Some applications of recollements to Gorenstein projective modules
We apply the technique of recollement to study the Gorenstein projective modules. First, we construct a recollement of defect Gorenstein categories for the upper triangular matrix algebras. Then we use the defect Gorenstein category to give a categorical interpretation of the Groenstein properties of the upper triangular matrix algebra obtained by X-W Chen, B. Xiong and P. Zhang respectively. Finally, as a generalization of the structure of the cluster-tilted algebra of type $A$, we define the gluing Nakayama algebra, and use recollement to describe its singularity category clearly.
1412.0910.tex
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\title[Some applications of recollements to Gorenstein projective modules]{Some applications of recollements to Gorenstein projective modules}



\author[Lu]{Ming Lu}
\address{Department of Mathematics, Sichuan University, Chengdu 610064, P.R.China}
\email{luming@scu.edu.cn}


\subjclass[2000]{18E30, 18E35}
\keywords{Gorenstein defect category, Gorenstein algebra, Gorenstein projective module, Recollement, Gluing Nakayama algebra}

\begin{abstract}
We apply the technique of recollement to study the Gorenstein projective modules. First, we construct a recollement of defect Gorenstein
categories for the upper triangular matrix algebras. Then we use the defect Gorenstein category to give a categorical interpretation of
the Groenstein properties of the upper triangular matrix algebra obtained by X-W Chen, B. Xiong and P. Zhang respectively.
Finally, as a generalization of the structure of the cluster-tilted algebra of type $A$, we define the gluing Nakayama algebra,
and use recollement to describe its singularity category clearly.
\end{abstract}

\maketitle

\section{Introduction}

In the study of B-branes on Landau-Ginzburg models in the framework of Homological Mirror Symmetry Conjecture, D. Orlov rediscovered the notion of singularity categories \cite{Or1,Or2,Or3}. The singularity category of an algebra $A$ is defined to be the Verdier quotient of the bounded derived category with respect to the thick subcategory formed by complexes isomorphic to those consisting of finitely generated projective modules. It measures the homological singularity of an algebra in the sense that an algebra $A$ has finite global dimension if and only if its singularity category vanishes.

In the meantime, the singularity category captures the stable homological features of an algebra \cite{Bu}. A fundamental result of R. Buchweitz \cite{Bu} and D. Happel \cite{Ha1} states that for a Gorenstein algebra $A$, the singularity category is triangle equivalent to the stable category of Gorenstein projective(also called Cohen-Macaulay) $A$-modules. Buchweitz's Theorem says that there is an exact embedding $\Phi:\underline{\Gp}A\rightarrow D_{sg}(A)$ given by $\Phi(M)=M$, where the second $M$ is the corresponding stalk complex at degree $0$, and $\Phi$ is an equivalence if and only if $A$ is Gorenstein. In general, to provide a categorical characterization of Gorenstein algebras, P. A. Bergh, D. A. J{\o}rgensen and S. Oppermann defined the Gorenstein defect category $D_{def}(A):=D_{sg}(A)/\Im \Phi$ and proved that $A$ is Gorenstein if and only if $D_{def}(A)=0$ \cite{BJO}.



Let $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$ with bimodule $_AM_B$. X-W. Chen proved that if $A,B$ are Gorenstein, then the upper triangular matrix algebra $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$ is Gorenstein if and only if $_AM$ and $M_B$ are finitely generated,
$\pd _A M<\infty$ and $\pd M_B<\infty$ \cite[Theorem 3.3]{Chen1}. Furthermore, the upper triangular matrix rings play very well to recollement.
A recollement of triangulated categories is a diagram
\[\xymatrix{
\mathcal{D}'\ar[r]^{i_*}&\mathcal{D}\ar@<2ex>[l]^{i^!}\ar@<-3ex>[l]_{i^*}\ar[r]^{j^*}&\mathcal{D}''
\ar@<2ex>[l]^{j_*}\ar@<-3ex>[l]_{j_!}
}
\]of triangulated categories and functors satisfying various conditions, which describes the middle term as being ``glued together" from a triangulated subcategory and another one. The recollement setup was first introduced by A. Beilinson, J. Bernstein and P. Deligne in \cite{BBD}, which plays an important role in  algebraic geometry and representation theory, see for instance \cite{MV, CPS1, CPS2, J, Ko, Zhang2}.


We know that $D^b(\Lambda)$ admits a recollement relative to $D^b(A)$ and $D^b(B)$ if $\pd M_B<\infty$ \cite{CPS2}.
P. Zhang proved that if $A$, $B$ are Gorenstein algebras and $_AM$, $M_B$ are projective, then $\underline{\Gp}\Lambda$ admits a recollement relative to $\underline{\Gp}A$ and $\underline{\Gp}B$. The author together with P. Liu generalized this, proved that $D_{sg}(\Lambda)$ admits a recollement relative to $D_{sg}(A)$ and $D_{sg}(B)$ if $\pd_A M<\infty$ and $\pd M_B<\infty$ for any Artin rings $A$ and $B$ \cite{LL}.
In \cite{LL}, the machinery of localization to the three categories in a recollement is introduced, and we also find conditions for the quotient categories to form a new recollement. We refer the reader to \cite{Kra} for localization theory of triangulated categories.






The aim of the present article is threefold. First is to construct a recollement of Gorenstein defect categories for upper triangular matrix algebra $\Lambda$.
\begin{theorem}
Let $A$ and $B$ be two finite-dimensional algebras, $_AM_B$ an $A\mbox{-}B$-bimodule such that $_AM$ and $M_B$ are finitely generated projective modules, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. If $\pd_B\Hom_A(M,A)<\infty$, then $D_{def}(\Lambda)$ admits a recollement relative to $D_{def}(A)$ and $D_{def}(B)$.
\end{theorem}
Second is to give a categorical interpretation of the Gorenstein property of upper triangular matrix algebra $\Lambda$.
\begin{theorem}
Let $A$ and $B$ be two finite-dimensional algebras, $_AM_B$ an $A\mbox{-}B$-bimodule, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. We also assume that $\pd _A M<\infty$ and $\pd M_B<\infty$. Then

(a) If $B$ is a Gorenstein algebra, then $D_{def}(\Lambda)\simeq D_{def}(A)$.

(b) If $A$ is a Gorenstein algebra and $\ind M_B<\infty$, then $D_{def}(\Lambda)\simeq D_{def}(B)$.
\end{theorem}

It is difficult to describe the exact subcategory of Gorenstein projective modules. In recent years, C. M. Ringel describe the Gorenstein projective modules over Nakayama algebra clearly \cite{Rin}.
Finally, as a generalization of the structure of the cluster-tilted algebra of type $A$,
we give the definition of gluing Nakayama algebra, see Definition \ref{definition gluing Nakayama algebra}.
We study their Gorenstein property, and get the following statement.
\begin{theorem}
For any gluing Nakayama algebra $\Gamma$, if the gluing components $A_1=kQ_1/I_1$, $A_2=kQ_2/I_2$, $\cdots$, $A_m=kQ_m/I_m$ of $\Gamma$ are Gorenstein, then
$$\underline{\Gproj}(\Gamma)\simeq \coprod_{i=1}^m\underline{\Gproj}(A_i).$$
\end{theorem}
As a corollary, we get the following. Let $\Lambda=kQ/I$ be a cluster-tilted algebra of type $A$. Then
$$\underline{\Gproj}(\Lambda)\simeq \coprod_{t(Q)} \underline{\mod} S_3,$$
where $t(Q)$ is the number of the triangles of $Q$, and $S_3$ is the selfinjective cluster-tilted algebra of type $A_3$. This result is gotten by M. Kalck in \cite{Ka}, see also \cite{CGLu}.


\vspace{0.2cm} \noindent{\bf Acknowledgments.}
The author thanks Professor Liangang Peng very much for his guidance and constant support.

\section{preliminary}
In this paper, we always assume that $k$ is a field and all algebras are finite-dimensional algebra over $k$ and modules are finitely generated, and all the categories we consider are $k$-linear with finite-dimensional Hom spaces.

