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European Journal of Mathematics

, Volume 4, Issue 3, pp 1100–1140 | Cite as

Ringel duality for certain strongly quasi-hereditary algebras

  • Martin Kalck
  • Joseph Karmazyn
Open Access
Research Article

Abstract

We study quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-hereditary algebras. As special cases, we obtain a Ringel duality formula for a family of strongly quasi-hereditary algebras arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander–Dlab–Ringel algebras, and for Eiriksson and Sauter’s nilpotent quiver algebras when the quiver has no sinks and no sources. We also recover Tan’s result that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual.

Keywords

Quasi-hereditary algebras Ringel duality Monomial algebras Knörrer invariant algebras 

Mathematics Subject Classification

16S50 16G10 

1 Introduction

Quasi-hereditary algebras form an important class of finite dimensional algebras with relations to Lie theory (this was the original motivation [10]) and exceptional sequences in algebraic geometry (see e.g. [9, 23]). Examples of quasi-hereditary algebras include blocks of category Open image in new window and Schur algebras.

Ringel duality [34] is a fundamental phenomenon in the theory of quasi-hereditary algebras, see for example [8, 13, 14, 15, 19, 22, 26, 28, 33] for (recent) work on this topic. For any quasi-hereditary algebra A there exists a quasi-hereditary algebra \(\mathfrak {R}(A)\), the Ringel dual of A, such that
$$\begin{aligned} A\text {-mod} \cong \mathfrak {R}(\mathfrak {R}(A))\text {-mod}. \end{aligned}$$
However, computing the Ringel dual of a quasi-hereditary algebra explicitly may not be straightforward. In this paper we introduce a new class of quasi-hereditary algebras that admit a uniform description of their Ringel duals, see Theorem 1.2.

Let us make this more precise. Let k be an algebraically closed field, and R be a finite dimensional monomial k-algebra, i.e. \(R=kQ/I\), where I is a two-sided ideal generated by paths in Q. For example Open image in new window , where I is a two-sided ideal generated by monomials in Open image in new window .

Definition 1.1

We call R ideally ordered, if for every primitive idempotent \(e \in R\) and every pair of monomials \(m, n \in eR\) there exists an epimorphism \(Rm \rightarrow Rn\) or an epimorphism \(Rn \rightarrow Rm\).

For an algebra R we consider the additive subcategory of all torsionless R-modulesdefine Open image in new window to be the direct sum of all indecomposable modules in Open image in new window up to isomorphism, and setFor submodules \(\Lambda \subset R\) we define the layer function Open image in new window and we call l the ideal layer function. For an ideally ordered algebra R the isomorphism classes of submodules \(\Lambda \subset R\) label the simple modules \(S(\Lambda )\) of Open image in new window and so the ideal layer function induces a partial ordering on the simple Open image in new window -modules: \(S(\Lambda _1) \leqslant S(\Lambda _2)\) \(\Leftrightarrow \) \(l(\Lambda _1) \leqslant l(\Lambda _2)\). We call this the ideal layer ordering.

The following is the main result of this paper and calculates the Ringel dual for algebras of the form Open image in new window . See Theorem 5.1 for a more detailed version.

Theorem 1.2

Let R be a finite dimensional ideally ordered monomial algebra. Then Open image in new window is quasi-hereditary with respect to the ideal layer ordering, has global dimension \(\leqslant 2\), and has Ringel dual \(E_{R^{\mathrm{op}}}^{\mathrm{op}}\):

Remark 1.3

As we were preparing to post this paper on the arXiv we became aware of the very recent paper [14] of Coulembier that had just appeared. This paper introduces a more general version of the Auslander–Dlab–Ringel construction and proves a Ringel duality formula in this setting. In particular, this generalises the Ringel duality formula of Conde and Erdmann [13] that we discuss below.

Our construction appears to be a special case that fits into this more general framework which, in particular, implies the Ringel duality formula of Theorem 1.2. However, the approach and proof in Coulembier’s work is different to the one in this paper. The work of Coulembier also seems to answer the questions we raise in Remark 5.3 (1) and at the end of Sect. 6.3 regarding the possibility of finding a more general framework in which a Ringel duality formula holds.

In light of this, the results of this paper can be thought of as providing a very explicit example of Coulembier’s Ringel duality formula, linking to several geometrically inspired examples such as Knörrer invariant algebras, and proving further properties that hold in our special case of the algebras Open image in new window such as being simultaneous left and right strongly quasi-hereditary for the same quasi-hereditary order and being left ultra strongly quasi-hereditary.

The class of ideally ordered monomial algebras includes many well known examples, and in many of these examples the endomorphism algebras \(E_{R}\) are also well understood.

Example 1.4

The following families of finite dimensional monomial algebras are ideally ordered.
  1. (0)

    Hereditary algebras.

     
  2. (1)

    The algebras Open image in new window for positive integers lm.

     
  3. (2)

    More generally, for Q a finite quiver, \(J \subseteq kQ\) the two-sided ideal generated by all arrows in Q, and \(m \geqslant 0\) the algebra Open image in new window is ideally ordered.

    To prove this, consider a monomial \(p \in eR\). There is a surjection \(Re \rightarrow Rp\) given by \(g \mapsto gp\) with kernel where i is minimal such that \(p \in J^i\). Hence for any monomial \(p \in eR\) there is an isomorphism \(Rp \cong Re/J^le\) for some Open image in new window . As a result, for any pair of monomials \(p,q \in eR\) the monomial ideals RpRq are isomorphic to some pair of quotient modules occurring in the chain of surjections
    $$\begin{aligned} Re \cong Re/J^{m}e \rightarrow Re/J^{m-1}e \rightarrow \cdots \rightarrow Re/J^{1}e. \end{aligned}$$
    Hence there is a surjection \(Rp \rightarrow Rq\) or \(Rq \rightarrow Rp\).
     
  4. (3)

    For every pair \(0<a<r\) of coprime integers the finite dimensional monomial Knörrer invariant algebra \(K_{r,a}\) is defined in [27, Definition 4.6], and the results of [27, Section 6.4] describe its monomial ideals and imply that it is ideally ordered. The definition of these algebras is recapped in Sect. 6.1.

     
  5. (4)

    Nakayama algebras, introduced in [31], are ideally ordered.

     
We give two constructions that can be used to produce ideally ordered monomial algebras.
  1. (5)
    Let R and K be ideally ordered monomial algebras and let \(_R M_K\) be an R-K-bimodule which is projective as R-module and as K-module. Then is an ideally ordered monomial algebra. Example 2.8 (a) shows that T need not be ideally ordered if we weaken the assumptions on \(_R M_K\).
     
  2. (6)

    If R is ideally ordered and \(e \in R\) is an arbitrary idempotent, then eRe is ideally ordered.

    Suppose that \(f \in eRe\) is a primitive idempotent and \(p,q \in feRe=fRe\) are monomials. Then f is a primitive idempotent in R, \(p,q \in fR\) are monomials, and as R is ideally ordered there is a surjection between Rp and Rq. Applying to this surjection of R-modules will produce the required surjection of eRe-modules between eRp and eRq since Open image in new window is exact. This shows eRe is ideally ordered.
     
We finish by exhibiting a local commutative monomial algebra which is not ideally ordered.
  1. (7)

    The algebra Open image in new window is not ideally ordered. To see this consider the ideals Rx and Ry.

     

We briefly discuss how these examples of ideally ordered monomial algebras R, and the algebras Open image in new window they define, relate to algebras and results in the literature.

1.1 Hille and Ploog’s algebras

The Ringel duality formula of Theorem 1.2, the definition of ideally ordered monomial algebras, and the construction of the algebras Open image in new window in this paper are all geometrically inspired. They were first observed in our previous work [27] for a class of quasi-hereditary algebras \(\Lambda _\alpha \) constructed by Hille and Ploog [24].

In more detail, the algebras \(\Lambda _{\varvec{\alpha }}\) arise from an exceptional collection of line bundles associated to a type A configuration of intersecting rational curves \(C_i\) in a rational, projective surface as illustrated in the picture below.The construction of \(\Lambda _{\varvec{\alpha }}\) (recapped in Sect. 6.1) depends on the order of the curves \(C_i\). Reversing the order of these curves, Hille and Ploog’s construction yields an algebra \(\Lambda _{\varvec{\alpha }^{\vee }}\).

It is natural to ask how the algebras \(\Lambda _{\varvec{\alpha }}\) and \(\Lambda _{\varvec{\alpha }^{\vee }}\) are related from a representation theoretic perspective. Our answer below is phrased in terms of Ringel duality.

Preposition 1.5

There is an isomorphism of algebras
$$\begin{aligned} \mathfrak {R}(\Lambda _{\varvec{\alpha }}) \cong \Lambda _{\varvec{\alpha }^{\vee }}^{\mathrm{op}}. \end{aligned}$$
(1)
In order to see that (1) is a special case of our main Theorem 1.2, we recall that there are isomorphisms of algebrasdescribed in [27, Section 6]. This is recalled in Proposition 6.7 and the discussion immediately beneath it. Here, \(K_{r, a}\) denotes a Knörrer invariant algebra, which is the ideally ordered monomial in Example 1.4 (3), and \(0< a < r\) are a pair of coprime integers depending on \(\varvec{\alpha }\).

We remark that in this setting the Ringel duality formula (1) also has an alternative proof, which is more geometric, see Proposition 1.5.

The aim of this paper was to find a more general representation theoretic framework extending the Ringel duality formula (1) to a larger class of (ultra) strongly quasi-hereditary algebras. In particular, the Knörrer invariant algebras are the original motivation for the ideally ordered condition.

Remark 1.6

The algebras \(\Lambda _{\varvec{\alpha }} \cong E_{K_{r, a}}\) and \(K_{r, a}\) were used to show a noncommutative version of Knörrer periodicity for cyclic quotient surface singularities in [27]. More precisely, it was proved there that the singularity category of a cyclic quotient surface singularity is equivalent to the singularity category of a corresponding Knörrer invariant algebra, generalising classical Knörrer’s periodicity for the polynomials \(x^n\) and \(x^n + y^2 +z^2\). The proof uses noncommutative resolutions and \(\Lambda _{\varvec{\alpha }}\cong E_{K_{r,a}}\) plays the role of a noncommutative resolution for \(K_{r,a}\).

1.2 Auslander–Dlab–Ringel and nilpotent quiver algebras

From a more representation theoretic viewpoint, a Ringel duality formula that looks similar to that of Theorem 1.2 was proved for Auslander–Dlab–Ringel algebras Open image in new window by Conde and Erdmann [13, Theorem A]. We define these algebras, recall Conde and Erdmann’s Ringel duality formula, and discuss the relationship between this result and the results of this paper in Sect. 6.3.

In particular, for the class of algebras Open image in new window in Example 1.4 (2) the corresponding algebras Open image in new window and Open image in new window coincide if Q has no sources.

Preposition 6.12

If Open image in new window for Q a finite quiver without sources and J the two-sided ideal generated by all arrows in Q, then there is an isomorphism of quasi-hereditary algebras Open image in new window .

We also prove that when Q has no sinks the ADR algebra coincides with the quiver nilpotent algebra \(N_m(Q)\) introduced by Eiriksson and Sauter [20], which is motivated via a quiver graded version of Richardson orbits and is recapped in Sect. 6.4.

