Auslander conditions and tilting-like cotorsion pairs

Jian Wanga, Yunxia Lia, Jinyong Wub and Jiangsheng Huc,d***Corresponding
author

aCollege of Science, Jinling Institute of Technology, Nanjing 211169, China

bDepartment of Mathematics, Wenzhou University, Wenzhou 325035, China

cSchool of Mathematics, Hangzhou Normal University, Hangzhou 311121, China

dSchool of Mathematics and Physics, Jiangsu University of Technology,

Changzhou 213001, China

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and
<EMAIL_ADDRESS>

Abstract

We study homological behavior of modules satisfying the Auslander condition.
Assume that $\mathcal{AC}$ is the class of left $R$-modules satisfying the
Auslander condition. It is proved that each cycle of an exact complex with
each term in $\mathcal{AC}$ belongs to $\mathcal{AC}$ for any ring $R$. As a
consequence, we show that for any left Noetherian ring $R$, $\mathcal{AC}$ is
a resolving subcategory of the category of left $R$-modules if and only if
${}_{R}R$ satisfies the Auslander condition if and only if each Gorenstein
projective left $R$-module belongs to $\mathcal{AC}$. As an application, we
prove that, for an Artinian algebra $R$ satisfying the Auslander condition,
$R$ is Gorenstein if and only if $\mathcal{AC}$ coincides with the class of
Gorenstein projective left $R$-modules if and only if
(${\mathcal{AC}^{<\infty}},(\mathcal{AC}^{<\infty})^{\bot}$) is a tilting-like
cotorsion pair if and only if (${\mathcal{AC}^{<\infty}},\mathcal{I}$) is a
tilting-like cotorsion pair, where $\mathcal{AC}^{<\infty}$ is the class of
left $R$-modules with finite $\mathcal{AC}$-dimension and $\mathcal{I}$ is the
class of injective left $R$-modules. This leads to some criteria for the
validity of the Auslander and Reiten conjecture which says that an Artinian
algebra satisfying the Auslander condition is Gorenstein.
Keywords: Auslander condition; Tilting-like cotorsion pair; Tilting cotorsion
pair; Gorenstein ring; Gorenstein projective module.
2020 Mathematics Subject Classification: 16E65; 16E10; 18G25; 16G10.

##  1\. Introduction

For a commutative Noetherian ring $R$, Bass proved in [7] that, it is a
Gorenstein ring (that is, the self-injective dimension of $R$ is finite) if
and only if the flat dimension of the $i$-th term in a minimal injective
coresolution of $R$ as an $R$-module is at most $i-1$ for any $i\geqslant 1$.
Auslander showed that this condition is left-right symmetric for a left and
right Noetherian ring $R$; see Fossum, Griffith and Reiten [16]. In this case,
$R$ is said to satisfy the _Auslander condition_ , which may be regarded as a
non-commutative version of commutative Gorenstein rings. Moreover, Auslander
and Reiten conjectured in [4] that an Artinian algebra satisfying the
Auslander condition is Gorenstein. The conjecture is of particular interest
due to its strong connection with several long-standing homological
conjectures such as the finitistic dimension conjecture, the Nakayama
conjecture and the Wakamatsu tilting conjecture (see [4, 8, 22]).

Cotorsion pairs were invented by Salce [28] in the category of abelian groups
and have been deeply studied in approximation theory of modules [19],
especially in the proof of the flat cover conjecture [10]. Motivated by the
fact that finitistic dimensions of an algebra can alternatively be computed by
Gorenstein projective dimension, Moradifar and $\check{\rm{S}}$aroch [27]
introduced the notion of tilting-like cotorsion pairs, which is a natural
generalization of tilting cotorsion pairs. These cotorsion pairs have been
studied in [27, 30], especially providing criteria for the validity of the
finitistic dimension conjecture and the Wakamatsu tilting conjecture.

The main aim of this paper is to investigate the homological behavior of
modules satisfying the Auslander condition. More precisely, we first
characterize when the class $\mathcal{AC}$ of left $R$-modules satisfying the
Auslander condition is resolving. As a result, we establish the connection
between the Auslander and Reiten conjecture mentioned above with the cotorsion
pair induced by the class $\mathcal{AC}^{<\infty}$ of left $R$-modules with
finite $\mathcal{AC}$-dimension.

To state our results precisely, we first introduce some notation and
definitions. Throughout this paper, $R$ is an associative ring with identity
and $R$-Mod is the category of left $R$-modules. Recall that a full
subcategory $\mathcal{C}$ of $R$-Mod is _resolving_ if it contains all
projective modules and is closed under direct summands, extensions and kernels
of surjective homomorphisms. The notion of a resolving subcategory was
introduced by Auslander and Bridger [3] in the study of totally reflexive
modules, which are also called modules of Gorenstein dimension zero or
finitely generated Gorenstein projective modules.

Let $M$ be a left $R$-module. We use

$0\to M\to E^{0}(M)\to E^{1}(M)\to\cdots\to E^{i}(M)\to\cdots$

to denote a minimal injective coresolution of $M$. Following [22], $M$ is said
to satisfy _the Auslander condition_ if the flat dimension of $E^{i}(M)$ is at
most $i$ for any $i\geqslant 0$. Denote by $\mathcal{AC}$ the class of left
$R$-modules satisfying the Auslander condition.

Recall that a ring $R$ is _Gorenstein_ [25] if it is left and right Noetherian
and $R$ has finite self-injective dimension on either side. Moreover, a
Gorenstein ring is called _Auslander-Gorenstein_ [11] if it satisfies the
Auslander condition. A left and right Noetherian ring $R$ is called
_Auslander-regular_ [11] if $R$ satisfies the Auslander condition and each
left $R$-module has finite projective dimension.

Our first main result, characterizing when $\mathcal{AC}$ is a resolving
subcategory of $R$-Mod, can be stated as follows.

######  Theorem 1.1.

Let $R$ be a left Noetherian ring. Then the following are equivalent:

  1. (1)

$\mathcal{AC}$ is a resolving subcategory of $R$-Mod.

  2. (2)

${}_{R}R$ satisfies the Auslander condition.

  3. (3)

Each Gorenstein projective left $R$-module satisfies the Auslander condition.

