A second note on homological systems

aJesús Efrén Pérez Terrazas, bLuis Fernando Tzec Poot

a,b Facultad de Matemáticas, Universidad Autónoma de Yucatán

<EMAIL_ADDRESS>
###### Abstract

Let $\left(\Delta;\Omega,\leq\right)$ be a b-homological system and
$\widetilde{\cal F}\left(\Delta\right)$ the category of the extended
$\Delta-$filtered modules. Here there is a proof that $\widetilde{\cal
F}\left(\Delta\right)$ is closed under direct summands.

$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$

$Keywords$ $and$ $phrases:$ homological system, preorder, extended filtered
modules.

2020 $Mathematics$ $Subject$ $Classification:$ $16G99,$ $16E99$

$\overline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$

##  1 Introduction

In the study of the representation type of the category of $\Delta-$filtered
modules of an homological system, we can use the notion of generic module (see
[2]), and then it becomes natural to introduce the notion of the category of
extended $\Delta-$filtered modules, and to understand some of its properties.

Here we give a proof of the closure under direct summands of the second
category, adapting the arguments of [5] to prove the closure under direct
summands of the first.

##  2 Pre-orders and filtrations

Let $C$ be a not empty set. The relation $\leq$ on $C$ is a _pre-order_ if it
is reflexive and transitive. In this case $\left(C,\leq\right)$ is a _pre-
ordered set._

The following claims are easy to prove and also are not hard to find in
internet:

######  Lemma 2.1.

  1. 1.

There is an equivalence relation on $C$ given by $c_{1}\sim c_{2}$ iff
$c_{1}\leq c_{2}$ and $c_{2}\leq c_{1}.$ Let $\pi:C\longrightarrow C/\sim$ be
the canonical induced suprajective function, and let us denote
$\pi(c)=\overline{c}.$ Then $C/\sim$ has a canonical partial order given by
$\overline{c_{1}}\leq\overline{c_{2}}$ iff $c_{1}\leq c_{2}.$ (See section 2.1
of [4])

  2. 2.

By the order-extension principle, [7], there is a total order (a
linearization) $\preceq$ on $C/\sim$ that extends $\leq.$

With respect to Lemma 2.1 let $\left(C,\leq\right)$ denote the pre-ordered
set, $\left(C/\sim,\leq\right)$ the partially ordered set and
$\left(C/\sim,\preceq\right)$ the totally ordered set.

######  Lemma 2.2.

Consider the setting of Lemma 2.1 and previous notation.

  1. 1.

If $c_{1}\leq c_{2}$ then $\overline{c_{1}}\leq\overline{c_{2}}$ and
$\overline{c_{1}}\preceq\overline{c_{2}}.$

  2. 2.

By the proof ot the order-extension principle, if $\overline{c_{1}}$ is not
related to $\overline{c_{2}}$ in the partially ordered set, then there is at
least one total order that extends $\leq$ such that
$\overline{c_{1}}\preceq\overline{c_{2}},$ and at least one total order that
extends $\leq$ such that $\overline{c_{2}}\preceq\overline{c_{1}}.$

######  Example 2.3.

Consider the set of natural numbers $\mathbb{N}$ and the partial order given
by $n\leq m$ iff “$n$ divides $m$”.

In this partial order 1 is the minimal element.

There are infinite ways to extend the previous partial order to a total order,
and here we want to consider some lexicographic orders:

1.- Let $q(n)$ be the number of prime factors, including repetitions, in a
factorization in primes of $n.$

2.- Choose an arbitrary total order $\preceq_{1}$ for those naturals with $q$
equal to one, some arbitrary total order $\preceq_{2}$ for those naturals with
$q$ equal to 2, etc.