Preposition 6.15

If Open image in new window for Q a finite quiver without sinks and J the two-sided ideal generated by all arrows in Q, then there is an isomorphism of quasi-hereditary algebras Open image in new window .

In particular, if \(R=kQ/J^m\) (as in Example 1.4 (2)) for a quiver with no sinks or sources, then Open image in new window and so Theorem 1.2 provides a Ringel duality formula for such nilpotent quiver algebras; see Corollary 6.17.

1.3 Nakayama and Auslander algebras

Several of the examples of ideally ordered monomial algebras above can be thought of as geometrically inspired by resolutions of singularities. Indeed, Examples 1.4 (1)–(4) can be thought of as different generalisations of the algebra Open image in new window .

Work of Dlab and Ringel [17] shows that every finite dimensional algebra admits a noncommutative ‘resolution’ by a quasi-hereditary algebra, and a generalisation of this result led to Iyama’s proof of the finiteness of Auslander’s representation dimension [25].

Such a resolution for finite dimensional algebras of finite representation type is provided by the Auslander algebra. This also occurs in more geometric contexts; the categorical resolutions considered by Kuznetsov and Lunts [30] use a construction motivated by Auslander algebras to resolve non-reduced schemes.

For R a finite dimensional algebra of finite representation type let Open image in new window denote the Auslander algebra of R, which we recall in Sect. 6.5.

Preposition 6.18

If R is an ideally ordered monomial algebra, then Open image in new window if and only if R is self-injective.

A particular example of a class of ideally ordered, monomial algebras of finite representation type are the Nakayama algebras (listed as Example 1.4 (4)).

Corollary 6.19

If R is self-injective Nakayama algebra, then Open image in new window .

In this setting Theorem 1.2 also generalises several known results in the literature, e.g. that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual, see [37].

Corollary 6.20

If R is a self-injective Nakayama algebra then Open image in new window .

For other related results see work by Baur et al. [6], Crawley-Boevey and Sauter [16] and Nguyen et al. [32].

1.4 Left and right strongly quasi-hereditary structure

A further special property of the quasi-hereditary algebras Open image in new window is that the ideal layer function simultaneously realises both a left and right strongly quasi-hereditary structure on the algebras.

Since Open image in new window is closed under kernels Open image in new window has global dimension 2, and it was recently shown by Tsukamoto [38] that this implies Open image in new window admits both a left strongly quasi-hereditary structure and a right strongly quasi-hereditary structure (for a possibly different order), building on earlier work of Dlab and Ringel, and Iyama.

In general the left and right strongly quasi-hereditary structures cannot be realised using the same order. Indeed, Tsukamoto shows that for Auslander algebras of representation-finite algebras (which all have global dimension 2) this is possible precisely if the underlying algebra is a Nakayama algebra.

As seen in the examples above, the class of quasi-hereditary algebras Open image in new window constructed from ideally ordered monomial algebras provides a larger class of such algebras.

1.5 Conventions

Throughout this paper k will denote an algebraically closed field. For paths \(p,q \in kQ\) in the path algebra of a quiver Q the composition pq will denote the path q followed by the path p. For R a Noetherian ring R-\(\mathrm{mod}\) will denote the category of finitely generated left R-modules, and for \(S \subset R\)-\(\mathrm{mod}\) we will define Open image in new window to be the additive subcategory generated by S: i.e. the smallest full subcategory of R-\(\mathrm{mod}\) containing S and closed under isomorphism, direct sums, and direct summands. In particular, the category of finitely generated projective R modules \(\text {proj}\)-R is equivalent to Open image in new window .

We recall the category of torsionless R-modules Open image in new window from the introduction, and now give a more general definition: for an R-module M we define the following subcategorywith corresponding module Open image in new window . Moreover, for an R-module M, we setand let Open image in new window denote the direct sum of all indecomposable objects in Open image in new window up to isomorphism.
We let \(\dagger \) denote the standard k-duality Open image in new window . For the injective cogenerator Open image in new window we define the category of divisible R-modulesand let Open image in new window denote the direct sum of all indecomposable objects in Open image in new window up to isomorphism.

2 Strongly quasi-hereditary algebras

In this section, we will give necessary and sufficient conditions for certain endomorphism algebras over ideally ordered monomial algebras to be left or right strongly quasi-hereditary.

We first recall the definition of a quasi-hereditary algebra. This needs some preparation. For a finite dimensional k-algebra A choose a labelling \(i \in I\) of the simple A-modules \(S_i\) up to isomorphism. A partial order \(\leqslant \) on the set I is called adapted if for each \(M \in A\)-\(\mathrm{mod}\) with top \(S_i\) and socle Open image in new window incomparable there exists some \(k>i\) or \(k>j\) such that \(S_k\) is a composition factor of M. In particular, total orderings are adapted. We denote the projective cover and injective envelope of the simple \(S_i\) by \(P_i\) and \(Q_i\) respectively.

Definition 2.1

Given a partial ordering \( \leqslant \) on the index set I, for \(i \in I\) the standard module \(\Delta _i\) is the maximal factor module of \(P_i\) whose composition series consists only of simple modules Open image in new window such that \(j \leqslant i\). Similarly, the costandard module \(\nabla _i\) is the maximal submodule of \(Q_i\) whose composition series consists only of simple modules Open image in new window such that \(j \leqslant i\).

The k-algebra A is quasi-hereditary with respect to an adapted partial ordering \(\leqslant \) if:
  1. (1)

    \(\text {End}_A(\Delta _i) \cong k\) for each \(i \in I\) and

     
  2. (2)

    A can be filtered by the standard modules under this ordering; i.e. there exists a series of A-modules Open image in new window such that each quotient \(M_{i-1}/M_i\) is isomorphic to a direct sum of standard modules.

     

The following terminology is due to Ringel [35]. We refer to the references and discussions in [35] for earlier work.

Definition 2.2

A quasi-hereditary algebra A is called left strongly quasi-hereditary if all standard modules have projective dimension at most 1. It is called right strongly quasi-hereditary if all costandard A-modules have injective dimension at most 1.

This is an equivalent characterisation of left/right strongly quasi-hereditary condition given in [35, Appendix A1]. The original definition, introduced in [35, Section 4], is in terms of a layer function.

Definition 2.3

A k-algebra A is left strongly quasi-hereditary with n layers if there is a layer function Open image in new window such that for any simple module s with projective cover P(s) there is an exact sequence
$$\begin{aligned} 0 \rightarrow R(s) \rightarrow P(s) \rightarrow \Delta (s) \rightarrow 0 \end{aligned}$$
such that
  1. (a)

    The module R(s) is the direct sum of projective covers \(P(s')\) of simple modules \(s'\) such that \(L(s')>L(s)\).

     
  2. (b)

    All simple factors \(s'\) of \(\mathsf {rad}\, \Delta (s)\) satisfy \(L(s')<L(s)\).

     
The layer function induces an ordering on the simple A-modules and the modules \(\Delta (s)\) are the standard modules for this strongly quasi-hereditary structure. Right strongly quasi-hereditary algebras are defined dually.
After some preparation, we introduce the class of endomorphism algebras which we are interested in. For the rest of this section we let R be a finite dimensional k-algebra. A submodule of the form \(Rp \subset R\) is a principal left ideal if \(p \in eR\) with \(e \in R\) a primitive idempotent. We introduce the additive subcategoryand we let Open image in new window denote the direct sum of all principal left ideals up to isomorphism. In this section we assume that Open image in new window is finitely generated and define Open image in new window .

The assumption on Open image in new window is satisfied for ideally ordered monomial algebras R due to Lemma 7.3 but does not hold for all finite dimensional algebras; e.g. if Open image in new window , then the ideals Open image in new window for \(\lambda \in \mathbb {C}\) give a \(\mathbb {C}\)-indexed set of ideals that are pairwise non-isomorphic as left modules.

Throughout the rest of the paper we will label the simple and projective Open image in new window -modules by the principal ideals of R, as we now explain. To do this we use the additive anti-equivalenceIt is clear that Open image in new window is a contravariant functor, and one can show that it is an additive anti-equivalence using that it maps the additive generator Open image in new window of Open image in new window to the additive generator Open image in new window of Open image in new window . Under this anti-equivalence the indecomposable summands \(\Lambda \) of Open image in new window are in 1-to-1 correspondence with indecomposable projective Open image in new window -modules, which we denote by \(P(\Lambda )\). The indecomposable projective modules \(P(\Lambda )\) are in 1-to-1 correspondence with simple Open image in new window -modules \(S(\Lambda )\) that occur as their heads (i.e, so that \(P(\Lambda ) \rightarrow S(\Lambda )\) is a projective cover). Hence the principal ideals \(\Lambda \subset R\) index the simple modules \(S(\Lambda )\) of Open image in new window . When given a partial ordering on the principal ideals, we use similar notation to label standard \(\Delta (\Lambda )\) and costandard \(\nabla (\Lambda )\) objects. This labelling allows to define the following layer function for the algebra Open image in new window .

Definition 2.4

Let R be a finite dimensional algebra. For principal left R-ideals \(\Lambda \), we define Open image in new window and we call l the ideal layer function. It induces a partial ordering on the principal left R-ideals, which we call the ideal ordering.

We will now determine when the ideal layer function induces a left or right strongly quasi-hereditary structure on Open image in new window by considering left and right minimal approximations with respect to the ideal ordering.

The notion of minimal approximation is common in representation theory; see [29] for a survey. A morphism \(\alpha :\Gamma \rightarrow \Lambda \) is a left approximation for a class of modules Open image in new window if Open image in new window and the induced morphism \({\mathrm{Hom}}_R(\Lambda ,C) \rightarrow {\mathrm{Hom}}_R(\Gamma ,C)\) is surjective for all Open image in new window . A morphism \(\Gamma \xrightarrow {\scriptscriptstyle \alpha \ } \Lambda \) is left minimal if any endomorphism \(\phi \) of \(\Lambda \) satisfying Open image in new window is an isomorphism. In particular, left minimal approximations are unique up to isomorphism.

Denote by Open image in new window the full subcategory of direct sums of principal left R-ideals \(\Lambda \) with \(l(\Lambda )>i\).

Lemma 2.5

Let \(\Gamma \) be a principal left ideal of layer \(\gamma \). There is a minimal left Open image in new window approximation Open image in new window of \(\mathrm{\Gamma }\).

Proof

It is well-known that \(\Gamma \) admits a left Open image in new window approximation \(\Phi :\Gamma \rightarrow \Lambda \). Indeed, this follows since there are only finitely many indecomposable objects in Open image in new window and since R is finite dimensional, see e.g. [5]. For the convenience of the reader, we recall the argument. We consider the modulewhere the sum is taken over all indecomposable objects M in Open image in new window (up to isomorphism). Then Open image in new window as each \({\mathrm{Hom}}_R(\Gamma ,M)\) is finite dimensional, Open image in new window is assumed to be finitely generated, and Open image in new window is closed under finite direct sums.
Choosing a basis Open image in new window ofdetermines a morphism \( \Phi :\Gamma \rightarrow \Lambda \) as the direct sum Open image in new window . One can check that \(\Phi \) is a left Open image in new window approximation.

The existence of a left approximation with a finite length target implies the existence of a minimal left approximation by, for example [4, Theorem I.2.4], which shows such a minimal approximation can be constructed from an approximation by projection onto a summand. Hence the existence of the approximation \(\Phi :\Gamma \rightarrow \Lambda \) ensures that a minimal left Open image in new window approximation \(\alpha _{\Gamma }:\Gamma \rightarrow \Gamma _{>\gamma }\) exists.\(\square \)

Definition 2.6

We say that Open image in new window has good left approximations iffor all principal left R-ideals \(\Gamma \).

