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Forum Mathematicum

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Volume 32, Issue 1

Issues

Representations of constant socle rank for the Kronecker algebra

Daniel Bissinger
  • Corresponding author
  • Mathematisches Seminar, Christian-Albrechts-UniversitΓ€t zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
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Published Online: 2019-09-11 | DOI:Β https://doi.org/10.1515/forum-2018-0143

Abstract

Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p-elementary abelian groups Er of rank r over a field of characteristic p>0, we introduce the notions of modules with constant d-radical rank and modules with constant d-socle rank for the generalized Kronecker algebra 𝒦r=k⁒Γr with rβ‰₯2 arrows and 1≀d≀r-1. We study subcategories given by modules with the equal d-radical property and the equal d-socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories in mod⁑𝒦r are of wild type. Then we use a natural functor 𝔉:mod⁑𝒦rβ†’mod⁑k⁒Er to transfer our results to mod⁑k⁒Er.

Keywords: Kronecker algebra; Auslander–Reiten theory; constant socle rank; wild representation type

MSC 2010: 16G20; 16G60; 16G70

Introduction

Let rβ‰₯2. Let k be an algebraically closed field of characteristic p>0, and let Er be a p-elementary abelian group of rank r. It is well known that the category of finite-dimensional k⁒Er-modules mod⁑k⁒Er is of wild type, whenever pβ‰₯3 or p=2 and r>2. Therefore, subclasses with more restrictive properties have been studied; in [5], the subclass of modules of constant rank CR⁒(Er) and modules with even more restrictive properties, called equal images property and equal kernels property, were introduced. Let γ€ˆx1,…,xr〉k be a complement of Rad2⁑(k⁒Er) in Rad⁑(k⁒Er), and set xΞ±:-βˆ‘i=1rΞ±i⁒xi for α∈kr. We say that M∈mod⁑k⁒Er has constant Jordan type if the Jordan canonical form of the nilpotent operator xΞ±M:Mβ†’M,m↦xΞ±β‹…m is independent of α∈krβˆ–{0}. If the image (kernel) of xΞ±M does not depend on Ξ±, we say that M has the equal images (kernels) property.

In [21], the author defined analogous categories CR, EIP and EKP in the context of the generalized Kronecker algebra 𝒦r, and in more generality for the generalized Beilinson algebra B⁒(n,r). Using a natural functor 𝔉:mod⁑𝒦rβ†’mod⁑k⁒Er with nice properties, she gave new insights into the categories of equal images and equal kernels modules for mod⁑k⁒Er of Loewy length ≀2. A crucial step is the description of CR, EIP and EKP in homological terms, involving a family β„™r-1-family of regular β€œtest”-modules.

Building on this approach, we show that the recently introduced modules [6] of constant socle rank and constant radical rank can be described in the same fashion. For 1≀d<r, we introduce modules of constant d-radical rank CRRd and constant d-socle rank CSRd in mod⁑𝒦r. More restrictive – and also easier to handle – are modules with the equal d-radical property ERPd and the equal d-socle property ESPd. For d=1, we have ESP1=EKP, ERP1=EIP and CSR1=CR=CRR1. Studying these classes in the hereditary module category mod⁑𝒦r allows us to use tools not available in mod⁑k⁒Er.

As a first step, we establish a homological characterization of CSRd, CRRd, ESPd and ERPd. We denote by Grd,r the Grassmanian of d-dimensional subspaces of kr. In generalization of [21], we define a Grd,r-family of β€œtest”-modules (XU)U∈Grd,r and show that the modules in this family can be described in purely combinatorial terms by being indecomposable of dimension vector (1,r-d). This allows us to construct many examples of modules of equal socle rank in mod⁑𝒦s for sβ‰₯3 by considering pullbacks along natural embeddings 𝒦r→𝒦s.

Since 𝒦r is a wild algebra for r>2 and every regular component in the Auslander–Reiten quiver of 𝒦r is of type ℀⁒A∞, it is desirable to find invariants that give more specific information about the regular components. It is shown in [21] that there are uniquely determined quasi-simple modules Mπ’ž and Wπ’ž in π’ž such that the cone (Mπ’žβ†’)βŠ†π’ž consisting of all modules lying on an oriented path starting in Mπ’ž satisfies (Mπ’žβ†’)=EKPβˆ©π’ž, and the cone (β†’Wπ’ž) consisting of all modules lying on an oriented path ending in Wπ’ž satisfies (β†’Wπ’ž)=EIPβˆ©π’ž. Using results on elementary modules, we generalize this statement for ESPd and ERPd. Our main results may be summarized as follows.

Theorem.

Let rβ‰₯3 and C be a regular component of the Auslander–Reiten quiver of Kr.

  • (a)

    For each 1≀i<r , the category Ξ”i:-ESPiβˆ–ESPi-1 is wild, where ESP0:-βˆ….

  • (b)

    For each 1≀i<r , there exists a unique quasi-simple module Mi in π’ž such that ESPiβˆ©π’ž=(Miβ†’).

  • (c)

    There exists at most one number 1<m⁒(π’ž)<r such that Ξ”m⁒(π’ž)βˆ©π’ž is non-empty. If such a number exists, Ξ”m⁒(π’ž)βˆ©π’ž is the ray starting in Mm⁒(π’ž).

If there is no such number as in (c), we set m⁒(π’ž)=1. An immediate consequence of (b) and (c) is that, for 1≀i≀j<r, we have Mi=Mj or τ⁒Mi=Mj. Moreover, statement (a) shows the existence of a lot of AR-components such that Ξ”iβˆ©π’ž is a ray, and for every such component, we have m⁒(π’ž)=i. With the dual result for ERP, we assign a number 1≀w⁒(π’ž)<r to each regular component π’ž, giving us the possibility to distinguish (r-1)2 different types of regular components.

Figure 1

Illustration of a regular component with m⁒(π’ž)β‰ 1. The shaded region on the right hand side is Ξ”1.

To prove statement (a), we exploit the fact that every regular module M in mod⁑𝒦r has self-extensions with dimk⁑Ext⁑(M,M)β‰₯2, by applying the process of simplification. This method was introduced in [17] and produces extension closed subcategories, whose objects may be filtered by pairwise orthogonal bricks. For a p-elementary abelian group Er of rank r over an algebraically closed field of characteristic p>0, mod⁑k⁒Er is the only such subcategory. We therefore use the functor 𝔉:mod⁑𝒦rβ†’mod⁑k⁒Er, whose essential image (the full subcategory of mod⁑k⁒Er formed by all modules isomorphic to modules of the form 𝔉⁑(M)) consists of all modules of Loewy length ≀2, to transfer our results to mod⁑k⁒Er. We denote by ESP2,d⁒(Er) the category of modules in mod⁑k⁒Er of Loewy length ≀2 with the equal d-socle property and arrive at the following result.

Corollary.

Let char⁑(k)>0, rβ‰₯3 and 1≀d<r. Then ESP2,d⁒(Er)βˆ–ESP2,d-1⁒(Er) has wild representation type, where ESP2,0⁒(Er):-βˆ….

For r=2, we consider the Beilinson algebra B⁒(3,2). The fact that B⁒(3,2) is a concealed algebra of type Q=1β†’2⇉3 allows us to apply the simplification process in mod⁑k⁒Q. We find a wild subcategory in the category of all modules in mod⁑B⁒(3,2) with the equal kernels property and conclude the following.

Corollary.

Assume that char⁑(k)=p>2; then the full subcategory of modules with the equal kernels property in mod⁑k⁒E2 and Loewy length ≀3 has wild representation type.

In particular, we generalize results by Benson [3] and Bondarenko and Lytvynchuk [4] concerning the wildness of various subcategories of k⁒Er-modules. We also construct examples of regular components π’ž such that each module in π’ž has constant d-socle rank, but no module in π’ž is GLr-stable in the sense of [6].

1 Preliminaries

Throughout this article, let k be an algebraically closed field and rβ‰₯2. If not stated otherwise, k is of arbitrary characteristic. We denote by Q=(Q0,Q1) a finite and connected quiver without oriented cycles. For an arrow Ξ±:xβ†’y∈Q1, we define s⁒(Ξ±)=x and t⁒(Ξ±)=y. We say that Ξ± starts in s⁒(Ξ±) and ends in t⁒(Ξ±). A (finite-dimensional) representation M=((Mx)x∈Q0,(M⁒(Ξ±))α∈Q1) over Q consists of vector spaces Mx and linear maps M⁒(Ξ±):Ms⁒(Ξ±)β†’Mt⁒(Ξ±) such that dimk⁑M:-βˆ‘x∈Q0dimk⁑Mx is finite. A morphism f:Mβ†’N between representations is a collection of linear maps (fz)z∈Q0 such that, for each arrow Ξ±:xβ†’y, there is a commutative diagram

The category of finite-dimensional representations over Q is denoted by rep⁑(Q), and kQ is the path algebra of Q with idempotents ex, x∈Q0. The k-algebra kQ is a finite-dimensional, associative, basic and connected k-algebra. Let mod⁑k⁒Q be the class of finite-dimensional kQ left modules. Given M∈mod⁑k⁒Q, we set Mx:-ex⁒M. The categories mod⁑k⁒Q and rep⁑(Q) are equivalent (see for example [1, Theorem III 1.6]). We will therefore switch freely between representations of Q and modules of kQ if one of the approaches seems more convenient for us. We assume that the reader is familiar with Auslander–Reiten theory and basic results on wild hereditary algebras. For a well written survey on the subjects, we refer to [1, 14, 13].

Definition.

Recall the definition of the dimension function

dimΒ―:mod⁑k⁒Qβ†’β„€Q0,M↦(dimk⁑Mx)x∈Q0.

If 0β†’Aβ†’Bβ†’Cβ†’0 is an exact sequence, then dim¯⁑A+dim¯⁑C=dim¯⁑B. The quiver Q defines a (non-symmetric) bilinear form

γ€ˆ-,-〉:β„€Q0Γ—β„€Q0β†’β„€,

given by ((xi),(yj))β†¦βˆ‘i∈Q0xi⁒yi-βˆ‘Ξ±βˆˆQ1xs⁒(Ξ±)⁒yt⁒(Ξ±). For the case that x,y are given by dimension vectors of quiver representations, there is another description of γ€ˆ-,-〉 known as the Euler–Ringel form [17]

γ€ˆdim¯⁑M,dim¯⁑N〉=dimk⁑Hom⁑(M,N)-dimk⁑Ext⁑(M,N).

We denote by q=qQ:β„€Q0β†’β„€ the corresponding quadratic form. A vector dβˆˆβ„€Q0 is called a real root if q⁒(d)=1 and an imaginary root if q⁒(d)≀0.

Denote by Ξ“r the r-Kronecker quiver, which is given by two vertices 1,2 and arrows Ξ³1,…,Ξ³r:1β†’2.

Figure 2

The Kronecker quiver Ξ“r.

We set 𝒦r:-k⁒Γr and P1:-𝒦r⁒e2, P2:-𝒦r⁒e1. P1 and P2 are the indecomposable projective modules of mod⁑𝒦r, dimk⁑Hom⁑(P1,P2)=r and dimk⁑Hom⁑(P2,P1)=0. As Figure 2 suggests, we write

dim¯⁑M=(dimk⁑M1,dimk⁑M2).

For example, dim¯⁑P1=(0,1) and dimk⁑P2=(1,r). The Coxeter matrix Φ and its inverse Φ-1 are

Ξ¦:-(r2-1-rr-1),Ξ¦-1=(-1r-rr2-1).

For M indecomposable, dim¯⁑τ⁒M=Φ⁒(dim¯⁑M) holds if M is not projective and dim¯⁑τ-1⁒M=Ξ¦-1⁒(dim¯⁑M) if M is not injective. The quadratic form q is given by q⁒(x,y)=x2+y2-r⁒x⁒y.

Figure 3

Auslander–Reiten quiver of 𝒦r.