Let us first recall the definition of recollement for triangulated categories from \cite{BBD}.

\begin{definition}\cite{BBD}
A recollement of triangulated categories is a diagram of triangulated
categories and triangulated functors
\[
\xymatrix{
\mathcal{D}'\ar[r]^{i_*=i_!}&\mathcal{D}\ar@<2ex>[l]^{i^!}\ar@<-3ex>[l]_{i^*}\ar[r]^{j^*=j^!}&\mathcal{D}''
\ar@<2ex>[l]^{j_*}\ar@<-3ex>[l]_{j_!} \quad(\dag)
}
\]
satisfying
\item[(R1)] $(i^*,i_*)$, $(i_*,i^!)$, $(j_!,j^*)$ and $(j^*,j_*)$ are adjoint pairs;
\item[(R2)] $i_*$, $j_!$ and $j_*$ are full embeddings;
\item[(R3)] $j^*i_*=0$ (and hence, by adjoint properties, $i^*j_!=0$ and $i^!j_*=0$);
\item[(R4)] Each object $X$ in $\cd$ determines distinguished triangles
\[i_!i^!X\to X\to j_*j^*X\to \cdot\ \text{and}\ j_!j^!X\to X\to i_*i^*X\to\cdot
\]in $\cd$, where the arrows to and from $X$ are counit and unit morphisms.
\end{definition}


Parallel to the recollement of triangulated categories, Franjou and Pirashvili defined recollement of abelian categories: Let $\cd,\cd',\cd''$ be abelian categories. The diagram $(\dag)$ of additive functors is an abelian category recollement of $\cd$ relative to $\cd'$ and $\cd''$, if $(R1),(R2)$ and $(R5)$ are satisfied, where
\begin{itemize}
\item[(R5)] $\Im i_*=\ker j^*$.
\end{itemize}

The following theorem is useful to construct recollements of quotient categories.
\begin{theorem}\cite{LL}\label{theorem 1}
Let \[\xymatrix{
\mathcal{D}'\ar[r]^{i_*}&\mathcal{D}\ar@<2ex>[l]^{i^!}\ar@<-3ex>[l]_{i^*}\ar[r]^{j^*}&\mathcal{D}''
\ar@<2ex>[l]^{j_*}\ar@<-3ex>[l]_{j_!}
}
\]be a recollement of triangulated categories and $\ct$ be an \'{e}paisse subcategory of $\cd$. Let $\mathcal{T}_1=i^*\mathcal{T}$ and $\mathcal{T}_2=j^*\mathcal{T}$.  If $i_*i^*\mathcal{T}\subseteq \mathcal{T}$ and $j_*j^*\mathcal{T}\subseteq \mathcal{T}$, then there exists a recollement of localizations
\[\xymatrix{\mathcal{D}'/\mathcal{T}_1\ar[r]^{\widetilde{i_*}}&\mathcal{D}/\mathcal{T}
\ar@<2ex>[l]^{\widetilde{i^!}}\ar@<-3ex>[l]_{\widetilde{i^*}}\ar[r]^{\widetilde{j^*}}&\mathcal{D}''/\mathcal{T}_2.
\ar@<2ex>[l]^{\widetilde{j_*}}\ar@<-3ex>[l]_{\widetilde{j_!}}  }
\]
\end{theorem}


\subsection{Gorenstein projective modules and Gorenstein algebras}
Let $A$ be an finite-dimensional $k$-algebra. Let $\mod A$ be the category of finitely generated left $A$-modules. With $D=\Hom_k(-,k)$ we denote the standard duality with respect to the ground field. Then $_AD(A_A)$ is an injective cogenerator for $\mod A$. For an arbitrary $A$-module $_AX$ we denote by $\pd_AX$ (resp. $\ind_AX$) the projective dimension (resp. the injective dimension) of the module $_AX$.

A complex $$P^\bullet:\cdots\rightarrow P^{-1}\rightarrow P^0\xrightarrow{d^0}P^1\rightarrow \cdots$$ of finitely generated projective $A$-modules is said to be \emph{totally acyclic} provided it is acyclic and the Hom complex $\Hom_A(P^\bullet,A)$ is also acyclic \cite{AM}.
An $A$-module $M$ is said to be (finitely generated) \emph{Gorenstein projective} provided that there is a totally acyclic complex $P^\bullet$ of projective $A$-modules such that $M\cong \Ker d^0$ \cite{EJ}. Dually, one can define \emph{Gorenstein injective} modules. We denote by $\Gproj A$ ($\Ginj A$) the full subcategory of $\mod A$ consisting of Gorenstein projective modules (resp. Gorenstein injective modules).


Let $\cx$ be a subcategory of $\mod A$. Then $^\bot\cx:=\{M|\Ext^i(M,X)=0, \mbox{ for all } X\in\cx, i\geq1\}$. Dually, we can define $\cx^\bot$. In particular, we define $^\bot A:=^\bot
(\mathrm{proj.} A)$.

\begin{lemma}
(a) \cite{Be}
\begin{eqnarray*}
\Gp(A)&=&\{M\in \Lambda\mbox{-}\mathrm{mod}\,|\,\exists \mbox{ an exact sequence }
0\rightarrow M\rightarrow T^0\xrightarrow{d^0}T^1\xrightarrow{d^1}\cdots, \\ &&\mbox{ with }T^i\in\proj A,\ker d^i\in\,^\bot\proj A,\forall i\geq0\}.
\end{eqnarray*}

(b) \cite{Zhang2} If $M$ is Gorenstein projective, then $\Ext^i_A(M,L)=0$, $\forall i>0$, for all $L$ of finite projective dimension.

(c) \cite{Zhang2} If $P^\bullet$ is a totally acyclic complex, then all $\Im d^i$ are Gorenstein projective; and any
truncations
$$\cdots\rightarrow P^i\rightarrow\Im d^i\rightarrow0,\quad 0\rightarrow\Im d^i\rightarrow P^{i+1}\rightarrow\cdots$$
and
$$0\rightarrow\Im d^i\rightarrow P^{i+1}\rightarrow\cdots\rightarrow P^j\rightarrow \Im d^j\rightarrow0,i<j$$
are $\Hom_A(-,\proj A)$-exact.
\end{lemma}

\begin{definition}\cite{AR1,AR2,Ha1}
A finite-dimensional algebra $A$ is called a Gorenstein algebra if $A$ satisfies $\pd_AD(A_A)<\infty$ and $\ind_AA<\infty$.
Given an $A$-module $X$. If $\Ext^i_A(X,A)=0$ for all $i>0$, then $X$ is called a Cohen-Macaulay module of $A$.
\end{definition}

Observe that for a Gorenstein algebra $A$, we have $\ind _AA=\ind A_A$, \cite[Lemma 6.9]{Ha1}; the common value is denoted by $\Gd A$. If $\Gd A\leq d$, we say that $A$ is \emph{$d$-Gorenstein}.





\begin{theorem}\cite{EJ}
Let $A$ be a Gorenstein algebra. Then

(1) If $P^\bullet$ is an exact sequence of projective left $A$-modules, then $\Hom_A(P^\bullet,A)$ is again an exact sequence of projective right $A$-modules.

(2) A module $G$ is Gorenstein projective if and only if there is an exact sequence $0\rightarrow G\rightarrow P^0\rightarrow P^1\rightarrow \cdots$ with each
$P^i$ projective.

(3) $\Gproj A=^\bot A$.
\end{theorem}
For a module $M$, take a short exact sequence $$0\rightarrow \Omega M\rightarrow P\rightarrow M\rightarrow0$$
with $P$ projective. The module $\Omega M$ is called a \emph{syzygy module} of $M$.
\begin{theorem}\cite{AM}
Let $A$ be an finite-dimensional algebra and let $d\geq0$. Then the following statements are equivalent:

(1) the algebra $A$ is $d$-Gorenstein;

(2) $\Gproj A=\Omega^d(\mod A)$;

(2') $\Ginj A=\Omega^{-d}(\mod A)$.

In this case, we have $\Gproj A= ^\bot A$.
\end{theorem}
So for a Gorenstein algebra, the definition of Cohen-Macaulay module coincides with the one of Gorenstein projective.

For an algebra $A$, we define the \emph{singularity category} of $A$ to be the quotient category $D_{sg}^b(A):=D^b(A)/K^b(\proj A)$ \cite{Ha1,Or1}.
\begin{theorem}\cite{Ha1}\label{theorem stable category of CM modules }
Let $A$ be a Gorenstein algebra. Then $\Gproj (A)$ is a Frobenius category with the projective modules as the projective-injective objects. $\underline{\Gproj}(A)$ is triangulated equivalent to the singularity category $D_{sg}(A)$ of $A$.
\end{theorem}





\section{Gorenstein defect category}
Let $A$ be an finite-dimensional algebra over $k$. Buchweitz's Theorem (\cite[Theorem 4.4.1]{Bu}) says that there is an exact embedding $\Phi:\underline{\Gp}(A)\hookrightarrow D_{sg}(A)$ given by $\Phi(M)=M$. To measure the difference of $\underline{\Gp}(A)$ and $D_{sg}(A)$, Bergh, J{\o}rgensen and Oppermann define the \emph{Gorenstein defect category} $D_{def}(A):=D_{sg}(A)/\Im(\Phi)$ \cite{BJO}.

Let $A$ and $B$ be finite-dimensional algebras, $_AM_B$ an $A\mbox{-}B$-bimodule, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. A left $\Lambda$-module is identified with a triple $\left( \begin{array}{cc} X\\Y \end{array}\right)_\phi$, where $X\in \mod A$, $Y\in \mod B$, and $\phi:M\otimes_BY\rightarrow X$ is a morphism of $A$-modules. A morphism $\left(\begin{array}{cc} X\\ Y \end{array}\right)_\phi\rightarrow \left(\begin{array}{cc} X' \\ Y' \end{array}\right)_{\phi'}$
is a pair $\left(\begin{array}{cc} f\\ g \end{array}\right)$, where $f\in \Hom_A(X,X')$, $g\in\Hom_B(Y,Y')$, such that $\phi'(\id_M\otimes g)=f\phi$.
Note that $\left(\begin{array}{cc} P \\ 0 \end{array}\right)$ and $\left(\begin{array}{cc} G(Q) \\ Q \end{array}\right)_{\id}$ are precisely the indecomposable projective modules, where $P$ and $Q$ are indecomposable projective as $A$ and $B$ modules respectively.

Dually, we have the description of right $\Lambda$-module via row vectors. Precisely, a right $\Lambda$-module is identified with $(X,Y)_{\psi}$, where $X_A$ is a right $A$-module and $Y_B$ is a right $B$-module, and $\psi:X\otimes_A M\rightarrow Y$ is a right $B$-module morphism.