Lemma 2.7

If R is an ideally ordered monomial algebra, then for a principal ideal \(\Gamma \) of layer \(\gamma \) the minimal left Open image in new window approximation is surjective. Hence when R is ideally ordered Open image in new window has good left approximations.

Proof

Since R is ideally ordered, we can use Lemma 7.3 to replace any principal R-ideal by an isomorphic monomial ideal wherever needed. In particular, without loss of generality let \(\Gamma =Rg\) (with \(g \in eR\) a monomial) be a principal left R-ideal of layer \(\gamma \).

A surjection from \(\Gamma \) to a principal ideal exists, \(\Gamma \rightarrow 0\) as 0 is a principal ideal. Using that R is finite dimensional there is a surjection to a principal ideal \(\Gamma _{>\gamma }\) which has maximal dimension among all principal ideals that admit surjections from \(\Gamma \). The existence of the surjection implies that \(\Gamma \) and \(\Gamma _{>\gamma }\) have the same head. In particular, we can assume that Open image in new window for a monomial \(n \in eR\). Using Lemma 7.1, the assignment \(g \mapsto n\) defines an R-linear surjection Open image in new window .

We now claim that \(\alpha _{\Gamma }\) is an approximation. To prove this we consider a principal ideal \(\Lambda \) and will show that the induced map \({\mathrm{Hom}}_R(\Gamma _{>\gamma }, \Lambda ) \rightarrow {\mathrm{Hom}}_R(\Gamma ,\Lambda )\) is a surjection. Take a morphism \(\beta \in {\mathrm{Hom}}_R(\Gamma ,\Lambda )\). We aim to show that \(\beta \) factors through \(\alpha _{\Gamma }\) and hence is the image of some morphism in \({\mathrm{Hom}}_R(\Gamma _{>\gamma }, \Lambda )\).

To see this, take the induced surjection \(\beta :\Gamma \rightarrow \mathsf {im}\,\beta \) and, as the image of a principal ideal in a principal ideal, \(\mathsf {im}\,\beta \cong Rm\) (with a monomial \(m \in eR\)) is a principal left R-ideal. Using the ideally ordered condition on R there is a surjection in at least one direction between \(\mathsf {im}\,\beta \) and \(\Gamma _{> \gamma }\). As \(\Gamma _{>\gamma }\) is a principal ideal of maximal dimension with a surjection from \(\Gamma \), it follows that Open image in new window and hence there is a surjection Open image in new window . Using Lemma 7.1, we can assume that \(\sigma \) is given by \(n \mapsto m\). Hence, the composition Open image in new window is a surjection defined by \(g \mapsto m\). Now Lemma 7.2 shows that the surjection \(\beta :\Gamma \rightarrow \mathsf {im}\,\beta \) factors over \(\pi \). In particular, \(\beta \) factors over Open image in new window . So Open image in new window is an approximation.

Finally, we claim that this approximation is minimal. To see this consider an endomorphism Open image in new window such that Open image in new window . Then as \(\alpha _{\Gamma }\) is a surjection it follows that \(\phi \) is a surjection, and hence an isomorphism.

By construction, Open image in new window for all \(\Gamma \) so Open image in new window has good left approximations. \(\square \)

We give examples showing that our results above apply beyond the class of ideally ordered monomial algebras.

Example 2.8

  1. (a)
    Consider the monomial algebra \(R=kQ/I\), where This is not ideally ordered since there are no surjections between Rb and Rc, however Open image in new window still has good left approximations. It is a short exercise to find the five isomorphism classes of indecomposable principal ideals and calculate their minimal left approximations. All but one of these minimal approximations are surjective, and the one which is not surjective has cokernel \(S_1\), the simple at vertex 1. There are no morphisms from \(S_1\) to any principal ideal, and hence Open image in new window has good left approximations.
     
  2. (b)
    Let \(n>0\) be an integer. Consider the non-monomial algebras \(R_n=kQ/I_n\) where Again, Open image in new window has good left approximations; it is a short exercise to find the \(n+3\) principal ideals and calculate that the minimal left approximation for each one is surjective.
     

Proposition 2.9

The algebra Open image in new window is left strongly quasi-hereditary with respect to the ideal layer function l if and only if Open image in new window has good left approximations with respect to l.

Proof

Assume Open image in new window has good left approximations Open image in new window . Using the condition on Open image in new window and applying Open image in new window yields a short exact sequence
$$\begin{aligned} 0 \rightarrow P(\Gamma _{>\gamma }) \xrightarrow {\iota (\Gamma )\,} P(\Gamma ) \rightarrow \Delta (\Gamma ) \rightarrow 0, \end{aligned}$$
(3)
where Open image in new window and \(\Delta (\Gamma )\) denotes the cokernel of \(\iota (\Gamma )\). We claim that the ideal layer function defines a left strongly quasi-hereditary structure on Open image in new window such that the \(\Delta (\Gamma )\) are standard modules. To see this we have to show that (3) satisfies conditions (a) and (b) outlined in Definition 2.3. Since all direct summands of \(P(\Gamma _{>\gamma })\) are of the form \(P(\Lambda )\) with \(l(\Lambda ) > \gamma \) condition (a) is satisfied by construction. Using the anti-equivalence Open image in new window - Open image in new window condition (b) translates to: every R-linear non-isomorphism \(\nu :\Gamma \rightarrow \Lambda \) with Open image in new window factors over Open image in new window . By definition of Open image in new window this holds for Open image in new window . If Open image in new window , then \(\nu \) cannot be surjective for otherwise it is an isomorphism since Open image in new window . Therefore, \(\mathsf {im}\,\nu \subsetneq \Lambda \) is a principal left R-ideal with \(l(\mathsf {im}\,\nu ) > l(\Lambda )=\gamma \). So \(\nu \) factors over Open image in new window .
To see the converse direction, assume Open image in new window does not have good left approximations. Then there exists a principal left R-ideal \(\Gamma \) such that Open image in new window . Assume that Open image in new window is quasi-hereditary with respect to the ideal layer function l and let \(\Delta (\Gamma )\) be the standard module corresponding to \(\Gamma \). Since Open image in new window is a minimal left Open image in new window approximationis the start of a minimal projective resolution of \(\Delta (\Gamma )\). By our choice of \(\Gamma \) the morphism Open image in new window is not injective. Hence \(\Delta (\Gamma )\) has projective dimension greater than 1 and, using Definition 2.2, A is not left strongly quasi-hereditary with respect to l in this case.\(\square \)

Remark 2.10

Assume that Open image in new window is quasi-hereditary with respect to the ideal layer function. One can show that as a set the standard module \(\Delta (\Gamma )\) is given by all (residue classes of) monomorphisms starting in \(\Gamma \). Indeed if \(\nu :\Gamma \rightarrow \Lambda \) is not a monomorphism then an argument along the lines of the proof of the proposition shows that \(\nu \) factors over Open image in new window and therefore corresponds to the zero element in \(\Delta (\Gamma )\).

Proposition 2.9 is related to [35, Theorem 5] by Ringel. He shows that for an R-module M there exists an R-module N such that Open image in new window is left strongly quasi-hereditary and all the indecomposable summands N are submodules of M. In particular, if M is an R-module such that all submodules are isomorphic to direct summands of M, then \(\text {End}_R(M)\) is left strongly quasi-hereditary. We will see in Theorem 5.1 that Open image in new window has this property if R is ideally ordered monomial. However, our proof of Theorem 5.1 uses Proposition 2.9, so we cannot apply Ringel’s result in our approach.

Now we look at the ‘dual’ side. First we ‘dualise’ Definition 2.6 using the same notation.

Definition 2.11

For every principal left ideal \(\Gamma \) there is a minimal right Open image in new window approximation Open image in new window with Open image in new window . We say that Open image in new window has good right approximations ifSince Open image in new window contains R as a direct summand this is equivalent to Open image in new window for all principal left R-ideals \(\Gamma \).

Example 2.12

  1. (a)

    Let R be a finite dimensional monomial algebra. Then Open image in new window has good right approximations. Indeed, let \(\Gamma \) be a principal left R ideal. Since R is monomial, \(\mathsf {rad}\,\Gamma \) is a direct sum of principal left ideals in Open image in new window and the natural inclusion \(\mathsf {rad}\,\Gamma \rightarrow \Gamma \) gives the desired minimal right approximation Open image in new window .

     
  2. (b)

    The algebra in Example 2.8 (b) does not have good right approximations: the minimal right approximation of the projective module \(P_1\) is Open image in new window and this has kernel \(S_4\).

     

The following result is proved dually to Proposition 2.9

Proposition 2.13

Open image in new window is right strongly quasi-hereditary with respect to the ideal layer function l if and only if Open image in new window has good right approximations. For example, this holds if R is finite dimensional monomial.

Combining Propositions 2.9 and 2.13 with Lemma 2.7 and Example 2.12 (a) yields the following theorem.

Theorem 2.14

If R is an ideally ordered monomial algebra, then Open image in new window is both left and right strongly quasi-hereditary with respect to the ordering induced by the ideal layer function.

We let Open image in new window and Open image in new window denote the full subcategories of Open image in new window -\(\text {mod}\) of objects filtered by standard and costandard modules respectively.

Remark 2.15

Assume that Open image in new window is quasi-hereditary with respect to the ideal layer function. Similarly to the case above, one can show that as a set a costandard module \(\nabla (\Lambda )\) is given by all surjections ending in \(\Lambda \). In particular, each costandard module has head \(S(\Pi )\) for some indecomposable projective R-module \(\Pi \) and Open image in new window .

Corollary 2.16

If Open image in new window has good right and left approximations, then Open image in new window is closed under submodules and Open image in new window is closed under quotients.

Proof

If Open image in new window has good left approximations, then Open image in new window is left strongly quasi-hereditary by Proposition 2.9, and hence all standard objects have projective dimension 1. By [35, Proposition A.1], all standard modules having projective dimension 1 is equivalent to Open image in new window being closed under quotients.

The analogous dual statement, using Proposition 2.13, shows that when Open image in new window has good right approximations then Open image in new window is closed under submodules.\(\square \)

3 The characteristic tilting module and Ringel duality

In the following section we first recall the characteristic tilting module T associated to a quasi-hereditary algebra. Then we show that our algebras Open image in new window are ultra strongly quasi-hereditary in the sense of Conde [12] and use this to determine a subcategory of the additive hull Open image in new window of T (Corollary 3.6). In the proof of our main Theorem 5.1 we show that these categories coincide for ideally ordered monomial algebras R and as a consequence establish our Ringel duality formula in this setup.

The following proposition can be found in Ringel [34], which is based on work of Auslander and Reiten [3] and Auslander and Buchweitz [2].

Proposition 3.1

Let A be a quasi-hereditary algebra. Then there exists a tilting module \(T \in A\)-\(\mathrm{mod}\) such thatwhere Open image in new window is the full subcategory of A-modules with filtrations by both standard and costandard modules.

Definition 3.2

A tilting module T occurring in Proposition 3.1 is called a characteristic tilting module. The Ringel dual \(\mathfrak {R}(A)\) of an algebra A is defined byfor T the basic characteristic tilting module consisting of one copy of each indecomposable module in Open image in new window up to isomorphism: i.e. we assume \(\mathfrak {R}(A)\) is a basic algebra.