Figure 3 shows the notation we use for the components 𝒫,ℐ in the Auslander–Reiten quiver of 𝒦r which contain the indecomposable projective modules P1,P2 and indecomposable injective modules I1,I2. The set of all other components is denoted by β„›.

Ringel has proven [18, Theorem 2.3] that every component in β„› is of type ℀⁒A∞ if rβ‰₯3 or a homogeneous tube ℀⁒A∞/γ€ˆΟ„γ€‰ if r=2. A module in such a component is called regular. An irreducible morphism in a regular component is injective if the corresponding arrow is uprising (see Figure 1 for rβ‰₯3) and surjective otherwise. A regular module M is called quasi-simple if the AR-sequence terminating in M has an indecomposable middle term. If M is quasi-simple in a regular component π’ž, there is an infinite chain (a ray) of irreducible monomorphisms

M=M⁒[1]β†’M⁒[2]β†’M⁒[3]β†’β‹―β†’M⁒[l]β†’β‹―

and an infinite chain (a coray) of irreducible epimorphisms

⋯⁒(l)⁒Mβ†’β‹―β†’(3)⁒Mβ†’(2)⁒Mβ†’(1)⁒M=M,

and for each regular module X, there are unique quasi-simple modules N,M and lβˆˆβ„• with (l)⁒M=X=N⁒[l]. The number l is called the quasi-length of X. We fix the orientation of each regular component in such a way that the quasi-simple modules form the bottom layer of the component (see Figure 1).

The indecomposable modules in 𝒫 are called preprojective modules and the modules in ℐ are called preinjective modules. Moreover, we call an arbitrary module preprojective (resp. preinjective, regular) if all its indecomposable direct summands are preprojective (resp. preinjective, regular). It is P in 𝒫 (I in ℐ) if and only if there is lβˆˆβ„•0 with Ο„l⁒P=Pi (Ο„-l⁒I=Ii) for i∈{1,2}. Recall that there are no homomorphisms from right to left [1, Corollary VIII.2.13]. To emphasize this result later on, we just write

Hom⁑(ℐ,𝒫)=0=Hom⁑(ℐ,β„›)=0=Hom⁑(β„›,𝒫).

Using the canonical equivalence ([1, Theorem III.1.6]) of categories mod⁑𝒦rβ‰…rep⁑(Ξ“r), we introduce the duality Ξ΄:mod⁑𝒦rβ†’mod⁑𝒦r by setting (δ⁒M)x:-(Mψ⁒(x))βˆ— and (δ⁒M)⁒(Ξ³i):-(M⁒(Ξ³i))βˆ— for each M∈rep⁑(Ξ“r), where ψ:{1,2}β†’{1,2} is the permutation of order 2. Note that δ⁒(Pi)=Ii for all iβˆˆβ„•. We state a simplified version of Kac’s theorem [12, Theorem 1.10] for the Kronecker algebra in combination with results on the quadratic form proven in [17, Lemma 2.3].

Theorem 1.1.

Let rβ‰₯2 and d∈N02.

  • (a)

    If d=dim¯⁑M for some indecomposable module M , then q⁒(d)≀1.

  • (b)

    If q⁒(d)=1 , then there exists a unique indecomposable module X with dim¯⁑X=d . In this case, X is preprojective or preinjective, and X is preprojective if and only if dimk⁑X1<dimk⁑X2.

  • (c)

    If q⁒(d)≀0 , then there exist infinitely many indecomposable modules Y with dim¯⁑Y=d and each Y is regular.

Since there is no pair (a,b)βˆˆβ„•02βˆ–{(0,0)} satisfying a2+b2-r⁒a⁒b=q⁒(a,b)=0 for rβ‰₯3, we conclude together with [1, Lemma VIII.2.7] the following.

Corollary 1.2.

Let M be an indecomposable Kr-module. Then Ext⁑(M,M)=0 if and only if M is preprojective or preinjective. If rβ‰₯3 and M is regular, then dimk⁑Ext⁑(M,M)β‰₯2.

Let modpf⁑𝒦r be the subcategory of all modules without non-zero projective direct summands and modif⁑𝒦r the subcategory of all modules without non-zero injective summands. Since 𝒦r is a hereditary algebra, the Auslander–Reiten translation Ο„:mod⁑𝒦rβ†’mod⁑𝒦r induces an equivalence from modpf⁑𝒦r to modif⁑𝒦r. In particular, if M and N are indecomposable with M,N not projective, we get Hom⁑(M,N)β‰…Hom⁑(τ⁒M,τ⁒N). The Auslander–Reiten formula [1, Theorem VI.2.13] simplifies to the following.

Theorem 1.3 ([13, Theorem 2.3]).

For X,Y in mod⁑Kr, there a functorial isomorphisms

Ext(X,Y)β‰…Hom(Y,Ο„X)βˆ—β‰…Hom(Ο„-1Y,X)βˆ—.

2 Modules of constant radical and socle rank

2.1 Elementary modules of small dimension

Let rβ‰₯3. The homological characterization in [21] uses an algebraic family of modules of projective dimension 1 for the Beilinson algebra B⁒(n,r) on n vertices. For n=2, we have B⁒(2,r)=𝒦r, and mod⁑𝒦r is a hereditary category. Hence every non-projective module is of projective dimension 1. In the following, we study the module family (XΞ±)α∈krβˆ–{0} for n=2. We will see later on that each non-zero proper submodule of XΞ± is isomorphic to a finite number of copies of P1, and XΞ± itself is regular. In particular, we do not find a short exact sequence 0β†’Aβ†’XΞ±β†’Bβ†’0 such that A and B are regular and non-zero. In the language of wild hereditary algebras, we therefore deal with elementary modules.

Definition ([16, Definition 1]).

A non-zero regular module E is called elementary if there is no short exact sequence 0→A→E→B→0 with A and B regular non-zero. In particular, elementary modules are indecomposable and quasi-simple.

Elementary modules are analogues of quasi-simple modules in the tame hereditary case (r=2). If X is a regular module, then X has a filtration 0=X0βŠ‚X1βŠ‚β‹―βŠ‚Xr=X such that Xi/Xi-1 is elementary for all 1≀i≀r and the elementary modules are the smallest class with that property. For basic results on elementary modules, used in this section, we refer to [16].

We are grateful to Otto Kerner for pointing out the following helpful lemma.

Lemma 2.1.1.

Let E be an elementary module and X,Y regular with non-zero morphisms f:Xβ†’E and g:Eβ†’Y. Then g∘fβ‰ 0. In particular, End⁑(E)=k.

Proof.

Since f is non-zero and Hom⁑(β„›,𝒫)=0=Hom⁑(ℐ,β„›), im⁑f is a regular non-zero submodule of E. Consequently, since E is elementary, coker⁑f is preinjective by [16, Proposition 1.3]; hence g cannot factor through coker⁑f. ∎

We use the theory on elementary modules to generalize [21, Corollary 2.7] in the hereditary case.

Proposition 2.1.2.

Let E be a non-empty family of elementary modules of bounded dimension, and put

𝒯⁒(β„°):-ker⁑Ext⁑(β„°,-)={M∈mod⁑𝒦r∣Ext⁑(E,M)=0⁒for all⁒Eβˆˆβ„°}.

Then the following statements hold.

  • (1)

    𝒯⁒(β„°) is closed under extensions, images and Ο„.

  • (2)

    𝒯⁒(β„°) contains all preinjective modules.

  • (3)

    For each regular component π’ž , the set 𝒯⁒(β„°)βˆ©π’ž forms a non-empty cone in π’ž , which consists of the predecessors of a uniquely determined quasi-simple module in π’ž , i.e. there is Wβˆˆπ’ž quasi-simple such that 𝒯(β„°)βˆ©π’ž=(β†’W):-{(i+1)Ο„lW∣i,lβˆˆβ„•0}.

Proof.

(1) Since Ext2=0, the category is closed under images and extensions. Let Mβˆˆπ’―β’(β„°); then the Auslander–Reiten formula yields 0=dimk⁑Hom⁑(M,τ⁒E) for all Eβˆˆβ„°. We first show that M is not preprojective. Assume to the contrary that P=M is preprojective. Let lβˆˆβ„•0 such that Ο„l⁒P is projective; then Ο„l⁒P=Pi for an i∈{1,2} and

0=dimk⁑Hom⁑(P,τ⁒E)=dimk⁑Hom⁑(Ο„l⁒P,Ο„l+1⁒E)=(dim¯⁑(Ο„l+1⁒E))3-i.

This is a contradiction since every non-sincere 𝒦r-module is semi-simple. Hence M is regular or preinjective.

Now we show τ⁒Mβˆˆπ’―β’(β„°). In view of the Auslander–Reiten formula, we get

dimk⁑Ext⁑(E,τ⁒M)=dimk⁑Hom⁑(M,E).

Let f:Mβ†’E be a morphism, and assume that fβ‰ 0. Since 0=Hom⁑(ℐ,β„›), the module M is not preinjective and therefore regular. As a regular module E has self-extensions (see Corollary 1.2), and therefore Eβˆ‰π’―β’(β„°). Hence 0β‰ dimk⁑Ext⁑(E,E)=dimk⁑Hom⁑(E,τ⁒E), and we find 0β‰ g∈Hom⁑(E,τ⁒E). Lemma 2.1.1 provides a non-zero morphism

M⁒→𝑓⁒E⁒→𝑔⁒τ⁒E.

We conclude 0β‰ dimk⁑Hom⁑(M,τ⁒E)=dimk⁑Ext⁑(E,M)=0, a contradiction. Hence

0=dimk⁑Hom⁑(M,E)=dimk⁑Ext⁑(E,τ⁒M) for all⁒Eβˆˆβ„°β’and⁒τ⁒Mβˆˆπ’―β’(β„°).

(2) The injective modules I1,I2 are contained in 𝒯⁒(β„°). Now apply (1).

(3) The existence of the cones can be shown as in [21, Theorem 3.3]. We sketch the proof. Let Xβˆˆπ’ž be a quasi-simple module, and denote the upper bound by L. By [13, Lemma 4.6, Proposition 10.5], we find n0βˆˆβ„• such that Ext⁑(Y,Ο„l⁒X)=0 for all lβ‰₯n0 and each regular representation Y with dimk⁑Y≀L and Ext⁑(E,Ο„-l⁒X)β‰ 0 for some Eβˆˆβ„°. In particular, we have Ο„l⁒Xβˆˆπ’―β’(β„°) and Ο„-l⁒Xβˆ‰π’―β’(β„°). Now (1) shows that 𝒯(β„°)βˆ©π’ž=(β†’M) for the uniquely determined quasi-simple module Mβˆˆπ’ž with Mβˆˆπ’―β’(β„°) and Ο„-1βˆ‰π’―β’(β„°). ∎

The next result follows by the Auslander–Reiten formula and duality since δ⁒(E) is elementary if and only if E is elementary.

Proposition 2.1.3.

Let E be a family of elementary modules of bounded dimension, and put

ℱ⁒(β„°):-{M∈mod⁑𝒦r∣Hom⁑(E,M)=0⁒for all⁒Eβˆˆβ„°}.

Then the following statements hold.

  • (1)

    ℱ⁒(β„°) is closed under extensions, submodules and Ο„-1.

  • (2)

    ℱ⁒(β„°) contains all preprojective modules.

  • (3)

    For each regular component π’ž , the set ℱ⁒(β„°)βˆ©π’ž forms a non-empty cone in π’ž , which consists of the successors of a uniquely determined quasi-simple module in π’ž.

Note that ℱ⁒(β„°) is a torsion-free class of some torsion pair (𝒯,ℱ⁒(β„°)) and 𝒯⁒(β„°) is the torsion class of some torsion pair (𝒯⁒(β„°),β„±) (see for example [1, Proposition VI.1.4]).

Lemma 2.1.4.

Let M,N be indecomposable modules with dim¯⁑M=(c,1),dim¯⁑N=(1,c), 1≀c<r. Then the following statements hold.

