# A representation embedding for algebras of infinite type

R. Bautista, E. Pérez, L. Salmerón

###### Abstract

We show that for any finite-dimensional algebra $\Lambda$ of infinite
representation type, over a perfect field, there is a bounded principal ideal
domain $\Gamma$ and a representation embedding from $\Gamma\hbox{-}{\rm mod}$
into $\Lambda\hbox{-}{\rm mod}$. As an application, we prove a variation of
the Brauer-Trall Conjecture II: finite-dimensional algebras of infinite-
representation type admit infinite families of non-isomorphic finite-
dimensional indecomposables with fixed endolength, for infinitely many
endolengths.

††2020 _Mathematics Subject Classification_ : 16G60, 16G20.†† _Keywords and
phrases_ : representation embedding, representation type, endolength,
differential tensor algebra, bocs, reduction functor.

##  1 Introduction

In this note, _$k$ denotes a perfect field, which will act centrally on every
algebra or bimodule we consider_. Given any $k$-algebra $\Lambda$, we denote
by $\Lambda\hbox{-}{\rm Mod}$ (resp. $\Lambda\hbox{-}{\rm mod}$) the category
of left $\Lambda$-modules (resp. finite-dimensional left $\Lambda$-modules).
We denote by $\Lambda\hbox{-}{\rm mod}_{\cal I}$ the full subcategory of
$\Lambda\hbox{-}{\rm mod}$ formed by the modules without injective direct
summands.

Recall that the _endolength_ of a $\Lambda$-module $M$, denoted by $\hbox{\rm
endol}\,(M)$, is by definition the length of the right ${\rm
End}_{\Lambda}(M)^{op}$-module $M$. We say that $\Lambda$ is _discrete_ (resp.
_$e$ -discrete_) iff for each dimension $d$ (resp. each endolength $d$) there
is only a finite number of distinct isoclasses of indecomposable finite-
dimensional $\Lambda$-modules $M$ with $\dim_{k}M=d$ (resp. with $\hbox{\rm
endol}\,(M)=d$).

In this article, we focus our attention on finite-dimensional algebras
$\Lambda$ with infinite representation type. Since the ground field is
perfect, this is equivalent to not being $e$-discrete, see [2]. So, for such
an algebra $\Lambda$ there is at least one number $d\in\mathbb{N}$ and an
infinite family of non-isomorphic indecomposable finite-dimensional
$\Lambda$-modules with endolength $d$.

######  Definition 1.1.

Given two $k$-algebras $\Gamma$ and $\Lambda$, a _representation embedding
from $\Gamma\hbox{-}{\rm mod}$ to $\Lambda\hbox{-}{\rm mod}$_ is a $k$-linear
functor $H:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$ which is exact,
preserves indecomposability, and reflects isomorphism classes.

######  Definition 1.2.

Given full subcategories ${\cal C}$ of $\Gamma\hbox{-}{\rm mod}$ and ${\cal
C^{\prime}}$ of $\Lambda\hbox{-}{\rm mod}$ we will say that a functor $H:{\cal
C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal C}^{\prime}$
_controls endolength of indecomposables_ iff there are some constants
$c,c^{\prime}\in\mathbb{N}$ such that

$\hbox{\rm endol}\,(N)\leq c\times\hbox{\rm endol}\,(H(N))\hbox{ \ and \
}\hbox{\rm endol}\,(H(N))\leq c^{\prime}\times\hbox{\rm endol}\,(N),$

for any indecomposable $N\in{\cal C}$.

######  Definition 1.3.

A $k$-algebra ${\mathtt{Q}}$ is called _minimal_ iff it is of one of the
following two types:

  1. 1.

${\mathtt{Q}}=T_{D_{1}\times D_{2}}(V)$, where $D_{1}$ and $D_{2}$ are finite-
dimensional division $k$-algebras and $V$ is a simple
$D_{1}\hbox{-}D_{2}$-bimodule.

  2. 2.

${\mathtt{Q}}=T_{D}(V)$, where $D$ is a finite-dimensional division
$k$-algebra and $V$ is a simple $D\hbox{-}D$-bimodule.

The _coefficient algebras_ of such a minimal algebra ${\mathtt{Q}}$ are
$D_{1}$ and $D_{2}$ in the first case, and $D$ in the second one.

The _coefficient algebras_ of a finite-dimensional algebra $\Lambda$ are, by
definition, the opposites of the endomorphism algebras of the simple
$\Lambda$-modules. In this note we prove the following result.

######  Theorem 1.4.

Let $\Lambda$ be a finite-dimensional $k$-algebra of infinite representation
type. Then, there is a functor which preserves indecomposables, reflects
isomorphism classes, and controls endolength of indecomposables of one of the
following two types:

  1. 1.

$H:{\mathtt{Q}}\hbox{-}{\rm mod}_{\cal I}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, where
${\mathtt{Q}}$ is a minimal algebra of the first type.

  2. 2.

$H:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, where
${\mathtt{Q}}$ is a minimal algebra of the second type.

In both cases, the algebra ${\mathtt{Q}}$ is of infinite representation type
and with coefficient algebras some of the coefficient algebras of $\Lambda$.
Moreover, the functor $H$ is of the form $Z\otimes_{\mathtt{Q}}-$, for some
$\Lambda\hbox{-}{\mathtt{Q}}$-bimodule $Z$ which is finitely generated by the
right.

From this, we derive:

######  Theorem 1.5.

Let $\Lambda$ be a finite-dimensional $k$-algebra of infinite representation
type. Then, there is a bounded principal ideal domain $\Gamma$ which is not
$e$-discrete and a functor

$G:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod},$

which preserves indecomposables, reflects isomorphism classes, and controls
endolength of indecomposables. The functor $G$ is of the form
$Z\otimes_{\Gamma}-$, for some $\Lambda\hbox{-}\Gamma$-bimodule $Z$ which is
finitely generated by the right.

Then, we will show:

######  Theorem 1.6.

Let $\Lambda$ be a finite-dimensional $k$-algebra of infinite representation
type. Then, there is a bounded principal ideal domain $\Gamma$ which is not
$e$-discrete and a representation embedding which controls endolength of
indecomposables

$G:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}.$

Moreover, the functor $G$ is of the form $Z\otimes_{\Gamma}-$, for some
$\Lambda\hbox{-}\Gamma$-bimodule $Z$ which is free of finite rank by the
right.

As an application of the preceding theorem, we will prove the following
variation of “the second Brauer-Thrall conjecture” involving endolength.

######  Definition 1.7.

We say that a $k$-algebra $\Sigma$ _satisfies EBTII_ if there is an infinite
sequence of natural numbers $d_{1}<d_{2}<\cdots$ such that for each one of
these numbers $d_{i}$ there is an infinite family $\\{M_{j}\\}_{j}$ of non-
isomorphic finite-dimensional indecomposable $\Sigma$-modules with $\hbox{\rm
endol}\,(M_{j})=d_{i}$.

######  Theorem 1.8.

Any finite-dimensional algebra $\Lambda$ of infinite representation type
satisfies EBTII.

This result is a consequence of the following. From (1.6), we have functors
$G:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$ which preserve
indecomposables, reflect isomorphism classes, and control endolength of
indecomposables, they preserve the property EBTII, by (7.6). These algebras
$\Gamma$ satisfy EBTII, as we shall verify in §7.

In view of the previous theorem, it is very natural to ask whether EBTII holds
for a wider class of representation-infinite artin algebras. The problem is
open even in the case of finite-dimensional algebras over a non-perfect
fields.

The proof of (1.4) relies on the theory of differential tensor algebras
(ditalgebras) and reduction functors first developed by the Kiev School of
representation theory, see [18] and [16] (see also [11]). For the general
background on ditalgebras and their module categories, we refer the reader to
[5]. The proofs of the results presented here are largely based on our
previous work [3].

As in [13], we study the phenomenon of existence of an infinite family of
indecomposables in $\Lambda\hbox{-}{\rm mod}$ with fixed endolength by
reducing with matrix problems methods to the case of hereditary algebras. In
[13], Crawley-Boevey uses the more general techniques of lift pairs which
apply to general artin algebras. Here, as in [3], we restrict ourselves to the
case of finite-dimensional algebras over perfect fields and use bocses
techniques which permit us to exploit the properties of reduction functors.

Representation embeddings have been of great interest in representation theory
of algebras, see for instance [7], [6], [12], or [20]. Suffice it to mention
that the definition of wildness requires the existence of such an embedding
from $k\langle x,y\rangle\hbox{-}{\rm mod}$. This work was inspired by
Bongartz article [6], which, under the assumption that the ground field is
algebraically closed, shows that for any finite-dimensional algebra $\Lambda$
of infinite representation type, there is a representation embedding
$k[x]\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$. Here we show that
some version of this holds in the more general situation of perfect fields.

##  2 Strategy for the proof of theorem (1.4)

We recall from [5] and [3], some basic definitions.

######  Definition 2.1.

Let ${\cal A}$ be a layered ditalgebra, with layer $(R,W)$, see [5, §4]. Given
$M\in{\cal A}\hbox{-}{\rm Mod}$, denote by $E_{M}:={\rm End}_{\cal A}(M)^{op}$
its endomorphism algebra. Then, $M$ admits a structure of
$R\hbox{-}E_{M}$-bimodule, where $m\cdot(f^{0},f^{1})=f^{0}(m)$, for $m\in M$
and $(f^{0},f^{1})\in E_{M}$. By definition, _the endolength of $M$_, denoted
by $\hbox{\rm endol}\,(M)$, is the length of $M$ as a right $E_{M}$-module.

As in the case of algebras, the ditalgebra ${\cal A}$ is called _$e$
-discrete_ iff, for each endolength $d$, there are only a finite number of
distinct isoclasses of finite-dimensional indecomposable ${\cal A}$-modules
with endolength $d$.

######  Definition 2.2.

Let ${\cal A}=(T,\delta)$ be a triangular ditalgebra, with layer $(R,W)$, over
the field $k$. Then, ${\cal A}$ is called _admissible_ iff $R\cong
D_{1}\times\cdots\times D_{n}$, for some finite-dimensional division
$k$-algebras $D_{1},\ldots,D_{n}$, and $W$ is finitely generated as an
$R\hbox{-}R$-bimodule. _The coefficient algebras of ${\cal A}$_ are the
algebras $D_{1},\dots,D_{n}$.

Since our ground field $k$ is perfect, any finite-dimensional $k$-algebra
$\Lambda$ splits over its radical, thus $\Lambda=S\oplus J$, where $J$ is the
radical of $\Lambda$. Then, we can consider the Drozd’s ditalgebra ${\cal D}$
of $\Lambda$, see [5, (19.1)], which is an admissible ditalgebra if and only
if $\Lambda$ is basic. In this case, the coefficient algebras of ${\cal D}$
coincide with those of $\Lambda$.

For the proof of our theorem (1.4), we can assume that $\Lambda$ is basic,
because Morita equivalences are representation embeddings which preserve
coefficient algebras and control endolength of indecomposables. The
composition of representations embeddings is a representation embedding, see
[6].

######  Remark 2.3.

If $\Lambda$ has infinite representation type, as we mentioned before, it is
not $e$-discrete, so there is an fixed endolength $d$ and an infinite family
$\\{M_{i}\\}_{i\in I}$ of finite-dimensional non-isomorphic indecomposable
$\Lambda$-modules with $\hbox{\rm endol}\,(M_{i})=d$.

Recall that we have the cokernel functor ${\rm Cok}:{\cal
P}(\Lambda)\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$ and the
equivalence $\Xi_{\Lambda}:{\cal D}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal P}^{1}(\Lambda)$, studied for instance
in [5, §18], whose composition functor permits to transfer many properties
from the category ${\cal D}\hbox{-}{\rm mod}$ to $\Lambda\hbox{-}{\rm mod}$.
Then, for instance by [5, (18.10)(4)], there is a family of non-isomorphic
indecomposable objects $\\{X_{i}\\}_{i\in I}$ in ${\cal P}^{2}(\Lambda)$ such
that ${\rm Cok}(X_{i})\cong M_{i}$, for all $i\in I$. So, for each $i\in I$,
we can choose $N_{i}\in{\cal D}\hbox{-}{\rm mod}$ with
$\Xi_{\Lambda}(N_{i})\cong X_{i}$. Thus, ${\rm Cok}\Xi_{\Lambda}(N_{i})\cong
M_{i}$ and, from [3, (4.4)], there is a natural number $d_{0}$ and an infinite
subset $I_{0}\subseteq I$ such that $\\{N_{i}\\}_{i\in I_{0}}$ is an infinite
family of finite-dimensional non-isomorphic indecomposable ${\cal D}$-modules
with $\hbox{\rm endol}\,(N_{i})=d_{0}$. So we have that ${\cal D}$ is not
$e$-discrete.

The strategy of the proof of our theorem (1.4) includes first the proof of the
following statement for admissible ditalgebras.

######  Theorem 2.4.

Let ${\cal A}$ be an admissible $k$-ditalgebra which is not $e$-discrete.
Then, there is a minimal algebra ${\mathtt{Q}}$ of infinite representation
type and a functor $F:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$ which preserves
indecomposables and reflects isomorphism classes. Moreover, the coefficient
algebras of ${\mathtt{Q}}$ are some of those of ${\cal A}$.

This theorem will be proved by a reduction procedure. Then, we apply it to the
Drozd’s ditalgebra ${\cal D}$ of $\Lambda$ and consider the composition of the
functor $F$ provided by (2.4) with ${\rm Cok}\Xi_{\Lambda}:{\cal
D}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$ to obtain the
functor we are looking for.

The proof of the last theorem will be done by induction using the notion of
$e$-norm of indecomposable finite-dimensional modules discussed in the
following.

##  3 Norms and reduction functors

Suppose that $(R,W)$ is the triangular layer of an admissible ditalgebra
${\cal A}$. Assume that $1=\sum_{i=1}^{n}e_{i}$ is the decomposition of the
unit of $R$ as a sum of central primitive orthogonal idempotents. Thus, with
the notation of (2.2), we have the finite-dimensional division $k$-algebra
$D_{i}=e_{i}D_{i}e_{i}$, for $i\in[1,n]$.