The notion of an ultra strongly quasi-hereditary algebras was introduced by Conde, see [12, Section 2.2.2].

Definition 3.3

A quasi-hereditary algebra A is left ultra strongly quasi-hereditary if a projective module \(P_i\) is filtered by costandard modules whenever the corresponding costandard module \(\nabla _i\) is simple.

Let Open image in new window be the idempotent corresponding to the direct summand R of Open image in new window . Note that \(e_0\) is primitive if and only if R is local. We have the following.

Proposition 3.4

Let R be a finite dimensional algebra. Assume that Open image in new window has good left approximations, so that Open image in new window is left strongly quasi-hereditary with respect to the ideal layer function l. Then the following conditions are equivalent:
  1. (a)

    Open image in new window is filtered by costandard objects.

     
  2. (b)

    \(\alpha _{\Gamma }:\Gamma \rightarrow \Gamma _{>\gamma }\) is surjective for all principal R-ideals \(\Gamma \).

     
  3. (c)

    Open image in new window is left ultra strongly quasi-hereditary.

     
If R is monomial then these conditions are equivalent to
  1. (d)

    R is ideally ordered.

     

Proof

We first show that (b) implies (a). By [34, Theorem 4], it suffices to show that Open image in new window for all principal left R ideals \(\Gamma \) and all primitive idempotents \(e_i \in R\). We can assume that \(\Delta (\Gamma )\) is not projective. Then applying Open image in new window to Open image in new window produces the projective resolution
$$\begin{aligned} 0 \rightarrow P(\Gamma _{>\gamma }) \xrightarrow {\iota (\Gamma )\,} P(\Gamma ) \rightarrow \Delta (\Gamma ) \rightarrow 0, \end{aligned}$$
and we have to show that every morphism \(P(\Gamma _{>\gamma }) \rightarrow P(Re_i)\) factors over \(\iota (\Gamma )\). Applying the anti-equivalence given in equation (2) translates this statement to: every morphism \(\varphi :Re_i \rightarrow \Gamma _{>\gamma }\) factors over Open image in new window . This holds since \(Re_i\) is projective and Open image in new window is surjective by assumption.

Conversely, if Open image in new window is not surjective for some principal ideal \(\Gamma \) then there exists Open image in new window . Since R is free there is an R-linear map \(R \rightarrow \Gamma _{>\gamma }\), \(1 \mapsto x\), which by construction does not factor over Open image in new window . In combination with the anti-equivalence and projective resolution above this shows Open image in new window and [34, Theorem 4] completes the proof that (a) implies (b).

That (a) is equivalent to (c) follows from the fact that \(\nabla (\Lambda )\) is simple if and only if \(\Lambda \) is projective, see Remark 2.15, and hence \(\nabla (\Lambda )\) simple implies \(P(\Lambda )\) is a direct summand of Open image in new window .

Let R be monomial. The implication (d) \(\Rightarrow \) (b) follows from Lemma 2.7. We now assume (b) and prove the converse.

Firstly, for any indecomposable principal ideal \(\Gamma \) the minimal left approximation Open image in new window is surjective by assumption (b), and we claim that \(\Gamma _{>\gamma }\) is indecomposable.

To show this take \(p \in eR\) for e a primitive idempotent and consider the principal ideal \(\Gamma \cong Rp\). Now suppose that there is a decomposition Open image in new window for some principal ideals \(Rq_i\). As \(\alpha _{\Gamma }\) is surjective, after relabelling we can assume that the image of p is Open image in new window and \(q_1 \ne 0\). As the morphism Open image in new window is surjective there must exist some \(r \in Re\) such that Open image in new window ; i.e. \(rq_1 =q_1\) and Open image in new window for \(j \geqslant 2\). As R is monomial, by considering the monomial of lowest degree occurring in \(q_1\) and \(rq_1 =q_1\) we can see that the degree 0 primitive idempotent e must occur in r. Then we can rewrite \(r=e+r'\) where all monomials occurring in \(r'\) have degree greater than 0. As a result, Open image in new window must be zero as Open image in new window so there can be no non-zero monomial of lowest degree occurring in Open image in new window . Hence Open image in new window for \(j\geqslant 2\), the decomposition is a trivial decomposition Open image in new window , and \(\Gamma _{> \gamma }\) is indecomposable.

This allows the successive construction of left \(\mathsf {pi}_{>k}\) approximations starting with the indecomposable principal ideal Re
$$\begin{aligned} Re \xrightarrow {\alpha _{Re}\,} Re_{>i_1} \xrightarrow {\alpha _{{Re}_{>i_1}}} Re_{>i_2} \xrightarrow {\alpha _{Re_{>i_2}}} \cdots \xrightarrow {\alpha _{Re_{>i_{n-1}}}} Re_{>i_n} \end{aligned}$$
where \(i_1=l(Re)\), Open image in new window , and Open image in new window is the minimal left Open image in new window approximation. Each Open image in new window is indecomposable, and the composition \(\alpha _k:Re \rightarrow Re_{>i_k}\) of the left approximations is again a left approximation.

We claim that any indecomposable principal ideal Rx with \(x \in eR\) is isomorphic to one of these successive approximations. To see this choose k to be maximal such that \(l(Rx) > i_k\). Then there is a surjection \(\pi :Re \rightarrow Rx\), and as Open image in new window this must factor through the left approximation \(\alpha _k:Re \rightarrow Re_{>i_k}\) by a surjection \(\phi :Re_{>i_k}\! \rightarrow Rx\). In particular, \(\dim Re_{>i_k} \!\geqslant \dim Rx\) so \(l(Re_{>i_k}) \leqslant l(Rx)\). But, by the definition of k, it is true that Open image in new window , hence it must be the case that \(l(Re_{>i_k}) = l(Rx)\) so \(\dim Re_{>i_k} \!=\dim Rx\) and hence the surjective morphism \(\phi \) is an isomorphism \(Re_{>i_k} \! \cong Rx\).

Finally, any pair Rx and Ry of principal ideals with \(x, y \in eR\) occur (up to isomorphism) in the successive approximation sequence, in which every morphism is surjective by assumption (b), and hence there is a surjection between them. This proves that the ideally ordered condition holds.\(\square \)

Example 3.5

The non-monomial algebra in Example 2.8 (b) satisfies the equivalent conditions (a), (b) and (c) of the theorem.

Corollary 3.6

Suppose that Open image in new window has both good left and right approximations. Then Open image in new window .

Proof

By the definition of a quasi-hereditary algebra every projective module is filtered by standard modules. Therefore, Open image in new window and by Proposition 3.4 (a), we also have Open image in new window . Now Corollary 2.16 yields Open image in new window and Open image in new window . This implies the claim.\(\square \)

Remark 3.7

In combination with Remark 2.15, we see that when Open image in new window has both good left and right approximations Open image in new window . For ideally ordered monomial algebras R, Theorem 5.1 (e) shows that Open image in new window holds as well.

Remark 3.8

Let \(R=R_2\) be the non-monomial algebra from Example 2.8 (b). The algebra Open image in new window is left ultra strongly quasi-hereditary with respect to the ideal layer function (in particular, Open image in new window is filtered by costandard modules) but not right strongly quasi-hereditary, so Open image in new window is not closed under subobjects. It turns out that there is precisely one indecomposable subobject of Open image in new window which is not filtered by standard modules. This module is also a quotient of Open image in new window and therefore Open image in new window . Restricting to the local submodules of Open image in new window yields the desired inclusion into Open image in new window in this case and can be used to show a version of the Ringel duality formula (10) in this example. Unfortunately, we do not know how to fit this example into a larger framework.

4 An equivalence from idempotents

In this section, we show that there is an equivalence of categorieswhere Open image in new window for a finite dimensional algebra R with Open image in new window finitely generated and \(e_0 \in A\) is the idempotent corresponding to the projection onto R.

To show this we recall several well-known lemmas.

Lemma 4.1

Let Open image in new window be an abelian category with Serre subcategory Open image in new window and let Open image in new window be the quotient functor. Then the restriction of q,is fully faithful. Here

Proof

This follows from the description of homomorphism spaces in the quotient category as colimits. Indeed for Open image in new window the colimit describing Open image in new window is taken over the single pair of subobjects (X, 0) and the quotient functor sends a morphism \(f:X \rightarrow Y\) to f.\(\square \)

The following lemma can be found in [21, Proposition 5.3 (b)]

Lemma 4.2

Let B be a noetherian ring and let \(e \in B\) be an idempotent. Then
$$\begin{aligned} F={\mathrm{Hom}}_B(Be, -):B\text {-}\mathrm{mod}\rightarrow eBe\text {-}\mathrm{mod}\end{aligned}$$
is an exact quotient functor with kernel \(B/BeB\text {-}\mathrm{mod}\). In particular, \(B/BeB\text {-}\mathrm{mod}\) is a Serre-subcategory in \(B\text {-}\mathrm{mod}\).

Corollary 4.3

In the notation of Lemma 4.2, we have Open image in new window .

Proof

Consider Open image in new window and \(M \in B/BeB\)-\(\mathrm{mod}\). Applying the right exact functor \({\mathrm{Hom}}_B(-,M)\) to the surjection \(Be \rightarrow N \rightarrow 0\) yields the injection \(0 \rightarrow {\mathrm{Hom}}_B(N,M) \rightarrow {\mathrm{Hom}}_B(Be,M)\). As B / BeB-\(\mathrm{mod}\) is the kernel of \({\mathrm{Hom}}_B(Be,-)\) and \(M \in B/BeB\)-\(\mathrm{mod}\) it follows that \({\mathrm{Hom}}_B(Be,M)=0\) and hence \({\mathrm{Hom}}_B(N,M)=0\).\(\square \)

From now on let Open image in new window for some finite dimensional algebra R, such that Open image in new window is finitely generated.

Lemma 4.4

In the notation of Sect. 3, we have \(\mathsf {soc} \, Ae_0 \subseteq S_0^{\oplus n}\) for some natural number n. Here Open image in new window is the semi-simple head of \(Ae_0\).

Proof

Indeed \(Ae_0\) consists of all R-homomorphisms Open image in new window . Let \(\Lambda \) be a principal left R-ideal. If \(R \rightarrow \Lambda \) is non-zero, then the composition with the canonical inclusion \(R \rightarrow \Lambda \rightarrow R\) is non-zero. Therefore every maximal sequence of non-zero morphisms starting in R ends in R, proving the claim.\(\square \)

Corollary 4.5

Open image in new window .

Proof

Assume that \(f :X \rightarrow U\) is a non-zero map, where U in Open image in new window and X in \(A/Ae_0A \text {-} \mathrm{mod}\). Lemma 4.4 implies that \(\mathsf {im}\,f\) contains a non-zero direct summand of \(S_0\). But \(\mathsf {im}\,f \in A/Ae_0A \text {-} \mathrm{mod}\) since X is contained in \(A/Ae_0A\text {-} \mathrm{mod}\). It follows that \(\mathsf {im}\,f\) has no submodule which is a direct summand of \(S_0\). A contradiction. So there is no non-zero morphism \(f:X \rightarrow U\).\(\square \)

The following statement is the main result of this section.

