# Applications of Gorenstein projective $\tau$-rigid modules

Hui Liu  H.Liu: Department of Mathematics, Nanjing University, Nanjing,
210093, P. R. China<EMAIL_ADDRESS>,  Xiaojin Zhang  X. Zhang:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou,
221116, P. R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Yingying
Zhang∗ Y. Zhang: Department of Mathematics, Huzhou University, Huzhou 313000,
Zhejiang Province, P.R.China<EMAIL_ADDRESS>
###### Abstract.

We first introduce the notion of $CM$-$\tau$-tilting free algebras as the
generalization of $CM$-free algebras and show the homological properties of
$CM$-$\tau$-tilting free algebras. Then we give a bijection between Gorenstein
projective $\tau$-rigid modules and certain modules by using an equivalence
established by Kong and Zhang. Finally, we give a partial answer to
Tachikawa’s first conjecture by using Gorenstein projective $\tau$-rigid
modules.

2020 Mathematics Subject Classification: 16G10, 18G25

Keywords: Gorenstein projective module, $\tau$-rigid module, $E$-rigid module,
self-injective algebra

$*$ is the corresponding author.

##  1\. Introduction

In 2014, Adachi, Iyama and Reiten [AIR] introduced $\tau$-tilting theory as a
generalization of tilting theory from the viewpoint of mutation. They showed
that $\tau$-tilting theory is closely related to silting theory [AI] and
cluster tilting theory [IY]. (support) $\tau$-tilting modules are the most
important objects in $\tau$-tilting theory. So it is interesting to study
(support) $\tau$-tilting modules for given algebras. For the recent
development on this topics, we refer to [AH, FGLZ, IZ1, IZ2, KK, PMH, W, WZ,
Z, Zh, Zi].

On the other hand, Gorenstein projective modules which form the main body of
Gorenstein homological algebra can be traced back to Auslander-Bridger’s
modules of $G$-dimension zero [AuB]. Gorenstein-projective modules are
particular important to the Gorenstein algebras [EJ2]. The definition of
Gorenstein projective modules over an arbitrary ring was given by Enochs and
Jenda [EJ1, EJ2]. From then on, Gorenstein projective modules have gained a
lot of attention in both homological algebra and the representation theory of
finite-dimensional algebras. In this paper, we focus on the finitely generated
Gorenstein projective modules over finite dimensional algebras over an
algebraically closed field. For the recent development on this topics, we
refer to [CSZ, HLXZ, RZ1, RZ2, RZ3, RZ4].

CM-finite algebras are important in both commutative algebra and
representation theory of finite dimensional algebras which were studied in [B,
C, LZ2]. In [XZ], Xie and Zhang introduced the notion of Gorenstein projective
$\tau$-rigid modules and studied the homological properties of Gorenstein
projective $\tau$-rigid modules. Moreover, they introduced the notion of
CM-$\tau$-tilting finite algebras as the generalization of both CM-finite
algebras and $\tau$-tilting finite algebras introduced in [DIJ]. We refer to
[LZh2] for the existence of non-projective Gorenstein projective
$\tau$-tilting modules. We refer to [GMZ] for the theory of $2$-term
Gorenstein silting complexes. Note that CM-free algebras are CM-finite
algebras. It is natural to define the class of CM-$\tau$-tilting free algebras
as a special case of the class of CM-$\tau$-tilting finite algebras. From now
on, let $\Lambda$ be a finite dimensional algebra over an algebraically closed
field. We call $\Lambda$ CM-$\tau$-tilting free if all finitely generated
Gorenstein projective $\tau$-rigid left $\Lambda$-modules are projective
modules. The following theorem is the first main result in this paper.

######  Theorem 1.1.

  1. Let $\Lambda$ be an algebra. Then we have the following

  2. (1)

$\Lambda$ is CM-$\tau$-tilting free if and only if the opposite algebra
$\Lambda^{o}$ is CM-$\tau$-tilting free.

  3. (2)

A Gorenstein algebra $\Lambda$ is CM-$\tau$-tilting free if and only if so is
the lower triangular matrix algebra $T_{2}(\Lambda)$.

For an algebra $\Lambda$ and $E$ a generator in the category of finitely
generated left $\Lambda$-modules. In [KZ], Kong and Zhang defined
$E$-Gorenstein projective modules and showed that there is an equivalence
between the set of finitely generated $E$-Gorenstein projective modules and
the set of Gorenstein projective modules over $\Gamma=({\rm End}E)^{op}$. As a
result, they showed that the CM-Auslander algebra of a CM-finite algebra is
indeed CM-free. In this paper, we use the equivalence defined by Kong and
Zhang to study Gorenstein projective $\tau$-rigid modules.

We introduce $E$-rigid modules as the generalization of $\tau$-rigid modules.
For an algebra $\Lambda$, we denote by $\Lambda$-$\mathsf{mod}\hskip
0.72229pt$ the category of finitely generated left $\Lambda$-modules. Let $E$
be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. Let
$M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and $E_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}E_{0}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}M\rightarrow 0$ an exact sequence with $f_{i}$ the
minimal right ${\rm add}E$-approximation of ${\rm Im}f_{i}$. We call $M$
$E$-rigid if ${\rm Hom}_{\Lambda}(f_{1},M)$ is surjective. Putting
$E=\Lambda$, one gets $\tau$-rigid modules. We call a module $M$
$E$-Gorenstein projective $E$-rigid if it is both $E$-Gorenstein projective
and $E$-rigid.

Denote by $\mathcal{EG}(E)$ the set of isomorphism classes of indecomposable
$E$-Gorenstein projecitve $E$-rigid modules in $\Lambda$-$\mathsf{mod}\hskip
0.72229pt$ and by $\mathcal{TG}(\Gamma)$ the set of isomorphism classes of
indecomposable Gorenstein projective $\tau$-rigid modules in
$\Gamma$-$\mathsf{mod}\hskip 0.72229pt$. Now we are in a position to state our
second main result in this paper.

######  Theorem 1.2.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and
$\Gamma=({\rm End}_{\Lambda}E)^{op}$. Then there is a bijection between
$\mathcal{EG}(E)$ and $\mathcal{TG}(\Gamma)$.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. We call
$\Lambda$ CM-$E$-free if every indecomposable $E$-Gorenstein projective
$E$-rigid module is a direct summand of $E$. Taking $E=\Lambda$, the
CM-$E$-free algebras are precisely CM-$\tau$-tilting free algebras above.

As an application, we have the following corollary.

######  Corollary 1.3.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. Then
$\Lambda$ is CM-$E$-free if and only if $\Gamma=({\rm End}E)^{op}$ is
CM-$\tau$-tilting free.

