# Categorical properties and homological conjectures for bounded extensions of
algebras

Yongyun Qin, Xiaoxiao Xu, Jinbi Zhang* and Guodong Zhou  Yongyun Qin, School
of Mathematics, Yunnan Key Laboratory of Modern Analytical Mathematics and
Applications, Yunnan Normal University, Kunming 650500, China.
<EMAIL_ADDRESS>Xiaoxiao Xu and Guodong Zhou, School of Mathematical
Sciences, Key Laboratory of Mathematics and Engineering Applications of
Ministry of Education, Shanghai Key Laboratory of PMMP, East China Normal
University, Shanghai 200241, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Jinbi Zhang, School of Mathematical Sciences, Anhui
University, Hefei 230601, China<EMAIL_ADDRESS>
###### Abstract.

An extension $B\subset A$ of finite dimensional algebras is bounded if the
$B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is
finite and ${\rm Tor}_{i}^{B}(A/B,(A/B)^{\otimes_{B}j})=0$ for all $i,j\geq
1$. We show that for a bounded extension $B\subset A$, the algebras $A$ and
$B$ are singularly equivalent of Morita type with level. Additively, under
some conditions, their stable categories of Gorenstein projective modules and
Gorenstein defect categories are equivalent, respectively. Some homological
conjectures are also investigated for bounded extensions, including Auslander-
Reiten conjecture, finististic dimension conjecture, Fg condition, Han’s
conjecture, and Keller’s conjecture. Applications to trivial extensions and
triangular matrix algebras are given.

In course of proofs, we give some handy criteria for a functor between module
categories induces triangle functors between stable categories of Gorenstein
projective modules and Gorenstein defect categories, which generalise some
known criteria and hence, might be of independent interest.

###### Key words and phrases:

Gorenstein defect category; Homological conjecture; Ring extension; Singular
equivalence of Morita type with level; Singularity category.
*Corresponding author

###### 2010 Mathematics Subject Classification:

16E35 16E30 16E40 16E65 16G50 16D20

###### Contents

  1. 1 Introduction
  2. 2 Bounded extensions
  3. 3 Singular equivalences of Morita type with level induced by bounded extensions
  4. 4 The stable categories of Gorenstein projective modules and the Gorenstein defect categories for bounded extensions
  5. 5 Homological conjectures for bounded extensions
  6. 6 Examples

##  1\. Introduction

Let $A$ be a finite dimensional algebra over a field ${\bf k}$. Let
$\text{{\rm Mod}}(A)$ be the category of left $A$-modules, and $\text{{\rm
mod}}(A)$ be the category of finitely generated left $A$-modules. Let
$\mathrm{proj}(A)$ (resp. $\mathrm{inj}(A)$) be the subcategory of finitely
generated projective (resp. injective) left $A$-modules. We denoted by ${\rm
K}(A)={\rm K}(\text{{\rm Mod}}(A))$ the homotopy category of complexes over
$\text{{\rm Mod}}(A)$, by $\mathrm{D}(A)=\mathrm{D}(\text{{\rm Mod}}(A))$ the
derived category of complexes over $\text{{\rm Mod}}(A)$, and by
$\mathrm{D}^{b}(\text{{\rm mod}}(A))$ the derived category of bounded
complexes over $\text{{\rm mod}}(A)$. The _singularity category_
$\mathrm{D}^{b}_{\mathrm{sg}}(A)$ of $A$ is the Verdier quotient

$\mathrm{D}^{b}_{\mathrm{sg}}(A):=\mathrm{D}^{b}(\text{{\rm
mod}}(A))/{\mathrm{K}}^{b}(\mbox{{\rm proj}}(A))$

of the bounded derived category $\mathrm{D}^{b}(\text{{\rm mod}}(A))$ with
respect to the subcategory of bounded complexes of projective modules; see [6,
35]. The singularity category 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 $\mathrm{D}^{b}_{\mathrm{sg}}(A)$ is trivial.

The singularity category also captures the stable homological features of an
algebra $A$ ([6]). Denote by $\mbox{{\rm{\text@underline{Gproj}}}}(A)$ the
stable category of finitely generated Gorenstein projective $A$-modules.
Following [6], there is a natural triangle functor
$F:\mbox{{\rm{\text@underline{Gproj}}}}(A)\rightarrow\mathrm{D}^{b}_{\mathrm{sg}}(A)$
which is always fully faithful, and $F$ is an equivalence if and only if $A$
is Gorenstein (see also [25]). Inspired by this, Bergh, Jørgensen, and
Oppermann ([5]) defined the _Gorenstein defect category_ as the Verdier
quotient

$\mathrm{D}^{b}_{\mathrm{def}}(A):=\mathrm{D}^{b}_{\mathrm{sg}}(A)/{\rm
Im}(F)$

and proved that $A$ is Gorenstein if and only if
$\mathrm{D}^{b}_{\mathrm{def}}(A)=0$. This means that the Gorenstein defect
category of $A$ measures how far the algebra $A$ is from being Gorenstein.

The essential image of the functor $F$ is also clear. Denote by
$\mathrm{D}^{b}_{\rm fGd}(A)$ the full subcategory of
$\mathrm{D}^{b}(\text{{\rm mod}}(A))$ formed by those complexes quasi-
isomorphic to bounded complexes of Gorenstein projective modules, that is,
$\mathrm{D}^{b}_{\rm fGd}(A)$ is the thick subcategory of
$\mathrm{D}^{b}(\text{{\rm mod}}(A))$ generated by $\mbox{{\rm Gproj}}(A)$,
see [31, Theorem 2.7]. The following triangle equivalence

$\mbox{{\rm{\text@underline{Gproj}}}}(A)\simeq\mathrm{D}^{b}_{\rm
fGd}(A)/{\mathrm{K}}^{b}(\mbox{{\rm proj}}(A))$

is well known, see, for instance, [6, Theorem 4.4.1], [36, Theorem 4], [12,
Lemma 4.1], or in a much more general setup [46, Theorem 3.7]. We will see
that the subcategory $\mathrm{D}^{b}_{\rm fGd}(A)$ play an important role in
this paper.

Two algebras $A$ and $B$ are called _singularly equivalent_ provided that
there is a triangle equivalence between $\mathrm{D}^{b}_{\mathrm{sg}}(A)$ and
$\mathrm{D}^{b}_{\mathrm{sg}}(B)$. Such an equivalence is called a _singular
equivalence_ between $A$ and $B$. In [44], Wang introduced the notation of
singular equivalence of Morita type with level, which is a generation of
singular equivalence of Morita type (see [13, 48, 14]). This equivalence
provides rich structural information, and plays an important role in the study
of singular equivalences and homological properties. For example, it was
verified in [44] and [9] that the finitistic conjecture and Keller’s
conjecture for singular Hochschild cohomology are invariant under singular
equivalences of Morita type with level. Thus we are interested in constructing
singular equivalences of Morita type with level.

We recall Wang’s definition:

######  Definition 1.1 ([44]).

Given two finite dimensional ${\bf k}$-algebras $A$ and $B$, and let $M$ be an
$A$-$B$-bimodule and $N$ be a $B$-$A$-bimodule. We say that
$(_{A}M_{B},{{}_{B}N_{A}})$ defines a singular equivalence of Morita type with
level $l$ for some $l\in\mathbb{N}$ if $M$ is projective as a left $A$-module
and as a right $B$-module, $N$ is projective as a left $B$-module and as a
right $A$-module, and the following conditions are satisfied:

$M\otimes_{B}N\cong\Omega^{l}_{A^{e}}(A)\mbox{ in
}\mbox{{\rm{\text@underline{mod}}}}(A^{e})\mbox{ and
}N\otimes_{A}M\cong\Omega^{l}_{B^{e}}(B)\mbox{ in
}\mbox{{\rm{\text@underline{mod}}}}(B^{e}),$

where $\Omega^{l}_{A^{e}}(-)$ and $\Omega^{l}_{B^{e}}(-)$ denote the syzygy
endofunctor of the stable module category of $A^{e}$ and $B^{e}$,
respectively.

In this situation, there is a triangle equivalence

$M\otimes_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\rightarrow\mathrm{D}^{b}_{\mathrm{sg}}(A)$

with quasi-inverse

$[l]\circ(N\otimes_{A}-):\mathrm{D}^{b}_{\mathrm{sg}}(A)\rightarrow\mathrm{D}^{b}_{\mathrm{sg}}(B).$

In [10], Chen, Liu, and Wang proved that a tensor functor with a bimodule
defines a singular equivalence Morita type with level under certain
conditions, and in [20], Dalezios proved that for certain Gorenstein algebras,
a singular equivalence induced from tensoring with a complex of bimodules
always induces a singular equivalence of Morita type with level. Qin in [38]
gave a complex version of Chen-Liu-Wang’s work, which extended Dalezios’
result to arbitrary algebra (not limited to Gorenstein algebra).

Given an algebra $A$, there is an exact sequence of triangulated categories:

$0\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(A)\rightarrow\mathrm{D}^{b}_{\mathrm{sg}}(A)\rightarrow\mathrm{D}^{b}_{\mathrm{def}}(A)\rightarrow
0.$

A natural and fundamental question is: When do two related algebras share the
same stable category of Gorenstein projective modules, singularity category
and Gorenstein defect category up to triangle equivalence? For partial answers
to this question, we refer to [7, 8, 31, 33, 39, 47]. In this paper, we also
give a partial answer, that is, we will compare the stable category of
Gorenstein projective modules, singularity category and Gorenstein defect
category between two algebras linked by a ring extension. From [19], an
extension $B\subset A$ is called left (resp. right) bounded if $A/B$ has
finite projective dimension as a $B$-$B$-bimodule, $A/B$ is projective as a
left (resp. right) $B$-module and some tensor power $(A/B)^{\otimes_{B}p}$ is
0. Inspired by this, we consider a more general situation:

######  Definition 1.2.

An extension $B\subset A$ of finite dimensional algebras is called bounded if

  * $(1)$

there exists $p\geq 1$ such that the tensor power $(A/B)^{\otimes_{B}p}$
vanishes;

  * $(2)$

$A/B$ has finite projective dimension as a $B$-$B$-bimodule;

  * $(3)$

${\rm Tor}_{i}^{B}(A/B,(A/B)^{\otimes_{B}j})=0$ for all $i,j\geq 1$.

Examples of bounded extensions in the sense of the above definition but not in
the sense of [19] are given in Proposition 2.3 and Example 2.4.

Let ${\rm Gproj}(A)^{\bot}:=\\{X\in\text{{\rm mod}}(A)\mid{\rm
Ext}_{A}^{i}(U,X)=0\mbox{ for all }U\in\mbox{{\rm Gproj}}(A),\,i\geq 1\\}$.
The first main result we obtain in this paper is as follows.

######  Theorem 1.3.

(Theorems 3.7, 3.8, 4.11 and 4.13) Let $B\subset A$ be a bounded extension.
Then

${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{sg}}(A):\mathrm{Res}^{A}_{B}=_{B}A\otimes_{A}-$

is an adjoint triangle equivalence which is in fact a singular equivalence of
Morita type with level between $A$ and $B$, and the functor
$\mathrm{Res}^{A}_{B}:\text{{\rm mod}}(A)\rightarrow\text{{\rm mod}}(B)$ is an
eventually homological isomorphism in the sense of [37].

Moreover, there are triangle equivalences

${}_{A}A\otimes_{B}-:\mbox{{\rm{\text@underline{Gproj}}}}(B)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(A)\mbox{
and
}_{A}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{def}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{def}}(A):\mathrm{Res}^{A}_{B}$

if one of the following two conditions holds:

  * $(1)$

$\mathbb{R}{\rm Hom}_{B}(A,B)$ is quasi-isomorphic to a bounded complex with
each term in ${\rm Gproj}(A)^{\bot}$;

  * $(2)$

${}_{B}B$ is a direct summand of ${}_{B}A$.