Proposition 4.6

The exact functor \(F={\mathrm{Hom}}_A(Ae_0, -)\) restricts to an additive equivalence

Proof

The equality on the right follows from the fact that Open image in new window . Since F is exact and maps an A-module M to \(e_0M\), the restriction is well-defined. We can apply Lemma 4.1 to \(q=F\) to deduce that F is fully faithful. Indeed, by Lemma 4.2, F is a quotient functor corresponding to the Serre subcategory \(A/Ae_0A\text {-}\mathrm{mod}\) and Corollaries 4.3 and 4.5 show that the required orthogonality conditions are satisfied.

It remains to show that F is essentially surjective. Let \(U \subseteq (e_0Ae_0)^{\oplus n}\) be generated by \(u_1, \ldots , u_n \in (e_0Ae_0)^n\). The \(u_i\) are elements of \((Ae_0)^n\). Let \(V \subseteq (Ae_0)^{\oplus n}\) be the A-submodule generated by the \(u_i\). One can check that \(F(V)=U\) and since \(e_0 u_i=u_i\) for all i V is a factor module of \((Ae_0)^{\oplus m}\) for some m. This shows that V is contained in Open image in new window and completes the proof. \(\square \)

5 Proof of Ringel duality formula

In this section we prove the following main result of this paper, which is an extended version of Theorem 1.2 stated in the introduction.

Theorem 5.1

Let R be a finite dimensional ideally ordered monomial algebra and Open image in new window . Then Open image in new window is quasi-hereditary and the Ringel duality formulaholds. More explicitly, where \(\dagger \) denotes the standard k-duality,and if we consider Open image in new window and Open image in new window as exact categories with split exact structures then this Ringel duality induces the derived equivalenceMoreover:
  1. (a)

    Every indecomposable submodule of \(R^n\) is isomorphic to a principal left ideal, every principal left ideal is isomorphic to a monomial ideal, and hence Open image in new window so Open image in new window .

     
  2. (b)

    The algebra Open image in new window is left and right strongly quasi-hereditary with respect to the ideal layer function. In particular, Open image in new window has global dimension at most 2. Moreover, it is left ultra strongly quasi-hereditary in the sense of Conde [12].

     
  3. (c)

    The ideal order is the unique order defining a quasi-hereditary structure on Open image in new window if R is local and satisfies the following condition: if there exists a surjection \(\Lambda \rightarrow \Gamma \) between principal left ideals, then there is an inclusion \(\Gamma \rightarrow \Lambda \).

     
  4. (d)

    Let T be the characteristic tilting module of Open image in new window and Open image in new window be the idempotent corresponding to R. Then there is an equality of subcategories Open image in new window . In other words, the indecomposable direct summands \(T_i\) of T are precisely those indecomposable Open image in new window -modules which are both quotients and submodules of the projective module Open image in new window .

     
  5. (e)
    We can describe the subcategories Open image in new window and Open image in new window of Open image in new window -\(\mathrm{mod}\) as follows:
     

Proof

We first prove the main Ringel duality formula, and in the process also prove (a) and (d). Let Open image in new window and let Open image in new window be the idempotent corresponding to R. By Corollary 3.6, we have an inclusionwhere T is the characteristic tilting module for Open image in new window . In combination with Proposition 4.6, we get an inclusionsince Open image in new window . Let p (respectively, \(p^{\mathrm{op}}\)) be the number of indecomposable direct summands of Open image in new window (respectively, Open image in new window ) By definition of Open image in new window , the number p also equals the number of simple Open image in new window -modules. Which in turn equals the number of indecomposable summands of T since T is tilting. Let s (respectively, \(s^{\mathrm{op}}\)) be the number of indecomposable direct summands of Open image in new window (respectively, Open image in new window ). By (8), \(s^{\mathrm{op}} \leqslant p\) (in particular, \(s^{\mathrm{op}}\) is finite). Moreover, Open image in new window implies \(p \leqslant s\). It follows from [36, Theorem 1.1] that \(s=s^{\mathrm{op}}\). Summing up, we have that Open image in new window . In particular, this yields equivalences Open image in new window , and therefore Open image in new window so proves (a). Moreover, using Open image in new window and Proposition 4.6 the inclusions (7) and (8) are equivalencesIn particular, this shows part (d).

By definition, the Ringel dual of Open image in new window is Open image in new window . Using Open image in new window we obtain Open image in new window . Under the standard k-duality the latter identifies with Open image in new window . This completes the proof of the main Ringel duality statement as given in formula (4). As a consequence we get the equivalence Open image in new window .

We now consider part (b). By part (a) we know Open image in new window , and as R is ideally ordered Theorem 2.14 implies that Open image in new window is both left and right strongly quasi-hereditary with respect to the ideal layer function. An algebra which is left and right strongly quasi-hereditary with respect to the same ideal layer function has global dimension at most two by [35, first Proposition in A.2]. Proposition 3.4 shows that Open image in new window is also left ultra strongly quasi-hereditary, and so completes the proof of statement (b).

We now prove (c). Let Open image in new window denote the number of simple Open image in new window -modules S that occur in a Jordan Hölder filtration of an Open image in new window -module M. If a partial ordering on I induces a quasi-hereditary structure, then \([\Delta _i,S_i]=1\) for all \(i \in I\); as k is algebraically closed this is equivalent to Open image in new window , see [18, Lemma 1.6].

Using the additional assumption in (c) that R is local, the ideally ordered condition produces a surjection between any two summands of Open image in new window (as all principal ideals are monomial by Lemma 7.3). Hence the ideal layer function induces an ordering on the summands of Open image in new window of the form \(\Lambda _0< \Lambda _1< \cdots < \Lambda _t\). Now consider another partial order that also produces a quasi-hereditary ordering.

We first prove that both orderings have the same maximal element. If \(\Lambda _i\) is maximal with respect to the new order, then the projective module Open image in new window is also a standard module in this order. If the new order gives rise to a quasi-hereditary structure then, as \(P_i\) is standard in this ordering, Open image in new window . As \(P_i\) is projective Open image in new window . Under the anti-equivalence Open image in new window , described in formula (2), this implies \(\dim \text {End}_{R}(\Lambda _i) =1\). Hence the identity morphism must equal socle projection so \(\Lambda _i\) is the simple R-module, which is unique as R is assumed to be local. The simple R-module is the largest summand \(\Lambda _t\) of Open image in new window under the ideal layer function ordering, and hence \(i=t\).

Secondly, we assume that the orderings match for \(k, k+1, \dots , t\), let Open image in new window be an immediate predecessor of \(\Lambda _k\) under the new order, and aim to show that \(j=k-1\). As R is ideally ordered there is a surjection between Open image in new window and Open image in new window (where Open image in new window exists as \(j<k\leqslant t\)). As they are labelled by the ideal layer function Open image in new window and there is a surjection Open image in new window . By the condition assumed in (c), the existence of this surjection implies an inclusion Open image in new window . Together these produce a non-trivial endomorphism Open image in new window which does not factor over \(\Lambda _i\) for \(i>j+1\). Using the anti equivalence Open image in new window again, this translates into a non-trivial endomorphism of Open image in new window that does not factor over \(P_i\) for \(i>j+1\). In particular, the standard object under the new order Open image in new window is the cokernel of a morphism Open image in new window where the summands of P are projective modules \(P_i\) such that \(i>j\) under the new ordering, see [18, Lemma 1.1\('\)]. If \(k \ne j+1\), then both the trivial endomorphism and the non-trivial endomorphism constructed above do not factor via P and hence Open image in new window . By considering the images of these morphisms we see Open image in new window . This would imply that the new ordering does not give a quasi-hereditary structure. Therefore \(j=k-1\).

Finally, by proceeding in this way we recover the ideal order and conclude that there is only one quasi-hereditary structure.

We show part (e). To prove (5), we explain the following chain of subcategoriesBy part (b), Open image in new window is right strongly quasi-hereditary. The first equality holds for all right strongly quasi-hereditary algebras, for example by a dual version of [35, Proposition A.1]. Using (9) and part (a), we see that Open image in new window so Open image in new window . The next inclusion follows from Open image in new window . The last inclusion holds for any right strongly quasi-hereditary algebra using that Open image in new window , which is closed under submodules as noted in Corollary 2.16. Using (9) and the fact that Open image in new window is left ultra strongly quasi-hereditary by part (b), dual arguments establish the following chain(the last inclusion was also shown in the proof of Corollary 3.6). This implies (6) and completes the proof of part (e). \(\square \)
For a monomial algebra R there is an equivalence of additive categories Open image in new window , and so Open image in new window is Morita equivalent toThis construction is considered in the general context of pre-radicals in Conde’s thesis. An additional special feature of the ideally ordered algebras is that Open image in new window , and this property does not hold for general monomial algebras. For example, consider the following example that was communicated to us by Xiao-Wu Chen.

Example 5.2

Let R be the path algebra of the following quiver with monomial relations.Then the left ideal Open image in new window is indecomposable but not principal.

Remark 5.3

We give several further remarks on this result.

(1) For the non-monomial algebra \(R=R_2\) in Example 2.8 (b), formula (4) from the theorem fails but the following Ringel duality formula holds:For ideally ordered monomial algebras this formula coincides with formula (4) above. Unfortunately, we were not able to find a more general setup where the formula (10) works.

Knörrer invariant algebras [27, Section 6.4.], see Example 1.4 (3) and Sect. 6.1, and truncated free algebras Open image in new window satisfy the additional condition imposed in (c).

(2) The statement that Open image in new window is related to Ringel’s [36, Remark before Corollary 2.2]. It would be interesting to see in what generality this equivalence holds.

We observe that it holds for Open image in new window where p runs over all paths of length 3, which is not ideally ordered but in which every principal left ideal is isomorphic to a monomial ideal. Indeed, in this case the equivalence is given by a tilting module which is obtained by mutating the characteristic tilting module (for the quasi-hereditary algebra structure defined by the ideal layer function) once.

(3) Consider Open image in new window , which is an ideally ordered finite dimensional local monomial algebra. Then there is a surjection \(Rx \rightarrow Ry\) but Ry does not include into Rx. One can check that the order \(R< Ry< Rx < Rx^2\) on indecomposable submodules of R defines a (left but not right strongly) quasi-hereditary structure on Open image in new window . In particular, in this case the ideal order is not the unique quasi-hereditary order.

(4) Part (c) can fail if R is not local (even if all the other conditions are satisfied). Indeed consider for example the algebra \(R=kQ/J^2\) whereand J is the ideal generated by all arrows. Then R is ideally ordered and for every surjection between principal left ideals \(\Gamma \rightarrow \Lambda \) there is an inclusion \(\Lambda \rightarrow \Gamma \). The order \(P_2< P_1 < S_1\) defines a quasi-hereditary structure on Open image in new window which is not left strongly quasi-hereditary. Hence, it differs from the quasi-hereditary structure defined by the ideal layer function (where \(P_2=P_1<S_1\)), and there is no unique quasi-hereditary structure in this case.

(5) It is true that R is ideally ordered iff \(R^{\mathrm{op}}\) is ideally ordered, and using this fact one can also prove the theorem without relying on Ringel’s result [36, Theorem 1.1].

6 Applications and examples

We discuss some relationships between Theorem 5.1 and several classes of algebras that have been studied in separate work.

6.1 Hille and Ploog’s algebras

The results of this paper were originally motivated by an investigation in [27] of a class of geometrically inspired quasi-hereditary algebras introduced by Hille and Ploog [24] for which the Ringel duality formula has a geometric interpretation, and we briefly recall this geometric setup and these algebras below.