In the discussion of [3] the following notion of norm is relevant.

######  Definition 3.1.

Assume that ${\cal A}$ is an admissible $k$-ditalgebra with layer $(R,W)$.
Then, for $M\in{\cal A}\hbox{-}{\rm mod}$, its _norm_ is the number

$||M||=\dim_{k}{\rm Hom}_{R}(W_{0}\otimes_{R}M,M).$

Then, the _length of $M$_ is $\ell_{R}(M)=\sum_{i=1}^{n}\ell_{D_{i}}(e_{i}M)$.
The _support of $M$_ is the set of idempotents $e_{i}$ with $e_{i}M\not=0$.
The ${\cal A}$-module $M$ is called _sincere_ if $e_{i}M\not=0$, for all
$i\in[1,n]$.

For $M\in{\cal A}\hbox{-}{\rm mod}$, we have

$||M||=\sum_{i,j}\dim_{k}(e_{i}W_{0}e_{j})\ell_{D_{i}}(e_{i}M)\ell_{D_{j}}(e_{j}M).$

In this note, when dealing with finite-dimensional indecomposable ${\cal
A}$-modules with finite endolength, we will use the following modified norm.

######  Definition 3.2.

Let $M$ be a $k$-finite-dimensional indecomposable ${\cal A}$-module and
denote by $E_{M}={\rm End}_{\cal A}(M)^{op}$. Then, the right $E_{M}$-module
$M$ has the direct sum decomposition $M=\bigoplus_{i=1}^{n}e_{i}M$, of right
$E_{M}$-modules. We get

$\hbox{\rm endol}\,(M)=\sum_{i=1}^{n}\ell_{E_{M}}(e_{i}M).$

From [5, (5.6)], ${\cal A}$ is a Roiter ditalgebra and, from [5, (5.12)], we
know that $E_{M}$ is a local finite-dimensional $k$-algebra. Since the field
$k$ is perfect, there is a division $k$-subalgebra $D_{M}$ of $E_{M}$ and a
$D_{M}\hbox{-}D_{M}$-bimodule decomposition $E_{M}=D_{M}\bigoplus{\rm
rad}E_{M}$.

So every finite-length right $E_{M}$-module $N$ is a finite-dimensional right
$D_{M}$-vector space and we have $\ell_{E_{M}}(N)=\ell_{D_{M}}(N)$. So,
$\hbox{\rm endol}\,(M)=\ell_{D_{M}}(M)$ and
$\ell_{E_{M}}(e_{i}M)=\ell_{D_{M}}(e_{i}M)$, for $i\in[1,n]$. Hence,

$\hbox{\rm endol}\,(M)=\sum_{i=1}^{n}\ell_{D_{M}}(e_{i}M).$

Let $c=c_{\cal A}$ be the least common multiple of $d_{1},d_{2},\ldots,d_{n}$,
where $d_{i}=\dim_{k}D_{i}$. So, for each $i$, there is a natural number
$c_{i}$ with $c=c_{i}d_{i}$. Then, define the _$e$ -norm_ of $M$ by

$||M||_{e}:=\sum_{i,j}c_{i}c_{j}\dim_{k}(e_{i}W_{0}e_{j})\ell_{D_{M}}(e_{i}M)\ell_{D_{M}}(e_{j}M).$

######  Remark 3.3.

From this definition, we get that $||M||_{e}\leq c^{2}\dim_{k}(W_{0})\hbox{\rm
endol}\,(M)^{2}$. So if there is an infinite family of non-isomorphic finite-
dimensional indecomposable ${\cal A}$-modules with endolength bounded by $q$,
then their $e$-norms are bounded by $c^{2}\dim_{k}(W_{0})q^{2}$.

If ${\cal A}$ is any admissible ditalgebra and $q\in\mathbb{N}$, we say that
${\cal A}$ is of _infinite $q$-type_ iff ${\cal A}$ admits an infinite family
of non-isomorphic finite-dimensional indecomposable ${\cal A}$-modules $M$
with $e$-norm $||M||_{e}=q$.

So, if ${\cal A}$ is not $e$-discrete, it is of infinite $q$-type for some
$q\in\mathbb{N}$.

######  Lemma 3.4.

Keep the notations of (3.1) and (3.2), for a given finite-dimensional
indecomposable ${\cal A}$-module $M$. Moreover, make $d_{M}=\dim_{k}D_{M}$,
then we have:

$||M||_{e}=\frac{c_{\cal A}^{2}}{d_{M}^{2}}||M||.$

###### Proof.

Recall that
$||M||=\sum_{i,j}\dim_{k}(e_{i}W_{0}e_{j})\ell_{D_{i}}(e_{i}M)\ell_{D_{j}}(e_{j}M)$
and notice that, for each $i\in[1,n]$, we have

$c\ell_{D_{i}}(e_{i}M)=c_{i}d_{i}\ell_{D_{i}}(e_{i}M)=c_{i}\dim_{k}e_{i}M=c_{i}d_{M}\ell_{D_{M}}(e_{i}M).$

∎

In the next statement we describe the behavior of the reduction functors
towards the $e$-norm of finite-dimensional indecomposables. We say that an
algebra $B$ is _an initial subalgebra of an admissible ditalgebra_ ${\cal A}$,
with layer $(R,W)$, if $B=T_{R}(W^{1}_{0})$, where $W_{0}^{1}$ is the smaller
term in the triangular filtration of $W_{0}$, see the reminder [3, (2.3)].

######  Lemma 3.5.

Assume that ${\cal A}$ is an admissible $k$-ditalgebra with layer $(R,W)$.
Suppose that $z\in\\{d,r,X\\}$ and ${\cal A}^{z}$ is the admissible ditalgebra
obtained from ${\cal A}$ by one of the basic reductions: deletion of an
idempotent (case $z=d$, as in [5, (8.17)]), regularization (case $z=r$, as in
[5, (8.19)]), or reduction by a $B$-module $X$, where $B$ is an initial
subalgebra of ${\cal A}$ and $X$ is a finite direct sum of pairwise non-
isomorphic finite-dimensional indecomposable $B$-modules $X_{1},\ldots,X_{t}$
with ${\rm End}_{B}(X_{i})^{op}/{\rm rad}{\rm End}_{B}(X_{i})^{op}$ isomorphic
to some of the coefficient algebras of ${\cal A}$ (case $z=X$, a particular
case of [3, (7.3)]). Consider also the associated full and faithful functor
$F^{z}:{\cal A}^{z}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$. Then, the
following holds for any finite-dimensional indecomposable $N\in{\cal
A}^{z}\hbox{-}{\rm mod}$.

  1. 1.

$||N||_{e}\leq||F^{d}(N)||_{e}$.

  2. 2.

$||N||_{e}<||F^{r}(N)||_{e}$, whenever $F^{r}(N)$ is sincere.

  3. 3.

$||N||_{e}<||F^{X}(N)||_{e}$, whenever $F^{X}(N)$ is sincere and
$W^{\prime}_{0}\not=0$.

###### Proof.

Recall that the layer $(R,W)$ of the admissible ditalgebra ${\cal A}$ is such
that $R=D_{1}\times\cdots\times D_{n}$ as in (2.2).

In order to show 1, for simplicity, assume that the idempotent to be deleted
is $e_{1}$, then the layer $(R^{d},W^{d})$ of ${\cal A}^{d}$ is such that
$R^{d}=D_{2}\times\cdots\times D_{n}$, so clearly $c_{{\cal A}^{d}}\leq
c_{\cal A}$ and, since $F^{d}$ is full and faithful, we get
$d_{N}=d_{F^{d}(N)}$. Moreover, since $||N||=||F^{d}(N)||$, we have

$||N||_{e}=\frac{c^{2}_{{\cal A}^{d}}}{d^{2}_{N}}||N||\leq\frac{c^{2}_{\cal
A}}{d^{2}_{F^{d}(N)}}||F^{d}(N)||=||F^{d}(N)||_{e}.$

For 2, we notice that the layer $(R^{r},W^{r})$ of ${\cal A}^{r}$ is such that
$R^{r}=R$, so $c_{{\cal A}^{r}}=c_{\cal A}$. Again, since $F^{r}$ is full and
faithful, we get $d_{N}=d_{F^{r}(N)}$. Moreover, since $||N||<||F^{r}(N)||$,
whenever $F^{r}(N)$ is sincere, as before, we have

$||N||_{e}=\frac{c^{2}_{{\cal A}^{r}}}{d^{2}_{N}}||N||<\frac{c^{2}_{\cal
A}}{d^{2}_{F^{r}(N)}}||F^{r}(N)||=||F^{r}(N)||_{e}.$

In the last case, the assumption implies that $c_{{\cal A}^{X}}\leq c_{\cal
A}$, and we can proceed as before, using that for $F^{X}(N)$ sincere we have
$||N||<||F^{X}(N)||$, see [3, (7.3)(1)]. ∎

##  4 Reduction to minimal algebras

In this section, we prove the following theorem (4.8), which is a stronger
version of (2.4). Its statement and proof use the following notions and
preliminary arguments. As in the case of algebras, we say that a functor
$F:{\cal A}^{\prime}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$, where ${\cal
A}^{\prime}$ and ${\cal A}$ are layered ditalgebras, _controls endolength of
indecomposables_ if there are some $c,c^{\prime}\in\mathbb{N}$ such that
$\hbox{\rm endol}\,(N)\leq c\times\hbox{\rm endol}\,(F(N))$ and $\hbox{\rm
endol}\,(F(N))\leq c^{\prime}\times\hbox{\rm endol}\,(N)$, for all
indecomposables $N\in{\cal A}^{\prime}\hbox{-}{\rm mod}$.

Recall that given any ditalgebra, ${\cal A}=(T,\delta)$, there is a canonical
embedding functor $L_{\cal A}:A\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$, where $A$ is the
subalgebra of $T$ formed by the homogeneous elements of degree zero. It maps
every $A$-module to itself and each morphism $f\in{\rm Hom}_{A}(M,N)$ is
mapped onto $(f,0)\in{\rm Hom}_{\cal A}(M,N)$, see [5, §2]. Clearly, the
functor $L_{\cal A}$ reflects indecomposability.

######  Definition 4.1.

We say that an admissible ditalgebra ${\cal Q}=(T,\delta)$ is _quasi-minimal_
iff $T$ has layer $(R,W)$ such that $R=D_{1}$ or $R=D_{1}\times D_{2}$, there
is an $R$-$R$-bimodule decomposition $W_{0}=W^{\prime}_{0}\oplus
W^{\prime\prime}_{0}$ is such that $W^{\prime}_{0}=e_{1}W^{\prime}_{0}e_{1}$
in the first case and $W^{\prime}_{0}=e_{2}W_{0}^{\prime}e_{1}$ in the second
one, and moreover, in any case, $W^{\prime}_{0}$ is a simple $R$-$R$-bimodule
with $\delta(W^{\prime}_{0})=0$.

Any such quasi-minimal ditalgebra ${\cal Q}$ determines a minimal algebra
${\mathtt{Q}}:=T_{R}(W^{\prime}_{0})$, which we call _the minimal algebra of
${\cal Q}$_.

######  Remark 4.2.

Quasi-minimal ditalgebras ${\cal Q}=(T,\delta)$ have the following important
feature. If $Q$ denotes the subalgebra of $T$ formed by the homogeneous
elements of degree 0, we have the corresponding canonical embedding functor
$L_{\cal Q}:Q\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal Q}\hbox{-}{\rm mod}$. Consider the
minimal algebra ${\mathtt{Q}}$ of ${\cal Q}$, the projection morphism of
algebras $\pi:Q=T_{R}(W)\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}T_{R}(W^{\prime}_{0})={\mathtt{Q}}$ , and
the corresponding _extension functor_ $E:{\mathtt{Q}}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Q\hbox{-}{\rm
Mod}$, which is given by scalar restriction through $\pi$, thus $E\cong
E({\mathtt{Q}})\otimes_{{\mathtt{Q}}}-$. Then, since
$\delta(W^{\prime}_{0})=0$ and $W^{\prime\prime}_{0}E(N)=0$, for any finite-
dimensional indecomposable ${\mathtt{Q}}$-module $N$, we have

$\mathbb{E}:={\rm End}_{{\cal Q}}(L_{\cal
Q}E(N))^{op}=\mathbb{E}_{N}^{0}\oplus\mathbb{E}_{N}^{1},$

where $\mathbb{E}_{N}^{0}$ and $\mathbb{E}_{N}^{1}$ are the subspaces of
$\mathbb{E}$ formed, respectively, by the endomorphisms of $E(N)$ of the form
$(f^{0},0)$, and by the endomorphisms of $E(N)$ of the form $(0,f^{1})$.
Moreover, $\mathbb{E}_{N}^{0}=L_{\cal Q}E({\rm End}_{{\mathtt{Q}}}(N)^{op})$.

Indeed, given any $g^{0}\in{\rm End}_{{\mathtt{Q}}}(N)^{op}$, from the
equality $W^{\prime\prime}_{0}E(N)=0$, we have $g^{0}(wm)=0=wg^{0}(m)$, for
all $w\in W^{\prime\prime}_{0}$ and $m\in E(N)$. Since $Q$ is generated by
$R$, $W^{\prime}_{0}$, and $W^{\prime\prime}_{0}$, we have that $g^{0}\in{\rm
End}_{Q}(E(N))^{op}$, so $L_{\cal Q}E(g^{0})=(g^{0},0)\in\mathbb{E}_{N}^{0}$.
Moreover, given $f=(f^{0},f^{1})\in\mathbb{E}$, the equality
$\delta(W^{\prime}_{0})=0$ implies that $f^{0}\in{\rm
End}_{{\mathtt{Q}}}(N)^{op}$. Thus, $(f^{0},0)\in\mathbb{E}^{0}_{N}$,
$(0,f^{1})=(f^{0},f^{1})-(f^{0},0)\in\mathbb{E}$, and
$\mathbb{E}_{N}^{0}=L_{\cal Q}E({\rm End}_{{\mathtt{Q}}}(N)^{op})$.

