# Generic modules for the category of filtered by standard modules for a
quasi-hereditary algebra

R. Bautista, E. Pérez and L. Salmerón

###### Abstract

Here we show that, given a quasi-hereditary finite-dimensional algebra
$\Lambda$ over an algebraically closed field, the category ${\cal F}(\Delta)$
of filtered by standard modules is tame if and only if, for any
$d\in\mathbb{N}$, there are only finitely many isomorphism classes of generic
$\Lambda$-modules adapted to ${\cal F}(\Delta)$ with endolength $d$. We study
the relationship between these generic modules and one-parameter families of
indecomposables in ${\cal F}(\Delta)$.

††2010 _Mathematics Subject Classification_ : 16G60, 16G70, 16G20.†† _Keywords
and phrases_ : differential tensor algebras, ditalgebras, quasi-hereditary
algebras, tame and wild algebras, reduction functors, generic modules.

##  1 Introduction

Denote by $k$ a fixed field and let $\Lambda$ be a finite-dimensional
$k$-algebra. Given a $\Lambda$-module $G$, recall that by definition the
_endolength_ of $G$ is its length as a right ${\rm
End}_{\Lambda}(M)^{op}$-module. The module $G$ is called _generic_ if it is
indecomposable, of infinite length as a $\Lambda$-module, but with finite
endolength. The algebra $\Lambda$ is called _generically tame_ if, for each
$d\in\mathbb{N}$, there is only a finite number of isoclasses of generic
$\Lambda$-modules with endolength $d$. This notion was introduced by W. W.
Crawley-Boevey in [9], where he proved that, in case $k$ is algebraically
closed, the class of generically tame algebras coincides with the class of
tame algebras. Recall that an algebra $\Lambda$, over an algebraically closed
field $k$, is tame if the pairwise non-isomorphic indecomposable modules in
each dimension can be parametrized by a finite number of one-parameter
families. Thus, the notion of generic tameness generalizes the notion of
tameness for arbitrary fields.

In order to be more precise, we recall some definitions from [1].

######  Definition 1.1.

For any $k$-algebra $R$ and any $R$-module $M\in R\hbox{-}{\rm Mod}$ we write
$E_{M}={\rm End}_{R}(M)^{op}$. Any $M\in R\hbox{-}{\rm Mod}$ is naturally a
right $E_{M}$-module. By definition, the _endolength_ of $M$, denoted by
$\hbox{\rm endol}\,(M)$, is its length as an $E_{M}$-module. We say that $M$
is _endofinite_ if it has finite endolength. The $R$-module $M$ is called
_pregeneric_ iff it is an endofinite indecomposable with infinite dimension
over $k$.

If $R$ is a finite-dimensional $k$-algebra, the notion of pregeneric
$R$-module coincides with the usual notion of generic $R$-module.

######  Definition 1.2.

Given a finite-dimensional $k$-algebra $R$ and a family of pairwise non-
isomorphic indecomposable finitely generated $R$-modules
$\\{\Delta_{i}\\}_{i\in[1,n]}$, we have the following associated categories.

Denote by ${\cal F}(\Delta)$ the full subcategory of $R\hbox{-}{\rm mod}$ with
objects the $R$-modules $M$ which admit a filtration

$0=M_{\ell}\subseteq M_{\ell-1}\subseteq\cdots\subseteq M_{1}\subseteq
M_{0}=M$

such that $M_{j}/M_{j+1}$ is isomorphic to a member of the family
$\\{\Delta_{i}\\}_{i\in[1,n]}$, for all $j\in[0,\ell-1]$. Denote by
$\widetilde{\cal F}(\Delta)$ the full subcategory of $R\hbox{-}{\rm Mod}$ with
objects the $R$-modules $M$ which admit a filtration

$0=M_{\ell}\subseteq M_{\ell-1}\subseteq\cdots\subseteq M_{1}\subseteq
M_{0}=M$

such that $M_{j}/M_{j+1}$ is isomorphic to a (possibly infinite) direct sum of
modules in the family $\\{\Delta_{i}\\}_{i\in[1,n]}$, for all
$j\in[0,\ell-1]$.

The main results of this article belong to the study of the category of
filtered by standard modules of hereditary algebras. We recall the definition,
see [11].

######  Definition 1.3.

A finite-dimensional algebra $\Lambda$ with $m$ simple modules (up to
isomorphism) is called _quasi-hereditary_ if there exist indecomposable
$\Lambda$-modules $\Delta_{1},\ldots,\Delta_{m}$ with the following
properties:

  1. 1.

${\rm End}_{\Lambda}(\Delta_{i})\cong k$, for each $i\in[1,m]$;

  2. 2.

${\rm Hom}_{\Lambda}(\Delta_{i},\Delta_{j})\not=0$ implies that $i\leq j$;

  3. 3.

${\rm Ext}^{1}_{\Lambda}(\Delta_{i},\Delta_{j})\not=0$ implies that $i<j$;

  4. 4.

$\Lambda\in{\cal F}(\Delta)$.

Here, $\Delta:=\\{\Delta_{1},\ldots,\Delta_{m}\\}$ is called _the set of
standard $\Lambda$-modules_

######  Definition 1.4.

Let $\Lambda$ be a quasi-hereditary algebra over an algebraically closed field
and consider the standard $\Lambda$-modules $\Delta_{1},\ldots,\Delta_{n}$.
Then,

  1. 1.

A $\Lambda$-module $G$ is called a _generic module for ${\cal F}(\Delta)$_ iff
$G$ is an endofinite indecomposable module with infinite length as a
$\Lambda$-module and $G\in\widetilde{\cal F}(\Delta)$.

  2. 2.

The category ${\cal F}(\Delta)$ is called _generically tame_ iff, for each
$d\in\mathbb{N}$, there is only a finite number of non-isomorphic generic
modules for ${\cal F}(\Delta)$ with endolength $d$.

The main results of this article are the following two statements.

######  Theorem 1.5.

Let $\Lambda$ be a quasi-hereditary algebra over an algebraically closed field
$k$ and consider its standard modules $\Delta_{1},\ldots,\Delta_{n}$. Assume
that ${\cal F}(\Delta)$ is not wild and that $G$ is a generic $\Lambda$-module
for ${\cal F}(\Delta)$, then following statements hold.

  1. 1.

There is a rational algebra $\Gamma_{G}$ and a
$\Lambda\hbox{-}\Gamma_{G}$-bimodule $Z_{G}$ such that as a right
$\Gamma_{G}$-module $Z_{G}$ is free of rank equal to the endolength of $G$. If
$Q_{G}$ is the field of fractions of $\Gamma_{G}$, then $G\cong
Z_{G}\otimes_{\Gamma_{G}}Q_{G}$.

  2. 2.

The functor $Z_{G}\otimes_{\Gamma_{G}}-:\Gamma_{G}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal
F}(\Delta)$ preserves isoclasses and indecomposability.

  3. 3.

The functor $Z_{G}\otimes_{\Gamma_{G}}-:\Gamma_{G}\hbox{-}{\rm
mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal F}(\Delta)$
preserves Auslander-Reiten sequences.

In the following, when we say that almost all objects in a class ${\cal C}$ of
modules satisfy some property, we mean that every object in this class has
this property, with the exception of those lying in a finite union of
isoclasses in ${\cal C}$.

######  Theorem 1.6.

Assume that $\Lambda$ is a quasi-hereditary algebra over an algebraically
closed field and consider its standard modules $\Delta_{1},\ldots,\Delta_{n}$.
Then, ${\cal F}(\Delta)$ is generically tame iff it is tame.

Moreover, if ${\cal F}(\Delta)$ is generically tame, for any $d\in\mathbb{N}$,
almost all indecomposable $\Lambda$-module $M\in{\cal F}(\Delta)$ with
$\dim_{k}M\leq d$ is of the form $M\cong Z_{G}\otimes_{\Gamma_{G}}N$, for some
generic module $G$ for ${\cal F}(\Delta)$ with $\hbox{\rm endol}\,(G)\leq d$
and some indecomposable $N\in\Gamma_{G}\hbox{-}{\rm mod}$, with the notation
of (1.5).

The proof of the preceding results rely on the construction of a directed
triangular weak ditalgebra $\underline{\cal A}(\Delta)$ associated to the
category ${\cal F}(\Delta)$, in such a way that ${\cal F}(\Delta)$ is
generically tame iff $\underline{\cal A}(\Delta)$ is pregenerically tame in
the sense of the following definition. The construction of $\underline{\cal
A}(\Delta)$ is due to Koenig, Külshammer, and Ovsienko, see [12]. A more
general version, with the terminology of this paper, can be found in [4]. We
will assume some familiarity with the language introduced in the first
sections of [3]. So $\underline{\cal A}\hbox{-}{\rm Mod}$ (resp.
$\underline{\cal A}\hbox{-}{\rm mod}$) denotes the category of modules (resp.
finite-dimensional modules) over an interlaced weak ditalgebra
$\underline{\cal A}$ and, given $M\in\underline{\cal A}\hbox{-}{\rm Mod}$, we
denote by ${\rm End}_{\underline{\cal A}}(M)$ its endomorphism algebra in
$\underline{\cal A}\hbox{-}{\rm Mod}$.

######  Definition 1.7.

Let $\underline{\cal A}=({\cal A},I)$ be an interlaced weak ditalgebra, with
layer $(R,W)$, see [3](4.1). Given $M\in\underline{\cal A}\hbox{-}{\rm Mod}$,
denote by $E_{M}:={\rm End}_{\underline{\cal A}}(M)^{op}$ the opposite of 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.

A module $M\in\underline{\cal A}\hbox{-}{\rm Mod}$ is called _pregeneric_ iff
$M$ is indecomposable, with finite endolength but with infinite dimension over
the ground field $k$. The interlaced weak ditalgebra $\underline{\cal A}$ is
called _pregenerically tame_ iff, for each natural number $d$, there are only
finitely many isoclasses of pregeneric $\underline{\cal A}$-modules of
endolength $d$.

The preceding notion extends the one introduced in [1]§2 for layered
ditalgebras. As remarked there, if $B$ is any $k$-algebra, we have the
corresponding regular ditalgebra ${\cal A}$ with layer $(B,0)$ and we can
identify canonically the categories ${\cal A}\hbox{-}{\rm Mod}$ with
$B\hbox{-}{\rm Mod}$. Thus, if $B$ is a finite-dimensional algebra, the
notions of _pregeneric module_ and _pregeneric tameness_ for the algebra $B$
coincide with the usual notions of _generic module_ and _generic tameness_.

######  Definition 1.8.

An _elementary bigraph_ $\mathbb{B}$ is a triple $\mathbb{B}=({\cal
P},\mathbb{B}_{0},\mathbb{B}_{1})$ formed by a finite set ${\cal
P}=\\{1,\ldots,n\\}$ of _points_ , a finite set $\mathbb{B}_{0}$ of _full
arrows_ , and a finite set $\mathbb{B}_{1}$ of _dashed arrows_. Each _arrow_
$\alpha\in\mathbb{B}_{0}\cup\mathbb{B}_{1}$ has a _starting point_
$s(\alpha)\in{\cal P}$ and a _terminal point_ $t(\alpha)\in{\cal P}$. An
elementary bigraph $\mathbb{B}$ is called _directed_ iff it admits no oriented
(non-trivial) cycle (composed by any kind of arrows).

Given an elementary bigraph $\mathbb{B}$ with $n$ points, we consider the
$k$-algebra $R$ defined as the product $R:=k\times k\times\cdots\times k$ of
$n$ copies of the ground field $k$. Denote by $e_{i}$ the idempotent of $R$
with 1 in its $i^{th}$ coordinate and 0 in the others. Then, we consider the
vector space $W_{0}$ (resp. $W_{1}$) as the vector space with basis
$\mathbb{B}_{0}$ (resp. $\mathbb{B}_{1}$). Make $W:=W_{0}\oplus W_{1}$. If we
define $e_{j}\alpha e_{i}=\delta_{i,s(\alpha)}\delta_{t(\alpha),j}\alpha$, we
get a natural structure of $R$-$R$-bimodule on the space $W$, and
$W=W_{0}\oplus W_{1}$ is an $R$-$R$-bimodule decomposition. Then, we have the
graded tensor algebra $T=T_{R}(W)$, with $[T]_{0}=T_{R}(W_{0})$ and
$[T]_{1}=[T]_{0}W_{1}[T]_{0}$, which is called _the $t$-algebra of the
elementary bigraph $\mathbb{B}$_.

Notice that $T$ can be identified with the _path algebra $k\mathbb{B}$ of the
bigraph $\mathbb{B}$_, with underlying vector space with basis the set of
paths (of any kind of arrows) of $\mathbb{B}$ (including one trivial path for
each point $i\in{\cal P}$). Each primitive idempotent $e_{i}$ of $R$ is
identified with the trivial path at the point $i$. The homogeneous elements
$[k\mathbb{B}]_{u}$ of degree $u$ are the linear combinations of paths
containing exactly $u$ dashed arrows.

A triangular interlaced weak ditalgebra $\underline{\cal A}=({\cal A},I)$, as
in [3](5.2), is called _directed_ iff the underlying $t$-algebra $T$ of ${\cal
A}=(T,\delta)$ is the tensor algebra of some directed elementary bigraph.

The proof of the main theorem (1.6) relies on the following result.

######  Theorem 1.9.

Assume that the ground field $k$ is algebraically closed. Then, every directed
triangular interlaced weak ditalgebra $\underline{\cal A}$ is pregenerically
tame iff it is tame.

The proof of this last statement will follow the line of reasoning of [3],
adapting the arguments to handle pregeneric modules. This strategy is similar
to the one used by Crawley-Boevey to transit from [8] to [9]. In order to
present a readable argument for the proof of our main results, we could not
avoid to recall many definitions and constructions from [9] and [3].

##  2 Endolength and pregeneric tameness

In the following lemmas, we recall elementary reduction procedures on
triangular interlaced weak ditalgebras studied in [3]§7, and we describe their
effect on the endolength. The first four of them correspond to the lemmas
(7.2), (7.3), (7.4), and (7.5) of [3]. For the statements on the endolength in
all of these cases, see the proof of [1](2.2).

Here, as in [1], given a layered interlaced weak ditalgebra $\underline{\cal
A}=({\cal A},I)$ with underlying weak ditalgebra ${\cal A}=(T,\delta)$ and
layer $(R,W)$, we denote by $A=[T]_{0}=T_{R}(W_{0})$ the subalgebra of
$T=T_{R}(W)$ formed by the degree-zero elements and, by $V=[T]_{1}=AW_{1}A$
the $A$-$A$-subbimodule of $T$ formed by the degree-one elements.

######  Proposition 2.1 (deletion of idempotents).

Suppose that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with layer $(R,W)$. Assume that $e\in R$ is a central idempotent of
$R$. Consider the canonical projections $\phi_{\bullet}:R\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}eRe$, and
$\phi_{r}:W_{r}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}eW_{r}e$, for $r\in[0,1]$. Make
$T^{d}:=T_{eRe}(eWe)$. Thus, we have a morphism of graded $k$-algebras
$\phi:T\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}T^{d}$ and we
have the ideal $I^{d}=\phi(I)$ of $A^{d}$. Then, there is a triangular
interlaced weak ditalgebra $\underline{\cal A}^{d}=({\cal A}^{d},I^{d})$ with
layer $(R^{d},W^{d})$ where $R^{d}=eRe$, $W^{d}_{0}=eW_{0}e$, and
$W^{d}_{1}=eW_{1}e$. The morphism $\phi:\underline{\cal A}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}^{d}$ of interlaced weak
ditalgebras induces a full and faithful functor
$F^{d}:=F_{\phi}:\underline{\cal A}^{d}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$ whose
image consists of the $\underline{\cal A}$-modules annihilated by $1-e$.
Moreover, the functor $F^{d}$ preserves endolength. Therefore,
$\underline{\cal A}^{d}$ is pregenerically tame whenever $\underline{\cal A}$
is so.

We say that $\underline{\cal A}^{d}$ is _obtained from $\underline{\cal A}$ by
deletion of the idempotent $1-e$_.

######  Proposition 2.2 (regularization).

Let $\underline{\cal A}=({\cal A},I)$ be a triangular interlaced weak
ditalgebra with underlying weak ditalgebra ${\cal A}=(T,\delta)$ and
triangular layer $(R,W)$. Assume that we have $R$-$R$-bimodule decompositions
$W_{0}=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$ and
$W_{1}=\delta(W^{\prime}_{0})\oplus W^{\prime\prime}_{1}$, make
$W^{\prime\prime}:=W_{0}^{\prime\prime}\oplus W^{\prime\prime}_{1}$. Consider
the identity map $\phi_{\bullet}:R\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}R$, the canonical projections
$\phi_{j}:W_{j}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}W^{\prime\prime}_{j}$, for $j\in[0,1]$, and
the tensor algebra $T^{r}=T_{R}(W^{\prime\prime})$. Then, we have a morphism
of graded algebras $\phi:T\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}T^{r}$ and the ideal $I^{r}=\phi(I)$ of
$A^{r}$. Then, there is a triangular interlaced weak ditalgebra
$\underline{\cal A}^{r}=({\cal A}^{r},I^{r})$ with layer $(R^{r},W^{r})$,
where $R^{r}=R$, $W_{0}^{r}=W^{\prime\prime}_{0}$, and
$W^{r}_{1}=W^{\prime\prime}_{1}$. The morphism $\phi:\underline{\cal
A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}^{r}$ of interlaced weak ditalgebras induces a full and faithful functor
$F^{r}:=F_{\phi}:\underline{\cal A}^{r}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$ which
preserves endolength. So $\underline{\cal A}^{r}$ is pregenerically tame
whenever $\underline{\cal A}$ is so. Moreover, if $\underline{\cal A}$ is a
Roiter interlaced weak ditalgebra, then $M\in\underline{\cal A}\hbox{-}{\rm
Mod}$ is isomorphic to an object in the image of $F^{r}$ iff ${\rm
Ker}\,\delta\cap W^{\prime}_{0}$ annihilates $M$. In particular, if this
intersection is zero, $F^{r}$ is an equivalence of categories.