3.- Given $n,m\in\mathbb{N}$ if $q(n)<q(m)$ then $n\preceq m,$ and if
$q(n)=q(m)=q$ then $n\preceq m$ iff $n\preceq_{q}m.$

######  Example 2.4.

Consider $n,m\in\mathbb{N}$ such that $n$ does not divide $m$ and $m$ does not
divide $n.$ If $m$ has more prime factors that $n$ then in any lexicographic
order of Example 2.3 we get $n\prec m.$ If $q(m)=q(n)=a$ then we can choose
$\preceq_{a}$ such that $m\prec n$ or $n\prec m.$

Now an idea to have $q(m)<q(n)$ and get $n\prec m.$

First observe that given an injective function
$\iota:A\longrightarrow\mathbb{N}$ the total order $\preceq$ induces a total
order on $A.$

Let $d=\left(n,m\right)$ be the maximal common divisor of $n$ and $m,$ and
$n=n^{\prime}d$ and $m=m^{\prime}d,$ where
$\left(n^{\prime},m^{\prime}\right)=1.$

Let $i:\mathbb{N}\longrightarrow\mathbb{N}$ be the function given by $i(a)=a$
if $m^{\prime}$ does not divide $a$ and $i(a)=n^{\prime}a$ if $m^{\prime}|a,$
and choose for the codomain a total order as in Example 2.3. It is easy to
verify that $i$ is injective and also that $i$ induces a total order $\preceq$
on $\mathbb{N}$ such that $n^{\prime}\prec m^{\prime}$ and $n\prec m,$ and
such that it also extends the partial order “divides to”.

The next definition is almost the same given in [4].

$R$ will denote a ring.

######  Definition 2.5.

A b-homological system $\left(\Delta;\Omega,\leq\right),$ consists of the
following:

HS1: A pre-ordered set $\left(\Omega,\leq\right).$

HS2: The set $\Delta=\left\\{\Delta_{\omega}\right\\}_{\omega\in\Omega},$
where $\Delta_{\omega}$ is a finitely generated $R-$module with local ring of
endomorphisms for any $\omega\in\Omega.$ Also $\omega\neq\omega^{\prime}$
implies $\Delta_{\omega}\ncong\Delta_{\omega^{\prime}}.$

HS3: If ${\rm
Hom}_{\Lambda}\left(\Delta_{\omega},\Delta_{\omega^{\prime}}\right)\neq 0$
then $\omega\leq\omega^{\prime}.$

HS4: If ${\rm
Ext}^{1}_{\Lambda}\left(\Delta_{\omega},\Delta_{\omega^{\prime}}\right)\neq 0$
then $\omega\leq\omega^{\prime}$ and it is not true that
$\omega^{\prime}\leq\omega.$

######  Remark 2.6.

Let $\pi:\Omega\longrightarrow\Omega/\sim$ be the canonical suprajective
function, $\left(\Omega/\sim,\leq\right)$ the induced partially ordered set
and $\left(\Omega/\sim,\preceq\right)$ a totally ordered set where $\preceq$
extends $\leq,$ as in Lemma 2.1.

By Lemma 2.2 and axiom HS4 if
$\overline{\omega^{\prime}}\preceq\overline{\omega}$ then ${\rm
Ext}^{1}_{\Lambda}\left(\Delta_{\omega},\Delta_{\omega^{\prime}}\right)=0,$
and by same lemma and axiom HS3 if
$\overline{\omega^{\prime}}\prec\overline{\omega}$ (i.e.
$\overline{\omega^{\prime}}\preceq\overline{\omega}$ and
$\overline{\omega^{\prime}}\neq\overline{\omega}$) then ${\rm
Hom}_{\Lambda}\left(\Delta_{\omega},\Delta_{\omega^{\prime}}\right)=0.$

Moreover, by Lemma 2.2 in the definition of b-homological system we can
substitute HS3 and HS4 for the following axioms:

HS3’: For any total order $\preceq$ that extends $\leq,$ if ${\rm
Hom}_{\Lambda}\left(\Delta_{\omega},\Delta_{\omega^{\prime}}\right)\neq 0$
then $\overline{\omega}\preceq\overline{\omega^{\prime}}.$

HS4’: For any total order $\preceq$ that extends $\leq,$ if ${\rm
Ext}^{1}_{\Lambda}\left(\Delta_{\omega},\Delta_{\omega^{\prime}}\right)\neq 0$
then $\overline{\omega}\prec\overline{\omega^{\prime}}.$

######  Definition 2.7.

Given the set $\Delta=\left\\{\Delta_{\omega}\right\\}_{\omega\in\Omega},$ as
in HS2, we denote by $\widetilde{\cal{F}}\left(\Delta\right)$ the full
subcategory of $\Lambda-$Mod of those $M$ having a
$\widetilde{\Delta}-$filtration; we mean for this a finite sequence of
submodules

$\\{0\\}=M_{0}\subsetneq M_{1}\subsetneq\cdots\subsetneq M_{s}=M$

such that
$M_{i}/M_{i-1}\cong\bigoplus_{\omega\in\Omega_{i}}\Delta_{\omega}^{\left(I_{\omega}\right)},$
where $\Omega_{i}$ is a finite subset of $\Omega$ and $I_{\omega}$ is an
arbitray set of indexes.

