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Volume 32, Issue 1

# Representations of constant socle rank for the Kronecker algebra

Daniel Bissinger
• Corresponding author
• Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
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Published Online: 2019-09-11 | DOI: 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 ${E}_{r}$ 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 ${\mathcal{𝒦}}_{r}=k{\mathrm{\Gamma }}_{r}$ with $r\ge 2$ arrows and $1\le d\le r-1$. 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 $\mathrm{mod}{\mathcal{𝒦}}_{r}$ are of wild type. Then we use a natural functor $𝔉:\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}k{E}_{r}$ to transfer our results to $\mathrm{mod}k{E}_{r}$.

MSC 2010: 16G20; 16G60; 16G70

## Introduction

Let $r\ge 2$. Let k be an algebraically closed field of characteristic $p>0$, and let ${E}_{r}$ be a p-elementary abelian group of rank r. It is well known that the category of finite-dimensional $k{E}_{r}$-modules $\mathrm{mod}k{E}_{r}$ is of wild type, whenever $p\ge 3$ or $p=2$ and $r>2$. Therefore, subclasses with more restrictive properties have been studied; in [5], the subclass of modules of constant rank $\mathrm{CR}\left({E}_{r}\right)$ and modules with even more restrictive properties, called equal images property and equal kernels property, were introduced. Let ${〈{x}_{1},\mathrm{\dots },{x}_{r}〉}_{k}$ be a complement of ${\mathrm{Rad}}^{2}\left(k{E}_{r}\right)$ in $\mathrm{Rad}\left(k{E}_{r}\right)$, and set ${x}_{\alpha }:-{\sum }_{i=1}^{r}{\alpha }_{i}{x}_{i}$ for $\alpha \in {k}^{r}$. We say that $M\in \mathrm{mod}k{E}_{r}$ has constant Jordan type if the Jordan canonical form of the nilpotent operator ${x}_{\alpha }^{M}:M\to M,m↦{x}_{\alpha }\cdot m$ is independent of $\alpha \in {k}^{r}\setminus \left\{0\right\}$. If the image (kernel) of ${x}_{\alpha }^{M}$ does not depend on α, we say that M has the equal images (kernels) property.

In [21], the author defined analogous categories $\mathrm{CR}$, $\mathrm{EIP}$ and $\mathrm{EKP}$ in the context of the generalized Kronecker algebra ${\mathcal{𝒦}}_{r}$, and in more generality for the generalized Beilinson algebra $B\left(n,r\right)$. Using a natural functor $𝔉:\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}k{E}_{r}$ with nice properties, she gave new insights into the categories of equal images and equal kernels modules for $\mathrm{mod}k{E}_{r}$ of Loewy length $\le 2$. A crucial step is the description of $\mathrm{CR}$, $\mathrm{EIP}$ and $\mathrm{EKP}$ in homological terms, involving a family ${ℙ}^{r-1}$-family of regular “test”-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\le d, we introduce modules of constant d-radical rank ${\mathrm{CRR}}_{d}$ and constant d-socle rank ${\mathrm{CSR}}_{d}$ in $\mathrm{mod}{\mathcal{𝒦}}_{r}$. More restrictive – and also easier to handle – are modules with the equal d-radical property ${\mathrm{ERP}}_{d}$ and the equal d-socle property ${\mathrm{ESP}}_{d}$. For $d=1$, we have ${\mathrm{ESP}}_{1}=\mathrm{EKP}$, ${\mathrm{ERP}}_{1}=\mathrm{EIP}$ and ${\mathrm{CSR}}_{1}=\mathrm{CR}={\mathrm{CRR}}_{1}$. Studying these classes in the hereditary module category $\mathrm{mod}{\mathcal{𝒦}}_{r}$ allows us to use tools not available in $\mathrm{mod}k{E}_{r}$.

As a first step, we establish a homological characterization of ${\mathrm{CSR}}_{d}$, ${\mathrm{CRR}}_{d}$, ${\mathrm{ESP}}_{d}$ and ${\mathrm{ERP}}_{d}$. We denote by ${\mathrm{Gr}}_{d,r}$ the Grassmanian of d-dimensional subspaces of ${k}^{r}$. In generalization of [21], we define a ${\mathrm{Gr}}_{d,r}$-family of “test”-modules ${\left({X}_{U}\right)}_{U\in {\mathrm{Gr}}_{d,r}}$ and show that the modules in this family can be described in purely combinatorial terms by being indecomposable of dimension vector $\left(1,r-d\right)$. This allows us to construct many examples of modules of equal socle rank in $\mathrm{mod}{\mathcal{𝒦}}_{s}$ for $s\ge 3$ by considering pullbacks along natural embeddings ${\mathcal{𝒦}}_{r}\to {\mathcal{𝒦}}_{s}$.

Since ${\mathcal{𝒦}}_{r}$ is a wild algebra for $r>2$ and every regular component in the Auslander–Reiten quiver of ${\mathcal{𝒦}}_{r}$ is of type $ℤ{A}_{\mathrm{\infty }}$, 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 ${M}_{\mathcal{𝒞}}$ and ${W}_{\mathcal{𝒞}}$ in $\mathcal{𝒞}$ such that the cone $\left({M}_{\mathcal{𝒞}}\to \right)\subseteq \mathcal{𝒞}$ consisting of all modules lying on an oriented path starting in ${M}_{\mathcal{𝒞}}$ satisfies $\left({M}_{\mathcal{𝒞}}\to \right)=\mathrm{EKP}\cap \mathcal{𝒞}$, and the cone $\left(\to {W}_{\mathcal{𝒞}}\right)$ consisting of all modules lying on an oriented path ending in ${W}_{\mathcal{𝒞}}$ satisfies $\left(\to {W}_{\mathcal{𝒞}}\right)=\mathrm{EIP}\cap \mathcal{𝒞}$. Using results on elementary modules, we generalize this statement for ${\mathrm{ESP}}_{d}$ and ${\mathrm{ERP}}_{d}$. Our main results may be summarized as follows.

#### Theorem.

Let $r\mathrm{\ge }\mathrm{3}$ and $\mathcal{C}$ be a regular component of the Auslander–Reiten quiver of ${\mathcal{K}}_{r}$.

• (a)

For each $1\le i , the category ${\mathrm{\Delta }}_{i}:-{\mathrm{ESP}}_{i}\setminus {\mathrm{ESP}}_{i-1}$ is wild, where ${\mathrm{ESP}}_{0}:-\mathrm{\varnothing }$.

• (b)

For each $1\le i , there exists a unique quasi-simple module ${M}_{i}$ in $\mathcal{𝒞}$ such that ${\mathrm{ESP}}_{i}\cap \mathcal{𝒞}=\left({M}_{i}\to \right)$.

• (c)

There exists at most one number $1 such that ${\mathrm{\Delta }}_{m\left(\mathcal{𝒞}\right)}\cap \mathcal{𝒞}$ is non-empty. If such a number exists, ${\mathrm{\Delta }}_{m\left(\mathcal{𝒞}\right)}\cap \mathcal{𝒞}$ is the ray starting in ${M}_{m\left(\mathcal{𝒞}\right)}$.

If there is no such number as in (c), we set $m\left(\mathcal{𝒞}\right)=1$. An immediate consequence of (b) and (c) is that, for $1\le i\le j, we have ${M}_{i}={M}_{j}$ or $\tau {M}_{i}={M}_{j}$. Moreover, statement (a) shows the existence of a lot of AR-components such that ${\mathrm{\Delta }}_{i}\cap \mathcal{𝒞}$ is a ray, and for every such component, we have $m\left(\mathcal{𝒞}\right)=i$. With the dual result for $\mathrm{ERP}$, we assign a number $1\le w\left(\mathcal{𝒞}\right) to each regular component $\mathcal{𝒞}$, giving us the possibility to distinguish ${\left(r-1\right)}^{2}$ different types of regular components.

Figure 1

Illustration of a regular component with $m\left(\mathcal{𝒞}\right)\ne 1$. The shaded region on the right hand side is ${\mathrm{\Delta }}_{1}$.

To prove statement (a), we exploit the fact that every regular module M in $\mathrm{mod}{\mathcal{𝒦}}_{r}$ has self-extensions with ${dim}_{k}\mathrm{Ext}\left(M,M\right)\ge 2$, 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 ${E}_{r}$ of rank r over an algebraically closed field of characteristic $p>0$, $\mathrm{mod}k{E}_{r}$ is the only such subcategory. We therefore use the functor $𝔉:\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}k{E}_{r}$, whose essential image (the full subcategory of $\mathrm{mod}k{E}_{r}$ formed by all modules isomorphic to modules of the form $𝔉\left(M\right)$) consists of all modules of Loewy length $\le 2$, to transfer our results to $\mathrm{mod}k{E}_{r}$. We denote by ${\mathrm{ESP}}_{2,d}\left({E}_{r}\right)$ the category of modules in $\mathrm{mod}k{E}_{r}$ of Loewy length $\le 2$ with the equal d-socle property and arrive at the following result.

#### Corollary.

Let $\mathrm{char}\mathit{}\mathrm{\left(}k\mathrm{\right)}\mathrm{>}\mathrm{0}$, $r\mathrm{\ge }\mathrm{3}$ and $\mathrm{1}\mathrm{\le }d\mathrm{<}r$. Then ${\mathrm{ESP}}_{\mathrm{2}\mathrm{,}d}\mathit{}\mathrm{\left(}{E}_{r}\mathrm{\right)}\mathrm{\setminus }{\mathrm{ESP}}_{\mathrm{2}\mathrm{,}d\mathrm{-}\mathrm{1}}\mathit{}\mathrm{\left(}{E}_{r}\mathrm{\right)}$ has wild representation type, where ${\mathrm{ESP}}_{\mathrm{2}\mathrm{,}\mathrm{0}}\mathit{}\mathrm{\left(}{E}_{r}\mathrm{\right)}\mathrm{:-}\mathrm{\varnothing }$.

For $r=2$, we consider the Beilinson algebra $B\left(3,2\right)$. The fact that $B\left(3,2\right)$ is a concealed algebra of type $Q=1\to 2⇉3$ allows us to apply the simplification process in $\mathrm{mod}kQ$. We find a wild subcategory in the category of all modules in $\mathrm{mod}B\left(3,2\right)$ with the equal kernels property and conclude the following.

#### Corollary.

Assume that $\mathrm{char}\mathit{}\mathrm{\left(}k\mathrm{\right)}\mathrm{=}p\mathrm{>}\mathrm{2}$; then the full subcategory of modules with the equal kernels property in $\mathrm{mod}\mathit{}k\mathit{}{E}_{\mathrm{2}}$ and Loewy length $\mathrm{\le }\mathrm{3}$ has wild representation type.

In particular, we generalize results by Benson [3] and Bondarenko and Lytvynchuk [4] concerning the wildness of various subcategories of $k{E}_{r}$-modules. We also construct examples of regular components $\mathcal{𝒞}$ such that each module in $\mathcal{𝒞}$ has constant d-socle rank, but no module in $\mathcal{𝒞}$ is ${\mathrm{GL}}_{r}$-stable in the sense of [6].

## 1 Preliminaries

Throughout this article, let k be an algebraically closed field and $r\ge 2$. If not stated otherwise, k is of arbitrary characteristic. We denote by $Q=\left({Q}_{0},{Q}_{1}\right)$ a finite and connected quiver without oriented cycles. For an arrow $\alpha :x\to y\in {Q}_{1}$, we define $s\left(\alpha \right)=x$ and $t\left(\alpha \right)=y$. We say that α starts in $s\left(\alpha \right)$ and ends in $t\left(\alpha \right)$. A (finite-dimensional) representation $M=\left({\left({M}_{x}\right)}_{x\in {Q}_{0}},{\left(M\left(\alpha \right)\right)}_{\alpha \in {Q}_{1}}\right)$ over Q consists of vector spaces ${M}_{x}$ and linear maps $M\left(\alpha \right):{M}_{s\left(\alpha \right)}\to {M}_{t\left(\alpha \right)}$ such that ${dim}_{k}M:-{\sum }_{x\in {Q}_{0}}{dim}_{k}{M}_{x}$ is finite. A morphism $f:M\to N$ between representations is a collection of linear maps ${\left({f}_{z}\right)}_{z\in {Q}_{0}}$ such that, for each arrow $\alpha :x\to y$, there is a commutative diagram

The category of finite-dimensional representations over Q is denoted by $\mathrm{rep}\left(Q\right)$, and kQ is the path algebra of Q with idempotents ${e}_{x}$, $x\in {Q}_{0}$. The k-algebra kQ is a finite-dimensional, associative, basic and connected k-algebra. Let $\mathrm{mod}kQ$ be the class of finite-dimensional kQ left modules. Given $M\in \mathrm{mod}kQ$, we set ${M}_{x}:-{e}_{x}M$. The categories $\mathrm{mod}kQ$ and $\mathrm{rep}\left(Q\right)$ 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

$\underset{¯}{\mathrm{dim}}:\mathrm{mod}kQ\to {ℤ}^{{Q}_{0}},M↦{\left({dim}_{k}{M}_{x}\right)}_{x\in {Q}_{0}}.$

If $0\to A\to B\to C\to 0$ is an exact sequence, then $\underset{¯}{\mathrm{dim}}A+\underset{¯}{\mathrm{dim}}C=\underset{¯}{\mathrm{dim}}B$. The quiver Q defines a (non-symmetric) bilinear form

$〈-,-〉:{ℤ}^{{Q}_{0}}×{ℤ}^{{Q}_{0}}\to ℤ,$

given by $\left(\left({x}_{i}\right),\left({y}_{j}\right)\right)↦{\sum }_{i\in {Q}_{0}}{x}_{i}{y}_{i}-{\sum }_{\alpha \in {Q}_{1}}{x}_{s\left(\alpha \right)}{y}_{t\left(\alpha \right)}$. For the case that $x,y$ are given by dimension vectors of quiver representations, there is another description of $〈-,-〉$ known as the Euler–Ringel form [17]

$〈\underset{¯}{\mathrm{dim}}M,\underset{¯}{\mathrm{dim}}N〉={dim}_{k}\mathrm{Hom}\left(M,N\right)-{dim}_{k}\mathrm{Ext}\left(M,N\right).$

We denote by $q={q}_{Q}:{ℤ}^{{Q}_{0}}\to ℤ$ the corresponding quadratic form. A vector $d\in {ℤ}^{{Q}_{0}}$ is called a real root if $q\left(d\right)=1$ and an imaginary root if $q\left(d\right)\le 0$.

Denote by ${\mathrm{\Gamma }}_{r}$ the r-Kronecker quiver, which is given by two vertices $1,2$ and arrows ${\gamma }_{1},\mathrm{\dots },{\gamma }_{r}:1\to 2$.

Figure 2

The Kronecker quiver ${\mathrm{\Gamma }}_{r}$.

We set ${\mathcal{𝒦}}_{r}:-k{\mathrm{\Gamma }}_{r}$ and ${P}_{1}:-{\mathcal{𝒦}}_{r}{e}_{2}$, ${P}_{2}:-{\mathcal{𝒦}}_{r}{e}_{1}$. ${P}_{1}$ and ${P}_{2}$ are the indecomposable projective modules of $\mathrm{mod}{\mathcal{𝒦}}_{r}$, ${dim}_{k}\mathrm{Hom}\left({P}_{1},{P}_{2}\right)=r$ and ${dim}_{k}\mathrm{Hom}\left({P}_{2},{P}_{1}\right)=0$. As Figure 2 suggests, we write

$\underset{¯}{\mathrm{dim}}M=\left({dim}_{k}{M}_{1},{dim}_{k}{M}_{2}\right).$

For example, $\underset{¯}{\mathrm{dim}}{P}_{1}=\left(0,1\right)$ and ${dim}_{k}{P}_{2}=\left(1,r\right)$. The Coxeter matrix Φ and its inverse ${\mathrm{\Phi }}^{-1}$ are

$\mathrm{\Phi }:-\left(\begin{array}{cc}\hfill {r}^{2}-1\hfill & \hfill -r\hfill \\ \hfill r\hfill & \hfill -1\hfill \end{array}\right),{\mathrm{\Phi }}^{-1}=\left(\begin{array}{cc}\hfill -1\hfill & \hfill r\hfill \\ \hfill -r\hfill & \hfill {r}^{2}-1\hfill \end{array}\right).$

For M indecomposable, $\underset{¯}{\mathrm{dim}}\tau M=\mathrm{\Phi }\left(\underset{¯}{\mathrm{dim}}M\right)$ holds if M is not projective and $\underset{¯}{\mathrm{dim}}{\tau }^{-1}M={\mathrm{\Phi }}^{-1}\left(\underset{¯}{\mathrm{dim}}M\right)$ if M is not injective. The quadratic form q is given by $q\left(x,y\right)={x}^{2}+{y}^{2}-rxy$.

Figure 3

Auslander–Reiten quiver of ${\mathcal{𝒦}}_{r}$.

Figure 3 shows the notation we use for the components $\mathcal{𝒫},\mathcal{ℐ}$ in the Auslander–Reiten quiver of ${\mathcal{𝒦}}_{r}$ which contain the indecomposable projective modules ${P}_{1},{P}_{2}$ and indecomposable injective modules ${I}_{1},{I}_{2}$. The set of all other components is denoted by $\mathcal{ℛ}$.

