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\title{When are the classes of Gorenstein modules (co)tilting?}

\alttitle{Quand les classes de modules de Gorenstein sont-elles (co)basculantes ?}

%%%1
\author{\firstname{Junpeng} \lastname{Wang}}

\address{Department of Mathematics, Northwest Normal University, Lanzhou 730070, People's Republic of China}

\email{wangjunpeng1218@163.com} 

%%%2
\author{\firstname{Zhongkui} \lastname{Liu}}

\address[1]{Department of Mathematics, Northwest Normal University, Lanzhou 730070, People's Republic of China}

\email{liuzk@nwnu.edu.cn} 

%%%3
\author{\firstname{Renyu} \lastname{Zhao}\IsCorresp}

\address[1]{Department of Mathematics, Northwest Normal University, Lanzhou 730070, People's Republic of China}

\email{zhaory@nwnu.edu.cn}


\thanks{This work is partially supported by the National Natural Science Foundation
of China (Grant no. 12361008, 12061061, 11861055, 11761060), the Natural Science Foundation of Gansu Province of China
(Grant no.~22JR5RA153) and the Foundation for Innovative Fundamental Research Group Project of Gansu Province (Grant no.~23JRRA684).}

\CDRGrant[National Natural Science Foundation of China]{12361008}
\CDRGrant[National Natural Science Foundation of China]{12061061}
\CDRGrant[National Natural Science Foundation of China]{11861055}
\CDRGrant[National Natural Science Foundation of China]{11761060}
\CDRGrant[Natural Science Foundation of Gansu Province of China]{22JR5RA153}
\CDRGrant[Foundation for Innovative Fundamental Research Group Project of Gansu Province]{23JRRA684}


\keywords{\kwd{Gorenstein projective (resp.\ injective and flat) module} \kwd{(co)tilting class}
\kwd{finitistic dimension conjecture} \kwd{(strongly) finite type}} 
 
 \altkeywords{\kwd{Module Gorenstein-projectif (respectivement $G$-injectif et $G$-plat)} \kwd{classe de (co)basculement} 
 \kwd{conjecture de la dimension finitiste} 
 \kwd{type (fortement) fini}}

\subjclass{18G25, 18E45, 16E60, 16E65} 

\begin{abstract}
For the class of Gorenstein projective (resp.\ injective and flat) modules,
we investigate and settle the questions when the middle class is tilting and the other ones are cotilting.
The applications have in three directions.
The first is to obtain the coincidence between the 1-tilting and silting property,
as well as the 1-cotilting and cosilting property of such classes respectively.
The second is to characterize Gorenstein modules via finitely generated modules, which provides a proof of that left Noetherian rings with finite
left Gorenstein global dimension satisfy First Finitistic Dimension Conjecture
and a result related to a question posed by Bazzoni in [J. Algebra 320 (2008) 4281-4299].
The last is to give some new characterizations of Dedekind and Pr\"{u}fer domains
and commutative Gorenstein Artin algebras
as well as general (possibly not commutative) Gorenstein rings and Ding--Chen rings.
\end{abstract}


\begin{altabstract}
Pour la classe des modules Gorenstein-projectifs (respectivement $G$-injectifs et $G$-plats), nous étudions et réglons les questions de savoir quand la seconde est basculante et les autres cobasculantes.
Les applications vont dans trois directions. La première est d'obtenir la coïncidence entre les propriétés de ces classes d'être, respectivement, 1-basculante et bousculante, ainsi que la propriété d'être 1-cobasculante et cobousculante. La deuxième consiste à caractériser les modules de Gorenstein via des modules finiment engendrés, ce qui prouve que les anneaux noethériens à gauche de dimension globale de Gorenstein gauche finie satisfont la première conjecture de la dimension finitiste et un résultat lié à une question posée par Bazzoni dans~%\cite[Baz2008] 
[J. Algebra 320 (2008) 4281-4299].
Le dernier objectif est de donner de nouvelles caractérisations des domaines de Dedekind et de Prüfer et des algèbres d'Artin de Gorenstein commutatives, ainsi que des anneaux de Gorenstein généraux (éventuellement non commutatifs) et des anneaux de Ding--Chen.
\end{altabstract}




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\begin{document}

\maketitle

\section{Introduction}

Enochs, Jenda and Torrecillas~\cite{EJ1995, EJT1993} introduced Gorenstein projective, injective and flat modules
for any ring and then established Gorenstein homological algebra. Such a relative homological algebra
has been developed rapidly during the past several decades and now becomes a rich theory. Tilting
theory has its origin as Morita equivalence for derived categories, and becomes an important tool
to deal with many famous conjecture in homological algebra and algebra representation theory, such
as ``Telescope Conjecture for Modules categories'' (see~\cite{AST2008}) and ``Finitistic Dimension Conjecture'' (see~\cite{AT2002}) and
so on. The goal of this manuscript is to investigate some connections between Gorenstein homological
algebra theory and tilting theory. Such a work can go back to the construction of Bass (co)tilting modules
over Gorenstein rings, and was extensively studied by Angeleri H\"{u}gel, Herbera, Trlifaj, Wang, Li, Hu, 
Di, Wei, Zhang, Chen, Moradifar, Saroch and Yassemi in~\cite{AHT2006,WLH2019,DWZC2017,MS2021,MY2018,Wei2018}.

Given a ring $R$, we denote by $\mathcal{GP}(R)$ (resp.\ $\mathcal{GI}(R)$, $\mathcal{GF}(R)$) the class of all Gorenstein projective (resp.\
injective and flat) left $R$-modules. For the corresponding classes of right $R$-modules, we use the notation
$\mathcal{GP}(R^{\op})$ and so on, where $R^{\op}$ is the oppositive ring of $R$. Recall that a class $\mathcal{X}$ of modules is \emph{tilting}
(resp.\ \emph{cotilting}) if $\mathcal{X}$ is an $n$-tilting (resp.\ $n$-cotilting) class for some nonnegative integer $n$, that is, there
is an $n$-tilting (resp.\ $n$-cotilting) module $T$ such that $\mathcal{X}=\{T\}^{\perp}$ (resp.\ $\mathcal{X}={^{\perp}\{T\}}$).
Such properties for a general class $\mathcal{X}$ were classified
by Angeleri H\"{u}gel, Trlifaj, G\"{o}bel and Bazzoni in~\cite{AHT2006, GT2006, Baz2008}.
In particular, the properties for the classes of Gorenstein modules were classified in~\cite{AHT2006}.
Let $R$ be a two-sided Noetherian ring. Then~\cite[Theorem~3.4]{AHT2006} proved that
the classes $\mathcal{GI}(R)$ and $\mathcal{GI}(R^{\op})$ are tilting if and only if
the class $\mathcal{GI}(R)$ is tilting and the class $\mathcal{GF}(R)$ is cotilting,
if and only if $R$ is Gorenstein.
If $R$ is an Artin algebra, then~\cite[Corollary~3.8]{AHT2006} showed that
the class $\mathcal{GP}(R)$ is cotilting if and only if $R$ is Gorenstein.
Motivated by these results, the following questions are posed naturally:
\begin{ques} \label{01}
When does the class $\mathcal{GP}(R)$ \emph{(}resp.\ $\mathcal{GF}(R)$\emph{)} form a cotilting class~?
\end{ques}

\begin{ques} \label{02}
When does the class $\mathcal{GI}(R)$ form a tilting class~?
\end{ques}

Our main results are as follows, which give a thorough-paced answer to Questions~\ref{01} and~\ref{02}.

\begin{theo}[see Theorems~\ref{8} and~\ref{9}] \label{03} 
 The following are equivalent for any ring $R$:
\begin{enumerate}[label=(\arabic*)]
 \item\label{theo3_1} The class $\mathcal{GP}(R)$ \emph{(}resp.\ $\mathcal{GF}(R)$\emph{)} is cotilting.
 \item\label{theo3_2} $R$ is a right coherent and left perfect (resp.\ right coherent) ring admitting finite left Gorenstein
global dimension \emph{(}resp.\ finite Gorenstein weak global dimension\emph{)}.
\end{enumerate}
\end{theo}


\begin{theo}[see Theorem~\ref{10}] \label{04} 
The following are equivalent for any ring $R$:
\begin{enumerate}[label=(\arabic*)]
 \item\label{theo4_1} The class $\mathcal{GI}(R)$ is tilting.
 \item\label{theo4_2} $R$ is a left Noetherian ring admitting finite left Gorenstein global dimension.
\end{enumerate}
\end{theo}

It is well-known that, injective and flat modules can be characterized by finitely generated modules over a ring.
As the first application of the preceding theorems,
we characterize Gorenstein projective (resp.\ injective and flat) modules by finitely generated modules having finite projective dimension
over left Noetherian rings with finite left Gorenstein global dimension (see Theorem~\ref{10-1} and Lemma~\ref{10-5}). On one hand,
the characterizations provides us a proof that any left Noetherian ring with finite left Gorenstein global dimension
satisfies ``First Finitistic Dimension Conjecture'' (see Corollary~\ref{11-16}).
It covers the same result for Gorenstein rings~\cite[Theorem~3.2]{AHT2006}
(see Christensen, Estrada, and Thompson~\cite[Remark~3.11]{CET2021} for the fact of that Gorenstein rings can be described as two-sided Noetherian rings with finite left Gorenstein global dimension
and see Example~\ref{11-0-1-1} for the existence of a left Noetherian ring with finite left Gorenstein global dimension which is not Gorenstein).
On the other hand, the characterizations provides us a result related to a question posed by Bazzoni in~\cite[Question~1(1)]{Baz2008}.


Silting modules was introduced by Angeleri H\"{u}gel, Marks and Vit\'{o}ria~\cite{AMV2016}, which provide a common generalization of
1-tilting modules and support $\tau$-tilting modules, and correspond bijectively to the two-terms silting complexes.
Cosilting modules, as the dual notion, was introduced by Breaz and Pop~\cite{BP2017}.
Recall that a class $\mathcal{X}$ of modules is \emph{silting} (resp.\ \emph{cosilting}) if there is a silting (resp. cosilting)
module $T$ such that $\mathcal{X}= \Gen T$ (resp.\ $\mathcal{X}= \Cogen T$).
Note from~\cite[Lemma~6.1.2]{GT2006} (resp.\ \cite[Lemma~8.2.2]{GT2006}) that
a class $\mathcal{X}$ is 1-tilting (resp.\ 1-cotilting) if and only if there is a 1-tilting (resp.\ 1-cotilting) module $T$
such that $\mathcal{X}= \Gen T$ (resp.\ $\mathcal{X}= \Cogen T$). Since the inclusions $\{1\text{-tilting modules}\} \subseteq \{\text{silting modules}\}$ and
$\{1\text{-cotilting classes}\} \subseteq \{\text{cosilting classes}\}$ are strict, it is a routine to check that the another one $\{1\text{-tilting classes}\} \subseteq \{\text{silting classes}\}$ and
$\{1\text{-cotilting classes}\} \subseteq \{\text{cosilting classes}\}$ are strict as well.
As the second application of the preceding theorems,
we completely settle the questions when the classes $\mathcal{GP}(R)$ and $\mathcal{GF}(R)$ are cosilting and when the class $\mathcal{GI}(R)$ is silting (see Propositions~\ref{11-4},~\ref{11-5} and~\ref{11-6}).
These results shows that the silting (resp.\ cosilting) and 1-tilting (resp.\ 1-cotilting) property for the classes $\mathcal{GP}(R)$ and $\mathcal{GF}(R)$ (resp.\ the class $\mathcal{GI}(R)$) coincide.




As an example of Gorenstein rings, Gorenstein Artin algebras play an important role in representation
theory of Artin algebras. On the other hand, many authors, such as Ding, Chen, Mao, Li and Gillespie~\cite{DC1996, DLM2009, Gil2010, Gil2017}, pointed out that Ding--Chen
rings are natural generalizations of Gorenstein rings. The third application of the theorems is to
give some characterizations of Gorenstein rings (including Gorenstein Artin algebras) and Ding--Chen
rings. Note that Corollary~\ref{05-1} below is a slight improvement of~\cite[Theorem~3.4]{AHT2006}.

\begin{coro}[see Theorem~\ref{11-0-0-0} and Corollary~\ref{11-0-1}] \label{05-1} 
A ring $R$ is Gorenstein if and only if both the classes $\mathcal{GI}(R)$ and $\mathcal{GI}(R^{\op})$ are tilting.
In particular, a commutative (or two-sided Noetherian) ring $R$ is Gorenstein if and only if the class $\mathcal{GI}(R)$ is tilting.
\end{coro}

\begin{coro}[see Theorem~\ref{11-0-2} and Corollary~\ref{11-0-3}] \label{05-2} A ring $R$ is Ding--Chen if and only if both the classes $\mathcal{GF}(R)$ and $\mathcal{GF}(R^{\op})$ are cotilting.
In particular, a commutative (or two-sided coherent) ring $R$ is Ding--Chen if and only if the class $\mathcal{GF}(R)$ is cotilting.
\end{coro}

\begin{coro}[see Theorem~\ref{11-0-4}] \label{05-3} 
A commutative ring $R$ is a Gorenstein Artin algebra if and only if the class $\mathcal{GP}(R)$ is cotilting.
\end{coro}

As shown in~\cite{GT2006}, both the (co)tilting modules and classes
over a Dedekind (resp.\ Pr\"{u}fer) domain have a nice description.
The last application of the theorems is to characterize Dedekind and Pr\"{u}fer domains using some special (co)tilting classes.



\begin{coro}[see Theorems~\ref{11-11} and~\ref{11-12}] \label{05-4}
Let $R$ be a domain. Then the following hold:
\begin{enumerate}[label=(\arabic*)]
 \item\label{coro8_1} $R$ is Dedekind if and only if the class $\mathcal{GI}(R)$ is 1-tilting and $\mathcal{GI}(R)=\mathcal{I}(R)$.
\item\label{coro8_2} $R$ is Pr\"{u}fer if and only if the class $\mathcal{GF}(R)$ is 1-cotilting and $\mathcal{GF}(R)=\mathcal{F}(R)$.
\end{enumerate}
Here $\mathcal{I}(R)$ (resp.\ $\mathcal{F}(R)$) denotes the class of all injective (resp.\ flat) left $R$-modules.
\end{coro}

We conclude this section by summarizing the contents of this paper. Section~\ref{Sec2} contains some notations, definitions and lemmas for use throughout this paper. Section~\ref{Sec3} is
devoted to proving Theorems~\ref{03} and~\ref{04}. Section~\ref{Sec4} gives some applications of Theorems~\ref{03} and~\ref{04}. %including





\section{Preliminaries}\label{Sec2}

Throughout this article, all rings $R$ are assumed to be associative rings with identity and all modules
are unitary. By an ``$R$-module'' we always mean a left $R$-module, for a right $R$-module, we view it as an $R^{\op}$-module, where $R^{\op}$ is the oppositive ring of $R$.
%by a ``module'', we mean a left or right $R$-module.

In this section we mainly recall some necessary notions and facts, which will be used in the paper.
Let $R$ be a ring. As usual, denote by $R\text{-}\Mod$ the class of all $R$-modules; by $\mathcal{P}(R)$ (resp.\ $\mathcal{I}(R)$
and $\mathcal{F}(R)$) its subclass of all projective (resp.\ injective and flat) $R$-modules; by $\pd_R(M)$ (resp.~\/$\id_R(M)$ and $\fd_R(M)$)
the projective (resp.\ injective and flat) dimension of an $R$-module $M$; by $\gldim(R)$ (resp.\ $\wgldim(R)$) the left global (resp.\ weak global) dimension of $R$. In addition, we write
$M^{+}=\Hom_R(M,\mathbb{Q}/\mathbb{Z})$.