  4. (4)

Each Ding projective left $R$-module satisfies the Auslander condition.

  5. (5)

Each Gorenstein flat left $R$-module satisfies the Auslander condition.

The proof of Theorem 1.1 relies heavily on the observation that each cycle of
an exact complex with each term in $\mathcal{AC}$ belongs to $\mathcal{AC}$
for any ring $R$; see Proposition 3.2. As a consequence of Theorem 1.1, we
prove that an Artinian algebra $R$ is Auslander-Gorenstein if and only if
$\mathcal{AC}$ coincides with the class of Gorenstein projective left
$R$-modules, which gives a criterion for the validity of the Auslander and
Reiten conjecture mentioned above; see Corollary 3.5.

Recall that a cotorsion pair $(\mathcal{A},\mathcal{B})$ of left $R$-modules
is said to be _tilting-like_ if $\mathcal{B}=(T\oplus S)^{\perp_{\infty}}$,
where $T$ is a tilting left $R$-module and $S$ is a strongly Gorenstein
projective left $R$-module; see [27, Definition 1.19]. Needless to say, for
$S=0$ we recover the tilting cotorsion pair induced by $T$.

Let $\mathcal{AC}^{<\infty}$ be the class of left $R$-modules with finite
$\mathcal{AC}$-dimension, i.e., for each $M$ in $\mathcal{AC}^{<\infty}$,
there is a non-negative integer $m$ and an exact sequence $0\rightarrow
P_{m}\rightarrow\cdots\rightarrow P_{0}\rightarrow M\rightarrow 0$ of left
$R$-modules with each $P_{i}$ in $\mathcal{AC}$. Assume that $R$ is a left
Noetherian ring such that ${}_{R}R$ satisfies the Auslander condition. It
follows from Theorem 1.1 that $\mathcal{AC}$ is a resolving subcategory of
$R$-Mod. In combination with a result due to Huang [21, Theorem 3.2], we
obtain that $\mathcal{AC}^{<\infty}$ is also resolving; see Corollary 3.6.

Our next main result characterizes when $\mathcal{AC}^{<\infty}$ can induce
tilting or tilting-like cotorsion pairs.

######  Theorem 1.2.

Let $R$ be a left and right Noetherian ring satisfying the Auslander
condition. Denote by $\mathcal{I}$ the class of injective left $R$-modules and
by $\mathcal{GI}$ the class of Gorenstein injective left $R$-modules. Then

  1. (1)

$R$ is a Gorenstein ring if and only if
(${\mathcal{AC}^{<\infty}},\mathcal{I}$) is a tilting-like cotorsion pair.

  2. (2)

The following conditions are equivalent:

    1. (a)

$R$ is an Auslander-regular ring.

    2. (b)

(${\mathcal{AC}^{<\infty}},\mathcal{I}$) is a tilting cotorsion pair.

    3. (c)

(${\mathcal{AC}^{<\infty}},\mathcal{GI}$) is a tilting cotorsion pair.

    4. (d)

(${\mathcal{AC}^{<\infty}},\mathcal{GI}$) is a tilting-like cotorsion pair.

  3. (3)

If $R$ is an Artinian algebra, then $R$ is Gorenstein if and only if
(${\mathcal{AC}^{<\infty}},(\mathcal{AC}^{<\infty})^{\bot}$) is a tilting-like
cotorsion pair.

  4. (4)

If $R$ is an Artinian algebra, then $R$ is Auslander-regular if and only if
$({\mathcal{AC}^{<\infty}},(\mathcal{AC}^{<\infty})^{\bot}$) is a tilting
cotorsion pair.

Remark that (1) and (3) in Theorem 1.2 together provide criteria for the
validity of the Auslander and Reiten conjecture mentioned above.

This paper is structured as follows. Section 2 contains notation and
definitions for use throughout this paper. In Section 3, we prove all results
mentioned in Introduction. In the course of our proofs, we prove that each
cycle of an exact complex with each term in $\mathcal{AC}$ belongs to
$\mathcal{AC}$ for any ring $R$; see Proposition 3.2.

##  2\. Preliminaries

In this section we give some notation and definitions which are used in the
paper.

###  2.1. Notation

For any left $R$-module $M$, fd${}_{R}M$ (resp. pd${}_{R}M$) is the flat
(resp. projective) dimension of $M$.

Let $\mathcal{C}$ be a full subcategory of $R$-Mod. Then we have
$\mathcal{C}^{\bot}=\\{M\in R{\text{-}}{\rm Mod}\ |\ \mbox{\rm
Ext}_{R}^{1}(C,M)=0\ {\rm for\ all}\ C\in\mathcal{C}\\},$
${}^{\bot}\mathcal{C}=\\{M\in R{\text{-}}{\rm Mod}\ |\ \mbox{\rm
Ext}_{R}^{1}(M,C)=0\ {\rm for\ all}\ C\in\mathcal{C}\\},$
$\mathcal{C}^{\bot_{\infty}}=\\{M\in R{\text{-}}{\rm Mod}\ |\ \mbox{\rm
Ext}_{R}^{i}(C,M)=0\ {\rm for\ all}\ C\in\mathcal{C}\ {\rm and\ all}\
i\geqslant 1\\}.$

For any left $R$-module $M$, we use $0\rightarrow M\rightarrow
E^{0}(M)\rightarrow\cdots\rightarrow E^{i}(M)\rightarrow\cdots$ to denote a
minimal injective coresolution of $M$.

Let
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{n}}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
be a complex of left $R$-modules. We will denote by $\ker(\partial_{n})$ the
$n$-th cycle.

###  2.2. Gorenstein modules

Following [14, 20], a left $R$-module $G$ is called _Gorenstein projective_ if
there is an exact sequence

$\mathbf{P}:\cdots\rightarrow P_{1}\rightarrow P_{0}\rightarrow
P^{0}\rightarrow P^{1}\rightarrow\cdots$

of projective left $R$-modules such that $G$ $\cong$ $\ker(P^{0}\rightarrow
P^{1})$ and $\mbox{\rm Hom}_{R}(\mathbf{P},Q)$ is exact for every projective
left $R$-module $Q$. In this case, the complex $\mathbf{P}$ is also called a
totally acyclic complex of projective left $R$-modules. Moreover, a left
$R$-module $M$ is called _strongly Gorenstein projective_ [9] if there is a
totally acyclic complex

$\cdots\rightarrow
P\xrightarrow{f}P\xrightarrow{f}P\xrightarrow{f}P\xrightarrow{f}\cdots$

of projective left $R$-modules such that $M$ $\cong$ $\ker(f)$.