Ringel has proven [18, Theorem 2.3] that every component in $\mathcal{ℛ}$ is of type $ℤ{A}_{\mathrm{\infty }}$ if $r\ge 3$ or a homogeneous tube $ℤ{A}_{\mathrm{\infty }}/〈\tau 〉$ 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 $r\ge 3$) 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 $\mathcal{𝒞}$, there is an infinite chain (a ray) of irreducible monomorphisms

$M=M\left[1\right]\to M\left[2\right]\to M\left[3\right]\to \mathrm{\cdots }\to M\left[l\right]\to \mathrm{\cdots }$

and an infinite chain (a coray) of irreducible epimorphisms

$\mathrm{\cdots }\left(l\right)M\to \mathrm{\cdots }\to \left(3\right)M\to \left(2\right)M\to \left(1\right)M=M,$

and for each regular module X, there are unique quasi-simple modules $N,M$ and $l\in ℕ$ with $\left(l\right)M=X=N\left[l\right]$. 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 $\mathcal{𝒫}$ are called preprojective modules and the modules in $\mathcal{ℐ}$ 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 $\mathcal{𝒫}$ (I in $\mathcal{ℐ}$) if and only if there is $l\in {ℕ}_{0}$ with ${\tau }^{l}P={P}_{i}$ (${\tau }^{-l}I={I}_{i}$) for $i\in \left\{1,2\right\}$. Recall that there are no homomorphisms from right to left [1, Corollary VIII.2.13]. To emphasize this result later on, we just write

$\mathrm{Hom}\left(\mathcal{ℐ},\mathcal{𝒫}\right)=0=\mathrm{Hom}\left(\mathcal{ℐ},\mathcal{ℛ}\right)=0=\mathrm{Hom}\left(\mathcal{ℛ},\mathcal{𝒫}\right).$

Using the canonical equivalence ([1, Theorem III.1.6]) of categories $\mathrm{mod}{\mathcal{𝒦}}_{r}\cong \mathrm{rep}\left({\mathrm{\Gamma }}_{r}\right)$, we introduce the duality $\delta :\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}{\mathcal{𝒦}}_{r}$ by setting ${\left(\delta M\right)}_{x}:-{\left({M}_{\psi \left(x\right)}\right)}^{\ast }$ and $\left(\delta M\right)\left({\gamma }_{i}\right):-{\left(M\left({\gamma }_{i}\right)\right)}^{\ast }$ for each $M\in \mathrm{rep}\left({\mathrm{\Gamma }}_{r}\right)$, where $\psi :\left\{1,2\right\}\to \left\{1,2\right\}$ is the permutation of order 2. Note that $\delta \left({P}_{i}\right)={I}_{i}$ for all $i\in ℕ$. 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 $r\mathrm{\ge }\mathrm{2}$ and $d\mathrm{\in }{\mathrm{N}}_{\mathrm{0}}^{\mathrm{2}}$.

• (a)

If $d=\underset{¯}{\mathrm{dim}}M$ for some indecomposable module M , then $q\left(d\right)\le 1$.

• (b)

If $q\left(d\right)=1$ , then there exists a unique indecomposable module X with $\underset{¯}{\mathrm{dim}}X=d$ . In this case, X is preprojective or preinjective, and X is preprojective if and only if ${dim}_{k}{X}_{1}<{dim}_{k}{X}_{2}$.

• (c)

If $q\left(d\right)\le 0$ , then there exist infinitely many indecomposable modules Y with $\underset{¯}{\mathrm{dim}}Y=d$ and each Y is regular.

Since there is no pair $\left(a,b\right)\in {ℕ}_{0}^{2}\setminus \left\{\left(0,0\right)\right\}$ satisfying ${a}^{2}+{b}^{2}-rab=q\left(a,b\right)=0$ for $r\ge 3$, we conclude together with [1, Lemma VIII.2.7] the following.

#### Corollary 1.2.

Let M be an indecomposable ${\mathcal{K}}_{r}$-module. Then $\mathrm{Ext}\mathit{}\mathrm{\left(}M\mathrm{,}M\mathrm{\right)}\mathrm{=}\mathrm{0}$ if and only if M is preprojective or preinjective. If $r\mathrm{\ge }\mathrm{3}$ and M is regular, then ${\mathrm{dim}}_{k}\mathit{}\mathrm{Ext}\mathit{}\mathrm{\left(}M\mathrm{,}M\mathrm{\right)}\mathrm{\ge }\mathrm{2}$.

Let ${\mathrm{mod}}_{\mathrm{pf}}{\mathcal{𝒦}}_{r}$ be the subcategory of all modules without non-zero projective direct summands and ${\mathrm{mod}}_{\mathrm{if}}{\mathcal{𝒦}}_{r}$ the subcategory of all modules without non-zero injective summands. Since ${\mathcal{𝒦}}_{r}$ is a hereditary algebra, the Auslander–Reiten translation $\tau :\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}{\mathcal{𝒦}}_{r}$ induces an equivalence from ${\mathrm{mod}}_{\mathrm{pf}}{\mathcal{𝒦}}_{r}$ to ${\mathrm{mod}}_{\mathrm{if}}{\mathcal{𝒦}}_{r}$. In particular, if M and N are indecomposable with $M,N$ not projective, we get $\mathrm{Hom}\left(M,N\right)\cong \mathrm{Hom}\left(\tau M,\tau N\right)$. The Auslander–Reiten formula [1, Theorem VI.2.13] simplifies to the following.

#### Theorem 1.3 ([13, Theorem 2.3]).

For $X\mathrm{,}Y$ in $\mathrm{mod}\mathit{}{\mathcal{K}}_{r}$, there a functorial isomorphisms

$\mathrm{Ext}\left(X,Y\right)\cong \mathrm{Hom}{\left(Y,\tau X\right)}^{\ast }\cong \mathrm{Hom}{\left({\tau }^{-1}Y,X\right)}^{\ast }.$

## 2.1 Elementary modules of small dimension

Let $r\ge 3$. The homological characterization in [21] uses an algebraic family of modules of projective dimension 1 for the Beilinson algebra $B\left(n,r\right)$ on n vertices. For $n=2$, we have $B\left(2,r\right)={\mathcal{𝒦}}_{r}$, and $\mathrm{mod}{\mathcal{𝒦}}_{r}$ is a hereditary category. Hence every non-projective module is of projective dimension 1. In the following, we study the module family ${\left({X}_{\alpha }\right)}_{\alpha \in {k}^{r}\setminus \left\{0\right\}}$ for $n=2$. We will see later on that each non-zero proper submodule of ${X}_{\alpha }$ is isomorphic to a finite number of copies of ${P}_{1}$, and ${X}_{\alpha }$ itself is regular. In particular, we do not find a short exact sequence $0\to A\to {X}_{\alpha }\to B\to 0$ 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 $0\to A\to E\to B\to 0$ 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 $0={X}_{0}\subset {X}_{1}\subset \mathrm{\cdots }\subset {X}_{r}=X$ such that ${X}_{i}/{X}_{i-1}$ is elementary for all $1\le i\le r$ 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 $X\mathrm{,}Y$ regular with non-zero morphisms $f\mathrm{:}X\mathrm{\to }E$ and $g\mathrm{:}E\mathrm{\to }Y$. Then $g\mathrm{\circ }f\mathrm{\ne }\mathrm{0}$. In particular, $\mathrm{End}\mathit{}\mathrm{\left(}E\mathrm{\right)}\mathrm{=}k$.

#### Proof.

Since f is non-zero and $\mathrm{Hom}\left(\mathcal{ℛ},\mathcal{𝒫}\right)=0=\mathrm{Hom}\left(\mathcal{ℐ},\mathcal{ℛ}\right)$, $\mathrm{im}f$ is a regular non-zero submodule of E. Consequently, since E is elementary, $\mathrm{coker}f$ is preinjective by [16, Proposition 1.3]; hence g cannot factor through $\mathrm{coker}f$. ∎

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

#### Proposition 2.1.2.

Let $\mathcal{E}$ be a non-empty family of elementary modules of bounded dimension, and put

$\mathcal{𝒯}\left(\mathcal{ℰ}\right):-\mathrm{ker}\mathrm{Ext}\left(\mathcal{ℰ},-\right)=\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathrm{Ext}\left(E,M\right)=0\mathit{\text{for all}}E\in \mathcal{ℰ}\right\}.$

Then the following statements hold.

• (1)

$\mathcal{𝒯}\left(\mathcal{ℰ}\right)$ is closed under extensions, images and τ.

• (2)

$\mathcal{𝒯}\left(\mathcal{ℰ}\right)$ contains all preinjective modules.

• (3)

For each regular component $\mathcal{𝒞}$ , the set $\mathcal{𝒯}\left(\mathcal{ℰ}\right)\cap \mathcal{𝒞}$ forms a non-empty cone in $\mathcal{𝒞}$ , which consists of the predecessors of a uniquely determined quasi-simple module in $\mathcal{𝒞}$ , i.e. there is $W\in \mathcal{𝒞}$ quasi-simple such that $\mathcal{𝒯}\left(\mathcal{ℰ}\right)\cap \mathcal{𝒞}=\left(\to W\right):-\left\{\left(i+1\right){\tau }^{l}W\mid i,l\in {ℕ}_{0}\right\}$.

#### Proof.

(1) Since ${\mathrm{Ext}}^{2}=0$, the category is closed under images and extensions. Let $M\in \mathcal{𝒯}\left(\mathcal{ℰ}\right)$; then the Auslander–Reiten formula yields $0={dim}_{k}\mathrm{Hom}\left(M,\tau E\right)$ for all $E\in \mathcal{ℰ}$. We first show that M is not preprojective. Assume to the contrary that $P=M$ is preprojective. Let $l\in {ℕ}_{0}$ such that ${\tau }^{l}P$ is projective; then ${\tau }^{l}P={P}_{i}$ for an $i\in \left\{1,2\right\}$ and

$0={dim}_{k}\mathrm{Hom}\left(P,\tau E\right)={dim}_{k}\mathrm{Hom}\left({\tau }^{l}P,{\tau }^{l+1}E\right)={\left(\underset{¯}{\mathrm{dim}}\left({\tau }^{l+1}E\right)\right)}_{3-i}.$

This is a contradiction since every non-sincere ${\mathcal{𝒦}}_{r}$-module is semi-simple. Hence M is regular or preinjective.

Now we show $\tau M\in \mathcal{𝒯}\left(\mathcal{ℰ}\right)$. In view of the Auslander–Reiten formula, we get

${dim}_{k}\mathrm{Ext}\left(E,\tau M\right)={dim}_{k}\mathrm{Hom}\left(M,E\right).$

Let $f:M\to E$ be a morphism, and assume that $f\ne 0$. Since $0=\mathrm{Hom}\left(\mathcal{ℐ},\mathcal{ℛ}\right)$, the module M is not preinjective and therefore regular. As a regular module E has self-extensions (see Corollary 1.2), and therefore $E\notin \mathcal{𝒯}\left(\mathcal{ℰ}\right)$. Hence $0\ne {dim}_{k}\mathrm{Ext}\left(E,E\right)={dim}_{k}\mathrm{Hom}\left(E,\tau E\right)$, and we find $0\ne g\in \mathrm{Hom}\left(E,\tau E\right)$. Lemma 2.1.1 provides a non-zero morphism

$M\stackrel{𝑓}{\to }E\stackrel{𝑔}{\to }\tau E.$

We conclude $0\ne {dim}_{k}\mathrm{Hom}\left(M,\tau E\right)={dim}_{k}\mathrm{Ext}\left(E,M\right)=0$, a contradiction. Hence

$0={dim}_{k}\mathrm{Hom}\left(M,E\right)={dim}_{k}\mathrm{Ext}\left(E,\tau M\right)\mathit{ }\text{for all}E\in \mathcal{ℰ}\text{and}\tau M\in \mathcal{𝒯}\left(\mathcal{ℰ}\right).$

(2) The injective modules ${I}_{1},{I}_{2}$ are contained in $\mathcal{𝒯}\left(\mathcal{ℰ}\right)$. Now apply (1).

(3) The existence of the cones can be shown as in [21, Theorem 3.3]. We sketch the proof. Let $X\in \mathcal{𝒞}$ be a quasi-simple module, and denote the upper bound by L. By [13, Lemma 4.6, Proposition 10.5], we find ${n}_{0}\in ℕ$ such that $\mathrm{Ext}\left(Y,{\tau }^{l}X\right)=0$ for all $l\ge {n}_{0}$ and each regular representation Y with ${dim}_{k}Y\le L$ and $\mathrm{Ext}\left(E,{\tau }^{-l}X\right)\ne 0$ for some $E\in \mathcal{ℰ}$. In particular, we have ${\tau }^{l}X\in \mathcal{𝒯}\left(\mathcal{ℰ}\right)$ and ${\tau }^{-l}X\notin \mathcal{𝒯}\left(\mathcal{ℰ}\right)$. Now $\left(1\right)$ shows that $\mathcal{𝒯}\left(\mathcal{ℰ}\right)\cap \mathcal{𝒞}=\left(\to M\right)$ for the uniquely determined quasi-simple module $M\in \mathcal{𝒞}$ with $M\in \mathcal{𝒯}\left(\mathcal{ℰ}\right)$ and ${\tau }^{-1}\notin \mathcal{𝒯}\left(\mathcal{ℰ}\right)$. ∎

The next result follows by the Auslander–Reiten formula and duality since $\delta \left(E\right)$ is elementary if and only if E is elementary.

#### Proposition 2.1.3.

Let $\mathcal{E}$ be a family of elementary modules of bounded dimension, and put

$\mathcal{ℱ}\left(\mathcal{ℰ}\right):-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathrm{Hom}\left(E,M\right)=0\mathit{\text{for all}}E\in \mathcal{ℰ}\right\}.$

Then the following statements hold.

• (1)

$\mathcal{ℱ}\left(\mathcal{ℰ}\right)$ is closed under extensions, submodules and ${\tau }^{-1}$.

• (2)

$\mathcal{ℱ}\left(\mathcal{ℰ}\right)$ contains all preprojective modules.

• (3)

For each regular component $\mathcal{𝒞}$ , the set $\mathcal{ℱ}\left(\mathcal{ℰ}\right)\cap \mathcal{𝒞}$ forms a non-empty cone in $\mathcal{𝒞}$ , which consists of the successors of a uniquely determined quasi-simple module in $\mathcal{𝒞}$.

Note that $\mathcal{ℱ}\left(\mathcal{ℰ}\right)$ is a torsion-free class of some torsion pair $\left(\mathcal{𝒯},\mathcal{ℱ}\left(\mathcal{ℰ}\right)\right)$ and $\mathcal{𝒯}\left(\mathcal{ℰ}\right)$ is the torsion class of some torsion pair $\left(\mathcal{𝒯}\left(\mathcal{ℰ}\right),\mathcal{ℱ}\right)$ (see for example [1, Proposition VI.1.4]).

#### Lemma 2.1.4.

Let $M\mathrm{,}N$ be indecomposable modules with $\underset{\mathrm{¯}}{\mathrm{dim}}\mathit{}M\mathrm{=}\mathrm{\left(}c\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{,}\underset{\mathrm{¯}}{\mathrm{dim}}\mathit{}N\mathrm{=}\mathrm{\left(}\mathrm{1}\mathrm{,}c\mathrm{\right)}$, $\mathrm{1}\mathrm{\le }c\mathrm{<}r$. Then the following statements hold.

• (a)

${\tau }^{z}M$ and ${\tau }^{z}N$ are elementary for all $z\in ℤ$ . Moreover, every proper factor of M is injective, and every proper submodule of N is projective.

• (b)

Every proper factor module of ${\tau }^{l}M$ is preinjective, and every proper submodule of ${\tau }^{-l}N$ is preprojective for $l\in {ℕ}_{0}$.

#### 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 τ-orbit are elementary. Since $\left(c,1\right)$ is an imaginary root of the quadratic form, M is regular. Now let $0\subset X\subset M$ be a proper submodule with dimension vector $\underset{¯}{\mathrm{dim}}X=\left(a,b\right)$. Then $b=1$ since $\mathrm{Hom}\left(\mathcal{ℐ},\mathcal{ℛ}\right)=0$. Hence $\underset{¯}{\mathrm{dim}}M/X=\left(c-a,0\right)$, and $M/X$ is injective.

(b) Let $l\ge 1$ and $0\to \mathrm{ker}f\to {\tau }^{l}M\to X\to 0$ be exact with X regular and non-zero. We have to show that ${\tau }^{l}M\cong X$. We assume that $\mathrm{ker}f\ne 0$. Since $\mathrm{Hom}\left(\mathcal{ℐ},\mathcal{ℛ}\right)=0$, we conclude that $\mathrm{ker}f$ has no indecomposable preinjective direct summand. Since ${\tau }^{-1}$ is right exact [1, Corollary VII.1.9], we get an exact sequence ${\tau }^{-1}\mathrm{ker}f\to M\to {\tau }^{-l}X\to 0$. We conclude with (a) that ${\tau }^{-l}X\cong M$; hence $X\cong {\tau }^{-l}M$. ∎

## 2.2 An algebraic family of test-modules

Let $r\ge 2$. Now we take a closer look at the modules ${\left({X}_{\alpha }\right)}_{\alpha \in {k}^{r}\setminus \left\{0\right\}}$. 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 $B\left(2,r\right)={\mathcal{𝒦}}_{r}$.

For $\alpha \in {k}^{r}$ and $M\in \mathrm{mod}{\mathcal{𝒦}}_{r}$, we define ${x}_{\alpha }:-{\alpha }_{1}{\gamma }_{1}+\mathrm{\cdots }+{\alpha }_{r}{\gamma }_{r}$ and denote by ${x}_{\alpha }^{M}:M\to M$ the linear operator associated to ${x}_{\alpha }$.

#### Definition.

For $\alpha \in {k}^{r}\setminus \left\{0\right\}$, the map $\overline{\alpha }:{〈{e}_{2}〉}_{k}={P}_{1}\to {P}_{2}$, ${e}_{2}↦{\alpha }_{1}{\gamma }_{1}+\mathrm{\cdots }+{\alpha }_{r}{\gamma }_{r}={x}_{\alpha }$ defines an embedding of ${\mathcal{𝒦}}_{r}$-modules and is just the left multiplication by ${x}_{\alpha }$. We now set ${X}_{\alpha }:-\mathrm{coker}\overline{\alpha }$.

