# defect invariant nakayama algebras

Emre SEN  ,  Gordana Todorov  and  Shijie Zhu

###### Abstract.

We show that for a given Nakayama algebra $\Theta$, there exist countably many
cyclic Nakayama algebras $\Lambda_{i}$, where $i\in\mathbb{N}$, such that the
syzygy filtered algebra of $\Lambda_{i}$ is isomorphic to $\Theta$ and we
describe those algebras $\Lambda_{i}$. We show, among these algebras, there
exists a unique algebra $\Lambda$ where the defects, representing the number
of indecomposable injective but not projective modules, remain invariant for
both $\Theta$ and $\Lambda$. As an application, we achieve the classification
of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant
Auslander-regular algebras of global dimension three. Specifically, by using
the Auslander-Iyama correspondence, we obtain cluster-tilting objects for
certain Nakayama algebras. Additionally, we introduce cosyzygy filteed
algebras and show that it is dual of syzygy filtered algebra.

###### Contents

  1. 0 Introduction
    1. 0.1 Organization of the paper
  2. 1 Preliminaries on Nakayama algebras
    1. 1.1 Nakayama algebras
    2. 1.2 The defect, socle-set, base-set, and filtered projectives
  3. 2 The $\varepsilon$-construction of Syzygy Filtered Algebras
    1. 2.1 Definition and Functorial properties
    2. 2.2 Relationship between defects of $\Lambda$ and its syzygy filtered algebra $\Theta$
    3. 2.3 Defect invariant Nakayama algebras
  4. 3 Reverse of $\varepsilon$-construction
    1. 3.1 Defect invariant case
    2. 3.2 General case, i.e. Not Defect-Invariant
  5. 4 Nakayama Algebras with higher homological dimensions
    1. 4.1 Properties of Injective Modules and Homological Dimensions
    2. 4.2 Proofs of Theorems B,C
    3. 4.3 Dominant Auslander-Gorenstein Algebras among Nakayama Algebras
    4. 4.4 Minimal Auslander-Gorenstein Algebras among Nakayama Algebras
    5. 4.5 Dominant Auslander-regular algebras among Nakayama Algebras
    6. 4.6 Higher Auslander Algebras among Nakayama Algebras
  6. 5 The duality of syzygy filtered algebras
    1. 5.1 Construction
    2. 5.2 The syzygy filtered algebra
    3. 5.3 The cosyzygy filtered algebra
    4. 5.4 Connecting Syzygy and Cosyzygy Filtered Algebras
    5. 5.5 A further question

††footnotetext: Keywords: Nakayama algebras, higher Auslander algebras,
minimal Auslander-Gorenstein algebras, dominant Auslander-Gorenstein algebras,
dominant Auslander-regular algebras, dominant dimension, global dimension,
Gorenstein dimension, wide subcategories.
MSC 2020: 16E05, 16E10,16G20

##  0\. Introduction

M. Auslander’s seminal result, known as the Auslander correspondence, brought
new perspectives to the representation theory of algebras: there exists a
bijection between the set of Morita-equivalence classes of representation-
finite Artin algebras $A$ and that of Artin algebras $\Gamma$ with
$\operatorname{gl\,\\!dim}\Gamma\leq 2\leq\operatorname{dom.\\!dim}\Gamma$
where $\operatorname{gl\,\\!dim}\Gamma$, $\operatorname{dom.\\!dim}\Gamma$
denote global and dominant dimensions of $\Gamma$ respectively. The bijection
is given by $A\rightarrow\Gamma:=\operatorname{End}_{A}(M)$ for an additive
generator $M$ of $\operatorname{mod-\\!}A$. Hence, representation theory of
$A$ can be carried into homological theory of $\Gamma$.

O. Iyama introduced the _higher Auslander correspondence_ in [13] such that an
Artin algebra $\Gamma$ satisfying the conditions

(0.1) $\displaystyle\operatorname{gl\,\\!dim}\Gamma\leq
d+1\leq\operatorname{dom.\\!dim}\Gamma$

for $d\geq 1$ can be realized bijectively as
$\Gamma:=\operatorname{End}_{A}({\mathcal{M}})$, where ${\mathcal{M}}$ is a
$d$-cluster-tilting object in the category of finitely generated $A$-modules
for an algebra $A$. Since then, it has become an active research field in the
representation theory of Artin algebras. One problem in this field is the
classification of $d$-cluster-tilting modules for a given class of algebras,
or the construction and characterization of higher Auslander algebras, i.e.,
algebras satisfying (0.1), within a given class of algebras.

We will focus the attention to Nakayama algebras: that is all indecomposable
modules are uniserial. Let $\Lambda$ be a cyclic Nakayama algebra and
${\mathcal{S}}(\Lambda)$ denote the complete set of representatives of
isomorphism classes of socles of indecomposable projective $\Lambda$-modules.
The syzygy filtered algebra $\bm{\varepsilon}(\Lambda)$, introduced in [28],
is the endomorphism algebra of projective covers of Auslander-Reiten
translates of elements of ${\mathcal{S}}(\Lambda)$, i.e.,

(0.2)
$\displaystyle\bm{\varepsilon}(\Lambda):=\operatorname{End}_{\Lambda}{\mathcal{P}}\quad\text{where}\quad{\mathcal{P}}=\bigoplus\limits_{S\in{\mathcal{S}}(\Lambda)}P(\tau
S).$

As highlighted by CM. Ringel in [25][Appendices B,C] and in [26], this
provides a strong tool to study the homological dimensions of $\Lambda$. Since
the syzygy filtered algebra is again a Nakayama algebra, it enables us to
relate the homological dimensions of the original algebra and syzygy filtered
algebras inductively. Moreover, the syzygy filtered algebra played a crucial
role in constructing cyclic Nakayama algebras that are higher Auslander
algebras in [29]. In this work, we establish the reverse of 0.2 by using
_defect_ of an algebra $\Lambda$, denoted by $\operatorname{def}\Lambda$,
which represents the count of indecomposable non-isomorphic injective but non-
projective modules [15].

We state our first result.

######  Theorem A.

Let $\Theta$ be a cyclic or linear (not necessarily connected) Nakayama
algebra of rank $n$, $\left\\{P_{1},\ldots,P_{n}\right\\}$ be the complete
ordered set of representatives of indecomposable projective $\Theta$-modules.
Let $w=(w_{1},\ldots,w_{n})$ be any integer sequence such that
$w_{i}\geq\operatorname{def}P_{i}$, where $\operatorname{def}P_{i}$ denotes
the number of proper injective quotients of $P_{i}$. Then,

  1. 1)

if $\sum w_{i}>\operatorname{def}\Theta$, then there are countably many cyclic
Nakayama algebras $\Lambda_{j}$ $j\geq 1$ such that
$\bm{\varepsilon}(\Lambda_{j})\cong\Theta$ and
$\operatorname{def}\Lambda_{j}=\sum w_{i}$,

  2. 2)

if $\sum w_{i}=\operatorname{def}\Theta$, then $\Lambda$ is unique.

Using Theorem A, we prove the following theorem about dominant dimensions of
Nakayama algebras.

######  Theorem B.

Let $\Lambda$, $\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$. Suppose that
$\operatorname{def}\Theta\neq 0$, then,

  1. 1)

if $\operatorname{def}\Lambda>\operatorname{def}\Theta$, then
$\operatorname{dom.\\!dim}\Lambda\leq 2$.

  2. 2)

if $\operatorname{def}\Lambda=\operatorname{def}\Theta$, then
$\operatorname{dom.\\!dim}\Lambda=\operatorname{dom.\\!dim}\Theta+2$.

|  |  |  |  |
---|---|---|---|---|---
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon^{-1}}$$\textstyle{\Lambda^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon^{-1}}$$\textstyle{\Lambda^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\varepsilon^{-1}}$$\scriptstyle{\varepsilon^{-1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon^{-1}}$Rank
of
Algebra:$\textstyle{\operatorname{rank}\Lambda}$$\textstyle{\operatorname{rank}\Lambda-\operatorname{def}\Lambda}$$\textstyle{\operatorname{rank}\Lambda-2\operatorname{def}\Lambda}$Dominant
Dimension:$\textstyle{k}$$\textstyle{k-2}$$\textstyle{k-4}$

Figure 1. Defect Invariant Nakayama Algebras

A direct consequence of Theorem A 2 and Theorem B 2 is: for a given Nakayama
algebra $\Theta$ with nonzero defect, there exists unique cyclic Nakayama
algebra $\Lambda$ such that
$\operatorname{dom.\\!dim}\Lambda=\operatorname{dom.\\!dim}\Theta+2$ (Figure
1). Hence, this enables us to construct Nakayama algebras characterized by
their dominant dimension. For instance, a finite-dimensional algebra is called
minimal Auslander-Gorenstein if its Gorenstein dimension is bounded by its
dominant dimension [14]. In the same vein, dominant Auslander-Gorenstein
algebras were introduced in [8], where the algebra is Gorenstein and the
injective dimension of each indecomposable projective module is bounded by its
dominant dimension. If it has finite global dimension, it is called dominant
Auslander-regular algebra. We illustrate them in the following chart 2.

|
---|---
Higher Auslander AlgebrasMinimal Auslander-Gorenstein
Algebras$\textstyle{\operatorname{gl\,\\!dim}A\leq\operatorname{dom.\\!dim}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{in.\\!dim}A\leq\operatorname{dom.\\!dim}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Dominant
Auslander-Regular AlgebrasDominant Auslander-Gorenstein
Algebras$\textstyle{\operatorname{in.\\!dim}P_{i}\leq\operatorname{dom.\\!dim}P_{i},\,\operatorname{in.\\!dim}P_{i}<\infty}$$\textstyle{\operatorname{in.\\!dim}P_{i}\leq\operatorname{dom.\\!dim}P_{i}}$$\textstyle{\forall
P_{i}\,\,\text{indecomposable projective}}$$\textstyle{\forall
P_{i}\,\,\text{indecomposable projective}}$ Figure 2. Nakayama algebras
characterized by dominant dimension

We prove the following:

######  Theorem C.

Let $\Lambda,\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$ and
$\operatorname{def}\Lambda=\operatorname{def}\Theta$. If $\Theta$ is a
dominant Auslander-Gorenstein algebra, $\Lambda$ is as well.

To provide a complete classification of Nakayama algebras with infinite global
dimension that are minimal Auslander-Gorenstein, it suffices to construct
$2$-Auslander-Gorenstein algebras as described in Section 4.4. Additionally,
we provide a complete classification of cyclic Nakayama algebras with global
dimensions $3$ and $4$ which are higher Auslander. Through this
classification, we determine which cyclic Nakayama algebras have $2$ and $3$
cluster tilting modules.

However, even for Nakayama algebras to classify all dominant Auslander-
Gorenstein algebras that are not minimal Auslander-Gorenstein, it turns out
that Theorem B 2 alone is not sufficient. Therefore, we focus on certain
classes of dominant Auslander-Gorenstein and dominant Auslander-regular
algebras. Based on Theorem B, we classify all cyclic dominant Auslander-
regular algebras of global dimension three in Section 4.5. On the other hand,
we give a construction of dominant Auslander-regular algebras of dominant
dimension one and two in Section 4.5. Additionally, by Theorem B 1, we show
that the dominant dimension of dominant Auslander-Gorenstein algebras of
infinite global dimension is even.

###  0.1. Organization of the paper

In Section 1, we recall the basic properties of Nakayama algebras. In Section
2, we provide details of the syzygy filtered algebra construction also
referred to as $\bm{\varepsilon}$-construction and study its functorial and
module theoretical properties. Then, in Section 3, we consider the reverse of
the $\bm{\varepsilon}$-construction and prove Theorem A. In Section 4, we
study the homological properties of algebras, prove Theorems B, C, and provide
a complete classification of certain classes of algebras. In Section 5, we
consider cosyzygy filtered algebras and demonstrate that they are dual to
syzygy filtered algebras.

##  1\. Preliminaries on Nakayama algebras

We recall basic properties of Nakayama algebras and the
$\bm{\varepsilon}$-construction. This is a process of constructing so called
“syzygy filtered algebra” which is closely related to the original algebra,
with respect to homological properties.

###  1.1. Nakayama algebras

Nakayama algebras are finite dimensional algebras for which all indecomposable
projective and injective modules are uniserial. The category of right finitely
generated modules over a Nakayama algebra $\Lambda$ will be denoted by
$\operatorname{mod-\\!}\Lambda$. Throughout the paper $\tau$ will denote
Auslander-Reiten translate.

######  Remark 1.1.1.

Let $\Lambda$ be a Nakayama algebra. Then $\Lambda\cong\mathbb{K}Q/I$, where
$Q$ is a quiver, cyclic or linear, with $n$ vertices, $\mathbb{K}Q$ is the
path algebra and $I$ is an admissible ideal of relations. We label the
vertices of quiver $Q$ by indices $1,2\cdots,n$, such that the simple modules
satisfy $S_{i+1}\cong\tau S_{i}$ for $1\leq i\leq n-1$, and $\tau
S_{n}=\begin{cases}0&\text{if $\Lambda$ is linear}\\\ S_{1}&\text{ if
$\Lambda$ is cyclic.}\end{cases}$

In fact, all the indecomposable modules over a Nakayama algebra are uniserial.
Hence the projective cover and injective envelope of any indecomposable module
is indecomposable. We will heavily use the following lemma about
indecomposable modules over Nakayama algebras. Of particular importance for
homological dimensions of Nakayama algebras are minimal projective and minimal
injective modules as defined below. All of these results are either already
proved in [27] and [28], or are not hard to check.

######  Definition 1.1.2.

An indecomposable projective module $P$ is called _minimal projective_ if
$\operatorname{rad}P$ is not projective. An indecomposable injective module
$I$ is called _minimal injective_ if $I/\operatorname{soc}I$ is not injective.

######  Lemma 1.1.3.

[Uniseriality] Let $\Lambda$ be a Nakayama algebra and let $M$ and $N$ be
indecomposable $\Lambda$-modules.

  1. 1)

If there exists $L\neq 0$ that is a submodule of both $M$ and $N$, then either
$M$ is a submodule of $N$ or $N$ is a submodule of $M$.

  2. 2)

If there exists $L\neq 0$ that is a quotient module of both $M$ and $N$, then
either $N$ is a quotient module of $M$ or $M$ is a quotient module of $N$.

######  Proposition 1.1.4.

Let $\Lambda$ be a Nakayama algebra.

  1. 1)

Let $0\to L\to M\to N\to 0$ be a non-split exact sequence of indecomposable
modules. Then $\operatorname{top}L\cong\tau\operatorname{soc}N$.

  2. 2)

Let $S$ be a simple module. The indecomposable projective module $P(S)$ is
non-injective if and only if $P(S)\cong\operatorname{rad}P(\tau^{-}S)$.

  3. 3)

Let $S$ be a simple module. The indecomposable injective module $I(S)$ is non-
projective if and only if $I(S)\cong I(\tau S)/\tau S$.

  4. 4)

Let $S$ be a simple $\Lambda$-module. Then $P(\tau S)$ is a projective-
injective $\Lambda$-module if and only if $P(S)$ is a minimal projective
module.

  5. 5)

Let $S$ be a simple $\Lambda$-module. Then $I(S)$ is a projective-injective
$\Lambda$-module if and only if $I(\tau S)$ is a minimal injective module.

######  Proposition 1.1.5.

Let $\Lambda$ be a cyclic Nakayama algebra. Then:

  1. 1)

$M\cong\operatorname{rad}P$ for some minimal projective module $P$ if and only
if $M\cong I/\operatorname{soc}I$ for some minimal injective module $I$.

  2. 2)

Let $S$ be a simple $\Lambda$-module. If $P(S)$ is projective-injective, and
$E$ is minimal injective quotient of $P(S)$ then
$E/\operatorname{soc}E\cong\operatorname{rad}(P(\tau^{-1}S))$.

####  1.1.1. Kupish Series $\&$ Gustafson Function

We recall Kupisch series and Gustafson function. For integers $x$ outside the
range $\\{1,\dots,n\\}$ we will use symbol $[x]$ to denote the integer $x$
(mod $n$) within the range $\\{1,\dots,n\\}$.

######  Definition 1.1.6.

Let $\Lambda\cong\mathbb{K}Q/I$ be Nakayama algebra where $Q$ is either cyclic
or connected linear quiver with vertices: $\\{1,\dots,n\\}$. The Gustafson
function is defined as: $g:\\{1,\dots,n\\}\to\\{1,\dots,n\\}$,
$g(i):=[i+\ell(P_{i})-1]$ where $\ell(P_{i})$ is the length of the projective
cover of the simple $S_{i}$.

######  Lemma 1.1.7.

Let $\Lambda$ be cyclic Nakayama algebra. Then:
(a) Gustafson map defines map on simple $\Lambda$-modules as
$\overline{g}(S)=\operatorname{soc}(P(S))$.
(b) Let $P$ and $P^{\prime}$ be indecomposable projective-injective
$\Lambda$-modules. Then:
$(P\cong
P^{\prime})\iff(\operatorname{top}P\cong\operatorname{top}P^{\prime}))\iff(\operatorname{soc}P\cong\operatorname{soc}P^{\prime})$.

######  Definition 1.1.8.

Let $\Lambda=\mathbb{K}Q/I$ be a connected Nakayama algebra where $Q$ is
either cyclic or linear quiver with $n$ vertices, labeled as in Definition
1.1.6. Let $\\{P_{1},\ldots,P_{n}\\}$ be the representatives of indecomposable
projectives. The associated Kupisch series for $\Lambda$ is
$(c_{1},\ldots,c_{n})$ where $c_{i}=\ell(P_{i})$.

######  Lemma 1.1.9.

Let $(c_{1},\ldots,c_{n})$ be sequence of positive integers. Then, this is an
admissible sequence for a Kupisch series of a Nakayama algebra, if there is a
cyclic permutations of these integers such that

  1. 1)

for each $1\leq i\leq n-1$, either $c_{i}=c_{i+1}$ or $c_{i}\leq c_{i+1}+1$.

  2. 2)

If $c_{n}\neq 1$, then either $c_{1}=c_{n}+1$ or $c_{n}\leq c_{1}$. If
$c_{n}=1$, then algebra is linear and there is no additional condition.

###  1.2. The defect, socle-set, base-set, and filtered projectives

We now recall the properties of some of the notions which are fundamental for
the rest of the paper.

######  Definition 1.2.1.

Let $P$ be an indecomposable projective module. Then _defect of $P$_, denoted
by $\operatorname{def}(P)$ is the number of injective proper quotients of $P$.
_Defect of $\Lambda$_, denoted by $\operatorname{def}(\Lambda)$ is the sum
$\Sigma\operatorname{def}(P)$ over a complete set of representatives of
isomorphism classes of indecomposable projective $\Lambda$-modules.

######  Definition 1.2.2.

Let $\Lambda$ be a cyclic Nakayama algebra. Define:
${\mathcal{S}}(\Lambda)$ is a complete set of representatives of isomorphism
classes of socles of indecomposable projective $\Lambda$-modules and call it
the _socle-set_.
${\mathcal{S}}^{\prime}(\Lambda):=\\{\tau S\ |\ S\in\mathcal{S}(\Lambda)\\}$
which is called the _top-set_ and the _base-set_ is defined as:
$\mathcal{B}(\Lambda):=\\{B\ |\
\operatorname{soc}B\in\mathcal{S}(\Lambda),\operatorname{top}B\in\mathcal{S}^{\prime}(\Lambda),B\text{
is of minimal length}\\}$.
$\mathcal{P}(\Lambda):=\\{P(S^{\prime})\ |\ \text{projective cover of
}S^{\prime}\in{\mathcal{S}}^{\prime}(\Lambda)\\}$=$\\{\text{filtered
projectives}\\}$ (Lemma 1.2.12).
${\mathcal{S}}_{f}(\Lambda):=\\{S\in\mathcal{S}(\Lambda)\ |\
S\cong\operatorname{soc}(P)\text{ for some filtered projective
}P\in{\mathcal{P}}(\Lambda)\\}$
${\mathcal{S}}_{nf}(\Lambda):=\\{S\in\mathcal{S}(\Lambda)\ |\
S\ncong\operatorname{soc}(P)\text{ for any filtered projective
}P\in{\mathcal{P}}(\Lambda)\\}$

######  Remark 1.2.3.

Let $\Lambda$ be a cyclic Nakayama algebra. Then:
(1) A simple module $S$ is in the socle-set ${\mathcal{S}}(\Lambda)$ if and
only if $S\cong\operatorname{soc}Q$ for some indecomposable projective-
injective module $Q$.
(2) A simple module $S$ is in the top-set ${\mathcal{S}}^{\prime}(\Lambda)$ if
and only if $S\cong\operatorname{soc}I$ for some minimal injective module $I$.
(3) Let $B\in\mathcal{B}(\Lambda)$. Then $B\neq 0$ and $B$ is a proper
submodule of a projective-injective.
(4) Let $E$ be a minimal injective. Then
$\operatorname{soc}E\in{\mathcal{S}}^{\prime}(\Lambda)$.