A left $R$-module $M$ is called _Ding projective_ [13, 18] if there is an
exact sequence

$\mathbf{P}:\cdots\rightarrow P_{1}\rightarrow P_{0}\rightarrow
P^{0}\rightarrow P^{1}\rightarrow\cdots$

of projective left $R$-modules with $M\cong\textrm{ker}(P^{0}\rightarrow
P^{1})$ such that $\mbox{\rm Hom}_{R}(\mathbf{P},F)$ is exact for every flat
left $R$-module $F$.

A left $R$-module $M$ is called _Gorenstein flat_ [15] if there is an exact
sequence

$\mathbf{F}:\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow
F^{0}\rightarrow F^{1}\rightarrow\cdots$

of flat left $R$-modules with $M\cong\textrm{ker}(F^{0}\rightarrow F^{1})$
such that $E\otimes_{R}\mathbf{F}$ is exact for every injective right
$R$-module $E$.

A left $R$-module $M$ is called _Gorenstein injective_ [14] if there exists an
exact sequence

$\mathbf{I}:\cdots\rightarrow I_{1}\rightarrow I_{0}\rightarrow
I^{0}\rightarrow I^{1}\rightarrow\cdots$

of injective left $R$-modules with $M\cong\textrm{ker}(I^{0}\rightarrow
I^{1})$ such that $\mbox{\rm Hom}_{R}(E,\mathbf{I})$ is exact for every
injective left $R$-module $E$.

In the following, we use $\mathcal{GP}$, $\mathcal{G}p$, $\mathcal{GF}$ and
$\mathcal{GI}$ to denote the classes of Gorenstein projective, finitely
generated Gorenstein projective, Gorenstein flat and Gorenstein injective left
$R$-modules, respectively.

###  2.3. Cotorsion pairs

Let $\mathcal{A}$ and $\mathcal{B}$ be classes in $R$-Mod. A pair
$(\mathcal{A},\mathcal{B})$ is called a _cotorsion pair_ [19, 28] if
$\mathcal{A}$ $=$ ${}^{\bot}\mathcal{B}$ and $\mathcal{B}$ $=$
$\mathcal{A}^{\bot}$. The class $\mathcal{A}\cap\mathcal{B}$ is called the
_kernel_ of the cotorsion pair $(\mathcal{A},\mathcal{B})$.

A cotorsion pair ($\mathcal{A}$, $\mathcal{B}$) is said to be _hereditary_
[19] if $\mbox{\rm Ext}_{R}^{i}(A,B)$ $=$ $0$ for all $i$ $\geqslant$ $1$, $A$
$\in$ $\mathcal{A}$ and $B$ $\in$ $\mathcal{B}$. This definition is equivalent
to the condition that if whenever $0\rightarrow A_{3}\rightarrow
A_{2}\rightarrow A_{1}\rightarrow 0$ is exact with $A_{2}$, $A_{1}$ $\in$
$\mathcal{A}$, then $A_{3}$ is also in $\mathcal{A}$, or equivalently, if
whenever $0\rightarrow B_{1}\rightarrow B_{2}\rightarrow B_{3}\rightarrow 0$
is exact with $B_{1}$, $B_{2}$ $\in$ $\mathcal{B}$, then $B_{3}$ is also in
$\mathcal{B}$.

A cotorsion pair $(\mathcal{A},\mathcal{B})$ is _complete_ [19] if, for each
left $R$-module $M$, there is an exact sequence $0\rightarrow M\rightarrow
B\rightarrow L\rightarrow 0$ with $B$ $\in$ $\mathcal{B}$ and $L$ $\in$
$\mathcal{A}$. This definition is equivalent to the condition that, for each
left $R$-module $M$, there is an exact sequence $0\rightarrow D\rightarrow
C\rightarrow M\rightarrow 0$ with $C$ $\in$ $\mathcal{A}$ and $D$ $\in$
$\mathcal{B}$.

A cotorsion pair $(\mathcal{A},\mathcal{B})$ is _generated by a set_ [19]
provided that there is a set $\mathcal{S}$ of left $R$-modules such that
$\mathcal{S}^{\bot}=\mathcal{B}$ $($i.e., $(\mathcal{A},\mathcal{B})$ $=$
$(^{\bot}(\mathcal{S}^{\bot}),\mathcal{S}^{\bot})$$)$. A cotorsion pair
generated by a set $\mathcal{S}$ is also called a cotorsion pair _cogenerated_
by $\mathcal{S}$ in some literature. Here, we use the terminology from [19].
If ($\mathcal{A},\mathcal{B}$) is a hereditary cotorsion pair generated by a
set $\mathcal{S}$, then there is a single module
$M:=\bigoplus_{A\in\mathcal{S}}A$ such that $M^{\bot_{\infty}}=\mathcal{B}$.

A left $R$-module $T$ is _tilting_ [2, 12] if $\mbox{\rm pd}_{R}T$ $<$
$\infty$, $T^{(\lambda)}$ $\in$ $T^{\bot_{\infty}}$ for every cardinal
$\lambda$ and there is an exact sequence $0\rightarrow R\rightarrow
T_{0}\rightarrow\cdots\rightarrow T_{r}\rightarrow 0$ with $T_{i}$ $\in$
$\mbox{\rm Add}T$ for all $0$ $\leqslant$ $i$ $\leqslant$ $r$, where $r$ is
the projective dimension of $T$ and Add$T$ is all direct summands of direct
sums of copies of $T$.