As the geometric background, consider a type \(A_n\) configuration of intersecting rational curves \(C_1, \dots , C_n\) in a smooth, rational, projective surface X with negative self-intersection numbers Open image in new window . Starting with this data, Hille and Ploog consider the full triangulated subcategorywhere we recall that Open image in new window denotes the line bundle occurring as the ideal sheaf of an effective divisor \(D \subset X\). Hille and Ploog show that this subcategory carries an (exact) tilting object \(\Lambda \). To do this they make use of universal (co)extensions, see [18] and also [23] for the special case of vector bundles on a rational surface. We briefly recall the definition in this setting.

Definition 6.1

Consider an ordered pair of vector bundles Open image in new window on a smooth projective rational surface X. Their universal (co)extension is defined to be the vector bundle occurring in the middle of the short exact sequencewhere both sequences are determined by the identity element in Open image in new window .
Hille and Ploog show thatis an exceptional sequence of line bundles and that iterated universal extension along this sequence produces a tilting bundle \(\Lambda \), see [24, Section 2]. This defines a corresponding algebrawhere we assume that \(\Lambda \) is taken to be a basic representative of the tilting object. These algebras are quasi-hereditary by construction.

We note that the algebra depends on the choice of consecutive ordering for the labelling of the curves and that there are two choices, \(C_1, \dots , C_n\) or \(C_n, \dots , C_1\), for the same geometric set up that produce two different algebras Open image in new window and Open image in new window . This phenomenon is explained by the following result.

Proposition 6.2

There is an isomorphism of algebras Open image in new window .

Remark 6.3

The algebra Open image in new window can in fact be realised in the form \(E_{R}\) where R is an ideally ordered monomial Knörrer invariant algebra, as we describe below. Then Proposition 6.2 is an consequence of Theorem 5.1. However, the following alternative, short, geometric proof was explained to us by Agnieszka Bodzenta; indeed it was the existence of a Ringel duality formula in this special case that inspired the representation-theoretic generalisation in this paper. Work of Bodzenta and Bondal also realises a Ringel duality associated to birational morphisms of smooth surfaces by gluing t-structures with reversed orderings, see [7].

Proof

Let X be a smooth, rational, projective surface containing a type \(A_n\) configuration of rational curves with self-intersection numbers Open image in new window . Consider the exceptional sequence \(\mathbb {E}\) in the Hom-finite abelian category Open image in new window . By definition, Open image in new window , where Open image in new window is obtained from \(\mathbb {E}\) by taking iterated universal extensions and by passing to a basic representative, see [27, Section 2.3]. On the other hand, taking iterated universal coextensions of \(\mathbb {E}\) yields Open image in new window (again we replace this by a basic version if necessary) and it follows from [18, paragraph above Proposition 3.1.] that there is an algebra isomorphismwhere Open image in new window denotes the Ringel dual of \(\text {End}_X(\Lambda )\). More precisely, since \(\mathbb {E}\) is standardisable, Dlab and Ringel [18, Theorem 2] show thatdefines an exact equivalence sending \(\mathbb {E}\) to the sequence of standard modules \(\Delta _{\Lambda _{\varvec{\alpha }}}\). By Ringel [34, p. 217 and Proposition 2], the characteristic tilting module \(T_{\Lambda _{\varvec{\alpha }}} \!\in \mathrm{mod}\text {-}\Lambda _{\varvec{\alpha }}\) is obtained from \(\Delta _{\Lambda _{\varvec{\alpha }}}\) by iterated universal coextensions (and passing to a basic module if necessary). In particular, the exact equivalence \({\mathrm{Hom}}_X(\Lambda , -)\) sends T to \(T_{\Lambda _{\varvec{\alpha }}}\). Combining this with definition of the Ringel dual we seeas claimed.
Now consider the dualityThen Open image in new window and \(T^{\ddagger }\) is obtained from this sequence by iterated universal extensions. By definition, Open image in new window . Since \(\ddagger \) is a duality, Open image in new window . In combination with (11) this completes the proof.\(\square \)

Remark 6.4

We note that there is a change in conventions for compositions of morphisms between this paper and [27]. This corresponds to exchanging algebras with their opposite algebras, or left modules with right modules. The effect this has on the quasi-hereditary structure and Ringel duality is as follows: if A is a quasi-hereditary algebra with defining layer function L and characteristic tilting module T, then \(T^{\dagger }\) is the characteristic tilting module for \(A^\mathrm{op}\) where \(\dagger :A\text {-}\mathrm{mod}\rightarrow A^{\mathrm{op}}\text {-}\mathrm{mod}\) denotes the standard k-duality and the layer function on \(A^{\mathrm{op}}\) on is \(L^{\dagger }\) defined by Open image in new window . In particular, \(\mathfrak {R}(A^{\mathrm{op}}) \cong \mathfrak {R}(A)^{\mathrm{op}}\).

We briefly recap how the algebras \(\Lambda \) defined by Hille and Ploog fit into the general setup of Theorem 5.1. To do so we recall the definition of the Hirzebruch–Jung continued fraction expansion, the Knörrer invariant algebras \(K_{r,a}\), and a description of the form \(\Lambda _{\varvec{\alpha }} \cong E_{K_{r,a}}\).

Definition 6.5

For coprime integers \(0<a<r\) the Hirzebruch–Jung continued fraction Open image in new window is the collection of integers \(\alpha _i \geqslant 2\) defined by

Definition 6.6

([27, Definitions 4.6, 6.20 and Corollary 6.27]) For coprime integers \(0<a<r\) the Knörrer invariant algebra \(K_{r,a}\) is defined to bewhere the parameters \(l \geqslant 1\) and \(\beta _i \geqslant 2\) are defined by the Hirzebruch–Jung continued fraction expansion Open image in new window for the fraction \(r/(r-a)\).

We recall that the results of [27, Section 6.4] describe the monomial ideal structure on \(K_{r,a}\), and in particular combining [27, Theorem 6.26] and [27, Propositions 6.22 and 6.24] yields the following result.

Proposition 6.7

The Knörrer invariant algebra \(K_{r,a}\) is an ideally order monomial algebra and there is an isomorphism of quasi-hereditary algebras Open image in new window where Open image in new window is defined by the Hirzebruch–Jung continued fraction expansion of r / a.

Suppose that Open image in new window  mod r. If Open image in new window , then Open image in new window . Similarly, if Open image in new window , then Open image in new window . Using this result it can be seen from the explicit definition of \(K_{r,a}\) that Open image in new window . As a result Open image in new window by Proposition 6.7, and hence Theorem 5.1 is a generalisation of Proposition 6.2.

6.2 Example of an application of the Ringel duality formula

In this section we consider as an example the pair of algebras \(\Lambda _{[3,2]}\) and \(\Lambda _{[2,3]}\). After giving explicit presentations, we discuss their relationship via Ringel duality, their construction from related Knörrer invariant algebras, and explicitly list the distinguished modules in their quasi-hereditary structures in order to verify the Ringel duality formula.

Firstly, by [27, Proposition 6.18] the algebras \(\Lambda _{[3,2]}\) and \(\Lambda _{[2,3]}\) can respectively be presented as the path algebra of the following quivers with relations:andSecondly, the Ringel duality formula of Proposition 6.2 states thatThirdly, by Proposition 6.7 the corresponding Knörrer invariant algebras areand these can be presented via the following monomial diagrams:
where the nodes represent the monomial basis of \(K_{r,a}\) with the root of the tree representing 1 and the arrows labelled i representing left multiplication by \(z_i\) of the node at the source equalling the node at the target. Using these monomial diagrams one can show that \(K_{[3,2]} \cong K_{[2,3]}^{\mathrm{op}}\) and to calculate all the left monomial ideals. The left monomial ideals for \(K_{[3,2]}\) are \(M_0 \cong (1)\), \(M_1 \cong (z_1)\) and \(M_2 \cong (z_2 z_1)\) and the left monomial ideals for \(K_{[2,3]}\) are \(N_0 \cong (1)\), \(N_1 \cong (z_1)\) and \(N_2 \cong (z_1^2)\). These can represented pictorially as subsets of the monomial diagrams by
It is explicit that Open image in new window and Open image in new window .
In order to explicitly verify the Ringel duality formula in this case we first describe the quasi-hereditary structure by calculating the projective \(P_i\), injective \(I_i\), standard \(\Delta _i\), costandard \(\nabla _i\), and characteristic tilting \(T_i\) objects for each algebra. We list these modules in the table below in terms of the simples, \(S_i\) notated by i, occurring in their composition series with the heads written at the top.Using these descriptions of the characteristic tilting modules, it is a short exercise to verify the Ringel duality formula by direct calculation:and

Remark 6.8

We observe some further properties of, and relations between, the modules in the tables above. These are all special cases of the general theory developed above.
(1)

If \(i \leqslant j\) in the partial order, then there is an inclusion Open image in new window (and a projection Open image in new window ). This holds for all left (respectively right) strongly quasi-hereditary algebras. In other words, in this situation it is a consequence of Theorem 2.14.

(2)

Every submodule of a standard module \(\Delta _i\) or a projective module \(P_i\) is filtered by standard modules. This is a consequence of Corollary 2.16. Dually, quotients of costandard modules \(\nabla _i\) or injective modules \(I_i\) are filtered by costandard modules, again by Corollary 2.16.

(3)

For both algebras the only simple costandard module is \(\nabla _0\). One can check that the corresponding projective modules \(P_0\) are filtered by costandard modules. This illustrates Proposition 3.4 in these cases.

\((3^\mathrm{op})\)

For both algebras the only simple standard module is \(\Delta _0\). The corresponding injective hulls \(I_0\) are not filtered by standard modules. In other words, the algebras \(\Lambda _{[3,2]}\) and \(\Lambda _{[2,3]}\) are not right ultra strongly quasi-hereditary.

(4)

The summands \(T_i\) of the characteristic tilting module are precisely those indecomposable modules which are both quotients and submodules of the projective module \(P_0\), see Theorem 5.1 (d). In particular, they have head \(S_0\) and a socle in Open image in new window .

6.3 Auslander–Dlab–Ringel algebras

Recent results of Conde–Erdmann [13], and work in Conde’s thesis, produce a Ringel duality formula similar to that of Theorem 5.1 for the class of Auslander–Dlab–Ringel (ADR) algebras.

Definition 6.9

Let R be a finite dimensional algebra of Loewy length \(L_R\). Define the additive subcategoryand let Open image in new window be the direct sum of indecomposable elements of the additive category Open image in new window up to isomorphism. Then the associated ADR algebra is defined to beThis is the basic algebra Morita equivalent to Open image in new window . In particular, the indecomposable modules in Open image in new window are exactly those of the form Open image in new window for e a primative idempotent and Open image in new window where Open image in new window is the Loewy length of Re.

Remark 6.10

We remark that the ADR algebra defined here is the opposite algebra of the ADR algebra defined by Conde and Erdmann in [13], however the effect on the quasi-hereditary structure is straightforward as is explained in Remark 6.4.

The ADR algebra Open image in new window is quasi-hereditary for the layer function Open image in new window ; this induces the partial orderingon indecomposable modules in Open image in new window . Indeed it is left ultra strongly quasi-hereditary (see [11, Section 5]), and Conde and Erdmann obtain the following Ringel duality formula for ADR algebras satisfying a regularity condition; we recall that a module is rigid if its radical and socle series coincide.