######  Definition 1.2.4.

Let $\Lambda\cong\mathbb{K}Q/I$ be cyclic Nakayama algebra, where $I$ is an
admissible ideal. Let $\\#\operatorname{rel}\Lambda=\\#\\{\text{irredundant
set of relations which generate }I\\}$.

######  Proposition 1.2.5.

Let $\Lambda$ be a cyclic Nakayama algebra. Then:

  1. 1)

$|{\mathcal{S}}(\Lambda)|=|{\mathcal{S}}^{\prime}(\Lambda)|=|{\mathcal{B}}(\Lambda)|=|{\mathcal{P}}(\Lambda)|=\\#\operatorname{rel}\Lambda$.

  2. 2)

$\\#\\{\text{isomorphism classes of projective-
injectives}\\}=\\#\operatorname{rel}\Lambda$

  3. 3)

$\\#\\{\text{isomorphism classes of minimal
projectives}\\}=\\#\operatorname{rel}\Lambda$

  4. 4)

$\\#\\{\text{isomorphism classes of minimal
injectives}\\}=\\#\operatorname{rel}\Lambda$

  5. 5)

$\\#\\{\text{isomorphism classes of projectives filtered by
}{\mathcal{B}}(\Lambda)\\}=\\#\operatorname{rel}\Lambda$.

######  Proposition 1.2.6.

$\operatorname{rank}\Lambda=\\#\operatorname{rel}\Lambda+\operatorname{def}{(\Lambda})$.

###### Proof.

The number of isoclasses of indecomposable injective $\Lambda$-modules is
equal to $\operatorname{rank}(\Lambda)$. The number of isoclasses of
projective-injectives is equal to $\\#\operatorname{rel}$. The number of
isoclasses of indecomposable injectives which are not projective is
$\operatorname{def}(\Lambda)$. ∎

######  Corollary 1.2.7.

A Nakayama algebra $\Lambda$ has zero defect if and only $\Lambda$ is
selfinjective.

###### Proof.

If $\Lambda$ is either selfinjective or semisimple, every projective
$\Lambda$-module is injective. Hence $\operatorname{def}\Lambda=0$ by
Definition 1.2.1. Assume $\operatorname{def}\Lambda=0$, then,

$\displaystyle\operatorname{def}\Lambda=0$
$\displaystyle\implies\operatorname{rank}\Lambda=\\#\operatorname{rel}\Lambda$
$\displaystyle\implies\operatorname{rank}\Lambda=\\#\\{\text{isomorphism
classes of projective-injectives}\\}$

by Propositions 1.2.6 and 1.2.5. So every projective $\Lambda$-module is
injective which makes $\Lambda$ a selfinjective algebra. ∎

As we will see in Section 2, the base-set corresponds to the set of simple
modules of the syzygy filtered algebras. We give some equivalent conditions
for modules being in the base-set ${\mathcal{B}}(\Lambda)$:

######  Proposition 1.2.8.

Let $\Lambda$ be a cyclic Nakayama algebra and $B$ an indecomposable module
with $\operatorname{soc}B\in{\mathcal{S}}(\Lambda)$. Then the following are
equivalent:

  1. (i)

$B\in{\mathcal{B}}(\Lambda)$.

  2. (ii)

$I(B)/B$ is non-injective and for each proper submodule $M\subseteq B$,
$I(B)/M$ is injective.

  3. (iii)

$I(B)/\operatorname{rad}B$ is minimal injective.

  4. (iv)

$\ell(B)=\operatorname{def}I(B)+1$.

  5. (v)

$\ell(B)$ equals the smallest natural number $n>0$ such that
$\tau^{-n}\operatorname{soc}B\in{\mathcal{S}}(\Lambda)$.

###### Proof.

Denote by $S=\operatorname{soc}B$ and $I=I(B)$.
$1)\implies 2)$ Let $B\in{\mathcal{B}}(\Lambda)$. Then $B\neq 0$ by
Remark1.2.3(3). Since $\operatorname{top}B\in\mathcal{S}^{\prime}(\Lambda)$ by
Definition1.2.2 it follows that
$\operatorname{soc}(I(B))/B=\tau^{-}(\operatorname{top}(B)$ by Uniseriality
lemma 1.1.3. Therefore $\operatorname{soc}(I(B)/B)\in\mathcal{S}(\Lambda)$.
Therefore $I(B)/B$ is a submodule of a projective-injective module. Then
$I(B)/B$ cannot be injective since that would imply that $I(B)/B$ is also
projective injective. As a projective it would be a summand of $I(B)$ and
therefore isomorphis to $I(B)$. This would imply that $B=0$ which is
contradiction to Remark 1.2.3. Therefore $I(B)/B$ is not injective.
Now, We want to show that $I(B)/M$ is injective for all proper submodules $M$
of $B$. Let $M\subsetneq B$ be of minimal length such that $I(B)/M$ is not
injective. Then $I(B)/\operatorname{rad}M$ is a minimal injective. Then
$\operatorname{soc}I(B)/\operatorname{rad}M\in\mathcal{S}^{\prime}(\Lambda)$.
Therefore
$\operatorname{top}M=\tau^{-}(\operatorname{top}\operatorname{rad}M)=\operatorname{soc}(I(B)/\operatorname{rad}M)\in\mathcal{S}^{\prime}(\Lambda)$.
This is contradiction to the fact that $M$ is proper submodule of $B$.
Therefore $I(B)/M$ is injective.
$2)\implies 3)\implies 4)$ are straightforward.
$4)\implies 1)$ Assume $\operatorname{def}(I)=d$. Then $I(\tau^{-i}S)$, for
$0\leq i\leq d$, are all the injective quotients of $I$. That is,
$I(\tau^{-i}S)$ are non-projective for $1\leq i\leq d$ and $I(\tau^{-d-1}S)$
is projective-injective. Hence
$\tau^{-i}S\not\in{\mathcal{S}}^{\prime}(\Lambda)$ for $0\leq i<d$ and
$\tau^{-d}S\in{\mathcal{S}}^{\prime}(\Lambda)$. Clearly, $B$ having
$\tau^{i}(S)$, $0\leq i\leq d$, as composition factors is a module in
${\mathcal{B}}(\Lambda)$.
5)$\implies$ 1) By Definition 1.2.2.
1)$\implies$ 5) Assume $B\in{\mathcal{B}}(\Lambda)$. So,
$\operatorname{soc}B\in{\mathcal{S}}(\Lambda)$,
$\operatorname{top}B\in{\mathcal{S}}^{\prime}(\Lambda)$ and
$\operatorname{top}B\cong\tau^{\ell(B)-1}\operatorname{soc}B$.
$B\in{\mathcal{B}}(\Lambda)$ implies all simple composition factors of
$B/\operatorname{soc}B$ are not an element of ${\mathcal{S}}(\Lambda)$. Hence
$\tau^{-1}\operatorname{top}B\cong\tau^{\ell(B)}\operatorname{soc}B\in{\mathcal{S}}(\Lambda)$
makes $\ell(B)$ minimal. ∎

######  Proposition 1.2.9.

Let $\Lambda$ be a cyclic Nakayama algebra. Then:

  1. 1)

Any element of the socle set $\mathcal{S}(\Lambda)$ is the socle of exactly
one element of ${\mathcal{B}}(\Lambda)$.

  2. 2)

Any element of the top set $\mathcal{S}^{\prime}(\Lambda)$ is the top of
exactly one element of ${\mathcal{B}}(\Lambda)$.

  3. 3)

The simple composition factors of distinct $B\in\mathcal{B}(\Lambda)$ are
disjoint.

  4. 4)

Distinct elements of the base-set are Hom-orthogonal i.e.
$\operatorname{Hom}_{\Lambda}\left(B,B^{\prime}\right)\cong 0$ when $B\ncong
B^{\prime}$ and $\operatorname{Hom}_{\Lambda}(B,B)\cong\mathbb{K}$.

  5. 5)

Each $B\in{\mathcal{B}}(\Lambda)$ is a submodule of exactly one indecomposable
projective-injective.

  6. 6)

$B\in{\mathcal{B}}(\Lambda)$ is simple $\Lambda$-module if and only if
$B\in{\mathcal{S}}^{\prime}(\Lambda)\cap{\mathcal{S}}(\Lambda)$.

  7. 7)

$\sum_{B\in{\mathcal{B}}(\Lambda)}\ell(B)=\operatorname{rank}\Lambda$.

######  Definition 1.2.10.

Let $\Lambda$ be cyclic Nakayama algebra and let ${\mathcal{B}}(\Lambda)$ be
the base-set as in Definition 1.2.2. Let $Filt({\mathcal{B}}(\Lambda)$ be the
full subcategory of $\operatorname{mod-\\!}\Lambda$ consisting of all modules
which have filtration by modules from ${\mathcal{B}}(\Lambda)$.

######  Remark 1.2.11.

Let $\Lambda$ be cyclic Nakayama algebra. Then:
(a) $Filt({\mathcal{B}}(\Lambda))$ is a wide subcategory of
$\operatorname{mod-\\!}\Lambda$ with modules in ${\mathcal{B}}(\Lambda)$ as
simple objects.
(b) Let $M\neq 0$ be an indecomposable $\Lambda$-module. Then $M$ is in
$\mathcal{B}(\Lambda)$ if and only if
$\operatorname{top}M\in\mathcal{S}^{\prime}(\Lambda)$ and
$\operatorname{soc}M\in\mathcal{S}(\Lambda)$.

######  Lemma 1.2.12.

Let $\Lambda$ be cyclic Nakayama algebra and let $P$ be a projective
$\Lambda$-module. The following characterizations of $P$ are equivalent:

  1. 1)

$P\in{\mathcal{P}}(\Lambda)$ (Definition 1.2.2), i.e. $P$ is filtered
projective

  2. 2)

$P=P(S)$, i.e. $P$ is projective cover for some
$S\in{\mathcal{S}}^{\prime}(\Lambda)$

  3. 3)

$P=P(B)$, i.e. $P$ is projective cover for some $B\in{\mathcal{B}}(\Lambda)$

  4. 4)

$P$ is filtered by ${\mathcal{B}}(\Lambda)$.

  5. 5)

$P$ is projective object in the wide subcategory
$Filt({\mathcal{B}}(\Lambda))$

We can interpret Propositions 1.1.4 and 1.2.8 in terms of Kupisch series.

######  Remark 1.2.13.

Let $(c_{1},c_{2},\ldots c_{n})$ be Kupisch series for Nakayama algebra
$\Lambda$. Then,

  1. 1)

for each $1\leq i\leq n-1$

    1. (a)

$c_{i}\geq 2$

    2. (b)

$c_{i}>c_{i+1}$ $\iff$ $P_{i}\supset P_{i+1}$ $\iff$$c_{i}=c_{i+1}+1$

    3. (c)

$c_{i}\leq c_{i+1}$ $\iff$ $P_{i}$ is minimal projective and $P_{i+1}$ is
projective-injective with $\operatorname{def}(P_{i+1})=c_{i+1}-c_{i}$

  2. 2)

$c_{n}=1$ $\iff$ $\Lambda$ is linear

######  Lemma 1.2.14.

Let $(c_{1},\ldots,c_{n})$ be Kupisch series of a Nakayama algebra $\Lambda$.
Then,

  1. 1)

$|{\mathcal{S}}(\Lambda)|=|\\{c_{i}\mid c_{i}\leq c_{i+1}\\}|$, and
$\operatorname{def}\Lambda=|\\{c_{i+1}\mid c_{i}>c_{c_{i+1}}\\}|$.

  2. 2)

The lengths of elements of the base-set ${\mathcal{B}}(\Lambda)$ is
$c_{i+1}-c_{i}+1$ provided that $c_{i}\leq c_{i+1}$.

###### Proof.

1) By Proposition 1.2.5, the cardinality of the socle set is equal to the
number of minimal projective $\Lambda$-modules. $P_{i}$ is minimal if and only
if $c_{i}=\ell(P_{i})\leq\ell(P_{i+1})=c_{i+1}$ by Remark 1.2.13, hence we get
desired equality. Since $\\{c_{i}\mid c_{i}\leq c_{i+1}\\}\cup\\{c_{i+1}\mid
c_{i}>c_{c_{i+1}}\\}=\\{c_{1},\ldots,c_{n}\\}$, we conclude
$\operatorname{def}\Lambda=|\\{c_{i+1}\mid c_{i}>c_{c_{i+1}}\\}|$ by
Proposition 1.2.6.
2) By Remark 1.2.13, $c_{i}\leq c_{i+1}$ implies that
$\operatorname{def}P_{i+1}=c_{i+1}-c_{i}$. By Proposition 1.2.8,
$\ell(B)=c_{i+1}-c_{i}+1$. ∎

######  Proposition 1.2.15.

Let $\Lambda$ be a cyclic Nakayama algebra and let $T$ be a simple
$\Lambda$-module. Let $P(T)$ and $P(\tau T)$ be projective covers,
$S=\operatorname{soc}(P(T))$ and $U=\operatorname{soc}(P(\tau T))$. Suppose
$P(T)$ is a minimal projective. Then $U\cong\tau^{d+1}S$ where
$d=\operatorname{def}(P(\tau T))$.

######  Proposition 1.2.16.

Let $\Lambda$ be a cyclic Nakayama algebra. Let $B,C\in{\mathcal{B}}(\Lambda)$
such that $\tau\operatorname{soc}B\cong\operatorname{top}C$. Then, there
exists a simple $\Lambda$-module $T$ such that $P(T)$ is projective-injective
$\Lambda$-module with submodule $C$ and $P(\tau^{-1}T)$ is minimal projective
with submodule $B$.

###### Proof.

By Proposition 1.2.9 5), there exists unique projective-injective module $P$
with submodule $C$. Let $T=\operatorname{top}P$. Then, by Proposition 1.1.4,
$P(\tau^{-1}T)$ is minimal projective. Let $E$ be minimal injective quotient
of $P$. Then, $P/C\cong E/\operatorname{soc}E$ is isomorphic to
$\operatorname{rad}P(\tau^{-1}T)$. Hence
$\operatorname{soc}P(\tau^{-1}T)\cong\operatorname{soc}B$ implies that $B$ is
submodule of $P(\tau^{-1}T)$ by Uniseriality lemma 1.1.3. ∎

######  Proposition 1.2.17.

Let $\Lambda$ be a cyclic Nakayama algebra. Let $B,C\in{\mathcal{B}}(\Lambda)$
such that $\tau\operatorname{soc}B\cong\operatorname{top}C$. If $S$ is a
simple module such that $P(S)$ and $P(\tau^{j}S)$ are projective
$\Lambda$-modules with $B$ and $C$ as submodules, respectively, then there
exist $k$ such that:

  1. 1)

$C$ is submodule of $P(\tau^{t}S)$ where $k\leq t\leq j$

  2. 2)

$B$ is submodule of $P(\tau^{s}S)$ where $1\leq s<k$

###### Proof.

By Proposition 1.2.16, there exists a simple $\Lambda$-module $T$ such that
$P(T)$ is projective-injective having $C$ as submodule. Let $T=\tau^{k}S$.
Since $C$ is submodule of $P(\tau^{j}S)$, by Uniseriality lemma 1.1.3,
$P(\tau^{j}S)\subseteq P(\tau^{k}S)=P(T)$. Therefore, every $P(\tau^{t}S)$
where $k\leq t\leq j$ have $C$ as submodule.

On the other hand, $P(\tau^{-1}T)=P(\tau^{k-1}S)$ is minimal projective module
by Proposition 1.1.4. By Proposition 1.2.16, $B$ is submodule of
$P(\tau^{k-1}S)$. Since $B$ is submodule of $P(S)$, by Uniseriality lemma
1.1.3, $P(\tau^{k-1}S)\subseteq P(S)$. Therefore, every $P(\tau^{s}S)$ where
$1\leq s<k$ have $B$ as submodule. ∎

##  2\. The $\varepsilon$-construction of Syzygy Filtered Algebras

We now recall the definition of $\bm{\varepsilon}$-construction which was
introduced in [27] and was fundamental in [28],[29]. The idea of this
construction is to select the certan important projective modules and consider
(a smaller algebra) of the endomorphisms of those modules. Those important
projective modules ${\mathcal{P}}(\Lambda)$ are precisely projective covers of
the top-set ${\mathcal{S}}^{\prime}(\Lambda)$ as in Definition 1.2.2. The new
constructed algebra is also called _Syzygy Filtered Algebra_ due to the fact
that the category of modules of that algebra is equivalent to the category of
all second syzygies [27].

###  2.1. Definition and Functorial properties

######  Definition 2.1.1.

Let $\bm{\varepsilon}(\Lambda)$ be the endomorphism algebra of the direct sum
of projective $\Lambda$-modules which are projective covers of elements of the
top set ${\mathcal{S}}^{\prime}(\Lambda)$, i.e.

$\displaystyle\mathcal{P}:=\bigoplus\limits_{S\in{\mathcal{S}}^{\prime}(\Lambda)}P(S)\quad\text{and}\quad\bm{\varepsilon}(\Lambda):=\operatorname{End}_{\Lambda}(\mathcal{P})=\operatorname{End}_{\Lambda}\left(\bigoplus\limits_{S\in{\mathcal{S}}^{\prime}(\Lambda)}P(S)\right).$

This construction fits perfectly the classical set up of the endomorphism
algebras and the functors between the corresponding module categories, which
we now recall.

######  Proposition 2.1.2.

Let $\Lambda$ be an artin algebra, $\emph{mod-}\Lambda$ the category of
finitely presented right $\Lambda$-modules, $P$ a projective $\Lambda$-module,
$\Gamma=\operatorname{End}_{\Lambda}(P)$ and
$\operatorname{Hom}_{\Lambda}(P,-):\emph{mod-}\Lambda\xrightarrow{}\emph{mod-}\Gamma$.
Then:

  1. 1)

The functor $\operatorname{Hom}_{\Lambda}(P,-)$ induces an equivalence of
categories $fp(\operatorname{add}P)\to\emph{mod-}\Gamma$, where
$fp(\operatorname{add}P)$ is the full subcategory of $\emph{mod-}\Lambda$
consisting of the modules which are finitely presented by
$\operatorname{add}P$.

  2. 2)

$(-\otimes_{\Gamma}P):\emph{mod-}\Gamma\xrightarrow{}\emph{mod-}\Lambda$ is
left adjoint to $\operatorname{Hom}_{\Lambda}(P,-)$, it is right exact, fully
faithful and induces inverse equivalence: $\emph{mod-}\Gamma\to
fp(\operatorname{add}P)$.

  3. 3)

The composition $\operatorname{Hom}_{\Lambda}(P,-)\circ(-\otimes_{\Gamma}P)$
is the identity of $\emph{mod-}\Gamma$.

  4. 4)

$(-\otimes_{\Gamma}P)$ preserves projective covers.

######  Proposition 2.1.3.

Let $\Lambda$ be cyclic Nakayama algebra and let $\bm{\varepsilon}(\Lambda)$
be the associated syzygy filtered algebra. Let
${\mathcal{P}}=\oplus_{S\in{\mathcal{S}}^{\prime}}P(S)$. Then
$Filt({\mathcal{B}}(\Lambda))=fp(\operatorname{add}{\mathcal{P}})$. Then there
is a commutative diagram:

(2.1) $\displaystyle\begin{aligned} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
26.81984pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\\!\operatorname{Hom}_{\Lambda}({\mathcal{P}},-)\
\ }$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 84.81984pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.81984pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 52.81984pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.81984pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 84.81984pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{mod-\\!}\bm{\varepsilon}(\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
33.53894pt\raise-6.875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.375pt\hbox{$\scriptstyle{(-\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}})\
\ \ }$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 16.88895pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.58527pt\raise-26.875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.375pt\hbox{$\scriptstyle{(-\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}})}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
26.81984pt\raise-30.17828pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
143.81648pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Filt({\mathcal{B}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
42.96962pt\raise-16.12pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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{}_{\cong\ \ }}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
94.28447pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{aligned}$

Then the following statements are true:

  1. i)

$\operatorname{Hom}_{\Lambda}({\mathcal{P}},-):Filt({\mathcal{B}}(\Lambda))\to\operatorname{mod-\\!}\bm{\varepsilon}(\Lambda)$
is an equivalence with inverse
$-\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}}$.