Following [19], a cotorsion pair $(\mathcal{A},\mathcal{B})$ of left
$R$-modules is called a _tilting cotorsion pair_ if $\mathcal{B}$ $=$
$T^{\bot_{\infty}}$, where $T$ is a tilting left $R$-module. Recall from [27,
Definition 1.19] that a cotorsion pair $(\mathcal{A},\mathcal{B})$ of left
$R$-modules is said to be _tilting-like_ if $\mathcal{B}=(T\oplus
S)^{\perp_{\infty}}$, where $T$ is a tilting left $R$-module and $S$ is a
strongly Gorenstein projective left $R$-module. It is clear that all tilting
cotorsion pairs are tilting-like cotorsion pairs. But the converse is not true
in general. For example, the module $M:$= $\mathbb{Z}/2\mathbb{Z}$ over the
quasi-Frobenius ring $\mathbb{Z}/4\mathbb{Z}$ is clearly a non-projective
strongly Gorenstein projective module, whence the hereditary cotorsion pair
generated by $M$ is tilting-like, but it cannot be tilting since $M$ does not
have finite projective dimension.

##  3\. Proofs of the statements

In this section, we prove all results mentioned in Introduction. We keep the
notation introduced in the previous sections.

###  3.1. Proof of Theorem 1.1

This subsection is devoted to the proof of Theorem 1.1. For this purpose, we
need some technical results.

Let $0\rightarrow M\rightarrow E^{0}(M)\rightarrow\cdots\rightarrow
E^{i}(M)\rightarrow\cdots$ be a minimal injective coresolution of a left
$R$-module $M$. If there is an exact sequence $0\rightarrow M\rightarrow
E^{0}\rightarrow\cdots\rightarrow E^{i}\rightarrow\cdots$ of left $R$-modules
with each $E^{i}$ injective, then one can check that $E^{i}(M)$ is a direct
summand of $E^{i}$ for all $i$ $\geqslant$ $0$. We will use this fact in the
following.

######  Lemma 3.1.

Let $R$ be a ring and $0\rightarrow M\rightarrow P\rightarrow N\rightarrow 0$
an exact sequence of left $R$-modules. Then the following are true:

  1. (1)

$E^{i}(P)$ is a direct summand of $E^{i}(M)\oplus E^{i}(N)$ for all $i$
$\geqslant$ $0$.

  2. (2)

$E^{i}(M)$ is a direct summand of $E^{i-1}(E^{0}(M)\oplus N)\oplus E^{i}(P)$
for all $i$ $\geqslant$ $1$.

###### Proof.

(1) By the Horseshoe Lemma, we have the following commutative diagram with
exact rows and columns

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{0}(M)\oplus
E^{0}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{0}(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

Repeating this procedure, we obtain an exact sequence

$0\rightarrow P\rightarrow E^{0}(M)\oplus E^{0}(N)\rightarrow\cdots\rightarrow
E^{i}(M)\oplus E^{i}(N)\rightarrow\cdots$

of left $R$-modules. Using the exact sequence, one can check that $E^{i}(P)$
is a direct summand of $E^{i}(M)\oplus E^{i}(N)$ for all $i$ $\geqslant$ $0$.

(2) Note that there is an exact sequence $0\rightarrow M\rightarrow
E^{0}(M)\rightarrow K\rightarrow 0$ of left $R$-modules. Hence we have the
following commutative diagram with exact rows and columns

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

Since $D$ $\cong$ $E^{0}(M)\oplus N$, there is an exact sequence $0\rightarrow
P\rightarrow E^{0}(M)\oplus N\rightarrow K\rightarrow 0$ of left $R$-modules.
Applying [23, Theorem 3.4], one can obtain an exact sequence

(3.1) $0\rightarrow E^{0}(P)\rightarrow E^{0}(E^{0}(M)\oplus N)\oplus
E^{1}(P)\rightarrow C\rightarrow 0$

of left $R$-modules, where $C$ comes from the exact sequence

(3.2) $0\rightarrow K\rightarrow C\rightarrow E^{1}(E^{0}(M)\oplus N)\oplus
E^{2}(P)\rightarrow\cdots\rightarrow E^{i}(E^{0}(M)\oplus N)\oplus
E^{i+1}(P)\rightarrow\cdots$

of left $R$-modules.

By the sequence (3.1), $C$ is a direct summand of $E^{0}(E^{0}(M)\oplus
N)\oplus E^{1}(P)$, and so $C$ is injective. Then $E^{0}(K)$ is a direct
summand of $C$. It follows that $E^{0}(K)$ is a direct summand of
$E^{0}(E^{0}(M)\oplus N)\oplus E^{1}(P)$. Since $C$ is injective, it follows
from the sequence (3.2) that $E^{i}(K)$ is a direct summand of
$E^{i}(E^{0}(M)\oplus N)\oplus E^{i+1}(P)$ for all $i$ $\geqslant$ $1$. Thus
$E^{i-1}(K)$ is a direct summand of $E^{i-1}(E^{0}(M)\oplus N)\oplus E^{i}(P)$
for all $i$ $\geqslant$ $1$. Since $E^{i}(M)$ $\cong$ $E^{i-1}(K)$, $E^{i}(M)$
is a direct summand of $E^{i-1}(E^{0}(M)\oplus N)\oplus E^{i}(P)$ for all $i$
$\geqslant$ $1$. ∎

The following is a key result to prove Theorem 1.1.

######  Proposition 3.2.

Let $R$ be a ring. Then each cycle of an exact complex with each term in
$\mathcal{AC}$ belongs to $\mathcal{AC}$.

###### Proof.

Suppose that there is an exact complex

$\textstyle{\mathbf{X}:\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{n}}$$\textstyle{P_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{n-1}}$$\textstyle{P_{n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

of left $R$-modules with each $P_{i}$ in $\mathcal{AC}$. For any cycle $M$ of
the exact complex $\mathbf{X}$, we need to prove that fd${}_{R}E^{i}(M)$
$\leqslant$ $i$ for all $i$ $\geqslant$ $0$. We shall prove it by induction on
$i$.

Let $M$ $\cong$ $\ker(\partial_{n})$ be the $n$-th cycle. Then there is an
exact sequence $0\rightarrow M\rightarrow P\rightarrow N\rightarrow 0$ with
$P$ $\cong$ $P_{n}$ and $N$ $\cong$ $\ker(\partial_{n-1})$.

For $i=0$, $E^{0}(M)$ is a direct summand of $E^{0}(P)$ by [1, Proposition
18.12]. Since $P$ is in ${\mathcal{AC}}$, $E^{0}(P)$ is flat. Thus $E^{0}(M)$
is flat, as desired.