Our ditalgebra ${\cal Q}$ is triangular, so, from [5, (5.4)],
$\mathbb{E}^{1}_{N}$ is a nilpotent ideal of $\mathbb{E}$, so we know that
$\mathbb{E}_{N}^{1}\subseteq{\rm Rad}\mathbb{E}$. Since $k$ is perfect, we
also have ${\rm End}_{{\mathtt{Q}}}(N)^{op}=K_{N}\oplus{\rm Rad}({\rm
End}_{{\mathtt{Q}}}(N)^{op})$, where $K_{N}$ is a division algebra. From the
preceding facts, we obtain the equality

${\rm End}_{{\cal Q}}(L_{\cal Q}E(N))^{op}=L_{\cal Q}E(K_{N})\oplus{\rm
Rad}({\rm End}_{{\cal Q}}(L_{\cal Q}E(N))^{op}).$

Thus, the functor ${\mathtt{L}}_{\mathtt{Q}}:=L_{\cal
Q}E:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal Q}\hbox{-}{\rm mod}$ preserves
indecomposables. It reflects isomorphism classes because, ${\cal Q}$ is a
Roiter ditalgebra, see [5, (5.8) and (5.6)]. We will keep the preceding
notation whenever we deal with quasi-minimal ditalgebras.

We have the following elementary statement relating dimension and endolength.

######  Lemma 4.3.

Let $\Sigma$ be any $k$-algebra (not necessarily finite-dimensional) and $M$
an indecomposable finite-dimensional $\Sigma$-module. Consider the local
algebra $E:={\rm End}_{\Sigma}(M)^{op}$ and the division algebra
$K_{M}:=E/{\rm Rad}E$. Then, we have

$\hbox{\rm endol}\,(M)=\dim_{k}M/\dim_{k}K_{M}.$

###### Proof.

Since $M$ is finite-dimensional, its length as an $E$-module is finite.
Consider a composition series $0=M_{0}\subset M_{1}\subset\cdots\subset
M_{l}=M$ for the right $E$-module $M$. Then, each simple $E$-module
$M_{i}/M_{i-1}$ is a simple $K_{M}$-module: a one-dimensional $K_{M}$-vector
space. So, we have $\dim_{k}(M_{i}/M_{i-1})=\dim_{k}K_{M}$, for each $i$.
Thus, $\dim_{k}M=l\dim_{k}K_{M}$. Our statement follows from this. ∎

######  Remark 4.4.

Similarly, given any admissible ditalgebra ${\cal A}$ and an indecomposable
$M\in{\cal A}\hbox{-}{\rm mod}$, if we consider the local algebra $E:={\rm
End}_{\cal A}(M)^{op}$ and the division algebra $K_{M}:=E/{\rm Rad}E$, we have
$\hbox{\rm endol}\,(M)=\dim_{k}M/\dim_{k}K_{M}$.

Given a quasi-minimal ditalgebra ${\cal Q}$, with the notation of (4.2), given
an indecomposable finite-dimensional ${\mathtt{Q}}$-module $M$, since the
functor ${\mathtt{L}}_{{\mathtt{Q}}}$ preserves dimension and
$K_{M}\cong{\mathtt{L}}_{{\mathtt{Q}}}(K_{M})\cong
K_{{\mathtt{L}}_{{\mathtt{Q}}}(M)}$, we obtain from (4.2) and (4.3) that
$\hbox{\rm
endol}\,({\mathtt{L}}_{{\mathtt{Q}}}(M))=\dim_{k}{\mathtt{L}}_{{\mathtt{Q}}}(M)/\dim_{k}K_{{\mathtt{L}}_{{\mathtt{Q}}}(M)}=\dim_{k}M/\dim_{k}K_{M}=\hbox{\rm
endol}\,(M)$. Thus, ${\mathtt{L}}_{{\mathtt{Q}}}:{\mathtt{Q}}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
Q}\hbox{-}{\rm mod}$ preserves endolength of indecomposable modules.

######  Definition 4.5.

Let ${\cal C}$ and ${\cal C^{\prime}}$ be full subcategories of finite-
dimensional modules over some algebras or some ditalgebras. Then, we say that
a functor $F:{\cal C}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal C}^{\prime}$ _controls dimension_ iff
there are constants $c,c^{\prime}\in\mathbb{N}$ such that

$\dim_{k}M\leq c\times\dim_{k}FM\hbox{ \ and \ }\dim_{k}FM\leq
c^{\prime}\times\dim_{k}M,$

for all $M\in{\cal C}$.

######  Remark 4.6.

From (4.3), we know that any functor $F:{\cal C}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal C}^{\prime}$ which controls dimension
and induces isomorphisms $K_{M}\cong K_{FM}$, for all indecomposable $M$,
controls endolength of indecomposables.

######  Lemma 4.7.

Let ${\cal A}$ be an admissible ditalgebra. Assume that ${\cal A}^{X}$ is the
layered ditalgebra obtained from ${\cal A}$ by reduction, using the $B$-module
$X$, where $B$ is an initial subalgebra of ${\cal A}$ and $X$ is a finite
direct sum of non-isomorphic finite-dimensional indecomposable $B$-modules.
Assume that $X$ is a complete admissible triangular $B$-module and let
$F^{X}:{\cal A}^{X}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm Mod}$ be the associated
functor, as in [5, 12.10]. Then, $F^{X}$ controls dimension and controls
endolength of indecomposables.

###### Proof.

The finite-dimensional algebra $\Gamma={\rm End}_{B}(X)^{op}$ admits the
splitting $\Gamma=S\oplus P$, where $P$ is the radical of $\Gamma$, because
the ground field is perfect. The semisimple algebra $S$ is basic because the
indecomposable direct summands of $X$ are pairwise non-isomorphic. Let us
denote by $f_{1},\ldots,f_{t}$ the orthogonal primitive central idempotents
given by the decomposition of the unit of $S$.

From [3, (2.7)], we already know that or all $N\in{\cal A}^{X}\hbox{-}{\rm
Mod}$, we have that $\hbox{\rm endol}\,(F^{X}(N))\leq\dim_{k}X\times\hbox{\rm
endol}\,(N)$.

It is clear that $\dim_{k}F^{X}(N)\leq\dim_{k}X\times\dim_{k}N$. Moreover, we
have

$\dim_{k}F^{X}(N)=\dim_{k}X\otimes_{S}N=\sum_{i}\dim_{k}Xf_{i}\otimes_{Sf_{i}}f_{i}N\geq\sum_{i}\dim_{k}f_{i}N=\dim_{k}N,$

because
$\dim_{k}Xf_{i}\otimes_{Sf_{i}}f_{i}M=\dim_{k}(\dim_{Sf_{i}}Xf_{i})f_{i}M\geq\dim_{k}f_{i}N,$
with $Xf_{i}\not=0$. So, the functor $F^{X}$ controls dimension, and we can
apply (4.6). ∎

######  Theorem 4.8.

For any admissible $k$-ditalgebra ${\cal A}$ not $e$-discrete, there is a
quasi-minimal ditalgebra ${\cal Q}$ with minimal algebra ${\mathtt{Q}}$ of
infinite representation type, and a faithful functor
$F:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$ which preserves
indecomposability, reflects isomorphism classes, and controls endolength of
indecomposables. The functor $F$ is the composition

${\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{{\mathtt{L}}_{{\mathtt{Q}}}}}{\cal
Q}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{G}}{\cal A}\hbox{-}{\rm mod},$

where $G$ is a composition of full and faithful reduction functors associated
to a finite sequence of reductions which transform ${\cal A}$ into ${\cal Q}$.
Moreover, the coefficient algebras of ${\mathtt{Q}}$ are some of those of
${\cal A}$.

###### Proof.

Assume that $R=D_{1}\times\cdots\times D_{n}$ and $1=\sum_{i=1}^{n}e_{i}$ is a
decomposition of the unit of $R$ as a sum of central ortogonal idempotents, so
each $e_{i}Re_{i}=D_{i}$ is a finite-dimensional $k$-algebra, say of dimension
$d_{i}=\dim_{k}D_{i}$. We will use the $e$-norm $||M||_{e}$ defined for
indecomposable modules in ${\cal A}\hbox{-}{\rm mod}$.

By definition, given $q\in\mathbb{N}$, _a $q$-infinite family of ${\cal
A}\hbox{-}{\rm mod}$_ is an infinite family ${\cal F}$ of non-isomorphic
finite-dimensional indecomposable ${\cal A}$-modules $M$ with common $e$-norm
$||M||_{e}=q$. Since ${\cal A}$ is not $e$-discrete, ${\cal A}\hbox{-}{\rm
mod}$ admits a $q$-infinite family ${\cal F}$.

If there are infinitely many ${\cal A}$-modules $M$ in the family ${\cal F}$,
with $e_{i}M=0$, for a fixed $i\in[1,n]$, we consider the ditalgebra ${\cal
A}^{d_{1}}$ obtained from ${\cal A}$ by deletion of the idempotent $e_{i}$ and
the associated full and faithful functor $F^{d_{1}}:{\cal
A}^{d_{1}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$. Then, ${\cal
A}^{d_{1}}$ is an admissible ditalgebra and we know that
$1\leq||N||_{e}\leq||F^{d_{1}}(N)||_{e}=q$, whenever $M\in{\cal F}$ and
$F^{d_{1}}(N)\cong M$. So there is a $q_{1}$-infinite family ${\cal
F}^{d_{1}}$ in ${\cal A}^{d_{1}}\hbox{-}{\rm mod}$ with $q_{1}\leq q$. In a
finite number of such deletion of idempotents, we obtain an admissible
ditalgebra ${\cal A}^{\prime}={\cal A}^{d_{1}\cdots d_{m}}$ and a
$q^{\prime}$-infinite family ${\cal F}^{\prime}$ of sincere modules in ${\cal
A}^{\prime}\hbox{-}{\rm mod}$ with $q^{\prime}\leq q$.

If there is a quasi-minimal ditalgebra ${\cal Q}$ and a functor
$F:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}^{\prime}\hbox{-}{\rm mod}$ as in
the statement of the theorem for the ditalgebra ${\cal A}^{\prime}$, then the
composition $F^{d_{1}}\cdots F^{d_{m}}F:{\mathtt{Q}}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
A}\hbox{-}{\rm mod}$ is the required functor for ${\cal A}$, because each
functor $F^{d_{i}}$ preserves endolength, see [3, (2.5)].

So we can assume that we already started with a $q$-infinite family of sincere
${\cal A}$-modules ${\cal F}$ and proceed to the proof of the theorem by
induction on $q$.

In case $q=1$, for each $M\in{\cal F}$, with the notation of (3.2), we have

$1=||M||_{e}=\sum_{i,j}c_{i}c_{j}\dim_{k}(e_{i}W_{0}e_{j})\ell_{D_{M}}(e_{i}M)\ell_{D_{M}}(e_{j}M).$

This implies that $n=1$ or $n=2$. If $n=1$, $R=D_{1}$ and
$\dim_{k}(e_{1}W_{0}e_{1})=1$. Moreover, $D_{1}=k$ and $A=T_{k}(W_{0})$ with
$\dim_{k}W_{0}=1$. Here $\delta(W_{0})\subseteq W_{1}$, because the triangular
filtration of $W_{0}$ is just $0\subseteq W_{0}$. If $\delta(W_{0})\not=0$,
since $W^{1}_{0}$ is a simple $R\hbox{-}R$-bimodule and $W_{0}^{1}\cap{\rm
Ker}\,\delta=0$, we can consider the admissible ditalgebra ${\cal A}^{r}$
obtained from ${\cal A}$ by regularization and the associated equivalence
functor $F^{r}:{\cal A}^{r}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$, see [5, (8.19)].
But, here, ${\cal A}^{r}$ has only one indecomposable module, so ${\cal A}$
would not be of infinite representation type. So, $\delta(W_{0})=0$, ${\cal
A}$ is quasi-minimal, and we can take ${\cal Q}={\cal A}$,
${\mathtt{Q}}=A\cong k[x]$, and $G=Id$. The functor
$F={\mathtt{L}}_{{\mathtt{Q}}}$ preserves indecomposables and reflects
isomorphisms by (4.2). It preserves endolength of indecomposables by (4.4).

In case $n=2$, we have $R=D_{1}\times D_{2}$ and, since, $\dim_{k}W_{0}=1$, we
get $D_{1}=D_{2}=k$. In this case, since $L_{\cal A}:A\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
A}\hbox{-}{\rm mod}$ reflects indecomposability and isomorphism classes, and
$A=T_{k\times k}(W_{0})$ is of finite representation type, then ${\cal A}$ is
also of finite representation type, which is not the case. Hence, this case
does not occur. So the base of the induction is established.

Now, assume $q\geq 2$ and that the statement of the theorem holds for every
admissible ditalgebra ${\cal A}^{\prime}$ with a $q^{\prime}$-infinite family
${\cal F}^{\prime}$ in ${\cal A}^{\prime}\hbox{-}{\rm mod}$ with
$q^{\prime}<q$. We will proceed as in the proof of [3, (7.5)].

Since $k$ is perfect, the algebra $R\otimes_{k}R^{op}$ is semisimple and we
can look at the additive triangular filtration $0=W^{0}_{0}\subseteq
W_{0}^{1}\subseteq\cdots\subseteq W_{0}^{s}=W_{0}$, see [5, (5.1)]. We can
assume that $W_{0}^{1}$ is a simple direct summand of the
$R\hbox{-}R$-bimodule $W_{0}$. The triangularity conditions imply that
$\delta(W_{0}^{1})\subseteq W_{1}$. Consider $R\hbox{-}R$-bimodule
decompositions $W_{0}=W_{0}^{1}\oplus W^{\prime\prime}_{0}$ and
$W_{1}=\delta(W^{1}_{0})\oplus W^{\prime\prime}_{1}$. We have the following
two possibilities.