  • (a)

    Ο„z⁒M and Ο„z⁒N are elementary for all zβˆˆβ„€ . Moreover, every proper factor of M is injective, and every proper submodule of N is projective.

  • (b)

    Every proper factor module of Ο„l⁒M is preinjective, and every proper submodule of Ο„-l⁒N is preprojective for lβˆˆβ„•0.

Proof.

We will give the proofs for M. The statements for N follow by duality.

(a) By [16, Lemma 1.1], M is elementary if and only if all elements in its Ο„-orbit are elementary. Since (c,1) is an imaginary root of the quadratic form, M is regular. Now let 0βŠ‚XβŠ‚M be a proper submodule with dimension vector dim¯⁑X=(a,b). Then b=1 since Hom⁑(ℐ,β„›)=0. Hence dim¯⁑M/X=(c-a,0), and M/X is injective.

(b) Let lβ‰₯1 and 0β†’ker⁑fβ†’Ο„l⁒Mβ†’Xβ†’0 be exact with X regular and non-zero. We have to show that Ο„l⁒Mβ‰…X. We assume that ker⁑fβ‰ 0. Since Hom⁑(ℐ,β„›)=0, we conclude that ker⁑f has no indecomposable preinjective direct summand. Since Ο„-1 is right exact [1, Corollary VII.1.9], we get an exact sequence Ο„-1⁒ker⁑fβ†’Mβ†’Ο„-l⁒Xβ†’0. We conclude with (a) that Ο„-l⁒Xβ‰…M; hence Xβ‰…Ο„-l⁒M. ∎

2.2 An algebraic family of test-modules

Let rβ‰₯2. Now we take a closer look at the modules (XΞ±)α∈krβˆ–{0}. Let us start this section by recalling some definitions from [21] and the construction of the module family. We use a slightly different notation since we are only interested in the case B⁒(2,r)=𝒦r.

For α∈kr and M∈mod⁑𝒦r, we define xΞ±:-Ξ±1⁒γ1+β‹―+Ξ±r⁒γr and denote by xΞ±M:Mβ†’M the linear operator associated to xΞ±.

Definition.

For α∈krβˆ–{0}, the map Ξ±Β―:γ€ˆe2〉k=P1β†’P2, e2↦α1⁒γ1+β‹―+Ξ±r⁒γr=xΞ± defines an embedding of 𝒦r-modules and is just the left multiplication by xΞ±. We now set XΞ±:-coker⁑α¯.

These modules are the β€œtest”-modules introduced in [21]. In fact, im⁑α¯ is a 1-dimensional submodule of P2 contained inside the radical rad⁑(P2) of the local module P2. From the definition, we get an exact sequence 0β†’P1β†’P2β†’XΞ±β†’0 and dim⁑XΞ±=(1,r)-(0,1)=(1,r-1). Since P2 is local with semi-simple radical rad⁑(P2)=P1r, it now seems natural to study embeddings P1dβ†’P2 for 1≀d<r and the corresponding cokernels. This motivates the next definition. We restrict ourselves to d<r since otherwise the cokernel is the simple injective module.

Definition.

Let and 1≀d<r. For T=(u1,…,ud)∈(kr)d, we define TΒ―:(P1)dβ†’P2 as the 𝒦r-linear map

T¯⁒(x)=βˆ‘i=1duΒ―iβˆ˜Ο€i⁒(x),

where πi:(P1)d→P1 denotes the projection onto the i-th coordinate.

The map TΒ― is injective if and only if T is linearly independent; then we have

dim¯⁑coker⁑T¯=dim¯⁑P2-d⁒dim¯⁑P1=(1,r-d),

and coker⁑TΒ― is indecomposable because P2 is local. Moreover, (1,r-d) is an imaginary root of q, and therefore coker⁑TΒ― is regular indecomposable and by Lemma 2.1.4 elementary. We define γ€ˆT〉:-γ€ˆu1,…,ud〉k.

Lemma 2.2.1.

Let T,S∈(kr)d such that dimkβ‘γ€ˆT〉=d=dimkβ‘γ€ˆS〉; then coker⁑TΒ―β‰…coker⁑SΒ― if and only if γ€ˆT〉=γ€ˆS〉.

Proof.

If γ€ˆT〉=γ€ˆS〉, then the definition of TΒ― and SΒ― implies im⁑TΒ―=im⁑SΒ―. Hence coker⁑TΒ―=P2/im⁑TΒ―=coker⁑SΒ―.

Now let γ€ˆSγ€‰β‰ γ€ˆT〉, and assume that 0β‰ Ο†:coker⁑TΒ―β†’coker⁑SΒ― is 𝒦r-linear. Since coker⁑SΒ― is local with radical P1r-d and Hom⁑(β„›,𝒫)=0, the map Ο† is surjective and therefore injective. Recall that P2 has {e1,Ξ³1,…,Ξ³r} as a basis. Let x∈P2 such that φ⁒(e1+im⁑TΒ―)=x+im⁑SΒ―. Since Ο† is 𝒦r-linear, we get

x+im⁑SΒ―=e1⁒φ⁒(e1+im⁑TΒ―)=e1⁒x+im⁑SΒ―

and hence x-e1⁒x∈im⁑SΒ―. Write x=μ⁒e1+βˆ‘i=1rΞΌi⁒γi; then

x-μ⁒e1=βˆ‘i=1rΞΌi⁒γi=x-e1⁒x∈im⁑S¯ and x+im⁑SΒ―=μ⁒e1+im⁑SΒ―.

The assumption γ€ˆSγ€‰β‰ γ€ˆT〉 yields y∈im⁑SΒ―βˆ–im⁑TΒ―βŠ†γ€ˆΞ³1,…,Ξ³r〉k. Then y+im⁑TΒ―β‰ 0 and

φ⁒(y+im⁑TΒ―)=y⁒φ⁒(e1+im⁑TΒ―)=μ⁒y⁒(e1+im⁑SΒ―)=μ⁒y+im⁑SΒ―=im⁑SΒ―,

a contradiction to the injectivity of Ο†. Hence Hom⁑(coker⁑TΒ―,coker⁑SΒ―)=0 and coker⁑T¯≇coker⁑SΒ―. ∎

Definition.

Let rβ‰₯2 and U∈Grd,r with basis T=(u1,…,ud). We define XU:-coker⁑TΒ―.

Remark.

XU is well defined (up to isomorphism) with dimension vector dim⁑XU=(1,r-d), and XU is elementary for rβ‰₯3 and quasi-simple for r=2.

For a module X, we define add⁑X as the category of summands of finite direct sums of X, and Qd denotes the set of isomorphism classes [M] of indecomposable modules M with dimension vector (1,r-d) for 1≀d<r.

Proposition 2.2.2.

Let M be indecomposable.

  • (a)

    If [M]∈Qd , then there exists U∈Grd,r with Mβ‰…XU.

  • (b)

    The map Ο†:Grd,rβ†’Qd, U↦[XU] is bijective.

  • (c)

    Let 1≀c≀d<r and [M]∈Qd . There is [N]∈Qc and an epimorphism Ο€:Nβ†’M.

Proof.

(a) Let 0⊊X⊊M be a submodule of M. Then XβŠ†rad⁑(M)=P1r-d, and X is in add⁑P1. It is

1=dimk⁑M1=dimk⁑Hom⁑(𝒦r⁒e1,M)=dimk⁑Hom⁑(P2,M),

so we find a non-zero map Ο€:P2β†’M. Since every proper submodule of M is in add⁑P1 and Hom⁑(P2,P1)=0, the map Ο€:P2β†’M is surjective and yields an exact sequence 0β†’P1dβ’β†’πœ„β’P2β’β†’πœ‹β’Mβ†’0. For 1≀i≀d, there exist uniquely determined elements Ξ²i,Ξ±i1,…,Ξ±ir∈k such that

ι⁒(gi⁒(e2))=Ξ²i⁒e1+Ξ±i1⁒γ1+β‹―+Ξ±ir⁒γr∈P2=γ€ˆΞ³i,e1∣1≀i≀r〉k,

where gi:P1β†’P1d denotes the embedding into the i-th coordinate. Since e2 is an idempotent with e2⁒γj=Ξ³j (1≀j≀r) and e2⁒e1=0, we get

Ξ±i1⁒γ1+β‹―+Ξ±ir⁒γr=e2⁒(ι∘gi⁒(e2))=(ι∘gi)⁒(e2β‹…e2)=(ι∘gi)⁒(e2)=Ξ²i⁒e1+Ξ±i1⁒γ1+β‹―+Ξ±ir⁒γr.

Hence Ξ²i=0. Now define Ξ±i:-(Ξ±i1,…,Ξ±ir), T:-(Ξ±1,…,Ξ±d) and U:-γ€ˆT〉. It is ΞΉ=TΒ―, and by the injectivity of ΞΉ, we conclude that T is linearly independent, and therefore U∈Grd,r. Now we conclude

XU=coker⁑T¯=coker⁑ι=M.

(b) This is an immediate consequence of (a) and Lemma 2.2.1.

(c) By (a), we find U in Grd,r with basis T=(u1,…,ud) such that XUβ‰…M. Let V be the subspace with basis S=(u1,…,uc). Then im⁑SΒ―βŠ†im⁑TΒ―, and we get an epimorphism Ο€:XV=P2/im⁑SΒ―β†’P2/im⁑TΒ―=XU, x+SΒ―β†’x+TΒ― with dim¯⁑XV=(1,r-c). ∎

As a generalization of xΞ±M:Mβ†’M, we introduce maps xTM:Mβ†’Md and yTM:Mdβ†’M for 1≀d<r and T∈(kr)d. Note that xTM=yTM if and only if d=1.

Definition.

Let 1≀d<r and T=(Ξ±1,…,Ξ±d)∈(kr)d. We denote by xTM and yTM the operators

xTM:Mβ†’Md,m↦(xΞ±1M⁒(m),…,xΞ±dM⁒(m)),yTM:Mdβ†’M,(m1,…,md)↦xΞ±1M⁒(m1)+…+xΞ±dM⁒(md).

It is im⁑xTMβŠ†M2βŠ•β‹―βŠ•M2, M2βŠ•β‹―βŠ•M2βŠ†ker⁑yTM and (xTM)βˆ—=yTδ⁒M since, for f=(f1,…,fd)∈(δ⁒M)d and m∈M, we have

(xTM)βˆ—β’(f)⁒(m)=(xTM)βˆ—β’(f1,…,fd)⁒(m)=βˆ‘i=1d(fi∘xΞ±iM)⁒(m)=βˆ‘i=1dfi(xΞ±i.m)=βˆ‘i=1d(xΞ±i.fi)(m)=yTδ⁒M⁒(f1,…,fd)⁒(m)=yTδ⁒M⁒(f)⁒(m).

Lemma 2.2.3.

Let 1≀d<r and U∈Grd,r. Every non-zero quotient Q of XU is indecomposable. Q is preinjective (injective) if dim¯⁑Q=(1,0) and regular otherwise.

Proof.

Since XU is regular, we conclude with Hom⁑(β„›,𝒫)=0 that every indecomposable non-zero quotient of XU is preinjective or regular. Let Q be such a quotient with dim¯⁑Q=(a,b) and Qβ‰ XU. Since XU is local with radical P1r-d and dimΒ―=(1,r-d), it follows (1,r-d)=(a,b)+(0,c) for some c>0. Hence a=1, and Q is an injective module if b=0. Otherwise, Q is also indecomposable since b>0 and Q=AβŠ•B with A,Bβ‰ 0 imply w.l.o.g. (dim¯⁑B)1=0. Hence B∈add⁑P1, which is a contradiction to Hom⁑(β„›,𝒫)=0. ∎

2.3 Modules for the generalized Kronecker algebra

In the following, we will give the definition of 𝒦r-modules (rβ‰₯2) with constant radical rank and constant socle rank.

Definition.

Let M be in mod⁑𝒦r and 1≀d<r.

  • (a)

    M has constant d-radical rank if the dimension of

    RadU⁑(M):-βˆ‘u∈UxuM⁒(M)βŠ†M2

    is independent of the choice of U∈Grd,r.