######  Lemma 2.3 (factoring out a direct summand of $W_{0}$).

Let $\underline{\cal A}=({\cal A},I)$ be a triangular interlaced weak
ditalgebra with underlying weak ditalgebra ${\cal A}=(T,\delta)$ and
triangular layer $(R,W)$. Assume that there is a decomposition of
$R$-$R$-bimodules $W_{0}=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$, such that
$W^{\prime}_{0}\subseteq I$ and $\delta(W^{\prime}_{0})\subseteq
AW^{\prime}_{0}V+VW^{\prime}_{0}A$. Make $T^{q}=T_{R}(W^{q})$, where
$W^{q}_{0}=W^{\prime\prime}_{0}$, $W^{q}_{1}=W_{1}$, and
$W^{q}=W_{0}^{q}\oplus W_{1}^{q}$. Then, there is a derivation $\delta^{q}$ on
$T^{q}$ such that ${\cal A}^{q}:=(T^{q},\delta^{q})$ is a weak ditalgebra with
triangular layer $(R,W^{q})$. The $t$-algebra
$A^{q}=T_{R}(W^{\prime\prime}_{0})$ can be identified with the quotient
algebra $A/I^{\prime}$, where $I^{\prime}=AW^{\prime}_{0}A$ is the ideal of
$A$ generated by $W^{\prime}_{0}$, so we can consider $I^{q}:=I/I^{\prime}$ as
an ideal of $A^{q}$. Then $\underline{\cal A}^{q}=({\cal A}^{q},I^{q})$ is a
triangular interlaced weak ditalgebra, and there is a morphism of interlaced
weak ditalgebras $\phi:\underline{\cal A}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}^{q}$, with
$\phi(I)=I^{q}$, which induces an endolength preserving equivalence of
categories $F^{q}:=F_{\phi}:\underline{\cal A}^{q}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}{\rm Mod}$. So, $\underline{\cal A}^{q}$ is pregenerically tame iff
$\underline{\cal A}$ is so.

######  Lemma 2.4 (absorption).

Let $\underline{\cal A}=({\cal A},I)$ be a triangular interlaced weak
ditalgebra with underlying weak ditalgebra ${\cal A}=(T,\delta)$ and
triangular layer $(R,W)$. Assume that there is a decomposition of
$R$-$R$-bimodules $W_{0}=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$ with
$\delta(W^{\prime}_{0})=0$. Then, we can consider another layer
$(R^{a},W^{a})$ for the same weak ditalgebra ${\cal A}$, where $R^{a}$ is the
subalgebra of $T$ generated by $R$ and $W^{\prime}_{0}$, thus we can identify
it with $T_{R}(W^{\prime}_{0})$, $W_{0}^{a}:=R^{a}W^{\prime\prime}_{0}R^{a}$,
and $W_{1}^{a}:=R^{a}W_{1}R^{a}$. We denote by ${\cal A}^{a}$ the same weak
ditalgebra ${\cal A}$ equipped with its new layer $(R^{a},W^{a})$; in
particular, we have $\delta^{a}=\delta$. The layer $(R^{a},W^{a})$ is
triangular and we obtain a triangular interlaced weak ditalgebra
$\underline{\cal A}^{a}=({\cal A}^{a},I^{a})$, with $I^{a}=I$. The identity is
a morphism of interlaced weak ditalgebras $\phi:\underline{\cal
A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}^{a}$, which induces an endolength preserving equivalence of categories
$F^{a}:=F_{\phi}:\underline{\cal A}^{a}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$, and
$\underline{\cal A}^{a}$ is pregenerically tame iff $\underline{\cal A}$ is
so.

We say that $\underline{\cal A}^{a}$ _is obtained from $\underline{\cal A}$ by
absorption of the bimodule $W^{\prime}_{0}$_.

In order to fix notation and terminology, we recall from [3](7.6) the
following two propositions.

######  Proposition 2.5.

Assume that ${\cal A}=(T,\delta)$ is a weak ditalgebra with layer $(R,W)$ such
that there is an $R$-$R$-bimodule decomposition $W_{0}=W^{\prime}_{0}\oplus
W^{\prime\prime}_{0}$ with $\delta(W^{\prime}_{0})=0$. Suppose that $X$ is an
_admissible_ $B$-module, where $B=T_{R}(W^{\prime}_{0})$. This means that the
algebra $\Gamma={\rm End}_{B}(X)^{op}$ admits a splitting $\Gamma=S\oplus P$,
where $S$ is a subalgebra of $\Gamma$, $P$ is an ideal of $\Gamma$, the direct
sum is an $S$-$S$-bimodule decomposition of $\Gamma$, and the right
$S$-modules $X$ and $P$ are finitely generated projective. By
$(x_{i},\nu_{i})_{i\in I}$ and $(p_{j},\gamma_{j})_{j\in J}$ we denote finite
dual basis of the right $S$-modules $X$ and $P$, respectively.

There is a comultiplication $\mu:P^{*}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}P^{*}\otimes_{S}P^{*}$ induced by the
multiplication of $P$, which is coassociative (see [5](11.7)). For $\gamma\in
P^{*}$, we have

$\mu(\gamma)=\sum_{i,j\in J}\gamma(p_{i}p_{j})\gamma_{j}\otimes\gamma_{i}.$

The action of $P$ on $X$ induces (see [5](10–11)) a morphism of
$S$-$R$-bimodules $\lambda:X^{*}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}P^{*}\otimes_{S}X^{*}$ and a morphism of
$R$-$S$-bimodules $\rho:X\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}X\otimes_{S}P^{*}$ with

$\lambda(\nu)=\sum_{i\in I,j\in
J}\nu(x_{i}p_{j})\gamma_{j}\otimes\nu_{i}\hbox{ \ and \ }\rho(x)=\sum_{j\in
J}xp_{j}\otimes\gamma_{j}.$

Make $\underline{W}_{0}=BW^{\prime\prime}_{0}B$ and
$\underline{W}_{1}=BW_{1}B$. Recall that we can identify $A=T_{R}(W_{0})$ with
$T_{B}(\underline{W}_{0})$ and $T$ with $T_{B}(\underline{W})$, where
$\underline{W}=\underline{W}_{0}\oplus\underline{W}_{1}$, see [5](12.2).

We have the $S$-$S$-bimodules

$\begin{matrix}W_{0}^{X}=X^{*}\otimes_{B}\underline{W}_{0}\otimes_{B}X=X^{*}\otimes_{R}W^{\prime\prime}_{0}\otimes_{R}X\\\
\hbox{ \ \ \, \ \ }\\\
W_{1}^{X}=(X^{*}\otimes_{B}\underline{W}_{1}\otimes_{B}X)\oplus
P^{*}=(X^{*}\otimes_{R}W_{1}\otimes_{R}X)\oplus P^{*}.\\\ \end{matrix}$

Consider the tensor algebra $T^{X}=T_{S}(W^{X})$, where $W^{X}=W_{0}^{X}\oplus
W_{1}^{X}$. Following [5](12.8), observe that for $\nu\in X^{*}$ and $x\in X$,
there is a linear map

$\sigma_{\nu,x}:T\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}T^{X}$

such that $\sigma_{\nu,x}(b)=\nu(bx)$, for $b\in B$, and given
$w_{1},w_{2},\ldots,w_{n}\in\underline{W}$, we have
$\sigma_{\nu,x}(w_{1}w_{2}\cdots w_{n})$ is given by

$\sum_{i_{1},i_{2},\ldots,i_{n-1}}\nu\otimes w_{1}\otimes
x_{i_{1}}\otimes\nu_{i_{1}}\otimes w_{2}\otimes
x_{i_{2}}\otimes\nu_{i_{2}}\otimes w_{3}\otimes\cdots\otimes
x_{i_{n-1}}\otimes\nu_{i_{n-1}}\otimes w_{n}\otimes x.$

There is a derivation $\delta^{X}$ on $T^{X}$ determined by
$\delta^{X}(\gamma)=\mu(\gamma)$, for $\gamma\in P^{*}$ and

$\delta^{X}(\nu\otimes w\otimes x)=\lambda(\nu)\otimes w\otimes
x+\sigma_{\nu,x}(\delta(w))+(-1)^{\deg w+1}\nu\otimes w\otimes\rho(x),$

for $w\in\underline{W}_{0}\cup\underline{W}_{1}$, $\nu\in X^{*}$, and $x\in
X$.

Then, ${\cal A}^{X}=(T^{X},\delta^{X})$ is a weak ditalgebra. If the layer
$(R,W)$ of ${\cal A}$ is triangular and $X$ is a triangular admissible
$B$-module, as in [5](14.6), then the layer $(S,W^{X})$ of ${\cal A}^{X}$ is
also triangular.

######  Proposition 2.6 (reduction with an admissible module $X$).

Let $\underline{\cal A}=({\cal A},I)$ be a triangular interlaced weak
ditalgebra with underlying weak ditalgebra ${\cal A}=(T,\delta)$ and
triangular layer $(R,W)$. Suppose that there is an $R$-$R$-bimodule
decomposition $W_{0}=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$ with
$\delta(W^{\prime}_{0})=0$ and that $X$ is a triangular admissible $B$-module,
where $B=T_{R}(W^{\prime}_{0})$. As usual, denote by $A^{X}=[T^{X}]_{0}$.
Consider the ideal $I^{X}$ of $A^{X}$ generated by the elements
$\sigma_{\nu,x}(h)$, for $\nu\in X^{*}$, $h\in I$, and $x\in X$. Then,

  1. 1.

The pair $\underline{\cal A}^{X}:=({\cal A}^{X},I^{X})$ is a triangular
interlaced weak ditalgebra;

  2. 2.

There is a functor $F^{X}:\underline{\cal A}^{X}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}{\rm Mod}$ such that for $M\in\underline{\cal A}^{X}\hbox{-}{\rm
Mod}$, the underlying $B$-module of $F^{X}(M)$ is $X\otimes_{S}M$ and

$a\cdot(x\otimes m)=\sum_{i}x_{i}\otimes\sigma_{{\nu_{i}},x}(a)m,$

for $a\in A$, $x\in X$, and $m\in M$. Moreover, given the morphism
$f=(f^{0},f^{1})\in{\rm Hom}_{\underline{\cal A}^{X}}(M,N)$, we have
$F^{X}(f)=(F^{X}(f)^{0},F^{X}(f)^{1})$ given by

$\begin{matrix}F^{X}(f)^{0}[x\otimes m]&=&x\otimes
f^{0}(m)+\sum_{j}xp_{j}\otimes f^{1}(\gamma_{j})[m]\\\
F^{X}(f)^{1}(v)[x\otimes m]&=&\sum_{i}x_{i}\otimes
f^{1}(\sigma_{\nu_{i},x}(v))[m]\hfill\\\ \end{matrix}$

for $v\in V$, $x\in X$, and $m\in M$.

  3. 3.

For any $N\in\underline{\cal A}\hbox{-}{\rm Mod}$ which is isomorphic as a
$B$-module to some $B$-module of the form $X\otimes_{S}M$, for some $M\in
S\hbox{-}{\rm Mod}$, we have $N\cong F^{X}(\overline{M})$ in $\underline{\cal
A}\hbox{-}{\rm Mod}$, for some $\overline{M}\in\underline{\cal
A}^{X}\hbox{-}{\rm Mod}$.

We also need to recall from [3] the following additional condition on the
admissible $B$-module $X$ which guarantees that $F^{X}$ is full and faithful.

######  Remark 2.7.

Assume that $B=T_{R}(W^{\prime}_{0})$ is a tensor algebra and let $X$ be an
admissible $B$-module. Thus, we have a splitting $\Gamma={\rm
End}_{B}(X)^{op}=S\oplus P$, as in (2.5). We have the ditalgebra $(B,0)$, with
trivial differential, and the ditalgebra $(B,0)^{X}=(T_{S}(P^{*}),\delta)$,
where $\delta$ is the differential determined by the comultiplication
$\mu:P^{*}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}P^{*}\otimes_{S}P^{*}$. Recall from [5],
that the admissible $B$-module $X$ is called _complete_ if the functor

$F^{X}:(B,0)^{X}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}B\hbox{-}{\rm Mod}$

is full and faithful. From (17.4), (17.5), and (17.11) of [5], we know that we
obtain complete admissible $B$-modules $X$ in the following cases:

  1. 1.

$X$ is a finite direct sum of non-isomorphic finite-dimensional
indecomposables in $B\hbox{-}{\rm mod}$;

  2. 2.

$X$ is the $B$-module obtained from the regular $S$-module $S$ by restriction
through a given epimorphism of $k$-algebras $\phi:B\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}S$;

  3. 3.

$X=X_{1}\oplus X_{2}$, where $X_{1}$ and $X_{2}$ are complete triangular
admissible $B$-modules such that ${\rm Hom}_{B}({\cal I}_{X_{i}},{\cal
I}_{X_{j}})=0$, for $i\not=j$, and ${\cal I}_{X_{i}}$ denotes the class of
$B$-modules of the form $X_{i}\otimes_{S}N$, for some $N\in S\hbox{-}{\rm
Mod}$.

All the complete admissible $B$-modules we shall consider in this paper are
constructed using 1, 2, and 3.

Again, from [3], we have the following.

######  Proposition 2.8.

Under the assumptions of (2.6), consider the ideal $I_{0}:=B\cap I$ of
$B=T_{R}(W_{0}^{\prime})$. Then, we can consider the triangular interlaced
weak ditalgebra $({\cal B},I_{0})$, where ${\cal B}=(B,0)$. Then, for any
complete admissible $B$-module $X$, we have a full and faithful functor

$F^{\prime X}:({\cal B}^{X},I_{0}^{X})\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}({\cal B},I_{0})\hbox{-}{\rm
Mod}=(B/I_{0})\hbox{-}{\rm Mod},$

and a full and faithful functor $F^{X}:\underline{\cal A}^{X}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}{\rm Mod}$.

######  Definition 2.9.

An interlaced weak ditalgebra $({\cal A},I)$ over the field $k$ is called
_wild_ iff there is an $A/I$-$k\langle x,y\rangle$-bimodule $Z$, which is free
of finite rank as a right $k\langle x,y\rangle$-module, such that the
composition functor

$k\langle x,y\rangle\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\ Z\otimes_{k\langle x,y\rangle}-\
}}A/I\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\ L_{({\cal A},I)}\ }}({\cal
A},I)\hbox{-}{\rm Mod}$

preserves isoclasses of indecomposables. Here $L_{({\cal A},I)}$ denotes the
canonical embedding functor mapping each morphism $f^{0}$ onto $(f^{0},0)$. In
this case, we say that _$Z$ produces the wildness of_ $({\cal A},I)$.

We recall from [3](7.13) the following statement.

######  Proposition 2.10.

Assume that a triangular interlaced weak ditalgebra $({\cal A}^{z},I^{z})$ is
obtained from a Roiter interlaced weak ditalgebra $({\cal A},I)$ by some of
the procedures described in this section, that is $z\in\\{d,r,q,a,X\\}$. Then,
$({\cal A}^{z},I^{z})$ is a Roiter interlaced weak ditalgebra. The associated
full and faithful functor

$F^{z}:({\cal A}^{z},I^{z})\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}({\cal A},I)\hbox{-}{\rm Mod}$

preserves isoclasses and indecomposables. Moreover, if $({\cal A}^{z},I^{z})$
is wild (with wildness produced by an $(A^{z}/I^{z})$-$k\langle
x,y\rangle$-bimodule $Z$) then so is $({\cal A},I)$ (with wildness produced by
the $(A/I)$-$k\langle x,y\rangle$-bimodule $F^{z}(Z)$).

######  Proposition 2.11.

Any pregenerically tame triangular interlaced weak ditalgebra $\underline{\cal
A}$ is not wild.

###### Proof.

The proof of [1](2.9) readily adapts to this situation. ∎

##  3 Bimodules and reduction functors

We recall some basic notions of [9], adapted to the language used here.

######  Definition 3.1.

Given an interlaced weak ditalgebra $\underline{\cal A}=({\cal A},I)$ and a
$k$-algebra $E$, an $\underline{\cal A}\hbox{-}E$-bimodule is an object
$M\in\underline{\cal A}\hbox{-}{\rm Mod}$, together with a $k$-algebra
homomorphism $\alpha_{M}:E\rightarrow{\rm End}_{\underline{\cal A}}(M)^{op}$.
If $N$ is another $\underline{\cal A}$-$E$-bimodule, then

${\rm Hom}_{\underline{\cal A}\hbox{-}E}(M,N)=\\{f\in{\rm
Hom}_{\underline{\cal A}}(M,N)\mid f\alpha_{M}(e)=\alpha_{N}(e)f,\hbox{ for
all }e\in E\\}.$

The category of $\underline{\cal A}$-$E$-bimodules, where the rule of
composition is the same as for $\underline{\cal A}\hbox{-}{\rm Mod}$, is
denoted by $\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$.

We shall denote $\underline{A}:=A/I$, where $A$ is the tensor algebra
generated by the degree zero elements in the underlying tensor algebra of
${\cal A}$. An $\underline{\cal A}$-$E$-bimodule $M$ is proper iff
$\alpha_{M}$ factors through the canonical embedding map

${\rm End}_{\underline{A}}(M)^{op}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal A}}(M)^{op}\hbox{
such that }f\mapsto(f,0).$

The full subcategory of $\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ formed
by the proper $\underline{\cal A}\hbox{-}E$-bimodules will be denoted by
$\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}_{p}$. The canonical embedding
functor $L_{\underline{\cal A}}:\underline{A}\hbox{-}{\rm
Mod}\rightarrow\underline{\cal A}\hbox{-}{\rm Mod}$, which maps each
$\underline{A}$-morphism $f$ to $(f,0)$, induces a functor
$L^{E}_{\underline{\cal A}}:\underline{A}\hbox{-}E\hbox{-}{\rm
Mod}\rightarrow\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$. Clearly, the
proper $\underline{\cal A}$-$E$-bimodules coincide with the images of the
$\underline{A}$-$E$-bimodules under the faithful functor
$L^{E}_{\underline{\cal A}}$, we will identify this class of objects with the
$\underline{A}$-$E$-bimodules.

The following lemma is shown as in the classical case, see [5](21.3).

