# Syzygy Filtrations of Cyclic Nakayama Algebras

Emre SEN  Department of Mathematics, University of Iowa, Iowa City, IA  emre-
<EMAIL_ADDRESS>
###### Abstract.

We introduce a method ”syzygy filtration” to give building blocks of syzygies
appearing in projective resolutions of indecomposable $\Lambda$-modules where
$\Lambda$ is a cyclic Nakayama algebra. We interpret homological invariants of
$\Lambda$ including left and right finitistic dimension, left and right
$\varphi$-dimension, Gorenstein dimension, dominant dimension and their upper
bounds in terms of this new filtration. For all of them, we obtain a unified
upper bound $2r$ where $r$ is the number of relations defining the algebra
$\Lambda$. We give a simpler proof of theorem relating Gorenstein dimension to
$\varphi$-dimension. Moreover we show that
$\varphi\dim\Lambda-\operatorname{fin.\\!dim}\Lambda\leq 1$.

To Professors Kiyoshi Igusa and Gordana Todorov in celebration of seventieth
birthdays

###### Contents

  1. 1 Introduction
  2. 2 $\varphi$-dimension for Artin algebras
  3. 3 Syzygy Filtered Algebras
    1. 3.1 Modules of syzygy filtered algebras
    2. 3.2 Higher Syzygy Filtered Algebras
  4. 4 Outcomes of Syzygy Filtrations
    1. 4.1 Results about $\varphi\dim$
    2. 4.2 Results about finitistic dimension
    3. 4.3 Results about Gorenstein Properties
    4. 4.4 A Detailed Example
  5. 5 Other Applications of Syzygy Filtrations
    1. 5.1 Dominant Dimension
    2. 5.2 Upper Bounds For Global Dimension
    3. 5.3 Right Finitistic and $\varphi$-Dimensions

##  1\. Introduction

For an artin algebra $A$, one of the major open question is the finitistic
dimension conjecture. It states that supremum of finite projective dimensions
of modules is finite i.e.

$\displaystyle\operatorname{fin.\\!dim}A<\infty$

where

$\displaystyle\operatorname{fin.\\!dim}A:=\sup\left\\{p\dim M|\,M\,\text{is A
module with}\,\,p\dim(M)<\infty\right\\}$

If representation dimension of $A$ is less than or equal to $3$ then the
conjecture holds, proof was given by K. Igusa and G. Todorov in [IT05]. In the
proof, they introduced a function for each module $M$ and its value is always
a finite number. Again one can consider supremum of those values and define:

$\displaystyle\varphi\dim A:=\sup\left\\{\varphi(M)|\,M\,\text{is an
A-mod}\right\\}$

It turns out that this new homological definition has applications since an
algebra is self injective if and only if $\varphi$-dimension of the algebra is
zero [LM18]. Later A. Garcia and R. Schiffler proved that if algebra is
Gorenstein than Gorenstein dimension is equal to $\varphi$-dimension of the
algebra. We think that possible numerical values $\varphi\dim$ itself can be
useful homological measure of artin algebras. In [Sen18] it was shown that for
a cyclic Nakayama algebra of infinite global dimension, $\varphi$-dimension is
always an even number. And for small values of $\varphi\dim$, conditions on
algebra were described. In the same work, to study $\varphi$-dimension, one
important tool was $\Delta$filtration and its useful properties. In this work,
we develop syzygy filtration method which is based on the filtrations
introduced at [Sen18]. Here, we apply syzygy filtrations iteratively and give
applications of it especially on homological invariants and their bounds for
cyclic Nakayama algebras. We prove:

######  Theorem 1.1.

Let $\Lambda$ be a cyclic Nakayama algebra. Then the difference of
$\varphi\dim\Lambda$ and $\operatorname{fin.\\!dim}\Lambda$ can be at most
$1$, i.e. $\varphi\dim\Lambda-\operatorname{fin.\\!dim}\Lambda\leq 1$.

######  Theorem 1.2.

Let $\Lambda$ be a cyclic Nakayama algebra. If $\Lambda$ is Gorenstein, then
Gorenstein dimension is equal to $\varphi$-dimension i.e.
$\operatorname{gor.\\!dim}\Lambda=\varphi\dim\Lambda$.

The second result appears in the literature [ES17], [LM18] etc. However we
give an elementary proof in the case of Nakayama algebras by new methods.
Briefly, we construct an algebra $\bm{\varepsilon}(\Lambda)$ 3.2 and its
modules will correspond to second syzygies of $\Lambda$ modules 3.6. Regarding
$\bm{\varepsilon}(\Lambda)$ with $gl\dim\Lambda=\infty$, we have:

######  Theorem 1.3.

  1. i)

$\Lambda$ is self injective if and only if
$\Lambda\cong\bm{\varepsilon}(\Lambda)$.

  2. ii)

$\varphi\dim\Lambda=\varphi\dim\bm{\varepsilon}(\Lambda)+2$, when
$\Lambda\ncong\bm{\varepsilon}(\Lambda)$.

  3. iii)

If $\Lambda$ is Gorenstein then $\bm{\varepsilon}(\Lambda)$ is also.

This enables us to use mathematical induction on homological dimensions, since
we show that $\bm{\varepsilon}(\Lambda)$ is also a Nakayama algebra in 3.9.

Another important homological dimension is dominant dimension of algebra $A$:
Let $0\mapsto A\mapsto I_{0}\mapsto I_{1}\ldots$ be minimal injective
resolution of regular module $A$. Then:

$\displaystyle\operatorname{dom.\\!dim}(A)=\sup\left\\{n|\,I_{i}\,\text{is
projective for}\,i=0,1,\ldots n\right\\}+1$

provided that $I_{0}$ is projective. Otherwise it is set to be zero. Marzinnik
studied dominant dimension of Nakayama algebras and showed that sharp upper
bound of dominant dimension is $2N-2$ where $N$ is the number of simple
modules [Mar18]. We produce that result by using syzygy filtrations, and the
crucial step is the reduction:

######  Proposition 1.4.

Let $\Lambda$ be cyclic Nakayama algebra with
$3\leq\operatorname{dom.\\!dim}\Lambda<\infty$. Then
$\operatorname{dom.\\!dim}\Lambda=\operatorname{dom.\\!dim}\bm{\varepsilon}(\Lambda)+2$.

Indeed, all homological measures we discussed so far share the same unified
upper bound:

######  Theorem 1.5.

Let $\Lambda$ be cyclic nonselfinjective Nakayama algebra defined by $r$ many
irredundant relations over $N$ vertices. Then, $\varphi\dim\Lambda$,
$\operatorname{fin.\\!dim}\Lambda$, $\operatorname{gor.\\!dim}\Lambda$,
$\operatorname{dom.\\!dim}\Lambda$ are bounded by $2r$. Moreover, the upper
bound is $2N-2$ since $r\leq N-1$.

Another application is the following result of [MM18]:

######  Corollary 1.6.

Let $\Lambda$ be a Nakayama algebra of $N$ nodes and with a simple module $S$
of even projective dimension. Choose $m$ minimal such that a simple $\Lambda$
module $S$ has projective dimension equal to $2m$. Then the global dimension
of $\Lambda$ is bounded by $N+m-1$.

In general left and right finitistic dimension of algebras are not same. In
literature, we could not find such results concerning Nakayama algebras. We
prove the following by using syzygy filtrations again:

######  Theorem 1.7.

Let $\Lambda$ be cyclic Nakayama algebra.

  1. i)

Left and right finitistic dimensions are same i.e.
$\operatorname{fin.\\!dim}\Lambda=\operatorname{fin.\\!dim}\Lambda^{op}$.

  2. ii)

Left and right $\varphi$-dimensions are same i.e.
$\varphi\dim\Lambda=\varphi\dim\Lambda^{op}$

The paper has the following structure: in the first section, we recall
definition of $\varphi$-dimension. In the second section, we introduce algebra
$\bm{\varepsilon}(\Lambda)$ properly, and its properties. In the remaining
sections, we show applications of syzygy filtered algebras to homological
dimensions including left-right finitistic dimension, left-right
$\varphi$-dimension, dominant dimension, Gorenstein dimension and global
dimension.

##  2\. $\varphi$-dimension for Artin algebras

Let $A$ be artin algebra. let $K_{0}$ be the abelian group generated by all
symbols $\left[X\right]$, where $X$ is a finitely generated $A$ module, modulo
the relations:

  1. i)

$\left[A_{1}\right]=\left[A_{2}\right]+\left[A_{3}\right]$ if $A_{1}\cong
A_{2}\oplus A_{3}$

  2. ii)

$\left[P\right]=0$ if $P$ is projective.

Then $K_{0}$ is the free abelian group generated by the isomorphism classes of
indecomposable finitely generated nonprojective $A$-modules. For any finitely
generated $A$-module $M$ let $L\left[M\right]:=\left[\Omega M\right]$ where
$\Omega M$ is the first syzygy of $M$. Since $\Omega$ commutes with direct
sums and takes projective modules to zero this gives a homomorphism
$L:K_{0}\mapsto K_{0}$. For every finitely generated $\Lambda$-module $M$ let
$\left\langle addM\right\rangle$ denote the subgroup of $K_{0}$ generated by
all the indecomposable summands of $M$, which is a free abelian group since it
is a subgroup of the free abelian group $K_{0}$. In other words, if
$M=\oplus_{i=1}^{m}{M_{i}}^{n_{i}}$ then $\left\langle
addM\right\rangle=\left\langle\\{[M_{i}]\\}_{i=1}^{m}\right\rangle$ and
$L^{t}(\left\langle
addM\right\rangle)=\left\langle\\{[\Omega^{t}M_{i}]\\}_{i=1}^{m}\right\rangle$.

######  Definition 2.1.

For a given module $M$, let $\varphi\left(M\right)$ be defined as:

$\varphi(M):=min\\{t\ |\ rank\left(L^{t}\langle
addM\rangle\right)=rank\left(L^{t+j}\langle addM\rangle\right)\text{ for
}\forall j\geq 1\\}.$

Notice that $\varphi(M)$ is finite for each module $M$, since rank has to
become stable at some step. For example, if projective dimension of $M$ is
finite, then $\varphi(M)=p\dim(M)$.

######  Definition 2.2.

Let $A$ be artin algebra. We define:

$\varphi\dim(A):=sup\\{\varphi(M)\ |\ M\in modA\\}.$

We will state few useful propositions about $\varphi$-dimension. Proofs and
other properties can be found in [Sen18].

######  Lemma 2.3.

Let $A$ be an artin algebra of finite representation type. Let
$\\{M_{1},\dots,M_{m}\\}$ be a complete set of representatives of isomorphism
classes of indecomposable $\Lambda$-modules. Then
$\varphi\dim(A)=\varphi(\oplus_{i=1}^{m}M_{i})$.

######  Remark 2.4.

If global dimension of algebra $A$ is finite, then
$\operatorname{gl\,\\!dim}(A)=\varphi\dim(A)$. Recall that
$\operatorname{fin.\\!dim}A=\sup\\{p\dim(X)\ |\
p\dim(X)<\infty,X\in\text{mod-}A\\}$, which implies
$\operatorname{fin.\\!dim}A\leq\varphi\dim A$.

##  3\. Syzygy Filtered Algebras

Let $\Lambda$ be a cyclic Nakayama algebra over $N\geq 2$ vertices given by
irredundant system relations
$\bm{\alpha}_{k_{2i}}\ldots\bm{\alpha}_{k_{2i-1}}=0$ where $1\leq i\leq r$ and
$k_{f}\in\left\\{1,2,\ldots,N\right\\}$ for cyclic quiver $\bm{Q}$ where each
arrow $\bm{\alpha}_{i}$, $1\leq i\leq N-1$ starts at the vertex $i$ and ends
at the vertex $i+1$ and $\bm{\alpha}_{N}$ starts at vertex $N$ and ends at
vertex $1$.