We have the following recollement of abelian categories.

\begin{theorem}\cite{Zhang2}\label{recollement of abelian categories}
Let $A$ and $B$ be finite-dimensional algebras, $_AM_B$ an $A\mbox{-}B$-bimodule, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. Denote by $G=M\otimes_B-$. We have the following recollement of abelian categories:
\[\xymatrix{
\mod A \ar[r]^{i_*}& \mod \Lambda
\ar@<2ex>[l]^{i^!}\ar@<-3ex>[l]_{i^*}\ar[r]^{j^*}& \mod B
\ar@<2ex>[l]^{j_*}\ar@<-3ex>[l]_{j_!} , }\]
where $i^*$ is given by $\left(\begin{array}{cc} X  \\ Y   \end{array} \right)_{\phi }\mapsto \Coker(\phi )$;
$i_*$ is given by $X \mapsto \left(\begin{array}{cc} X  \\ 0  \end{array} \right)$; $i^!$ is given by $\left(\begin{array}{cc} X  \\ Y   \end{array} \right)_{\phi }\mapsto X $;
$j_!$ is given by $Y  \mapsto\left(\begin{array}{cc} G(Y ) \\ Y   \end{array} \right)_{\Id}$;
$j^*$ is given by $\left(\begin{array}{cc} X  \\ Y   \end{array} \right)_{\phi }\mapsto Y $;
$j_*$ is given by $ Y \mapsto \left(\begin{array}{cc} 0 \\ Y   \end{array} \right)$.
\end{theorem}





\begin{lemma}\cite{chen2}\label{lemma functor to derived functor}
Let $F_1:\ca\rightarrow\cb$ be an exact functor between abelian categories which has an exact right adjoint $F_2$. Then the pair $(D^b(F_1),D^b(F_2))$ is adjoint, where $D^b(F_1)$ is the induced functor from $D^b(\ca)$ to $D^b(\cb)$ ($D^b(F_2)$ is defined similarly). Moreover, if $F_1$ is fully faithful, then so is $D^b(F_1)$.
\end{lemma}

The following well-known result is very helpful.

\begin{theorem}\cite{CPS2}\label{recollement of derived categories}
Let $A$ and $B$ be finite-dimensional algebras, $_AM_B$ an $A\mbox{-}B$-bimodule such that $_AM$ and $M_B$ are projective, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$.
Then we have the following recollement:
\[\xymatrix{
D^b(A) \ar[r]^{D^b(i_*)}& D^b(\Lambda)
\ar@<2ex>[l]^{D^b(i^!)}\ar@<-3ex>[l]_{\L(i^*)}\ar[r]^{D^b(j^*)}& D^b(B)
\ar@<2ex>[l]^{D^b(j_*)}\ar@<-3ex>[l]_{D^b(j_!)}  }\]
where $\L(i^*), D^b(i_*),D^b(i^!),D^b(j_!),D^b(j^*),D^b(j_*)$ are the derived functors of these in Theorem \ref{recollement of abelian categories}.
\end{theorem}



Now we prove the main results in this section.

\begin{theorem}\label{theorem recollement of Gorenstein defect categories}
Let $A$ and $B$ be finite-dimensional algebras, $_AM_B$ a finitely generated $A\mbox{-}B$-bimodule such that $_AM$ and $M_B$ are projective, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. If $$\pd_B\Hom_A(M,A)<\infty,$$ then $D_{def}(\Lambda)$ admits a recollement relative to $D_{def}(A)$ and $D_{def}(B)$.
\end{theorem}
\begin{proof}
By Buchweitz's theorem, we identify the stable category of Gorenstein projective objects with a thick subcategory of the singularity category.

By \cite[Theorem 2.5]{LL}, we get the following recollement:
\[\xymatrix{
D_{sg}(A) \ar[r]^{\tilde{i}_*}& D_{sg}(\Lambda)
\ar@<2ex>[l]^{\tilde{i}^!}\ar@<-3ex>[l]_{\tilde{i}^*}\ar[r]^{\tilde{j}^*}& D_{sg}(B)\quad (*)
\ar@<2ex>[l]^{\tilde{j}_*}\ar@<-3ex>[l]_{\tilde{j}_!}  }\]
where $\tilde{i}^!$, $\tilde{i}_*$, $\tilde{i}^!$, $\tilde{j}_!$, $\tilde{j}^*$ and $\tilde{j}_*$ are induced by the six structure functors in Theorem \ref{recollement of derived categories}.



Denote by $G$ the functor $M\otimes_B-$, and $F$ the functor $\Hom_A(M,-)$. Then $G$ and $F$ are exact functors since $_AM$ and $M_B$ are projective.

For any $ \left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi}\in\underline{\Gp}\Lambda$, we have the following exact sequence:
\begin{equation}\label{equation coprojecitve resolution of (X,Y)'}
0\rightarrow \left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi}\rightarrow \left(\begin{array}{cc}P_0\oplus G(Q_0)\\ Q_0   \end{array}  \right)_{\phi_0}\xrightarrow{d^0}\left(\begin{array}{cc} P_1\oplus G(Q_1)\\ Q_1   \end{array}  \right)_{\phi_1}\xrightarrow{d^1}\cdots,\end{equation}
$$ \mbox{ with }\left(\begin{array}{cc} P_i\oplus G(Q_i)\\ Q_i   \end{array}  \right)_{\phi_i}\in\proj(\Lambda),\ker d^i\in\,^\bot\proj(\Lambda),\forall \,i\geq0.
$$
where $P_i\in \proj A,Q_i\in \proj B$. It is easy to see that $\phi:G(Y)\rightarrow X$ is monic since $\phi_0$ is monic.
Apply $j^*$ to the exact sequence (\ref{equation coprojecitve resolution of (X,Y)'}), since
$$j^*\left(\begin{array}{cc} P_i\\ Q_i   \end{array}  \right)_{\phi_i}=Q_i \in\proj B$$
for all $i\geq0$, we get an exact sequence
$$
0\rightarrow Y\rightarrow Q_0\xrightarrow{j^*d^0} Q_1\xrightarrow{j^*d^1}\cdots.
$$
In fact, $\ker (j^* d^i)=j^*(\ker d^i)$ for any $i\geq 0$. Note that $j^*$ is exact and preserves projective, we get that $$\Ext^k_B(j^*\ker d^i,Q)\cong \Ext^k_{\Lambda}(\ker d^i,j_*Q)=\Ext^k_\Lambda(\ker d^i, \left( \begin{array}{ccc} 0\\ Q \end{array} \right))=0,\forall k>0$$
since $\pd  \left( \begin{array}{ccc} 0\\ Q \end{array} \right)\leq1$ and $\ker d^i$ is Gorenstein projective.
So $\tilde{j}^*\left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi} \in\underline{\Gp}B$.

Denote $\ker d^i$ in the exact sequence (\ref{equation coprojecitve resolution of (X,Y)'}) by $\left( \begin{array}{cc} X_i\\ Y_i  \end{array} \right)_{\psi_i}$. Then $\ker d^i\in \Gp \Lambda$, we also get that $\psi_i$ is monic.
Applying $i^*$ to the exact sequence (\ref{equation coprojecitve resolution of (X,Y)'}), we get an exact sequence since $\phi$ and $\psi_i$ are monic:
$$0\rightarrow \Coker\phi \rightarrow P_0\xrightarrow{i^*d^0} P_1\xrightarrow{i^*d^1}\cdots.$$
Note that $\ker (i^*d^i)=i^*(\ker d^i)$, for $i\geq 0$. Let $$T^i:\cdots\rightarrow T^i_1\rightarrow T^i_0\rightarrow \ker d^i\rightarrow0$$
be a projective resolution of $\ker d^i$. Applying $i^*$ to it, we get a projective resolution $i^*(T^i)$ of $i^*\ker d^i$.
So $$\Ext^k _{\Lambda}(\ker d^i,i_*P)\cong H^k\Hom_{\Lambda}(T^i, i_*P)\cong H^k\Hom_A(i^*T^i,P)\cong \Ext^k_A(i^*\ker d^i,P)$$
for any $k\geq0$ and $P\in \proj A$.
Since $i_*P=\left(\begin{array}{cc}P\\ 0   \end{array}  \right)$ is projective, we get that $\Ext^k _{\Lambda}(\ker d^i,i_*P)\cong0$ for $k\geq1$, and then
$\Ext^k_A(i^*\ker d^i,P)=0$. So $\tilde{i}^* (\left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi})\in \underline{\Gp}A$.