For an $R$-module $M$, denote by $\Add M$ (resp.\ $\Prod M$) the class of all $R$-modules
that are isomorphic to a direct summand of a copy coproduct (resp.\ product) of $M$; by $\Gen M$ (resp.\
$\Cogen M$) is the class formed by all $R$-modules that are isomorphic to the epimorphic
images of some $R$-module in $\Add M$ (resp.\ to the submodules of some $R$-module in $\Prod M$).


\subsection{Cotorsion pairs and induced dimensions}

Let $R$ be a ring and $\mathcal{X}, \mathcal{Y}$ classes of $R$-modules. A pair
$(\mathcal{X}, \mathcal{Y})$ is called a \emph{cotorsion pair} if $\mathcal{X}^{\perp}=\mathcal{Y}$ and ${^{\perp}\mathcal{Y}}=\mathcal{X}$.
Here $\mathcal{X}^{\perp}=\{M\in R\text{-}\Mod\mid \Ext_R^{1}(X,M)=0, \forall X\in\mathcal{X}\}$, and ${^{\perp}\mathcal{Y}}$ is defined dually.
A cotorsion pair $(\mathcal{X}, \mathcal{Y})$ is said to be \emph{hereditary} if
$\Ext_R^{n}(X,Y)=0$ for all $X \in \mathcal{X}$, $Y \in \mathcal{Y}$ and $n \geq 1$.
A cotorsion pair $(\mathcal{X}, \mathcal{Y})$ is called \emph{complete} if
for any $M \in R\text{-}\Mod$, there are exact sequences of $R$-modules $0 \to Y \to X \to M \to 0$ and
$0 \to M \to Y' \to X' \to 0$ with $X, X' \in \mathcal{X}$ and~$Y, Y' \in \mathcal{Y}$. A cotorsion pair $(\mathcal{X},\mathcal{Y})$ is said to be \emph{cogenerated} (resp.\ \emph{generated}) \emph{by a set}
 if there is a set $\mathcal{S}$ of $R$-modules in $\mathcal{X}$ (resp.\ $\mathcal{Y}$) such that $\mathcal{S}^\perp=\mathcal{Y}$ (resp.\ $^\perp\mathcal{S}=\mathcal{X}$).

Given an $R$-module $M$ and a class $\mathcal{X}$ of $R$-modules, a \emph{special $\mathcal{X}$-preenvelope} of $M$ is defined as a monic
homomorphism $\alpha : M \to X$ with $X \in \mathcal{X}$ and $\Coker\alpha \in {^{\perp}\mathcal{X}}$.
A class $\mathcal{X}$ is said to be \emph{special preenveloping} if every $R$-module has a special $\mathcal{X}$-preenvelope.
A class $\mathcal{X}$ is \emph{injectively coresolving} if it is closed under extensions and cokernels of monic morphisms, and $I(R)\in \mathcal{X}$.
Dually, we have the definitions of
\emph{special} $\mathcal{X}$-\emph{precover} and that $\mathcal{X}$ is \emph{special precovering} (resp.\ \emph{projectively resolving}).

For an $R$-module $M$ and a class $\mathcal{X}$ of $R$-modules, the $\mathcal{X}$-projective dimension of $M$, denoted by $\mathcal{X}\text{-}\pd_R(M)$,
is defined as follows:
\[
\begin{aligned}
&\mathcal{X}\text{-}\pd_R(M)=\inf\{n\in \mathbb{N}\mid \text{there~is~an~exact~sequence~of}~R\text{-}\text{modules}\\
&~0\to X_n \to \cdots \to X_1 \to X_0 \to M\to 0,~\text{where each} ~X_i\in\mathcal{X}\}.
\end{aligned}
\]
If no such an exact sequence exists, then we set $\mathcal{X}\text{-}\pd_R(M)=\infty$.
Dually, we have the definition of $\mathcal{X}$-\emph{injective dimension} of $M$, $\mathcal{X}\text{-}\id_R(M)$.

Next two lemmas give some characterizations of relative projective dimensions of
modules, which are applied for the proof of Lemma~\ref{2}.

\begin{lemma} \label{06-1}
Let $\mathcal{X}$ be a class of $R$-modules such that $(\mathcal{X}, \mathcal{X}^{\perp})$ is a complete and hereditary cotorsion pair. Then the following are equivalent for any $R$-module $M$ and any integer $n\geq0$:
\begin{enumerate}[label=(\arabic*)]
 \item\label{lemma9_1} $\mathcal{X}\text{-}\pd_R(M)\leq n$.
 \item\label{lemma9_2} There is an exact sequence of $R$-modules $0\to X_n\to X_{n-1}\to\cdots\to X_1\to X_0\to M \to 0$ with each $X_i\in \mathcal{X}$.
 \item\label{lemma9_3} For any exact sequence of $R$-modules $0\to K\to X_{n-1}\to\cdots\to X_1\to X_0\to M \to 0$, if each $X_i$ is in $\mathcal{X}$, then so is $K$.
 \item\label{lemma9_4} $\Ext_R^{n+1}(M,Y)=0$ for all $Y \in \mathcal{X}^{\perp}$.
\end{enumerate}
\end{lemma}

\begin{proof}\ 

\begin{proof}[\mathversion{bold}$\ref{lemma9_1}\Leftrightarrow\ref{lemma9_2}$] 
It is clear.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma9_2}\Rightarrow\ref{lemma9_3}$] Note that $\mathcal{X}$ is
projectively resolving and closed under arbitrary direct sums and summands,
as $(\mathcal{X}, \mathcal{X}^{\perp})$ forms a complete and hereditary cotorsion pair. So the result follows from~\cite[Lemma~3.12]{AB1969}.
\let\qed\relax
\end{proof}


\begin{proof}[\mathversion{bold}$\ref{lemma9_3}\Rightarrow\ref{lemma9_4}$] Consider the exact sequence $0\to K\to P_{n-1}\to\cdots\to P_1\to P_0\to M \to 0$ of $R$-modules
with each $P_i$ projective. Then by~\ref{lemma9_3} $K$ is in $\mathcal{X}$ since so is each $P_i$. Now, for any $Y \in \mathcal{X}^{\perp}$,~by dimension shifting one has
$0=\Ext_R^{1}(K,Y)\cong\Ext_R^{n+1}(M,Y)$.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma9_4}\Rightarrow\ref{lemma9_2}$] Consider
an exact sequence $0\to X_n\to X_{n-1}\to\cdots\to X_1\to X_0\to M \to 0$ of $R$-modules with $X_i\in \mathcal{X}$ for all $0\leq i\leq n-1$
(one can choose that all such $X_i$ projective). Then, for any $Y \in \mathcal{X}^{\perp}$,~by~\ref{lemma9_4} and dimension shifting one has
$\Ext_R^{1}(X_n, Y)\cong\Ext_R^{n+1}(M,Y)=0$. Thus, $X_n \in \mathcal{X}$ since $(\mathcal{X}, \mathcal{X}^{\perp})$ is a cotorsion pair.
\end{proof}
\let\qed\relax
\end{proof}


\begin{lemma} \label{06-7-2}
Let $\mathcal{X}$ be a class of $R$-modules such that $(\mathcal{X}, \mathcal{X}^{\perp})$ is a complete and hereditary cotorsion pair. Then the following are equivalent:
\begin{enumerate}[label=(\arabic*)]
 \item\label{lemma10_1} $\sup\{\mathcal{X}\text{-}\pd_R(M)\mid M\in R\text{-}\Mod\}<\infty$.
 \item\label{lemma10_2} Every $R$-module in $\mathcal{X}^{\perp}$ has finite injective dimension.
 \item\label{lemma10_3} $\mathcal{X}\text{-}\pd_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{lemma}

\begin{proof}\ 

\begin{proof}[\mathversion{bold}$\ref{lemma10_1}\Rightarrow\ref{lemma10_3}$] It is obvious. 
\let\qed\relax
\end{proof}


\begin{proof}[\mathversion{bold}$\ref{lemma10_2}\Leftrightarrow\ref{lemma10_3}$] It follows
from Lemma~\ref{06-1}.
\let\qed\relax
\end{proof}


\begin{proof}[\mathversion{bold}$\ref{lemma10_3}\Rightarrow\ref{lemma10_1}$] We first claim that
\[
\mathcal{X}\text{-}\pd_R\left(\coprod_{j\in J}M_j\right)=\sup
\{\mathcal{X}\text{-}\pd_R(M_j)\mid j\in J\}
\]
for any family $\{M_j\mid j\in J\}$ of $R$-modules.
Indeed, let $\sup\{\mathcal{X}\text{-}\pd_R(M_j)\mid j\in J\}=m<\infty$. Then $\mathcal{X}\text{-}\pd_R(M_j)\leq m$ for each $j\in J$.
By Lemma~\ref{06-1}, there exists an exact sequence of $R$-modules $0\to X_{m,j}\to \cdots\to X_{1,j}\to X_{0,j}\to M_j \to 0$
with each $X_{i,j}\in \mathcal{X}$.
This induces another exact sequence of $R$-modules $0\to \coprod_{j\in J}X_{m,j}\to \cdots\to \coprod_{j\in J}X_{1,j}\to \coprod_{j\in J}X_{0,j}\to \coprod_{j\in J}M_j \to 0$
with each $\coprod_{j\in J}X_{i,j}\in \mathcal{X}$ as $\mathcal{X}$ is closed under arbitrary direct sums.
Hence, $\mathcal{X}\text{-}\pd_R(\coprod_{j\in J}M_j)\leq m$ by Lemma~\ref{06-1}.
Conversely,
let $\mathcal{X}\text{-}\pd_R(\coprod_{j\in J}M_j)=m<\infty$. Then Lemma~\ref{06-1} yields that $\Ext_R^{m+1}(\coprod_{j\in J}M_j, Y)=0$ for all $Y\in \mathcal{X}^{\perp}$.
So, using the isomorphism $\Ext_R^{m+1}(\coprod_{j\in J}M_j, Y)\cong \prod_{j\in J}\Ext_R^{m+1}(M_j, Y)$ one has $\Ext_R^{m+1}(M_j, Y)=0$ for all $j\in J$.
Thus, $\sup\{\mathcal{X}\text{-}\pd_R(M_j)\mid j\in J\}\leq m$ by Lemma~\ref{06-1}. This shows the~claim.

Now assume that $\mathcal{X}\text{-}\pd_R(M)<\infty$ for any $R$-module $M$. If $\sup\{\mathcal{X}\text{-}\pd_R(M)\mid M\in R\text{-}\Mod\}=\infty$,
then there is an $R$-module $M_n$
such that $\mathcal{X}\text{-}\pd_R(M_n)\geq n$ for any integer $n\geq0$.
This leads to a contradiction:
\[
\infty = \sup\{\mathcal{X}\text{-}\pd_R(M_n)\ |\ n\in \mathbb{N}\} \overset{\text{the above claim}} {=\!=\!=\!=\!=\!=\!=} \mathcal{X}\text{-}\pd_R(\oplus_{n\in \mathbb{N}} M_n) < \infty.
\]
Consequently, $\sup\{\mathcal{X}\text{-}\pd_R(M)\mid M\in R\text{-}\Mod\}<\infty$.
\end{proof}
\let\qed\relax
\end{proof}


Lemmas~\ref{06-7-3} and~\ref{06-7-4} below are the dual of Lemmas~\ref{06-1} and~\ref{06-7-2} respectively,
which are applied for the proof of Lemma~\ref{3}.

\begin{lemma} \label{06-7-3}
Let $\mathcal{X}$ be a class of $R$-modules such that $(^{\perp}\mathcal{X}, \mathcal{X})$ is a complete and hereditary cotorsion pair. Then the following are equivalent for any $R$-module $M$ and any integer $n\geq0$:
\begin{enumerate}[label=(\arabic*)]
 \item\label{lemma11_1} $\mathcal{X}\text{-}\id_R(M)\leq n$.
 \item\label{lemma11_2} There is an exact sequence of $R$-modules $0\to M \to X^{0}\to X^{1}\to \cdots\to X^{n} \to 0$ with each $X^{i}\in \mathcal{X}$.
 \item\label{lemma11_3} For any exact sequence of $R$-modules $0\to M \to X^{0}\to X^{1}\to \cdots\to X^{n-1}\to C\to 0$, if each $X^{i}$ is in $\mathcal{X}$, then so is $C$.
 \item\label{lemma11_4} $\Ext_R^{n+1}(Y,M)=0$ for all $Y \in {^{\perp}\mathcal{X}}$.
\end{enumerate}
\end{lemma}

\goodbreak
\begin{lemma} \label{06-7-4}
Let $\mathcal{X}$ be a class of $R$-modules such that $(^{\perp}\mathcal{X}, \mathcal{X})$ is a complete and hereditary cotorsion pair. Then the following are equivalent:
\begin{enumerate}[label=(\arabic*)]
 \item\label{lemma12_1} $\sup\{\mathcal{X}\text{-}\id_R(M)\mid M\in R\text{-}\Mod\}<\infty$.
 \item\label{lemma12_2} Every $R$-module in $^{\perp}\mathcal{X}$ has finite projective dimension.
 \item\label{lemma12_3} $\mathcal{X}\text{-}\id_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{lemma}



\subsection{Gorenstein homological modules and dimensions}
An $R$-module $M$ is said to be
\emph{Gorenstein~projective}~\cite{EJ1995} if there exists a
Hom$_{R}(-,\mathcal{P}(R))$-exact exact sequence of projective $R$-modules
$\cdots\to P_1\to P_0\to 
P^{0}\to P^{1}\to \cdots$
such that
$M\cong\im (P_0\to P^{0})$.
Dually we have the definition of \emph{Gorenstein injective}~\cite{EJ1995} $R$-modules.
An $R$-module $M$ is said to be \emph{Gorenstein flat}~\cite{EJT1993} if
there exists an exact sequence of flat $R$-modules
$\cdots\to F_1\to F_0\to 
F^{0}\to F^{1}\to \cdots$
such that
$M\cong \im (F_0\to F^{0})$
and that remains exact whenever the functor $I\otimes_{R}-$ is applied for any injective $R^{\op}$-module $I$.

We denote by~$\mathcal{GP}(R)$ (resp.\ $\mathcal{GI}(R)$ and $\mathcal{GF}(R)$) the class of all Gorenstein projective (resp.\ injective and flat) $R$-modules;
by~$\Gpd_R(M)$ (resp.\ $\Gid_R(M)$ and $\Gfd_R(M)$) the Gorenstein projective (resp.\ injective and flat) dimension of an $R$-module $M$, that is,
$\mathcal{GP}(R)$-projective (resp.\ $\mathcal{GI}(R)$-injective and $\mathcal{GF}(R)$-projective) dimension of $M$.