Here we list irredundant system of relations and denote it by REL:

(3.1)
$\displaystyle\bm{\alpha}_{k_{2}}\ldots\bm{\alpha}_{k_{1}+1}\bm{\alpha}_{k_{1}}=0$
$\displaystyle\bm{\alpha}_{k_{4}}\ldots\bm{\alpha}_{k_{3}+1}\bm{\alpha}_{k_{3}}=0$
$\displaystyle\vdots$
$\displaystyle\bm{\alpha}_{k_{2r}}\ldots\bm{\alpha}_{k_{2r-1}+1}\bm{\alpha}_{k_{2r-1}}=0$

It is clear that $\Lambda$ is bound quiver algebra
$\faktor{k\bm{Q}}{\left<REL\right>}$.

Projective modules can be described with respect to socles. Simply:

(3.2) $\displaystyle P_{k_{1}}\hookrightarrow\ldots\hookrightarrow
P_{(k_{2r-1})+1}\quad\text{ have simple }S_{k_{2}}\text{ as their socle}$
$\displaystyle P_{k_{3}}\hookrightarrow\ldots\hookrightarrow
P_{k_{1}+1}\quad\text{ have simple }S_{k_{4}}\text{ as their socle}$
$\displaystyle\vdots\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ $\displaystyle
P_{k_{2r-1}}\hookrightarrow\ldots\hookrightarrow P_{(k_{2r-3})+1}\quad\text{
have simple }S_{k_{2r}}\text{ as their socle}$

Let ${\mathcal{S}}(\Lambda)$ be the complete set of representatives of socles
of projective modules over $\Lambda$ i.e.
${\mathcal{S}}(\Lambda)=\left\\{S_{k_{2}},S_{k_{4}},\ldots,S_{k_{2r}}\right\\}$.
Similarly, let ${\mathcal{S}}^{\prime}(\Lambda)$ be the complete set of
representatives of simple modules such that they are indexed by one cyclically
larger indices of ${\mathcal{S}}(\Lambda)$ i.e.
${\mathcal{S}}^{\prime}(\Lambda)=\left\\{S_{k_{2}+1},S_{k_{4}+1},\ldots,S_{k_{2r}+1}\right\\}$.

Now we define the following base set ${\mathcal{B}}(\Lambda)$, its elements
are nonzero and minimal traces of projectives in cyclic order and indexed by
${\mathcal{S}}^{\prime}(\Lambda)$ i.e.

$\displaystyle{\mathcal{B}}(\Lambda):=\left\\{\Delta_{1}\cong\begin{vmatrix}S_{k_{2r}+1}\\\
\vdots\\\
S_{k_{2}}\end{vmatrix}\\!,\Delta_{2}\cong\begin{vmatrix}S_{k_{2}+1}\\\
\vdots\\\
S_{k_{4}}\end{vmatrix}\\!,..,\Delta_{j}\cong\begin{vmatrix}S_{k_{2(j-1)}+1}\\\
\vdots\\\
S_{k_{2j}}\end{vmatrix}\\!,..,\Delta_{r}\cong\begin{vmatrix}S_{k_{2r-2}+1}\\\
\vdots\\\ S_{k_{2r}}\end{vmatrix}\\!\right\\}$

######  Remark 3.1.

Details and properties of modules filtered by ${\mathcal{B}}(\Lambda)$ can be
found in [Sen18]. Here, instead of $\bf\Delta$, we use
${\mathcal{B}}(\Lambda)$. Let $M$ be an indecomposable nonprojective $\Lambda$
module. If its second syzygy is nontrivial, then $\Omega^{2}(M)$ has unique
filtration by ${\mathcal{B}}(\Lambda)$.

Now we are ready to construct new algebra which is filtered by modules of
${\mathcal{B}}(\Lambda)$.

######  Definition 3.2.

Let $\bm{\varepsilon}(\Lambda)$ be an endomorphism algebra of direct sum of
projective modules over $\Lambda$ indexed by
${\mathcal{S}}^{\prime}(\Lambda)$, i.e.

(3.3)
$\displaystyle\bm{\varepsilon}(\Lambda):=End_{\Lambda}\left(\bigoplus\limits_{i\in{\mathcal{S}}^{\prime}(\Lambda)}P_{i}\right)$

Since $\bm{\varepsilon}(\Lambda)$ is related to ${\mathcal{B}}(\Lambda)$
filtered modules, we call it _syzygy filtered algebra_. We will prove that it
is a Nakayama algebra.

######  Lemma 3.3.

Let $P_{x},P_{y},P_{z}$ be three indecomposable projective modules from
${\mathcal{S}}^{\prime}(\Lambda)$ and maps $f,g,h$ satisfying $f=h\circ g$:

|
---|---
$\textstyle{P_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{f}$$\textstyle{P_{y}}$$\textstyle{P_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$

Then, the sequence

$\displaystyle
0\mapsto\operatorname{coker}g\mapsto\operatorname{coker}f\mapsto\operatorname{coker}h\mapsto
0$

is exact.

###### Proof.

Observe that $P_{x},P_{y},P_{z}$ are uniserial. We have the following exact
sequences:

$\displaystyle 0\mapsto\operatorname{Im}f\mapsto
P_{y}\mapsto\operatorname{coker}f\mapsto 0$ $\displaystyle
0\mapsto\operatorname{Im}h\mapsto P_{y}\mapsto\operatorname{coker}h\mapsto 0$
$\displaystyle
0\mapsto\operatorname{Im}(hog)\mapsto\operatorname{Im}h\mapsto\operatorname{coker}g\mapsto
0$

The last sequence exists because both $\operatorname{Im}f$ and
$\operatorname{Im}h$ have the same socle which is socle of $P_{y}$.
Furthermore, $\operatorname{Im}h\subset\operatorname{Im}f$ is impossible
because $h(\operatorname{Im}g)=\operatorname{Im}f$ is given. We show that the
quotient is $\operatorname{coker}g$ because: consider a simple module $x$ from
composition series of $\faktor{\Im h}{\operatorname{Im}f}$, this implies $x$
cannot be in the image of $g$, since $x$ is not subquotient of
$\operatorname{Im}f$. However, $x$ is subquotient of $\operatorname{Im}h$
therefore subquotient of $\faktor{P_{z}}{\operatorname{Im}g}$, which is simply
$\operatorname{coker}g$.
One can show that $\operatorname{coker}g$ is submodule of
$\operatorname{coker}f$ since $\operatorname{Im}h\subset P_{y}$ and
$\operatorname{Im}f\subset\operatorname{Im}h$ we get:

$\displaystyle\operatorname{Im}f\hookrightarrow\operatorname{Im}h\hookrightarrow
P_{y}$ $\displaystyle\iff$
$\displaystyle\faktor{\operatorname{Im}h}{\operatorname{Im}f}\hookrightarrow\faktor{P_{y}}{\operatorname{Im}f}$
$\displaystyle\iff$

$\operatorname{coker}g\cong\faktor{\operatorname{Im}h}{\operatorname{Im}f}\subset\faktor{P_{y}}{\operatorname{Im}f}\cong\operatorname{coker}f$

Now :

$\displaystyle\frac{\operatorname{coker}f}{\operatorname{coker}g}\cong\frac{\faktor{P_{y}}{\operatorname{Im}f}}{\faktor{\operatorname{Im}h}{\operatorname{Im}(hog)}}\cong\frac{\faktor{P_{y}}{\operatorname{Im}f}}{\faktor{\operatorname{Im}h}{\operatorname{Im}(f)}}\cong\frac{P_{y}}{\operatorname{Im}h}\cong\operatorname{coker}h$

Claim follows. ∎

######  Corollary 3.4.

The map $f$ from $P_{k_{2i}+1}$ to $P_{k_{2i-2}+1}$ with
$\operatorname{coker}f\in{\mathcal{B}}(\Lambda)$ for any $i$ is unfactorizable
i.e. cannot be factored thorough another projective $\Lambda$ module different
than $P_{k_{2i}+1}$ and $P_{k_{2i-2}+1}$.

###### Proof.

Assume that there exist maps $g,h$ and a projective module
$P\in{\mathcal{S}}^{\prime}(\Lambda)$ such that $f=hog$. By lemma 3.3, we
obtain the following the module structure:

$\displaystyle\operatorname{coker}f\cong\begin{vmatrix}\operatorname{coker}g\\\
\operatorname{coker}h\end{vmatrix}$

Tops of the modules $\operatorname{coker}h$, $\operatorname{coker}g$ are in
${\mathcal{S}}^{\prime}(\Lambda)$. Similarly, socles of the modules
$\operatorname{coker}h$, $\operatorname{coker}g$ are in
${\mathcal{S}}(\Lambda)$. So $\operatorname{coker}h$ and
$\operatorname{coker}g$ have unique ${\mathcal{B}}(\Lambda)$ filtration. This
forces that $\operatorname{coker}f$ cannot be an element of
${\mathcal{B}}(\Lambda)$ since it is an extension of elements of
${\mathcal{B}}(\Lambda)$ . Contradiction.
So either $\operatorname{coker}g\cong\operatorname{coker}f$ or
$\operatorname{coker}h\cong\operatorname{coker}f$, so $f$ can not be factored.
∎

######  Proposition 3.5.

Indecomposable $\bm{\varepsilon}(\Lambda)$ modules are uniserial.

###### Proof.

By the corollary 3.4, nontrivial unfactorizable maps between projectives of
${\mathcal{S}}^{\prime}(\Lambda)$ is given by :

$\displaystyle...\mapsto P_{k_{2i}+1}\mapsto P_{k_{2i-2}+1}\mapsto
P_{k_{2i-4}+1}\mapsto\ldots$

Since there are finitely many indecomposable modules, if we regard each
projective a point, quiver is either direct sums of ${\mathbb{A}}$ type or
cyclic quiver. In both cases, indecomposable modules are uniserial. ∎

###  3.1. Modules of syzygy filtered algebras

We give other equivalent formulations for ${\mathcal{B}}(\Lambda)$ and modules
of $\bm{\varepsilon}(\Lambda)$.

Let $Filt({\mathcal{B}}(\Lambda))$ be category of ${\mathcal{B}}(\Lambda)$
filtered $\Lambda$ modules. A module $U\in{\mathcal{B}}(\Lambda)$ if and only
if $U$ has minimal projective presentation:

(3.6)

where $P_{a},P_{b}\in Filt({\mathcal{B}}(\Lambda))$ and $f$ cannot be factored
through a projective module in $Filt({\mathcal{B}}(\Lambda))$ i.e.
unfactorizable 3.4.

We can describe modules over syzygy filtered algebra in the following way. Let
${\mathcal{P}}:=\bigoplus_{i\in{\mathcal{S}}^{\prime}(\Lambda)}P_{i}$. Then
$\bm{\varepsilon}(\Lambda)=End_{\Lambda}{\mathcal{P}}$ is finite dimensional
algebra and ${\mathcal{P}}$ is left $\bm{\varepsilon}(\Lambda)$ module. If $X$
is right $\Lambda$ module, then
$\operatorname{Hom}_{\Lambda}({\mathcal{P}},X)$ is right
$\bm{\varepsilon}(\Lambda)$ module. Simple $\bm{\varepsilon}(\Lambda)$ modules
are of the form $\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Delta_{i})$ where
$\Delta_{i}\in{\mathcal{B}}(\Lambda)$ 3. For simplicity, we denote functor
$\operatorname{Hom}_{\Lambda}({\mathcal{P}},-)$ by $H$. We get:

|  |
---|---|---
$\textstyle{\text{mod-}\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H}$$\textstyle{\text{mod-}\bm{\varepsilon}(\Lambda)}$$\textstyle{Filt({\mathcal{B}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\subset}$

Restriction of $H$ onto $Filt({\mathcal{B}}(\Lambda))$ gives equivalence of
categories mod-$\bm{\varepsilon}(\Lambda)$ and $Filt({\mathcal{B}}(\Lambda))$.

######  Remark 3.6.

Any $\bm{\varepsilon}(\Lambda)$ module is isomorphic to $H(X)$ for some $X\in
Filt({\mathcal{B}}(\Lambda))$. Simple modules are of the form $H(\Delta_{i})$.
Therefore, by abuse of terminology we call ${\mathcal{B}}(\Lambda)$ filtered
$\Lambda$ modules $\bm{\varepsilon}(\Lambda)$ modules.

