ∩ δi (M ) for every N and M with N ⊆ M . In particular, δi ◦ δj = δmin{i,j} . Moreover, the following hold for M in F (∆): (1) if i ≤ ∆. ssl M , then ∆. ssl (M/δi (M )) = ∆. ssl M − i; (2) if N is a submodule of M , then ∆. ssl N ≤ ∆. ssl M ; (3) if i ≤ ∆. ssl M then δi (M ) is the largest ∆-filtered submodule N of M such that ∆. ssl N = i. When the underlying RUSQ algebra is the ADR algebra, there is further information about the ∆-semisimple filtrations. Theorem 3.3 ([7, Lemma 4.3, Theorem 4.4]). Let M be in mod A. Then the RA module N := HomA (G, M ) lies in F (∆) and the socle series of M determines the ∆-semisimple filtration of N . More precisely, δi (N ) = HomA (G, Soci M ) , for all i, and ∆. ssl N = LL(M ). Moreover, if Soci M/ Soci−1 M = M δi (N ) /δi−1 (N ) = ∆ (xθ , i) . θ∈Θ L θ∈Θ Lxθ , then 10 TERESA CONDE AND KARIN ERDMANN Quotients of socle series and quotients of ∆-semisimple filtrations will occur frequently, and to reduce necessary symbols, we will use the following notation: for any module M , any module N in F (∆) and i ≥ 1 we write Soci M := Soci M/ Soci−1 M and δ i (N ) := δi (N ) /δi−1 (N ) . 3.2.2. Preliminary results. We now investigate the structure of the RA -modules for algebras A such that LL(Pi ) = L for all i. Then, the minimal elements in the poset (Λ, ✂) are precisely all (k, L) for 1 ≤ k ≤ n. Soon we will also study the situation when all Pi are rigid, that is, when their radical series and their socle series coincide. Lemma 3.4. Let A be such that LL(Pi ) = L for all i, 1 ≤ i ≤ n. Then T (k, L) = Lk,L = ∆ (k, L) , for 1 ≤ k ≤ n. Proof. Recall the description of the indecomposable tilting modules in Lemma 2.2. The composition factor Lk,l has multiplicity one in both ∆ (k, l) and T (k, l), and the composition factors of these two modules are of the form Li,j , with (i, j)✂(k, l). The lemma follows from the fact that each (k, L) is a minimal element in (Λ, ✂).  Lemma 3.5. Let A be such that LL(Pi ) = L for all i, 1 ≤ i ≤ n. Let M be in F (∆) and let l be the smallest integer such that (M : ∆(k, l)) 6= 0 for some k. Then ∆. ssl M ≤ L − l + 1. Proof. We use downwards induction on l. If l = L, then all the factors in a ∆filtration of M are of the form ∆(k, L) for some k. Since two standard modules of this form have no nontrivial extensions, it follows that M is a direct sum of standard modules, so that M = δ1 (M ) and ∆. ssl M = 1. Let l < L. Assume the claim holds for modules N where a minimal l′ with (N : ∆(k, l′ )) 6= 0 is such that l < l′ ≤ L. Consider M as in the statement of the lemma. There is an exact sequence 0 δ1 (M ) M M/δ1 (M ) 0 . Note that Ext1RA (∆ (k, l) , ∆ (i, j)) = 0 for any i, k and j ≥ l. As a consequence, any ∆(k, l) which occurs in a ∆-filtration of M must occur in δ1 (M ) since it has no nontrivial extensions with any other standard module which may appear in a ∆-filtration of M . Therefore, a minimal l′ with (M/δ1 (M ) : ∆(k, l′ )) 6= 0 satisfies l < l′ . By the induction hypothesis, M/δ1 (M ) has ∆-semisimple length at most L − l′ + 1. Proposition 3.2 implies that ∆. ssl M ≤ L − l′ + 1 + 1 ≤ L − l + 1.  Proposition 3.6. Let A be such that LL(Pi ) = L for all i, 1 ≤ i ≤ n. Then, for every (k, l) in Λ, we have T (k, l) ⊆ HomA (G, SocL−l+1 Qk ) = δL−l+1 (Qk,L ) = δL−l+1 (T (k, 1)) . Proof. According to Theorem 2.4, we have that T (k, 1) = Qk,L = HomA (G, Qk ). Theorem 3.3 implies that HomA (G, SocL−l+1 Qk ) = δL−l+1 (Qk,L ) = δL−l+1 (T (k, 1)) . By part 4 of Theorem 2.5, it follows that T (k, l) ⊆ T (k, 1). By Lemma 3.5, ∆. ssl T (k, l) ≤ L − l + 1. According to part 3 of Proposition 3.2, this shows that T (k, l) is contained in δL−l+1 (T (k, 1)).  THE RINGEL DUAL OF THE ADR ALGEBRA 11 By Proposition 3.6, if all the projectives in mod A have the same Loewy length, then T (k, l) is a submodule of δL−l+1 (Qk,l ) for every (k, l) in Λ. If additionally all projectives Pi are rigid, then a ∆-filtration of T (k, l) has the same number of factors as a ∆-filtration of δL−l+1 (Qk,L ). Proposition 3.7. Suppose that A satisfies LL(Pi ) = L for all i, 1 ≤ i ≤ n. Assume that the projectives Pi are rigid. Then the monic HomRA (Pi,L , T (k, l)) HomRA (Pi,L , δL−l+1 (Qk,L )) induced by the inclusion T (k, l) ⊆ δL−l+1 (Qk,L ) is an isomorphism. In particular, the modules T (k, l) and δL−l+1 (Qk,L ) are filtered by the same number of standard modules. Proof. Recall that the K-module HomRA (Pi,L , δL−l+1 (Qk,L )) is isomorphic to the module HomA (Pi , SocL−l+1 Qk ) via the functor HomA (G, −). So consider a morphism f : Pi −→ SocL−l+1 Qk and the corresponding map f∗ = HomA (G, f ). To prove the statement, we must show that the image of f∗ is contained in T (k, l). According to [6, Lemma 5.7], it is enough to prove that all the composition factors of Im f∗ are of the form Lx,y with (x, y) ⋫ (k, l), that is, with y ≥ l. The module SocL−l+1 Qk has Loewy length L − l + 1, hence RadL−l+1 Pi ⊆ Ker f . Now, Pi is rigid and therefore RadL−l+1 Pi = Socl−1 Pi . Thus   δl−1 (Pi,L ) = HomA G, RadL−l+1 Pi ⊆ Ker f∗ . By Theorem 3.3 , the quotient Pi,L /δl−1 (Pi,L ) is only filtered by standard modules ∆(s, t) with t ≥ l, and hence all composition factors of Im f∗ are of the form Lx,y with y ≥ l. The last assertion in the statement of the proposition follows from part 6 of Theorem 2.5.  