As a refinement of the usual global (resp.\ weak global) dimension of rings, Gorenstein global (resp.\ Gorenstein weak global) dimension of rings is defined as follows:

\begin{defi} \label{06-2}\ \relax

\begin{enumerate}[label=(\arabic*)]
 \item\label{defi13_1} 
 For any ring $R$, its left (resp.\ right) Gorestein global dimension, denoted by Ggldim$(R)$ (resp.\ Ggldim$(R^{\mathrm{\op}})$), is defined via the following formula
\[
\begin{aligned}
\sup\{\Gpd_R(M)\mid M\in R\text{-}\Mod\}=&\Ggldim(R)=\sup\{\Gid_R(M)\mid M\in R\text{-}\Mod\} \\
(\text{resp}.~\sup\{\Gpd_R(M)\mid M\in R^{\op}\text{-} \Mod\}=&\Ggldim(R^{\op })=\sup\{\Gid_R(M)\mid M\in R^{\op }\text{-}\Mod\}).
\end{aligned}
\]

\item\label{defi13_2} We say that $R$ admits \emph{finite left Gorenstein global dimension} (resp.\ \emph{finite right Gorenstein global dimension}) if $\Ggldim(R)<\infty$ (resp.\ $\Ggldim(R^{\op })<\infty$).
\end{enumerate}
\end{defi}


\begin{defi} \label{06-3}\ \relax

\begin{enumerate}[label=(\arabic*)]
 \item\label{defi14_1} For any ring $R$, its Gorestein weak global dimension, denoted by Gwgldim$(R)$, is defined via the following formula
\begin{center}$\begin{aligned}
\sup\{\Gfd_R(M)\mid M\in R\text{-Mod}\}=\Gwgldim(R)=\sup\{\Gfd_R(M)\mid M\in R^{\op }\text{-Mod}\}.\\
\end{aligned}$\end{center}

\item\label{defi14_2} We say that $R$ admits \emph{finite Gorenstein weak global dimension} if $\Gwgldim(R)<\infty$. 
\end{enumerate}
\end{defi}


\begin{remas} \label{06-3-1}\ \relax

\begin{enumerate}[label=(\arabic*)]
 \item\label{rema15_1} For any ring $R$, the equality
\[
\sup\{\Gpd_R(M)\mid M\in R\text{-Mod}\}=\sup\{\Gid_R(M)\mid M\in R\text{-Mod}\}
\]
was proved by Bennis and Mahdou in~\cite{BM2010} and by Emmanouil in~\cite{Emm2012} using different method.

 \item\label{rema15_2} For any ring $R$, the equality
\[
\begin{aligned}
\sup\{\Gfd_R(M)\mid M\in R\text{-Mod}\}=\sup\{\Gfd_R(M)\mid M\in R^{\op }\text{-Mod}\}
\end{aligned}
\]
was proved in~\cite{CET2021}.
Note from \v{S}aroch and \v{S}t'ov\'{\i}\v{c}ek~\cite[Theorem~4.11]{SS2020} that any ring is \emph{left and right GF-closed} (i.e., the class of all Gorenstein flat left or right $R$-modules is closed under extensions).
Such an equality can be also obtained by Bouchiba~\cite[Theorem~6(2)]{Bou2015}, as noted in~\cite{WZ2022}.
\end{enumerate}
\end{remas}

It is well-known that $\wgldim(R)\leq \gldim(R)$ for any ring $R$.
The corresponding inequality 
\[
\Gwgldim(R)\leq \Ggldim(R)
\]
 for a right coherent ring $R$ was proved in~\cite[Corollary~3.5]{CET2021}.
Recently, the firstly named author and coauthors~\cite{WYZ2022} improved the result.

\begin{lemma}[{\cite[Theorem~3.7 and Remark~3.12]{WYZ2022}}] \label{07}  Let $R$ be a ring.

\begin{enumerate}[label=(\arabic*)]
 \item\label{lemma16_1} There is an inequality ~$\Gwgldim(R)\leq \Ggldim(R).$
 \item\label{lemma16_2} The equality $\Gwgldim(R)=\Ggldim(R)$
(resp.\ $\Gwgldim(R)=\Ggldim(R^{op})$) holds true if $R$ is left perfect (resp.\ right perfect).
\end{enumerate}
\end{lemma}






We end this section by the next lemma.

\begin{lemma} \label{06-6} Let $R$ be a right coherent ring or a ring with $\Gwgldim(R)<\infty$ (in particular the case $\Ggldim(R)<\infty$). Then
$\mathcal{GP}(R)=\mathcal{GF}(R)$ if and only if $R$ is left perfect.
\end{lemma}
\begin{proof} The ``only if'' part holds by~\cite[Proposition~3.1]{CET2023}.
For the ``if'' part, we suppose that $R$ is left perfect. Then one has $\mathcal{P}(R)=\mathcal{F}(R)$.
If $R$ is a right coherent ring, then $\mathcal{GP}(R)=\mathcal{GF}(R)$ holds by~\cite[Proposition~3.5]{Wz2017};
in the other case, i.e.,
$R$ is a ring with $\Gwgldim(R)<\infty$, the equality follows by~\cite[Theorems~2.3 and~2.9]{WZ2022}.
\end{proof}


\section{(Co)tilting classes and Gorenstein modules}\label{Sec3}





In this section, we will
give a thorough-paced answer to Questions 1 and 2 (from the introduction).
We start with the following definitions.

\begin{defi} \label{1-0} {\rm A class $\mathcal{X}$ of $R$-modules is called \emph{definable} if $\mathcal{X}$ is closed under pure submodules, direct
products and direct limits.}
\end{defi}

\begin{rema} \label{1-0-0} {\rm Note that a class $\mathcal{X}$ of $R$-modules is definable if and only if
it is closed under products, pure epimorphic images and pure submodules.
Indeed, if $\mathcal{X}$ is definable, then clearly it is closed under products and pure submodules.
It is also closed under pure epimorphic images by~\cite[Proposition~4.3(3)]{Baz2008}.
Conversely, suppose that $\mathcal{X}$ is closed under products, pure epimorphic images and pure submodules.
Since any direct limit of a family of $R$-modules is a pure epimorphic image of the direct sum of such a family of $R$-modules and
any direct sum of a family of $R$-modules is a pure submodule of the direct product of such a family of $R$-modules,
$\mathcal{X}$ is also closed under direct limits, and hence is definable.}
\end{rema}




\begin{defi} \label{1-1} {\rm An $R$-module $M$ is said to be \emph{of type FP}$_\infty$~\cite{AHT2006, BGH2014} or \emph{compact}~\cite{Baz2008} if $M$ possesses a projective resolution consisting of finitely generated $R$-modules.}
\end{defi}

\begin{defi} \label{1-2} {\rm A class $\mathcal{X}$ of $R$-modules is of~\emph{finite type}~if~$\mathcal{X}={^{\perp}(\mathcal{S}^{\perp}})$, where $\mathcal{S}$ is a set of
 $R$-modules of type FP$_\infty$. If furthermore the set $\mathcal{S}$ consists of $R$-modules of type FP$_\infty$ with finite projective dimension,
 then we call that the class $\mathcal{X}={^{\perp}(\mathcal{S}^{\perp}})$ is of \emph{strongly finite type}}.
\end{defi}

\begin{remas} \label{1-2-0} \ 

\begin{enumerate}[label=(\arabic*)]
 \item\label{rema22_1} Clearly any class $\mathcal{X}$ of $R$-modules of strongly finite type is of finite type.
Note that there exists a class of $R$-modules of finite type which is not of strongly finite type.
We also note that Bazzoni, G\"{o}bel and Trlifaj in~\cite{Baz2008, GT2006} called that a class $\mathcal{X}$ is ``of finite type'', is just of strongly finite type in our sense.

\item\label{rema22_2} Let $\mathcal{X}$ be a class of $R$-modules which is of finite type. Then there is a complete and hereditary cotorsion pair $(\mathcal{X}, \mathcal{X}^{\perp})$
cogenerated by some set $\mathcal{S}$ of $R$-modules of type FP$_\infty$. The cotorsion pair is said to be of~\emph{finite type}. Similarly, we have the notion of that
a cotorsion pair is of ~\emph{strongly finite type}.
\end{enumerate}
\end{remas}


\begin{defi} \label{1-3} \ 

\begin{enumerate}[label=(\arabic*)]
 \item\label{defi23_1} An $R$-module $T$ is called \emph{tilting} if there is an integer $n\geq0$ such that $T$ is an
$n$-tilting $R$-module, that is, $T$ satisfies the following:

\begin{enumerate}[label=(T\arabic*)]
 \item\label{defi23_T1} $\pd_R(M)\leq n$.
 \item\label{defi23_T2} $\Ext_R^{i\geq1}(T,T^{(J)})=0$ for all set $J$.
 \item\label{defi23_T3} There is an exact sequence of $R$-modules $0\to R \to X^{0}\to X^{1}\to \cdots\to X^{n} \to 0$ with each $X^{i}\in \Add T$.
\end{enumerate}

\item\label{defi23_2} A class $\mathcal{X}$ of $R$-modules is called \emph{tilting} if $\mathcal{X}$ is an $n$-tilting class for some $n\geq0$,
that is, there is an $n$-tilting $R$-module $T$ such that $\mathcal{X}=\{T\}^{\perp}$.
\end{enumerate}
\end{defi}

\begin{defi} \label{1-4} \ 

\begin{enumerate}[label=(\arabic*)]
\item\label{defi24_1} An $R$-module $T$ is called \emph{cotilting} if there is an integer $n\geq0$ such that $T$ is an
$n$-cotilting $R$-module, that is, $T$ satisfies the following:

\begin{enumerate}[label=(CT\arabic*)]
\item\label{defi24_CT1} 
$\id_R(M)\leq n$.

\item\label{defi24_CT2} 
 $\Ext_R^{i\geq1}(T^{J},T)=0$ for all set $J$.
 
\item\label{defi24_CT3} 
 There is an exact sequence of $R$-modules $0\to X_n\to \cdots\to X_1\to X_0\to W \to 0$ with each $X_{i}\in \Prod T$ and $W$ an injective cogenerator.
\end{enumerate}

\item\label{defi24_2} A class $\mathcal{X}$ of $R$-modules is called \emph{cotilting} if $\mathcal{X}$ is an $n$-cotilting class for some $n\geq0$,
that is, there is an $n$-cotilting $R$-module $T$ such that $\mathcal{X}={^{\perp}\{T\}}$.
\end{enumerate}
\end{defi}

\begin{remas} \label{1-5} \ 

\begin{enumerate}[label=(\arabic*)]
\item\label{rema25_1}
Obviously 0-tilting (resp.\ 0-cotilting) $R$-modules coincide with projective generators (resp.\ injective cogenerators).
Thus, 0-tilting and 0-cotilting classes of $R$-modules are just $R$-Mod.

\item\label{rema25_2}
It is known from~\cite[Theorems~5.1.14 and~8.1.9]{GT2006} that any tilting or cotilting class is definable.

\item\label{rema25_3}
Following~\cite[Proposition~3.7(1)]{Baz2008}, we know
that a class $\mathcal{X}$ (and hence a cotorsion pair $(\mathcal{X},\mathcal{X}^{\perp})$) of $R$-modules is tilting if and only if it is of strongly finite type.

\item\label{rema25_4}
If given a tilting (resp.\ cotilting class $\mathcal{X}$), then we have a complete and hereditary cotorsion pair
$(\mathcal{X}, \mathcal{X}^{\perp})$ (resp.\ $(^{\perp}\mathcal{X}, \mathcal{X})$) which is cogenerated (resp.\ generated) by the tilting (resp.\ cotilting) $R$-module
$T$. The corresponding cotorsion pair $(\mathcal{X}, \mathcal{X}^{\perp})$ (resp.\ $(^{\perp}\mathcal{X}, \mathcal{X})$) is called \emph{tilting} (resp.\ \emph{cotilting}).
\end{enumerate}
\end{remas}


Let $n\geq 0$ be an integer. Recall from~\cite{Baz2008} that a class $\mathcal{X}$ of $R$-modules is \emph{closed under}~$n$\emph{-submodules}
(resp.\ \emph{closed under}~$n$\emph{-images}) provided that any $R$-module $M$ is in $\mathcal{X}$ whenever there is an exact
sequence $0 \to M \to X_{0} \to X_{1} \to \cdots \to X_{n-1}$ (resp.\ $X_{n-1} \to\cdots \to X_1 \to X_0 \to M \to 0$) of
$R$-modules with each $X_i\in \mathcal{X}$.

\looseness-1
By virtue of~\cite[Theorem~6.1]{Baz2008}, we know that a class $\mathcal{X}$ of $R$-modules is cotilting if and only if $\mathcal{X}$ is is definable,
projectively resolving and there is an integer $n\geq 0$ such that $\mathcal{X}$ is closed under $n$-submodules; we
know from~\cite[Theorem~6.1]{Baz2008} that a definable class $\mathcal{X}$ of $R$-modules is tilting if and only if $\mathcal{X}$ is injective coresolving,
special preenveloping and there is an integer $n\geq 0$ such that $\mathcal{X}$ is closed under $n$-images. These
classification results for (co)tilting classes and their proof lead us to obtain the next two lemmas, which
play an important role in the proof Theorems~\ref{8},~\ref{9} and~\ref{10}.

\begin{lemma} \label{2} Let $R$ be a ring and $\mathcal{X}$ a class of $R$-modules. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
\item\label{rema26_1}
 $\mathcal{X}$ is cotilting.
 
\item\label{rema26_2} $\mathcal{X}$ is definable, projectively resolving and $\sup\{\mathcal{X}\text{-}\pd_R(M)\mid M\in R\text{-}\Mod\}<\infty$.

\item\label{rema26_3} $\mathcal{X}$ is definable, projectively resolving and $\mathcal{X}\text{-}\pd_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{lemma}

\begin{proof} Let $\mathcal{X}$ be a cotilting or definable, and projectively resolving class of $R$-modules. Then $\mathcal{X}$ is
special precovering by the proof of~\cite[Theorem~6.1]{Baz2008}. In other words, there is a complete and hereditary
cotorsion pair $(\mathcal{X}, \mathcal{X}^{\perp})$ in this case. Thus, $\ref{rema26_2}\Leftrightarrow\ref{rema26_3}$ holds by
Lemma~\ref{06-7-2}. Moreover, combining~\cite[Theorem~6.1]{Baz2008} with the implication $\ref{rema26_1}\Leftrightarrow\ref{rema26_3}$ in Lemma~\ref{06-7-2}, one can obtain\break \hbox{$\ref{rema26_1}\Leftrightarrow\ref{rema26_2}$}.
\end{proof}

\begin{lemma} \label{3} Let $R$ be a ring and $\mathcal{X}$ a class of $R$-modules. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
\item\label{lemma27_1} $\mathcal{X}$ is tilting.

 \item\label{lemma27_2} $\mathcal{X}$ is definable, injectively coresolving, special preenveloping and there is a positive integer $n$ such that $\mathcal{X}$ is closed under $n$-images.

 \item\label{lemma27_3} $\mathcal{X}$ is definable, injectively coresolving, special preenveloping and $\mathcal{X}\text{-}\id_R(M)<\infty$ for any $R$-module $M$.

 \item\label{lemma27_4} $\mathcal{X}$ is definable, injectively coresolving, special preenveloping and $\sup\{\mathcal{X}\text{-}\pd_R(M)\mid M\in R\text{-}$ $\Mod\}<\infty$.
\end{enumerate}
\end{lemma}

\begin{proof} Note from Remak~\ref{1-5}\ref{rema25_2} that any tilting class of $R$-modules is always definable. So $\ref{lemma27_1}\Leftrightarrow\ref{lemma27_2}$ holds by~\cite[Theorem~6.3]{Baz2008}.
Now let $\mathcal{X}$ be a class satisfying any one of~\ref{lemma27_2},~\ref{lemma27_3} and~\ref{lemma27_4}. Then there is a complete and hereditary
cotorsion pair $(^{\perp}\mathcal{X}, \mathcal{X})$. Thus, $\ref{lemma27_3}\Leftrightarrow\ref{lemma27_4}$
follows from Lemma
\ref{06-7-4} and
$\ref{lemma27_2}\Leftrightarrow\ref{lemma27_3}$ comes from Lemma~\ref{06-7-3}.
\end{proof}

\begin{remas} \label{3-1} \ 

\begin{enumerate}[label=(\arabic*)]
\item\label{rema28_1} Note that the condition ``$\mathcal{X}$ is tilting'' (resp.\ ``$\mathcal{X}$ is cotilting'') implies
the one ``$\mathcal{X}\text{-}\pd_R(M)<\infty$ for any $R$-module $M$'' (resp.\ ``$\mathcal{X}\text{-}\id_R(M)<\infty$ for any $R$-module $M$'')
was proved in~\cite[Lemma~2.2(a)]{AC2001} (resp.\ \cite[Lemma~2.2(b)]{AC2001}).