These modules are the “test”-modules introduced in [21]. In fact, $\mathrm{im}\overline{\alpha }$ is a 1-dimensional submodule of ${P}_{2}$ contained inside the radical $\mathrm{rad}\left({P}_{2}\right)$ of the local module ${P}_{2}$. From the definition, we get an exact sequence $0\to {P}_{1}\to {P}_{2}\to {X}_{\alpha }\to 0$ and $dim{X}_{\alpha }=\left(1,r\right)-\left(0,1\right)=\left(1,r-1\right)$. Since ${P}_{2}$ is local with semi-simple radical $\mathrm{rad}\left({P}_{2}\right)={P}_{1}^{r}$, it now seems natural to study embeddings ${P}_{1}^{d}\to {P}_{2}$ for $1\le d and the corresponding cokernels. This motivates the next definition. We restrict ourselves to $d since otherwise the cokernel is the simple injective module.

#### Definition.

Let and $1\le d. For $T=\left({u}_{1},\mathrm{\dots },{u}_{d}\right)\in {\left({k}^{r}\right)}^{d}$, we define $\overline{T}:{\left({P}_{1}\right)}^{d}\to {P}_{2}$ as the ${\mathcal{𝒦}}_{r}$-linear map

$\overline{T}\left(x\right)=\sum _{i=1}^{d}{\overline{u}}_{i}\circ {\pi }_{i}\left(x\right),$

where ${\pi }_{i}:{\left({P}_{1}\right)}^{d}\to {P}_{1}$ denotes the projection onto the i-th coordinate.

The map $\overline{T}$ is injective if and only if T is linearly independent; then we have

$\underset{¯}{\mathrm{dim}}\mathrm{coker}\overline{T}=\underset{¯}{\mathrm{dim}}{P}_{2}-d\underset{¯}{\mathrm{dim}}{P}_{1}=\left(1,r-d\right),$

and $\mathrm{coker}\overline{T}$ is indecomposable because ${P}_{2}$ is local. Moreover, $\left(1,r-d\right)$ is an imaginary root of q, and therefore $\mathrm{coker}\overline{T}$ is regular indecomposable and by Lemma 2.1.4 elementary. We define $〈T〉:-{〈{u}_{1},\mathrm{\dots },{u}_{d}〉}_{k}$.

#### Lemma 2.2.1.

Let $T\mathrm{,}S\mathrm{\in }{\mathrm{\left(}{k}^{r}\mathrm{\right)}}^{d}$ such that ${\mathrm{dim}}_{k}\mathit{}\mathrm{〈}T\mathrm{〉}\mathrm{=}d\mathrm{=}{\mathrm{dim}}_{k}\mathit{}\mathrm{〈}S\mathrm{〉}$; then $\mathrm{coker}\mathit{}\overline{T}\mathrm{\cong }\mathrm{coker}\mathit{}\overline{S}$ if and only if $\mathrm{〈}T\mathrm{〉}\mathrm{=}\mathrm{〈}S\mathrm{〉}$.

#### Proof.

If $〈T〉=〈S〉$, then the definition of $\overline{T}$ and $\overline{S}$ implies $\mathrm{im}\overline{T}=\mathrm{im}\overline{S}$. Hence $\mathrm{coker}\overline{T}={P}_{2}/\mathrm{im}\overline{T}=\mathrm{coker}\overline{S}$.

Now let $〈S〉\ne 〈T〉$, and assume that $0\ne \phi :\mathrm{coker}\overline{T}\to \mathrm{coker}\overline{S}$ is ${\mathcal{𝒦}}_{r}$-linear. Since $\mathrm{coker}\overline{S}$ is local with radical ${P}_{1}^{r-d}$ and $\mathrm{Hom}\left(\mathcal{ℛ},\mathcal{𝒫}\right)=0$, the map φ is surjective and therefore injective. Recall that ${P}_{2}$ has $\left\{{e}_{1},{\gamma }_{1},\mathrm{\dots },{\gamma }_{r}\right\}$ as a basis. Let $x\in {P}_{2}$ such that $\phi \left({e}_{1}+\mathrm{im}\overline{T}\right)=x+\mathrm{im}\overline{S}$. Since φ is ${\mathcal{𝒦}}_{r}$-linear, we get

$x+\mathrm{im}\overline{S}={e}_{1}\phi \left({e}_{1}+\mathrm{im}\overline{T}\right)={e}_{1}x+\mathrm{im}\overline{S}$

and hence $x-{e}_{1}x\in \mathrm{im}\overline{S}$. Write $x=\mu {e}_{1}+{\sum }_{i=1}^{r}{\mu }_{i}{\gamma }_{i}$; then

$x-\mu {e}_{1}=\sum _{i=1}^{r}{\mu }_{i}{\gamma }_{i}=x-{e}_{1}x\in \mathrm{im}\overline{S}\mathit{ }\text{and}\mathit{ }x+\mathrm{im}\overline{S}=\mu {e}_{1}+\mathrm{im}\overline{S}.$

The assumption $〈S〉\ne 〈T〉$ yields $y\in \mathrm{im}\overline{S}\setminus \mathrm{im}\overline{T}\subseteq {〈{\gamma }_{1},\mathrm{\dots },{\gamma }_{r}〉}_{k}$. Then $y+\mathrm{im}\overline{T}\ne 0$ and

$\phi \left(y+\mathrm{im}\overline{T}\right)=y\phi \left({e}_{1}+\mathrm{im}\overline{T}\right)=\mu y\left({e}_{1}+\mathrm{im}\overline{S}\right)=\mu y+\mathrm{im}\overline{S}=\mathrm{im}\overline{S},$

a contradiction to the injectivity of φ. Hence $\mathrm{Hom}\left(\mathrm{coker}\overline{T},\mathrm{coker}\overline{S}\right)=0$ and $\mathrm{coker}\overline{T}\ncong \mathrm{coker}\overline{S}$. ∎

#### Definition.

Let $r\ge 2$ and $U\in {\mathrm{Gr}}_{d,r}$ with basis $T=\left({u}_{1},\mathrm{\dots },{u}_{d}\right)$. We define ${X}_{U}:-\mathrm{coker}\overline{T}$.

#### Remark.

${X}_{U}$ is well defined (up to isomorphism) with dimension vector $dim{X}_{U}=\left(1,r-d\right)$, and ${X}_{U}$ is elementary for $r\ge 3$ and quasi-simple for $r=2$.

For a module X, we define $\mathrm{add}X$ as the category of summands of finite direct sums of X, and ${Q}^{d}$ denotes the set of isomorphism classes $\left[M\right]$ of indecomposable modules M with dimension vector $\left(1,r-d\right)$ for $1\le d.

#### Proposition 2.2.2.

Let M be indecomposable.

• (a)

If $\left[M\right]\in {Q}^{d}$ , then there exists $U\in {\mathrm{Gr}}_{d,r}$ with $M\cong {X}_{U}$.

• (b)

The map $\phi :{\mathrm{Gr}}_{d,r}\to {Q}^{d}$, $U↦\left[{X}_{U}\right]$ is bijective.

• (c)

Let $1\le c\le d and $\left[M\right]\in {Q}^{d}$ . There is $\left[N\right]\in {Q}^{c}$ and an epimorphism $\pi :N\to M$.

#### Proof.

(a) Let $0⊊X⊊M$ be a submodule of M. Then $X\subseteq \mathrm{rad}\left(M\right)={P}_{1}^{r-d}$, and X is in $\mathrm{add}{P}_{1}$. It is

$1={dim}_{k}{M}_{1}={dim}_{k}\mathrm{Hom}\left({\mathcal{𝒦}}_{r}{e}_{1},M\right)={dim}_{k}\mathrm{Hom}\left({P}_{2},M\right),$

so we find a non-zero map $\pi :{P}_{2}\to M$. Since every proper submodule of M is in $\mathrm{add}{P}_{1}$ and $\mathrm{Hom}\left({P}_{2},{P}_{1}\right)=0$, the map $\pi :{P}_{2}\to M$ is surjective and yields an exact sequence $0\to {P}_{1}^{d}\stackrel{𝜄}{\to }{P}_{2}\stackrel{𝜋}{\to }M\to 0$. For $1\le i\le d$, there exist uniquely determined elements ${\beta }_{i},{\alpha }_{i}^{1},\mathrm{\dots },{\alpha }_{i}^{r}\in k$ such that

$\iota \left({g}_{i}\left({e}_{2}\right)\right)={\beta }_{i}{e}_{1}+{\alpha }_{i}^{1}{\gamma }_{1}+\mathrm{\cdots }+{\alpha }_{i}^{r}{\gamma }_{r}\in {P}_{2}={〈{\gamma }_{i},{e}_{1}\mid 1\le i\le r〉}_{k},$

where ${g}_{i}:{P}_{1}\to {P}_{1}^{d}$ denotes the embedding into the i-th coordinate. Since ${e}_{2}$ is an idempotent with ${e}_{2}{\gamma }_{j}={\gamma }_{j}$ ($1\le j\le r$) and ${e}_{2}{e}_{1}=0$, we get

${\alpha }_{i}^{1}{\gamma }_{1}+\mathrm{\cdots }+{\alpha }_{i}^{r}{\gamma }_{r}={e}_{2}\left(\iota \circ {g}_{i}\left({e}_{2}\right)\right)=\left(\iota \circ {g}_{i}\right)\left({e}_{2}\cdot {e}_{2}\right)=\left(\iota \circ {g}_{i}\right)\left({e}_{2}\right)={\beta }_{i}{e}_{1}+{\alpha }_{i}^{1}{\gamma }_{1}+\mathrm{\cdots }+{\alpha }_{i}^{r}{\gamma }_{r}.$

Hence ${\beta }_{i}=0$. Now define ${\alpha }_{i}:-\left({\alpha }_{i}^{1},\mathrm{\dots },{\alpha }_{i}^{r}\right)$, $T:-\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{d}\right)$ and $U:-〈T〉$. It is $\iota =\overline{T}$, and by the injectivity of ι, we conclude that T is linearly independent, and therefore $U\in {\mathrm{Gr}}_{d,r}$. Now we conclude

${X}_{U}=\mathrm{coker}\overline{T}=\mathrm{coker}\iota =M.$

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

(c) By (a), we find U in ${\mathrm{Gr}}_{d,r}$ with basis $T=\left({u}_{1},\mathrm{\dots },{u}_{d}\right)$ such that ${X}_{U}\cong M$. Let V be the subspace with basis $S=\left({u}_{1},\mathrm{\dots },{u}_{c}\right)$. Then $\mathrm{im}\overline{S}\subseteq \mathrm{im}\overline{T}$, and we get an epimorphism $\pi :{X}_{V}={P}_{2}/\mathrm{im}\overline{S}\to {P}_{2}/\mathrm{im}\overline{T}={X}_{U}$, $x+\overline{S}\to x+\overline{T}$ with $\underset{¯}{\mathrm{dim}}{X}_{V}=\left(1,r-c\right)$. ∎

As a generalization of ${x}_{\alpha }^{M}:M\to M$, we introduce maps ${x}_{T}^{M}:M\to {M}^{d}$ and ${y}_{T}^{M}:{M}^{d}\to M$ for $1\le d and $T\in {\left({k}^{r}\right)}^{d}$. Note that ${x}_{T}^{M}={y}_{T}^{M}$ if and only if $d=1$.

#### Definition.

Let $1\le d and $T=\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{d}\right)\in {\left({k}^{r}\right)}^{d}$. We denote by ${x}_{T}^{M}$ and ${y}_{T}^{M}$ the operators

${x}_{T}^{M}:M\to {M}^{d},$$m↦\left({x}_{{\alpha }_{1}}^{M}\left(m\right),\mathrm{\dots },{x}_{{\alpha }_{d}}^{M}\left(m\right)\right),$${y}_{T}^{M}:{M}^{d}\to M,$$\left({m}_{1},\mathrm{\dots },{m}_{d}\right)↦{x}_{{\alpha }_{1}}^{M}\left({m}_{1}\right)+\mathrm{\dots }+{x}_{{\alpha }_{d}}^{M}\left({m}_{d}\right).$

It is $\mathrm{im}{x}_{T}^{M}\subseteq {M}_{2}\oplus \mathrm{\cdots }\oplus {M}_{2}$, ${M}_{2}\oplus \mathrm{\cdots }\oplus {M}_{2}\subseteq \mathrm{ker}{y}_{T}^{M}$ and ${\left({x}_{T}^{M}\right)}^{\ast }={y}_{T}^{\delta M}$ since, for $f=\left({f}_{1},\mathrm{\dots },{f}_{d}\right)\in {\left(\delta M\right)}^{d}$ and $m\in M$, we have

$\begin{array}{cc}\hfill {\left({x}_{T}^{M}\right)}^{\ast }\left(f\right)\left(m\right)& ={\left({x}_{T}^{M}\right)}^{\ast }\left({f}_{1},\mathrm{\dots },{f}_{d}\right)\left(m\right)=\sum _{i=1}^{d}\left({f}_{i}\circ {x}_{{\alpha }_{i}}^{M}\right)\left(m\right)\hfill \\ & =\sum _{i=1}^{d}{f}_{i}\left({x}_{{\alpha }_{i}}.m\right)=\sum _{i=1}^{d}\left({x}_{{\alpha }_{i}}.{f}_{i}\right)\left(m\right)\hfill \\ & ={y}_{T}^{\delta M}\left({f}_{1},\mathrm{\dots },{f}_{d}\right)\left(m\right)={y}_{T}^{\delta M}\left(f\right)\left(m\right).\hfill \end{array}$

#### Lemma 2.2.3.

Let $\mathrm{1}\mathrm{\le }d\mathrm{<}r$ and $U\mathrm{\in }{\mathrm{Gr}}_{d\mathrm{,}r}$. Every non-zero quotient Q of ${X}_{U}$ is indecomposable. Q is preinjective (injective) if $\underset{\mathrm{¯}}{\mathrm{dim}}\mathit{}Q\mathrm{=}\mathrm{\left(}\mathrm{1}\mathrm{,}\mathrm{0}\mathrm{\right)}$ and regular otherwise.

#### Proof.

Since ${X}_{U}$ is regular, we conclude with $\mathrm{Hom}\left(\mathcal{ℛ},\mathcal{𝒫}\right)=0$ that every indecomposable non-zero quotient of ${X}_{U}$ is preinjective or regular. Let Q be such a quotient with $\underset{¯}{\mathrm{dim}}Q=\left(a,b\right)$ and $Q\ne {X}_{U}$. Since ${X}_{U}$ is local with radical ${P}_{1}^{r-d}$ and $\underset{¯}{\mathrm{dim}}=\left(1,r-d\right)$, it follows $\left(1,r-d\right)=\left(a,b\right)+\left(0,c\right)$ 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 $Q=A\oplus B$ with $A,B\ne 0$ imply w.l.o.g. ${\left(\underset{¯}{\mathrm{dim}}B\right)}_{1}=0$. Hence $B\in \mathrm{add}{P}_{1}$, which is a contradiction to $\mathrm{Hom}\left(\mathcal{ℛ},\mathcal{𝒫}\right)=0$. ∎

## 2.3 Modules for the generalized Kronecker algebra

In the following, we will give the definition of ${\mathcal{𝒦}}_{r}$-modules $\left(r\ge 2\right)$ with constant radical rank and constant socle rank.

#### Definition.

Let M be in $\mathrm{mod}{\mathcal{𝒦}}_{r}$ and $1\le d.

• (a)

M has constant d-radical rank if the dimension of

${\mathrm{Rad}}_{U}\left(M\right):-\sum _{u\in U}{x}_{u}^{M}\left(M\right)\subseteq {M}_{2}$

is independent of the choice of $U\in {\mathrm{Gr}}_{d,r}$.

• (b)

M has constant d-socle rank if the dimension of

${\mathrm{Soc}}_{U}\left(M\right):-\left\{m\in M\mid {x}_{u}^{M}\left(M\right)=0\text{for all}u\in U\right\}$$=\bigcap _{u\in U}\mathrm{ker}\left({x}_{u}^{M}\right)\supseteq {M}_{2}$

is independent of the choice of $U\in {\mathrm{Gr}}_{d,r}$.

• (c)

M has the equal d-radical property if ${\mathrm{Rad}}_{U}\left(M\right)$ is independent of the choice of $U\in {\mathrm{Gr}}_{d,r}$

• (d)

M has the equal d-socle property if ${\mathrm{Soc}}_{U}\left(M\right)$ is independent of the choice of $U\in {\mathrm{Gr}}_{d,r}$

#### Definition.

Let $1\le d. We define

• (a)

${\mathrm{ESP}}_{d}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid {M}_{2}={\mathrm{Soc}}_{U}\left(M\right)\text{for all}U\in {\mathrm{Gr}}_{d,r}\right\}$,

• (b)

${\mathrm{ERP}}_{d}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid {M}_{2}={\mathrm{Rad}}_{U}\left(M\right)\text{for all}U\in {\mathrm{Gr}}_{d,r}\right\}$,

• (c)

${\mathrm{CSR}}_{d}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \text{there exists}c\in {ℕ}_{0}\text{such that}{dim}_{k}{\mathrm{Soc}}_{U}\left(M\right)=c\text{for all}U\in {\mathrm{Gr}}_{d,r}\right\}$,

• (d)

${\mathrm{CRR}}_{d}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \text{there exists}c\in {ℕ}_{0}\text{such that}{dim}_{k}{\mathrm{Rad}}_{U}\left(M\right)=c\text{for all}U\in {\mathrm{Gr}}_{d,r}\right\}$.

#### Lemma 2.3.1.

Let M be indecomposable and not simple.

• (a)

M has the equal d -socle property if and only if $M\in {\mathrm{ESP}}_{d}$.

• (b)

M has the equal d -radical property if and only if $M\in {\mathrm{ERP}}_{d}$.