Applying ${i}^!$ to the exact sequence (\ref{equation coprojecitve resolution of (X,Y)'}), we get an exact sequence:
$$0\rightarrow X\rightarrow P_0 \oplus G(Q_0)\xrightarrow{i^!d_0} P_1\oplus G(Q_1)\xrightarrow{i^!d_1}\cdots.$$
In fact $i^!$ admits a right exact functor $i_?:\mod A\rightarrow \mod \Lambda$ given by $X\mapsto \left( \begin{array}{cc} X\\ F(X) \end{array} \right)_{\psi_X}$, where $\psi_X=\alpha^{-1}_{X,F(X)}(\Id_{F(X)})$.
For any projective $A$-module $P$,
there are two exact sequences:
$$0\rightarrow \left( \begin{array}{ccc} P\\ 0 \end{array} \right)_{0} \rightarrow \left( \begin{array}{ccc} P\\ F(P) \end{array} \right)_{\psi_P}\rightarrow \left( \begin{array}{ccc} 0\\ F(P) \end{array} \right)_{0}\rightarrow0,$$
$$0\rightarrow \left( \begin{array}{ccc} GF(P)\\ 0 \end{array} \right)_{0} \rightarrow \left( \begin{array}{ccc} GF(P)\\ F(P) \end{array} \right)_{\Id}\rightarrow \left( \begin{array}{ccc} 0\\ F(P) \end{array} \right)_{0}\rightarrow0.$$
It is easy to see that
$\pd(\left( \begin{array}{ccc} GF(P)\\ 0 \end{array} \right)_{0})= \pd_A GF(P)\leq \pd_B F(P)$ since $G$ is exact and preserves projective, and $\pd\left( \begin{array}{ccc} GF(P)\\ F(P) \end{array} \right)_{\Id}= \pd_B F(P)$. By the assumption $$\pd_B\Hom_A(M,A)<\infty,$$ we get that $$\pd \left( \begin{array}{ccc} 0\\ F(P) \end{array} \right)_{0}<\infty$$ and then $$\pd\left( \begin{array}{ccc} P\\ F(P) \end{array} \right)_{\psi_P}<\infty .$$
Since $i^!$ is exact and preserves projectives, similar to $i^*$, we get that
$$\Ext^k_B(i^!\ker d^i,P)\cong \Ext^k_{\Lambda}(\ker d^i,i_?P)=\Ext^k_\Lambda(\ker d^i, \left( \begin{array}{ccc} P\\ F(P) \end{array} \right)_{\psi_P})=0, k\geq1.$$
Thus $\tilde{i}^!(\left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi})\in\underline{\Gp}A$.

Similarly, we can check that $\tilde{i}_*(\underline{\Gp}A)\subseteq \underline{\Gp}\Lambda$. Then
\begin{equation}
\tilde{i}^*(\underline{\Gp}(\Lambda))=\underline{\Gp}(A)
\end{equation} by $\tilde{i}^*\tilde{i}_*\simeq \Id$. From it, we get that
\begin{equation}
\tilde{i}_*\tilde{i}^*(\underline{\Gp}(\Lambda))\subseteq \underline{\Gp}(\Lambda)
\end{equation}
and
\begin{equation}
\tilde{i}_*\tilde{i}^!(\underline{\Gp}(\Lambda))\subseteq \underline{\Gp}(\Lambda).
\end{equation}

By (R4) and $\underline{\Gp}\Lambda$ is an \'{e}paisse subcategory of $D_{def}(\Lambda)$, we get that
\begin{equation}
\tilde{j}_*\tilde{j}^*(\underline{\Gp}(\Lambda))\subseteq \underline{\Gp}(\Lambda).
\end{equation}



On the other hand, for any $Y\in\underline{\Gp}B$,
we have the following exact sequence:
\begin{equation}
0\rightarrow Y\rightarrow Q^0\xrightarrow{d^0}Q^1\xrightarrow{d^1}\cdots, \mbox{ with }Q^i\in\proj B,\ker d^i\in\,^\bot\proj B,\forall i\geq0.
\end{equation}
Apply ${j}_!$ to it, we get the following exact sequence:
$$0\rightarrow \left(\begin{array}{cc} G(Y)\\ Y   \end{array}  \right)_{\Id}\rightarrow \left(\begin{array}{cc}G(Q_0)\\ Q_0   \end{array}  \right)_{\Id}\xrightarrow{j_!d^0}\left(\begin{array}{cc} G(Q^1)\\ Q_1   \end{array}  \right)_{\Id}\xrightarrow{j_!d^1}\cdots.$$
For any projective object $\left( \begin{array}{cc} P\\ Q \end{array}\right)_\phi\in \mod \Lambda$, we have
$$\Ext^k_{\Lambda}(j_!\ker d^i,\left( \begin{array}{cc} P\\ Q \end{array}\right)_\phi)\cong \Ext^k_{B}(\ker d^i,j^*\left( \begin{array}{cc} P\\ Q \end{array}\right)_\phi )=\Ext^k_B(\ker d^i, Q)=0,\forall k>0.$$
So $\tilde{j}_!(Y)\in\underline{\Gp}\Lambda$. In fact, we have proved that
\begin{equation}
\tilde{j}^*(\underline{\Gp}\Lambda)=\underline{\Gp}B
\end{equation}
since
$\tilde{j}^*\tilde{j}_!\simeq\Id$.

From the Equalities (2), (3), (5), (7) and Theorem \ref{theorem 1}, we get that $D_{def}(\Lambda)$ admits a recollement relative to $D_{def}(A)$ and $D_{def}(B)$.
\end{proof}

\begin{corollary}
Let $A$ be an finite-dimensional algebras. Then $D_{def}(\left( \begin{array}{cc} A&A\\0&A \end{array}\right))$ admits a recollement relative to $D_{def}(A)$ and $D_{def}(A)$.
\end{corollary}

For any finite-dimensional algebra $A$, we have that $D_{def}(A)$ is zero if and only if $A$ is Gorenstein \cite{BJO}. X-W. Chen proved the following: if $A,B$ are Gorenstein, then the upper triangular matrix algebra $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$ is Gorenstein if and only if $_AM$ and $M_B$ are finitely generated,
$\pd _A M<\infty$ and $\pd M_B<\infty$ \cite[Theorem 3.3]{Chen1}. Following Chen, B. Xiong and P. Zhang
 proved that if $\pd _A M<\infty$ and $\pd M_B<\infty$, then $\Lambda$ is Gorenstein if and only if $A$ and $B$ are Gorenstein \cite[Thoerem 2.2]{XZ}. In the following, we will try to interpret these results by the Gorentein defect categories.

\begin{lemma}\cite{EJ}\label{lemma projective injective dimension of Gorenstein algebra}
Let $R$ be a Gorenstein ring, $N$ a left $R$-module. Then $N$ has finite projective dimension if and only if $N$ has finite injective dimension.
\end{lemma}

\begin{theorem}\label{proposition Gorenstein defect equivalent}
Let $A$ and $B$ be finite-dimensional algebras, $_AM_B$ an $A\mbox{-}B$-bimodule, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. We also assume that $\pd _A M<\infty$ and $\pd M_B<\infty$.

(a) If $B$ is a Gorenstein algebra, then $D_{def}(\Lambda)\simeq D_{def}(A)$.

(b) If $A$ is a Gorenstein algebra and $\ind_B M_B<\infty$, then $D_{def}(\Lambda)\simeq D_{def}(B)$.
\end{theorem}
\begin{proof}
(a) By \cite{LL}, we get that $D_{sg}(\Lambda)$ admits a recollement relative to $D_{sg}(A)$ and $D_{sg}(B)$ as the diagram $(*)$ in the proof of Theorem \ref{theorem recollement of Gorenstein defect categories} shows, where the six structure functors are induced by the derived functors of the ones in Theorem \ref{recollement of abelian categories}.
In fact, $i_*,i^!,j^*,j_*$ are exact.

For any $X\in \underline{\Gp}(A)$, there is an exact sequence:
\begin{equation}\label{equation coprojective resolution of X}
0\rightarrow X \rightarrow P_0\xrightarrow{d^0}P_1\xrightarrow{d^1}\cdots,\end{equation}
$$ \mbox{ with }P_i\in\proj A,\ker d^i\in\,^\bot\proj A,\forall i\geq0.
$$
Applying $i_*$ to the exact sequence (\ref{equation coprojective resolution of X}), we get the following exact sequence:
\begin{equation}
0\rightarrow \left(\begin{array}{cc} X\\ 0   \end{array}  \right)_{\phi}\rightarrow \left(\begin{array}{cc}P_0\\ 0   \end{array}  \right)\xrightarrow{i_*d^0}\left(\begin{array}{cc} P_1\\ 0   \end{array}  \right)\xrightarrow{i_*d^1}\cdots.\end{equation}
Note that $\left(\begin{array}{cc}P_i\\ 0   \end{array}  \right)$ are projective for $i\geq0$.
For any projective $\Lambda$-module $U$, $i^!U\cong P\oplus M\otimes_BQ$ for some projective $A$-module $P$, and projective $B$-module $Q$. Since $\pd M_B<\infty$, we get that $\pd M\otimes_BQ<\infty$, so
$$\Ext^k_\Lambda(i_*\ker d_i, U)\cong \Ext_A^k(\ker d_i,i^! U)\cong \Ext_A^k(\ker d_i,P\oplus M\otimes_BQ)=0,\forall k>0.$$
Thus $\tilde{i}_*(\underline{\Gp}(A))\subseteq \underline{\Gp}(\Lambda)$.
So $\tilde{i}_*$ induces a full functor from $D_{def}(A)$ to $D_{def}(\Lambda)$, which is also denoted by $\tilde{i}_*$.
In the following, we will prove that $\tilde{i}_*: D_{def}(A)\rightarrow D_{def}(\Lambda)$ is full faithful and dense.

