Symmetric subcategories, tilting modules and derived recollements

Hongxing Chen and Changchang Xi∗

†† ∗ Corresponding author. Email<EMAIL_ADDRESS>Fax: 0086 10
68903637.††2010 Mathematics Subject Classification: Primary 18E30, 16G10,
13B30; Secondary 16S10, 13E05.††Keywords: Derived category; Exact category;
Homotopy limit; Recollement; Symmetric subcategory; Tilting modules.

###### Abstract

For any good tilting module $T$ over a ring $A$, there exists an $n$-symmetric
subcategory $\mathscr{E}$ of a module category such that the derived category
of the endomorphism ring of $T$ is a recollement of the derived categories of
$\mathscr{E}$ and $A$ in the sense of Beilinson-Bernstein-Deligne. Thus the
kernel of the total left-derived tensor functor induced by the tilting module
is triangle equivalent to the derived category of $\mathscr{E}$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Semi-orthogonal decompositions, recollements and homotopy (co)limits
    2. 2.2 Derived categories of exact categories
    3. 2.3 Derived functors of the module categories of rings
  3. 3 Symmetric subcategories of abelian categories
  4. 4 Tilting modules and derived recollements
    1. 4.1 Derived functors induced by good tilting modules
    2. 4.2 Fully faithful triangle functors between derived categories
    3. 4.3 Proofs of main result and its corollaries

##  1 Introduction

In representation theory of algebras and groups, finitely generated tilting
modules have successfully been applied to understanding different aspects of
algebraic and homological structure and properties of finite groups, finite-
dimensional algebras and modules (for example, see [9, 14, 17, 18, 27]), while
infinitely generated tilting modules not only have involved many important
modules, such as adic modules, Fuchs divisible modules, generic modules, and
Prüfer modules, but also have been of significant interest in studying derived
categories and equivalences of general algebras and rings (see [1, 3, 4, 10,
11, 31]). In this general context, a lot of results emerge with completely
different features. So, the theory of infinitely generated tilting modules
renews our view on finitely generated tilting modules, and gives us also
surprising information about the whole tilting theory.

Let us first recall the definition of tilting modules.

######  Definition 1.1.

[9, 18, 23, 16, 1] Let $n\geq 0$ be a natural number and $A$ a unitary ring. A
left $A$-module $T$ is called an _$n$ -tilting_ $A$-module if the three
conditions hold:

$(T1)$ There is an exact sequence of $A$-modules:

$0\longrightarrow P_{n}\longrightarrow\cdots\longrightarrow
P_{1}\longrightarrow P_{0}\longrightarrow{}_{A}T\longrightarrow 0$

with all $P_{i}$ projective $A$-modules, that is, ${\rm proj.dim}(_{A}T)\leq
n$.

$(T2)$ For all $j\geq 1$ and nonempty sets $\alpha$, ${\rm
Ext}^{j}_{A}(T,T^{(\alpha)})=0$, where $T^{(\alpha)}$ stands for the direct
sum of $\alpha$ copies of $T$.

$(T3)$ There exists an exact sequence of $A$-modules

$0\longrightarrow{}_{A}A\longrightarrow T_{0}\longrightarrow
T_{1}\longrightarrow\cdots\longrightarrow T_{n}\longrightarrow 0,$

where $T_{i}$ is isomorphic to a direct summand of a direct sum of copies of
$T$ for $0\leq i\leq n$.

Recall that an $n$-tilting $A$-module $T$ is said to be _good_ (see [4]) if
each $T_{i}$ in $(T3)$ is isomorphic to a direct summand of the direct sum of
finitely many copies of $T$.

Given an arbitrary tilting module $T^{\prime}$, one can always find a good
tilting module $T$ such that $T$ and $T^{\prime}$ generate the same full
subcategories in $A\mbox{{\rm-Mod}}$, the category of all left $A$-modules,
that is, $T$ and $T^{\prime}$ are equivalent (see [4]).

For an exact category $\mathscr{E}$, we denote by ${\mathscr{D}}(\mathscr{E})$
the unbounded derived category of $\mathscr{E},$ and for a ring $A$, we denote
by ${\mathscr{D}}(A)$ the unbounded derived module category of $A$, that is,
${\mathscr{D}}(A)={\mathscr{D}}(A\mbox{{\rm-Mod}}).$

The purpose of this article is to prove the following derived recollement
theorem for arbitrary good tilting modules over rings. For the definition of
$n$-symmetric subcategories, we refer to Section 3.

######  Theorem 1.2.

Let $A$ be a ring, $T$ a good $n$-tilting $A$-module, and $B$ the endomorphism
ring of ${}_{A}T$. Then there exist an $n$-symmetric subcategory $\mathscr{E}$
of $B\mbox{{\rm-Mod}}$ and a recollement of unbounded derived categories

$\textstyle{{\mathscr{D}}(\mathscr{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{{\mathscr{D}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $G$ is the total left-derived tensor functor defined by ${}_{A}T_{B}$.
Moreover, this recollement restricts to a recollement of bounded-above derived
categories.

Actually, for a tilting $A$-module $T$ of projective dimension $n$, the
$n$-symmetric subcategory in Theorem 1.2 can precisely be described as

$\mathscr{E}:=\\{Y\in B\mbox{{\rm-Mod}}\mid{\rm
Tor}_{i}^{B}(T,Y)=0\mbox{\;for\, all \,}i\geq 0\\}.$

Recall that Happel’s Theorem ([17], see also [14]) for finitely generated
tilting modules ensures a triangle equivalence of the derived categories of
the rings $A$ and $B$, while for an arbitrary infinitely generated tilting
module, one cannot get a derived equivalence but a recollement of triangulated
categories in which one category seems difficult to understand (see [4] or
Theorem 4.1). In [10, 11] special efforts are made to describe this category
as a derived module category for good tilting modules with special properties.
As pointed in [11], this does not always happen. So it remains mysterious what
a general form of Happel’s Theorem looks like for arbitrary infinitely
generated tilting modules. Now, Theorem 1.2 may serve as a counterpart of
Happel’s Theorem for infinitely generated good tilting modules. Furthermore,
compared with the main results in [31] and [5], Theorem 1.2 looks more
intrinsic and the derived category of the symmetric subcategory $\mathscr{E}$
seems much simpler than the derived category of a differential graded algebra.

A crucial point in the proof of Theorem 1.2 is to show that the kernel of the
total left-derived tensor functor ${{}_{A}}T\otimes^{\rm\mathbb{L}}_{B}-$ can
be realised as the derived category of the $n$-symmetric subcategory
$\mathscr{E}$. When $\mathscr{E}$ is an abelian subcategory, this can be shown
by analysing cohomologies of a special complex of $B$-modules which is a
compact generator for the kernel (for example, see [2, 10, 5, 11]). In
general, however, $\mathscr{E}$ is not always an abelian subcategory. Thus the
methods developed in the foregoing-mentioned papers do not work anymore
because cohomologies of complexes over $\mathscr{E}$ do not necessarily belong
to $\mathscr{E}$. To circumvent this obstacle here, we introduce two triangle
functors from ${\mathscr{D}}(B)$ to ${\mathscr{D}}(\mathscr{E})$ and from
${\mathscr{D}}(B)$ to ${\mathscr{D}}(\mathscr{E}^{\bot})$ (see Section 4.1 for
more details), where $\mathscr{E}^{\bot}$ is the right perpendicular
subcategory to $\mathscr{E}$ in $B\mbox{{\rm-Mod}}$. We then realise the
kernel of ${{}_{A}}T\otimes^{\rm\mathbb{L}}_{B}-$ as derived category of the
symmetric subcategory $\mathscr{E}$ of $B$-Mod through a bridge of a
$t$-structure induced from the tilting module. Notably, the method in this
article sheds some new light on when the kernel of
${{}_{A}}T\otimes^{\rm\mathbb{L}}_{B}-$ is triangle equivalent to the derived
module category of a ring, that is, when a good titling module $T$ is
homological (see [11]). Also, the method establishes a connection of
homological tilting modules with derived decompositions of module categories
(see [13] or Section 4.3 for Definition). We may state these observations as a
corollary.

######  Corollary 1.3.

The following are equivalent for a good tilting $A$-module $T$:

$(1)$ $T$ is a homological tilting module.

$(2)$ $\mathscr{E}$ is an abelian subcategory of $B\mbox{{\rm-Mod}}$.

$(3)$ $H^{m}({\rm Hom}_{A}(P^{\bullet},A)\otimes_{B}T)=0$ for all $m\geq 2$,
where $P^{\bullet}$ is a deleted projective resolution of ${{}_{A}}T$, and
$H^{m}$ is the $m$-th cohomology functor.

$(4)$ $(\mathscr{E},\mathscr{E}^{\bot})$ is a derived decomposition of the
abelian category $B\mbox{{\rm-Mod}}$, where $\mathscr{E}^{\bot}:=\\{Y\in
B\mbox{{\rm-Mod}}\mid{\rm Ext}^{n}_{B}(X,Y)=0,\forall\;X\in\mathscr{E},n\geq
0\\}$.

The equivalence of $(1)$ and $(2)$ is known in [5, Proposition 6.2], while the
equivalence of (1) and (3) is proved in [11, Theorem 1.1]. Only the condition
(4) is new.

Finally, we point out that Theorem 1.2 can be applied to special rings to get
recollemnts of bounded derived categories.

######  Corollary 1.4.

If $A$ is a left coherent ring (that is, every finitely generated left ideal
of $A$ is finitely presented) and $T$ is a good tilting $A$-module with
$B:={\rm End}_{A}(T)$, then, for $*\in\\{b,+,-,\emptyset\\}$, there exists a
recollement of derived categories

$\textstyle{\mathscr{D}^{*}(\mathscr{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}^{*}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{\mathscr{D}^{*}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $\mathscr{E}$ is a symmetric subcategory of $B\mbox{{\rm-Mod}}$ and $G$
is the total left-derived tensor functor defined by ${}_{A}T_{B}$.

This article is organised as follows. In Section 2, we recall definitions,
notations and basic facts needed for proofs. In Section 3, we introduce
$m$-symmetric subcategories for each natural number $m$ and give methods to
construct such subcategories. In Section 4, we prove the main result, Theorem
1.2, and its corollaries.

Acknowledgement. The research work of both authors was partially supported by
the National Natural Science Foundation of China (12031014) and the Beijing
Natural Science Foundation (1192004).

##  2 Preliminaries

In this section, we briefly recall some definitions, basic facts and notation
used in this paper. For unexplained notation employed in this paper, we refer
the reader to [10, 11] and the references therein.

###  2.1 Semi-orthogonal decompositions, recollements and homotopy (co)limits

Let $\mathcal{C}$ be an additive category.

A full subcategory of $\mathcal{C}$ is always assumed to be closed under
isomorphisms. For an object $X\in\mathcal{C}$, we write ${\rm add}(X)$ for the
full subcategory of $\mathcal{C}$ consisting of all direct summands of finite
coproducts of copies of $X$. If $\mathcal{C}$ admits coproducts (that is,
coproducts indexed over sets exist in ${\mathcal{C}}$), we write ${\rm
Add}(X)$ for the full subcategory of $\mathcal{C}$ consisting of all direct
summands of coproducts of copies of $X$. Dually, we write ${\rm Prod}(X)$ for
the full subcategory of $\mathcal{C}$ consisting of all direct summands of
products of copies of $X$ if $\mathcal{C}$ admits products (that is, products
indexed over sets exist in ${\mathcal{C}}$).

We denote the composition of two morphisms $f:X\to Y$ and $g:Y\to Z$ in
$\mathcal{C}$ by $fg$. We write $f^{*}$ for ${\rm Hom}_{\mathcal{C}}(Z,f):{\rm
Hom}_{\mathcal{C}}(Z,X)\rightarrow{\rm Hom}_{\mathcal{C}}(Z,Y)$ and $f_{*}$
for ${\rm Hom}_{\mathcal{C}}(f,Z):{\rm Hom}_{\mathcal{C}}(Y,Z)\rightarrow{\rm
Hom}_{\mathcal{C}}(X,Z)$. While for two functors $F:\mathcal{C}\to\mathcal{D}$
and $G:\mathcal{D}\to\mathcal{E}$, the composition of $F$ and $G$ is denoted
by $GF$ which is a functor from $\mathcal{C}$ to $\mathcal{E}$. The kernel and
image of $F$ are defined as ${\rm Ker}(F):=\\{X\in\mathcal{C}\mid FX\simeq
0\\}$ and ${\rm
Im}(F):=\\{Y\in\mathcal{D}\mid\exists\,X\in\mathcal{C},\;FX\simeq Y\\}$,
respectively. Thus ${\rm Ker}(F)$ and ${\rm Im}(F)$ are closed under
isomorphisms in $\mathcal{C}$.

For a complex $X^{\bullet}=(X^{i},d_{X}^{i})_{i\in\mathbb{Z}}$ over
$\mathcal{C}$, the morphism $d_{X}^{i}:X^{i}\rightarrow X^{i+1}$ is called a
_differential_ of $X^{\bullet}$. For simplicity, we sometimes write
$(X^{i})_{i\in\mathbb{Z}}$ for $X^{\bullet}$ without mentioning $d^{i}_{X}$.
For an integer $n$, we denote by $X^{\bullet}[n]$ the complex by shifting $n$
degrees, namely $(X^{\bullet}[n])^{i}=X^{n+i}$ and
$d^{i}_{X^{\bullet}[n]}=(-1)^{n}d^{n+i}_{X}$.

Let ${\mathscr{C}}(\mathcal{C})$ be the category of all complexes over
$\mathcal{C}$ with chain maps, and ${\mathscr{K}}(\mathcal{C})$ the homotopy
category of ${\mathscr{C}}(\mathcal{C})$. As usual, we denote by
${\mathscr{C}}^{-}(\mathcal{C})$ the category of bounded-above complexes over
$\mathcal{C}$, and by ${\mathscr{K}}^{-}(\mathcal{C})$ the homotopy category
of ${\mathscr{C}}^{-}(\mathcal{C})$. Note that ${\mathscr{K}}(\mathcal{C})$
and ${\mathscr{K}}^{-}(\mathcal{C})$ are triangulated categories. For a
triangulated category, its shift functor is denoted by $[1]$ universally.

Now, let $\mathcal{A}$ be an abelian category and
$X^{\bullet}\in{\mathscr{C}}(\mathcal{A})$. For $n\in\mathbb{Z}$, there are
two left-truncated complexes

$X{{}^{\leq n}}:\;\;\cdots\longrightarrow
X^{n-3}\xrightarrow{d_{X}^{n-3}}X^{n-2}\xrightarrow{d_{X}^{n-2}}X^{n-1}\xrightarrow{d_{X}^{n-1}}X^{n}\longrightarrow
0,$ $\tau^{\leq n}X^{\bullet}:\;\;\cdots\longrightarrow
X^{n-3}\xrightarrow{d_{X}^{n-3}}X^{n-2}\xrightarrow{d_{X}^{n-2}}X^{n-1}\xrightarrow{d_{X}^{n-1}}{\rm
Ker}(d_{X}^{n})\longrightarrow 0,$

together with canonical chain maps $X^{\bullet}\to X{{}^{\leq n+1}}\to
X{{}^{\leq n}}$ and $\tau^{\leq n}X^{\bullet}\to\tau^{\leq n+1}X^{\bullet}\to
X^{\bullet}$. Dually, right-truncated complexes $X^{\geq n}$ and $\tau^{\geq
n}X^{\bullet}$ (by taking cokernels on the left) can be defined. Further,
there are bi-truncated complexes for a pair of integers $(n,m)$ with $n<m$:

$X^{[n,m)}:\;\;0\longrightarrow
X^{n}\xrightarrow{d_{X}^{n}}X^{n+1}\longrightarrow\cdots\xrightarrow{d_{X}^{m-2}}X^{m-1}\xrightarrow{d_{X}^{m-1}}{\rm
Ker}(d_{X}^{m})\longrightarrow 0,$ $X^{(n,m]}:\;\;0\longrightarrow{\rm
Coker}(d_{X}^{n-1})\xrightarrow{\overline{d_{X}^{n}}}X^{n+1}\longrightarrow\cdots\xrightarrow{d_{X}^{m-2}}X^{m-1}\xrightarrow{d_{X}^{m-1}}X^{m}\longrightarrow
0.$

where $\overline{d_{X}^{n}}$ is induced from $d_{X}^{n}$.

Let $H^{n}(X^{\bullet})$ be the cohomology of $X^{\bullet}$ in degree $n$.
Then $H^{n}(-)$ is a functor from $\mathscr{C}(\mathcal{A})$ to $\mathcal{A}$
for all $n$.