#### Proof.

(a) Assume that M is in ${\mathrm{ESP}}_{d}$, and let $W:-{\mathrm{Soc}}_{U}\left(M\right)$ for $U\in {\mathrm{Gr}}_{d,r}$. Denote by ${e}_{1},\mathrm{\dots },{e}_{r}\in {k}^{r}$ the canonical basis vectors. We find a k-complement of ${\bigcap }_{i=1}^{r}\mathrm{ker}\left({x}_{{e}_{i}}^{M}\right)\cap {M}_{1}$ in ${M}_{1}$, say $K\subseteq {M}_{1}$. Then M decomposes into submodules $M=\left(K+{M}_{2}\right)\oplus {\bigcap }_{i=1}^{r}\mathrm{ker}\left({x}_{{e}_{i}}^{M}\right)\cap {M}_{1}$. Since M is not simple, we have $0\ne {M}_{2}$ and conclude $\left\{0\right\}={\bigcap }_{i=1}^{r}\mathrm{ker}\left({x}_{{e}_{i}}^{M}\right)\cap {M}_{1}$, i.e. ${\bigcap }_{i=1}^{r}\mathrm{ker}\left({x}_{{e}_{i}}^{M}\right)={M}_{2}$ (see also [10, Lemma 5.1.1]). Denote by $S\left(d\right)$ the set of all subsets of $\left\{1,\mathrm{\dots },r\right\}$ of cardinality d. Then

$\bigcap _{S\in S\left(d\right)}\bigcap _{j\in S}\mathrm{ker}\left({x}_{{e}_{j}}^{M}\right)=\bigcap _{i=1}^{r}\mathrm{ker}\left({x}_{{e}_{i}}^{M}\right)={M}_{2}.$

Since ${〈{e}_{j}\mid j\in S〉}_{k}\in {\mathrm{Gr}}_{d,r}$ and $M\in {\mathrm{ESP}}_{d}$, we get ${\bigcap }_{j\in S}\mathrm{ker}\left({x}_{{e}_{j}}^{M}\right)=W$ and hence ${M}_{2}={\bigcap }_{S\in S\left(d\right)}W=W={\mathrm{Soc}}_{U}\left(M\right)$.

(b) Let $M\in {\mathrm{ERP}}_{d}$, $U\in {\mathrm{Gr}}_{d,r}$ and $W:-{\mathrm{Rad}}_{U}\left(M\right)$. Since M is not simple, it is [10, Lemma 5.1.1] ${\sum }_{i=1}^{r}{x}_{{e}_{i}}^{M}={M}_{2}$ and hence

$W=\sum _{S\in S\left(d\right)}\sum _{j\in S}{x}_{{e}_{j}}^{M}\left(M\right)=\sum _{i=1}^{r}{x}_{{e}_{i}}^{M}\left(M\right)={M}_{2}.\mathit{∎}$

#### Remark.

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

$\mathrm{EKP}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid {M}_{2}=\mathrm{ker}\left({x}_{\alpha }^{M}\right)\text{for all}\alpha \in {k}^{r}\setminus \left\{0\right\}\right\},$$\mathrm{EIP}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid {M}_{2}=\mathrm{im}\left({x}_{\alpha }^{M}\right)\text{for all}\alpha \in {k}^{r}\setminus \left\{0\right\}\right\},$$\mathrm{CR}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \text{there exists}c\in {ℕ}_{0}\text{such that}c={dim}_{k}\mathrm{ker}\left({x}_{\alpha }^{M}\right)\text{for all}\alpha \in {k}^{r}\setminus \left\{0\right\}\right\}.$

Note that ${\mathrm{CRR}}_{1}=\mathrm{CR}={\mathrm{CSR}}_{1}$, ${\mathrm{ERP}}_{1}=\mathrm{EIP}$, ${\mathrm{ESP}}_{1}=\mathrm{EKP}$, and for $U\in {\mathrm{Gr}}_{d,r}$ with basis $\left({u}_{1},\mathrm{\dots },{u}_{d}\right)$, we have ${\mathrm{Rad}}_{U}\left(M\right)={\sum }_{i=1}^{d}{x}_{{u}_{i}}^{M}\left(M\right)=\mathrm{im}{y}_{\left({u}_{1},\mathrm{\dots },{u}_{d}\right)}^{M}$ and ${\mathrm{Soc}}_{U}\left(M\right)={\bigcap }_{i=1}^{d}\mathrm{ker}\left({x}_{{u}_{i}}^{M}\right)=\mathrm{ker}{x}_{\left({u}_{1},\mathrm{\dots },{u}_{d}\right)}^{M}$. We restrict the definition to $d since ${\mathrm{Gr}}_{r,r}=\left\{{k}^{r}\right\}$, and therefore every module in $\mathrm{mod}{\mathcal{𝒦}}_{r}$ is of constant r-socle and r-radical rank.

#### Lemma 2.3.2.

Let M be in $\mathrm{mod}\mathit{}{\mathcal{K}}_{r}$ and $\mathrm{1}\mathrm{\le }d\mathrm{<}r$ .

• (a)

$M\in {\mathrm{CSR}}_{d}$ if and only if $\delta M\in {\mathrm{CRR}}_{d}$.

• (b)

$M\in {\mathrm{ESP}}_{d}$ if and only if $\delta M\in {\mathrm{ERP}}_{d}$.

#### Proof.

Note that ${\mathrm{Rad}}_{U}\left(\delta M\right)=\mathrm{im}\left({y}_{T}^{\delta M}\right)=\mathrm{im}{\left({x}_{T}^{M}\right)}^{\ast }\cong {\left(\mathrm{im}{x}_{T}^{M}\right)}^{\ast }$ and hence

$\begin{array}{cc}\hfill M-{dim}_{k}{\mathrm{Soc}}_{U}\left(M\right)& ={dim}_{k}M-{dim}_{k}\mathrm{ker}{x}_{T}^{M}={dim}_{k}\mathrm{im}{x}_{T}^{M}\hfill \\ & ={dim}_{k}{\left(\mathrm{im}{x}_{T}^{M}\right)}^{\ast }={dim}_{k}{\mathrm{Rad}}_{U}\left(\delta M\right).\hfill \end{array}$

Hence $M\in {\mathrm{CSR}}_{d}$ if and only if $\delta M\in {\mathrm{CRR}}_{d}$. Moreover, M in ${\mathrm{ESP}}_{d}$ if and only if ${\mathrm{Soc}}_{U}\left(M\right)={M}_{2}$, and hence ${dim}_{k}{\mathrm{Rad}}_{U}\left(\delta M\right)={dim}_{k}{M}_{1}={dim}_{k}{\left(\delta M\right)}_{2}$. ∎

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

#### Proposition 2.3.3.

Let $\mathrm{1}\mathrm{\le }d\mathrm{<}r\mathrm{\in }\mathrm{N}$. Then

${\mathrm{ESP}}_{d}=\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathrm{Hom}\left({X}_{U},M\right)=0\mathit{\text{for all}}U\in {\mathrm{Gr}}_{d,r}\right\},$${\mathrm{CSR}}_{d}=\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathit{\text{there exists}}c\in {ℕ}_{0}\mathit{\text{such that}}{dim}_{k}\mathrm{Hom}\left({X}_{U},M\right)=c\mathit{\text{for all}}U\in {\mathrm{Gr}}_{d,r}\right\}.$

#### Proof.

Let $U\in {\mathrm{Gr}}_{d,r}$ with basis $T=\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{d}\right)$. Consider the short exact sequence

$0\to {\left({P}_{1}\right)}^{d}\stackrel{\overline{T}}{\to }{P}_{2}\to {X}_{U}\to 0.$

Application of $\mathrm{Hom}\left(-,M\right)$ yields

$0\to \mathrm{Hom}\left({X}_{U},M\right)\to \mathrm{Hom}\left({P}_{2},M\right)\stackrel{{\overline{T}}^{\ast }}{\to }\mathrm{Hom}\left({P}_{1}^{d},M\right)\to \mathrm{Ext}\left({X}_{U},M\right)\to 0.$

Moreover, let

$f:\mathrm{Hom}\left({P}_{2},M\right)\to {M}_{1},$$g\to g\left({e}_{1}\right),$$g:\mathrm{Hom}\left({P}_{1}^{d},M\right)\to {M}_{2}^{d},$$h↦\left(h\circ {\iota }_{1}\left({e}_{2}\right),\mathrm{\dots },h\circ {\iota }_{d}\left({e}_{2}\right)\right)$

be the natural isomorphisms, where ${\iota }_{i}:{P}_{1}\to {P}_{1}^{d}$ denotes the embedding into the i-th coordinate. Let ${\pi }_{{M}_{2}^{d}}:{M}^{d}\to {M}_{2}^{d}$ be the natural projection. The equality $g\circ {\overline{T}}^{\ast }={\pi }_{{M}_{2}^{d}}\circ x_{T}^{M}{}_{|{M}_{1}}\circ f$ holds since both maps are applied to a homomorphism. Hence ${dim}_{k}\mathrm{ker}\left({\pi }_{{M}_{2}^{d}}\circ x_{T}^{M}{}_{|{M}_{1}}:{M}_{1}\to {M}_{2}^{d}\right)={dim}_{k}\mathrm{ker}\left({\overline{T}}^{\ast }\right)={dim}_{k}\mathrm{Hom}\left({X}_{U},M\right)$. Now let $c\in {ℕ}_{0}$. We conclude

$\begin{array}{cc}\hfill {dim}_{k}\mathrm{Hom}\left({X}_{U},M\right)=c& {\stackrel{dim}{⇔}}_{k}\mathrm{ker}\left({\pi }_{{M}_{2}^{d}}\circ x_{T}^{M}{}_{|{M}_{1}}\right)=c\stackrel{\mathrm{im}{x}_{T}^{M}\subseteq {M}_{2}^{d}}{⇔}{dim}_{k}\mathrm{ker}\left(x_{T}^{M}{}_{|{M}_{1}}\right)=c\hfill \\ & \stackrel{{M}_{2}\subseteq \mathrm{ker}\left({x}_{T}^{M}\right)}{⇔}{dim}_{k}\mathrm{ker}\left({x}_{T}^{M}\right)=c+{dim}_{k}{M}_{2}\hfill \\ & \stackrel{{\mathrm{Soc}}_{U}\left(M\right)=\mathrm{ker}\left({x}_{T}^{M}\right)}{⇔}{dim}_{k}{\mathrm{Soc}}_{U}\left(M\right)=c+{dim}_{k}{M}_{2}.\hfill \end{array}$

This finishes the proof for ${\mathrm{CSR}}_{d}$. Moreover, note that $c=0$ together with Lemma 2.3.1 yields

$\begin{array}{cc}\hfill M\in {\mathrm{ESP}}_{d}& ⇔\text{there exists}W\le M\text{such that}{\mathrm{Soc}}_{U}\left(M\right)=W\text{for all}U\in {\mathrm{Gr}}_{d,r}\hfill \\ & ⇔{\mathrm{Soc}}_{U}\left(M\right)={M}_{2}\text{for all}U\in {\mathrm{Gr}}_{d,r}⇔{dim}_{k}{\mathrm{Soc}}_{U}\left(M\right)=0+{dim}_{k}{M}_{2}\text{for all}U\in {\mathrm{Gr}}_{d,r}\hfill \\ & ⇔\mathrm{Hom}\left({X}_{U},M\right)=0\text{for all}U\in {\mathrm{Gr}}_{d,r}.\mathit{∎}\hfill \end{array}$

Since $\tau \circ \delta =\delta \circ {\tau }^{-1}$, the next result follows from the Auslander–Reiten formula and Lemma 2.3.2.

#### Proposition 2.3.4.

Let $\mathrm{1}\mathrm{\le }d\mathrm{<}r\mathrm{\in }\mathrm{N}$. Then

${\mathrm{ERP}}_{d}=\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathrm{Ext}\left(\delta \tau {X}_{U},M\right)=0\mathit{\text{for all}}U\in {\mathrm{Gr}}_{d,r}\right\},$${\mathrm{CRR}}_{d}=\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathit{\text{there exists}}c\in {ℕ}_{0}\mathit{\text{such that}}{dim}_{k}\mathrm{Ext}\left(\delta \tau {X}_{U},M\right)=c\mathit{\text{for all}}U\in {\mathrm{Gr}}_{d,r}\right\}.$

#### Remark.

For $d=1$, we have $U={〈\alpha 〉}_{k}$ with $\alpha \in {k}^{r}\setminus \left\{0\right\}$, ${X}_{U}\cong {X}_{\alpha }$, and [21, Proposition 3.1] yields $\delta \tau {X}_{U}\cong {X}_{U}$. 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\le d and $V\in {\mathrm{Gr}}_{d,r}$, the module ${X}_{V}$ is in ${\mathrm{CSR}}_{d+1}\setminus {\mathrm{CSR}}_{d}$. If not stated otherwise, we assume from now on that $r\ge 3$.

In view of Proposition 2.1.3, Lemma 2.1.4 and the definitions of ${\mathrm{ESP}}_{d}$ and ${\mathrm{ERP}}_{d}$, we immediately get the following proposition.

#### Proposition 2.3.5.

Let $\mathrm{1}\mathrm{\le }d\mathrm{<}r$ and $\mathcal{C}$ a regular component of ${\mathcal{K}}_{r}$.

• (a)

${\mathrm{ESP}}_{1}\subseteq {\mathrm{ESP}}_{2}\subseteq \mathrm{\cdots }\subseteq {\mathrm{ESP}}_{r-1}$ and ${\mathrm{ERP}}_{1}\subseteq {\mathrm{ERP}}_{2}\subseteq \mathrm{\cdots }\subseteq {\mathrm{ERP}}_{r-1}$.

• (b)

${\mathrm{ESP}}_{d}$ is closed under extensions, submodules and ${\tau }^{-1}$ . Moreover, ${\mathrm{ESP}}_{d}$ contains all preprojective modules, and ${\mathrm{ESP}}_{d}\cap \mathcal{𝒞}$ forms a non-empty cone in $\mathcal{𝒞}$ , i.e. there is a quasi-simple module ${M}_{d}\in \mathcal{𝒞}$ such that

${\mathrm{ESP}}_{d}\cap \mathcal{𝒞}=\left({M}_{d}\to \right):-\left\{{\tau }^{-l}{M}_{d}\left[i+1\right]\mid i,l\in {ℕ}_{0}\right\}.$

• (c)

${\mathrm{ERP}}_{d}$ is closed under extensions, images and τ . Moreover, ${\mathrm{ERP}}_{d}$ contains all preinjective modules, and ${\mathrm{ERP}}_{d}\cap \mathcal{𝒞}$ forms a non-empty cone in $\mathcal{𝒞}$ , i.e. there is a quasi-simple module ${W}_{d}\in \mathcal{𝒞}$ such that

${\mathrm{ERP}}_{d}\cap \mathcal{𝒞}=\left(\to {W}_{d}\right):-\left\{\left(i+1\right){\tau }^{l}{W}_{d}\mid i,l\in {ℕ}_{0}\right\}.$

#### Definition.

For $1\le i, we set ${\mathrm{\Delta }}_{i}:-{\mathrm{ESP}}_{i}\setminus {\mathrm{ESP}}_{i-1}$ and ${\nabla }_{i}:-{\mathrm{ERP}}_{i}\setminus {\mathrm{ESP}}_{i-1}$, where ${\mathrm{ESP}}_{0}=\mathrm{\varnothing }={\mathrm{ERP}}_{0}$.

The next result suggests that, for each regular component $\mathcal{𝒞}$ and $1, only a small part of vertices in $\mathcal{𝒞}$ corresponds to modules in ${\mathrm{\Delta }}_{i}$. Nonetheless, we will see in Section 4 that, for $1\le i, the categories ${\mathrm{\Delta }}_{i}$ and ${\nabla }_{i}$ are of wild type.

#### Proposition 2.3.6.

Let $r\mathrm{\ge }\mathrm{3}$. Let $\mathcal{C}$ be a regular component and ${M}_{i}\mathrm{,}{W}_{i}$ ($\mathrm{1}\mathrm{\le }i\mathrm{<}r$) in $\mathcal{C}$ the uniquely determined quasi-simple modules such that ${\mathrm{ESP}}_{i}\mathrm{\cap }\mathcal{C}\mathrm{=}\mathrm{\left(}{M}_{i}\mathrm{\to }\mathrm{\right)}$ and ${\mathrm{ERP}}_{i}\mathrm{\cap }\mathcal{C}\mathrm{=}\mathrm{\left(}\mathrm{\to }{W}_{i}\mathrm{\right)}$.

• (a)

There exists at most one number $1 such that ${\mathrm{\Delta }}_{m\left(\mathcal{𝒞}\right)}\cap \mathcal{𝒞}$ is non-empty. If such a number exists, then ${\mathrm{\Delta }}_{m\left(\mathcal{𝒞}\right)}\cap \mathcal{𝒞}=\left\{{M}_{m\left(\mathcal{𝒞}\right)}\left[l\right]\mid l\ge 1\right\}$.

• (b)

There exists at most one number $1 such that ${\nabla }_{w\left(\mathcal{𝒞}\right)}\cap \mathcal{𝒞}$ is non-empty. If such a number exists, then ${\nabla }_{w\left(\mathcal{𝒞}\right)}\cap \mathcal{𝒞}=\left\{\left(l\right){W}_{w\left(\mathcal{𝒞}\right)}\mid l\ge 1\right\}$.

#### Proof.

(a) By Proposition 2.3.5, there are ${n}_{1},\mathrm{\dots },{n}_{r-1}\in {ℕ}_{0}$ such that

$0={n}_{1}\le {n}_{2}\le \mathrm{\cdots }\le {n}_{r-1}\mathit{ }\text{and}\mathit{ }{M}_{i}={\tau }^{{n}_{i}}{M}_{1}\mathit{ }\text{for all}i\in \left\{1,\mathrm{\dots },r-1\right\}.$

We will show that either $0={n}_{1}=\mathrm{\cdots }={n}_{r-1}$ or that there exists a uniquely determined $1 such that ${n}_{i}>{n}_{i-1}$.