We give another proof about uniseriality of $\bm{\varepsilon}(\Lambda)$
modules. Notice that, since indecomposable $\Lambda$ modules are uniserial, in
particular modules in $Filt({\mathcal{B}}(\Lambda))$ are. For all $X\in
Filt({\mathcal{B}}(\Lambda))$, the ${\mathcal{B}}(\Lambda)$ radical filtration
is equivalent to composition series:

$\displaystyle X=X_{n}\supset X_{n-1}\supset\ldots\supset X_{1}\supset
X_{0}=0$

i.e. each $X_{i}\in Filt({\mathcal{B}}(\Lambda))$ and
$\faktor{X_{i}}{X_{i-1}}\in{\mathcal{B}}(\Lambda)$. $H$ is exact, and it is
clear that

(3.7) $\displaystyle
H(\operatorname{Ext}_{\Lambda}(X,Y))\cong\operatorname{Ext}_{\bm{\varepsilon}(\Lambda)}(H(X),H(Y))$

By using 3.7, we get composition series of $H(X)$ in
$\bm{\varepsilon}(\Lambda)$:

$\displaystyle H(X)=H(X_{n})\supset H(X_{n-1})\supset\ldots\supset
H(X_{1})\supset X_{0}=0$

Quotient of two consecutive modules is simple $\bm{\varepsilon}(\Lambda)$
module and it is equivalent to radical filtration. Therefore each
indecomposable $\bm{\varepsilon}(\Lambda)$ module is uniserial [ARS97]

######  Proposition 3.7.

If global dimension of $\Lambda$ is infinite, then $\bm{\varepsilon}(\Lambda)$
is cyclic and its global dimension is infinite.

###### Proof.

There is $\Lambda$-module $M$ with $p\dim M=\infty$. By remark 3.1,
$\Omega^{2}(M)$ has ${\mathcal{B}}(\Lambda)$ filtration. This makes
$H(\Omega^{2}(M))$ $\bm{\varepsilon}(\Lambda)$-module by 3.6. This implies,
global dimension of $\bm{\varepsilon}(\Lambda)$ is infinite since projective
resolution of $\Omega^{2}(M)$ in $\Lambda$ can be carried into resolution:
$..H(P_{3})\mapsto H(P_{2})\mapsto H(\Omega^{2}(M))$ in
$\bm{\varepsilon}(\Lambda)$ by remark 3.6. Its quiver has to be cyclic since
global dimension of ${\mathbb{A}}$ type quiver is always finite. ∎

######  Remark 3.8.

If underlying quiver is of type $\mathbb{A}$, then algebra is called _linear
Nakayama algebra_.

######  Theorem 3.9.

$\bm{\varepsilon}(\Lambda)$ is Nakayama algebra and defining relations of
$\bm{\varepsilon}(\Lambda)$ are determined by ${\mathcal{B}}(\Lambda)$: any
indecomposable $\bm{\varepsilon}(\Lambda)$ projective module $H(P)$ satisfying
$rad_{\bm{\varepsilon}(\Lambda)}H(P)$ is nonprojective in
$\bm{\varepsilon}(\Lambda)$ gives the relation:

(3.8)
$\displaystyle\beta_{i_{k}}\circ\beta_{i_{k-1}}\circ\dots\circ\beta_{i_{2}}\circ\beta_{i_{1}}=0$

where $i_{j}$ is the index of $\bm{\varepsilon}(\Lambda)$-top of
$\bm{\varepsilon}(\Lambda)$ module
$rad^{j-1}_{\bm{\varepsilon}(\Lambda)}(H(P))$, $1\leq j\leq k$. In particular
$i_{1}$ is index of top of $H(P)$ and $i_{k}$ is index of socle of $H(P)$.

###### Proof.

$\bm{\varepsilon}(\Lambda)$ is Nakayama algebra by result 3.5.
By remark 3.6, elements of ${\mathcal{B}}(\Lambda)$ can be viewed as simple
modules of Nakayama algebra $\bm{\varepsilon}(\Lambda)$. In other words:
Complete set of representatives of simple $\bm{\varepsilon}(\Lambda)$ modules
is the set $\left\\{H(\Delta_{1}),\ldots,H(\Delta_{r})\right\\}$, i.e. $H$
applied elements of ${\mathcal{B}}(\Lambda)$ For any cyclic Nakayama algebra,
composition series of indecomposable minimal (i.e. its radical is not
projective) projective module in each class gives the defining relations.
Since $\bm{\varepsilon}(\Lambda)$ is Nakayama algebra, proof follows. ∎

######  Proposition 3.10.

For any cyclic Nakayama algebra $\Lambda$,
$\Lambda\cong\bm{\varepsilon}(\Lambda)$ if and only if $\Lambda$ is self-
injective.

###### Proof.

Assume that $\Lambda$ is self-injective. Then:

$\displaystyle\bm{\varepsilon}(\Lambda):=End\left(\bigoplus\limits_{i\in{\mathcal{S}}^{\prime}(\Lambda)}P_{i}\right)=End\left(\bigoplus\limits_{1\leq
i\leq N}P_{i}\right)\cong\Lambda$

since, ${\mathcal{S}}^{\prime}(\Lambda)$ contains indices of all simples over
$\Lambda$.
For the other direction, assume that $\Lambda\cong\bm{\varepsilon}(\Lambda)$.
By theorem 3.9, $\bm{\varepsilon}(\Lambda)$ is Nakayama algebra, this implies
the following equality:

(3.9)
$\displaystyle\left\\{k_{1},\ldots,k_{n}\right\\}=\left\\{k_{2}+1,k_{4}+1\ldots\right\\}$

up to cyclic permutation because of classes of projective modules 3.2. It is
enough check those indices by the following observation: if two Nakayama
algebras are isomorphic then number of simple modules are same. Now we can
choose $k_{2}+1=k_{3}$, $k_{3}+1=k_{4}$ etc. So ${\mathcal{B}}(\Lambda)$
contains all simple $\Lambda$ modules, this is equivalent to selfinjectivity
of $\Lambda$. ∎

###  3.2. Higher Syzygy Filtered Algebras

By the theorem 3.9 and proposition 3.10, we can construct higher syzygy
filtrations:

######  Definition 3.11.

Let $\Lambda$ be a cyclic Nakayama algebra and $\bm{\varepsilon}(\Lambda)$ is
syzygy filtered algebra. $n$th syzygy filtered algebra
$\bm{\varepsilon}^{n}(\Lambda)$ is an endomorphism algebra of direct sum of
projective modules which are indexed depending on $(n-1)$th syzygy algebra
i.e.

(3.10)
$\displaystyle\bm{\varepsilon}^{n}(\Lambda):=End_{\bm{\varepsilon}^{n-1}(\Lambda)}\left(\bigoplus\limits_{i\in{\mathcal{S}}^{\prime}(\bm{\varepsilon}^{n-1}(\Lambda))}P_{i}\right)$

provided that $\bm{\varepsilon}^{n-1}(\Lambda)$ is cyclic non self-injective
Nakayama algebra.

We can interpret indecomposable $\bm{\varepsilon}^{n}(\Lambda)$ modules in
terms of ${\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))$ filtered
$\bm{\varepsilon}^{n-1}(\Lambda)$ modules as we discussed in subsection 3.1.

Let $Filt({\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))$ be category of
${\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))$ filtered
$\bm{\varepsilon}^{n-1}(\Lambda)$ modules. A module
$U\in{\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))$ if and only if $U$ has
minimal projective presentation:

(3.13)

where $P_{a},P_{b}\in Filt({\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda)))$
and $f$ cannot be factored through a projective module in
$Filt({\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda)))$ i.e. unfactorizable
3.4.

We can describe modules over higher syzygy filtered algebra in the following
way. Let
${\mathcal{P}}(\bm{\varepsilon}^{n-1}(\Lambda)):=\bigoplus\limits_{i\in{\mathcal{S}}^{\prime}(\bm{\varepsilon}^{n-1}(\Lambda))}P_{i}$.
Then
$\bm{\varepsilon}^{n}(\Lambda):=End_{\bm{\varepsilon}^{n-1}(\Lambda)}\left({\mathcal{P}}(\bm{\varepsilon}^{n-1}(\Lambda))\right)$
is finite dimensional algebra and
${\mathcal{P}}(\bm{\varepsilon}^{n-1}(\Lambda))$ is left
$\bm{\varepsilon}^{n}(\Lambda)$ module. If $X$ is right
$\bm{\varepsilon}^{n-1}(\Lambda)$ module, then
$\operatorname{Hom}_{\Lambda}({\mathcal{P}}(\bm{\varepsilon}^{n-1}(\Lambda)),X)$
is right $\bm{\varepsilon}^{n}(\Lambda)$ module. Simple
$\bm{\varepsilon}^{n}(\Lambda)$ modules are of the form
$\operatorname{Hom}_{\bm{\varepsilon}^{n-1}(\Lambda)}({\mathcal{P}}(\bm{\varepsilon}^{n-1}(\Lambda)),B)$
where $B\in{\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))$. For simplicity, we
denote functor
$\operatorname{Hom}_{\bm{\varepsilon}^{n-1}(\Lambda)}({\mathcal{P}}(\bm{\varepsilon}^{n-1}(\Lambda)),-)$
by $H^{n-1}$. We get:

|  |  |
---|---|---|---
$\textstyle{\text{mod-}\bm{\varepsilon}^{n-1}(\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{n-1}}$$\textstyle{\text{mod-}\bm{\varepsilon}^{n}(\Lambda)}$$\textstyle{Filt({\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\subset}$

Restriction of $H^{n-1}$ onto
$Filt({\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda)))$ gives equivalence of
categories mod-$\bm{\varepsilon}^{n}(\Lambda)$ and
$Filt({\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))$. In particular, this
shows that indecomposable modules over $\bm{\varepsilon}^{n}(\Lambda))$ are
uniserial, hence it is Nakayama algebra.

######  Remark 3.12.

Any $\bm{\varepsilon}^{n}(\Lambda)$ module is isomorphic to $H^{n-1}(X)$ for
some $X\in Filt({\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda)))$. Simple
modules are of the form $H(B)$ where
$B\in{\mathcal{B}}(\bm{\varepsilon}^{n-1}(\Lambda))$. Moreover, for each
$1\leq j\leq n$ categories below are equivalent:

$\textstyle{Filt({\mathcal{B}}(\bm{\varepsilon}^{j}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bm{\varepsilon}^{j+1}(\Lambda)}$

via
$H^{j}=\operatorname{Hom}_{\bm{\varepsilon}^{j}(\Lambda)}\left({\mathcal{P}}(\bm{\varepsilon}^{j}(\Lambda)),-\right)$.
Notice that, If $\Omega^{2j+2}(M)$ is nontrivial module where
$M\in\text{mod-}\Lambda$, then it has
${\mathcal{B}}(\bm{\varepsilon}^{j}(\Lambda)$ filtration. By abuse of
terminology we view ${\mathcal{B}}(\bm{\varepsilon}^{j}(\Lambda))$ filtered
$\Lambda$ modules as modules over $\bm{\varepsilon}^{n}(\Lambda)$. Another
useful fact is $H^{j}$ is exact, therefore projective resolution
$\ldots\mapsto P_{1}\mapsto P_{0}\mapsto M$ of $M$ in $\Lambda$ gives
projective resolution in $\bm{\varepsilon}^{j+1}(\Lambda)$ provided that
$\Omega^{2j+2}(M)$ is nontrivial i.e.

(3.14) $\displaystyle\dots\longrightarrow H^{j}(P_{2j+3})\longrightarrow
H^{j}(P_{2j+2})\longrightarrow H^{j}(\Omega^{2j+2}(M))\longrightarrow 0$

##  4\. Outcomes of Syzygy Filtrations

Here we give some results on Nakayama algebras by using syzygy filtrations.

###  4.1. Results about $\varphi\dim$

######  Theorem 4.1.

If global dimension of cyclic not selfinjective Nakayama algebra $\Lambda$ is
infinite, then $\varphi\dim\Lambda=\varphi\dim\bm{\varepsilon}(\Lambda)+2$.