In course of proof of the above result, we obtain some handy criteria for a
functor between module categories to induce triangle functors between stable
categories of Gorenstein projective modules, which is our second main result.

######  Criterion 1.4 (Criteria 4.8 and 4.12).

Let $X$ be an $\Gamma$-$\Lambda$-bimodule over two finite dimensional ${\bf
k}$-algebras $\Lambda$ and $\Gamma$ such that $X$ has finite projective
dimension both as left $\Gamma$-moudle and as right $\Lambda$-module. Assume
that $\mathbb{R}{\rm Hom}_{\Gamma}(X,\Gamma)$ is quasi-isomorphic to a bounded
complex with each term in ${\rm Gproj}(\Lambda)^{\bot}$, or that there is an
integer $t$ such that $\Omega^{t}(X\otimes_{\Lambda}M)\in{}^{\perp}{\Gamma}$
for any $M\in{}^{\perp}\Lambda$. Then $X\otimes^{\mathbb{L}}_{\Lambda}-$
induces a triangle functor from $\mathrm{D}^{b}_{\rm fGd}(\Lambda)$ to
$\mathrm{D}^{b}_{\rm fGd}({\Gamma})$, and hence a triangle functor from
$\mbox{{\rm{\text@underline{Gproj}}}}(\Lambda)\simeq\mathrm{D}^{b}_{\rm
fGd}(\Lambda)/{\mathrm{K}}^{b}(\mbox{{\rm proj}}(\Lambda))$ to
$\mbox{{\rm{\text@underline{Gproj}}}}(\Gamma)\simeq\mathrm{D}^{b}_{\rm
fGd}(\Gamma)/{\mathrm{K}}^{b}(\mbox{{\rm proj}}(\Gamma))$.

This result is of independent interest as it generalises some known criteria
appeared in the litterature; see [27] and [34, Theorem I (ii)].

We first apply Theorem 1.3 to several homological conjectures, which are still
open now. These conjectures play important roles in representation theory of
finite dimensional algebras and homological algebra. Our third main result
concerns reductions of some homological conjectures via bounded extensions.

######  Theorem 1.5.

(Theorem 5.1 and 5.3) Let $B\subset A$ be a bounded extension. Then the
following statements hold:

  * $(1)$

$B$ satisfies the finitistic dimension conjecture if and only if so does $A$.

  * $(2)$

$B$ satisfies Han’s conjecture if and only if so does $A$.

  * $(3)$

$B$ satisfies Keller’s conjecture if and only if so does $A$.

If, in addition, $\mathbb{R}{\rm Hom}_{B}(A,B)$ is perfect as left $A$-module
or ${}_{B}B$ is a direct summand of ${}_{B}A$, then

  * $(4)$

$B$ satisfies Auslander-Reiten’s conjecture if and only if so does $A$.

  * $(5)$

$B$ satisfies the Fg condition for the Hochschild cohomology if and only if so
does $A$.

Note that the bounded extension in our definition is less restrictive than the
left or right bounded extension introduced by Cibils, Lanzilotta, Marcos, and
Solotar[19] (but the present definition was also suggested by them [17], see
Remark 2.2 (2)). Consequently, Theorem 1.5 (2) extends the main result of
[19].

Moreover, we apply Theorem 1.3 to trivial extensions and triangular matrix
algebras, and get several reduction methods on singularity categories and
Gorenstein defect categories; see Propositions 6.4, 6.6, and Examples 6.5,
6.7. In particular, since the arrow removal operation yields a bounded split
extension, we reobtain [24, Theorem A (ii)] and the main result of [23] in
Example 6.5.

##  2\. Bounded extensions

In this section, we introduce bounded extensions and provide some examples.

Let $B$ be a finite dimensional algebra over a fixed field ${\bf k}$ and $M$
be a $B$-$B$-bimodule. Define $M^{\otimes_{B}0}=B$ and
$M^{\otimes_{B}(i+1)}=M\otimes_{B}M^{\otimes_{B}i}$ for $i\geq 0$. We say that
$M$ is _$B$ -tensor nilpotent_ if $M^{\otimes_{B}n}=0$ for some $n\geq 1$.

######  Definition 2.1.

An extension $B\subset A$ of finite dimensional algebras is called bounded if

  * $(1)$

$A/B$ is $B$-tensor nilpotent;

  * $(2)$

$A/B$ has finite projective dimension as a $B$-$B$-bimodule;

  * $(3)$

${\rm Tor}_{i}^{B}(A/B,(A/B)^{\otimes_{B}j})=0$ for all $i,j\geq 1$.

######  Remark 2.2.

  * $(1)$

It follows from [32, Proposition 2.4] that ${\rm
Tor}_{i}^{B}(A/B,(A/B)^{\otimes_{B}j})=0$ for all $i,j\geq 1$ if and only if
${\rm Tor}_{i}^{B}((A/B)^{\otimes_{B}j},A/B)=0$ for all $i,j\geq 1$.

  * $(2)$

This definition of bounded extensions is not essentially new and has in fact
been suggested by Cibils, Lanzilotta, Marcos and Solotar in [17].

Now, we give some examples of extensions which satisfy our definition, but not
the definition of Cibils, Lanzilotta, Marcos, and Solotar in [19], where they
required that $A/B$ is projective as a left or right $B$-module.

Let $Q=(Q_{0},Q_{1})$ be a quiver with $Q_{0}$ the set of vertices and $Q_{1}$
the set of arrows between vertices. The source and target of an arrow
$\alpha\in Q_{1}$ are denoted by $s(\alpha)$ and $t(\alpha)$, respectively.
Note that the composition of two arrows $\alpha,\beta\in Q_{1}$ is written as
$\beta\alpha$, where $\alpha$ comes first and then $\beta$ follows. Suppose
that $Q$ is a finite acyclic quiver, that is, a quiver with finitely many
vertices and finitely many arrows without loops and oriented cycles. Then the
path algebra ${\bf k}Q$ of $Q$ is a finite dimensional ${\bf k}$-algebra of
global dimension $1$. As usual, let $e_{i}$ be the primitive idempotent
element in ${\bf k}Q$ corresponding to the vertex $i\in Q_{0}$, and let $S(i)$
and $T(i)$ stand for the simple left $B$-module and the simple right
$B$-module corresponding to the vertex $i\in Q_{0}$, respectively.

######  Proposition 2.3.

Let $B={\bf k}Q$ be a finite quiver algebra without oriented cycle and $u\neq
v\in Q_{0}$. Define $M:=S(u)\otimes_{{\bf k}}T(v)$ and let $A:=B\ltimes M$ be
the trivial extension of $B$ by $M$. Suppose that there are arrows originating
from $u$ and arrows terminating at $v$. If there doesn’t exist paths from $u$
to $v$, then $B\subset A$ is a bounded extension with
${}_{B}M\not\in\mbox{{\rm proj}}(B)$ and $M_{B}\not\in\mbox{{\rm proj}}(B^{\rm
op})$.

###### Proof.

It is clear that $A/B=M$. Since the algebra $B$ has finite global dimension,
then $B^{e}$ also have it and the $B$-$B$-bimodules are of finite projective
dimension, in particular $M$ is of finite projective dimension as a
$B$-$B$-bimodule. Moreover, it follows from $u\neq v$ that
$T(v)\otimes_{B}S(u)=0$ and then $M\otimes_{B}M=0$. Thus $M$ is $B$-tensor
nilpotent as a $B^{e}$-module. Let

$0\rightarrow\mathop{\bigoplus}_{\alpha\in
Q_{1}\atop{s(\alpha)=u}}Be_{t(\alpha)}\rightarrow Be_{u}\rightarrow
S(u)\rightarrow 0$

be a projective resolution of $S(u)$ as a left $B$-module. Since there doesn’t
exist paths from $u$ to $v$ and $u\neq v$, we get that
$T(v)\otimes_{B}Be_{t(\alpha)}\cong T(v)e_{t(\alpha)}=0$ for $\alpha\in Q_{1}$
with $s(\alpha)=u$, and $T(v)\otimes_{B}Be_{u}\cong T(v)e_{u}=0$. Note that
$M\cong S(u)$ as a left $B$-modules and $M\cong T(v)$ as a right $B$-modules.
Applying the functor $T(v)\otimes_{B}-$ to the above projective resolution, we
get that ${\rm Tor}^{B}_{i}(M,M)=0$ for all $i\geq 1$. Since
$M\otimes_{B}M=0$, we have that ${\rm Tor}^{B}_{i}(M,M^{\otimes_{B}j})=0$ for
all $i,j\geq 1$. Hence $B\subset A$ is a bounded extension. Since there are
arrows starting $u$ and arrows ending $v$, ${}_{B}S(u)$ and $T(v)_{B}$ are not
projective. Therefore, ${}_{B}M\cong S(u)\not\in\mbox{{\rm proj}}(B)$ and
$M_{B}\cong T(v)\not\in\mbox{{\rm proj}}(B^{\rm op})$. ∎

A concrete example can be constructed in the spirit of the above result.

######  Example 2.4.

Let $B$ be the following quiver algebra over ${\bf k}$

$1\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}2\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}3\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}4.$

Define $M:=S(3)\otimes_{{\bf k}}T(2)$ and $A:=B\ltimes M$. Then it follows
from Proposition 2.3 that $B\subset A$ is a bounded extension and
${}_{B}M\not\in\mbox{{\rm proj}}(B)$ and $M_{B}\not\in\mbox{{\rm proj}}(B^{\rm
op})$. Hence, the extension $B\subset A$ satisfies our condition in Definition
2.1, but not the definition of [19].

##  3\. Singular equivalences of Morita type with level induced by bounded
extensions

In this section, we are concerned about the singular equivalences induced by
bounded extensions.

For our proof of Theorem 1.3, we need the following three lemmas whose easy
proofs are left to the reader.

######  Lemma 3.1.

Let $A$, $B$ and $C$ be finite dimensional algebras, $M$ be an $A$-$B$
bimodule and $N$ be a $B$-$C$ bimodule. If ${\rm Tor}_{i}^{B}(M,N)=0$ for any
$i>0$, then $M\otimes_{B}^{\mathbb{L}}N\cong M\otimes_{B}N$ in
$\mathrm{D}^{b}(\text{{\rm mod}}(A\otimes_{{\bf k}}C^{op}))$.

######  Lemma 3.2.

Let $B\subset A$ be an extension of finite dimensional algebras and denote
$M=A/B$. Then the following statements hold:

  * $(1)$

The functor
$A\otimes_{B}^{\mathbb{L}}-\otimes_{B}^{\mathbb{L}}A:\mathrm{D}^{b}(\text{{\rm
mod}}(B^{e}))\rightarrow\mathrm{D}^{b}(\text{{\rm mod}}(A^{e}))$ sends
${\mathrm{K}}^{b}(\mbox{{\rm proj}}(B^{e}))$ to $\mathrm{K}^{b}(\mbox{{\rm
proj}}(A^{e}))$.

  * $(2)$

If ${\rm pd}(_{B}M_{B})<\infty$, then the functor
${}_{B}M\otimes_{B}^{\mathbb{L}}-:\mathrm{D}^{b}(\text{{\rm
mod}}(B^{e}))\rightarrow\mathrm{D}^{b}(\text{{\rm mod}}(B^{e}))$ sends
${\mathrm{K}}^{b}(\mbox{{\rm proj}}(B^{e}))$ to ${\mathrm{K}}^{b}(\mbox{{\rm
proj}}(B^{e}))$.