The last part of Proposition 3.7 suggests that the total number of ∆-quotients is an important invariant in this setting. If M is in F (∆), we denote the total number of ∆-quotients of M by r(M ). In this context, we shall refer to r(M ) as the rank of M ∈ F (∆). By part 6 of Theorem 2.5, the rank of M ∈ F (∆) is equal to the total number of composition factors of M which are of the form Lk,lk as k varies. Using Proposition 3.7 it is possible to compute the ∆-semisimple length of all tilting modules T (k, l) in the case when all the projectives in mod A are rigid and have the same Loewy length. Lemma 3.8. Suppose that A satisfies LL(Pi ) = L for all i, 1 ≤ i ≤ n, and assume that the projectives are rigid. Then ∆. ssl T (k, l) = min{L − l + 1, LL(Qk )}, for (k, l) ∈ Λ. In particular, if LL(Qi ) = L for all i, then ∆. ssl T (k, l) = L − l + 1 for all (k, l) ∈ Λ. In order to prove Lemma 3.8, we will apply the following general principle. Lemma 3.9. Let (B, Φ, ⊑) be a RUSQ algebra, and let M be in F (∆). Assume that N is a submodule of M satisfying r(N ) = r(M ). There is a canonical monic 12 TERESA CONDE AND KARIN ERDMANN L δ i (N ) −→ δ i (M ) for every i. If δ i (M ) ∼ = ω∈Ωi ∆(xω , yω ) (for some index set Ωi ), then M ∆(xω , yω′ ) δ i (N ) ∼ = ω∈Ωi where yω ≤ yω′ for all ω. In particular, ∆. ssl M = ∆. ssl N . Remark 3.10. Let (B, Φ, ⊑) be a RUSQ algebra. Recall from §3.2.1 that the functors δi satisfy δi (N ) = N ∩ δi (M ) for every N and M with N ⊆ M . This implies that there is a well-defined monomorphism N/δi (N ) −→ M/δi (M ), mapping n + δi (N ) to n + δi (M ). Proof of Lemma 3.9. Note that δi (N ) ⊆ δi (M ). Using part 6 of Theorem 2.5, we deduce that r(δi (N )) ≤ r(δi (M )). By Remark 3.10, we also conclude that r(N/δi (N )) ≤ r(M/δi (M )). Since r (M ) = r (δi (M )) + r (M/δi (M )) ≥ r (δi (N )) + r (N/δi (N )) = r(N ) = r(M ), it follows that r (δi (M )) = r (δi (N )) and r (M/δi (M )) = r (N/δi (N )) for all i. As a consequence, we deduce that r (δ i (M )) = r (δ i (N )) for all i. Therefore, the canonical monic in Remark 3.10 restricts to a monic δ i (N ) −→ δ i (M ) between ∆semisimple modules which must satisfy the claim in the statement of the lemma.  Proof of Lemma 3.8. By Proposition 3.7, T (k, l) is contained in δL−l+1 (Qk,L ) and they have the same rank. According to Lemma 3.9, we must have ∆. ssl T (k, l) = ∆. ssl δL−l+1 (Qk,L ). Using Theorem 3.3, we conclude that the ∆-semisimple length of Qk,L = HomA (G, Qk ) is LL(Qk ). From Proposition 3.2, we deduce that the ∆semisimple length of δL−l+1 (Qk,L ) is given by min{L − l + 1, LL(Qk )}.  We now go one step further and fully describe, in Theorem 3.12, the ∆-semisimple filtration of T (k, l) in terms of the socle series of Qk in the case when all the projectives in mod A are rigid and have the same Loewy length. For the proof of Theorem 3.12, the following result will be useful. Lemma 3.11. Let (B, Φ, ⊑) be a RUSQ algebra, and let M be in F (∆), with ∆. ssl M = m ≥ 2. Suppose that 2 ≤ i ≤ m, and let π be a split epic mapping the ∆-semisimple module δi (M ) /δi−1 (M ) onto a summand ∆ (k, l). Then the epic δi (M ) /δi−2 (M ) δ i (M ) π ∆ (k, l) does not split. Proof. Denote the canonical epic δi (M ) /δi−2 (M ) −→ δ i (M ) by ̟. Suppose, by contradiction, that π◦̟ splits. Then δi (M ) /δi−2 (M ) ∼ = Ker(π◦̟)⊕∆ (k, l). Note that Ker ̟ ⊆ Ker(π ◦ ̟), and that Ker ̟ ∼ = δ i−1 (M ). As a consequence, there is a monic from Ker ̟ ⊕ ∆ (k, l) to δi (M ) /δi−2 (M ), so Soc(δ i−1 (M )) ⊕ Lk,lk can be embedded in Soc(δi (M ) /δi−2 (M )). Using the first statement in Proposition 3.2, is easy to check that δ i−1 (M ) = δ(δi (M ) /δi−2 (M )). Since Soc N = Soc δ (N ) for any B-module N (see [7, §§3.2.2]), then Soc(δi (M ) /δi−2 (M )) = Soc(δ i−1 (M )), which leads to a contradiction.  