###### Proof.

Since indecomposable modules over $\bm{\varepsilon}(\Lambda)$ are second
syzygies of modules over $\Lambda$ by remark 3.6, and by 3.14 with $j=0$
difference is exactly $2$, theorem follows. ∎

The following three results were proven in [Sen18]. We state them in terms of
syzygy filtered algebra. We give new proof of the second theorem by applying
syzygy filtrations.

######  Theorem 4.2.

[Sen18] Let $\Lambda$ be a cylic Nakayama algebra of infinite global
dimension.

  1. 1)

$\varphi\dim\Lambda=0$ if and only if $\Lambda$ is self-injective algebra.

  2. 2)

$\varphi\dim\Lambda\neq 1$.

  3. 3)

$\varphi\dim\Lambda=2$ if and only if $\bm{\varepsilon}(\Lambda)$ is self-
injective and $\Lambda$ is not.

######  Theorem 4.3.

Assume that global dimension of cyclic Nakayama algebra $\Lambda$ is infinite.
Then $\varphi\dim\Lambda$ is even. Furthermore, the upper bound of
$\varphi\dim\Lambda$ is $2r$, where $r=|{\mathcal{B}}(\Lambda)|$.

###### Proof.

By the above theorem 4.1, in each step $\varphi$-dimension reduces by two.
Assume to the contrary that there exists $d$ such that
$\varphi\dim\bm{\varepsilon}^{d}(\Lambda)=1$. But by above result 4.2, this is
impossible.
To prove the second part, observe that number of simple modules of
$\bm{\varepsilon}^{i}(\Lambda)$ is given by
$|{\mathcal{B}}(\bm{\varepsilon}^{i-1}(\Lambda))|$. To get an upper bound, in
each reduction 4.1, number of simple modules should decrease by one. Since
$|{\mathcal{B}}(\Lambda)|=r$, at most $r$-many reduction is possible.
$\bm{\varepsilon}^{r}(\Lambda)$ is self injective, hence upper bound for
$\varphi\dim$ is $2r$. ∎

###  4.2. Results about finitistic dimension

First we state and prove theorems regarding possible small values of
finitistic dimension and $\varphi$-dimension. Then we will show reduction
method for finitistic dimension as we did in theorem 4.1 for
$\varphi$-dimension.

We start with the following observation:

######  Proposition 4.4.

If $\varphi\dim\Lambda=2$, then
$\varphi\dim\Lambda-\operatorname{fin.\\!dim}\Lambda\leq 1$.

###### Proof.

Since $\varphi\dim\Lambda=2$, by theorem 4.2 $\Lambda$ is not self-injective.
This implies that there exist projectives $P_{x}$, $P_{y}$ over $\Lambda$ such
that $P_{y}$ is proper submodule of $P_{x}$. So there exists exact sequence:

(4.1) $\displaystyle 0\mapsto P_{y}\mapsto
P_{x}\mapsto\faktor{P_{x}}{P_{y}}\mapsto 0$

So $p\dim_{\Lambda}\left(\faktor{P_{x}}{P_{y}}\right)=1$, this implies
$\operatorname{fin.\\!dim}\Lambda\geq 1$.
Moreover, by remark 2.4, we conclude that
$\varphi\dim\Lambda-\operatorname{fin.\\!dim}\Lambda\leq 1$. ∎

######  Proposition 4.5.

$\operatorname{fin.\\!dim}\Lambda=\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)+2$,
provided that finitistic dimension of $\Lambda$ is greater than $2$.

###### Proof.

There exists a $\Lambda$-module $M$ such that $p\dim
M=\operatorname{fin.\\!dim}\Lambda$ and easily $p\dim M=p\dim\Omega^{2}(M)+2$.
It is enough to show that $\Omega^{2}(M)$ is nonprojective. This makes
$H(\Omega^{2}(M))$ an indecomposable nonprojective $\bm{\varepsilon}(\Lambda)$
module. So
$\operatorname{fin.\\!dim}\Lambda\geq\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)+2$
Choose $\bm{\varepsilon}(\Lambda)$ module $M^{\prime}$, satisfying
$\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)=p\dim M^{\prime}$. There
exists module $M$ such that $H(\Omega^{2}(M))\cong M^{\prime}$ by remark 3.1.
Equality follows.
If we assume $\Omega^{2}(M)$ projective for all $\Lambda$ modules $M$, global
dimension is $2$, which we excluded. The only remaining case is:
$\operatorname{fin.\\!dim}\Lambda=2$ and $gl\dim\Lambda=\infty$. We get
$\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)=0$. This forces that all
simple $\bm{\varepsilon}(\Lambda)$ modules have periodic resolution. Otherwise
the following sequence is exact : $0\mapsto P_{x+1}\mapsto P_{x}\mapsto
S_{x}\mapsto 0$, where $S_{x}$ is simple module. This occurs when
$\bm{\varepsilon}(\Lambda)$ is self-injective. That case was analyzed in 3.10.
Its finitistic dimension is $0$, hence proposition holds. ∎

We placed condition: finitistic dimension greater than 2, because there are
examples such that $\operatorname{fin.\\!dim}\Lambda=1$ but
$\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)=0$, therefore claim cannot
hold. A small example is: $N=3$ and relations
$\alpha_{2}\alpha_{1}=0,\alpha_{1}\alpha_{3}\alpha_{2}=0$. We study
generalities of them in 5.14.

Now we restate theorem 1.1 and give proof.

######  Theorem 4.6.

If $\Lambda$ is a cyclic Nakayama algebra then
$\varphi\dim\Lambda-\operatorname{fin.\\!dim}\Lambda\leq 1$

###### Proof.

If global dimension of $\Lambda$ is finite, statement is true, since
$gl\dim\Lambda=\varphi\dim\Lambda=\operatorname{fin.\\!dim}\Lambda$. So, we
assume that global dimension is infinite.
We can apply syzygy filtration algorithm recursively to reach a self injective
algebra in finitely many steps which we assume $d$. Hence
$\bm{\varepsilon}^{d}(\Lambda)$ is a self injective algebra but
$\bm{\varepsilon}^{d-1}(\Lambda)$ is not. By theorem 4.1 this implies
$\varphi\dim\Lambda=2d$, $\varphi\dim\bm{\varepsilon}^{d}(\Lambda)=0$ and
$\varphi\dim\bm{\varepsilon}^{d-1}(\Lambda)=2$. By theorem 4.4 we get

(4.2) $\displaystyle
1\leq\operatorname{fin.\\!dim}\bm{\varepsilon}^{d-1}(\Lambda)\leq 2$

On the other hand we have
$2+\operatorname{fin.\\!dim}\bm{\varepsilon}^{k}(\Lambda)=\operatorname{fin.\\!dim}\bm{\varepsilon}^{k-1}(\Lambda)$
for any $1\leq k\leq d-1$ given at 4.5. This implies:

$\displaystyle 1+2d-2\leq\operatorname{fin.\\!dim}\Lambda\leq 2+2d-2$
$\displaystyle\varphi\dim\Lambda-1\leq\operatorname{fin.\\!dim}\Lambda\leq\varphi\dim\Lambda$

We get $\varphi\dim\Lambda-\operatorname{fin.\\!dim}\Lambda\leq 1$ ∎

###  4.3. Results about Gorenstein Properties

We begin with definition of Gorenstein dimension. By using duality and
representation type of Nakayama algebras, instead of injective dimension of
$\Lambda$, we prefer to study projective dimensions of injective modules. Then
we state and prove theorems regarding Gorenstein homological properties.

######  Definition 4.7.

If projective resolution of injective modules are finite, algebra is called
_Gorenstein_. Supremum of projective dimensions of injective modules is called
_Gorenstein dimension_ , and denoted by $\operatorname{gor.\\!dim}$.

First, we need possible small values of Gorenstein dimension. Then we apply
reduction method as we did in $\varphi$-dimension and finitistic dimension.

######  Lemma 4.8.

If $\Lambda$ is cyclic Gorenstein Nakayama algebra, then
$\operatorname{gor.\\!dim}\Lambda\neq 1$.

###### Proof.

If $\Lambda$ is self-injective, by definition,
$\operatorname{gor.\\!dim}\Lambda=0$. If $\Lambda$ is not self injective,
there exists a projective-injective module $PI$ such that its quotient is
injective $I$ and

$\displaystyle 0\mapsto soc(PI)\mapsto PI\mapsto I\mapsto 0$

$\Lambda$ is cyclic, simple modules are not projective, $p\dim I\geq 2$, so
$\operatorname{gor.\\!dim}\Lambda\neq 1$. ∎

######  Lemma 4.9.

Let $\Delta_{i}$ be an element of ${\mathcal{B}}(\Lambda)$ such that it is
submodule of indecomposable projective injective $\Lambda$ module $PI$ and it
has proper submodule $X$ i.e.

$\displaystyle X\hookrightarrow\Delta_{i}\hookrightarrow PI$

Then the quotient $\faktor{PI}{X}$ is an injective $\Lambda$ module.

###### Proof.

We have the following set up: socles of $X$, $\Delta_{i}$ and $PI$ are
isomorphic moreover it is from the set ${\mathcal{S}}(\Lambda)$, recall
definition of ${\mathcal{B}}(\Lambda)$ and construction 3.2. Assume that
$\Delta_{i-1}$ is another element of ${\mathcal{B}}(\Lambda)$ satisfying
$\operatorname{Ext}^{1}_{\Lambda}(\Delta_{i-1},\Delta_{i})$ is indecomposable,
in other words socle of $\Delta_{i-1}$ and top of $\Delta_{i}$ are
consecutive. It is clear that $X$ cannot be an _element_ of
${\mathcal{B}}(\Lambda)$ because $\Delta_{i}$ is from the base set
${\mathcal{B}}(\Lambda)$.

For simplicity we denote it quotient by $Q=\faktor{PI}{X}$. Assume to the
contrary that $Q$ is not injective module. Therefore it has to be proper
submodule of its injective envelope i.e. $Q\hookrightarrow\Sigma(Q)$

Let $P^{\prime}$ be projective cover of the injective envelope of the quotient
$Q$ i.e. $P^{\prime}=P\left(\Sigma(Q)\right)$. We know that socle of
$P^{\prime}$ is in ${\mathcal{S}}(\Lambda)$ because of 3.2. Here we eliminate
impossible choices for the socle of $P^{\prime}$:

  * •

$\Delta_{i}$ cannot be submodule of $P^{\prime}$ because this violates
projective-injectivity of $PI$. In particular socle of $P^{\prime}$ cannot be
socle of $X$.

  * •

Extension $\operatorname{Ext}^{1}_{\Lambda}(\Delta_{i},K)$ cannot be a
submodule of $PI$ where $K=\ker\left(P^{\prime}\mapsto\Sigma(Q)\right)$,
because it violates projective injectivity of $PI$ again.

  * •

Socle of $P^{\prime}$ cannot be socle of $\Delta_{i-1}$, because
$\faktor{\Delta_{i}}{X}$ is subquotient of $P^{\prime}$.

It is impossible to find socle of projective cover $P^{\prime}$. This creates
contradiction. Otherwise, if $Q$ is injective then :

$\displaystyle P\left(\Sigma(Q)\right)=P\left(Q\right)=Q$

and socle of $PI$ is isomorphic to socle of $\Delta_{i}$. ∎

######  Proposition 4.10.

For every nonprojective ${\bm{\varepsilon}}(\Lambda)$ injective module
${}_{\bm{\varepsilon}}I$, there exists injective $\Lambda$ module $I$ such
that $H(\Omega^{2}(I))\cong\\!_{\bm{\varepsilon}}I$

###### Proof.

Assume that ${}_{\bm{\varepsilon}}I$ is a nontrivial nonprojective
${\bm{\varepsilon}}(\Lambda)$ injective module. By remark 3.6, there is
${\mathcal{B}}(\Lambda)$ filtered module $I\Delta$ such that
$H(I\Delta)=\\!_{\bm{\varepsilon}}I$. Moreover, by remark 3.1, $I\Delta$ can
be realized as second syzygy. Therefore we get exact sequence of $\Lambda$
modules:

$\displaystyle 0\mapsto I\Delta\mapsto P\mapsto\faktor{P}{I\Delta}\mapsto 0$

$I\Delta$ has ${\mathcal{B}}(\Lambda)$ filtration, assume that $\Delta_{x}$ is
its ${\mathcal{B}}(\Lambda)$-top and $\Delta_{y}$ be its
${\mathcal{B}}(\Lambda)$-socle where we choose ordering
$\left\\{\Delta_{x-1},\Delta_{x},\Delta_{x+1},\ldots,\Delta_{y-1},\Delta_{y},\Delta_{y+1}\ldots\right\\}$
in ${\mathcal{B}}(\Lambda)$, i.e
$\operatorname{Ext}^{1}(\Delta_{i},\Delta_{i+1})\ncong 0$. Observe that:

  * •

${\mathcal{B}}(\Lambda)$-socle of ${\mathcal{B}}(\Lambda)$ filtered projective
cover of $\Delta_{x}$ i.e. $P(\Delta_{x})$ is either $\Delta_{y+1}$ or
cyclically greater than $\Delta_{y}$.