_Case 1: $\delta(W_{0}^{1})\not=0$._

Since $W^{1}_{0}$ is a simple $R\hbox{-}R$-bimodule and $W_{0}^{1}\cap{\rm
Ker}\,\delta=0$, we can consider the admissible ditalgebra ${\cal A}^{r}$
obtained from ${\cal A}$ by regularization and the associated equivalence
functor $F^{r}:{\cal A}^{r}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$, see [5, (8.19)].
For any indecomposable $N\in{\cal A}^{r}\hbox{-}{\rm mod}$ with $F^{r}(N)\cong
M\in{\cal F}$, we have $||N||_{e}<||M||_{e}=q$, by (3.5)(2). Then, there is a
family ${\cal F}^{r}$ of such ${\cal A}^{r}$-modules $N$ which is a
$q^{\prime}$-infinite family in ${\cal A}^{r}\hbox{-}{\rm mod}$ with
$q^{\prime}<q$. So, we can apply the induction hypothesis to ${\cal A}^{r}$ to
derive the existence of a quasi-minimal ditalgebra ${\cal Q}$ with minimal
algebra ${\mathtt{Q}}$ and a functor $F:{\mathtt{Q}}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
A}^{r}\hbox{-}{\rm mod}$ as in the statement of the theorem, so the
composition $F^{r}F:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$ is the wanted
functor for ${\cal A}$. Here, the functor $F^{r}$ preserves endolength, see
[3, (2.6)].

_Case 2: $\delta(W_{0}^{1})=0$_.

Since $W_{0}^{1}$ is a simple $R\hbox{-}R$-bimodule, we get
$W_{0}^{1}=e_{j}W_{0}^{1}e_{i}$, for some $i,j\in[1,n]$, and we have two
subcases.

_Case 2.a:_ If $i=j$, for simplicity, we can assume that $i=1$. Then, we get
$T_{R}(W_{0}^{1})=T_{D_{1}}(W_{0}^{1})\times D_{2}\times\cdots\times D_{n}$.
Consider the admissible ditalgebra ${\cal A}^{d}$ obtained from ${\cal A}$ by
deletion of the idempotent $e=1-e_{1}$ and the corresponding full and faithful
functor $F^{d}:{\cal A}^{d}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm Mod}$. Then, ${\cal
Q}:={\cal A}^{d}$ is a quasi-minimal ditalgebra, with minimal algebra
${\mathtt{Q}}=T_{D_{1}}(W_{0}^{1})$ of infinite representation type, and the
composition of functors

${\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{{\mathtt{L}}_{{\mathtt{Q}}}}}{\cal
Q}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F^{d}}}{\cal A}\hbox{-}{\rm mod}$

is the required functor $F$ for ${\cal A}$, in this case.

_Case 2.b:_ If $i\not=j$, for simplicity, we assume that $i=1$ and $j=2$, and
we get $B:=T_{R}(W_{0}^{1})=T_{D_{1}\times D_{2}}(W_{0}^{1})\times
D_{3}\times\cdots\times D_{n}$.

If $B$ is of infinite representation type, so is the algebra $T_{D_{1}\times
D_{2}}(W_{0}^{1})$. Then, we can consider the ditalgebra ${\cal A}^{d}$
obtained from ${\cal A}$ by deletion of the idempotent $e=1-e_{1}-e_{2}$ and
the associated full and faithful functor $F^{d}:{\cal A}^{d}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
A}\hbox{-}{\rm Mod}$. Then, ${\cal Q}:={\cal A}^{d}$ is a quasi-minimal
ditalgebra, with minimal algebra ${\mathtt{Q}}=T_{D_{1}\times
D_{2}}(W_{0}^{1})$ of infinite representation type, and the composition of
functors

${\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{{\mathtt{L}}_{{\mathtt{Q}}}}}{\cal
Q}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F^{d}}}{\cal A}\hbox{-}{\rm mod}$

is the required functor $F$ for ${\cal A}$.

On the other hand, if $B$ is of finite representation type, so is the algebra
$B_{0}:=T_{D_{1}\times D_{2}}(W_{0}^{1})$. Let $X_{1},\ldots,X_{l}$ be a
complete set of representatives of the isomorphism classes of the
indecomposables in $B_{0}\hbox{-}{\rm mod}$. Since $B_{0}$ is hereditary of
finite representation type, then the division algebras $D_{X_{i}}={\rm
End}_{B_{0}}(X_{i})^{op}/{\rm rad}{\rm End}_{B_{0}}(X_{i})^{op}$ coincide with
some $D_{1}$ or $D_{2}$ (this follows for instance from [1, VIII.1: (1.5) and
(1.13)], see also [13]).

Consider the $B$-module $X=X_{1}\oplus\cdots\oplus X_{l}\oplus eR$ and the
admissible ditalgebra ${\cal A}^{X}$ obtained from ${\cal A}$ by reduction by
the module $X$ and the associated full and faithful functor $F^{X}:{\cal
A}^{X}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$, as in (3.5). The
layer of ${\cal A}^{X}$ has the form $(R^{X},W^{X})$ with
$R^{X}=D_{X_{1}}\times\cdots\times D_{X_{l}}\times D_{3}\times\cdots\times
D_{n}$. From (3.5), whenever $F^{X}(N)\cong M\in{\cal F}$, we have
$||N||_{e}<||M||_{e}=q$. But from [3, (7.3)(2)], we know that $F^{X}$ is
dense. So, there in an infinite family in ${\cal F}^{X}=\\{N\mid
F^{X}(N)\in{\cal F}\\}$, which shows that the admissible ditalgebra ${\cal
A}^{X}$ is of $q^{\prime}$-infinite type for some $q^{\prime}<q$. Then, we can
apply the induction hypothesis to ${\cal A}^{X}$ and obtain a quasi-minimal
ditalgebra ${\cal Q}$ and a functor $F:{\mathtt{Q}}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
A}^{X}\hbox{-}{\rm mod}$ as in the statement of the theorem, so the
composition $F^{X}F:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm mod}$ gives the wanted
functor for ${\cal A}$. Here, the functor $F^{X}$ controls endolength of
indecomposables by (4.8).

We have constructed, in each case, a functor $F:{\mathtt{Q}}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
A}\hbox{-}{\rm mod}$ which preserves indecomposability, reflects isomorphism
classes, and controls endolength of indecomposables. ∎

The following lemma is very useful. It can be applied, to Morita equivalences.

######  Lemma 4.9.

Given two algebras $\Gamma$ and $Q$, assume that $\Psi:\Gamma\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Q\hbox{-}{\rm mod}$
is a full and faithful functor of the form $P\otimes_{\Gamma}-$, for some
$Q\hbox{-}\Gamma$-bimodule $P$ that is finitely generated projective by the
right. Then, the functor $\Psi$ controls endolength.

###### Proof.

Take $N\in\Gamma\hbox{-}{\rm mod}$ and set $E:={\rm End}_{\Gamma}(N)^{op}$. We
know that for some natural number $n$, there is a surjective morphism
$\Gamma^{n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}P$ of right
$\Gamma$-modules. Therefore, we have a surjective morphism of right
$E$-modules

$N^{n}\cong\Gamma^{n}\otimes_{\Gamma}N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}P\otimes_{\Gamma}N.$

Therefore, $\ell_{E}(P\otimes_{\Gamma}N)\leq
n\times\ell_{E}(N)=n\times\hbox{\rm endol}\,(N)$.

Now, the action of $E$ on the right $E$-module $P\otimes_{\Gamma}N$ determined
by the right $E$-module $N$ is such that $(p\otimes n)f=p\otimes f(n)$, for
$p\in P$ and $n\in N$. The algebra ${\rm End}_{Q}(P\otimes N)^{op}$ acts on
the right of $P\otimes_{\Gamma}N$ by the usual rule and determines a right
action of $E$ by restriction using the isomorphism of algebras
$E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{Q}(P\otimes N)^{op}$ provided by the full and faithful functor $\Psi$.
This action of $E$ on $P\otimes_{\Gamma}N$ coincides with the previous one
and, hence, we obtain $\hbox{\rm
endol}\,(P\otimes_{\Gamma}N)\leq\ell_{E}(P\otimes_{\Gamma}N)$.

If $0=N_{t}\subseteq\cdots\subseteq N_{1}\subseteq N_{0}=N$ is a composition
series for the ${\rm End}_{\Gamma}(N)^{op}$-module $N$, since $P$ is a
projective right $\Gamma$-module, we have a filtration
$0=P\otimes_{\Gamma}N_{t}\subseteq\cdots\subseteq
P\otimes_{\Gamma}N_{1}\subseteq P\otimes_{\Gamma}N_{0}=P\otimes_{\Gamma}N$ of
the $Q$-module $P\otimes_{\Gamma}N$, which is in fact a filtration of the
${\rm End}_{Q}(P\otimes_{\Gamma}N)^{op}$-module $P\otimes_{\Gamma}N$. Hence
$\hbox{\rm endol}\,(N)\leq\hbox{\rm endol}\,(P\otimes_{\Gamma}N)$. ∎

Proof of Theorem (1.4): As remarqued just before (2.3), we can assume that
$\Lambda$ is basic because Morita equivalences are representation embeddings
which control endolength of indecomposables, see (4.9).

From (2.3), we can apply (4.8) to the Drozd’s ditalgebra ${\cal D}$ of
$\Lambda$ to obtain a quasi-minimal ditalgebra ${\cal Q}$, with minimal
algebra ${\mathtt{Q}}$ of infinite representation type, and a composition of
functors $F$ of the form

${\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{{\mathtt{L}}_{{\mathtt{Q}}}}}{\cal
Q}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{G}}{\cal D}\hbox{-}{\rm mod},$

as in (4.8), which preserves indecomposability, reflects isomorphism classes,
and controls endolength of indecomposables.

From [5, (22.7)], we obtain that $GE({\mathtt{Q}})$ is a
$D\hbox{-}{\mathtt{Q}}$-bimodule which is finitely generated by the right and
that $F\cong L_{\cal D}(GE({\mathtt{Q}})\otimes_{\mathtt{Q}}-)$, where the
argument is the functor
$GE({\mathtt{Q}})\otimes_{\mathtt{Q}}-:{\mathtt{Q}}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}D\hbox{-}{\rm
mod}$. Here $D$ denotes the subalgebra of the underlying tensor algebra of
${\cal D}$ formed by the elements of degree 0.

Consider the usual equivalence functor $\Xi_{\Lambda}:{\cal D}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
P}^{1}(\Lambda)$ and the cokernel funtor ${\rm Cok}:{\cal
P}^{1}(\Lambda)\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, see [5, §18 and
§19]. Denote by $Z_{0}={\rm Cok}\Xi_{\Lambda}(D)$ the transition bimodule
associated to $\Lambda$, as in [5, (22.18)]. Then, the
$\Lambda\hbox{-}{\mathtt{Q}}$-bimodule $Z:=Z_{0}\otimes_{D}GE({\mathtt{Q}})$
is finitely generated by the right. Denote by $H$ the composition of functors

${\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F}}{\cal D}\hbox{-}{\rm
mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\Xi_{\Lambda}}}{\cal
P}^{1}(\Lambda)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{{\rm
Cok}}}\Lambda\hbox{-}{\rm mod}.$

Then, by [5, (22.18)], we know that it is naturally isomorphic to

${\rm Cok}\Xi_{\Lambda}L_{\cal D}(GE({\mathtt{Q}})\otimes_{\mathtt{Q}}-)\cong
Z_{0}\otimes_{D}GE({\mathtt{Q}})\otimes_{\mathtt{Q}}-=Z\otimes_{\mathtt{Q}}-.$

Now, if ${\mathtt{Q}}$ is a minimal algebra of the second type, we have, as in
[3, Claim 1 in the proof of (8.2)], that every $M\in{\mathtt{Q}}\hbox{-}{\rm
mod}$ satisfies that $\Xi_{\Lambda}F(M)\in{\cal P}^{2}(\Lambda)$. Thus, in
this case, the functor $H$ preserves indecomposables and reflects isomorphism
classes.

If ${\mathtt{Q}}$ is a minimal algebra of the first type, we have as in [3,
Claim 2 in the proof of (8.2)], that for any non-injective indecomposable
$M\in{\mathtt{Q}}\hbox{-}{\rm mod}$ we have $\Xi_{\Lambda}F(M)\in{\cal
P}^{2}(\Lambda)$. So, in this case $H$ preserves indecomposables which are not
injective (since it is a composition of functors which preserve non-
injectivity). Moreover, the restriction ${\mathtt{Q}}\hbox{-}{\rm mod}_{\cal
I}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm
mod}$ of $H$, to the full subcategory of ${\mathtt{Q}}$-modules without
injective direct summands, reflects isomorphism classes. Indeed, if
$M,N\in{\mathtt{Q}}\hbox{-}{\rm mod}_{\cal I}$, then $\Xi_{\Lambda}F(M)$ and
$\Xi_{\Lambda}F(N)$ have no injective direct summand. Thus, from [5, (18.10)],
we get that $H(M)\cong H(N)$ implies that $M\cong N$.

We know that $F$ controls endolength and, from [3, (4.4)], we get that ${\rm
Cok}\Xi_{\Lambda}$ controls endolength, then so does their composition $H$.
$\hbox{ }\hfill\square$

######  Remark 4.10.

Under the assumptions of theorem (1.4), assume moreover that $\Lambda$ is such
that $\Lambda/{\rm rad}\Lambda$ is a finite product of copies of $k$. In this
case, the minimal algebras appearing there have their coefficient algebras
isomorphic to $k$. So, there is no such minimal algebra of the first type of
infinite representation type, and the only such minimal algebra of the second
type of infinite representation type is $k[x]$. Then, we get a functor
$H:k[x]\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, which preserves
indecomposables and reflects isomorphism classes, given by tensoring with a
bimodule $\,{}_{\Lambda}Z_{k[x]}$ which is finitely generated over $k[x]$.
From here we obtain a representation embedding
$H^{\prime}:k[x]_{h}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, given by
tensoring with a bimodule $\,{}_{\Lambda}Z^{\prime}_{k[x]_{h}}$ which is
finitely generated projective over $k[x]_{h}$. This is the case if the field
$k$ is algebraically closed and $\Lambda$ is basic, see Thm. 6 of [6].