######  Lemma 3.3 (compare with [11, Section 4]).

Let $B\subset A$ be an extension of finite dimensional algebras. If ${\rm
Tor}_{i}^{B}(A/B,(A/B)^{\otimes_{B}j})=0$ for each $i,j\geq 1$, then the
following statements hold:

  * $(1)$

${\rm Tor}_{i}^{B}(A,A)=0$ for each $i\geq 1$;

  * $(2)$

${\rm Tor}_{i}^{B}((A/B)^{\otimes_{B}j},A)=0$ for each $i,j\geq 1$;

  * $(3)$

${\rm Tor}_{i}^{B}(A,(A/B)^{\otimes_{B}j}\otimes_{B}A)=0$ for each $i,j\geq
1$.

The following lemma will be crucial in the proof of Theorem 3.7.

######  Lemma 3.4.

Let $B\subset A$ be an extension of finite dimensional algebras and denote
$M=A/B$. Assume that ${\rm pd}(_{B}M_{B})<\infty$ and that ${\rm
Tor}_{i}^{B}(M,M^{\otimes_{B}j})=0$ for all $i,j\geq 1$. Then
$A\otimes_{B}M^{\otimes_{B}j}\otimes_{B}A\in{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(A^{e}))$ for any $j\geq 1$.

###### Proof.

Since ${\rm Tor}_{i}^{B}(M,M^{\otimes_{B}j})=0$ for each $i,j\geq 1$, it
follows from Lemma 3.1 that

$M\otimes_{B}M\otimes_{B}\cdots\otimes_{B}M\cong
M\otimes_{B}^{\mathbb{L}}M\otimes_{B}^{\mathbb{L}}\cdots\otimes_{B}^{\mathbb{L}}M\in\mathrm{D}^{b}(\text{{\rm
mod}}(B^{e})),$

which belongs to ${\mathrm{K}}^{b}(\mbox{{\rm proj}}(B^{e}))$ by ${\rm
pd}(_{B}M_{B})<\infty$ and by Lemma 3.2 (2). Therefore,
$M^{\otimes_{B}j}\in{\mathrm{K}}^{b}(\mbox{{\rm proj}}(B^{e}))$ for any $j\geq
1$, and it follows from Lemma 3.2 (1) that

$A\otimes_{B}^{\mathbb{L}}M^{\otimes_{B}j}\otimes_{B}^{\mathbb{L}}A\in{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(A^{e})).$

On the other hand, Lemma 3.3 (2), Lemma 3.3 (3) and Lemma 3.1 yield that

$A\otimes_{B}^{\mathbb{L}}M^{\otimes_{B}j}\otimes_{B}^{\mathbb{L}}A\cong
A\otimes_{B}M^{\otimes_{B}j}\otimes_{B}A.$

Therefore,
$A\otimes_{B}M^{\otimes_{B}j}\otimes_{B}A\in{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(A^{e}))$ for any $j\geq 1$. ∎

######  Lemma 3.5.

[34, Theorem 3.1 and Proposition 3.7] Let $B\subset A$ be an extension of
finite dimensional algebras. If ${\rm pd}(A_{B})<\infty$ and ${\rm
pd}({}_{B}A)<\infty$, then there is an adjoint pair

${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{sg}}(A):\mathrm{Res}^{A}_{B}=_{B}A\otimes_{A}-.$

Assume, moreover, that ${\rm pd}({{}_{B}M_{B}})<\infty$ with $M=A/B$, then
${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\to\mathrm{D}^{b}_{\mathrm{sg}}(A)$
is fully faithful.

######  Remark 3.6.

Let us remark on the proof of the first statement of the above result.
Consider the adjoint triple between derived categories

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Since ${\rm pd}(A_{B})<\infty$ and ${\rm pd}({}_{B}A)<\infty$, this adjoint
triple extends one step upwards and downwards, respectively. In this case, by
[1, Lemma 2.5], the three functors: the left adjoint of
$A\otimes_{B}^{\mathbb{L}}-$, $A\otimes_{B}^{\mathbb{L}}-$ and
$\mathrm{Res}^{A}_{B}$ restrict to ${\mathrm{K}}^{b}(\rm proj)$, by [1, Lemma
2.8] and [34, Lemma 3.2], the three functors: $A\otimes_{B}^{\mathbb{L}}-$,
$\mathrm{Res}^{A}_{B}$ and the right adjoint of $\mathrm{Res}^{A}_{B}$
restrict to $\mathrm{D}^{b}(\rm mod)$, hence, both
$A\otimes_{B}^{\mathbb{L}}-$ and $\mathrm{Res}^{A}_{B}$ restrict to
$\mathrm{D}^{b}_{\mathrm{sg}}$, (see [35, Lemma 1.2]).

This kind of restriction results are rather useful.

Now we will show that for a bounded extension $A\subset B$, the adjoint pair
is an adjoint equivalence. Furthermore, this singular equivalence is of
special form, singular equivalence of Morita type with level in the sense of
Wang [44]. This constitutes the first part of Theorem 1.3 in the introduction.

######  Theorem 3.7.

Let $B\subset A$ be a bounded extension. Then

${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{sg}}(A):\mathrm{Res}^{A}_{B}$

is an adjoint equivalence which induces a singular equivalence of Morita type
with level between $A$ and $B$.

###### Proof.

Denote $M=A/B$. Since ${\rm pd}({}_{B}M_{B})<\infty$, we get that ${\rm
pd}(M_{B})<\infty$ and ${\rm pd}({}_{B}M)<\infty$. Now the exact sequence of
$B$-$B$-bimodules $0\rightarrow B\rightarrow A\rightarrow M\rightarrow 0$
shows that ${\rm pd}(A_{B})<\infty$ and ${\rm pd}({}_{B}A)<\infty$. By Lemma
3.5, there is an adjoint pair

${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{sg}}(A):\mathrm{Res}^{A}_{B}$

and
${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\to\mathrm{D}^{b}_{\mathrm{sg}}(A)$
is fully faithful.

What we add here is that

${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{sg}}(A):\mathrm{Res}^{A}_{B}$

is an adjoint equivalence which are in fact singular equivalences of Morita
type with level. By [20, Theorem 3.6] and [38, Proposition 3.3] it suffices to
show that the natural morphisms $B\rightarrow GF(B)$ and $FG(A)\rightarrow A$
are isomorphisms in $\mathrm{D}^{b}_{\mathrm{sg}}(B^{e})$ and
$\mathrm{D}^{b}_{\mathrm{sg}}(A^{e})$, respectively, where
$F={{}_{A}}A\otimes^{\mathbb{L}}_{B}-$ and $G=\mathrm{Res}^{A}_{B}$.

We observe that the mapping cone of $B\rightarrow GF(B)$ is $M$, and the
assumption ${\rm pd}(_{B}M_{B})<\infty$ yields that $B\cong GF(B)$ in
$\mathrm{D}^{b}_{\mathrm{sg}}(B^{e})$.This follows also from the fact that
${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\to\mathrm{D}^{b}_{\mathrm{sg}}(A)$
is fully faithful.

By Lemma 3.3 (1) and Lemma 3.1, we have that

$FG(A)=A\otimes_{B}^{\mathbb{L}}A\cong
A\otimes_{B}A\in\mathrm{D}^{b}(\text{{\rm mod}}(A^{e})),$

and then mapping cone of $FG(A)\rightarrow A$ is isomorphic to the complex

$0\rightarrow A\otimes_{B}A\rightarrow A\rightarrow 0$

induced by the product of $A$. Since $M^{\otimes_{B}p}=0$ for some integer
$p$, it follows from [18, Proposition 2.3] that there is a long exact sequence
of $A$-$A$-bimodules

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{B}M^{\otimes_{B}{(p-1)}}\otimes_{B}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{B}M\otimes_{B}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{B}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the last non-zero differential is the product of $A$. Therefore, the
mapping cone of $FG(A)\rightarrow A$ is isomorphic to the complex

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{B}M^{\otimes_{B}{(p-1)}}\otimes_{B}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{B}M\otimes_{B}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

in $\mathrm{D}^{b}(\text{{\rm mod}}(A^{e}))$. Now by Lemma 3.4,
$A\otimes_{B}M^{\otimes_{B}j}\otimes_{B}A\in{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(A^{e}))$ for any $j\geq 1$, and thus $FG(A)\cong A$ in
$\mathrm{D}^{b}_{\mathrm{sg}}(A^{e})$. ∎

A functor $F:\mathcal{B}\rightarrow\mathcal{C}$ between abelian categories is
called a $t$-eventually homological isomorphism if there is an integer $t$
such that for every $j>t$, there is an isomorphism ${\rm
Ext}^{j}_{\mathcal{B}}(X,Y)\cong{\rm Ext}^{j}_{\mathcal{C}}(FX,FY)$ for all
objects $X,Y\in\mathcal{B}$. This notion was introduced in [37] and was used
widely to reduce homological properties of algebras [24, 37, 40].

######  Theorem 3.8.

Let $B\subset A$ be a bounded extension. Then the functor
$\mathrm{Res}^{A}_{B}:\text{{\rm mod}}(A)\rightarrow\text{{\rm mod}}(B)$ is a
$t$-eventually homological isomorphism for some integer $t$.

###### Proof.

Let $F={{}_{A}}A\otimes^{\mathbb{L}}_{B}-$ and $G=\mathrm{Res}^{A}_{B}$. Let
$\eta:FG\rightarrow 1_{D(A)}$ be the counit of the adjoint pair

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For any $X,Y\in\text{{\rm mod}}(A)$, there is a canonical triangle

$\textstyle{FG(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{X}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Cone}(\eta_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in $D(A)$. Applying ${\rm Hom}_{D(A)}(-,Y[i])$, we get a long exact sequence

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom}_{D(A)}({\rm
Cone}(\eta_{X}),Y[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom}_{D(A)}(X,Y[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom}_{D(A)}(FG(X),Y[i])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots(\star).}$

Since $FG(X)\cong FG(A)\otimes^{\mathbb{L}}_{A}X$, it follows from the
commutative diagram

$\textstyle{FG(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{X}}$$\scriptstyle{\cong}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\rm
Cone}(\eta_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{FG(A)\otimes^{\mathbb{L}}_{A}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{A}\otimes^{\mathbb{L}}_{A}X}$$\textstyle{A\otimes^{\mathbb{L}}_{A}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Cone}(\eta_{A})\otimes^{\mathbb{L}}_{A}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

that ${\rm Cone}(\eta_{X})\cong{\rm Cone}(\eta_{A})\otimes^{\mathbb{L}}_{A}X$
in $D(A)$. On the other hand, it follows from the proof of Theorem 3.7 that
${\rm Cone}(\eta_{A})\in{\mathrm{K}}^{b}(\mbox{{\rm proj}}(A^{e}))$. Hence,
there is a bounded complex of projective $A$-$A$ bimodules

$P^{\bullet}:0\rightarrow P^{-t}\longrightarrow\cdots\longrightarrow
P^{s}\longrightarrow 0$

such that $P^{\bullet}\cong{\rm Cone}(\eta_{A})$ in $D(A^{e})$. Therefore, the
complex ${\rm Cone}(\eta_{X})\cong P^{\bullet}\otimes_{A}X$ is of the form

$0\rightarrow P^{-t}\otimes_{A}X\longrightarrow\cdots\longrightarrow
P^{s}\otimes_{A}X\longrightarrow 0,$

where each ${}_{A}P^{i}\otimes_{A}X\in\mathrm{proj}A$ since
$P^{i}\in\mathrm{proj}A^{e}$. It follows that

${\rm Hom}_{D(A)}({\rm Cone}(\eta_{X}),Y[i])\cong{\rm
Hom}_{K(A)}(P^{\bullet}\otimes_{A}X,Y[i]),$

which is equal to zero for any $i>t$. Therefore, the long exact sequence
$(\star)$ yields that

${\rm Hom}_{D(A)}(X,Y[i])\cong{\rm Hom}_{D(A)}(FG(X),Y[i])\cong{\rm
Hom}_{D(B)}(GX,GY[i])$

for any $i>t$. Therefore, ${\rm Ext}_{A}^{i}(X,Y)\cong{\rm
Ext}_{B}^{i}(GX,GY)$ for any $i>t$ and any $X,Y\in\text{{\rm mod}}(A)$. This
shows that $G$ is a $t$-eventually homological isomorphism. ∎

##  4\. The stable categories of Gorenstein projective modules and the
Gorenstein defect categories for bounded extensions

In this section, we will compare the stable categories of Gorenstein
projective modules and the Gorenstein defect categories between two algebras
linked by a ring extension.