######  Lemma 3.2.

Assume that $E$ is a $k$-algebra and $\underline{\cal A}$, $\underline{\cal
A}^{\prime}$ are interlaced weak ditalgebras. Then any functor
$F:{\underline{\cal A}^{\prime}}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$ induces
a functor

$F_{E}:{\underline{\cal A}^{\prime}}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod},$

which maps any object $M\in\underline{\cal A}^{\prime}\hbox{-}E\hbox{-}{\rm
Mod}$ onto $F(M)=F_{E}(M)$, where the structure of $\underline{\cal
A}$-$E$-bimodule on $F(M)$ is given by the composition morphism

$\alpha_{F(M)}:E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\alpha_{M}}}{\rm End}_{\underline{\cal
A}^{\prime}}(M)^{op}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F}}{\rm End}_{\underline{\cal
A}}(F(M))^{op}.$

Moreover, if $F$ is full and faithful, then $F_{E}$ is so. In this case, if an
$\underline{\cal A}\hbox{-}E$-bimodule $M$ is isomorphic as an
$\underline{\cal A}$-module to $F(N)$, for some $\underline{\cal
A^{\prime}}$-module $N$, then $N$ admits a natural structure of
$\underline{\cal A^{\prime}}$-$E$-bimodule such that $M\cong F_{E}(N)$ in
$\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$.

######  Definition 3.3.

Let $\underline{\cal A}$ be a interlaced weak ditalgebra with layer $(R,W)$
and $E$ any $k$-algebra. Given any $\underline{\cal A}\hbox{-}E$-bimodule $M$,
the composition of $\alpha_{M}:E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal A}}(M)^{op}$ with
the projection $\pi_{M}:{\rm End}_{\underline{\cal
A}}(M)^{op}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{R}(M)^{op}$, which maps $(f^{0},f^{1})$ onto $f^{0}$, determines a
structure of $R\hbox{-}E$-bimodule on $M$. We will denote by $\ell_{E}(M)$ the
length of this right $E$-module.

The full subcategory of $\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ formed
by the finite $E$-length bimodules is denoted by $\underline{\cal
A}\hbox{-}E\hbox{-}{\rm mod}$ and its intersection with $\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}_{p}$ by $\underline{\cal A}\hbox{-}E\hbox{-}{\rm
mod}_{p}$.

Suppose that $\underline{\cal A}$ and $\underline{\cal A^{\prime}}$ are
layered interlaced weak ditalgebras and $E$ is a $k$-algebra. Then we say that
a functor $F_{E}:\underline{\cal A}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}^{\prime}\hbox{-}E\hbox{-}{\rm Mod}$ is length controlling iff $\ell_{E}(M)$
finite implies $\ell_{E}(F(M))$ finite, and, furthermore,
$\ell_{E}(M)\leq\ell_{E}(F(M))$, for all $M\in\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$. If $\ell_{E}(M)=\ell_{E}(F(M))$, for all $M$,
then $F$ is called length preserving.

######  Lemma 3.4.

Let $\phi:\underline{\cal A}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}^{\prime}$ be a morphism
of triangular interlaced weak ditalgebras and $F_{\phi}:\underline{\cal
A}^{\prime}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$ the
associated restriction functor. Then, for any $k$-algebra $E$, the induced
functor $F^{E}_{\phi}:\underline{\cal A}^{\prime}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ is length preserving and it sends proper
bimodules to proper bimodules.

###### Proof.

The proof of [5](21.6) immediately adapts to this situation. ∎

######  Lemma 3.5.

Assume that $\underline{\cal A}$ is a triangular interlaced weak ditalgebra,
that $\underline{\cal A}^{d}$ is obtained by deletion of an idempotent $e$,
and let $F^{d}:\underline{\cal A}^{d}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$ be the
associated functor, as in (2.1). Then, for any $k$-algebra $E$, we have:

  1. 1.

$F^{d}_{E}:\underline{\cal A}^{d}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ is a length preserving functor. It induces an
equivalence from $\underline{\cal A}^{d}\hbox{-}E\hbox{-}{\rm Mod}$ to the
full subcategory of $\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ formed by
the bimodules $M$ with $eM=0$.

  2. 2.

$F^{d}_{E}$ restricts to an equivalence from $\underline{\cal
A}^{d}\hbox{-}E\hbox{-}{\rm Mod}_{p}$ to the full subcategory of
$\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}_{p}$ formed by the proper
bimodules $M$ satisfying $eM=0$.

###### Proof.

It follows from (3.4), because $F^{d}=F_{\phi}$, where $\phi:\underline{\cal
A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}^{d}$ is the morphism constructed in (2.1). ∎

######  Proposition 3.6.

Let $\underline{\cal A}$ be a triangular interlaced weak ditalgebra with
triangular layer $(R,W)$. Assume that $\underline{\cal A}^{q}$ is obtained
from $\underline{\cal A}$ factoring out a direct summand of $W_{0}$, as in
(2.3), and consider the associated functor $F^{q}:\underline{\cal
A}^{q}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$. Then,
for any $k$-algebra $E$, we have:

  1. 1.

$F^{q}_{E}:\underline{\cal A}^{q}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ is a length preserving equivalence.

  2. 2.

The functor $F^{q}_{E}$ restricts to an equivalence $F_{E}^{q}:\underline{\cal
A}^{q}\hbox{-}E\hbox{-}{\rm Mod}_{p}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}E\hbox{-}{\rm
Mod}_{p}$.

###### Proof.

It follows from (3.4), because $F^{q}=F_{\phi}$, where $\phi:\underline{\cal
A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}^{q}$ is the morphism constructed in (2.3). ∎

######  Lemma 3.7.

Let $\underline{\cal A}=({\cal A},I)$ be a triangular interlaced weak
ditalgebra and $X$ a complete triangular admissible $B$-module, where
$B=T_{R}(W^{\prime}_{0})$ as in (2.6). Consider the weak ditalgebra
$\underline{\cal A}^{X}$ and the associated functor $F^{X}:\underline{\cal
A}^{X}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$. Assume
that for $M\in\underline{\cal A}\hbox{-}{\rm Mod}$ there is an $S$-module $N$
such that $X\otimes_{S}N\cong M$ in $B\hbox{-}{\rm Mod}$. Then, there is
$\overline{N}\in\underline{\cal A}^{X}\hbox{-}{\rm Mod}$ with
$F^{X}(\overline{N})\cong M$ in $\underline{\cal A}\hbox{-}{\rm Mod}$.

Moreover, given a $k$-algebra $E$, if $M\in\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}_{p}$, $N\in S\hbox{-}E\hbox{-}{\rm Mod}$, and
$X\otimes_{S}N\cong M$ in $B\hbox{-}E\hbox{-}{\rm Mod}$, then $\overline{N}$
admits a natural structure of proper $\underline{\cal
A}^{X}\hbox{-}E$-bimodule with underlying $\underline{\cal A}^{X}$-module $N$
and such that $F^{X}_{E}(\overline{N})\cong M$ in $\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}_{p}$.

###### Proof.

Similar to the proof of [5](25.5), now using (2.6)(3). ∎

######  Proposition 3.8.

Assume that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with layer $(R,W)$. Under the assumptions of (2.6), consider the
ideal $I_{0}:=B\cap I$ of $B=T_{R}(W_{0}^{\prime})$. Assume that
$\underline{\cal A}^{X}$ is the triangular interlaced weak ditalgebra obtained
from $\underline{\cal A}$ by reduction, using a complete triangular admissible
$B$-module $X$, thus $\underline{\cal A}^{X}$ has layer $(S,W^{X})$, where
$\Gamma={\rm End}_{B}(X)^{op}=S\oplus P$. Assume that $S$ is a minimal
algebra, as defined in (3.9). Consider the associated full and faithful
functor $F^{X}:\underline{\cal A}^{X}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$, as in
[5](12.10). Denote by $\mu(X)$ the number of generators in a set of generators
of the right $S$-module $X$ with minimal cardinality. Then, the following
holds.

  1. 1.

For all $N\in{\cal A}^{X}\hbox{-}{\rm Mod}$, we have that $\hbox{\rm
endol}\,(F^{X}(N))\leq\mu(X)\times\hbox{\rm endol}\,(N)$. Then
$\underline{\cal A}^{X}$ is pregenerically tame, whenever $\underline{\cal A}$
is so.

  2. 2.

For any $k$-algebra $E$, the associated functor $F^{X}_{E}:\underline{\cal
A}^{X}\hbox{-}E\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}E\hbox{-}{\rm
Mod}$ is length controlling, full and faithful. For any $N\in\underline{\cal
A}^{X}\hbox{-}E\hbox{-}{\rm Mod}$, the inequality
$\ell_{E}(F^{X}_{E}(N))\leq\mu(X)\times\ell_{E}(N)$ holds.

###### Proof.

The argument is essentially the same as [2](4.7), an adaptation of the proof
of [5](25.7), we sketch it here.

We first prove the second item. Take an $\underline{\cal
A}^{X}\hbox{-}E$-bimodule $N$, with bimodule structure
$\alpha_{N}:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{\underline{\cal A}^{X}}(N)^{op}$, so the $\underline{\cal
A}\hbox{-}E$-bimodule $F_{X}(N)$, has bimodule structure
$\alpha_{F^{X}(N)}:E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal
A}}(F^{X}(N))^{op}$, as in (3.2).

As in the proof of [5](25.7), it can be shown that $\ell_{E}(X\otimes_{S}N)$
coincides with the length of the right $E$-module $X\otimes_{S}N$, where $E$
acts as usual on the right tensor factor $N$. Since $S$ is a minimal algebra,
from this it follows that $\ell_{E}(N)\leq\ell_{E}(F^{X}_{E}(N))$, see the
argument in the proof of [5](25.7).

Moreover, from an epimorphism $S^{\mu(X)}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}X$ of right $S$-modules, we obtain an
epimorphism of right $E$-modules $S^{\mu(X)}\otimes_{S}N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}X\otimes_{S}N$. Therefore, we obtain
$\ell_{E}(F^{X}(N))=\ell_{E}(X\otimes_{S}N)\leq\ell_{E}(N^{\mu(X)})=\mu(X)\times\ell_{E}(N)$,
as wanted.

For the first item, given $N\in\underline{\cal A}^{X}\hbox{-}{\rm Mod}$, make
$E:={\rm End}_{\underline{\cal A}^{X}}(N)^{op}$. Then, we have the
$\underline{\cal A}^{X}\hbox{-}E$-bimodule $N$, with bimodule structure
$\alpha_{N}:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{\underline{\cal A}^{X}}(N)^{op}$ given by the identity map.

Since $F^{X}$ is full and faithful, we have the isomorphism of algebras
$E\cong{\rm End}_{\underline{\cal A}}(F^{X}(N))^{op}$ induced by $F^{X}$. It
provides, by restriction, a structure of right $E$-module on $F_{X}(N)$.
Clearly, $\hbox{\rm endol}\,(N)=\ell_{E}(N)$ and $\hbox{\rm
endol}\,(F^{X}(N))=\ell_{E}(F^{X}(N))$. So, the formula in 1 follows from 2.

Finally, if we assume that $\underline{\cal A}^{X}$ is not pregenerically
tame, we have an infinite family of pairwise non-isomorphic pregeneric
$\underline{\cal A}^{X}$-modules with bounded endolength. Then, applying the
full and faithful functor $F^{X}$ to them, we obtain an infinite family of
pairwise non-isomorphic pregeneric $\underline{\cal A}$-modules with bounded
endolength. Hence, the interlaced weak ditalgebra $\underline{\cal A}$ is not
pregenerically tame. ∎

######  Definition 3.9.

Recall that a _minimal algebra_ $R$ is a finite product of algebras
$R=R_{1}\times\cdots\times R_{n}$, where each $R_{i}$ is either a rational
$k$-algebra or is isomorphic the field $k$. Assume ${\cal A}$ is a triangular
weak ditalgebra with layer $(R,W)$, where $R$ is a minimal algebra. Then, a
_source point_ $i_{0}$ of ${\cal A}$ is a centrally primitive idempotent
$e_{i_{0}}$ of $R$ such that $e_{i_{0}}W=0$ and $Re_{i_{0}}=ke_{i_{0}}$.

The following special type of weak ditalgebra, introduced in [3], plays a
relevant role in the construction of the reduction functors used in the proof
of our main results.

######  Definition 3.10.

A _stellar weak ditalgebra_ ${\cal A}$ is a weak ditalgebra with triangular
layer $(R,W)$, with $R$ a minimal algebra, such that there is a source point
$i_{0}$ of ${\cal A}$, and $W_{0}=W_{0}e_{i_{0}}$. So, we can consider the
points ${i_{1}},\ldots,{i_{s}}$ of ${\cal A}$ with
$e_{i_{1}}W_{0}e_{i_{0}}\not=0,\ldots,e_{i_{s}}W_{0}e_{i_{0}}\not=0$. The
algebra $A^{\prime}$ generated by $R^{\prime}=\prod_{j=0}^{s}Re_{i_{j}}$ and
$W^{\prime}_{0}=\sum_{j=1}^{s}e_{i_{j}}W_{0}e_{i_{0}}$ will be called the
_star_ of ${\cal A}$. The points $e_{i_{0}},\ldots,e_{i_{s}}$ are called the
_points of the star_ , and $e_{i_{0}}$ is its _center_.

Notice that if ${\cal A}$ is a stellar weak ditalgebra with layer $(R,W)$,
then we have $A=R\oplus W_{0}$.

######  Lemma 3.11.

Let $\underline{\cal A}=({\cal A},I)$ be a stellar triangular interlaced weak
ditalgebra with triangular layer $(R,W)$. Moreover, assume that $I\subseteq
W_{0}$ and take any $d\in\mathbb{N}$. Then, there is a stellar triangular
interlaced weak ditalgebra $\underline{\cal A}^{Y}=({\cal A}^{Y},I^{Y})$ with
triangular layer $(S,W^{Y})$, such that $I^{Y}$ is a direct summand of
$W_{0}^{Y}$ as an $S$-$S$-bimodule. Let $F^{Y}:\underline{\cal
A}^{Y}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$ be the
associated functor constructed in [3](8.4) and let $E$ be any $k$-algebra.
Then the following holds.

  1. 1.

The functor $F_{E}^{Y}:\underline{\cal A}^{Y}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ induces a length controlling equivalence to a
full subcategory of $\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ which
contains all the $\underline{\cal A}\hbox{-}E$-bimodules with length $\leq d$.

  2. 2.

If $E$ is a division $k$-algebra, $F^{Y}_{E}$ induces an equivalence from
$\underline{\cal A}^{Y}\hbox{-}E\hbox{-}{\rm Mod}_{p}$ to a full subcategory
of $\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ which contains all the
proper $\underline{\cal A}\hbox{-}E$-bimodules with length $\leq d$.

###### Proof.

Fix $d\in\mathbb{N}$, consider the functor $F^{Y}:\underline{\cal
A}^{Y}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$
constructed in [3](8.4) and adopt the notation of its proof. Thus,
$R=\prod_{i\in[0,n]}R_{i}e_{i}$, $[0,n]=J\uplus J^{\prime}$, where
$Re_{i}=ke_{i}$, for $j\in J$, and $Re_{j}=k[x]_{g_{j}(x)}e_{j}$, for $j\in
J^{\prime}$; $e=\sum_{i\in J}e_{i}$ and $e^{\prime}=\sum_{i\in
J^{\prime}}e_{i}$. Moreover, $\underline{\cal A}^{Y}$ is stellar triangular
interlaced weak ditalgebra, and we have the algebra $C=\prod_{j\in
J^{\prime}}e_{j}R_{j}/\langle h_{j}(x)^{d}\rangle$, which admits only a finite
number of isoclasses of indecomposable finite-dimensional $C$-modules
represented by the $C$-modules $Z_{1},\ldots,Z_{u}$. The polynomial
$h_{j}(x)\in k[x]$ is fixed for each $j\in J^{\prime}$. Consider
$Z:=\oplus_{s=1}^{u}Z_{s}$ and the $R$-module

$Y=Z\oplus\left[\oplus_{j\in J^{\prime}}e_{j}(R_{j})_{h_{j}(x)}\right]\oplus
Re.$

Then, we have the splitting ${\rm End}_{R}(Y)^{op}=S\oplus P$, where

$S=\left[\prod_{s=1}^{u}kf_{s}\right]\times\left[\prod_{j\in
J^{\prime}}e_{j}(R_{j})_{h_{j}(x)}\right]\times Re,$

$f_{s}\in{\rm End}_{R}(Y)^{op}$ is the idempotent corresponding to the
indecomposable direct summand $Z_{s}$ of $Y$, and $P={\rm rad}{\rm
End}_{R}(Z)^{op}$.

From (3.8), we already know that $F_{E}^{Y}:\underline{\cal
A}^{Y}\hbox{-}E\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}E\hbox{-}{\rm
Mod}$ is a length controlling, full and faithful functor. Then, its
restriction to proper bimodules $F_{E}^{Y}:\underline{\cal
A}^{Y}\hbox{-}E\hbox{-}{\rm Mod}_{p}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}E\hbox{-}{\rm
Mod}_{p}$ is full and faithful too.

Let us show that the functor $F_{E}^{Y}$ is dense on a full subcategory of
$\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ which contains all bimodules
with length $\leq d$. The argument is essentially the same given in [9](2.11)
(see also [5](25.9)). We only have to adapt it to the more general case of a
simultaneous unravellings at all the points $e_{j}$ with $j\in J^{\prime}$,
with associated polynomials $h_{j}(x)$, in the general case of triangular
interlaced weak ditalgebras. We only give a sketch.

For $j\in J^{\prime}$, make $H_{j}:=e_{j}R_{j}=k[x]_{g_{j}(x)}$ and consider
the prime divisors $\pi_{j,1},\ldots,\pi_{j,u_{j}}$ of $h_{j}(x)$. Then, write
$Z_{j,s,a}:=H_{j}/\pi_{j,s}^{a}$, for $s\in[1,u_{j}]$ and $a\in[1,d]$. So
$\\{Z_{j,s,a}\\}_{s,a}$ is a complete set of representatives of the isoclasses
of the indecomposable $H_{j}/\langle h_{j}(x)^{d}\rangle$-modules. When we
consider the union of all these families of indecomposables, for $j\in
J^{\prime}$, we recover the family of indecomposabe $C$-modules
$Z_{1},\ldots,Z_{u}$. Accordingly, we replace the preceding notation
$f_{1},\ldots,f_{u}$ for the idempotents associated to $Z_{1},\ldots,Z_{u}$,
respectively, by the more descriptive one which uses $f_{j,s,a}$ to denote the
idempotent in $S$ corresponding to the indecomposable $Z_{j,s,a}$.