  • (b)

    M has constant d-socle rank if the dimension of

    SocU⁑(M):-{m∈M∣xuM⁒(M)=0⁒for all⁒u∈U}=β‹‚u∈Uker⁑(xuM)βŠ‡M2

    is independent of the choice of U∈Grd,r.

  • (c)

    M has the equal d-radical property if RadU⁑(M) is independent of the choice of U∈Grd,r

  • (d)

    M has the equal d-socle property if SocU⁑(M) is independent of the choice of U∈Grd,r

Definition.

Let 1≀d<r. We define

  • (a)

    ESPd:-{M∈mod⁑𝒦r∣M2=SocU⁑(M)⁒for all⁒U∈Grd,r},

  • (b)

    ERPd:-{M∈mod⁑𝒦r∣M2=RadU⁑(M)⁒for all⁒U∈Grd,r},

  • (c)

    CSRd:-{M∈mod⁑𝒦r∣there exists⁒cβˆˆβ„•0⁒such that⁒dimk⁑SocU⁑(M)=c⁒for all⁒U∈Grd,r},

  • (d)

    CRRd:-{M∈mod⁑𝒦r∣there exists⁒cβˆˆβ„•0⁒such that⁒dimk⁑RadU⁑(M)=c⁒for all⁒U∈Grd,r}.

Lemma 2.3.1.

Let M be indecomposable and not simple.

  • (a)

    M has the equal d -socle property if and only if M∈ESPd.

  • (b)

    M has the equal d -radical property if and only if M∈ERPd.

Proof.

(a) Assume that M is in ESPd, and let W:-SocU⁑(M) for U∈Grd,r. Denote by e1,…,er∈kr the canonical basis vectors. We find a k-complement of β‹‚i=1rker⁑(xeiM)∩M1 in M1, say KβŠ†M1. Then M decomposes into submodules M=(K+M2)βŠ•β‹‚i=1rker⁑(xeiM)∩M1. Since M is not simple, we have 0β‰ M2 and conclude {0}=β‹‚i=1rker⁑(xeiM)∩M1, i.e. β‹‚i=1rker⁑(xeiM)=M2 (see also [10, Lemma 5.1.1]). Denote by S⁒(d) the set of all subsets of {1,…,r} of cardinality d. Then

β‹‚S∈S⁒(d)β‹‚j∈Sker⁑(xejM)=β‹‚i=1rker⁑(xeiM)=M2.

Since γ€ˆej∣j∈S〉k∈Grd,r and M∈ESPd, we get β‹‚j∈Sker⁑(xejM)=W and hence M2=β‹‚S∈S⁒(d)W=W=SocU⁑(M).

(b) Let M∈ERPd, U∈Grd,r and W:-RadU⁑(M). Since M is not simple, it is [10, Lemma 5.1.1] βˆ‘i=1rxeiM=M2 and hence

W=βˆ‘S∈S⁒(d)βˆ‘j∈SxejM⁒(M)=βˆ‘i=1rxeiM⁒(M)=M2.∎

Remark.

For the benefit of the reader, we recall the definitions of the classes CR, EKP and EIP given in [21].

EKP:-{M∈mod⁑𝒦r∣M2=ker⁑(xΞ±M)⁒for all⁒α∈krβˆ–{0}},EIP:-{M∈mod⁑𝒦r∣M2=im⁑(xΞ±M)⁒for all⁒α∈krβˆ–{0}},CR:-{M∈mod⁑𝒦r∣there exists⁒cβˆˆβ„•0⁒such that⁒c=dimk⁑ker⁑(xΞ±M)⁒for all⁒α∈krβˆ–{0}}.

Note that CRR1=CR=CSR1, ERP1=EIP, ESP1=EKP, and for U∈Grd,r with basis (u1,…,ud), we have RadU⁑(M)=βˆ‘i=1dxuiM⁒(M)=im⁑y(u1,…,ud)M and SocU⁑(M)=β‹‚i=1dker⁑(xuiM)=ker⁑x(u1,…,ud)M. We restrict the definition to d<r since Grr,r={kr}, and therefore every module in mod⁑𝒦r is of constant r-socle and r-radical rank.

Lemma 2.3.2.

Let M be in mod⁑Kr and 1≀d<r .

  • (a)

    M∈CSRd if and only if δ⁒M∈CRRd.

  • (b)

    M∈ESPd if and only if δ⁒M∈ERPd.

Proof.

Note that RadU(Ξ΄M)=im(yTδ⁒M)=im(xTM)βˆ—β‰…(imxTM)βˆ— and hence

M-dimk⁑SocU⁑(M)=dimk⁑M-dimk⁑ker⁑xTM=dimk⁑im⁑xTM=dimk⁑(im⁑xTM)βˆ—=dimk⁑RadU⁑(δ⁒M).

Hence M∈CSRd if and only if δ⁒M∈CRRd. Moreover, M in ESPd if and only if SocU⁑(M)=M2, and hence dimk⁑RadU⁑(δ⁒M)=dimk⁑M1=dimk⁑(δ⁒M)2. ∎

For the proof of the following proposition, we use the same methods as in [21, Theorem 2.5].

Proposition 2.3.3.

Let 1≀d<r∈N. Then

ESPd={M∈mod⁑𝒦r∣Hom⁑(XU,M)=0⁒for all⁒U∈Grd,r},CSRd={M∈mod⁑𝒦r∣there exists⁒cβˆˆβ„•0⁒such that⁒dimk⁑Hom⁑(XU,M)=c⁒for all⁒U∈Grd,r}.

Proof.

Let U∈Grd,r with basis T=(Ξ±1,…,Ξ±d). Consider the short exact sequence

0β†’(P1)d⁒→T¯⁒P2β†’XUβ†’0.

Application of Hom⁑(-,M) yields

0β†’Hom⁑(XU,M)β†’Hom⁑(P2,M)⁒→TΒ―βˆ—β’Hom⁑(P1d,M)β†’Ext⁑(XU,M)β†’0.

Moreover, let

f:Hom⁑(P2,M)β†’M1,gβ†’g⁒(e1),g:Hom⁑(P1d,M)β†’M2d,h↦(h∘ι1⁒(e2),…,h∘ιd⁒(e2))

be the natural isomorphisms, where ΞΉi:P1β†’P1d denotes the embedding into the i-th coordinate. Let Ο€M2d:Mdβ†’M2d be the natural projection. The equality g∘TΒ―βˆ—=Ο€M2d∘xTM|M1∘f holds since both maps are applied to a homomorphism. Hence dimk⁑ker⁑(Ο€M2d∘xTM|M1:M1β†’M2d)=dimk⁑ker⁑(TΒ―βˆ—)=dimk⁑Hom⁑(XU,M). Now let cβˆˆβ„•0. We conclude

dimk⁑Hom⁑(XU,M)=c⇔dimk⁒ker⁑(Ο€M2d∘xTM|M1)=c⁒⇔im⁑xTMβŠ†M2d⁒dimk⁑ker⁑(xTM|M1)=c⇔M2βŠ†ker⁑(xTM)⁒dimk⁑ker⁑(xTM)=c+dimk⁑M2⇔SocU⁑(M)=ker⁑(xTM)⁒dimk⁑SocU⁑(M)=c+dimk⁑M2.

This finishes the proof for CSRd. Moreover, note that c=0 together with Lemma 2.3.1 yields

M∈ESPd⇔there exists⁒W≀M⁒such that⁒SocU⁑(M)=W⁒for all⁒U∈Grd,r⇔SocU(M)=M2for allU∈Grd,r⇔dimkSocU(M)=0+dimkM2for allU∈Grd,r⇔Hom⁑(XU,M)=0⁒for all⁒U∈Grd,r.∎

Since Ο„βˆ˜Ξ΄=Ξ΄βˆ˜Ο„-1, the next result follows from the Auslander–Reiten formula and Lemma 2.3.2.

Proposition 2.3.4.

Let 1≀d<r∈N. Then

ERPd={M∈mod⁑𝒦r∣Ext⁑(δ⁒τ⁒XU,M)=0⁒for all⁒U∈Grd,r},CRRd={M∈mod⁑𝒦r∣there exists⁒cβˆˆβ„•0⁒such that⁒dimk⁑Ext⁑(δ⁒τ⁒XU,M)=c⁒for all⁒U∈Grd,r}.

Remark.

For d=1, we have U=γ€ˆΞ±γ€‰k with α∈krβˆ–{0}, XUβ‰…XΞ±, and [21, Proposition 3.1] yields δ⁒τ⁒XUβ‰…XU. However, this identity holds if and only if d=1. From Proposition 2.3.3 and Lemma 2.1.4, it follows immediately that, for 1≀d<r-1 and V∈Grd,r, the module XV is in CSRd+1βˆ–CSRd. If not stated otherwise, we assume from now on that rβ‰₯3.

In view of Proposition 2.1.3, Lemma 2.1.4 and the definitions of ESPd and ERPd, we immediately get the following proposition.

Proposition 2.3.5.

Let 1≀d<r and C a regular component of Kr.

  • (a)

    ESP1βŠ†ESP2βŠ†β‹―βŠ†ESPr-1 and ERP1βŠ†ERP2βŠ†β‹―βŠ†ERPr-1.

  • (b)

    ESPd is closed under extensions, submodules and Ο„-1 . Moreover, ESPd contains all preprojective modules, and ESPdβˆ©π’ž forms a non-empty cone in π’ž , i.e. there is a quasi-simple module Mdβˆˆπ’ž such that

    ESPdβˆ©π’ž=(Mdβ†’):-{Ο„-lMd[i+1]∣i,lβˆˆβ„•0}.

  • (c)

    ERPd is closed under extensions, images and Ο„ . Moreover, ERPd contains all preinjective modules, and ERPdβˆ©π’ž forms a non-empty cone in π’ž , i.e. there is a quasi-simple module Wdβˆˆπ’ž such that

    ERPdβˆ©π’ž=(β†’Wd):-{(i+1)Ο„lWd∣i,lβˆˆβ„•0}.

Definition.

For 1≀i<r, we set Ξ”i:-ESPiβˆ–ESPi-1 and βˆ‡i:-ERPiβˆ–ESPi-1, where ESP0=βˆ…=ERP0.

The next result suggests that, for each regular component π’ž and 1<i<r, only a small part of vertices in π’ž corresponds to modules in Ξ”i. Nonetheless, we will see in Section 4 that, for 1≀i<r, the categories Ξ”i and βˆ‡i are of wild type.

Proposition 2.3.6.

Let rβ‰₯3. Let C be a regular component and Mi,Wi (1≀i<r) in C the uniquely determined quasi-simple modules such that ESPi∩C=(Miβ†’) and ERPi∩C=(β†’Wi).

  • (a)

    There exists at most one number 1<m⁒(π’ž)<r such that Ξ”m⁒(π’ž)βˆ©π’ž is non-empty. If such a number exists, then Ξ”m⁒(π’ž)βˆ©π’ž={Mm⁒(π’ž)⁒[l]∣lβ‰₯1}.

  • (b)

    There exists at most one number 1<w⁒(π’ž)<r such that βˆ‡w⁒(π’ž)βˆ©π’ž is non-empty. If such a number exists, then βˆ‡w⁒(π’ž)βˆ©π’ž={(l)⁒Ww⁒(π’ž)∣lβ‰₯1}.

Proof.

(a) By Proposition 2.3.5, there are n1,…,nr-1βˆˆβ„•0 such that

0=n1≀n2≀⋯≀nr-1 and Mi=Ο„ni⁒M1 for all⁒i∈{1,…,r-1}.

We will show that either 0=n1=β‹―=nr-1 or that there exists a uniquely determined 1<i<r such that ni>ni-1.