  * •

${\mathcal{B}}(\Lambda)$-socle of filtered projective cover of $\Delta_{x-1}$
i.e. $P(\Delta_{x-1})$ is $\Delta_{y-1}$ or cyclically less than
$\Delta_{y-1}$.

For details of cyclic ordering we refer to [Sen18]. The table 1 illustrates
it. The second item is crucial, otherwise $H(I\Delta)$ cannot be injective
epsilon module.

Quotient $Q=\faktor{P}{I\Delta}$ cannot have ${\mathcal{B}}(\Lambda)$
filtration by the second item. But socle of the quotient is isomorphic to
socle of $\Delta_{x-1}$. Notice that this socle is in the set
${\mathcal{S}}(\Lambda)$, therefore there exists projective injective module
PI such that $quotient$ is its submodule by 3.2. Therefore we get another
exact sequence:

$\displaystyle 0\mapsto Q\mapsto PI\mapsto\faktor{PI}{Q}\mapsto 0$

We have: $Q$ is proper submodule of $\Delta_{x-1}$ and $\Delta_{x-1}$ is
submodule of projective injective module $PI$. By the lemma 4.9, we conclude
that $\faktor{PI}{Q}$ is injective $\Lambda$ module.

Therefore $H\left(\Omega^{2}\left(I\right)\right)$ is injective
$\bm{\varepsilon}(\Lambda)$ module where $I=\faktor{PI}{Q}$ is injective
$\Lambda$ module.

Table 1. Comparison of projective modules and $I\Delta$

:  $P(\Delta_{x})$ $I\Delta$ $P(\Delta_{x-1})$ ordering $\Delta_{x-1}$
$\Delta_{x}$ $\Delta_{x}$ $\vdots$ $\vdots$ $\vdots$ $\Delta_{y-1}$ $\uparrow$
c. less $\Delta_{y}$ $\Delta_{y}$ $\Delta_{y+1}$ $\downarrow$ c. greater

∎

We can apply the above proposition to Gorenstein Nakayama algebra $\Lambda$.
Every nonprojective injective $\Lambda$ module has finite projective
dimension, by remark 3.1 and proposition 4.10, every nonprojective injective
$\bm{\varepsilon}(\Lambda)$-module has finite projective dimension. Moreover,
$\operatorname{gor.\\!dim}\Lambda=\operatorname{gor.\\!dim}\bm{\varepsilon}(\Lambda)+2$.
We put them together below:

######  Corollary 4.11.

If $\Lambda$ is Gorenstein, then $\bm{\varepsilon}(\Lambda)$ is also
Gorenstein. In other words, being Gorenstein is homological invariant with
respect to syzygy filtrations

######  Theorem 4.12.

Let $\Lambda$ be Gorenstein and cyclic Nakayama algebra. Then
$\varphi\dim\Lambda$ is equal to $\operatorname{gor.\\!dim}\Lambda$.

###### Proof.

If global dimension of $\Lambda$ is finite, then it is equal to both
$\varphi\dim\Lambda$ and $\operatorname{gor.\\!dim}\Lambda$, it is trivial
case. So, we analyze the case of infinite global dimension.
There exists $d$ such that $\bm{\varepsilon}^{d}(\Lambda)$ is self-injective
but $\bm{\varepsilon}^{d-1}(\Lambda)$ is not. Since $\Lambda$ is Gorenstein,
by proposition 4.10 and following corollary, $\bm{\varepsilon}^{k}(\Lambda)$
is Gorenstein for $1\leq k\leq d$. In particular,
$\bm{\varepsilon}^{d-1}(\Lambda)$ is of Gorenstein dimension $2$. This implies
that $\operatorname{gor.\\!dim}\bm{\varepsilon}^{d}(\Lambda)=0$ which makes
$\bm{\varepsilon}^{d}(\Lambda)$ self-injective. By proposition 3.10,
$\varphi\dim\Lambda=2$. They are equal. By results 4.1, 4.10,
$\varphi\dim\Lambda=\operatorname{gor.\\!dim}\Lambda$. ∎

In theorem 4.12, we showed the equality of $\varphi$-dimension and Gorenstein
dimension of cyclic Nakayama algebras. We combine it with the result 4.3 to
get:

######  Corollary 4.13.

If $\Lambda$ is Gorenstein with infinite global dimension, then
$\operatorname{gor.\\!dim}\Lambda$ is even.

Gorenstein properties of Nakayama algebras were studied in the paper of Ringel
[Rin13]. Proposition 6 of the same work states that under some conditions,
Gorenstein dimension is an even number. Above corollary can be viewed as a
generalization of it.

###  4.4. A Detailed Example

######  Example 4.14.

We want to show reduction processes described in 4.1 and 4.10 on syzygy
filtered algebras 3.11.
Let $\Lambda$ be cyclic Nakayama algebra with $N=5$ vertices and irredundant
system of relations:

$\displaystyle\alpha_{5}\alpha_{4}\alpha_{3}\alpha_{2}\alpha_{1}=0$
$\displaystyle\alpha_{1}\alpha_{5}\alpha_{4}\alpha_{3}\alpha_{2}=0$
$\displaystyle\alpha_{3}\alpha_{2}\alpha_{1}\alpha_{5}\alpha_{4}\alpha_{3}=0$
$\displaystyle\alpha_{4}\alpha_{3}\alpha_{2}\alpha_{1}\alpha_{5}\alpha_{4}=0$

The only injective but nonprojective $\Lambda$ module is
$I_{2}=\begin{vmatrix}S_{3}\\\ S_{4}\\\ S_{5}\\\ S_{1}\\\ S_{2}\end{vmatrix}$.
Its projective dimension is $6$ by the projective resolution:

|  |
---|---|---
$\textstyle{P_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}}$$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{radP_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{radP_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

It can be shown that $\Omega^{2}(I)$, $\Omega^{4}(I)$ are injective
nonprojective $\bm{\varepsilon}(\Lambda)$ and $\bm{\varepsilon}^{2}(\Lambda)$
modules respectively. We get $\operatorname{gor.\\!dim}\Lambda=6$. We present
the results in the following tableaux. For notational simplicity, we used
different letters to simple modules in the syzygy filtered algebras. The
modules $\omega_{1},\omega_{2}$ correspond to $\Lambda$ modules
$\begin{vmatrix}S_{1}\end{vmatrix},\begin{vmatrix}radP_{1}\end{vmatrix}$.
Notice that they are $\Omega$-periodic $\Lambda$ modules. Recall that the
sequence of integers $(c_{1},\ldots,c_{n})$ is called Kupisch series of
Nakayama algebra if $c_{i}=l(P_{i})$ i.e. length of projective module $P_{i}$.
We point out that this method is different than retractions described in
[CY14], one can compare Kupisch series of retrations and Kupisch series in the
table.

Algebra | Kupisch | ${\mathcal{S}}\left(-\right)$ | ${\mathcal{S}}^{\prime}\left(-\right)$ | ${\mathcal{B}}\left(-\right)$ | Notes
---|---|---|---|---|---
$\Lambda$ | $(5,5,6,6,6)$ | $S_{1},S_{3},S_{4},S_{5}$ | $S_{1},S_{2},S_{4},S_{5}$ | $\left\\{\begin{vmatrix}S_{1}\end{vmatrix},\begin{vmatrix}S_{2}\\\ S_{3}\end{vmatrix},\begin{vmatrix}S_{4}\end{vmatrix},\begin{vmatrix}S_{5}\end{vmatrix}\right\\}$ | $\varphi\dim\Lambda=6$
$\bm{\varepsilon}(\Lambda)$ | $(4,4,5,5)$ | ${\mathcal{E}}_{1},{\mathcal{E}}_{3},{\mathcal{E}}_{4}$ | ${\mathcal{E}}_{1},{\mathcal{E}}_{2},{\mathcal{E}}_{4}$ | $\left\\{\begin{vmatrix}{\mathcal{E}}_{1}\end{vmatrix},\begin{vmatrix}{\mathcal{E}}_{2}\\\ {\mathcal{E}}_{3}\end{vmatrix},\begin{vmatrix}{\mathcal{E}}_{4}\end{vmatrix}\right\\}$ | $\varphi\dim\bm{\varepsilon}(\Lambda)=4$
$\bm{\varepsilon}^{2}(\Lambda)$ | $(3,3,4)$ | $\delta_{1},\delta_{3}$ | $\delta_{1},\delta_{2}$ | $\left\\{\begin{vmatrix}\delta_{1}\end{vmatrix},\begin{vmatrix}\delta_{2}\\\ \delta_{3}\end{vmatrix}\right\\}$ | $\varphi\dim\bm{\varepsilon}^{2}(\Lambda)=2$
$\bm{\varepsilon}^{3}(\Lambda)$ | $(2,2)$ | $\omega_{1},\omega_{2}$ | $\omega_{1},\omega_{2}$ | $\left\\{\begin{vmatrix}\omega_{1}\end{vmatrix},\begin{vmatrix}\omega_{2}\end{vmatrix}\right\\}$ | $\varphi\dim\bm{\varepsilon}^{3}(\Lambda)=0$

##  5\. Other Applications of Syzygy Filtrations

###  5.1. Dominant Dimension

We study dominant dimension here in terms of syzygy filtered algebras.
Consider algebra $A$ and its minimal injective resolution:
$0\mapsto_{A}A\mapsto I_{0}\mapsto I_{2}\ldots$ Dominant dimension of $A$ is:

$\displaystyle\operatorname{dom.\\!dim}A=\sup\left\\{n|\,I_{i}\,\text{is
projective for }i=0,1,\ldots n\right\\}+1$

We start with lemma:

######  Lemma 5.1.

Assume that a projective-injective $\Lambda$ module $I$ has
${\mathcal{B}}(\Lambda)$ filtration. Then $H(I)$ is projective-injective
$\bm{\varepsilon}(\Lambda)$ module.