The modules in $\widetilde{\cal{F}}\left(\Delta\right)$ will be called
$\Delta-$filtered extended modules.

A _slim_ $\widetilde{\Delta}-$filtration of $M$ is a finite sequence of
submodules

$\\{0\\}=H_{0}\subsetneq H_{1}\subsetneq\cdots\subsetneq H_{t}=M$

such that
$H_{j}/H_{j-1}\cong\Delta_{\omega_{j}}^{\left(I_{\omega_{j}}\right)},$ where
$\omega_{j}\in\Omega$ and $I_{\omega_{j}}$ is an arbitrary set of indexes.

For a given $\widetilde{\Delta}-$filtration $F$ of $M$ we denote by
$\ell_{\omega}\left(F\right)=\left|I_{\omega_{j_{1}}}\right|+\cdots+\left|I_{\omega_{j_{a}}}\right|,$
where $\left\\{\omega_{j_{1}},\ldots,\omega_{j_{a}}\right\\}$ are all the
$\omega_{j}$ equal to $\omega.$

######  Remark 2.8.

It is immediate that $\widetilde{\cal{F}}\left(\Delta\right)$ is closed under
isomorphisms, extensions, and contains any zero object.

Applying the Correspondence Theorem for modules it is easy to verify that for
a $\widetilde{\Delta}-$filtration $F$ of
$M\in\widetilde{\cal{F}}\left(\Delta\right)$ it is possible to obtain through
refinement a slim $\widetilde{\Delta}-$filtration $\underline{F},$ and clearly
$\ell_{\omega}\left(F\right)=\ell_{\omega}\left(\underline{F}\right)$ for all
$\omega\in\Omega.$

######  Example 2.9.

Let $\Lambda$ be an Artin $C-$algebra and
$\Delta=\left\\{P_{1},P_{2},\ldots,P_{m}\right\\}$ a complete list, without
repetitions, of representants of isomorphism classes of indecomposable
projective $\Lambda-$modules. Let $\Omega=\left\\{1,2,\ldots,m\right\\}$ and
let $i$ be related to $j$ if there is a non-zero morphism
$f:P_{i}\longrightarrow P_{j};$ the transitive closure of this relation is a
pre-order $\leq$ on $\Omega.$ It is easy to verify that
$\left(\Delta;\Omega,\leq\right)$ is a b-homological system, and of course
$\widetilde{\cal{F}}\left(\Delta\right)$ is the subcategory of $\Lambda-$Mod
of the projectives modules.

######  Example 2.10.

Let $\Lambda$ be a quasi-hereditary algebra and let
$\Delta=\left\\{S_{1},S_{2},\ldots,S_{m}\right\\}$ be a complete list, without
repetitions, of representants of isomorphism classes of simple
$\Lambda-$modules. It is known that there is a total order $\leq$ on
$\Omega=\left\\{1,2,\ldots,m\right\\}$ such that
$\left(\Delta;\Omega,\leq\right)$ is a b-homological system.

######  Example 2.11.

In [8] there are examples of b-homological systems with $\Delta$ infinite: see
the definition of page 16 and Theorem 1 of page 17.

In the rest of the section we assume a given $\left(\Delta;\Omega,\leq\right)$
b-homological system and we will use the notation of Remark 2.6.