Next, we recall the definition of semi-orthogonal decompositions of
triangulated categories (for example, see [20, Chapter 11]). Note that semi-
orthogonal decompositions are also termed hereditary torsion pairs (see [7,
Chapter I.2]) and closely related to Bousfield localizations [24, Section
9.1]) and $t$-structures of triangulated categories (see [6]).

######  Definition 2.1.

Let $\mathscr{D}$ be a triangulated category. A pair
$(\mathscr{X},\mathscr{Y})$ of full subcategories $\mathscr{X}$ and
$\mathscr{Y}$ of $\mathscr{D}$ is called a _semi-orthogonal decomposition_ of
$\mathscr{D}$ if

$(1)$ $\mathscr{X}$ and $\mathscr{Y}$ are triangulated subcategories of
$\mathscr{D}$.

$(2)$ ${\rm Hom}_{\mathscr{D}}(X,Y)=0$ for all $X\in\mathscr{X}$ and
$Y\in\mathscr{Y}$.

$(3)$ For each object $D\in\mathscr{D}$, there exists a triangle

$X_{D}\stackrel{{\scriptstyle
f_{D}}}{{\longrightarrow}}D\stackrel{{\scriptstyle
g^{D}}}{{\longrightarrow}}Y^{D}\longrightarrow X_{D}[1]$

in $\mathscr{D}$ such that $X_{D}\in\mathscr{X}$ and $Y^{D}\in\mathscr{Y}$.

The following is well known (for example, see [7, Chapter I.2]) and will be
used later.

######  Lemma 2.2.

Let $(\mathscr{X},\mathscr{Y})$ be a semi-orthogonal decomposition of
$\mathscr{D}$. The following statements hold.

$(1)$ The inclusion ${\bf i}:\mathscr{X}\to\mathscr{D}$ has a right adjoint
${\bf R}:\mathscr{D}\to\mathscr{X}$ given by $D\mapsto X_{D}$ for each
$D\in\mathscr{D}$, such that $f_{D}$ is the counit adjunction morphism of $D$.
Dually, the inclusion ${\bf j}:\mathscr{Y}\to\mathscr{D}$ has a left adjoint
${\bf L}:\mathscr{D}\to\mathscr{Y}$ given by $D\mapsto Y^{D}$ for each
$D\in\mathscr{D}$ such that $g^{D}$ is the unit adjunction morphism of $D$.

$(2)$ ${\rm Ker}({\bf L})=\mathscr{X}$ and ${\bf L}$ induces a triangle
equivalence $\overline{\bf L}:\mathscr{D}/\mathscr{X}\to\mathscr{Y}$ of which
a quasi-inverse is the composition of ${\bf j}$ with the localization functor
$\mathscr{D}\to\mathscr{D}/\mathscr{X}$. Dually, ${\rm Ker}({\bf
R})=\mathscr{Y}$ and ${\bf R}$ induces a triangle equivalence $\overline{\bf
R}:\mathscr{D}/\mathscr{Y}\to\mathscr{X}$ of which a quasi-inverse is the
composition of ${\bf i}$ with the localization functor
$\mathscr{D}\to\mathscr{D}/\mathscr{Y}$.

Semi-orthogonal decompositions are intimately connected with recollements of
triangulated categories introduced by Beilinson, Bernstein and Deligne in [6]
for understanding derived categories of perverse sheaves over singular spaces.
Later, this has widely been used in representation theories of Lie algebras
and associative algebras (for example, see [15, 19, 3, 12]).

######  Definition 2.3.

[6] Let $\mathcal{D},\mathcal{D^{\prime}}$ and $\mathcal{D^{\prime\prime}}$ be
triangulated categories. The category $\mathcal{D}$ is called a _recollement_
of $\mathcal{D^{\prime}}$ and $\mathcal{D^{\prime\prime}}$ (or there is a
recollement among $\mathcal{D}^{\prime\prime},\mathcal{D}$ and
$\mathcal{D}^{\prime}$) if there are six triangle functors

$\textstyle{\mathcal{D^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}=i_{!}}$$\textstyle{\mathcal{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{!}=j^{*}}$$\scriptstyle{i^{!}}$$\scriptstyle{i^{*}}$$\textstyle{\mathcal{D^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\scriptstyle{j_{!}}$

satisfying the four properties:

$(1)$ The four pairs $(i^{*},i_{*}),(i_{!},i^{!}),(j_{!},j^{!})$ and
$(j^{*},j_{*})$ of functors are adjoint.

$(2)$ The three functors $i_{*},j_{*}$ and $j_{!}$ are fully faithful.

$(3)$ $i^{!}j_{*}=0$ (and thus also $j^{!}i_{!}=0$ and $i^{*}j_{!}=0$).

$(4)$ There are two triangles for each object $D$ in $\mathcal{D}$:

$i_{!}i^{!}(D)\longrightarrow D\longrightarrow j_{*}j^{*}(D)\longrightarrow
i_{!}i^{!}(D)[1],$ $j_{!}j^{!}(D)\longrightarrow D\longrightarrow
i_{*}i^{*}(D)\longrightarrow j_{!}j^{!}(D)[1],$

where $i_{!}i^{!}(D)\rightarrow D$ and $j_{!}j^{!}(D)\rightarrow D$ are counit
adjunction morphisms, and where $D\rightarrow j_{*}j^{*}(D)$ and $D\rightarrow
i_{*}i^{*}(D)$ are unit adjunction morphisms.

If $\mathcal{D}$ is a recollement of $\mathcal{D^{\prime}}$ and
$\mathcal{D^{\prime\prime}}$, then the pairs $\big{(}{\rm Im}(j_{!}),{\rm
Im}(i_{*})\big{)}$ and $\big{(}{\rm Im}(i_{*}),{\rm Im}(j_{*})\big{)}$ are
semi-orthogonal decomositions of $\mathscr{D}$.

Next, we recall the definition of homotopy colimits and limits in triangulated
categories.

######  Definition 2.4.

[8, 24] Let $\mathscr{D}$ be a triangulated category such that countable
coproducts exist in $\mathscr{D}$. Let

$X_{0}\stackrel{{\scriptstyle
f_{0}}}{{\longrightarrow}}X_{1}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}X_{2}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
f_{n}}}{{\longrightarrow}}X_{n+1}\longrightarrow\cdots$

be a sequence of objects and morphisms in $\mathscr{D}$. The _homotopy
colimit_ of this sequence, denoted by $\underrightarrow{{\rm
Hocolim}}(X_{n})$, is given, up to non-canonical isomorphism, by the triangle

$\bigoplus_{n\geq
1}X_{n}\stackrel{{\scriptstyle(1-f_{*})}}{{\longrightarrow}}\bigoplus_{n\geq
1}X_{n}\longrightarrow\underrightarrow{{\rm
Hocolim}}(X_{n})\longrightarrow\bigoplus_{n\geq 1}X_{n}[1]$

where the morphism $(1-f_{*})$ is induced by $({\rm
Id}_{X_{i}},-f_{i}):X_{i}\to X_{i}\oplus X_{i+1}\subseteq\bigoplus_{n\geq
1}X_{n}$ for all $i\in\mathbb{N}$.

Dually, the _homotopy limit_ , denoted by $\underleftarrow{{\rm Holim}}$, of a
sequence of objects and morphisms in a triangulated category with countable
products can be defined.

Now, we consider homotopy colimits and limits in derived categories of abelian
categories. Let $\mathcal{A}$ be an abelian category. Recall that
$\mathcal{A}$ _satisfies $\rm{AB4}$_ if coproducts indexed over sets exist in
$\mathcal{A}$ and coproducts of short exact sequences in $\mathcal{A}$ are
exact. Dually, $\mathcal{A}$ _satisfies_ $\rm{AB4^{\prime}}$ if products
indexed over sets exist in $\mathcal{A}$ and products of short exact sequences
in $\mathcal{A}$ are exact. A typical example of abelian categories satisfying
both $\rm{AB4}$ and $\rm{AB4^{\prime}}$ is the module category of a ring.

A full subcategory $\mathcal{B}$ of the abelian category $\mathcal{A}$ is
called an _abelian subcategory_ of $\mathcal{A}$ if $\mathcal{B}$ is an
abelian category and the inclusion $\mathcal{B}\to\mathcal{A}$ is an exact
functor between abelian categories. This is equivalent to saying that
$\mathcal{B}$ is closed under taking kernels and cokernels in $\mathcal{A}$.

The following result is taken from [25, Lemma 6.1] and its dual statement.

######  Lemma 2.5.

Let $\mathcal{A}$ be an abelian category satisfying $AB4$ and $AB4^{\prime}$.
For a complex $X^{\bullet}\in{\mathscr{C}}(\mathcal{A})$, there are
isomorphisms in ${\mathscr{K}}(\mathcal{A})$ (and also in
${\mathscr{D}}(\mathcal{A})$)

$X^{\bullet}\simeq\underrightarrow{{\rm
Hocolim}}(X^{\geq-n})\simeq\underrightarrow{{\rm
Hocolim}}(X^{[-n,n+1)})\simeq\underleftarrow{{\rm Holim}}(X^{\leq
n})\simeq\underleftarrow{{\rm Holim}}(X^{(-n,n+1]}),$

where $n$ runs over all natural numbers.

Finally, we mention a general fact on adjoint pairs of functors.

######  Lemma 2.6.

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ and
$G:\mathcal{D}\rightarrow\mathcal{C}$ be two functors of categories such that
$(F,G)$ forms a adjoint pair.

$(1)$ If $G$ is fully faithful, then the counit $\epsilon:F\circ G\rightarrow
Id_{\mathcal{D}}$ is an isomorphism.

$(2)$ $G$ gives rise to an equivalence $\mathcal{D}\rightarrow{\rm Im}(G)$
with the quasi-inverse given by the restriction of $F$ to ${\rm Im}(G)$. In
particular, if $C\in{\rm Im}(G)$, then the unit $\eta_{C}:C\rightarrow GF(C)$
is an isomorphism.

###  2.2 Derived categories of exact categories

An _exact category_ (in the sense of Quillen) is by definition an additive
category endowed with a class of conflations closed under isomorphism and
satisfying certain axioms (for example, see [21, Section 4]). When the
additive category is an abelian category, the class of conflations coincides
with the class of short exact sequences.

Let $\mathcal{E}$ be an exact category and $\mathcal{F}$ a full subcategory of
$\mathcal{E}$. Suppose that $\mathcal{F}$ is _closed under extensions_ in
$\mathcal{E}$, that is, for any conflation $0\to X\to Y\to Z\to 0$ in
$\mathcal{E}$ with both $X,Z\in\mathcal{F}$, we have $Y\in\mathcal{F}$. Then
$\mathcal{F}$, endowed with the conflations in $\mathcal{E}$ having their
terms in $\mathcal{F}$, is an exact category, and the inclusion
$\mathcal{F}\subseteq\mathcal{E}$ is a fully faithful exact functor. In this
case, $\mathcal{F}$ is called a _fully exact subcategory_ of $\mathcal{E}$
(see [21, Section 4]).

From now on, let $\mathcal{A}$ be an abelian category and $\mathcal{E}$ a
fully exact subcategory of $\mathcal{A}$. A complex
$X^{\bullet}\in{\mathscr{C}}(\mathcal{E})$ is said to be _strictly exact_ if
it is exact in ${\mathscr{C}}(\mathcal{A})$ and all of its boundaries belong
to $\mathcal{E}$. Let $\mathscr{K}_{\rm{ac}}(\mathcal{E})$ be the full
subcategory of ${\mathscr{K}}(\mathcal{E})$ consisting of those complexes
which are isomorphic to strictly exact complexes. Then
$\mathscr{K}_{\rm{ac}}(\mathcal{E})$ is a full triangulated subcategory of
${\mathscr{K}}(\mathcal{E})$ closed under direct summands. The _unbounded
derived category_ of $\mathcal{E}$, denoted by ${\mathscr{D}}(\mathcal{E})$,
is defined to be the Verdier quotient of ${\mathscr{K}}(\mathcal{E})$ by
$\mathscr{K}_{\rm{ac}}(\mathcal{E})$. Similarly, the bounded-below, bounded-
above and bounded derived categories ${\mathscr{D}}^{+}(\mathcal{E})$,
${\mathscr{D}}^{-}(\mathcal{E})$ and ${\mathscr{D}}^{b}(\mathcal{E})$ can be
defined. Observe that the canonical functor
$\mathscr{D}^{*}(\mathcal{E})\to{\mathscr{D}}(\mathcal{E})$ is fully faithful
for $*\in\\{+,-,b\\}$.

If $F:\mathcal{E}_{1}\to\mathcal{E}_{2}$ is an additive functor of exact
categories, then $F$ induces a functor
$K(F):\mathscr{K}(\mathcal{E}_{1})\to\mathscr{K}(\mathcal{E}_{2})$ of homotopy
categories. Further, if $F$ is an exact functor, that is, $F$ sends
conflations in $\mathcal{E}_{1}$ to the ones in $\mathcal{E}_{2}$, then $F$
induces a functor
$D(F):\mathscr{D}(\mathcal{E}_{1})\to\mathscr{D}(\mathcal{E}_{2})$ of derived
categories.

Let $\mathcal{F}$ and $\mathcal{E}$ be fully exact subcategories of
$\mathcal{A}$ with $\mathcal{F}\subseteq\mathcal{E}$. Then $\mathcal{F}$ can
be regarded as a fully exact subcategory of $\mathcal{E}$. We consider the
following three conditions:

$(a)$ If $0\to X\to Y\to Z\to 0$ is an exact sequence in $\mathcal{A}$ with
$X\in\mathcal{E}$ and $Y,Z\in\mathcal{F}$, then $X\in\mathcal{F}$.

$(b)$ Each exact sequence $0\to E_{1}\to E_{0}\to F\to 0$ in $\mathcal{A}$
with $E_{1},E_{0}\in\mathcal{E}$ and $F\in\mathcal{F}$ can be completed into
an exact commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with $F_{1},F_{0}\in\mathcal{F}$.

$(c)$ There is a natural number $n$ such that, for each object
$E\in\mathcal{E}$, there is a long exact sequence in $\mathcal{A}$:

$0\longrightarrow F_{n}\stackrel{{\scriptstyle
f_{n}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}F_{1}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}F_{0}\stackrel{{\scriptstyle
f_{0}}}{{\longrightarrow}}E\longrightarrow 0$

with $F_{i}\in\mathcal{F}$ and ${\rm Im}(f_{i})\in\mathcal{E}$ for all $0\leq
i\leq n$.

The following result describes impacts of these conditions on derived
categories.

######  Lemma 2.7.

$(1)$ If $(a)$ and $(b)$ hold, then the inclusion
$\mathcal{F}\subseteq\mathcal{E}$ induces a fully faithful triangle functor
${\mathscr{D}}^{-}(\mathcal{F})\to{\mathscr{D}}^{-}(\mathcal{E})$.

$(2)$ Suppose that $\mathcal{E}$ is closed under direct summands in
$\mathcal{A}$. If $(a)$ and $(c)$ hold, then the inclusion
$\mathcal{F}\subseteq\mathcal{E}$ induces a triangle equivalence
${\mathscr{D}}(\mathcal{F})\rightarrow{\mathscr{D}}(\mathcal{E})$ which
restricts to an equivalence
$\mathscr{D}^{*}(\mathcal{F})\rightarrow\mathscr{D}^{*}(\mathcal{E})$ for any
$*\in\\{+,-,b\\}$.

Proof. $(1)$ follows from the dual of [21, Theorem 12.1] (see also [26,
Proposition A.2.1]), while $(2)$ follows from [26, Proposition A.5.6].
$\square$

If all arrows in $(a),(b)$ and $(c)$ are reversed, we get dual conditions of
$(a),(b)$ and $(c)$, respectively. So the dual version of Lemma 2.7 holds
true.

###  2.3 Derived functors of the module categories of rings

Let $R$ be a (unitary associative) ring. The full subcategories of projective
and injective $R$-modules are denoted by $R\mbox{{\rm-Proj}}$ and
$R\mbox{{\rm-Inj}}$, respectively. If $M$ is an $R$-module and $I$ is a
nonempty set, then $M^{(I)}$ and $M^{I}$ denote the direct sum and product of
$I$ copies of $M$, respectively. The projective dimension and the endomorphism
ring of $M$ are denoted by ${\rm proj.dim}({{}_{R}}M)$ and ${\rm End}_{R}(M)$,
respectively.