###### Proof.

Assume that $H(I)$ is projective but not injective $\bm{\varepsilon}(\Lambda)$
module. Since it is not injective, it should have an injective envelope
$I^{\prime}$ in $\bm{\varepsilon}(\Lambda)$. Consider the exact sequence:

$\displaystyle 0\longrightarrow H(I)\hookrightarrow
I^{\prime}\longrightarrow\faktor{I^{\prime}}{H(I)}\longrightarrow 0$

This implies
$l_{\bm{\varepsilon}(\Lambda)}(H(I))<l_{\bm{\varepsilon}(\Lambda)}(I^{\prime})$
where $l$ denotes length of module and this is equivalent to
$l_{\Lambda}(H(I))<l_{\Lambda}(I^{\prime})$ which is not possible, since $I$
is projective injective module in $\Lambda$. ∎

The key observation is the following:

######  Proposition 5.2.

Assume that dominant dimension of cyclic Nakayama algebra $\Lambda$ is greater
or equal than $3$. Then we have:

$\displaystyle\operatorname{dom.\\!dim}\Lambda=\operatorname{dom.\\!dim}\bm{\varepsilon}(\Lambda)+2$

###### Proof.

Let dominant dimension of algebra be $n+1$, and coming from the projective
module $P$ i.e. consider the following injective resolution of projective
module $P$:

$\displaystyle P\mapsto I_{0}\mapsto I_{1}\mapsto\ldots\mapsto I_{n}\ldots$

with all $I_{i}$’s are projective for $i=0,1\ldots,n$. Since those injective
modules are projective, the projective resolution of the cosyzygy module
$\Sigma^{n+1}(P$) is given by the same sequence of maps. Therefore we get:

$\displaystyle\Omega^{k}(\Sigma^{n+1}P)\cong\Sigma^{n-k+1}(P)$

Therefore all injective modules $I_{i}$, $i=0,\ldots,n-2$ have
${\mathcal{B}}(\Lambda)$ filtration by remark 3.1. This is to say, all
$H(I_{i})$ are $\bm{\varepsilon}(\Lambda)$ projective modules. Now it is
enough to prove that they are also injective $\bm{\varepsilon}(\Lambda)$
modules. But by lemma 5.1 this follows. Hence dominant dimension of $H(P)$ in
$\bm{\varepsilon}(\Lambda)$ differs from dominant dimension in $\Lambda$ by
$2$ i.e.
$\operatorname{dom.\\!dim}\Lambda=\operatorname{dom.\\!dim}\bm{\varepsilon}(\Lambda)+2$.
∎

Now we can apply induction:

######  Theorem 5.3.

If $\Lambda$ is cyclic nonselfinjective Nakayama algebra then
$\varphi\dim\Lambda\geq\operatorname{dom.\\!dim}\Lambda$.

###### Proof.

There are two possible proofs. First, we state shorter one: the inequalities
$\operatorname{dom.\\!dim}\Lambda\leq\operatorname{fin.\\!dim}\Lambda$ and
$\operatorname{fin.\\!dim}\Lambda\leq\varphi\dim\Lambda$ implies the result.
Second approach is similar to what we did in reduction processes as in 4.5,
4.1, by using proposition 5.2 iteratively. ∎

######  Corollary 5.4.

Dominant dimension of cyclic nonselfinjective Nakayama algebra is bounded by
$2r$, where $2r=|{\mathcal{B}}(\Lambda)|$ .

###### Proof.

By above theorem and theorem 4.3,
$\operatorname{dom.\\!dim}\Lambda\leq\varphi\dim\Lambda\leq 2r$. ∎

Indeed, it was proven in [Mar18].

###  5.2. Upper Bounds For Global Dimension

We give another proof of the main theorem of [MM18] by using syzygy filtration
method.

######  Theorem 5.5.

[MM18] Let $\Lambda$ be a Nakayama algebra with a simple module $S$ of even
projective dimension. Choose $m$ minimal such that a simple $\Lambda$ module
has projective dimension equal to $2m$. Then the global dimension of $\Lambda$
is bounded by $N+m-1$ where $N$ is the number of vertices of $\Lambda$

To prove it, we need to analyze behavior of cyclic Nakayama algebras of finite
global dimension under higher syzygy filtrations described in 3.11. Before
doing this, we state and prove all auxiliary results.

######  Lemma 5.6.

Auxiliary results:

  1. 1)

Global dimension of linear Nakayama algebra which was defined in 3.8 is
bounded by $N-1$, where $N$ is the number of vertices.

  2. 2)

For any connected linear Nakayama algebra of global dimension greater or equal
than $2$, there exists a simple module $S$, such that $p\dim S=2$.

  3. 3)

Consider the following projective resolution of indecomposable uniserial
modules:

|  |
---|---|---
$\textstyle{P\Omega(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{PX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{\Omega^{2}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\Omega^{2}(X)$ is projective module. If all submodules of
$\Omega^{2}(X)$ are projective then there exists a simple module $S$ such that
$p\dim S=2$.

###### Proof.

  1. 1-)

We will prove it by induction. If $N=2$, either algebra is hereditary or
semisimple, in both cases global dimensions are smaller or equal than $1$.
Inductive step is the following: assume that global dimension of oriented
linear quiver of $N$ nodes is bounded by $N-1$. Consider the oriented linear
quiver we obtained by adding vertex $x$ to the source vertex. We have:

$\displaystyle p\dim M_{x}=p\dim\Omega(M_{x})+1$

where $M_{x}$ is any module with simple top $x$. The top of $\Omega(M_{x})$ is
different than $x$, hence it is a module over unextended quiver. Therefore its
projective dimension is bounded by $N-1$. This forces that $p\dim M_{x}\leq N$
which we want to show.

  2. 2-)

Let $1,2,\ldots,x$ be indices of simple modules and $\alpha_{i}$ denote path
from vertex $i$ to vertex $i+1$. $P_{x}$ is simple projective module. Now
$P_{x-1}$ is not simple, since algebra is connected. Hence socle of $P_{x-1}$
is $P_{x}$. There exists a projective module $P$ whose socle is $x-1$, by
connectedness again.
Now projective dimension of $soc(\operatorname{coker}(P_{x-1}\mapsto P))$ is
$2$.

  3. 3-)

If $X$ is simple module, choose $S\cong X$. Otherwise, $p\dim top(X)=2$.

∎

######  Proposition 5.7.

Let $\Lambda$ be cyclic Nakayama algebra with a simple module $S$ satisfying
$p\dim S=2$. Then $\bm{\varepsilon}(\Lambda)$ is linear Nakayama algebra.

###### Proof.

It is enough to show that ${\mathcal{B}}(\Lambda)$ contains a projective
$\Lambda$-module i.e for some $i$, $\Delta_{i}$ is projective. Without loss of
a generality, let $S\cong S_{k_{1}}$. The second syzygy of $S$ is isomorphic
to $\operatorname{coker}(P_{k_{4}+1}\mapsto P_{k_{2}+1})$. This is of minimal
length by comparison of dimensions of modules in the resolution, which makes
it an element of ${\mathcal{B}}(\Lambda)$. ∎

######  Lemma 5.8.

Let $m>1$ be minimal such that $S$ is a simple module with $p\dim S=2m$. Then
$\bm{\varepsilon}^{m-1}(\Lambda)$ is cyclic Nakayama algebra.

###### Proof.

Existence of $S$ with even projective dimension implies that global dimension
of $\Lambda$ is finite by a result of Madsen [Mad05]. By theorem 3.9 and
definition 3.11, it is enough to prove that $\bm{\varepsilon}^{m-1}(\Lambda)$
is cyclic.
Assume that it is linear. By minimality of $m$, projective covers of
$H(\Omega^{2m-2}(S)),H(\Omega^{2m-1}(S))$ and $H(\Omega^{2m}(S))$ are
projective $\bm{\varepsilon}^{m-1}(\Lambda)$-modules. Therefore global
dimension of $\bm{\varepsilon}^{m-1}(\Lambda)$ is greater or equal than $2$.
We can take the connected subalgebra containing those projectives. Moreover,
simple modules of $\bm{\varepsilon}^{m-1}(\Lambda)$ are in one to one
correspondence with ${\mathcal{B}}(\bm{\varepsilon}^{m-2}(\Lambda))$ by 3.12.
By part 2) of lemma 5.6, there exists a simple module whose projective
dimension is $2$, contadicting the minimality of $m$. Therefore
${\mathcal{B}}(\bm{\varepsilon}^{m-2}(\Lambda))$ cannot contain a projective
module which implies cyclicity of $\bm{\varepsilon}^{m-1}(\Lambda)$. Since for
any distinct projective modules of $\bm{\varepsilon}^{m-1}(\Lambda)$,
cokernels between ${\mathcal{B}}(\bm{\varepsilon}^{m-2}(\Lambda))$ filtered
projectives are nontrivial, recall corollary 3.4. ∎

######  Lemma 5.9.

Let $m>1$ be minimal such that $S$ is a simple module with $p\dim S=2m$. Then
$\bm{\varepsilon}^{m}(\Lambda)$ is linear Nakayama algebra.

###### Proof.

Simple modules of $\bm{\varepsilon}^{m}(\Lambda)$ are in one to one
correspondence with ${\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))$ by 3.12.
It is enough to show that there exists a projective
$\bm{\varepsilon}^{m}(\Lambda)$-module $H(P)$ such that $P$ is in
${\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))$. This implies there is no
other ${\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))$ filtered projective
$P^{\prime}$ such that $P^{\prime}\mapsto P$ nontrivially. Otherwise, cokernel
of that map would be in ${\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))$ which
contradicts description of ${\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))$
modules by minimal presentation 3.13.
We claim that $\Omega^{2m}(S)$ is in
${\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))$. Assume that this is not the
case. Notice that $\Omega^{2m}(S)$ has
${\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))$ filtration. Simple
$\bm{\varepsilon}^{m}(\Lambda)$ modules appearing in the filtration of
$\Omega^{2m}(S)$ cannot be projective, otherwise by part 3) of lemma 5.6, we
get a simple module of projective dimension $2$, contradicting with minimality
of $m$.
Since global dimension is finite, at some stage, any resolution has to stop.
By part 3) of lemma 5.6, $m>1$ cannot be minimal. As a result,
$H(\Omega^{2m}(S))$ has to be a simple projective module in
$\bm{\varepsilon}^{m}(\Lambda)$, this implies linearity of Nakayama algebra. ∎

Here we give proof of result 5.5:

###### Proof.

By lemmas 5.8, 5.9, $\bm{\varepsilon}^{m}(\Lambda)$ is linear Nakayama algebra
and $\bm{\varepsilon}^{j}(\Lambda)$ is cyclic Nakayama algebra for any $1\leq
j\leq m-1$. Therefore, construction of $\bm{\varepsilon}^{m}(\Lambda)$ is
obtained by definition 3.11, and by reduction described in proposition 4.5 we
get:

$\displaystyle gl\dim\Lambda=2m+gl\dim\bm{\varepsilon}^{m}(\Lambda)$

Since $\bm{\varepsilon}^{m}(\Lambda)$ linear Nakayama algebra, by part 1) in
lemma 5.6, it is bounded by number of vertices minus one. Indeed, number of
its vertices is given by $|{\mathcal{B}}(\bm{\varepsilon}^{m-1}(\Lambda))|$
which is the number of irredundant system of relations defining
$\bm{\varepsilon}^{m-1}(\Lambda)$. We denote it by $r_{m-1}$. Therefore:

$\displaystyle gl\dim\Lambda=2m+gl\dim\bm{\varepsilon}^{m}(\Lambda)$
$\displaystyle gl\dim\Lambda\leq 2m+r_{m-1}-1$

Since we applied reduction $m$ times, the maximum possible number of vertices
$r_{m-1}$ for $\bm{\varepsilon}^{m}(\Lambda)$ is $N-m$, hence we get:

$\displaystyle gl\dim\Lambda\leq 2m+r_{m-1}-1\leq 2m+N-m-1=N+m-1$

∎

######  Corollary 5.10.

Let $\Lambda$ be cyclic Nakayama algebra of finite global dimension. Then
$gl\dim\Lambda\leq 2N-2$ where $N$ is the number of vertices.

###### Proof.

We can apply construction described in 3.11 at most $r$ times. By proposition
4.5 and inequality $r\leq N-1$, $gl\dim\Lambda\leq 2(N-1)$, which is the bound
obtained by Gustafson in [Gus85]. ∎

We want to emphasis that number of relations helps to lower the bound obtained
in [MM18]. Notice that global dimension attains the bound if and only if
$r=N-1$ and $r_{i}=r_{i-1}-1$ for all $1\leq i\leq m-1$. If we take $r=N$,
$\Lambda$ becomes self-injective algebra.

######  Example 5.11.

Let $\Lambda$ be cyclic Nakayama algebra of $N=8$ vertices with relations:

$\displaystyle\alpha_{4}\alpha_{3}=0,\hskip
28.45274pt\alpha_{6}\alpha_{5}=0,\hskip
28.45274pt\alpha_{2}\alpha_{1}\alpha_{8}=0$

By simple computation, $m=1$, and global dimension is $2$. So it is smaller
than $r=3$.

###  5.3. Right Finitistic and $\varphi$-Dimensions

We define $\operatorname{fin.\\!dim}\Lambda^{op}$ as finitistic dimension of
the algebra $\Lambda$ with respect to injective resolutions i.e.

