Loc}(\underline{M})$ and show ${\rm\underline{Hom}}_{A}(X,Y)=0$ for any
$Y\in\mathscr{Y}$.

Actually, it follows from $\underline{M}^{\bot>0}=\underline{M^{\bot>0}}$ that
$X\in M^{\bot>0}$. Note that $X$ lies in $\mathscr{S}$. Let
$\mu^{\bullet}_{X}:M^{\bullet}_{X}\to X$ be an ${\rm Add}(M)$-resolution of
$X$. It follows from Proposition 3.8(1) that ${\rm
Hom}_{{\mathscr{K}}(A)}({\rm Con}(\mu^{\bullet}_{X}),{\rm
Con}(\pi^{\bullet}_{Y}))=0$ for any $Y\in\mathscr{Y}$. Since
${\rm\underline{Hom}}_{A}(M,\Omega_{A}^{n}(Y))={\rm\underline{Hom}}_{A}(M,Y[-n])=0$
for any $n\geq 0$, the chain map $\pi^{\bullet}_{Y}:P^{\bullet}_{Y}\to Y$ is
an ${\rm Add}(M)$-resolution of $Y$. Thus each homomorphism $f:X\to Y$ in
$A\mbox{{\rm-Mod}}$ can be lifted to a chain map $f^{\bullet}:{\rm
Con}(\mu^{\bullet}_{X})\to{\rm Con}(\pi^{\bullet}_{Y})$ in ${\mathscr{K}}(A)$.
From ${\rm Hom}_{{\mathscr{K}}(A)}({\rm Con}(\mu^{\bullet}_{X}),{\rm
Con}(\pi^{\bullet}_{Y}))=0$, we see that $f$ factorizes through the projective
module $P_{Y}^{0}$. Thus $f=0$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$ and
${\rm\underline{Hom}}_{A}(X,Y)=0$.

By definition, $\mathscr{Y}=\underline{M}^{\bot\leq 0}$. It follows from ${\rm
add}(\nu_{A}(M))={\rm add}({{}_{A}}M)$ and $(\lozenge)$ that
$\mathscr{Y}={{}^{\bot\geq-1}}\underline{M}$. Since
$\mathscr{H}=\mathscr{X}\cap\mathscr{Y}[1]$ and
$\mathscr{Y}[1]=\underline{M}^{\bot<0}$, one gets
$\mathscr{H}=\underline{M}^{\bot\neq 0}\cap{\rm
Loc}(\underline{M})=\underline{M}^{\bot\neq 0}\cap\underline{\mathscr{S}}$.
$\square$

###  4.1 Restrictions of recollements

In this subsection we consider the restriction of the recollement in Corollary
3.15 to relative stable categories. This leads to a part of the proof of
Theorem 1.5.

We begin with the following preparation.

######  Lemma 4.2.

$(1)$
$\underline{M}^{\bot}={{}^{\bot}}\underline{M},\quad\underline{\mathscr{G}}=\underline{M}^{\bot\neq
0,-1}={{}^{\bot\neq
0,-1}}\underline{M}\quad\mbox{and}\quad\mathscr{H}\subseteq\underline{\mathscr{G}\cap\mathscr{S}}\subseteq\mathscr{X}.$

$(2)$ Let $\pi:\underline{\mathscr{G}}\to\mathscr{C}$ be the canonical
quotient functor. Then the following are true.

$(a)$ The composition
$\underline{M}^{\bot}\hookrightarrow\underline{\mathscr{G}}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathscr{C}$
is a fully faithful triangle functor. In particular, the image of this
composition is a full triangulated subcategory of $\mathscr{C}$.

$(b)$ The composition
$\mathscr{H}\hookrightarrow\underline{\mathscr{G}\cap\mathscr{S}}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}(\mathscr{G}\cap\mathscr{S})/[M]$
is a fully faithful additive functor.

Proof. (1) Since ${\rm add}(\nu_{A}(M))={\rm add}({{}_{A}}M)$, $(1)$ follows
from $(\lozenge)$ and Proposition 4.1.