Let M be in π’ž, and assume Mβˆ‰ESP1. In the following, we show that τ⁒Mβˆ‰ESPr-1. There exists α∈krβˆ–{0} with Hom⁑(XU,M)β‰ 0 for U=γ€ˆΞ±γ€‰k. Hence we find a non-zero map f:τ⁒XU→τ⁒M. Consider an exact sequence 0β†’P1r-2β†’XUβ†’Nβ†’0. Then dim¯⁑N=(1,1), and by Lemma 2.2.3, N is indecomposable. By Proposition 2.2.2, there exists V∈Grr-1,r with XV≅δ⁒N. Since δ⁒XU=τ⁒XU (see [21, Proposition 3.1]), we get a non-zero morphism g:XV→τ⁒XU and by Lemma 2.1.1 a non-zero morphism

XV⁒→𝑔⁒τ⁒XU⁒→𝑓⁒τ⁒M.

Therefore, τ⁒Mβˆ‰ESPr-1 by Proposition 2.3.3.

Now assume that niβ‰ nj for some i and j. Then, in particular, M1β‰ Mr-1. Hence nr-1β‰₯1. By definition, we have M:-τ⁒M1βˆ‰ESP1, and the above considerations yield τ⁒(τ⁒M1)=τ⁒Mβˆ‰ESPr-1. Therefore, 1≀nr-1<2 since ESPr-1βˆ©π’ž is closed under Ο„-1. Therefore, nr-1=1 and Mr-1=τ⁒M1. We conclude that there is a uniquely determined 1<i<r such that ni>ni-1, and in this case, ni=ni-1+1. Now we set m⁒(π’ž):-i.

(b) This follows by duality. ∎

We state two more results that follow from Proposition 2.3.3 and will be needed later on. The first one is a generalization of [21, Lemma 3.5] and follows with the same arguments.

Lemma 2.3.7.

Let 0→A→B→C→0 be an almost split sequence such that two modules of the sequence are of constant d-socle rank. Then the third module also has constant d-socle rank.

Definition.

Let rβ‰₯2 and 1≀d<r, and let 𝔛d,r:-{XU∣U∈Grd,r}. Let 𝔛d,rβŠ₯ be the right orthogonal category 𝔛d,rβŠ₯={M∈mod⁑𝒦r∣Hom⁑(XU,M)=0⁒for all⁒U∈Grd,r}, and let 𝔛d,rβŠ₯ be the left orthogonal category 𝔛d,rβŠ₯:-{M∈mod⁑𝒦r∣Hom⁑(M,XU)=0⁒for all⁒U∈Grd,r}. Then we set 𝔛¯d,r:-𝔛d,rβŠ₯βˆ©π”›d,rβŠ₯.

Note that every module in 𝔛¯d,r is regular by Proposition 2.3.5.

Lemma 2.3.8.

Let rβ‰₯3 and 1≀d<r, and let M be quasi-simple regular in a component C such that M∈XΒ―d,r. Then every module in C has constant d-socle rank.

Proof.

Let V∈Grd,r. We have shown in Proposition 2.3.5 that the set

{N∣Hom⁑(XV,N)=0}βˆ©π’žβ€ƒ(resp.⁒{N∣Ext⁑(XV,N)=0}βˆ©π’žβ’)

is closed under Ο„-1 (resp. Ο„). Since 0=dimk⁑Hom⁑(M,XV)=dimk⁑Ext⁑(XV,τ⁒M), we have Ext⁑(XV,Ο„l⁒M)=0 for lβ‰₯1. The Euler–Ringel form yields

0=dimk⁑Ext⁑(XV,Ο„l⁒M)=-γ€ˆdim¯⁑XV,dim¯⁑τl⁒M〉+dim¯⁑Hom⁑(XV,Ο„l⁒M).

Since γ€ˆdim¯⁑XV,dim¯⁑τl⁒M〉=γ€ˆ(1,r-d),dim¯⁑M〉 is independent of V, Ο„l⁒M has constant d-socle rank. On the other hand, Hom⁑(XV,M)=0 implies that Ο„-q⁒M has constant d-socle rank for all qβ‰₯0. It follows that each quasi-simple module in π’ž has constant d-socle rank. Now apply Lemma 2.3.7. ∎

3 Process of simplification and applications

3.1 Representation type

Denote by Ξ›:-k⁒Q the path algebra of a connected, wild quiver Q. We use the notation introduced in [14]. Recall that a module M is called brick if End⁑(M)=k, and two modules M,N are called orthogonal if we have Hom⁑(M,N)=0=Hom⁑(N,M).

Definition.

Let 𝒳 be a non-empty class of pairwise orthogonal bricks in mod⁑Λ. The full subcategory ℰ⁒(𝒳) is by definition the class of all modules Y in mod⁑Λ with an 𝒳-filtration, that is, a chain

0=Y0βŠ‚Y1βŠ‚β‹―βŠ‚Yn-1βŠ‚Yn=Y

with Yi/Yi-1βˆˆπ’³ for all 1≀i≀n.

In [17, Theorem 1.2], the author shows that ℰ⁒(𝒳) is an exact abelian subcategory of mod⁑Λ, closed under extensions, and 𝒳 is the class of all simple modules in ℰ⁒(𝒳). In particular, a module M in ℰ⁒(𝒳) is indecomposable if and only if it is indecomposable in mod⁑k⁒Q.

Proposition 3.1.1.

Let rβ‰₯3, and let XβŠ†mod⁑Kr be a non-empty class of pairwise orthogonal bricks with self-extensions (and therefore regular).

  • (a)

    Every module in ℰ⁒(𝒳) is regular.

  • (b)

    Every regular component π’ž contains at most one module of ℰ⁒(𝒳).

  • (c)

    Every indecomposable module Nβˆˆβ„°β’(𝒳) is quasi-simple in mod⁑𝒦r.

  • (d)

    ℰ⁒(𝒳) is a wild subcategory of mod⁑𝒦r.

Proof.

(a) and (b) are proven in [15, Lemma 1.1, Proposition 1.4] for any wild hereditary algebra, and (c) follows by [15, Proposition 1.4] and the fact that every regular brick in mod⁑𝒦r is quasi-simple [13, Proposition 9.2]. Let Mβˆˆπ’³. Then we have t:-dimk⁑Ext⁑(M,M)β‰₯2 by Corollary 1.2. Due to [11, Section 7] and [15, Remark 1.4], the category ℰ⁒({M})βŠ†β„°β’(𝒳) is equivalent to the category of finite-dimensional modules over the power-series ring kβ’γ€ˆγ€ˆX1,…,Xt〉〉 in non-commuting variables X1,…,Xt. Since tβ‰₯2, the category ℰ⁒({M})βŠ†β„°β’(𝒳) is wild, and also ℰ⁒(𝒳). ∎

We will use the above result to prove the existence of numerous components such that all of its vertices correspond to modules of constant d-socle rank. By duality, all results also follow for constant radical rank. As a by-product, we verify the wildness of EKP=ESP1 and EIP=ERP1. Using the functor 𝔉:mod⁑𝒦rβ†’mod⁑k⁒Er, we show the wildness of the corresponding full subcategories in mod2⁑k⁒Er of Er-modules of Loewy length ≀2.

3.2 Passage between 𝒦r and 𝒦s

Let 2≀r<sβˆˆβ„•. Denote by infrs:mod⁑𝒦rβ†’mod⁑𝒦s the functor that assigns to a 𝒦r-module M the module infrs⁑(M) with the same underlying vector space so that the action of e1,e2,Ξ³1,…,Ξ³r on infrs⁑(M) stays unchanged and all other arrows act trivially on infrs⁑(M). Moreover, let ΞΉ:𝒦r→𝒦s be the natural k-algebra monomorphism given by ι⁒(ei)=ei for i∈{1,2} and ι⁒(Ξ³j)=Ξ³j. Then each 𝒦s-module N becomes a 𝒦r-module Nβˆ— via pullback along ΞΉ. Denote the corresponding functor by resrs:mod⁑𝒦sβ†’mod⁑𝒦r. In the following, r,s will be fixed, so we suppress the index and write just inf and res.

Lemma 3.2.1.

Let 2≀r<s∈N. The functor inf:mod⁑Krβ†’mod⁑Ks is fully faithful and exact. The essential image of inf is a subcategory of mod⁑Ks closed under factors and submodules. Moreover, inf⁑(M) is indecomposable if and only if M is indecomposable in mod⁑Kr.

Proof.

Clearly, inf is fully faithful and exact. Now let M∈mod⁑𝒦r, and let UβŠ†inf⁑(M) be a submodule. Then Ξ³i (i>r) acts trivially on U, and hence the pullback res⁑(U)-:Uβˆ— is a 𝒦r-module with inf⁑(Uβˆ—)=inf∘res⁑(U)=U. Now let V∈mod⁑𝒦s, and let f∈Hom𝒦s⁑(inf⁑(M),V) be an epimorphism. Let v∈V and m∈M such that f⁒(m)=v. It follows Ξ³i⁒v=Ξ³i⁒f⁒(m)=f⁒(Ξ³i⁒m)=0 for all i>r. This shows that inf⁑(Vβˆ—)=inf∘res⁑(V)=V.

Since inf is fully faithful, we have End𝒦s⁑(inf⁑(M))β‰…End𝒦r⁑(M). Hence End𝒦s⁑(inf⁑(M)) is local if and only if End𝒦r⁑(M) is local. ∎

Statement (a) of the following lemma is stated in [7, Proposition 3.1] without proof.

Lemma 3.2.2.

Let 2≀r<s, and let M be an indecomposable Kr-module that is not simple. The following statements hold.

  • (a)

    inf⁑(M) is regular and quasi-simple.

  • (b)

    inf⁑(M)βˆ‰CSRm for all m∈{1,…,s-r}.

Proof.

(a) Write dim¯⁑M=(a,b)βˆˆβ„•0Γ—β„•0. Since M is not simple, a⁒bβ‰ 0 and q⁒(dim¯⁑M)=a2+b2-r⁒a⁒b≀1. It follows

q⁒(dim¯⁒inf⁑(M))=a2+b2-s⁒a⁒b=a2+b2-r⁒a⁒b-(s-r)⁒a⁒b≀1-(s-r)⁒a⁒b<1.

Hence q⁒(dim¯⁒inf⁑(M))≀0, and inf⁑(M) is regular.

Assume that inf⁑(M) is not quasi-simple; then inf⁑(M)=U⁒[i] for U quasi-simple with iβ‰₯2. By Lemma 3.2.1, we have U⁒[i-1]=inf⁑(A) and Ο„-1⁒U⁒[i-1]=inf⁑(B) for some A,B indecomposable in mod⁑𝒦r. Fix an irreducible monomorphism f:inf⁑(A)β†’inf⁑(M). Since inf is full, we find g:Aβ†’M with inf⁑(g)=f. The faithfulness of inf implies that g is an irreducible monomorphism g:Aβ†’M. By the same token, there exists an irreducible epimorphism Mβ†’B. As all irreducible morphisms in 𝒫 are injective and all irreducible morphisms in ℐ are surjective, M is located in a ℀⁒A∞ component. It follows that τ⁒B=A in mod⁑𝒦r. Let dim¯⁑B=(c,d); then the Coxeter matrices for 𝒦r and 𝒦s yield

((r2-1)⁒c-r⁒d,r⁒c-d)=dim¯⁑τ⁒B=dim¯⁑A=dim¯⁒inf⁑(A)=dim¯⁑τ⁒inf⁑(B)=((s2-1)⁒c-s⁒d,s⁒c-d).

This is a contradiction since s≠r.

(b) Denote by {e1,…,es} the canonical basis of ks. Let 1≀m≀s-r, and set U:-γ€ˆer+1,…,er+m〉k. Then SocU⁑(M)=β‹‚i=1mker⁑(xer+iM)=M. Let j∈{1,…,r} such that Ξ³j acts non-trivially on M. Let V∈Grm,s such that ej∈V. Then SocV⁑(inf⁑(M))β‰ M, and M does not have constant m-socle rank. ∎

Proposition 3.2.3.

Let 2≀r<s∈N and 1≀d<r, and let M be an indecomposable and non-simple Kr-module. Then the following statements hold.