For $i$ $\geqslant$ $1$, assume that fd${}_{R}E^{i-1}(M)\leqslant i-1$ for any
cycle $M$ of the complex $\mathbf{X}$. We will prove that fd${}_{R}E^{i}(M)$
$\leqslant$ $i$. Note that $E^{0}(E^{0}(M)\oplus N)$ $\cong$ $E^{0}(M)\oplus
E^{0}(N)$. One can check that $E^{i-1}(E^{0}(M)\oplus N)$ $\cong$ $E^{i-1}(N)$
for all $i\geqslant 2$. Since $M$ and $N$ are cycles of the exact complex
$\mathbf{X}$, fd${}_{R}E^{i-1}(E^{0}(M)\oplus N)$ $\leqslant$ $i-1$ by the
inductive hypothesis. By assumption, $P$ is in ${\mathcal{AC}}$. It follows
that fd${}_{R}E^{i}(P)$ $\leqslant$ $i$. Then fd${}_{R}(E^{i-1}(E^{0}(M)\oplus
N)\oplus E^{i}(P))$ $\leqslant$ $i$. By Lemma 3.1, each $E^{i}(M)$ is a direct
summand of $E^{i-1}(E^{0}(M)\oplus N)\oplus E^{i}(P)$. It follows that
fd${}_{R}E^{i}(M)$ $\leqslant$ $i$. This completes the proof. ∎

Now we give the proof of Theorem 1.1.

Proof of Theorem 1.1. (1) $\Rightarrow$ (2), (3) $\Rightarrow$ (2), (4)
$\Rightarrow$ (2) and (5) $\Rightarrow$ (2) are clear.

(2) $\Rightarrow$ (1). Since $R$ is left Noetherian and $R$ is in
$\mathcal{AC}$ as a left $R$-module, all projective left $R$-modules are in
$\mathcal{AC}$ by (2) and [22, Theorem 4.9]. Let $0\rightarrow
C_{1}\rightarrow C_{2}\rightarrow C_{3}\rightarrow 0$ be an exact sequence of
left $R$-modules. Then there exists an exact sequence

$0\rightarrow C_{1}\rightarrow C_{2}\rightarrow C_{3}\oplus C_{3}\rightarrow
C_{3}\oplus C_{3}\rightarrow\cdots$

of left $R$-modules. We choose an exact sequence

$\cdots\rightarrow P_{n}\rightarrow\cdots\rightarrow P_{0}\rightarrow
C_{1}\rightarrow 0$

of left $R$-modules with each $P_{i}$ projective. Thus we obtain an exact
complex

$\mathbf{X}:\cdots\rightarrow P_{n}\rightarrow\cdots\rightarrow
P_{0}\rightarrow C_{2}\rightarrow C_{3}\oplus C_{3}\rightarrow C_{3}\oplus
C_{3}\rightarrow\cdots$

with $C_{1}$ a cycle of the exact complex $\mathbf{X}$. If $C_{2}$ and $C_{3}$
are in $\mathcal{AC}$, then each term of the exact complex $\mathbf{X}$
belongs to $\mathcal{AC}$ by noting that $\mathcal{AC}$ is closed under finite
direct sums and all projective left $R$-modules are in $\mathcal{AC}$. Hence
$C_{1}$ is in $\mathcal{AC}$ by Proposition 3.2. By Lemma 3.1, $\mathcal{AC}$
is closed under extensions. One can check that $\mathcal{AC}$ is closed under
direct sums. Then $\mathcal{AC}$ is closed under direct summands by [20,
Proposition 1.4]. So $\mathcal{AC}$ is resolving, as desired.

(1) $\Rightarrow$ (3). Note that each Gorenstein projective left $R$-module is
a cycle of an exact complex of projective left $R$-modules. Thus (3) follows
from (1) and Proposition 3.2.

(1) $\Rightarrow$ (4). The proof is similar to that (1) $\Rightarrow$ (3).

(2) $\Rightarrow$ (5). Note that all flat left $R$-modules are in
$\mathcal{AC}$ by (2) and [22, Theorem 4.9]. Actually, each Gorenstein flat
left $R$-module is a cycle of an exact complex of flat left $R$-modules. Thus
(5) follows from Proposition 3.2. $\Box$

Let $n$ be a non-negative integer. Recall that a left $R$-module $G$ has
_finite Gorenstein flat dimension at most $n$_ if there is an exact sequence
$0\rightarrow G_{n}\rightarrow\cdots\rightarrow G_{0}\rightarrow G\rightarrow
0$ of left $R$-modules with all $G_{i}$ Gorenstein flat (see [20]). If there
is no need to refer the number $n$, we shall simply say that $G$ has finite
Gorenstein flat dimension. In the following, we denote by $\mathcal{A}c$ the
class of finitely generated left $R$-modules satisfying the Auslander
condition.

######  Corollary 3.3.

Let $R$ be Auslander-Gorenstein. Then the following are true:

  1. (1)

$\mathcal{AC}$ $=$ $\mathcal{GF}$.

  2. (2)

$\mathcal{A}c$ $=$ $\mathcal{G}p$.

###### Proof.

(1) By Theorem 1.1, $\mathcal{GF}$ $\subseteq$ $\mathcal{AC}$. Next we check
$\mathcal{GF}$ $\supseteq$ $\mathcal{AC}$. Suppose that $M$ is a left
$R$-module in $\mathcal{AC}$. Since $R$ is Gorenstein, the injective dimension
of $R$ as a left $R$-module is finite. Let $n$ be the injective dimension.
Since $M$ is in $\mathcal{AC}$, there is an exact sequence $0\rightarrow
M\rightarrow E^{0}(M)\rightarrow\cdots\rightarrow E^{n-1}(M)\rightarrow
L\rightarrow 0$ of left $R$-modules such that fd${}_{R}E^{i}(M)$ $\leqslant$
$i$ for all $i$ $\geqslant$ $0$. By [15, Theorem 12.3.1], the Gorenstein flat
dimension of $L$ is at most $n$. It follows from [20, Theorem 3.14] that
$\mbox{\rm Tor}_{n+k}^{R}(E,L)$ $=$ $0$ for each injective right $R$-module
$E$ and all $k$ $\geqslant$ $1$. Using dimension shifting, one can check that
$\mbox{\rm Tor}_{k}^{R}(E,M)$ $=$ $0$ for each injective right $R$-module $E$
and all $k$ $\geqslant$ $1$. Note that the Gorenstein flat dimension of $M$ is
finite by [15, Theorem 12.3.1]. Hence $M$ is Gorenstein flat by [20, Theorem
3.14].