THE RINGEL DUAL OF THE ADR ALGEBRA 13 Recall that the ∆-semisimple filtration of the projectives Pi,j is determined by the radical series of Pi when Pi is rigid: this is a consequence of Theorem 3.3. Theorem 3.12. Suppose that A satisfies LL(Pi ) = L for all i, 1 ≤ i ≤ n, and assume that the projectives are rigid. Let (k, l) ∈ Λ, and suppose that the socle layers of Qk are M L xω , Soci Qk ∼ = ω∈Ωk i for i = 1, . . . , LL(Qk ). Then δ i (T (k, l)) ∼ = M ∆ (xω , l + i − 1) , ω∈Ωk i for i = 1, . . . , ∆. ssl T (k, l). We outline the strategy of the proof of Theorem 3.12. Throughout, lL k = L for all k, and all projectives are assumed to be rigid. Suppose that Soci Qk = ω∈Ωk Lxω . i L ∆ (xω , l + i − 1), we proceed in two In order to prove that δ i (T (k, l)) ∼ = ω∈Ωk i steps. L (1) We show that δ i (T (k, l)) ∼ = ω∈Ωki ∆ (xω , yω,l ) with l + i − 1 ≤ yω,l ≤ L, for all 1 ≤ i ≤ ∆. ssl T (k, l). In particular, the rank r(δi (T (k, l))) is constant on δi (T (k, l)) as l varies, for all 1 ≤ i ≤ ∆. ssl T (k, l). (2) We show that yω,l = l + i − 1 for all ω ∈ Ωki . Proof of Theorem 3.12. Fix k and l, with 1 ≤ Lk ≤ n and 1 ≤ l ≤ L. Assume that the socle series of Qk is given by Soci Qk = ω∈Ωk Lxω . We prove the statement i of the theorem by induction on i, 1 ≤ i ≤ ∆. ssl T (k, l). Note that δ1 (T (k, l)) = ∆ (k, l). As Soc Qk = Lk , then |Ωk1 | = 1 and xω = k for ω ∈ Ωk1 , thus the claim holds trivially for i = 1. So let i be such that 2 ≤ i ≤ ∆. ssl T (k, l) and suppose, by induction, that M δ i−1 (T (k, l)) = ∆ (xω , l + i − 2) . ω∈Ωk i−1 We wish to describe δ i (T (k, l)). Step 1. Recall that LT (k, 1) = Qk,L = HomA (G, Qk ). By Theorem 3.3, δ i (T (k, 1)) is isomorphic to ω∈Ωk ∆ (xω , i). By Proposition 3.7, T (k, l) ⊆ δL−l+1 (T (k, 1)) i and r(T (k, l)) = r (δL−l+1 (T (k, 1))). By Lemma 3.9 (using Proposition 3.2), there is a canonical monic mapping δ i (T (k, l)) −→ δ i (T (k, 1)), and δ i (T (k, l)) is isomorphic to M ∆ (xω , yω,l ) , ω∈Ωk i with i ≤ yω,l ≤ L. We want to show that yω,l ≥ l + i − 1 for every ω. Take ω ′ ∈ Ωki so that the integer yω′ ,l is minimal. If yω′ ,l ≤ l + i − 2, then, by induction, (xω′ , yω′ ,l ) 6⊳ (x, y) for every standart module ∆(x, y) filtering the quotient δi (T (k, l))/δi−2 (T (k, l)). Thus, the canonical epic δi (T (k, l)) /δi−2 (T (k, l)) δ i (T (k, l)) ∆ (xω′ , yω′ ,l ) splits. This contradicts Lemma 3.11, therefore yω,l ≥ l + i − 1 for every ω. 14 TERESA CONDE AND KARIN ERDMANN L Step 2. Let U := ω∈Ωk ∆ (xω , l + i − 1), which is a submodule of δ i (T (k, 1)) and i let ι : U −→ δ i (T (k, 1)) be the inclusion map. Take ν : δi (T (k, 1)) −→ δ i (T (k, 1)) to be the canonical epic with kernel δi−1 (T (k, 1)) and consider a projective cover M π : P := Pxω ,l+i−1 −→ U. ω∈Ωk i There is a morphism f∗ : P −→ δi (T (k, 1)) such that ι ◦ π = ν ◦ f∗ . We claim that the image of f∗ is contained in T (k, l). For this we need the characterisation of T (k, l) in [6, Lemma 5.7]: this is the largest submodule of T (k, 1) such that all composition factors are of the form Lx,y with y ≥ l. Recall that f∗ = HomA (G, f ) where M f: Pxω / Radl+i−1 Pxω −→ Soci Qk . ω∈Ωk i L The image of f has Loewy length at most i, thus ω∈Ωk Radi Pxω / Radl+i−1 Pxω i is mapped to zero, and consequently   M Radi Pxω / Radl+i−1 Pxω  ⊆ Ker f∗ . HomA G, ω∈Ωk i Using that the projective indecomposable A-modules are rigid, together with Theorem 3.3, this can be rewritten as δl−1 (P ) ⊆ Ker f∗ . Therefore, the image of f∗ is a quotient of P/δl−1 (P ). From Theorem 3.3, we deduce that Im f∗ has only composition factors of the form Lx,y with y ≥ l. Hence Im f∗ ⊆ T (k, l) by a previous observation. As a consequence, Im f∗ ⊆ T (k, l) ∩ δi (T (k, 1)) = δi (T (k, l)) . Using that δi−1 (N ) = N ∩ δi−1 (M ) for every N and M with N ⊆ M , one deduces that restriction of ν to δi (T (k, l)) factors through the canonical monic δ i (T (k, l)) −→ δ i (T (k, 1)). Hence Im(ν ◦ f∗ ) can be embedded in δ i (T (k, l)). Since ι ◦ π = ν ◦ f∗ , it follows that δ i (T (k, l)) has a submodule isomorphic to U . By the conclusion of Step 1, we must have δ i (T (k, l)) ∼  = U. 3.3. Proof of Theorem A. We finally prove the main result of this paper. Theorem A. Suppose that A satisfies LL(Pi ) = LL(Qi ) = L for all i, 1 ≤ i ≤ n. Moreover, suppose that all projectives Pi and all injectives Qi are rigid. Then R (RA ) ∼ = (RAop )op . = SA ∼ The two key ingredients for the proof of Theorem A (Propositions 3.13 and 3.15) rely on the description of the ∆-semisimple filtration of the tilting modules given in Theorem 3.12. Recall that the underlying algebra A is an Artin K-algebra. Therefore the ADR algebra RA is also an Artin K-algebra, and HomRA (X, Y ) lies in mod K for X and Y in mod RA . Proposition 3.13. Suppose that A satisfies LL(Pi ) = LL(Qi ) = L for all i, 1 ≤ i ≤ n, and assume that all projectives Pi and all injectives Qi are rigid. Then the K-modules HomRA (T (k, l), T (i, j)) and HomRA (δL−l+1 (Qk,L ), δL−j+1 (Qi,L )) have the same (Jordan–Hölder) length. THE RINGEL DUAL OF THE ADR ALGEBRA 15 Proof. Denote the length of a module M in mod K by l(M ). We will use the notation in the statement of Theorem 3.12 to describe the socle layers of the injective indecomposable module Qk . We start by determining the value of l(HomRA (T (k, l), T (i, j))). Using that the functor HomRA (−, T (i, j)) preserves exact sequences in F (∆) (see [15, Corollary 4]), together with Theorem 3.12, we deduce that ∆.ssl T (k,l) l (HomRA (T (k, l) , T (i, j))) = X y=1 X l (HomRA (∆ (xω , l + y − 1) , T (i, j))) ω∈Ωk y Lemma 3.8 implies that ∆. ssl T (k, l) = L − l + 1. By Theorem 2.5, the module T (i, j) is filtered by the costandard modules ∇ (i, j) , ∇ (i, j + 1) , . . . , ∇ (i, L). Using again that Ext1RA (F (∆), F (∇)) = 0, we get L−l+1 X X l (HomRA (∆ (xω , l + y − 1) , T (i, j))) y=1 ω∈Ωk y = L−l+1 X L X X l (HomRA (∆ (xω , l + y − 1) , ∇ (i, z))) . y=1 ω∈Ωk z=j y Note that L−l+1 X L X X l (HomRA (∆ (xω , l + y − 1) , ∇ (i, z))) y=1 ω∈Ωk z=j y = L−l+1 X L X X δ(xω ,l+y−1),(i,z) l (EndRA (∆ (xω , l + y − 1))) L−l+1 X L X X δ(xω ,l+y−1),(i,z) l (EndA (Lxω )) y=1 ω∈Ωk z=j y = y=1 ω∈Ωk z=j y = L−l+1 X X δxω ,i l (EndA (Lxω )) . y=max{j−l,0}+1 ω∈Ωk y In here the second equality will follow from Lemma 3.14, and the third equality follows by analysing the values taken by the Kronecker delta. Now we calculate l(HomRA (δL−l+1 (Qk,L ), δL−j+1 (Qi,L ))). Observe that HomRA (δL−l+1 (Qk,L ) , δL−j+1 (Qi,L )) = HomRA (HomA (G, SocL−l+1 Qk ) , HomA (G, SocL−j+1 Qi )) ∼ = HomA (SocL−l+1 Qk , SocL−j+1 Qi ) , where the first equality follows from Proposition 3.6 and the second identity is due to the fact that HomA (G, −) is a fully faithful functor. Any map f : SocL−l+1 Qk −→ SocL−j+1 Qi is such that LL(Im f ) ≤ L − j + 1, so f must factor through the largest quotient of SocL−l+1 Qk whose Loewy length is at most L − j + 1. That is, f factors 16 TERESA CONDE AND KARIN ERDMANN through the module SocL−l+1 Qk / RadL−j+1 (SocL−l+1 Qk ) = SocL−l+1 Qk / Socmax{L−l+1−(L−j+1),0} Qk , where the equality follows from the rigidity of Qk . So the canonical epic SocL−l+1 Qk −→ SocL−l+1 Qk / Socmax{j−l,0} Qk induces an isomorphism of K-modules
HomA SocL−l+1 Qk / Socmax{j−l,0} Qk , SocL−j+1 Qi ∼ = HomA (SocL−l+1 Qk , SocL−j+1 Qi ) . Notice that both SocL−l+1 Qk / Socmax{j−l,0} Qk and SocL−j+1 Qi are modules over A/ RadL−j+1 A. In fact, SocL−j+1 Qi is an injective in mod(A/ RadL−j+1 A). Thus the restriction of HomA (−, SocL−j+1 Qi ) to mod(A/ RadL−j+1 A) yields an exact functor. Therefore l (HomRA (δL−l+1 (Qk,L ) , δL−j+1 (Qi,L ))) = l HomA SocL−l+1 Qk / Socmax{j−l,0} Qk , SocL−j+1 Qi = L−l+1 X X l (HomA (Lxω , SocL−j+1 Qi )) L−l+1 X X l (HomA (Lxω , Li )) L−l+1 X X δxω ,i l (EndA (Lxω )) ,

y=max{j−l,0}+1 ω∈Ωk y = y=max{j−l,0}+1 ω∈Ωk y = y=max{j−l,0}+1 ω∈Ωk y which shows that HomRA (δL−l+1 (Qk,L ), δL−j+1 (Qi,L )) and HomRA (T (k, l), T (i, j)) have the same length over K.  Lemma 3.14. Let (B, Φ, ⊑) be a RUSQ algebra (over K). Then EndB (∆ (i, j + 1)) ∼ = EndB (∆ (i, j)) = HomB (∆ (i, j + 1) , ∆ (i, j)) ∼ as K-modules. If B is the ADR algebra RA of an Artin algebra A then the modules above are isomorphic to EndA (Li ). Proof. Consider the short exact sequence in mod B 0 ∆ (i, j + 1) ∆ (i, j) Li,j 0. By applying the functor HomB (∆ (i, j + 1) , −) to this exact sequence we deduce that EndB (∆ (i, j + 1)) ∼ = HomB (∆ (i, j + 1) , ∆ (i, j)). Using HomB (−, ∆ (i, j)), we get an exact sequence 0 EndB (∆ (i, j)) HomB (∆ (i, j + 1) , ∆ (i, j)) Ext1B (Li,j , ∆ (i, j)) . Note that Ext1B (Li,j , ∆ (i, j)) = 0. If this was not the case, there would exist a module M with socle Li,li , having a unique composition factor of type Lx,lx and satisfying [M : Li,j ] = 2. According to parts 6 and 7 of Theorem 2.5, M would have THE RINGEL DUAL OF THE ADR ALGEBRA 17 to be a standard module. This cannot happen as [M : Li,j ] = 2. This shows that the K-modules EndB (∆ (i, j)) and HomB (∆ (i, j + 1) , ∆ (i, j)) are isomorphic. For the claim about RA , recall that ∆ (i, 1) = HomA (G, Li ). Since the functor  HomA (G, −) is fully faithful, EndRA (∆ (i, 1)) ∼ = EndA (Li ). By Proposition 3.6, the RA -module T (k, l) is contained in δL−l+1 (Qk,L ). We will show that the maps in HomRA (δL−l+1 (Qk,L ) , δL−j+1 (Qi,L )) give rise to maps in HomRA (T (k, l) , T (i, j)) via restriction. This is the final piece needed to prove Theorem A. Proposition 3.15. Suppose that A satisfies LL(Pi ) = LL(Qi ) = L for all i, 1 ≤ i ≤ n, and assume that all projectives Pi and all injectives Qi are rigid. Consider a morphism f∗ : δL−l+1 (Qk,L ) −→ δL−j+1 (Qi,L ) . Then f∗ (T (k, l)) ⊆ T (i, j). Proof. Because HomA (G, −) is a full functor, then f∗ = HomA (G, f ) for a map f : SocL−l+1 Qk −→ SocL−j+1 Qi in mod A. Note that Ker f ⊇ RadL−j+1 (SocL−l+1 Qk ) = Socmax{j−l,0} Qk as Qk is a rigid module. Write z := max{j − l, 0}. Since T (k, l) ⊆ δL−l+1 (Qk,L ) and z ≤ L − l + 1, then δz (T (k, l)) ⊆ δz (δL−l+1 (Qk,L )) = δz (Qk,L ) . Observe that, δz (Qk,L ) = HomA (G, Socz Qk ) ⊆ HomA (G, Ker f ) = Ker f∗ , so δz (T (k, l)) is contained in the kernel of f∗ |T (k,l) . In other words, the module f∗ (T (k, l)) is isomorphic to a quotient of T (k, l)/δz (T (k, l)). Theorem 3.12 implies that all composition factors of T (k, l)/δz (T (k, l)) are of the form Lx,y , with y ≥ l + z ≥ j. Therefore all composition factors of f∗ (T (k, l)) are of the form Lx,y , with (x, y) 6⊲ (i, j). By Lemma 5.7 in [6], the module f∗ (T (k, l)) must be contained in T (i, j).  