_Proof of (1):_ It is enough to see that any $M\in\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ with length $\leq d$ is isomorphic in
$\underline{\cal A}\hbox{-}{\rm Mod}$ to $F^{Y}(N)$, for some
$N\in\underline{\cal A}^{Y}\hbox{-}{\rm Mod}$, and then we can apply (3.2).
Clearly, $\ell_{E}(e_{j}M)\leq d$, for all $j$. Since $x\in Re_{j}$, we have
that multiplication by $h_{j}(x)$ on $e_{j}M$ is an endomorphism of
$E$-modules, so by Fitting’s Lemma, we have a decomposition of $E$-modules
$e_{j}M=M_{j^{*}}\oplus M_{j}$, which is also a decomposition of
$k[x]$-modules, where $h_{j}(x)$ acts invertibly on $M_{j^{*}}$ and
$h_{j}(x)^{d}$ annihilates $M_{j}$. So, $M_{j}$ is a $k[x]/\langle
h_{j}(x)^{d}\rangle$-module and admits a decomposition of the form
$M_{j}=\bigoplus_{s=1}^{u_{i}}\bigoplus_{a=1}^{d}M_{j}(s,a)$, where
$M_{j}(s,a)$ is isomorphic to a possibly infinite direct sum of copies of
$k[x]/\langle\pi_{j,s}^{a}\rangle$. Then, we have $M_{j}(s,a)\cong
k[x]/\pi^{a}_{j,s}k[x]\otimes_{k}V_{j}(s,a)$, for an appropriate $k$-vector
space $V_{j}(s,a)$. Recall that we have isomorphisms of $k$-algebras
$H_{j}/H_{j}\pi_{j,s}^{a}\cong k[x]/\pi_{j,s}^{a}k[x]$.

Then, given $M\in\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$, we can
consider the $S$-module $N$ defined by the following: For $j\in J^{\prime}$,
make $e_{j}N:=M_{j_{*}}$ and, for $s\in[1,u_{j}]$ and $a\in[1,d]$, make
$f_{j,s,a}N:=V_{j}(s,a)$; and, for $j\in J$, make $e_{j}N:=e_{j}M$. Notice
that we have isomorphisms of $R\hbox{-}E$-bimodules
$Ye_{j}\otimes_{Se_{j}}e_{j}N\cong(H_{j})_{h_{j}(x)}\otimes_{(H_{j})_{h_{j}(x)}}M_{j_{*}}\cong
M_{j_{*}}$, for $j\in J^{\prime}$; $Ye_{j}\otimes_{Se_{j}}e_{j}N\cong
Re_{j}\otimes_{Re_{j}}e_{j}M\cong e_{j}M$, for $j\in J$; and, finally,
$Yf_{j,s,a}\otimes_{Sf_{j,s,a}}f_{j,s,a}N\cong
H_{j}/\pi^{a}_{j,s}H_{j}\otimes_{k}V_{j}(s,a)\cong M_{j}(s,a).$

Hence, we have an isomorphism of $R$-modules

$\begin{matrix}M&=&\bigoplus_{j\in J^{\prime}}e_{j}M\oplus\bigoplus_{j\in
J}e_{j}M\hfill\\\ &\cong&\bigoplus_{j\in
J^{\prime}}[(Y\otimes_{S}e_{j}N)\oplus(\bigoplus_{s=1}^{u_{i}}\bigoplus_{a=1}^{d}Y\otimes_{S}f_{j,s,a}N)]\oplus\bigoplus_{j\in
J}e_{j}N,\hfill\\\ \end{matrix}$

that is we have an isomorphism of $R$-modules $Y\otimes_{S}N\cong M$, and we
can complete the proof of (1), using (3.7).

_Proof of (2):_ Assume that $E$ is a division algebra and proceed as in the
first part of the proof of [5](25.9), using Crawley-Boevey’s argument of
[9](2.11), adapted to the multiple unravelling situation in the context of
triangular interlaced weak ditalgebras, using again (3.7). ∎

######  Lemma 3.12.

Let ${\cal A}$ be a stellar ditalgebra with triangular layer $(R,W)$ and such
that $W_{1}$ is freely generated by some finite directed subset. Then, given
any $d\in\mathbb{N}$, there is a stellar seminested ditalgebra ${\cal A}^{X}$,
as in [3](8.6), and a full and faithful functor

$\begin{matrix}F^{X}:{\cal A}^{X}\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&{\cal A}\hbox{-}{\rm Mod}\end{matrix}$

such that, for any $k$-algebra $E$, we have:

  1. 1.

The functor $F^{X}_{E}:{\cal A}^{X}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ induces a length controlling equivalence to a
full subcategory of ${\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ which contains all
the ${\cal A}\hbox{-}E$-bimodules with length $\leq d$.

  2. 2.

If $E$ is a division $k$-algebra, $F^{X}_{E}$ induces an equivalence from
${\cal A}^{X}\hbox{-}E\hbox{-}{\rm Mod}_{p}$ to a full subcategory of ${\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ which contains all the proper ${\cal
A}\hbox{-}E$-bimodules with length $\leq d$.

###### Proof.

This is similar to the proof of (3.11). We apply the corresponding arguments
to the functor $F^{X}:{\cal A}^{X}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}\hbox{-}{\rm Mod}$ constructed in
[3](8.7). ∎

######  Theorem 3.13.

Let $\underline{\cal A}=({\cal A},I)$ be a stellar triangular interlaced weak
ditalgebra with triangular layer $(R,W)$, with star center $e_{0}\not\in I$.
Let $E$ be any $k$-algebra and take $d\in\mathbb{N}$. Then, there is a
seminested ditalgebra ${\cal A}^{\prime}$ and a composition of full and
faithful length controlling reduction functors

$F_{E}:({\cal A}^{\prime},0)\hbox{-}E\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}({\cal A},I)\hbox{-}E\hbox{-}{\rm Mod}$

such that any $M\in({\cal A},I)\hbox{-}E\hbox{-}{\rm Mod}$ with length
$\ell_{E}(M)\leq d$ is of the form $F_{E}(N)\cong M$ for some
$N\in\underline{\cal A}^{\prime}\hbox{-}{\rm Mod}$. Moreover, if $E$ is a
division algebra and $M$ is proper, then $N$ can be taken to be proper.

###### Proof.

The idea is to follow the proof of [3](8.9), replacing dimensions by lengths
and applying (3.11), (3.6), and (3.12), instead of (8.4), (8.5), and (8.7) of
[3], respectively. ∎

##  4 Reductions with sources and restrictions

In this section we reconsider reduction procedures and functors described in
more specific situations which consider a source $e_{0}$ of a triangular
interlaced weak ditalgebra $\underline{\cal A}$. We follow the presentation of
the statements adding the behavior on bimodules.

######  Definition 4.1.

Assume that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with layer $(R,W)$ where $R$ is a minimal algebra. Assume that
$1=\sum_{i=0}^{n}e_{i}$ is the decomposition of the unit of $R$ as sum of
primitive orthogonal idempotents. Then, the idempotent $e_{0}$ is called a
_source of $\underline{\cal A}$_ iff $e_{0}W=0$, $e_{0}R=ke_{0}$, and
$e_{0}\not\in I$.

From now on, the layers $(R,W)$ of the triangular interlaced weak ditalgebras
we consider always have an $R$ which is a minimal algebra.

######  Lemma 4.2 (detachement of a source).

Assume that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with source idempotent $e_{0}$. Let $(R,W)$ be the triangular layer
of ${\cal A}=(T,\delta)$. Make $f:=1-e_{0}=\sum_{i=1}^{n}e_{i}$, then consider
the tensor algebra $T^{\circleddash}:=T_{R}(fWf)$, which can be considered as
a subalgebra of $T=T_{R}(W)$ invariant under $\delta$. Denote by
$\delta^{\circleddash}$ the restriction of $\delta$ to $T^{\circleddash}$.
Then, we obtain a new weak ditalgebra ${\cal
A}^{\circleddash}=(T^{\circleddash},\delta^{\circleddash})$ with layer
$(R^{\circleddash},W^{\circleddash}):=(R,fWf)$, which is also triangular. The
inclusion $fWf\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}W$
determines an embedding of algebras $T^{\circleddash}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}T$ and, moreover, an embedding of weak
ditalgebras ${\cal A}^{\circleddash}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}$.

Notice that $fTf$ is an algebra with unit $f$ and we have a surjective
morphism of algebras $\phi:T\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}fTf$ given by $\phi(t)=ftf$, for $t\in T$.
Observe also that $\phi\delta(t)=\delta^{\circleddash}\phi(t)$, for $t\in T$,
and

$T^{\circleddash}=ke_{0}\times fTf.$

The space $I^{\circleddash}=fIf$ is then an ideal of
$A^{\circleddash}:=[T^{\circleddash}]_{0}$ and $\underline{\cal
A}^{\circleddash}=({\cal A}^{\circleddash},I^{\circleddash})$ is a triangular
interlaced weak ditalgebra. We say that $\underline{\cal A}^{\circleddash}$ is
obtained from $\underline{\cal A}$ by _detachment of the source $e_{0}$_. We
have that $\underline{\cal A}^{\circleddash}$ is a Roiter interlaced weak
ditalgebra whenever $\underline{\cal A}$ is so.

Moreover, we have the restriction functor ${\rm Res}:\underline{\cal
A}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}^{\circleddash}\hbox{-}{\rm Mod}$ such that ${\rm Res}(M)=fM$ and, for
$h\in{\rm Hom}_{\underline{\cal A}}(M,N)$, its restriction is defined by ${\rm
Res}(h^{0},h^{1})=({\rm Res}(h^{0}),{\rm Res}(h^{1}))$, where ${\rm
Res}(h^{0}):fM\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}fN$ is
the restriction of the map $h^{0}$ and ${\rm
Res}(h^{1}):fVf\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
Hom}_{k}(fM,fN)$ is obtained from the restriction of
$h^{1}:V\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
Hom}_{k}(M,N)$. Then, for each $k$-algebra $E$, we have an induced functor

${\rm Res}_{E}:\underline{\cal A}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod},$

such that $\ell_{E}({\rm Res}_{E}(M))\leq\ell_{E}(M)$, for each
$M\in\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$.

###### Proof.

See [3](9.2) and (3.2). ∎

Recall the following, from [3](9.3).

######  Lemma 4.3.

Assume that $\underline{\cal A}$ is a triangular interlaced weak ditalgebra
with source idempotent $e_{0}$. Consider the triangular interlaced weak
ditalgebra $\underline{\cal A}^{\circleddash}$ obtained from $\underline{\cal
A}$ by detachement of the source $e_{0}$, the interlaced weak ditalgebra
$\underline{\cal A}^{\circleddash d_{0}}$ obtained from $\underline{\cal
A}^{\circleddash}$ by deletion of the idempotent $e_{0}$, and the interlaced
weak ditalgebra $\underline{\cal A}^{d_{0}}$ obtained directly from
$\underline{\cal A}$ by deletion of the idempotent $e_{0}$. Then,
$\underline{\cal A}^{\circleddash d_{0}}=\underline{\cal A}^{d_{0}}$.
Moreover, if $\underline{\cal A}$ is not wild, neither is $\underline{\cal
A}^{\circleddash}$.

The following series of lemmas show that, given a triangular interlaced weak
ditalgebra $\underline{\cal A}$ with source $e_{0}$, whenever we can perform a
reduction of type $z\in\\{d,r,q,a,X\\}$ on $\underline{\cal A}^{d_{0}}$, then
we can perform corresponding reductions of the same type $z$ on
$\underline{\cal A}^{\circleddash}$ and on $\underline{\cal A}$.

######  Lemma 4.4 (deletion of idempotents with source).

Assume that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with source idempotent $e_{0}$. Let $(R,W)$ be the triangular layer
of ${\cal A}=(T,\delta)$. Let $e=e_{0}+\sum_{s=1}^{t}e_{j_{s}}$ with
${j_{s}}\not=0$, for all $s$. Consider the triangular interlaced weak
ditalgebra $\underline{\cal A}^{d}$ obtained from $\underline{\cal A}$ by
deletion of the idempotent $1-e$, as in (2.1).

Then, $e_{0}$ is a source idempotent in $\underline{\cal A}^{d}$, so can also
consider the triangular interlaced weak ditalgebra $\underline{\cal
A}^{\circleddash d}$, obtained from $\underline{\cal A}^{\circleddash}$ by
deletion of the same idempotent $1-e$. Then, there is a canonical isomorphism
of interlaced weak ditalgebras $\underline{\cal A}^{\circleddash d}$ and
$\underline{\cal A}^{d\circleddash}$, which allows us to identify them and,
for each $k$-algebra $E$, we have the following commutative diagram of
functors

$\begin{matrix}\underline{\cal A}^{d}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{d}}}&\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}\\\ \downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}\\\ \underline{\cal
A}^{d\circleddash}\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash d}}}&\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod}.\\\ \end{matrix}$

Moreover, if $M\in\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ is such that
${\rm Res}_{E}(M)\cong F_{E}^{\circleddash d}(N^{\prime})$, for some
$N^{\prime}\in\underline{\cal A}^{d\circleddash}\hbox{-}E\hbox{-}{\rm Mod}$,
then $M\cong F_{E}^{d}(N)$, for some $N\in\underline{\cal
A}^{d}\hbox{-}E\hbox{-}{\rm Mod}$.

###### Proof.

It follows from [3](9.4) and (3.2). ∎

######  Lemma 4.5 (regularization with source).

Assume that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with source idempotent $e_{0}$. Let $(R,W)$ be the triangular layer
of ${\cal A}=(T,\delta)$. Suppose that we have $R$-$R$-bimodule decompositions
$W_{0}=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$ and
$W_{1}=\delta(W^{\prime}_{0})\oplus W^{\prime\prime}_{1}$. Consider the
triangular interlaced weak ditalgebra $\underline{\cal A}^{r}$ obtained by
regularization of the bimodule $W^{\prime}_{0}$, as in (2.2).

Then, $e_{0}$ is a source idempotent in $\underline{\cal A}^{r}$. We
furthermore assume that $W^{\prime}_{0}e_{0}=0$. Thus, if we take $f=1-e_{0}$,
we get $fW^{\prime}_{0}f=W^{\prime}_{0}$, $fW_{0}f=W^{\prime}_{0}\oplus
fW^{\prime\prime}_{0}f$, and $fW_{1}f=\delta(W^{\prime}_{0})\oplus
fW^{\prime\prime}_{1}f$. Then, we can consider the triangular interlaced weak
ditalgebra $\underline{\cal A}^{\circleddash r}$ obtained from
$\underline{\cal A}^{\circleddash}$ by regularization of the same bimodule
$W^{\prime}_{0}$.

There is a canonical isomorphism of interlaced weak ditalgebras
$\underline{\cal A}^{\circleddash r}$ and $\underline{\cal
A}^{r\circleddash}$, which allows us to identify them and, for each
$k$-algebra $E$, we have the following commutative diagram of functors

$\begin{matrix}\underline{\cal A}^{r}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{r}}}&\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}\\\ \downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}\\\ \underline{\cal
A}^{r\circleddash}\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash r}}}&\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod}.\\\ \end{matrix}$

Moreover, if $\underline{\cal A}$ is a Roiter interlaced weak ditalgebra and
$M\in\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ is such that ${\rm
Res}_{E}(M)\cong F_{E}^{\circleddash r}(N^{\prime})$, for some
$N^{\prime}\in\underline{\cal A}^{r\circleddash}\hbox{-}E\hbox{-}{\rm Mod}$,
then $M\cong F_{E}^{r}(N)$, for some $N\in\underline{\cal
A}^{r}\hbox{-}E\hbox{-}{\rm Mod}$.

###### Proof.

It follows from [3](9.5) and (3.2). ∎

######  Lemma 4.6 (reduction by factoring out a direct summand of $W_{0}$ with
source).

Let $\underline{\cal A}=({\cal A},I)$ be a triangular interlaced weak
ditalgebra with source $e_{0}$ and let $(R,W)$ be the triangular layer of
${\cal A}=(T,\delta)$. Make $f=1-e_{0}$. Assume that there is a decomposition
$fW_{0}f=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$ as $fR$-$fR$-bimodules
such that $W^{\prime}_{0}\subseteq fIf$ and $\delta(W^{\prime}_{0})\subseteq
A^{\circleddash}W^{\prime}_{0}fVf+fVfW^{\prime}_{0}A^{\circleddash}$. Thus
$W_{0}=W_{0}^{\prime}\oplus(W^{\prime\prime}_{0}\oplus W_{0}e_{0})$ is a
decomposition of $R$-$R$-bimodules with $W^{\prime}_{0}\subseteq I$ and
$\delta(W^{\prime}_{0})\subseteq AW^{\prime}_{0}V+VW^{\prime}_{0}A$. Then, we
can form the triangular interlaced weak ditalgebra $\underline{\cal A}^{q}$,
determined by this last decomposition, as in (2.3). Then, $e_{0}$ is a source
idempotent in $\underline{\cal A}^{q}$ and we can form the interlaced weak
ditalgebra $\underline{\cal A}^{q\circleddash}$. We can also form the
triangular interlaced weak ditalgebra $\underline{\cal A}^{\circleddash}$ and
then consider the interlaced weak ditalgebra $\underline{\cal A}^{\circleddash
q}$ associated to the decomposition $fW_{0}f=W_{0}^{\prime}\oplus
W^{\prime\prime}_{0}$. Then, there is a canonical isomorphism of interlaced
weak ditalgebras which allows us to identify $\underline{\cal
A}^{q\circleddash}$ with $\underline{\cal A}^{\circleddash q}$, and, for each
$k$-algebra $E$, we have a commutative square of functors

$\begin{matrix}\underline{\cal A}^{q}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{q}}}&\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}\\\ \downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}\\\ \underline{\cal
A}^{q\circleddash}\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash q}}}&\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod}.\\\ \end{matrix}$

###### Proof.

It follows from [3](9.6). ∎

######  Lemma 4.7 (absorption of a loop with source).

Let $\underline{\cal A}=({\cal A},I)$ be a triangular interlaced weak
ditalgebra with source $e_{0}$, where ${\cal A}=(T,\delta)$ admits the
triangular layer $(R,W)$, with $Re_{i}=ke_{i}$ for some idempotent
$e_{i}\not=e_{0}$. Make $f=1-e_{0}$. Assume that there is a decomposition of
$fR$-$fR$-bimodules $fW_{0}f=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$ with
$W^{\prime}_{0}\cong e_{i}R\otimes_{k}Re_{i}$ and $\delta(W^{\prime}_{0})=0$.
Then, we can consider the triangular interlaced weak ditalgebra
$\underline{\cal A}^{\circleddash}$ and the triangular interlaced weak
ditalgebra $\underline{\cal A}^{\circleddash a}$ obtained by the absorption of
the $R$-$R$-subbimodule $W^{\prime}_{0}$, as in (2.4). We also have the
decomposition of $R$-$R$-bimodules
$W_{0}=W^{\prime}_{0}\oplus(W^{\prime\prime}_{0}\oplus W_{0}e_{0})$, and we
can consider the triangular interlaced weak ditalgebra $\underline{\cal
A}^{a}$ obtained by absorption of the $R$-$R$-bimodule $W^{\prime}_{0}$, where
$e_{0}$ is a source. Consider the triangular interlaced weak ditalgebra
$\underline{\cal A}^{a\circleddash}$, then $\underline{\cal A}^{\circleddash
a}=\underline{\cal A}^{a\circleddash}$ and, for each $k$-algebra $E$, there is
a commutative square of functors

$\begin{matrix}\underline{\cal A}^{a}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{a}}}&\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}\\\ \downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}\\\ \underline{\cal
A}^{a\circleddash}\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash a}}}&\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod}.\\\ \end{matrix}$

###### Proof.

It follows from [3](9.7). ∎

######  Lemma 4.8 (reduction with an admissible module with source).

Assume that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with source idempotent $e_{0}$. Let $(R,W)$ be the triangular layer
of ${\cal A}=(T,\delta)$. Write $f=1-e_{0}$ and assume that we have a
decomposition $fW_{0}f=W^{\prime}_{0}\oplus W^{\prime\prime}_{0}$ of
$fR$-$fR$-bimodules with $\delta(W^{\prime}_{0})=0$. Notice that we have the
decomposition of $R$-$R$-bimodules $W_{0}=W^{\prime}_{0}\oplus
W^{\prime\prime}_{0}\oplus W_{0}e_{0}$, with $\delta(W^{\prime}_{0})=0$.