(2) By [15, Corollary 10.3.11], a finitely generated left $R$-module is
Gorenstein projective if and only if it is Gorenstein flat. By (1), we obtain
the desired result. ∎

######  Remark 3.4.

It follows from [7] that a commutative Gorenstein ring is Auslander-
Gorenstein. Thus, $\mathcal{AC}$ $=$ $\mathcal{GF}$ and $\mathcal{A}c$ $=$
$\mathcal{G}p$ over any commutative Gorenstein ring $R$.

Let $\cdots\rightarrow P_{i}\xrightarrow{f_{i}}\cdots\rightarrow
P_{0}\xrightarrow{f_{0}}M\rightarrow 0$ be a projective resolution of a left
$R$-module $M$. The module $\mbox{\rm im}{(f_{i})}$ is called an _$i$ -th
syzygy_ of $M$, denoted by $\Omega^{i}M$ ($\Omega^{0}M$ $=$ $M$).

Let $n$ be a non-negative integer. The _Gorenstein projective dimension_ ,
$\mbox{\rm Gpd}_{R}G$, of a left $R$-module $G$ is defined by declaring that
$\mbox{\rm Gpd}_{R}G$ $\leqslant$ $n$ if, and only if there is an exact
sequence $0\rightarrow G_{n}\rightarrow\cdots\rightarrow G_{0}\rightarrow
G\rightarrow 0$ of left $R$-modules with all $G_{i}$ Gorenstein projective
(see [20]). We will say that $G$ has finite Gorenstein projective dimension if
there is no need to refer the number $n$.

Note that $\mbox{\rm Gpd}_{R}M$ $\leqslant n$ if and only if $\Omega^{n}(M)$
is Gorenstein projective (see [20, Proposition 2.7]). Moreover, if the
projective dimension of $M$ is finite, then Gpd${}_{R}M$ $=$ pd${}_{R}M$ (see
[20, Proposition 2.27]).

######  Corollary 3.5.

Let $R$ be an Artinian algebra. Then the following are equivalent:

  1. (1)

$R$ is Auslander-Gorenstein.

  2. (2)

$\mathcal{GP}$ $=$ $\mathcal{AC}$.

  3. (3)

$\mathcal{G}p$ $=$ $\mathcal{A}c$.

###### Proof.

(1) $\Rightarrow$ (2) follows from Corollary 3.3 since $\mathcal{GP}$ $=$
$\mathcal{GF}$ over an Artinian algebra.

(2) $\Rightarrow$ (3) is clear.

(3) $\Rightarrow$ (1). Note that $R$ is in $\mathcal{A}c$ as a left $R$-module
by (3). Let

$0\rightarrow R\rightarrow E^{0}(R)\rightarrow\cdots\rightarrow
E^{i}(R)\rightarrow E^{i+1}(R)\rightarrow\cdots$

be a minimal injective coresolution of $R$. Then there is a non-negative
integer $n$ such that pd${}_{R}E^{i}(R)$ $\leqslant$ $n$ for all $i$
$\geqslant$ $0$ by noting that an Artinian algebra has only finite number of
non-isomorphic indecomposable finitely generated injective left $R$-modules.
Using the minimal injective coresolution of $R$, we can obtain another exact
complex

$\cdots\rightarrow R\oplus R\rightarrow R\oplus R\rightarrow
E^{0}(R)\rightarrow\cdots\rightarrow E^{i}(R)\rightarrow
E^{i+1}(R)\rightarrow\cdots.$

Using the Horseshoe Lemma, we obtain an exact complex

$\mathbf{X}:\cdots\rightarrow\Omega^{n}(R\oplus
R)\rightarrow\Omega^{n}(R\oplus
R)\rightarrow\Omega^{n}(E^{0}(R))\rightarrow\cdots\rightarrow\Omega^{n}(E^{i}(R))\rightarrow\Omega^{n}(E^{i+1}(R))\rightarrow\cdots$

of left $R$-modules with

(3.3)
$\ker(\Omega^{n}(E^{i}(R))\rightarrow\Omega^{n}(E^{i+1}(R)))\cong\Omega^{n}(\ker(E^{i}(R)\rightarrow
E^{i+1}(R)))$

for all $i$ $\geqslant$ $0$. By the proof above, $\Omega^{n}(E^{i}(R))$ is
projective for all $i\geqslant 0$. Note that all syzygies in the exact complex
$\mathbf{X}$ can be taken to be finitely generated. Thus each term in the
exact complex $\mathbf{X}$ is in $\mathcal{A}c$ by (3). Applying Proposition
3.2, ker$(\Omega^{n}(E^{i}(R))\rightarrow\Omega^{n}(E^{i+1}(R)))$ is in
$\mathcal{A}c$ for all $i$ $\geqslant$ $0$. It follows from (3) that
ker$(\Omega^{n}(E^{i}(R))\rightarrow\Omega^{n}(E^{i+1}(R)))$ is Gorenstein
projective for all $i$ $\geqslant$ $0$. Using the isomorphism (3.3), one can
obtain that ker$(E^{i}(R)\rightarrow E^{i+1}(R))$ has finite Gorenstein
projective dimension at most $n$ for all $i$ $\geqslant$ $0$. Let $L_{i}$ $=$
ker$(E^{i}(R)\rightarrow E^{i+1}(R))$ for all $i$ $\geqslant$ $0$. Then
Gpd${}_{R}L_{n+1}$ $\leqslant$ $n$, and so Ext${}_{R}^{n+1}(L_{n+1},R)$ $=$
$0$ by [20, Theorem 2.20]. Using dimension shifting, one can check that
Ext${}_{R}^{1}(L_{n+1},L_{n})$ $=$ $0$. It follows that the exact sequence
$0\rightarrow L_{n}\rightarrow E^{n}(R)\rightarrow L_{n+1}\rightarrow 0$
splits. Then $L_{n}$ is injective, and therefore $R$ has finite injective
dimension at most $n$ as a left $R$-module. So $R$ is Auslander-Gorenstein by
[4, Corollary 5.5(b)]. ∎

######  Corollary 3.6.

If $R$ is a left Noetherian ring such that ${}_{R}R$ satisfies the Auslander
condition, then $\mathcal{AC}^{<\infty}$ is a resolving subcategory of
$R$-Mod.