  2. ii)

Let $M\in\emph{mod-}\bm{\varepsilon}(\Lambda)$. Then
$M\cong\operatorname{Hom}_{\Lambda}({\mathcal{P}},X)$ for some $X\in
Filt({\mathcal{B}}(\Lambda))$.

  3. iii)

Let $S\in\emph{mod-}\bm{\varepsilon}(\Lambda)$ be simple. Then
$S\cong\operatorname{Hom}_{\Lambda}({\mathcal{P}},B)$ for some
$B\in{\mathcal{B}}(\Lambda)$.

  4. iv)

Let $P\in\emph{mod-}\bm{\varepsilon}(\Lambda)$ be projective. Then
$P\cong\operatorname{Hom}_{\Lambda}({\mathcal{P}},\tilde{P})$ where
$\tilde{P}$ is ${\mathcal{B}}(\Lambda)$-filtered projective $\Lambda$-module.

  5. v)

If $P$ is projective-injective $\Lambda$-module and $P\in
Filt({\mathcal{B}}(\Lambda))$, then
$\operatorname{Hom}_{\Lambda}({\mathcal{P}},P)$ is projective-injective
$\bm{\varepsilon}(\Lambda)$-module.

  6. vi)

Let $X\in Filt{{\mathcal{B}}}(\Lambda)$. Then
$X\cong(M\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}})$ for some
$M\in\emph{mod-}\bm{\varepsilon}(\Lambda)$.

  7. vii)

Let $B\in{{\mathcal{B}}(\Lambda)}$. Then
$B\cong(S\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}})$ for some simple
$S\in\emph{mod-}\bm{\varepsilon}(\Lambda)$.

  8. viii)

Let $\tilde{P}$ be ${\mathcal{B}}(\Lambda)$-filtered projective
$\Lambda$-module. Then
$\tilde{P}\cong(P\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}})$ for some
projective $P\in\emph{mod-}\bm{\varepsilon}(\Lambda)$.

######  Notation 2.1.4.

Let $\bm{\varepsilon}(\Lambda)$ be the syzygy filtered algebra of a cyclic
Nakayama algebra $\Lambda$. Let $M$ be a $\bm{\varepsilon}(\Lambda)$-module.
We will often use notation
$\Delta(M):=(M\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}})\in
Filt({\mathcal{B}}({\Lambda}))$, i.e., we will denote the functor
$(-\otimes_{\bm{\varepsilon}(\Lambda)}{\mathcal{P}})$ by $\Delta$.

###  2.2. Relationship between defects of $\Lambda$ and its syzygy filtered
algebra $\Theta$

It was the reverse of $\bm{\varepsilon}$-construction that was used in [29]
for the description of Nakayama algebras which are higher Auslander algebras
and in the Section 4 it will be used also to describe families of algebras
satisfying higher homological properties.

Here we consider cyclic Nakayama algebras $\Lambda$ such that their syzygy
filtered algebras $\bm{\varepsilon}(\Lambda)$ are isomorphic to a given
Nakayama algebra $\Theta$.

In general for any Nakayama algebra $\Lambda$ particularly important sets of
modules which will be used here, are: socle-set ${\mathcal{S}}(\Lambda)$, top-
set ${\mathcal{S}}^{\prime}(\Lambda)$, base-set ${\mathcal{B}}(\Lambda)$,
filtered projectives ${\mathcal{P}}(\Lambda)$, socles of filtered projectives
${\mathcal{S}}_{f}(\Lambda)$, the other socles of projectives
${\mathcal{S}}_{nf}(\Lambda)$ (Definition 1.2.2). Also, the notion of defect
(Def.1.2.1) and relations (Def.1.2.4) are used. In addition, there is the
basic relation
$\operatorname{rank}(\Lambda)=\\#\operatorname{rel}(\Lambda)+\operatorname{def}\Lambda$
(Prop.1.2.6).

######  Proposition 2.2.1.

Let $\Lambda$ and $\Theta$ be cyclic Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$.

  1. 1)

$\\#\operatorname{rel}(\Lambda)=\operatorname{rank}(\Theta)$

  2. 2)

$|{\mathcal{S}}_{f}(\Lambda)|=\\#\operatorname{rel}(\Theta)$

  3. 3)

$|{\mathcal{S}}_{nf}(\Lambda)|=\operatorname{def}(\Theta)$

  4. 4)

$\operatorname{def}(\Lambda)=$ number of non-filtered $\Lambda$-projectives

  5. 5)

$\operatorname{def}(\Lambda)=t+\operatorname{def}(\Theta)+r$, where
$t=|\\{\text{non-filt.proj.inj. }Q\ |\
\operatorname{soc}(Q)\in{\mathcal{S}}_{f}(\Lambda)\\}|$ and $r=|\\{\text{non-
filt.proj non-inj}\\}|$.

###### Proof.

1) This follows since
$\\#\operatorname{rel}(\Lambda)=|{\mathcal{P}}(\Lambda)|$ by Prop.1.2.5 and
$\Theta=\operatorname{End}_{\Lambda}(\oplus_{P\in{\mathcal{P}}(\Lambda)}P)$.

2) By Def.1.2.2, ${\mathcal{S}}_{f}(\Lambda)=\\{S\in{\mathcal{S}}\ |\
S\cong\operatorname{soc}(P)\text{ for some }P\in{\mathcal{P}}(\Lambda)\\}$,
i.e. these are socles of filtered projectives. Let
${\mathcal{B}}_{f}(\Lambda):=\\{B\in{\mathcal{B}}(\Lambda)\ |\
\operatorname{soc}(B)\in{\mathcal{S}}_{f}(\Lambda)\\}$. Any element of the
socle set $\mathcal{S}(\Lambda)$ is the socle of exactly one element of
${\mathcal{B}}(\Lambda)$ by Prop.1.2.2. Therefore
$|{\mathcal{S}}_{f}(\Lambda)|=|{\mathcal{B}}_{f}(\Lambda)|$ and the functor
Hom${}_{\Lambda}({\mathcal{P}},-)$ gives a bijection between
${\mathcal{B}}_{f}(\Lambda)$ and the socle set ${\mathcal{S}}(\Theta)$. Again,
using Prop.1.2.5 it follows that
$|{\mathcal{S}}(\Theta)|=\\#\operatorname{rel}(\Theta)$.

3) Since
${\mathcal{S}}_{nf}(\Lambda)={\mathcal{S}}(\Lambda)\backslash{\mathcal{S}}_{f}(\Lambda)$
it follows that
$|{\mathcal{S}}_{nf}(\Lambda)|=|{\mathcal{S}}(\Lambda)|-|{\mathcal{S}}_{f}(\Lambda)|$
=
$\\#\operatorname{rel}(\Lambda)-\\#\operatorname{rel}(\Theta)=\operatorname{rank}(\Theta)-\\#\operatorname{rel}(\Theta)=\operatorname{def}(\Theta)$.

4) By Def.1.2.1 it follows that $\operatorname{def}(\Lambda)=\\#\\{$injective,
non-projective $\Lambda$-modules$\\}$ and also
$\operatorname{def}(\Lambda)=\\#\\{$projective, non-injective
$\Lambda$-modules$\\}$. So
$\operatorname{def}(\Lambda)=\operatorname{rank}(\Lambda)-\\#\operatorname{rel}(\Lambda)=$

$|\\{\text{projective }\Lambda\text{-modules}\\}|-|\\{\text{filtered
projective }\Lambda\text{-modules}\\}|=|\\{\text{non-filtered projective
}\Lambda\text{-modules}\\}|.$

5) It follows from 4) that $\operatorname{def}(\Lambda)=|\\{\text{non-filtered
projective }\Lambda\text{-modules}\\}|=$

$|\\{\text{non-filt proj-inj }\Lambda\text{-modules}\\}|+|\\{\text{non-filt
proj non-inj }\Lambda\text{-modules}\\}|=$ $|\\{\text{socles of non-filt proj-
inj}\\}|+|\\{\text{non-filt proj non-inj }\Lambda\text{-modules}\\}|.$

Now, we will use the fact that $|\\{\text{socles of non-filt proj-inj}\\}|=$

$|\\{S\in{\mathcal{S}}_{f}(\Lambda),\text{ socle of non-filt proj-
inj}\\}|+|\\{S\in{\mathcal{S}}_{nf}(\Lambda),\text{ socle of non-filt proj-
inj}\\}|.$

Notice that $\\{S\in{\mathcal{S}}_{nf}(\Lambda),\text{ socle of non-filt proj-
inj}\\}={\mathcal{S}}_{nf}(\Lambda)$ since $S\in{\mathcal{S}}_{nf}(\Lambda)$
can only be socle for a non-filtered projective. So this summand is equal to
$\operatorname{def}(\Theta)$ by 3). Let

$t:=|\\{\text{non-filt.proj.inj. }Q\ |\
\operatorname{soc}(Q)\in{\mathcal{S}}_{f}(\Lambda)\\}|\text{ and let
}r:=|\\{\text{non-filt.proj non-inj}\\}|.$
$\operatorname{def}(\Lambda)=t+\operatorname{def}(\Theta)+r.$

∎

######  Notation 2.2.2.

We will use term _sequence_ to mean cyclic ordering within AR-quiver - more
precisely: _sequence of simples_ means ordering of simples in the $\tau$-orbit
according to the power of $\tau$, _sequence of projectives_ means ordering of
projectives induced by the ordering of their tops, _sequence of $\Delta(S)$_
means ordering induced by ordering of the simples $S$.

######  Remark 2.2.3.

Let $\Lambda$ be a cyclic Nakayama algebra. Let $B\in{\mathcal{B}}(\Lambda)$.
Then any projective $P(Z)$ between $P(\tau(\operatorname{soc}(B)))$ and
$P(\operatorname{top}(B))$, not including $P(\operatorname{top}(B))$, is non-
filtered projective since it’s top $Z\notin{\mathcal{S}}^{\prime}$, so $P(Z)$
cannot be filtered, i.e. for any composition factor $Z$ of
$\operatorname{rad}B$, $P(Z)$ cannot be filtered.

######  Proposition 2.2.4.

Let $\Lambda$ and $\Theta$ be cyclic Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$. Let $T$ be a simple $\Theta$-module,
$P(T)$ and $P(\tau T)$ be projective covers in mod $\Theta$. Let $\Delta T$ be
the corresponding $\Lambda$-module. Then $\ell_{\Lambda}(\Delta
T)\geq\operatorname{def}_{\Theta}(P(\tau T))+1$ ($\ell$ is the length of the
module).

###### Proof.

Case 1. Suppose $P(T)$ is not a minimal projective in mod-$\Theta$. Then
$P(\tau T)=\operatorname{rad}P(T)$ and therefore $\operatorname{def}(P(\tau
T))=0$ since only projective-injectives might have nonzero defect. Since
$\ell(\Delta(T))\geq 1$, it follows that
$\ell(\Delta(T))\geq\operatorname{def}P(\tau T)+1$.
Case 2. Suppose $P(T)$ is a minimal projective in mod-$\Theta$. Then $P(\tau
T)$ is projective injective by Prop.1.1.5. We will use the equivalence
$\Delta=(-\otimes{\mathcal{P}}):\operatorname{mod-\\!}\Theta\to
Filt({\mathcal{B}}(\Lambda))$ and Prop.1.2.15. Let
$S=\operatorname{soc}(P(T))$ and $U=\operatorname{soc}(P(\tau T))$. Then there
are inclusions $\Delta(U)\to\Delta P(\tau T)=P(\Delta(\tau T))$ and
$\Delta(S)\to\Delta P(T)=P(\Delta(T))$. Here we used the fact that $\Delta$
preserves projective covers (Prop.2.1.2). Let $d:=\operatorname{def}(P(\tau
T))$. Then $U\cong\tau^{d+1}S$ by Prop.1.2.15, and
$\Delta(U)\cong\Delta(\tau^{d+1}S)$. Consider the sequence of non-isomorphic
modules from ${\mathcal{B}}(\Lambda)$:

$\\{\Delta(U),\ \Delta(\tau^{-1}U),\
\Delta(\tau^{-2}U),\dots,\Delta(\tau^{-(d-1)}U),\ \Delta(\tau^{-(d)}U),\
\Delta(\tau^{-(d+1)}U)\cong\Delta(S)\\}.$

Each of $\Delta(\tau^{-i})$ for $1\leq i\leq(d+1)$ must be a submodule of some
projective-injective $P(Z_{i})$ for some simple module $Z_{i}$. Also
$\Delta(U)$ is a submodule of $P(\Delta(\tau T))$ and $\Delta(S)$ is a
submodule of $P(\Delta(T))$. This defines a sequence of non-isomorphic
projectives:

$\\{P(\Delta(\tau T)),\ P(Z_{1}),\ P(Z_{2}),\ \dots,P(Z_{d-1}),\ P(Z_{d}),\
P(Z_{d+1})=P(\operatorname{top}(\Delta(T)))\\}$

which further defines a sequence of non-isomorphic simple modules:

$\\{\operatorname{top}(\Delta(\tau T),\ Z_{1},\ Z_{2},\ \dots,Z_{d-1},\
Z_{d},\ Z_{d+1}=\operatorname{top}(\Delta(T)\\}.$

The simples $\\{Z_{1},Z_{2},\dots,Z_{d-1},Z_{d}\\}$ are among the composition
factors of $\operatorname{rad}(\Delta(T))$. Therefore
$d\leq\ell(\operatorname{rad}(\Delta(T)))$ and therefore
$\operatorname{def}(P(\tau T))+1=d+1\leq\ell(\Delta T)$. ∎

The filtered projective $\Lambda$-modules are well understood: they are the
images of projective $\Theta$-modules under the functor
$\Delta=(-\otimes_{\Theta}{\mathcal{P}})$: essentially all simple
$\Theta$-factors $S$ get replaced by $\Lambda$-modules $\Delta(S)$. While the
non-filtered projective $\Lambda$-modules which are directly related to the
$\operatorname{def}(P(\tau T))$ were already introduced in the previous proof
as $P(Z_{i})$, the following proposition will describe more precisely
additional non-filtered projective $\Lambda$-modules which exist when
$d=\operatorname{def}(P(\tau T))<\ell(\Delta(T))$.

######  Proposition 2.2.5.

Let $\Lambda$ and $\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$. Let $T$ be a simple $\Theta$-module,
$P(T)$ and $P(\tau T)$ be projective covers, $S=\operatorname{soc}(P(T))$ and
$U=\operatorname{soc}(P(\tau T))$. Suppose $d=\operatorname{def}(P(\tau
T))<\ell(\Delta(T))=m$. Then there is a sequence of indices
$\\{a_{1},a_{2},\dots,a_{d}\\}\subset\\{1,\ 2,\dots,m-1\\}$ such that

  1. 1)

$\Delta(\tau^{-i}U)\hookrightarrow P(Z_{i})$ for non-filtered projective-
injective $\Lambda$-modules $P(Z_{i})$ where
$Z_{i}=(\tau^{-a_{i}}(\operatorname{top}(\Delta(\tau T))))$ are simple
$\Lambda$-modules for $1\leq i\leq d$.

  2. 2)

For each $a_{i}\leq j\leq a_{i+1}$ there is a non-filtered projective non-
injective $\Lambda$-module $P(Y_{j})$ with the simple top $Y_{j}$ which fits
in the sequence $\\{\dots,Z_{i},Y_{j},Z_{{i+1}},\dots\\}$. Each such
projective module $P(Y_{j})$ has $\Delta(\tau^{-(i+1)}U)$ as a submodule.

  3. 3)

For each $0\leq j\leq a_{1}$ there is a non-filtered projective
$\Lambda$-module $P(Y_{j})$ with the simple top $Y_{j}$ which fits in the
sequence $\\{\operatorname{top}(\Delta(\tau T),Y_{j},Z_{{1}},\dots\\}$. Each
such projective module $P(Y_{j})$ has $\Delta(\tau^{-1}U)$ as a submodule.

  4. 4)

For each $a_{d}\leq j\leq m-1$ there is a non-filtered projective
$\Lambda$-module $P(Y_{j})$ with the simple top $Y_{j}$ which fits in the
sequence $\\{Z_{{d}},Y_{j},\dots,\operatorname{top}(\Delta(\tau T)\\}$. Each
such projective module $P(Y_{j})$ has $\Delta(\tau^{-1}S)$ as a submodule.

###### Proof.

1) Since $d=\operatorname{def}(P(\tau T))$, we recall from the previous proof
that there are $d$ non-isomorphic elements of the base-set
${\mathcal{B}}(\Lambda)$ given as $\\{\Delta(\tau^{-i}U\\}_{i=1}^{d}$, the
projective-injective modules $\\{P(Z_{i})\\}_{i=1}^{d}$ which contain
$\\{\Delta(\tau^{-i}U\\}_{i=1}^{d}$ as submodules for each $1\leq i\leq d$.
Since the simple modules $\\{Z_{i}\\}_{i=1}^{d}$ are composition factors of
$\operatorname{rad}(\Delta(T))$, they are not in ${\mathcal{S}}^{\prime}(C)$
and therefore the projectives $\\{P(Z_{i})\\}_{i=1}^{d}$ are not filtered.

Furthermore, the simple modules $\\{Z_{i}\\}_{i=1}^{d}$ are among the simple
composition factors of $\Delta(T)$. We will denote $\widetilde{\tau
T}\\!\\!:=\operatorname{top}(\Delta(\tau T)))$ and
$\widetilde{T}\\!\\!:=\operatorname{top}(\Delta(T))$. Notice that
$\operatorname{soc}\Delta(T)=\tau^{-1}(\widetilde{\tau T})$ and
$\operatorname{top}(\Delta(T))=\tau^{-m}(\widetilde{\tau T})$ So composition
factors of $\Delta(T)$ are

$\\{\tau^{-1}(\widetilde{\tau T}),\ \tau^{-2}(\widetilde{\tau T}),\ \dots\
,\tau^{-(m-1)}(\widetilde{\tau T})\\}.$

Therefore there is a sequence of indices
$\\{a_{1},a_{2},\dots,a_{d}\\}\subset\\{1,\ 2,\dots,m-1\\}$ such that
$Z_{i}=\tau^{-a_{i}}(\widetilde{\tau
T})=\tau^{-a_{i}}(\operatorname{top}(\Delta(\tau T))\text{ for
}i\in\\{1,\dots,d\\}$.

2) Suppose $a_{i}\leq j\leq a_{i+1}$. Let $Y_{j}=\tau^{-j}(\widetilde{\tau
T})$ and let $P(Y_{j})$ be the projective cover. Since each of
$Y_{j}=\tau^{-j}(\widetilde{\tau T})$ is a composition factor of
$\operatorname{rad}(\Delta T)$, the $Y_{j}$’s are not in
${\mathcal{S}}^{\prime}(C)$ and therefore the projectives $P(Y_{j})$ are not
filtered.

Notice that
$\tau\operatorname{soc}\Delta(\tau^{-(i+1)}U)\cong\operatorname{top}\Delta(\tau^{-i}U)$,
and there exists projectives $P(Z_{i})$ and $P(Z_{i+1})$ having
$\Delta(\tau^{-i}U)$ and $\Delta(\tau^{-(i+1)}U)$ as submodules respectively.
By Proposition 1.2.17, $P(Y_{j})$ has either $\Delta(\tau^{-i}U)$ or
$\Delta(\tau^{-(i+1)}U)$ as submodule. Hence, we can choose $a_{i}$ so that
$P(Z_{i})$ is minimal projective, which makes $\Delta(\tau^{-(i+1)}U)$
submodule of $P(Y_{j}))$ by Proposition 1.2.16.

3) Suppose $0\leq j\leq a_{1}$. Let $Y_{j}=\tau^{-j}(\widetilde{\tau T})$ and
let $P(Y_{j})$ be the projective cover. Since each of
$Y_{j}=\tau^{-j}(\widetilde{\tau T})$ is a composition factor of
$\operatorname{rad}(\Delta T)$, the $Y_{j}$’s are not in
${\mathcal{S}}^{\prime}(C)$ and therefore the projectives $P(Y_{j})$ are not
filtered.