Consider the $k$-subalgebra $\underline{B}=T_{fR}(W^{\prime}_{0})$ of $fTf$
and the $k$-subalgebra $B=T_{R}(W^{\prime}_{0})=ke_{0}\times\underline{B}$ of
$T^{\circleddash}$. Suppose that $\underline{X}$ is a complete triangular
admissible $\underline{B}$-module, thus we have a splitting
$\underline{\Gamma}={\rm End}_{\underline{B}}(\underline{X})^{op}=S\oplus P$,
where $S$ is a subalgebra of $\underline{\Gamma}$, $P$ is an ideal of
$\underline{\Gamma}$, and $\underline{X}$ and $P$ are finitely generated
projective right $S$-modules.

Now, consider the $B$-module $X=ke_{0}\oplus\underline{X}$. Then, we have a
splitting $\Gamma={\rm End}_{B}(X)^{op}=S^{\circleddash}\oplus P$, where
$S^{\circleddash}=ke_{0}\times S$ is a subalgebra of $\Gamma$, $P$ is an ideal
of $\Gamma$, and $X$ and $P$ are finitely generated projective right
$S^{\circleddash}$-modules. Then, $X$ is a complete triangular admissible
$B$-module.

Then, we can form the triangular interlaced weak ditalgebra $\underline{\cal
A}^{X}$ obtained from $\underline{\cal A}$ by reduction using the admissible
$B$-module $X$, as in (2.6). Then, $e_{0}$ is a source idempotent in
$\underline{\cal A}^{X}$ and we can form the triangular interlaced weak
ditalgebra $\underline{\cal A}^{X\circleddash}$. We can also form the
triangular interlaced weak ditalgebra $\underline{\cal A}^{\circleddash X}$
obtained from $\underline{\cal A}^{\circleddash}$ by reduction using the same
$B$-module $X$.

Moreover, there is a canonical isomorphism of interlaced weak ditalgebras
$\underline{\cal A}^{\circleddash X}$ and $\underline{\cal
A}^{X\circleddash}$, which allows us to identify them and, for each
$k$-algebra $E$, we have the following commutative diagram of functors with
full and faithful rows

$\begin{matrix}\underline{\cal A}^{X}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{X}}}&\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}\\\ \downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle{\rm Res}_{E}$}}$\hss}\\\ \underline{\cal
A}^{X\circleddash}\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash X}}}&\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod}.\\\ \end{matrix}$

Moreover, if $M\in\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}$ is such that
${\rm Res}_{E}(M)\cong F_{E}^{\circleddash X}(N^{\prime})$, for some
$N^{\prime}\in\underline{\cal A}^{X\circleddash}\hbox{-}E\hbox{-}{\rm Mod}$,
then $M\cong F^{X}(N)$, for some $N\in\underline{\cal
A}^{X}\hbox{-}E\hbox{-}{\rm Mod}$.

###### Proof.

It follows from [3](9.8) and (3.2). ∎

##  5 Reduction to minimal ditalgebras

In this section we show that the study of bimodules with bounded length over
any non-wild directed interlaced weak ditalgebra $\underline{\cal A}$, can be
reduced to the study of the bimodules over a minimal ditalgebra obtained from
$\underline{\cal A}$ by a finite number of reductions.

######  Lemma 5.1.

Assume that $\underline{\cal A}=({\cal A},I)$ is a triangular interlaced weak
ditalgebra with source $e_{0}$. Denote by $\underline{\cal A}^{d_{0}}$ the
triangular interlaced weak ditalgebra obtained from $\underline{\cal A}$ by
deletion of the idempotent $e_{0}$. Consider, as before, the idempotent
$f=1-e_{0}$. Then,

  1. 1.

We have $\underline{\cal A}^{\circleddash}=\underline{\cal
E}\times\underline{\cal A}^{d_{0}}$, with $\underline{\cal E}=({\cal E},0)$,
where ${\cal E}=(ke_{0},0)$ is the one-point ditalgebra determined by the
source $e_{0}$ of ${\cal A}$.

  2. 2.

There is an equivalence of categories $G:\underline{\cal
A}^{\circleddash}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal E}\hbox{-}{\rm
Mod}\times\underline{\cal A}^{d_{0}}\hbox{-}{\rm Mod}$ given by
$G(M)=(e_{0}M,fM)$ and, for $g=(g^{0},g^{1})\in{\rm Hom}_{\underline{\cal
A}^{\circleddash}}(M,N)$, we have
$G(g)=[(g^{0}_{0},0),(g^{0}_{f},g^{1}_{f})]$, where $g^{0}=g_{0}^{0}\oplus
g_{f}^{0}$ is the canonical decomposition of $g^{0}$, and
$g_{f}^{1}:fVf\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
Hom}_{k}(fM,fN)$ denotes the restriction of the $A^{\circleddash}$-bimodule
morphism $g^{1}:fVf\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
Hom}_{k}(M,N)$.

Let us write ${\cal S}:=\underline{\cal E}\hbox{-}{\rm Mod}={\cal
E}\hbox{-}{\rm Mod}=ke_{0}\hbox{-}{\rm Mod}$ and, for any $k$-algebra $E$,
make ${\cal S}^{E}:=\underline{\cal E}\hbox{-}E\hbox{-}{\rm
Mod}=ke_{0}\hbox{-}E\hbox{-}{\rm Mod}$. Then, $G$ induces a length preserving
equivalence

$G_{E}:\underline{\cal A}^{\circleddash}\hbox{-}E\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal
S}^{E}\times\underline{\cal A}^{d_{0}}\hbox{-}E\hbox{-}{\rm Mod}.$

  3. 3.

Under the assumptions of (4.8), we have $\underline{\cal
A}^{Xd_{0}}=\underline{\cal A}^{d_{0}\underline{X}}$. Moreover, for any
$k$-algebra $E$, we have the commutative diagram

$\begin{matrix}\underline{\cal A}^{X\circleddash}\hbox{-}E\hbox{-}{\rm
Mod}&=&\underline{\cal A}^{\circleddash X}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash X}}}&\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod}\\\ \hbox to0.0pt{\hss$\vbox{\hbox{
$\scriptstyle G_{E}^{X}$}}$}\Bigg{\downarrow}&&&&\hbox
to0.0pt{\hss$\vbox{\hbox{ $\scriptstyle G_{E}$}}$}\Bigg{\downarrow}\\\ {\cal
S}^{E}\times\underline{\cal A}^{Xd_{0}}\hbox{-}E\hbox{-}{\rm Mod}&=&{\cal
S}^{E}\times\underline{\cal A}^{d_{0}\underline{X}}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{1\times
F_{E}^{\underline{X}}}}&{\cal S}^{E}\times\underline{\cal
A}^{d_{0}}\hbox{-}E\hbox{-}{\rm Mod}.\\\ \end{matrix}$

This implies that, given $d\in\mathbb{N}$, if the functor
$F_{E}^{\underline{X}}$ covers any bimodule $M$ with $\ell_{E}(M)\leq d$, so
does $F_{E}^{\circleddash X}$.

  4. 4.

For $z\in\\{d,r,q,a\\}$, assume that we are respectively under the assumptions
of (4.4), (4.5), (4.6), and (4.7). Then, for any $k$-algebra $E$, we get a
commutative diagram of functors

$\begin{matrix}\underline{\cal A}^{z\circleddash}\hbox{-}E\hbox{-}{\rm
Mod}&=&\underline{\cal A}^{\circleddash z}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash z}}}&\underline{\cal
A}^{\circleddash}\hbox{-}E\hbox{-}{\rm Mod}\\\ \hbox to0.0pt{\hss$\vbox{\hbox{
$\scriptstyle G_{E}^{z}$}}$}\Bigg{\downarrow}&&&&\hbox
to0.0pt{\hss$\vbox{\hbox{ $\scriptstyle G_{E}$}}$}\Bigg{\downarrow}\\\ {\cal
S}^{E}\times\underline{\cal A}^{zd_{0}}\hbox{-}E\hbox{-}{\rm Mod}&=&{\cal
S}^{E}\times\underline{\cal A}^{d_{0}z}\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{1\times
F_{E}^{z}}}&{\cal S}^{E}\times\underline{\cal A}^{d_{0}}\hbox{-}E\hbox{-}{\rm
Mod}.\\\ \end{matrix}$

This implies that, given $d\in\mathbb{N}$, if $F_{E}^{z}$ covers any bimodule
$M$ with $\ell_{E}(M)\leq d$, so does $F_{E}^{\circleddash z}$.

###### Proof.

The equivalence $G$ is constructed in [3](10.1). It is clear that $G_{E}$ is a
length preserving functor. The rest of the statements follow from [3](10.1). ∎

######  Theorem 5.2.

Assume that the ground field $k$ is algebraically closed. Assume
$\underline{\cal A}=({\cal A},I)$ is a directed triangular interlaced weak
ditalgebra, as in (1.8), where $I$ is a proper ideal of $A$. Suppose that
$\underline{\cal A}$ is not wild and take $d\in\mathbb{N}$. Then, there is a
finite sequence of reductions

$\underline{\cal A}=({\cal A},I)\mapsto({\cal
A}^{z_{1}},I^{z_{1}})\mapsto\cdots\mapsto({\cal A}^{z_{1}z_{2}\cdots
z_{t}},I^{z_{1}z_{2}\cdots z_{t}})$

of type $z_{i}\in\\{a,d,r,q,X\\}$ such that ${\cal A}^{z_{1}z_{2}\cdots
z_{t}}$ is a minimal ditalgebra, we have $I^{z_{1}z_{2}\cdots z_{t}}=0$ and,
for any $k$-algebra $E$ and every $({\cal A},I)\hbox{-}E$-bimodule $M$ with
$\ell_{E}(M)\leq d$ has the form $M\cong F_{E}^{z_{1}}\cdots
F_{E}^{z_{t}}(N)$, for some $({\cal A}^{z_{1}\cdots z_{t}},I^{z_{1}\cdots
z_{t}})\hbox{-}E$-bimodule $N$. Moreover, if $E$ is a division algebra and $M$
is a proper bimodule, then $N$ can be chosen to be proper.

###### Proof.

The proof adapts the line of reasoning of the proof of [3](10.2), to our
situation here: now we consider bimodules and look at the length instead of
the dimension of the bimodules. Let $(R,W)$ be the layer of ${\cal A}$. By
assumption $R$ is a product of fields. If there is some idempotent $e_{i}\in
I\cap R$ we can consider the triangular interlaced weak ditalgebra $({\cal
A}^{d},I^{d})$ obtained by deletion of the idempotent $e_{i}$ and, for any
$k$-algebra $E$, the equivalence functor $F_{E}^{d}:({\cal
A}^{d},I^{d})\hbox{-}E\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}({\cal A},I)\hbox{-}E\hbox{-}{\rm Mod}$
which is length preserving. We have that $({\cal A}^{d},I^{d})$ is again a
non-wild directed triangular interlaced weak ditalgebra. Since we have a
surjective morphism of algebras $\phi:A\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}A^{d}$ and $\phi(I)=I^{d}$, we know that
$I^{d}$ is a proper ideal of $A^{d}$. Then, by applying this procedure a
finite number of times, if necessary, we can assume that no idempotent $e_{i}$
belongs to $I$. Then, since ${\cal A}$ is directed, we can choose a source
point $e_{0}$ of $({\cal A},I)$, so $Re_{0}=ke_{0}$, $e_{0}W=0$, and
$e_{0}\not\in I$.

We will proceed by induction on the number of points of the interlaced weak
ditalgebra $({\cal A},I)$. At the base of the induction, if the source point
$e_{0}$ is the unique point of ${\cal A}$, then ${\cal A}$ admits no arrows,
so ${\cal A}$ is a minimal ditalgebra, and we are done.

Assume $n\in\mathbb{N}$, with $n>1$. Suppose that the theorem holds for any
non-wild directed triangular interlaced weak ditalgebra $({\cal
A}^{\prime},I^{\prime})$ with $m<n$ points, where $I^{\prime}$ is a proper
ideal of $A^{\prime}$. Suppose that $({\cal A},I)$ is a non-wild directed
triangular interlaced weak ditalgebra with source $e_{0}$ and $n$ points,
where $I$ is a proper ideal of $A$, and take $d\in\mathbb{N}$. Then we can
consider the non-wild directed triangular interlaced weak ditalgebra $({\cal
A}^{\circleddash},I^{\circleddash})$. If we consider the ditalgebra obtained
from $({\cal A}^{\circleddash},I^{\circleddash})$ by deletion of the
idempotent $e_{0}$, we obtain a directed triangular interlaced weak ditalgebra
$({\cal A}^{\circleddash d_{0}},I^{\circleddash d_{0}})$. From (4.3), we know
that $({\cal A}^{\circleddash d_{0}},I^{\circleddash d_{0}})=({\cal
A}^{d_{0}},I^{d_{0}})$. We can apply our induction hypothesis to $({\cal
A}^{d_{0}},I^{d_{0}})$ and to $d\in\mathbb{N}$ to obtain a finite sequence of
reductions

$({\cal A}^{d_{0}},I^{d_{0}})\mapsto({\cal
A}^{d_{0}z_{1}},I^{d_{0}z_{1}})\mapsto\cdots\mapsto({\cal
A}^{d_{0}z_{1}z_{2}\cdots z_{t}},I^{d_{0}z_{1}z_{2}\cdots z_{t}})$

of type $z_{i}\in\\{a,d,r,q,X\\}$ such that ${\cal A}^{d_{0}z_{1}z_{2}\cdots
z_{t}}$ is a minimal ditalgebra, we have $I^{d_{0}z_{1}z_{2}\cdots z_{t}}=0$
and, for any $k$-algebra $E$, each $({\cal
A}^{d_{0}},I^{d_{0}})\hbox{-}E$-bimodule $M^{\prime\prime}$ with
$\ell_{E}(M^{\prime\prime})\leq d$ has the form $M^{\prime\prime}\cong
F_{E}^{z_{1}}\cdots F_{E}^{z_{t}}(N^{\prime\prime})$, for some
$N^{\prime\prime}\in({\cal A}^{d_{0}z_{1}\cdots z_{t}},I^{d_{0}z_{1}\cdots
z_{t}})\hbox{-}E\hbox{-}{\rm Mod}$.

Since we have $I^{d_{0}z_{1}z_{2}\cdots z_{t}}=0$, we get $I^{\circleddash
z_{1}z_{2}\cdots z_{t}}=0$. From (5.1), we have the commutative diagram

$\begin{matrix}({\cal A}^{\circleddash z_{1}\cdots z_{t}},I^{\circleddash
z_{1}\cdots z_{t}})\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash
z_{t}}}}&\cdots&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash z_{1}}}}&({\cal
A}^{\circleddash},I^{\circleddash})\hbox{-}E\hbox{-}{\rm Mod}\\\ \hbox
to0.0pt{\hss$\vbox{\hbox{ $\scriptstyle G_{E}^{z_{1}\cdots
z_{t}}$}}$}\Bigg{\downarrow}&&&&\hbox to0.0pt{\hss$\vbox{\hbox{ $\scriptstyle
G_{E}$}}$}\Bigg{\downarrow}\\\ {\cal S}^{E}\times({\cal A}^{d_{0}z_{1}\cdots
z_{t}},I^{d_{0}z_{1}\cdots z_{t}})\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{1\times
F_{E}^{z_{t}}}}&\cdots&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{1\times F_{E}^{z_{1}}}}&{\cal
S}^{E}\times({\cal A}^{d_{0}},I^{d_{0}})\hbox{-}E\hbox{-}{\rm Mod},\\\
\end{matrix}$

where the vertical arrows correspond to length-preserving equivalences of
categories, and the fact that any $({\cal
A}^{\circleddash},I^{\circleddash})\hbox{-}E$-bimodule $M^{\prime}$ with
$\ell_{E}(M^{\prime})\leq d$, has the form $M^{\prime}\cong
F_{E}^{\circleddash z_{1}}\cdots F_{E}^{\circleddash z_{t}}(N^{\prime})$, for
some $N^{\prime}\in({\cal A}^{\circleddash z_{1}\cdots z_{t}},I^{\circleddash
z_{1}\cdots z_{t}})\hbox{-}E\hbox{-}{\rm Mod}$.