Homological conjectures are hot in both homological algebra and representation
theory of finite dimensional algebras. The most famous one of these
conjectures is the Nakayama Conjecture which states that for an algebra
$\Lambda$ if every term of its minimal injective resolution is projective,
then $\Lambda$ is self-injective. To study the Nakayama conjecture, Tachikawa
[T] divided Nakayama conjecture into the following two conjectures which are
well-known as Tachikawa’s first conjecture and Tachikawa’s second conjecture.

######  Conjecture 1.4.

Let $\Lambda$ be an algebra.

  1. (1)

If ${\rm Ext}_{\Lambda}^{i}(\mathbb{D}\Lambda,\Lambda)=0$ for $i\geq 1$, then
$\Lambda$ is a self-injective algebra.

  2. (2)

Let $\Lambda$ be a self-injective algebra and
$M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. If ${\rm
Ext}_{\Lambda}^{i}(M,M)=0$ for $i\geq 1$, then $M$ is projective.

The following is the third main result in this paper, which gives a partial
answer to Tachikawa’s first conjecture.

######  Theorem 1.5.

Let $\Lambda$ be an algebra. Then the following are equivalent.

  1. (1)

$\Lambda$ is self-injective.

  2. (2)

$\mathbb{D}\Lambda$ is semi-Gorenstein projective $\tau$-rigid.

  3. (3)

Every cotilting module is semi-Gorenstein projective $\tau$-rigid.

  4. (4)

$\mathbb{D}\Lambda$ is Gorenstein projective.

  5. (5)

Every coltilting module is Gorenstein projective.

  6. (6)

$\Lambda$ is semi-Gorenstein injective $\tau^{-1}$-rigid.

  7. (7)

Every tilting module is semi-Gorenstein injective $\tau^{-1}$-rigid.

  8. (8)

$\Lambda$ is Gorenstein-injective.

  9. (9)

Every tilting module is Gorenstein injective.

Throughout this paper, we assume that all algebras are finite dimensional
algebras over an algebraically closed field and all modules are finitely
generated left modules. Denote by $\tau$ (resp. $\tau^{-1}$) the Auslander-
Reiten translation functor (resp. the inverse of the Auslander-Reiten
translation functor.)

##  2\. Preliminaries

In this section, we recall some preliminaries on Gorenstein projecitve
modules, $\tau$-tilting modules and lower triangular matrix algebras. Let $K$
be an algebraically closed field. We denote by
$\mathbb{D}:=\operatorname{Hom}\nolimits_{K}(-,K)$ the K-duality. Let
$\Lambda$ be a finite dimensional algebra over an algebraically closed field
$K$. We denote by $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ the category of
finitely generated left A-modules. Denote by $\mathcal{P}(\Lambda)$ (resp.
$\mathcal{I}(\Lambda)$) the category of finitely generated projective (resp.
injective) left $\Lambda$-modules.

###  2.1. Gorenstein projecitve modules

Firstly, we need the definition of Gorenstein projective modules and the
definition of Gorenstein injective modules in [EJ1, EJ2].

######  Definition 2.1.

Let $\Lambda$ be an algebra and $M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$.

  1. (1)

$M$ is called Gorenstein projective, if there is an exact sequence
$\cdots\rightarrow P_{-1}\rightarrow P_{0}\rightarrow P_{1}\rightarrow\cdots$
in $\mathcal{P}(\Lambda)$, which stays exact under
$\operatorname{Hom}\nolimits_{\Lambda}(-,\Lambda)=(-)^{\ast}$, such that
$M\simeq\mathop{\rm Im}\nolimits(P_{-1}\rightarrow P_{0})$.

  2. (2)

$M$ is called Gorenstein injective, if there is an exact sequence
$\cdots\rightarrow I_{-1}\rightarrow I_{0}\rightarrow I_{1}\rightarrow\cdots$
in $\mathcal{I}(\Lambda)$, which stays exact under
$\operatorname{Hom}\nolimits_{\Lambda}(\mathbb{D}\Lambda,-)$, such that
$M\simeq\mathop{\rm Im}\nolimits(I_{-1}\rightarrow I_{0})$.

Denote by $\mathcal{GP}(\Lambda)$ the category of all finitely generated
Gorenstein projective left $\Lambda$-modules and by
$\underline{\mathcal{GP}(\Lambda)}$ the stable category of
$\mathcal{GP}(\Lambda)$ which modulo projective modules. The following
properties of Gorenstein projective modules in [AuR] are well-known.

######  Proposition 2.2.

Let $\Lambda$ be an algebra.

  1. (1)

$(-)^{*}=\operatorname{Hom}\nolimits_{\Lambda}(-,\Lambda):\mathcal{GP}(\Lambda)\rightarrow\mathcal{GP}(\Lambda^{\mathrm{op}})$
is a duality.

  2. (2)

$\Omega^{1}:\underline{\mathcal{GP}(\Lambda)}\rightarrow\underline{\mathcal{GP}(\Lambda)}$
is an equivalence of categories.

  3. (3)

$\mathop{\rm
Tr}\nolimits:\underline{\mathcal{GP}(\Lambda)}\rightarrow\underline{\mathcal{GP}(\Lambda^{\mathrm{op}})}$
is a duality.

We also need the following property of tensor product of Gorenstein projective
modules [HLXZ, Proposition 2.6].

######  Proposition 2.3.

Let $A$ and $B$ be two algebras. Let $M\in A$-$\mathsf{mod}\hskip 0.72229pt$
and $N\in B$-$\mathsf{mod}\hskip 0.72229pt$ be Gorenstein projective modules.
Then $M\otimes N\in(A\otimes B)$-$\mathsf{mod}\hskip 0.72229pt$ is Gorenstein
projective.

###  2.2. $\tau$-tilting modules

Let $\Lambda$ be an algebra. For a module $M\in\Lambda$-$\mathsf{mod}\hskip
0.72229pt$, denote by $|M|$ the number of pairwise non-isomorphic
indecomposable summands of $M$. Now we recall the definitions of $\tau$-rigid
modules (resp. $\tau^{-1}$-rigid) and $\tau$-tilting modules (resp.
$\tau^{-1}$-tilting) from [S] and [AIR].

######  Definition 2.4.

Let $\Lambda$ be an algebra and $M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$.

  1. (1)

We call $M$ $\tau$-rigid if $\operatorname{Hom}\nolimits_{\Lambda}(M,\tau
M)=0$. Moreover, $M$ is called a $\tau$-tilting module if $M$ is $\tau$-rigid
and $|M|=|\Lambda|$.

  2. (2)

We call $M$ support $\tau$-tilting if there exists an idempotent $e$ of
$\Lambda$ such that $M$ is a $\tau$-tilting $\Lambda/(e)$-module.