Notice that
$\tau\operatorname{soc}\Delta(\tau^{-1}U)\cong\operatorname{top}\Delta(U)$,
and there exists projectives $P(Z_{1})$ and $P(\Delta(\tau T))$ having
$\Delta(\tau^{-1}U)$ and $\Delta(U)$ as submodules respectively. By
Proposition 1.2.17, $P(Y_{j})$ has either $\Delta(\tau^{-1}U)$ or $\Delta(U)$
as submodule. Hence, we can choose $a_{1}$ so that $P(Z_{1})$ is minimal
projective, which makes $\Delta(\tau^{-1}U)$ submodule of $P(Y_{j}))$ by
Proposition 1.2.16.

4) Similar to 3). ∎

######  Remark 2.2.6.

Suppose $\bm{\varepsilon}(\Lambda)\cong\Theta$. The image under $\Delta$ of a
minimal projective $\Theta$-module, need not be a minimal projective
$\Lambda$-module. Also, the image under $\Delta$ of a projective-injective
$\Theta$-module, need not be a projective-injective $\Lambda$-module.
$\Lambda$-module.

###  2.3. Defect invariant Nakayama algebras

######  Definition 2.3.1.

Let $\Lambda$ be a cyclic Nakayama algebra and $\bm{\varepsilon}(\Lambda)$ its
syzygy filtered algebra. We call $\Lambda$ defect invariant if
$\operatorname{def}\Lambda=\operatorname{def}\bm{\varepsilon}(\Lambda)$.

We want to emphasize that:

######  Proposition 2.3.2.

Let $\Lambda$ be a cyclic Nakayama algebra, and
$\Theta:=\bm{\varepsilon}(\Lambda)$. Then $\Lambda$ is defect invariant if and
only if both of the following hold:
(i) For each simple module $S\notin{\mathcal{S}}^{\prime}(\Lambda)$, $P(S)$ is
projective-injective;
(ii) $\operatorname{Hom}_{\Lambda}({\mathcal{P}},P)$ is a projective-injective
$\Theta$-module if and only if $P$ is a filtered projective-injective
$\Lambda$-module.

###### Proof.

It follows from Proposition 2.2.1 that

$\operatorname{def}(\Lambda)=t+\operatorname{def}(\Theta)+r,$

where $t=|\\{\text{non-filt.proj.inj. }Q\ |\
\operatorname{soc}(Q)\in{\mathcal{S}}_{f}(\Lambda)\\}|$ and $r=|\\{\text{non-
filt.proj non-inj}\\}|$. So,
$\operatorname{def}(\Lambda)=\operatorname{def}(\Theta)$ if and only if
$r=|\\{\text{non-filt.proj non-inj}\\}|=0$ and $t=|\\{\text{non-filt.proj.inj.
}Q\ |\ \operatorname{soc}(Q)\in{\mathcal{S}}_{f}(\Lambda)\\}|=0$.

(i) Since $r=0$ it follows that all non-filtered projectives must be
injective.

(ii) Since $t=0$ it follows that all non-filtered projective-injectives must
have socles in ${\mathcal{S}}_{nf}(\Lambda)$. Therefore the number of non-
filtered projective injective $\Lambda$-modules is
$|{\mathcal{S}}_{nf}(\Lambda)|=\operatorname{def}(\Theta)$ by Prop.2.2.1.

$\\#\operatorname{rel}(\Lambda)=$
$|\\{\text{proj.inj.}\Lambda\text{-modules}\\}|=$
$|\\{\text{filt.proj.inj.}\Lambda\text{-modules}\\}|+|\\{\text{non-
filt.proj.inj.}\Lambda\text{-modules}\\}|$
Hom${}_{\Lambda}({\mathcal{P}},-)$ gives a bijection between
$\\{\text{filt.proj.}\Lambda\text{-modules}\\}$ and
$\\{\text{proj.}\Theta\text{-modules}\\}$.
Hom${}_{\Lambda}({\mathcal{P}},-)$ maps
$\\{\text{filt.proj.inj.}\Lambda\text{-modules}\\}$ to
$\\{\text{proj.inj.}\Theta\text{-modules}\\}$. Therefore
$|\\{\text{filt.proj.inj.}\Lambda\text{-modules}\\}|\leq|\\{\text{proj.inj.}\Theta\text{-modules}\\}|$
and
$\\#\operatorname{rel}(\Lambda)=|\\{\text{filt.proj.inj.}\Lambda\text{-modules}\\}|+|\\{\text{non-
filt.proj.inj.}\Lambda\text{-modules}\\}|\leq$
$|\\{\text{proj.inj.}\Theta\text{-modules}\\}|+|\\{\text{non-
filt.proj.inj.}\Lambda\text{-modules}\\}|=$
$\\#\operatorname{rel}(\Theta)+\operatorname{def}(\Theta)=\operatorname{rank}(\Theta)$.
Since $\\#\operatorname{rel}(\Lambda)=\operatorname{rank}(\Theta)$ it follows
that
$|\\{\text{filt.proj.inj.}\Lambda\text{-modules}\\}|=|\\{\text{proj.inj.}\Theta\text{-modules}\\}|$
Therefore: $\operatorname{Hom}_{\Lambda}({\mathcal{P}},Q)$ is a projective-
injective $\Theta$-module if and only if $Q$ is a filtered projective-
injective $\Lambda$-module, as claimed. ∎

######  Corollary 2.3.3.

Let $\Lambda$ be a cyclic Nakayama algebra, and
$\Theta:=\bm{\varepsilon}(\Lambda)$. If $\Lambda$ is defect invariant, then
each non-filtered projective module $P$ is also a minimal projective module.

If $\Lambda$ is the defect invariant reverse of given algebra $\Theta$, then
Propositions 2.2.4 and 2.2.5 determine the length of each indecomposable
projective $\Lambda$-module. Here we present them as corollaries.

######  Corollary 2.3.4.

Let $\Lambda$ and $\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$. If
$\operatorname{def}\Lambda=\operatorname{def}\Theta$, then
$\ell_{\Lambda}(\Delta(S))=\operatorname{def}_{\Theta}(P(\tau S))+1$.

###### Proof.

By Proposition 2.2.4, $\ell(\Delta(T))\geq\operatorname{def}_{\Theta}(P(\tau
T))+1$ for every simple $\Theta$-module $T$. Without loss of generality,
assume that the simple module $T_{1}$ satisfies

$\displaystyle\ell(\Delta(T_{1}))>\operatorname{def}_{\Theta}(P(\tau
T_{1}))+1,$

i.e., strict inequality. For $2\leq i\leq\operatorname{rank}\Theta$, let

$\displaystyle\ell(\Delta(T_{i}))\geq\operatorname{def}_{\Theta}(P(\tau
T_{i}))+1.$

The sum of all inequalities is

$\displaystyle\sum\limits^{\operatorname{rank}\Theta}_{i=1}\ell_{\Lambda}(\Delta(T_{i}))>\sum\limits^{\operatorname{rank}\Theta}_{i=1}(\operatorname{def}_{\Theta}(P(\tau
T_{i}))+1).$

The left-hand side is $\operatorname{rank}\Lambda$ by Proposition 1.2.9. The
right-hand side is $\operatorname{def}\Theta+\operatorname{rank}\Theta$. We
get
$\operatorname{rank}\Lambda>\operatorname{rank}\Theta+\operatorname{def}\Theta$.
By Proposition 2.2.1,
$\operatorname{rank}\Lambda=\operatorname{rank}\Theta+\operatorname{def}\Lambda$,
which implies $\operatorname{def}\Lambda>\operatorname{def}\Theta$,
contradicting the assumption in the statement. Hence, any simple
$\Theta$-module $T$ satisfies
$\ell(\Delta(T))=\operatorname{def}_{\Theta}(P(\tau T))+1$. ∎

######  Corollary 2.3.5.

Let $\Lambda$ be a cyclic Nakayama algebra and $\Theta=\bm{\epsilon}(\Lambda)$
be its syzygy filtered algebra. Then
$\operatorname{def}\Lambda\geq\operatorname{def}\Theta$. The equality holds if
and only if $\operatorname{def}_{\Lambda}(I(\Delta
T))=\operatorname{def}_{\Theta}P(\tau T)$, for each simple $\Theta$-module
$T$.

###### Proof.

The first statement follows from Proposition 2.2.1. If we assume
$\operatorname{def}\Lambda=\operatorname{def}\Theta$, then by Corollary 2.3.4
we get $\ell_{\Lambda}(\Delta(T))=\operatorname{def}_{\Theta}(P(\tau T))+1$.
By Proposition 1.2.8, we conclude that
$\operatorname{def}_{\Lambda}(I(\Delta(T)))=\operatorname{def}_{\Theta}(P(\tau
T))$. ∎

######  Corollary 2.3.6.

Let $\Lambda$ be a cyclic Nakayama algebra, and
$\Theta:=\bm{\varepsilon}(\Lambda)$. If $\Lambda$ is defect invariant, then
the length of $P(\Delta(S))\in Filt({\mathcal{B}}(\Lambda))$, where $S$ is
simple $\Theta$-module, is given by

(2.2)
$\displaystyle\ell_{\Lambda}\left(P(\Delta(S))\right)=\ell_{\Theta}(P(S))+\sum\limits^{\ell_{\Theta}(P(S))}_{i=1}\operatorname{def}_{\Theta}(P(\tau^{i}S)).$

###### Proof.

$\displaystyle\ell_{\Lambda}\left(P(\Delta(S))\right)$
$\displaystyle=\sum\limits_{i=0}^{\ell_{\Theta}(P(S))-1}\ell_{\Lambda}(\Delta(\tau^{i}S))$
$\displaystyle=\sum\limits_{i=0}^{\ell_{\Theta}(P(S))-1}(\operatorname{def}_{\Theta}(P(\tau^{i+1}S)+1)$
$\displaystyle=\sum\limits^{\ell_{\Theta}(P(S))}_{i=1}(\operatorname{def}_{\Theta}(P(\tau^{i}S))+1)$
$\displaystyle=\ell_{\Theta}(P(S))+\sum\limits^{\ell_{\Theta}(P(S))}_{i=1}\operatorname{def}_{\Theta}(P(\tau^{i}S)).$

∎

##  3\. Reverse of $\varepsilon$-construction

So far we have assumed the existence of both Nakayama algebras $\Lambda$ and
$\Theta$ where $\Theta=\varepsilon(\Lambda)$. In this section we will start
with a Nakayama algebra $\Theta$ and construct an admissible sequence of
integers that will serve as Kupish series for an algebra $\Lambda$ such that
$\varepsilon(\Lambda)=\Theta$.

###  3.1. Defect invariant case

An outline of the process of creating admissible sequence of integers for a
Kupish series, which will be done properly in the next proposition, is as
follows:

We start with a Nakayama algebra $\Theta$ and a complete set of
representatives of indecomposable projective $\Theta$-modules
$\\{P_{1},\dots,P_{n}\\}$. Next we consider a sequence of integers
${\mathcal{K}}=\left(c_{1},\ldots,c_{N}\right)$ and show that this sequence is
a Kupish series for a Nakayama algebra $\Lambda$ providing that it satisfies
four conditions: 1), 2), 3), 4) as stated in the Proposition 3.1.1. Before
proving this proposition we will explain the role or purpose of each of these
conditions (as they are numbered in the proposition)..

  1. 1)

$N=n+\operatorname{def}\Theta$ is a statement that $N$, the number of
indecomposable projective $\Lambda$-modules has to be larger by
$\operatorname{def}\Theta$, from $n$, which is the number of indecomposable
projective $\Theta$-modules. This is necessary according to 2.2.1 and 1.2.6.

  2. 2)

${\mathcal{N}}=(c_{z_{1}},c_{z_{2}},\ldots,c_{z_{n}})$ will be the lengths of
the filtered projective $\Lambda$-modules. These lengths are described in
terms of defects of the original projective $\Theta$-modules. By corollaries
2.3.5 and 2.3.6, each element of the base-set of $\mathcal{B}(\Lambda)$ can be
expressed in terms of the defects of projectives of $\Theta$. Consequently the
lengths of all filtered $\Lambda$-projective can be described in terms of
defects of projective $\Theta$-modules.

  3. 3)

This is the statement that the number of non-filtered $\Lambda$-projectives is
determined by the defect of $\Theta$-projectives.

  4. 4)

This is numerical condition which describes lengths of non-filtered
$\Lambda$-projectives. Here proposition 2.2.5 is used.

######  Proposition 3.1.1.

Let $\Theta$ be a Nakayama algebra and $\left\\{P_{1},\ldots,P_{n}\right\\}$
the complete set of representatives of isomorphism classes of indecomposable
projective $\Theta$-modules. Let ${\mathcal{K}}=(c_{1},\ldots,c_{N})$ be a
sequence of positive integers which satisfy the following four conditions:

  1. 1)

$N=n+\operatorname{def}\Theta$.

  2. 2)

${\mathcal{N}}=(c_{z_{1}},c_{z_{2}},\ldots,c_{z_{n}})$ is a subsequence of
${\mathcal{K}}$ such that

(3.1) $\displaystyle
c_{z_{i}}=\ell(P_{i})+\sum\limits^{\ell(P_{i})}_{j=1}(\operatorname{def}(P_{\left[i+j\right]}).$

  3. 3)

The number of terms in ${\mathcal{K}}$ between $c_{z_{i}}$ and $c_{z_{i+1}}$
is equal to $\operatorname{def}(P_{i+1})$. The number of terms in
${\mathcal{K}}$ between $c_{1}$ and $c_{z_{1}}$ is equal to
$\ell(P_{1})-a_{0}-2$ where $a_{0}$ is the number of terms in ${\mathcal{K}}$
between $c_{z_{n}}$ and $c_{N}$.

  4. 4)

Let $z_{i}<a_{1}<a_{2}<\cdots<a_{\operatorname{def}(P_{i+1})}<z_{i+1}$ be
indices of subsequence of ${\mathcal{K}}$. Let
$c_{a_{m}}=\ell(P_{i})+\sum\limits^{\ell(P_{i})+m}_{j=1}\operatorname{def}(P_{\left[i+j\right]})$
for each $a_{m}$.

Then, ${\mathcal{K}}$ is an admissible sequence for a Kupisch series of a
cyclic Nakayama algebra.

To verify a given sequence of positive integers $(c_{1},\ldots,c_{N})$ forms a
Kupisch series of a cyclic Nakayama algebra, we need to check, by Lemma 1.1.9,
that for each $1\leq i\leq N$, either $c_{i}=c_{i+1}+1$ or $c_{i}\leq
c_{i+1}$. Therefore, to prove Proposition 3.1.1, we compute all possible
values of $c_{i}-c_{i+1}$ for $c_{i},c_{c_{i+1}}\in{\mathcal{K}}$ in the
following lemmas.

######  Lemma 3.1.2.

Let $c_{z_{i}}$, $c_{z_{i+1}}$ be consecutive elements of ${\mathcal{K}}$.
Recall that

$\displaystyle
c_{z_{i}}=\ell(P_{i})+\sum\limits^{\ell(P_{i})}_{j=1}\left(\operatorname{def}(P_{\left[i+j\right]})\right),\quad\
c_{z_{i+1}}=\ell(P_{i+1})+\sum\limits^{\ell(P_{i+1})}_{j=1}\left(\operatorname{def}(P_{\left[i+j+1\right]})\right).$

Then,

  1. 1)

$c_{z_{i}}-c_{z_{i+1}}=1$ if and only if $\ell(P_{i})=1+\ell(P_{i+1})$.

  2. 2)

$\ell(P_{i})\leq\ell(P_{i+1})$ implies

$\displaystyle
c_{z_{i}}-c_{z_{i+1}}=-\sum\limits^{\ell(P_{i+1})}_{j=\ell(P_{i})}\operatorname{def}P_{[i+j+1]}=-\sum\limits^{1+\operatorname{def}(P_{i+1})}_{j=1}\left(\operatorname{def}(P_{\left[i+\ell(P_{i})+j\right]})\right).$

###### Proof.

1) We start with $\ell(P_{i})=1+\ell(P_{i+1})$. Then,

$\displaystyle c_{z_{i}}$
$\displaystyle=\ell(P_{i})+\sum\limits^{\ell(P_{i})}_{j=1}\operatorname{def}(P_{\left[i+j\right]})$
$\displaystyle=1+\ell(P_{[i+1]})+\sum\limits^{1+\ell(P_{i+1})}_{j=1}\operatorname{def}(P_{[i+j]})$
$\displaystyle=1+\ell(P_{[i+1]})+\operatorname{def}(P_{[i+1]})+\sum\limits^{1+\ell(P_{i+1})}_{j=2}\operatorname{def}(P_{[i+j]})$
$\displaystyle=1+\ell(P_{[i+1]})+\operatorname{def}(P_{[i+1]})+\sum\limits^{\ell(P_{i+1})}_{j=1}\operatorname{def}(P_{[i+j+1]})$
$\displaystyle=1+\operatorname{def}(P_{[i+1]})+\ell(P_{[i+1]})+\sum\limits^{\ell(P_{i+1})}_{j=1}\operatorname{def}(P_{[i+j+1]})$
$\displaystyle=1+\operatorname{def}(P_{[i+1]})+c_{z_{i+1}}=1+c_{z_{i+1}}.$

Since $\ell(P_{i})=1+\ell(P_{i+1})$, $\operatorname{rad}P_{i}\cong P_{i+1}$
implies that $\operatorname{def}(P_{i+1})=0$.