Theorem 6.11

Let R be an Artin algebra with Loewy length L. If all projective and injective indecomposable R-modules are rigid with Loewy length L, thenThat is, the Ringel dual of Open image in new window is isomorphic to the opposite algebra of Open image in new window .

This formula looks very similar to the formula in Theorem 5.1 of this paper. However, in general Open image in new window and there does not appear to be any reason to think the overlap is large.

For example, ADR algebras are not left and right strongly hereditary in general and so not all ADR algebras are in the Open image in new window algebra class. Moreover, it can be seen that Hille and Ploog’s algebras are not always ADR algebras. Indeed, in the example of Sect. 6.2 the modules Open image in new window are straightforward to calculate from the monomial diagrams, and the additive category generated by such objects can be seen to coincide with the additive category Open image in new window for \(R=K_{[3,2]}\) so Open image in new window but not for Open image in new window where Open image in new window .

Indeed, the results of Conde and Erdmann also only describe the Ringel dual of an ADR algebra when the dual is also an ADR algebra. However, as can be seen in the example of Sect. 6.2, there are examples of ADR algebras of the form Open image in new window whose dual is not an ADR algebra but whose Ringel dual can still be described by Theorem 5.1: for \(R=K_{[3,2]}\) and Open image in new window Indeed, it is also straightforward to calculate the socle and radical filtrations in this example and hence clear to see that \(K_{[3,2]}\) is rigid whereas \(K_{[2,3]}\) is not.

Whilst these classes of algebras may not be related in general, there are cases which fall into both classes of algebras. Recall the monomial algebras Open image in new window of Example 1.4 (2) which are ideally ordered and for which Open image in new window . In particular, in this case Open image in new window is a corner algebra of Open image in new window : i.e. there is an idempotent Open image in new window such that Open image in new window .

Proposition 6.12

Let Q be a finite quiver without sources and J be the two-sided ideal generated by all arrows in Q. Then Open image in new window is an ideally ordered monomial algebra and there is an isomorphism of quasi-hereditary algebras Open image in new window .

Proof

The algebra R has Loewy length m and, as noted in Example 1.4 (2), any monomial ideal is isomorphic to Open image in new window for some \(l=1, \dots , m\) and some primitive idempotent \(e \in R\), hence R is ideally ordered and Open image in new window . As R is ideally ordered Open image in new window by Theorem 5.1 (a), and hence to show that Open image in new window it is sufficient to show that Open image in new window .

To show this consider an indecomposable object of Open image in new window . This is necessarily of the form Open image in new window for some primitive idempotent \(e_i\) corresponding to a vertex \(i \in Q\) and integer \(l=1, \dots , m\). As Q has no sources it follows that there exists a series of arrows \(j_{m-l} \xrightarrow {a_{m-l}\,} \cdots \xrightarrow {a_3\,} j_2 \xrightarrow {a_2\,} j_{1} \xrightarrow {a_1\,} i\) such that the path Open image in new window induces a homomorphism Open image in new window of indecomposable projective R-modules. By construction this has kernel Open image in new window , and hence there is an inclusion Open image in new window . In particular Open image in new window , and hence Open image in new window . Hence Open image in new window .

Whilst the layer functions defining the quasi-hereditary structures on Open image in new window and Open image in new window are not identical in general, we claim that the corresponding orderings do induce the same standard modules and hence the same quasi-hereditary structure on Open image in new window . To show this we let \(P_{i,l}\) denote the projective Open image in new window -module Open image in new window and \(S_{i,l}\) denote its simple quotient. We recall the order for Open image in new window is defined by Open image in new window and the order for Open image in new window is defined by Open image in new window . In particular, both orderings induce strongly quasi-hereditary structures, and hence for both orderings there are short exact sequences defining the respective standard modulesfor each projective module \(P_{i,l}\), see Definition 2.3. Hence to show that the two orderings induce the same quasi-hereditary structure it is sufficient to show that the projective submodules Open image in new window of \(P_{i,l}\) appearing in (12) are the same for both orderings. For this we note that under the additive anti-equivalencean Open image in new window -module Open image in new window is a proper submodule of \(P_{i,l}\) if and only if the corresponding R-module Open image in new window is a proper quotient of Open image in new window . This in turn is equivalent to Open image in new window and is also equivalent to \(i=j\) and \(k<l\). This shows that the two orderings induce the same quasi-hereditary structure.\(\square \)

It is a natural question whether it is possible to find an expanded class of algebras with a more general Ringel duality formula that encompasses both Theorems 5.1 and 6.11.

6.4 Nilpotent quiver algebras

The nilpotent quiver algebras introduced by Eiriksson and Sauter [20, Section 3] are a class of quasi-hereditary algebras.

Definition 6.13

Let \(Q=(Q_0,Q_1)\) be a finite quiver. For \(s\in \mathbb {Z}_{>0}\) the nilpotent quiver algebra is defined to bewhere \(Q^{(s)}\) is the staircase quiver \(Q^{(s)}\) defined by having vertices \(i_l\) for \(i \in Q_0\) and Open image in new window and arrows
and where \(J \subset kQ^{(s)}\) is the two-sided ideal generated by the relations

Remark 6.14

We remark again that the nilpotent quiver algebra defined here is the opposite algebra of the nilpotent quiver algebra defined by Eiriksson and Sauter in [20], however the effect on the quasi-hereditary structure is straightforward as is explained in Remark 6.4.

It follows from [20, Proposition 3.15] that all nilpotent quiver algebras \(N_s(Q)\) are right strongly quasi-hereditary and left ultra strongly quasi-hereditary for the quasi-hereditary structure determined by the layer function Open image in new window .

In particular, for \(R=kQ/J^m\) the ADR and nilpotent quiver algebras are related as follows.

Proposition 6.15

Let Q be a finite quiver, J the two-sided ideal generated by all arrows in Q, and m a positive integer. Then there is an isomorphism of quasi-hereditary algebrasif and only if all projective \(kQ/J^m\)-modules have Loewy length m: i.e. Q contains no sinks and m is arbitrary or \(m=1\) and Q is arbitrary.

Proof

Let \(R=kQ/J^m\), and let \(e_i \in R\) for \(i \in Q_0\) denote the primitive idempotents corresponding to vertices of Q. Up to isomorphism, the indecomposable modules in Open image in new window are exactly Open image in new window for \(1 \leqslant l \leqslant L(Re_i)\) and \(i \in Q_0\), where \(L(Re_i)\) is the Loewy length of the projective \(Re_i\).

In particular, the maximal Loewy length of a projective module in Open image in new window is m and so the maximum possible number of non-isomorphic indecomposables in Open image in new window is \(m|Q_0|\). But \(|Q^{(m)}_0|=m|Q_0|\), so for Open image in new window to be isomorphic to \(N_{m}(Q)\) it is necessary that all projective R-modules have Loewy length m.

Now suppose that all projective R-modules do have Loewy length m and consider the algebra Open image in new window . We start by labelling the indecomposable module in Open image in new window corresponding to Open image in new window by \(i_l\) and hence label the corresponding primitive idempotent by \(e_{i_l}\). There are indecomposable modules \(i_{l}\) for \(i \in Q_0\) and Open image in new window , matching the definition of the vertices in the staircase quiver \(Q^{(m)}\).

We now want to produce a morphism Open image in new window , and to do this we consider the morphisms between the indecomposable modules in Open image in new window . Firstly, there are surjections Open image in new window which we label by arrows Open image in new window for \(i \in Q_0\) and Open image in new window .

Secondly, an arrow \(a:i \rightarrow j \in Q_1\) corresponds to a morphism of projectives Open image in new window and for each l this induces a morphism Open image in new window with kernel Open image in new window which in turn induces an injective morphismfor each Open image in new window . We label these morphisms by Open image in new window for \(a \in Q_1\) and Open image in new window . In particular, the morphisms described here match the arrows of the staircase quiver \(Q^{(m)}\) under the identification \(a_l=\rho (a)_l\) and Open image in new window . In particular, an arrow \(a:i \rightarrow j\) in Q corresponds to a morphism Open image in new window which induces morphismswhere the relations Open image in new window and Open image in new window hold.
This allows us to define a morphism from the path algebra of the staircase algebra \( kQ^{(m)}\) to Open image in new window byand, as the relations imposed on \(kQ^{(m)}\) by \(N_m(Q)\) are mapped to 0, this induces a morphismWe will now show that \(\Phi \) is surjective, and then calculate the dimensions of \(N_m(Q)\) and Open image in new window to show that it is an isomorphism.
Suppose that Open image in new window is a morphismfor some \(i,j \in Q_0\) and Open image in new window . There is a surjection Open image in new window and so f gives a morphism Open image in new window . There is also a surjection Open image in new window , and as \(Re_i\) is projective this induces a uniquely defined morphism Open image in new window such thatAs a morphism between projective modules, the morphism Open image in new window corresponds to an element Open image in new window . In particular, \(g= \sum _p \lambda _p p \in kQ/J^m=R\) for scalars \(\lambda _p\) and homogeneous paths p from j to i in \(kQ/J^m=R\) corresponding to morphisms Open image in new window .
We now work with one indecomposable path p, corresponding to a morphism Open image in new window , and suppose that p consists of Open image in new window arrows Open image in new window for \(a_i \in Q_1\). We define a corresponding path in \(N_m(Q)\) from \(i_{l-n}\) to \(j_l\) byfor Open image in new window . Similarly, we define the path in \(N_m(Q)\)for \(i \in Q_0\) and Open image in new window such that \(\Phi ( (\pi _{i,l}) ) \cong \pi _{i,l}\). Then the morphism p factors over its kernel, which is Open image in new window , soHenceUsing the relations in \(N_m(Q)\) we can rearrange this expression aswhere we note that if g is non-zero then \(k-n \leqslant l\) and henceand henceReturning to the morphism \(g = \sum \lambda _p p\) we see thatand we can now conclude thatwhere |p| is the length of the path p, but \(\pi _{i,l}\) is surjective and henceWe conclude that \(\Phi \) is a surjection, and we now show that this surjective morphism is in fact an isomorphism by calculating the dimensions of \(N_m(Q)\) and Open image in new window .
We first calculate the dimension of Open image in new window by calculating the dimension of the morphisms between any two indecomposables in Open image in new window . As shown above, a morphism in Open image in new window of the form Open image in new window is induced by a particular element in \(kQ/J^m\) corresponding to a morphism of projective modules Open image in new window . Such elements are spanned by the paths, and we now calculate the morphisms in Open image in new window that are induced by such path in \(R=kQ/J^m\). These will give a basis for the morphisms Open image in new window . A path \(p:j \rightarrow i \in kQ/J^m = R\) of length |p| (under the length grading on Q) induces the morphism Open image in new window which composes to give a non-zero morphism Open image in new window if and only if \(|p|<k\). In turn, this descends to give a non-zero morphism Open image in new window if and only if Open image in new window , which occurs if and only if \(l \geqslant k-|p|\). As such there are isomorphisms of vector spacesWe then calculate the dimension of \(N_m(Q)\) by counting the number of paths between any two vertices. Using the explicit description of \(N_m(Q)\) above, any path in \(N_m(Q)\) corresponds to the composition of arrows of type \(a_l\) and arrows of type Open image in new window , these commute Open image in new window , and Open image in new window . Using these relations any non-zero path can be rearranged such that all the Open image in new window type arrows occur in the path before the \(a_l\) type arrows. That is: a path from \(i_l\) to \(j_k\) in \(N_m(Q)\) exactly corresponds to the path Open image in new window in \(N_m(Q)\) induced by a path Open image in new window from j to i in Q of length |p| pre-composed with \(l-k+|p|\) arrows of Open image in new window typeso that the induced path is from \(i_l\) to \(j_k\). However, the path is non-zero if and only if the number of type Open image in new window arrows is greater than or equal to 0 and strictly less than l, and it follows thatHenceIt follows that the surjective homomorphism Open image in new window is in fact an isomorphism as Open image in new window . Hence Open image in new window .