######  Definition 2.12.

Let $\underline{F}=\left\\{H_{0}\subsetneq H_{1}\subsetneq\cdots\subsetneq
H_{t}\right\\}$ be a slim $\widetilde{\Delta}-$filtration of
$M\in\widetilde{\cal{F}}\left(\Delta\right).$ The _order vector of
$\underline{F}$_ is defined as
$h\left(\underline{F}\right)=\left(\overline{\omega}_{1},\overline{\omega}_{2},\ldots,\overline{\omega}_{t}\right),$
where $H_{j}/H_{j-1}\cong\Delta_{\omega_{j}}^{\left(I_{\omega_{j}}\right)}.$

######  Lemma 2.13.

Let $A_{i},$ with $i\in I,$ and $B_{j},$ with $j\in J,$ be $R-$modules such
that $A_{i}$ is f.g. for any $i.$ Then for arbitrary sets of indexes $S_{i}$
and $T_{j}$ and $n\in\mathbb{N}\cup\\{0\\}$ there is a canonical isomorphism:

${\rm Ext}^{n}_{\Lambda}\left(\bigoplus_{i\in
I}A_{i}^{\left(S_{i}\right)},\bigoplus_{j\in
J}B_{j}^{\left(T_{j}\right)}\right)\cong\prod_{i\in I}\left[\bigoplus_{j\in
J}\left({\rm
Ext}^{n}_{\Lambda}\left(A_{i},B_{j}\right)\right)^{\left(T_{j}\right)}\right]^{S_{i}}$

In particular, if ${\rm Ext}^{n}_{\Lambda}\left(A_{i},B_{j}\right)=0$ for each
$i$ and each $j$ then

${\rm Ext}^{n}_{\Lambda}\left(\bigoplus_{i\in
I}A_{i}^{\left(S_{i}\right)},\bigoplus_{j\in
J}B_{j}^{\left(T_{j}\right)}\right)=0$

Proof: By Theorem 7.13 of [6] there are isomorphisms

${\rm Ext}^{n}_{\Lambda}\left(\bigoplus_{i\in
I}A_{i}^{\left(S_{i}\right)},\bigoplus_{j\in
J}B_{j}^{\left(T_{j}\right)}\right)\cong\prod_{i\in I}{\rm
Ext}^{n}_{\Lambda}\left(A_{i}^{\left(S_{i}\right)},\bigoplus_{j\in
J}B_{j}^{\left(T_{j}\right)}\right)\cong\prod_{i\in I}\left[\left({\rm
Ext}^{n}_{\Lambda}\left(A_{i},\bigoplus_{j\in
J}B_{j}^{\left(T_{j}\right)}\right)\right)\right]^{S_{i}}$

For $i\in I$ it is assumed that $A_{i}$ is f.g., an so it is known the
existence of the following isomorphisms

${\rm Ext}^{n}_{\Lambda}\left(A_{i},\bigoplus_{j\in
J}B_{j}^{\left(T_{j}\right)}\right)\cong\bigoplus_{j\in J}{\rm
Ext}^{n}_{\Lambda}\left(A_{i},B_{j}^{\left(T_{j}\right)}\right)\cong\bigoplus_{j\in
J}{\rm Ext}^{n}_{\Lambda}\left(A_{i},B_{j}\right)^{\left(T_{j}\right)}$

$\hfill\square$

######  Proposition 2.14.

Let $M\in\widetilde{\cal{F}}\left(\Delta\right)\setminus\\{0\\}$ and
$\underline{F}=\left\\{H_{0}\subsetneq H_{1}\subsetneq\cdots\subsetneq
H_{t}\right\\}$ be a slim $\widetilde{\Delta}-$filtration of $M.$

Then there exists a slim $\widetilde{\Delta}-$filtration
$\underline{F}^{\prime}=\left\\{H^{\prime}_{0}\subsetneq
H^{\prime}_{1}\subsetneq\cdots\subsetneq H^{\prime}_{t}\right\\}$ of $M$ with
$\ell_{\omega}\left(\underline{F}\right)=\ell_{\omega}\left(\underline{F}^{\prime}\right),$
for each $\omega\in\Omega,$ and such that
$h\left(\underline{F}^{\prime}\right)=\left(\overline{\omega^{\prime}_{1}},\overline{\omega^{\prime}_{2}},\ldots,\overline{\omega^{\prime}_{t}}\right)$
satisfies
$\overline{\omega^{\prime}_{t}}\preceq\overline{\omega^{\prime}_{t-1}}\preceq\cdots\preceq\overline{\omega^{\prime}_{1}}.$

Proof: Let us assume for some $j_{0}\in\left\\{1,2,\ldots,t-1\right\\}$ that
$\overline{\omega}_{j_{0}}\prec\overline{\omega}_{j_{0}+1}.$