A full subcategory $\mathcal{T}$ of $R\mbox{{\rm-Mod}}$ is called a _thick
subcategory_ if it is closed under direct summands in $R\mbox{{\rm-Mod}}$ and
has the _two out of three property_ : for any short exact sequence $0\to X\to
Y\to Z\to 0$ in $R\mbox{{\rm-Mod}}$ with two terms in $\mathcal{T}$, the third
term belongs to $\mathcal{T}$ as well.

In the sequel, we simply write ${\mathscr{C}}(R)$, ${\mathscr{K}}(R)$ and
${\mathscr{D}}(R)$ for ${\mathscr{C}}(R\mbox{{\rm-Mod}})$,
${\mathscr{K}}(R\mbox{{\rm-Mod}})$ and ${\mathscr{D}}(R\mbox{{\rm-Mod}})$,
respectively, and regard $R\mbox{{\rm-Mod}}$ as the subcategory of
${\mathscr{D}}(R)$ consisting of all stalk complexes concentrated in degree
zero.

Let ${\mathscr{K}}(R)_{P}$ (respectively, ${\mathscr{K}}(R)_{I}$) be the
smallest full triangulated subcategory of ${\mathscr{K}}(R)$, which contains
all the bounded-above (respectively, bounded-below) complexes of projective
(respectively, injective) $R$-modules, and is closed under arbitrary direct
sums (respectively, direct products). Since ${\mathscr{K}}(R\mbox{{\rm-
Proj}})$ satisfies theses two properties, we have
${\mathscr{K}}(R)_{P}\subseteq{\mathscr{K}}(R\mbox{{\rm-Proj}})$. Let
$\mathscr{K}_{\rm{ac}}(R)$ be the full subcategory of ${\mathscr{K}}(R)$
consisting of all exact complexes. Then
$({\mathscr{K}}(R){{}_{P}},\mathscr{K}_{\rm{ac}}(R))$ forms a semi-orthogonal
decomposition of ${\mathscr{K}}(R)$. Let
${}_{p}(-):{\mathscr{K}}(R)\to{\mathscr{K}}(R){{}_{P}}$ be a right adjoint of
the inclusion ${\mathscr{K}}(R)_{P}\to{\mathscr{K}}(R)$. Then, by Lemma 2.2,
the functor ${}_{p}(-)$ induces a triangle equivalence
${\mathscr{D}}(R)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\mathscr{K}}(R){{}_{P}}$
of which a quasi-inverse is the composition of the inclusion
${\mathscr{K}}(R)_{P}\to{\mathscr{K}}(R)$ with the localization functor
${\mathscr{K}}(R)\to{\mathscr{D}}(R)$. Moreover, for each $X^{\bullet}$ in
${\mathscr{K}}(R)$, the associated counit adjunction morphism
$\alpha_{X^{\bullet}}:{{}_{p}}X^{\bullet}\to X^{\bullet}$ is a quasi-
isomorphism. Thus $\alpha_{X^{\bullet}}$ or simply ${}_{p}X^{\bullet}$ is
called a _projective resolution_ of $X^{\bullet}$ in ${\mathscr{D}}(R)$. For
example, if $X$ is an $R$-module, then ${}_{p}X$ can be chosen as a deleted
projective resolution of ${}_{R}X$. Dually,
$(\mathscr{K}_{\rm{ac}}(R),{\mathscr{K}}(R){{}_{I}})$ is a semi-orthogonal
decomposition of ${\mathscr{K}}(R)$. In particular, there exists a quasi-
isomorphism $\beta_{X^{\bullet}}:X^{\bullet}\to{{}_{i}}X^{\bullet}$ in
${\mathscr{K}}(R)$ with ${{}_{i}}X^{\bullet}\in{\mathscr{K}}(R)_{I}$. The
complex ${}_{i}X^{\bullet}$ is called the _injective resolution_ of
$X^{\bullet}$ in ${\mathscr{D}}(R)$. In particular, ${\rm
Hom}_{{\mathscr{K}}(R)}(P^{\bullet},X^{\bullet})\simeq{\rm
Hom}_{{\mathscr{D}}(R)}(P^{\bullet},X^{\bullet})$ and ${\rm
Hom}_{{\mathscr{K}}(R)}(X^{\bullet},I^{\bullet})\simeq{\rm
Hom}_{{\mathscr{D}}(R)}(X^{\bullet},I^{\bullet})$ for
$P^{\bullet}\in\mathscr{K}^{-}(R\mbox{{\rm-
Proj}}),I^{\bullet}\in\mathscr{K}^{+}(R\mbox{-Inj})$ and
$X^{\bullet}\in{\mathscr{K}}(R).$

Now, let $S$ be another ring. For a triangle functor
$F:{\mathscr{K}}(R)\to{\mathscr{K}}(S)$, its _total left-derived functor_
${\mathbb{L}}F:{\mathscr{D}}(R)\to{\mathscr{D}}(S)$ is defined by
$X^{\bullet}\mapsto F(_{p}X^{\bullet})$, and its _total right-derived functor_
${\mathbb{R}}F:{\mathscr{D}}(R)\to{\mathscr{D}}(S)$ is defined by
$X^{\bullet}\mapsto F(_{i}X^{\bullet})$. Further, if $F(X^{\bullet})$ is exact
whenever $X^{\bullet}$ is exact, then $F$ induces a triangle functor
$D(F):{\mathscr{D}}(R)\to{\mathscr{D}}(S)$, $X^{\bullet}\mapsto
F(X^{\bullet})$. In this case, up to natural isomorphism,
${\mathbb{L}}F={\mathbb{R}}F=D(F)$, and $D(F)$ is called the _derived functor_
of $F$.

Given a complex $M^{\bullet}$ of $R$-$S$-bimodules, we denote by
${\mathbb{R}}{\rm Hom}_{R}(M^{\bullet},-)$ and
$M^{\bullet}\otimes^{\rm\mathbb{L}}_{S}-$ the total right-derived functor of
${\rm Hom}^{\bullet}_{R}(M^{\bullet},-)$ and the total left-derived functor of
$M^{\bullet}\otimes^{\bullet}_{S}-$, respectively. Then
$\big{(}M^{\bullet}\otimes^{\rm\mathbb{L}}_{S}-,{\mathbb{R}}{\rm
Hom}_{R}(M^{\bullet},-)\big{)}$ is an adjoint pair of triangle functors. In
case of $Y\in S\mbox{{\rm-Mod}}$ and $X\in R\mbox{{\rm-Mod}}$, we write
$M^{\bullet}\otimes_{S}Y$ and ${\rm Hom}_{R}(M^{\bullet},\,X)$ for
$M^{\bullet}\otimes^{\bullet}_{S}Y$ and ${\rm
Hom}^{\bullet}_{R}(M^{\bullet},\,X)$, respectively.

##  3 Symmetric subcategories of abelian categories

Now we introduce $n$-symmetric subcategories of an ableian category for $n\geq
0$.

Recall that an abelian category is _complete_ (respectively, _cocomplete_) if
it has products (respectively, coproducts) indexed over sets; and _bicomplete_
if it is complete and cocomplete.

######  Definition 3.1.

Let $n\in\mathbb{N}$, and let $\mathcal{A}$ be a bicomplete abelian category.
An additive full subcategory $\mathcal{B}$ of $\mathcal{A}$ is said to be _$n$
-symmetric_ if

$(1)$ $\mathcal{B}$ is closed under extensions, products and coproducts.

$(2)$ Given an exact sequence $0\rightarrow X\rightarrow
M_{n}\rightarrow\cdots\rightarrow M_{1}\rightarrow M_{0}\rightarrow
Y\rightarrow 0$ in $\mathcal{A}$, we have $X,Y\in\mathcal{B}$ whenever all
$M_{i}\in\mathcal{B}$.

######  Remark 3.2.

Let $\mathcal{B}$ be an additive full subcategory of a bicomplete abelian
category $\mathcal{A}$.

$(1)$ If $\mathcal{B}$ is an $n$-symmetric subcategory of $\mathcal{A}$, then
$\mathcal{B}$ is an exact, thick subcategory of $\mathcal{A}$. It is also
$(n+1)$-symmetric.

$(2)$ If $\mathcal{B}_{i}$ is an $m_{i}$-symmetric subcategories of
$\mathcal{A}$ for $i=1,2$ then $\mathcal{B}_{1}\cap\mathcal{B}_{2}$ is a
max$\\{m_{1},m_{2}\\}$-symmetric subcategory of $\mathcal{A}$.

$(3)$ If $\mathcal{B}$ is closed under extensions and satisfies Definition
3.1(2), then $\mathcal{B}$ is an $n$-wide subcategory of $\mathcal{A}$ in the
sense of [22, Definition 4.1].

$(4)$ $\mathcal{B}$ is $0$-symmetric if and only if $\mathcal{B}$ is a Serre
subcategory (that is, closed under subobjects, quotient objects and
extensions) closed under coproducts and products if and only $\mathcal{B}$ is
a localizing subcategory (that is, a Serre subcategory closed under
coproducts) closed under products. $\mathcal{B}$ is $1$-symmetric if and only
if $\mathcal{B}$ is an abelian subcategory closed under extensions, coproducts
and products.

In particular, if $R$ is a ring and $\mathcal{A}=R\mbox{{\rm-Mod}}$, then a
full subcategory $\mathcal{B}$ of $\mathcal{A}$ is $0$-symmetric if and only
if there is an ideal $I$ of $R$ with $I^{2}=I$ such that
$\mathcal{B}=(R/I)\mbox{{\rm-Mod}}$ = $\\{M\in R\mbox{{\rm-Mod}}\mid IM=0\\}$.
This follows from [28, Proposition 6.12]. Moreover, a full subcategory
$\mathcal{B}$ of $\mathcal{A}$ is $1$-symmetric if and only if there is a ring
epimorphism $\lambda:R\rightarrow S$ with ${\rm Tor}_{1}^{R}(S,S)=0$ such that
$\mathcal{B}$ coincides with the image of the induced fully faithful functor
$\lambda_{*}:S\mbox{{\rm-Mod}}\rightarrow R\mbox{{\rm-Mod}}$, that is,
$\mathcal{B}={\rm Im}(\lambda_{*})\simeq S\mbox{{\rm-Mod}}.$ This can be seen
by [10, Lemma 2.1].

######  Proposition 3.3.

Let $\mathcal{A}$ and $\mathcal{C}$ be bicomplete abelian categories
satisfying AB4 and AB4′, and let
$F:{\mathscr{D}}(\mathcal{A})\rightarrow{\mathscr{D}}(\mathcal{C})$ be a
triangle functor commuting with coproducts and products. If there are integers
$s\leq r$ such that $H^{i}(FX)=0$ for all $X\in\mathcal{A}$, $r<i$, or $i<s$,
then $\mathscr{E}:=\mathcal{A}\cap{\rm Ker}(F)$ is an $(r-s)$-symmetric
subcategory of $\mathcal{A}$.

Proof. It is easy to see that $\mathscr{E}$ is an additive subcategory and
closed under extensions, coproducts and products. Let $n:=r-s$ and let
$0\rightarrow X\rightarrow M_{n}\rightarrow\cdots\rightarrow M_{1}\rightarrow
M_{0}\rightarrow Y\rightarrow 0$ be an exact sequence in $\mathcal{A}$ with
$M_{j}\in\mathscr{E}$ for all $j$. By assumption, we have $H^{i}(F(Y))=0$ for
$i>r$ or $i<s$. Now, let $s\leq i\leq r$. Then $n+1+i\geq n+1+s=r+1$. It
follows from $F(M_{j})=0$ for $0\leq j\leq n$ that $FY\simeq F(X)[n+1]$ in
${\mathscr{D}}(\mathcal{C}).$ This implies $H^{i}(F(Y))\simeq
H^{i}(F(X)[n+1])\simeq H^{n+1+i}(F(X))$ for $i\in\mathbb{Z}$. In particular,
$H^{i}(F(Y))=0$ by assumption. Thus $H^{i}(F(Y))=0$ for all $i\in\mathbb{Z}$
and $Y\in{\rm Ker}(F)$. This also shows $X\in{\rm Ker}(F)$. Thus $\mathscr{E}$
is an $n$-symmetric subcategory of $\mathcal{A}$. $\square$

As an application of Proposition 3.3, we consider the module categories of
rings.

######  Example 3.4.

Let $R$ and $S$ be rings, and let ${{}_{R}}M_{S}$ be an $R$-$S$-bimodule.

(1) If $M_{S}$ has a finite projective resolution of length $n$ by finitely
generated projective right $S$-modules, that is, there is an extact sequence
$0\rightarrow Q_{n}\rightarrow\cdots\rightarrow Q_{1}\rightarrow
Q_{0}\rightarrow M_{S}\rightarrow 0$ with all $Q_{j}$ being finitely generated
projective right $S$-modules, then ${\rm
Ker}(M\otimes^{\rm\mathbb{L}}_{S}-)\cap S\mbox{{\rm-Mod}}$ is an $n$-symmetric
subcategory of $S\mbox{{\rm-Mod}}$.

(2) If ${{}_{R}}M$ has a finite projective resolution of length $n$ by
finitely generated projective modules, then ${\rm Ker}(\mathbb{R}{\rm
Hom}_{R}(M,-))\cap R\mbox{{\rm-Mod}}$ is an $n$-symmetric subcategory of
$R\mbox{{\rm-Mod}}$.

Note that ${\rm Ker}(M\otimes^{\rm\mathbb{L}}_{S}-)\cap S\mbox{{\rm-Mod}}$ =
$\\{Y\in S\mbox{{\rm-Mod}}\mid{\rm Tor}_{i}^{S}(M,Y)=0,\forall i\geq 0\\}$ and
${\rm Ker}(\mathbb{R}{\rm Hom}_{R}(M,-))\cap R\mbox{{\rm-Mod}}$ = $\\{X\in
R\mbox{{\rm-Mod}}\mid{\rm Ext}^{i}_{R}(M,X)=0,\forall i\geq 0\\}$.

(3) Let ${}_{A}T$ be a good $n$-tilting $A$-module with $B:={\rm End}_{A}(T)$.
Then $\mathscr{E}:={\rm Ker}(T\otimes^{\rm\mathbb{L}}_{B}-)\cap{B\mbox{{\rm-
Mod}}}$ is always an $n$-symmetric subcategory of $B\mbox{{\rm-Mod}}$. It
follows from [3, Theorem 5.12 and Theorem 7.5] that $\mathscr{E}$ is
$0$-symmetric if and only if ${}_{A}T$ is pure-projective if and only if the
heart of the $t$-structure induced from the tilting module ${}_{A}T$ is a
Grothendieck category. Furthermore, it follows from [11, Theorem 1.1] that
$\mathscr{E}$ is $1$-symmetric if and only if ${}_{A}T$ is a homological
tilting module.

A further discussion on $n$-symmetric subcategories will be done in a
forthcoming article.

##  4 Tilting modules and derived recollements

This entire section is devoted to a proof of Theorem 1.2. We first make a
couple of preparations

###  4.1 Derived functors induced by good tilting modules

Throughout this section, $A$ denotes a unitary ring, $T$ a good $n$-tilting
$A$-module with $(T1),(T2)$, and $(T3)$ for a natural number $n$, and $B:={\rm
End}_{A}(T)$. Further, we define

$G:={{}_{A}}T\otimes^{\rm\mathbb{L}}_{B}-:\,{\mathscr{D}}(B)\longrightarrow{\mathscr{D}}(A)\quad\mbox{and}\quad
H:={\rm\mathbb{R}}{\rm
Hom}\,_{A}(T,-):\,{\mathscr{D}}(A)\longrightarrow{\mathscr{D}}(B),$

to be the total left- and right-derived functors of ${{}_{A}}T{{}_{B}}$,
respectively. We write by

$\eta^{\prime}:Id_{B\mbox{{\rm-Mod}}}\longrightarrow{\rm
Hom}_{A}(T,T\otimes_{B}-)\quad\mbox{and}\quad\eta:Id_{{\mathscr{D}}(B)}\longrightarrow
H\circ G$

the unit adjunctions associated with the adjoint pairs $(T\otimes_{B}-,{\rm
Hom}_{A}(T,-))$ and $(G,H)$, respectively. Recall that the restriction of the
localization functor ${\mathscr{K}}(B)\to{\mathscr{D}}(B)$ to
${\mathscr{K}}(B)_{p}$ is a triangle equivalence. A quasi-inverse of this
equivalence is denoted by

${}_{p}(-):{\mathscr{D}}(B)\longrightarrow{\mathscr{K}}(B)_{P}.$

Without loss of generality, we assume that ${{}_{p}}M$ is a deleted projective
resolution of $M$ for each $B$-module $M$.

The derived categories ${\mathscr{D}}(A)$ and ${\mathscr{D}}(B)$ are related
by the following recollement (see [4, Theorem 2.2]). This is used in [11,
Lemma 5.2] to understand when a tilting module is homological.