  3. (3)

We call $M$ $\tau^{-1}$-rigid if
$\operatorname{Hom}\nolimits_{\Lambda}(\tau^{-1}M,M)=0$. Moreover, $M$ is
called a $\tau^{-1}$-tilting module if $M$ is $\tau^{-1}$-rigid and
$|M|=|\Lambda|$.

  4. (4)

We call $M$ support $\tau^{-1}$-tilting if $M$ is a $\tau^{-1}$-tilting
$\Lambda/(e)$-module for some idempotent $e$ of $\Lambda$.

The following result in [AIR, Proposition 2.4(c)] is very essential in this
paper.

######  Proposition 2.5.

Let $M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and
$P_{1}(M)\stackrel{{\scriptstyle f}}{{\rightarrow}}P_{0}(M)\rightarrow
M\rightarrow 0$ a minimal projective presentation of $M$. Then $M$ is
$\tau$-rigid if and only if $\operatorname{Hom}\nolimits_{\Lambda}(f,M)$ is
epic.

###  2.3. Lower triangular matrix algebras

Let $A$ and $B$ be rings and let ${{}_{B}}M_{A}$ be a $B$-$A$-bimodule. Then
the lower triangular matrix ring $\left(\begin{smallmatrix}A&0\\\
M&B\end{smallmatrix}\right)$ is the ring with addition
$\left(\begin{smallmatrix}r&0\\\
m&s\end{smallmatrix}\right)+\left(\begin{smallmatrix}r^{\prime}&0\\\
m^{\prime}&s^{\prime}\end{smallmatrix}\right)=\left(\begin{smallmatrix}r+r^{\prime}&0\\\
m+m^{\prime}&s+s^{\prime}\end{smallmatrix}\right)$ and product
$\left(\begin{smallmatrix}r&0\\\
m&s\end{smallmatrix}\right)\left(\begin{smallmatrix}r^{\prime}&0\\\
m^{\prime}&s^{\prime}\end{smallmatrix}\right)=\left(\begin{smallmatrix}rr^{\prime}&0\\\
mr^{\prime}+sm^{\prime}&ss^{\prime}\end{smallmatrix}\right).$ A left
$\left(\begin{smallmatrix}A&0\\\ M&B\end{smallmatrix}\right)$-module $(X,Y,f)$
is decided uniquely by a morphism $f:M\otimes_{A}X\rightarrow Y$ of left
$B$-modules with $X\in A$-mod and $Y\in B$-mod. Throughout this paper, we
assume that both $A$ and $B$ are finite-dimensional algebras over an
algebraically closed field $K$ and $M$ is finitely generated in both $B$-mod
and $A^{o}$-mod. Putting $A=B=M$, one gets the lower triangular matrix algebra
$T_{2}(A)$.

We also need the following facts on the projective modules over
$\left(\begin{smallmatrix}A&0\\\ M&B\end{smallmatrix}\right)$ [AuRS, III
Proposition 2.5].

######  Proposition 2.6.

Let $\Lambda=\left(\begin{smallmatrix}A&0\\\ M&B\end{smallmatrix}\right)$ be a
lower triangular matrix algebra. Then the indecomposable projective modules in
$\Lambda$-mod are of the form $(P,M\otimes P,I_{M\otimes P})$ and of the form
$(0,Q,0)$, where $P$ is indecomposable projective in $A$-mod and $Q$ is
indecomposable projective in $B$-mod.

##  3\. CM-$\tau$-tilting free algebras

In this section, we first introduce CM-$\tau$-tilting free algebras which are
the generalization of CM-free algebras and local algebras. Then we study the
homological properties of these algebras and give a construction of
CM-$\tau$-tilting free algebras by using $T_{2}$-extension.

We start with the definition of Gorenstein projective $\tau$-rigid modules.

######  Definition 3.1.

Let $\Lambda$ be an algebra and $M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$.
$M$ is called Gorenstein projective $\tau$-rigid if it is both Gorenstein
projective and $\tau$-rigid.

Recall from [XZ] that an algebra $\Lambda$ is called CM-$\tau$-tilting finite
if there are only finite indecomposable Gorenstein projective $\tau$-rigid
modules in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ up to isomorphism. In the
following we introduce a special class of CM-$\tau$-tilting finite algebras
called CM-$\tau$-free algebras.

######  Definition 3.2.

Let $\Lambda$ be an algebra. We call $\Lambda$ CM-$\tau$-tilting free if all
Gorenstein projective $\tau$-rigid modules are projective.

It is obvious that CM-free algebras are CM-$\tau$-tilting free. While the
converse is not true. In the following we give an example due to Ringel and
Zhang [RZ1, Example 6.1] to show this.

######  Example 3.3.

Let $k$ be a infinite field and $q\in k$ and $q\neq 0$. Let $\Lambda$ be a
$6$-dimensional local algebra $\Lambda(q)$.

  1. (1)

There are infinite many Gorenstein projective modules of dimension $3$ and
hence $\Lambda$ is not CM-free.

  2. (2)

All $\tau$-rigid modules in $\mathsf{mod}\hskip 0.72229pt\Lambda$ are
projective since $\Lambda$ is local. Therefore, $\Lambda$ is CM-$\tau$-tilting
free.

In the following we show the symmetric properties of CM-$\tau$-tilting free
algebras.

######  Proposition 3.4.

Let $\Lambda$ be an algebra. $\Lambda$ is CM-$\tau$-tilting free if and only
if $\Lambda^{o}$ is CM-$\tau$-tilting free.

###### Proof.

By [AIR, Theorem 2.14], one gets that for any $\tau$-rigid module $M$ without
projective direct summands, $M$ is $\tau$-rigid if and only if so is ${\rm
Tr}M$.

We only show the necessity since the sufficiency is similar. Suppose that
$\Lambda$ is CM-$\tau$-tilting free. On the contrary, suppose that
$\Lambda^{o}$ is not CM-$\tau$-tilting free. Then there is a indecomposable
Gorenstein projective $\tau$-rigid module
$N\in\Lambda^{o}$-$\mathsf{mod}\hskip 0.72229pt$, then one gets ${\rm
Tr}N\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ is a Gorenstein projective
$\tau$-rigid module which is not projective by Proposition 2.2, a
contradiction. ∎

An algebra $\Lambda$ is Gorenstein if $\mathop{\rm
id}\nolimits_{\Lambda}\Lambda<\infty$ and $\mathop{\rm
id}\nolimits\Lambda_{\Lambda}<\infty$. A Gorenstein algebra $\Lambda$ is
$n$-Gorenstein if $\mathop{\rm id}\nolimits_{\Lambda}\Lambda\leq n$. In the
following, we classify Gorenstein CM-free algebras which are well-known.

######  Proposition 3.5.

Let $\Lambda$ be an $n$-Gorenstein algebra. If $\Lambda$ is CM-free, then
$\Lambda$ is of global dimension at most $n$.