Then, from HS4, Remark 2.6 and Lemma 2.13 the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{j_{0}}/H_{j_{0}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{j_{0}+1}/H_{j_{0}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{j_{0}+1}/H_{j_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

splits, so there exists a submodule $\overline{N}$ of
$H_{j_{0}+1}/H_{j_{0}-1}$ such that $\overline{N}\cong H_{j_{0}+1}/H_{j_{0}}$
and $\left(H_{j_{0}+1}/H_{j_{0}-1}\right)/\overline{N}\cong
H_{j_{0}}/H_{j_{0}-1}.$

Let $p:H_{j_{0}+1}\longrightarrow H_{j_{0}+1}/H_{j_{0}-1}$ be the canonical
epimorphism and $N$ the pre-image of $\overline{N}$ under $p.$

The following is a slim $\widetilde{\Delta}-$filtration of $M:$

$\underline{F}_{1}=\left\\{H_{0}\subsetneq H_{1}\subsetneq\cdots\subsetneq
H_{j_{0}-1}\subsetneq N\subsetneq H_{j_{0}+1}\subsetneq\cdots\subsetneq
H_{t}\right\\}$

It is easy to verify that
$\ell_{\omega}\left(\underline{F}\right)=\ell_{\omega}\left(\underline{F}_{1}\right)$
for each $\omega,$ and its order vector is equal to

$\left(\overline{\omega}_{1},\ldots,\overline{\omega}_{j_{0}-1},\overline{\omega}_{j_{0}+1},\overline{\omega}_{j_{0}},\overline{\omega}_{j_{0}+2},\ldots,\overline{\omega}_{t}\right)$

We can repeat this process a finite number of steps in order to reach a slim
$\widetilde{\Delta}-$filtration with a descending order vector.
$\hfill\square$

The Proposition 2.14 provides another $\widetilde{\Delta}-$filtration; let
$M\in\widetilde{\cal{F}}\left(\Delta\right)\setminus\\{0\\}$ and
$\underline{F}=\left\\{H_{0}\subsetneq H_{1}\subsetneq\cdots\subsetneq
H_{t}\right\\}$ be a slim $\widetilde{\Delta}-$filtration of $M$ such that
$h\left(\underline{F}\right)=\left(\overline{\omega_{1}},\overline{\omega_{2}},\ldots,\overline{\omega_{t}}\right)$
satisfies
$\overline{\omega_{t}}\preceq\overline{\omega_{t-1}}\preceq\cdots\preceq\overline{\omega_{1}}.$

Assume $\overline{\omega_{j_{0}}}=\overline{\omega_{j_{0}+1}}$ then we have an
exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{j_{0}}/H_{j_{0}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{j_{0}+1}/H_{j_{0}-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{j_{0}+1}/H_{j_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and as before, by HS4, Remark 2.6 and Lemma 2.13 this exact sequence splits,
so we get
$H_{j_{0}+1}/H_{j_{0}-1}\cong\Delta_{\omega_{j_{0}}}^{\left(I_{\omega_{j_{0}}}\right)}\oplus\Delta_{\omega_{j_{0}+1}}^{\left(I_{\omega_{j_{0}+1}}\right)}.$

We can apply this idea to obtain a $\widetilde{\Delta}-$filtration

$\\{0\\}=W_{u_{a+1}}\subsetneq W_{u_{a}}\subsetneq\cdots\subsetneq
W_{u_{1}}=M$

where $u_{s}\in\Omega/\sim$ for $s\in\left\\{a,a-1,\ldots,2,1\right\\},$
$u_{1}\prec u_{2}\prec\cdots\prec u_{a-1}\prec u_{a},$ and (notice $u_{a+1}$
is just a tag)
$W_{u_{s}}/W_{u_{s+1}}\cong\displaystyle\bigoplus_{\begin{subarray}{c}\omega\in\Omega\\\
\overline{\omega}=u_{s}\end{subarray}}\Delta_{\omega}^{\left(I_{\omega}\right)},$
and in this sum for almost all index set is true $I_{\omega}=\emptyset,$ but
at least one is not empty.

######  Definition 2.15.

A $\widetilde{\Delta}-$filtration as above will be called an _ordered_
$\widetilde{\Delta}-$filtration.