Proof of Theorem A. Consider the morphism of Artin K-algebras     M M δj (Qi,L ) −→ EndRA  ϕ : EndRA  T (i, j) = R (RA ) op , (i,j)∈Λ (i,j)∈Λ L L which sends each map g ∈ EndRA ( ni=1 L j=1 δL−j+1 (Qi,L )) to the corresponding L L restriction to ni=1 L T (i, j). According to Proposition 3.15, ϕ is well defined. j=1 Moreover, if g 6= 0 then ϕ(g) 6= 0, as the modules δL−j+1 (Qi,L ) have simple socle. So ϕ is an injective morphism of K-algebras, and in particular, a monomorphism of modules in mod K. Proposition 3.13 implies that ϕ is a bijection. As δj (Qi,L ) = HomA (G, Socj Qi ), then     n M L M M Socj Qi  = (SA )op , δj (Qi,L ) ∼ EndRA  = EndA  (i,j)∈Λ i=1 j=1 using that HomA (G, −) is a fully faithful functor. Thus the algebras R(RA ) and SA are isomorphic. The identity SA ∼ = (RAop )op was established in Subsection 3.1.  18 TERESA CONDE AND KARIN ERDMANN Remark 3.16. Note that the class of connected selfinjective algebras with radical cube zero but radical square nonzero satisfies the conditions of Theorem A. This class contains several important examples and was studied in [13]. Remark 3.17. Let A be an Artin algebra satisfying A ∼ = Aop . Suppose further that A has rigid projectives and injectives, and assume they all have the same Loewy length. Then, by Theorem A (3) R (RA ) ∼ = (RA ) op . = (RAop ) op ∼ = SA ∼ In particular, the identity (3) holds when A is a block of weight 1 of a symmetric group algebra. According to [17], blocks of symmetric group algebras of weight 2 must also satisfy the identity (3) when the base field has characteristic p > 2. Take any group algebra KG where G is a finite p-group and K is a field of characteristic p. In this setting, KG is a local symmetric algebra, and the only projective indecomposable KG-module is rigid: this follows from Jennings’ Theorem (see [14], and also Theorem 3.14.6 and Corollary 3.14.7 in [3]). Thus, the identity (3) holds for A = KG. Finally, note that (3) also holds when A is a preprojective algebra of type An . 4. Cartan matrices and multiplicities In this section we describe and compare the Cartan matrices of the algebras RA , R (RA ) and SA . Our ultimate goal is to demonstrate that the assumptions in the statement of Theorem A are, in a certain sense, the minimal requirements for this result to hold. To make our arguments simpler, we will work, throughout this section, with finite-dimensional K-algebras A satisfying dim EndA (L) = 1 for every simple module L. We start by setting some notation. If M is a module, we write [M ] for its image in the Grothendieck group G0 (A). Recall that a complete list of pairwise nonisomorphic simple A-modules Li , with i = 1, . . . , n, gives rise to the Z-basis {[Li ] : i = 1, . . . , n} of G0 (A). The Cartan matrix of A will be denoted by C(A). This is the n×n matrix whose column with label j has the composition factors of the projective Pj . Here the integer n corresponds to the number of isomorphism classes of simple A-modules. The entry ij of C(A) is given by [Pj : Li ]. Due to our assumptions about the simple A-modules, we have that [Pj : Li ] = [Qi : Lj ], so the composition factors of the injective incomposable A-module Qi are recorded in the ith row of C(A). Using Lemma 3.14, our assumptions about A, and basic properties of quasihereditary algebras, it is not difficult to conclude that the ADR algebra RA of A still satisfies dim EndRA (Li,j ) = 1 for every (i, j) in Λ. By similar arguments, the corresponding algebras R (RA ) and SA also satisfy the respective condition on simple modules. 4.1. The Cartan matrix of RA . The column of C(RA ) associated with the label (k, l) encodes the composition factors of the projective Pk,l , whereas the row with label (i, j) describes the composition factors of Qi,j . THE RINGEL DUAL OF THE ADR ALGEBRA 19 4.1.1. Projective RA -modules. Each Pk,l is filtered by standard modules. According to Theorem 3.3, the multiplicity of ∆ (i, j) in Pk,l corresponds to the multiplicity of the simple A-module Li in the j th socle layer of Pk / Radl Pk . That is, (Pk,l : ∆ (i, j)) = [Socj (Pk / Radl Pk ) : Li ]. We know the composition factors of the modules ∆ (i, j) (these are Li,j , . . . Li,li ). Hence the Cartan matrix C(RA ) is completely determined by the socle series of the radical quotients of the projective A-modules. The converse of the previous statement is also true: the socle quotients of Pk / Radl Pk can be read off from the Cartan matrix of RA . To see this, note that [Pk,l : Li,j ] counts the number of factors ∆ (i, y), 1 ≤ y ≤ j, in a ∆-filtration of Pk,l . Therefore, (4) [Pk,l : Li,j ] − [Pk,l : Li,j−1 ] = (Pk,l : ∆ (i, j)) = [Socj (Pk / Radl Pk ) : Li ], for j > 1. We also deduce the following identity (5) [Pk,l : Li,j ] = j X (Pk,l : ∆ (i, y)) = [Socj (Pk / Radl Pk ) : Li ]. y=1 4.1.2. Injective RA -modules. By parts 4 and 5 of Theorem 2.5, Qi,j has a ∇filtration with quotients ∇(i, y) for 1 ≤ y ≤ j, each of these occurring exactly once. Therefore, the composition factors of the costandard RA -modules can be totally described in terms of the rows of C(RA ). Namely, [Qi,1 ] =[∇(i, 1)] [Qi,2 ] =[∇(i, 1)] + [∇(i, 2)] .. . [Qi,li ] =[∇(i, 1)] + [∇(i, 2)] + · · · + [∇(i, li )]. Lemma 4.1. We have that [∇ (i, j)] = [Qi,j ] − [Qi,j−1 ] for 1 < j ≤ li and [Qi,1 ] = [∇(i, 1)], that is, for j > 1, the composition factors of ∇ (i, j) can be computed by subtracting the row (i, j − 1) from the row (i, j) of C(RA ). We can also describe the composition factors of the tilting RA -modules T (i, j) using the Cartan matrix of RA . By part 5 of Theorem 2.5, [T (i, j)] = [Qi,li ] − [Qi,j−1 ]. That is, one can compute the composition factors of T (i, j) by taking the difference of two rows in C(RA ). 4.2. The Cartan matrix of R (RA ). The Cartan matrix C(R (RA )) of R (RA ) has entries
′ ′ ′ (6) [Pk,l : L′i,j ] = dim HomR(RA ) Pi,j , Pk,l = dim HomRA (T (i, j) , T (k, l)) . Note that dim HomRA (T (i, j) , T (k, 1)) = dim HomRA (T (i, j) , Qk,lk ) = [T (i, j) : Lk,lk ]. 20 TERESA CONDE AND KARIN ERDMANN Using that Ext1RA (F (∆) , F (∇)) vanishes, together with Theorem 2.5, we deduce that dim HomRA (T (i, j) , T (k, l)) = dim HomRA (T (i, j) , Qk,lk ) − dim HomRA (T (i, j) , Qk,l−1 ) = [T (i, j) : Lk,lk ] − [T (i, j) : Lk,l−1 ] for l > 1. That is, the entries of C(R(RA )) are given by ′ [Pk,l : L′i,j ] = [T (i, j) : Lk,lk ] − [T (i, j) : Lk,l−1 ] for l > 1. The identity Qi,j−1 ∼ = Qi,li /T (i, j) implies the following result. Corollary 4.2. We have ′ (7) [Pk,l : L′i,j ] = [Qi,li : Lk,lk ] − [Qi,j−1 : Lk,lk ] − [Qi,li : Lk,l−1 ] + [Qi,j−1 : Lk,l−1 ], where Qi,0 := 0 and [N : Lk,0 ] := 0 for N in mod R(RA ). In particular, the Cartan matrix of R (RA ) is determined by the Cartan matrix of RA . Remark 4.3. The result above can be stated more generally for RUSQ (and dually for LUSQ) algebras. In fact, if B is a RUSQ algebra satisfying dim EndB (L) = 1 for every simple module L, then the Cartan matrix of R(B) is determined by the Cartan matrix of B via the formula in (7). To deduce that the Cartan matrix of R(B) is determined by the Cartan matrix of B for B a LUSQ algebra, note that: B op is RUSQ, C(B op ) = C(B)T and R (B op ) ∼ = R (B) op . 4.3. The Cartan matrix of SA . The Cartan matrix of SA has entries SA SA SA , Pk,l [Pk,l ) = dim HomA (Socj Qi , Socl Qk ) . : LSi,jA ] = dim HomSA (Pi,j Observe that HomA (Socj Qi , Socl Qk ) ∼ = HomA (Socj Qi / Radl (Socj Qi ) , Socl Qk ), and Socl Qk is an injective indecomposable A/ Radk A-module. Therefore, we have the following formula: (8) SA [Pk,l : LSi,jA ] = [Socj Qi / Radl (Socj Qi ) : Lk ]. 4.4. Comparing C(R(RA )) with C(SA ). In Subsection 3.1, we have looked at some facts hinting at a relationship between the quasihereditary algebras (R(RA ), Λ, ✂op ) and (SA , ΛAop , ✂Aop ). In particular, we have seen that it would be reasonable to require that LL(Pi ) = LL(Qi ) = li for all i, in order to have an isomorphism between R(RA ) and SA . This requirement would at least assure that |Λ| = |ΛAop |. We have also seen that it ′ would be natural to map an idempotent ξ(i,j) in R(RA ) associated with the label (i, j) to an idempotent ε[i,li −j+1] in SA associated with the label [i, li − j + 1]. For this correspondence to yield an isomorphism, the numbers ′ ′ ′ = [Pk,l : L′i,j ] R(RA )ξ(k,l) dim ξ(i,j) SA dim ε[i,li −j+1] SA ε[k,lk −l+1] = [Pk,l : LSi,lAi −j+1 ] k −l+1 should coincide. That is, the entry (i, j)(k, l) of C(R(RA )) must match with the entry [i, li − j + 1][k, lk − l + 1] of C(SA ) for every i,k,j and l. Our aim is to show that the assumptions in the statement of Theorem A are somehow minimal. For this, consider the conditions: THE RINGEL DUAL OF THE ADR ALGEBRA 21 (B1): LL(Pi ) = LL(Qi ) = li for all 1 ≤ i ≤ n; SA ′ (B2): [Pk,l : L′i,j ] = [Pk,l : LSi,lAi −j+1 ] for all 1 ≤ i, k ≤ n, 1 ≤ j ≤ li and k −l+1 1 ≤ l ≤ lk . We wish to prove the following result. Theorem B. Let A be a finite-dimensional connected K-algebra. Suppose that dim EndA (Li ) = 1 for every i, and assume that (B1) and (B2) hold. Then: (1) all the Loewy lengths li are the same (i.e. li = lk for all i and k); (2) each projective Pi is rigid; (3) each injective Qi is rigid. We prove this theorem in a number of steps. Lemma 4.4. Assume (B1) and (B2). Then for all i we have T (i, li ) = ∆ (i, li ) = ∇ (i, li ) = Li,li . Proof. According to part 5 of Theorem 2.5, we have T (i, li ) = ∇ (i, li ). We want to show that ∇ (i, li ) is a simple module. Note that [∇(i, li ) : Li,j ] = 0 for j 6= li , as (i, li ) ✁ (i, j) for j 6= li . If the module T (i, li ) = ∇ (i, li ) is not simple, then it has some factor ∆ (k, l) for k 6= i. That is, HomA (T (i, li ), T (k, l)) 6= 0 for some k 6= i. By (6), (B2) and (8), we have then that Lk occurs in Soc1 Qi , which is a contradiction because i 6= k.  