##  5 Reduction to principal ideal domains

In this work, we use the term uniserial subcategory in the following sense.

######  Definition 5.1.

Let ${\cal U}$ be a full abelian subcategory of the category
$\Sigma\hbox{-}{\rm mod}$ of finite-dimensional modules for some $k$-algebra
$\Sigma$. Then ${\cal U}$ is called _a uniserial subcategory of
$\Sigma\hbox{-}{\rm mod}$_ iff every indecomposable module in ${\cal U}$ has a
unique composition series in ${\cal U}$, and all the simple factors in this
composition series are isomorphic to one simple object of the category ${\cal
U}$.

######  Lemma 5.2.

Let $\Gamma$ be a bounded principal ideal domain and let ${\cal P}$ be a
complete set of representatives of the non-similar atoms of $\Gamma$. Then,
the category of finite-length $\Gamma$-modules is a direct sum of uniserial
subcategories $\\{{\cal U}_{p}\\}_{p\in{\cal P}}$, where each ${\cal U}_{p}$
contains a unique simple object $E_{1}^{p}$.

Moreover, we have almost split sequences of the form:

$\begin{matrix}\xi_{1}:\ E^{p}_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}E^{p}_{2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}E^{p}_{1},\\\ \,\\\ \zeta_{n}:\
E^{p}_{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}E^{p}_{n+1}\oplus
E^{p}_{n-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}E^{p}_{n},\hbox{ for }n\geq 2.\\\
\end{matrix}$

###### Proof.

The category of finite-length $\Gamma$-modules is well known. For instance,
from [3, (6.5)], we have the following. For each atom $p\in\Gamma$, consider
the corresponding simple $\Gamma$-module $S_{p}:=\Gamma/\Gamma p$. Then, for
each $i\in\mathbb{N}$, up to isomorphism, there is a unique indecomposable
$\Gamma$-module $E_{i}^{p}$ with length $i$ and all composition factors
isomorphic to $S_{p}$. The family $\\{E_{i}^{p}\mid i\in\mathbb{N},p\in{\cal
P}\\}$ is a complete set of representatives of the isoclasses of the
indecomposable $\Gamma$-modules of finite length. They are all finite-
dimensional.

For $p\in{\cal P}$, consider the full subcategory ${\cal U}_{p}$ of
$\Gamma\hbox{-}{\rm mod}$ formed by the modules $M\in\Gamma\hbox{-}{\rm mod}$
which admit a filtration

$0=M_{0}\subset M_{1}\subset\cdots\subset M_{n-1}\subset M_{n}=M,$

where $M_{j}/M_{j-1}\cong S_{p}$, for all $j\in[1,n]$. Thus, $\\{E_{i}^{p}\mid
i\in\mathbb{N}\\}$ are the representatives of the indecomposables in ${\cal
U}_{p}$. Moreover, in the proof of [3, (6.5)], it is shown that ${\rm
Hom}_{\Gamma}(E_{i}^{p},E_{j}^{q})=0$, whenever $p$ and $q$ are not similar
atoms, that is whenever $p\not=q$ in ${\cal P}$.

It remains to show that ${\cal U}_{p}$ is uniserial. Let $M$ be an
indecomposable $\Gamma$-module in ${\cal U}_{p}$. Then, it has a composition
series

$0=M_{0}\subset M_{1}\subset\cdots\subset M_{n-1}\subset M_{n}=M,$

where $M_{i}/M_{i-1}\cong S_{p}$. Here $\ell_{\Gamma}(M_{i})=i$, for all
$i\in[1,n]$.

_We will show that $M_{j}$ is the unique submodule of $M$ with length $j$._

We proceed by induction on the length $n=\ell_{\Gamma}(M)$.

If $n=1$, we have that $M=M_{1}$ is simple and our claim is obvious. Assume
that our claim holds for indecomposable modules in ${\cal U}_{p}$ with length
$s$ such that $s<n$. We have an exact sequence

$0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{f}}M_{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{g}}S_{p}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0.$

Let $S$ be a simple submodule of $M$ and $h:S\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n}$ the corresponding inclusion. Since
$M$ is indecomposable, we know that $gh:S\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}S_{p}$ is not an isomorphism, so $gh=0$ and
$S\subseteq M_{n-1}$. Similarly, working with the given filtration of
$M_{n-1}$, we show that $S\subseteq M_{n-2}$. In a finite number of steps, we
get $S\subseteq M_{1}$. Hence $S=M_{1}$. We have verified that the socle of
$M$ is the simple $M_{1}$.

Consider now a submodule $N$ of $M_{n}$ with length $j>1$. Since ${\rm
soc}(M)=M_{1}$, we know that $M_{1}={\rm soc}(N)\subset N$. In particular, the
submodules $M_{1},\ldots,M_{n-1}$ are all indecomposable. We have the quotient
$\eta:M_{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n}/M_{1}$, and the submodules
$\eta(M_{j})=M_{j}/M_{1}$ and $\eta(N)=N/M_{1}$ with length $j-1$ of the
module $M/M_{1}\in{\cal U}_{p}$, which has length $n-1$.

From [3, (6.5)], we know there is an irreducible morphism
$g:M_{n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{n-1}$ in
$\Gamma\hbox{-}{\rm mod}$, which must be surjective. Its kernel has length 1,
so it is a simple submodule of $M_{n}$, and it coincides with $M_{1}$. It
follows that $M/M_{1}\cong M_{n-1}$ is indecomposable. Then, by induction
hypothesis $N/M_{1}=M_{j}/M_{1}$ and, so, $M_{j}=N$, as we wanted to verify. ∎

######  Proposition 5.3.

Let $Q$ be a finite-dimensional hereditary algebra of infinite representation
type. Then, its preprojective components, as well as its preinjective
components, are $e$-discrete.

###### Proof.

We only consider the case of a preprojective component ${\cal P}$, the other
case is tackled dually. If $M\in{\cal P}$, there is a projective module
$P_{M}\in{\cal P}$ and a non-negative number $\nu(M)$, such that
$(DTr)^{\nu(M)}M=P_{M}$. We also know, that ${\rm End}_{Q}(M)\cong{\rm
End}_{Q}(P_{M})$ is a division algebra, see [1, VIII.1.(1.5)]. Therefore, from
(4.3), we obtain $\hbox{\rm endol}\,(M)=\dim_{k}M/\dim_{k}{\rm
End}_{Q}(P_{M})$. This implies that there is only a finite number of
indecomposables in ${\cal P}$ with the same endolength iff there is only a
finite number of indecomposables in ${\cal P}$ with the same dimension. That
is ${\cal P}$ is $e$-discrete iff it is discrete.

So, we have to show that there are only finitely many indecomposables in
${\cal P}$ with the same dimension. For this, we can assume that $Q$ is basic
and consider a decomposition of the unit $1=\sum_{i=1}^{n}e_{i}$ of $Q$, as a
sum of primitive orthogonal idempotents, denote by
$\underline{\dim}(M)=(\dim_{k}e_{1}M,\ldots,\dim_{k}e_{n}M)$, _the dimension
vector of_ $M$. Recall that given $M,N\in{\cal P}$, we have that $M\cong N$
iff their their dimension vectors coincide, see [1, VIII.2.(2.3)]. Then,
consider the homological quadratic form $q$ of the hereditary algebra $Q$,
which satisfies

$q(\underline{\dim}(M))=\dim_{k}{\rm End}_{Q}(M)-\dim_{k}{\rm
Ext}_{Q}^{1}(M,M).$

Since $M$ is preprojective, we have ${\rm Ext}_{Q}^{1}(M,M)=0$, see [1,
VIII.1.(1.7)]. Therefore, we get that for any $M\in{\cal P}$,
$q(\underline{\dim}(M))\in\\{d_{1},\ldots,d_{n}\\}$, where
$d_{i}:=\dim_{k}{\rm End}_{Q}(Qe_{i})$, for $i\in[1,n]$. There are only
finitely many possible dimension vectors $\underline{\dim}(M)$ satisfying
this, so ${\cal P}$ is discrete. ∎

The following statement is similar to [3, (6.8)].

######  Proposition 5.4.

If ${\mathtt{Q}}$ is a generically tame minimal algebra of infinite
representation type, then there is a bounded principal ideal domain $\Gamma$
which is not $e$-discrete and an exact full and faithful functor
$\Psi:\Gamma\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm Mod}$. The functor
$\Psi$ restricts to a representation embedding
$\Psi^{\prime}:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm mod}_{\cal I}$.

###### Proof.

This proof is an adaptation of the proof of [3, (6.8)]. If ${\mathtt{Q}}$ is
of the second type in (1.3). From [3, (6.2)], we can assume that
${\mathtt{Q}}=D[x,s]$, that is the skew polynomial algebra with coefficients
in $D$, for some automorphism $s:D\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}D$. It is well known that $\Gamma:=D[x,s]$
is a bounded principal ideal domain (see [17, (3.15)]) and taking as $\Psi$
the identity functor, we see, from (6.6), that our proposition holds in this
case.

The generically tame minimal algebras ${\mathtt{Q}}$ of infinite
representation type of the first type in (1.3) can be described as follows,
see [3, (6.6)]:

  1. (i)

${\mathtt{Q}}$ is the matrix algebra $\begin{pmatrix}F&0\\\ M&G\end{pmatrix}$,
where $F$ and $G$ are finite-dimensional division $k$-algebras and $M$ is a
simple $G\hbox{-}F$-bimodule where the field $k$ acts centrally. Moreover,
$\dim_{G}M=2=\dim M_{F}$; or

  2. (ii)

${\mathtt{Q}}$ is the matrix algebra $\begin{pmatrix}F&0\\\ M&G\end{pmatrix}$,
where $F$ and $G$ are finite-dimensional division $k$-algebras and $M$ is a
simple $G\hbox{-}F$-bimodule where the field $k$ acts centrally. Moreover,
$\dim_{G}M=4$ and $\dim M_{F}=1$, or $\dim_{G}M=1$ and $\dim M_{F}=4$.

In each one of these cases, it is shown in the proof of [3, (6.8)], using
previous work by V. Dlab and M. Ringel [15]&[14], and Crawley-Boevey [10],
that there is a bounded principal ideal domain $\Gamma$ and an exact full and
faithful functor $\Psi:\Gamma\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm Mod}$ mapping
indecomposable $\Gamma$-modules of finite length onto regular
${\mathtt{Q}}$-modules. Moreover, ${\mathtt{Q}}\hbox{-}{\rm
reg}\cong\Psi(\Gamma\hbox{-}{\rm mod})\coprod{\cal U}$, where ${\cal U}$ is a
uniserial subcategory of ${\mathtt{Q}}\hbox{-}{\rm mod}$ generated by a simple
regular module. Furthermore, from (7.3), we see that, for each
$d\in\mathbb{N}$, there is, up to isomorphism, at most one indecomposable in
${\cal U}$, having endolength $d$.

As we already know (by [2] or [12, (8.6)]), since ${\mathtt{Q}}$ has infinite
representation type, it is not $e$-discrete. In fact, we know that
${\mathtt{Q}}\hbox{-}{\rm reg}$ is not $e$-discrete, as a consequence of
(5.3). Thus, there is some $d\in\mathbb{N}$ and an infinite family of non-
isomorphic indecomposable modules $\\{\Psi(M(i))\\}_{i}$ in
${\mathtt{Q}}\hbox{-}{\rm reg}$ with endolength $\hbox{\rm
endol}\,({\Psi(M(i))})=d$. Since $\Psi$ controls endolength, see (4.9), there
is a subfamily $\\{M(j)\\}_{j}$ of $\\{M(i)\\}_{i}$ of non-isomorphic
indecomposables in $\Gamma\hbox{-}{\rm mod}$ with common endolength, and
$\Gamma$ is not $e$-discrete.

The statement on the restriction of $\Psi$ is due to the following. If
$N\in\Gamma\hbox{-}{\rm mod}$ is such that $\Psi(N)$ admits an indecomposable
injective direct summand $I$, say $\Psi(N)=I\oplus C$, from [5, (29.5)], we
know that there is an indecomposable direct summand $I^{\prime}$ of $N$ such
that $\Psi(I^{\prime})\cong I$. Now, by the description of almost split
sequences in $\Gamma\hbox{-}{\rm mod}$, see [3, (6.5)], we know that
$I^{\prime}$ appears at the left of an almost split sequence $\xi$. Then,
$\Psi(\xi)$ is an exact sequence in ${\mathtt{Q}}\hbox{-}{\rm mod}$ which must
split. Then, $\xi$ also splits, a contradiction. So, the functor $\Psi$
restricts to a functor $\Psi^{\prime}:\Gamma\hbox{-}{\rm
mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm mod}_{\cal I}$, as
wanted. ∎

Notice that in the preceding proposition, if ${\mathtt{Q}}$ is a minimal
algebra of type 2 or of type 1.(ii), the principal ideal domain $\Gamma$ is
centrally bounded.

If ${\mathtt{Q}}$ is a minimal algebra of the first type with infinite
representation type, then there are only two possibilities: either its
quadratic form is positive semidefinite (so ${\mathtt{Q}}$ is tame hereditary)
or its quadratic form is indefinite (so ${\mathtt{Q}}$ is wild hereditary).

######  Lemma 5.5.

If ${\mathtt{Q}}$ is a wild hereditary minimal algebra of the first type, then
there is a centrally bounded principal ideal domain $\Gamma$ which is not
$e$-discrete and an exact full and faithful functor $\Psi:\Gamma\hbox{-}{\rm
mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm mod}_{\cal I}$.