(5.1)
$\displaystyle\operatorname{fin.\\!dim}\Lambda^{op}:=\sup\left\\{\operatorname{in.\\!dim}_{\Lambda}M|\,\,\operatorname{in.\\!dim}_{\Lambda}M<\infty,\,\,M\in\text{mod-}\Lambda\right\\}$

In general left and right finitistic dimensions of an algebra can be
different. But in the case of Nakayama algebras, we prove that they are same.
To do this, we need dual constructions i.e. classes of injective modules and
injective base set as we did in 3 and 3.2 for projective modules.

Injective $\Lambda$ modules are characterized by their tops using the
relations given in 3.1:

$\displaystyle
P_{(k_{2r-1})+1}=I_{k_{2}}\twoheadrightarrow\ldots\twoheadrightarrow
I_{k_{4}+1}\quad\text{ have simple }S_{k_{2r-1}+1}\text{ as their top}$
$\displaystyle P_{k_{1}+1}=I_{k_{4}}\twoheadrightarrow\ldots\twoheadrightarrow
I_{k_{2}+1}\quad\text{ have simple }S_{k_{1}+1}\text{ as their top}$
$\displaystyle\qquad\qquad\qquad\vdots\qquad\qquad\qquad\qquad\quad\quad\quad$
$\displaystyle
P_{(k_{2r-3})+1}=I_{k_{2r}}\twoheadrightarrow\ldots\twoheadrightarrow
I_{k_{4}+1}\quad\quad\text{ have simple }S_{k_{2r-3}+1}\text{ as their top}$

######  Definition 5.12.

let $\Lambda$ be a cyclic Nakayama algebra defined by the irredundant system
of $r$ relations REL 3.1. For each $j\in\\{1,\dots,r\\}$ let $\nabla_{j}$ be a
shortest indecomposable uniserial module with $soc\nabla_{j}\cong
S_{k_{2j+1}}$ and $top\nabla_{j}\cong S_{k_{2j-1}+1}$. Let
${\bf\nabla}=\\{\nabla_{1},\nabla_{2},..,\nabla_{j},..,\nabla_{r}\\}$, i.e. a
set of shortest modules of the following form:

${\bf\nabla}:=\left\\{\nabla_{1}\cong\begin{vmatrix}S_{k_{1}+1}\\\ \vdots\\\
S_{k_{3}}\end{vmatrix}\\!,\nabla_{2}\cong\begin{vmatrix}S_{k_{3}+1}\\\
\vdots\\\
S_{k_{5}}\end{vmatrix}\\!,..,\nabla_{j}\cong\begin{vmatrix}S_{k_{2j-1}+1}\\\
\vdots\\\
S_{k_{2j+1}}\end{vmatrix}\\!,..,\nabla_{r}\cong\begin{vmatrix}S_{k_{2r-1}+1}\\\
\vdots\\\ S_{k_{1}}\end{vmatrix}\\!\right\\}.$

We will use term $\bf\nabla$-module for any module which has filtration by
$\bf\nabla$.

One can repeat all the process of the paper about projective resolutions and
${\mathcal{B}}(\Lambda)$ filtered algebras for injective resolutions and
$\nabla$ filtered algebras. We do not want to progress in those directions
however the following statements can be concluded by using duality between
$\Lambda$ projective modules and $\Lambda$ injective modules:

######  Remarks 5.13.

  1. i)

Let
$\eta(\Lambda):=End_{\Lambda}\left(\bigoplus\limits_{i\in{\mathcal{T}}(\Lambda)}I_{i}\right)$
i.e. cosyzygy filtered algebra where
${\mathcal{T}}:=\left\\{S_{k_{1}},S_{k_{3}},\ldots,S_{k_{2r-1}}\right\\}$

  2. ii)

If the second cosyzygy $\Sigma^{2}(M)$ of $M$ is not trivial, then it has
$\nabla$ filtration.

  3. iii)

Dual $\varphi$ function can be constructed as:

$\varphi_{R}(M):=min\\{t\ |\ rank\left(DL^{t}\langle
addM\rangle\right)=rank\left(DL^{t+j}\langle addM\rangle\right)\text{ for
}\forall j\geq 1\\}.$

where $DL[M]:=[\Sigma M]$ and gives map $DK_{0}\mapsto DK_{0}$, where $DK_{0}$
is abelian group generated by all symbols $[X]$ modulo relations:

     * •

$[A_{1}]=[A_{2}]+[A_{3}]$ if $A_{1}\cong A_{2}\oplus A_{3}$

     * •

$[I]=0$ if $I$ is injective.

  4. iv)

Therefore we can define $\varphi\dim\Lambda^{op}$ as:

(5.2)
$\displaystyle\varphi\dim\Lambda^{op}:=\sup\left\\{\varphi_{R}(M)|\,\,\text{for
all}\,M\in\text{mod-}\Lambda\right\\}$

  5. v)

Let $\Lambda$ be cyclic nonselfinjective Nakayama algebra.
$\varphi\dim\Lambda^{op}=2$ if and only if $\eta(\Lambda)$ is selfinjective.

  6. vi)

Another result we need is dual of the theorem 1.1: If $\Lambda$ is Nakayama
algebra then
$\varphi\dim\Lambda^{op}-\operatorname{fin.\\!dim}\Lambda^{op}\leq 1$.

We prove that

######  Proposition 5.14.

$\operatorname{fin.\\!dim}\Lambda=1$ if and only if
$\operatorname{fin.\\!dim}\Lambda^{op}=1$.

###### Proof.

We start with algebra $\Lambda$ satisfying
$\operatorname{fin.\\!dim}\Lambda=1$. This implies that any indecomposable
module $M$ of projective dimension one gives rise to short exact sequence:

$\displaystyle P\mapsto P^{\prime}\mapsto M$

We claim that socle of $M$ cannot be socle of any projective module, otherwise
we get a module of projective dimension $2$ by the following resolution:

|  |
---|---|---
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\faktor{P^{\prime\prime}}{M}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We can detect possible socles of modules of projective dimension of one: they
can be any simple indexed by $\left\\{1,2,\ldots,N\right\\}$ except
$\left\\{k_{1},k_{3},\ldots,k_{2r+1}\right\\}$ by looking the classes of
projective modules 3.2. But this forces that socles of projective modules have
to be $\left\\{k_{1},k_{3},\ldots,k_{2r-1}\right\\}$ and together with
description 3.2:

$\displaystyle\left\\{k_{1},k_{3},\ldots,k_{2r+1}\right\\}=\left\\{k_{2},k_{4},\ldots,k_{2r}\right\\}\iff$
(5.3) $\displaystyle{\mathcal{B}}(\Lambda)=\nabla$

Assume that there exists an injective resolution of indecomposable $\Lambda$
module $M$:

---
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{2}(M)\cong
I_{2}}$

We use the following arguments: Injective module $I_{2}$ has $\nabla$ (and
${\mathcal{B}}(\Lambda)$ by 5.3) filtration. Since
$\operatorname{fin.\\!dim}\Lambda=1$, in particular $\varphi\dim\Lambda=2$,
all projective modules in $Filt({\mathcal{B}}(\Lambda))$ have the same length
i.e. $\bm{\varepsilon}(\Lambda)$ is self injective. This forces $I_{2}$ to be
a projective-injective $\Lambda$ module. However this creates contradiction,
because a projective injective module cannot be a syzygy of another module.
There is no $\Lambda$ module whose injective dimension is finite and greater
than one. Therefore $\operatorname{fin.\\!dim}\Lambda^{op}=1$. By repeating
the similar arguments and using duality between projective and injective
modules only if part can be shown. ∎

######  Theorem 5.15.

For cyclic Nakayama algebras left and right finitistic dimensions are same
i.e. $\operatorname{fin.\\!dim}\Lambda=\operatorname{fin.\\!dim}\Lambda^{op}$.

###### Proof.

if global dimension is finite, it is a well known result. Therefore we
consider the case of infinite global dimension.
We will prove it by induction on the dimension. Previous proposition 5.14
verifies that if $n=1$ and $n=2$, $\operatorname{fin.\\!dim}\Lambda=1$ if and
only if $\operatorname{fin.\\!dim}\Lambda^{op}=1$ and
$\operatorname{fin.\\!dim}\Lambda=2$ if and only if
$\operatorname{fin.\\!dim}\Lambda^{op}=2$ (combined with 4.4 ). Assume that
$\operatorname{fin.\\!dim}\Lambda=\operatorname{fin.\\!dim}\Lambda^{op}=d$ for
all $d\leq n$. Now, let $\operatorname{fin.\\!dim}\Lambda=n+1$. We get:

$\displaystyle\operatorname{fin.\\!dim}\Lambda=n+1\iff\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)=n-1,\text{by
prop\,\ref{reductionfindim}}$
$\displaystyle\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)=n-1\iff\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)^{op}=n\\!-\\!1\,\,\text{by
induction hypothesis}$
$\displaystyle\operatorname{fin.\\!dim}\bm{\varepsilon}(\Lambda)^{op}=n-1\iff\operatorname{fin.\\!dim}\Lambda^{op}=n+1\,\,\text{by
dual prop\,\ref{reductionfindim}}$

∎

######  Theorem 5.16.

For cyclic Nakayama algebras left and right $\varphi$-dimensions are same i.e.
$\varphi\dim\Lambda=\varphi\dim\Lambda^{op}$.

###### Proof.

It is enough to consider global dimension is infinite.
By theorem 1.1, there are two possibilities:

$\displaystyle\varphi\dim\Lambda=\operatorname{fin.\\!dim}\Lambda$
$\displaystyle\varphi\dim\Lambda=1+\operatorname{fin.\\!dim}\Lambda$

Theorem above 5.15,and equalities of left and right finitistic dimensions
imply:

$\displaystyle\varphi\dim\Lambda=\varphi\dim\Lambda^{op}$

∎

In [BMR], authors show equalities of left and right $\varphi$-dimension for
truncated path algebras.

We can put together the results 1.1 , 4.12, 5.16, 5.15 and 5.4 into one
theorem:

######  Theorem 5.17.

Let $\Lambda$ be cyclic nonselfinjective Nakayama algebra defined by $r$ many
irredundant system of relations over $N$ vertices. Then, $\varphi\dim\Lambda$,
$\operatorname{fin.\\!dim}\Lambda$, $\operatorname{gor.\\!dim}\Lambda$,
$\operatorname{dom.\\!dim}\Lambda$ are bounded by $2r$ where
$r=|{\mathcal{B}}(\Lambda)|$. Moreover, the upper bound is $2N-2$ since $r\leq
N-1$.

######  Example 5.18.

We reconsider this example given in [Sen18]. Let $\Lambda$ be cyclic Nakayama
algebra on $N$ vertices with Kupisch series $(2N+1,2N+1,\ldots,2N+1,2N)$. This
has $r=N-1$ and $\varphi\dim(\Lambda)=2r=2N-2$. Indeed, one can check that, it
is Gorenstein and
$\operatorname{fin.\\!dim}\Lambda=\varphi\dim\Lambda=\operatorname{gor.\\!dim}\Lambda=\operatorname{dom.\\!dim}\Lambda=\operatorname{fin.\\!dim}\Lambda^{op}=\varphi\dim\Lambda^{op}=2r=2N-2$.