###### Proof.

Since $\Lambda$ is $n$-Gorenstein, then for any
$M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$, $\Omega^{n}M$ is Gorenstein
projective by [AuR, Theorem 4.1]. Since $\Lambda$ is CM-free, then
$\Omega^{n}M$ is projective, and hence $\mathop{\rm pd}\nolimits M\leq n$. The
assertion holds. ∎

It is natural to ask the following question.

######  Question 3.6.

Is a Gorenstein CM-$\tau$-tilting free algebra local or of finite global
dimension?

We give a negative answer to the Question 3.6 by giving the following counter-
example.

######  Example 3.7.

Let $\Lambda$ be an algebra given by the quiver

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{2}$

with the relation $\alpha^{2}=0$.

  1. (1)

$\Lambda$ is a $1$-Gorenstein algebra with infinite global dimension which is
not local.

  2. (2)

The support $\tau$-tilting quiver of $\Lambda$ is as follows:

$\textstyle{\begin{smallmatrix}&1\\\ 1&&2\\\
2&\end{smallmatrix}\oplus\begin{smallmatrix}2\end{smallmatrix}}$$\textstyle{\begin{smallmatrix}2\end{smallmatrix}}$$\textstyle{\begin{smallmatrix}&1\\\
1&&2\\\ 2&\end{smallmatrix}\oplus\begin{smallmatrix}1\\\ 1\\\
2\end{smallmatrix}}$$\textstyle{\begin{smallmatrix}1\\\
1\end{smallmatrix}\oplus\begin{smallmatrix}1\\\ 1\\\
2\end{smallmatrix}}$$\textstyle{\begin{smallmatrix}1\\\
1\end{smallmatrix}}$$\textstyle{\begin{smallmatrix}0\end{smallmatrix}}$

And hence $\Lambda$ is a CM-$\tau$-tilting free algebra.

In the following we try to construct more examples on CM-$\tau$-tilting free
algebras. We need the following results in [LZ1, Theorem 1.1] and [LZh,
Corollary 3.4]

######  Proposition 3.8.

Let $\Lambda$ be a Gorenstein algebra.

  1. (1)

A $T_{2}(\Lambda)$-module $(X,Y,\phi)$ with $\phi:X\rightarrow Y$ is
Gorenstein-projective if and only if $X,Y$ and ${\rm Coker}\phi$ are
Gorenstein-projective modules in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and
$\phi:X\rightarrow Y$ is injective.

  2. (2)

If a $T_{2}(\Lambda)$-module $(X,Y,\phi)$ with $\phi:X\rightarrow Y$ is
$\tau$-rigid, then $X$ is $\tau$-rigid and ${\rm Coker\phi}$ is $\tau$-rigid.

We also need the following property on the tensor product of $\tau$-rigid
modules [LZh2, Proposition 3.3].

######  Proposition 3.9.

Let $A$ and $B$ be two algebras. If $P\in A$-$\mathsf{mod}\hskip 0.72229pt$ is
a projective module and $M\in B$-$\mathsf{mod}\hskip 0.72229pt$ is a
$\tau$-rigid module, then $P\otimes M\in(A\otimes B)$-$\mathsf{mod}\hskip
0.72229pt$ is a $\tau$-rigid module.

Now we are in a position to show the $T_{2}$-extension of a Gorenstein
CM-$\tau$-tilting free algebra $\Lambda$. As a result, we can construct a
class of Gorenstein CM-$\tau$-tilting free algebras.

######  Theorem 3.10.

Let $\Lambda$ be a Gorenstein algebra. Then the lower triangular matrix
algebra $T_{2}(\Lambda)$ is a CM-$\tau$-tilting free algebra if and only if
$\Lambda$ is a CM-$\tau$-tilting free algebra.

###### Proof.

$\Leftarrow$ For a $T_{2}(\Lambda)$-module $(X,Y,\phi)$, assume that
$(X,Y,\phi)$ is Gorenstein projective $\tau$-rigid. By Proposition 3.8(1),
$X,Y$ and ${\rm Coker}\phi$ are Gorenstein-projective modules in
$\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and $\phi:X\rightarrow Y$ is
injective. Then by Proposition 3.8(2), $X$ is $\tau$-rigid and ${\rm
Coker\phi}$ is $\tau$-rigid. Since $\Lambda$ is CM-$\tau$-tilting free, then
the fact both $X$ and ${\rm Coker}\phi$ are Gorenstein projective $\tau$-rigid
implies that both $X$ and ${\rm Coker}\phi$ are projective. Since
$0\rightarrow X\rightarrow Y\rightarrow{\rm Coker}\phi\rightarrow 0$ is an
exact sequence, one gets that the sequence splits, and hence $(X,Y,\phi)$ is a
direct sum of $(X,X,I_{X})$ and $(0,{\rm Coker}\phi,0)$. Then one gets the
assertion.

$\Rightarrow$ On the contrary, suppose that $\Lambda$ is not CM-$\tau$-tilting
free. Then we get a non-projective Gorenstein projective $\tau$-rigid module
$M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. Since $T_{2}(\Lambda)\simeq
T_{2}(K)\otimes\Lambda$, we get $T_{2}(K)\otimes M$ is Gorenstein projective
and $\tau$-rigid by Proposition 2.3 and Proposition 3.9. Since $M$ is non-
projective, one gets that $T_{2}(K)\otimes M$ is non-projective by [LZh2,
Lemma 2.7]. Note that $T_{2}(\Lambda)$ is CM-$\tau$-tilting free, a
contradiction. ∎

As a straight application, we get the following corollary.

######  Corollary 3.11.

Let $\Lambda$ be an algebra. Then $\Lambda$ is Gorenstein CM-$\tau$-tilting
free if and only if so is the lower triangular matrix algebra
$T_{2}(\Lambda)$.

###### Proof.

By Theorem 3.10 it suffices to show that $\Lambda$ is Gorenstein if and only
if so ia $T_{2}(\Lambda)$. This follows from the facts $T_{2}(\Lambda)\simeq
T_{2}(K)\otimes\Lambda$ and $\rm{id}_{\Lambda}T_{2}(\Lambda)={\rm
id}_{\Lambda}\Lambda+1$. ∎

##  4\. $E$-rigid modules

In this section, we introduce the notion of $E$-Gorenstein projective and
$E$-rigid modules, and show the bijection between Gorenstein projective
$\tau$-rigid modules and $E$-Gorenstein projective $E$-rigid modules. Let
$\Lambda$ be an algebra and $E$ a generator in $\Lambda$-$\mathsf{mod}\hskip
0.72229pt$. In the following we recall the definition of $E$-Gorenstein
projective modules from [KZ].