2) Now we assume that $\ell(P_{i})\leq\ell(P_{i+1})$. By Remark 1.2.13,
$\ell(P_{i+1})-\ell(P_{i})=\operatorname{def}(P_{i+1})$. So,

$\displaystyle c_{z_{i+1}}$
$\displaystyle=\ell(P_{i+1})+\sum\limits^{\ell(P_{i+1})}_{j=1}\operatorname{def}(P_{\left[i+1+j\right]})=\sum\limits^{\ell(P_{i+1})}_{j=1}\left(\operatorname{def}(P_{\left[i+1+j\right]})+1\right)$
$\displaystyle=\sum\limits^{\ell(P_{i})-1}_{j=1}\left(\operatorname{def}(P_{\left[i+1+j\right]})+1\right)+\sum\limits^{1+\operatorname{def}(P_{i+1})}_{j=1}\left(\operatorname{def}(P_{\left[i+\ell(P_{i})+j\right]})+1\right)$

On the other hand,

$\displaystyle c_{z_{i}}$
$\displaystyle=\ell(P_{i})+\sum\limits^{\ell(P_{i})}_{j=1}\operatorname{def}(P_{\left[i+j\right]})=1+\operatorname{def}(P_{i+1})+\sum\limits^{\ell(P_{i})-1}_{j=1}\left(\operatorname{def}(P_{\left[i+j+1\right]})+1\right)$

It follows that

$\displaystyle c_{z_{i}}-c_{z_{i+1}}$
$\displaystyle=1+\operatorname{def}(P_{i+1})-\left(\sum\limits^{1+\operatorname{def}(P_{i+1})}_{j=1}\left(\operatorname{def}(P_{\left[i+\ell(P_{i})+j\right]})+1\right)\right)$
$\displaystyle=-\sum\limits^{1+\operatorname{def}(P_{i+1})}_{j=1}\operatorname{def}(P_{\left[i+\ell(P_{i})+j\right]})=-\sum\limits^{\ell(P_{i+1})}_{j=\ell(P_{i})}\operatorname{def}(P_{[i+j+1]})$

since there are $1+\operatorname{def}(P_{i+1})$-many terms with summand one. ∎

######  Lemma 3.1.3.

Let $c_{a_{m}},c_{a_{m+1}}\in{\mathcal{K}}-{\mathcal{N}}$ such that
$a_{m}+1=a_{m+1}$. Then,

$\displaystyle
c_{a_{m}}-c_{a_{m+1}}=\operatorname{def}(P_{[i+\ell(P_{i})+m+1]}).$

Let $c_{a_{1}}\notin{\mathcal{N}}$, $c_{z_{i}}\in{\mathcal{N}}$ such that
$c_{j}=c_{z_{i}},c_{j+1}=c_{a_{1}}$. Then,

$\displaystyle
c_{z_{i}}-c_{a_{1}}=-\operatorname{def}(P_{[i+\ell(P_{i})+j]}).$

Let $c_{a_{m}}\notin{\mathcal{N}}$, $c_{z_{i+1}}\in{\mathcal{N}}$ such that
$c_{j}=c_{a_{m}},c_{j+1}=c_{z_{i+1}}$. Then,

$\displaystyle
c_{a_{m}}-c_{z_{i+1}}=-\operatorname{def}(P_{[i+\ell(P_{i+1})+1]}).$

###### Proof.

By Proposition 3.1.1 4),

$\displaystyle c_{a_{m}}$
$\displaystyle=\ell(P_{i})+\sum\limits^{\ell(P_{i})+m}_{j=1}\operatorname{def}(P_{\left[i+j\right]})$
$\displaystyle=-m+\sum\limits^{\ell(P_{i})}_{j=1}\left(\operatorname{def}(P_{\left[i+j\right]})+1\right)+\sum\limits^{m}_{j=1}\left(\operatorname{def}P_{[i+\ell(P_{i})+j]}+1\right)$

Similarly,

$\displaystyle c_{a_{m+1}}$
$\displaystyle=\ell(P_{i})+\sum\limits^{\ell(P_{i})+m+1}_{j=1}\operatorname{def}(P_{\left[i+j\right]})$
$\displaystyle=-(m+1)+\sum\limits^{\ell(P_{i})}_{j=1}\left(\operatorname{def}(P_{\left[i+j\right]})+1\right)+\sum\limits^{m+1}_{j=1}\left(\operatorname{def}P_{[i+\ell(P_{i})+j]}+1\right)$

Therefore the difference
$c_{a_{m}}-c_{a_{m+1}}=-m+(m+1)-\operatorname{def}(P_{[i+\ell(P_{i})+m+1]})=\operatorname{def}(P_{[i+\ell(P_{i})+m+1]})$.
For the second equality, we can use the following expressions:

$\displaystyle
c_{z_{i}}=\sum\limits^{\ell(P_{i})}_{j=1}\left(\operatorname{def}(P_{\left[i+j\right]})+1\right)$
$\displaystyle
c_{a_{1}}=-1+\sum\limits^{\ell(P_{i})}_{j=1}\left(\operatorname{def}(P_{\left[i+j\right]})+1\right)+\left(\operatorname{def}P_{[i+\ell(P_{i})+j]}+1\right).$

Therefore $c_{z_{i}}-c_{a_{1}}=-\operatorname{def}(P_{[i+\ell(P_{i})+j]})$.
For the third equality, notice that $m=\operatorname{def}(P_{i+1})$. We get

$\displaystyle
c_{a_{m}}=-\operatorname{def}(P_{i+1})+\sum\limits^{\ell(P_{i})}_{j=1}\left(\operatorname{def}(P_{\left[i+j\right]})+1\right)+\sum\limits^{\operatorname{def}(P_{i+1})}_{j=1}\left(\operatorname{def}P_{[i+\ell(P_{i})+j]}+1\right)$

On the other hand,

$\displaystyle c_{z_{i+1}}=$
$\displaystyle\sum\limits^{\ell(P_{i+1})}_{j=1}\left(\operatorname{def}(P_{\left[i+1+j\right]})+1\right)$
$\displaystyle=$
$\displaystyle\sum\limits^{\ell(P_{i})}_{j=1}\left(\operatorname{def}(P_{\left[i+1+j\right]})+1\right)+\sum\limits^{\operatorname{def}(P_{i+1})}_{j=1}\left(\operatorname{def}(P_{\left[i+\ell(P_{i})+j+1\right]})+1\right)$

The difference is

$\displaystyle-\operatorname{def}(P_{[i+\ell(P_{i})+\operatorname{def}(P_{i+1})+1]})=-\operatorname{def}(P_{[i+\ell(P_{i+1})+1]})$

∎

###### Proof of Proposition 3.1.1.

To verify whether a given sequence of integers
${\mathcal{K}}=(c_{1},\ldots,c_{n})$ forms a Kupisch series of a cyclic
Nakayama algebra, we need to check that

  1. i)

each $c_{i}\geq 2$

  2. ii)

and if $c_{i}>c_{i+1}$, then $c_{i}-c_{i+1}=1$.

If $c_{i}\leq c_{i+1}$, $c_{i+1}$ can take any positive integer value, so it
satisfies admissibility. By lemmas 3.1.2 and 3.1.3, the case $c_{j}>c_{j+1}$
only happens when $c_{j}=c_{z_{i}}$ and $c_{j}=c_{z_{i+1}}$ with
$\ell(P_{i})=1+\ell(P_{i+1})$. It follows that the difference is one, so
${\mathcal{K}}$ is admissible sequence.
Each $c_{j}\in{\mathcal{K}}$ has at least $\ell(P_{j^{\prime}})$-many nonzero
summands. If $\Theta$ is cyclic Nakayama algebra, then $\ell(P_{i})\geq 2$ for
any $i$ which makes $c_{j}\geq 2$. Let $\Theta$ be a linear Nakayama algebra.
For simple projective $T_{i}$, the corresponding $c_{i}$ is
$\ell(P(T_{i+1}))+\sum\limits^{\ell(P(T_{i+1}))}_{j=1}\operatorname{def}(P_{[i+j]})$.
Since $\operatorname{def}P(T_{i+1})\neq 0$, we get $c_{i}\geq 2$.

∎

######  Proposition 3.1.4.

Let $\Theta$ be a Nakayama algebra of rank $n$. Let ${\mathcal{K}}$ be the
admissible sequence constructed from $\Theta$ in proposition 3.1.1. If a
cyclic Nakayama algebra $\Lambda$ is given by admissible sequence
${\mathcal{K}}$, then, $\bm{\varepsilon}(\Lambda)\cong\Theta$ and
$\operatorname{def}\Lambda=\operatorname{def}\Theta$.

###### Proof.

In proposition 3.1.1, we construct the sequence ${\mathcal{K}}$ for any given
Nakayama algebra $\Theta$, and proved that it is admissible. Let $\Lambda$ be
the algebra given by such ${\mathcal{K}}$. Let
$\left\\{Q_{1},\ldots,Q_{N}\right\\}$ be the complete set of representatives
of isomorphism classes of indecomposable $\Lambda$-modules and
$\ell(Q_{i})=c_{i}\in{\mathcal{K}}$. We divide the proof into the following
steps.

  1. 1)

$|{\mathcal{S}}(\Lambda)|=n$.

  2. 2)

$\operatorname{def}\Lambda=\operatorname{def}\Theta$.

  3. 3)

${\mathcal{B}}(\Lambda)$-filtered indecomposable projective $\Lambda$-modules
are $Q_{z_{1}},\ldots,Q_{z_{n}}$.

1) By lemma 1.2.14, $|{\mathcal{S}}(\Lambda)|=|\\{c_{i}\mid c_{i}\leq
c_{i+1},1\leq i\leq N\\}$ where $c_{N+1}=c_{1}$. There are two types of
indecomposable projective $\Lambda$-modules, either
$\ell(Q_{i})=c_{i}\notin{\mathcal{N}}$ or $\ell(Q_{i})=c_{i}\in{\mathcal{N}}$
where ${\mathcal{N}}\subset{\mathcal{K}}$ by the construction given in
proposition 3.1.1. We have the following subsets

$\displaystyle\\{c_{i}\mid c_{i}\leq c_{i+1},\,1\leq i\leq N\\}=\\{c_{j}\mid
c_{j}\leq c_{j+1},\,c_{j}\notin{\mathcal{N}}\\}\cup\\{c_{z_{i}}\mid
c_{z_{i}}\leq c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}.$

By modifying lemma 3.1.3 for new indices, any $c_{j}\notin{\mathcal{N}}$
satisfies $c_{j}\leq c_{j+1}$. Therefore the cardinality of the first set is
$|{\mathcal{K}}|-|{\mathcal{N}}|=\operatorname{def}\Theta$.

Now, we consider the partition of the set $\\{c_{z_{i}}\mid
c_{z_{i}}\in{\mathcal{N}}\\}$ of size $n=|{\mathcal{N}}|$ into

$\displaystyle\\{c_{z_{i}}\mid c_{z_{i}}\leq
c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}\cup\\{c_{z_{i+1}}\mid
c_{z_{i}}>c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}.$

We claim that the cardinality of the first subset is
$n-\operatorname{def}\Theta$. Because, by lemma 3.1.2, $c_{z_{i}}>c_{z_{i+1}}$
if and only if $\ell(P_{i})=1+\ell(P_{i+1})$. By lemma 1.2.14,
$|\\{\ell(P_{i+1})\,|\,\ell(P_{i})>\ell(P_{i+1})\\}|=\operatorname{def}\Theta$
which makes the size of the first set $n-\operatorname{def}\Theta$. Therefore

$\displaystyle|{\mathcal{S}}(\Lambda)|$ $\displaystyle=|\\{c_{j}\mid c_{j}\leq
c_{j+1},\,j\notin{\mathcal{N}}\\}|+|\\{c_{z_{i}}\mid c_{z_{i}}\leq
c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}|$
$\displaystyle=\operatorname{def}\Theta+\left(n-\operatorname{def}\Theta\right)=n.$

2) By Poposition 1.2.6,
$\operatorname{rank}\Lambda=\operatorname{def}\Lambda+\\#\operatorname{rel}\Lambda$.
By the previous step, $|{\mathcal{S}}(\Lambda)|=n=\\#\operatorname{rel}$.
Since $\operatorname{rank}\Lambda=|{\mathcal{K}}|=n+\operatorname{def}\Theta$,
claim follows.
3) Notice that there are $n$ ${\mathcal{B}}(\Lambda)$-filtered indecomposable
projective $\Lambda$-modules, since
$|{\mathcal{B}}(\Lambda)|=|{\mathcal{S}}(\Lambda)|=n$. The lengths of elements
of ${\mathcal{B}}(\Lambda)$ are $c_{i+1}-c_{i}+1$ provided that $c_{i}\leq
c_{i+1}$ by lemma 1.2.14. In lemmas 3.1.2 and 3.1.3, we calculated all
possible differences of $c_{i}-c_{i+1}$ which are always of the form
$-\operatorname{def}(P_{j})$ when $c_{i}\leq c_{i+1}$. Moreover, by
proposition 3.1.1 2), for any $1\leq i\leq n$, $\operatorname{def}(P_{i})+1$
is a summand of $Q_{z_{i}}$. Therefore, every $Q_{z_{i}}$ is
${\mathcal{B}}(\Lambda)$-filtered. Because there are $n$-many $Q_{z_{i}}$,
hence we get all.

Now we can show that $\bm{\varepsilon}(\Lambda)\cong\Theta$. It is enough to
compute dimensions of projective $\bm{\varepsilon}(\Lambda)$-modules, since
indecomposable $\bm{\varepsilon}(\Lambda)$-modules are uniserial, i.e.
dimension and length of indecomposable modules are equal. Therefore,
$\dim_{\mathbb{K}}\operatorname{Hom}_{A}\left(\bigoplus\limits_{1\leq i\leq
n}Q_{z_{i}},Q_{z_{i}}\right)=\ell_{{\mathcal{B}}(\Lambda)}(Q_{z_{i}})=\ell(P_{i})$,
since the number of elements ${\mathcal{B}}(\Lambda)$ appearing in each
$Q_{z_{i}}$ is $\ell(P_{i})$. Hence Kupisch series of
$\bm{\varepsilon}(\Lambda)$ is equal to Kupisch series of $\Theta$ upto cyclic
permutation. ∎

###  3.2. General case, i.e. Not Defect-Invariant

Let $\Theta$ be a Nakayama algebra of rank $n$, and let
$\\{P_{1},P_{2},\ldots,P_{n}\\}$ be the ordered set of indecomposable
projective $\Theta$-modules. In this subsection, we will construct cyclic
Nakayama algebra $\Lambda$ such that $\varepsilon(\Lambda)\cong\Theta$ and
$\operatorname{def}(\Lambda)>\operatorname{def}(\Theta)$.

######  Definition 3.2.1.

For any finite sequence of integers $(c_{1},\ldots,c_{n})$ we define the
distance between $c_{i}$ and $c_{j}$ with $j>i$ as
$\min\\{[j-i+1],[i-j+1]\\}$, denoted by $\operatorname{dist}(c_{i},c_{j})$.

We start with a Nakayama algebra $\Theta$ and a complete set of
representatives of indecomposable projective $\Theta$-modules
$\\{P_{1},\dots,P_{n}\\}$. Next we consider a sequence of integers
${\mathcal{K}}=\left(c_{1},\ldots,c_{N}\right)$ and show that this sequence is
a Kupish series for a Nakayama algebra $\Lambda$ providing that it satisfies
four conditions: 1), 2), 3), 4) as stated in the Proposition 3.1.1. Before
proving this proposition we will explain the role or purpose of each of these
conditions (as they are numbered in the proposition)..

Now we present the technical proposition, which enables us to construct a
cyclic Nakayama algebra for a given connected Nakayama algebra $\Theta$ and a
vector $w=(w_{1},\dots,w_{n})$ such that $w_{i}\geq\operatorname{def}(P_{i})$.
By 1.2.6, the rank of $\Lambda$ is $\operatorname{rank}\Theta+\sum_{i}w_{i}$
which is at part 1). Each $w_{i}+1$ will correspond to length of filtered
modules which is hidden in part 2). For non-filtered $\Lambda$ modules, by
2.2.5, there are $\operatorname{def}P_{i+1}$-many possible socles and
$w_{i+1}$ many nonisomorphic tops, which we describe the lengths in part 4).
There are $w_{i}$ non-filtered projective modules between filtered projectives
which is in part 3). Below is the proposition.

######  Proposition 3.2.2.

Let $\Theta$ be a Nakayama algebra $\\{P_{1},\ldots,P_{n}\\}$ be the complete
set of representatives of isomorphism classes of indecomposable projective
$\Theta$-modules. Let $w\in\mathbb{R}^{n}$ be a vector satisfying
$w_{i}\geq\operatorname{def}(P_{i})$. Let ${\mathcal{K}}=(c_{1},\ldots,c_{N})$
be a sequence of positive integers which satisfy the following conditions:

  1. 1)

$N=n+\sum^{n}_{i=1}w_{i}$

  2. 2)

${\mathcal{N}}=(c_{z_{1}},c_{z_{2}},\ldots,c_{z_{n}})$ is a subsequence of
${\mathcal{K}}$ such that

(3.2) $\displaystyle
c_{z_{i}}=\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)$

  3. 3)

The number of terms in ${\mathcal{K}}$ between $c_{z_{i}}$ and $c_{z_{i+1}}$
is equal to $w_{i+1}$. The number of terms in ${\mathcal{K}}$ between $c_{1}$
and $c_{z_{1}}$ is equal to $w_{1}-a_{0}-2$ where $a_{0}$ is the number of
terms in ${\mathcal{K}}$ between $c_{z_{n}}$ and $c_{N}$.

  4. 4)

For any term $c_{a_{j}}\in{\mathcal{K}}-{\mathcal{N}}$ which satisfies
${z_{i}}<{a_{1}}<\cdots<{a_{w_{i+1}}}<{z_{i+1}}$, there is a subsequence of
indices
$\\{b_{1},b_{2},\dots,b_{\operatorname{def}(P_{i+1})}\\}\subset\\{a_{1},\
a_{2},\dots,a_{w_{i+1}}\\}$ such that

(3.3) $\displaystyle
c_{b_{m}}=-\operatorname{dist}(c_{b_{m}},c_{z_{i}})+\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)+\sum\limits^{m}_{j=1}\left(w_{[i+\ell(P_{i})+j]}+1\right)$

Any term $c_{b_{m}}\in{\mathcal{K}}-{\mathcal{N}}$ with index either
$z_{n}<b_{m}\leq N$ or $1\leq b_{m}<z_{1}$ is given by

(3.4) $\displaystyle
c_{b_{m}}=-\operatorname{dist}(c_{b_{m}},c_{z_{n}})+\sum\limits^{\ell(P_{n})}_{j=1}\left(w_{\left[j\right]}+1\right)+\sum\limits^{m}_{j=1}\left(w_{[\ell(P_{n})+j]}+1\right)$

  5. 5)

For each $b_{t}<a_{m}<b_{t+1}$, we set

(3.5) $\displaystyle
c_{a_{m}}=-\operatorname{dist}(c_{a_{m}},c_{z_{i}})+\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)+\sum\limits^{t}_{j=1}\left(w_{[i+\ell(P_{i})+j]}+1\right)$

  6. 6)

For each $z_{i}<a_{m}<b_{1}$, we set

(3.6) $\displaystyle
c_{a_{m}}=-\operatorname{dist}(c_{a_{m}},c_{z_{i}})+\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)$

  7. 7)

For each $b_{\operatorname{def}P_{i+1}}<a_{m}<z_{i+1}$, we set

(3.7) $\displaystyle
c_{a_{m}}=-\operatorname{dist}(c_{a_{m}},c_{z_{i}})+\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)+\sum\limits^{\operatorname{def}P_{i+1}}_{j=1}\left(w_{[i+\ell(P_{i})+j]}+1\right)$

The sequence ${\mathcal{K}}$ satisfying all these conditions forms an
admissible sequence of a cyclic Nakayama algebra.

To verify whether a given sequence of integers
${\mathcal{K}}=(c_{1},\ldots,c_{n})$ forms a Kupisch series of a cyclic
Nakayama algebra, we need to check that

  1. i)

each $c_{i}\geq 2$

  2. ii)

and if $c_{i}>c_{i+1}$, then $c_{i}-c_{i+1}=1$.

If $c_{i}\leq c_{i+1}$, $c_{i+1}$ can take any positive integer value, so it
satisfies admissibility. Hence, to prove proposition 3.2.2, we need to compute
all possible $c_{i}-c_{i+1}$ for $c_{i},c_{c_{i+1}}\in{\mathcal{K}}$ which are
stated in the following two lemmas 3.2.3 and 3.2.4.

######  Lemma 3.2.3.

Let $c_{z_{i}}$ and $c_{z_{i+1}}$ be consecutive elements of ${\mathcal{K}}$.
Then,

  1. 1)

$c_{z_{i}}-c_{z_{i+1}}=1$ if and only if $\ell(P_{i})=1+\ell(P_{i+1})$

  2. 2)

$c_{z_{i}}-c_{z_{i+1}}<0$ if and only if $\ell(P_{i})=\ell(P_{i+1})$.

In the latter case, we have

(3.8) $\displaystyle c_{z_{i}}-c_{z_{i+1}}=-w_{[i+\ell(P_{i})+1]}.$

###### Proof.

Notice that $c_{z_{i}}$ and $c_{z_{i+1}}$ are consecutive elements of
${\mathcal{K}}$ if and only if $w_{i+1}=0$ by construction of ${\mathcal{K}}$
in proposition 3.2.2 3).