###### Proof.

By Theorem 1.1, $\mathcal{AC}$ is a resolving subcategory of $R$-Mod. To prove
that $\mathcal{AC}^{<\infty}$ is a resolving subcategory of $R$-Mod, we only
need to check that $\mathcal{AC}^{<\infty}$ is closed under direct summands by
[21, Theorem 3.2]. Note that $\mathcal{AC}^{<\infty}$ $=$ $\bigcup_{s\geqslant
0}\mathcal{AC}^{\leqslant s}$, where $\mathcal{AC}^{\leqslant s}$ is the class
of left $R$-modules with finite $\mathcal{AC}$-dimension at most $s$, i.e.,
for each $M$ in $\mathcal{AC}^{\leqslant s}$, there is an exact sequence
$0\rightarrow P_{s}\rightarrow\cdots\rightarrow P_{0}\rightarrow M\rightarrow
0$ of left $R$-modules with each $P_{i}$ in $\mathcal{AC}$. Applying [21,
Theorem 3.2], we obtain that $\mathcal{AC}^{\leqslant s}$ is closed under
extensions and kernels of surjective homomorphisms for all $s\geqslant 0$.
Since $\mathcal{AC}^{\leqslant s}$ is closed under direct sums, it follows
from [20, Proposition 1.4] that $\mathcal{AC}^{\leqslant s}$ is closed under
direct summands. This completes the proof. ∎

###  3.2. Proof of Theorem 1.2

In the following, we will give the proof of Theorem 1.2. Recall that the _left
(little) finitistic dimension_ of a ring $R$ is the supremum of projective
dimensions of (finitely generated) left $R$-modules of finite projective
dimension. It is conjectured that the left little finitistic dimension of each
Artinian algebra is finite; see [6] and [5, Conjectures]. This is the
finitistic dimension conjecture.

######  Lemma 3.7.

Let $R$ be a ring and ($\mathcal{A},\mathcal{B}$) a tilting-like cotorsion
pair. If all left $R$-modules with finite projective dimension are in
$\mathcal{A}$, then the left finitistic dimension of $R$ is finite. Moreover,
if $R$ is left and right Noetherian satisfying the Auslander condition, then
injective dimension of $R$ as a left $R$-module is finite.

###### Proof.

Let ($\mathcal{A},\mathcal{B}$) be a tilting-like cotorsion pair. Then there
is a tilting left $R$-module $T$ and a strongly Gorenstein projective left
$R$-module $S$ such that $\mathcal{B}$ $=$ $(T\oplus S)^{\bot_{\infty}}$. Let
$n$ be the projective dimension of $T$. By [30, Theorem 1.1], each left
$R$-module in $\mathcal{A}$ has finite Gorenstein projective dimension at most
$n$. Let $M$ be any left $R$-module with finite projective dimension. Then $M$
is in ${\mathcal{A}}$ by assumption. Hence the Gorenstein projective dimension
of $M$ is at most $n$, and so the projective dimension of $M$ is at most $n$.
It follows that the left finitistic dimension of $R$ is finite at most $n$,
which implies that the left little finitistic dimension of $R$ is finite. If
$R$ is left and right Noetherian satisfying the Auslander condition, then the
injective dimension of $R$ as a left $R$-module is finite by [24, Corollary
5.3]. ∎

Following [26], a full subcategory $\mathcal{C}$ of $R$-Mod has the _two out
of three property_ if for any exact sequence $0\rightarrow A\rightarrow
B\rightarrow C\rightarrow 0$ of left $R$-modules with two terms in
$\mathcal{C}$, then the third one is also in $\mathcal{C}$.

######  Lemma 3.8.

Let $R$ be a left and right Noetherian ring satisfying the Auslander
condition. Then the following are equivalent:

  1. (1)

$R$ is Gorenstein.

  2. (2)

(${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is a tilting-like cotorsion pair.

  3. (3)

There is a tilting-like cotorsion pair ($\mathcal{A},\mathcal{B}$) such that
all left $R$-modules with finite projective dimension are in $\mathcal{A}$ and
all injective left $R$-modules are in the kernel of the cotorsion pair.

###### Proof.

(1) $\Rightarrow$ (2). By [29, Theorem 5.6],
(${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is a complete hereditary cotorsion
pair generated by a set of left $R$-modules. Then there is a left $R$-module
$M$ in ${{}^{\bot}\mathcal{GI}}$ such that $M^{\bot}$ $=$ $\mathcal{GI}$. Note
that $M^{\bot_{\infty}}$ $=$ $M^{\bot}$ since
(${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is a hereditary cotorsion pair. Then
$M^{\bot_{\infty}}$ $=$ $\mathcal{GI}$. One can check that the kernel of
(${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is the class $\mathcal{I}$ of
injective left $R$-modules. Since $R$ is Gorenstein, each injective left
$R$-module has finite projective dimension at most $n$ by [15, Theorem
9.1.10], where $n$ is the injective dimension of the left $R$-module $R$. Note
that $\mathcal{I}$ is closed under direct sums since $R$ is a left Noetherian
ring. By [15, Theorem 12.3.1], each left $R$-module in
${{}^{\bot}\mathcal{GI}}$ has finite Gorenstein projective dimension at most
$n$. Thus (${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is a tilting-like cotorsion
pair by [30, Theorem 1.1].

(2) $\Rightarrow$ (3). It is clear that all projective left $R$-modules are in
${{}^{\bot}\mathcal{GI}}$. By [29, Lemma 5.4], ${{}^{\bot}\mathcal{GI}}$ has
two out of three property. Thus all left $R$-modules with finite projective
dimension are in ${{}^{\bot}\mathcal{GI}}$. By assumption,
(${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is a tilting-like cotorsion pair.
Note that the kernel of (${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is the class
of injective left $R$-modules. This yields that
(${{}^{\bot}\mathcal{GI}},\mathcal{GI}$) is a desired cotorsion pair
satisfying the conditions in (3).