Let M be in $\mathcal{𝒞}$, and assume $M\notin {\mathrm{ESP}}_{1}$. In the following, we show that $\tau M\notin {\mathrm{ESP}}_{r-1}$. There exists $\alpha \in {k}^{r}\setminus \left\{0\right\}$ with $\mathrm{Hom}\left({X}_{U},M\right)\ne 0$ for $U={〈\alpha 〉}_{k}$. Hence we find a non-zero map $f:\tau {X}_{U}\to \tau M$. Consider an exact sequence $0\to {P}_{1}^{r-2}\to {X}_{U}\to N\to 0$. Then $\underset{¯}{\mathrm{dim}}N=\left(1,1\right)$, and by Lemma 2.2.3, N is indecomposable. By Proposition 2.2.2, there exists $V\in {\mathrm{Gr}}_{r-1,r}$ with ${X}_{V}\cong \delta N$. Since $\delta {X}_{U}=\tau {X}_{U}$ (see [21, Proposition 3.1]), we get a non-zero morphism $g:{X}_{V}\to \tau {X}_{U}$ and by Lemma 2.1.1 a non-zero morphism

${X}_{V}\stackrel{𝑔}{\to }\tau {X}_{U}\stackrel{𝑓}{\to }\tau M.$

Therefore, $\tau M\notin {\mathrm{ESP}}_{r-1}$ by Proposition 2.3.3.

Now assume that ${n}_{i}\ne {n}_{j}$ for some i and j. Then, in particular, ${M}_{1}\ne {M}_{r-1}$. Hence ${n}_{r-1}\ge 1$. By definition, we have $M:-\tau {M}_{1}\notin {\mathrm{ESP}}_{1}$, and the above considerations yield $\tau \left(\tau {M}_{1}\right)=\tau M\notin {\mathrm{ESP}}_{r-1}$. Therefore, $1\le {n}_{r-1}<2$ since ${\mathrm{ESP}}_{r-1}\cap \mathcal{𝒞}$ is closed under ${\tau }^{-1}$. Therefore, ${n}_{r-1}=1$ and ${M}_{r-1}=\tau {M}_{1}$. We conclude that there is a uniquely determined $1 such that ${n}_{i}>{n}_{i-1}$, and in this case, ${n}_{i}={n}_{i-1}+1$. Now we set $m\left(\mathcal{𝒞}\right):-i$.

(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 $\mathrm{0}\mathrm{\to }A\mathrm{\to }B\mathrm{\to }C\mathrm{\to }\mathrm{0}$ 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 $r\ge 2$ and $1\le d, and let ${𝔛}_{d,r}:-\left\{{X}_{U}\mid U\in {\mathrm{Gr}}_{d,r}\right\}$. Let ${𝔛}_{d,r}^{\perp }$ be the right orthogonal category ${𝔛}_{d,r}^{\perp }=\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathrm{Hom}\left({X}_{U},M\right)=0\text{for all}U\in {\mathrm{Gr}}_{d,r}\right\}$, and let ${}^{\perp }𝔛_{d,r}$ be the left orthogonal category ${}^{\perp }𝔛_{d,r}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathrm{Hom}\left(M,{X}_{U}\right)=0\text{for all}U\in {\mathrm{Gr}}_{d,r}\right\}$. Then we set ${\overline{𝔛}}_{d,r}:-{𝔛}_{d,r}^{\perp }\cap {}^{\perp }𝔛_{d,r}$.

Note that every module in ${\overline{𝔛}}_{d,r}$ is regular by Proposition 2.3.5.

#### Lemma 2.3.8.

Let $r\mathrm{\ge }\mathrm{3}$ and $\mathrm{1}\mathrm{\le }d\mathrm{<}r$, and let M be quasi-simple regular in a component $\mathcal{C}$ such that $M\mathrm{\in }{\overline{\mathrm{X}}}_{d\mathrm{,}r}$. Then every module in $\mathcal{C}$ has constant d-socle rank.

#### Proof.

Let $V\in {\mathrm{Gr}}_{d,r}$. We have shown in Proposition 2.3.5 that the set

$\left\{N\mid \mathrm{Hom}\left({X}_{V},N\right)=0\right\}\cap \mathcal{𝒞}\mathit{ }\text{(resp.}\left\{N\mid \mathrm{Ext}\left({X}_{V},N\right)=0\right\}\cap \mathcal{𝒞}\text{)}$

is closed under ${\tau }^{-1}$ (resp. $\tau \right)$. Since $0={dim}_{k}\mathrm{Hom}\left(M,{X}_{V}\right)={dim}_{k}\mathrm{Ext}\left({X}_{V},\tau M\right)$, we have $\mathrm{Ext}\left({X}_{V},{\tau }^{l}M\right)=0$ for $l\ge 1$. The Euler–Ringel form yields

$0={dim}_{k}\mathrm{Ext}\left({X}_{V},{\tau }^{l}M\right)=-〈\underset{¯}{\mathrm{dim}}{X}_{V},\underset{¯}{\mathrm{dim}}{\tau }^{l}M〉+\underset{¯}{\mathrm{dim}}\mathrm{Hom}\left({X}_{V},{\tau }^{l}M\right).$

Since $〈\underset{¯}{\mathrm{dim}}{X}_{V},\underset{¯}{\mathrm{dim}}{\tau }^{l}M〉=〈\left(1,r-d\right),\underset{¯}{\mathrm{dim}}M〉$ is independent of V, ${\tau }^{l}M$ has constant d-socle rank. On the other hand, $\mathrm{Hom}\left({X}_{V},M\right)=0$ implies that ${\tau }^{-q}M$ has constant d-socle rank for all $q\ge 0$. It follows that each quasi-simple module in $\mathcal{𝒞}$ has constant d-socle rank. Now apply Lemma 2.3.7. ∎

## 3.1 Representation type

Denote by $\mathrm{\Lambda }:-kQ$ 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 $\mathrm{End}\left(M\right)=k$, and two modules $M,N$ are called orthogonal if we have $\mathrm{Hom}\left(M,N\right)=0=\mathrm{Hom}\left(N,M\right)$.

#### Definition.

Let $\mathcal{𝒳}$ be a non-empty class of pairwise orthogonal bricks in $\mathrm{mod}\mathrm{\Lambda }$. The full subcategory $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is by definition the class of all modules Y in $\mathrm{mod}\mathrm{\Lambda }$ with an $\mathcal{𝒳}$-filtration, that is, a chain

$0={Y}_{0}\subset {Y}_{1}\subset \mathrm{\cdots }\subset {Y}_{n-1}\subset {Y}_{n}=Y$

with ${Y}_{i}/{Y}_{i-1}\in \mathcal{𝒳}$ for all $1\le i\le n$.

In [17, Theorem 1.2], the author shows that $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is an exact abelian subcategory of $\mathrm{mod}\mathrm{\Lambda }$, closed under extensions, and $\mathcal{𝒳}$ is the class of all simple modules in $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$. In particular, a module M in $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is indecomposable if and only if it is indecomposable in $\mathrm{mod}kQ$.

#### Proposition 3.1.1.

Let $r\mathrm{\ge }\mathrm{3}$, and let $\mathcal{X}\mathrm{\subseteq }\mathrm{mod}\mathit{}{\mathcal{K}}_{r}$ be a non-empty class of pairwise orthogonal bricks with self-extensions (and therefore regular).

• (a)

Every module in $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is regular.

• (b)

Every regular component $\mathcal{𝒞}$ contains at most one module of $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$.

• (c)

Every indecomposable module $N\in \mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is quasi-simple in $\mathrm{mod}{\mathcal{𝒦}}_{r}$.

• (d)

$\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is a wild subcategory of $\mathrm{mod}{\mathcal{𝒦}}_{r}$.

#### 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 $\mathrm{mod}{\mathcal{𝒦}}_{r}$ is quasi-simple [13, Proposition 9.2]. Let $M\in \mathcal{𝒳}$. Then we have $t:-{dim}_{k}\mathrm{Ext}\left(M,M\right)\ge 2$ by Corollary 1.2. Due to [11, Section 7] and [15, Remark 1.4], the category $\mathcal{ℰ}\left(\left\{M\right\}\right)\subseteq \mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is equivalent to the category of finite-dimensional modules over the power-series ring $k〈〈{X}_{1},\mathrm{\dots },{X}_{t}〉〉$ in non-commuting variables ${X}_{1},\mathrm{\dots },{X}_{t}$. Since $t\ge 2$, the category $\mathcal{ℰ}\left(\left\{M\right\}\right)\subseteq \mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is wild, and also $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$. ∎

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 $\mathrm{EKP}={\mathrm{ESP}}_{1}$ and $\mathrm{EIP}={\mathrm{ERP}}_{1}$. Using the functor $𝔉:\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}k{E}_{r}$, we show the wildness of the corresponding full subcategories in ${\mathrm{mod}}_{2}k{E}_{r}$ of ${E}_{r}$-modules of Loewy length $\le 2$.

## 3.2 Passage between ${\mathcal{𝒦}}_{r}$ and ${\mathcal{𝒦}}_{s}$

Let $2\le r. Denote by ${inf}_{r}^{s}:\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}{\mathcal{𝒦}}_{s}$ the functor that assigns to a ${\mathcal{𝒦}}_{r}$-module M the module ${inf}_{r}^{s}\left(M\right)$ with the same underlying vector space so that the action of ${e}_{1},{e}_{2},{\gamma }_{1},\mathrm{\dots },{\gamma }_{r}$ on ${inf}_{r}^{s}\left(M\right)$ stays unchanged and all other arrows act trivially on ${inf}_{r}^{s}\left(M\right)$. Moreover, let $\iota :{\mathcal{𝒦}}_{r}\to {\mathcal{𝒦}}_{s}$ be the natural k-algebra monomorphism given by $\iota \left({e}_{i}\right)={e}_{i}$ for $i\in \left\{1,2\right\}$ and $\iota \left({\gamma }_{j}\right)={\gamma }_{j}$. Then each ${\mathcal{𝒦}}_{s}$-module N becomes a ${\mathcal{𝒦}}_{r}$-module ${N}^{\ast }$ via pullback along ι. Denote the corresponding functor by ${\mathrm{res}}_{r}^{s}:\mathrm{mod}{\mathcal{𝒦}}_{s}\to \mathrm{mod}{\mathcal{𝒦}}_{r}$. In the following, $r,s$ will be fixed, so we suppress the index and write just $inf$ and $\mathrm{res}$.

#### Lemma 3.2.1.

Let $\mathrm{2}\mathrm{\le }r\mathrm{<}s\mathrm{\in }\mathrm{N}$. The functor $\mathrm{inf}\mathrm{:}\mathrm{mod}\mathit{}{\mathcal{K}}_{r}\mathrm{\to }\mathrm{mod}\mathit{}{\mathcal{K}}_{s}$ is fully faithful and exact. The essential image of $\mathrm{inf}$ is a subcategory of $\mathrm{mod}\mathit{}{\mathcal{K}}_{s}$ closed under factors and submodules. Moreover, $\mathrm{inf}\mathit{}\mathrm{\left(}M\mathrm{\right)}$ is indecomposable if and only if M is indecomposable in $\mathrm{mod}\mathit{}{\mathcal{K}}_{r}$.

#### Proof.

Clearly, $inf$ is fully faithful and exact. Now let $M\in \mathrm{mod}{\mathcal{𝒦}}_{r}$, and let $U\subseteq inf\left(M\right)$ be a submodule. Then ${\gamma }_{i}$ ($i>r$) acts trivially on U, and hence the pullback $\mathrm{res}\left(U\right)-:{U}^{\ast }$ is a ${\mathcal{𝒦}}_{r}$-module with $inf\left({U}^{\ast }\right)=inf\circ \mathrm{res}\left(U\right)=U$. Now let $V\in \mathrm{mod}{\mathcal{𝒦}}_{s}$, and let $f\in {\mathrm{Hom}}_{{\mathcal{𝒦}}_{s}}\left(inf\left(M\right),V\right)$ be an epimorphism. Let $v\in V$ and $m\in M$ such that $f\left(m\right)=v$. It follows ${\gamma }_{i}v={\gamma }_{i}f\left(m\right)=f\left({\gamma }_{i}m\right)=0$ for all $i>r$. This shows that $inf\left({V}^{\ast }\right)=inf\circ \mathrm{res}\left(V\right)=V$.

Since $inf$ is fully faithful, we have ${\mathrm{End}}_{{\mathcal{𝒦}}_{s}}\left(inf\left(M\right)\right)\cong {\mathrm{End}}_{{\mathcal{𝒦}}_{r}}\left(M\right)$. Hence ${\mathrm{End}}_{{\mathcal{𝒦}}_{s}}\left(inf\left(M\right)\right)$ is local if and only if ${\mathrm{End}}_{{\mathcal{𝒦}}_{r}}\left(M\right)$ is local. ∎

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

#### Lemma 3.2.2.

Let $\mathrm{2}\mathrm{\le }r\mathrm{<}s$, and let M be an indecomposable ${\mathcal{K}}_{r}$-module that is not simple. The following statements hold.

• (a)

$inf\left(M\right)$ is regular and quasi-simple.

• (b)

$inf\left(M\right)\notin {\mathrm{CSR}}_{m}$ for all $m\in \left\{1,\mathrm{\dots },s-r\right\}$.

#### Proof.

(a) Write $\underset{¯}{\mathrm{dim}}M=\left(a,b\right)\in {ℕ}_{0}×{ℕ}_{0}$. Since M is not simple, $ab\ne 0$ and $q\left(\underset{¯}{\mathrm{dim}}M\right)={a}^{2}+{b}^{2}-rab\le 1$. It follows

$q\left(\underset{¯}{\mathrm{dim}}inf\left(M\right)\right)={a}^{2}+{b}^{2}-sab={a}^{2}+{b}^{2}-rab-\left(s-r\right)ab\le 1-\left(s-r\right)ab<1.$

Hence $q\left(\underset{¯}{\mathrm{dim}}inf\left(M\right)\right)\le 0$, and $inf\left(M\right)$ is regular.

Assume that $inf\left(M\right)$ is not quasi-simple; then $inf\left(M\right)=U\left[i\right]$ for U quasi-simple with $i\ge 2$. By Lemma 3.2.1, we have $U\left[i-1\right]=inf\left(A\right)$ and ${\tau }^{-1}U\left[i-1\right]=inf\left(B\right)$ for some $A,B$ indecomposable in $\mathrm{mod}{\mathcal{𝒦}}_{r}$. Fix an irreducible monomorphism $f:inf\left(A\right)\to inf\left(M\right)$. Since $inf$ is full, we find $g:A\to M$ with $inf\left(g\right)=f$. The faithfulness of $inf$ implies that g is an irreducible monomorphism $g:A\to M$. By the same token, there exists an irreducible epimorphism $M\to B$. As all irreducible morphisms in $\mathcal{𝒫}$ are injective and all irreducible morphisms in $\mathcal{ℐ}$ are surjective, M is located in a $ℤ{A}_{\mathrm{\infty }}$ component. It follows that $\tau B=A$ in $\mathrm{mod}{\mathcal{𝒦}}_{r}$. Let $\underset{¯}{\mathrm{dim}}B=\left(c,d\right)$; then the Coxeter matrices for ${\mathcal{𝒦}}_{r}$ and ${\mathcal{𝒦}}_{s}$ yield

$\left(\left({r}^{2}-1\right)c-rd,rc-d\right)=\underset{¯}{\mathrm{dim}}\tau B=\underset{¯}{\mathrm{dim}}A=\underset{¯}{\mathrm{dim}}inf\left(A\right)=\underset{¯}{\mathrm{dim}}\tau inf\left(B\right)=\left(\left({s}^{2}-1\right)c-sd,sc-d\right).$

This is a contradiction since $s\ne r$.

(b) Denote by $\left\{{e}_{1},\mathrm{\dots },{e}_{s}\right\}$ the canonical basis of ${k}^{s}$. Let $1\le m\le s-r$, and set $U:-{〈{e}_{r+1},\mathrm{\dots },{e}_{r+m}〉}_{k}$. Then ${\mathrm{Soc}}_{U}\left(M\right)={\bigcap }_{i=1}^{m}\mathrm{ker}\left({x}_{{e}_{r+i}}^{M}\right)=M$. Let $j\in \left\{1,\mathrm{\dots },r\right\}$ such that ${\gamma }_{j}$ acts non-trivially on M. Let $V\in {\mathrm{Gr}}_{m,s}$ such that ${e}_{j}\in V$. Then ${\mathrm{Soc}}_{V}\left(inf\left(M\right)\right)\ne M$, and M does not have constant m-socle rank. ∎

#### Proposition 3.2.3.

Let $\mathrm{2}\mathrm{\le }r\mathrm{<}s\mathrm{\in }\mathrm{N}$ and $\mathrm{1}\mathrm{\le }d\mathrm{<}r$, and let M be an indecomposable and non-simple ${\mathcal{K}}_{r}$-module. Then the following statements hold.

• (a)

If $M\in {}^{\perp }𝔛_{d,r}$ , then $inf\left(M\right)\in {}^{\perp }𝔛_{d+s-r,s}$.

• (b)

If $M\in {𝔛}_{d,r}^{\perp }$ , then $inf\left(M\right)\in {𝔛}_{d+s-r,s}^{\perp }$.

• (c)

If $M\in {\overline{𝔛}}_{d,r}$ , then $inf\left(M\right)$ is contained in a regular component $\mathcal{𝒞}$ with $\mathcal{𝒞}\subseteq {\mathrm{CSR}}_{d+s-r}$.