\item\label{rema28_2} According to~\cite[Proposition~7.2]{Baz2008}, we know that ``special preenveloping'' in~\ref{lemma27_2},~\ref{lemma27_3} and
\ref{lemma27_4} of Lemma~\ref{3} can not be omitted.
\end{enumerate}
\end{remas}

We know from~\cite[Proposition~4.13]{SS2020} that the class $\mathcal{GF}(R)$ is definable if and only if it is closed under products,
and that these equivalent conditions for $R$ deduces that $R$ is right coherent.
The next three results can be viewed as a continuation of such facts.



%By the next three results, we concern when the class $\mathcal{GP}$ (resp.\ $\mathcal{GI}$, $\mathcal{GF}$) is definable,
%which plays an important role to obtain Theorems 3.13, 3.14 and 3.15, repectively.


\begin{lemma} \label{4} Let $R$ be a ring with $\Gwgldim(R)<\infty$. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
\item\label{lemma29_1} The class $\mathcal{GF}(R)$ is definable.
 \item\label{lemma29_2} The class $\mathcal{GF}(R)$ is closed under arbitrary direct products.
 \item\label{lemma29_3} $R$ is right coherent.
\end{enumerate}
\end{lemma}

\begin{proof} Note that
the implication $\ref{lemma29_1} \Leftrightarrow\ref{lemma29_2}\Rightarrow\ref{lemma29_3}$ holds by~\cite[Proposition~4.13]{SS2020}.

Let's prove $\ref{lemma29_3}\Rightarrow\ref{lemma29_2}$.
Let $\{(G_j)_{j\in J}\}$ be a family of Gorenstein flat $R$-modules. Then for each $j\in J$, there is an
exact sequence of $R$-modules
\[
0\to G_j\to F^{0}_j\to F^{1}_j\to \cdots
\]
with each $F^{i}_j\in \mathcal{F}(R)$.
Hence, one can obtain an exact sequence
\[
0\to \prod_{j \in J} G_j\to \prod_{j \in J} F^{0}_j\to \prod_{j \in J} F^{1}_j\to \cdots
\]
of $R$-modules with each $\prod_{j \in J} F^{i}_j\in \mathcal{F}(R)$ since $R$ is right coherent.
Therefore, $\prod_{j \in J} G_j$ is \hbox{Gorenstein} flat by~\cite[Theorem~2.9]{WZ2022}.
\end{proof}

\begin{lemma} \label{5} Consider the following conditions for a ring $R$:
\begin{enumerate}[label=(\arabic*)]
\item\label{lemma30_1} The class $\mathcal{GP}(R)$ is definable.
 \item\label{lemma30_2} The class $\mathcal{GP}(R)$ is closed under arbitrary direct products.
 \item\label{lemma30_3} $R$ is right coherent and left perfect.
 \item\label{lemma30_4} $R$ is a right coherent ring such that $\mathcal{GP}(R)=\mathcal{GF}(R)$.
 \item\label{lemma30_5} The class $\mathcal{GF}(R)$ is definable and $\mathcal{GP}(R)=\mathcal{GF}(R)$.
\end{enumerate}
Then $\ref{lemma30_5}\Rightarrow~\ref{lemma30_1}\Rightarrow\ref{lemma30_2}\Rightarrow\ref{lemma30_3}\Leftrightarrow\ref{lemma30_4}$ and $\ref{lemma30_1}${\rm --}$\ref{lemma30_5}$ are equivalent if\/ $\Gwgldim(R)<\infty$.
\end{lemma}

\goodbreak
\begin{proof}\ 

\begin{proof}[\mathversion{bold}$\ref{lemma30_5}\Rightarrow~\ref{lemma30_1}\Rightarrow\ref{lemma30_2}$] It is clear. 
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma30_3}\Leftrightarrow\ref{lemma30_4}$] It holds by Lemma~\ref{06-6}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma30_2}\Rightarrow\ref{lemma30_3}$] Suppose that the class $\mathcal{GP}(R)$ is closed under arbitrary direct products.
Then of course $\prod_{j \in J} P_j$ is Gorenstein projective for any family $\{(P_j)_{j\in J}\}$ of projective $R$-modules.
Thus, by the definition, there is a short exact sequence of $R$-modules
\[
0\to \prod_{j\in J}P_j\to Q\to G\to0
\]
with $Q$ projective and $G$ Gorenstein projective.
Thus $\Ext_R^{1}(G,\prod_{j\in J}P_j)\cong \prod_{j\in J}\Ext_R^{1}(G,P_j)=0$.
This induces that the short sequence is split, and so $\prod_{j\in J}P_j$ is projective. It follows that $R$ is right coherent and left perfect.
\let\qed\relax
\end{proof}

Now let $\Gwgldim(R)<\infty$. In order to see that $\ref{lemma30_1}${\rm --}$\ref{lemma30_5}$ are equivalent, it remains to show: 

\begin{proof}[\mathversion{bold}$\ref{lemma30_3}\Rightarrow\ref{lemma30_5}$]
For this, we assume that $R$ is right coherent and left perfect. Then $\mathcal{GP}(R)=\mathcal{GF}(R)$ by Lemma~\ref{06-6}.
Besides, the class $\mathcal{GF}(R)$ is definable due to Lemma~\ref{4}.\end{proof}
\let\qed\relax
\end{proof}


Let $R$ be a left coherent ring and $M$ an $R^{\op}$-module. Then there is a $\Hom_{R^{\op}}(-,\mathcal{F}(R^{\op}))$-exact complex
\[
\cdots\to F_1 \to F_0 \to M \to 0
\]
with all $F_i \in \mathcal{F}(R^{\op})$
and a $\Hom_{R^{\op}}(-,\mathcal{F}(R^{\op}))$-exact complex
\[
0\to M\to F^{0}\to F^{1} \to\cdots
\]
with all $F^{i} \in \mathcal{F}(R^{\op})$
(these complexes are called \emph{left and right} $\mathcal{F}(R^{\op})$\emph{-resolution} of $M$ respectively,
here we need not require the exactness of the complexes). It follows from~\cite[Definition~8.2.13]{EJ2000}
that $\Hom_{R^{\op}}(-,-)$ is \emph{left balanced} by $\mathcal{F}(R^{\op})\times \mathcal{F}(R^{\op})$.
This may construct left derived functors of $\Hom_{R^{\op}}(-,-)$,
denoted by $\Ext_m^{\mathcal{F}(R^{\op})}(-,-)$,
which can be computed using a right $\mathcal{F}(R^{\op})$-resolution of the first variable
or a left $\mathcal{F}(R^{\op})$-resolution of the second variable.


\begin{prop} \label{6} Consider the following conditions for a ring $R$:
\begin{enumerate}[label=(\arabic*)]
\item\label{prop31_1} The class $\mathcal{GI}(R)$ is definable.
 \item\label{prop31_2} The class $\mathcal{GI}(R)$ is closed under pure submodules and pure epimorphic images.
 \item\label{prop31_3} The class $\mathcal{GI}(R)$ is closed under arbitrary sums.
 \item\label{prop31_4} $R$ is left Noetherian.
 \item\label{prop31_5} An $R$-module $M\in \mathcal{GI}(R)$ if and only if $M^{+}\in\mathcal{GF}(R^{\op})$.
 \item\label{prop31_6} An $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if so is $M^{++}$.
\end{enumerate}
Then $\ref{prop31_6}\Leftarrow \ref{prop31_5}\Rightarrow\ref{prop31_2}\Leftrightarrow \ref{prop31_1}\Rightarrow\ref{prop31_3}\Rightarrow\ref{prop31_4}$ and $\ref{prop31_1}${\rm --}$\ref{prop31_6}$ are equivalent if $\Gwgldim(R)<\infty$.
\end{prop}

\begin{proof}\ 

\begin{proof}[\mathversion{bold}$\ref{prop31_5}\Rightarrow\ref{prop31_2}$] Assume that an $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if $M^{+}$ is in $\mathcal{GF}(R^{\op})$.
To see~\ref{prop31_2}, let $0 \to A \to N \to B \to 0$ be a pure short exact sequence of $R$-modules with $N$ Gorenstein
injective. Then there is a split short exact sequence of $R^{\op}$-modules
$0 \to B^{+} \to N^{+} \to A^{+} \to 0$.
By the ``only if part'' of the assumption, $N^{+}$ is in $\mathcal{GF}(R^{\op})$. Note from~\cite[Corollary~2.6]{Ben2009} that the class $\mathcal{GF}(R^{\op})$ is closed under
any direct summands since any ring is (left and) right GF-closed by~\cite[Theorem~4.11]{SS2020}.
Thus, both $A$ and $B$ are in $\mathcal{GF}(R^{\op})$.
It then follows from the ``if part'' of the assumption that $A$ and $B$ are Gorenstein injective.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop31_2}\Leftrightarrow\ref{prop31_1}$] It holds by Remark~\ref{1-0-0} since the class $\mathcal{GI}(R)$ is always closed under arbitrary direct products (see~\cite[Theorem~2.6]{Hol2004}).
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop31_1}\Rightarrow\ref{prop31_3}$] It is clear since any direct sum is a certain direct limits.
\let\qed\relax
\end{proof}


\begin{proof}[\mathversion{bold}$\ref{prop31_3}\Rightarrow\ref{prop31_4}$] Suppose that the class $\mathcal{GI}(R)$ is closed under arbitrary direct sums. Then of course $\coprod_{j\in J}I_j$
is Gorenstein injective for any family $\{(I_j)_{j\in J}\}$ of injective $R$-modules. Thus, by the definition, there is a
short exact sequence of $R$-modules
\[
0 \to G \to H \to \coprod_{j\in J}I_j \to 0
\]
with $H$ injective and $G$ Gorenstein injective. It follows that $\Ext_R^{1}(\coprod_{j\in J}I_j, G)\cong \prod_{j\in J}\Ext_R^{1}(I_j,G)=0$.
This gives that the short sequence is split, and so, $\coprod_{j\in J}I_j$ is injective.
Thus, $R$ is left Noetherian.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop31_5}\Rightarrow \ref{prop31_6}$] Assume that an $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if $M^{+}$ is in $\mathcal{GF}(R^{\op})$. Then
$R$ is left Noetherian by what we have proved (i.e. $\ref{prop31_5}\Rightarrow\ref{prop31_2}\Leftrightarrow \ref{prop31_1}\Rightarrow\ref{prop31_3}\Rightarrow\ref{prop31_4}$). In particular, $R$ is left coherent.
So, for any $R$-module $M$, one has $M \in \mathcal{GI}(R)$ if and only if $M^{++} \in \mathcal{GI}(R)$ since $M^{++} \in \mathcal{GI}(R)$ $\Leftrightarrow$ $M^{+} \in \mathcal{GF}(R^{\op })$ by~\cite[Theorem~3.6]{Hol2004}.


Now let $\Gwgldim(R)<\infty$. In order to prove that $\ref{prop31_1}${\rm --}$\ref{prop31_6}$ are equivalent, it remains to show the
implications $\ref{prop31_4}\Rightarrow\ref{prop31_5}$ and $\ref{prop31_6}\Rightarrow\ref{prop31_5}$.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop31_6}\Rightarrow\ref{prop31_5}$] Suppose that $R$ is a ring over which any $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if so is $M^{++}$.
Since $R$ is a right GF-closed ring with $\Gwgldim(R)=\Gwgldim(R^{\op })<\infty$,
\cite[Theorem~4]{Bou2015} yields that, for any $R$-module $M$, $M^{++}\in \mathcal{GI}(R)$ if and only if $M^{+}\in \mathcal{GF}(R^{\op})$. Now the result follows.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop31_4}\Rightarrow\ref{prop31_5}$] Suppose that $R$ is left Noetherian. For any $M \in \mathcal{GI}(R)$, there is an
exact sequence of $R$-modules
\[
 \cdots \to I_1 \to I_0 \to M \to 0
\]
with each $I_i$ injective.
This yields another exact sequence of $R^{\op}$-modules
\[
0 \to M^{+} \to (I_0)^{+} \to(I_1)^{+}\to\cdots
\]
with all $(I_i)^{+}$ flat since $R$ is left Noetherian.
Thus, $M^{+}$ is Gorenstein flat by~\cite[Theorem~2.9]{WZ2022} and the assumption $\Gwgldim(R)<\infty$.
Conversely, let $M^{+} \in \mathcal{GF}(R^{\op})$. Note that $R$ is left Noetherian.~%\stackrel{\delta_C^{m-1}}\to 
By~\cite[Theorem~3.6]{Hol2004} there is a $\Hom_{R^{\op}}(-,\mathcal{F}(R^{\op}))$-exact exact sequence of $R^{\op}$-modules
\[
\mathbf{F}= 0 \to M^{+} \to F^{0}\to F^{1}\to \cdots
\]
with all $F^{i}$ flat.
Since $\mathcal{I}(R)$ is covering, there exists a $\Hom_{R}(\mathcal{I}(R),-)$-exact complex of $R$-modules
\[
\mathbf{E} = \cdots \to I_1 \to I_0 \to M \to0
\]
with all $I_i$ injective.
This enables us to obtain a $\Hom_{R^{\op}}(-,\mathcal{F}(R^{\op}))$-exact complex of $R^{\op}$-modules
\[
\mathbf{E}^{+}=0 \to M^{+}\to (I_0)^{+}\to (I_1)^{+}\to \cdots
\]
with all $(I_i)^{+}$ flat. Using the facts $\Ext^{\mathcal{F}(R^{\op})}_{i\geq1}(R_R,M^{+})=0$
 and $\Ext^{\mathcal{F}(R^{\op})}_{0}(R_R,M^{+})\cong M^{+}$ which are
guaranteed by the exact sequence $\mathbf{F}$, we get that the complex $\mathbf{E}^{+}$ is exact, and then so is $\mathbf{E}$. Thus, $M \in \mathcal{GI}(R)$
by~\cite[Theorem~2.6]{WZ2022} and the assumption $\Gwgldim(R)<\infty$.
\end{proof}
\let\qed\relax
\end{proof}


\begin{rema} \label{7} {\rm Recall from Li, Wang, Geng and Hu~\cite{L-H2021} that a ring $R$ is \emph{left Gorenstein hereditary} if $\Ggldim(R)\leq1$ (so $\Gwgldim(R)\leq1$ by Lemma~\ref{07}\ref{lemma16_1}).
According to~\cite[Theorem~1.2]{L-H2021}, we know that a left Gorenstein hereditary ring is left Noetherian if and only if an
$R$-module $M$ is Gorenstein injective if and only $M^{+}$ is in $\mathcal{GF}(R^{\op})$. Note that this result is a special
case of Proposition~\ref{6}.
}
\end{rema}

We are now in a position to give our main theorems below, which answer Questions~\ref{01} and~\ref{02} (from
the introduction) thoroughly.