The next proposition will be crucial in the proof of part 1 of Theorem B. Proposition 4.5. Assume that (B1) and (B2) hold for A and suppose that i 6= k. If Ext1A (Li , Lk ) 6= 0 then lk ≤ li . Proof. By assumption, we must have [Soc2 Qk / Rad(Soc2 Qk ) : Li ] 6= 0. Using (8), (B2) and (6), we deduce that HomRA (T (k, lk − 1), T (i, li )) 6= 0. From Lemma 4.4, it follows that [T (k, lk − 1) : Li,li ] 6= 0. Parts 4 and 5 of Theorem 2.5 imply that either [T (k, lk ) : Li,li ] 6= 0, or [T (k, lk − 1)/T (k, lk ) : Li,li ] 6= 0, where the quotient T (k, lk − 1)/T (k, lk ) is isomorphic to ∇(k, lk − 1). Using that i 6= k, together with Lemma 4.4, we conclude that [T (k, lk ) : Li,li ] = 0. Thus [∇(k, lk − 1) : Li,li ] 6= 0. As a consequence, (Pi,li : ∆(k, lk − 1)) 6= 0, using Brauer– Humphreys reciprocity for quasihereditary algebras (see [11, Lemma 2.5]). The identity (4) implies that [Soclk −1 Pi : Lk ] 6= 0. So Pi has Loewy length at least lk − 1. In fact, as i 6= k, one deduces that li > lk − 1, or equivalently lk ≤ li .  A key argument in proof of Theorem B is the fact that the conditions (B1) and (B2) are ‘symmetric’. We give an informal explanation for this phenomenon. The axioms (B1) and (B2) are saying that C(R(RA )) and C(SA ) coincide (up to a suitable simultaneous permutation of rows and columns). As explained in Remark 4.3, the matrix C(R(B)) is determined by C(B) when B is a LUSQ algebra. So the Cartan matrices of the algebras R(R(RA )) ∼ = RA and R(SA ) should still coincide when (B1) and (B2) hold for A. Recall the discussion in Subsection 3.1. Note that RA ∼ = R(RAop )op , = R((RAop )op ) ∼ = (SAop )op and R(SA ) ∼ 22 TERESA CONDE AND KARIN ERDMANN therefore C(SAop ) = C(RA )T and C(R(RAop )) = C(R(SA ))T . It seems then natural that (B1) and (B2) hold for the underlying algebra A if and only if (B1) and (B2) hold for Aop . Lemma 4.6. Let A be a finite-dimensional K-algebra, and let dim EndA (Li ) = 1 for every i. Assume that (B1) and (B2) hold for A. Then Aop is a finiteop ) = 1 for every i. Moreover, the dimensional K-algebra satisfying dim EndAop (LA i conditions (B1) and (B2) hold for Aop . Proof. The first part of the statement of the lemma is evident. It is also clear that Aop satisfies (B1). We show that (B2) holds for Aop . Using Corollary 4.2, duality, and condition (B2) for A, we get R(RAop ) [Pk,l R(RAop ) : Li,j RA = [Qi,l i op ] RA RA RA RA ] − [Qi,l : Lk,l ] − [Qi,j−1 : Lk,l i k k op op op op RA RA RA ] : Lk,l−1 ] + [Qi,j−1 : Lk,l−1 op op op SA SA SA SA A A = [Pi,l : LSk,lAk ] − [Pi,j−1 : LSk,lAk ] − [Pi,l : LSk,l−1 ] + [Pi,j−1 : LSk,l−1 ] i i ′ ′ ′ ′ = [Pi,1 : L′k,1 ] − [Pi,l : L′k,1 ] − [Pi,1 : L′k,lk −l+2 ] + [Pi,l : L′k,lk −l+2 ]. i −j+2 i −j+2 By applying Corollary 4.2 to the last expression, it follows that R(RAop ) [Pk,l R(RAop ) : Li,j ] =[Qk,lk : Li,li ] − ([Qk,lk : Li,li ] − [Qk,lk : Li,li −j+1 ]) − ([Qk,lk : Li,li ] − [Qk,lk −l+1 : Li,li ]) + [Qk,lk : Li,li ] − [Qk,lk −l+1 : Li,li ] − [Qk,lk : Li,li −j+1 ] + [Qk,lk −l+1 : Li,li −j+1 ] SA SA ]. : Li,l =[Qk,lk −l+1 : Li,li −j+1 ] = [Pk,l i −j+1 k −l+1 op op This concludes the proof of the proposition.  Corollary 4.7. Assume that (B1) and (B2) hold for A and suppose that i 6= k. If Ext1A (Li , Lk ) 6= 0 then lk = li . Proof. The inequality lk ≤ li follows by applying Proposition 4.5 to A. According to Lemma 4.6, (B1) and (B2) also hold for Aop . Moreover, note that op Aop Ext1Aop (LA ) 6= 0. The inequality li ≤ lk then follows by applying Proposik , Li tion 4.5 to Aop .  We finally prove Theorem B. Proof of Theorem B. We start by showing that all li must be equal. For every distinct i and k in {1, . . . , n} there exists a sequence (i1 , . . . , im ) with i1 = i, im = k and 1 ≤ ix ≤ n, satisfying the following property: for each 1 ≤ x < m, either Ext1A (Lix , Lix+1 ) 6= 0 or Ext1A (Lix+1 , Lix ) 6= 0. Part 1 is then an easy consequence of Corollary 4.7. Let now L be the common Loewy length of all projectives and injectives. Note that ′ [Pk,1 : L′i,j ] = [Qi,L : Lk,L ] − [Qi,j−1 : Lk,L ] = [Pk,L : Li,L ] − [Pk,L : Li,j−1 ] = [SocL (Pk / RadL Pk )/ Socj−1 (Pk / RadL Pk ) : Li ] = [Pk / Socj−1 Pk : Li ]. THE RINGEL DUAL OF THE ADR ALGEBRA 23 Here, we have used Corollary 4.2, duality, and the identity in (5). Similarly, SA A [Pk,L : LSi,L−j+1 ] = [SocL−j+1 Qi : Lk ] = [Pk / RadL−j+1 Pk : Li ], where the first equality follows from (8), and the second equality follows by duality for A/ RadL−j+1 A. By (B2), the multiplicities [Pk / Socj−1 Pk : Li ] and [Pk / RadL−j+1 Pk : Li ] coincide for every i, k and j. As a consequence, the modules Socj−1 Pk and RadL−j+1 Pk must have the same Jordan-Hölder length. Since RadL−j+1 Pk is contained in Socj−1 Pk , then these modules are actually equal. Therefore Pk is a rigid module for every k. Observe that Aop is a finite-dimensional connected K-algebra. Moreover, Aop op ) = 1 for every i. According to Corollary 4.7, Aop also satisfies dim EndAop (LA i satisfies axioms (B1) and (B2). Thus, by the previous, the projective Aop -modules op PkA = D(Qk ) must be rigid. As a consequence, every injective indecomposable A-module is rigid.  