######  Theorem 4.1.

[4] There exists a recollement of triangulated categories:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.8896pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm
Ker}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
53.03282pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.03282pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathscr{D}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
93.70938pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\raise
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0.0pt\raise
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0.0pt\raise
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In particular, $({\rm Ker}(G),{\rm Im}(H))$ is a semi-orthogonal decomposition
of ${\mathscr{D}}(B)$.

Observe that the modules ${{}_{A}}T$ and $T_{B}$ have finite projective
dimension. In fact, $T_{B}$ has a finite projective resolution by finitely
generated projective right $B$-modules since the tilting module ${}_{A}T$ is
good. The next result is deduced from [30, Theorem 10.5.9, Corollary 10.5.11].

######  Lemma 4.2.

$(1)$ Let $\alpha_{Y^{\bullet}}:{{}_{p}}Y^{\bullet}\to Y^{\bullet}$ be the
projective resolution of $Y^{\bullet}$ in ${\mathscr{D}}(B)$. If ${\rm
Tor}^{B}_{j}(T,Y^{i})=0$ for all $i\in\mathbb{Z}$ and $j\geq 1$, then
$T\otimes_{B}\alpha_{Y^{\bullet}}:G(Y^{\bullet})=T\otimes_{B}{{}_{p}}Y^{\bullet}\rightarrow
T\otimes_{B}Y^{\bullet}$ is an isomorphism in ${\mathscr{D}}(A)$.

$(2)$ Let $\beta_{X^{\bullet}}:X^{\bullet}\to{{}_{i}}X^{\bullet}$ be the
injective resolution of $X^{\bullet}$ in ${\mathscr{D}}(A)$. If ${\rm
Ext}_{A}^{j}(T,X^{i})=0$ for all $i\in\mathbb{Z}$ and $j\geq 1$, then ${\rm
Hom}_{A}(T,\beta_{X^{\bullet}}):{\rm Hom}_{A}(T,X^{\bullet})\rightarrow{\rm
Hom}_{A}(T,{{}_{i}}X^{\bullet})=H(X^{\bullet})$ is an isomorphism in
${\mathscr{D}}(B)$.

Combining Theorem 4.1 with Lemma 4.2, we have the lemma.

######  Lemma 4.3.

$(1)$ For each $P\in B\mbox{{\rm-Proj}}$, the unit adjunction morphism
$\eta_{P}^{\prime}:P\to{\rm Hom}_{A}(T,T\otimes_{B}P)$ is injective with ${\rm
Coker}(\eta_{P}^{\prime})\in\mathscr{E}$.

$(2)$ For each $M^{\bullet}\in{\mathscr{D}}(B)$, there is a commutative
diagram of triangles in ${\mathscr{D}}(B)$:

$\textstyle{{\rm
Coker}(\eta^{\prime}_{{{}_{p}}M^{\bullet}})[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}_{p}}M^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}_{{{}_{p}}M^{\bullet}}}$$\scriptstyle{\simeq}$$\scriptstyle{\alpha_{M^{\bullet}}}$$\textstyle{{\rm
Hom}_{A}(T,T\otimes_{B}{{}_{p}}M^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{{\rm
Hom}_{A}(T,\beta_{T\otimes_{B}{{}_{p}}M^{\bullet}})}$$\textstyle{{\rm
Coker}(\eta^{\prime}_{{{}_{p}}M^{\bullet}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Coker}(\eta^{\prime}_{{{}_{p}}M^{\bullet}})[-1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{M}}$$\textstyle{H\circ
G(M^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Coker}(\eta^{\prime}_{{{}_{p}}M^{\bullet}})}$

where
$\eta^{\prime}_{{{}_{p}}M^{\bullet}}:=(\eta_{{{}_{p}}M^{n}}^{\prime})_{n\in\mathbb{Z}}$
and ${\rm Coker}(\eta^{\prime}_{{{}_{p}}M^{\bullet}})\in{\rm Ker}(G)$.

$(3)$

${\rm
Ker}(G)=\\{\overline{Y}^{\bullet}\in{\mathscr{D}}(B)\mid\overline{Y}^{\bullet}\simeq
Y^{\bullet}{\rm\;in\;}{\mathscr{D}}(B){\rm\;with\;}Y^{i}\in\mathscr{E}{\rm\;for\
all\;}i\in\mathbb{Z}\\}\quad\mbox{and}$ ${\rm
Im}(H)=\\{\overline{Z}^{\bullet}\in{\mathscr{D}}(B)\mid\overline{Z}^{\bullet}\simeq
Z^{\bullet}\mbox{\; in \;}{\mathscr{D}}(B)\mbox{\; with\;}Z^{i}\in{\rm
Hom}_{A}(T,{\rm Add}(T))\mbox{\;for\, all \,}i\in\mathbb{Z}\\},$

where ${\rm Hom}_{A}(T,{\rm Add}(T)):=\\{{\rm Hom}_{A}(T,T^{\prime})\in
B\mbox{{\rm-Mod}}\mid T^{\prime}\in{\rm Add}({{}_{A}}T)\\}$.

Proof. $(1)$ Since the canonical map ${\rm Hom}_{A}(T,T)^{(\alpha)}\to{\rm
Hom}_{A}(T,T^{(\alpha)})$ is injective for any nonempty set $\alpha$, the map
$\eta_{P}^{\prime}$ is injective for any free $B$-module $P$, and therefore
$\eta_{P}^{\prime}$ is injective for any projective $B$-module $P$. By Lemma
4.2, $H\circ G(P)\simeq{\rm Hom}_{A}(T,T\otimes_{B}P)$. It follows from
Theorem 4.1 that there exists a triangle

$X^{\bullet}_{P}\longrightarrow
P\stackrel{{\scriptstyle\eta_{P}^{\prime}}}{{\longrightarrow}}{\rm
Hom}_{A}(T,T\otimes_{B}P)\longrightarrow X^{\bullet}_{P}[1]$

in ${\mathscr{D}}(B)$ with $X^{\bullet}_{P}\in{\rm Ker}(G)$. As
$\eta_{P}^{\prime}$ is injective, we have ${\rm
Coker}(\eta_{P}^{\prime})\simeq X^{\bullet}_{P}[1]\in{\rm Ker}(G)$, and
therefore ${\rm Coker}(\eta_{P}^{\prime})\in\mathscr{E}$.

$(2)$ This follows from Theorem 4.1, Definition 2.3(4) and Lemma 4.2(2)
because
${{}_{p}}M^{\bullet}\in{\mathscr{K}}(B)_{P}\subseteq{\mathscr{K}}(B\mbox{{\rm-
Proj}})$ and ${\rm Ext}_{A}^{j}(T,T^{(\alpha)})=0$ for $j\geq 1$ and nonempty
sets $\alpha$.

$(3)$ This is shown in [10, Proposition 4.6] for good $1$-tilting modules, but
the proof there also works for good $n$-tilting modules. $\square$

Thanks to [3, Theorem 3.5], there exists a t-structure in ${\mathscr{D}}(A)$
associated with ${{}_{A}}T$ such that its heart is given by

$\mathcal{H}:=\\{X\in{\mathscr{D}}(A)\mid{\rm
Hom}_{{\mathscr{D}}(A)}(T,X[n])=0\mbox{\;for\, all \,}n\neq 0\\}.$

Then $\mathcal{H}$ contains ${\rm Add}(_{A}T)$ and is an abelian category of
which projective objects are isomorphic in ${\mathscr{D}}(A)$ to modules in
${\rm Add}({{}_{A}}T)$.

######  Lemma 4.4.

[3, Proposition 5.5] The following are true.

$(1)$ The restriction of the functor $H$ to $\mathcal{H}$ yields an exact and
fully faithful functor

$H_{T}={\rm Hom}_{\mathcal{H}}(T,-):\mathcal{H}\longrightarrow B\mbox{{\rm-
Mod}}.$

$(2)$ ${\rm Im}(H_{T})={\rm Im}(H)\cap B\mbox{{\rm-Mod}}$.

$(3)$ $H_{T}$ has a left adjoint $F:B\mbox{{\rm-Mod}}\to\mathcal{H}$ given by
the composition of the restriction of $G$ to $B\mbox{{\rm-Mod}}$ with the left
adjoint of the inclusion $\mathcal{H}\to\mathscr{U}$, where
$\mathscr{U}:=\\{X\in{\mathscr{D}}(A)\mid{\rm
Hom}_{{\mathscr{D}}(A)}(T,X[n])=0\mbox{\;for\,}n>0\\}.$

Let $\mathcal{Y}:={\rm Im}(H_{T})$. Then $H_{T}$ induces an equivalence
$\mathcal{H}\simeq\mathcal{Y}$. We have the following.

######  Corollary 4.5.

$(1)$ The restriction of $H_{T}$ to ${\rm Add}({{}_{A}}T)$ coincides with the
one of ${\rm Hom}_{A}(T,-)$ to ${\rm Add}(_{A}T)$, and the restriction of $F$,
defined in Lemma 4.4(3), to $B\mbox{{\rm-Proj}}$ coincides with the one of
$T\otimes_{B}-$ to $B\mbox{{\rm-Proj}}$.

$(2)$ $\mathcal{Y}$ is an abelian subcategory of $B\mbox{{\rm-Mod}}$ closed
under isomorphisms, extensions and direct products. Moreover, a $B$-module $M$
is a projective object of $\mathcal{Y}$ if and only if $M\simeq{\rm
Hom}_{A}(T,T^{\prime})$ for some $T^{\prime}\in{\rm Add}({{}_{A}}T)$.

Proof. (1) If $P\in B\mbox{{\rm-Proj}}$, then $G(P)=T\otimes_{B}P\in{\rm
Add}({{}_{A}}T)\subseteq\mathscr{U}$. Now, Corollary 4.5 follows from Lemma
4.4(1) and $(3)$.

(2) By Lemma 4.4(1), we have the equivalence
$H_{T}:\mathcal{H}\rightarrow\mathcal{Y}$, while the projective objects of
$\mathcal{H}$ are just objects of ${\rm Add}(_{A}T)$, up to ismorphism in
${\mathscr{D}}(A)$. Hence (2) follows. $\square$

Now, let

$\mathscr{E}:=\\{Y\in B\mbox{{\rm-Mod}}\mid{\rm
Tor}_{i}^{B}(T,Y)=0\mbox{\;for\, all \,}i\geq 0\\}=B\mbox{{\rm-Mod}}\cap{\rm
Ker}(G).$

In general, $\mathscr{E}$ is not an abelian subcategory of $B\mbox{{\rm-
Mod}}$. Nevertheless, we have the property.

######  Lemma 4.6.

$(1)$ $\mathscr{E}$ is an $n$-symmetric subcategory of $B\mbox{{\rm-Mod}}$.
Particularly, $\mathscr{E}$ is a thick subcategory of $B\mbox{{\rm-Mod}}$.

$(2)$ If a $B$-module $M$ is quasi-isomorphic in ${\mathscr{C}}(B)$ to a
complex with all terms in $\mathscr{E}$, then $M$ belongs to $\mathscr{E}$.

$(3)$ Let $\mathscr{E}_{0}:=\\{{\rm Coker}(\eta^{\prime}_{P})\mid P\in
B\mbox{{\rm-Proj}}\\}$. Then a $B$-module $M\in\mathscr{E}$ if and only if
there exists an exact sequence $0\to M\to E_{\infty}\to E^{1}\to 0$ of
$B$-modules such that $E^{1}\in\mathscr{E}_{0}$ and $E_{\infty}$ admits a long
exact sequence

$\cdots\longrightarrow E_{i}\longrightarrow\cdots\longrightarrow
E_{2}\longrightarrow E_{1}\longrightarrow E_{\infty}\longrightarrow 0$

with $E_{i}\in\mathscr{E}_{0}$ for all $i\geq 1$.

Proof. $(1)$ follows from Example 3.4(3). $(2)$ is due to the characterization
of ${\rm Ker}(G)$ in Lemma 4.3(3).

(3) The sufficiency of $(3)$ follows from $(1)$. To show the necessity of
$(3)$, we take $M\in\mathscr{E}$. Then $G(M)=0$. Let
${}_{p}M:=(P^{-i})_{i\in\mathbb{N}}$ be a deleted projective resolution of
$M$. By Lemma 4.3(2), $M[1]\simeq{\rm Coker}(\eta^{\prime}_{{{}_{p}}M})$ in
${\mathscr{D}}(B)$. Define $E_{i}:={\rm Coker}(\eta^{\prime}_{P^{-i}})$ for
each $i\in\mathbb{N}$. Then ${\rm Coker}(\eta^{\prime}_{{{}_{p}}M})$ has the
form $\cdots\to E_{i}\to E_{i-1}\to\cdots\to E_{2}\stackrel{{\scriptstyle
d_{2}}}{{\longrightarrow}}E_{1}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}E_{0}\to 0$, where $E_{i}$ is of the degree $(-i)$.
Since $P^{-i}\in B\mbox{{\rm-Proj}}$, we have $E_{i}\in\mathscr{E}_{0}$ by
Lemma 4.3(1). Note that $H^{i}({\rm Coker}(\eta^{\prime}_{{{}_{p}}M}))\simeq
H^{i+1}(M)=0$ for any $i\neq-1$ and $H^{-1}({\rm
Coker}(\eta^{\prime}_{{{}_{p}}M}))\simeq H^{0}(M)=M$. This implies that
$d_{1}$ is surjective and the kernel of the map ${\rm Coker}(d_{2})\to E_{0}$
induced from $d_{1}$ is isomorphic to $M$. Now, if $E_{\infty}:={\rm
Coker}(d_{2})$ and $E^{1}:=E_{0}$, then the necessity of $(3)$ holds.
$\square$

The category $\mathscr{E}$ can be regarded as a fully exact subcategory of
$B\mbox{{\rm-Mod}}$. Moreover, a complex
$X^{\bullet}\in{\mathscr{C}}(\mathscr{E})$ is strictly exact (see Section 2.2)
if and only if it is exact in ${\mathscr{C}}(B)$ by Lemma 4.6(2). This implies
that $\mathscr{K}_{\rm{ac}}(\mathscr{E})$ is the full triangulated subcategory
of ${\mathscr{K}}(\mathscr{E})$ consisting of exact complexes. Moreover, by
Lemma 4.6(1), $\mathscr{K}_{\rm{ac}}(\mathscr{E})$ is closed under arbitrary
direct sums and products in ${\mathscr{K}}(B)$, and therefore in
${\mathscr{K}}(\mathscr{E})$. Recall that the _unbounded derived category_
${\mathscr{D}}(\mathscr{E})$ of $\mathscr{E}$ is defined to be the Verdier
quotient of ${\mathscr{K}}(\mathscr{E})$ by
$\mathscr{K}_{\rm{ac}}(\mathscr{E})$. Then ${\mathscr{D}}(\mathscr{E})$ has
direct sums and products, and the localization functor
${\mathscr{K}}(\mathscr{E})\to{\mathscr{D}}(\mathscr{E})$ preserves direct
sums and products by [8, Lemma 1.5] and its dual statement.

Let

$i:\mathscr{E}\longrightarrow B\mbox{{\rm-Mod}}\quad\mbox{and}\quad
j:\mathcal{Y}\longrightarrow B\mbox{{\rm-Mod}}$

be the inclusions. They are exact functors between exact categories and
automatically induce derived functors between derived categories:

$D(i):{\mathscr{D}}(\mathscr{E})\longrightarrow{\mathscr{D}}(B)\quad\mbox{and}\quad
D(j):{\mathscr{D}}(\mathcal{Y})\longrightarrow{\mathscr{D}}(B).$

Moreover, by Lemma 4.3(1), there are another two additive functors between
additive categories:

${\rm Hom}_{A}(T,T\otimes_{B}-):B\mbox{{\rm-
Proj}}\longrightarrow\mathcal{Y}:\;\;X\mapsto{\rm Hom}_{A}(T,T\otimes_{B}X)$
${\rm Coker}(\eta^{\prime}_{-}):B\mbox{{\rm-
Proj}}\longrightarrow\mathscr{E}:\;\;X\mapsto{\rm Coker}(\eta^{\prime}_{X})$

for $X\in B\mbox{{\rm-Proj}}$. Here we fix an exact sequence

$0\longrightarrow
X\stackrel{{\scriptstyle\eta_{X}^{\prime}}}{{\longrightarrow}}{\rm
Hom}_{A}(T,T\otimes_{B}X)\longrightarrow{\rm
Coker}(\eta^{\prime}_{X})\longrightarrow 0$

for each $X$. Since ${\mathscr{K}}(B)_{P}$ is a triangulated subcategory of
${\mathscr{K}}(B\mbox{{\rm-Proj}})$, we can define the following derived
functors $\Psi:{\mathscr{D}}(B)\rightarrow{\mathscr{D}}(\mathcal{Y})$ and
$\Phi:{\mathscr{D}}(B)\rightarrow{\mathscr{D}}(\mathscr{E})$ between derived
categories of exact categories, where

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with $Q_{\mathcal{Y}}$ and $Q_{\mathscr{E}}$ the localization functors. By
Lemma 4.3(2), there is a commutative diagram of natural transformations among
triangle endofunctors of ${\mathscr{D}}(B)$:

$(\sharp)\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
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This yields a commutative diagram of triangles in ${\mathscr{D}}(B)$ if
applied to an object in ${\mathscr{D}}(B)$.