\goodbreak
\begin{theo} \label{8} The following are equivalent for any ring $R$:
\begin{enumerate}[label=(\arabic*)]
 \item\label{theo33_1} The class $\mathcal{GF}(R)$ is cotilting.
 \item\label{theo33_2} $R$ is a right coherent ring with $\Gwgldim(R)<\infty$.
 \item\label{theo33_3} $R$ is a right coherent ring such that $\Gfd_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{theo}

\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{theo33_1}\Rightarrow\ref{theo33_2}$] Assume that $\mathcal{GF}(R)$ is cotilting. Then by Lemma~\ref{2}, $\mathcal{GF}(R)$ is
definable and admits $\Gwgldim(R)=\sup\{\Gfd_R(M)\mid M~\text{is an}~R\text{-module}\}<\infty$. Furthermore, $R$ is right coherent by Lemma~\ref{4}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo33_2}\Rightarrow\ref{theo33_1}$] Suppose that $R$ is a right coherent ring with $\Gwgldim(R)<\infty$.
Then $\mathcal{GF}(R)$ is definable by Lemma~\ref{4}. Note that $\mathcal{GF}(R)$ is always projectively resolving via~\cite[Corollary~4.12]{SS2020}. Thus,
Lemma~\ref{2} yields that $\mathcal{GF}(R)$ is cotilting.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo33_2}\Leftrightarrow\ref{theo33_3}$] According to~\cite[Corollary~4.12]{SS2020}, we know that $(\mathcal{GF}(R), \mathcal{GF}(R)^{\perp})$ is a complete and hereditary cotorsion pair.
Thus the result comes from Lemma~\ref{06-7-2}.
\end{proof}
\let\qed\relax
\end{proof}

\begin{theo} \label{9} The following are equivalent for any ring $R$:

\begin{enumerate}[label=(\arabic*)]
 \item\label{theo34_1} The class $\mathcal{GP}(R)$ is cotilting.
 \item\label{theo34_2} $R$ is a right coherent and left perfect ring with $\Ggldim(R)<\infty$.
 %\item $R$ is a right coherent and left perfect ring with $\Gwgldim(R)<\infty$.
 \item\label{theo34_3} The class $\mathcal{GF}(R)$ is cotilting and $\mathcal{GP}(R)=\mathcal{GF}(R)$.
 \item\label{theo34_4} $R$ is a right coherent and left perfect ring such that $\Gpd_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{theo}

\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{theo34_3}\Rightarrow\ref{theo34_1}$] It is trivial.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo34_1}\Rightarrow\ref{theo34_2}$] Assume that $\mathcal{GP}(R)$ is cotilting. Then by Lemma~\ref{2}, $\mathcal{GP}(R)$ is definable and admits
$\Ggldim(R)=\sup\{\Gpd_R(M)\mid M~\text{is an}~R\text{-module}\}<\infty$, and so $\Gwgldim(R)\leq \Ggldim(R)<\infty$ by Lemma~\ref{07}\ref{lemma16_1}.
At the same time, $R$ is right coherent and left perfect by Lemma~\ref{5}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo34_2}\Rightarrow\ref{theo34_3}$] Suppose that $R$ is a right coherent and left perfect ring with $\Ggldim(R)<\infty$.
This happens if and only if $R$ is a right coherent and left perfect ring with $\Gwgldim(R)<\infty$ by Lemma~\ref{07}\ref{lemma16_2}.
So $\mathcal{GF}(R)$ is cotilting due to Theorem~\ref{8}. Moreover, the equality $\mathcal{GP}(R)=\mathcal{GF}(R)$ holds by Lemma~\ref{06-6}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo34_2}\Leftrightarrow\ref{theo34_4}$] Suppose that $R$ is a right coherent and left perfect ring. Then $\mathcal{GP}(R)=\mathcal{GF}(R)$ by Lemma~\ref{06-6}.
Now~\cite[Corollary~4.12]{SS2020} yields that $(\mathcal{GP}(R), \mathcal{GP}(R)^{\perp})$ is a complete and hereditary cotorsion pair.
Hence the result follows from Lemma~\ref{06-7-2}.
\end{proof}
\let\qed\relax
\end{proof}

\begin{theo} \label{10} The following are equivalent for any ring $R$:

\begin{enumerate}[label=(\arabic*)]
 \item\label{theo35_1} The class $\mathcal{GI}(R)$ is tilting.
 \item\label{theo35_2} $R$ is left Noetherian ring with $\Ggldim(R)<\infty$.
 \item\label{theo35_3} The class $\mathcal{GF}(R^{\op})$ is a cotilting class and an $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if so is $M^{++}$.
 \item\label{theo35_4} $R$ is a left Noetherian ring such that $\Gid_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{theo}

\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{theo35_1}\Rightarrow\ref{theo35_2}$] Assume that $\mathcal{GI}(R)$ is tilting.
Then by Lemma~\ref{3}, $\mathcal{GI}(R)$ is definable and
$\Ggldim(R)=\sup\{\Gid_R(M)\mid M~\text{is an}~R\text{-module}\}<\infty$, which deduces that $\Gwgldim(R)<\infty$ by Lemma~\ref{07}\ref{lemma16_1}. Note that $R$ is also left Noetherian by Proposition~\ref{6}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo35_2}\Rightarrow\ref{theo35_3}$] Suppose that $R$ is a left Noetherian ring with $\Ggldim(R)<\infty$.
This happens if and only if $R$ is a left Noetherian ring with $\Gwgldim(R)=\Gwgldim(R^{\op})<\infty$ by~\cite[Theorem~7]{Bou2015}.
It follows from Theorem~\ref{8} that $\mathcal{GF}(R^{\op})$ is cotilting.
Furthermore, the other assertion in~\ref{theo35_3} holds by Proposition~\ref{6}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo35_3}\Rightarrow\ref{theo35_1}$] Suppose that the following two conditions hold:
\begin{enumerate}[label=(\Roman*)]
 \item\label{proof_theo35_I} The class $\mathcal{GF}(R^{\op})$ is cotilting,\quad\text{and}
 \item\label{proof_theo35_II} An $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if $M^{++}$ is in $\mathcal{GI}(R)$.
\end{enumerate}
By Theorem~\ref{8}, the condition~\ref{proof_theo35_I} yields that $R$ is left coherent ring with $\Gwgldim(R)=\Gwgldim$ $(R^{\op})<\infty$.
Then according to Proposition~\ref{6}, the condition~\ref{proof_theo35_II} yields that $R$ is also left Noetherian. Whence, $\mathcal{GI}(R)$ is definable again by Proposition~\ref{6}.
Notice further that $\mathcal{GI}(R)$ is always special preenveloping and injectively coresolving by~\cite[Theorem~5.6]{SS2020}, it is tilting due to Lemma~\ref{3}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo35_2}\Leftrightarrow\ref{theo35_4}$] It holds by Lemma~\ref{06-7-4} since $(^{\perp}\mathcal{GI}(R), \mathcal{GI}(R))$ is a complete and hereditary cotorsion pair (see\cite[Theorem~5.6]{SS2020}).
\end{proof}
\let\qed\relax
\end{proof}

\begin{remas} \label{10-0}
Let $n\geq0$ be an integer. By the proofs in Theorems~\ref{8},~\ref{9} and~\ref{10}, one see
that
\begin{enumerate}[label=(\arabic*)]
 \item\label{rema36_1} The class $\mathcal{GF}(R)$ is $n$-cotilting if and only if $R$ is a right coherent ring with $\Gwgldim(R)\leq n$.
 \item\label{rema36_2} The class $\mathcal{GP}(R)$ is $n$-cotilting if and only if $R$ is a right coherent and left perfect ring with $\Ggldim(R)\leq n$.
 \item\label{rema36_3} The class $\mathcal{GI}(R)$ is $n$-tilting if and only if $R$ is a left Noetherian ring with $\Ggldim(R)\leq n$ if and only if the class $\mathcal{GF}(R^{\op})$ is $n$-cotilting and an $R$-module is in $\mathcal{GI}(R)$ if and only if so is $M^{++}$.
\end{enumerate}

\end{remas}

\section{Applications}\label{Sec4}

This section is divided into four subsections, by which some applications of Theorems~\ref{8},~\ref{9} and~\ref{10}
are given.

\subsection{Characterizations of Gorenstein rings and Ding--Chen rings}

Recall that a ring $R$ is \emph{Gorenstein}~\cite{Iwa1979}
(resp.\ \emph{Ding--Chen}~\cite{DC1996, Gil2010}) if $R$ is an $n$-Gorenstein ring (resp.\ $n$-FC ring) for some nonnegative integer $n$, i.e., $R$
is a two-sided Noetherian (resp.\ two-sided coherent) ring with self-injective (resp.\ self-FP-injective)
dimension at most $n$ on both sides.
In particular, 0-Gorenstein ring and 0-FC ring is just\emph{ QF ring} and \emph{FC ring}, respectively.
Recall that an Artin algebra $R$ is \emph{Gorenstein} if it is Gorenstein as a ring.
In particular, 0-Gorenstein Artin algebra is exactly \emph{self-injective Artin algebra}.

As the first application of Theorems~\ref{8},~\ref{9} and~\ref{10},
this subsection is devoted to give some new characterizations for Gorenstein rings (including Gorenstein Artin algebras) and Ding--Chen rings. Firstly, we characterize Gorenstein rings via the class $\mathcal{GI}(-)$ being tilting.





\begin{theo} \label{11-0-0-0} The following are equivalent for any ring $R$:
\begin{enumerate}[label=(\arabic*)]
 \item\label{theo37_1} $R$ is Gorenstein.
 \item\label{theo37_2} $R$ is a right Noetherian ring such that the class $\mathcal{GI}(R)$ is tilting.
 \item\label{theo37_3} $R$ is a left Noetherian ring such that the class $\mathcal{GI}(R^{\op})$ is tilting.
 \item\label{theo37_4} Both the classes $\mathcal{GI}(R)$ and $\mathcal{GI}(R^{\op})$ are tilting.
\end{enumerate}
In particular, a commutative ring $R$ is Gorenstein if and only if the class $\mathcal{GI}(R)$ is tilting.
\end{theo}

\begin{proof}
According to Theorem~\ref{10}, we know that the condition~\ref{theo37_2} (resp.\ \ref{theo37_3})
happens if and and only if $R$ is a two-sided Noetherian ring with $\Ggldim(R)<\infty$ (resp.\ $\Ggldim(R^{\op})<\infty$),
and that the condition~\ref{theo37_4} happens if and and only if $R$ is a two-sided Noetherian ring with $\Ggldim(R)<\infty$ and $\Ggldim(R^{\op})<\infty$.
Thus, the implication $\ref{theo37_1}\Leftrightarrow\ref{theo37_2}$ (resp.\ $\ref{theo37_1}\Leftrightarrow\ref{theo37_3}$ and $\ref{theo37_1}\Leftrightarrow\ref{theo37_4}$) follows from~\cite[Remark~3.11]{CET2021}.

The last statement is an immediate consequence of the implication $\ref{theo37_1}\Leftrightarrow\ref{theo37_4}$.
\end{proof}

\begin{coro} \label{11-0-1} Let $R$ be a two-sided Noetherian ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{coro38_1} $R$ is Gorenstein.
 \item\label{coro38_2} Any one of the classes $\mathcal{GI}(R)$ and $\mathcal{GI}(R^{\op})$ is tilting.
 \item\label{coro38_3} Any one of the classes $\mathcal{GF}(R)$ and $\mathcal{GF}(R^{\op})$ is cotilting.
\end{enumerate}
\end{coro}

\begin{proof}\


\begin{proof}[\mathversion{bold}$\ref{coro38_1}\Leftrightarrow\ref{coro38_2}$] It holds by the proof of $\ref{theo37_1}\Leftrightarrow\ref{theo37_4}$ in Theorem~\ref{11-0-0-0}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{coro38_1}\Leftrightarrow\ref{coro38_3}$]
Using the assumption and Theorem~\ref{8}, one has the condition~\ref{coro38_3}
happens if and and only if $\Gwgldim(R)<\infty$ or $\Gwgldim(R^{\op})<\infty$.
Thus, the results follows by~\cite[Remark~3.11]{CET2021} (see also~\cite[Lemma~1.2]{WZ2022}).
\end{proof}
\let\qed\relax
\end{proof}

As mentioned in the introduction,
Angeleri H\"{u}gel, Herbera and Trlifaj in~\cite[Theorem~3.4]{AHT2006} proved that a two-sided Noetherian ring $R$ is Gorenstein
if and only if both the classes $\mathcal{GI}(R)$ and $\mathcal{GI}(R^{\op})$ are tilting,
equivalently, if and only if the class $\mathcal{GI}(R)$ is tilting and the class $\mathcal{GF}(R)$ is cotilting.
Note that both Corollary~\ref{11-0-1} and the equivalence of $\ref{theo37_1}\Leftrightarrow~\ref{theo37_4}$ in Theorem~\ref{11-0-0-0} provide a slight improvement of~\cite[Theorem~3.4]{AHT2006}.
On the other hand, the coming example shows that a general ring $R$ satisfying that the class $\mathcal{GI}(R)$ is tilting and the class $\mathcal{GF}(R)$ is cotilting may not be Gorenstein.

\begin{exam} \label{11-0-1-1}{\rm Let $S=\left(
 \begin{smallmatrix}
 \mathbb{Z} & \mathbb{Q} \\
 0 & \mathbb{Q} \\
 \end{smallmatrix}
 \right).
$
Then by Small~\cite{Small1965}, $S$ is right Noetherian but not left Noetherian ring with $\gldim(S)=1$ and $\gldim(S^{\op})=2$.
Now we consider the ring $R=S^{\op}$. It is then seen that $R$ is left Noetherian but not right Noetherian ring with $\gldim(R)=2$ and $\gldim(R^{\op})=1$.
Note that $\gldim(R^{\op})=1$ shows that $R$ is right hereditary, and hence right coherent.
Thus, $R$ is left Noetherian and right coherent ring with $\Ggldim(R)\leq \gldim(R)=2$ and $\Gwgldim(R)\leq \wgldim(R)\leq \gldim(R^{\op})=1$.
It follows from Theorems~\ref{8} and~\ref{10} that the class $\mathcal{GI}(R)$ is tilting and the class $\mathcal{GF}(R)$ is cotilting.
However, $R$ is not Gorenstein since it is not right Noetherian.
}
\end{exam}

Secondly, we characterize Ding--Chen rings via the class $\mathcal{GF}(-)$ being cotilting.

\begin{theo} \label{11-0-2} Let $R$ be a ring. Then the following are equivalent:
\begin{enumerate}[label=(\arabic*)]
 \item\label{theo40_1} $R$ is Ding--Chen.
 \item\label{theo40_2} $R$ is a left coherent ring such that the class $\mathcal{GF}(R)$ is cotilting.
 \item\label{theo40_3} $R$ is a right coherent ring such that the class $\mathcal{GF}(R^{\op})$ is cotilting.
 \item\label{theo40_4} Both the classes $\mathcal{GF}(R)$ and $\mathcal{GF}(R^{\op})$ are cotilting.
\end{enumerate}
In particular, a commutative ring $R$ is Gorenstein if and only if the class $\mathcal{GF}(R)$ is cotilting.
\end{theo}

\begin{proof} According to Theorem~\ref{8}, we know that the condition\ref{theo40_2} (resp.\ \ref{theo40_3})
happens if and and only if $R$ is a two-sided coherent ring with $\Gwgldim(R)<\infty$ (resp.\ $\Gwgldim(R^{\op})<\infty$),
and that the condition~\ref{theo40_4} happens if and and only if $R$ is a two-sided coherent ring with $\Gwgldim(R)<\infty$ and $\Gwgldim(R^{\op})<\infty$.
Thus, the implication $\ref{theo40_1}\Leftrightarrow\ref{theo40_2}$ (resp.\ $\ref{theo40_1}\Leftrightarrow\ref{theo40_3}$ and $\ref{theo40_1}\Leftrightarrow\ref{theo40_4}$) follows from~\cite[Remark~3.11]{CET2021}.