We start with $\ell(P_{i})=1+\ell(P_{i+1})$. Then,

$\displaystyle c_{z_{i}}$
$\displaystyle=\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)$
$\displaystyle=1+(w_{i+1})+\sum\limits^{\ell(P_{i+1})+1}_{j=2}\left((w_{\left[i+j\right]})+1\right)$
$\displaystyle=1+(w_{i+1})+\sum\limits^{\ell(P_{i+1})}_{j=1}\left((w_{\left[i+j+1\right]})+1\right)$
$\displaystyle=1+(w_{i+1})+c_{z_{i+1}}=1+c_{z_{i+1}}.$

The difference $c_{z_{i}}-c_{z_{i+1}}$ is

(3.9) $\displaystyle c_{z_{i}}-c_{z_{i+1}}$
$\displaystyle=\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)-\sum\limits^{\ell(P_{i+1})}_{j=1}\left(w_{\left[i+j+1\right]}+1\right)$
(3.10)
$\displaystyle=1+w_{[i+1]}-\sum\limits^{\ell(P_{i+1})}_{j=\ell(P_{i})}(w_{[i+j+1]}+1)$
(3.11)
$\displaystyle=1-\sum\limits^{\ell(P_{i+1})}_{j=\ell(P_{i})}(w_{[i+j+1]}+1)$

Therefore, $c_{z_{i}}-c_{z_{i+1}}=1$ implies that the second summand is zero.
This is possible when $\ell(P_{i})+i=\ell(P_{i+1})+i+1$, so
$\ell(P_{i})=1+\ell(P_{i+1})$. ∎

Now we assume that $\ell(P_{i})\leq\ell(P_{i+1})$. By remark 1.2.13,
$\ell(P_{i+1})-\ell(P_{i})=\operatorname{def}(P_{i+1})$. So,

$\displaystyle c_{z_{i+1}}$
$\displaystyle=\sum\limits^{\ell(P_{i+1})}_{j=1}\left((w_{\left[i+1+j\right]})+1\right)$
$\displaystyle=\sum\limits^{\ell(P_{i})-1}_{j=1}\left((w_{\left[i+1+j\right]})+1\right)+\sum\limits^{1+\operatorname{def}(P_{i+1})}_{j=1}\left((w_{\left[i+\ell(P_{i})+j\right]})+1\right)$

On the other hand,

$\displaystyle c_{z_{i}}$
$\displaystyle=\sum\limits^{\ell(P_{i})}_{j=1}\left((w_{\left[i+j\right]})+1\right)=1+(w_{i+1})+\sum\limits^{\ell(P_{i})-1}_{j=1}\left((w_{\left[i+j+1\right]})+1\right)$

It follows that

$\displaystyle c_{z_{i}}-c_{z_{i+1}}$
$\displaystyle=1+(w_{i+1})-\left(\sum\limits^{1+\operatorname{def}(P_{i+1})}_{j=1}\left((w_{\left[i+\ell(P_{i})+j\right]})+1\right)\right)$
$\displaystyle=1-\sum\limits^{1+\operatorname{def}(P_{i+1})}_{j=1}(w_{\left[i+\ell(P_{i})+j\right]}+1)=-w_{[i+\ell(P_{i})+1]}\leq
0$

since $w_{i+1}=0$ and this implies that $\operatorname{def}P_{i+1}=0$. Hence
we get $c_{z_{i}}-c_{z_{i+1}}\leq 0$.

######  Lemma 3.2.4.

Let $c_{a_{m}},c_{a_{m+1}}\in{\mathcal{N}}$ such that $a_{m}+1=a_{m+1}$. Then,

  1. 1)

if there exist indices $b_{t}<a_{m}<a_{m+1}<b_{t}$, then
$c_{a_{m}}-c_{a_{m+1}}=1$

  2. 2)

if there exist indices $z_{i}<a_{m}<a_{m+1}<b_{1}$, then
$c_{a_{m}}-c_{a_{m+1}}=1$

  3. 3)

if there exist indices $b_{t}=a_{m}<a_{m+1}<b_{t+1}$, then
$c_{a_{m}}-c_{a_{m+1}}=1$

  4. 4)

if $a_{m}=b_{t}$, $a_{m+1}=b_{t+1}$, then
$c_{a_{m}}-c_{a_{m+1}}=-w_{[i+\ell(P_{i})+t+1]}$

  5. 5)

if $a_{m}<a_{m+1}=b_{t+1}$, then
$c_{a_{m}}-c_{a_{m+1}}=--w_{[i+\ell(P_{i})+t+1]}$.

###### Proof.

For the first three cases, the difference $c_{a_{m}}-c_{a_{m+1}}$ is

(3.12)
$\displaystyle-\operatorname{dist}(c_{a_{m}},c_{z_{i}})+\operatorname{dist}(c_{a_{m+1}},c_{z_{i}})=a_{m+1}-a_{m}=1.$

Let $c_{b_{t}}$, $c_{b_{t+1}}$ be consecutive in ${\mathcal{K}}$, i.e.,
$b_{t+1}=b_{t}+1$. Then,

$\displaystyle c_{b_{t}}-c_{b_{t+1}}=$
$\displaystyle-\operatorname{dist}(c_{b_{t}},c_{z_{i}})+\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)+\sum\limits^{t}_{j=1}\left(w_{[i+\ell(P_{i})+j]}+1\right)$
$\displaystyle+\operatorname{dist}(c_{b_{t+1}},c_{z_{i}})-\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)-\sum\limits^{t+1}_{j=1}\left(w_{[i+\ell(P_{i})+j]}+1\right)$
$\displaystyle=-w_{[i+\ell(P_{i})+t+1]}$

Let $c_{a_{m}}$ and $c_{b_{t+1}}$ be consecutive in ${\mathcal{K}}$, i.e.,
$b_{t+1}=a_{m+1}$. Then,

$\displaystyle
c_{a_{m}}=-\operatorname{dist}(c_{a_{m}},c_{z_{i}})+\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)+\sum\limits^{t}_{j=1}\left(w_{[i+\ell(P_{i})+j]}+1\right)$
$\displaystyle
c_{b_{t+1}}=-\operatorname{dist}(c_{b_{t+1}},c_{z_{i}})+\sum\limits^{\ell(P_{i})}_{j=1}\left(w_{\left[i+j\right]}+1\right)+\sum\limits^{t+1}_{j=1}\left(w_{[i+\ell(P_{i})+j]}+1\right)$

the difference is $-w_{[i+\ell(P_{i})+t+1]}$ ∎

Now, we state the proof of proposition 3.2.2.

###### Proof.

We want to show that ${\mathcal{K}}$ is an admissible sequence of a cyclic
Nakayama algebra. Notice that each $c_{i}$ has at least $\ell(P_{j})$ many
summands, so $c_{i}\geq 2$ for all $i$. In lemmas 3.2.4 and 3.2.3, we
calculated all possible differences, and show that $c_{i}>c_{i+1}$ implies
that $c_{i}=c_{i+1}+1$. Hence, ${\mathcal{K}}$ is an admissible sequence. ∎

######  Proposition 3.2.5.

Let $\Theta$ be either a cyclic Nakayama algebra or a connected linear
Nakayama algebra of rank $n$. Let ${\mathcal{K}}$ be the admissible sequence
constructed from $\Theta$ as in proposition 3.2.2. If a cyclic Nakayama
algebra $\Lambda$ is given by admissible sequence ${\mathcal{K}}$, then,
$\bm{\varepsilon}(\Lambda)\cong\Theta$ and
$\operatorname{def}\Lambda=|w|\geq\operatorname{def}\Theta$.

###### Proof.

In proposition 3.1.1, we construct the sequence ${\mathcal{K}}$ for any given
cyclic Nakayama algebra $\Theta$, and proved that it is admissible. Let
$\Lambda$ be the algebra given by such ${\mathcal{K}}$. Let
$\left\\{Q_{1},\ldots,Q_{N}\right\\}$ be the complete set of representatives
of isomorphism classes of indecomposable $\Lambda$-modules and
$\ell(Q_{i})=c_{i}\in{\mathcal{K}}$. We divide the proof into the following
steps.

  1. 1)

$|{\mathcal{S}}(\Lambda)|=n$

  2. 2)

$\operatorname{def}\Lambda=|w|$

  3. 3)

${\mathcal{B}}(\Lambda)$-filtered indecomposable projective $\Lambda$-modules
are $Q_{z_{1}},\ldots,Q_{z_{n}}$.

First, we show that the cardinality of the socle set of $\Lambda$ is $n$,
i.e., $|{\mathcal{S}}(\Lambda)|=n$. By lemma 1.2.14,
$|{\mathcal{S}}(\Lambda)|=|\\{c_{i}\mid c_{i}\leq c_{i+1},1\leq i\leq N\\}$
where $c_{N+1}=c_{1}$. There are two types of indecomposable projective
$\Lambda$-modules, either $\ell(Q_{i})=c_{i}\notin{\mathcal{N}}$ or
$\ell(Q_{i})=c_{i}\in{\mathcal{N}}$ where ${\mathcal{N}}\subset{\mathcal{K}}$
by the construction given in proposition 3.2.2. We have the following subsets

$\displaystyle\\{c_{i}\mid c_{i}\leq c_{i+1},\,1\leq i\leq N\\}=\\{c_{j}\mid
c_{j}\leq c_{j+1},\,c_{j}\notin{\mathcal{N}}\\}\cup\\{c_{z_{i}}\mid
c_{z_{i}}\leq c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}.$

The cardinality of the first subset is $\operatorname{def}\Theta$. Because, by
lemma 3.2.4, $c_{j}\leq c_{j+1}$ if $c_{j+1}$ is of the form $c_{b_{t}}$ for
some $z_{i}<b_{t}<z_{i+1}$. In total, there are
$\sum_{i}\operatorname{def}(P_{i})$-many such terms. Hence the cardinality is
$\operatorname{def}\Theta$.

Now, we consider the partition of the set $\\{c_{z_{i}}\mid
c_{z_{i}}\in{\mathcal{N}}\\}$ of size $n=|{\mathcal{N}}|$ into

$\displaystyle\\{c_{z_{i}}\mid c_{z_{i}}\leq
c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}\cup\\{c_{z_{i+1}}\mid
c_{z_{i}}>c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}.$

We claim that the cardinality of the first subset is
$n-\operatorname{def}\Theta$. Because, by lemma 3.2.3, $c_{z_{i}}>c_{z_{i+1}}$
if and only if $\ell(P_{i})=1+\ell(P_{i+1})$. By lemma 1.2.14,
$|\\{\ell(P_{i+1})\,|\,\ell(P_{i})>\ell(P_{i+1})\\}|=\operatorname{def}\Theta$
which makes the size of the first set $n-\operatorname{def}\Theta$. Therefore

$\displaystyle|{\mathcal{S}}(\Lambda)|$ $\displaystyle=|\\{c_{j}\mid c_{j}\leq
c_{j+1},\,j\notin{\mathcal{N}}\\}|+|\\{c_{z_{i}}\mid c_{z_{i}}\leq
c_{z_{i+1}},\,c_{z_{i}}\in{\mathcal{N}}\\}|$
$\displaystyle=\operatorname{def}\Theta+\left(n-\operatorname{def}\Theta\right)=n.$

Secondly, we compute the defect of $\Lambda$, i.e.,
$\operatorname{def}\Lambda=|w|$. By proposition 1.2.6,
$\operatorname{rank}\Lambda=\operatorname{def}\Lambda+\\#\operatorname{rel}\Lambda$.
Together with the previous claim and proposition 2.2.1,
$|{\mathcal{S}}(\Lambda)|=n=\\#\operatorname{rel}\Lambda$. Since
$\operatorname{rank}\Lambda=|{\mathcal{K}}|=n+|w|$, claim follows.

Finally, We show that ${\mathcal{B}}(\Lambda)$-filtered indecomposable
projective $\Lambda$-modules are $Q_{z_{1}},\ldots,Q_{z_{n}}$. Notice that
there are $n$ ${\mathcal{B}}(\Lambda)$-filtered indecomposable projective
$\Lambda$-modules, since
$|{\mathcal{B}}(\Lambda)|=|{\mathcal{S}}(\Lambda)|=n$. The lengths of elements
of ${\mathcal{B}}(\Lambda)$ are $c_{i+1}-c_{i}+1$ provided that $c_{i}\leq
c_{i+1}$ by lemma 1.2.14. In lemmas 3.2.3 and 3.2.4, we calculated all
possible differences of $c_{i}-c_{i+1}$ which are always of the form
$-w_{[i]}$ when $c_{i}\leq c_{i+1}$. Moreover, by proposition 3.2.2 2), for
any $1\leq i\leq n$, $w_{[i]}+1$ is a summand of $Q_{z_{i}}$. Therefore, every
$Q_{z_{i}}$ is ${\mathcal{B}}(\Lambda)$-filtered. Because there are $n$-many
$Q_{z_{i}}$, hence we get all.

Now we can show that $\bm{\varepsilon}(\Lambda)\cong\Theta$. It is enough to
compute dimensions of projective $\bm{\varepsilon}(\Lambda)$-modules, since
indecomposable $\bm{\varepsilon}(\Lambda)$-modules are uniserial, i.e.,
dimension and length of indecomposable modules are equal. Therefore,

$\displaystyle\dim_{\mathbb{K}}\operatorname{Hom}_{\Lambda}\left(\bigoplus\limits_{1\leq
i\leq
n}Q_{z_{i}},Q_{z_{i}}\right)=\ell_{{\mathcal{B}}(\Lambda)}(Q_{z_{i}})=\ell(P_{i})$

since the number of elements ${\mathcal{B}}(\Lambda)$ appearing in each
$Q_{z_{i}}$ is $\ell(P_{i})$. Hence Kupisch series of
$\bm{\varepsilon}(\Lambda)$ is equal to Kupisch series of $\Theta$ upto cyclic
permutation. ∎

######  Example 3.2.6.

Let $\Theta$ be given by Kupisch series $(2,2,1,3,2,1)$. Then, defect
invariant reverse $\Lambda$ is given by $(2,4,3,3,3,4,3,2,2)$. If we choose
$w=(1,1,0,3,0,0)$, then

$\displaystyle(3,\underline{2},4,3,\underline{3},\underline{3},4,3,2,\underline{4})$
$\displaystyle(3,\underline{2},4,3,\underline{3},\underline{3},4,3,2,\underline{3})$
$\displaystyle(3,\underline{5},4,3,\underline{3},\underline{3},4,3,2,\underline{4})$
$\displaystyle(3,\underline{5},4,3,\underline{3},\underline{3},4,3,2,\underline{3})$

are Kupisch series of nonisomorphic Nakayama algebras $\Lambda_{i}$ such that
$\bm{\varepsilon}(\Lambda_{i})\cong\Theta$ where $\underline{c_{i}}$ is the
length of non-filtered projective $\Lambda_{i}$-modules.

##  4\. Nakayama Algebras with higher homological dimensions

In this section, our goal is to establish results concerning specific
homological dimensions of Nakayama algebras. Initially, we revisit some
statements proved in [29]. Afterward, we analyze algebras characterized by
dominant dimensions in the subsequent subsections. To clarify what we mean by
algebras characterized by dominant dimension, we introduce a comprehensive
definition here:

######  Definition 4.0.1.

Let $A$ be a finite dimensional algebra over field $\mathbb{K}$.

  1. 1)

$A$ is called _dominant Auslander-Gorenstein_ if injective dimension of each
indecomposable projective $A$-module is bounded by its dominant dimension [8].

  2. 2)

$A$ is called _dominant Auslander-regular_ if $A$ is dominant Auslander-
Gorenstein algebra of finite global dimension [8].

  3. 3)

$A$ is called _minimal Auslander-Gorenstein_ algebra, if it is dominant
Auslander-Gorenstein and nonzero injective dimensions of projective modules
are equal [14].

  4. 4)

$A$ is called _higher Auslander algebra_ if it is minimal Auslander-Gorenstein
algebra with finite global dimension [13].

This section is organized as follows. First, we review some statements from
[28] and [29] regarding injective objects. Next, we present proofs for
Theorems B and C. In the subsequent sections, we systematically study algebras
as defined in 4.0.1.

###  4.1. Properties of Injective Modules and Homological Dimensions

######  Lemma 4.1.1.

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

$\displaystyle X\hookrightarrow B\hookrightarrow P$

Then the quotient $P/X$ is an injective $\Lambda$-module.

######  Lemma 4.1.2.

If $I$ is an indecomposable injective non-projective $\Lambda$-module, then
$\Omega^{1}(I)$ is a proper submodule of an element $B$ of the base-set
${\mathcal{B}}(\Lambda)$.

######  Proposition 4.1.3.

If $I$ is an indecomposable injective non-projective
$\bm{\varepsilon}(\Lambda)$-module, then there exists an injective non-
projective $\Lambda$-module $J$ such that
$\operatorname{Hom}_{\Lambda}({\mathcal{P}},{\Omega^{2}(J)})\cong I$.

######  Proposition 4.1.4.

Let $\Lambda$ be a cyclic Nakayama algebra such that
$\operatorname{dom.\\!dim}\Lambda\geq 3$. Then, there is a bijection between
indecomposable injective non-projective $\Lambda$-modules $I$ and
indecomposable injective non-projective $\bm{\varepsilon}(\Lambda)$-modules
$J$ via the correspondence $I=\Sigma^{2}(\Delta(J))$ and
$\Delta(J)=\Omega^{2}(I)$.

######  Corollary 4.1.5.

Let $\Lambda$ be a cyclic Nakayama algebra such that
$\operatorname{dom.\\!dim}\Lambda\geq 3$. Then,
$\operatorname{def}\Lambda=\operatorname{def}\bm{\varepsilon}(\Lambda)$.

######  Proposition 4.1.6.

If $I_{1},I_{2},\ldots,I_{m}$ are indecomposable nonisomorphic injective
$\Lambda$-modules satisfying $\Omega^{2}(I_{i})\cong\Omega^{2}(I_{j})$ for all
$1\leq i\neq j\leq m$, then projective covers $P(I_{1}),$
$P(I_{2}),\ldots,P(I_{m})$ are isomorphic. In particular
$\operatorname{top}I_{i}\cong\operatorname{top}I_{j}$ and
$\operatorname{top}P(\Omega^{1}(I_{i}))\ncong\operatorname{top}P(\Omega^{1}(I_{j}))$
for all $1\leq i\neq j\leq m$.

######  Definition 4.1.7.

[4] Let $A$ be an artin algebra. The dominant dimension of an $A$-module $M$,
which we denote by $\operatorname{dom.\\!dim}_{A}M$, is the maximum integer
$t$ (or $\infty$) having the property that if $0\rightarrow M\rightarrow
I_{0}\rightarrow I_{1}\rightarrow\cdots\rightarrow I_{t}\rightarrow\cdots$ is
a minimal injective resolution of $M$, then $I_{j}$ is projective for all
$j<t$ (or $\infty$). Dominant dimension of $A$ is

(4.1)
$\displaystyle\operatorname{dom.\\!dim}A=\inf\left\\{\operatorname{dom.\\!dim}P\,|\,P\text{
is projective non-injective }A\,\text{module}\right\\}.$

Because for any artin algebra $\Lambda$
$\operatorname{dom.\\!dim}\Lambda=\operatorname{dom.\\!dim}\Lambda^{op}$ holds
[18], 4.1 is equivalent to

(4.2) $\displaystyle\operatorname{dom.\\!dim}A=\inf\left\\{\operatorname{co-
dom.\\!dim}I\,|\,I\text{ is injective non-projective
}A\,\text{module}\right\\}.$

where $\operatorname{co-dom.\\!dim}_{A}I$ is the maximum integer $t$ (or
$\infty$) having the property that if $\cdots\rightarrow
P_{t}\rightarrow\cdots\rightarrow P_{0}\rightarrow I\rightarrow 0$ is a
minimal projective resolution of injective module $I$, then $P_{j}$ is
injective for all $j<t$ (or $\infty$). Global dimension of algebra $A$ is the
supremum of projective dimensions of all finitely generated $A$-modules:

$\displaystyle\operatorname{gl\,\\!dim}A=\sup\left\\{\operatorname{p\\!\dim}M|\,M\in\operatorname{mod-\\!}A\right\\}$

where mod-$A$ denotes the category of finitely generated $A$-modules. We call
an algebra Gorenstein if the projective dimensions of injective modules are
finite.

Proposition 2.1.3 gives a connection between projective dimensions of
$\Lambda$ and $\bm{\varepsilon}(\Lambda)$ modules in the following way.

######  Proposition 4.1.8.

[28] If X is an indecomposable ${\mathcal{B}}(\Lambda)$-filtered
$\Lambda$-module which is not projective-injective, then there exist a module
$M$ such that $X\cong\Omega^{2}(M)$. Furthermore, $M$ can be chosen so that,
in its minimal projective presentation $P_{1}\rightarrow P_{0}\rightarrow
M\rightarrow 0$ both $P_{0}$ and $P_{1}$ are projective-injective. In
particular $X\rightarrow P_{0}\rightarrow P_{1}$ is the injective
copresentation of $X$ and $M=\Omega^{-2}X$.