######  Definition 4.1.

Let $\dots\rightarrow E_{-1}\stackrel{{\scriptstyle
d_{-1}}}{{\rightarrow}}E_{0}\stackrel{{\scriptstyle
d_{0}}}{{\rightarrow}}E_{1}\rightarrow\dots\ \ \ \ \ (*)$ be an exact sequence
with $E_{i}\in\mathop{\rm add}\nolimits E$ which are still exact by applying
the functors ${\rm Hom}_{\Lambda}(-,E)$ and ${\rm Hom}_{\Lambda}(E,-)$. Then
we call $M\simeq{\rm Imd_{0}}$ an $E$-Gorenstein projective module. Denote by
$\mathcal{G}(E)$ the subcategory of $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$
consisting of all $E$-Gorenstein projective modules.

For an algebra $\Gamma$, denote by $\mathcal{G}(\Gamma)$ the category of
finitely generated Gorenstein projective left $\Lambda$-modules. The following
property of $E$-Gorenstein projective modules is essential [KZ, Theorem 4.3].

######  Proposition 4.2.

Let $\Lambda$ be an algebra and $E$ a generator in
$\Lambda$-$\mathsf{mod}\hskip 0.72229pt$, and $\Gamma=({\rm
End}_{\Lambda}E)^{op}$. Then ${\rm Hom}_{\Lambda}(E,-)$ induces an equivalence
between $\mathcal{G}(E)$and $\mathcal{G}(\Gamma)$.

In the following, we introduce the notion of $E$-rigid modules which are the
generalization of $\tau$-rigid modules.

######  Definition 4.3.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and
$M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. Let
$E_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}E_{0}\stackrel{{\scriptstyle
f_{0}}}{{\rightarrow}}M\rightarrow 0$ be an exact sequence with $f_{i}$ the
minimal right $\mathop{\rm add}\nolimits E$-approximation of $\mathop{\rm
Im}\nolimits f_{i}$ (that is, a minimal $\mathop{\rm add}\nolimits
E$-presentation of $M$). If the map ${\rm Hom}_{\Lambda}(f_{1},M)$ is
surjective, then we call $M$ an $E$-rigid module. In particular, if
$E=\Lambda$, then $E$-rigid modules are precisely $\tau$-rigid modules.

We give the following example on $E$-rigid modules.

######  Example 4.4.

Let $\Lambda=KQ$ be given by the quiver $Q:1\rightarrow 2\rightarrow 3$. Let
$E_{1}=P(1)\oplus P(2)\oplus P(3)\oplus I(2)$ be a generator in
$\Lambda$-$\mathsf{mod}\hskip 0.72229pt$.

  1. (1)

$E_{1}$ is $E_{1}$-rigid but not rigid and hence not $\tau$-rigid since we
have the following non-split exact sequence:

$0\rightarrow P(3)\rightarrow P(1)\rightarrow I(2)\rightarrow 0.$

  2. (2)

The following is a minimal $\mathop{\rm add}\nolimits E_{1}$-presentation of
$S(1):$

$P(2)\rightarrow I(2)\rightarrow S(1)\rightarrow 0.$

Hence $S(1)$ is $E_{1}$-rigid but not in $\mathop{\rm add}\nolimits E_{1}$.

In the following we show the property on $E$-rigid modules which generalizes
[AIR, Proposition 2.5].

######  Proposition 4.5.

Let $M,E\in$ $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and $E$ a generator. Let
$E_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}E_{0}\stackrel{{\scriptstyle
f_{0}}}{{\rightarrow}}M\rightarrow 0$ be a minimal $\mathop{\rm add}\nolimits
E$-presentation of $M$. If $M$ is $E$-rigid, then $E_{0}$ and $E_{1}$ have no
non-zero direct summands in common.

###### Proof.

We show that any homomorphism $s:E_{1}\rightarrow E_{0}$ is in the radical.
One gets a homomorphism $f_{0}s:E_{1}\rightarrow M$. Since $M$ is $E$-rigid,
by Proposition 2.5 there exists a homomorphism $t:E_{0}\rightarrow M$ such
that $tf_{1}=f_{0}s$. Since $f_{0}$ is a minimal right $\mathop{\rm
add}\nolimits E$-approximation of $M$, then there is a homomorphism
$u:E_{0}\rightarrow E_{0}$ such that $t=f_{0}u$. So one gets
$f_{0}(s-uf_{1})=0$ which implies ${\rm Im}(s-uf_{1})\in{\rm Im}f_{1}$. Since
$E_{1}\rightarrow{\rm Im}f_{1}$ is a minimal right $\mathop{\rm add}\nolimits
E$-approximation, then there exists a homomorphism $v:E_{1}\rightarrow E_{1}$
such that $s=uf_{1}+f_{1}v$. We have the following commutative diagram:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.37915pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ }$}}}}}}}{\hbox{\kern 32.37915pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.97981pt\raise-14.81194pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{v}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
8.37917pt\raise-30.69229pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
40.7583pt\raise-19.31888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{s}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.7583pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
55.0555pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{f_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 73.15413pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.15413pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
81.53328pt\raise-19.31888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15279pt\hbox{$\scriptstyle{t}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
81.53328pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
60.88194pt\raise-23.82582pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{u}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
49.13747pt\raise-30.70047pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
95.83882pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 113.92911pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
113.92911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
154.72076pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
154.72076pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-8.37915pt\raise-38.63776pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.28886pt\raise-32.52666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
32.37915pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.37915pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
55.0555pt\raise-32.52666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
73.13745pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.13745pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
116.82494pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
116.82494pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern
154.72076pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\
}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

The assertion follows from the fact $f_{1}$ is in the radical. ∎

######  Definition 4.6.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. We call
$M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ $E$-Gorenstein projective
$E$-rigid if it is both $E$-Gorenstein projective and $E$-rigid.

Denote by $\mathcal{EG}(E)$ the set of isomorphism classes of indecomposable
$E$-Gorenstein projecitve $E$-rigid modules in $\Lambda$-$\mathsf{mod}\hskip
0.72229pt$ and by $\mathcal{TG}(\Gamma)$ the set of isomorphism classes of
indecomposable Gorenstein projective $\tau$-rigid modules in
$\Gamma$-$\mathsf{mod}\hskip 0.72229pt$. Now we are in a position to state our
main results in this section.

######  Theorem 4.7.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ and
$\Gamma=({\rm End}_{\Lambda}E)^{op}$. Then there is a bijection between
$\mathcal{EG}(E)$ and $\mathcal{TG}(\Gamma)$.

###### Proof.

We show that the functor ${\rm Hom}_{\Lambda}(E,-)$ is the desired map. We
divide the proof into several steps.