The last statement is an immediate consequence of the implication $\ref{theo40_1}\Leftrightarrow\ref{theo40_4}$.
\end{proof}

\goodbreak

Note that the proof of Theorem~\ref{11-0-2} implies that

\begin{coro} \label{11-0-3} Let $R$ be a two-sided coherent ring. Then the following are equivalent:
\begin{enumerate}[label=(\arabic*)]
 \item\label{coro41_1} $R$ is Ding--Chen.
 \item\label{coro41_2} Any one of the classes $\mathcal{GF}(R)$ and $\mathcal{GF}(R^{\op})$ is cotilting.
\end{enumerate}
\end{coro}

Finally, we characterize Gorenstein Artin algebras via the class $\mathcal{GP}(-)$ being cotilting.


\begin{theo} \label{11-0-4} Let $R$ be a commutative ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{theo41_1} $R$ is a Gorenstein Artin algebra.
 \item\label{theo41_2} The class $\mathcal{GP}(R)$ is cotilting.
 \item\label{theo41_3} The class $\mathcal{GF}(R)$ is cotilting and $\mathcal{GF}(R)=\mathcal{GP}(R)$.
 \item\label{theo41_4} The class $\mathcal{GI}(R)$ forms a tilting class such that an $R$-module $M$ is in $\mathcal{GP}(R)$ if and only if $M^{+}$ is in $\mathcal{GI}(R)$.
\end{enumerate}
\end{theo}

\begin{proof}\  

\begin{proof}[\mathversion{bold}$\ref{theo41_3}\Rightarrow\ref{theo41_2}$] It is clear.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo41_2}\Rightarrow\ref{theo41_1}$] Suppose that the class $\mathcal{GP}(R)$ is cotilting.
Then by Theorem~\ref{9}, this happens if and only if $R$ is a right coherent and left perfect ring with $\Ggldim(R)<\infty$.
Note that the right coherence and left perfectness of $R$ imply that the class $\mathcal{P}(R)$ is closed under direct products.
It follows from~\cite[Theorem~3.4]{Chase1960} that $R$ is Artinian since $R$ is commutative, and hence, $R$ is an Artin algebra.
Note from~\cite[Remark~3.11]{CET2021} that any Artin algebra with finite Gorenstein weak global dimension is Gorenstein.
In particular, $R$ is a Gorenstein Artin algebra.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo41_1}\Rightarrow\ref{theo41_4}$] Assume that $R$ is a Gorenstein Artin algebra.
Then $R$ is a two-sided Artinian (hence a two-sided Noetherian) ring with $\Ggldim(R)<\infty$.
It follows from Theorem~\ref{10} that the class $\mathcal{GI}(R)$ is tilting.
Meanwhile, as $R$ is right coherent and left perfect, one has $\mathcal{GF}(R)=\mathcal{GP}(R)$ by Lemma~\ref{06-6}.
Consequently,~\cite[Theorem~3.6]{Hol2004} yields that an $R$-module $M$ is in $\mathcal{GP}(R)$ if and only if $M^{+}$ is in $\mathcal{GI}(R^{\op})=\mathcal{GI}(R)$ (as $R$ is commutative).
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo41_4}\Rightarrow\ref{theo41_3}$] Suppose that the following two conditions hold:
\begin{enumerate}[label=(\Roman*)]
 \item\label{proof_theo42_I} The class $\mathcal{GI}(R)$ is tilting,\quad\text{and}
 \item\label{proof_theo42_II} An $R$-module is in $\mathcal{GP}(R)$ if and only if $M^{+}$ is in $\mathcal{GI}(R)$.
\end{enumerate}
Since $R$ is commutative, it follows from Theorems~\ref{10} and~\ref{8} that the condition~\ref{proof_theo42_I} induces that the class $\mathcal{GF}(R)=\mathcal{GF}(R^{\op})$ is cotilting,
and that $R$ is (right) coherent.
Thus, combining~\cite[Theorem~3.6]{Hol2004} with the condition~\ref{proof_theo42_II}, one has $\mathcal{GF}(R)=\mathcal{GP}(R)$.
This completes the proof.
\end{proof}
\let\qed\relax
\end{proof}

Let $n\geq0$ be an integer and $R$ a perfect and $n$-FC ring. If $n=0$, then $R$ is 0-Gorenstein by~\cite[Corollary~3.7]{DC1993}.
Otherwise, for $n\geq1$, let $R=\left(
 \begin{smallmatrix}
 \mathbb{Q} & \mathbb{R} \\
 0 & \mathbb{Q} \\
 \end{smallmatrix}
 \right).
$
Then~\cite[Example~3.4]{Wz2017} showed that $R$ is a perfect and hereditary (hence perfect and 1-FC) ring which is
not $n$-Gorenstein for any $n\geq0$. We note that such a ring $R$ is not commutative.

The following corollary shows that any commutative perfect and $n$-FC rings are always Gorenstein.

\begin{coro} \label{11-0-5} Let $R$ be a commutative ring. Then the following are equivalent:
\begin{enumerate}[label=(\arabic*)]
 \item\label{coro43_1} $R$ is a Gorenstein Artin algebra.
 \item\label{coro43_2} $R$ is Ding--Chen and perfect.
\end{enumerate}
\end{coro}

\goodbreak
\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{coro43_1}\Rightarrow\ref{coro43_2}$] It is obvious.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{coro43_2}\Rightarrow\ref{coro43_1}$] Assume that $R$ is Ding--Chen and perfect.
Note that $R$ is commutative, it suffices to show that the class $\mathcal{GF}(R)$ is cotilting and $\mathcal{GF}(R)=\mathcal{GP}(R)$ by Theorem~\ref{11-0-4}.
On one hand, the class $\mathcal{GF}(R)$ is cotilting by Theorem~\ref{11-0-2} since $R$ is Ding--Chen.
On the other hand, the equality $\mathcal{GF}(R)=\mathcal{GP}(R)$ holds by Lemma~\ref{06-6} as $R$ is right coherent and left perfect.
Thus, the result follows.
\end{proof}
\let\qed\relax
\end{proof}

We end this subsection with some remarks.

\begin{rema} \label{11-0-6}
Let $n\geq0$ be an integer. By the proofs in Theorems~\ref{11-0-0-0},~\ref{11-0-2} and~\ref{11-0-4}, as well as Corollaries~\ref{11-0-1} and~\ref{11-0-3}, one can see
that
\begin{enumerate}[label=(\arabic*)]
 \item\label{rema44_1} A ring $R$ is $n$-Gorenstein if and only if both the classes $\mathcal{GI}(R)$ and $\mathcal{GI}(R^{\op})$ are $n$-tilting.
 In particular, a commutative (or two-sided Noetherian) ring $R$ is $n$-Gorenstein if and only if the class $\mathcal{GI}(R)$ is $n$-tilting.
 \item\label{rema44_2} A ring $R$ is $n$-FC if and only if both the classes $\mathcal{GF}(R)$ and $\mathcal{GF}(R^{\op})$ are $n$-cotilting.
 In particular, a commutative (or two-sided coherent) ring $R$ is $n$-FC if and only if the class $\mathcal{GF}(R)$ is $n$-cotilting.
 \item\label{rema44_3} A commutative ring $R$ is an $n$-Gorenstein Artin algebra if and only if the class $\mathcal{GP}(R)$ is $n$-cotilting.
\end{enumerate}
\end{rema}


\subsection{Characterizations of Gorenstein modules via finitely generated modules}

It is well-known that injective (resp.\ flat) modules can be characterized via finitely generated modules by vanishing of the functor $\Ext$ (resp.\ Tor).
As the second application of Theorems~\ref{8},~\ref{9} and~\ref{10}, we will obtain a Gorenstein version of the characterizations (see Theorem~\ref{10-1} and Lemma~\ref{10-5}).
As a result, we prove that left Noetherian rings with finite
left Gorenstein global dimension satisfy First Finitistic Dimension Conjecture (see Corollary~\ref{11-16}),
and prove a result related to a question posed by Bazzoni~\cite[Question~1(1)]{Baz2008}~(see Theorem~\ref{10-6}).




Let us firstly consider the characterizations of Gorenstein injective modules via finitely generated modules by vanishing of the functor $\Ext$.
Let $R$ be a Gorenstein ring. Then Enochs and Jenda~\cite[Theorem~2.5]{EJ2004} proved that an $R$-module $M$ is Gorenstein injective if and only if $\Ext_R^{i}(L,M)=0$ for all $i >0$ and all countably generated $R$-modules $L$ with $\pd_R(L)<\infty$.~\cite[Corollary~3.5(1)]{AHT2006} tells us that this characterization of Gorenstein injective modules can be relaxed as ``an $R$-module $M$ is Gorenstein injective if and only if $\Ext_R^{i}(L,M)=0$ for all $i >0$ and all finitely generated $R$-modules $L$ with $\pd_R(L)<\infty$''.

\looseness-1
In what follows, we denote by $\hat{\mathcal{P}}$ the class consisting of all $R$-modules with finite projective dimensions.
By the proof of~\cite[Corollary~7.1.13(a)]{GT2006} and by noting from Chen~\cite[Lemma~5.1]{Chen2010} that there is a hereditary and complete cotorsion pair $(\hat{\mathcal{P}},\mathcal{GI}(R))$ whenever $R$ is of $\Ggldim(R)<\infty$, one can see that, to obtain the above characterization, the Gorenstein condition can be relaxed to ``left Noetherian rings with finite left Gorenstein global dimension''.
The added value of the next result is to show that, in order to obtain the characterization ``for all $R$-module $M$, $M$ is Gorenstein injective if and only if $\Ext_R^{i}(L,M)=0$ for all $i >0$ and all finitely generated $R$-modules $L$ with $\pd_R(L)<\infty$'', the ring $R$ must be left Noetherian rings with finite left Gorenstein global dimension.

\begin{theo} \label{10-1}
The following are equivalent for any ring $R$:
\begin{enumerate}[label=(\arabic*)]
 \item\label{theo45_1} $R$ is a left Noetherian ring with $\Ggldim(R)<\infty$.
\item\label{theo45_2} An $R$-module $M$ is Gorenstein injective if and only if $\Ext_R^{i}(L,M)=0$ for all $i >0$ and all $R$-modules $L$ of type FP$_\infty$ with $\pd_R(L)<\infty$.
\item\label{theo45_3} An $R$-module $M$ is Gorenstein injective if and only if $\Ext_R^{i}(L,M)=0$ for all $i >0$ and all finitely generated $R$-modules $L$ with $\pd_R(L)<\infty$.
\item\label{theo45_4} An $R$-module $M$ is Gorenstein injective if and only if $\Ext_R^{i}(L,M)=0$ for all $i >0$ and all finitely presented $R$-modules $L$ with $\pd_R(L)<\infty$.
 \end{enumerate}

Furthermore, if any one of the above conditions is satisfied, then the cotorsion pair $(\hat{\mathcal{P}},\mathcal{GI}(R))$ is of strongly finite type.
\end{theo}

\begin{proof}For any ring $R$, note from~\cite[Theorem~5.6]{SS2020} that $(^{\perp}\mathcal{GI}(R),\mathcal{GI}(R))$ forms a hereditary and complete cotorsion pair.
So, the condition~\ref{theo45_1} is equivalent to that ``the complete and hereditary cotorsion pair $(^{\perp}\mathcal{GI}(R),\mathcal{GI}(R))$ is of strongly finite type''.
It follows from Remark~\ref{1-5}\ref{rema25_3} that the condition~\ref{theo45_2} is further equivalent to that ``the class $\mathcal{GI}(R)$ is tilting''.
Thus, $~\ref{theo45_1}\Leftrightarrow \ref{theo45_2}$ holds by Theorem~\ref{10}.

On the other hand, one concludes from~\cite[Lemma~9.2.7]{CFH2012} that $\Ext_R^{i\geq1}(L,G)=0$ for all Gorenstein injective $R$-modules $G$ and all $R$-modules $L$ with $\pd_R(L)<\infty$.
Meanwhile, it is trivial that any modules of type FP$_\infty$ is finite presented and that any finite presented modules are finitely generated.
Whence, one has $~\ref{theo45_2}\Leftrightarrow\ref{theo45_3}\Leftrightarrow\ref{theo45_4}$.

Now suppose that $R$ is a left Noetherian ring with $\Ggldim(R)<\infty$.
Then the proof of $~\ref{theo45_1}\Leftrightarrow \ref{theo45_2}$ above shows that the complete and hereditary cotorsion pair $(^{\perp}\mathcal{GI}(R),\mathcal{GI}(R))$ is of strongly finite type.
But~\cite[Lemma~5.1]{Chen2010} tells us that $^{\perp}\mathcal{GI}(R)=\hat{\mathcal{P}}$.
\end{proof}

In what follows, for any class $\mathcal{X}$ of $R$-modules, we denote by $\mathcal{X}^{<\omega}$ the subclass of $\mathcal{X}$ consisting of all modules of type FP$_\infty$.