First, we prove that $\tilde{i}^* (\underline{\Gp}(\Lambda))\subseteq\underline{\Gp}(A)$. For any $ \left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi}\in\underline{\Gp}(\Lambda)$, we have the following exact sequence:
\begin{equation}\label{equation coprojecitve resolution of (X,Y)}
0\rightarrow \left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi}\rightarrow \left(\begin{array}{cc}P_0\oplus M\otimes_B Q_0\\ Q_0   \end{array}  \right)_{\phi_0}\xrightarrow{d^0}\left(\begin{array}{cc} P_1\oplus M\otimes_B Q_1\\ Q_1   \end{array}  \right)_{\phi_1}\xrightarrow{d^1}\cdots,\end{equation}
$$ \mbox{ with }\left(\begin{array}{cc} P_i\oplus M\otimes_B Q_i\\ Q_i   \end{array}  \right)_{\phi_i}\in\proj\Lambda,\ker d^i\in\,^\bot\proj\Lambda,\forall \,i\geq0.
$$
In fact, $\phi_i=\left(\begin{array}{cc} 0\\ \Id \end{array}  \right)$.
So $\phi: M\otimes_BY\rightarrow X$ is monic.
Applying $i^*$ to the exact sequence (\ref{equation coprojecitve resolution of (X,Y)}), we get an exact sequence:
$$0\rightarrow \Coker \phi\rightarrow P_0\xrightarrow{i^*d^0} P_1\xrightarrow{i^*d^1} \cdots.$$
Note that $\ker (i^*d^i)=i^*(\ker d^i)$, for $i\geq 0$.
So $$\Ext_A^k(\ker (i^*d^i),P)\cong \Ext_A^k(i^*(\ker d^i),P)\cong \Ext_\Lambda^k(\ker d^i,i_*P)=0,\forall k>0$$
follows from that $i_*P=\left(\begin{array}{cc}P\\ 0   \end{array}  \right)$ is projective.
So $\tilde{i}^* (\left(\begin{array}{cc} X\\ Y   \end{array}  \right)_{\phi})\in \underline{\Gp}(A)$.

For any $X_1,X_2\in D_{sg}(A)$, and $f:X_1\rightarrow X_2$, if
$$\tilde{i}_*(f)=\left(\begin{array}{cc}f\\ 0   \end{array}  \right):\left(\begin{array}{cc}X_1\\0   \end{array}  \right) \rightarrow\left(\begin{array}{cc}X_2\\ 0   \end{array}  \right)$$
is zero in $ D_{def}(\Lambda)$, then it factors through some object $\left(\begin{array}{cc}X\\ Y   \end{array}  \right)_\phi\in \underline{\Gp}(\Lambda)$.
Applying $\tilde{i}^*$, we get that $f:X_1\rightarrow X_2$ factors through $\Coker\phi=i^*(\left(\begin{array}{cc}X\\ Y   \end{array}  \right)_\phi)\in \underline{\Gp}(A)$. It follows that  $\tilde{i}_*: D_{def}(A)\rightarrow D_{def}(\Lambda)$ is full faithful.


For any $Y\in\underline{\Gp}(B)$,
we have the following exact sequence:
\begin{equation}\label{equation coprojecitve resolution of Y}
0\rightarrow Y\rightarrow Q^0\xrightarrow{d^0}Q^1\xrightarrow{d^1}\cdots, \mbox{ with }Q^i\in\proj B,\ker d^i\in\,^\bot\proj B,\forall i\geq0.
\end{equation}
Since $\pd M_B<\infty$, we get that $\ind _BDM<\infty$. Then $\pd_BDM<\infty$ by Lemma \ref{lemma projective injective dimension of Gorenstein algebra}.
We get that $D\mathrm{Tor}^B_1(M,\ker d^i)\cong \Ext_B^1(\ker d^i, DM)\cong0$.
So
$$
0\rightarrow M\otimes_B Y\rightarrow M\otimes_BQ^0\rightarrow M\otimes_BQ^1\rightarrow\cdots$$
is also exact. Therefore, applying $j_!$ to the exact sequence (\ref{equation coprojecitve resolution of Y}), we get an exact sequence:
$$0\rightarrow \left(\begin{array}{cc} M\otimes_B Y\\ Y   \end{array}  \right)_{\Id}\rightarrow \left(\begin{array}{cc}M\otimes_BQ^0\\ Q_0   \end{array}  \right)_{\Id}\xrightarrow{j_!d^0}\left(\begin{array}{cc} M\otimes_BQ^1\\ Q_1   \end{array}  \right)_{\Id}\xrightarrow{j_!d^1}\cdots.$$
Now, $\left(\begin{array}{cc} M\otimes_BQ^i\\ Q_i   \end{array}  \right)_{\Id}$ are projective.
For any projective object $\left( \begin{array}{cc} P\\ Q \end{array}\right)_\phi\in\mod \Lambda$, we have $Q$ is projective as $B$-module, so
$$\Ext^k_{\Lambda}(j_!\ker d^i,\left( \begin{array}{cc} P\\ Q \end{array}\right)_\phi)\cong \Ext^k_{B}(\ker d^i,j^*\left( \begin{array}{cc} P\\ Q \end{array}\right)_\phi )=\Ext^k_B(\ker d^i, Q)=0,\forall k>0.$$
So $\tilde{j}_!(D_{sg}(B))=\tilde{j}_!(\underline{\Gp}(B))\subseteq\underline{\Gp}(\Lambda)$ for $B$ is a Gorenstein algebra.

For any object $T\in D_{sg}(\Lambda)$,
we have the following distinguished triangle in $D_{sg}(\Lambda)$:
$$ \tilde{j}_!\tilde{j}^*T\rightarrow T \rightarrow \tilde{i}_*\tilde{i}^*T \rightarrow\cdot.$$
But $\tilde{j}_!\tilde{j}^*T\in \underline{\Gp}(\Lambda)$ and then
$T\cong \tilde{i}_*\tilde{i}^*T$ in $D^b_{def}(\Lambda)$, so $\tilde{i}_*: D_{def}(A)\rightarrow D_{def}(\Lambda)$ is dense.
Therefore, $D_{def}(A)\simeq D_{def}(\Lambda)$.

For (b), by the proof of (a), we get that $$\tilde{i}_*(D_{sg}(A))=\tilde{i}_*(\underline{\Gp}(A))\subseteq \underline{\Gp}(\Lambda).$$
Because $\ind_BM_B<\infty$, from the proof of (a), we also get that $\tilde{j}_!(\underline{\Gp}(B) )\subseteq \underline{\Gp}(\Lambda)$.
So $\tilde{j}_!$ induces a functor from $D_{def}(B)$ to $D_{def}(\Lambda)$, which is also denoted by $\tilde{j}_!$.

Similar to the proof of Theorem \ref{theorem recollement of Gorenstein defect categories}, we get that $\tilde{j}^*( \underline{\Gp}(\Lambda))\subseteq \underline{\Gp}(B)$. From these, we can prove that $\tilde{j}_!:D_{def}(B)\rightarrow D_{def}(\Lambda)$ is full faithful and dense similar to the proof of (a).


\end{proof}

\begin{corollary}
Let $A$ and $B$ be finite-dimensional algebras, $_AM_B$ an $A\mbox{-}B$-bimodule, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. We also assume that $\pd _A M<\infty$ and $\pd M_B<\infty$.

(a) If $B$ is a Gorenstein algebra, then $\Lambda$ is Gorenstein if and only if $A$ is.

(b) If $A$ is a Gorenstein algebra, $\Lambda$ is Gorenstein if and only if $B$ is.
\end{corollary}
\begin{proof}
By Theorem \ref{proposition Gorenstein defect equivalent}, (a) is trivial.

For (b), if $B$ is Gorenstein, then $\ind M_B<\infty$ since $\pd M_B<\infty$, so $D_{def}(\Lambda)\simeq0$, then $\Lambda$ is Gorenstein.
On the other hand, because $\pd M_B<\infty$, we get that $\pd(0,M)=\pd M_B<\infty$. Since $\Lambda$ is Gorenstein, then $\ind  (0,M)<\infty$. Note that the injective $\Lambda$-modules are precisely of form $(J,0)\oplus(\Hom_B(M,I),I)$, where $J,I$ are injective $A$-module and $B$-module respectively.
Let $$0\rightarrow (0,M)\rightarrow (J_0\oplus \Hom_B(M,I_0),I_0)_{\psi_0}\rightarrow \cdots\rightarrow (J_r\oplus \Hom_B(M,I_r),I_r)_{\psi_r}\rightarrow 0$$
be a minimal injective resolution of $(0,M)$.
Then $I_i,J_i$ are injective as $B$-module and $A$-module respectively. Furthermore, it is easy to see that
$$0\rightarrow M\rightarrow I_0\rightarrow\cdots \rightarrow I_r\rightarrow0$$
is also exact. So $\ind M_B<\infty$. By Theorem \ref{proposition Gorenstein defect equivalent}, we get that $D_{def}(B)\simeq D_{def}(\Lambda)\simeq0$. So $B$ is also Gorenstein.

\end{proof}

\section{Gluing of Nakayama algebras}
Let $Q$ be a finite quiver. For any arrow $\alpha$ in $Q$, we denote by $s(\alpha),t(\alpha)$ the source and the target of $\alpha$ respectively. The path algebra $kQ$ of $Q$ is an associative algebra with an identity. If $I$ is an admissible ideal of $kQ$, the pair $(Q,I)$ is said to be a bound quiver. The quotient algebra $kQ/I$ is said to be the algebra of the bound quiver $(Q,I)$, or simply, a bound quiver algebra.