## References

  * [ARS97] Maurice Auslander, Idun Reiten, and Sverre O Smalo.  Representation theory of Artin algebras, volume 36.  Cambridge university press, 1997.
  * [BMR] Marcos Barrios, Gustavo Mata, and Gustavo Rama.  Igusa-todorov $\phi$ function for truncated path algebras.  Algebras and Representation Theory, pages 1–13.
  * [CY14] Xiao-Wu Chen and Yu Ye.  Retractions and gorenstein homological properties.  Algebras and Representation Theory, 17(3):713–733, Jun 2014.
  * [ES17] Ana Garcia Elsener and Ralf Schiffler.  On syzygies over 2-calabi–yau tilted algebras.  Journal of Algebra, 470:91–121, 2017.
  * [Gus85] William H Gustafson.  Global dimension in serial rings.  Journal of Algebra, 97(1):14–16, 1985.
  * [IT05] Kiyoshi Igusa and Gordana Todorov.  On the finitistic global dimension conjecture for artin algebras.  Representations of algebras and related topics, 45:201–204, 2005\.
  * [LM18] Marcelo Lanzilotta and Gustavo Mata.  Igusa–todorov functions for artin algebras.  Journal of Pure and Applied Algebra, 222(1):202–212, 2018.
  * [Mad05] Dag Madsen.  Projective dimensions and nakayama algebras.  Representations of algebras and related topics, pages 247–265, 2005\.
  * [Mar18] René Marczinzik.  Upper bounds for the dominant dimension of nakayama and related algebras.  Journal of Algebra, 496:216–241, 2018.
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  * [Sen18] Emre Sen.  The $\varphi\dim$ of cyclic nakayama algebras.  arXiv preprint arXiv:1806.01449, 2018.


# Primordial Black Hole Clusters and their Evolution

M.Yu. Khlopov
Centre for Cosmoparticle Physics ”Cosmion”
National Research Nuclear University
”Moscow Engineering Physics Institute”, 115409 Moscow, Russia
APC laboratory 10, rue Alice Domon et Léonie Duquet
75205 Paris Cedex 13, France
<EMAIL_ADDRESS>
N.A. Chasnikov
National Research Nuclear University
”Moscow Engineering Physics Institute”, 115409 Moscow, Russia
<EMAIL_ADDRESS>
###### Abstract

A possibility of pregalactic seeds of the Active Galactic Nuclei can be a
nontrivial cosmological consequence of particle theory. Such seeds can appear
as Primordial Black Hole (PBH) clusters, formed in the succession of phase
transitions with spontaneous and then manifest breaking of the global U(1)
symmetry. If the first phase transition takes place at the inflationary stage,
a set of massive closed walls may be formed at the second phase transition and
the collapse of these closed walls can result in formation of PBH clusters. We
present the results of our studies of the evolution of such PBH Clusters.

##  1 Introduction

Primordial Black Holes (PBHs) are a very sensitive cosmological probe for
physics phenomena occurring in the early Universe. They could be formed by
many different mechanisms, reflecting the fundamental structure of particle
theory and nonhomogeneity of very early Universe. Here after a brief review of
mechanisms of PBH formation we consider a nontrivial possibility of clusters
of massive PBHs. The evolution of such clusters can provide pregalactic seeds
of Active Galactic Nuclei (AGN) and we discuss various aspects of this
evolution in the present paper.

##  2 PBH Formation

Primordial Black Holes can be formed in many different ways [1, 2], such as
initial density inhomogeneities, first order and non-equilibrium second order
phase transitions, etc. Let us give a brief review of these possibilities.

###  2.1 PHB formation in initial density inhomogeneities

A probability for fluctuation of 1 for metric fluctuations distributed
according to Gaussian law with dispersion
$\left\langle\delta^{2}\right\rangle\ll 1$ is determined by exponentially
small tail of high amplitude part of this distribution. In non-Gaussian
fluctuations, this process can be more suppressed [3]. In the space, described
by

$p=\gamma\epsilon,0\leq\gamma\leq 1$ (1)

equation of state a probability to form black hole from fluctuation within
cosmological horizon is given by [4, 5]

$W_{PBH}=e^{-\frac{\gamma^{2}}{2\left\langle\delta^{2}\right\rangle}}$ (2)

It provides exponential sensitivity of PBH spectrum to softening of equation
of state in early Universe ($\gamma\rightarrow 0$) or to increase of
ultraviolet part of spectrum of density fluctuations ($\delta^{2}\rightarrow
1$). These phenomena can appear as cosmological consequence of particle theory
(see [4, 5] for review of this and some other mechanisms of PBH formation and
for references).

###  2.2 PBH from non-equilibrium second order phase transition

The mechanism of PBH formation in the non-equilibrium second order phase
transition is of special interest, since it can provide formation of massive
and even Supermassive PBHs. PBHs are produced in this mechanism by self-
collapsing of closed domain walls. If there are two vacuum states of a system,
there are two possibilities to populate that states in the early Universe:
under the usual circumstances of thermal phase transition the Universe
contains both states populated with equal probability. The other possibility
is beyond the pure thermodynamical equilibrium condition, when the two vacuum
states are populated with islands of the less probable vacuum, surrounded by
the sea of another, more preferable, vacuum.

It is necessary to redefine effectively the correlation length of the scalar
field that drives a phase transition and consequently the formation of
topological defects and the only necessary ingredient for that is the
existence of an effectively flat direction(s), along which the scalar
potential vanishes during inflation.

The background deSitter fluctuations of such effectively massless scalar field
could provide non-equilibrium redefinition of correlation length and give rise
to the islands of one vacuum in the sea of another one. In spite of such
redefinition the phase transition itself takes place deeply in the Friedman-
Robertson-Walker (FRW) epoch. After the phase transition two vacua are
separated by a wall, and such a closed wall, separating the island with the
less probable vacuum, can be very large.

At some moment after crossing horizon the walls start shrinking due to surface
tension. As a result, if the wall does not release the significant fraction of
its energy in the form of outward scalar waves, almost the whole energy of
such closed wall can be concentrated in a small volume within its
gravitational radius what is the necessary condition for PBH formation.

The mass spectrum of the PBHs which can be created by such a way depends on
the scalar field potential which parameterizes the flat direction during
inflation and triggers the phase transition at the FRW stage.

We consider the Universe that, due to the existence of an inflaton, goes
through a period of inflation and then settles down to the standard FRW
geometry. Then we introduce a complex scalar field $\varphi$, not the
inflaton, with a large radial mass $\sqrt{\lambda f}>H_{i}$ that has got
Mexican hat potential

$V(\varphi)=\lambda\left(\left|\varphi\right|^{2}-\frac{f^{2}}{2}\right),$ (3)

which provides the U(1) symmetry spontaneous breaking in the period of
inflation, corresponding to the scales of the modern cosmological horizon.
Therefore we deal only with the phase of that complex field
$\theta=\frac{\varphi}{f}$, which parameterizes potential

$V=\Lambda^{4}\left(1-cos\frac{\phi}{f}\right)$ (4)

Under this condition we come to the conclusion, that the correlation length of
second order phase transition with spontaneously broken U(1) symmetry exceeds
the present cosmological horizon, and all global U(1) strings are beyond our
horizon. If we assume $m\ll H_{i}$ then this implies that during inflation the
potential energy of field $\varphi$ is much smaller than the cosmological
friction term what justifies neglecting the potential until the Universe goes
deeply into the FRW phase. During inflation and long afterward, $H_{i}$ is
very large (by assumption) compared to the potential (2). It follows that we
can drop the gradient term in the equation of motion [6]

$\ddot{\theta}+3H\dot{\theta}+\frac{dV}{d\theta}=0$ (5)

and resulting equation is solved by $\theta_{0}=\theta_{Nmax}$ , where
$\theta_{Nmax}$ is an arbitrary constant. In the standard assumption, our
present horizon has been nucleated at the $N_{max}$ e-folds before the end of
inflationary epoch, being embedded in an enormous inflation horizon, created
by exponential blow up of a single casual horizon. It follows that
$\theta_{Nmax}$ will be the same over the inter inflationary horizon. Without
loss of generality, we put $\theta_{Nmax}<\pi$ and considering the quantum
fluctuations of the phase $\theta$ at the deSitter background. There are
quantum fluctuations produced on the vacuum state of $\theta$ due to the
boundary conditions of deSitter space. These fluctuations are sometimes
referred to as contribution to the “Hawking temperature” of deSitter space
but, there are no true thermal effects. It makes the dynamics of phase
$\theta$ strongly non-equilibrium leading to the non-thermal distribution of
scales populated with different vacuums in the postinflationary Universe. The
average amplitude of such fluctuations for massless field generated during
each time interval $H_{{}_{i}}^{-1}$ is $\delta\theta=\frac{H_{i}}{2\pi f}$.
The total number of steps during time interval $\Delta t$ is given by
$N=H_{i}\Delta t$ \- looks like the one-dimensional Brownian motion. Each
domain is characterized by average phase value
$\theta_{Nmax-1}=\theta_{Nmax}\pm\delta\theta$.

In the half of these domains the phases evolve toward $\pi$ while in the other
domains they move toward zero. This process is duplicated in each volume of
size $H^{-1}$ during next e-fold. Now at any given scale $l=k^{-1}$ the size
of distribution of the phase value $\theta$ can be described by Gaussian law
[7]

$P(\theta_{l},l)=\frac{1}{\sqrt{2\pi}\sigma_{l}}exp\left(-\frac{(\theta_{Nmax}-\theta_{l})^{2}}{2\sigma_{l}^{2}}\right)$
(6)

It is recommended for more information to address the papers [1, 6, 8, 9].

###  2.3 Initial PBH Mass spectrum

Initial mass spectrum $n\left(m,t=0\right)$ depends of parameter $f$ and
$\Lambda$. [6] In addition, is a numerical solution. There is another way: one
can describe this system by Ito’s equation. One-dimensional Brownian motion in
the terms of stochastic equations is an Ornstein-Uhlenbeck process. [10, 11]
Using this mathematical framework, one can find the analytical solution of
Ito’s equation and obtain initial mass spectrum as an analytical formula
$n_{0}=n_{f,\Lambda}\left(m,t=0\right)$. This method is in the process of
development.

##  3 Clusters of PHBs

According to the $2^{nd}$ order phase transitions mechanism, PBH appears as a
sufficiently large cluster, which could collapse into one large Super Massive
Black Hole (SMBH) - the Active Galactic Nucleus (AGN) of the future galaxy.

###  3.1 PBH Cluster dynamics

By analogy with the star cluster [12] with the difference that the black holes
can merge into one, the following processes are significant:

  * •

BH collisions $\rightarrow BH$ merging and as a result $N_{BH}\rightarrow 1$

  * •

Flying-out BH from cluster $\rightarrow$ reducing the mass of the cluster [13,
14]

  * •

Dynamical friction $\rightarrow$ lower Maxwell distribution [15]

Dynamical friction mostly contributes into the
$\left\langle\sigma\nu\right\rangle$ of the collision process. [15] The
equation describing the dynamics of the BH Cluster is a modification of
Smoluchowski (or Kolmogorov-Feller) equation and runs as follows

$\begin{split}n{}^{\prime}=\int_{0}^{M}n\left(m-\mu,t\right)\left\langle\sigma\nu\right\rangle_{m-\mu,m}d\mu\\\
-n\left(m,t\right)\left(\int_{0}^{\infty}n\left(\mu,t\right)\left\langle\sigma\nu\right\rangle_{\mu,m}d\mu+\int_{0}^{\infty}n\left(\mu,t\right)\Lambda\left(m,\mu\right)\mu^{2}d\mu\right),\end{split}$
(7)

where M is total initial cluster mass, $\Lambda\left(m,\mu\right)$ one can
find in [14]. Let us consider numerical solution of that equation for
arbitrary initial parameters:
Numerical solution of that equation shows, that there is the “trend” to the
decrease of BHs with larger masses. However, the numerical solution is
indispensable because of the unknown initial parameters $f$ and $\Lambda$ of
the model. Exact analytical solution of that equation is overly precise and is
a very nontrivial exercise. If a single SMBH is supposed to be the result, one
needs to get a solution of that equation as

$n(m,t)=\delta(m-m_{SMBH})\chi(t-t_{gen})$ (8)

Substituting this partial solution into the equation of the PBH cluster
dynamics, one can obtain the timescale of the process and the mass of the
resulting SMBH as a functionals of initial conditions:

$m_{SMBH}=F\left[n\left(m,t=0\right)\right]$ (9)
$t_{gen}=G\left[n(m,t=0)\right]$ (10)

The calculated values of $t_{gen}$ and $m_{SMBH}$ can be confronted with the
observational data, putting constraints on the fundamental physical scales $f$
and $\Lambda$.

## Acknowledgements

We are grateful to the Organizers of XVI Bled Workshop for hospitality and
fruitful and creative atmosphere of discussions.

## References

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