Now, let $D(H_{T}):{\mathscr{D}}(\mathcal{H})\to{\mathscr{D}}(B)$ be the
derived functor of $H_{T}$, and let
$\mathbb{L}F:{\mathscr{D}}(B)\to{\mathscr{D}}(\mathcal{H})$ be the total left-
derived functor of $F$. Since $(F,H_{T})$ is an adjoint pair by Lemma 4.4(3),
we see that $(\mathbb{L}F,D(H_{T}))$ is an adjoint pair.

######  Lemma 4.7.

$(1)$ Let
$\overline{H_{T}}:{\mathscr{D}}(\mathcal{H})\to{\mathscr{D}}(\mathcal{Y})$ be
the equivalence induced by $H_{T}$. Then

$D(H_{T})=D(j)\circ\overline{H_{T}},\;\;\Psi=\overline{H_{T}}\circ\mathbb{L}F\;\;\mbox{and}\;\;{\rm
Im}(D(j))={\rm Im}(H).$

Moreover, $\big{(}\Psi,D(j)\big{)}$ is an adjoint pair and $D(j)$ is a fully
faithful functor.

$(2)$ Let $\kappa:{\rm Ker}(G)\rightarrow{\mathscr{D}}(B)$ and $\nu:{\rm
Im}(H)\rightarrow{\mathscr{D}}(B)$ be the inclusion functors. Then
$D(i)=\kappa\circ\overline{D(i)}$ and $D(j)=\nu\circ\overline{D(j)}$, namely
the commutative diagrams exist:

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$(3)$ ${\rm Ker}(D(i))=0$, ${\rm Ker}(\Phi)={\rm Im}(H)$, and $\Phi$ commutes
with direct products. Moreover, $\Phi$ induces a triangle functor

$\overline{\Phi}:\;\;{\mathscr{D}}(B)/{\rm
Im}(H)\longrightarrow{\mathscr{D}}(\mathscr{E})$

which commutes with direct products.

$(4)$ The two compositions

${\rm
Ker}(G)\stackrel{{\scriptstyle\kappa}}{{\longrightarrow}}{\mathscr{D}}(B)\stackrel{{\scriptstyle
Q}}{{\longrightarrow}}{\mathscr{D}}(B)/{\rm
Im}(H)\quad\mbox{and}\quad{\mathscr{D}}(B)/{\rm
Im}(H)\stackrel{{\scriptstyle\overline{\Phi}[-1]}}{{\longrightarrow}}{\mathscr{D}}(\mathscr{E})\stackrel{{\scriptstyle\overline{D(i)}}}{{\longrightarrow}}{\rm
Ker}(G)$

are quasi-inverse triangle equivalences, where $Q$ denotes the localization
functor.

Proof. $(1)$ Clearly, $H_{T}$ induces an equivalence
$H^{\prime}_{T}:\mathcal{H}\to\mathcal{Y}$, its derived functor is denoted by
$\overline{H}_{T}:{\mathscr{D}}(\mathcal{H})\rightarrow{\mathscr{D}}(\mathcal{Y})$.
Hence $H_{T}=j\circ H^{\prime}_{T}$. Since all functors in this equality are
exact, we have $D(H_{T})=D(j)\circ\overline{H_{T}}$. Note that $\mathbb{L}F$
is the composition of the functors:

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By Corollary 4.5, the restriction of
$F:{\mathscr{K}}(B)\to{\mathscr{K}}(\mathcal{H})$ to
${\mathscr{K}}(B\mbox{{\rm-Proj}})$ coincides with the one of
${{}_{A}}T\otimes_{B}-$, and has its image in ${\mathscr{K}}({\rm
Add}({{}_{A}}T))$, while the restriction of
$H_{T}:{\mathscr{K}}(\mathcal{H})\to{\mathscr{K}}(\mathcal{Y})$ to
${\mathscr{K}}({\rm Add}({{}_{A}}T))$ coincides with the one of ${\rm
Hom}_{A}(T,-)$. It follows that $\Psi=\overline{H_{T}}\circ\mathbb{L}F$. Since
$\overline{H_{T}}$ is an equivalence, $(\Psi,D(j))$ is an adjoint pair and
${\rm Im}(D(j))={\rm Im}(D(H_{T}))$. By Lemma 4.3(3), ${\rm
Im}(H)\subseteq{\rm Im}(D(H_{T}))$.

To show ${\rm Im}(D(H_{T}))\subseteq{\rm Im}(H)$, we apply the technique of
homotopy limits in derived categories.

Let $M^{\bullet}\in{\mathscr{C}}(B)$. By Lemma 2.5,
$M^{\bullet}\simeq\underleftarrow{{\rm Holim}}(M^{(-n,n+1]})$ in
${\mathscr{D}}(B)$ where $n\in\mathbb{N}$. Suppose
$M^{\bullet}\in{\mathscr{C}}(\mathcal{Y})$. Then
$M^{(-n,n+1]}\in{\mathscr{C}}^{b}(\mathcal{Y})$ by Corollary 4.5(2). Further,
by Theorem 4.1, ${\rm Im}(H)$ is a triangulated subcategory of
${\mathscr{D}}(B)$ closed under direct products. It follows from
$\mathcal{Y}\subseteq{\rm Im}(H)$ that $M^{(-n,n+1]}\in{\rm Im}(H)$ and
further $M^{\bullet}\in{\rm Im}(H)$. This shows ${\rm
Im}(D(H_{T}))\subseteq{\rm Im}(H)$, Thus ${\rm Im}(H)={\rm Im}(D(H_{T}))={\rm
Im}(D(j))$.

Now, we show that the counit adjunction $\phi:\mathbb{L}F\circ
D(H_{T})\to\rm{Id}_{{\mathscr{D}}(\mathcal{H})}$ is an isomorphism. This
implies that both $D(H_{T})$ and $D(j)$ are fully faithful functors since
$\overline{H_{T}}$ is an equivalence.

In fact, the adjoint pair $(F,H_{T})$ of additive functors induces an adjoint
pair $(K(F),K(H_{T}))$ of functors between ${\mathscr{K}}(\mathcal{H})$ and
${\mathscr{K}}(B)$. Let $\phi^{\prime}:K(F)\circ K(H_{T})\to
Id_{{\mathscr{K}}(\mathcal{H})}$ be its counit adjunction, and let
$\psi^{\prime}:{\rm Id}_{{\mathscr{K}}(B)}\to K(H_{T})\circ K(F)$ be its unit
adjunction. Now, let $X\in{\mathscr{D}}(\mathcal{H})$,
$Q^{\bullet}:={{}_{p}}(K(H_{T})(X))$ and $\alpha_{Q^{\bullet}}:Q^{\bullet}\to
K(H_{T})(X)$ be the projective resolution of $K(H_{T})(X)$ in
${\mathscr{D}}(B)$. Then $\phi_{X}:\mathbb{L}F\circ D(H_{T})(X)\to X$ is given
by the composition of $K(F)(\alpha_{Q^{\bullet}})$ with
$\phi^{\prime}_{X}:K(F)\circ K(H_{T})(X)\to X$. Since $H_{T}$ is fully
faithful by Lemma 4.4(1),
$K(H_{T}):{\mathscr{K}}(\mathcal{H})\to{\mathscr{K}}(B)$ is fully faithful.
This implies that $\phi^{\prime}_{X}$ is an isomorphism by Lemma 2.6(1). Thus
$\phi_{X}$ is an isomorphism in ${\mathscr{D}}(\mathcal{H})$ if and only if
$K(F)(\alpha_{Q^{\bullet}})$ is a quasi-isomorphism. Since
$H_{T}:\mathcal{H}\to B\mbox{{\rm-Mod}}$ is exact and fully faithful by Lemma
4.4(1), for each $U\in\mathscr{K}(\mathcal{H})$, we see that
$U\in\mathscr{K}_{\rm{ac}}(\mathcal{H})$ if and only if
$K(H_{T})(U)\in\mathscr{K}_{\rm{ac}}(B)$. Consequently,
$K(F)(\alpha_{Q^{\bullet}})$ is a quasi-isomorphism if and only if so is
$K(H_{T})\circ K(F)(\alpha_{Q^{\bullet}})$. Since
$Q^{\bullet}\in{\mathscr{K}}(B)_{P}\subseteq{\mathscr{K}}(B\mbox{{\rm-
Proj}})$, we get $K(H_{T})K(F)(Q^{\bullet})={\rm
Hom}_{A}(T,T\otimes_{B}Q^{\bullet})$ and
$\psi^{\prime}_{Q^{\bullet}}=\eta^{\prime}_{Q^{\bullet}}$. Moreover, $H\circ
G(Q^{\bullet})\simeq{\rm Hom}_{A}(T,T\otimes_{B}Q^{\bullet})$ in
${\mathscr{D}}(B)$ by Lemma 4.3(2), and therefore there exists a diagram

$\textstyle{Q^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{Q^{\bullet}}}$$\textstyle{Q^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{Q^{\bullet}}}$$\scriptstyle{\psi_{Q^{\bullet}}^{\prime}}$$\textstyle{K(H_{T})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\psi_{H_{T}(X)}^{\prime}}$$\textstyle{H\circ
G(Q^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{K(H_{T})\circ
K(F)(Q^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;K(H_{T})\circ
K(F)(\alpha_{Q^{\bullet}})\;\;}$$\textstyle{K(H_{T})\circ K(F)\circ
K(H_{T})(X).}$

in which the left square is commutative in ${\mathscr{D}}(B)$ and the right
square is commutative in ${\mathscr{K}}(B)$. Note that the isomorphism
$\psi_{H_{T}(X)}^{\prime}$ follows from Lemma 2.6(2) and the fact that
$K(H_{T})(X)$ lies in ${\rm Im}(K(H_{T}))$. Since ${\rm Im}(H)={\rm
Im}(D(H_{T}))$ by $(1)$, we have $Q^{\bullet}\simeq K(H_{T})(X)\in{\rm
Im}(H)$. Recall that $H$ is fully faithful. Thus $\eta_{Q^{\bullet}}$ is an
isomorphism in ${\mathscr{D}}(B)$, and further, $K(H_{T})\circ
K(F)(\alpha_{Q^{\bullet}})$ is an isomorphism in ${\mathscr{D}}(B)$. As
$K(H_{T})\circ K(F)(\alpha_{Q^{\bullet}})$ is represented by a chain map, it
is a quasi-isomorphism. This shows that $\phi$ is a natural isomorphism.

$(2)$ This follows from $(1)$ and Lemma 4.3(3).

$(3)$ By Lemma 4.6(2), $X^{\bullet}\in{\mathscr{C}}(\mathscr{E})$ is strictly
exact if and only if it is exact in ${\mathscr{C}}(B)$. This implies ${\rm
Ker}(D(i))=0$, and therefore ${\rm Ker}(D(i)\circ\Phi)={\rm Ker}(\Phi)$. By
$(1)$ and $(\sharp)$, we have ${\rm Im}(H)={\rm Ker}(\Phi)$. Thus $\Phi$
induces a triangle functor $\overline{\Phi}:{\mathscr{D}}(B)/{\rm
Im}(H)\rightarrow{\mathscr{D}}(\mathscr{E})$. Since ${\rm
Im}(H)\subseteq{\mathscr{D}}(B)$ is closed under direct products,
${\mathscr{D}}(B)/{\rm Im}(H)$ has direct products and the localization
functor $Q$ in $(4)$ commutes with direct products by the dual statement of
[8, Lemma 1.5]. As ${\mathscr{D}}(B)$ and ${\mathscr{D}}(B)/{\rm Im}(H)$ have
the same objects, $\overline{\Phi}$ commutes with direct products if and only
if so does $\Phi$. Now let $\\{M_{i}\\}_{i\in I}$ be a family of objects in
${\mathscr{D}}(B)$ together with the projections $\pi_{i}:\prod_{i\in
I}M_{i}\to M_{i}$, where $I$ is a non-empty set. To prove that the canonical
morphism $\pi:\Phi(\prod_{i\in I}M_{i})\to\prod_{i\in I}\Phi(M_{i})$ in
${\mathscr{D}}(\mathscr{E})$ induced from $\\{\Phi(\pi_{i})\\}_{i\in I}$ is an
isomorphism, it suffices to show that $D(i)(\pi)$ is an isomorphism in
${\mathscr{D}}(B)$. The reason reads as follows: $\pi$ can be embedded into a
triangle $\Phi(\prod_{i\in I}M_{i})\to\prod_{i\in I}\Phi(M_{i})\rightarrow
W\rightarrow\Phi(\prod_{i\in I}M_{i})[1]$ in ${\mathscr{D}}(\mathscr{E})$, and
if $D(i)(\pi)$ is an isomorphism, then $D(i)(W)=0$. It follows from ${\rm
Ker}(D(i))=0$ that $W=0$ and $\pi$ is an isomorphism in
${\mathscr{D}}(\mathscr{E})$.

Since $D(i)$ commutes with direct products by Lemma 4.6(1), it is enough to
show that the canonical morphism $D(i)\circ\Phi(\prod_{i\in
I}M_{i})\to\prod_{i\in I}(D(i)\circ\Phi(M_{i}))$ in ${\mathscr{D}}(B)$ induced
from $\\{D(i)\circ\Phi(\pi_{i})\\}_{i\in I}$ is an isomorphism, that is,
$D(i)\circ\Phi$ commutes with direct products. But this follows from the
diagram $(\sharp)$ and the fact that $H$ and $G$ commute with direct products.
Thus $\Phi$ and therefore $\overline{\Phi}$ commute with direct products.

$(4)$ By Theorem 4.1, the pair $({\rm Ker}(G),{\rm Im}(H))$ is a semi-
orthogonal decomposition of ${\mathscr{D}}(B)$. By $(3)$, ${\rm
Ker}(\overline{D(i)}\circ\Phi[-1])={\rm Ker}(\Phi)={\rm Im}(H)$. It follows
from the diagram $(\sharp)$ and Lemma 2.2 that
$\overline{D(i)}\circ\Phi[-1]:{\mathscr{D}}(B)\to{\rm Ker}(G)$ is a right
adjoint of $\kappa$ and induces a triangle equivalence
$\overline{D(i)}\circ\overline{\Phi}[-1]:{\mathscr{D}}(B)/{\rm Im}(H)\to{\rm
Ker}(G)$. Thus $(4)$ holds. $\square$

As a consequence of Theorem 4.1 and Lemma 4.7(1), we re-obtain [3, Theorem
4.5] which was proved by employing the theory of derivators in [29].

######  Corollary 4.8.

There is a triangle equivalence
${\mathscr{D}}(\mathcal{H})\to{\mathscr{D}}(A)$ given by the composition of
the equivalence $D(H_{T}):{\mathscr{D}}(\mathcal{H})\to{\rm Im}(H)$ with a
quasi-inverse ${\rm Im}(H)\to{\mathscr{D}}(A)$ of the equivalence
$H:{\mathscr{D}}(A)\to{\rm Im}(H)$.

######  Remark 4.9.

The functors $H$ and $G$ can be restricted to functors between bounded-above
derived categories since both ${{}_{A}}T$ and $T_{B}$ have finite projective
dimension. This implies that $\big{(}{\mathscr{D}}^{-}(B)\cap{\rm
Ker}(G),{\mathscr{D}}^{-}(B)\cap{\rm Im}(H)\big{)}$ is a semi-orthogonal
decomposition of ${\mathscr{D}}^{-}(B)$. Moreover, by definition, $\Phi$ and
$\Psi$ can also be restricted to functors between bounded-above derived
categories:

$\Phi^{-}:\;\;{\mathscr{D}}^{-}(B)\to{\mathscr{D}}^{-}(\mathscr{E})\quad\mbox{and}\quad\Psi^{-}:\;\;{\mathscr{D}}^{-}(B)\to{\mathscr{D}}^{-}(\mathcal{Y}).$

Thus Lemma 4.7 hold true for bounded-above derived categories.