Throughout this section, let $\Lambda$ and $\Gamma$ be two finite dimensional
${\bf k}$-algebras and let $X$ be a finite dimensional
$\Gamma$-$\Lambda$-bimodule such that $X$ has finite projective dimension both
as left $\Gamma$-module and as right $\Lambda$-module.

We will consider the following question which generalises the situation
considered in [34]:

######  Question 4.1.

When does the functor $X\otimes_{\Lambda}-:\text{{\rm
mod}}(\Lambda)\rightarrow\text{{\rm mod}}(\Gamma)$ induce a triangle functor
from $\mbox{{\rm{\text@underline{Gproj}}}}(\Lambda)$ to
$\mbox{{\rm{\text@underline{Gproj}}}}(\Gamma)$?

Some remarks are in order. Following the spirit of Remark 3.6, under the
condition that ${\rm pd}(_{\Gamma}X),{\rm pd}(X_{\Lambda})<\infty$, the
adjoint pair between derived categories

$\textstyle{\mathrm{D}(\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X\otimes_{\Lambda}^{{\mathbb{L}}}-}$$\textstyle{\mathrm{D}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathbb{R}}{\rm
Hom}_{\Gamma}(X,-)}$

extends one step upwards and one step downwards and we obtain an adjoint
quadruple:

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where

$X^{{\rm tr}_{\Gamma}}={\mathbb{R}}{\rm Hom}_{\Gamma}(X,\Gamma),X^{{\rm
tr}_{\Lambda^{{\rm op}}}}={\mathbb{R}}{\rm Hom}_{\Lambda^{{\rm
op}}}(X,\Lambda).$

By [1, Lemma 2.5], the two functors $X^{{\rm tr}_{\Lambda^{{\rm
op}}}}\otimes_{\Gamma}^{{\mathbb{L}}}-$ and
$X\otimes_{\Lambda}^{{\mathbb{L}}}-\simeq{\mathbb{R}}{\rm Hom}(X^{{\rm
tr}_{\Lambda^{{\rm op}}}},-)$ restrict to ${\mathrm{K}}^{b}(\rm proj)$, by [1,
Lemma 2.8] and [34, Lemma 3.2] (or more generally fo differential graded
algebras, [28, Proposition 2.5]) the two functors
$X\otimes_{\Lambda}^{{\mathbb{L}}}-\simeq{\mathbb{R}}{\rm Hom}(X^{{\rm
tr}_{\Lambda^{{\rm op}}}},-)$ and ${\mathbb{R}}{\rm Hom}_{\Gamma}(X,-)\simeq
X^{{\rm tr}_{\Gamma}}\otimes_{\Gamma}^{{\mathbb{L}}}-$ restrict to
$\mathrm{D}^{b}(\rm mod)$, hence,
$X\otimes_{\Lambda}^{{\mathbb{L}}}-\simeq{\mathbb{R}}{\rm Hom}(X^{{\rm
tr}_{\Lambda^{{\rm op}}}},-)$ restricts to $\mathrm{D}^{b}_{\mathrm{sg}}$,
(see [35, Lemma 1.2]).

We give several criteria in order to answer Question 4.1.

The first criterion is equivalent to asking that the adjoint quadruple can be
extended further one step downwards.

######  Criterion 4.2 (Compare with [34, Propisition 3.5]).

If $X^{{\rm tr}_{\Gamma}}=\mathbb{R}{\rm Hom}_{\Gamma}(X,\Gamma)$ is perfect
as complex of $\Lambda$-modules, then the triangle functor
$X\otimes_{\Lambda}^{{\mathbb{L}}}-\simeq{\mathbb{R}}{\rm Hom}(X^{{\rm
tr}_{\Lambda^{{\rm op}}}},-)$ restricts to the stable categories of Gorenstein
projective modules.

###### Proof.

This can be deduced from Hu and Pan [27] and also is made explicit in [34] in
case that there is a ring homomorphism $f:\Lambda\to\Gamma$ and
$X={}_{\Gamma}\Gamma_{\Lambda}$.

In fact, the condition that $X^{{\rm tr}_{\Gamma}}=\mathbb{R}{\rm
Hom}_{\Gamma}(X,\Gamma)$ is perfect as complex of $\Lambda$-modules implies
that the adjoint quadruple:

$\textstyle{\mathrm{D}(\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X\otimes_{\Lambda}^{{\mathbb{L}}}-\simeq{\mathbb{R}}{\rm
Hom}(X^{{\rm tr}_{\Lambda^{{\rm op}}}},-)}$$\scriptstyle{{\mathbb{R}}{\rm
Hom}_{\Lambda}(X^{{\rm
tr}_{\Gamma}},-)}$$\textstyle{\mathrm{D}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X^{{\rm
tr}_{\Lambda^{{\rm
op}}}}\otimes_{\Gamma}^{{\mathbb{L}}}-}$$\scriptstyle{{\mathbb{R}}{\rm
Hom}_{\Gamma}(X,-)\simeq X^{{\rm
tr}_{\Gamma}}\otimes_{\Gamma}^{{\mathbb{L}}}-}$

can be extended further one step downwards, because ${\mathbb{R}}{\rm
Hom}_{\Lambda}(X^{{\rm tr}_{\Gamma}},-)\simeq(X^{{\rm tr}_{\Gamma}})^{{\rm
tr}_{\Lambda}}\otimes_{\Lambda}^{\mathbb{L}}-$ has a right adjoint
${\mathbb{R}}{\rm Hom}_{\Lambda}((X^{{\rm tr}_{\Gamma}})^{{\rm
tr}_{\Lambda}},-)$. The functor

$G:={\mathbb{R}}{\rm Hom}_{\Gamma}(X,-)\simeq X^{{\rm
tr}_{\Gamma}}\otimes_{\Gamma}^{{\mathbb{L}}}-:\mathrm{D}^{b}(\text{{\rm
mod}}(\Gamma))\to\mathrm{D}^{b}(\text{{\rm mod}}(\Lambda))$

sends ${\mathrm{K}}^{b}(\mbox{{\rm proj}}(\Gamma))$ to
${\mathrm{K}}^{b}(\mbox{{\rm proj}}(\Lambda))$. Moreover, since ${\rm
pd}(X_{\Lambda})<\infty$, a suitable shift of
$F:=X\otimes_{\Lambda}^{{\mathbb{L}}}-$ is nonnegative in the sense of [27,
Definition 4.1]. By [27, Theorem 5.3], $F$ restricts to the stable categories
of Gorenstein projective modules. ∎

The condition that $X^{{\rm tr}_{\Gamma}}=\mathbb{R}{\rm
Hom}_{\Gamma}(X,\Gamma)$ is perfect as complex of $\Lambda$-modules is strong.
We will introduce two alternative conditions in the sequel, see Criteria 4.8
and 4.12.

We need several notations. For an algebra $A$, we denote by $\Omega_{A}(-)$
the syzygy endofunctor of the stable module category of $A$. Define

${}^{\perp}A:=\\{X\in\text{{\rm mod}}(A)\ |\ {\rm Ext}_{A}^{i}(X,A)=0\mbox{
for all }i>0\\}.$

Notice that a module in this subcategory is called semi-Gorenstein projective
by Ringel and Zhang [41]. Denote

${\rm Gproj}(A)^{\bot}:=\\{X\in\text{{\rm mod}}(A)\mid{\rm
Ext}_{A}^{i}(U,X)=0\mbox{ for all }U\in\mbox{{\rm Gproj}}(A),\,i\geq 1\\}.$

The second criterion concerns a right exact functor between module categories
which, under certain conditions, sends Gorenstein projective modules to
Gorenstein projective modules.

######  Criterion 4.3.

If ${}_{\Gamma}X$ is projective and ${\rm Hom}_{\Gamma}(X,\Gamma)\in{\rm
Gproj}(\Lambda)^{\bot}$, then $X\otimes_{\Lambda}-:\text{{\rm
mod}}(\Lambda)\rightarrow\text{{\rm mod}}(\Gamma)$ sends Gorenstein projective
modules to Gorenstein projective modules and $X\otimes_{\Lambda}U\cong
X\otimes_{\Lambda}^{\mathbb{L}}U\in\mathrm{D}^{b}(\text{{\rm mod}}(\Gamma))$
for any $U\in{\rm Gproj}(\Lambda)$. In this case, there is an induced triangle
functor
$X\otimes_{\Lambda}-:\mbox{{\rm{\text@underline{Gproj}}}}(\Lambda)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(\Gamma)$.

###### Proof.

Let $U\in{\rm Gproj}(\Lambda)$. By definition, there is a complete resolution
$P^{\bullet}$ of $U$. Since $X$ is projective left $\Gamma$-module, thus
$X\otimes_{\Lambda}P^{i}$ is projective left $\Gamma$-module,
$i\in\mathbb{Z}$. As $P^{\bullet}$ is exact we see that
${\mathrm{H}}^{i}(X\otimes_{\Lambda}P^{\bullet})={\rm
Tor}_{j-i}^{\Lambda}(X,B^{j+1}(P^{\bullet}))$ for any $j>i$. Since the
projective dimension of $X$ as right $\Lambda$-module is assumed to be finite,
we can always choose $j$ sufficiently big so that this ${\rm Tor}$ group
vanishes. Thus $X\otimes_{\Lambda}P^{\bullet}$ is exact, which also shows that
${\rm Tor}_{i}^{A}(X,U)=0$ for $i>0$. By Lemma 3.1,
${}_{\Gamma}X\otimes^{\mathbb{L}}_{\Lambda}U\cong{}_{\Gamma}X\otimes_{\Lambda}U\in\mathrm{D}^{b}(\Lambda)$
for $U\in{\rm Gproj}(\Lambda)$.

Note that

${\rm Hom}_{\Gamma}(X\otimes_{\Lambda}P^{\bullet},\Gamma)\cong{\rm
Hom}_{\Lambda}(P^{\bullet},{\rm Hom}_{\Gamma}(X,\Gamma))$

Since ${\rm Hom}_{\Gamma}(X,\Gamma)\in{\rm Gproj}(\Lambda)^{\bot}$, thus ${\rm
Hom}_{\Lambda}(P^{\bullet},{\rm Hom}_{\Gamma}(X,\Gamma))$ and ${\rm
Hom}_{\Gamma}(X\otimes_{\Lambda}P^{\bullet},\Gamma)$ are exact.

This shows that $X\otimes_{\Lambda}U$ is still Gorenstein projective.

For the second statement, it remains to show that the additive functor
$X\otimes_{\Lambda}-:\mbox{{\rm{\text@underline{Gproj}}}}(\Lambda)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(\Gamma)$
is a triangle functor.