Recall that for any ring $R$, the \emph{big finitistic dimension} of $R$ is defined as
\[
\FPD(R)=\sup\{\pd_R(M)\mid M~\text{is an}~R\text{-module with}~\pd_R(M)<\infty\},
\]
and the \emph{little finitistic dimension} of $R$ is defined as
\[
\fpd(R)=\sup\{\pd_R(M)\mid M~\text{is a finitely generated}~R\text{-module with}~\pd_R(M)<\infty\}.
\]
For a ring $R$, ``First Finitistic Dimension Conjecture'' and ``Second Finitistic Dimension Conjecture''
calim $\FPD(R)=\fpd(R)$ and $\fpd(R)<\infty$ respectively.
It is a famous result that ``First Finitistic Dimension Conjecture'' and ``Second Finitistic Dimension Conjecture'' vanish for Gorenstein rings (see~\cite[Theorem~3.2]{AHT2006} and~\cite[Theorem~7.1.12]{GT2006}).
The next corollary shows that the Gorenstein condition can be relaxed to ``left Noetherian rings with finite left Gorenstein global dimension'' and to ``rings with finite left Gorenstein global dimension'', respectively.
Note that a ring $R$ is Gorenstein if and only if $R$ is a two-sided Noetherian ring with $\Ggldim(R)<\infty$ (see~\cite[Remark~3.11]{CET2021}); see Example~\ref{11-0-1-1} for the existence of a left Noetherian ring $R$ with $\Ggldim(R)<\infty$ which is not Gorenstein.








\begin{coro} \label{11-16} Let $R$ be a ring with $\Ggldim(R)<\infty$. Then ``Second Finitistic Dimension Conjecture'' vanishes.
If $R$ is also left Noetherian, then ``First Finitistic Dimension Conjecture'' vanishes as well.
\end{coro}

\begin{proof} Let $R$ be a left Noetherian ring with $\Ggldim(R)<\infty$ and $\hat{\mathcal{P}}$ be as above.
Then by Theorem~\ref{10-1},
the pair $(\hat{\mathcal{P}},\mathcal{GI}(R))$ is of strongly finite type.
It follows that $\hat{\mathcal{P}}={^{\perp}}((\hat{\mathcal{P}}^{<\omega})^{\perp})$.
Thus,~\cite[Corollary~3.2.4]{GT2006} yields that every $R$-module $L$ with $\pd_R(L)<\infty$
(i.e., $L\in \hat{\mathcal{P}}$) is a summand of some $R$-module in $\hat{\mathcal{P}}^{<\omega}$.
Note that $\hat{\mathcal{P}}^{<\omega}$ is just
the class of all finitely generated $R$-modules with finite projective dimension since $R$ is left Noetherian.
Hence, $\FPD(R)\leq \fpd(R)$ and so $\FPD(R)= \fpd(R)$.
\end{proof}


Let $R$ be a left Noetherian ring with $\Ggldim(R)<\infty$.
By Theorem~\ref{10-1},
Gorenstein injective $R$-modules can be characterized via finitely generated modules by vanishing of the functor $\Ext$.
Now we consider the behaviors of Gorenstein projective and Gorenstein flat modules.


\begin{lemma} \label{10-5}
Let $R$ be a left Noetherian ring with $\Ggldim(R^{\op})<\infty$. Then the following are equivalent for any $R^{\op}$-module $N$:

\begin{enumerate}[label=(\arabic*),series=lemma47]
 \item\label{lemma47_1} $N$ is Gorenstein flat.
 \item\label{lemma47_2} $\Tor_{i}^R(N,L)=0$ for all $i > 0$ and all finite generated $R$-modules $L$ with $\pd_R(L)<\infty$.
\item\label{lemma47_3} $\Ext_R^{i}(L, N^{+})=0$ for all $i > 0$ and all finite generated $R$-modules $L$ with $\pd_R(L)<\infty$.
\item\label{lemma47_4} $\Tor^{R}_i(N^{++},L)=0$ for all $i > 0$ and all finite generated $R$-modules $L$ with $\pd_R(L)<\infty$.
\end{enumerate}
Furthermore, if $R$ is also a left Noetherian and right perfect ring with $\Ggldim(R^{\op})<\infty$,
then the above conditions are equivalent to
\begin{enumerate}[label=(\arabic*),resume=lemma47]
 \item\label{lemma47_5} $M$ is Gorenstein projective.
\end{enumerate}
\end{lemma}

\goodbreak
\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{lemma47_1}\Leftrightarrow\ref{lemma47_3}$] Since $R$ is left Notherian, one has an $R^{\op}$-module $N$ is Gorenstein flat
if and only if $N^{+}$ is Gorenstein injective by~\cite[Theorem~3.6]{Hol2004}.
So the result holds by Theorem~\ref{10-1} as \hbox{$\Ggldim(R)<\infty$}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma47_2}\Leftrightarrow\ref{lemma47_3}$] It holds by the isomorphism $\Tor_{i}^R(N,L)^{+}\cong \Ext_R^{i}(L, N^{+})$ and the faithful property of the functor $(-,-)^{+}$.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma47_3}\Leftrightarrow\ref{lemma47_4}$] It follows from the isomorphism $\Ext_R^{i}(L, N^{+})^{+}\cong \Tor^{R}_i(N^{++},L)$ (since $R$ is left Noetherian) and the faithful property of the functor $(-,-)^{+}$.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma47_1}\Leftrightarrow\ref{lemma47_5}$]
Suppose that $R$ is a left Noetherian and left perfect ring with $\Ggldim(R)<\infty$ .
Then the equivalence follows from Lemma~\ref{06-6}.
\end{proof}
\let\qed\relax
\end{proof}



Note that there are dual notions of that classes of modules are of (strongly) finite type.
Recall that a class $\mathcal{X}$ of $R^{\op}$-modules is of \emph{cofinite type} (resp.\ \emph{strongly cofinite type})
if there exists a set $\mathcal{S}$ consists of $R^{\op}$-modules of type FP$_\infty$ (resp.\ with finite projective dimension)
such that $\mathcal{X}=\{M\in R^{\op}\text{-}\Mod\mid \Tor_{i\geq1}^{R}(M,S)=0, \forall S\in \mathcal{S}\}$
(we refer to the readers that Bazzoni, G\"{o}bel and Trlifaj in~\cite{Baz2008, GT2006} called that a class $\mathcal{X}$ is ``of cofinite type'', is just of strongly cofinite type in our sense).
Let $\mathcal{X}$ be a class of $R^{\op}$-modules which is of strongly cofinite type.
Then according to~\cite[Definition~8.1.11 and Proposition~8.1.12]{GT2006}, $\mathcal{X}$ is always cotilting.
However, there exists a cotilting class which is not of strongly cofinite type (see~\cite[Example~8.2.13]{GT2006}).
Furthermore, it is an open question whether left Noetherian rings admits cotilting classes of right modules which are not of strongly cofinite type (see~\cite[Question~1(1)]{Baz2008}).
The next result shows that such a question has an affirmative answer in the Gorenstein homological algebra.

\begin{theo} \label{10-6}
Let $R$ be a left Noetherian ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{lemma48_1} The class $\mathcal{GF}(R^{\op})$ \emph{(}resp.\ $\mathcal{GP}(R^{\op})$\emph{)} is cotilting.
 \item\label{lemma48_2} The class $\mathcal{GF}(R^{\op})$ \emph{(}resp.\ $\mathcal{GP}(R^{\op})$\emph{)} is of strongly cofinite type.
\end{enumerate}
\end{theo}

\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{lemma48_2}\Rightarrow\ref{lemma48_1}$] It holds by~\cite[Definition~8.1.11 and Proposition~8.1.12]{GT2006}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{lemma48_1}\Rightarrow\ref{lemma48_2}$] Suppose that the class $\mathcal{GF}(R^{\op})$ \emph{(}resp.\ $\mathcal{GP}(R^{\op})$\emph{)} is cotilting.
Then, in view of Theorem~\ref{8} (resp.\ \ref{9}), this will happen if and only if $R$ is a ring with $\Gwgldim(R^{\op})<\infty$
(resp.\ a right perfect ring with $\Ggldim(R^{\op})<\infty$) as $R$ is left Noetherian.
Notice further that $\Ggldim(R^{\op})=\Gwgldim(R^{\op})$ by~\cite[Theorem~7]{Bou2015}, and that an $R$-module is of type FP$_\infty$ if and only if it is finitely generated, again since $R$ is left Noetherian.
Thus, the result follows from Lemma~\ref{10-5}.
\end{proof}
\let\qed\relax
\end{proof}


By virtue of~\cite[Theorem~2.2]{AHT2006} (or~\cite[Theorems~5.2.23 and~8.1.14]{GT2006}),
we know that, over any ring $R$, there is a bijective correspondence between tilting
classes of $R$-modules (resp.\ class of $R^{\op}$-modules of strongly cofinite type),
and resolving subcategories $\mathcal{S}$ consisting of those $R$-modules of type FP$_\infty$ with finite projective dimension.
The correspondence is given by the mutually inverse assignments
\[
\mathcal{X}\mapsto (^{\perp}\mathcal{X})^{<\omega}~\text{and}~\mathcal{S}\mapsto\mathcal{S}^{\perp}~~(\emph{resp.}\quad\mathcal{X}\mapsto (^{\top}\mathcal{X})^{<\omega}~\text{and}~\mathcal{S}\mapsto\mathcal{S}^{\top}).
\]
Here $\mathcal{S}^{\top}= \{M\in R^{\op}\text{-}\Mod\mid \Tor_{i\geq1}^{R}(M,S)=0, \forall S\in \mathcal{S}\}$ and $^{\top}\mathcal{X}$ is defined by dually.


Let $(\mathcal{X},\mathcal{Y})$ be a complete and hereditary cotorsion pair of $R$-modules.
Recall that $(\mathcal{X},\mathcal{Y})$ is a \emph{projective cotorsion pair} (resp.\ an \emph{injective cotorsion pair}) if $(\mathcal{X}\cap\mathcal{Y})=\mathcal{P}(R)$ (resp.\ $(\mathcal{X}\cap\mathcal{Y})=\mathcal{I}(R)$).
Gillespie in~\cite{Gil2016} studied the lattices of projective and injective cotorsion pairs respectively.
According to~\cite[Theorems~5.2 and~5.4]{Gil2016} we know that $\mathcal{X}\subseteq \mathcal{GP}(R)$ (resp.\ $\mathcal{Y}\subseteq \mathcal{GI}(R)$) whenever $(\mathcal{X},\mathcal{Y})$ is a projective cotorsion pair (resp.\ an injective cotorsion pair). In other words, in the lattices of projective (resp.\ injective) cotorsion pairs, the one induced by Gorenstein projective (resp.\ Gorenstein injective) modules is a maximal element.

Motivated by Gillespie's results, it is natural to consider which role does the class of Gorenstein projective (resp.\ Gorenstein injective) modules play in the collections of tilting (resp.\ cotilting) classes.
We end the subsection by the next result, building from the above facts in~\cite[Theorem~2.2]{AHT2006} (or~\cite[Theorems~5.2.23 and~8.1.14]{GT2006}), which shows that, under some certain conditions,
in the lattice of tilting (resp.\ cotilting) classes, the class of Gorenstein projective (resp.\ Gorenstein injective) modules is a minimal element.


\begin{prop} \label{10-7}
Let $R$ be a ring.

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop49_1} If the class $\mathcal{GI}(R)$ is tilting, then it is the smallest tilting class, in the sense of that $\mathcal{GI}(R)\subseteq \mathcal{X}$ for all tilting class of $R$-modules.

\item\label{prop49_2} If the class $\mathcal{GF}(R^{\op})$ (resp.\ $\mathcal{GP}(R^{\op})$) is cotilting over a left Noetherian ring $R$, then it is the smallest class of strongly cofinite type in the similar sense.
\end{enumerate}
\end{prop}
\begin{proof}\ 
\begin{proof}[\ref{prop49_1}] 
By Theorem~\ref{10}, the class $\mathcal{GI}(R)$ is tilting if and only if $R$ is a left Noetherian ring with $\Ggldim(R)<\infty$.
Then the result holds by~\cite[Theorems~5.2.23]{GT2006} and Theorem~\ref{10-1}.
\let\qed\relax
\end{proof}

\begin{proof}[\ref{prop49_2}] 
 Suppose that $R$ is left Noetherian. By Theorem~\ref{10-6} and its proof, one has
the class $\mathcal{GF}(R^{\op})$ (resp.\ $\mathcal{GP}(R^{\op})$) is cotilting if and only if the class $\mathcal{GF}(R^{\op})$ (resp.\ $\mathcal{GP}(R^{\op})$) is of strongly cofinite type,
or equivalently, if and only if $R$ is a (resp.\ right perfect) ring with $\Ggldim(R)<\infty$.
Then the result is an immediate consequence of~\cite[Theorems~8.1.14]{GT2006} and Lemma~\ref{10-5}.
\end{proof}
\let\qed\relax
\end{proof}


\subsection{(Co)tilting property for the classes of classical homological modules}

In this subsection, as
the third application of Theorems~\ref{8},~\ref{9} and~\ref{10}, we will consider when the classes $\mathcal{P}(R)$ and $\mathcal{F}(R)$
are cotilting and when the class $\mathcal{I}(R)$ is tilting as follows.


\begin{prop} \label{11-1} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop50_1} The class $\mathcal{F}(R)$ is cotilting.
 \item\label{prop50_2} The class $\mathcal{GF}(R)$ is cotilting and $\mathcal{F}(R)=\mathcal{GF}(R)$.
 \item\label{prop50_3} $R$ is a right coherent ring with $\wgldim(R)<\infty$.
 \item\label{prop50_4} $R$ is a right coherent ring such that $\fd_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{prop}


\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{prop50_2}\Rightarrow\ref{prop50_1}$] It is trivial.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop50_1}\Rightarrow\ref{prop50_3}$] Assume that the class $\mathcal{F}(R)$ is cotilting. Then by Lemma~\ref{2}, $\mathcal{F}(R)$ is definable and
$\wgldim(R)=\sup\{\fd_R(M)\mid M~\text{is an}~R\text{-module}\}<\infty$. In particular, $\mathcal{F}(R)$ is closed under arbitrary direct products, and hence $R$ is also right
coherent.
\let\qed\relax
\end{proof}



\begin{proof}[\mathversion{bold}$\ref{prop50_3}\Rightarrow\ref{prop50_2}$] Suppose that $R$ is a right coherent ring with $\wgldim(R)<\infty$. Then of course $R$ is a right
coherent ring with $\Gwgldim(R)<\infty$. It follows from Theorem~\ref{8} that the class $\mathcal{GF}(R)$ is cotilting.
Meanwhile, $\wgldim(R)<\infty$ implies that $\mathcal{F}(R)=\mathcal{GF}(R)$ (see the proof of~\cite[Proposition~2.2(2)]{Ben2009}).
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop50_3}\Leftrightarrow\ref{prop50_4}$] It holds by Lemma~\ref{06-7-2}, using the complete hereditary cotorsion pair $(\mathcal{F}\Mk (R),\mathcal{F}\Mk (R)^{\perp}\mk)$.
\end{proof}
\let\qed\relax
\end{proof}

\begin{prop} \label{11-2} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop51_1} The class $\mathcal{P}(R)$ is cotilting.
 \item\label{prop51_2} The class $\mathcal{GP}(R)$ is cotilting and $\mathcal{P}(R)=\mathcal{GP}(R)$.
 \item\label{prop51_3} $R$ is a right coherent and left perfect ring with $\gldim(R)<\infty$.
 \item\label{prop51_4} $R$ is a right coherent and left perfect ring such that $\pd_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{prop}

\goodbreak
\begin{proof}\   


\begin{proof}[\mathversion{bold}$\ref{prop51_2}\Rightarrow\ref{prop51_1}$] It is obvious.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop51_1}\Rightarrow\ref{prop51_3}$] Assume that the class $\mathcal{P}(R)$ is cotilting. Then by Lemma~\ref{2}, $\mathcal{P}(R)$ is definable and
$\gldim(R)=\sup\{\pd_R(M)\mid M~\text{is an}~R\text{-module}\}<\infty$. In particular, $\mathcal{P}(R)$ is closed under arbitrary direct products, and hence $R$ is also right
coherent and left perfect.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop51_3}\Rightarrow\ref{prop51_2}$] Suppose that $R$ is a right coherent and left perfect ring with $\gldim(R)<\infty$. Then of course $R$ is a right
coherent and left perfect ring with $\Ggldim(R)<\infty$. It follows from Theorem~\ref{9} that the class $\mathcal{GP}(R)$ is cotilting.
Meanwhile, $\gldim(R)<\infty$ implies that $\mathcal{GP}(R)=\mathcal{P}(R)$ (see the proof of~\cite[Proposition~2.27)]{Hol2004}).
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop51_3}\Leftrightarrow\ref{prop51_4}$] It holds by Lemma~\ref{06-7-2}, using the trivial cotorsion pair $(\mathcal{P}(R),R\text{-Mod})$.
\end{proof}
\let\qed\relax
\end{proof}

\begin{prop} \label{11-3} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop52_1} The class $\mathcal{I}(R)$ is tilting.
 \item\label{prop52_2} The class $\mathcal{GI}(R)$ is cotilting and $\mathcal{I}(R)=\mathcal{GI}(R)$.
 \item\label{prop52_3} $R$ is left Noetherian ring with $\gldim(R)<\infty$.
 \item\label{prop52_4} $R$ is a left Noetherian ring such that $\fd_R(M)<\infty$ for any $R$-module $M$.
\end{enumerate}
\end{prop}
\begin{proof} By dual of Proposition~\ref{11-2}, we leave it to the readers.\end{proof}

\subsection{(Co)silting property for classes of classical and Gorensrein homological modules}

As the last application of Theorems~\ref{8},~\ref{9} and~\ref{10}, we will consider when the classes $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, $\mathcal{P}(R)$ and $\mathcal{F}(R)$
are cosilting and when the classes $\mathcal{GI}(R)$ and $\mathcal{I}(R)$ are silting, which induces some characterizations of Dedekind and Pr\"{u}fer domains.

\begin{defi} \label{11-3-0}\ 

\begin{enumerate}[label=(\arabic*)]
 \item\label{defi53_1} For a morphism $\sigma$ between projective $R$-modules, we denote by $\mathcal{D}_{\sigma}$ the class of $R$-modules
\[
\mathcal{D}_{\sigma}=\{M \in R\text{-}\Mod\mid \Hom_R(\sigma, M)~\mathrm{is~surjective}\}.
\]

\item\label{defi53_2} An $R$-module T is \emph{silting} if it admits a projective presentation $P_1
\stackrel{\sigma}\to P_0 \to T \to 0$ such that
$\Gen T = \mathcal{D}_{\sigma}$. The class $\Gen T$ is then called a \emph{silting class} of $R$-modules.