In this section, we shall freely identify the representations of $(Q,I)$ with left modules over $kQ/I$.

An algebra $A$ is called a Nakayama algebra if it is both right and left serial. That is, $A$ is a Nakayama algebra if and only if every indecomposable projective $A$-module and every indecomposable injective $A$-module are uniserial.
\begin{lemma}\label{lemma classification of Nakayama algebra}
A basic and connected algebra $A$ is a Nakayama algebra if and only if its ordinary quiver $Q_A$ is one of the following two quivers:

(a) Nakayama quiver of Type I: $$\circ_1\leftarrow \circ_2\leftarrow \circ_3\leftarrow\cdots \leftarrow \circ_{n-1}\leftarrow \circ_n.$$


(b)

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\vspace{3cm}
\end{lemma}


For any Nakayama algebra $A=kQ_A/I$, we always assume that any path of length greater than $n$ is zero in $A$. Furthermore, we assume that the Nakayama algebra $A$ is Gorenstein.

As a generalization of the structure of the cluster-tilted algebra of type $A$, we give the definition of gluing Nakayama algebra as follows.

\begin{definition}\label{definition gluing Nakayama algebra}
Let $A_1=kQ_1/I_1$, $A_2=kQ_2/I_2$, $\cdots$, $A_n=kQ_m/I_m$ be $m$ Nakayama algebras. We define a gluing quiver of $Q_1,Q_2,\dots,Q_m$ inductively.

Given two vertices $v_1\in Q_1$ and $v_2\in Q_2$, we define a connected quiver $Q_{1,2}$ from $Q_1,Q_2$ by identifying $v_1$ to $v_2$. This is called a gluing quiver of $Q_1$ and $Q_2$.

Let $v_{1,2}\in Q_{1,2}$ and $v_3\in Q_3$ be two vertices. We define a connected quiver $Q_{1,2,3}$ from $Q_{1,2}$ and $Q_3$ by identifying $v_{1,2}$ to $v_3$.
Inductively, we can define a quiver $Q=Q_{1,\dots,m}$ from $Q_{1,\dots,m-1}$ and $Q_m$.

In this way, $Q$ is called the gluing quiver of
$Q_1,Q_2,\dots,Q_m$ and $A=kQ/I$ is called the gluing Nakayama algebra, where the ideal $I$ is generated by $\{I_1,I_2,\dots,I_m\}$; $Q_1,Q_2,\dots,Q_m$ are called to be the gluing components of $Q$; $A_1,A_2,\dots,A_m$ are called to be the gluing components of $A$.
\end{definition}


\begin{example}
(1) Let $Q_1,Q_2,\dots,Q_m$ be $m$ oriented cyclic quivers where $Q_i$ has $n_i$ vertices, $1\leq i\leq m$.
Given any vertex $v_i\in Q_i$ for all $1\leq i\leq m$. The gluing quiver $Q$ is defined by identifying all the vertices $v_i$ to a vertex. This is called the simple gluing Nakayama quiver of type II, denoted by $(Q,v)$, where the vertex $v$ of $Q$ is glued by $v_i,1\leq i\leq m$, and called the glued vertex.
The special case of $m=2$ is showed in the following Figure 2.



(2) Let $(Q_1,v_1),(Q_2,v_2)$ be two simple gluing Nakayama quiver of type II, and $Q_0$ be a Nakayama quiver of type I
as Lemma \ref{lemma classification of Nakayama algebra} (a) shows. The gluing quiver $Q$ from $Q_0,Q_1,Q_2$,
is defined to be the quiver by identifying the vertices $v_1\in Q_1$ and $1\in Q_0$,
and the vertices $v_2\in Q_2$ and $n\in Q_0$. The special case of $Q_0$ is of type $A_2$ is showed in the following Figure 3.
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\multiput(85,-12)(0,2){13}{\circle*{0.5}}
\put(84,-16){\vector(-1,-1){8}}
\put(75,-25){\circle{2}}
\put(73.5,-25.5){\vector(-3,-1){12}}
\put(60,-30){\circle{2}}
\put(58.5,-29.5){\vector(-3,1){12}}
\put(45,-25){\circle{2}}
\put(43.5, -24){\vector(-1,1){8}}


\put(35,-15){\circle{2}}
\put(34.4,-14){\vector(-1,3){4.2}}



\put(30.5,1.5){\vector(1,3){4}}

\put(35,15){\circle{2}}
\put(36,16){\vector(1,1){8}}

\put(45,25){\circle{2}}

\put(46.5,25.5){\vector(3,1){12}}
\put(31,0){$v$}

\put(-50,-40){Figure 2. The simple gluing Nakayama quiver }

\put(-24,-47){of Type II with two component.}
\end{picture}

\end{center}
\vspace{2cm}

\setlength{\unitlength}{0.5mm}
\begin{center}
\begin{picture}(100,60)(0,-20)
\qbezier(-5,0)(0,30)(40,10)
\qbezier(-5,0)(0,-30)(40,-10)
\qbezier(40,10)(55,0)(40,-10)
\put(40,0){\circle*{2}}

\qbezier(70,10)(55,0)(70,-10)
\qbezier(70,10)(110,30)(115,0)
\qbezier(70,-10)(110,-30)(115,0)
\put(70,0){\circle*{2}}
\put(40,0){\vector(1,0){29}}

\put(33,-5){$v_2$}
\put(72,-5){$v_1$}
\put(10,-5){$Q_2$}
\put(90,-5){$Q_1$}

\put(-40,-28){Figure 3. The gluing quiver from two simple gluing }

\put(-11,-35){ Nakayama quiver of type II and one of type I. }
\end{picture}
\vspace{0.7cm}
\end{center}


\end{example}



\begin{lemma}\cite{Chen1}\label{Gorenstein dimension of upper triangular matrix algebra}
Let $A$ and $B$ be two Gorenstein algebras, $_AM_B$ an $A\mbox{-}B$-bimodule such that $_AM$ and $M_B$ are projective, and $\Lambda=\left( \begin{array}{cc} A&M\\0&B \end{array}\right)$. Then $\Lambda$ is Gorenstein. In particular, if $\Gd A\neq\Gd B$, then $\Gd (\Lambda)=\max\{\Gd A,\Gd B\}$;
Otherwise, $\Gd ( \Lambda)\leq \Gd A+1$.
\end{lemma}
\begin{proof}
The proof of this lemma is similar to \cite[Theorem 3.3]{Chen1}.

Let $G=M\otimes_B-$, $F=\Hom_A(M,-)$. They are exact functors.
For any indecomposable injective $\Lambda$-module, it is of form $ \left( \begin{array}{cc} 0\\ I\end{array}\right)$ or $\left( \begin{array}{cc}  J\\ F(J) \end{array}\right)_{\psi_J}$ where $I$ and $J$ are injective $B$-module and $A$-module respectively.
For $\left( \begin{array}{cc} 0\\ I\end{array}\right)$, we know that $\pd_B I\leq \Gd B$.
There is an exact sequence:
$$0\rightarrow \left( \begin{array}{cc} G(I)\\ 0\end{array}\right)\rightarrow\left( \begin{array}{cc} G(I)\\ I\end{array}\right)_{\id}
\rightarrow \left( \begin{array}{cc} 0\\ I\end{array}\right) \rightarrow0.$$
It is easy to see that $$\pd_\Lambda \left( \begin{array}{cc} G(I)\\ I\end{array}\right)_{\id}=\pd_B I\leq \Gd B$$ and
$$\pd_\Lambda\left( \begin{array}{cc} G(I)\\ 0\end{array}\right)=\pd_A G(I)\leq \pd_B I\leq \Gd B$$ since $G$ is exact and preserves projectives.
So $\pd_\Lambda\left( \begin{array}{cc} 0\\ I\end{array}\right)\leq \Gd B+1$.

For $\left( \begin{array}{cc}  J\\ F(J) \end{array}\right)_{\psi_J}$, there is an exact sequence:
$$0\rightarrow \left( \begin{array}{cc}J\\ 0\end{array}\right)\rightarrow\left( \begin{array}{cc} J\\ F(J)\end{array}\right)_{\psi_J}
\rightarrow \left( \begin{array}{cc} 0\\ F(J)\end{array}\right) \rightarrow0.$$
we get that $\pd_\Lambda \left( \begin{array}{cc}J\\ 0\end{array}\right)=\pd_AJ\leq \Gd A$ and
$$ \pd_\Lambda \left( \begin{array}{cc} 0\\ F(J)\end{array}\right)\leq \Gd B+1$$
since $F(J)$ is injective as $B$-module.
So $$\pd_\Lambda \left( \begin{array}{cc} J\\ F(J)\end{array}\right)_{\psi_J}\leq \max\{\Gd A,\Gd B+1\}.$$
Therefor, the projective dimension of any injective $\Lambda$-module is less than $$ \max\{\Gd A,\Gd B+1\}.$$
Dually, we can prove that for any indecomposable projective $\Lambda$-module, its injective dimension is less than $\max\{ \Gd A+1,\Gd B\}$. Then $\Lambda$ is Gorenstein. If $\Gd A\neq \Gd B$, then $\Gd \Lambda \leq \max\{\Gd A,\Gd B\}$. Conversely, it is easy to see that $\max\{\Gd A,\Gd B\}\leq \Gd \Lambda$.
So $\Gd\Lambda=\max\{\Gd A,\Gd B\}$.