Moreover, $({\cal A}^{z_{1}\cdots z_{t}\circleddash},I^{z_{1}\cdots
z_{t}\circleddash})=({\cal A}^{\circleddash z_{1}\cdots z_{t}},I^{\circleddash
z_{1}\cdots z_{t}})$ and, from the lemmas of §4, we have a commutative diagram

$\begin{matrix}({\cal A}^{z_{1}\cdots z_{t}},I^{z_{1}\cdots
z_{t}})\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{z_{t}}}}&\cdots&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{z_{1}}}}&({\cal
A},I)\hbox{-}E\hbox{-}{\rm Mod}\\\ \hbox to0.0pt{\hss$\vbox{\hbox{
$\scriptstyle{\rm Res}_{E}$}}$}\Bigg{\downarrow}&&&&\hbox
to0.0pt{\hss$\vbox{\hbox{ $\scriptstyle{\rm Res}_{E}$}}$}\Bigg{\downarrow}\\\
({\cal A}^{z_{1}\cdots z_{t}\circleddash},I^{z_{1}\cdots
z_{t}\circleddash})\hbox{-}E\hbox{-}{\rm Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash
z_{t}}}}&\cdots&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{\circleddash z_{1}}}}&({\cal
A}^{\circleddash},I^{\circleddash})\hbox{-}E\hbox{-}{\rm Mod}\end{matrix}$

where ${\cal A}^{z_{1}\cdots z_{t}\circleddash}$ is a minimal ditalgebra,
$I^{z_{1}\cdots z_{t}\circleddash}=0$, and any $({\cal
A}^{\circleddash},I^{\circleddash})\hbox{-}E$-bimodule with
$\ell_{E}(M^{\prime})\leq d$ is of the form $F_{E}^{\circleddash z_{1}}\cdots
F_{E}^{\circleddash z_{t}}(N^{\prime})\cong M^{\prime}$. If $M\in({\cal
A},I)\hbox{-}E\hbox{-}{\rm Mod}$ and $\ell_{E}(M)\leq d$, then $\ell_{E}({\rm
Res}_{E}(M))\leq d$, and ${\rm Res}_{E}(M)\cong F_{E}^{\circleddash
z_{1}}\cdots F_{E}^{\circleddash z_{t}}(N^{\prime})$. Thus, for some
$N\in({\cal A}^{z_{1}\cdots z_{t}},I^{z_{1}\cdots z_{t}})\hbox{-}E\hbox{-}{\rm
Mod}$, we have $F_{E}^{z_{1}}\cdots F_{E}^{z_{t}}(N)\cong M$. Moreover,
$\ell_{E}(N)\leq d$. For the sake of simplicity, make $({\cal
A}^{\prime},I^{\prime}):=({\cal A}^{z_{1}\cdots z_{t}},I^{z_{1}\cdots
z_{t}})$.

Here, we have that ${\cal A}^{\prime\circleddash}$ is a minimal ditalgebra,
$I^{\prime\circleddash}=0$, and $({\cal A}^{\prime},I^{\prime})$ is a
triangular interlaced weak ditalgebra with source $e_{0}$. If
$(R^{\prime},W^{\prime})$ denotes the layer of ${\cal A}^{\prime}$, say with
$R^{\prime}=R^{\prime}_{0}\times\cdots\times R^{\prime}_{m}$, and we make
$f=1-e_{0}=\sum_{j=1}^{m}e_{j}\in R^{\prime}$, we know that
$\bigoplus_{i,j=1}^{m}e_{j}W^{\prime}_{0}e_{i}=fW^{\prime}_{0}f=0$. Thus,
$W^{\prime}_{0}e_{0}=W^{\prime}_{0}$, and we know that $({\cal
A}^{\prime},I^{\prime})$ is a stellar triangular interlaced weak ditalgebra
with star center $e_{0}$. Then, apply (3.13) to $d$, to obtain a composition
of full and faithful reduction functors

$\begin{matrix}({\cal A}^{\prime\prime},0)\hbox{-}E\hbox{-}{\rm
Mod}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{y_{s}}}}&\cdots&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F_{E}^{y_{1}}}}&({\cal
A}^{\prime},I^{\prime})\hbox{-}E\hbox{-}{\rm Mod}\\\ \end{matrix}$

such that ${\cal A}^{\prime\prime}$ is a non-wild seminested ditalgebra and
any $N\in({\cal A}^{\prime},I^{\prime})\hbox{-}E\hbox{-}{\rm Mod}$ with
$\ell_{E}(N)\leq d$ is of the form $F_{E}^{y_{1}}\cdots F_{E}^{y_{s}}(L)\cong
N$ for some $L\in({\cal A}^{\prime\prime},0)\hbox{-}E\hbox{-}{\rm Mod}$.
Moreover, $\ell_{E}(L)\leq d$. Then, from [5](28.22), there is a minimal
ditalgebra ${\cal B}$ and a composition of full and faithful reduction
functors $F_{E}:{\cal B}\hbox{-}E\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal A}^{\prime\prime}\hbox{-}E\hbox{-}{\rm
Mod}$ such that for any ${\cal A}^{\prime\prime}\hbox{-}E$-bimodule $L$ with
$\ell_{E}(L)\leq d$, there is a ${\cal B}\hbox{-}E$-bimodule $H$ such that
$F_{E}(H)\cong L$.

Thus, given $M\in({\cal A},I)\hbox{-}E\hbox{-}{\rm Mod}$ with $\ell_{E}(M)\leq
d$, we obtain

$M\cong F_{E}^{z_{1}}\cdots F_{E}^{z_{t}}F_{E}^{y_{1}}\cdots
F_{E}^{y_{s}}F_{E}(H).$

∎

######  Definition 5.3.

If ${\cal B}$ is a minimal ditalgebra with layer $(R,W)$, we have the
decomposition of the unit element of $R$ as a sum $1=\sum_{i=1}^{s}e_{i}$ of
primitive orthogonal idempotents. If $Be_{i}\not=k$, we denote by $Q_{i}$ the
field of fractions of the rational $k$-algebra $Be_{i}$, considered as a
$B$-module.

######  Theorem 5.4.

Assume that the field $k$ is algebraically closed and that $\underline{\cal
A}$ is a directed triangular interlaced weak ditalgebra, as in (1.8). Suppose
that $\underline{\cal A}$ is not wild and take $d\in\mathbb{N}$. Consider the
functor $F:{\cal B}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$
associated to the composition of reductions of (5.2), where ${\cal B}$ is a
minimal ditalgebra. Denote by $B$ the underlying algebra of degree zero
elements of ${\cal B}$. The following holds.

  1. 1.

The $\underline{A}\hbox{-}B$-bimodule $\overline{T}:=F(B)$ is finitely
generated by the right and the composition

$B\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{L_{\cal B}}}{\cal B}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{F}}\underline{\cal
A}\hbox{-}{\rm Mod}$

is naturally isomorphic to $L_{\underline{\cal A}}(\overline{T}\otimes_{B}-)$.
So, $F(N)\cong\overline{T}\otimes_{B}N$, for any $N\in{\cal B}\hbox{-}{\rm
Mod}$. If $N$ is a proper ${\cal B}\hbox{-}E$-bimodule, for some $k$-algebra
$E$, then we have an isomorphism of proper $\underline{A}\hbox{-}E$-bimodules
$F_{E}(N)\cong\overline{T}\otimes_{B}(N_{E})$.

  2. 2.

The $\underline{\cal A}$-modules of the form
$G\cong\overline{T}\otimes_{B}Q_{i}$ are pregeneric.

  3. 3.

If $M$ is an indecomposable $\underline{\cal A}$-module with $\hbox{\rm
endol}\,(M)\leq d$, there is an indecomposable ${\cal B}$-module $N$ with
finite endolength such that $F(N)\cong M$.

Moreover, if $M$ is pregeneric, we have ${\rm End}_{\underline{\cal
A}}(M)^{op}=\overline{Q}_{M}\bigoplus{\rm rad}{\rm End}_{\underline{\cal
A}}(M)^{op}$, for some subalgebra $\overline{Q}_{M}\cong k(x)$ and the ideal
${\rm rad}{\rm End}_{\underline{\cal A}}(M)^{op}$ is nilpotent.

  4. 4.

If $M$ is pregeneric, the $B$-module $N$ is generic and has a natural
structure of $B\hbox{-}k(x)$-bimodule. The morphism of algebras
$\mu:k(x)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{\underline{\cal A}}(\overline{T}\otimes_{B}N)^{op}$ given by
$\mu(q)=(id_{\overline{T}}\otimes\mu_{q},0)$ gives $M$ the structure of a
proper $\underline{A}\hbox{-}k(x)$-bimodule $\overline{T}\otimes_{B}N\cong
F(N)\cong M$ such that $\hbox{\rm endol}\,(M)=\dim_{k(x)}M$.

###### Proof.

(1) follows from [3](7.12). The proof of $(2)$ is similar to the proof of
[9](4.3), because $F_{k(x)}$ is length controlling.

(3)&(4): Here, $M$ is an $\underline{\cal A}\hbox{-}E$-bimodule with finite
length $\ell_{E}(M)$, where $E={\rm End}_{\underline{\cal A}}(M)^{op}$ and its
bimodule structure map $\alpha_{M}:E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal A}}(M)^{op}$ is
the identity. From (5.2), we know that there is a ${\cal B}\hbox{-}E$-bimodule
$N$ with finite length $\ell_{E}(N)$ such that $F_{E}(N)\cong M$. For the sake
of simplicity, we can assume that $F_{E}(N)=M$. The bimodule structure
$\alpha_{N}:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{\cal B}(N)^{op}$ of $N$ is given by the composition

$E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\alpha_{M}}}{\rm
End}_{\underline{\cal A}}(M)^{op}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F^{-1}}}{\rm End}_{{\cal B}}(N)^{op},$

so $\alpha_{N}$ is an isomorphism.

Since $E\cong{\rm End}_{\cal B}(N)^{op}$ and $\ell_{E}(N)$ is finite, the
module $N\in{\cal B}\hbox{-}{\rm Mod}$ has finite endolength. By [5](31.6),
the module $N$ over the minimal algebra $B$ has finite endolength.

If $M$ is finite-dimensional, so is $N$. If $M$ is infinite-dimensional, $N$
has the same property, so, $N$ is a pregeneric $B$-module.

For each point $i$ of ${\cal B}$, we have that $Be_{i}=ke_{i}$ or
$\Gamma_{i}=Be_{i}\not=ke_{i}$. In the last case, we have $Q_{i}\cong k(x)$
the field of fractions of $\Gamma_{i}$, considered as a $\Gamma_{i}$-module.
It is known that the $B$-modules $Q_{i}$ provide a complete set of
representatives of the isomorphism classes of the pregeneric $B$-modules, see
[5](31.3). So, we can assume that $N\cong Q_{i}$, for some $i$. Moreover, by
[5](31.6), we have

${\rm End}_{{\cal B}}(N)^{op}=Q_{N}\bigoplus{\rm rad}{\rm End}_{{\cal
B}}(N)^{op},$

for some subalgebra $Q_{N}$ of ${\rm End}_{{\cal B}}(N)^{op}$ such that
$Q_{N}\cong k(x)$ and the ideal ${\rm rad}{\rm End}_{{\cal B}}(N)^{op}$ is
nilpotent. Moreover, $N$ is a proper ${\cal B}\hbox{-}k(x)$-bimodule. Then
$M=F(N)$ is a proper $\underline{\cal A}\hbox{-}k(x)$-bimodule and, since
$F:{\rm End}_{{\cal B}}(N)^{op}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal A}}(M)^{op}$ is
an isomorphism, we obtain that

${\rm End}_{\underline{\cal A}}(M)^{op}=\overline{Q}_{M}\bigoplus{\rm rad}{\rm
End}_{\underline{\cal A}}(M)^{op},$

where $\overline{Q}_{M}:=F(Q_{N})$ is a subalgebra of ${\rm
End}_{\underline{\cal A}}(M)^{op}$ such that $\overline{Q}_{M}\cong k(x)$ and
the ideal ${\rm rad}{\rm End}_{\underline{\cal A}}(M)^{op}$ is nilpotent. From
this fact follows that $\hbox{\rm endol}\,(M)=\dim_{k(x)}M$. ∎

Proof of (1.9) We know that $\underline{\cal A}$ is tame iff it is not wild,
by [3](11.4). From (2.11), we know that if $\underline{\cal A}$ is
pregenerically tame, then it is not wild. So, it remains to show that if
$\underline{\cal A}$ is not wild, then it is pregenerically tame. So, take
$d\in\mathbb{N}$ and a pregeneric $\underline{\cal A}$-module $G$ with
$\hbox{\rm endol}\,(G)\leq d$. Apply (5.4) to $d$ to obtain a functor $F:{\cal
B}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\underline{\cal A}\hbox{-}{\rm Mod}$, and a
generic ${\cal B}$-bimodule $G^{\prime}$ with $F(G^{\prime})\cong G$. From
[5](31.3), there are only finitely possible pregeneric ${\cal B}$-modules
$G^{\prime}$ up to isomorphism. It follows that $\underline{\cal A}$ is
pregenerically tame. $\hfill\square$

##  6 On the category of modules for a special $\underline{\cal A}$

In this section, we consider first a _special triangular interlaced weak
ditalgebra $\underline{\cal A}=({\cal A},I)$_. That is such that its layer
$(S,W)$ satisfies: $S$ is a finite product of copies of the field $k$, $W$ is
finite-dimensional and so is $\underline{A}=A/I$, where $A=T_{S}(W_{0})$.

It is known that, in this case, the category $\underline{\cal A}\hbox{-}{\rm
Mod}$ admits an exact structure ${\cal E}$ where the conflations are the
composable pairs of morphisms $M\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{f}}E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{g}}N$ such that $gf=0$ and the sequence
$0\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{f^{0}}}E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{g^{0}}}N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0$ is exact in $S\hbox{-}{\rm Mod}$, see
[4]§10.

The _right algebra of $\underline{\cal A}$_ is the finite-dimensionl
$k$-algebra $\Gamma:={\rm End}_{\underline{\cal A}}(\underline{A})^{op}$. The
composition $\underline{A}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{\underline{A}}(\underline{A})^{op}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal
A}}(A)^{op}=\Gamma$, is an embedding of $k$-algebras, so $\Gamma$ is naturally
an $\underline{A}$-algebra. The functor $H:\underline{\cal A}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Gamma\hbox{-}{\rm
Mod}$, given by $H={\rm Hom}_{\underline{\cal A}}(\underline{A},-)$ is an
exact functor, because $\underline{A}$ is ${\cal E}$-projective, see [6] and
[4]§11.

######  Lemma 6.1.

Let $\underline{\cal A}$ be a special interlaced weak ditalgebra, as before.
Then the functor $H={\rm Hom}_{\underline{\cal A}}(\underline{A},-)$ preserves
arbitrary direct sums. More precisely, for any family $\\{M_{i}\\}_{i\in I}$
of $\underline{\cal A}$-modules there is an isomorphism of $\Gamma$-modules

$\Theta:{\rm Hom}_{\underline{\cal A}}(\underline{A},\bigoplus_{i\in
I}M_{i})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\bigoplus_{i\in
I}{\rm Hom}_{\underline{\cal A}}(\underline{A},M_{i})$

such that $f\longmapsto\sum_{i\in I}\overline{\pi}_{i}f$, where
$\overline{\pi}_{i}:\bigoplus_{i\in I}M_{i}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{i}$ denotes the projection on the factor
$M_{i}$ in $\underline{\cal A}\hbox{-}{\rm Mod}$. This is a natural
isomorphism.

###### Proof.

Among other properties, we will verify in a moment that the sum $\sum_{i\in
I}\overline{\pi}_{i}f$ is finite.

First recall that whenever $Q$ is a finitely generated $R$-module over some
algebra $R$, we have the morphism $\theta:{\rm Hom}_{R}(Q,\bigoplus_{i\in
I}M_{i})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\bigoplus_{i\in
I}{\rm Hom}_{R}(Q,M_{i})$, such that $f\longmapsto\sum_{i\in I}\pi_{i}f$,
where $\pi_{i}:\bigoplus_{i\in I}M_{i}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{i}$ denotes the projection on the factor
$M_{i}$. Indeed, since $Q$ is finitely generated, the image of each morphism
$f$ lies in a finite subsum $\bigoplus_{i\in I_{f}}M_{i}$ of $\bigoplus_{i\in
I}M_{i}$, the sum $\sum_{i}\pi_{i}f$ is finite, and we have the commutative
square

$\begin{matrix}{\rm Hom}_{R}(Q,\bigoplus_{i\in I}M_{i})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&\bigoplus_{i\in I}{\rm Hom}_{R}(Q,M_{i})\\\
\hbox to0.0pt{\hss$\vbox{\hbox{
$\scriptstyle$}}$}\Bigg{\uparrow}&&\Bigg{\uparrow}\hbox to0.0pt{$\vbox{\hbox{
$\scriptstyle$}}$\hss}\\\ {\rm Hom}_{R}(Q,\bigoplus_{i\in
I_{f}}M_{i})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\cong}}&\bigoplus_{i\in I_{f}}{\rm
Hom}_{R}(Q,M_{i}),\end{matrix}$

where the horizontal arrows have the same recipe, the right vertical arrow is
an inclusion and the frst vertical arrow is induced by an inclusion. This
implies that the first row is an isomorphism too.