Indeed, since
${}_{\Gamma}X\otimes^{\mathbb{L}}_{\Lambda}U\cong{}_{\Lambda}X\otimes_{\Gamma}U\in\mathrm{D}^{b}(\Lambda)$
for $U\in{\rm Gproj}(\Lambda)$, the functor
${}_{\Gamma}X\otimes^{\mathbb{L}}_{\Lambda}-$ induces a triangle functor from
$\mathrm{D}^{b}_{\rm fGd}(\Gamma)$ to $\mathrm{D}^{b}_{\rm fGd}(\Lambda)$. We
have thus a commutative diagram

$\textstyle{\mbox{{\rm{\text@underline{Gproj}}}}(\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}_{\Gamma}X\otimes_{\Lambda}-}$$\scriptstyle{\simeq}$$\textstyle{\mbox{{\rm{\text@underline{Gproj}}}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{D}^{b}_{\rm
fGd}(\Lambda)/{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}_{\Gamma}X\otimes_{\Lambda}^{\mathbb{L}}-}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{D}^{b}_{\rm
fGd}(\Gamma)/{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(\Gamma))}$$\textstyle{\mathrm{D}^{b}_{\mathrm{sg}}(\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}_{\Gamma}X\otimes_{\Lambda}^{\mathbb{L}}-}$$\textstyle{\mathrm{D}^{b}_{\mathrm{sg}}(\Gamma)}$

Therefore,
$X\otimes_{\Lambda}-:\mbox{{\rm{\text@underline{Gproj}}}}(\Lambda)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(\Gamma)$
is a triangle functor. ∎

######  Remark 4.4.

Although Criterion 4.3 is somehow trivial, it generalises [34, Proposition
3.4] in which the authors consider the situation where there is a ring
homomorphism $f:\Lambda\to\Gamma$ and $X={}_{\Gamma}\Gamma_{\Lambda}$.

######  Remark 4.5.

In the hyposthese of Criterion 4.3, if we assume that ${}_{\Gamma}X$ is
projective, then ${\rm Hom}_{\Gamma}(X,\Gamma)\in{\rm Gproj}(\Lambda)^{\bot}$
is equivalent to saying that $X\otimes_{\Lambda}U$ belongs to
${}^{\perp}\Gamma$ whenever $U\in{\rm Gproj}(\Lambda)$. In fact, for $G\in{\rm
Gproj}(\Lambda)$ and a deleted projective resolution $Q^{\bullet}$ of $G$,
$X\otimes_{\Lambda}Q^{\bullet}$ is exact which is a projective resolution of
$X\otimes_{\Lambda}G$. By the isomorphism of complexes

${\rm Hom}_{\Lambda}(Q^{\bullet},{\rm Hom}_{\Gamma}(X,\Gamma))\cong{\rm
Hom}_{\Gamma}(X\otimes_{\Lambda}Q^{\bullet},\Gamma),$

for any $i>0$, there is an isomorphism

${\rm Ext}_{\Lambda}^{i}(G,{\rm Hom}_{\Gamma}(X,\Gamma))\cong{\rm
Ext}_{\Gamma}^{i}(X\otimes_{\Lambda}G,\Gamma).$

The equivalence deduces from the above isomorphisms.

This following example verifies the hypothesis of Criterion 4.3, but not that
of Criterion 4.2.

######  Example 4.6.

Let $\Gamma$ be a finite dimensional k-algebra and $\Lambda=kQ/I$ be the
algebra where $Q$ is the quiver

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

and $I=\langle\beta^{2},\beta\alpha\rangle$. Let $X=\Gamma\otimes_{{\bf
k}}e_{1}\Lambda$. It is clearly that ${}_{\Gamma}X_{\Lambda}$ is a projective
bimodule, thus ${}_{\Gamma}X,X_{\Lambda}$ are projective, and ${\rm
Hom}_{\Gamma}(X,\Gamma)\cong{\rm Hom}_{\bf k}(e_{1}\Lambda,{\bf
k})\otimes_{{\bf k}}\Gamma\in{\rm Gproj}(\Lambda)^{\bot}.$ But ${\rm Hom}_{\bf
k}(e_{1}\Lambda,{\bf k})$ has infinite projective dimension as left
$\Lambda$-module. Hence, ${\mathbb{R}}{\rm Hom}_{\Gamma}(X,\Gamma)={\rm
Hom}_{\Gamma}(X,\Gamma)$ falls into ${\rm Gproj}(\Lambda)^{\bot}$, but not
perfect as $\Lambda$-module.

######  Lemma 4.7.

Let $\Sigma$ be a finite dimensional ${\bf k}$-algebra, $M\in\text{{\rm
mod}}(\Sigma)$. If $M$ is quasi-isomorphic to a bounded complex with each term
in ${\rm Gproj}(\Sigma)^{\bot}$, then $M\in{\rm Gproj}(\Sigma)^{\bot}$.

###### Proof.

Let $M$ quasi-isomorphic to

$L^{\bullet}:0\rightarrow L^{s}\rightarrow L^{s+1}\rightarrow\cdots\rightarrow
L^{t}\rightarrow 0$

with $L^{j}\in{\rm Gproj}(\Sigma)^{\bot},\forall j$. For any $U\in{\rm
Gproj}(\Sigma)$, let $(Q^{\bullet},\partial^{\bullet})$ be a complete
resolution of $U$ with $U=\ker(\partial^{0})$. Let
$K^{i}:=\ker(\partial^{i})$. Then $K^{i}\in\mbox{{\rm Gproj}}(\Sigma)$. We
have

$\displaystyle{\rm Ext}_{\Sigma}^{p}(U,M)$ $\displaystyle={\rm
Ext}_{\Sigma}^{j+p}(K^{j},M)$ $\displaystyle={\rm
Hom}_{\mathrm{D}^{b}(\text{{\rm mod}}(\Sigma))}(K^{j},M[j+p])$
$\displaystyle\cong{\rm Hom}_{\mathrm{D}^{b}(\text{{\rm
mod}}(\Sigma))}(K^{j},L^{\bullet}[j+p]).$

Due to $K^{j}\in\mbox{{\rm Gproj}}(\Sigma)$, ${\rm
Ext}^{i}_{\Sigma}(K^{j},L^{q})=0$ for $i\geq 1$ and $s\leq q\leq t$. By [29,
Lemma 1.6], ${\rm Hom}_{\mathrm{D}^{b}(\text{{\rm
mod}}(\Sigma))}(K^{j},L^{\bullet}[j+p])=0$ for $j\geq t-p+1$. Then ${\rm
Ext}_{\Sigma}^{p}(U,M)=0$ for each $p\geq 1$, that is, $M\in{\rm
Gproj}(\Sigma)^{\bot}$. ∎

The following result, which is our third criterion, can be considered as a
derived version of Criterion 4.3.

######  Criterion 4.8.

Assume that $\mathbb{R}{\rm Hom}_{\Gamma}(X,\Gamma)$ is quasi-isomorphic to a
bounded complex with each term in ${\rm Gproj}(\Lambda)^{\bot}$. Then the
functor $X\otimes_{\Lambda}-:\text{{\rm mod}}(\Lambda)\rightarrow\text{{\rm
mod}}(\Gamma)$ induces $X\otimes^{\mathbb{L}}_{\Lambda}-:\mathrm{D}^{b}_{\rm
fGd}(\Lambda)\rightarrow\mathrm{D}^{b}_{\rm fGd}(\Gamma)$.

###### Proof.

By [1, Lemma 2.8], the functor $X\otimes_{\Lambda}-:\text{{\rm
mod}}(\Lambda)\rightarrow\text{{\rm mod}}(\Gamma)$ induces
$X\otimes^{\mathbb{L}}_{\Lambda}-:\mathrm{D}^{b}(\text{{\rm
mod}}(\Lambda))\rightarrow\mathrm{D}^{b}(\text{{\rm mod}}(\Gamma))$. It is
enough to prove that $X\otimes_{\Lambda}U\in\mathrm{D}^{b}_{\rm fGd}(\Gamma)$
for any $U\in{\rm Gproj}(\Lambda)$.

Take a projective resolution $P^{\bullet}$ of $X$ as
$\Gamma\otimes_{k}\Lambda^{op}$-module:

$\cdots\rightarrow P^{i}\stackrel{{\scriptstyle
d^{i}}}{{\rightarrow}}P^{i+1}\stackrel{{\scriptstyle
d^{i+1}}}{{\rightarrow}}\cdots\rightarrow P^{-1}\stackrel{{\scriptstyle
d^{-1}}}{{\rightarrow}}P^{0}\rightarrow{{}_{\Gamma}X_{\Lambda}}\rightarrow 0$

with $P^{-1}={{}_{\Gamma}X_{\Lambda}}$. Denote

$P_{\leq 0}^{\bullet}=(\cdots\rightarrow P^{i}\stackrel{{\scriptstyle
d^{i}}}{{\rightarrow}}P^{i+1}\stackrel{{\scriptstyle
d^{i+1}}}{{\rightarrow}}\cdots\rightarrow P^{-1}\stackrel{{\scriptstyle
d^{-1}}}{{\rightarrow}}P^{0}\rightarrow 0)$

and for some $l\geq 1$,

$P_{[-l+1,0]}^{\bullet}=(0\rightarrow P^{-l+1}\rightarrow
P^{-l+2}\rightarrow\cdots\rightarrow P^{-1}\stackrel{{\scriptstyle
d^{-1}}}{{\rightarrow}}P^{0}\rightarrow 0)$

Let $l:=\mathrm{max}\\{{\rm pd}(_{\Gamma}X),{\rm pd}(X_{\Lambda})\\}$ and
$N:=\ker(d^{-l+1})$. Then both ${}_{\Gamma}N$ and $N_{\Lambda}$ are
projective. For each $U\in\mbox{{\rm Gproj}}(\Lambda)$, by the first statement
of Criterion 4.3, the complex $P^{\bullet}\otimes_{\Lambda}U$ is exact, and
${{}_{\Gamma}X}\otimes_{\Lambda}U\cong{}_{\Gamma}X\otimes^{\mathbb{L}}_{\Lambda}U\cong
P^{\bullet}_{\leq 0}\otimes_{\Lambda}U$ in $\mathrm{D}^{b}(\text{{\rm
mod}}(\Gamma))$, which is also quasi-isomorphic to

$0\rightarrow N\otimes_{\Lambda}U\rightarrow
P^{-l+1}\otimes_{A}U\rightarrow\cdots\rightarrow
P^{-1}\otimes_{\Lambda}U\rightarrow P^{0}\otimes_{\Lambda}U\rightarrow 0,$

where $P^{i}\otimes_{\Lambda}U$ is a projective $\Gamma$-module for $-l+1\leq
i\leq 0$. Now we claim that ${\rm Hom}_{\Gamma}(N,\Gamma)\in{\rm
Gproj}(\Lambda)^{\bot}$, and then by Criterion 4.3 we get
$N\otimes_{\Lambda}U\in{\rm Gproj}(\Gamma)$, thus $X\otimes_{\Lambda}U$ is
quasi-isomorphic to a complex in $\mathrm{D}^{b}_{\rm fGd}(\Gamma)$. This
shows that $X\otimes^{\mathbb{L}}_{\Lambda}-$ sends a Gorenstein projective
module into $\mathrm{D}^{b}_{\rm fGd}(\Gamma)$, which proves the statement.

Now we show that ${\rm Hom}_{\Gamma}(N,\Gamma)\in{\rm Gproj}(\Lambda)^{\bot}$.