(a) ${\rm Hom}_{\Lambda}(E,-)$ is a well-defined map from $\mathcal{EG}(E)$ to
$\mathcal{TG}(\Gamma)$.

By Proposition 4.2, one gets that ${\rm Hom}_{\Lambda}(E,M)$ is indecomposable
in $\mathcal{G}(\Gamma)$ for any indecomposable $M\in\mathcal{G}(E)$. For any
$M\in\mathcal{EG}(E)$, we get a minimal $\mathop{\rm add}\nolimits E$
presentation

$E_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}E_{0}\stackrel{{\scriptstyle
f_{0}}}{{\rightarrow}}M\rightarrow 0.\ \ \ \ \ \ \ (1)$

Applying the functor ${\rm Hom}_{\Lambda}(E,-)$ to $(1)$, we get the following
minimal projective presentation of ${\rm Hom}(E,M)$:

${\rm
Hom}_{\Lambda}(E,E_{1})\stackrel{{\scriptstyle{{}^{*}f_{1}}}}{{\rightarrow}}{\rm
Hom}_{\Lambda}(E,E_{0})\stackrel{{\scriptstyle{{}^{*}f_{0}}}}{{\rightarrow}}{\rm
Hom}_{\Lambda}(E,M)\rightarrow 0,\ \ \ \ \ \ \ \ (2)$

where ${}^{*}f_{i}={\rm Hom}_{\Lambda}(E,f_{i})$ for $i=0,1$.

Note that $M\in\mathcal{EG}(E)$, applying the functor ${\rm
Hom}_{\Lambda}(-,M)$ to $(1)$, one gets the following exact sequence

${\rm
Hom}_{\Lambda}(E_{0},M)\stackrel{{\scriptstyle{f_{1}}^{\prime}}}{{\rightarrow}}{\rm
Hom}_{\Lambda}(E_{1},M)\rightarrow 0.$

Applying the functor ${\rm Hom}_{\Gamma}(-,{\rm Hom}_{\Lambda}(E,M))$ to
${}^{*}f_{1}$, one gets the following complex

${\rm Hom}_{\Gamma}({\rm Hom}_{\Lambda}(E,E_{0}),{\rm
Hom}_{\Lambda}(E,M))\stackrel{{\scriptstyle h}}{{\rightarrow}}{\rm
Hom}_{\Gamma}({\rm Hom}_{\Lambda}(E,E_{1}),{\rm Hom}_{\Lambda}(E,M)).$

By Proposition 4.2, one gets the following commutative diagram:

$\textstyle{{\rm
Hom}_{\Lambda}(E_{0},M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{f_{1}}^{\prime}}$$\scriptstyle{\cong}$$\textstyle{{\rm
Hom}_{\Lambda}(E_{1},M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{{\rm
Hom}_{\Gamma}({\rm Hom}_{\Lambda}(E,E_{0}),{\rm
Hom}_{\Lambda}(E,M))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\rm
Hom}_{\Gamma}({\rm Hom}_{\Lambda}(E,E_{1}),{\rm
Hom}_{\Lambda}(E,M))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

which implies that $h$ is surjective. That is, ${\rm Hom}_{\Lambda}(E,M)$ is
in $\mathcal{TG}(\Gamma)$ by Proposition 2.5.

(b) ${\rm Hom}_{\Lambda}(E,-)$ is injective.

Let $M_{1},M_{2}$ be in $\mathcal{EG}(E)$ such that ${\rm
Hom}_{\Lambda}(E,M_{1})\simeq{\rm Hom}_{\Lambda}(E,M_{2})$. By Proposition
4.2, one gets that $M_{1}\simeq M_{2}$.

(c) ${\rm Hom}_{\Lambda}(E,-)$ is surjective.

For any $N\in\mathcal{TG}(\Gamma)$, we get a minimal projective presentation
of $N$:

$Q_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}Q_{0}\stackrel{{\scriptstyle
f_{0}}}{{\rightarrow}}N\rightarrow 0.\ \ \ \ \ (3)$

By Proposition 4.2, we get the following minimal $\mathop{\rm add}\nolimits
E$-presentation of $M$:

$E_{1}\stackrel{{\scriptstyle
g_{1}}}{{\rightarrow}}E_{0}\stackrel{{\scriptstyle
g_{0}}}{{\rightarrow}}M\rightarrow 0\ \ \ \ \ \ (4)$

such that $M\in\mathcal{G}(E)$, ${\rm Hom}_{\Lambda}(E,M)\simeq N$, ${\rm
Hom}_{\Lambda}(E,E_{i})\simeq Q_{i}$ and ${\rm Hom}_{\Lambda}(E,g_{i})\simeq
f_{i}$ for $i=0,1$.

Note that $N\in\mathcal{TG}(\Gamma)$, applying the functor ${\rm
Hom}_{\Lambda}(-,N)$ to $f_{1}$, one gets the following exact sequence

${\rm
Hom}_{\Lambda}(Q_{0},N)\stackrel{{\scriptstyle{f_{1}}^{{}^{\prime}}}}{{\rightarrow}}{\rm
Hom}_{\Lambda}(Q_{1},N)\rightarrow 0.$

Similarly, applying the functor ${\rm Hom}_{\Lambda}(-,M)$ to $g_{1}$, one has
the following complex

${\rm
Hom}_{\Lambda}(E_{0},M)\stackrel{{\scriptstyle{g_{1}}^{{}^{\prime}}}}{{\rightarrow}}{\rm
Hom}_{\Lambda}(E_{1},M).$

By using Proposition 4.2 again, one gets the following commutative diagram:

$\textstyle{{\rm
Hom}_{\Gamma}(Q_{0},N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{f_{1}}^{\prime}}$$\scriptstyle{\cong}$$\textstyle{{\rm
Hom}_{\Gamma}(Q_{1},N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{{\rm
Hom}_{\Lambda}(E_{0},M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{g_{1}}^{\prime}}$$\textstyle{{\rm
Hom}_{\Lambda}(E_{1},M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

which implies that ${g_{1}}^{\prime}$ is surjective, that is $M$ is $E$-rigid.
And hence $M\in\mathcal{EG}(E)$. ∎

In the following we show the following application of Theorem 4.7.

######  Definition 4.8.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$.

  1. (1)

We call $\Lambda$ CM-E-free if every object in $\mathcal{EG}(E)$ is a direct
summand of $E$. Take $E=\Lambda$, the CM-$E$-free algebras are precisely
CM-$\tau$-tilting free algebras in Section 3.

  2. (2)

We call $\Lambda$ CM-E-finite if the number of objects in $\mathcal{EG}(E)$ is
finite. Take $E=\Lambda$, the CM-$E$-finite algebras are precisely
CM-$\tau$-tilting finite algebras in Section 3.

As an application, we have the following corollary.