  • (a)

    If Mβˆˆπ”›d,rβŠ₯ , then inf⁑(M)βˆˆπ”›d+s-r,sβŠ₯.

  • (b)

    If Mβˆˆπ”›d,rβŠ₯ , then inf⁑(M)βˆˆπ”›d+s-r,sβŠ₯.

  • (c)

    If Mβˆˆπ”›Β―d,r , then inf⁑(M) is contained in a regular component π’ž with π’žβŠ†CSRd+s-r.

Proof.

By definition, it is 1≀d+s-r<s. Now fix V∈Grd+s-r,s, and note that

dim¯⁑XV=(1,s-(d+s-r))=(1,r-d),

which is the dimension vector of every 𝒦r-module XU for U∈Grd,r.

(a) Assume that Hom⁑(inf⁑(M),XV)β‰ 0, and let 0β‰ f:inf⁑(M)β†’XV. By Lemmata 3.2.1 and 2.1.4, the 𝒦s-module inf⁑(M) is regular and every proper submodule of XV is preprojective. Hence f is surjective onto XV. Again, Lemma 3.2.1 yields Z∈mod⁑𝒦r indecomposable with dim¯⁑Z=(1,r-d)=dim¯⁑XV such that XV=inf⁑(Z). By Proposition 2.2.2, there exists U∈Grd,r with Z=XU. Since inf is fully faithful, it follows 0=Hom⁑(M,XU)β‰…Hom⁑(inf⁑(M),inf⁑(XU))=Hom⁑(inf⁑(M),XV)β‰ 0, a contradiction.

(b) Assume that Hom⁑(XV,inf⁑(M))β‰ 0, and let f:XVβ†’inf⁑(M) be non-zero. Since inf⁑(M) is regular indecomposable, the module im⁑fβŠ†inf⁑(M) is not injective, and Lemma 2.2.3 yields that im⁑f is indecomposable and regular. As im⁑f is a submodule of inf⁑(M), there exists an indecomposable module Z∈mod⁑𝒦r with inf⁑(Z)=im⁑f. Since im⁑f is not simple, we have dim¯⁑im⁑f=(1,r-c) for 1≀r-c≀r-d. Hence Z=XU for U∈Grc,r, and by Proposition 2.2.2 (d), there exists W∈Grd,r and an epimorphism Ο€:XWβ†’XU. We conclude with 0β‰ Hom⁑(im⁑f,inf⁑(M))=Hom⁑(inf⁑(XU),inf⁑(M))β‰…Hom⁑(XU,M) and the surjectivity of Ο€:XWβ†’XU that Hom⁑(XW,M)β‰ 0, a contradiction to the assumption.

(c) By Lemma 3.2.2, the module inf⁑(M) is quasi-simple in a regular component and satisfies the conditions of Lemma 2.3.8 for q:-d+s-r by (a) and (b). ∎

Examples.

The following two examples will be helpful later on.

  • (1)

    Let r=3. Ringel has shown that the representation F=(k2,k2,F⁒(Ξ³1),F⁒(Ξ³2),F⁒(Ξ³3)) with the linear maps F⁒(Ξ³1)=i⁒dk2, F⁒(Ξ³2)⁒(a,b)=(b,0) and F⁒(Ξ³3)⁒(a,b)=(0,a) is elementary. Let E be the corresponding 𝒦3-module. Then dim¯⁑E=(2,2), and it is easy to see that every indecomposable submodule of E has dimension vector (0,1) or (1,2). In particular, Hom⁑(W,E)=0 for each indecomposable module with dimension vector dim¯⁑W=(1,1). Assume now that f:Eβ†’W is non-zero; then f is surjective since every proper submodule of W is projective. Since E is elementary, ker⁑f is a preprojective module with dimension vector (1,1), a contradiction. Hence Eβˆˆπ”›Β―2,3.

  • (2)

    Recall that ESP1=EKP and ERP1=EIP. Given a regular component π’ž, there are unique quasi-simple modules Mπ’ž and Wπ’ž in π’ž such that EIPβˆ©π’ž=(β†’Wπ’ž) and EKPβˆ©π’ž=(Mπ’žβ†’). The width 𝒲⁒(π’ž)βˆˆβ„€ is defined [21, Theorem 3.3] as the unique integer satisfying τ𝒲⁒(π’ž)+1⁒Mπ’ž=Wπ’ž. In fact, it is shown that 𝒲⁒(π’ž)βˆˆβ„•0, and an example of a regular component π’ž with 𝒲⁒(π’ž)=0 and End⁑(Mπ’ž)=k is given. Since XU≅δ⁒τ⁒XU for U∈Gr1,r (see [21, Theorem 3.1]), we conclude for an arbitrary regular component π’ž that

    𝒲⁒(π’ž)=0⇔τ⁒Mπ’ž=Wπ’žβ‡”Mπ’žβˆˆEKP⁒and⁒τ⁒Mπ’žβˆˆEIP⇔Hom(XU,Mπ’ž)=0=Ext(XU,Ο„Mπ’ž)for allU∈Gr1,r by Propositions 2.3.3 and 2.3.4⇔Hom⁑(XU,Mπ’ž)=0=Hom⁑(Mπ’ž,XU)⁒for all⁒U∈Gr1,r⇔Mπ’žβˆˆπ”›Β―1,r.

Lemma 3.2.4.

Let sβ‰₯3 and 2≀d<s. Then there exists a regular module Ed with the following properties.

  • (a)

    Ed is a (quasi-simple) brick in mod⁑𝒦s.

  • (b)

    Edβˆˆπ”›Β―d,s.

  • (c)

    There exist V,W∈Gr1,s with Hom⁑(XV,Ed)=0β‰ Hom⁑(XW,Ed).

Proof.

We start by considering s=3 and d=2. Pick the elementary module Ed:-E from the preceding example. E is a brick, and Eβˆˆπ”›Β―d,s. Set Ξ±:-(1,0,0), Ξ²:-(0,1,0)∈k3 and V:-γ€ˆΞ±γ€‰k, W:-γ€ˆΞ²γ€‰k. By the definition of E, we have

dimk⁑ker⁑xΞ±E=2β‰ 3=dimk⁑ker⁑xΞ²E,

and therefore

dimk⁑Hom⁑(XV,Ed)=0β‰ 1=dimk⁑Hom⁑(XW,Ed).

Now let s>3. If d=s-1, consider Ed:-inf3s⁑(E). In view of Proposition 3.2.3, we have Edβˆˆπ”›Β―2+s-3,s=𝔛¯d,s Moreover, inf⁑(E) is a brick in mod⁑𝒦s and for the canonical basis vectors e1,e2∈ks and V=γ€ˆe1〉k, W:-γ€ˆe2〉k we get as before

dimk⁑Hom⁑(XV,inf⁑(E))=0β‰ 1=Hom⁑(XW,inf⁑(E)).

Now let 1<d<s-1. Set r:-1+s-dβ‰₯3, consider a regular component for 𝒦r with 𝒲⁒(π’ž)=0 such that Mπ’ž is a brick and set M:-Mπ’ž. Then Mβˆˆπ”›Β―1,r, and Proposition 3.2.3 yields Ed:-inf⁑(M)βˆˆπ”›Β―1+s-(1+s-d),s=𝔛¯d,s. Since M is a brick, inf⁑(M) is a brick in mod⁑𝒦s. Recall that Hom⁑(XU,M)=0 for all U∈Grd,r implies that, viewing M as a representation, the linear map M⁒(Ξ³1):M1β†’M2 corresponding to Ξ³1 is injective. Since the map is not affected by inf, inf⁑(M)⁒(Ξ³1):M1β†’M2 is also injective. Therefore, we conclude for the first basis vector e1∈ks and V:-γ€ˆe1〉k that 0=Hom⁑(XV,inf⁑(M)). By Lemma 3.2.2, we find W∈Gr1,s with 0β‰ Hom⁑(XW,inf⁑(M)). ∎

3.3 Numerous components lying in CSRd

In this section, we use the simplification method to construct a family of regular components such that every vertex in such a regular component corresponds to a module in CSRd. By the next result, it follows that π’³βŠ†π”›Β―d,r implies ℰ⁒(𝒳)βŠ†π”›Β―d,r.

Lemma 3.3.1 ([13, Lemma 1.9]).

Let X,Y be modules with Hom⁑(X,Y) non-zero. If X and Y have filtrations

X=X0βŠƒX1βŠƒβ‹―βŠƒXrβŠƒXr+1=0,Y=Y0βŠƒY1βŠƒβ‹―βŠƒYsβŠƒYs+1=0,

then there are i,j with Hom⁑(Xi/Xi+1,Yj/Yj+1)β‰ 0.

For a regular module Mβˆˆπ’¦r, denote by π’žM the regular component that contains M.

Proposition 3.3.2.

Let 1≀d<r, and let X be a family of pairwise orthogonal bricks in XΒ―d,r. Then

Ο†:ind⁑ℰ⁒(𝒳)β†’β„›,Mβ†¦π’žM

is an injective map such that, for each component C in im⁑φ, we have CβŠ†CSRd. Here ind⁑E⁒(X) denotes the category of a chosen set of representatives of non-isomorphic indecomposable objects of mod⁑Kr in E.

Proof.

Since each module in 𝔛¯d,r is regular, Proposition 3.1.1 implies that every module N∈ind⁑ℰ⁒(𝒳) is contained in a regular component π’žM and is quasi-simple. By Lemma 3.3.1, the module N satisfies Hom⁑(XU,N)=0=Hom⁑(N,XU) for all U∈Grd,r. But now Lemma 2.3.8 implies that every module in π’žM has constant d-socle rank. The injectivity of Ο† follows immediately from Proposition 3.1.1. ∎

Corollary 3.3.3.

There exists an infinite set Ω of regular components such that, for all C∈Ω,

  • (a)

    𝒲⁒(π’ž)=0 , in particular, every module in π’ž has constant rank,

  • (b)

    π’ž does not contain any bricks.

Proof.

Let π’ž be a regular component that contains a brick and 𝒲⁒(π’ž)=0 (such a component exists by the example above). Let M:-Mπ’ž; then Mβˆˆπ”›Β―1,r. Apply Proposition 3.3.2 with 𝒳={M}, and set Ξ©:-imβ‘Ο†βˆ–{π’žM}. Let Nβˆˆβ„°β’(𝒳)βˆ–{M} be indecomposable. N is quasi-simple in π’žN and has a {M}-filtration 0=N0βŠ‚β‹―βŠ‚Nl=N with lβ‰₯2 and N1=M=Nl/Nl-1. Hence Nβ†’Nl/Nl-1β†’N1β†’N is a non-zero homomorphism that is not injective. Therefore, N is not a brick. This finishes the proof since every regular brick in mod⁑𝒦r is quasi-simple [13, Proposition 9.2] and End⁑(Ο„l⁒N)β‰…End⁑(N)β‰ k for all lβˆˆβ„€. ∎

Now we apply our results on the simplification method to modules Ed constructed in Lemma 3.2.4.

Definition ([6, Proposition 3.6]).

Denote with GLr the group of invertible rΓ—r-matrices which acts on βŠ•i=1rk⁒γi via g.Ξ³j=βˆ‘i=1rgi⁒j⁒γi for 1≀j≀r, g∈GLr. For g∈GLr, let Ο†g:𝒦r→𝒦r be the algebra homomorphism with Ο†g⁒(e1)=e1, Ο†g⁒(e2)=e2 and Ο†g⁒(Ξ³i)=g.Ξ³i, 1≀i≀r. For a 𝒦r-module M, denote the pullback of M along Ο†g by M(g). The module M is called GLr-stable if M(g)β‰…M for all g∈GLr.

Theorem 3.3.4.

Let 2≀d<r; then there exists a wild full subcategory EβŠ†mod⁑Kr and an injection

Ο†d:ind⁑ℰ→ℛ,Mβ†¦π’žM,

such that, for each component C in im⁑φd, we have CβŠ†CSRd and no module in C is GLr-stable.