###  4.2 Fully faithful triangle functors between derived categories

In this section we develop properties of triangle functors needed in our
proofs.

By the diagram $(\sharp)$, there exists a triangle in ${\mathscr{D}}(B)$ for
each $M\in B\mbox{{\rm-Mod}}$:

$(\diamondsuit)\quad D(i)\circ\Phi[-1](M)\longrightarrow M\longrightarrow
D(j)\circ\Psi(M)\longrightarrow D(i)\circ\Phi(M),$

with $\Psi(M)\in{\mathscr{C}}^{-}(\mathcal{Y})$,
$\Phi(M)\in{\mathscr{C}}^{-}(\mathscr{E})$ and $\Psi(M)^{i}=0=\Phi(M)^{i}$ for
all $i\geq 1$. Taking homologies of the triangle, we get a $5$-term exact
sequence of $B$-modules:

$\varepsilon_{M}:\quad 0\longrightarrow
Y_{M}\stackrel{{\scriptstyle\epsilon^{-2}_{M}}}{{\longrightarrow}}X_{M}\stackrel{{\scriptstyle\epsilon^{-1}_{M}}}{{\longrightarrow}}M\stackrel{{\scriptstyle\epsilon^{0}_{M}}}{{\longrightarrow}}Y^{M}\longrightarrow
X^{M}\longrightarrow 0,$

where
$Y_{M}:=H^{-1}\big{(}D(j)\circ\Psi(M)\big{)},\;X_{M}:=H^{-1}\big{(}D(i)\circ\Phi(M)\big{)},\;Y^{M}:=H^{0}\big{(}D(j)\circ\Psi(M)\big{)}$
and $X^{M}:=H^{0}\big{(}D(i)\circ\Phi(M)\big{)}.$ In the following, we
consider the full subcategories of $B\mbox{{\rm-Mod}}$:

$\mathcal{E}:=\\{M\in B\mbox{{\rm-Mod}}\mid X^{M}=0\\},$ $\mathcal{X}:=\\{X\in
B\mbox{{\rm-Mod}}\mid{\rm Ext}_{B}^{n}(X,Y)=0,\,Y\in\mathcal{Y},\,n=0,1\\},$
${{}^{\bot}}\mathcal{Y}:=\\{X\in B\mbox{{\rm-Mod}}\mid{\rm
Ext}_{B}^{n}(X,Y)=0,\,Y\in\mathcal{Y},\,n\geq 0\\},$
$\mathscr{E}^{\bot}:=\\{Y\in B\mbox{{\rm-Mod}}\mid{\rm
Ext}_{B}^{n}(X,Y)=0,\,X\in\mathscr{E},\,n\geq 0\\},$ ${\rm
Coker}(\mathscr{E}):=\\{{\rm Coker}(f)\in B\mbox{{\rm-Mod}}\mid f:E_{1}\to
E_{0}\;\mbox{with}\;E_{1},E_{0}\in\mathscr{E}\\}.$

Then $\mathcal{X}$ is closed under extensions and cokernels in $B\mbox{{\rm-
Mod}}$, and both $\mathscr{E}^{\bot}$ and ${{}^{\bot}}\mathcal{Y}$ are thick
subcategories of $B\mbox{{\rm-Mod}}$. Moreover, we have the following result
which generalizes [3, Lemma 5.8].

######  Lemma 4.10.

$\mathscr{E}^{\bot}=\mathcal{Y}$ and $\mathscr{E}={{}^{\bot}}\mathcal{Y}$.

Proof. We have $\mathscr{E}\subseteq{\rm Ker}(G)$ and
$\mathcal{Y}\subseteq{\rm Im}(H)$. Since ${\rm
Hom}_{{\mathscr{D}}(B)}(X^{\bullet},Y^{\bullet})=0$ for any
$X^{\bullet}\in{\rm Ker}(G)$ and $Y^{\bullet}\in{\rm Im}(H)$ by Theorem 4.1,
we have $\mathcal{Y}\subseteq\mathscr{E}^{\bot}$ and
$\mathscr{E}\subseteq{{}^{\bot}}\mathcal{Y}$. To show
$\mathscr{E}^{\bot}\subseteq\mathcal{Y}$, we first show that if
$Z\in\mathscr{E}^{\bot}$, then ${\rm Hom}_{{\mathscr{D}}(B)}(E^{\bullet},Z)=0$
for all $E^{\bullet}\in{\mathscr{C}}^{-}(\mathscr{E})$.

Indeed, let $\mathscr{D}_{Z}$ be the full subcategory of ${\mathscr{D}}(B)$
consisting of all complexes $U^{\bullet}$ such that ${\rm
Hom}_{{\mathscr{D}}(B)}(U^{\bullet}[n],Z)$ = $0$ for all $n\in\mathbb{Z}$.
Then $\mathscr{D}_{Z}$ is a triangulated subcategory of ${\mathscr{D}}(B)$
which is closed under direct sums and contains $\mathscr{E}$. Thus homotopy
colimits in ${\mathscr{D}}(B)$ of sequences in $\mathscr{D}_{Z}$ belong to
$\mathscr{D}_{Z}$ by Definition 2.4. By the first isomorphism in Lemma 2.5,
each complex in ${\mathscr{C}}^{-}(\mathscr{E})$ can be obtained in
${\mathscr{D}}(B)$ from bounded complexes in ${\mathscr{C}}^{b}(\mathscr{E})$
by taking homotopy colimits. Hence
${\mathscr{C}}^{-}(\mathscr{E})\subseteq\mathscr{D}_{Z}$, and therefore ${\rm
Hom}_{{\mathscr{D}}(B)}(E^{\bullet},Z)=0$ for all
$E^{\bullet}\in{\mathscr{C}}^{-}(\mathscr{E})$.

Since $\Phi(Z)[-1]\in{\mathscr{C}}^{-}(\mathscr{E})$, we have ${\rm
Hom}_{{\mathscr{D}}(B)}(\Phi(Z)[-1],Z)=0$. It follows from the triangle
$(\diamondsuit)$ that $D(j)\circ\Psi(Z)\simeq Z\oplus D(i)\circ\Phi(Z)$. As
${\rm Hom}_{{\mathscr{D}}(B)}(D(i)\circ\Phi(Z),D(j)\circ\Psi(Z))=0$, we get
$D(i)\circ\Phi(Z)=0$ and $Z\simeq D(j)\circ\Psi(Z)\in{\rm Im}(H)$. Thus
$Z\in{\rm Im}(H)\cap B\mbox{{\rm-Mod}}=\mathcal{Y}$ by Lemma 4.4(2). This
shows $\mathscr{E}^{\bot}\subseteq\mathcal{Y}$, and therefore
$\mathscr{E}^{\bot}=\mathcal{Y}$.

Since $\mathcal{Y}$ is an abelian category of $B\mbox{{\rm-Mod}}$ by Corollary
4.5(2), it is closed under cokernels in $B\mbox{{\rm-Mod}}$. By the last
isomorphism in Lemma 2.5, each complex in ${\mathscr{C}}^{-}(\mathscr{Y})$ can
be obtained in ${\mathscr{D}}(B)$ from bounded complexes in
${\mathscr{C}}^{b}(\mathscr{Y})$ by taking homotopy limits. Similarly, we show
that if $Z\in{{}^{\bot}}\mathcal{Y}$, then ${\rm
Hom}_{{\mathscr{D}}(B)}(Z,Y^{\bullet})=0$ for any
$Y^{\bullet}\in{\mathscr{C}}^{-}(\mathcal{Y})$. Then the inclusion
${{}^{\bot}}\mathcal{Y}\subseteq\mathscr{E}$ follows from $(\diamondsuit)$ and
$\Psi(Z)\in{\mathscr{C}}^{-}(\mathcal{Y})$. Thus
$\mathscr{E}={{}^{\bot}}\mathcal{Y}$. $\square$

Next, we study properties of $\varepsilon_{M}$ and $\mathcal{E}$.

######  Lemma 4.11.

The following statements hold true for $M\in B\mbox{{\rm-Mod}}$.

$(1)$ $Y_{M},Y^{M}\in\mathcal{Y}$ and $X^{M}\in{\rm Coker}(\mathscr{E})$. If
$M\in\mathcal{E}$, then $X_{M}\in{\rm Coker}(\mathscr{E})$.

$(2)$ $\mathcal{X}={\rm Coker}(\mathscr{E})=\\{M\in\mathcal{E}\mid
Y_{M}=0=Y^{M}\\}$ and $\mathcal{Y}=\\{M\in\mathcal{E}\mid X_{M}=0\\}$.

$(3)$ $\mathcal{E}$ is closed under extensions and quotients in $B\mbox{{\rm-
Mod}}$.

Proof. $(1)$ Since $\mathcal{Y}$ is an abelian category of $B\mbox{{\rm-Mod}}$
by Corollary 4.5(2), it is clear that $H^{i}(\Psi(M))\in\mathcal{Y}$ for any
$i\in\mathbb{Z}$. Moreover, there is a right exact sequence
$\Phi(M)^{-1}\stackrel{{\scriptstyle d^{-1}}}{{\longrightarrow}}\Phi(M)^{0}\to
X^{M}\to 0$ with $\Phi(M)^{-1},\Phi(M)^{0}\in\mathscr{E}$. This shows
$X^{M}\in{\rm Coker}(\mathscr{E})$.

Let $M\in\mathcal{E}$. Then $X^{M}=0$, which means that the homomorphism
$d^{-1}$ is surjective. Since $\mathscr{E}$ is closed under kernels of
surjective homomorphisms in $B\mbox{{\rm-Mod}}$ by Lemma 4.6(1), we have ${\rm
Ker}(d^{-1})\in\mathscr{E}$. Similarly, there is a right exact sequence
$\Phi(M)^{-2}\to{\rm Ker}(d^{-1})\to X_{M}\to 0$ such that
$\Phi(M)^{-2}\in\mathscr{E}$. Thus $X_{M}\in{\rm Coker}(\mathscr{E})$.

$(2)$ Since $\mathcal{X}$ is closed under cokernels in $B\mbox{{\rm-Mod}}$ and
$\mathscr{E}\subseteq\mathcal{X}$, we have ${\rm
Coker}(\mathscr{E})\subseteq\mathcal{X}$.

Let $M\in\mathcal{X}$. Since $Y^{M}\in\mathcal{Y}$ by $(1)$, ${\rm
Hom}_{B}(M,Y^{M})=0$ and $Y^{M}\simeq X^{M}$. As $X^{M}$ is a quotient of
$\Phi(M)^{0}\in\mathscr{E}$, it follows from Lemma 4.10 that ${\rm
Hom}_{B}(X^{M},Y^{M})=0$. This forces $X^{M}=0=Y^{M}$, and thus
$M\in\mathcal{E}$. Since $Y_{M}\in\mathcal{Y}$, we have ${\rm
Ext}_{B}^{1}(M,Y_{M})=0$. Then $M\simeq Y_{M}\oplus X_{M}$. Note that ${\rm
Hom}_{B}(M,Y_{M})=0$ for $M\in\mathcal{X}$ and $Y_{M}\in\mathcal{Y}$.
Consequently, $Y_{M}=0$ and $M\simeq X_{M}\in{\rm Coker}(\mathscr{E})$. Thus
$\mathcal{X}\subseteq\\{M\in\mathcal{E}\mid Y_{M}=0=Y^{M}\\}$. Conversely, if
$N\in\mathcal{E}$ and $Y_{N}=0=Y^{N}$, then $N\simeq X_{N}\in{\rm
Coker}(\mathscr{E})$ by $(1)$.

If $N\in\mathcal{E}$ with $X_{N}=0$, then $N\simeq Y^{N}\in\mathcal{Y}$. Since
$\mathcal{Y}\subseteq{\rm Im}(H)$, it follows from Lemma 4.7(1) that
$N\in\mathcal{Y}$ if and only if $\Phi(N)=0$ in ${\mathscr{D}}(B)$. This
implies the last equality in $(2)$.

$(3)$ Let $0\to U\to V\to W\to 0$ be an exact sequence of $B$-modules. Then
this sequence induces a triangle

$D(i)\circ\Phi(U)\longrightarrow D(i)\circ\Phi(V)\longrightarrow
D(i)\circ\Phi(W)\longrightarrow D(i)\circ\Phi(U)[1]$

in ${\mathscr{D}}(B)$. Taking homology $H^{0}$ of this triangle yields an
exact sequence $X^{U}\to X^{V}\to X^{W}\to H^{1}(D(i)\circ\Phi(U))$ in
$B\mbox{{\rm-Mod}}$. By the definition of $\Phi$, $\Phi(U)$ is a complex with
$0$ at all positive degree, this implies $H^{1}(D(i)\circ\Phi(U))=0$. Hence
$\mathcal{E}$ is closed under extensions and quotients in $B\mbox{{\rm-Mod}}$.
$\square$

######  Corollary 4.12.

$(1)$ For each $M\in\mathcal{E}$, there exists a $4$-term exact sequence of
$B$-modules:

$0\longrightarrow
Y_{M}\stackrel{{\scriptstyle\varepsilon_{M}^{-2}}}{{\longrightarrow}}X_{M}\stackrel{{\scriptstyle\varepsilon_{M}^{-1}}}{{\longrightarrow}}M\stackrel{{\scriptstyle\varepsilon_{M}^{0}}}{{\longrightarrow}}Y^{M}\longrightarrow
0$

with $Y_{M},Y^{M}\in\mathcal{Y}$ and $X_{M}\in\mathcal{X}$.

$(2)$ $\varepsilon_{M}^{-1}$ and $\varepsilon_{M}^{0}$ in (1) give rise to
isomorphisms of abelian groups for $X\in\mathcal{X}$ and $Y\in\mathcal{Y}$:

$(\varepsilon_{M}^{-1})^{*}:{\rm
Hom}_{B}(X,X_{M})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm
Hom}_{B}(X,M)\quad\mbox{and}\quad(\varepsilon_{M}^{0})_{*}:{\rm
Hom}_{B}(Y^{M},Y)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm
Hom}_{B}(M,Y).$

$(3)$ The inclusion $\mathcal{X}\to\mathcal{E}$ has a right adjoint
$r:\mathcal{E}\to\mathcal{X}$ given by $M\mapsto X_{M}$ for any
$M\in\mathcal{E}$. Moreover, if $0\to M_{1}\to M_{2}\to M_{3}\to 0$ is an
exact sequence in $B\mbox{{\rm-Mod}}$ with $M_{i}\in\mathcal{E}$ for $1\leq
i\leq 3$, then $r(M_{1})\to r(M_{2})\to r(M_{3})\to 0$ is exact.

Proof. $(1)$ We have an exact sequence $\epsilon_{M}$. Since
$M\in\mathcal{E}$, $\epsilon_{M}$ can be shortened into $4$-term sequence with
the desired properties by Lemma 4.11(1)-(2).

(2) Since $X\in\mathcal{X}$ and $Y_{M},Y^{M}\in\mathcal{Y}$, we have ${\rm
Hom}_{B}(X,Y^{M})={\rm Hom}_{B}(X,Y_{M})={\rm Ext}_{B}^{1}(X,Y_{M})=0$. It
then follows from the exact sequence in (1) that $(\varepsilon_{M}^{-1})^{*}$
is an isomorphism. This implies the first isomorphism in $(2)$. Similarly,
$(\varepsilon_{M}^{0})_{*}$ is an isomorphism.

(3) If $M\in\mathcal{X}$, then $\varepsilon_{M}^{-1}$ is an isomorphism by the
proof of Lemma 4.11(2). Now, the isomorphism $(\varepsilon_{M}^{-1})^{*}$ in
$(2)$ gives an adjunction isomorphism of adjoint pairs of the functors.

Applying the functor $D(i)\circ\Phi:{\mathscr{D}}(B)\to{\mathscr{D}}(B)$ to
the exact sequence $0\to M_{1}\to M_{2}\to M_{3}\to 0$ in $B\mbox{{\rm-Mod}}$,
we obtain a triangle

$D(i)\circ\Phi(M_{1})\longrightarrow D(i)\circ\Phi(M_{2})\longrightarrow
D(i)\circ\Phi(M_{3})\longrightarrow D(i)\circ\Phi(M_{1})[1].$

If we take $H^{-1}$ on the triangle, then we gain an exact sequence
$r(M_{1})\to r(M_{2})\to r(M_{3})\to X^{M_{1}}$. Now, it follows from
$M_{1}\in\mathcal{E}$ that $X^{M_{1}}=0$, and therefore the second part of
$(3)$ holds. $\square$

The following property of $\mathcal{E}$ is crucial in establishing an
equivalence of derived categories of exact categories in Lemma 4.16 below.