\item\label{defi53_3} For a morphism $\tau$ between injective $R$-modules, we denote by $\mathcal{C}_\tau$ the class of $R$-modules
\[
\mathcal{C}_\tau= \{M \in R\text{-}\Mod\mid \Hom_R(M, \tau)~\mathrm{is~surjective}\}.
\]

\item\label{defi53_4} An $R$-module C is cosilting if it admits an injective copresentation $0 \to T \to E_0
\stackrel{\tau}\to E_1$ such that
$\Cogen T=\mathcal{C}_\tau $. The class $\Cogen T$ is then called a \emph{cosilting class} of $R$-modules.
\end{enumerate}
\end{defi}

Recall that a class $\mathcal{X}$ of $R$-modules is \emph{torsion} (resp.\ \emph{torsionfree}) if $\mathcal{X}$ is closed under epimorphic
images, extensions and coproducts (resp.\ under submodules, extensions and products).
We know from~\cite[Corollary~3.5 and Proposition~3.10]{AMV2016} (see also~\cite[Remarks in p.~4135]{AH2017}) that any silting class of $R$-modules is definable and torsion;
from~\cite[Corollary~3.9]{A2018} that a class $\mathcal{X}$ of $R$-modules is cosilting if and only if $\mathcal{X}$ is definable and torsionfree.

\looseness-1
It was shown in~\cite[Lemma~6.1.2]{GT2006} (resp.\ \cite[Lemma~8.2.2]{GT2006}) that
a class $\mathcal{X}$ of modules is 1-tilting (resp.\ 1-cotilting) if and only if there is a 1-tilting (resp.\ 1-cotilting) module $T$
such that $\mathcal{X}= \Gen T$ (resp.\ $\mathcal{X}= \Cogen T$). By~\cite[Example~2.4(1) and~(3)]{A2018} (resp.\ \cite[Example~3.3(a) and~(c)]{BP2017}) the inclusions $\{1\text{-tilting modules}\} \subseteq \{\text{silting modules}\}$ and $\{1\text{-cotilting modules}\} \subseteq \{\text{cosilting modules}\}$ are strict.
It is then a routine to check that
the inclusions $\{1\text{-tilting classes of modules}\} \subseteq \{\text{silting classes of modules}\}$ and $\{1\text{-cotilting}$ $\text{classes of modules}\} \subseteq \{\text{cosilting classes of modules}\}$ are strict as well.


However, we will show that the silting (resp.\ cosilting) and 1-tilting (resp.\ 1-cotilting) property of the class $\mathcal{GI}(R)$ (resp.\ the classes $\mathcal{GP}(R)$ and $\mathcal{GF}(R)$) coincide.


\begin{prop} \label{11-4} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop54_1} The class $\mathcal{GF}(R)$ is cosilting.
 \item\label{prop54_2} The class $\mathcal{GF}(R)$ is 1-cotilting.
 \item\label{prop54_3} $R$ is a right coherent ring with $\Gwgldim(R)\leq1$.
\end{enumerate}
\end{prop}

\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{prop54_3}\Rightarrow\ref{prop54_2}$] It follows from Remark~\ref{10-0}\ref{rema36_1}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop54_2}\Rightarrow\ref{prop54_1}$] It is obvious.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop54_1}\Rightarrow\ref{prop54_3}$] Assume that the class $\mathcal{GF}(R)$ is cosilting. Then by~\cite[Corollary~3.9]{A2018}, the class $\mathcal{GF}(R)$ is definable and torsionfree. Thus, one has 
\[
\Gwgldim(R)=\sup\{\Gfd_R(M)\mid M~\text{is an}~R\text{-module}\}\leq1
\]
 since $\mathcal{GF}(R)$ is closed under submodules. Furthermore, $R$ is right coherent by Lemma~\ref{4}.
\end{proof}
\let\qed\relax
\end{proof}

\begin{prop} \label{11-5} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop55_1} The class $\mathcal{GP}(R)$ is cosilting.
 \item\label{prop55_2} The class $\mathcal{GF}(R)$ is cosilting and $\mathcal{GP}(R)=\mathcal{GF}(R)$.
 \item\label{prop55_3} The class $\mathcal{GP}(R)$ is 1-cotilting.
 \item\label{prop55_4} The class $\mathcal{GF}(R)$ is 1-cotilting and $\mathcal{GP}(R)=\mathcal{GF}(R)$.
 \item\label{prop55_5} $R$ is a right coherent and left perfect ring with $\Ggldim(R)\leq1$.
\end{enumerate}
\end{prop}

\begin{proof}\  


\begin{proof}[\mathversion{bold}$\ref{prop55_3} \Leftrightarrow\ref{prop55_4}\Leftrightarrow\ref{prop55_5}$] It follows from Remark~\ref{10-0}\ref{rema36_2}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop55_4}\Rightarrow\ref{prop55_2}\Rightarrow\ref{prop55_1}$] It is clear.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop55_1}\Rightarrow\ref{prop55_5}$] Assume that the class $\mathcal{GP}(R)$ is cosilting.
Then by~\cite[Corollary~3.9]{A2018}, the class $\mathcal{GP}(R)$ is definable and torsionfree.
So one gets that
\[
\Ggldim(R)=\sup\{\Gpd_R(M)\mid M~\text{is an}~R\text{-module}\}\leq1
\]
since $\mathcal{GP}(R)$ is closed under submodules.
It follows from Lemma~\ref{07}\ref{lemma16_1} that $\Gwgldim(R)\leq 1$.
Thus, $R$ is
right coherent and left perfect due to Lemma~\ref{5}.
\end{proof}
\let\qed\relax
\end{proof}



\begin{prop} \label{11-6} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop56_1} The class $\mathcal{GI}(R)$ is silting.
 \item\label{prop56_2} The class $\mathcal{GF}(R^{\op})$ is cosilting and an $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if so is $M^{++}$.
 \item\label{prop56_3} The class $\mathcal{GI}(R)$ is 1-tilting.
 \item\label{prop56_4} The class $\mathcal{GF}(R^{\op})$ is 1-cotilting and an $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if so is $M^{++}$.
 \item\label{prop56_5} $R$ is a left Noetherian ring with $\Ggldim(R)\leq1$.
\end{enumerate}
\end{prop}

\begin{proof}\  


\begin{proof}[\mathversion{bold}$\ref{prop56_3}\Leftrightarrow\ref{prop56_5}\Leftrightarrow\ref{prop56_5}$] It follows from Remark~\ref{10-0}\ref{rema36_3}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop56_4}\Rightarrow\ref{prop56_2}$ and $\ref{prop56_3}\Rightarrow\ref{prop56_1}$] These implications are trivial.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop56_1}\Rightarrow\ref{prop56_5}$] Assume that the class $\mathcal{GI}(R)$ is silting. Then by~\cite[Corollary~3.9]{A2018}, the class $\mathcal{GI}(R)$ is definable and torsion.
Now, one obtains that
\[
\Ggldim(R)=\sup\{\Gid_R(M)\mid M~\text{is~an}~R\text{-module}\}\leq1
\]
as the class $\mathcal{GI}(R)$ is closed under epimorphic images.
Again, $\Gwgldim(R)\leq 1$ via Lemma~\ref{07}\ref{lemma16_1}.
Now $R$ is also left Noetherian by Proposition~\ref{6}. 
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{prop56_2}\Rightarrow\ref{prop56_5}$] Suppose that the class $\mathcal{GF}(R^{\op})$ is cosilting and an $R$-module $M$ is in $\mathcal{GI}(R)$ if and only if so is $M^{++}$.
According to Proposition~\ref{11-4}, the first statement of the assumption yields that $R$ is a left coherent ring with $\Gwgldim(R)=\Gwgldim(R^{\op})\leq1$.
Now the second statement of the assumption yields that $R$ is left Noetherian by Proposition~\ref{6}.
In addition, we conclude that $\Ggldim(R)=\Gwgldim(R)\leq1$ by~\cite[Theorem~7]{Bou2015}.
\end{proof}
\let\qed\relax
\end{proof}

Using Propostions~\ref{11-1},~\ref{11-2} and~\ref{11-3}, and applying the arguments used in the proof of Propositions
\ref{11-4},~\ref{11-5} and~\ref{11-6}, respectively, one can obtain


\begin{prop} \label{11-7} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop57_1} The class $\mathcal{F}(R)$ is cosilting.
 \item\label{prop57_2} The class $\mathcal{F}(R)$ is 1-cotilting.
 \item\label{prop57_3} $R$ is a right coherent ring with $\wgldim(R)\leq1$.
\end{enumerate}
\end{prop}

\begin{prop} \label{11-8} Let $R$ be a ring. Then the following are equivalent:
\begin{enumerate}[label=(\arabic*)]
 \item\label{prop58_1} The class $\mathcal{P}(R)$ is cosilting.
 \item\label{prop58_2} The class $\mathcal{P}(R)$ is 1-cotilting.
 \item\label{prop58_3} $R$ is a right coherent and left perfect ring with $\gldim(R)\leq1$.
\end{enumerate}
\end{prop}

\begin{prop} \label{11-9} Let $R$ be a ring. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{prop59_1} The class $\mathcal{I}(R)$ is silting.
 \item\label{prop59_2} The class $\mathcal{I}(R)$ is 1-tilting.
 \item\label{prop59_3} $R$ is a left Noetherian ring with $\gldim(R)\leq1$.
\end{enumerate}
\end{prop}

Recall that a ring $R$ is \emph{Dedekind} (resp.\ Pr\"{u}fer) if $R$ is a hereditary (resp.\ semi-hereditary) domain.
Here a (possibly not communicative) \emph{hereditary} (resp.\ \emph{semi-hereditary}) ring is defined as the one such that every
left and right ideal (resp.\ finitely generated left and right ideal) is projective.
It is well-known that any hereditary (resp.\ semi-hereditary) ring $R$ has $\gldim(R)\leq1$ (resp.\ $\wgldim(R)\leq1$).
\goodbreak
According to~\cite[Theorems~6.2.15, 6.2.19, 6.2.22, 8.2.9 and~8.2.12]{GT2006}, we know that both the (co)tilting modules and classes
over a Dedekind (resp.\ Pr\"{u}fer) domain have a nice description.
We will end the paper by characterizing Dedekind (resp.\ Pr\"{u}fer) domain using some special (co)tilting classes.


\begin{theo}\label{11-11} Let $R$ be a domain. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{theo60_1} $R$ is Pr\"{u}fer.
 \item\label{theo60_2} The class $\mathcal{F}(R)$ is 1-cotilting.
 \item\label{theo60_3} The class $\mathcal{F}(R)$ is cosilting.
 \item\label{theo60_4} The class $\mathcal{GF}(R)$ is 1-cotilting and $\mathcal{GF}(R)=\mathcal{F}(R)$.
 \item\label{theo60_5} The class $\mathcal{GF}(R)$ is cosilting and $\mathcal{GF}(R)=\mathcal{F}(R)$.
\end{enumerate}
\end{theo}

\begin{proof}\ 

\begin{proof}[\mathversion{bold}$\ref{theo60_4} \Rightarrow\ref{theo60_2}$ and $\ref{theo60_5}\Rightarrow\ref{theo60_3}$] These implications  are trivial.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo60_2}\Rightarrow\ref{theo60_3}$] It holds by Proposition~\ref{11-7}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo60_3}\Rightarrow\ref{theo60_1}$] Suppose that $R$ is a domain such that the class $\mathcal{F}(R)$ is cosilting.
Then by Proposition~\ref{11-7}, $R$ is a coherent domain with $\wgldim(R)\leq1$ , and so $R$ is a semi-hereditary domain by~\cite[Theorem~4.67]{Lam2012}.
Thus, $R$ is Pr\"{u}fer.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo60_1}\Rightarrow\ref{theo60_4}$ and $\ref{theo60_1}\Rightarrow\ref{theo60_5}$] Assume that $R$ is a Pr\"{u}fer domain.
Then $\wgldim(R)\leq1$, which implies that $\mathcal{GF}(R)=\mathcal{F}(R)$.
In addition, $R$ is coherent by~\cite[Theorem~4.67]{Lam2012}.
So the class $\mathcal{GF}(R)$ is 1-cotilting (resp.\ cosilting) due to Proposition~\ref{11-7}.
\end{proof}
\let\qed\relax
\end{proof}

\begin{theo}\label{11-12} Let $R$ be a domain. Then the following are equivalent:

\begin{enumerate}[label=(\arabic*)]
 \item\label{theo61_1} $R$ is Dedekind.
 \item\label{theo61_2} The class $\mathcal{I}(R)$ is 1-tilting.
 \item\label{theo61_3} The class $\mathcal{I}(R)$ is silting.
 \item\label{theo61_4} The class $\mathcal{GI}(R)$ is 1-tilting and $\mathcal{GI}(R)=\mathcal{I}(R)$.
 \item\label{theo61_5} The class $\mathcal{GI}(R)$ is silting and $\mathcal{GI}(R)=\mathcal{I}(R)$.
\end{enumerate}
\end{theo}

\begin{proof}\ 


\begin{proof}[\mathversion{bold}$\ref{theo61_4}\Rightarrow\ref{theo61_2}$ and $\ref{theo61_5}\Rightarrow\ref{theo61_3}$] These implications  are trivial.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo61_2}\Rightarrow\ref{theo61_3}$] It follows by Proposition~\ref{11-9}.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo61_3}\Rightarrow\ref{theo61_1}$] Suppose that $R$ is a domain such that the class $\mathcal{I}(R)$ is silting.
Then by Proposition~\ref{11-9}, $R$ is a domain with $\gldim(R)\leq1$, and so $R$ is a hereditary domain by~\cite[Theorem~4.23]{Rot1979}.
That is, $R$ is Dedekind.
\let\qed\relax
\end{proof}

\begin{proof}[\mathversion{bold}$\ref{theo61_1}\Rightarrow\ref{theo61_4}$ and $\ref{theo61_1}\Rightarrow\ref{theo61_5}$] Suppose that $R$ is a Dedekind domain.
Then $\gldim(R)\leq1$, which implies that $\mathcal{GI}(R)=\mathcal{I}(R)$. Furthermore, $R$ is Noetherian by~\cite[Corollary~4.26]{Rot1979}.
Consequently, the class $\mathcal{GI}(R)$ is 1-tilting (resp.\ silting) via Proposition~\ref{11-9}.
\end{proof}
\let\qed\relax
\end{proof}


\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no affiliations other than their research organizations.



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