If $\Gd A=\Gd B$, then it is easy to see that $\Gd \Lambda\leq \Gd A+1$. \end{proof}

Let $A=kQ'/I'$ and $B=kQ''/I''$. Let $Q$ be the quiver glued by $Q',Q''$ as the following picture shows:
\setlength{\unitlength}{0.5mm}
\begin{center}
\begin{picture}(120,60)(0,-20)
\qbezier(-5,0)(0,30)(40,10)
\qbezier(-5,0)(0,-30)(40,-10)
\qbezier(40,10)(55,0)(40,-10)
\put(40,0){\circle*{2}}

\qbezier(70,10)(55,0)(70,-10)
\qbezier(70,10)(110,30)(115,0)
\qbezier(70,-10)(110,-30)(115,0)
\put(70,0){\circle*{2}}
\put(40,0){\vector(1,0){29}}

\put(33,-5){$w$}
\put(72,-5){$v$}
\put(10,-5){$Q''$}
\put(90,-5){$Q'$}
\put(-40,-28){Figure 4. The gluing quiver $Q$ from $Q'$ and $Q''$ }

\end{picture}
\vspace{0.7cm}
\end{center}

Let $\Lambda=kQ/I$, where $I$ is the ideal generated by $I'$ and $I''$.
Then $\Lambda$ is the upper matrix algebra $\left(\begin{array}{cc} A & M\\ 0&B \end{array}\right)$, where $_AM$ and $M_B$ are projective.
If $A$ and $B$ are Gorenstein algebras, then $\Lambda$ is a Gorenstein algebra.
\begin{lemma}\cite{CGLu}\label{lemma recollement splits}
Let $A=kQ'/I'$ and $B=kQ''/I''$. Let $Q$ be the quiver glued by $Q',Q''$ as the Figure $4$ shows. Let $\Lambda=kQ/I$, where $I$ is the ideal generated by $I'$ and $I''$. Then $$\underline{\Gproj}(\Lambda)\simeq \underline{\Gproj}(A)\coprod\underline{\Gproj}(B).$$
\end{lemma}


For the remaining of this section, let $A=kQ_A/I_1$ and $B=kQ_B/I_2$ be two simple gluing Nakayama algebras, $\Lambda$ be the bound quiver algebra $kQ_\Lambda/I_\Lambda$ as Figure 3 shows. We also let $\Gamma=kQ_\Gamma/I_\Gamma$ be the simple gluing Nakayama algebra from $A$ and $B$ by identifying vertices $v_1$ and $v_2$. Denote by $v\in Q_\Gamma$ the glued vertex.

Since $B$ is a subalgebra of $\Gamma$, every $\Gamma$-module is naturally a $B$-module, there exists an exact functor $j:\mod \Gamma\rightarrow \mod B$.



We define the functor $j_\lambda: \mod B\rightarrow \mod \Gamma$ as follows. Given any $Y\in \mod B$, it corresponds to a representation $(Y_i,\phi_\alpha)$ of $(Q_B,I_2)$. If the linear space $Y_v$ has dimension $l$, define a representation $(N_j,\psi_\beta)$ of $(Q_\Lambda,I_\Lambda)$, with $N_j=Y_j$ if $j$ is a vertex of $Q_B$, with $N_j=k^l$ if $j$ is a vertex of $Q_A$ and there is a nonzero path from $v$ to $j$;
with $\psi_\beta=\phi_\beta$ if $\beta$ is an arrow of $Q_B$, with $\psi_\beta=\id$ if $\beta$ is an arrow of $Q_A$ and $N_{s(\beta)}=k^l=N_{t(\beta)}$, otherwise zero. $(N_j,\psi_\beta)$ corresponds to a module $N\in\mod \Lambda$.
We define $j_\lambda(Y):=N$, it acts on morphisms naturally. In fact, $j_\lambda$ is an exact fully faithful functor and $(j_\lambda, j)$ is an adjoint pair.

We define another functor $j_\rho: \mod B\rightarrow \mod \Gamma$ as follows. Given any $Y\in \mod B$, it corresponds to a representation $(Y_i,\phi_\alpha)$ of $(Q_B,I_2)$. If the linear space $Y_v$ has dimension $l$, define a representation $(L_j,\psi_\beta)$ of $(Q_\Lambda,I_\Lambda)$, with $L_j=Y_j$ if $j$ is a vertex of $Q_B$, with $L_j=k^l$ if $j$ is a vertex of $Q_A$ and there is a nonzero path from $j$ to $v$;
with $\psi_\beta=\phi_\beta$ if $\beta$ is an arrow of $Q_B$, with $\psi_\beta=\id$ if $\beta$ is an arrow of $Q_A$ and $N_{s(\beta)}=k^l=N_{t(\beta)}$, otherwise zero. $(L_j,\psi_\beta)$ corresponds to a module $L\in\mod \Lambda$.
We define $j_\rho(Y):=L$, it acts on morphisms naturally. In fact, $j_\rho$ is an exact fully faithful functor and $(j, j_\rho)$ is an adjoint pair.

It is easy to see that $j_\lambda,j$ preserve projective objects, and $j_\rho$ preserves injective objects.


Dually, we can define $i:\mod \Gamma \rightarrow \mod A$ as $j$; $i_\lambda,i_\rho:\mod A\rightarrow \mod \Gamma$ as $j_\lambda,j_\rho$.

Similar to the upper triangular matrix algebra, we have the following lemma.
\begin{lemma}\label{lemma simple gluing Nakayama algebra Gorenstein}
Keep the notations as above. If $A$ and $B$ are Gorenstein algebras, then $\Gamma$ is also a Gorenstein algebra. In particular, $\Gd (\Gamma)\leq\max\{1,\Gd A,\Gd B\}$.
\end{lemma}
\begin{proof}
Let $_\Gamma I=(I_j,\psi_\beta)$ be an indecomposable injective module of $\Gamma$.
It corresponds to a vertex $w\in Q_\Gamma$. Without losing generality, we assume that $w\in Q_A$.
Let $_A J$ be the indecomposable injective module of $A$ corresponding to $w$. Since $A$ is Gorenstein, $\pd_A J<\infty$.
Let \begin{equation}\label{equation projective of J}
0\rightarrow P^r\rightarrow\cdots\rightarrow P^1\xrightarrow{d^1} P^0\xrightarrow{d^0} _AJ\rightarrow0
\end{equation}
be a minimal projective resolution of $_AI$.
If $_AJ_v=0$, then $i_\lambda (_AJ)=_\Gamma I$.
Applying $i_\lambda$ to (\ref{equation projective of J}), we get a projective resolution of $_\Gamma I$:
\begin{equation}\label{equation projective of I}
0\rightarrow i_\lambda(P^r)\rightarrow \cdots\rightarrow i_\lambda(P^1)\rightarrow i_\lambda(P^0)\rightarrow _\Gamma I\rightarrow0
\end{equation}
since $i_\lambda$ is an exact functor which also preserves projective objects. So $\pd _\Gamma I\leq\Gd A$.

If $_AJ_v\neq0$, let $_BR$ be the indecomposable injective module of $B$ corresponding to the vertex $v_2$(also vertex $v$ in $Q_\Gamma$).
Since $B$ is also Gorenstein, let
\begin{equation}\label{equation projective of R}
0\rightarrow Q^s\rightarrow\cdots\rightarrow Q^1\xrightarrow{e^1} Q^0\xrightarrow{e^0} _BR\rightarrow0
\end{equation}
be the minimal projective resolution of $R$.
Then there is an exact sequence
\begin{equation}
0\rightarrow i_\lambda(\ker d^0)\oplus j_\lambda(\ker e^0) \oplus _\Gamma T\rightarrow i_\lambda(P^0)\oplus j_\lambda(Q^0) \rightarrow _\Gamma I\rightarrow0,
\end{equation}
where $_\Gamma T$ is the indecomposable projective module of $\Gamma$ corresponding to the vertex $v$.
Since $\pd i_\lambda(\ker d^0) \leq r-1$, $\pd j_\lambda(\ker e^0) \leq s-1$, we get
$\pd _\Gamma I\leq \max\{1, \Gd A,\Gd B\}$.


Dually, we can prove that every projective $\Gamma$-module has finite injective dimension. So $\Gamma$ is Gorenstein, and from the proof, we also get that
$\Gd (\Gamma)\leq\max\{1,\Gd A,\Gd B\}$.
\end{proof}

\begin{lemma}\label{lemma the singularity category of simple gluing Nakayama algebras}
Keep the notations as above. If $A$ and $B$ are Gorenstein algebras, then $\underline{\Gproj}(\Gamma)\simeq \underline{\Gproj}(A)\coprod\underline{\Gproj}(B)$.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma simple gluing Nakayama algebra Gorenstein}, we know that $\Gamma$ is also Gorenstein.
From Theorem \ref{theorem stable category of CM modules }, we also get that $\underline{\Gproj}(\Gamma)\simeq D_{sg}(\Gamma)$, $\underline{\Gproj}(A)\simeq D_{sg}(A)$, $\underline{\Gproj}(B)\simeq D_{sg}(B)$.