######  Lemma 4.13.

If $M$ is an injective $B$-module, then $M\in\mathcal{E}$.

To show Lemma 4.13, we need to determine the image of an injective cogenerator
for $B\mbox{{\rm-Mod}}$ under $H\circ G$. Let

$(-)^{\vee}:={\rm
Hom}_{\mathbb{Z}}(-,\mathbb{Q}/\mathbb{Z}):\;\;\mathbb{Z}\mbox{{\rm-
Mod}}\longrightarrow\mathbb{Z}\mbox{{\rm-Mod}}.$

Then $(-)^{\vee}$ induces an exact functor $B^{\rm op}\mbox{{\rm-Mod}}\to
B\mbox{{\rm-Mod}}$ and $B^{\vee}$ is an injective cogenerator for
$B\mbox{{\rm-Mod}}$. Further, we define two natural transformations:

$\theta:T\otimes_{B}{\rm Hom}_{A}(-,T)^{\vee}\to{\rm
Hom}_{A}(-,A)^{\vee}:\;\;A\mbox{{\rm-Mod}}\longrightarrow A\mbox{{\rm-Mod}},$
$\rho:{\rm Hom}_{A}(-,A)\otimes_{A}T\longrightarrow{\rm
Hom}_{A}(-,T):\;\;A\mbox{{\rm-Mod}}\to B^{\rm op}\mbox{{\rm-Mod}}$

given by

$\theta_{X}:\;{{}_{A}}T\otimes_{B}{\rm Hom}_{A}(X,T)^{\vee}\longrightarrow{\rm
Hom}_{A}(X,A)^{\vee}:\;\;t\otimes\sigma\mapsto[f\mapsto(f(\cdot t))\sigma],$
$\rho_{X}:{\rm Hom}_{A}(X,A)\otimes_{A}T\longrightarrow{\rm
Hom}_{A}(X,T):\;\;f\otimes t\mapsto f(\cdot t),$

where $X\in A\mbox{{\rm-Mod}},t\in T,\sigma\in{\rm
Hom}_{A}(X,T)^{\vee},f\in{\rm Hom}_{A}(X,A)$ and $(\cdot t)\in{\rm
Hom}_{A}(A,T)$ is the right multiplication by $t$.

######  Lemma 4.14.

$(1)$ $\theta$ is a natural isomorphism.

$(2)$ If $X\in A\mbox{{\rm-Proj}}$, then

$\quad(i)$ ${\rm Tor}_{j}^{B}(T,{\rm Hom}_{A}(X,T)^{\vee})=0$ for all $j\geq
1$.

$\quad(ii)$ $\rho_{X}$ is injective and ${\rm Tor}_{j}^{A}({\rm
Hom}_{A}(X,A),T)=0$ for all $j\geq 1$.

Proof. $(1)$ For $U\in A\mbox{{\rm-Mod}}$ and $g\in{\rm Hom}_{A}(U,T)$, we
define

$\theta_{U,\,X}:\;{\rm Hom}_{A}(U,T)\otimes_{B}{\rm
Hom}_{A}(X,T)^{\vee}\longrightarrow{\rm
Hom}_{A}(X,U)^{\vee}:\;\;g\otimes\sigma\mapsto[f\mapsto(fg)\sigma]$

Clearly, $\theta_{A,\,X}=\theta_{X}$ under the identification of ${\rm
Hom}_{A}(A,T)$ with $T$. Moreover, if $U\in{\rm add}(T)$, then
$\theta_{U,\,X}$ is an isomorphism. Note that $T_{B}$ has a finitely generated
projective resolution of length at most $n$:

$(\ast)\quad 0\longrightarrow{\rm
Hom}_{A}(T_{n},T)\longrightarrow\cdots\longrightarrow{\rm
Hom}_{A}(T_{1},T)\longrightarrow{\rm Hom}_{A}(T_{0},T)\longrightarrow
T_{B}\longrightarrow 0$

with $T_{j}\in{\rm add}(_{A}T)$ for all $0\leq j\leq n$. Now, we can construct
the following exact commutative diagram:

$\textstyle{{\rm Hom}_{A}(T_{1},T)\otimes_{B}{\rm
Hom}_{A}(X,T)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{T_{1},\,X}}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom}_{A}(T_{0},T)\otimes_{B}{\rm
Hom}_{A}(X,T)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{T_{0},\,X}}$$\scriptstyle{\simeq}$$\textstyle{T\otimes_{B}{\rm
Hom}_{A}(X,T)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{X}}$$\textstyle{0}$$\textstyle{{\rm
Hom}_{A}(X,T_{1})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom}_{A}(X,T_{0})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom}_{A}(X,A)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Thus $\theta_{X}$ is an isomorphism. This shows $(1)$.

(2) If $X$ is projective, then the sequence

$0\longrightarrow{\rm
Hom}_{A}(X,T_{n})^{\vee}\longrightarrow\cdots\longrightarrow{\rm
Hom}_{A}(X,T_{1})^{\vee}\longrightarrow{\rm
Hom}_{A}(X,T_{0})^{\vee}\longrightarrow{\rm
Hom}_{A}(X,A)^{\vee}\longrightarrow 0$

is exact. Now $(i)$ follows if we apply $-\otimes_{B}{\rm
Hom}_{A}(X,T)^{\vee}$ to $(\ast)$.

Note that $(ii)$ holds if and only if the canonical map
$A^{\alpha}\otimes_{A}T\to T^{\alpha}$ is injective and ${\rm
Tor}^{A}_{j}(A^{\alpha},T)=0$ for any nonempty set $\alpha$ and for any $j\geq
1$. The latter follows from [11, Lemmas 2.5 and 2.4(3)]. $\square$

######  Lemma 4.15.

Let $P^{\bullet}:0\to P_{n}\to\cdots\to P_{1}\to P_{0}\to 0$ be the deleted
projective resolution of ${{}_{A}}T$ (see Definition 1.1), where $P_{i}$ are
of degree $-i$ for $0\leq i\leq n$. Then

$(1)$ $G(B^{\vee})\simeq{\rm Hom}_{A}(P^{\bullet},A)^{\vee}$ in
${\mathscr{D}}(A)$.

$(2)$ Let $\rho^{\bullet}:=(\rho_{P_{i}})_{0\leq i\leq n}:\;\;{\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T\longrightarrow{\rm
Hom}_{A}(P^{\bullet},T).$ Then $\rho^{\bullet}$ is an injective chain map and
there is a commutative diagram of triangles in ${\mathscr{D}}(B)$:

$\textstyle{{\rm
Coker}(\rho^{\bullet})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Hom}_{A}(P^{\bullet},T)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\bullet}{{}^{\vee}}}$$\scriptstyle{\simeq}$$\textstyle{({\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Coker}(\rho^{\bullet})^{\vee}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Coker}(\rho^{\bullet})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{B^{\vee}}}$$\textstyle{H\circ
G(B^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\;\;\;{\rm
Coker}(\rho^{\bullet})^{\vee}[1].}$

Proof. $(1)$ By $(T1)$ and $(T2)$ in Definition 1.1, there is an exact
sequence of $B$-modules

$0\longrightarrow{\rm
Hom}_{A}(P_{n},T)^{\vee}\longrightarrow\cdots\longrightarrow{\rm
Hom}_{A}(P_{1},T)^{\vee}\longrightarrow{\rm
Hom}_{A}(P_{0},T)^{\vee}\longrightarrow B^{\vee}\longrightarrow 0,$

where $P_{i}\in A\mbox{{\rm-Proj}}$ for $0\leq i\leq n$. This means that ${\rm
Hom}_{A}(P^{\bullet},T)^{\vee}$ is quasi-isomorphic to $B^{\vee}$. Therefore
$G(B^{\vee})\simeq G({\rm Hom}_{A}(P^{\bullet},T)^{\vee})$ in
${\mathscr{D}}(A)$. Moreover, by Lemma 4.14(2), ${\rm Tor}_{j}^{B}(T,{\rm
Hom}_{A}(P_{i},T)^{\vee})=0$ for any $j\geq 1$. This yields $G({\rm
Hom}_{A}(P^{\bullet},T)^{\vee})\simeq T\otimes_{B}{\rm
Hom}_{A}(P^{\bullet},T)^{\vee}$ by Lemma 4.2(1). Since $T\otimes_{B}{\rm
Hom}_{A}(P^{\bullet},T)^{\vee}\simeq{\rm Hom}_{A}(P^{\bullet},A)^{\vee}$ by
Lemma 4.14(1), we have $G(B^{\vee})\simeq{\rm Hom}_{A}(P^{\bullet},A)^{\vee}$
in ${\mathscr{D}}(A)$.

$(2)$ For $X\in A\mbox{{\rm-Proj}}$ and $j\geq 1$, we have ${\rm
Ext}_{A}^{j}(T,{\rm Hom}_{A}(X,A)^{\vee})\simeq{\rm Tor}_{j}^{A}({\rm
Hom}_{A}(X,A),T)^{\vee}=0$ by Lemma 4.14(3). Since $P_{i}\in A\mbox{{\rm-
Proj}}$ for all $0\leq i\leq n$, Lemma 4.2(2) implies that

${\rm Hom}_{A}(T,\beta_{{\rm Hom}_{A}(P^{\bullet},A)^{\vee}}):\;\;{\rm
Hom}_{A}(T,{\rm Hom}_{A}(P^{\bullet},A)^{\vee})\longrightarrow H({\rm
Hom}_{A}(P^{\bullet},A)^{\vee})$

is an isomorphism in ${\mathscr{D}}(B)$. Consequently, there are isomorphisms
in ${\mathscr{D}}(B)$:

$H\circ G(B^{\vee})\simeq H({\rm Hom}_{A}(P^{\bullet},A)^{\vee})\simeq{\rm
Hom}_{A}(T,{\rm Hom}_{A}(P^{\bullet},A)^{\vee})\simeq({\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T)^{\vee}.$

Clearly, $\rho^{\bullet}$ is injective by Lemma 4.14(3). Thus the sequence

$0\longrightarrow{\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T\stackrel{{\scriptstyle\rho^{\bullet}}}{{\longrightarrow}}{\rm
Hom}_{A}(P^{\bullet},T)\longrightarrow{\rm
Coker}(\rho^{\bullet})\longrightarrow 0$

is exact, and yields an triangle in ${\mathscr{D}}(B^{op})$

${\rm Coker}(\rho^{\bullet})[-1]\longrightarrow{\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T\stackrel{{\scriptstyle\rho^{\bullet}}}{{\longrightarrow}}{\rm
Hom}_{A}(P^{\bullet},T)\longrightarrow{\rm Coker}(\rho^{\bullet})$

and a triangle in ${\mathscr{D}}(B)$

${\rm Coker}(\rho^{\bullet})^{\vee}\longrightarrow{\rm
Hom}_{A}(P^{\bullet},T)^{\vee}\stackrel{{\scriptstyle\rho^{\bullet}{{}^{\vee}}}}{{\longrightarrow}}({\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T)^{\vee}\longrightarrow{\rm
Coker}(\rho^{\bullet})^{\vee}[1].$

Now, it follows from this triangle that (2) can be deduced by the commutative
diagram in ${\mathscr{D}}(B)$:

$\textstyle{{\rm
Hom}_{A}(P^{\bullet},T)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}_{{\rm
Hom}_{A}(P^{\bullet},T)^{\vee}}}$$\scriptstyle{\simeq}$$\textstyle{{\rm
Hom}_{A}\big{(}T,T\otimes_{B}{\rm
Hom}_{A}(P^{\bullet},T)^{\vee}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{({\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{B^{\vee}}}$$\textstyle{H\circ
G(B^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\;\;\;H\circ
G(B^{\vee})}$

where the composition of the first arrow is $\rho^{\bullet}{{}^{\vee}}$.
$\square$

Proof of Lemma 4.13: Let $M$ be an injective $B$-module. Then $M\in{\rm
Prod}(B^{\vee})$. Since $X^{M}=H^{0}(D(i)\circ\Phi(M))$ and since the functors
$D(i)$, $\Phi$ and $H^{0}$ commute with direct products, we have $X^{M}\in{\rm
Prod}(X^{B^{\vee}})$. By the diagram $(\sharp)$ and Lemma 4.15(2),
$D(i)\circ\Phi(B^{\vee})\simeq{\rm Coker}(\rho^{\bullet})^{\vee}[1]$ in
${\mathscr{D}}(B)$. Since $H^{0}\big{(}{\rm
Coker}(\rho^{\bullet})^{\vee}[1]\big{)}=\big{(}H^{-1}({\rm
Coker}(\rho^{\bullet}))\big{)}^{\vee}=0$, we have $X^{B^{\vee}}=0$. Thus
$X^{M}=0$. $\square$

######  Lemma 4.16.

$(1)$ The inclusion $\mathcal{E}\subseteq B\mbox{{\rm-Mod}}$ induces a
triangle equivalence ${\mathscr{D}}(\mathcal{E})\rightarrow{\mathscr{D}}(B)$
which restricts to an equivalence
$\mathscr{D}^{*}(\mathcal{E})\rightarrow\mathscr{D}^{*}(B)$ for any
$*\in\\{+,-,b\\}$.

$(2)$ The inclusion $\mathscr{E}\subseteq\mathcal{E}$ induces a fully faithful
triangle functor
${\mathscr{D}}^{-}(\mathscr{E})\to{\mathscr{D}}^{-}(\mathcal{E})$.

Proof. $(1)$ By Lemma 4.11(3), $\mathcal{E}$ is closed under extensions and
quotients in $B\mbox{{\rm-Mod}}$. In particular, $\mathcal{E}$ is a fully
exact subcategory of $B\mbox{{\rm-Mod}}$, which is closed under direct
summands. By Lemma 4.15, $\mathcal{E}$ contains all injective $B$-modules.
Thus $\mathcal{E}$ satisfies the two properties:

$(i)$ If $0\to X\to Y\to Z\to 0$ is exact in $B\mbox{{\rm-Mod}}$ with
$Y\in\mathcal{E}$, then $Z\in\mathcal{E}$.

$(ii)$ For each $B$-module $M$, there exists a short exact sequence $0\to M\to
E_{0}\to E_{1}\to 0$ in $B\mbox{{\rm-Mod}}$ such that $E_{0}$ is injective and
$E_{1}\in\mathcal{E}$.

Now, $(1)$ follows from the dual statement of Lemma 2.7(2) with
$\mathcal{A}=B\mbox{{\rm-Mod}}$.

$(2)$ By Lemma 4.6(1), if $0\to X\to Y\to Z\to 0$ is exact in $B\mbox{{\rm-
Mod}}$ with $Z\in\mathscr{E}$, then $X\in\mathscr{E}$ if and only if
$Y\in\mathscr{E}$. This implies that $\mathscr{E}$ is a fully exact
subcategory of $B\mbox{{\rm-Mod}}$. Clearly,
$\mathscr{E}\subseteq\mathcal{X}\subseteq\mathcal{E}$ by Lemma 4.10 and Lemma
4.11(2).

Let $0\to E_{1}\to E_{0}\stackrel{{\scriptstyle g}}{{\longrightarrow}}F\to 0$
be an exact sequence in $B\mbox{{\rm-Mod}}$ with $E_{1},E_{0}\in\mathcal{E}$
and $F\in\mathscr{E}$. By Corollary 4.12(3), we can apply the functor
$r:\mathcal{E}\to\mathcal{X}$ to $g$ and obtain a surjective map
$r(g):r(E_{0})\to r(F)$. Moreover, the composition of $r(g)$ with the counit
map $\varepsilon_{F}^{-1}:r(F)\to F$ coincides with the composition of the
counit map $\varepsilon_{E_{0}}^{-1}:r(E_{0})\to E_{0}$ with $g$. Note that
$\varepsilon_{F}^{-1}$ is an isomorphism for
$F\in\mathscr{E}\subseteq\mathcal{X}$. Since $r(E_{0})\in\mathcal{X}={\rm
Coker}(\mathscr{E})$ by Lemma 4.11(2), there exists a surjective map
$f:F_{0}\to r(E_{0})$ with $F_{0}\in\mathscr{E}$. Let
$h:=f\varepsilon_{E_{0}}^{-1}$. Then
$hg=f\varepsilon_{E_{0}}^{-1}g=fr(g)\varepsilon_{F}^{-1}:F_{0}\to F$, and
therefore $hg$ is surjective. Let $F_{1}:={\rm Ker}(hg)$. Since
$F_{0},F\in\mathscr{E}$, we have $F_{1}\in\mathscr{E}$. Moreover, there is an
exact commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{hg}$$\scriptstyle{h}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Now, $(2)$ follows from Lemma 2.7(1). $\square$

Finally, we prove the following result which is crucial to the proof of
Theorem 1.2.