######  Remark 4.1.9.

The evaluation functor $\operatorname{Hom}_{\Lambda}({\mathcal{P}},-)$ is
exact, because ${\mathcal{P}}$ is a projective module. If $M\in\
Filt({\mathcal{B}}(\Lambda))$ has projective resolution

$\displaystyle\cdots\rightarrow P_{2}\rightarrow P_{1}\rightarrow
P_{0}\rightarrow M\rightarrow 0$

then the projective resolution of
$X=\operatorname{Hom}_{\Lambda}({\mathcal{P}},M)\in\operatorname{mod-\\!}\bm{\varepsilon}(\Lambda)$
is given by:

$\displaystyle\cdots\rightarrow\operatorname{Hom}_{\Lambda}({\mathcal{P}},P_{2})\rightarrow\operatorname{Hom}_{\Lambda}({\mathcal{P}},P_{1})\rightarrow\operatorname{Hom}_{\Lambda}({\mathcal{P}},P_{0})\rightarrow
X\rightarrow 0$

Each $\operatorname{Hom}_{\Lambda}({\mathcal{P}},P_{i})$, $i\geq 0$ is a
projective $\bm{\varepsilon}(\Lambda)$-module. Any nonsplit exact sequence

$\displaystyle 0\rightarrow X_{2}\rightarrow X_{1}\rightarrow X_{0}\rightarrow
0$

in $\operatorname{mod-\\!}\bm{\varepsilon}(\Lambda)$ can be carried into
$\operatorname{mod-\\!}\Lambda$ by the categorical equivalence, i.e.,

$\displaystyle 0\rightarrow\Delta X_{2}\rightarrow\Delta
X_{1}\rightarrow\Delta X_{0}\rightarrow 0$

is exact in $\operatorname{mod-\\!}\Lambda$ where each
$X_{i}\cong\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Delta X_{i})$.

######  Remark 4.1.10.

Let $\Lambda$ be a cyclic Nakayama algebra. We have the following reductions:

  1. 1)

$\operatorname{gl\,\\!dim}\Lambda=\operatorname{gl\,\\!dim}{\bm{\varepsilon}}(\Lambda)+2$
if $\operatorname{gl\,\\!dim}\Lambda\geq 2$ [28]

  2. 2)

$\operatorname{dom.\\!dim}\Lambda=\operatorname{dom.\\!dim}{\bm{\varepsilon}}(\Lambda)+2$
if $\operatorname{dom.\\!dim}\Lambda\geq 3$ [28]

  3. 3)

$\operatorname{fin.\\!dim}\Lambda=\operatorname{fin.\\!dim}{\bm{\varepsilon}}(\Lambda)+2$
if $\varphi$-$\dim\Lambda\geq 2$ [28]

  4. 4)

$\operatorname{del}\Lambda=\operatorname{del}{\bm{\varepsilon}}(\Lambda)+2$ if
$\varphi$-$\dim\Lambda\geq 2$ [30].

  5. 5)

$\varphi$-$\dim\Lambda=\varphi$-$\dim{\bm{\varepsilon}}(\Lambda)+2$ if
$\varphi$-$\dim\Lambda\geq 2$ [27]

where $\operatorname{fin.\\!dim}$, $\operatorname{del}$, $\varphi$-$\dim$
stand for finitistic dimension, delooping level and Igusa-Todorov function
[12]. All of these dimensions share a unified sharp upper bound, $2r$, where
$r=|{\mathcal{S}}(\Lambda)|$.

######  Remark 4.1.11.

Let $\Lambda$ be a cyclic Nakayama algebra. If $\Delta
S\in{\mathcal{B}}(\Lambda)$ has projective submodule, then
$\bm{\varepsilon}(\Lambda)$ is not cyclic.

######  Lemma 4.1.12.

Let $\Lambda$, $\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$. If $\Theta$ has simple component, then
$\operatorname{dom.\\!dim}\Lambda\leq 2$.

###### Proof.

  1. 1)

$\Lambda$ is cyclic, so there exists $\Delta\in{\mathcal{B}}(\Lambda)$ such
that $\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Delta)$ is simple component
of $\Theta$, $\Delta$ has no nontrivial extension with any other element of
the base-set and there is a nonfiltered projective $\Lambda$-module $P$ such
that $\Delta\subset P$.

  2. 2)

Now, $P/\Delta$ is submodule of an element of the base-set. Hence there exists
an injective $\Lambda$-module $I$ such that $\Omega^{1}(I)\cong P/\Delta$, and
$\Omega^{2}(I)\cong\Delta$. So, $\operatorname{p\\!\dim}I=2$, which forces
that dominant dimension is at most two.

∎

###  4.2. Proofs of Theorems B,C

The relationship between dominant dimensions of $\Lambda$ and $\Theta$ where
$\bm{\varepsilon}(\Lambda)\cong\Theta$ is stated below.

######  Proposition 4.2.1.

Let $\Lambda$, $\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$ and
$\operatorname{def}\Lambda=\operatorname{def}\Theta\neq 0$. Then, for any
injective non-projective $\Theta$-module $I$, there exists unique injective
non-projective $\Lambda$-module $J$ such that
$I\cong\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Omega^{2}(J))$. Moreover,

  1. 1)

$\operatorname{co-dom.\\!dim}_{\Lambda}J=2+\operatorname{co-
dom.\\!dim}_{\Theta}I$,

  2. 2)

$\operatorname{p\\!\dim}_{\Lambda}J=\operatorname{p\\!\dim}_{\Theta}I+2$.

###### Proof.

By proposition 4.1.3, there exists exact sequence for every injective non-
projective $\Theta$-module $I$ such that

$\displaystyle\Delta I\rightarrow P_{1}\rightarrow P_{0}\rightarrow
J\rightarrow 0$

where $J$ is an injective $\Lambda$-module and
$I\cong\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Omega^{2}(J))=\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Delta
I)$. $P_{0}$ is projective-injective module since it is projective cover of
injective module $J$. By lemma 4.1.2, $\Omega^{1}(J)$ is not an element of
${\mathcal{B}}(\Lambda)$, hence $P_{1}=P(\Omega^{1}(J))$ is projective
$\Lambda$-module which is not ${\mathcal{B}}(\Lambda)$-filtered. By
proposition 3.1.1, $P_{1}$ is projective-injective and minimal projective.
Therefore,

$\displaystyle\operatorname{co-dom.\\!dim}_{\Lambda}J=2+\operatorname{co-
dom.\\!dim}_{\Lambda}\Delta I=2+\operatorname{co-dom.\\!dim}_{\Theta}I,$
$\displaystyle\operatorname{p\\!\dim}_{\Lambda}J=2+\operatorname{p\\!\dim}_{\Lambda}\Delta
I=2+\operatorname{p\\!\dim}_{\Theta}I.$

∎

Here, we give the proof of Theorem B.

######  Proposition 4.2.2.

Let $\Lambda$, $\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$. Then,

  1. 1)

if $\operatorname{def}\Lambda>\operatorname{def}\Theta$, then
$\operatorname{dom.\\!dim}\Lambda\leq 2$.

  2. 2)

if $\operatorname{def}\Lambda=\operatorname{def}\Theta$, then
$\operatorname{dom.\\!dim}\Lambda=2+\operatorname{dom.\\!dim}\Theta$.

###### Proof.

The first statement follows from the contra-positive of proposition 4.1.4. For
the second statement, we use proposition 4.2.1.

$\displaystyle\operatorname{co-dom.\\!dim}\Lambda$
$\displaystyle=\inf\left\\{\operatorname{co-dom.\\!dim}I|\,I\,\text{is
injective}\right\\}$ $\displaystyle=2+\inf\left\\{\operatorname{co-
dom.\\!dim}\Omega^{2}(I)|\,I\,\text{is injective}\right\\}$
$\displaystyle=2+\inf\left\\{\operatorname{co-
dom.\\!dim}_{\Theta}(\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Omega^{2}(I))|\,I\,\text{is
injective}\right\\}$ $\displaystyle=2+\inf\left\\{\operatorname{co-
dom.\\!dim}_{\Theta}I^{\prime}|\,I^{\prime}\,\text{is
injective}\,\Theta\text{-module}\right\\}$ $\displaystyle=2+\operatorname{co-
dom.\\!dim}\Theta$

Since co-dominant dimension and dominant dimension are equal by [18], we
conclude that
$\operatorname{dom.\\!dim}\Lambda=2+\operatorname{dom.\\!dim}\Theta$. ∎

######  Remark 4.2.3.

Let $\Lambda,\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$ and
$\operatorname{def}\Lambda=\operatorname{def}\Theta$. If $\Theta$ is
selfinjective, then $\Lambda\cong\Theta$.

We prove Theorem C.

######  Proposition 4.2.4.

Let $\Lambda,\Theta$ be Nakayama algebras such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$ and
$\operatorname{def}\Lambda=\operatorname{def}\Theta$. If $\Theta$ is dominant
Auslander-Gorenstein algebra, $\Lambda$ is as well.

###### Proof.

By remark 4.2.3, if $\Theta$ is selfinjective, then $\Lambda\cong\Theta$ is a
dominant Auslander-Gorenstein algebra. If $\Theta$ is not selfinjective, then
by proposition 4.2.1, for every $I$, there exists unique indecomposable
injective $\Lambda$-module $J$ such that
$\operatorname{Hom}_{\Lambda}({\mathcal{P}},\Omega^{2}(J))\cong I$. Moreover,

$\displaystyle\operatorname{co-dom.\\!dim}_{\Lambda}J=2+\operatorname{co-
dom.\\!dim}_{\Theta}I,\,\,\operatorname{p\\!\dim}_{\Lambda}J=2+\operatorname{p\\!\dim}_{\Theta}I.$

$\Theta$ is a dominant Auslander-regular algebra, so every indecomposable
injective $\Theta$-module $I$ satisfy
$\operatorname{p\\!\dim}_{\Theta}I\leq\operatorname{co-dom.\\!dim}_{\Theta}I$.
Therefore,

$\displaystyle\operatorname{p\\!\dim}_{\Lambda}J=2+\operatorname{p\\!\dim}_{\Theta}I\leq
2+\operatorname{co-dom.\\!dim}_{\Theta}I=\operatorname{co-
dom.\\!dim}_{\Lambda}J$

implies that $\Lambda$ is also dominant Auslander-Gorenstein algebra. ∎

###  4.3. Dominant Auslander-Gorenstein Algebras among Nakayama Algebras

Aim of this section is to prove the following:

######  Proposition 4.3.1.

Let $\Lambda$ be a cyclic Nakayama algebra of infinite global dimension. If
$\Lambda$ is a dominant Auslander-Gorenstein algebra, then dominant dimension
of $\Lambda$ is even.

###### Proof.

We begin by considering the case where $\Lambda$ is self-injective, which
implies a dominant dimension of zero. Thus, we can assume that $\Lambda$ is
not self-injective.

Next, assuming the contrary, suppose that the dominant dimension of $\Lambda$
is $2k+1$. Consequently, the dominant dimension of
$\bm{\varepsilon}^{k}(\Lambda)$ is one by remark 4.1.10. Moreover,
$\bm{\varepsilon}^{k}(\Lambda)$ is a dominant Auslander-Gorenstein algebra by
Proposition 4.2.1.

There exists an injective $\bm{\varepsilon}^{k}(\Lambda)$-module $I$ with
$\operatorname{p\\!\dim}I=\operatorname{co-dom.\\!dim}I=1$. Hence,
$\Omega^{1}(I)$ is a projective module. By lemma 4.1.2, there exists
$\Delta\in{\mathcal{B}}(\bm{\varepsilon}^{k}(\Lambda))$ with
$\Omega^{1}(I)\subseteq\Delta$. According to Remark 4.1.11, $\Delta$ is a
projective module, which implies that the global dimensions of
$\bm{\varepsilon}^{k}(\Lambda)$ and $\Lambda$ are finite, contradicting the
assumption that $\operatorname{gl\,\\!dim}\Lambda=\infty$. Therefore dominant
dimension of $\Lambda$ is even. ∎

######  Remark 4.3.2.

Let $\Theta$ be a minimal Auslander-Gorenstein algebra of infinite global
dimension. Then, it is possible to construct a dominant Auslander-Gorenstein
algebra $\Lambda$ such that $\bm{\varepsilon}(\Lambda)\cong\Theta$ and
$\operatorname{def}\Lambda\neq\operatorname{def}\Theta$. We give an example.
Let $\Lambda$ be given by Kupisch series $(4,4,3,3,2,2,2,2,2,2,2,3,4,4,4,4)$.
One can verify that dominant dimensions of indecomposable projective non-
injective modules are $2,4,4$. $\bm{\varepsilon}(\Lambda)$ is given by Kupisch
series $(4,4,3,3,2,2,2,2,2,2,2,2,2,3)$ and it is $2$-Auslander-Gorenstein
algebra with defect two.

###  4.4. Minimal Auslander-Gorenstein Algebras among Nakayama Algebras

In this subsection, we provide a classification of minimal Auslander-
Gorenstein algebras. We recall that:

######  Definition 4.4.1.

[14] An algebra $A$ is called minimal Auslander-Gorenstein if the Gorenstein
dimension is bounded by the dominant dimension. $A$ is called $d$-Auslander-
Gorenstein, if it is minimal Auslander-Gorenstein with injective dimension
$d$.

######  Proposition 4.4.2.

Let $\Lambda$ be a cyclic Nakayama algebra which is $d$-Auslander-Gorenstein
where $d\geq 3$. Then, $\bm{\varepsilon}(\Lambda)$ is $(d-2)$-Auslander-
Gorenstein algebra.

###### Proof.

Dominant dimension of $\Lambda$ is $d$, $d\geq 3$. So, by corollary 4.1.5,
$\operatorname{def}\Lambda=\operatorname{def}\bm{\varepsilon}(\Lambda)$. By
proposition 4.2.2, $\operatorname{dom.\\!dim}\bm{\varepsilon}(\Lambda)=d-2$.
By remark 4.1.10, $\operatorname{gor.\\!dim}\bm{\varepsilon}(\Lambda)=d-2$,
hence $\bm{\varepsilon}(\Lambda)$ is $(d-2)$-Auslander-Gorenstein algebra. ∎

When global dimension is finite, a minimal Auslander-Gorenstein algebra
becomes a higher Auslander algebra, which we was studied in [29]. Therefore,
we consider Nakayama algebras of infinite global dimension. By [27],
Gorenstein dimension of minimal Auslander-Gorenstein algebra is even. To give
complete classification of cyclic Nakayama algebras which are minimal
Auslander-Gorenstein, it is enough to construct $2$-Auslander-Gorenstein
algebras by proposition above.

We start with the following observation:

######  Lemma 4.4.3.

Let $\Lambda$ be a $2$-Auslander-Gorenstein algebra of infinite global
dimension. Then, any $\Delta_{i}\in{\mathcal{B}}(\Lambda)$ satisfy
$\ell(\Delta_{i})\leq 2$.

###### Proof.

Assume that there exists at least one $\Delta_{i}$ module with a length
greater than or equal to three. Denote the simple composition factors of such
$\Delta_{i}$ as $X_{1},\ldots,X_{t}$, where $t\geq 3$.

According to Lemma 4.1.1, every proper radical of $\Delta_{i}$ is the first
syzygy of an injective module. Notice that
$\operatorname{top}\Omega^{2}(I(X_{j}))\cong\tau\operatorname{soc}\Delta_{i}$
for any $1\leq j\leq t-1$.

Additionally, considering that every second syzygy of injective modules are
projective due to the 2-Auslander-Gorenstein property of $\Lambda$, it follows
that all $\Omega^{2}(I(X_{i}))$ are isomorphic to the same projective module.
This implies that $P(\Omega^{1}(I(X_{2})))$ is the radical of
$P(\Omega^{1}(I(X_{1})))$. Consequently, the former projective module cannot
be injective.

Therefore, we can conclude that there exists an injective module with
injective dimension two and dominant dimension one. As a result, $\Lambda$
cannot be a minimal Auslander-Gorenstein algebra.

∎

######  Corollary 4.4.4.

Let $\Lambda$ be a $2$-Auslander-Gorenstein algebra of infinite global
dimension. Then, defect of any indecomposable projective $\Lambda$-module can
be at most one.

###### Proof.

By the previous lemma, for any $\Delta_{i}\in{\mathcal{B}}(\Lambda)$, we have
$\ell(\Delta_{i})\leq 2$. By remark 1.2.3, $\Delta_{i}$ is a submodule of a
projective-injective module $P$. If $\ell(\Delta_{i})=1$, then defect of $P$
is zero, because the quotient $P/\Delta_{i}$ is radical of $P(\tau^{-1}S)$
where $S\cong\operatorname{top}P$. If $\ell(\Delta_{i})=2$, then defect of $P$
is one by lemma 4.1.1. ∎

######  Corollary 4.4.5.

If $\Lambda$ is a minimal Auslander-Gorenstein algebra of infinite global
dimension, then defect of projective modules are either one or zero.

###### Proof.

According to [28], $\Lambda$ has an even Gorenstein dimension, denoted as
$2d$. Let $P$ be an indecomposable projective module with injective quotients
$I_{1},\ldots,I_{t}$, where $t\geq 3$. There exists a bijection between these
injective modules and the injectives of the syzygy filtered algebra,
established via the functor
$\operatorname{Hom}_{\Lambda}(\mathcal{P},\Omega^{2}(-))$. By induction, we
obtain
$t=\operatorname{def}\operatorname{Hom}_{\Lambda}(\mathcal{P},\Omega^{2d-2}(P))$.
As $\bm{\varepsilon}^{2d-2}(\Lambda)$ is a $2$-Auslander-Gorenstein algebra,
by Corollary 4.4.4, we conclude that $t\leq 2$. ∎

####  4.4.1. Constrution of $2$-Auslander-Gorenstein Algebras among Nakayama
Algebras

We start with a selfinjective Nakayama algebra $\Theta$ and a complete set of
representatives of indecomposable projective $\Theta$-modules
$\\{P_{1},\dots,P_{n}\\}$. Next we consider a sequence of integers
${\mathcal{K}}=\left(c_{1},\ldots,c_{N}\right)$ and show that the sequence is
a Kupish series for a Nakayama algebra $\Lambda$ providing that it satisfies
four conditions: 1), 2), 3), 4) as stated in the Proposition 4.4.6. Before
proving this proposition we will explain the role or purpose of each of these
conditions.

  1. 1)

$N=n+d$ is a statement that the number of indecomposable projective
$\Lambda$-modules can be increased by any $1\leq d\leq 2n$, from the number of
indecomposable projective $\Theta$-modules.

  2. 2)

${\mathcal{N}}=(c_{z_{1}},c_{z_{2}},\ldots,c_{z_{n}})$ will be the lengths of
the filtered projective $\Lambda$-modules. These lengths are described in
terms of integer valued function $f$ wih codomain $\\{1,2\\}$.

  3. 3)

This is the statement that the number of nonfiltered $\Lambda$-projectives is
determined by function $f$.

  4. 4)

This is numerical condition which describes lengths of non-filtered
$\Lambda$-projectives.

######  Proposition 4.4.6.

Let $\Theta$ be a selfinjective Nakayama algebra of rank $n$. Let
$\left\\{P_{1},\ldots,P_{n}\right\\}$ be the complete set of representatives
of isomorphism classes of indecomposable projective $\Theta$-modules. Let $f$
be a function on simple $\Theta$-modules taking integer values either one or
two. Consider the finite sequence of integers
${\mathcal{K}}=\left(c_{1},\ldots,c_{N}\right)$ satisfying

  1. 1)

$N=n+d$ for some integer $d$ satisfying $1\leq d\leq 2n-1$

  2. 2)

There exists a subsequence
${\mathcal{N}}=(c_{z_{1}},c_{z_{2}},\ldots,c_{z_{n}})$ of ${\mathcal{K}}$ such
that

(4.3) $\displaystyle c_{z_{i}}=\sum\limits^{\ell(P_{i})}_{j=1}f(S_{[j+i-1]})$

where $[j]$ denote the least positive residue of $j$ modulo $n$ when $j>0$ and
$[n]=n$, and $f(S_{j})$ is integer valued function either one or two.

  3. 3)

Let $c_{z_{i}},c_{z_{i+1}}$ be consecutive elements of ${\mathcal{N}}$ where
$1\leq i\leq n-1$. Then, in ${\mathcal{K}}$ the number of terms in between
$c_{z_{i}}$ and $c_{z_{i+1}}$ is either zero or one. If $c_{z_{i}}$ and
$c_{z_{i+1}}$ are consecutive in ${\mathcal{K}}$, then $f(S_{i})=1$. Else,
$f(S_{i})=2$. The number of terms between $c_{z_{n}}$ and $c_{N}$ is $b$ if
and only if the number of terms between $c_{1}$ and $c_{z_{1}}$ is $1-b$.

  4. 4)

Any term $c_{a_{m}}\in{\mathcal{K}}-{\mathcal{N}}$ which satisfies
${z_{i}}<{a_{m}}<{z_{i+1}}$ is given by

(4.4) $\displaystyle
c_{a_{m}}=1+\sum\limits^{\ell(P_{i+1})}_{j=1}f(S_{[j+i]})$

Any term $c_{a_{m}}\in{\mathcal{K}}-{\mathcal{N}}$ with index either
$z_{n}<a_{m}\leq N$ or $1\leq a_{m}<z_{1}$ is given by

(4.5) $\displaystyle c_{a_{m}}=1+\sum\limits^{\ell(P_{1})}_{j=1}f(S_{[j]})$

The sequence ${\mathcal{K}}$ satisfying all these conditions forms an
admissible sequence of a cyclic Nakayama algebra $\Lambda$ which is
$2$-Auslander-Gorenstein with defect $d$.