###### Proof.

From [12, (8.2) and (8.4)], there is a finite field extension $K$ of $k$ and a
${\mathtt{Q}}\hbox{-}K\langle x,y\rangle$-bimodule $T$ which is finitely
generated projective over $K\langle x,y\rangle$ and such that the tensor
product functor

$F:=T\otimes_{K\langle x,y\rangle}-:K\langle x,y\rangle\hbox{-}{\rm
Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm Mod}$

is full and faithful. Let $\phi:K\langle x,y\rangle\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}K[x]$ be the canonical projection and
consider the corresponding restriction functor $H:K[x]\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}K\langle
x,y\rangle\hbox{-}{\rm Mod}$. We have that $H\cong K[x]\otimes_{K[x]}-$, where
the $K\langle x,y\rangle\hbox{-}K[x]$-bimodule $K[x]$ is a left $K\langle
x,y\rangle$-module by restriction through $\phi$. Then, the composition
$FH:K[x]\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm Mod}$ is exact,
full and faithful, and can be realized as a tensor product by a bimodule which
is finitely generated projective by the right.

Since $T$ is finitely generated by the right, the functor $F$ restricts to the
subcategories of finite-dimensional modules, and so does the functor $H$, and
their composition $FH:K[x]\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm mod}$. Take the
principal ideal domain $\Gamma:=K[x]$ and notice, as in the proof of the last
lemma, that ${\rm Im}\,FH$ is contained in ${\mathtt{Q}}\hbox{-}{\rm
mod}_{\cal I}$, so $\Psi:=FH$ restricts to the desired exact full and faithful
functor. Here $\Gamma$ is not $e$-discrete, for instance because it is
particular case of (6.6). ∎

######  Lemma 5.6.

Assume that ${\mathtt{Q}}=T_{D}(V)$ is a minimal algebra of the second type.
Then, there is a centrally bounded principal ideal domain $\Gamma$ which is
not $e$-discrete and a full and faithful exact functor
$\Psi:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm mod}$.

###### Proof.

We just follow the proof of [3, (6.2)]. There are two cases.

_Case 1: $\dim_{D}V\geq 2$._ Here, by [12, (8.2) and (8.5)], there is a finite
field extension $K$ of $k$ and a ${\mathtt{Q}}\hbox{-}K\langle
x,y\rangle$-bimodule $Z$, which is free of finite rank over $K\langle
x,y\rangle$ such that the tensor product $Z\otimes_{K\langle
x,y\rangle}-:K\langle x,y\rangle\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\mathtt{Q}}\hbox{-}{\rm Mod}$ is full and
faithful, and we proceed as in the proof of (5.5) with $\Gamma=K[x]$, which is
not $e$-discrete.

_Case 2: $\dim_{D}V=1$._ Here, we know as mentioned in the proof of [3,
(6.2)], that ${\mathtt{Q}}$ is a skew polynomial algebra $D[x,s]$, for some
automorphism $s:D\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}D$.
This case was already considered in (5.4). We can take $\Gamma={\mathtt{Q}}$,
which is a centrally bounded principal ideal domain by [3, (6.4)]. ∎

Proof of Theorem (1.5): By (1.4), there is a minimal algebra ${\mathtt{Q}}$ of
infinite representation type and a functor $H$ of one of the following types:

  1. 1.

$H:{\mathtt{Q}}\hbox{-}{\rm mod}_{\cal I}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, when
${\mathtt{Q}}$ is a minimal algebra of the first type, or

  2. 2.

$H:{\mathtt{Q}}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, when
${\mathtt{Q}}$ is of the second type.

Here the functor $H$ is of the form $Z\otimes_{\mathtt{Q}}-$, for some
$\Lambda\hbox{-}{\mathtt{Q}}$-bimodule $Z$, which is finitely generated by the
right.

There are a few possibilities for ${\mathtt{Q}}$, considered in the preceding
lemmas (5.4), (5.6), and (5.5). In each case, there is a bounded principal
ideal domain $\Gamma$, which is not $e$-discrete, and a functor $\Psi$ which
is exact full and faithful, and we consider its composition
$G:=H\Psi:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$. The functor $G$
controls endolength of indecomposables because $\Psi$ and $H$ do so, by (4.9)
and (1.4). $\square$

##  6 The representation embedding

In this section we derive our main result (1.6), from the preceding work. For
this, we consider Ore localizations $S^{-1}\Gamma$, for some special
denominator subset $S$ of a principal ideal domain $\Gamma$, see [19, §3.1].

######  Remark 6.1.

Recall that any principal ideal domain $\Gamma$ is a unique factorization
domain in the following sense. Any non-zero non-unit element $a\in\Gamma$ is a
product $a=p_{1}\cdots p_{s}$ where $p_{1},\ldots,p_{s}$ are atoms of $\Gamma$
and for any other such factorization $a=q_{1}\cdots q_{t}$ as product of
atoms, we have $s=t$ and there is a permutation $\sigma\in S_{t}$ such that
$q_{i}$ is similar to $p_{\sigma(i)}$, for each $i$.

######  Lemma 6.2.

Let $\Gamma$ be a bounded principal ideal domain and let $b\in\Gamma$ be such
that $\Gamma b=b\Gamma\not=\Gamma$. Consider the denominator set
$S=\\{1,b,b^{2},\ldots\\}$. We have the epimorphism of rings
$\nu:\Gamma\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}S^{-1}\Gamma$ and the corresponding
restriction functor $F:S^{-1}\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\hbox{-}{\rm mod}$. Let $p$ be an atom
of $\Gamma$ which is not similar to any of the atomic factors of $b$, then
every indecomposable $\Gamma$-module $M$ with composition factors isomorphic
to $\Gamma/\Gamma p$ is of the form $M\cong F(N)$, for some $N\in
S^{-1}\Gamma\hbox{-}{\rm mod}$.

###### Proof.

Let $M\in\Gamma\hbox{-}{\rm mod}$ be indecomposable with all composition
factors isomorphic to $\Gamma/\Gamma p$. We will show by induction on the
length of $M$ that the linear map $\mu_{b}:M\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M$ given by multiplication by $b$ is a
linear isomorphism. This is enough, because we know that the structure of $M$
as a $\Gamma$-module is given by the morphism of $k$-algebras
$\mu:\Gamma\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{k}(M)$, mapping each $a\in\Gamma$ onto the linear map $\mu_{a}$, given
by multiplication by $a$. If $\mu_{b}$ is invertible, then $\mu_{s}$ is
invertible for each $s\in S$. Thus by the universal property of localizations,
we obtain a map $\hat{\mu}:S^{-1}\Gamma\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{k}(M)$, such that
$\hat{\mu}\nu=\mu$, that is a structure of $S^{-1}\Gamma$-module $N$ with
underlying vector space $M$. Thus, $F(N)=M$.

If we assume that $M$ is simple, then $M=\Gamma m$ with $m\in M$ and, if
$bm=0$, we would have $b\in\Gamma p$, which is not the case by (6.1). Hence,
$M=\Gamma m=\Gamma bm=b\Gamma m$. Then, the linear map
$\mu_{b}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ given by
multiplication by $b$ is surjective. Since $M$ is finite-dimensional, we
obtain that $\mu_{b}$ is a linear isomorphism. Then, our claim holds for
indecomposable $\Gamma$-modules of length 1.

Now, assume that the indecomposable module $M$ has length $l>1$ and consider
its composition series $0=M_{0}\subset M_{1}\subset\cdots\subset M_{l}=M$ in
$\Gamma\hbox{-}{\rm mod}$. By induction hypothesis, multiplication by $b$
determines linear automorphisms on each $M_{i}$, with $i<l$. From the proof of
(5.2), we get an exact sequence of $\Gamma$-modules $0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{l-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0,$ where $M_{l-1}$ is indecomposable. Then,
we have the following commutative diagram of linear maps

$\begin{matrix}0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{l-1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0\\\ &&\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle$}}$}\big{\downarrow}&&\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle$}}$}\big{\downarrow}&&\hbox
to0.0pt{\hss$\vbox{\hbox{$\scriptstyle$}}$}\big{\downarrow}&&\\\
0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{l-1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0,\\\ \end{matrix}$

where the vertical arrows are given by multiplication by $b$. By induction
hypothesis, the left and right vertical arrows are linear isomorphisms, then
so is the central one. ∎

Proof of theorem (1.6): By (1.5), there is a bounded principal ideal domain
$\Gamma$ which is not $e$-discrete and a functor

$G:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod},$

which preserves indecomposables, reflects isomorphism classes, and controls
endolength. The functor $G$ is of the form $Z\otimes_{\Gamma}-$, for some
$\Lambda\hbox{-}\Gamma$-bimodule $Z$ which is finitely generated by the right.

The right $\Gamma$-module $Z$ decomposes as a direct sum $Z=T\oplus P$, where
$T$ is a torsion finitely generated $\Gamma$-module and $P$ is a free
$\Gamma$-module of finite rank, see [9, (1.4.4)]. If $T=0$, we are done.

Assume that $T\not=0$, then $T$ decomposes by [8, (1.5.5)] as a finite direct
sum of indecomposables of the form $E_{i}^{p}$. Then, the annihilator of $T$
is a proper ideal of $\Gamma$ which is of the form $b\Gamma=\Gamma b$, for
some non-unit element $b\in\Gamma$.

We know that each $E_{i}^{p}$ has annihilator $q_{p}^{i}\Gamma$, where
$q_{p}\Gamma$ is the annihilator of $E_{1}^{p}$. Then, $T$ is annihilated by
the product of the elements $q_{p}^{i}$, corresponding to the indecomposables
$E_{i}^{p}$ appearing in the decomposition of $T$. Thus, $b\not=0$.

Now, we consider the multiplicative subset $S=\\{1,b,b^{2},\ldots\\}$ of
$\Gamma$, which is a right denominator set and a left denominator set of
$\Gamma$. The Ore localization $S^{-1}\Gamma\cong\Gamma S^{-1}$ is a bounded
principal ideal domain. The epimorphism $\nu:\Gamma\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}S^{-1}\Gamma$ induces the full and faithful
restriction functor $F:S^{-1}\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\hbox{-}{\rm mod}$ which is isomorphic
to $S^{-1}\Gamma\otimes-$. Thus, the composition $GF:S^{-1}\Gamma\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm
mod}$ preserves indecomposables, reflects isomorphism classes, and is
isomorphic to $Z\otimes_{\Gamma}S^{-1}\Gamma\otimes-$. We claim that
$Z\otimes_{\Gamma}S^{-1}\Gamma$ is a free right $S^{-1}\Gamma$-module. For
this, it will be enough to show that $T\otimes_{\Gamma}S^{-1}\Gamma=0$. Since
$Tb=0$, and $b$ is invertible in $S^{-1}\Gamma$, we have
$0=Tb\otimes_{\Gamma}S^{-1}\Gamma=T\otimes_{\Gamma}bS^{-1}\Gamma=T\otimes_{\Gamma}S^{-1}\Gamma$.

The principal ideal domains $S^{-1}\Gamma$ appearing before are not
$e$-discrete. Indeed, the atoms of $S^{-1}\Gamma$ correspond to those non-
similar to the atomic factors of $b$. Notice that, from (6.2), we obtain that
the restriction functor $F:S^{-1}\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\hbox{-}{\rm mod}$ identifies the
category $S^{-1}\Gamma\hbox{-}{\rm mod}$ with the full subcategory
$\Gamma\hbox{-}{\rm mod}_{\cal Z}$ of $\Gamma\hbox{-}{\rm mod}$ formed by all
the $\Gamma$-modules without direct summands isomorphic to $E_{i}^{p}$, with
$i\in\mathbb{N}$ and $p$ an atomic factor of $b$. Thus, since $\Gamma$ is not
$e$-discrete, neither is $S^{-1}\Gamma$. $\square$

######  Remark 6.3.

In the case where the ground field is the field $\mathbb{R}$ of real numbers
it is possible to make more explicit statements. In order to state them, we
fix some notation: we consider the field $\mathbb{C}$ of complex numbers and
the real quaternions $\mathbb{H}$; the skew polynomial algebras
$\mathbb{R}[x]$, $\mathbb{C}[x]$, $\mathbb{H}[x]$, and $\mathbb{C}[x,\sigma]$,
where $\sigma$ is the complex conjugation; and the principal ideal domain
$\mathbb{D}=\mathbb{R}[x,y]/\langle y^{2}+x^{2}+1\rangle$.

Let us describe the $\mathbb{R}$-algebras which have a leading role in our
arguments. This description relies heavily on the important work by Dlab and
Ringel on hereditary algebras, we cite [4], but refer the reader to its
bibliography. In this case, up to isomorphism, the generically tame finite-
dimensional minimal algebras of infinite-representation type are
$\begin{pmatrix}\mathbb{R}&0\\\ \mathbb{H}&\mathbb{H}\\\ \end{pmatrix}$ and
$\begin{pmatrix}\mathbb{H}&0\\\ \mathbb{H}&\mathbb{R}\\\ \end{pmatrix}$, see
[4, (9.5)]. The skew polynomial $\mathbb{R}$-algebras, are, up to isomorphism,
the following $\mathbb{R}[x]$, $\mathbb{C}[x]$, $\mathbb{H}[x]$, and
$\mathbb{C}[x,\sigma]$. These are the minimal algebras $Q$ which may appear in
(4.8) or (1.4). The bounded principal ideal domains that appear in theorem
(1.5) are $\mathbb{R}[x]$, $\mathbb{C}[x]$, $\mathbb{H}[x]$,
$\mathbb{C}[x,\sigma]$, and $\mathbb{D}$. Here, we can see [4, (9.6)] as an
illustration of (5.4). The bounded principal ideal domains in (1.6), are Ore
localizations of the preceding ones.

The wild hereditary case, addressed in (5.5) and (5.6), yields only the
principal ideal domains $\mathbb{R}[x]$ and $\mathbb{C}[x]$. The algebra
$\mathbb{H}[x]$ may only appear in (1.4) if $\mathbb{H}$ is one of the
coefficient algebras of $\Lambda$, see the proof of (4.8)Case 2.a.