######  Proposition 4.17.

The functor $\overline{D(i)}:{\mathscr{D}}(\mathscr{E})\to{\rm Ker}(G)$ is a
triangle equivalence.

Proof. Since $D^{-}(i):{\mathscr{D}}^{-}(\mathscr{E})\to{\mathscr{D}}^{-}(B)$
is the composition of the induced functor
${\mathscr{D}}^{-}(\mathscr{E})\to{\mathscr{D}}^{-}(\mathcal{E})$ with the one
${\mathscr{D}}^{-}(\mathcal{E})\to{\mathscr{D}}^{-}(B)$, Lemma 4.16 implies
that $D^{-}(i)$ is fully faithful. By Lemmas 4.3(3) and 4.6(2), ${\rm
Im}(D^{-}(i))={\mathscr{D}}^{-}(B)\cap{\rm Ker}(G)$. Consequently, the
restriction

$\overline{D^{-}(i)}:{\mathscr{D}}^{-}(\mathscr{E})\longrightarrow{\mathscr{D}}^{-}(B)\cap{\rm
Ker}(G)$

of the functor $\overline{D(i)}$ is a triangle equivalence. Moreover, by Lemma
4.7(4) and Remark 4.9, the two compositions:

$(\ast\ast)\quad{\mathscr{D}}^{-}(B)\cap{\rm
Ker}(G)\stackrel{{\scriptstyle\kappa^{-}}}{{\longrightarrow}}{\mathscr{D}}^{-}(B)\stackrel{{\scriptstyle
Q^{-}}}{{\longrightarrow}}{\mathscr{D}}^{-}(B)/({\mathscr{D}}^{-}(B)\cap{\rm
Im}(H))\quad\mbox{and}$ ${\mathscr{D}}^{-}(B)/({\mathscr{D}}^{-}(B)\cap{\rm
Im}(H))\stackrel{{\scriptstyle\overline{\Phi^{-}}[-1]}}{{\longrightarrow}}{\mathscr{D}}^{-}(\mathscr{E})\stackrel{{\scriptstyle\overline{D^{-}(i)}}}{{\longrightarrow}}{\mathscr{D}}^{-}(B)\cap{\rm
Ker}(G)$

are quasi-inverse triangle equivalences, where - denotes universally the
restriction of the involved functors to bounded-above derived categories. This
implies that $\overline{\Phi^{-}}[-1]$ is a triangle equivalence.

Next, we show that the inclusion ${\mathscr{D}}^{-}(B)\to{\mathscr{D}}(B)$
induces a fully faithful functor

${\mathscr{D}}^{-}(B)/({\mathscr{D}}^{-}(B)\cap{\rm
Im}(H))\longrightarrow{\mathscr{D}}(B)/{\rm Im}(H).$

By the dual statement of [21, Lemma 10.3], it suffices to show that each
morphism $f^{\bullet}:M^{\bullet}\to Y^{\bullet}$ in ${\mathscr{D}}(B)$ with
$M^{\bullet}\in{\mathscr{D}}^{-}(B)$ and $Y^{\bullet}\in{\rm Im}(H)$
factorizes through an object $Z^{\bullet}\in{\mathscr{D}}^{-}(B)\cap{\rm
Im}(H)$. Since ${\mathscr{K}}^{-}(B\mbox{{\rm-Proj}})$ is equivalent to
${\mathscr{D}}^{-}(B)$, we can assume that
$M^{\bullet}\in{\mathscr{C}}^{-}(B\mbox{{\rm-Proj}})$ and $f^{\bullet}$ is
represented by a chain map. As ${\rm Im}(H)={\rm Im}(D(j))$ by Lemma 4.7(1),
we have $Y^{\bullet}\in{\mathscr{C}}(\mathcal{Y})$ up to isomorphism in
${\mathscr{D}}(B)$. Then $f^{\bullet}$ factorizes through the left truncated
complex $\tau^{\leq m}Y^{\bullet}:\;\cdots\to Y^{i}\to Y^{i+1}\to\cdots\to
Y^{m-1}\to{\rm Ker}(d_{Y}^{m})\to 0$ of $Y^{\bullet}$ at some degree $m$.
Since $\mathcal{Y}$ is an abelian subcategory of $B\mbox{{\rm-Mod}}$, ${\rm
Ker}(d_{Y}^{m})\in\mathcal{Y}$ and $\tau^{\leq
m}Y^{\bullet}\in{\mathscr{C}}^{-}(\mathcal{Y})$, and therefore $\tau^{\leq
m}Y^{\bullet}\in{\mathscr{D}}^{-}(B)\cap{\rm Im}(H)$.

Now, we construct the commutative diagram of functors:

$\textstyle{{\mathscr{D}}^{-}(\mathscr{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{D^{-}(i)}}$$\scriptstyle{\simeq}$$\textstyle{{\mathscr{D}}^{-}(B)\cap{\rm
Ker}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa^{-}}$$\textstyle{{\mathscr{D}}^{-}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q^{-}}$$\textstyle{{\mathscr{D}}^{-}(B)/({\mathscr{D}}^{-}(B)\cap{\rm
Im}(H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\Phi^{-}}[-1]}$$\scriptstyle{\simeq}$$\textstyle{{\mathscr{D}}^{-}(\mathscr{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{D}}(\mathscr{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{D(i)}}$$\textstyle{{\rm
Ker}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{{\mathscr{D}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\textstyle{{\mathscr{D}}(B)/{\rm
Im}(H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\Phi}[-1]}$$\textstyle{{\mathscr{D}}(\mathscr{E}).}$

Let $\Theta:{\mathscr{D}}(\mathscr{E})\to{\mathscr{D}}(\mathscr{E})$ be the
composition of all functors in the bottom row of the diagram. Then the
restriction $\Theta^{-}$ of $\Theta$ to ${\mathscr{D}}^{-}(\mathscr{E})$ is
the composition of the functors in the top row of the diagram. Since
$\overline{D^{-}(i)}$ is an equivalence, $\Theta^{-}$ is naturally isomorphic
to the identity functor of ${\mathscr{D}}^{-}(\mathscr{E})$ by $(\ast\ast)$.

Further, $\Theta$ commutes with direct products. This is obtained by showing
that each component of $\Theta$ commutes with direct products. In fact, since
$\mathscr{E}$ is closed under products in $B\mbox{{\rm-Mod}}$ by Lemma 4.6(1),
$D(i)$ commutes with direct products. Clearly, the inclusion $\kappa$
preserves direct products. As ${\rm Im}(H)$ is a triangulated subcategory of
${\mathscr{D}}(B)$ closed under products by Theorem 4.1, it follows from the
dual result of [8, Lemma 1.5] that $Q$ commutes with direct products. Finally,
$\overline{\Phi}[-1]$ commutes with direct products by Lemma 4.7(3).

Next, we show that $\Theta$ is dense, and therefore $\overline{\Phi}[-1]$ is
dense.

Let $X^{\bullet}\in{\mathscr{C}}(\mathscr{E})$. Then $X^{\leq
n}\in{\mathscr{C}}^{-}(\mathscr{E})$ for $n\geq 0$ and
$X^{\bullet}\simeq\underleftarrow{{\rm Holim}}(X^{\leq n})$ in
${\mathscr{K}}(B)$ by Lemma 2.5. Recall that ${\mathscr{K}}(\mathscr{E})$ is a
triangulated subcategory of ${\mathscr{K}}(B)$ closed under direct products by
Lemma 4.6(1). This implies $X^{\bullet}\simeq\underleftarrow{{\rm
Holim}}(X^{\leq n})$ in ${\mathscr{K}}(\mathscr{E})$ and thus also in
${\mathscr{D}}(\mathscr{E})$ because the localization functor
${\mathscr{K}}(\mathscr{E})\to{\mathscr{D}}(\mathscr{E})$ preserves direct
products. As $\Theta$ commutes with products, it commutes with homotopy limits
in ${\mathscr{D}}(\mathscr{E})$. Consequently,
$\Theta(X^{\bullet})=\Theta\big{(}\underleftarrow{{\rm Holim}}(X^{\leq
n})\big{)}\simeq\underleftarrow{{\rm Holim}}\big{(}\Theta(X^{\leq n})\big{)}$.
Since the restriction of $\Theta$ to ${\mathscr{D}}^{-}(\mathscr{E})$ is
isomorphic to the identity functor of ${\mathscr{D}}^{-}(\mathscr{E})$, it
follows from $X^{\leq n}\in{\mathscr{D}}^{-}(\mathscr{E})$ that
$\underleftarrow{{\rm Holim}}\big{(}\Theta(X^{\leq
n})\big{)}\simeq\underleftarrow{{\rm Holim}}(X^{\leq n})\simeq X^{\bullet}$ in
${\mathscr{D}}(\mathscr{E})$. Thus $\Theta(X^{\bullet})\simeq X^{\bullet}$ in
${\mathscr{D}}(\mathscr{E})$. This shows that $\Theta$ is dense.

Finally, we show that $\overline{D(i)}$ is a triangle equivalence.

By Lemma 4.3(3), the functor $\overline{D(i)}$ is dense. By Lemma 4.7(4), the
composition of $\overline{\Phi}[-1]$ with $\overline{D(i)}$ is a triangle
equivalence. Since $\overline{\Phi}[-1]$ is dense, $\overline{D(i)}$ is full.
Moreover, by Lemma 4.7(2), ${\rm Ker}(\overline{D(i)})=0$. In other words,
$\overline{D(i)}$ sends nonzero objects of ${\mathscr{D}}(\mathscr{E})$ to
nonzero objects of ${\rm Ker}(G)$. It is known in [27, p.446] that a triangle
functor between triangulated categories is faithful if it is full and sends
nonzero objects to nonzero objects. This implies that $\overline{D(i)}$ is
faithful, and thus an equivalence. $\square$

###  4.3 Proofs of main result and its corollaries

We keep all notation introduced in the previous sections.

Proof of Theorem 1.2. Since $\overline{D(i)}$ is a triangle equivalence by
Proposition 4.17, we can replace the left term ${\rm Ker}(G)$ of the
recollement in Theorem 4.1 with ${\mathscr{D}}(\mathscr{E})$ and obtain the
recollement of the form

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.11113pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathscr{D}}(\mathscr{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
21.2598pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{D(i)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 48.25435pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
48.25435pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathscr{D}}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
88.93091pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{G}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 111.90108pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
21.11417pt\raise-20.89996pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\Phi[-1]}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
6.73749pt\raise-5.50076pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
14.10565pt\raise 10.00838pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
111.90108pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathscr{D}}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
88.48903pt\raise-19.79163pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
69.78076pt\raise-5.50076pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
89.48555pt\raise 20.40274pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.61389pt\hbox{$\scriptstyle{j_{!}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
77.75409pt\raise 10.28725pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

In this recollement, the functors $D(i)$, $\Phi[-1]$, $H$ and $G$ can be
restricted to functors between $\mathscr{D}^{-}$-derived categories by Remark
4.9. So, to show that this recollement restricts to $\mathscr{D}^{-}$-derived
categories, it suffices to show that the left adjoint
$j_{!}:{\mathscr{D}}(A)\to{\mathscr{D}}(B)$ of $G$ restricts to a functor
${\mathscr{D}}^{-}(A)\to{\mathscr{D}}^{-}(B)$.

For any $m\in\mathbb{Z}$ and $X\in{\mathscr{D}}(A)$, there are isomorphisms of
abelian groups:

$({\dagger})\quad{\rm Hom}_{{\mathscr{D}}(A)}(X,G(B^{\vee})[m])\simeq{\rm
Hom}_{{\mathscr{D}}(B)}(j_{!}(X),B^{\vee}[m])\simeq{\rm
Hom}_{{\mathscr{K}}(B)}(j_{!}(X),B^{\vee}[m])$ $\;\;\quad\simeq
H^{m}(j_{!}(X)^{\vee})\simeq(H^{-m}(j_{!}(X)))^{\vee}.$

Let $X\in{\mathscr{D}}^{-}(A)$. Since $G(B^{\vee})\in{\mathscr{D}}^{b}(A)$ by
Lemma 4.15, there is an integer $m_{X}$ such that ${\rm
Hom}_{{\mathscr{D}}(A)}(X,$ $G(B^{\vee})[m])=0$ for all $m\leq m_{X}$. Observe
that the functor $(-)^{\vee}$ sends nonzero modules to nonzero modules. Thus
$H^{-m}(j_{!}(X))=0$ for all $m\leq m_{X}$. This implies
$j_{!}(X)\in{\mathscr{D}}^{-}(B)$. $\square$

Recall that a ring homomorphism $\lambda:B\to C$ is called a _ring
epimorphism_ if the restriction functor $\lambda_{*}:C\mbox{{\rm-Mod}}\to
B\mbox{{\rm-Mod}}$ is fully faithful. A ring epimorphism $\lambda:B\to C$ is
said to be _homological_ if ${\rm Tor}^{B}_{m}(C,C)=0$ for all $m\geq 1$. This
is also equivalent to saying that the induced derived functor
$D(\lambda_{*}):{\mathscr{D}}(C)\to{\mathscr{D}}(B)$ is fully faithful.
Following [11], a good tilting module ${{}_{A}}T$ is said to be _homological_
if there exists a homological ring epimorphism $\lambda:B\to C$ such that
$D(\lambda_{*})$ induces an equivalence
${\mathscr{D}}(C)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm
Ker}(G)$ of triangulated categories.

To prove Corollary 1.3, we recall the definition of derived decompositions of
abelian categories in [13].

######  Definition 4.18.

[13, Definition 1.1] Let $\mathcal{A}$ be an abelian category, and let
$\mathcal{B}$ and $\mathcal{C}$ be full subcategories of $\mathcal{A}$. The
pair $(\mathcal{B},\mathcal{C})$ is called a _derived decomposition_ of
$\mathcal{A}$ if both $\mathcal{B}$ and $\mathcal{C}$ are abelian
subcategories of $\mathcal{A}$ such that

$(1)$ the inclusions $\mathcal{B}\subseteq\mathcal{A}$ and
$\mathcal{C}\subseteq\mathcal{A}$ induce fully faithful functors
${\mathscr{D}}^{b}(\mathcal{B})\hookrightarrow{\mathscr{D}}^{b}(\mathcal{A})$
and
${\mathscr{D}}^{b}(\mathcal{C})\hookrightarrow{\mathscr{D}}^{b}(\mathcal{A})$,
respectively.

$(2)$
$\big{(}{\mathscr{D}}^{b}(\mathcal{B}),{\mathscr{D}}^{b}(\mathcal{C})\big{)}$
is a semi-orthogonal decomposition of ${\mathscr{D}}^{b}(\mathcal{A})$.

Proof of Corollary 1.3. The equivalence of (1) and (2) is shown in [5,
Proposition 6.2], while the one of (1) and (3) is proved in [11, Theorem 1.1].
Here, we prefer to giving a new and shorter proofs of these facts by the
results in the previuos sections.

$(1)\Rightarrow(2)$: If $T$ is homological, then, by definition, the
associated homological ring epimorphism $\lambda:B\to C$ induces an
equivalence from $C\mbox{{\rm-Mod}}$ to $\mathscr{E}$. Thus $\mathscr{E}$ is
an abelian subcategory of $B\mbox{{\rm-Mod}}$.

$(2)\Rightarrow(1)$: Suppose $(2)$ holds. Then $\mathscr{E}$ is closed under
direct sums, direct products, kernels and cokernels by Lemma 4.6(1). It
follows that there exists a ring epimorphism $\lambda:B\to C$ such that
$\lambda_{*}$ induces an equivalence $C\mbox{{\rm-
Mod}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathscr{E}$ of abelian
categories (for example, see [10, Lemma 2.1]). Since $\overline{D(i)}$ is
fully faithful by Proposition 4.17, the functor $D(\lambda_{*})$ is also fully
faithful. Thus $\lambda$ is homological. Further, it follows from ${\rm
Im}(\overline{D(i)})={\rm Ker}(G)$ that $D(\lambda_{*})$ induces an
equivalence from ${\mathscr{D}}(C)$ to ${\rm Ker}(G)$. This shows $(1)$.

$(2)\Rightarrow(3)$: Let $\rho^{\bullet}:{\rm
Hom}_{A}(P^{\bullet},A)\otimes_{A}T\to{\rm Hom}_{A}(P^{\bullet},T)$ be the
chain map defined in Lemma 4.15(2). Associated with $B^{\vee}$, there exists
the triangle in ${\mathscr{D}}(B):$

$(\bigtriangleup):\;X^{\bullet}[-1]\to B^{\vee}\to Y^{\bullet}\to X^{\bullet}$