Applying the functor $\mathbb{R}{\rm
Hom}_{\Gamma}(-,\Gamma):D(\Gamma\otimes_{{\bf k}}\Lambda^{\rm op})\rightarrow
D(\Lambda)$ to the triangle in $D(\Gamma\otimes_{{\bf k}}\Lambda^{\rm op})$:

$N[l-1]\rightarrow P^{\bullet}_{[-l+1,0]}\rightarrow X\rightarrow N[l],$

we have the following triangle in $D(\Lambda)$:

$\mathbb{R}{\rm Hom}_{\Gamma}(X,\Gamma)\rightarrow\mathbb{R}{\rm
Hom}_{\Gamma}(P^{\bullet}_{[-l+1,0]},\Gamma)\rightarrow\mathbb{R}{\rm
Hom}_{\Gamma}(N,\Gamma[-l+1])\rightarrow\mathbb{R}{\rm
Hom}_{\Gamma}(X,\Gamma)[1].$

Since $P^{i}$ is a projective $\Gamma\otimes_{{\bf k}}\Lambda^{\rm op}$-module
for each $-l+1\leq i\leq 0$, ${\rm
Hom}_{\Gamma}(P^{i},\Gamma)\in\mathrm{inj}(\Lambda)$ and $\mathbb{R}{\rm
Hom}_{\Gamma}(P^{\bullet}_{[-l+1,0]},\Gamma)\cong{\rm
Hom}_{\Gamma}(P^{\bullet}_{[-l+1,0]},\Gamma)\in\mathrm{D}^{b}(\text{{\rm
mod}}(\Lambda))$ has all terms lying in $\mathrm{inj}(\Lambda)$. By
assumption, $\mathbb{R}{\rm Hom}_{\Gamma}(X,\Gamma)$ has all terms in ${\rm
Gproj}(\Lambda)^{\bot}$. As $\mathrm{inj}(\Lambda)\subseteq{\rm
Gproj}(\Lambda)^{\bot}$, $\mathbb{R}{\rm Hom}_{\Gamma}(N,\Gamma[-l+1])$ is
quasi-isomorphic to a bounded complex with each term in ${\rm
Gproj}(\Lambda)^{\bot}$. Since $N$ is projective left $\Gamma$-module, then
${\rm Hom}_{\Gamma}(N,\Gamma)\cong\mathbb{R}{\rm Hom}_{\Gamma}(N,\Gamma)$ in
$\mathrm{D}^{b}(\text{{\rm mod}}(\Lambda))$. By Lemma 4.7, ${\rm
Hom}_{\Gamma}(N,\Gamma)\in{\rm Gproj}(\Lambda)^{\bot}$.

∎

######  Remark 4.9.

Since $\Lambda$ and hence ${\rm proj}(\Lambda)$ belong to ${\rm
Gproj}(\Lambda)^{\bot}$, if $\mathbb{R}{\rm Hom}_{\Gamma}(X,\Gamma)$ is
perfect as complex of $\Lambda$-modules, $\mathbb{R}{\rm
Hom}_{\Gamma}(X,\Gamma)$ is quasi-isomorphic to a bounded complex with each
term in ${\rm Gproj}(\Lambda)^{\bot}$. So Criterion 4.8 generalises Criterion
4.2.

Note, however, that if $\Lambda$ is Gorenstein, ${\rm Gproj}(\Lambda)^{\bot}$
is exactly the subcategory of modules of finite projective dimension and the
thick subcategory generated by ${\rm Gproj}(\Lambda)^{\bot}$ inside
$\mathrm{D}^{b}(\text{{\rm mod}}(\Lambda))$ is exactly
${\mathrm{K}}^{b}(\mbox{{\rm proj}}(\Lambda))$.

Now we apply Criteria 4.3 and 4.8 to bounded extensions. We begin with an
observation which follows from [34, Theorem I (iii) or Proposition 3.7].

######  Proposition 4.10.

Let $B\subset A$ be a bounded extension. Then
${}_{A}A\otimes_{B}-:\mbox{{\rm{\text@underline{Gproj}}}}(B)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(A)$
is fully faithful.

######  Theorem 4.11.

Let $B\subset A$ be a bounded extension. Suppose that $\mathbb{R}{\rm
Hom}_{B}(A,B)$ is quasi-isomorphic to a bounded complex with each term in
${\rm Gproj}(A)^{\bot}$. Then

${}_{A}A\otimes_{B}-:\mbox{{\rm{\text@underline{Gproj}}}}(B)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(A)\mbox{
and
}{}_{A}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{def}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{def}}(A):\mathrm{Res}^{A}_{B}$

are equivalences.

###### Proof.

Since ${}_{A}A$ projective as left A-module and ${\rm pd}(_{B}A)$, ${\rm
pd}(A_{B})<\infty$, it follows from Criteria 4.3 (with
$\Lambda=B,\Gamma=A,{}_{\Gamma}X_{\Lambda}={}_{A}A_{B}$) that the functor
${}_{A}A\otimes_{B}-:\text{{\rm mod}}(B)\rightarrow\text{{\rm mod}}(A)$
induces a functor
${}_{A}A\otimes_{B}-:\mbox{{\rm{\text@underline{Gproj}}}}(B)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(A)$
and thus ${}_{A}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\rm
fGd}(B)/{\mathrm{K}}^{b}(\mbox{{\rm proj}}(B))\rightarrow\mathrm{D}^{b}_{\rm
fGd}(A)/{\mathrm{K}}^{b}(\mbox{{\rm proj}}(A))$.

Since $A_{A}$ is projective, ${\rm pd}(_{B}A)<\infty$ and $\mathbb{R}{\rm
Hom}_{B}(A,B)$ is quasi-isomorphic to a bounded complex with each term in
${\rm Gproj}(A)^{\bot}$, by Criteria 4.8 (with
$\Lambda=A,\Gamma=B,{}_{\Gamma}X_{\Lambda}={}_{B}A_{A}$) , we have the functor
$\mathrm{Res}^{A}_{B}={}_{B}A\otimes^{\mathbb{L}}_{A}-$ restricts to a
triangle functor from $\mathrm{D}^{b}_{\rm fGd}(A)$ to $\mathrm{D}^{b}_{\rm
fGd}(B)$.

We obtain an induced commutative diagram:

$\textstyle{\mbox{{\rm{\text@underline{Gproj}}}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}_{A}A\otimes_{B}-}$$\scriptstyle{\simeq}$$\textstyle{\mbox{{\rm{\text@underline{Gproj}}}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{D}^{b}_{\rm
fGd}(B)/{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}_{A}A\otimes^{\mathbb{L}}_{B}-}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{D}^{b}_{\rm
fGd}(A)/{\mathrm{K}}^{b}(\mbox{{\rm
proj}}(A))}$$\scriptstyle{\mathrm{Res}^{A}_{B}}$$\textstyle{\mathrm{D}^{b}_{\mathrm{sg}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}_{A}A\otimes^{\mathbb{L}}_{B}-}$$\textstyle{\mathrm{D}^{b}_{\mathrm{sg}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}^{A}_{B}}$$\textstyle{\mathrm{D}^{b}_{\mathrm{def}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}_{A}A\otimes^{\mathbb{L}}_{B}-}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{D}^{b}_{\mathrm{def}}(A)}$$\scriptstyle{\mathrm{Res}^{A}_{B}}$

By Theorem 3.7,

${{}_{A}}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{sg}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{sg}}(A):\mathrm{Res}^{A}_{B}$

form an adjoint equivalence which imply that all the horizontal functors are
equivalences.

∎

Now we will give the last criteria from a different point of view.

######  Criterion 4.12.

Assume there is an integer $t$ such that
$\Omega^{t}(X\otimes_{\Lambda}U)\in{}^{\perp}{\Gamma}$ for any
$U\in{}^{\perp}\Lambda$. Then $X\otimes^{\mathbb{L}}_{\Lambda}-$ induces a
triangle functor from $\mathrm{D}^{b}_{\rm fGd}(\Lambda)$ to
$\mathrm{D}^{b}_{\rm fGd}({\Gamma})$.

###### Proof.

This can be proved similarly as [40, Lemma 3.8]. For the convenience of the
reader, we include here a proof.

As in the proof of Criterion 4.8, take a projective resolution $P^{\bullet}$
of $X$ as $\Gamma\otimes_{k}\Lambda^{op}$-module:

$\cdots\rightarrow P^{i}\stackrel{{\scriptstyle
d^{i}}}{{\rightarrow}}P^{i+1}\stackrel{{\scriptstyle
d^{i+1}}}{{\rightarrow}}\cdots\rightarrow P^{-1}\stackrel{{\scriptstyle
d^{-1}}}{{\rightarrow}}P^{0}\rightarrow{{}_{\Gamma}X_{\Lambda}}\rightarrow 0$

and take $N=\Omega^{l}(_{\Gamma}X_{\Lambda})$ for $l>>0$. Then $N$ is
projective as left $\Gamma$-module and right $\Lambda$-module. For any
$U\in{\rm Gproj}(\Lambda)$, $P^{\bullet}\otimes_{\Lambda}U$ is exact, so the
sequence

$0\rightarrow{}_{\Gamma}N\otimes_{\Lambda}U\rightarrow{}_{\Gamma}P^{-l+1}\otimes_{A}U\rightarrow\cdots\rightarrow{}_{\Gamma}P^{-1}\otimes_{\Lambda}U\rightarrow{}_{\Gamma}P^{0}\otimes_{\Lambda}U\rightarrow{}_{\Gamma}X\otimes_{\Lambda}U\rightarrow
0$

is exact. Hence,
${}_{\Gamma}N\otimes_{\Lambda}U\cong\Omega^{l}({}_{\Gamma}X\otimes_{\Lambda}U)\oplus{}_{\Gamma}V$
with ${}_{\Gamma}V\in\mbox{{\rm proj}}({\Gamma})$, and
${}_{\Gamma}X\otimes^{\mathbb{L}}_{\Lambda}U\cong{{}_{\Gamma}X}\otimes_{\Lambda}U\in\mathrm{D}^{b}(\text{{\rm
mod}}(\Gamma))$, which is quasi-isomorphic to

$0\rightarrow{}_{\Gamma}N\otimes_{\Lambda}U\rightarrow{}_{\Gamma}P^{-l+1}\otimes_{A}U\rightarrow\cdots\rightarrow{}_{\Gamma}P^{-1}\otimes_{\Lambda}U\rightarrow{}_{\Gamma}P^{0}\otimes_{\Lambda}U\rightarrow
0.$

To show that this complex falls into $\mathrm{D}^{b}_{\rm fGd}({\Gamma})$,
since ${}_{\Gamma}P^{j}\otimes_{\Lambda}U$ is projective for $j=-l+1,\dots,0$,
it remains to prove that when $l>t$, ${}_{\Gamma}N\otimes_{\Lambda}U$ belongs
to $\mbox{{\rm Gproj}}(\Gamma)$.

First, since ${}^{\perp}{\Gamma}$ is closed under taking syzygies and
$\Omega^{t}({}_{\Gamma}X\otimes_{\Lambda}U)\in{}^{\perp}{\Gamma}$, when $l>t$,
${}_{\Gamma}N\otimes_{\Lambda}U\cong\Omega^{l}({}_{\Gamma}X\otimes_{\Lambda}U)\oplus{}_{\Gamma}V$
with ${}_{\Gamma}V\in\mbox{{\rm proj}}({\Gamma})$ will belong to
${}^{\perp}{\Gamma}$.

Decompose a complete $\Lambda$-projective resolution
$(Q^{\bullet},\partial^{\bullet})$ of $U$ with $U=\ker(\partial^{0})$ into
short exact sequence $0\rightarrow U^{i}\rightarrow Q^{i}\rightarrow
U^{i+1}\rightarrow 0$ for all $i\in\mathbb{Z}$ with
$U^{i}:=\ker(\partial^{i})$. Since $N_{\Lambda}$ is projective, we have exact
sequences:

$0\rightarrow{}_{\Gamma}N\otimes_{\Lambda}U^{i}\rightarrow{}_{\Gamma}N\otimes_{\Lambda}Q^{i}\rightarrow{}_{\Gamma}N\otimes_{\Lambda}U^{i+1}\rightarrow
0.$

Since all $U^{i}$ are Gorenstein projective,
${}_{\Gamma}N\otimes_{\Lambda}U^{i}\in{}^{\perp}{\Gamma}$. As all
${}_{\Gamma}N\otimes_{\Lambda}Q^{i}$ are projective, we can form a complete
resolution of $N\otimes_{\Lambda}U$, hence $N\otimes_{\Lambda}U\in\mbox{{\rm
Gproj}}(\Gamma)$.