######  Corollary 4.9.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. Then
$\Lambda$ is CM-$E$-free if and only if $\Gamma=({\rm End}E)^{op}$ is
CM-$\tau$-tilting free.

###### Proof.

It is a straight result of Theorem 4.8. ∎

We end this section with the following question(compare [KZ, Theorem 4.5]).

######  Question 4.10.

Let $E$ be a generator in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$. Let
$\Lambda$ be CM-E-finite with $F$ the additive generator of $E$-Gorenstein
projective $E$-rigid modules. Is $\Gamma^{\prime}=({\rm End}(F))^{op}$ a
CM-$\tau$-tilting free algebra?

##  5\. Tachikawa’s Conjecture

In this section, we try to characterize self-injective algebras in terms of
Gorenstein projective $\tau$-rigid modules. As a result, we can give a partial
answer to Tachikawa’s first conjecture.

Recall from [RZ1] that a module $M\in\Lambda$-$\mathsf{mod}\hskip 0.72229pt$
is called semi-Gorenstein projective if ${\rm Ext}_{\Lambda}^{i}(M,\Lambda)=0$
for any $i\geq 1$. Dually, we call a module
$N\in$$\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ semi-Gorenstein injective if
${\rm Ext}_{\Lambda}^{i}(\mathbb{D}\Lambda,N)=0$ for $i\geq 1$. The
Tachikawa’s first conjecture is equivalent to that for an algebra $\Lambda$,
if $\mathbb{D}\Lambda$ is semi-Gorenstein projective, then $\Lambda$ is self-
injective.

To show the main result in this section, we need the following result in
[XZZ].

######  Proposition 5.1.

Let $\Lambda$ be an algebra. The following are equivalent.

  1. (1)

$\Lambda$ is a $1$-Gorenstein algebra.

  2. (2)

$\mathbb{D}\Lambda$ is $\tau$-rigid

  3. (3)

$\Lambda$ is $\tau^{-1}$-rigid.

Note that a self-injective algebra should be a $1$-Gorenstein algebra, by
Proposition 5.1 it is natural to study Tachikawa’s first conjecture in terms
of $\tau$-rigid modules. In the following we give our main result in this
section which is a partial answer to Tachikawa’s first conjecture.

######  Theorem 5.2.

Let $\Lambda$ be an algebra. Then the following are equivalent.

  1. (1)

$\Lambda$ is self-injective.

  2. (2)

$\mathbb{D}\Lambda$ is semi-Gorenstein projective $\tau$-rigid.

  3. (3)

Every cotilting module is semi-Gorenstein projective $\tau$-rigid.

  4. (4)

$\mathbb{D}\Lambda$ is Gorenstein projective.

  5. (5)

Every coltilting module is Gorenstein projective.

  6. (6)

$\Lambda$ is semi-Gorenstein-injective $\tau^{-1}$-rigid.

  7. (7)

Every tilting module is semi-Gorenstein injective $\tau^{-1}$-rigid.

  8. (8)

$\Lambda$ is Gorenstein-injective.

  9. (9)

Every tilting module is Gorenstein-injective.

###### Proof.

(a) Since $(1)\Leftrightarrow(4)$ is from the definitions of Gorenstein
projective modules and self-injective algebras, we only show
$(1)\Leftrightarrow(2)$,$(2)\Leftrightarrow(3)$, and $(4)\Leftrightarrow(5)$.

$(1)\Rightarrow(2)$ Since $\Lambda$ is self-injective, then
$\mathbb{D}\Lambda$ is projective and hence semi-Gorenstein projective and
$\tau$-rigid.

$(2)\Rightarrow(1)$ Since $\mathbb{D}\Lambda$ is $\tau$-rigid and
$|\mathbb{D}\Lambda|=|\Lambda|$, one gets that $\mathbb{D}\Lambda$ is a
$\tau$-tilting module. Note that $\mathbb{D}\Lambda$ is faithful, then
$\mathbb{D}\Lambda$ is a tilting module. Since $\mathbb{D}\Lambda$ is semi-
Gorenstein projective, then the fact $\rm{pd}\mathbb{D}\Lambda\leq 1$ implies
that $\mathbb{D}\Lambda$ is projective. The assertion holds.

$(3)\Rightarrow(2)$ This is obvious.

$(2)\Rightarrow(3)$ By $(2)\Leftrightarrow(1)$, $\Lambda$ is self-injective,
and hence $\Lambda$ is the unique cotilting module in $\mathsf{mod}\hskip
0.72229pt\Lambda$. The assertion holds.

$(5)\Rightarrow(4)$ This is obvious.

$(4)\Rightarrow(5)$ By (4) one gets that $\Lambda$ is self-injective, and
hence $\Lambda$ is the unique cotilting module in $\mathsf{mod}\hskip
0.72229pt\Lambda$.

(b) Since $(1)\Leftrightarrow(8)$ is clear, we only need to show
$(1)\Leftrightarrow(6)$, $(6)\Leftrightarrow(7)$ and $(8)\Leftrightarrow(9)$.

$(1)\Rightarrow(6)$ Since $\Lambda$ is self-injective, then
$\Lambda\simeq\mathbb{D}\Lambda$ is injective, and hence semi-Gorenstein
injective and $\tau^{-1}$-rigid.

$(6)\Rightarrow(1)$ Since $\Lambda$ is $\tau^{-1}$-rigid, by Proposition 5.1,
one gets that $\Lambda$ is $1$-Gorenstein. Note that $\Lambda$ is semi-
Gorenstein injective, one gets ${\rm
Ext}_{\Lambda}^{i}(\mathbb{D}\Lambda,\Lambda)=0$ for $i\geq 1$, which implies
that $\Lambda$ is self-injective.

$(7)\Rightarrow(6)$ This is obvious.

$(6)\Rightarrow(7)$ By $(6)\Leftrightarrow(1)$, $\Lambda$ is self-injective.
Then the unique tilting module in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ is
$\mathbb{D}\Lambda$, the assertion holds.

$(9)\Rightarrow(8)$ This is obvious.

$(8)\Rightarrow(9)$ By $(8)\Leftrightarrow(1)$, $\Lambda$ is self-injective.
Then the unique tilting module in $\Lambda$-$\mathsf{mod}\hskip 0.72229pt$ is
$\mathbb{D}\Lambda$, the assertion holds. ∎

######  Remark 5.3.

By Theorem 5.2, one gets that the fact $\mathbb{D}\Lambda$ is semi-Gorenstein
projective $\tau$-rigid is equivalent to the fact $\mathbb{D}\Lambda$ is
Gorenstein projective. To prove Tachikawa’s Conjecture, it is meaningful to
show that the fact $\mathbb{D}\Lambda$ is semi-Gorenstein projective implies
the fact $\mathbb{D}\Lambda$ is $\tau$-rigid.