(3) $\Rightarrow$ (1). By Lemma 3.7, $R$ has finite injective dimension as a
left $R$-module. Since ($\mathcal{A},\mathcal{B}$) is a tilting-like cotorsion
pair, each left $R$-module in the kernel has finite projective dimension by
[30, Theorem 1.1]. Note that the left $R$-module
Hom${}_{\mathbb{Z}}(R,\mathbb{Q}/\mathbb{Z})$ is injective, where $\mathbb{Q}$
is the additive group of rational numbers and $\mathbb{Z}$ is the additive
group of integers. Then Hom${}_{\mathbb{Z}}(R,\mathbb{Q}/\mathbb{Z})$ is in
the kernel of the cotorsion pair ($\mathcal{A},\mathcal{B}$) by assumption.
Hence Hom${}_{\mathbb{Z}}(R,\mathbb{Q}/\mathbb{Z})$ has finite projective
dimension. It follows that $R$ has finite injective dimension as a right
$R$-module. Thus $R$ is Gorenstein. ∎

We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2. (1) Assume that $R$ is Gorenstein. Let
$\mathcal{GF}{{}^{<\infty}}$ be the class of left $R$-modules with finite
Gorenstein flat dimension. Then $\mathcal{AC}^{<\infty}$ $=$
$\mathcal{GF}^{<\infty}$ by Corollary 3.3. By [15, Theorem 12.3.1],
$\mathcal{GF}^{<\infty}$ $=$ $R$-Mod, and so $\mathcal{AC}^{<\infty}$ $=$
$R$-Mod. Let $n$ be the injective dimension of the left $R$-module $R$ and
$\mathcal{GP}{{}^{\leqslant n}}$ the class of left $R$-modules with finite
Gorenstein projective dimension at most $n$. Then $R$-Mod $=$
$\mathcal{GP}^{\leqslant n}$ by [15, Theorem 12.3.1]. It is well known that
($R$-Mod, $\mathcal{I}$) is a complete hereditary cotorsion pair and there is
a left $R$-module $M$ such that $M^{\bot_{\infty}}$ $=$ $\mathcal{I}$. By [30,
Proposition 4.8 and Theorem 1.1], (${\mathcal{AC}^{<\infty}},\mathcal{I}$) is
a tilting-like cotorsion pair.

Conversely, assume that (${\mathcal{AC}^{<\infty}},\mathcal{I}$) is a tilting-
like cotorsion pair. Note that all left $R$-modules with finite projective
dimension are in ${\mathcal{AC}^{<\infty}}$ and the kernel of
(${\mathcal{AC}^{<\infty}},\mathcal{I}$) is $\mathcal{I}$. Then $R$ is
Gorenstein by Lemma 3.8.

(2) (a) $\Rightarrow$ (b). Note that (${\mathcal{AC}^{<\infty}},\mathcal{I}$)
is a tilting-like cotorsion pair by (1). Since $R$ is Auslander-regular, one
can check that there is a non-negative integer $n$ such that each left
$R$-module has finite projective dimension at most $n$. So
(${\mathcal{AC}^{<\infty}},\mathcal{I}$) is a tilting cotorsion pair by [19,
Corollary 13.20].

(b) $\Rightarrow$ (c). To prove (c), it suffices to show $\mathcal{GI}$ $=$
$\mathcal{I}$. Since (${\mathcal{AC}^{<\infty}},\mathcal{I}$) is a tilting
cotorsion pair, it follows from [19, Lemma 13.10] that each left $R$-module
has finite projective dimension. By [17, Corollary 2.9], $\mathcal{GI}$ $=$
$\mathcal{I}$, as desired.

(c) $\Rightarrow$ (d) is clear.

(d) $\Rightarrow$ (a). By assumption,
(${\mathcal{AC}^{<\infty}},\mathcal{GI}$) is a tilting-like cotorsion pair.
Applying Lemma 3.8, $R$ is Gorenstein. Then ${\mathcal{AC}^{<\infty}}$ $=$
$R$-Mod by (1). Thus each left $R$-module has finite projective dimension by
[15, Remark 11.2.3], and therefore $R$ is Auslander-regular.

(3) The “only if” part follows from (1). For the “if” part, by Theorem 1.1,
all left $R$-modules with finite projective dimension are in
${\mathcal{AC}^{<\infty}}$. By Lemma 3.7, $R$ has finite injective dimension
as a left $R$-module. So $R$ is Gorenstein by [4, Corollary 5.5(b)].

(4) The “only if” part follows from (2). For the “if” part, assume that
(${\mathcal{AC}^{<\infty}},(\mathcal{AC}^{<\infty})^{\bot}$) is a tilting
cotorsion pair, it is also a tilting-like cotorsion pair. Since $R$ is
Gorenstein by (3), it follows from [15, Theorem 12.3.1] that each left
$R$-module has finite Gorenstein projective dimension. Note that each left
$R$-module in ${\mathcal{AC}^{<\infty}}$ has finite projective dimension by
[19, Lemma 13.10] and all Gorenstein projective left $R$-modules are in
$\mathcal{AC}$ by Theorem 1.1. It follows that each Gorenstein projective left
$R$-module has finite projective dimension, and so each Gorenstein projective
left $R$-module is projective. Thus each left $R$-module has finite projective
dimension. So $R$ is Auslander-regular. $\Box$

ACKNOWLEDGEMENTS

This research was partially supported by the Natural Science Foundation of the
Jiangsu Higher Education Institutions of China (19KJB110012, 21KJB110003),
NSFC (12201264, 12171206), Jinling Institute of Technology of China
(jit-b-201638, jit-fhxm-201707), the Natural Science Foundation of Jiangsu
Province (BK20211358) and Jiangsu 333 Project. The authors thank Professor
Zhaoyong Huang from Nanjing University for helpful discussions on parts of
this paper. Also, the authors appreciate the referee for reading the paper
carefully and for many suggestions on mathematics and English expressions.

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