#### Proof.

By definition, it is $1\le d+s-r. Now fix $V\in {\mathrm{Gr}}_{d+s-r,s}$, and note that

$\underset{¯}{\mathrm{dim}}{X}_{V}=\left(1,s-\left(d+s-r\right)\right)=\left(1,r-d\right),$

which is the dimension vector of every ${\mathcal{𝒦}}_{r}$-module ${X}_{U}$ for $U\in {\mathrm{Gr}}_{d,r}$.

(a) Assume that $\mathrm{Hom}\left(inf\left(M\right),{X}_{V}\right)\ne 0$, and let $0\ne f:inf\left(M\right)\to {X}_{V}$. By Lemmata 3.2.1 and 2.1.4, the ${\mathcal{𝒦}}_{s}$-module $inf\left(M\right)$ is regular and every proper submodule of ${X}_{V}$ is preprojective. Hence f is surjective onto ${X}_{V}$. Again, Lemma 3.2.1 yields $Z\in \mathrm{mod}{\mathcal{𝒦}}_{r}$ indecomposable with $\underset{¯}{\mathrm{dim}}Z=\left(1,r-d\right)=\underset{¯}{\mathrm{dim}}{X}_{V}$ such that ${X}_{V}=inf\left(Z\right)$. By Proposition 2.2.2, there exists $U\in {\mathrm{Gr}}_{d,r}$ with $Z={X}_{U}$. Since $inf$ is fully faithful, it follows $0=\mathrm{Hom}\left(M,{X}_{U}\right)\cong \mathrm{Hom}\left(inf\left(M\right),inf\left({X}_{U}\right)\right)=\mathrm{Hom}\left(inf\left(M\right),{X}_{V}\right)\ne 0$, a contradiction.

(b) Assume that $\mathrm{Hom}\left({X}_{V},inf\left(M\right)\right)\ne 0$, and let $f:{X}_{V}\to inf\left(M\right)$ be non-zero. Since $inf\left(M\right)$ is regular indecomposable, the module $\mathrm{im}f\subseteq inf\left(M\right)$ is not injective, and Lemma 2.2.3 yields that $\mathrm{im}f$ is indecomposable and regular. As $\mathrm{im}f$ is a submodule of $inf\left(M\right)$, there exists an indecomposable module $Z\in \mathrm{mod}{\mathcal{𝒦}}_{r}$ with $inf\left(Z\right)=\mathrm{im}f$. Since $\mathrm{im}f$ is not simple, we have $\underset{¯}{\mathrm{dim}}\mathrm{im}f=\left(1,r-c\right)$ for $1\le r-c\le r-d$. Hence $Z={X}_{U}$ for $U\in {\mathrm{Gr}}_{c,r}$, and by Proposition 2.2.2 (d), there exists $W\in {\mathrm{Gr}}_{d,r}$ and an epimorphism $\pi :{X}_{W}\to {X}_{U}$. We conclude with $0\ne \mathrm{Hom}\left(\mathrm{im}f,inf\left(M\right)\right)=\mathrm{Hom}\left(inf\left({X}_{U}\right),inf\left(M\right)\right)\cong \mathrm{Hom}\left({X}_{U},M\right)$ and the surjectivity of $\pi :{X}_{W}\to {X}_{U}$ that $\mathrm{Hom}\left({X}_{W},M\right)\ne 0$, a contradiction to the assumption.

(c) By Lemma 3.2.2, the module $inf\left(M\right)$ 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 $F=\left({k}^{2},{k}^{2},F\left({\gamma }_{1}\right),F\left({\gamma }_{2}\right),F\left({\gamma }_{3}\right)\right)$ with the linear maps $F\left({\gamma }_{1}\right)=i{d}_{{k}^{2}}$, $F\left({\gamma }_{2}\right)\left(a,b\right)=\left(b,0\right)$ and $F\left({\gamma }_{3}\right)\left(a,b\right)=\left(0,a\right)$ is elementary. Let E be the corresponding ${\mathcal{𝒦}}_{3}$-module. Then $\underset{¯}{\mathrm{dim}}E=\left(2,2\right)$, and it is easy to see that every indecomposable submodule of E has dimension vector $\left(0,1\right)$ or $\left(1,2\right)$. In particular, $\mathrm{Hom}\left(W,E\right)=0$ for each indecomposable module with dimension vector $\underset{¯}{\mathrm{dim}}W=\left(1,1\right)$. Assume now that $f:E\to W$ is non-zero; then f is surjective since every proper submodule of W is projective. Since E is elementary, $\mathrm{ker}f$ is a preprojective module with dimension vector $\left(1,1\right)$, a contradiction. Hence $E\in {\overline{𝔛}}_{2,3}$.

• (2)

Recall that ${\mathrm{ESP}}_{1}=\mathrm{EKP}$ and ${\mathrm{ERP}}_{1}=\mathrm{EIP}$. Given a regular component $\mathcal{𝒞}$, there are unique quasi-simple modules ${M}_{\mathcal{𝒞}}$ and ${W}_{\mathcal{𝒞}}$ in $\mathcal{𝒞}$ such that $\mathrm{EIP}\cap \mathcal{𝒞}=\left(\to {W}_{\mathcal{𝒞}}\right)$ and $\mathrm{EKP}\cap \mathcal{𝒞}=\left({M}_{\mathcal{𝒞}}\to \right)$. The width $\mathcal{𝒲}\left(\mathcal{𝒞}\right)\in ℤ$ is defined [21, Theorem 3.3] as the unique integer satisfying ${\tau }^{\mathcal{𝒲}\left(\mathcal{𝒞}\right)+1}{M}_{\mathcal{𝒞}}={W}_{\mathcal{𝒞}}$. In fact, it is shown that $\mathcal{𝒲}\left(\mathcal{𝒞}\right)\in {ℕ}_{0}$, and an example of a regular component $\mathcal{𝒞}$ with $\mathcal{𝒲}\left(\mathcal{𝒞}\right)=0$ and $\mathrm{End}\left({M}_{\mathcal{𝒞}}\right)=k$ is given. Since ${X}_{U}\cong \delta \tau {X}_{U}$ for $U\in {\mathrm{Gr}}_{1,r}$ (see [21, Theorem 3.1]), we conclude for an arbitrary regular component $\mathcal{𝒞}$ that

$\begin{array}{cc}\hfill \mathcal{𝒲}\left(\mathcal{𝒞}\right)=0& ⇔\tau {M}_{\mathcal{𝒞}}={W}_{\mathcal{𝒞}}\hfill \\ & ⇔{M}_{\mathcal{𝒞}}\in \mathrm{EKP}\text{and}\tau {M}_{\mathcal{𝒞}}\in \mathrm{EIP}\hfill \\ & ⇔\mathrm{Hom}\left({X}_{U},{M}_{\mathcal{𝒞}}\right)=0=\mathrm{Ext}\left({X}_{U},\tau {M}_{\mathcal{𝒞}}\right)\text{for all}U\in {\mathrm{Gr}}_{1,r} \text{by Propositions 2.3.3 and 2.3.4}\hfill \\ & ⇔\mathrm{Hom}\left({X}_{U},{M}_{\mathcal{𝒞}}\right)=0=\mathrm{Hom}\left({M}_{\mathcal{𝒞}},{X}_{U}\right)\text{for all}U\in {\mathrm{Gr}}_{1,r}\hfill \\ & ⇔{M}_{\mathcal{𝒞}}\in {\overline{𝔛}}_{1,r}.\hfill \end{array}$

#### Lemma 3.2.4.

Let $s\mathrm{\ge }\mathrm{3}$ and $\mathrm{2}\mathrm{\le }d\mathrm{<}s$. Then there exists a regular module ${E}_{d}$ with the following properties.

• (a)

${E}_{d}$ is a (quasi-simple) brick in $\mathrm{mod}{\mathcal{𝒦}}_{s}$.

• (b)

${E}_{d}\in {\overline{𝔛}}_{d,s}$.

• (c)

There exist $V,W\in {\mathrm{Gr}}_{1,s}$ with $\mathrm{Hom}\left({X}_{V},{E}_{d}\right)=0\ne \mathrm{Hom}\left({X}_{W},{E}_{d}\right)$.

#### Proof.

We start by considering $s=3$ and $d=2$. Pick the elementary module ${E}_{d}:-E$ from the preceding example. E is a brick, and $E\in {\overline{𝔛}}_{d,s}$. Set $\alpha :-\left(1,0,0\right)$, $\beta :-\left(0,1,0\right)\in {k}^{3}$ and $V:-{〈\alpha 〉}_{k}$, $W:-{〈\beta 〉}_{k}$. By the definition of E, we have

${dim}_{k}\mathrm{ker}{x}_{\alpha }^{E}=2\ne 3={dim}_{k}\mathrm{ker}{x}_{\beta }^{E},$

and therefore

${dim}_{k}\mathrm{Hom}\left({X}_{V},{E}_{d}\right)=0\ne 1={dim}_{k}\mathrm{Hom}\left({X}_{W},{E}_{d}\right).$

Now let $s>3$. If $d=s-1$, consider ${E}_{d}:-{inf}_{3}^{s}\left(E\right)$. In view of Proposition 3.2.3, we have ${E}_{d}\in {\overline{𝔛}}_{2+s-3,s}={\overline{𝔛}}_{d,s}$ Moreover, $inf\left(E\right)$ is a brick in $\mathrm{mod}{\mathcal{𝒦}}_{s}$ and for the canonical basis vectors ${e}_{1},{e}_{2}\in {k}^{s}$ and $V={〈{e}_{1}〉}_{k}$, $W:-{〈{e}_{2}〉}_{k}$ we get as before

${dim}_{k}\mathrm{Hom}\left({X}_{V},inf\left(E\right)\right)=0\ne 1=\mathrm{Hom}\left({X}_{W},inf\left(E\right)\right).$

Now let $1. Set $r:-1+s-d\ge 3$, consider a regular component for ${\mathcal{𝒦}}_{r}$ with $\mathcal{𝒲}\left(\mathcal{𝒞}\right)=0$ such that ${M}_{\mathcal{𝒞}}$ is a brick and set $M:-{M}_{\mathcal{𝒞}}$. Then $M\in {\overline{𝔛}}_{1,r}$, and Proposition 3.2.3 yields ${E}_{d}:-inf\left(M\right)\in {\overline{𝔛}}_{1+s-\left(1+s-d\right),s}={\overline{𝔛}}_{d,s}$. Since M is a brick, $inf\left(M\right)$ is a brick in $\mathrm{mod}{\mathcal{𝒦}}_{s}$. Recall that $\mathrm{Hom}\left({X}_{U},M\right)=0$ for all $U\in {\mathrm{Gr}}_{d,r}$ implies that, viewing M as a representation, the linear map $M\left({\gamma }_{1}\right):{M}_{1}\to {M}_{2}$ corresponding to ${\gamma }_{1}$ is injective. Since the map is not affected by $inf$, $inf\left(M\right)\left({\gamma }_{1}\right):{M}_{1}\to {M}_{2}$ is also injective. Therefore, we conclude for the first basis vector ${e}_{1}\in {k}^{s}$ and $V:-{〈{e}_{1}〉}_{k}$ that $0=\mathrm{Hom}\left({X}_{V},inf\left(M\right)\right)$. By Lemma 3.2.2, we find $W\in {\mathrm{Gr}}_{1,s}$ with $0\ne \mathrm{Hom}\left({X}_{W},inf\left(M\right)\right)$. ∎

## 3.3 Numerous components lying in ${\mathrm{CSR}}_{d}$

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 ${\mathrm{CSR}}_{d}$. By the next result, it follows that $\mathcal{𝒳}\subseteq {\overline{𝔛}}_{d,r}$ implies $\mathcal{ℰ}\left(\mathcal{𝒳}\right)\subseteq {\overline{𝔛}}_{d,r}$.

#### Lemma 3.3.1 ([13, Lemma 1.9]).

Let $X\mathrm{,}Y$ be modules with $\mathrm{Hom}\mathit{}\mathrm{\left(}X\mathrm{,}Y\mathrm{\right)}$ non-zero. If X and Y have filtrations

$X={X}_{0}\supset {X}_{1}\supset \mathrm{\cdots }\supset {X}_{r}\supset {X}_{r+1}=0,Y={Y}_{0}\supset {Y}_{1}\supset \mathrm{\cdots }\supset {Y}_{s}\supset {Y}_{s+1}=0,$

then there are $i\mathrm{,}j$ with $\mathrm{Hom}\mathit{}\mathrm{\left(}{X}_{i}\mathrm{/}{X}_{i\mathrm{+}\mathrm{1}}\mathrm{,}{Y}_{j}\mathrm{/}{Y}_{j\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{\ne }\mathrm{0}$.

For a regular module $M\in {\mathcal{𝒦}}_{r}$, denote by ${\mathcal{𝒞}}_{M}$ the regular component that contains M.

#### Proposition 3.3.2.

Let $\mathrm{1}\mathrm{\le }d\mathrm{<}r$, and let $\mathcal{X}$ be a family of pairwise orthogonal bricks in ${\overline{\mathrm{X}}}_{d\mathrm{,}r}$. Then

$\phi :\mathrm{ind}\mathcal{ℰ}\left(\mathcal{𝒳}\right)\to \mathcal{ℛ},M↦{\mathcal{𝒞}}_{M}$

is an injective map such that, for each component $\mathcal{C}$ in $\mathrm{im}\mathit{}\phi$, we have $\mathcal{C}\mathrm{\subseteq }{\mathrm{CSR}}_{d}$. Here $\mathrm{ind}\mathit{}\mathcal{E}\mathit{}\mathrm{\left(}\mathcal{X}\mathrm{\right)}$ denotes the category of a chosen set of representatives of non-isomorphic indecomposable objects of $\mathrm{mod}\mathit{}{\mathcal{K}}_{r}$ in $\mathcal{E}$.

#### Proof.

Since each module in ${\overline{𝔛}}_{d,r}$ is regular, Proposition 3.1.1 implies that every module $N\in \mathrm{ind}\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is contained in a regular component ${\mathcal{𝒞}}_{M}$ and is quasi-simple. By Lemma 3.3.1, the module N satisfies $\mathrm{Hom}\left({X}_{U},N\right)=0=\mathrm{Hom}\left(N,{X}_{U}\right)$ for all $U\in {\mathrm{Gr}}_{d,r}$. But now Lemma 2.3.8 implies that every module in ${\mathcal{𝒞}}_{M}$ has constant d-socle rank. The injectivity of φ follows immediately from Proposition 3.1.1. ∎

#### Corollary 3.3.3.

There exists an infinite set Ω of regular components such that, for all $\mathcal{C}\mathrm{\in }\mathrm{\Omega }$,

• (a)

$\mathcal{𝒲}\left(\mathcal{𝒞}\right)=0$ , in particular, every module in $\mathcal{𝒞}$ has constant rank,

• (b)

$\mathcal{𝒞}$ does not contain any bricks.

#### Proof.

Let $\mathcal{𝒞}$ be a regular component that contains a brick and $\mathcal{𝒲}\left(\mathcal{𝒞}\right)=0$ (such a component exists by the example above). Let $M:-{M}_{\mathcal{𝒞}}$; then $M\in {\overline{𝔛}}_{1,r}$. Apply Proposition 3.3.2 with $\mathcal{𝒳}=\left\{M\right\}$, and set $\mathrm{\Omega }:-\mathrm{im}\phi \setminus \left\{{\mathcal{𝒞}}_{M}\right\}$. Let $N\in \mathcal{ℰ}\left(\mathcal{𝒳}\right)\setminus \left\{M\right\}$ be indecomposable. N is quasi-simple in ${\mathcal{𝒞}}_{N}$ and has a $\left\{M\right\}$-filtration $0={N}_{0}\subset \mathrm{\cdots }\subset {N}_{l}=N$ with $l\ge 2$ and ${N}_{1}=M={N}_{l}/{N}_{l-1}$. Hence $N\to {N}_{l}/{N}_{l-1}\to {N}_{1}\to N$ is a non-zero homomorphism that is not injective. Therefore, N is not a brick. This finishes the proof since every regular brick in $\mathrm{mod}{\mathcal{𝒦}}_{r}$ is quasi-simple [13, Proposition 9.2] and $\mathrm{End}\left({\tau }^{l}N\right)\cong \mathrm{End}\left(N\right)\ne k$ for all $l\in ℤ$. ∎

Now we apply our results on the simplification method to modules ${E}_{d}$ constructed in Lemma 3.2.4.

#### Definition ([6, Proposition 3.6]).

Denote with ${\mathrm{GL}}_{r}$ the group of invertible $r×r$-matrices which acts on ${\oplus }_{i=1}^{r}k{\gamma }_{i}$ via $g.{\gamma }_{j}={\sum }_{i=1}^{r}{g}_{ij}{\gamma }_{i}$ for $1\le j\le r$, $g\in {\mathrm{GL}}_{r}$. For $g\in {\mathrm{GL}}_{r}$, let ${\phi }_{g}:{\mathcal{𝒦}}_{r}\to {\mathcal{𝒦}}_{r}$ be the algebra homomorphism with ${\phi }_{g}\left({e}_{1}\right)={e}_{1}$, ${\phi }_{g}\left({e}_{2}\right)={e}_{2}$ and ${\phi }_{g}\left({\gamma }_{i}\right)=g.{\gamma }_{i}$, $1\le i\le r$. For a ${\mathcal{𝒦}}_{r}$-module M, denote the pullback of M along ${\phi }_{g}$ by ${M}^{\left(g\right)}$. The module M is called ${\mathrm{GL}}_{r}$-stable if ${M}^{\left(g\right)}\cong M$ for all $g\in {\mathrm{GL}}_{r}$.

#### Theorem 3.3.4.

Let $\mathrm{2}\mathrm{\le }d\mathrm{<}r$; then there exists a wild full subcategory $\mathcal{E}\mathrm{\subseteq }\mathrm{mod}\mathit{}{\mathcal{K}}_{r}$ and an injection

${\phi }_{d}:\mathrm{ind}\mathcal{ℰ}\to \mathcal{ℛ},M↦{\mathcal{𝒞}}_{M},$

such that, for each component $\mathcal{C}$ in $\mathrm{im}\mathit{}{\phi }_{d}$, we have $\mathcal{C}\mathrm{\subseteq }{\mathrm{CSR}}_{d}$ and no module in $\mathcal{C}$ is ${\mathrm{GL}}_{r}$-stable.