This completes the proof. ∎

######  Theorem 4.13.

Let $B\subset A$ be a bounded extension. Suppose that ${}_{B}B$ is a direct
summand of ${}_{B}A$. Then

${}_{A}A\otimes_{B}-:\mbox{{\rm{\text@underline{Gproj}}}}(B)\rightarrow\mbox{{\rm{\text@underline{Gproj}}}}(A)\mbox{
and
}_{A}A\otimes^{\mathbb{L}}_{B}-:\mathrm{D}^{b}_{\mathrm{def}}(B)\rightleftharpoons\mathrm{D}^{b}_{\mathrm{def}}(A):\mathrm{Res}^{A}_{B}$

are equivalences. In particular, these two equivalences exist if the extension
${}_{B}B\subset A$ is split or ${}_{B}(A/B)$ is projective.

###### Proof.

Note that $\mathrm{Res}^{A}_{B}={{}_{B}A}\otimes_{A}-$ is exact and ${\rm
pd}(_{B}\mathrm{Res}^{A}_{B}(A))<\infty$. Now we claim that there is an
integer $t$ such that $\Omega^{t}(\mathrm{Res}^{A}_{B}(U))\in{}^{\perp}B$ for
any $U\in{}^{\perp}A$, and then Criterion 4.12 can be applied. The
equivalences can be shown as in the last paragraph of the proof of Theorem
4.11.

Let us show the claim that there is an integer $t$ such that
$\Omega^{t}(\mathrm{Res}^{A}_{B}(U))\in{}^{\perp}B$ for any $U\in{}^{\perp}A$.

Since ${}_{B}B$ is a direct summand of ${}_{B}A$, then for any
$U\in{}^{\perp}A$ and $i>0$, ${\rm Ext}_{B}^{i}(\mathrm{Res}^{A}_{B}(U),B)$ is
a direct summand of ${\rm Ext}_{B}^{i}(\mathrm{Res}^{A}_{B}(U),_{B}A)$. On the
other hand, it follows from Theorem 3.8 that the functor
$\mathrm{Res}^{A}_{B}$ is a $t$-eventually homological isomorphism for some
integer $t$. Therefore, for any $i>t$, we have isomorphisms

${\rm Ext}_{B}^{i}(\mathrm{Res}^{A}_{B}(U),_{B}A)={\rm
Ext}_{B}^{i}(\mathrm{Res}^{A}_{B}(U),\mathrm{Res}^{A}_{B}(A))\cong{\rm
Ext}_{A}^{i}(U,A),$

which is equal to zero since $U\in{}^{\perp}A$. Hence, for any $i>t$ we have
${\rm Ext}_{B}^{i}(\mathrm{Res}^{A}_{B}(U),B)=0$, and for any $j>0$,

${\rm Ext}_{B}^{j}(\Omega^{t}(\mathrm{Res}^{A}_{B}(U)),B)={\rm
Ext}_{B}^{j+t}(\mathrm{Res}^{A}_{B}(U),B)=0.$

This shows that $\Omega^{t}(\mathrm{Res}^{A}_{B}(U))\in{}^{\perp}B$.

∎

##  5\. Homological conjectures for bounded extensions

In this section, we use the results of Section 3 to deal with several
homological conjectures for bounded extensions, which are still open. They are
stated as follows.

The finitistic dimension conjecture [4]: _The finitistic dimension of $A$
which is the supremum of the projective dimensions of finitely generated
$A$-modules with finite projective dimension, is finite._

Han’s conjecture [26]: _If the Hochschild homology groups $\mathrm{HH}_{n}(A)$
vanish for all sufficiently large $n$, then its global dimension is finite._

Keller’s conjecture [30]: _Let ${\bf S}_{{\rm dg}}(A)$ be the dg enhancement
of the singularity category $\mathrm{D}^{b}_{\mathrm{sg}}(A)$. Then there is
an isomorphism in the homotopy category of $B_{\infty}$-algebras_

${\mathrm{C}}^{*}_{{\mathrm{sg}}}(A,A)\cong{\mathrm{C}}^{*}({\bf S}_{{\rm
dg}}(A),{\bf S}_{{\rm dg}}(A)),$

_where ${\mathrm{C}}^{*}_{{\mathrm{sg}}}(A,A)$ is the singular Hochschild
cohomology complex [45] of $A$, and ${\mathrm{C}}^{*}({\bf S}_{{\rm
dg}}(A),{\bf S}_{{\rm dg}}(A))$ is the Hochschild cochain complex of ${\bf
S}_{{\rm dg}}(A)$. _

Auslander and Reiten’s Conjecture [2]: _A finitely generated $A$-module $X$
satisfying ${\rm Ext}_{A}^{i}(X,X\oplus A)=0$ for all $i\geq 1$ must be
projective, or equivalently, a generator $M$ of $A$ with ${\rm
Ext}_{A}^{i}(M,M)=0$ for all $i\geq 1$ is necessarily projective._

From [22, 43], an algebra $A$ is said to satisfy the Fg condition if the
Hochschild cohomology ring $HH^{*}(A)$ is Noetherian and the Yoneda algebra
${\rm Ext}_{A}^{*}(A/{\rm rad}A,A/{\rm rad}A)$ is a finitely generated
$HH^{*}(A)$-module.

######  Theorem 5.1.

Let $B\subset A$ be a bounded extension. Then the following statements hold:

  * $(1)$

$B$ satisfies the finitistic dimension conjecture if and only if so does $A$.

  * $(2)$

$B$ satisfies Han’s conjecture if and only if so does $A$.

  * $(3)$

$B$ satisfies Keller’s conjecture if and only if so does $A$.

###### Proof.

By Theorem 3.7, we know that $A$ and $B$ are singularly equivalent of Morita
type with level, which induces an equivalence of singularity categories, thus
the finiteness of the global dimension is preserved. Note that this
equivalence preserves the finitistic dimension conjecture [44], Hochschild
homology [44] and Keller’s conjecture [9]. Thus the statements (1)-(3) follow.
∎

######  Remark 5.2.

The assumption in Theorem 5.1 regarding the bounded extension is less
restrictive than that imposed on the left or right bounded extension as
introduced in [19]. Consequently, Theorem 5.1(2) extends the main result
presented in [19]. Additionally, the approach employed to establish Theorem
5.1(2) differs entirely from that adopted in [19], where the Jacobi-Zariski
long nearly exact sequence is utilized.

######  Theorem 5.3.

Let $B\subset A$ be a bounded extension. Assume that ${}_{A}A^{{\rm
tr}_{B}}=\mathbb{R}{\rm Hom}_{B}({}_{B}A_{A},{}_{B}B)$ is perfect as left
$A$-module or $B\subset A$ is split. Then the following statements hold:

  * $(1)$

$B$ is Gorenstein if and only if so does $A$.

  * $(2)$

$B$ satisfies the Fg condition if and only if so does $A$.

  * $(3)$

$B$ satisfies Auslander-Reiten’s conjecture if and only if so does $A$.

###### Proof.

(1) Since ${\rm proj}(A)$ belong to ${\rm Gproj}(A)^{\bot}$, it follows from
Theorem 4.11 and Theorem 4.13 that
$\mathrm{D}^{b}_{\mathrm{def}}(B)\simeq\mathrm{D}^{b}_{\mathrm{def}}(A)$.
Therefore, $B$ is Gorenstein if and only if so does $A$.

(2) Assume that $B$ satisfies the Fg condition. Then it follows from [22,
Theorem 1.5 (a)] that $B$ is a Gorenstein algebra, and so does $A$ by (1). On
the other hand, it follows from Theorem 3.7 that there is a singular
equivalence of Morita type with level between $A$ and $B$. Therefore, $A$
satisfies the Fg condition by [42, Theorem 7.4], and similarly, if $A$
satisfies the Fg condition then so does $B$.

(3) By the exact sequence $0\rightarrow B\rightarrow A\rightarrow M\rightarrow
0$, since the projective dimension of $M$ as $B$-$B$-bimodule is finite, ${\rm
pd}(_{B}A)$ and ${\rm pd}(A_{B})$ are finite. Then we have the following
adjoint 5-tuple of triangle functors

where $A^{{\rm tr}_{B^{\rm op}}}=\mathbb{R}{\rm Hom}_{B^{\rm
op}}(_{A}A_{B},B)$, and $(_{A}A\otimes^{L}_{B}-,_{B}A\otimes_{A}-)$ induces
mutually inverse singular equivalences between $A$ and $B$. Since
${}_{B}A\otimes_{A}-$ preserves ${\mathrm{K}}^{b}(\mathrm{proj})$ and
coproducts, we get that ${}_{B}A\otimes_{A}-$ preserves
${\mathrm{K}}^{b}(\mathrm{Proj})$. Then it follows from [15, Corollary 3.5]
that Auslander-Reiten’s conjecture holds for $B$ if it holds for $A$. Now we
will prove the converse statement under the following two cases.

Case 1. Assume the extension $B\subset A$ is split. Then by the proof of
Theorem 4.13, there is an integer $t$ such that
$\Omega^{t}(\mathrm{Res}^{A}_{B}(U))\in{}^{\perp}B$ for any $U\in{}^{\perp}A$.
Therefore, there is a commutative diagram

$\textstyle{\underline{{}^{\bot}A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega^{t}\circ\mathrm{Res}^{A}_{B}}$$\textstyle{\underline{{}^{\bot}B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{sg}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[-t]\circ\mathrm{Res}^{A}_{B}}$$\textstyle{D_{sg}(B)}$

where $[1]$ is the shift functor on singularity category,
$\underline{{}^{\perp}A}$ is the additive quotient category of ${}^{\perp}A$
modulo projective modules and the vertical maps are natural full embedding,
see [15, Lemma 2.1] for example. Since
$[-t]\circ\mathrm{Res}^{A}_{B}:D_{sg}(A)\rightarrow D_{sg}(B)$ is an
equivalence, it follows from [15, Lemma 3.3] that Auslander-Reiten’s
conjecture holds for $A$ if it holds for $B$.

Case 2. Assume $A^{{\rm tr}_{B}}=\mathbb{R}{\rm Hom}_{B}({}_{B}A_{A},{}_{B}B)$
is perfect as left $A$-module. Then the functor $\mathbb{R}{\rm
Hom}_{B}({}_{B}A_{A},-)\cong{}_{A}A^{{\rm tr}_{B}}\otimes_{B}^{\mathbb{L}}-$
preserves perfect complexes and the diagram extends downwards one step
further. Thus we complete the proof by [15, Corollary 3.7].

∎

##  6\. Examples

In this section, we apply Theorems 3.7, 3.8, 4.11 and 4.13 to many examples,
and we get several reduction methods on singularity categories and Gorenstein
defect categories.

Let $B\subset A$ be an algebra extension which splits. We will describe the
construction of split algebra extensions. Let $i:B\to A$ be the algebra
inclusion and $q:A\to B$ be the retraction. Denote $M={\rm Ker}(q)$ and
$j:M\to A$ be the inclusion. Then there is a linear map $p:A\to M$ such that
$j\circ p+i\circ q={\rm Id}_{A}$, that is, we have the splitting:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{0}$

and we have the vector space isomorphism $A\cong B\oplus M$. Since $q:A\to B$
is an algebra homomorphism, $M$ is a $B$-$B$-bimodule. The multiplication on
$A$ induces, via the vector space isomorphism $A\cong B\oplus M$, a
multiplication on $M$: $\cdot:M\otimes_{B}M\to M$ which is also a
$B$-$B$-bimodule homomorphism. In this case, the multiplication on $B\oplus M$
will be given by

$(b,x)(b^{\prime},x^{\prime})=(bb^{\prime},bx^{\prime}+xb^{\prime}+x\cdot
x^{\prime}).$

This motivated the following definition as well as a complete description of
split algebra extensions.