The following proposition is probably known, but we could not find any
reference.

######  Proposition 6.4.

Let $\Gamma$ be a principal ideal domain which is not a field and such that
its center $Z$ is again a principal ideal domain. Then if the set of maximal
ideals of $Z$ has cardinality $C$, any complete set ${\cal P}$ of
representatives of the non-similar atoms of $\Gamma$ has at least the same
cardinality $C$.

###### Proof.

First notice that any non-trivial bilateral element $b\in\Gamma$ has the
property that any factor in a factorization of $b$ in $\Gamma$ is a left
divisor and a right divisor of $b$. Indeed, if $b=ac$ in $\Gamma$, we have
$bc=c^{\prime}b$, for some $c^{\prime}\in\Gamma$, thus
$ac^{2}=bc=c^{\prime}b=c^{\prime}ac$, and therefore, $b=ac=c^{\prime}a$. Thus,
any left divisor $a$ of $b$ is a right divisor of $b$. Similarly, one shows
that any right divisor of $b$ is also a left divisor of $b$. Now, assume that
$b=apc$, a factorization in $\Gamma$. As before, we obtain $b=(pc)a^{\prime}$
and $b=c^{\prime}(ap)$, for some $a^{\prime},c^{\prime}\in\Gamma$, as claimed.

Now, consider the collection ${\cal M}(\Gamma)$ of maximal (two sided) ideals
of $\Gamma$ and let us exhibit an injective map $\alpha:{\cal
M}(\Gamma)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal P}$.
Given $\Gamma b\in{\cal M}(\Gamma)$, by [17, (1.2.19)], the atomic factors of
$b$ belong to the same similarity class. We can assume that some atomic factor
$p$ of $b$ belongs to ${\cal P}$ and set $\alpha(\Gamma b)=p$. As remarked
before, $p$ is a right factor of $b$, so $\Gamma b\subseteq{\rm
Ann}_{\Gamma}(\Gamma/\Gamma p)$. Since $\Gamma/\Gamma p\not=0$, we get $\Gamma
b={\rm Ann}_{\Gamma}(\Gamma/\Gamma p)$.

Assume that $\Gamma b^{\prime}$ is a maximal ideal of $\Gamma$ different from
$\Gamma b$ and $p^{\prime}$ is an atomic factor of $b^{\prime}$ with $p$
similar to $p^{\prime}$. Thus $\Gamma/\Gamma p\cong\Gamma/\Gamma p^{\prime}$,
and their annihilators coincide, which is not possible. So, $\alpha$ is an
injective map.

It remains to show that there is an injection $\beta:{\cal
M}(Z)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal M}(\Gamma)$,
where ${\cal M}(Z)$ denotes the collection of maximal ideals of $Z$. We may
assume that the principal ideal domain $Z$ is not a field. Assume that $Zz$ is
a maximal ideal of $Z$. Then, $z$ has a factorization $z=a_{1}a_{2}\cdots
a_{s}$, where $a_{1},\ldots,a_{s}\in\Gamma$ are bilateral elements and $\Gamma
a_{s}$ is a maximal ideal of $\Gamma$. We choose such factor $a_{s}$ of $z$
and define $\beta(Zz)=\Gamma a_{s}$. Now, assume that $Zz^{\prime}$ is another
maximal ideal of $Z$ with $Zz^{\prime}\not=Zz$, factorize
$z^{\prime}=a^{\prime}_{1}a^{\prime}_{2}\cdots a^{\prime}_{t}$ as before and
let us see that $\Gamma a_{s}\not=\Gamma a^{\prime}_{t}$. If $\Gamma
a_{s}=\Gamma a^{\prime}_{t}$, then $a_{s}=ua^{\prime}_{t}$, for some unit
$u\in\Gamma$. By the first paragraph of this proof, $a_{s}$ divides by the
right $z$ and $z^{\prime}$. But since $Zz+Zz^{\prime}=Z$, we have
$1=cz+c^{\prime}z^{\prime}$ which implies that $a_{s}$ divides $1$, a
contradiction. ∎

######  Lemma 6.5.

Let $\Gamma$ be a principal ideal domain with center $Z$. Assume that $\Gamma$
is finitely generated as a $Z$-module, say by $m$ elements. Let $\Gamma b$ be
a maximal ideal of $\Gamma$ and set $I=Z\cap\Gamma b$. Then, for any atom
$p\in\Gamma$ dividing $b$, we have

$\hbox{\rm endol}\,(\Gamma/\Gamma p)\leq m\times\ell_{Z}(Z/I),$

where $\ell_{Z}$ denotes the length as a $Z$-module.

###### Proof.

There is a surjective morphism of $Z$-modules $Z^{m}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma$, which induces a surjective morphism
$(Z/I)^{m}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Gamma/\Gamma
b$, so $\ell_{Z}(\Gamma/\Gamma b)\leq m\times\ell_{Z}(Z/I)$.

The morphism of algebras $\rho:Z\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\Gamma}(\Gamma/\Gamma b)^{op}$
which maps any $z$ onto the right multiplication by $z$ on $\Gamma/\Gamma b$
induces on $\Gamma/\Gamma b$ the natural action of $Z$ on $\Gamma/\Gamma b$.
This implies that any composition series for the ${\rm
End}_{\Gamma}(\Gamma/\Gamma b)^{op}$-module $\Gamma/\Gamma b$ is a $Z$-module
filtration of the $Z$-module $\Gamma/\Gamma b$. Therefore, we have $\hbox{\rm
endol}\,(\Gamma/\Gamma b)\leq\ell_{Z}(\Gamma/\Gamma b)$.

Since $\Gamma b$ is a maximal ideal of $\Gamma$, we get that $\Gamma/\Gamma b$
is a simple ring. As recalled before, all atomic factors of $b$ are similar.
Thus $\Gamma/\Gamma b$ has a simple left ideal. Hence, $\Gamma/\Gamma b\cong
M_{n}(D)$, for some $n\in\mathbb{N}$ and a finite-dimensional division
$k$-algebra $D$. Moreover, we can take $D={\rm End}_{\Gamma}(S)^{op}$, where
$S$ is, up to isomorphism, the unique simple left $\Gamma/\Gamma b$-module.
Here, $S=M_{1\times n}(D)$ is a simple left $M_{n}(D)$-module with the usual
matrix multiplication. So there is an isomorphism $D^{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}S$ of right $D$-modules.

As before, we have $\hbox{\rm endol}\,(\,_{\Gamma/\Gamma
b}S)\leq\ell_{D}(S_{D})=n$. Since $p$ is an atom, the left $\Gamma/\Gamma
b$-module $\Gamma/\Gamma p$ is simple and we have $\Gamma/\Gamma p\cong S$. We
have $\Gamma/\Gamma b$ is isomorphic to $S^{n}$ and we know that for a finite
direct sum, the endolength of the summands is always bounded by the endolength
of the sum. Then,

$\hbox{\rm endol}\,(\Gamma/\Gamma p)=\hbox{\rm endol}\,(S)\leq\hbox{\rm
endol}\,(\Gamma/\Gamma b)\leq\ell_{Z}(\Gamma/\Gamma b)\leq
m\times\ell_{Z}(Z/I).$

∎

######  Remark 6.6.

It is known that the principal ideal domain $\Gamma=D[x,s]$ of twisted
polynomials over a finite-dimensional division algebra is not $e$-discrete,
see [12, (8.6)]. This can also be derived from the preceding two statements as
follows.

From [17, (1.1.22)], the center of $\Gamma$ has the form $Z=F[t]$, a
polynomial algebra over a finite field extension $F$ of $k$, and $\Gamma$ is
finitely generated over $Z$. So $Z$ is a principal ideal domain with
infinitely many prime ideals.

Recall from, the proof of (6.4), that $\Gamma$ admits an infinite set of non-
similar atoms ${\cal P}$. Each $p\in{\cal P}$ appears as an atomic factor of
some two-sided element $b\in\Gamma$ such that $\Gamma b$ is a maximal ideal of
$\Gamma$. Since $\Gamma b$ is maximal, it is a prime ideal of $\Gamma$, so
$I=Z\cap\Gamma b$ is a prime (thus maximal) ideal of $Z$, and $Z/I$ is a
simple $Z$-module. Therefore, from (6.5), we obtain that $\hbox{\rm
endol}\,(\Gamma/\Gamma p)\leq m\times\ell_{Z}(Z/I)=m$, where $m$ is the number
of generators of the $Z$-module $\Gamma$.

Thus, there are infinitely many simple $\Gamma$-modules of the form
$\Gamma/\Gamma p$, which are in fact all finite-dimensional, with the same
endolength, and $\Gamma$ is not $e$-discrete.

##  7 The property EBTII

In this section, we study some useful relations between the endolengths of
indecomposable modules filtered by the same simple object and the property
EBTII for bounded principal ideal domains.

######  Proposition 7.1.

Assume that ${\cal U}$ is a uniserial subcategory of $\Sigma\hbox{-}{\rm mod}$
for some $k$-algebra $\Sigma$ and that we have a family $\\{M_{n}\mid
n\in\mathbb{N}\\}$ of representatives of the isoclasses of the indecomposable
$\Sigma$-modules in ${\cal U}$. Moreover, assume that we have almost split
sequences in the category ${\cal U}$ of the form:

$\begin{matrix}\xi_{1}:\ M_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{1},\\\ \,\\\ \zeta_{n}:\
M_{n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{n+1}\oplus
M_{n-1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{n},\hbox{ for
}n\geq 2.\\\ \end{matrix}$

Set $E_{M_{n}}={\rm End}_{\Sigma}(M_{n})^{op}$ and $K_{M_{n}}={\rm
End}_{\Sigma}(M_{n})^{op}/{\rm Rad}{\rm End}_{\Sigma}(M_{n})^{op}$. Then,
there is an isomorphism of $k$-algebras $K_{M_{n}}\cong K_{M_{1}}=E_{M_{1}}$,
for all $n\in\mathbb{N}$.

###### Proof.

Assume that $n\geq 2$. From our assumptions, we get that there is a chain of
injective irreducible morphisms

$M_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\alpha_{1}}}M_{2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\cdots\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\alpha_{n-1}}}M_{n}$

and a chain of surjective irreducible morphisms

$M_{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\beta_{n-1}}}M_{n-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\cdots\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\beta_{1}}}M_{1},$

such that
$\beta_{1}\alpha_{1}=0;\alpha_{1}\beta_{1}=\beta_{2}\alpha_{2};\,\cdots;\beta_{i}\alpha_{i}=\alpha_{i-1}\beta_{i-1};\cdots$.
Since $M_{n}$ is uniserial, we have that each $f\in{\rm End}_{\Sigma}(M_{n})$
induces a commutative diagram

$\begin{matrix}0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{n-1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\alpha_{n-1}}}&M_{n}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\hat{\beta}_{n}}}&M_{1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0\\\ &&\big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle f_{n-1}$}}$\hss}&&\big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle f$}}$\hss}&&\big{\downarrow}\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle f_{1}$}}$\hss}&&\\\ 0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{n-1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\alpha_{n-1}}}&M_{n}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\hat{\beta}_{n}}}&M_{1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0\\\ \end{matrix}$

where $\hat{\beta}_{n}=\beta_{1}\beta_{2}\cdots\beta_{n-1}$. The recipe
$f\longmapsto f_{1}$ defines a morphism of algebras
$\varphi_{n}:E_{M_{n}}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}E_{M_{1}}$. Since $E_{M_{1}}$ is a division
algebra and ${\rm Rad}E_{M_{n}}$ consists of nilpotent morphisms, the map
$\varphi_{n}$ induces an injective morphism of division algebras
$\overline{\varphi}_{n}:K_{M_{n}}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}K_{M_{1}}$. It only remains to show that
$\overline{\varphi}_{n}$ is surjective or, equivalently, that $\varphi_{n}$ is
surjective. We proceed to a proof by induction on $n$.

If $n=2$, we have an almost split sequence

$\begin{matrix}0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\alpha_{1}}}&M_{2}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\beta_{1}}}&M_{1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0.\\\ \end{matrix}$

Given any morphism $f_{1}:M_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{1}$, the composition
$f_{1}\beta_{1}:M_{2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{1}$ is not an isomorphism. Hence, there
is a morphism $f:M_{2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{2}$ such that
$\beta_{1}f=f_{1}\beta_{1}$. So $\varphi_{2}$ is surjective.

Assume that $n\geq 3$ and that $\varphi_{n-1}$ is surjective, and let us show
that $\varphi_{n}$ is surjective. Consider the almost split sequence

$\begin{matrix}0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M_{n-1}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\ (\alpha_{n-1},\beta_{n-2})^{t}\
}}&M_{n}\oplus M_{n-2}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\ (\beta_{n-1},\alpha_{n-2})\
}}&M_{n-1}&\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}&0.\\\
\end{matrix}$

Given $f_{1}\in E_{M_{1}}$, by induction hypothesis, there is some $f_{n-1}\in
E_{M_{n-1}}$ with $f_{1}\hat{\beta}_{n-1}=\hat{\beta}_{n-1}f_{n-1}$. Consider
the composition $f_{n-1}\beta_{n-1}:M_{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n-1}$, which is not an isomorphism, thus
there is a morphism $h:M_{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n}\oplus M_{n-2}$ such that
$(\beta_{n-1},\alpha_{n-2})h=f_{n-1}\beta_{n-1}$. Denote the matrix components
of $h$ as $h=(f_{n},g)^{t}$. So, we get
$\beta_{n-1}f_{n}+\alpha_{n-2}g=f_{n-1}\beta_{n-1}$. Therefore, we have

$\hat{\beta}_{n-1}\beta_{n-1}f_{n}+\hat{\beta}_{n-1}\alpha_{n-2}g=\hat{\beta}_{n-1}f_{n-1}\beta_{n-1}=f_{1}\hat{\beta}_{n-1}\beta_{n-1}=f_{1}\hat{\beta}_{n}.$

Moreover, we have

$\hat{\beta}_{n-1}\alpha_{n-2}g=\beta_{1}\cdots\beta_{n-2}\alpha_{n-2}g=\beta_{1}\cdots\beta_{n-3}\alpha_{n-3}\beta_{n-3}g=\beta_{1}\alpha_{1}\beta_{1}\cdots\beta_{n-3}g=0.$

Finally, we obtain
$\hat{\beta}_{n}f_{n}=\hat{\beta}_{n-1}\beta_{n-1}f_{n}=f_{1}\hat{\beta}_{n}$,
so $\varphi_{n}$ is surjective. ∎

Given a bounded principal ideal domain $\Gamma$, we can apply this proposition
to the uniserial subcategories ${\cal U}_{p}$ which form $\Gamma$-mod in
(5.2), see [3, (6.5)]. So we have the following.