Acknowledgement The authors want to thank Prof. Zhaoyong Huang and Prof. Zhi-
Wei Li for useful discussions and suggestions. The authors are supported by
the NSFC(Nos. 12171207, 12201211, 12371038).

## References

  * [AIR] T. Adachi, O. Iyama and I. Reiten, $\tau$-tilting theory, Compos. Math, 150(3)(2014), 415-452.
  * [AH] T. Aihara and T. Honma, $\tau$-tilting finite triangular matrix algebras, J. Pure Appl. Algebra, 225(2)(2021), 106785.
  * [AI] T. Aihara and O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc, 85(3)(2012), 633-668.
  * [AuB] M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society, 1969.
  * [AuR] M. Auslander and I. Reiten, Cohen-Macaulay and Gorenstein Artin algebras, Representation Theory of Finite Groups and Finite-Dimensional Algebras, Birkhäuser Basel, 95(1991), 221-245.
  * [AuRS] M. Auslander, I. Reiten and S. O. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math, vol. 36, Cambridge Univ. Press, 1995.
  * [B] A. Beligiannis, On algebras of finite Cohen-Macaulay type, Adv. Math, 226(2)(2011), 1973-2019.
  * [C] X. W. Chen, An Auslander-type result for Gorenstein-projective modules, Adv. Math, 218 (2008), 2043-2050.
  * [CE] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, 1956.
  * [CSZ] X. W. Chen, D. W. Shen and G. D. Zhou, The Gorenstein-projective modules over a monomial algebra. Proc. Roy. Soc. Edinburgh Sect. A, 148(6)(2018), 1115-1134.
  * [DIJ] L. Demonet, O. Iyama and G. Jasso, $\tau$-tilting finite algebras, bricks and $g$-vectors, Int. Math. Res. Not, 3(2019), 852-892.
  * [EJ1] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z, 220(1995), 611-633.
  * [EJ2] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra. de Gruyter Exp.Math, vol. 30, Walter de Gruyter Co., 2000.
  * [FGLZ] C. J. Fu, S. F. Geng, P. Liu and Y. Zhou, On support $\tau$-tilting graphs of gentle algebras, J. Algebra, 628(2023), 189-211.
  * [GMZ] N. Gao, J. Ma and C. H. Zhang, Gorenstein silting modules and Gorenstein projective modules, arXiv:2209.00520.
  * [HLXZ] W. Hu, X. H. Luo, B. L. Xiong and G. D. Zhou, Gorenstein projective bimodules via monomorphism categories and filtration categories, J. Pure Appl. Algebra, 223(3)(2019),1014-1039.
  * [IY] O. Iyama and Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math, 172(2008), 117-168.
  * [IZ1] O. Iyama and X. J. Zhang, Tilting modules over Auslander-Gorenstein algebras, Pacific J. Math, 298(2)(2019), 399-416.
  * [IZ2] O. Iyama and X. J. Zhang, Classifying $\tau$-tilting modules over the Auslander algebra of of $K[x]/(x^{n})$, J. Math. Soc. Japan, 72(3)(2020), 731-764.
  * [K] S. Kvamme, Gorenstein projective objects in functor categories, Nagoya Math. J, 240(2020), 1-41.
  * [KK] R. Koshio, and Y. Kozakai, On support $\tau$-tilting modules over blocks covering cyclic blocks, J. Algebra, 580(2021), 84-103.
  * [KZ] F. Kong and P. Zhang, From CM-finite to CM-free, J. Pure Appl. Algebra, 220(2016), 782-801.
  * [LZ1] Z. W. Li and P. Zhang, Gorenstein algebras of finite Cohen-Macaulay type, Adv. Math, 218(2010), 728-734.
  * [LZ2] Z. W. Li and P. Zhang, A construction of Gorenstein-projective modules, J. Algebra, 323(2010), 1802-1812.
  * [LZh] Z. W. Li and X. J. Zhang, $\tau$-tilting modules over trivial extensions, Internat J. Algebra Comput, 32(3)(2022), 617-628.
  * [LZh2] Z. W. Li and X. J. Zhang, A construction of Gorenstein projective $\tau$-tilting modules, Colloq Math, 171(1)(2023), 103-112.
  * [PMH] Y. Y. Peng, X. Ma and Z. Y. Huang, $\tau$-tilting modules over triangular matrix artin algebras, Internat J. Algebra Comput, 31(4)(2021), 639-661.
  * [RZ1] C. M. Ringel and P. Zhang, Gorenstein projective modules and Semi-Gorenstein projective modules, Algebra Number Theory, 14(2020), 1-36.
  * [RZ2] C. M. Ringel and P. Zhang, Gorenstein projective modules and Semi-Gorenstein projective modules II, J Pure Appl. Algebra, 224(2020), 106248.
  * [RZ3] C. M. Ringel and P. Zhang, On modules $M$ such that both $M$ and $M^{*}$ are semi-Gorenstein-projective, Algebr. Represent. Theory, 24(4)(2021), 1125-1140.
  * [RZ4] C. M. Ringel and P. Zhang, Gorenstein-projective modules over short local algebras, J. Lond. Math. Soc, 106(2)(2022), 528-589.
  * [S] S. O. Smalø, Torsion theory and tilting modules, Bull. Lond. Math. Soc, 16(1984), 518-522.
  * [T] H. Tachikawa, Quasi-Frobenius Rings and Generalizations, LNM, vol.351, Springer, Berlin-Heidelberg-New York, 1973.
  * [W] Q. Wang, On $\tau$-tilting finiteness of the Schur algebra, J. Pure Appl. Algebra, 226(1) (2022), 106818.
  * [WZ] Q. Wang and Y. Y. Zhang, On $\tau$-tilting modules over trivial extensions of gentle tree algebras, arXiv: 2204.06418.
  * [XZZ] Z. Z. Xie, L. B. Zan and X. J. Zhang, Three results for $\tau$-rigid modules, Rocky Mountain J. Math, 49(8)(2019), 2791-2808.
  * [XZ] Z. Z. Xie and X. J. Zhang, A bijection theorem for Gorenstein projective $\tau$-tilting modules, to appear in Journal of Algebra and its Applications, arXiv: 2109. 01248.
  * [Z] X. J. Zhang, Self-orthogonal $\tau$-tilting modules and tilting modules, J Pure Appl. Algebra, 226(3)(2022), 10860.
  * [Zh] Y. Y. Zhang, Gluing support $\tau$-tilting modules via symmetric ladders of height 2, to appear in Sci. China Math, arXiv: 2105.09636.
  * [Zi] S. Zito, $\tau$-tilting finite cluster-tilted algebras, Proc. Edinb. Math. Soc, 63(4)(2020), 950-955.