5. Ringel selfdual ADR algebras We wish to characterise Ringel selfdual ADR algebras. First, the notion of Ringel selfduality must be rigorously defined. We say that two quasihereditary algebras (B, Φ, ⊑) and (C, Ψ, 4) are equivalent if the respective categories F (∆) and F (∆C ) are equivalent. A quasihereditary algebra (B, Φ, ⊑) is Ringel selfdual if the algebras (B, Φ, ⊑) and (R(B), Φ, ⊑ op ) are equivalent. It is not unusual for a quasiherederitary algebra (B, Φ, ⊑) to be Ringel selfdual. This phenomenon is frequently observed in quasihereditary algebras and highest weight categories arising from the theory of semisimple Lie algebras and algebraic groups. Therefore, it is natural to ask which quasihereditary algebras are Ringel selfdual. As pointed out in [16, Appendix, A.2], if (B, Φ, ⊑) is both right and left strongly quasihereditary, then B has global dimension at most 2. For this reason, one should not expect that right strongly quasihereditary algebras are often Ringel selfdual. In particular, one should not expect that RA is Ringel selfdual. We give necessary and sufficient conditions for an ADR algebra to be Ringel selfdual. Theorem C. The algebra (RA , Λ, ✂) is Ringel selfdual if and only if A is a selfinjective Nakayama algebra. Proof. If RA is Ringel selfdual, then the indecomposable tilting modules over RA must coincide with the indecomposable tilting modules over R(RA ). The RA -modules Pi,1 , i = 1. . . . , n, form a complete list of projective indecomposable modules isomorphic to a standard module (see [6], Propositions 3.1 and 3.4). ′ ∼ Consider now the Ringel dual R(RA ) of RA . By Theorem 2.6, Pi,1 = T ′ (i, li ) ′ has a unique ∆ -filtration, given by ′ ′ ′ ⊂ · · · ⊂ Pi,j ⊂ · · · ⊂ Pi,1 = T ′ (i, li ) , 0 ⊂ Pi,l i ′ ′ ′ ′ ′ ∼ /Pi,j+1 . The modules Pi,l , i = 1. . . . , n, Pi,j /Pi,j+1 = Pi,1 = ∆′ (i, j) and T ′ (i, j) ∼ i form a complete list of projective modules isomorphic to a standard module. By the previous observation about the algebra RA , there must be a bijective correspondence between the labels (i, 1) in mod RA and the labels (k, lk ) in mod R(RA ). 24 TERESA CONDE AND KARIN ERDMANN Consequently, each tilting RA -module T (i, 1) = Qi,li must correspond bijectively ′ to some tilting R(RA )-module of the form T ′ (k, lk ) = Pk,1 . By the involutive properties of the Ringel duality (see the proof of Theorem 6 and Lemma 7 in [15]), ′ coincides then with some each projective R(RA )-module HomRA (T, T (k, 1)) ∼ = Pk,1 ′ ′ projective RA -module HomR(RA ) (T , T (x, lx )) ∼ = Px,lx . Therefore, each injective RA -module Qi,li is isomorphic to some projective RA -module of type Px,lx . By definition, we have Px,lx = HomA (G, Px ), and by Theorem 2.4, Qi,li is isomorphic to HomA (G, Qi ). Since the functor HomA (G, −) is fully faithful, it follows that Qi ∼ = Px . Thus, A must be a selfinjective algebra. According to previous observations we also know that gl. dim RA ≤ 2. Therefore, Rad A lies in add G by Proposition 2 in [18]. Since A is selfinjective, the property Rad A ∈ add G implies that the projective(-injective) indecomposable A-modules are uniserial. So A is a selfinjective Nakayama algebra. The converse is a well-known result, and a proof can be found in [20]. Alternatively, note that every connected selfinjective Nakayama algebra A satisfies the assumptions in the statement of Theorem A, hence we have a structure-preserving isomorphism between R(RA ) and SA . Now observe that SA ∼ = RA as A is a selfinjective Nakayama algebra. Thus, the ADR algebra of a connected selfinjective Nakayama algebra is Ringel selfdual. Note that ADR algebras and Ringel duals are well behaved with respect to the direct product of algebras, that is RA1 ×A2 ∼ = R(B1 ) × R(B2 ). Using that an arbi= RA1 × RA2 and R(B1 × B2 ) ∼ trary selfinjective Nakayama algebra is the direct product of connected selfinjective Nakayama algebras, we deduce that the ADR algebra of a selfinjective Nakayama algebra is Ringel selfdual.  Remark 5.1. Note that if A is a Nakayama algebra, then the ADR algebra of A coincides with its Auslander algebra. References 1. M. Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177– 268; ibid. 1 (1974), 269–310. 2. M. Auslander, I. Reiten, and S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997, Corrected reprint of the 1995 original. 3. D. J. Benson, Representations and cohomology. 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