Let $W$ be the ordered $\widetilde{\Delta}-$filtration of above; it is clear
that $\ell_{\omega}\left(W\right)=\ell_{\omega}\left(\underline{F}\right).$

######  Lemma 2.16.

Let $\\{0\\}=W_{u_{a+1}}\subsetneq W_{u_{a}}\subsetneq\cdots\subsetneq
W_{u_{1}}=M$ and $\\{0\\}=W_{u^{\prime}_{b+1}}\subsetneq
W^{\prime}_{u^{\prime}_{b}}\subsetneq\cdots\subsetneq
W^{\prime}_{u^{\prime}_{1}}=N$ be ordered $\widetilde{\Delta}-$filtrations.
Let $f:W_{u_{a}}\longrightarrow N$ be a morphism. If $u^{\prime}_{b}\prec
u_{a}$ then ${\rm Im}f=\\{0\\}.$ If ${\rm Im}f\neq\\{0\\}$ then ${\rm
Im}f\subseteq W^{\prime}_{u^{\prime}_{c}},$ where $u^{\prime}_{c}$ is the
first element of
$\left\\{u^{\prime}_{1},u^{\prime}_{2},\ldots,u^{\prime}_{b-1},u^{\prime}_{b}\right\\}$
such that $u_{a}\preceq u^{\prime}_{c}.$

Proof: Let $\pi_{1}:W^{\prime}_{u^{\prime}_{1}}\longrightarrow
N/W^{\prime}_{u^{\prime}_{2}}$ be the canonical epimorphism. By HS3, Remark
2.6 and Lemma 2.13, if $u^{\prime}_{1}\prec u_{a}$ then $\pi_{1}f=0,$ and so
${\rm Im}f\subseteq W^{\prime}_{u^{\prime}_{2}}.$ The statement can be proved
repeating the argument. $\hfill\square$

######  Proposition 2.17.

Let $M\in\widetilde{\cal{F}}\left(\Delta\right)\setminus\\{0\\}.$ There exists
an unique ordered $\widetilde{\Delta}-$filtration of $M.$

Proof: The existence of at least one ordered $\widetilde{\Delta}-$filtration
was established in the lines before Definition 2.15.

Let $\\{0\\}=W_{u_{a+1}}\subsetneq W_{u_{a}}\subsetneq\cdots\subsetneq
W_{u_{1}}=M$ and $\\{0\\}=W_{u_{a+1}}\subsetneq
W^{\prime}_{u^{\prime}_{b}}\subsetneq\cdots\subsetneq
W^{\prime}_{u^{\prime}_{1}}=M$ be ordered $\widetilde{\Delta}-$filtrations.

Consider the canonical inclusion $\iota:W_{u_{a}}\longrightarrow M.$ By Lemma
2.16 we get $u_{a}\preceq u^{\prime}_{b}.$ In a similar way
$u^{\prime}_{b}\preceq u_{a}$ and so $u_{a}=u^{\prime}_{b}.$

Then, applying again Lemma 2.16 to both canonical inclusions, we get
$W_{u_{a}}=W^{\prime}_{u^{\prime}_{b}}.$

We can repeat previous arguments in $M/W_{u_{a}}$ in order to show
$u_{a-1}=u^{\prime}_{b-1}$ and
$W_{u_{a-1}}/W_{u_{a}}=W^{\prime}_{u^{\prime}_{b-1}}/W^{\prime}_{u^{\prime}_{b}},$
so $W_{u_{a-1}}=W^{\prime}_{u^{\prime}_{b-1}}.$

Inductively it follows $a=b,$ $u_{j}=u^{\prime}_{j}$ and
$W_{u_{j}}=W^{\prime}_{u_{j}},$ for $j\in\left\\{a,a-1,\ldots,2,1\right\\}.$
$\hfill\square$

######  Corollary 2.18.

  1. 1.

Let $M\in\widetilde{\cal{F}}\left(\Delta\right)$ and $F$ and $F^{\prime}$ be
$\widetilde{\Delta}-$filtrations of $M.$ Then
$\ell_{\omega}\left(F\right)=\ell_{\omega}\left(F^{\prime}\right)$ for each
$\omega\in\Omega.$ It follows that the cardinality of factors isomorphic to
$\Delta_{\omega}$ are well defined for $M,$ i.e. $\ell_{\omega}(M)$ makes
sense.

  2. 2.