By the construction above, we get that the functors $j,j_\lambda,j_\rho$ induce functors on the bounded derived categories, also denoted by the same notations.
It is easy to see that $j_\lambda$ and $j$ preserve projective objects, they induces functors on the singularity categories, denoted by $\tilde{j}_\lambda,\tilde{j}$.

Given any projective object $Q\in\mod B$. Since $B$ is Gorenstein, it has a finite injective resolution:
$0\rightarrow Q\rightarrow I^0\rightarrow I^1\rightarrow\cdots \rightarrow I^t\rightarrow0$.
Since $j_\rho$ is exact and preserves injective objects, we get that $j_\rho(Q)$ has finite injective dimension as $\Gamma$-module,
but $\Gamma$ is also Gorenstein, it follows that $j_\rho(Q)$ has finite projective dimension.
So $j_\rho$ maps $K^b(\proj B)$ to $K^b(\proj \Gamma)$, and then it induces a functor on the singularity categories, denoted by $\tilde{j}_\rho$.

Similarly, $i_\lambda,i$ induces functors on the singularity categories, denoted by $\tilde{i}_\lambda$, $\tilde{i}$.

In particular, it is easy to see that $\tilde{j}_\lambda$, $\tilde{j}_\rho$ and $\tilde{i}_\lambda$ are fully faithful, since $\tilde{j}\tilde{j}_\lambda\simeq \id$, etc..
In the following, we prove that $\Im(\tilde{i}_\lambda)=\ker \tilde{j}$.
For any module $X\in\mod A$, it corresponds to a representation $(X_i,\phi_\alpha)$. Then $j\circ i_\lambda(X)= _BR^{\oplus r}$, where $_BR$ is the indecomposable projective module corresponding to the vertex $v_2$, $r=\dim X_{v_1}$.
So $ \tilde{j}\circ\tilde{i}_\lambda=0$.

On the other hand, for any module $Y\in\mod \Gamma$, if $j(Y)\simeq0$ in $K^b(\proj B)$, then $j(Y)$ has finite projective dimension as $B$-module.
Let $Q^0\xrightarrow{e^0} j(Y)$ be the projective cover of $j(Y)$.
Similarly, let $P^0\xrightarrow{d^0}i(Y)$ be the projective cover of $i(Y)$.
Then there exists the following exact sequence:
\begin{equation}
0\rightarrow j_\lambda (\ker e^0) \oplus i_\lambda (\ker d^0)\oplus T^{\oplus s} \rightarrow j_\lambda(Q^0) \oplus i_\lambda(P^0)\rightarrow Y\rightarrow0,
\end{equation}
where $T$ is the indecomposable projective module corresponding to $v$ and $s=\dim  Y_v$.
Then $Y\simeq \Sigma(i_\lambda(\ker d^0))$ in $D_{sg}^b(\Gamma)$ since $j_\lambda(\ker e^0)$ has finite projective dimension, where $\Sigma$ is the suspension functor.

By \cite[Theorem 1.1]{CPS2}, we get that there is a recollement:

\[\xymatrix{D_{sg}(A)\ar[r]^{\tilde{i}_\lambda}&D_{sg}(\Gamma)
\ar@<2ex>[l]^{\tilde{i}}\ar@<-3ex>[l]_{\tilde{i}_\mu}\ar[r]^{\tilde{j}}&D_{sg}(B),
\ar@<2ex>[l]^{\tilde{j}_\rho}\ar@<-3ex>[l]_{\tilde{j}_\lambda}  }
\]
where $\tilde{i}_\mu$ is the left adjoint functor of $\tilde{i}_\lambda$.

So for any $X\in D_{sg}(\Gamma)$, there exist a distinguished triangle
\begin{equation}\label{equation triangle of X}
\tilde{i}_\lambda\tilde{i} X\rightarrow X\rightarrow \tilde{j}_\rho\tilde{j}X \xrightarrow{h} \Sigma\tilde{i}_\lambda\tilde{i} X.
\end{equation}
We claim that $h=0$ for any $X$.
Since $D_{sg}(\Gamma)\simeq \underline{\Gproj}(\Gamma)$, we assume that $X\in \Gproj \Gamma\subseteq \mod\Gamma$.
Let $\dim X_v=r$ and $I$ the indecomposable injective $A$-module corresponding to $v_1$. Then it is easy to see that $h$ factors through $_\Gamma i_\lambda(I)^{\oplus r}$.
By $A$ is Gorenstein, we get that $\pd_A I<\infty$, and then $ \pd_\Gamma i_\lambda(I)<\infty$ since $i_\lambda$ is exact and preserves projective modules.
So $h=0$. It implies the triangle (\ref{equation triangle of X}) splits, so $X\cong X_1\oplus X_2$, where $X_1=\tilde{i}_\lambda\tilde{i} X\in \Im(\tilde{i}_\lambda)$, $X_2=\tilde{j}_\rho\tilde{j}X\in\Im(\tilde{j}_\rho)$. In fact, $\Hom_{D_{sg}(\Gamma)}(\Im( \tilde{i}_\lambda), \Im(\tilde{j}_\rho))=0$; Similar to the above, we can get that
$\Hom_{D_{sg}(\Gamma)}(\Im(\tilde{j}_\rho),\Im( \tilde{i}_\lambda))=0$. Thus $D_{sg}(\Gamma)\simeq D_{sg}(A)\coprod D_{sg}(B)$.
It follows that $\underline{\Gproj}(\Gamma)\simeq \underline{\Gproj}(A)\coprod\underline{\Gproj}(B)$.
\end{proof}

\begin{proposition}\label{proposition Gorenstein of the gluing Nakayama algebras}
For any gluing Nakayama algebra $\Gamma$, if the gluing components $A_1=kQ_1/I_1$, $A_2=kQ_2/I_2$, $\cdots$, $A_m=kQ_m/I_m$ of $\Gamma$ are Gorenstein, then
$\Gamma$ is Gorenstein.
\end{proposition}

\begin{proof}
We prove it by inducting on $m$. By Lemma \ref{Gorenstein dimension of upper triangular matrix algebra}, we assume that all the gluing components are of type II.

If $m=1$, it is obvious.
We assume that for any gluing Nakayama algebra $A'$ with $m-1$ gluing components is Gorenstein.
For $n$, let $A$ be the gluing Nakayama algebra with the gluing components $A_1=kQ_1/I_1$, $A_2=kQ_2/I_2$, $\cdots$, $A_{m-1}=kQ_{m-1}/I_{m-1}$.
Let $B=kQ_m/I_m$.
With the same proof as Lemma \ref{lemma simple gluing Nakayama algebra Gorenstein}, we can get $\Gamma$ is Gorenstein.
\end{proof}

\begin{corollary}
For any gluing Nakayama algebra $\Gamma$, if the gluing components $A_1=kQ_1/I_1$, $A_2=kQ_2/I_2$, $\cdots$, $A_m=kQ_m/I_m$ of $\Gamma$ are selfinjective, then
$\Gamma$ is Gorenstein with the Gorenstein dimension at most $1$.
\end{corollary}
\begin{proof}
It follows from Lemma \ref{Gorenstein dimension of upper triangular matrix algebra} and Lemma \ref{lemma simple gluing Nakayama algebra Gorenstein} inductively.
\end{proof}
\begin{theorem}\label{theorem stable category of gluing Nakayama algebra}
For any gluing Nakayama algebra $\Gamma$, if the gluing components $A_1=kQ_1/I_1$, $A_2=kQ_2/I_2$, $\cdots$, $A_m=kQ_m/I_m$ of $\Gamma$ are Gorenstein, then
$$\underline{\Gproj}(\Gamma)\simeq \coprod_{i=1}^m\underline{\Gproj}(A_i).$$
\end{theorem}
\begin{proof}
We prove it by induction on $m$.
If $m=1$, it is obvious.
For $m$, if there exist $A_i$ such that it is of type I, then by \cite[Lemma 3.5]{CGLu} and the inductive hypothesis, we get the conclusion immediately.
If all of $A_i$ are of type II, by Lemma \ref{lemma the singularity category of simple gluing Nakayama algebras}, we also get the conclusion immediately.
\end{proof}
\begin{corollary}
For any gluing Nakayama algebra $\Gamma$, if the gluing components $A_1=kQ_1/I_1$, $A_2=kQ_2/I_2$, $\cdots$, $A_m=kQ_m/I_m$ of $\Gamma$ are selfinjective, then
$$\underline{\Gproj}(\Gamma)\simeq \coprod_{i=1}^m\underline{\mod}(A_i).$$
\end{corollary}



\begin{corollary}\cite{Ka,CGLu}
Let $kQ/I$ be a cluster-tilted algebra of type $A$. Then
$$\underline{\Gproj}(kQ/I)\simeq \coprod_{t(Q)} \underline{\mod} S_3,$$
where $t(Q)$ is the number of the triangles of $Q$, and $S_3$ is the selfinjective cluster-tilted algebra of type $A_3$.
\end{corollary}
\begin{proof}
From \cite{CCS1, BV}, we know that cluster-tilted algebras of type $A$ are gluing Nakayama algebras, so it follows from Theorem \ref{theorem stable category of gluing Nakayama algebra}
immediately.
\end{proof}


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\end{document}

1412.0910.extracted_bib
{AR1 M. Auslander and I. Reiten. Application of contravariantly finite subcategories, Adv. Math. 86 (1991), 111-152.
1412.0910.extracted_bib
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