######  Corollary 7.2.

Let $\Gamma$ be a bounded principal ideal domain. For any indecomposable
finite-dimensional $\Gamma$-module $M$, set $E_{M}={\rm End}_{\Gamma}(M)^{op}$
and $K_{M}={\rm End}_{\Gamma}(M)^{op}/{\rm Rad}{\rm End}_{\Gamma}(M)^{op}$.
Let $S$ denote the simple socle of $M$, see (5.2). Then, there is an
isomorphism of $k$-algebras $K_{M}\cong K_{S}=E_{S}$.

######  Proposition 7.3.

Assume that ${\cal U}$ is a uniserial subcategory of $\Sigma\hbox{-}{\rm mod}$
satisfying the assumptions of (7.1). Then, if $M_{n}$ is an indecomposable
$\Sigma$-module in ${\cal U}$ with length $n$ and socle $M_{1}$, we have
$\hbox{\rm endol}\,(M_{n})=n\times\hbox{\rm endol}\,(M_{1})$.

Thus, given $d\in\mathbb{N}$, the category ${\cal U}$ has at most one
indecomposable with endolength $d$.

###### Proof.

From the last proposition, we know that $\dim_{k}K_{M_{n}}=\dim_{k}K_{M_{1}}$,
Then, from (4.3), we have

$\begin{matrix}\hbox{\rm
endol}\,(M_{n})&=&\dim_{k}M_{n}/\dim_{k}K_{M_{n}}\hfill\\\
&=&(n\times\dim_{k}M_{1})/\dim_{k}K_{M_{1}}\hfill\\\
&=&n\times\dim_{k}M_{1}/\dim_{k}K_{M_{1}}=n\times\hbox{\rm
endol}\,(M_{1}).\hfill\\\ \end{matrix}$

∎

######  Corollary 7.4.

Assume that $\widetilde{\cal U}$ is a full subcategory of $\Sigma\hbox{-}{\rm
mod}$, for some $k$-algebra $\Sigma$, such that $\widetilde{\cal
U}=\coprod_{p\in{\cal P}}{\cal U}_{p}$, where each ${\cal U}_{p}$ is a
uniserial subcategory satisfying the assumptions of (7.1). Then, if
$\widetilde{\cal U}$ is not $e$-discrete, the index set ${\cal P}$ is infinite
and $\widetilde{\cal U}$ satisfies EBTII.

###### Proof.

If the subcategory $\widetilde{\cal U}$ is not $e$-discrete, there is some
number $d$ and an infinite family $\\{M(i)\\}_{i}$ of non-isomorphic finite-
dimensional indecomposable $\Sigma$-modules in $\widetilde{\cal U}$ with
common endolength $\hbox{\rm endol}\,(M(i))=d$. Denote by $S(i)$ the simple
socle of each $\Sigma$-module $M(i)$ and by $n_{i}$ its length. For all $i$,
we have

$d=\hbox{\rm endol}\,(M(i))=n_{i}\times\hbox{\rm endol}\,(S(i)),$

and the endolength of all these simple modules is bounded by $d$. If
$S(i)\cong S(j)$, for some $i$, we would have

$n_{i}\times\hbox{\rm endol}\,(S(i))=\hbox{\rm endol}\,(M(i))=d=\hbox{\rm
endol}\,(M(j))=n_{j}\times\hbox{\rm endol}\,(S(j)).$

Thus, $M(i)$ and $M(j)$ have the same length and the same socle, which implies
that they are isomorphic, thus $i=j$. So, there is some $d_{1}<d$ and an
infinite family of non-isomorphic simple $\Sigma$-modules $S(i)$ with the same
endolength $d_{1}$.

Hence, for each one of these simples $S(i)$, the indecomposable module
$N_{n}(i)$ with length $n$ and socle $S(i)$ has endolength $\hbox{\rm
endol}\,(N_{n}(i))=n\times d_{1}$. The infinite family of non-isomorphic
finite-dimensional indecomposable $\Sigma$-modules $\\{N_{n}(i)\\}_{i}$ in
$\widetilde{\cal U}$ with endolength $n\times d_{1}$ can be constructed for
any $n$. ∎

######  Corollary 7.5.

Let $\Gamma$ be a bounded principal ideal domain. If $M_{n}$ is the
indecomposable $\Gamma$-module with length $n$ and socle $M_{1}$, we have

$\hbox{\rm endol}\,(M_{n})=n\times\hbox{\rm endol}\,(M_{1}).$

Moreover, if $\Gamma$ is not $e$-discrete it satisfies satisfies EBTII.

###### Proof.

It follows from [3, (6.5)], (7.4), and (7.2). ∎

######  Lemma 7.6.

Assume that a functor $G:\Sigma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$ preserves
indecomposables, reflects isomorphism classes, and controls endolength of
indecomposables. Then, if EBTII holds for $\Sigma\hbox{-}{\rm mod}$, it also
holds for $\Lambda\hbox{-}{\rm mod}$.

###### Proof.

By assumption, there are $c,c^{\prime}\in\mathbb{N}$ such that, for any
indecomposable $M\in\Sigma\hbox{-}{\rm mod}$, we have

$\hbox{\rm endol}\,(M)\leq c\times\hbox{\rm endol}\,(GM)\hbox{ \ and \
}\hbox{\rm endol}\,(GM)\leq c^{\prime}\times\hbox{\rm endol}\,(M),$

and there are infinitely many endolengths $d$’s such that for each one of
these $d$’s there are infinitely many non-isomorphic finite-dimensional
indecomposable $\Sigma$-modules with the same endolength $d$.

If we start with such an infinite family $\\{M_{i}\\}_{i}$ in
$\Sigma\hbox{-}{\rm mod}$ with $\hbox{\rm endol}\,(M)=d_{1}$, we obtain an
infinite subfamily $\\{GM_{j}\\}_{j}$ of the family of images of
$\\{GM_{i}\\}_{i}$ with the same endolength $\hbox{\rm
endol}\,(GM_{j})=d^{\prime}_{1}\leq c^{\prime}d_{1}$, because $\hbox{\rm
endol}\,(GM_{i})\leq c^{\prime}\times\hbox{\rm endol}\,(M_{i})$.

Now, by assumption, we can choose a sufficiently large endolength $d_{2}$ (say
with $d_{2}>cd^{\prime}_{1}$) for which there is an infinite family of non-
isomorphic finite-dimensional indecomposables $\\{N_{i}\\}_{i}$ in
$\Sigma\hbox{-}{\rm mod}$ with $\hbox{\rm endol}\,(N_{i})=d_{2}$. As before,
we obtain an infinite subfamily $\\{GN_{j}\\}_{j}$ of the family of images
$\\{GN_{i}\\}_{i}$ with common endolength $\hbox{\rm
endol}\,(GN_{j})=d^{\prime}_{2}\leq c^{\prime}d_{2}$, again because $\hbox{\rm
endol}\,(GN_{i})\leq c^{\prime}\times\hbox{\rm endol}\,(N_{i})$. Since $G$
controls endolength of indecomposables, we obtain

$cd^{\prime}_{1}<d_{2}=\hbox{\rm endol}\,(N_{i})\leq c\times\hbox{\rm
endol}\,(GN_{i})=cd^{\prime}_{2}.$

Thus, $d^{\prime}_{1}<d^{\prime}_{2}$. We can iterate the preceding procedure
to construct inductively endolengths $d^{\prime}_{1}<d^{\prime}_{2}<\cdots$
and families of non-isomorphic finite-dimensional indecomposables in
$\Lambda\hbox{-}{\rm mod}$ with the same endolength $d^{\prime}_{i}$, for each
$d^{\prime}_{i}$. ∎

Proof of Theorem (1.8): From (1.6), we have a representation embedding
$G:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$, where $\Gamma$ is
a bounded principal ideal domain, which is not $e$-discrete. The algebra
$\Gamma$ satisfies EBTII as a consequence of (7.5). Our theorem follows from
(7.6), because $G$ controls endolength of indecomposables. $\square$

In the following proposition we show that the formula relating the endolength
of indecomposables in $\Gamma\hbox{-}{\rm mod}$ described in (7.5), is
preserved by the exact embedding $G:\Gamma\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm
mod}$ of (1.6).

######  Proposition 7.7.

Let $G:\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm mod}$ be the
representation embedding of (1.6). Consider a family of indecomposable
$\Gamma$-modules $\\{M_{n}\\}_{n\in\mathbb{N}}$ such that each $M_{n}$ has
socle $M_{1}$ and length $n$. Then, we have

$\hbox{\rm endol}\,(G(M_{n}))=n\times\hbox{\rm endol}\,(G(M_{1})).$

###### Proof.

By construction, the functor $G$ is the following composition

$\Gamma\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\Psi}}{\mathtt{Q}}\hbox{-}{\rm
mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{{\mathtt{L}}_{{\mathtt{Q}}}}}{\cal
Q}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F}}{\cal D}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\Xi}}{\cal
P}^{1}(\Lambda)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{{\rm
Cok}}}\Lambda\hbox{-}{\rm mod},$

where ${\cal Q}$ is a quasi-minimal ditalgebra, ${\mathtt{Q}}$ is its minimal
algebra, $\Psi$ is a full and faithful functor, $F$ is a composition of
reduction functors, so it is full and faithful, and ${\rm Cok}$ is a full
functor.

_Claim 1: There is an isomorphism of $k$-algebras $K_{G(M_{n})}\cong
K_{G(M_{1})}$._

We know that ${\rm End}_{\Gamma}(M_{n})^{op}=K_{M_{n}}\oplus{\rm Rad}{\rm
End}_{\Gamma}(M_{n})^{op}$. Since $\Psi$ is full and faithful, we have ${\rm
End}_{\mathtt{Q}}(\Psi(M_{n}))^{op}=\Psi(K_{M_{n}})\oplus{\rm Rad}{\rm
End}_{\mathtt{Q}}(\Psi(M_{n}))^{op}$, for all $n$. Moreover, from (4.2), we
know that ${\rm End}_{\cal
Q}({\mathtt{L}}_{{\mathtt{Q}}}(\Psi(M_{n})))^{op}={\mathtt{L}}_{{\mathtt{Q}}}(K_{\Psi(M_{n})})\oplus{\rm
Rad}{\rm End}_{\cal Q}({\mathtt{L}}_{{\mathtt{Q}}}(\Psi(M_{n})))^{op}$. Since
the functors $F$ and $\Xi$ are full and faithful, we get

${\rm End}_{{\cal P}^{1}(\Lambda)}(\Xi
F{\mathtt{L}}_{{\mathtt{Q}}}(\Psi(M_{n})))^{op}=\Xi
F{\mathtt{L}}_{{\mathtt{Q}}}\Psi(K_{M_{n}})\oplus{\rm Rad}{\rm End}_{{\cal
P}^{1}(\Lambda)}(\Xi F{\mathtt{L}}_{{\mathtt{Q}}}\Psi(M_{n}))^{op}.$

Using that the functor ${\rm Cok}$ is full, we obtain

${\rm End}_{\Lambda}(G(M_{n}))^{op}={\rm Cok}\Xi
F{\mathtt{L}}_{{\mathtt{Q}}}\Psi(K_{M_{n}})\oplus{\rm Rad}{\rm
End}_{\Lambda}({\rm Cok}\Xi F{\mathtt{L}}_{{\mathtt{Q}}}\Psi(M_{n}))^{op}.$

Hence, ${\rm Cok}\Xi F{\mathtt{L}}_{{\mathtt{Q}}}\Psi(K_{M})$ is a division
algebra isomorphic to $K_{G(M_{n})}$. Thus, for each $n$, we have
$K_{M_{n}}\cong K_{G(M_{n})}$, for all $n$. From (7.2), we know that
$K_{M_{n}}\cong K_{M_{1}}$. Thus the formula of our Claim 1 holds.

_Claim 2: We have $\dim_{k}G(M_{n})=n\times\dim_{k}G(M_{1})$._

Indeed, for $n\geq 2$, we have exact sequences in $\Gamma\hbox{-}{\rm mod}$ of
the form

$0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{n}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0,$

which are mapped by the exact functor $G$ on the exact sequences

$0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}G(M_{n-1})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}G(M_{n})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}G(M_{1})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0$

in $\Lambda\hbox{-}{\rm mod}$. The the announced formula follows by an easy
induction from the additivity of dimension for exact sequences.

Finally, using the preceding Claim 1, Claim 2, and (4.3), we have

$\begin{matrix}\hbox{\rm
endol}\,(G(M_{n}))&=&\dim_{k}G(M_{n})/\dim_{k}K_{G(M_{n})}\hfill\\\
&=&n\times\dim_{k}G(M_{1})/\dim_{k}K_{G(M_{1})}\hfill\\\ &=&n\times\hbox{\rm
endol}\,(G(M_{1})).\hfill\\\ \end{matrix}$

∎

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R. Bautista
Centro de Ciencias Matemáticas
Universidad Nacional Autónoma de México
Morelia, México
<EMAIL_ADDRESS>

E. Pérez
Facultad de Matemáticas
Universidad Autónoma de Yucatán
Mérida, México
<EMAIL_ADDRESS>

L. Salmerón
Centro de Ciencias Matemáticas
Universidad Nacional Autónoma de México
Morelia, México
<EMAIL_ADDRESS>