Then, we have isomorphisms

$\begin{matrix}{\rm Hom}_{S}(\underline{A},\bigoplus_{i\in I}M_{i})\times{\rm
Hom}_{\underline{A}\hbox{-}\underline{A}}(\underline{V},{\rm
Hom}_{k}(\underline{A},\bigoplus_{i\in I}M_{i}))\\\ \Bigg{\downarrow}\hbox
to0.0pt{$\vbox{ \hbox{$\scriptstyle\theta_{1}\times\theta_{2*}$}}$\hss}\\\
\bigoplus_{i\in I}{\rm Hom}_{S}(\underline{A},M_{i})\times{\rm
Hom}_{\underline{A}\hbox{-}\underline{A}}(\underline{V},\bigoplus_{i\in I}{\rm
Hom}_{k}(\underline{A},M_{i}))\\\ \Bigg{\downarrow}\hbox to0.0pt{$\vbox{
\hbox{$\scriptstyle id\times\theta_{3}$}}$\hss}\\\ \bigoplus_{i\in I}{\rm
Hom}_{S}(\underline{A},M_{i})\times\bigoplus_{i\in I}{\rm
Hom}_{\underline{A}\hbox{-}\underline{A}}(\underline{V},{\rm
Hom}_{k}(\underline{A},M_{i})).\\\ \end{matrix}$

Let us show that the composition $\Theta$ these isomorphisms restricts to an
isomorphism as in the statement of this lemma. So we take $f\in{\rm
Hom}_{\underline{\cal A}}(\underline{A},\bigoplus_{i\in I}M_{i})$ and let us
show that $\Theta(f)\in\bigoplus_{i\in I}{\rm Hom}_{\underline{\cal
A}}(\underline{A},M_{i})$. We have

$\begin{matrix}\Theta(f)=\Theta(f^{0},f^{1})&=&(\theta_{1}(f^{0}),\theta_{3}(\theta_{2*}(f^{1})))\hfill\\\
&=&(\sum_{i}\pi_{i}f^{0},\sum_{j\in I}p_{j}\theta_{2*}(f^{1}))\hfill\\\
&=&(\sum_{i}\pi_{i}f^{0},\sum_{j}p_{j}\theta_{2}f^{1}),\hfill\\\ \end{matrix}$

where each $p_{j}:\bigoplus_{i\in I}{\rm
Hom}_{k}(\underline{A},M_{i})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{k}(\underline{A},M_{j})$ is the
canonical projection. Thus, for $v\in\underline{V}$, we have
$\sum_{j}\sum_{i}p_{j}\pi_{i}f^{1}(v)=\sum_{i}\pi_{i}f^{1}(v)$. Since $f$ is a
morphism in $\underline{\cal A}\hbox{-}{\rm Mod}$, for $a,m\in\underline{A}$,
we have

$af^{0}(m)=f^{0}(am)+f^{1}(\delta(a))[m].$

Then, we have
$\sum_{i}\pi_{i}af^{0}(m)=\sum_{i}\pi_{i}f^{0}(am)+\sum_{i}\pi_{i}f^{1}(\delta(a))[m]$.
Since each $\pi_{i}$ is a morphism of $\underline{A}$-modules, we get
$\sum_{i}a\pi_{i}f^{0}(m)=\sum_{i}\pi_{i}f^{0}(am)+\sum_{i}\pi_{i}f^{1}(\delta(a))[m]$
or, equivalently,

$a\Theta(f)^{0}[m]=\Theta(f)^{0}(am)+\Theta(f)^{1}(\delta(a))[m].$

So, $\Theta:{\rm Hom}_{\underline{\cal A}}(\underline{A},\bigoplus_{i\in
I}M_{i})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\bigoplus_{i\in
I}{\rm Hom}_{\underline{\cal A}}(\underline{A},M_{i})$ is a natural
isomorphism. Moreover, it is easy to see that indeed we have
$\Theta(f)=\sum_{i\in I}\overline{\pi}_{i}f$. ∎

######  Lemma 6.2.

The functor $H:\underline{\cal A}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\hbox{-}{\rm Mod}$ is exact, full and
faithful.

###### Proof.

We can adapt the argument of the proof of [5](7.12), using that $H$ commutes
with direct sums by (6.1). We use the commutative square of isomorphisms

$\begin{matrix}{\rm Hom}_{\underline{\cal
A}}(\underline{A}^{(I)},\underline{A}^{(J)})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{F}}&{\rm
Hom}_{\Gamma}(F[\underline{A}^{(I)}],F[\underline{A}^{(J)}])\hfill\\\
{\parallel}&\,&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\cong$}}$\hss}\\\ {\rm Hom}_{\underline{\cal
A}}(\underline{A}^{(I)},\underline{A}^{(J)})&&{\rm
Hom}_{\Gamma}(F[\underline{A}]^{(I)},F[\underline{A}]^{(J)})\\\
\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\cong$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\cong$}}$\hss}\\\ \prod_{I}{\rm
Hom}_{\underline{\cal A}}(\underline{A},\underline{A}^{(J)})&&\prod_{I}{\rm
Hom}_{\Gamma}(F[\underline{A}],F[\underline{A}]^{(J)})\\\ \downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\cong$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\cong$}}$\hss}\\\ \prod_{I}\coprod_{J}{\rm
Hom}_{\underline{\cal A}}(\underline{A},\underline{A})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\prod_{I}\coprod_{J}F^{|}}}&\prod_{I}\coprod_{J}{\rm
Hom}_{\Gamma}(F[\underline{A}],F[\underline{A}]).\\\ \end{matrix}$

∎

######  Lemma 6.3.

For each $M\in\underline{A}\hbox{-}{\rm Mod}$, there is a natural isomorphism

$\sigma_{M}:\Gamma\otimes_{\underline{A}}M\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{\underline{\cal
A}}(\underline{A},L_{\underline{\cal A}}(M)).$

So the funtor $H$ restricts to an equivalence of exact categories
$H:\underline{\cal A}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal I}$. where $\widetilde{\cal
I}$ denotes the full subcategory of $\Gamma\hbox{-}{\rm Mod}$ of modules
induced from $\underline{A}\hbox{-}{\rm Mod}$, that is by the class of
$\Gamma$-modules isomorphic to some $\Gamma\otimes_{\underline{A}}N$, for some
$N\in\underline{A}\hbox{-}{\rm Mod}$.

###### Proof.

The proof of [4](11.5) can be adapted to this situation, using again (6.1) and
that $\Gamma\otimes_{\underline{A}}-:\underline{A}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Gamma\hbox{-}{\rm
Mod}$ is exact. See also, [7]§8. ∎

######  Lemma 6.4.

The category $\widetilde{\cal I}$ is closed under extensions and under direct
summands.

###### Proof.

Consider the exact equivalence $H:\underline{\cal A}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal
I}$. In order to show that $\widetilde{\cal I}$ is closed under extensions,
take an exact sequence

$0\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{f}}E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{g}}N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0$

in $\Gamma\hbox{-}{\rm Mod}$ with $M,N\in\widetilde{\cal I}$. Then
$M\cong\Gamma\otimes_{\underline{A}}M^{\prime}$ and
$N\cong\Gamma\otimes_{\underline{A}}N^{\prime}$, and we may assume that these
isomorphisms are equalities. Consider an exact sequence

$0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}K^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{s^{0}}}P^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{t^{0}}}N^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0$

in $\underline{A}\hbox{-}{\rm Mod}$ with $P^{\prime}$ projective. It
determines a conflation $K^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{s}}P^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{t}}N^{\prime}$ in $\underline{\cal
A}\hbox{-}{\rm Mod}$ and, applying $H$, we get an exact sequence

$0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}H(K^{\prime})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{H(s)}}H(P^{\prime})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{H(t)}}H(N^{\prime})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0$

in $\Gamma\hbox{-}{\rm Mod}$ with $H(P^{\prime})$ projective. Then we have a
commutative diagram

$\begin{matrix}0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&H(K^{\prime})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{H(s)}}&H(P^{\prime})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{H(t)}}&H(N^{\prime})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0\\\ &&\Bigg{\downarrow}\hbox
to0.0pt{$\vbox{ \hbox{$\scriptstyle h$}}$\hss}&&\Bigg{\downarrow}\hbox
to0.0pt{$\vbox{ \hbox{$\scriptstyle$}}$\hss}&&{\parallel}\\\
0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{f}}&E&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{g}}&N&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0.\\\ \end{matrix}$

Now, there is some morphism $h^{\prime}:K^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}$ in $\underline{\cal
A}\hbox{-}{\rm Mod}$ with $H(h^{\prime})=h$. Since $\underline{\cal
A}\hbox{-}{\rm Mod}$ is an exact category, we have a pushout diagram in
$\underline{\cal A}\hbox{-}{\rm Mod}$ of the form

$\begin{matrix}K^{\prime}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{s}}&P^{\prime}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{t}}&N^{\prime}\\\ \Bigg{\downarrow}\hbox
to0.0pt{$\vbox{ \hbox{$\scriptstyle
h^{\prime}$}}$\hss}&&\Bigg{\downarrow}\hbox to0.0pt{$\vbox{
\hbox{$\scriptstyle$}}$\hss}&&{\parallel}\\\ M^{\prime}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{f^{\prime}}}&E^{\prime}&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{g^{\prime}}}&N^{\prime}.\\\ \end{matrix}$

Applying the functor $H$ to this diagram, we obtain a pushout diagram in
$\Gamma\hbox{-}{\rm Mod}$

$\begin{matrix}0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&H(K^{\prime})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\ H(s)\
}}&H(P^{\prime})&\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\
H(t)\ }}&H(N^{\prime})&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0\\\ &&\Bigg{\downarrow}\hbox
to0.0pt{$\vbox{ \hbox{$\scriptstyle h$}}$\hss}&&\Bigg{\downarrow}\hbox
to0.0pt{$\vbox{ \hbox{$\scriptstyle$}}$\hss}&&{\parallel}\\\
0&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&M&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{\ H(f^{\prime})\
}}&H(E^{\prime})&\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\
H(g^{\prime})\ }}&N&\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}&0.\\\ \end{matrix}$

By the uniqueness of pushouts, we obtain $E\cong H(E^{\prime})$ and
$E\in\widetilde{\cal I}$.

Now, in order to show that $\widetilde{\cal I}$ is closed under direct
summands, take $M\in\widetilde{\cal I}$. We may assume that $M=H(M^{\prime})$.
Take a direct summand $N$ of $M$, consider the inclusion
$i:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$, the projection
$p:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$, and the
idempotent $e=ip:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$,
which satisfy $pi=id_{N}$. Then, there is an idempotent
$e^{\prime}:M^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}$ in $\underline{\cal
A}\hbox{-}{\rm Mod}$ with $F(e^{\prime})=e$. Since $\underline{\cal A}$ is a
Roiter interlaced weak ditalgebra, idempotents split in $\underline{\cal
A}\hbox{-}{\rm Mod}$, see [3](5.7), so there is an $\underline{\cal A}$-module
$N^{\prime}$ and morphisms $q:M^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}N^{\prime}$ and
$j:N^{\prime}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}$, such that $jq=e^{\prime}$ and
$qj=id_{N^{\prime}}$ in $\underline{\cal A}\hbox{-}{\rm Mod}$. The images
$H(q):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}H(N^{\prime})$
and $H(j):H(N^{\prime})\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M$ satisfy that
$H(j)H(q)=H(e^{\prime})=e=pi$ and $H(q)H(j)=id_{H(N^{\prime})}$. This implies
that $H(N^{\prime})\cong N$, indeed we have the isomorphism
$pH(j):H(N^{\prime})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$
with inverse $H(q)i:N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}H(N^{\prime})$. ∎

######  Theorem 6.5.

If $\underline{\cal A}$ is a special triangular interlaced weak ditalgebra,
then it is a Roiter interlaced weak ditalgebra and the category
$\underline{\cal A}\hbox{-}{\rm mod}$ has a natural exact structure ${\cal
E}_{\underline{\cal A}}$, see [4](10.11). The corresponding exact category
$(\underline{\cal A}\hbox{-}{\rm mod},{\cal E}_{\underline{\cal A}})$ has
almost split conflations.

###### Proof.

The proof of [5](7.18) can be adapted to this situation, using the equivalence
$H:\underline{\cal A}\hbox{-}{\rm mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\cal I}$, where ${\cal I}$ is denotes the
full subcategory of $\Gamma\hbox{-}{\rm mod}$ of modules induced from
$\underline{A}\hbox{-}{\rm mod}$. ∎

######  Lemma 6.6.

Let $\underline{\cal A}$ be a special triangular interlaced weak ditalgebra,
with layer $(R,W)$. For any non ${\cal E}_{\underline{\cal A}}$-projective
indecomposable object $M\in\underline{\cal A}\hbox{-}{\rm mod}$, there is an
almost split conflation $N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M$. Thus, the indecomposable object $N$ is
uniquely determined up to isomorphism. We call $N$ the _translate_ of $M$ and
denote it by $\tau(M)$. Under our assumptions, there is a number
$c_{\underline{\cal A}}$ such that, for any non ${\cal E}_{\underline{\cal
A}}$-projective indecomposable $M\in\underline{\cal A}\hbox{-}{\rm mod}$, the
following inequality holds:

$\dim_{k}\tau(M)\leq c_{\underline{\cal A}}\dim_{k}M.$

###### Proof.

The proof of [5](32.5) can be adapted to this situation. ∎

######  Theorem 6.7.

Let $\underline{\cal A}$ be a non-wild special triangular interlaced weak
ditalgebra. Thus, $(\underline{\cal A}\hbox{-}{\rm mod},{\cal
E}_{\underline{\cal A}})$ has almost split conflations and a translation
$\tau$. Then, for every dimension $d$, almost all $d$-dimensional
indecomposable modules $M\in\underline{\cal A}\hbox{-}{\rm mod}$ satisfy
$\tau(M)\cong M$.

###### Proof.

The proof of [5](32.6) can be adapted to this situation, using (6.6). ∎

##  7 Coverage theorems for $\Delta$-filtered modules

In this section we consider the directed interlaced weak ditalgebra
$\underline{\cal A}=\underline{\cal A}(\Delta)$ associated to the quasi-
hereditary algebra $\Lambda$ and its standard modules
$\Delta_{1},\ldots,\Delta_{n}$, see for instance [4](4.20). We also consider
the right algebra $\Gamma$ of $\underline{\cal A}$.

######  Definition 7.1.

Let $\Gamma$ be the right algebra of $\underline{\cal A}$. Consider the family
of $\Gamma$-modules
$\\{\Delta^{\prime}_{j}=\Gamma\otimes_{\underline{A}}S_{j}\\}_{j\in[1,n]}$,
where $S_{1},\ldots,S_{n}$ is a complete set of representatives of the
isoclasses of the simple $\underline{A}$-modules. Then, we have the full
subcategory $\widetilde{\cal F}(\Delta^{\prime})$ of $\Gamma\hbox{-}{\rm
Mod}$, as in (1.2).

######  Proposition 7.2.

With the preceding notation, we have $\widetilde{\cal I}=\widetilde{\cal
F}(\Delta^{\prime})$. So, we have an exact equivalence $H:\underline{\cal
A}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal I}=\widetilde{\cal
F}(\Delta^{\prime})$, see (6.2) and (6.3).

###### Proof.

We show first that any $M\in\widetilde{\cal I}$ admits a filtration

$0=M_{\ell}\subseteq M_{\ell-1}\subseteq\cdots\subseteq M_{1}\subseteq
M_{0}=M$

such that $M_{i}/M_{i+1}$ is isomorphic to a (possibly infinite) direct sum of
modules in the family
$\\{\Delta^{\prime}_{j}=\Gamma\otimes_{\underline{A}}S_{j}\\}_{j\in[1,n]}$.

For this, we may assume that $M=\Gamma\otimes_{\underline{A}}M^{\prime}$, for
some $\underline{A}$-module $M^{\prime}$. Since $\underline{A}$ is finite-
dimensional, we have a filtration of $M^{\prime}$ of the form

$0=M^{\prime}_{\ell}\subseteq M^{\prime}_{\ell-1}\subseteq\cdots\subseteq
M^{\prime}_{1}\subseteq M^{\prime}_{0}=M^{\prime},$

with $M^{\prime}_{i}=J^{i}M^{\prime}$, where $J$ denotes the Jacobson radical
of the finite-dimensional algebra $\underline{A}$. Since $\underline{A}/J$ is
semisimple, each $\underline{A}/J$-module $L$ is semisimple. So, we have
$L\cong\bigoplus_{i=1}^{n}S_{i}\otimes_{k}V_{i},$ where $V_{1},\ldots,V_{n}$
are vector spaces over the field $k$ (not necessarily finite-dimensional).

The preceding filtration gives rise to exact sequences

$\begin{matrix}0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}_{\ell}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}L^{\prime}_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0\\\ 0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}_{\ell-2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}L^{\prime}_{\ell-2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0\\\ \cdots\\\ 0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M^{\prime}_{0}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}L^{\prime}_{0}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0\\\ \end{matrix}$

with $L^{\prime}_{0},\ldots,L^{\prime}_{\ell-1}$ semisimple. Applying the
exact functor $\Gamma\otimes_{\underline{A}}-$ we obtain exact sequences

$\begin{matrix}0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}M^{\prime}_{\ell}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}M^{\prime}_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}L^{\prime}_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0\\\ 0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}M^{\prime}_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}M^{\prime}_{\ell-2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}L^{\prime}_{\ell-2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0\\\ \cdots\\\ 0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}M^{\prime}_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}M^{\prime}_{0}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}L^{\prime}_{0}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0.\\\ \end{matrix}$

Then, the module $\Gamma\otimes_{\underline{A}}M^{\prime}_{0}=M$ admits a
filtration with factors of the form $\Gamma\otimes_{\underline{A}}L$, where
$L\cong\bigoplus_{i=1}^{n}S_{i}\otimes_{k}V_{i}$ is a semisimple
$\underline{A}$-module. So, these factors have the form

$\Gamma\otimes_{\underline{A}}L\cong\Gamma\otimes_{\underline{A}}\bigoplus_{i=1}^{n}S_{i}\otimes_{k}V_{i}\cong\bigoplus_{i=1}^{n}\Gamma\otimes_{\underline{A}}S_{i}\otimes_{k}V_{i}=\bigoplus_{i=1}^{n}\Delta^{\prime}_{i}\otimes_{k}V_{i}.$

In order to prove the converse, assume that a $\Gamma$-module
$M\in\widetilde{F}(\Delta^{\prime})$ has a filtration as in the begining of
this proof. Since direct sums of $\Gamma$-modules of the family
$\\{\Gamma\otimes_{\underline{A}}S_{i}\\}_{i\in[1,n]}$ have the form
$\Gamma\otimes_{\underline{A}}L^{\prime}$, where $L^{\prime}$ is a semisimple
$\underline{A}$-module, we have exact sequences

$\begin{matrix}0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{\ell}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}L^{\prime}_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0\\\ 0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{\ell-1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{\ell-2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}L^{\prime}_{\ell-2}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0\\\ \cdots\\\ 0\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{1}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}M_{0}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}L^{\prime}_{0}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}0.\\\ \end{matrix}$

From the first sequence, we get that
$M_{\ell-1}\cong\Gamma\otimes_{\underline{A}}L^{\prime}_{\ell-1}$. From the
second one, using that $\widetilde{\cal I}$ is closed under extensions, by
(6.4), we get that $M_{\ell-2}\in\widetilde{\cal I}$. Proceeding inductively,
in a finite number of steps, we get that $M=M_{0}\in\widetilde{\cal I}$. ∎

We know that there is a Morita equivalence $\Omega:\Gamma\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Lambda\hbox{-}{\rm
Mod}$ such that $\Omega(\Delta^{\prime}_{i})\cong\Delta_{i}$, for $i\in[1,n]$,
see [4](12.9). The existence of the interlaced weak ditalgebra
$\underline{\cal A}(\Delta)$ and the equivalence $\Omega$ constructed in [12]
play a crucial role in our arguments.