Proof.

Fix 2≀d<r, and let Ed be as in Lemma 3.2.4 with V,W∈Gr1,r and Hom⁑(XV,Ed)=0β‰ Hom⁑(XW,Ed). Set 𝒳:-{Ed}, and let Mβˆˆβ„°β’(𝒳). By Proposition 3.3.2, we get an injective map

Ο†d:ind⁑ℰ⁒(𝒳)β†’β„›,Mβ†’π’žM

such that each component π’ž in im⁑φd satisfies π’žβŠ†CSRd.

Moreover, ℰ⁒(𝒳) is a wild full subcategory of mod⁑𝒦r by Proposition 3.1.1. Let Mβˆˆβ„°β’(𝒳) be indecomposable. Then M has a filtration 0=Y0βŠ‚Y1βŠ‚β‹―βŠ‚Ym with Yl/Yl-1=Ed for all 1≀l≀m. By Lemma 3.3.1, we have 0=Hom⁑(XU,M), and since Ed=Y1βŠ†M, we conclude 0β‰ Hom⁑(XW,M). This proves that M does not have constant 1-socle rank. Therefore, π’žM contains a module that is not of constant 1-socle rank. By [6, Proposition 3.6], the module M is not GLr-stable. Assume that π’žM contains an GLr-stable module N. Since g∈G acts as an auto-equivalence on mod⁑𝒦r (see also [9, Lemma 2.2]), we conclude that g sends the Auslander–Reiten sequence 0β†’Xβ†’Eβ†’Nβ†’0 to the Auslander–Reiten sequence 0β†’Xgβ†’Egβ†’Nβ†’0. Hence Xgβ‰…X and Egβ‰…E for all g∈GLr, and therefore X and E are GLr-stable. If E is not indecomposable, we write E=E1βŠ•E2 with E1,E2 indecomposable such that the quasi-lengths ql⁑(E1),ql⁑(E2) satisfy ql⁑(E1)=ql⁑(E2)-2. We get dimk⁑E2>dimk⁑E1 and therefore (E2)gβ‰…E2 and (E1)gβ‰…E1. Hence every direct summand in the Auslander–Reiten sequence is GLr-stable. Now one can easily conclude that every module in π’žM is GLr-stable, a contradiction since M is not GLr-stable. ∎

3.4 Components lying almost completely in CSRd

The following definition and two lemmata are a generalization of [22, Definition 4.7, Proposition 4.13] and [21, Proposition 3.7]. We sketch the proof of Lemma 3.4.1.

Definition.

Let M be an indecomposable 𝒦r-module, 1≀d<r and U∈Grd,r. M is called U-trivial if

dimk⁑Hom⁑(XU,M)=dimk⁑M1.

Note that the sequence 0β†’P1r-dβ†’P2β†’XUβ†’0 and left-exactness of Hom⁑(-,M) imply that

dimk⁑Hom⁑(XU,M)≀dimk⁑M1.

Lemma 3.4.1.

Let M be a regular U-trivial module. If M is not elementary, then

Ext⁑(XV,τ⁒M)=0=Hom⁑(XV,Ο„-1⁒M) for all⁒V∈Grd,r.

Proof.

Assume that Ext⁑(XV,τ⁒M)β‰ 0; then we find an epimorphism f:Mβ†’XV and an exact sequence 0β†’ker⁑fβ†’Mβ†’XVβ†’0. Note that dimk⁑Hom⁑(XU,ker⁑f)≀dimk⁑(ker⁑f)1<dimk⁑M1=Hom⁑(XU,M). We apply Hom⁑(XU,-) and conclude that fβˆ—:Hom⁑(XU,M)β†’Hom⁑(XU,XV),g↦f∘g is non-zero. In particular, we have 0β‰ Hom⁑(XU,XV) and therefore U=V. Let h∈Hom⁑(XU,M) such that f∘hβ‰ 0. Since XU is a brick, we conclude that f is an isomorphism and Mβ‰…XU is elementary.

Assume that Ο„-1⁒Mβˆ‰ESPdβŠ‡ESP1. Consider γ€ˆΞ±γ€‰k=W∈Gr1,r together with an epimorphism p:XWβ†’XU (see Proposition 2.2.2). We conclude with dimk⁑M1β‰₯dimk⁑Hom⁑(XW,M)β‰₯dimk⁑Hom⁑(XU,M)=dimk⁑M1 that M is W-trivial. Now the equation dimk⁑Hom⁑(XW,M)+dimk⁑M2=dimk⁑ker⁑(xΞ±M) (which follows from the proof of Proposition 2.3.3) shows that xΞ± acts as zero on M. Hence xΞ± acts as zero on δ⁒M, and is W-trivial with δ⁒τ-1⁒Mβˆ‰Ξ΄β’(ESP1)=ERP1. Hence we find Z∈Gr1,r such that 0β‰ Ext⁑(δ⁒τ⁒XZ,δ⁒τ-1⁒M). Since δ⁒τ⁒XZβ‰…XZ (see [21, Proposition 3.1]) and Ο„βˆ˜Ξ΄=Ξ΄βˆ˜Ο„-1, we conclude that 0β‰ Ext⁑(δ⁒τ⁒XZ,δ⁒τ-1⁒M)β‰…Ext⁑(XZ,τ⁒δ⁒M). Now the above arguments show that δ⁒Mβ‰…XW and therefore M≅δ⁒XW≅τ⁒XW. ∎

Lemma 3.4.2.

Let M be regular quasi-simple in a regular component C such that

Ext⁑(XU,τ⁒M)=0=Hom⁑(XU,Ο„-1⁒M) for all⁒U∈Grd,r.

If M does not have constant d-socle rank, then a module X in C has constant d-socle rank if and only if X is in (β†’Ο„M)βˆͺ(Ο„-1Mβ†’).

Corollary 3.4.3.

Let 3≀r<s and 1≀d<r, and let b:-d+s-r and 1≀l≀s-r. Let M be an indecomposable Kr-module in XΒ―d,r that is not elementary. Denote by C the regular component that contains inf⁑(M).

  • (a)

    Every module in π’ž has constant b -socle rank.

  • (b)

    Nβˆˆπ’ž has constant l -socle rank if and only if N∈(β†’Ο„inf(M))βˆͺ(Ο„-1inf(M)β†’).

Proof.

(a) is an immediate consequence of Proposition 3.2.3.

(b) Consider the indecomposable projective module P2=𝒦r⁒e1 in mod⁑𝒦r. We get

Hom⁑(inf⁑(P2),inf⁑(M))β‰…Hom⁑(P2,M)=M1=inf⁑(M)1.

Since

dim¯⁒inf⁑(P2)=(1,r)=(1,s-(s-r)),

we find W∈Grs-r,s with inf⁑(P2)=XW. Now let 1≀l≀s-r. By Proposition 2.2.2, there is U∈Grl,s and an epimorphism Ο€:XUβ†’XW. Let {f1,…,fq} be a basis of Hom⁑(inf⁑(P2),inf⁑(M)). Since Ο€ is surjective, the set {f1⁒π,…,fq⁒π}βŠ†Hom⁑(XU,inf⁑(M)) is linearly independent. Hence

q≀dimk⁑Hom⁑(XU,inf⁑(M))≀dimk⁑inf⁑(M)1=q

holds, and inf⁑(M) is U-trivial.

Since M is not elementary, inf⁑(M) is not elementary, and therefore Lemma 3.4.1 yields that

Ext⁑(XW,τ⁒inf⁑(M))=0=Hom⁑(XW,Ο„-1⁒inf⁑(M)) for all⁒W∈Grl,s.

By Lemma 3.2.2, the module inf⁑(M) does not have the constant l-socle rank for 1≀l≀s-r. Note that M is regular, and therefore inf⁑(M) is a quasi-simple module. Now apply Lemma 3.4.2. ∎

Figure 4

Regular component containing inf⁑(Mπ’ž).

Example.

Let rβ‰₯3, and let π’ž be a regular component with 𝒲⁒(π’ž)=0 such that Mπ’ž is not a brick (see Corollary 3.3.3) and in particular not elementary. Then Mπ’žβˆˆπ”›Β―1,r, and we can apply Corollary 3.4.3. Figure 4 shows the regular component π’Ÿ of 𝒦s containing inf⁑(Mπ’ž). Every module in π’Ÿ has constant b:-1+s-r socle rank. But for 1≀q≀s-r, a module in this component has constant q-socle rank if and only if it lies in the shaded region.

4 Wild representation type

4.1 Wildness of strata

As another application of the simplification method and the inflation functor infrs:mod⁑𝒦rβ†’mod⁑𝒦s, we get the following result.

Theorem 4.1.1.

Let sβ‰₯3 and 1≀d≀s-1. Then Ξ”d=ESPdβˆ–ESPd-1βŠ†mod⁑Ks is a wild subcategory, where ESP0:-βˆ….

Proof.

For d=1, consider a regular component π’ž for 𝒦s that contains a brick F. By Proposition 2.3.5, we find a module E in the Ο„-orbit of F that is in ESP1 and set 𝒳:-{E}. Then E is brick since Hom⁑(E,E)β‰…Hom⁑(F,F)=k and dimk⁑Ext⁑(E,E)β‰₯2 by Corollary 1.2. Therefore, ℰ⁒(𝒳) is wild category (see Proposition 3.1.1). As ESP1 is closed under extensions, it follows ℰ⁒(𝒳)βŠ†ESP1. Note that this case does only require the application of Proposition 3.1.1.

Now let d>1 and r:-s-d+1β‰₯2. Consider the projective indecomposable 𝒦r-module P:-P2 with dim¯⁑P=(1,r). By Lemma 3.2.2, inf⁑(P) is a regular quasi-simple module in mod⁑𝒦s with

dimk⁑Ext⁑(inf⁑(P),inf⁑(P))β‰₯2.

Since P is in ESP1, we have 0=Hom⁑(XU,P) for all U∈Gr1,r. Hence Proposition 3.2.3 implies

0=Hom⁑(XU,inf⁑(P)) for all⁒U∈Gr1+s-r=Grd,s

so that inf⁑(P) is in ESPd.

Let 𝒳:-{inf⁑(P)}; then ℰ⁒(𝒳) is a wild category, and since ESPd is extension closed, it follows ℰ⁒(𝒳)βŠ†ESPd. Since dim¯⁒inf⁑(P)=(1,s-d+1), we find V∈Grd-1,s (see Proposition 2.2.2) with

inf⁑(P)=XV and 0β‰ End⁑(inf⁑(P))=Hom⁑(XV,inf⁑(P)).

That means inf⁑(P)βˆ‰ESPd-1. Since ESPd-1 is closed under submodules, we have ℰ⁒(𝒳)∩ESPd-1=βˆ…. Hence ℰ⁒(𝒳)βŠ†ESPdβˆ–ESPd-1. ∎

Remarks.

Let us collect the following observations.

  • (i)

    Note that all indecomposable modules in the wild category ℰ⁒(𝒳) are quasi-simple in mod⁑𝒦s and ℰ⁒(𝒳)βŠ†ESPdβˆ–ESPd-1.

  • (ii)

    For 1≀d<r, we define EKPd:-{M∈mod⁑𝒦r∣Hom⁑(δ⁒τ⁒XU,M)=0⁒for all⁒U∈Grd,r}. One can show that M∈EKPd if and only if yTM|M1d:M1dβ†’M is injective for all linearly independent tuples T in (kr)d. From the definitions, we get a chain of proper inclusions ESPr-1βŠƒESPr-2βŠƒβ‹―βŠƒESP1=EKP1βŠƒEKP2βŠƒβ‹―βŠƒEKPr-1. By adapting the preceding proof, it follows that EKPr-1 is wild. Moreover, it can be shown that, for each regular component π’ž, the set (EKP1βˆ–EKPr-1)βˆ©π’ž is empty or forms a ray.