Before proving this, we need the following lemmas.

######  Lemma 4.4.7.

Let $c_{t}=c_{z_{i}},c_{t+1}=c_{z_{i+1}}$. Then $c_{t}\leq c_{t+1}$ and
$c_{t+1}-c_{t}=f(S_{[\ell(P_{i+1})]})-1$

###### Proof.

The difference

$\displaystyle
c_{z_{i}}-c_{z_{i+1}}=\sum\limits^{\ell(P_{i})}_{j=1}f(S_{[j+i-1]})-\sum\limits^{\ell(P_{i+1})}_{j=1}f(S_{[j+i]})$
$\displaystyle=f(S_{[i]})+\sum\limits^{\ell(P_{i})}_{j=2}f(S_{[j+i-1]})-\left(\sum\limits^{\ell(P_{i+1})-1}_{j=1}f(S_{[j+i]})+f(S_{[\ell(P_{i+1})]})\right)$
$\displaystyle=f(S_{[i]})+\sum\limits^{\ell(P_{i})-1}_{j=1}f(S_{[j+i]})-\left(\sum\limits^{\ell(P_{i+1})-1}_{j=1}f(S_{[j+i]})+f(S_{[\ell(P_{i+1})]})\right)$
$\displaystyle=f(S_{[i]})+\left(\sum\limits^{\ell(P_{i})-1}_{j=1}f(S_{[j+i]})-\sum\limits^{\ell(P_{i+1})-1}_{j=1}f(S_{[j+i]})\right)-f(S_{[\ell(P_{i+1})]})$
$\displaystyle=f(S_{[i]})-f(S_{[\ell(P_{i+1})]})=1-f(S_{[\ell(P_{i+1})]})\leq
0$

since $\ell(P_{i})=\ell(P_{i+1})$. $f(S_{i})=1$, and $f(S_{[\ell(P_{i+1})]})$
is either one or two, hence the difference is less than or equal to zero. ∎

######  Lemma 4.4.8.

Let $c_{t}=c_{z_{i}}$, $c_{t+1}\notin{\mathcal{N}}$. Then,
$c_{t}-c_{t+1}=1-f(S_{[\ell(P_{i+1})]})\leq 0$. Let $c_{t}\notin{\mathcal{N}}$
and $c_{t+1}=c_{z_{i+1}}$. Then $c_{t}-c_{t+1}=1$.

###### Proof.

The second statement follows from proposition 4.4.6 where we set the
difference to one. For the first statement we have:

$\displaystyle
c_{z_{i}}-c_{t+1}=\sum\limits^{\ell(P_{i})}_{j=1}f(S_{[j+i-1]})-\left(1+\sum\limits^{\ell(P_{i+1})}_{j=1}f(S_{[j+i]})\right)$
$\displaystyle=f(S_{[i]})+\sum\limits^{\ell(P_{i})}_{j=2}f(S_{[j+i-1]})-\left(1+\sum\limits^{\ell(P_{i+1})-1}_{j=1}f(S_{[j+i]})+f(S_{[\ell(P_{i+1})]})\right)$
$\displaystyle=f(S_{[i]})+\sum\limits^{\ell(P_{i})-1}_{j=1}f(S_{[j+i]})-\left(1+\sum\limits^{\ell(P_{i+1})-1}_{j=1}f(S_{[j+i]})+f(S_{[\ell(P_{i+1})]})\right)$
$\displaystyle=f(S_{[i]})+\left(\sum\limits^{\ell(P_{i})-1}_{j=1}f(S_{[j+i]})-\sum\limits^{\ell(P_{i+1})-1}_{j=1}f(S_{[j+i]})\right)-1-f(S_{[\ell(P_{i+1})]})$
$\displaystyle=f(S_{[i]})-1-f(S_{[\ell(P_{i+1})]})=1-f(S_{[\ell(P_{i+1})]})$

Since $f(S_{i})=2$, the difference is less than or equal to zero. ∎

###### Proof.

To verify whether a given sequence of integers
${\mathcal{K}}=(c_{1},\ldots,c_{n})$ forms a Kupisch series of a cyclic
Nakayama algebra, we need to check that

  1. i)

each $c_{i}\geq 2$

  2. ii)

and if $c_{i}>c_{i+1}$, then $c_{i}-c_{i+1}=1$.

If $c_{i}\leq c_{i+1}$, $c_{i+1}$ can take any positive integer value, so it
satisfies admissibility. By lemmas 4.4.8 and 4.4.7, the case $c_{j}>c_{j+1}$
only happens when $c_{t}\notin{\mathcal{N}}$ and $c_{t+1}=c_{z_{i+1}}$. It
follows that the difference is one, so ${\mathcal{K}}$ is admissible sequence.
In particular, each $c_{j}\in{\mathcal{K}}$ has at least $\ell(P_{i})$-many
nonzero summands. $\Lambda$ is a cyclic selfinjective Nakayama algebra, so
$\ell(P_{i})\geq 2$ for any $i$ which makes $c_{j}\geq 2$. This shows
${\mathcal{K}}$ is an admissible sequence. ∎

Let $\left\\{Q_{1},\ldots,Q_{N}\right\\}$ be the complete set of
representatives of isomorphism classes of indecomposable $\Lambda$-modules and
$\ell(Q_{i})=c_{i}\in{\mathcal{K}}$. We divide the proof into the following
steps.

######  Claim 4.4.8.1.

$\operatorname{def}\Lambda=d$ and $|{\mathcal{S}}(\Lambda)|=n$

###### Proof.

By lemma 1.2.14, $\operatorname{def}\Lambda=|\\{c_{i+1}\mid
c_{i}>c_{i+1},1\leq i\leq N\\}|$ and $|{\mathcal{S}}(A)|=|\\{c_{i}\mid
c_{i}\leq c_{i+1},1\leq i\leq N\\}$ where $c_{N+1}=c_{1}$. By lemmas 3.1.3 and
3.1.2, $c_{i}>c_{i+1}$ if $c_{i}\notin{\mathcal{N}}$. There are exactly $d$ of
them, so defect of $\Lambda$ is $d$, and it follows
$|{\mathcal{S}}(\Lambda)|=N-d=n$. ∎

######  Claim 4.4.8.2.

${\mathcal{B}}(\Lambda)$-filtered indecomposable projective $\Lambda$-modules
are $Q_{z_{1}},\ldots,Q_{z_{n}}$.

###### Proof.

Notice that there are $n$-many ${\mathcal{B}}(\Lambda)$-filtered
indecomposable projective $A$-modules, since
$|{\mathcal{B}}(\Lambda)|=|{\mathcal{S}}(\Lambda)|=n$. The lengths of elements
of ${\mathcal{B}}(\Lambda)$ are $c_{i+1}-c_{i}+1$ provided that $c_{i}\leq
c_{i+1}$ by lemma 1.2.14. In lemmas 3.1.2 and 3.1.3, we calculated all
possible differences of $c_{i}-c_{i+1}$ which are always of the form
$1-f(S_{j})$ when $c_{i}\leq c_{i+1}$. Moreover, by the construction in
proposition 4.4.6, for any $1\leq i\leq n$, $f(S_{i})+1$ is a summand of
$Q_{z_{i}}$. Therefore, every $Q_{z_{i}}$ is
${\mathcal{B}}(\Lambda)$-filtered. Because there are $n$-many $Q_{z_{i}}$,
hence we get all. ∎

###### Proof.

Now we can show that $\bm{\varepsilon}(\Lambda)\cong\Theta$.
$\dim_{\mathbb{K}}\operatorname{Hom}_{\Lambda}\left(\bigoplus\limits_{1\leq
i\leq
n}Q_{z_{i}},Q_{z_{i}}\right)=\ell_{{\mathcal{B}}(\Lambda)}(Q_{z_{i}})=\ell(P_{i})$,
since the number of elements ${\mathcal{B}}(\Lambda)$ appearing in each
$Q_{z_{i}}$ is $\ell(P_{i})$. Hence Kupisch series of
$\bm{\varepsilon}(\Lambda)$ is equal to Kupisch series of $\Theta$ upto cyclic
permutation. ∎

######  Claim 4.4.8.3.

$\Lambda$ is 2-Auslander-Gorenstein.

###### Proof.

Projective but not injective $\Lambda$ modules are $Q_{i}$ such that
$\ell(Q_{i})\in{\mathcal{N}}$, $\ell(Q_{i-1})\in{\mathcal{K}}-{\mathcal{N}}$
and $\ell(Q_{i-2})\in{\mathcal{N}}$. Consider the injective resolution of
$Q_{i}$

$\displaystyle 0\rightarrow Q_{i}\rightarrow Q_{i-1}\rightarrow
S_{i-1}\rightarrow 0.$

Notice that $S_{i-1}$ is simple proper submodule of an element of the base-set
${\mathcal{B}}(\Lambda)$, since $\operatorname{coker}g\in
Filt({\mathcal{B}}(\Lambda))$ where $g:Q_{i}\rightarrow Q_{i-2}$ and
$S_{i-1}\subset\operatorname{coker}g$. Therefore injective envelope of
$S_{i-1}$ is projective-injective $\Lambda$-module, and by lemma 4.1.1
$\Sigma^{2}(Q_{i})\cong\Sigma^{1}(S_{i-1})$ is an injective non-projective
$\Lambda$-module. Hence, injective dimension of $\Lambda$ is two. In
particular, $Q_{i-1}$ and $I(S_{i-1})$ are projective-injective modules, which
implies $\operatorname{dom.\\!dim}\Lambda=2$. ∎

Now, we can apply the main theorem to $2$-Auslander-Gorenstein algebras to get
all $2k$-Auslander-Gorenstein algebras $k\geq 2$. Here we give some examples.

######  Example 4.4.9.

Let $\Theta$ be given by Kupisch series $(2,2)$. Then algebras given by
$(3,2,2)$, $(5,5,4,4),(6,6,6,5,5)$ are minimal Auslander-Gorenstein algebras.

######  Example 4.4.10.

Let $\Theta$ be selfinjective algebra of rank $n$ and length $m$. 2-Auslander-
Gorenstein algebra $\Lambda$ of defect $n$ which satisfies
$\bm{\varepsilon}(\Lambda)\cong\Theta$ is given by the Kupisch series

$\displaystyle((2m,2m+1)^{m})$

######  Example 4.4.11.

Let $\Theta$ be selfinjective algebra of rank $n$ and length $m$. 2-Auslander-
Gorenstein algebra $\Lambda$ of defect $n-1$ which satisfies
$\bm{\varepsilon}(\Lambda)\cong\Theta$ is given by the Kupisch series

$\displaystyle\left((2m,2m+1)^{n-m-1},2m,(2m,2m-1)^{m}\right)\quad$
$\displaystyle\text{if}\,n\geq m+1$
$\displaystyle\left(f(P_{1}),1+f(P_{2}),f(P_{2}),\ldots,f(P_{n-1}),1+f(P_{n}),f(P_{n})\right)\quad$
$\displaystyle\text{if}\,n<m$

where $f(P_{i})=[P_{i}:S_{n}]+2\sum\limits_{j\neq n}[P_{i}:S_{j}]$.

######  Example 4.4.12.

Here we present a maple code to generate all 2-Auslander-Gorenstein algebras
among Nakayama algebras from a given selfinjective Nakayama algebra $\Theta$.

    # Takes kupisch series and gives ith projective object of Theta

    proj := proc(length, order, rank)
    [seq(irem(j-1, rank)+1, j = order .. length+order-1))];end proc;

    # calculates index of a simple module S in projective module P in Theta

    weight := proc(P::list, S);
        tot := 0;
        for i from 1 to nops(P) do
            if P[i] = S then tot := tot + 1   fi;
        od; tot; end proc;

     # creates length of ith filtered projective of Lambda
     for a given defect vector v

     newproj:=proc(i,m,n,v::list)
     PP:=proj(m,i,n); tot:=0;
     for j from 1 to nops(v) do
     tot:=tot+weight(PP,v[j]);
     od; tot+m;end proc;

    # creates kupisch series of Lambda for a given defect vector v

    kupischSeries:=proc(m,n,v::list);
    L:=[];
    for i from 1 to n do
    if member(i,v)=true then L:=[op(L),newproj(i,m,n,v),1+newproj(i+1,m,n,v)]
    else L:=[op(L),newproj(i,m,n,v)] fi;
    od;end proc;

######  Example 4.4.13.

Let $\Theta$ be a radical square zero selfinjective algebra. For vector
$[1,1,0,\ldots,0]$, 2-Auslander-Gorenstein algebras are

$\displaystyle[4,5,4,5]$ $\displaystyle[4,4,3,4,3]$
$\displaystyle[4,4,3,3,2,3]$ $\displaystyle[4,4,3,3,2,2,3]$
$\displaystyle[4,4,3,3,2,2,2,3]$ $\displaystyle[4,4,3,3,2,2,2,2,3]$
$\displaystyle[4,4,3,3,2,2,2,2,2,3]$ $\displaystyle[4,4,3,3,2,2,2,2,2,2,3]$
$\displaystyle[4,4,3,3,2,2,2,2,2,2,2,3]$
$\displaystyle[4,4,3,3,2,2,2,2,2,2,2,2,3]$

Let $\Theta$ be a selfinjective Nakayama algebra where $\ell(P_{i})=5$ for all
projetives. For vector $(1,0,1,1,\ldots,0)$, all defect invariant algebras are

$\displaystyle[9,9,8,9,10,9,10]$ $\displaystyle[8,9,8,8,9,8,9,8]$
$\displaystyle[8,8,7,8,8,7,8,7,8]$ $\displaystyle[8,8,7,7,8,7,7,6,7,8]$
$\displaystyle[8,8,7,7,7,6,7,6,6,7,8]$
$\displaystyle[8,8,7,7,7,6,6,5,6,6,7,8]$
$\displaystyle[8,8,7,7,7,6,6,5,5,6,6,7,8]$

###  4.5. Dominant Auslander-regular algebras among Nakayama Algebras

####  4.5.1. Construction of dominant Auslander-regular algebras of dominant
dimension one

######  Lemma 4.5.1.

Let $\Theta$ be a connected linear Nakayama algebra of rank $n$. Let
$v_{\Theta}=(\operatorname{def}P_{1},\operatorname{def}P_{2},\ldots,\operatorname{def}P_{n})$
be its defect vector. Then, for any vector
$w=(\operatorname{def}P_{1}+m,\operatorname{def}P_{2},\ldots,\operatorname{def}P_{n})$
with $m\geq 1$, there exists cyclic Nakayama algebra $\Lambda$ such that
$\bm{\varepsilon}(\Lambda)\cong\Theta$, $\operatorname{def}\Lambda=|d|$ and
$\operatorname{dom.\\!dim}\Lambda=1$.

######  Corollary 4.5.2.

If $\Theta$ is dominant Auslander-regular algebra, then $\Lambda$ is as well.

###### Proof.

We apply proposition 3.2.2 for the given vector $w$ in the following way. Let
$\operatorname{def}P_{1}=t$. Hence, $\ell(\Delta(S_{n}))=m+t$. Let
$X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{t}$ be the ordered composition factors of
$\operatorname{rad}\Delta(S_{n})$, with
$\operatorname{top}\operatorname{rad}\Delta(S_{i})\cong
X_{1},\operatorname{soc}\operatorname{rad}\Delta(S_{n})\cong Y_{t}$. We choose
the subsequence in part 4) of proposition 3.2.2 so that

  1. 1)

$z_{n}<a_{1}<a_{2}<\cdots<a_{m+t}<z_{1}$,

  2. 2)

$b_{1}=a_{m+1}$, $b_{2}=a_{m+2}$, …, $b_{t}=a_{m+t}$.

This makes $P_{b_{i}}\cong P(Y_{i})$ and each of them are projective-injective
and minimal projective modules of $\Lambda$. For the remaining indices, the
corresponding projective modules are set to be

$\displaystyle P(X_{m})\subset P(X_{m-1})\subset\cdots\subset P(X_{1})\subset
P(\Delta(S_{n})).$

By proposition 3.2.5, $\bm{\varepsilon}(\Lambda)\cong\Theta$. Notice that
dominant dimension of $\Lambda$ is one, since each $P(X_{j})$ is submodule of
$P(\Delta(S_{i}))\in{\mathcal{B}}(\Lambda)$, by lemma 4.1.1, they are first
syzygies of injective modules. ∎

####  4.5.2. Construction of dominant Auslander-regular algebras of dominant
dimension two

######  Lemma 4.5.3.

Let $\Theta$ be a connected linear Nakayama algebra of rank $n$. Let
$v_{\Theta}=(\operatorname{def}P_{1},\operatorname{def}P_{2},\ldots,\operatorname{def}P_{n})$
be its defect vector. Then, for the vector
$w=(\operatorname{def}P_{1},\operatorname{def}P_{2},\ldots,\operatorname{def}P_{n},1)$,
there exists cyclic Nakayama algebra $\Lambda$ such that
$\bm{\varepsilon}(\Lambda)\cong\Theta\oplus\mathbb{A}_{1}$,
$\operatorname{def}\Lambda=1+\operatorname{def}\Theta$ and
$\operatorname{dom.\\!dim}\Lambda=2$.

###### Proof.

The vector $w$ describes lengths of elements of ${\mathcal{B}}(\Lambda)$,
i.e., $\ell(\Delta(S_{i}))\cong 1+\operatorname{def}P_{i+1}$. In particular,
$\ell(\Delta(S_{n}))=2$ and $\ell(\Delta(S_{n+1}))=1+\operatorname{def}P_{1}$.
We apply proposition 3.2.2, and set $P(\Delta(S_{n+1}))$ as submodule of
$P(\operatorname{soc}\Delta(S_{n}))$ which makes it projective-injective
module. Dominant dimension of $\Lambda$ is two, since
$\operatorname{soc}\Delta(S_{n})$ is the first syzygy of an injective module,
and its projective dimension is one. $\Lambda$ is a dominant Auslander-regular
algebra, since for other injective objects we get $\operatorname{co-
dom.\\!dim}_{\Lambda}J=\operatorname{co-dom.\\!dim}_{\Theta}I+2$. ∎

######  Example 4.5.4.

Let $\Theta$ be given by Kupisch series $(2,3,2,2,1)$. Then, $\Lambda$ is
given by $(3,3,3,2,3,2,4,3)$ and
$\bm{\varepsilon}(\Lambda)\cong\Theta\oplus\mathbb{A}$. Notice that projective
module of length four corresponds to $P(\operatorname{soc}\Delta(S_{5}))$ in
the proof above.

####  4.5.3. Global dimension three

Here, we construct cyclic Nakayama algebras of global dimension three, which
are dominant Auslander-regular algebras.

Let $\Lambda$ be a cyclic Nakayama algebra of global dimension $3$. Then,
$\operatorname{gl\,\\!dim}\bm{\varepsilon}(\Lambda)\leq 1$. Therefore
$\bm{\varepsilon}(\Lambda)$ is a linear Nakayama algebra with components that
are either linearly oriented $\mathbb{A}$-type or simple $\mathbb{A}_{1}$.
Thus, we consider a Nakayama algebra $\Theta\cong\bm{\varepsilon}(\Lambda)$
such that

(4.6)
$\displaystyle\Theta\cong\mathbb{A}_{n_{1}}\oplus\mathbb{A}_{n_{2}}\oplus\cdots\oplus\mathbb{A}_{n_{k}}$

with $n_{i}\geq 1$, and at least one $n_{j}\geq 2$.

######  Remark 4.5.5.

By definition, defect of simple component is zero. However, in many cases it
behaves like $\mathbb{A}_{2}$ algebra. That’s why we define _virtual defect_
of simple components as one. Hence, if $n\geq 2$, then defect vector of
$\mathbb{A}_{n}$ is $(n-1,0,\ldots,0)$ and defect vector of $\mathbb{A}_{1}$
is $(1)$. In [29], defect of simple components set to one.

######  Proposition 4.5.6.

Let $\Lambda$ be a cyclic Nakayama algebra given by Kupisch series