In [9], the results on generic modules for a finite-dimensional algebra
$\Lambda$ are obtained from the corresponding objects of ${\cal D}\hbox{-}{\rm
Mod}$, where ${\cal D}$ is the Drozd’s ditalgebra of $\Lambda$. Here, the
results on generic modules for ${\cal F}(\Delta)$ are obtained from the
properties of the corresponding objects of $\underline{\cal
A}(\Delta)\hbox{-}{\rm Mod}$. The passage between both categories is done with
the help of the equivalences $H$ and $\Omega$. In the following we study first
their effect on endolength and then transfer the results of (5.4) onto ${\cal
F}(\Delta)$.

######  Lemma 7.3.

Let $\Gamma$ be the right algebra of $\underline{\cal A}$ and consider the
exact equivalence $H:\underline{\cal A}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal F}(\Delta^{\prime})$. Then,
for any $N\in\underline{\cal A}\hbox{-}{\rm Mod}$, we have

$\hbox{\rm endol}\,(H(N))\leq\dim_{k}\Gamma\times\hbox{\rm endol}\,(N).$

For any $k$-algebra $E$, the functor $H$ induces a functor
$H_{E}:\underline{\cal A}\hbox{-}E\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal
F}(\Delta^{\prime})\hbox{-}E$, where $\widetilde{\cal
F}(\Delta^{\prime})\hbox{-}E$ denotes the category of
$\Gamma\hbox{-}E$-bimodules $M$ which lie in $\widetilde{\cal
F}(\Delta^{\prime})$ as left $\Gamma$-modules. For any $N\in\underline{\cal
A}\hbox{-}{\rm Mod}$, if $E:={\rm End}_{\Gamma}(H(N))^{op}$ then $N$ has a
natural structure of $\underline{\cal A}\hbox{-}E$-bimodule and the following
inequality holds:

$\ell_{E}(H(N))\leq\dim_{k}\Gamma\times\ell_{E}(N).$

###### Proof.

If $b=\dim_{k}\Gamma$, there is an epimorphism
$\underline{A}^{b}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma$ of right $\underline{A}$-modules.
Given $N\in\underline{\cal A}\hbox{-}{\rm Mod}$ make $E:={\rm
End}_{\Gamma}(H(N))^{op}$, we have $N\in\underline{\cal
A}\hbox{-}E\hbox{-}{\rm Mod}$ with bimodule structure
$\alpha_{N}:=H^{-1}:E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal A}}(N)^{op}$.
Then, there is an epimorphism
$\underline{A}^{b}\otimes_{\underline{A}}N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\Gamma\otimes_{\underline{A}}N$ of right
$E$-modules. Hence,

$\ell_{E}(H(N))=\ell_{E}(\Gamma\otimes_{\underline{A}}N)\leq
b\times\ell_{E}(N).$

Since $H$ is full and faithful, we have the first equality in

$\hbox{\rm
endol}\,(H(N))=\ell_{E}(H(N))=\ell_{E}(\Gamma\otimes_{\underline{A}}N)\leq
b\times\ell_{E}(N)=b\times\hbox{\rm endol}\,(N).$

∎

######  Lemma 7.4.

Let $\Gamma$ be the right algebra of $\underline{\cal A}$ and consider the
exact equivalence $H:\underline{\cal A}\hbox{-}{\rm Mod}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal F}(\Delta^{\prime})$. If $E$
is any $k$-algebra and $N$ is a proper $\underline{\cal A}\hbox{-}E$-bimodule,
then the map $\sigma_{N}:\Gamma\otimes_{\underline{A}}N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}H(N)$ is an isomorphism of right
$E$-modules, where the compared $E$-module structures are given by
$\Gamma\otimes_{\underline{A}}(N_{E})$ and

$E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\alpha_{N}}}{\rm
End}_{\underline{A}}(N)^{op}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{H}}{\rm End}_{\Gamma}(H(N))^{op}.$

As a consequence, we get $\ell_{E}(N)\leq\ell_{E}(H(N))$.

###### Proof.

Once we know that $H(N)\cong\Gamma\otimes_{\underline{A}}(N_{E})$, we have
$\ell_{E}(N)\leq\ell_{E}(H(N))$. Indeed, we know that $\underline{A}$ is a
direct summand of the right $\underline{A}$-module $\Gamma$, see for instance
the proof of [4](11.10). Hence, we get

$\ell_{E}(N)=\ell_{E}(\underline{A}\otimes_{\underline{A}}N)\leq\ell_{E}(\Gamma\otimes_{\underline{A}}N)=\ell_{E}(H(N)).$

In order to prove the first statement, consider the structure of proper
$\underline{\cal A}\hbox{-}E$-bimodule $\alpha_{N}:E\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}{\rm End}_{\underline{\cal A}}(N)^{op}$ of
$N$, which is such that $\alpha_{N}(e)=(\alpha_{N}(e)^{0},0)$, for all $e\in
E$. Recall that, for any $n\in N$ and $e\in E$, we have
$ne=\alpha_{N}(e)^{0}[n]$. The structure map $\alpha_{H(N)}$ for the
$\Gamma\hbox{-}E$-bimodule $H(N)$ is given by the composition

$E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\alpha_{N}}}{\rm
End}_{\underline{\cal A}}(N)^{op}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits^{H}}{\rm End}_{\Gamma}(H(N))^{op}.$

We can transfer the structure of right $E$-module of $H(N)$ onto
$\Gamma\otimes_{\underline{A}}N$ using the isomorphism
$\sigma_{N}:\Gamma\otimes_{\underline{A}}N\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}H(N)$, by the formula

$(f\otimes n)e:=\sigma_{N}^{-1}(\sigma_{N}(f\otimes n)e),\hbox{ for
}f\in\Gamma,n\in N,e\in E,$

so $\sigma_{N}$ becomes an isomorphism of right $E$-modules. So, it will be
enough to show that $(f\otimes n)e=f\otimes ne$, for all $n\in N$ and $e\in
E$. Equivalently, that

$\sigma_{N}(f\otimes ne)=\sigma_{N}(f\otimes n)e.$

With the notation of [4](11.5), we have $\sigma_{N}(f\otimes ne)=f_{ne}$ and

$\begin{matrix}\sigma_{N}(f\otimes n)e&=&f_{n}e\hfill\\\
&=&\alpha_{H(N)}(e)(f_{n})\hfill\\\ &=&(H\circ\alpha_{N})(e)[f_{n}]\hfill\\\
&=&H(\alpha_{N}(e))[f_{n}]=\alpha_{N}(e)\circ f_{n}.\hfill\\\ \end{matrix}$

So, we need to show that $f_{ne}=\alpha_{N}(e)\circ f_{n}$. We start by
looking at the first components. Since $N$ is a proper $\underline{\cal
A}\hbox{-}E$-bimodule, we have $\,{}_{\underline{A}}N_{E}$ so, for
$a\in\underline{A}$, we get

$\begin{matrix}(\alpha_{N}(e)\circ f_{n})^{0}(a)&=&(\alpha_{N}(e)^{0}\circ
f_{n}^{0})(a)\hfill\\\
&=&\alpha_{N}(e)^{0}(f^{0}_{n}(a))=(f^{0}(a)n)e=f^{0}(a)(ne)=f^{0}_{ne}(a).\hfill\\\
\end{matrix}$

Now, look at the second components and recall that $\alpha_{N}(e)^{1}=0$,
because $N$ is a proper $\underline{\cal A}\hbox{-}E$-bimodule. Then, for
$v\in V$, we have

$\begin{matrix}(\alpha_{N}(e)\circ
f_{n})^{1}(v)[a]&=&\alpha_{N}(e)^{0}f_{n}^{1}(v)[a]\hfill\\\
&=&(f_{n}^{1}(v)[a])e=(f^{1}(v)[a]n)e=(f^{1}(v)[a])(ne)=f^{1}_{ne}(v)[a].\hfill\\\
\end{matrix}$

So, indeed $f_{ne}=\alpha_{N}(e)\circ f_{n}$, as we wanted to show. ∎

######  Lemma 7.5.

Assume that $\underline{\cal A}$ is not wild, let $\Gamma$ be its right
algebra, and consider the exact equivalence $H:\underline{\cal A}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal
F}(\Delta^{\prime})$. If $M$ is an indecomposable in $\underline{\cal
A}\hbox{-}{\rm Mod}$ with finite endolength, then the $\Gamma$-module
$H(M)\cong\Gamma\otimes_{\underline{A}}M$ is an indecomposable $\Gamma$-module
with finite endolength. Moreover, we have

$\hbox{\rm endol}\,(M)\leq\hbox{\rm endol}\,(H(M)).$

###### Proof.

If $M$ is finite-dimensional, we get $\hbox{\rm
endol}\,(M)=\dim_{k}M=\dim_{k}\underline{A}\otimes_{\underline{A}}M\leq\dim_{k}\Gamma\otimes_{\underline{A}}M=\dim_{k}H(M)=\hbox{\rm
endol}\,(H(M))$.

If $M$ is infinite-dimensional, then $M$ is a pregeneric $\underline{\cal
A}$-module. From (5.4), it is a proper $\underline{\cal
A}\hbox{-}k(x)$-bimodule. Then, $M$ is an
$\underline{A}\hbox{-}k(x)$-bimodule, that is an
$\underline{A}\otimes_{k}k(x)$-module. Then, $\Gamma\otimes_{\underline{A}}M$
has a structure of a $\Gamma\hbox{-}k(x)$-bimodule, that is of a left
$\Gamma\otimes_{k}k(x)$-module.

Make $\underline{A}^{k(x)}:=\underline{A}\otimes_{k}k(x)$ and
$\Gamma^{k(x)}:=\Gamma\otimes_{k}k(x)$. Notice that there is an isomorphism of
$\Gamma\hbox{-}k(x)$-bimodules:
$\Gamma\otimes_{\underline{A}}M\cong\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M$,
through this isomorphism we identify both bimodules.

From (5.4), we know that the $\underline{A}^{k(x)}$-module $M$ is finite-
dimensional over $k(x)$, so it has finite length and so, it admits a
filtration

$0=M_{l}\subseteq M_{l-1}\subseteq\cdots\subseteq M_{1}\subseteq M_{0}=M,$

where each factor $M_{i}/M_{i+1}$ is isomorphic to some simple
$\underline{A}^{k(x)}$-module, so to some $S_{j}\otimes_{k}k(x)$. Then, we
have a filtration of the $\Gamma^{k(x)}$-module
$\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M$ of the form

$0=\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M_{l}\subseteq\cdots\subseteq\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M_{1}\subseteq\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M,$

where the factors have the form

$\Delta^{\prime}_{j}\otimes_{k}k(x)\cong\Gamma\otimes_{\underline{A}}S_{j}\otimes_{k}k(x)\cong\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}(S_{j}\otimes_{k}k(x)).$

The multiplicity of the simple $\underline{A}^{k(x)}$-module
$S_{j}\otimes_{k}k(x)$ in the given filtration for $M$ is $\dim_{k(x)}e_{j}M$,
hence the multiplicity of the $\Gamma^{k(x)}$-module
$\Delta^{\prime}_{j}\otimes_{k}k(x)$ in the considered tensored filtration for
$\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M$ is the same number. Hence,

$\dim_{k(x)}\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M=\sum_{i}\dim_{k(x)}e_{i}M\times\dim_{k(x)}(\Delta^{\prime}_{i}\otimes_{k}k(x)).$

Therefore,

$\hbox{\rm
endol}\,(\Gamma\otimes_{\underline{A}}M)=\dim_{k(x)}\Gamma^{k(x)}\otimes_{\underline{A}^{k(x)}}M=\sum_{i}\dim_{k(x)}e_{i}M\times\dim_{k}(\Delta^{\prime}_{i}).$

Then, we get

$\begin{matrix}\hbox{\rm
endol}\,(M)&=&\dim_{k(x)}M=\sum_{i}\dim_{k(x)}e_{i}M\hfill\\\
&\leq&\sum_{i}\dim_{k(x)}e_{i}M\times\dim_{k}(\Delta^{\prime}_{i})=\hbox{\rm
endol}\,(\Gamma\otimes_{\underline{A}}M),\hfill\\\ \end{matrix}$

as claimed. ∎

######  Proposition 7.6.

Consider the exact equivalence $H:\underline{\cal A}\hbox{-}{\rm
Mod}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\widetilde{\cal
F}(\Delta^{\prime})$. Take $N\in\underline{\cal A}\hbox{-}{\rm Mod}$ and
consider $M:=H(N)\in\widetilde{\cal F}(\Delta^{\prime})$. Then, if $M$ is a
generic $\Gamma$-module, we have that $N$ is a pregeneric $\underline{\cal
A}$-module.

###### Proof.

Make $E:={\rm End}_{\underline{\cal A}}(N)^{op}$, then $N$ is an
$\underline{\cal A}\hbox{-}E$-bimodule and $M=H(N)$ is a
$\Gamma\hbox{-}E$-bimodule. The action of $E$ by the right on $N$ is given by
$n\phi=\phi^{0}(n)$, for $n\in N$ and $\phi\in E$, and the action of $E$ by
the right on $M$ is given by $h\phi=H(\phi)[h]=\phi\circ h$, the composition
in $\underline{\cal A}\hbox{-}{\rm Mod}$, for $h\in{\rm Hom}_{\underline{\cal
A}}(\underline{A},N)=H(N)=M$ and $\phi\in E$.

Notice that the right ${\rm End}_{\Gamma}(M)^{op}$-module $M$ is a right
$E$-module by restriction using the isomorphism of algebras
$H:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm
End}_{\Gamma}(M)^{op}$. This action coincides with the preceding one. We have
that $\ell_{E}(M)=\hbox{\rm endol}\,(M)$, which is finite by assumption. We
will show that $\ell_{E}(N)$ is finite too. Since $\hbox{\rm
endol}\,(N)=\ell_{E}(N)$, we will get from this that $N$ is a pregeneric
$\underline{\cal A}$-module, as claimed.

Make $\mathfrak{r}:={\rm rad}E$. From [10](4.2;4.4), we know that $E\cong{\rm
End}_{\Gamma}(M)^{op}$ is a local ring and that $\mathfrak{r}$ is nilpotent.
Thus, for some natural number $l$, we have $\mathfrak{r}^{l}=0$. So, we have a
filtration of right $E$-modules

$0=M_{l}=M\mathfrak{r}^{l}\subseteq
M_{l-1}=M\mathfrak{r}^{l-1}\subseteq\cdots\subseteq
M_{1}=M\mathfrak{r}^{1}\subseteq M_{0}=M,$

and we can consider the division $k$-algebra $D:=E/\mathfrak{r}$. Each
$M_{i}/M_{i+1}$ is a $D$-vector space. Since $\ell_{E}(M)$ is finite, we know
that each $M_{i}/M_{i+1}$ is a $D$-vector space with finite dimension, say
$d_{i+1}$. Consider $d:=\max\\{d_{1},\ldots,d_{l}\\}$.

We also have a filtration of right $E$-modules

$0=N_{l}=N\mathfrak{r}^{l}\subseteq
N_{l-1}=N\mathfrak{r}^{l-1}\subseteq\cdots\subseteq
N_{1}=N\mathfrak{r}^{1}\subseteq N_{0}=N,$

where each quotient $N_{i}/N_{i+1}$ is a $D$-vector space. We will show that

$\dim_{D}(N_{i}/N_{i+1})\leq d,\hbox{ for each }i.$

So, we fix $i\in[1,l]$. Then, $N_{i}/N_{i+1}$ is generated by classes modulo
$N_{i+1}$ of elements of the form $n\rho$, with $n\in N$ and
$\rho\in\mathfrak{r}^{i}$. In order to prove our claim, it is enough to show
that the classes of any set with $d+1$ such elements
$n_{1}\rho_{1},\ldots,n_{d+1}\rho_{d+1}$ are $D$-linearly dependent in
$N_{i}/N_{i+1}$.

For each $j\in[1,d+1]$ and $n_{j}$ as before, we have the morphism of
$\underline{A}$-modules $h_{j}^{0}:\underline{A}\smash{\mathop{\hbox
to20.0pt{\rightarrowfill}}\limits}N$ with $h_{j}^{0}(1)=n_{j}$, which
determines the morphism $h_{j}=(h_{j}^{0},0)\in{\rm Hom}_{\underline{\cal
A}}(\underline{A},N)=H(N)=M$.

Since, $h_{1}\rho_{1},\ldots,h_{d+1}\rho_{d+1}\in M\mathfrak{r}^{i}=M_{i}$,
their classes modulo $M_{i+1}$ are $D$-linearly dependent. So, there are
elements $\phi_{1},\ldots,\phi_{d+1}\in E$ such that their classes modulo
$\mathfrak{r}$ are not all zero, and
$g:=h_{1}\rho_{1}\phi_{1}+\cdots+h_{d+1}\rho_{d+1}\phi_{d+1}\in
M_{i+1}=M\mathfrak{r}^{i+1}$.

The element $g\in M\mathfrak{r}^{i+1}$ is a finite sum of elements of the form
$h\zeta$, where $h\in M=H(N)$ and $\zeta=\zeta_{1}\zeta_{2}\cdots\zeta_{i+1}$,
where $\zeta_{1},\ldots,\zeta_{i+1}\in\mathfrak{r}$.

Then, $g^{0}(1)$ is a finite sum of elements of the form
$\zeta^{0}_{i+1}\cdots\zeta^{0}_{2}\zeta^{0}_{1}h^{0}(1)$, which implies that
$g^{0}(1)\in N\mathfrak{r}^{i+1}=N_{i+1}$.

But, we also have

$\begin{matrix}(h_{1}\rho_{1}\phi_{1}+\cdots+h_{d+1}\rho_{d+1}\phi_{d+1})^{0}(1)&=&\phi^{0}_{1}\rho^{0}_{1}h^{0}_{1}(1)+\cdots+\phi_{d+1}^{0}\rho_{d+1}^{0}h_{d+1}^{0}(1)\hfill\\\
&=&(n_{1}\rho_{1})\phi_{1}+\cdots+(n_{d+1}\rho_{d+1})\phi_{d+1}.\hfill\\\
\end{matrix}$

Therefore, we get
$(n_{1}\rho_{1})\phi_{1}+\cdots+(n_{d+1}\rho_{d+1})\phi_{d+1}=g^{0}(1)\in