#### Proof.

Fix $2\le d, and let ${E}_{d}$ be as in Lemma 3.2.4 with $V,W\in {\mathrm{Gr}}_{1,r}$ and $\mathrm{Hom}\left({X}_{V},{E}_{d}\right)=0\ne \mathrm{Hom}\left({X}_{W},{E}_{d}\right)$. Set $\mathcal{𝒳}:-\left\{{E}_{d}\right\}$, and let $M\in \mathcal{ℰ}\left(\mathcal{𝒳}\right)$. By Proposition 3.3.2, we get an injective map

${\phi }_{d}:\mathrm{ind}\mathcal{ℰ}\left(\mathcal{𝒳}\right)\to \mathcal{ℛ},M\to {\mathcal{𝒞}}_{M}$

such that each component $\mathcal{𝒞}$ in $\mathrm{im}{\phi }_{d}$ satisfies $\mathcal{𝒞}\subseteq {\mathrm{CSR}}_{d}$.

Moreover, $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is a wild full subcategory of $\mathrm{mod}{\mathcal{𝒦}}_{r}$ by Proposition 3.1.1. Let $M\in \mathcal{ℰ}\left(\mathcal{𝒳}\right)$ be indecomposable. Then M has a filtration $0={Y}_{0}\subset {Y}_{1}\subset \mathrm{\cdots }\subset {Y}_{m}$ with ${Y}_{l}/{Y}_{l-1}={E}_{d}$ for all $1\le l\le m$. By Lemma 3.3.1, we have $0=\mathrm{Hom}\left({X}_{U},M\right)$, and since ${E}_{d}={Y}_{1}\subseteq M$, we conclude $0\ne \mathrm{Hom}\left({X}_{W},M\right)$. This proves that M does not have constant 1-socle rank. Therefore, ${\mathcal{𝒞}}_{M}$ contains a module that is not of constant 1-socle rank. By [6, Proposition 3.6], the module M is not ${\mathrm{GL}}_{r}$-stable. Assume that ${\mathcal{𝒞}}_{M}$ contains an ${\mathrm{GL}}_{r}$-stable module N. Since $g\in G$ acts as an auto-equivalence on $\mathrm{mod}{\mathcal{𝒦}}_{r}$ (see also [9, Lemma 2.2]), we conclude that g sends the Auslander–Reiten sequence $0\to X\to E\to N\to 0$ to the Auslander–Reiten sequence $0\to {X}^{g}\to {E}^{g}\to N\to 0$. Hence ${X}^{g}\cong X$ and ${E}^{g}\cong E$ for all $g\in {\mathrm{GL}}_{r}$, and therefore X and E are ${\mathrm{GL}}_{r}$-stable. If E is not indecomposable, we write $E={E}_{1}\oplus {E}_{2}$ with ${E}_{1},{E}_{2}$ indecomposable such that the quasi-lengths $\mathrm{ql}\left({E}_{1}\right),\mathrm{ql}\left({E}_{2}\right)$ satisfy $\mathrm{ql}\left({E}_{1}\right)=\mathrm{ql}\left({E}_{2}\right)-2$. We get ${dim}_{k}{E}_{2}>{dim}_{k}{E}_{1}$ and therefore ${\left({E}_{2}\right)}^{g}\cong {E}_{2}$ and ${\left({E}_{1}\right)}^{g}\cong {E}_{1}$. Hence every direct summand in the Auslander–Reiten sequence is ${\mathrm{GL}}_{r}$-stable. Now one can easily conclude that every module in ${\mathcal{𝒞}}_{M}$ is ${\mathrm{GL}}_{r}$-stable, a contradiction since M is not ${\mathrm{GL}}_{r}$-stable. ∎

## 3.4 Components lying almost completely in ${\mathrm{CSR}}_{d}$

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 ${\mathcal{𝒦}}_{r}$-module, $1\le d and $U\in {\mathrm{Gr}}_{d,r}$. M is called U-trivial if

${dim}_{k}\mathrm{Hom}\left({X}_{U},M\right)={dim}_{k}{M}_{1}.$

Note that the sequence $0\to {P}_{1}^{r-d}\to {P}_{2}\to {X}_{U}\to 0$ and left-exactness of $\mathrm{Hom}\left(-,M\right)$ imply that

${dim}_{k}\mathrm{Hom}\left({X}_{U},M\right)\le {dim}_{k}{M}_{1}.$

#### Lemma 3.4.1.

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

$\mathrm{Ext}\left({X}_{V},\tau M\right)=0=\mathrm{Hom}\left({X}_{V},{\tau }^{-1}M\right)\mathit{ }\mathit{\text{for all}}V\in {\mathrm{Gr}}_{d,r}.$

#### Proof.

Assume that $\mathrm{Ext}\left({X}_{V},\tau M\right)\ne 0$; then we find an epimorphism $f:M\to {X}_{V}$ and an exact sequence $0\to \mathrm{ker}f\to M\to {X}_{V}\to 0$. Note that ${dim}_{k}\mathrm{Hom}\left({X}_{U},\mathrm{ker}f\right)\le {dim}_{k}{\left(\mathrm{ker}f\right)}_{1}<{dim}_{k}{M}_{1}=\mathrm{Hom}\left({X}_{U},M\right)$. We apply $\mathrm{Hom}\left({X}_{U},-\right)$ and conclude that ${f}_{\ast }:\mathrm{Hom}\left({X}_{U},M\right)\to \mathrm{Hom}\left({X}_{U},{X}_{V}\right),g↦f\circ g$ is non-zero. In particular, we have $0\ne \mathrm{Hom}\left({X}_{U},{X}_{V}\right)$ and therefore $U=V$. Let $h\in \mathrm{Hom}\left({X}_{U},M\right)$ such that $f\circ h\ne 0$. Since ${X}_{U}$ is a brick, we conclude that f is an isomorphism and $M\cong {X}_{U}$ is elementary.

Assume that ${\tau }^{-1}M\notin {\mathrm{ESP}}_{d}\supseteq {\mathrm{ESP}}_{1}$. Consider ${〈\alpha 〉}_{k}=W\in {\mathrm{Gr}}_{1,r}$ together with an epimorphism $p:{X}_{W}\to {X}_{U}$ (see Proposition 2.2.2). We conclude with ${dim}_{k}{M}_{1}\ge {dim}_{k}\mathrm{Hom}\left({X}_{W},M\right)\ge {dim}_{k}\mathrm{Hom}\left({X}_{U},M\right)={dim}_{k}{M}_{1}$ that M is W-trivial. Now the equation ${dim}_{k}\mathrm{Hom}\left({X}_{W},M\right)+{dim}_{k}{M}_{2}={dim}_{k}\mathrm{ker}\left({x}_{\alpha }^{M}\right)$ (which follows from the proof of Proposition 2.3.3) shows that ${x}_{\alpha }$ acts as zero on M. Hence ${x}_{\alpha }$ acts as zero on $\delta M$, and is W-trivial with $\delta {\tau }^{-1}M\notin \delta \left({\mathrm{ESP}}_{1}\right)={\mathrm{ERP}}_{1}$. Hence we find $Z\in {\mathrm{Gr}}_{1,r}$ such that $0\ne \mathrm{Ext}\left(\delta \tau {X}_{Z},\delta {\tau }^{-1}M\right)$. Since $\delta \tau {X}_{Z}\cong {X}_{Z}$ (see [21, Proposition 3.1]) and $\tau \circ \delta =\delta \circ {\tau }^{-1}$, we conclude that $0\ne \mathrm{Ext}\left(\delta \tau {X}_{Z},\delta {\tau }^{-1}M\right)\cong \mathrm{Ext}\left({X}_{Z},\tau \delta M\right)$. Now the above arguments show that $\delta M\cong {X}_{W}$ and therefore $M\cong \delta {X}_{W}\cong \tau {X}_{W}$. ∎

#### Lemma 3.4.2.

Let M be regular quasi-simple in a regular component $\mathcal{C}$ such that

$\mathrm{Ext}\left({X}_{U},\tau M\right)=0=\mathrm{Hom}\left({X}_{U},{\tau }^{-1}M\right)\mathit{ }\mathit{\text{for all}}U\in {\mathrm{Gr}}_{d,r}.$

If M does not have constant d-socle rank, then a module X in $\mathcal{C}$ has constant d-socle rank if and only if X is in $\mathrm{\left(}\mathrm{\to }\tau M\mathrm{\right)}\mathrm{\cup }\mathrm{\left(}{\tau }^{\mathrm{-}\mathrm{1}}M\mathrm{\to }\mathrm{\right)}$.

#### Corollary 3.4.3.

Let $\mathrm{3}\mathrm{\le }r\mathrm{<}s$ and $\mathrm{1}\mathrm{\le }d\mathrm{<}r$, and let $b\mathrm{:-}d\mathrm{+}s\mathrm{-}r$ and $\mathrm{1}\mathrm{\le }l\mathrm{\le }s\mathrm{-}r$. Let M be an indecomposable ${\mathcal{K}}_{r}$-module in ${\overline{\mathrm{X}}}_{d\mathrm{,}r}$ that is not elementary. Denote by $\mathcal{C}$ the regular component that contains $\mathrm{inf}\mathit{}\mathrm{\left(}M\mathrm{\right)}$.

• (a)

Every module in $\mathcal{𝒞}$ has constant b -socle rank.

• (b)

$N\in \mathcal{𝒞}$ has constant l -socle rank if and only if $N\in \left(\to \tau inf\left(M\right)\right)\cup \left({\tau }^{-1}inf\left(M\right)\to \right)$.

#### Proof.

(a) is an immediate consequence of Proposition 3.2.3.

(b) Consider the indecomposable projective module ${P}_{2}={\mathcal{𝒦}}_{r}{e}_{1}$ in $\mathrm{mod}{\mathcal{𝒦}}_{r}$. We get

$\mathrm{Hom}\left(inf\left({P}_{2}\right),inf\left(M\right)\right)\cong \mathrm{Hom}\left({P}_{2},M\right)={M}_{1}=inf{\left(M\right)}_{1}.$

Since

$\underset{¯}{\mathrm{dim}}inf\left({P}_{2}\right)=\left(1,r\right)=\left(1,s-\left(s-r\right)\right),$

we find $W\in {\mathrm{Gr}}_{s-r,s}$ with $inf\left({P}_{2}\right)={X}_{W}$. Now let $1\le l\le s-r$. By Proposition 2.2.2, there is $U\in {\mathrm{Gr}}_{l,s}$ and an epimorphism $\pi :{X}_{U}\to {X}_{W}$. Let $\left\{{f}_{1},\mathrm{\dots },{f}_{q}\right\}$ be a basis of $\mathrm{Hom}\left(inf\left({P}_{2}\right),inf\left(M\right)\right)$. Since π is surjective, the set $\left\{{f}_{1}\pi ,\mathrm{\dots },{f}_{q}\pi \right\}\subseteq \mathrm{Hom}\left({X}_{U},inf\left(M\right)\right)$ is linearly independent. Hence

$q\le {dim}_{k}\mathrm{Hom}\left({X}_{U},inf\left(M\right)\right)\le {dim}_{k}inf{\left(M\right)}_{1}=q$

holds, and $inf\left(M\right)$ is U-trivial.

Since M is not elementary, $inf\left(M\right)$ is not elementary, and therefore Lemma 3.4.1 yields that

$\mathrm{Ext}\left({X}_{W},\tau inf\left(M\right)\right)=0=\mathrm{Hom}\left({X}_{W},{\tau }^{-1}inf\left(M\right)\right)\mathit{ }\text{for all}W\in {\mathrm{Gr}}_{l,s}.$

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

Figure 4

Regular component containing $inf\left({M}_{\mathcal{𝒞}}\right)$.

#### Example.

Let $r\ge 3$, and let $\mathcal{𝒞}$ be a regular component with $\mathcal{𝒲}\left(\mathcal{𝒞}\right)=0$ such that ${M}_{\mathcal{𝒞}}$ is not a brick (see Corollary 3.3.3) and in particular not elementary. Then ${M}_{\mathcal{𝒞}}\in {\overline{𝔛}}_{1,r}$, and we can apply Corollary 3.4.3. Figure 4 shows the regular component $\mathcal{𝒟}$ of ${\mathcal{𝒦}}_{s}$ containing $inf\left({M}_{\mathcal{𝒞}}\right)$. Every module in $\mathcal{𝒟}$ has constant $b:-1+s-r$ socle rank. But for $1\le q\le s-r$, a module in this component has constant q-socle rank if and only if it lies in the shaded region.

## 4.1 Wildness of strata

As another application of the simplification method and the inflation functor ${inf}_{r}^{s}:\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}{\mathcal{𝒦}}_{s}$, we get the following result.

#### Theorem 4.1.1.

Let $s\mathrm{\ge }\mathrm{3}$ and $\mathrm{1}\mathrm{\le }d\mathrm{\le }s\mathrm{-}\mathrm{1}$. Then ${\mathrm{\Delta }}_{d}\mathrm{=}{\mathrm{ESP}}_{d}\mathrm{\setminus }{\mathrm{ESP}}_{d\mathrm{-}\mathrm{1}}\mathrm{\subseteq }\mathrm{mod}\mathit{}{\mathcal{K}}_{s}$ is a wild subcategory, where ${\mathrm{ESP}}_{\mathrm{0}}\mathrm{:-}\mathrm{\varnothing }$.

#### Proof.

For $d=1$, consider a regular component $\mathcal{𝒞}$ for ${\mathcal{𝒦}}_{s}$ that contains a brick F. By Proposition 2.3.5, we find a module E in the τ-orbit of F that is in ${\mathrm{ESP}}_{1}$ and set $\mathcal{𝒳}:-\left\{E\right\}$. Then E is brick since $\mathrm{Hom}\left(E,E\right)\cong \mathrm{Hom}\left(F,F\right)=k$ and ${dim}_{k}\mathrm{Ext}\left(E,E\right)\ge 2$ by Corollary 1.2. Therefore, $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is wild category (see Proposition 3.1.1). As ${\mathrm{ESP}}_{1}$ is closed under extensions, it follows $\mathcal{ℰ}\left(\mathcal{𝒳}\right)\subseteq {\mathrm{ESP}}_{1}$. Note that this case does only require the application of Proposition 3.1.1.

Now let $d>1$ and $r:-s-d+1\ge 2$. Consider the projective indecomposable ${\mathcal{𝒦}}_{r}$-module $P:-{P}_{2}$ with $\underset{¯}{\mathrm{dim}}P=\left(1,r\right)$. By Lemma 3.2.2, $inf\left(P\right)$ is a regular quasi-simple module in $\mathrm{mod}{\mathcal{𝒦}}_{s}$ with

${dim}_{k}\mathrm{Ext}\left(inf\left(P\right),inf\left(P\right)\right)\ge 2.$

Since P is in ${\mathrm{ESP}}_{1}$, we have $0=\mathrm{Hom}\left({X}_{U},P\right)$ for all $U\in {\mathrm{Gr}}_{1,r}$. Hence Proposition 3.2.3 implies

$0=\mathrm{Hom}\left({X}_{U},inf\left(P\right)\right)\mathit{ }\text{for all}U\in {\mathrm{Gr}}_{1+s-r}={\mathrm{Gr}}_{d,s}$

so that $inf\left(P\right)$ is in ${\mathrm{ESP}}_{d}$.

Let $\mathcal{𝒳}:-\left\{inf\left(P\right)\right\}$; then $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is a wild category, and since ${\mathrm{ESP}}_{d}$ is extension closed, it follows $\mathcal{ℰ}\left(\mathcal{𝒳}\right)\subseteq {\mathrm{ESP}}_{d}$. Since $\underset{¯}{\mathrm{dim}}inf\left(P\right)=\left(1,s-d+1\right)$, we find $V\in {\mathrm{Gr}}_{d-1,s}$ (see Proposition 2.2.2) with

$inf\left(P\right)={X}_{V}\mathit{ }\text{and}\mathit{ }0\ne \mathrm{End}\left(inf\left(P\right)\right)=\mathrm{Hom}\left({X}_{V},inf\left(P\right)\right).$

That means $inf\left(P\right)\notin {\mathrm{ESP}}_{d-1}$. Since ${\mathrm{ESP}}_{d-1}$ is closed under submodules, we have $\mathcal{ℰ}\left(\mathcal{𝒳}\right)\cap {\mathrm{ESP}}_{d-1}=\mathrm{\varnothing }$. Hence $\mathcal{ℰ}\left(\mathcal{𝒳}\right)\subseteq {\mathrm{ESP}}_{d}\setminus {\mathrm{ESP}}_{d-1}$. ∎

#### Remarks.

Let us collect the following observations.

• (i)

Note that all indecomposable modules in the wild category $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ are quasi-simple in $\mathrm{mod}{\mathcal{𝒦}}_{s}$ and $\mathcal{ℰ}\left(\mathcal{𝒳}\right)\subseteq {\mathrm{ESP}}_{d}\setminus {\mathrm{ESP}}_{d-1}$.