######  Definition 6.1 ([3, Definition 2.3]).

Let $B$ be algebra and $M$ a $B$-$B$-bimodule. The bimodule ${}_{B}M_{B}$ is
called a bimodule algebra if there is a $B$-$B$-bimodule homomorphism

$\cdot:M\otimes_{B}M\to M$

subject to the following properties:

$(x\cdot y)\cdot z=x\cdot(y\cdot z),x,y,z\in M.$

######  Proposition 6.2 ([3, Proposition 2.4]).

Let $B$ be algebra and $M$ a $B$-$B$-bimodule algebra with product
$\cdot:M\otimes_{B}M\to M$. Then the vector space $A=B\oplus M$ has an algebra
structure given by

$(b,x)(b^{\prime},x^{\prime})=(bb^{\prime},bx^{\prime}+xb^{\prime}+x\cdot
x^{\prime}),b,b^{\prime}\in B,x,x^{\prime}\in M.$

Moreover, the inclusion $B\to A,b\mapsto(b,0)$ is an algebra extension which
splits via $q:A\to B,(b,x)\mapsto b$.

Conversely, each split algebra extension $B\subset A$ is given as above.

######  Proposition 6.3.

Let $B\subset A=B\oplus M$ be a split algebra extension. Then it is a bounded
extension if and only if ${\rm pd}(_{B}M_{B})<\infty$, $M^{\otimes_{B}p}=0$
for some integer $p$ and ${\rm Tor}_{i}^{B}(M,M^{\otimes_{B}j})=0$ for each
$i,j\geq 1$. Hence, Theorems 3.7, 3.8, and 4.13 apply.

Let us specify Proposition 6.3 for trivial extensions and further to
triangular matrix algebras.

Let $B$ be a finite dimensional algebra and $M$ a finite dimensional
$B$-$B$-bimodule. The trivial extension $A=B\ltimes M$ is by definition
$B\oplus M$ and the multiplication is given by

$(b,m)(b^{\prime},m^{\prime})=(bb^{\prime},mb^{\prime}+bm^{\prime}),b,b^{\prime}\in
B,m,m^{\prime}\in M.$

Obviously, the algebra inclusion $B\hookrightarrow A$ defined by
$b\mapsto(b,0)$ has a retraction $A\rightarrow B$ given by $(b,m)\mapsto b$.
Therefore, the extension $B\subset A$ splits with
${}_{B}(A/B)_{B}={}_{B}M_{B}$. Trivial extensions of algebras are exactly
split algebra extensions with trivial product on the bimodule algebras.

######  Proposition 6.4.

The trivial extension $B\subset A=B\ltimes M$ is a bounded extension if and
only if ${\rm pd}(_{B}M_{B})<\infty$, $M^{\otimes_{B}p}=0$ for some integer
$p$ and ${\rm Tor}_{i}^{B}(M,M^{\otimes_{B}j})=0$ for each $i,j\geq 1$. Hence,
Theorems 3.7, 3.8, and 4.13 apply.

######  Example 6.5.

Let $A$ be an admissible quotient ${\bf k}Q/I$ of a path algebra ${\bf k}Q$
over a field ${\bf k}$. Consider an arrow $\alpha$ in $Q$ such that $\alpha$
does not occur in a minimal generating set of $I$. The arrow removal algebra
$B=A/\langle\overline{\alpha}\rangle$ of $A$ was investigated in [16, 21, 23,
24]. Indeed, it follows from [24, Theorem A] that $A=B\ltimes M$ where $M$ is
projective as a $B$-$B$-bimodule and $M\otimes_{B}M=0$. Therefore, the
extension $B\subset A$ is a bounded split extension. Applying Theorem 3.7, 3.8
and Theorem 5.3(1)(2), we reobtain [24, Theorem A (ii)] and the main result of
[23]. We mention that the Auslander-Reiten conjecture, Gorenstein defect
categories and stable categories of Gorenstein projective modules between $A$
and $B$ were investigated in [40, Corollary I], which can also be deduced by
Theorem 4.13 and Theorem 5.3(3).

Let $\Lambda$ and $\Gamma$ be two finite dimensional algebras,
${}_{\Gamma}W_{\Lambda}$ be a $\Gamma$-$\Lambda$-bimodule and
$A=\left(\begin{smallmatrix}\Lambda&0\\\
{{}_{\Gamma}W_{\Lambda}}&\Gamma\end{smallmatrix}\right)$ be a triangular
matrix algebra. In [33], Lu proved that if $W$ is projective as a
$\Gamma$-$\Lambda$-bimodule, then there are triangle equivalences

$\mathrm{D}^{b}_{\mathrm{sg}}(A)\simeq\mathrm{D}^{b}_{\mathrm{sg}}(\Lambda)\coprod\mathrm{D}^{b}_{\mathrm{sg}}(\Gamma)\
\mathrm{and}\
\mathrm{D}^{b}_{\mathrm{def}}(A)\simeq\mathrm{D}^{b}_{\mathrm{def}}(\Lambda)\coprod\mathrm{D}^{b}_{\mathrm{def}}(\Gamma)$

(see the proof of [33, Proposition 4.2 and Theorem 4.4]). Now we will
generalize these results as follows.

######  Proposition 6.6.

Keep the above notations. If ${\rm pd}({}_{\Gamma}W_{\Lambda})<\infty$, then
there are triangle equivalences:

$\mathrm{D}^{b}_{\mathrm{sg}}(A)\simeq\mathrm{D}^{b}_{\mathrm{sg}}(\Lambda)\coprod\mathrm{D}^{b}_{\mathrm{sg}}(\Gamma),\mbox{{\rm{\text@underline{Gproj}}}}(A)\simeq\mbox{{\rm{\text@underline{Gproj}}}}(\Lambda)\coprod\mbox{{\rm{\text@underline{Gproj}}}}(\Gamma),\
\mbox{and}\
\mathrm{D}^{b}_{\mathrm{def}}(A)\simeq\mathrm{D}^{b}_{\mathrm{def}}(\Lambda)\coprod\mathrm{D}^{b}_{\mathrm{def}}(\Gamma).$

###### Proof.

It is clear that

$A=\left(\begin{smallmatrix}\Lambda&0\\\
{{}_{\Gamma}W_{\Lambda}}&\Gamma\end{smallmatrix}\right)\cong(\Lambda\times\Gamma)\ltimes
W,W\mathop{\otimes}\limits_{\Gamma\times\Lambda}W=0,\ \mathrm{and}\ {\rm
Tor}_{i}^{\Lambda\times\Gamma}(W,W)=0,\ \mathrm{for}\ i\geq 1.$

Therefore, the statement follows from Proposition 6.4. ∎

Now we will illustrate our results by an example.

######  Example 6.7.

Let $\Lambda={\bf k}Q/I$ be the algebra where $Q$ is the quiver

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

and $I=\langle\gamma^{2},\alpha\beta\rangle$. We write the concatenation of
paths from the right to the left. Let $\Gamma=\Lambda/\Lambda\alpha\Lambda$.
Then $\Gamma$ is the quiver

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\textstyle{2}$

with relation $\\{\gamma^{2}\\}$, and $\Gamma$ is a subalgebra of $\Lambda$.
Therefore, $\Lambda\alpha\Lambda$ can be viewed as a $\Gamma$-$\Gamma$
bimodule, and there is an algebra isomorphism
$\Lambda\cong\Gamma\ltimes\Lambda\alpha\Lambda$ mapping
$\overline{\varepsilon}_{i}$ to $(\overline{\varepsilon}_{i},0)$,
$\overline{\gamma}$ to $(\overline{\gamma},0)$, $\overline{\beta}$ to
$(\overline{\beta},0)$ and $\overline{\alpha}$ to $(0,\overline{\alpha})$,
where $\varepsilon_{i}$ is the trivial path at $i$. It is easy to check that
$\Lambda\alpha\Lambda_{\Gamma}\cong(e_{2}\Gamma/{\rm rad}(e_{2}\Gamma))^{4}$
and ${}_{\Gamma}\Lambda\alpha\Lambda\cong\Gamma e_{1}$. Therefore, we get
$\Lambda\alpha\Lambda\otimes_{\Gamma}\Lambda\alpha\Lambda=0$ and ${\rm
Tor}_{i}^{\Gamma}(\Lambda\alpha\Lambda,\Lambda\alpha\Lambda)=0$ for each
$i\geq 1$. Now we claim that ${\rm
pd}_{\Gamma^{e}}(\Lambda\alpha\Lambda)<\infty$, and then we get triangle
equivalences
$\mathrm{D}^{b}_{\mathrm{sg}}(\Lambda)\simeq\mathrm{D}^{b}_{\mathrm{sg}}(\Gamma)$
and
$\mathrm{D}^{b}_{\mathrm{def}}(\Lambda)\simeq\mathrm{D}^{b}_{\mathrm{def}}(\Gamma)$
by Proposition 6.4. Indeed, the enveloping algebra $\Gamma^{e}$ has the
following quiver

$\textstyle{1\times
1^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\otimes
1^{\mathrm{op}}}$$\scriptstyle{1\otimes\gamma^{\mathrm{op}}}$$\scriptstyle{\beta\otimes
1^{\mathrm{op}}}$$\scriptstyle{1\otimes\beta^{\mathrm{op}}}$$\textstyle{2\times
1^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2\otimes\gamma^{\mathrm{op}}}$$\scriptstyle{2\otimes\beta^{\mathrm{op}}}$$\textstyle{1\times
2^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\otimes
2^{\mathrm{op}}}$$\scriptstyle{\gamma\otimes
2^{\mathrm{op}}}$$\textstyle{2\times 2^{\mathrm{op}}}$

with relations $\\{(\gamma\otimes
1^{\mathrm{op}})^{2},(1\otimes\gamma^{\mathrm{op}})^{2},(2\otimes\gamma^{\mathrm{op}})^{2},(\gamma\otimes
2^{\mathrm{op}})^{2},(2\otimes\beta^{\mathrm{op}})(\beta\otimes
2^{\mathrm{op}})-(\beta\otimes
1^{\mathrm{op}})(1\otimes\beta^{\mathrm{op}}),(\gamma\otimes
1^{\mathrm{op}})(1\otimes\beta^{\mathrm{op}})-(1\otimes\beta^{\mathrm{op}})(\gamma\otimes
2^{\mathrm{op}}),(2\otimes\gamma^{\mathrm{op}})(\beta\otimes
1^{\mathrm{op}})-(\beta\otimes
1^{\mathrm{op}})(1\otimes\gamma^{\mathrm{op}}),(\gamma\otimes
1^{\mathrm{op}})(1\otimes\gamma^{\mathrm{op}})-(1\otimes\gamma^{\mathrm{op}})(\gamma\otimes
1^{\mathrm{op}})\\}$. The $\Gamma^{e}$-module $\Lambda\alpha\Lambda$ is given
by the following representation:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)}$$\scriptstyle{\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right)}$$\textstyle{k^{2}\ .}$

There is an exact sequence

$0\rightarrow\Gamma^{e}e_{1\times
1^{\mathrm{op}}}\rightarrow\Gamma^{e}e_{1\times
2^{\mathrm{op}}}\rightarrow\Lambda\alpha\Lambda\rightarrow 0$

of $\Gamma^{e}$-modules, and thus ${\rm
pd}(_{\Gamma^{e}}\Lambda\alpha\Lambda)=1$.

Acknowledgments. The first author was supported by the National Natural
Science Foundation of China (No. 12061060, 11961007), by the project of Young
and Middle-aged Academic and Technological leader of Yunnan (Grant No.
202305AC160005) and by Yunnan Key Laboratory of Modern Analytical Mathematics
and Applications (No. 202302AN360007). The second and the fourth authors were
supported by the National Natural Science Foundation of China (No. 12071137),
by Key Laboratory of Mathematics and Engineering Applications (Ministry of
Education), by Shanghai Key Laboratory of PMMP (No. 22DZ2229014) and by
Fundamental Research Funds for the Central Universities.

Declaration of interests. The authors have no conflicts of interest to
disclose.

Data availability. No new data were created or analyzed in this study.

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