Let $L,N\in\widetilde{\cal{F}}\left(\Delta\right)$ and $0\rightarrow
L\rightarrow M\rightarrow N\rightarrow 0$ be an exact sequence. Then
$\ell_{\omega}\left(M\right)=\ell_{\omega}\left(L\right)+\ell_{\omega}\left(N\right)$
for each $\omega\in\Omega.$

Proof:

1.- As pointed out in Remark 2.8 $F$ induces a slim
$\widetilde{\Delta}-$filtration $\underline{F}$ and $F^{\prime}$ induces a
slim $\widetilde{\Delta}-$filtration $\underline{F^{\prime}},$ and by
Proposition 2.17 they both induce the same ordered
$\widetilde{\Delta}-$filtration $W$ of $M.$

Then, by that remark and the way to obtain $W$ we have

$\ell_{\omega}\left(F\right)=\ell_{\omega}\left(\underline{F}\right)=\ell_{\omega}\left(Z\right)=\ell_{\omega}\left(\underline{F^{\prime}}\right)=\ell_{\omega}\left(F^{\prime}\right)$

for any $\omega.$

2.- It is a direct consequence of item (1).

$\hfill\square$

######  Lemma 2.19.

Let $M=\bigoplus_{i\in I}G_{i}$ where $G_{i}$ is f.g and has local ring of
endomorphisms for each $i.$ Assume $M=L\oplus N.$ Then there exist
decompositions in direct sums $L=\bigoplus_{j\in I^{\prime}}L_{j}$ and
$\bigoplus_{t\in I^{\prime\prime}}N_{t}$ such that:

  1. 1.

For any $j$ there exists an $i_{j}\in I$ such that $L_{j}\cong G_{i_{j}}.$

  2. 2.

For any $t$ there exists an $i_{t}\in I$ such that $N_{t}\cong G_{i_{t}}.$

  3. 3.

Let $G$ be an $R-$module with local ring of endomorphisms and $\ell_{G}(M)$
the cardinal of the subset $I_{G}$ of $I,$ where $i\in I_{G}$ iff $G_{i}\cong
G.$ Then $\ell_{G}(M)=\ell_{G}(L)+\ell_{G}(N).$

Proof: By the Crawley-J$\o$nsson-Warfield Theorem (26.5 of [1]) and a review
of its proof, we get items (1) and (2).

The third item follows by Azumaya’s Decomposition Theorem (12.6 of [1]).
$\hfill\square$

For the next proof recall that given $L,N\in R-$Mod, the _trace_ ${\rm
tr}_{L}\left(N\right)$ of $L$ on $N$ is the sum of all the images of the
homomorphisms from $L$ to $N.$

######  Corollary 2.20.

$\widetilde{\cal{F}}\left(\Delta\right)$ is closed under direct summands.

Proof: Let $M\in\widetilde{\cal{F}}\left(\Delta\right)$ and $M=L\oplus N.$

Consider the ordered $\widetilde{\Delta}-$filtration of $M$ with the notation
of Proposition 2.17.

By Lemma 2.16 we have the identity $W_{a}={\rm tr}_{W_{a}}\left(M\right),$ and
by additivity of the trace, ${\rm tr}_{W_{a}}\left(M\right)={\rm
tr}_{W_{a}}\left(L\right)\oplus{\rm tr}_{W_{a}}\left(N\right).$

By Lemma 2.19 we get that ${\rm tr}_{W_{a}}\left(L\right)=L_{a}\subseteq L$
and ${\rm tr}_{W_{a}}\left(N\right)=N_{a}\subseteq N$ are in
$\widetilde{\cal{F}}\left(\Delta\right).$

By same Lemma
$\ell_{\omega}\left(W_{a}\right)=\ell_{\omega}\left(L_{a}\right)+\ell_{\omega}\left(N_{a}\right)$
for each $\omega.$

We can repeat this argument for the quotients
$M/W_{a}\cong\left(L/L_{a}\right)\oplus\left(N/N_{a}\right)$ in order to get
larger submodules of $L$ and $N$ that are in
$\widehat{\cal{F}}\left(\Delta\right).$

Repeating the procedure we obtain ordered $\widetilde{\Delta}-$filtrations of
$L$ and $N,$ so they both belong to $\widetilde{\cal{F}}\left(\Delta\right).$
$\hfill\square$

## References

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