We will use the following result later on to prove the wildness of the subcategory in mod⁑k⁒E2 consisting of modules of Loewy length 3 and the equal kernels property. We denote by B⁒(3,2) the Beilinson algebra with 3 vertices and 2 arrows.

Proposition 4.1.2.

Let EKP⁒(3,2) be the full subcategory of modules in mod⁑B⁒(3,2) with the equal kernels property (see [21, Definition 2.1, Theorem 2.5]). The category EKP⁒(3,2) is of wild representation type.

Proof.

Consider the path algebra A of the extended Kronecker quiver Q=1β†’2⇉3. Since the underlying graph of Q is not a Dynkin or Euclidean diagram, the algebra A is of wild representation type by [8]. It is known that there exists a preprojective tilting module T in mod⁑A with End⁑(T)β‰…B⁒(3,2); see for example [20] or [23, Section 4]. We sketch the construction. The start of the preprojective component of A is illustrated in Figure 5, and the direct summands of T are marked with a dot.

Figure 5

Start of the preprojective component of A.

One can check that T=P⁒(1)βŠ•Ο„-2⁒P⁒(1)βŠ•Ο„-2⁒P⁒(3) is a tilting module. Since preprojective components are standard [19, Proposition 2.4.11], one can show that End⁑(T) is given by the quiver in Figure 6, bound by the relation Ξ±2⁒α1+Ξ²2⁒β1. Moreover, it follows from the description as a quiver with relations that End⁑(T)β‰…B⁒(3,2).

Figure 6

Ordinary quiver of End⁑(T).

Since A is hereditary, it follows that the algebra B⁒(3,2) is a concealed algebra [1, Definition 4.6]. By [2, Theorem XVIII.5.1], the functor Hom⁑(T,-):mod⁑Aβ†’mod⁑B⁒(3,2) induces an equivalence G between the regular categories add⁑ℛ⁒(A) and add⁑ℛ⁒(B⁒(3,2)), and we have an isomorphism between the two Grothendieck groups f:K0⁒(A)β†’K0⁒(B⁒(3,2)) with dim¯⁑G⁒(M)=dim¯⁑Hom⁑(T,M)=f⁒(dim¯⁑M) for all M∈mod⁑A. Now we make use of a homological characterization of the class EKP⁒(3,2) given in [21, Theorem 2.5]: for each α∈k2βˆ–{0}, there exist certain indecomposable B⁒(3,2)-modules XΞ±0,1, XΞ±1,1 such that

EKP⁒(3,2)={M∈mod⁑B⁒(3,2)∣Hom⁑(XΞ±0,1βŠ•XΞ±1,1,M)=0⁒for all⁒α∈k2βˆ–{0}}.

The modules XΞ±0,1, XΞ±1,1 arise as cokernels of embeddings similar to the embeddings studied in Section 2.2. We do not need the exact definition of XΞ±1. Let us show that each XΞ±1 is regular. Clearly, 0β‰ HomB⁒(3,2)⁑(XΞ±1,Z) for Z∈{XΞ±0,1,XΞ±1,1}, so Z is not in EKP⁒(3,2). Moreover, the equality (see [22, Proposition 3.14])

Ο„B⁒(3,2)⁒XΞ±1,1β‰…D⁒XΞ±3-1-1-1,1=D⁒XΞ±0,1

holds (D denotes a certain duality on mod⁑B⁒(3,2)). Since XΞ±0,1 is not in EKP⁒(3,2), we conclude that Ο„B⁒(3,2)⁒XΞ±1,1 is not in EIP⁒(3,2). Since EIP⁒(3,2) is closed under Ο„B⁒(3,2), we conclude that XΞ±1,1 is not in EIP⁒(3,2). The assumption XΞ±0,1∈EIP⁒(3,2) yields that Ο„B⁒(3,2)⁒XΞ±1,1 is in EKP⁒(3,2). Since EKP⁒(3,2) is closed under Ο„B⁒(3,2)-1, we get that XΞ±1,1 is in EKP⁒(3,2), a contradiction. Therefore, XΞ±0,1,XΞ±1,1 are not in EIP⁒(3,2)βˆͺEKP⁒(3,2) and by [21, Corollary 2.7] regular, so XΞ±1 is a regular module as well.

Moreover, dim¯⁑XΞ±1 is independent of Ξ±. Hence we find for each β∈krβˆ–{0} a regular indecomposable module UΞ² in mod⁑A with G⁒(UΞ²)=XΞ²1 and dim¯⁑UΞ²=dim¯⁑UΞ± for all α∈krβˆ–{0}. Now let M be in mod⁑A a regular brick with dimk⁑Ext⁑(M,M)β‰₯2 (see [15, Proposition 5.1]). By the dual version of [13, Lemma 4.6], we find lβˆˆβ„•0 with Hom⁑(UΞ±,Ο„-l⁒M)=0 for all α∈krβˆ–{0}. Set N:-Ο„-l⁒M and 𝒳:-{N}. N is a regular brick with dimk⁑Ext⁑(N,N)=dimk⁑Ext⁑(M,M)β‰₯2, and therefore ℰ⁒(𝒳) is a wild category in add⁑ℛ⁒(A) (see [15, Proposition 1.4]). By Lemma 3.3.1, we have Hom⁑(UΞ±,L)=0 for all L in ℰ⁒(𝒳) and all α∈krβˆ–{0}. Hence 0=Hom⁑(UΞ±,L)=Hom⁑(G⁒(UΞ±),G⁒(L))=Hom⁑(XΞ±1,L) for all α∈krβˆ–{0}. This shows that the essential image of ℰ⁒(𝒳) under G is a wild subcategory contained in EKP⁒(3,2). ∎

4.2 The module category of Er

Throughout this section, we assume that char⁑(k)=p>0 and rβ‰₯2. Moreover, let Er be a p-elementary abelian group of rank r with generating set {g1,…,gr}. For xi:-gi-1, we get an isomorphism

k⁒Erβ‰…k⁒[X1,…,Xr]/(X1p,…,Xrp)

of k-algebras by sending Xi to xi for all i. We recall the definition of the functor 𝔉:mod⁑𝒦rβ†’mod⁑k⁒Er introduced in [21]. Given a module M, 𝔉⁑(M) is by definition the vector space M and

xi.m:-Ξ³iβ‹…m=Ξ³iβ‹…m1+Ξ³iβ‹…m2=Ξ³iβ‹…m1,

where mi=eiβ‹…m. Moreover, 𝔉 is the identity map on morphisms, that is, 𝔉⁑(f):𝔉⁑(M)→𝔉⁑(N), 𝔉⁑(f)⁒(m)=f⁒(m) for all f:Mβ†’N.

Definition ([6, Definition 2.1]).

Let 𝕍:-γ€ˆx1,…,xr〉kβŠ†rad⁑(k⁒Er). For U in Grd,𝕍 with basis u1,…,ud and a k⁒Er-module M, we set

RadU⁑(M):-βˆ‘u∈Uuβ‹…M=βˆ‘i=1duiβ‹…M,SocU⁑(M):-{m∈M∣uβ‹…m=0⁒for all⁒u∈U}=β‹‚i=1d{m∈M∣uiβ‹…m=0}.

Definition ([6, Definition 3.1]).

Let M∈mod⁑k⁒Er and 1≀d<r.

  • (a)

    M has constant d-Rad rank ( d-Soc rank, respectively) if the dimension of RadU⁑(M) (SocU⁑(M), respectively) is independent of the choice of U∈Grd,𝕍.

  • (b)

    M has the equal d-Rad property ( d-Soc property, respectively) if RadU⁑(M) (SocU⁑(M), respectively) is independent of the choice of U∈Grd,𝕍.

Proposition 4.2.1.

Let M be a non-simple indecomposable Kr-module, and let 1≀d<r.

  • (a)

    M is in CSRd if and only if 𝔉⁑(M) has constant d - Soc rank.

  • (b)

    M is in ESPd if and only if 𝔉⁑(M) has the equal d - Soc property.

Proof.

We fix 𝕍:-γ€ˆx1,…,xr〉kβŠ†rad⁑(k⁒Er). In the following, we denote for u∈rad⁑(k⁒Er) with l⁒(u):Mβ†’M the induced linear map on M. Let U∈Grd,𝕍 with basis {u1,…,ud}, write uj=βˆ‘i=1rΞ±ji⁒xi for all 1≀j≀d, and set Ξ±j=(Ξ±j1,…,Ξ±jr). Then T:-(Ξ±1,…,Ξ±d) is linearly independent, and

ker⁑(l⁒(uj))=ker⁑(βˆ‘i=1rΞ±ji⁒l⁒(xi))=ker⁑(βˆ‘i=1rΞ±ji⁒γi)=ker⁑(xΞ±jM).

It follows

SocU⁑(𝔉⁑(M))=β‹‚i=1dker⁑(l⁒(ui))=β‹‚i=1dker⁑(xΞ±iM)=Socγ€ˆT〉⁑(M).

Hence M∈CSRd implies that 𝔉⁑(M) has constant d-Soc rank.

Now assume that T=(Ξ±1,…,Ξ±d) is linearly independent, and set uj:-βˆ‘i=1rΞ±ji⁒xi. Then

U:-γ€ˆu1,…,udγ€‰βˆˆGrd,𝕍 and Socγ€ˆT〉⁑(T)=SocU⁑(𝔉⁑(M)).

We have shown that M is in CSRd if and only if 𝔉⁑(M) has constant d-Soc rank. The other equivalence follows in the same fashion. ∎

For 1≀d<r, we denote by ESP2,d⁒(Er) the category of modules in mod⁑k⁒Er of Loewy length ≀2 with the equal d-Soc property. As an application of Section 4.1, we get a generalization of [3, Theorem 5.6.12] and [4, Theorem 1].

Corollary 4.2.2.

Let char⁑(k)>0, rβ‰₯3 and 1≀d≀r-1. Then ESP2,d⁒(Er)βˆ–ESP2,d-1⁒(Er) has wild representation type.

Proof.

Let 1≀c<r. By [21, Proposition 2.3] and Proposition 4.2.1, a restriction of 𝔉 to ESPc induces a faithful exact functor

𝔉c:ESPcβ†’mod2⁑k⁒Er

that reflects isomorphisms and with essential image ESP2,c⁒(Er). Let β„°βŠ†ESPdβˆ–ESPd-1 be a wild subcategory. Since 𝔉d-1 and 𝔉d reflect isomorphisms, we have 𝔉⁑(E)∈ESP2,d⁒(Er)βˆ–ESP2,d-1⁒(Er) for all Eβˆˆβ„°. Hence the essential image of β„° under 𝔉 is a wild category. ∎

Corollary 4.2.3.

Assume that char⁑(k)=p>2; then the full subcategory of modules with the equal kernels property in mod⁑k⁒E2 and Loewy length ≀3 is of wild representation type.

Proof.

By [21, Proposition 2.3] (n=3≀p, r=2), the functor 𝔉EKP⁒(3,2):mod⁑B⁒(3,2)β†’mod3⁑k⁒E2 is a representation embedding with essential image in EKP⁒(E2). ∎

Acknowledgements

The results of this article are part of my doctoral thesis, which I have written at the University of Kiel. I would like to thank my advisor Rolf Farnsteiner for fruitful discussions, his continuous support and helpful comments on an earlier version of this paper. I also would like to thank the whole research team for the very pleasant working atmosphere and the encouragement throughout my studies. Furthermore, I thank Otto Kerner for answering my questions on hereditary algebras and giving helpful comments, and Claus Michael Ringel for sharing his insights on elementary modules for the Kronecker algebra. I would like to thank the anonymous referee for the detailed comments.

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About the article


Received: 2018-06-14

Revised: 2019-07-11

Published Online: 2019-09-11

Published in Print: 2020-01-01


Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: DFG priority program SPP 1388

Partly supported by the DFG priority program SPP 1388 β€œDarstellungstheorie”.


Citation Information: Forum Mathematicum, Volume 32, Issue 1, Pages 23–43, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI:Β https://doi.org/10.1515/forum-2018-0143.

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