• (ii)

For $1\le d, we define ${\mathrm{EKP}}_{d}:-\left\{M\in \mathrm{mod}{\mathcal{𝒦}}_{r}\mid \mathrm{Hom}\left(\delta \tau {X}_{U},M\right)=0\text{for all}U\in {\mathrm{Gr}}_{d,r}\right\}$. One can show that $M\in {\mathrm{EKP}}_{d}$ if and only if $y_{T}^{M}{}_{|{M}_{1}^{d}}:{M}_{1}^{d}\to M$ is injective for all linearly independent tuples T in ${\left({k}^{r}\right)}^{d}$. From the definitions, we get a chain of proper inclusions ${\mathrm{ESP}}_{r-1}\supset {\mathrm{ESP}}_{r-2}\supset \mathrm{\cdots }\supset {\mathrm{ESP}}_{1}={\mathrm{EKP}}_{1}\supset {\mathrm{EKP}}_{2}\supset \mathrm{\cdots }\supset {\mathrm{EKP}}_{r-1}$. By adapting the preceding proof, it follows that ${\mathrm{EKP}}_{r-1}$ is wild. Moreover, it can be shown that, for each regular component $\mathcal{𝒞}$, the set $\left({\mathrm{EKP}}_{1}\setminus {\mathrm{EKP}}_{r-1}\right)\cap \mathcal{𝒞}$ is empty or forms a ray.

We will use the following result later on to prove the wildness of the subcategory in $\mathrm{mod}k{E}_{2}$ consisting of modules of Loewy length 3 and the equal kernels property. We denote by $B\left(3,2\right)$ the Beilinson algebra with 3 vertices and 2 arrows.

#### Proposition 4.1.2.

Let $\mathrm{EKP}\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{,}\mathrm{2}\mathrm{\right)}$ be the full subcategory of modules in $\mathrm{mod}\mathit{}B\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{,}\mathrm{2}\mathrm{\right)}$ with the equal kernels property (see [21, Definition 2.1, Theorem 2.5]). The category $\mathrm{EKP}\mathit{}\mathrm{\left(}\mathrm{3}\mathrm{,}\mathrm{2}\mathrm{\right)}$ is of wild representation type.

#### Proof.

Consider the path algebra A of the extended Kronecker quiver $Q=1\to 2⇉3$. 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 $\mathrm{mod}A$ with $\mathrm{End}\left(T\right)\cong B\left(3,2\right)$; 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 $T=P\left(1\right)\oplus {\tau }^{-2}P\left(1\right)\oplus {\tau }^{-2}P\left(3\right)$ is a tilting module. Since preprojective components are standard [19, Proposition 2.4.11], one can show that $\mathrm{End}\left(T\right)$ is given by the quiver in Figure 6, bound by the relation ${\alpha }_{2}{\alpha }_{1}+{\beta }_{2}{\beta }_{1}$. Moreover, it follows from the description as a quiver with relations that $\mathrm{End}\left(T\right)\cong B\left(3,2\right)$.

Figure 6

Ordinary quiver of $\mathrm{End}\left(T\right)$.

Since A is hereditary, it follows that the algebra $B\left(3,2\right)$ is a concealed algebra [1, Definition 4.6]. By [2, Theorem XVIII.5.1], the functor $\mathrm{Hom}\left(T,-\right):\mathrm{mod}A\to \mathrm{mod}B\left(3,2\right)$ induces an equivalence G between the regular categories $\mathrm{add}\mathcal{ℛ}\left(A\right)$ and $\mathrm{add}\mathcal{ℛ}\left(B\left(3,2\right)\right)$, and we have an isomorphism between the two Grothendieck groups $f:{K}_{0}\left(A\right)\to {K}_{0}\left(B\left(3,2\right)\right)$ with $\underset{¯}{\mathrm{dim}}G\left(M\right)=\underset{¯}{\mathrm{dim}}\mathrm{Hom}\left(T,M\right)=f\left(\underset{¯}{\mathrm{dim}}M\right)$ for all $M\in \mathrm{mod}A$. Now we make use of a homological characterization of the class $\mathrm{EKP}\left(3,2\right)$ given in [21, Theorem 2.5]: for each $\alpha \in {k}^{2}\setminus \left\{0\right\}$, there exist certain indecomposable $B\left(3,2\right)$-modules ${X}_{\alpha }^{0,1}$, ${X}_{\alpha }^{1,1}$ such that

$\mathrm{EKP}\left(3,2\right)=\left\{M\in \mathrm{mod}B\left(3,2\right)\mid \mathrm{Hom}\left({X}_{\alpha }^{0,1}\oplus {X}_{\alpha }^{1,1},M\right)=0\text{for all}\alpha \in {k}^{2}\setminus \left\{0\right\}\right\}.$

The modules ${X}_{\alpha }^{0,1}$, ${X}_{\alpha }^{1,1}$ arise as cokernels of embeddings similar to the embeddings studied in Section 2.2. We do not need the exact definition of ${X}_{\alpha }^{1}$. Let us show that each ${X}_{\alpha }^{1}$ is regular. Clearly, $0\ne {\mathrm{Hom}}_{B\left(3,2\right)}\left({X}_{\alpha }^{1},Z\right)$ for $Z\in \left\{{X}_{\alpha }^{0,1},{X}_{\alpha }^{1,1}\right\}$, so Z is not in $\mathrm{EKP}\left(3,2\right)$. Moreover, the equality (see [22, Proposition 3.14])

${\tau }_{B\left(3,2\right)}{X}_{\alpha }^{1,1}\cong D{X}_{\alpha }^{3-1-1-1,1}=D{X}_{\alpha }^{0,1}$

holds (D denotes a certain duality on $\mathrm{mod}B\left(3,2\right)$). Since ${X}_{\alpha }^{0,1}$ is not in $\mathrm{EKP}\left(3,2\right)$, we conclude that ${\tau }_{B\left(3,2\right)}{X}_{\alpha }^{1,1}$ is not in $\mathrm{EIP}\left(3,2\right)$. Since $\mathrm{EIP}\left(3,2\right)$ is closed under ${\tau }_{B\left(3,2\right)}$, we conclude that ${X}_{\alpha }^{1,1}$ is not in $\mathrm{EIP}\left(3,2\right)$. The assumption ${X}_{\alpha }^{0,1}\in \mathrm{EIP}\left(3,2\right)$ yields that ${\tau }_{B\left(3,2\right)}{X}_{\alpha }^{1,1}$ is in $\mathrm{EKP}\left(3,2\right)$. Since $\mathrm{EKP}\left(3,2\right)$ is closed under ${\tau }_{B\left(3,2\right)}^{-1}$, we get that ${X}_{\alpha }^{1,1}$ is in $\mathrm{EKP}\left(3,2\right)$, a contradiction. Therefore, ${X}_{\alpha }^{0,1},{X}_{\alpha }^{1,1}$ are not in $\mathrm{EIP}\left(3,2\right)\cup \mathrm{EKP}\left(3,2\right)$ and by [21, Corollary 2.7] regular, so ${X}_{\alpha }^{1}$ is a regular module as well.

Moreover, $\underset{¯}{\mathrm{dim}}{X}_{\alpha }^{1}$ is independent of α. Hence we find for each $\beta \in {k}^{r}\setminus \left\{0\right\}$ a regular indecomposable module ${U}_{\beta }$ in $\mathrm{mod}A$ with $G\left({U}_{\beta }\right)={X}_{\beta }^{1}$ and $\underset{¯}{\mathrm{dim}}{U}_{\beta }=\underset{¯}{\mathrm{dim}}{U}_{\alpha }$ for all $\alpha \in {k}^{r}\setminus \left\{0\right\}$. Now let M be in $\mathrm{mod}A$ a regular brick with ${dim}_{k}\mathrm{Ext}\left(M,M\right)\ge 2$ (see [15, Proposition 5.1]). By the dual version of [13, Lemma 4.6], we find $l\in {ℕ}_{0}$ with $\mathrm{Hom}\left({U}_{\alpha },{\tau }^{-l}M\right)=0$ for all $\alpha \in {k}^{r}\setminus \left\{0\right\}$. Set $N:-{\tau }^{-l}M$ and $\mathcal{𝒳}:-\left\{N\right\}$. N is a regular brick with ${dim}_{k}\mathrm{Ext}\left(N,N\right)={dim}_{k}\mathrm{Ext}\left(M,M\right)\ge 2$, and therefore $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ is a wild category in $\mathrm{add}\mathcal{ℛ}\left(A\right)$ (see [15, Proposition 1.4]). By Lemma 3.3.1, we have $\mathrm{Hom}\left({U}_{\alpha },L\right)=0$ for all L in $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ and all $\alpha \in {k}^{r}\setminus \left\{0\right\}$. Hence $0=\mathrm{Hom}\left({U}_{\alpha },L\right)=\mathrm{Hom}\left(G\left({U}_{\alpha }\right),G\left(L\right)\right)=\mathrm{Hom}\left({X}_{\alpha }^{1},L\right)$ for all $\alpha \in {k}^{r}\setminus \left\{0\right\}$. This shows that the essential image of $\mathcal{ℰ}\left(\mathcal{𝒳}\right)$ under G is a wild subcategory contained in $\mathrm{EKP}\left(3,2\right)$. ∎

## 4.2 The module category of ${E}_{r}$

Throughout this section, we assume that $\mathrm{char}\left(k\right)=p>0$ and $r\ge 2$. Moreover, let ${E}_{r}$ be a p-elementary abelian group of rank r with generating set $\left\{{g}_{1},\mathrm{\dots },{g}_{r}\right\}$. For ${x}_{i}:-{g}_{i}-1$, we get an isomorphism

$k{E}_{r}\cong k\left[{X}_{1},\mathrm{\dots },{X}_{r}\right]/\left({X}_{1}^{p},\mathrm{\dots },{X}_{r}^{p}\right)$

of k-algebras by sending ${X}_{i}$ to ${x}_{i}$ for all i. We recall the definition of the functor $𝔉:\mathrm{mod}{\mathcal{𝒦}}_{r}\to \mathrm{mod}k{E}_{r}$ introduced in [21]. Given a module M, $𝔉\left(M\right)$ is by definition the vector space M and

${x}_{i}.m:-{\gamma }_{i}\cdot m={\gamma }_{i}\cdot {m}_{1}+{\gamma }_{i}\cdot {m}_{2}={\gamma }_{i}\cdot {m}_{1},$

where ${m}_{i}={e}_{i}\cdot m$. Moreover, $𝔉$ is the identity map on morphisms, that is, $𝔉\left(f\right):𝔉\left(M\right)\to 𝔉\left(N\right)$, $𝔉\left(f\right)\left(m\right)=f\left(m\right)$ for all $f:M\to N$.

#### Definition ([6, Definition 2.1]).

Let $𝕍:-{〈{x}_{1},\mathrm{\dots },{x}_{r}〉}_{k}\subseteq \mathrm{rad}\left(k{E}_{r}\right)$. For U in ${\mathrm{Gr}}_{d,𝕍}$ with basis ${u}_{1},\mathrm{\dots },{u}_{d}$ and a $k{E}_{r}$-module M, we set

${\mathrm{Rad}}_{U}\left(M\right):-\sum _{u\in U}u\cdot M=\sum _{i=1}^{d}{u}_{i}\cdot M,$${\mathrm{Soc}}_{U}\left(M\right):-\left\{m\in M\mid u\cdot m=0\text{for all}u\in U\right\}=\bigcap _{i=1}^{d}\left\{m\in M\mid {u}_{i}\cdot m=0\right\}.$

#### Definition ([6, Definition 3.1]).

Let $M\in \mathrm{mod}k{E}_{r}$ and $1\le d.

• (a)

M has constant d-$\mathrm{Rad}$ rank ( d-$\mathrm{Soc}$ rank, respectively) if the dimension of ${\mathrm{Rad}}_{U}\left(M\right)$ (${\mathrm{Soc}}_{U}\left(M\right)$, respectively) is independent of the choice of $U\in {\mathrm{Gr}}_{d,𝕍}$.

• (b)

M has the equal d-$\mathrm{Rad}$ property ( d-$\mathrm{Soc}$ property, respectively) if ${\mathrm{Rad}}_{U}\left(M\right)$ (${\mathrm{Soc}}_{U}\left(M\right)$, respectively) is independent of the choice of $U\in {\mathrm{Gr}}_{d,𝕍}$.

#### Proposition 4.2.1.

Let M be a non-simple indecomposable ${\mathcal{K}}_{r}$-module, and let $\mathrm{1}\mathrm{\le }d\mathrm{<}r$.

• (a)

M is in ${\mathrm{CSR}}_{d}$ if and only if $𝔉\left(M\right)$ has constant d - $\mathrm{Soc}$ rank.

• (b)

M is in ${\mathrm{ESP}}_{d}$ if and only if $𝔉\left(M\right)$ has the equal d - $\mathrm{Soc}$ property.

#### Proof.

We fix $𝕍:-{〈{x}_{1},\mathrm{\dots },{x}_{r}〉}_{k}\subseteq \mathrm{rad}\left(k{E}_{r}\right)$. In the following, we denote for $u\in \mathrm{rad}\left(k{E}_{r}\right)$ with $l\left(u\right):M\to M$ the induced linear map on M. Let $U\in {\mathrm{Gr}}_{d,𝕍}$ with basis $\left\{{u}_{1},\mathrm{\dots },{u}_{d}\right\}$, write ${u}_{j}={\sum }_{i=1}^{r}{\alpha }_{j}^{i}{x}_{i}$ for all $1\le j\le d$, and set ${\alpha }_{j}=\left({\alpha }_{j}^{1},\mathrm{\dots },{\alpha }_{j}^{r}\right)$. Then $T:-\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{d}\right)$ is linearly independent, and

$\mathrm{ker}\left(l\left({u}_{j}\right)\right)=\mathrm{ker}\left(\sum _{i=1}^{r}{\alpha }_{j}^{i}l\left({x}_{i}\right)\right)=\mathrm{ker}\left(\sum _{i=1}^{r}{\alpha }_{j}^{i}{\gamma }_{i}\right)=\mathrm{ker}\left({x}_{{\alpha }_{j}}^{M}\right).$

It follows

${\mathrm{Soc}}_{U}\left(𝔉\left(M\right)\right)=\bigcap _{i=1}^{d}\mathrm{ker}\left(l\left({u}_{i}\right)\right)=\bigcap _{i=1}^{d}\mathrm{ker}\left({x}_{{\alpha }_{i}}^{M}\right)={\mathrm{Soc}}_{〈T〉}\left(M\right).$

Hence $M\in {\mathrm{CSR}}_{d}$ implies that $𝔉\left(M\right)$ has constant d-$\mathrm{Soc}$ rank.

Now assume that $T=\left({\alpha }_{1},\mathrm{\dots },{\alpha }_{d}\right)$ is linearly independent, and set ${u}_{j}:-{\sum }_{i=1}^{r}{\alpha }_{j}^{i}{x}_{i}$. Then

$U:-〈{u}_{1},\mathrm{\dots },{u}_{d}〉\in {\mathrm{Gr}}_{d,𝕍}\mathit{ }\text{and}\mathit{ }{\mathrm{Soc}}_{〈T〉}\left(T\right)={\mathrm{Soc}}_{U}\left(𝔉\left(M\right)\right).$

We have shown that M is in ${\mathrm{CSR}}_{d}$ if and only if $𝔉\left(M\right)$ has constant d-$\mathrm{Soc}$ rank. The other equivalence follows in the same fashion. ∎

For $1\le d, we denote by ${\mathrm{ESP}}_{2,d}\left({E}_{r}\right)$ the category of modules in $\mathrm{mod}k{E}_{r}$ of Loewy length $\le 2$ with the equal d-$\mathrm{Soc}$ 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 $\mathrm{char}\mathit{}\mathrm{\left(}k\mathrm{\right)}\mathrm{>}\mathrm{0}$, $r\mathrm{\ge }\mathrm{3}$ and $\mathrm{1}\mathrm{\le }d\mathrm{\le }r\mathrm{-}\mathrm{1}$. Then ${\mathrm{ESP}}_{\mathrm{2}\mathrm{,}d}\mathit{}\mathrm{\left(}{E}_{r}\mathrm{\right)}\mathrm{\setminus }{\mathrm{ESP}}_{\mathrm{2}\mathrm{,}d\mathrm{-}\mathrm{1}}\mathit{}\mathrm{\left(}{E}_{r}\mathrm{\right)}$ has wild representation type.

#### Proof.

Let $1\le c. By [21, Proposition 2.3] and Proposition 4.2.1, a restriction of $𝔉$ to ${\mathrm{ESP}}_{c}$ induces a faithful exact functor

${𝔉}_{c}:{\mathrm{ESP}}_{c}\to {\mathrm{mod}}_{2}k{E}_{r}$

that reflects isomorphisms and with essential image ${\mathrm{ESP}}_{2,c}\left({E}_{r}\right)$. Let $\mathcal{ℰ}\subseteq {\mathrm{ESP}}_{d}\setminus {\mathrm{ESP}}_{d-1}$ be a wild subcategory. Since ${𝔉}_{d-1}$ and ${𝔉}_{d}$ reflect isomorphisms, we have $𝔉\left(E\right)\in {\mathrm{ESP}}_{2,d}\left({E}_{r}\right)\setminus {\mathrm{ESP}}_{2,d-1}\left({E}_{r}\right)$ for all $E\in \mathcal{ℰ}$. Hence the essential image of $\mathcal{ℰ}$ under $𝔉$ is a wild category. ∎

#### Corollary 4.2.3.

Assume that $\mathrm{char}\mathit{}\mathrm{\left(}k\mathrm{\right)}\mathrm{=}p\mathrm{>}\mathrm{2}$; then the full subcategory of modules with the equal kernels property in $\mathrm{mod}\mathit{}k\mathit{}{E}_{\mathrm{2}}$ and Loewy length $\mathrm{\le }\mathrm{3}$ is of wild representation type.

#### Proof.

By [21, Proposition 2.3] ($n=3\le p$, $r=2$), the functor ${𝔉}_{\mathrm{EKP}\left(3,2\right)}:\mathrm{mod}B\left(3,2\right)\to {\mathrm{mod}}_{3}k{E}_{2}$ is a representation embedding with essential image in $\mathrm{EKP}\left({E}_{2}\right)$. ∎

## 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

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 “Darstellungstheorie”.

Citation Information: Forum Mathematicum, Volume 32, Issue 1, Pages 23–43, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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