Further, under this isomorphism the layer functions defining the quasi-hereditary structures on \(N_m(Q)\) and Open image in new window are identified and hence this is an isomorphism of quasi-hereditary algebras.\(\square \)

Example 6.16

We give a brief example of Proposition 6.15. Consider the quiverand let J denote the two-sided ideal generated by all arrows. Define Open image in new window , and then we present the two algebras \(N_3(Q)\) and Open image in new window .
Firstly, the algebra \(N_3(Q)\) is defined to be the path algebra of the quiver with relationsSecondly, we consider the indecomposable modules in Open image in new window . There are six classes and we list them and a basis for all injective or surjective maps between them below.This describes Open image in new window and matches the path algebra with relations description of \(N_3(Q)\) above.

Combining Proposition 6.15 with Proposition 6.12 and Theorem 5.1 (or Theorem 6.11) instantly gives the following corollary.

Corollary 6.17

If Q is a finite quiver without sinks or sources and m is a positive integer, then there are isomorphisms of quasi-hereditary algebrasIn particular, the Ringel dual for a nilpotent quiver algebra without sinks or sources is determined by the formula
$$\begin{aligned} \mathfrak {R}(N_m(Q)) \cong N_m(Q^{\mathrm{op}})^{\mathrm{op}}\!. \end{aligned}$$

We note that if Q is a finite quiver with no sinks but with sources then \(E_{kQ/J^m}^{\mathsf {ADR}} \cong N_m(Q)\) but Open image in new window (if \(m>1\)) as \(kQ^{\mathrm{op}}\) contains sinks. In particular, Proposition 6.15 and Theorem 6.11 cannot be used to strengthen the Ringel duality formula of Corollary 6.17 to all quivers with no sources.

6.5 Auslander and Nakayama algebras

For a finite dimensional algebra R of finite representation type we define Open image in new window , where the sum is taken over all indecomposable \(M \in R\text {-}\mathrm{mod}\) up to isomorphism, and the Auslander algebra is defined to be

Proposition 6.18

If R is an ideally ordered monomial algebra, then Open image in new window if and only if R is self-injective.

Proof

If Open image in new window , hence every injective R-module I embeds into \(R^n\). Therefore, I is a direct summand of \(R^n\), hence projective, and hence R is self-injective.

Conversely, if R is self-injective, then every injective R-module embeds into \(R^n\) for some n and hence every injective module is also a projective module. Then every object in \(R\text {-}\mathrm{mod}\) is a submodule of an injective R-module, hence of a projective R-module, hence Open image in new window and Open image in new window .\(\square \)

The Nakayama algebras, introduced in [31], are a well known class of finite dimensional algebras with finite representation type; see e.g. [4, Theorem VI.2.1]. Recall that a self-injective Nakayama algebra is of the form \(kC_n/J^m\) where \(C_n\) is an oriented cycle with n vertices and J is the ideal generated by all arrows, see e.g. [1, Theorem 32.4] for the a description of the underlying quiver of a general Nakayama algebra. In particular, the self-injective Nakayama algebras are ideally ordered monomial algebras.

Corollary 6.19

If R is a self-injective Nakayama algebra, then Open image in new window .

It follows from the explicit description \(R=kC_n/J^m\) that Open image in new window by Corollary 6.17 and so this corollary recovers the well known explicit description of the Auslander algebras of self-injective Nakayama algebra Open image in new window in terms of quivers with relations. As Open image in new window and \(N_m(C_n) \cong N_m(C_n^{\mathrm{op}})^{\mathrm{op}}\) the Ringel duality formula recovers the result of [37] that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual for the ideal layer function.

Corollary 6.20

For a self-injective Nakayama algebra R, Open image in new window .

Remark 6.21

In order to give another perspective on Proposition 6.18, and Corollaries 6.19 and 6.20, we recall that self-injective finite dimensional monomial algebras R are Nakayama algebras. To see this, we have to show that the quiver Q underlying R is a union of oriented lines and oriented cycles. In other words, at every vertex of Q there is at most one incoming and at most one outgoing arrow. Assume that there is a vertex i with more than one outgoing arrow. Then, as R is monomial, the corresponding indecomposable projective R-module \(P_i\) does not have a simple socle - in particular, \(P_i\) is not injective contradicting our assumption that R is self-injective. A dual argument shows that Q does not have vertices with more than one incoming arrow.

7 Appendix: Results on finite dimensional monomial algebras

In this section we collect some technical results on finite dimensional monomial algebras \(R=kQ/I\) (where I is generated by a collection of paths in Q). We will use the term ‘monomial’ to mean a monomial expression in the generators (i.e. arrows and lazy paths) of such an algebra.

Lemma 7.1

Let R be a monomial algebra and \(n, m \in R\) monomials. If there exists a surjection \(\phi :Rm \rightarrow Rn\), then the map \(Rm \rightarrow Rn\) defined by \(m \mapsto n\) is R-linear.

Proof

It suffices to show that \(\mathrm {ann}_R(m)\) is contained in \(\mathrm {ann}_R(n)\). Take \(r \in R\) with \(rm=0\), and we aim to show that \(rn=0\). We write \(r = \sum \lambda _i r_i\) with monomials \(r_i\) and non-zero scalars \(\lambda _i\). Since R is monomial, it follows that Open image in new window for all i. The existence of a surjection \(\phi :Rm \rightarrow Rn\) implies \(m, n \in eR\) for some primitive idempotent \(e \in R\) and that there exist \(s,t \in R\) such that Open image in new window and \(\phi (m)=sn\). In particular, \(tsn=n\) and so Open image in new window for some non-zero scalars \(\mu _i\) and distinct monomials \(s_i \ne e\). Therefore Open image in new window , and so as R is monomial it follows that all monomials that make up \(r_i s n\) are 0. In particular, \(r_i \mu _0 e n=\mu _0 r_i n =0\). This implies that \(r_in=0\) for all i, and hence \(rn=0\) so \(\mathrm {ann}_R(m) \subset \mathrm {ann}_R(n)\) finishing the proof.\(\square \)

Lemma 7.2

Let \(m, n \in R\) be monomials. If R is ideally ordered, then every surjection \(Rm \rightarrow Rn\) factors over \(\pi :Rm \rightarrow Rn\), \(m \mapsto n\).

Proof

Let \(\psi :Rm \rightarrow Rn\) be an surjection. In particular, \(m, n \in eR\) for some primitive idempotent \(e \in R\) and there exist \(s,t \in R\) such that \(\psi (m)=sn\) and Open image in new window . It follows that \(tsn=n\), so \(s=\lambda _0 e + \sum \lambda _i s_i \in eRe\) for non-zero scalars \(\lambda _i\) and distinct monomials \(s_i \ne e\). Hence \(sn=\lambda _0 n + \sum \lambda _i s_i n\). In particular, \(R s_i n \subsetneq Rn\), and since R is ideally ordered there exists surjections \(Rn \rightarrow R s_i n\) which, using Lemma 7.1, we can assume are defined by \(n \mapsto s_i n\). Denote the composition of such a surjection with the inclusion \(R s_i n \subseteq Rn\) by \(\varphi _i\) and define \(\varphi :Rn \rightarrow Rn\) as \(\varphi = \lambda _0 \mathrm {id} + \sum \lambda _i \varphi _i\). Then \(\varphi (n)=sn\) and therefore \(\psi = \varphi \pi \) factors as claimed.\(\square \)

Lemma 7.3

Let \(p \in eR\) for a primitive idempotent \(e \in R\). If R is ideally ordered, then the principal left ideal Rp is isomorphic to a principal ideal Rm, for a monomial \(m \in eR\).

Proof

Since R is monomial, we may write p as linear combination of monomials Open image in new window with \(\lambda _i\) non-zero scalars and \(p_i \in eR\) monomials. Since R is ideally ordered we may assume that the \(p_i\) are labelled in such a way that \(Rp_1 \rightarrow Rp_2 \rightarrow \cdots \rightarrow Rp_t\) are surjections.

We now wish to rewrite p so that none of the \(p_i\) can be expressed in the form \(np_1\) for a monomial n. To do this, let I index the \(p_i\) such that there is a monomial \(r_{i}\) with \(p_{i}=r_{i}p_1\) for \(i \in I\). Then we define Open image in new window and Open image in new window . As Open image in new window it follows that \(s= \lambda _1 e+r\) is a unit in eRe and there exists \(t \in eRe\) such that \(st=e\). In particular, \(Rtp =Rp\). Then we rewrite Open image in new window for some non-zero scalars Open image in new window and monomials Open image in new window . For each Open image in new window there is some \(p_i\) such that Open image in new window by their definition, and hence there are surjections Open image in new window for all j. As \(Rtp \cong Rp\) we now work with tp rather than p and tp has the property that there are no Open image in new window with Open image in new window for a monomial n.

We claim that \(Rtp \cong Rp_1\), hence \(Rp \cong Rp_1\). As there are surjections Open image in new window there are surjections Open image in new window , Open image in new window by Lemma 7.1. Let Open image in new window be the composition of such a surjection with the canonical inclusion Open image in new window and let \(\iota :Rp_1 \rightarrow R\) be the canonical inclusion. Define \(\psi :Rp_1 \rightarrow R\) by Open image in new window . Then Open image in new window so \(\mathsf {im}\,\psi = Rtp\). Hence \(\psi \) defines a surjective morphism \(\phi :Rp_1 \rightarrow Rtp\).

We must now check that this morphism is also injective. If \(\psi (rp_1)=0\), then Open image in new window . As R is monomial if \(rp_1\) is non-zero there must exist monomials \(n,m \in R\) such that Open image in new window for some j, and if this occurs either Open image in new window or Open image in new window for submonomials \(m'\) and \(n'\) neither equal e. The first case cannot occur as this implies Open image in new window which contradicts the existence of a surjection Open image in new window . The second situation also cannot occur as the construction of the Open image in new window above ensured none were of this form. Hence \(rp_1=0\) so the morphism is also injective and \(Rp_1 \cong Rtp \cong Rp\).\(\square \)

Notes

Acknowledgements

We thank Teresa Conde for interesting discussions about this work and about relations to her thesis. We are grateful to Karin Erdmann for pointing us to Ringel’s paper which simplifies the proof of our main result and adds another perspective to this work. We also thank Agnieszka Bodzenta who, in particular, explained to us the proof of Proposition 6.2 and Xiao-Wu Chen who shared with us Example 5.2. We would also like to thank Ögmunder Eiriksson, Julian Külshammer, Daiva Pučinskaitė, Špela Špenko, and Michael Wemyss for interesting and helpful discussions and David Ploog for pointing out misprints in an earlier version. We would also particularly like to thank the anonymous referee, whose many useful comments have greatly improved the paper.

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Authors and Affiliations

  1. 1.FreiburgGermany
  2. 2.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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