(2) If $X\in\underline{M}^{\bot}$, then $X\in{{}^{\bot}}\underline{M}$ by
$(1)$. In this case,
${\rm\underline{Hom}}_{A}(M,X)=0={\rm\underline{Hom}}_{A}(X,M)$, and therefore
${\rm\underline{Hom}}_{A}(M_{0},X)=0={\rm\underline{Hom}}_{A}(X,M_{0})$ for
$M_{0}\in{\rm Add}(M)$. This implies that $\Omega_{A}(X)\simeq\Omega_{M}(X)$,
$\Omega_{A}^{-}(X)\simeq\Omega_{M}^{-}(X)$ in $\mathscr{D}$ and
${\rm\underline{Hom}}_{A}(X,X^{\prime})=\underline{{\rm
Hom}}_{M}(X,X^{\prime})$ for any $A$-module $X^{\prime}$. Thus the composition
in $(a)$ is fully faithful. It is a triangle functor since
$\underline{M}^{\bot}$ is a full triangulated subcategory of
$A\mbox{{\rm-{\text@underline{Mod}}}}$. This shows $(a)$.

Now, let $X\in\mathscr{H}$. Then ${\rm\underline{Hom}}_{A}(M,X[-1])=0$. Since
$D{\rm\underline{Hom}}_{A}(M,X[-1])\simeq{\rm\underline{Hom}}_{A}(X,\nu_{A}(M))$
by $(\lozenge)$ and ${\rm add}(\nu_{A}(M))={\rm add}(M)$, we have
${\rm\underline{Hom}}_{A}(X,M)=0$. It then follows from ${\rm
Add}(\underline{M})={\rm Prod}(\underline{M})$ that
${\rm\underline{Hom}}_{A}(X,M^{\prime})=0$ for $M^{\prime}\in{\rm Add}(M)$.
Thus ${\rm\underline{Hom}}_{A}(X,Y)={\rm\underline{Hom}}_{M}(X,Y)$ for any
$A$-module $Y$. This implies $(b)$. $\square$

######  Proposition 4.3.

The recollement in Corollary 3.15 induces a recollement of triangulated
categories:

$\textstyle{\underline{M}^{\bot}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\circ{\rm
inc}}$$\textstyle{\mathscr{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Psi^{\prime}}}$$\scriptstyle{\widetilde{\Phi}}$$\scriptstyle{\widetilde{\Psi}}$$\textstyle{(\mathscr{G}\cap\mathscr{S})/[M]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\Phi^{\prime\prime}}$$\scriptstyle{{\rm
inc}}$

Proof. We first show that the recollement in Corollary 3.15 can be restricted
to a “recollement” of additive categories with six additive functors:

$(\sharp)\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.67778pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\underline{M}^{\bot}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
17.27716pt\raise 5.33751pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.33751pt\hbox{$\scriptstyle{{\rm inc}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.821pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.821pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\underline{\mathscr{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
17.88272pt\raise-7.78885pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\Psi^{\prime}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
2.61102pt\raise-3.00279pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
64.11482pt\raise 6.61111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\Phi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 86.96422pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
2.61102pt\raise-3.00279pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\ignorespaces\ignorespaces{\hbox{\kern
19.29599pt\raise 20.98077pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\Psi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
7.67703pt\raise 7.82858pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
86.96422pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\underline{\mathscr{G}\cap\mathscr{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
63.26486pt\raise-8.45274pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.94722pt\hbox{$\scriptstyle{\Phi^{\prime\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
49.81187pt\raise-3.00279pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
49.81187pt\raise-3.00279pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
62.4204pt\raise 19.73747pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.33751pt\hbox{$\scriptstyle{{\rm inc}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
52.82066pt\raise 6.16234pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces$

which satisfy the conditions $(1)$-$(3)$ in Definition 2.2. Obviously, ${\rm
inc}:\underline{M}^{\bot}\to\underline{\mathscr{G}}$ has left and right
adjoints which are the restriction of the functors $\widetilde{\Psi}$ and
$\widetilde{\Psi^{\prime}}$ in Corollary 3.15 to $\underline{\mathscr{G}}$,
respectively. Now, we claim that
$\widetilde{\Phi}(\underline{\mathscr{G}})\subseteq\underline{\mathscr{G}\cap\mathscr{S}}$
and
$\Phi^{\prime\prime}(\underline{\mathscr{G}\cap\mathscr{S}})\subseteq\underline{\mathscr{G}}$.
Then $({\rm inc},\widetilde{\Phi})$ and
$(\widetilde{\Phi},\Phi^{\prime\prime})$ in $(\sharp)$ are adjoint pairs.

By Corollary 3.13, $\underline{\mathscr{S}}={\rm Loc}(\underline{M})$. By
Theorem 3.14, each $A$-module $X$ is endowed with a triangle
$\Psi(X)[-1]\to\Phi(X)\to X\to\Psi(X)$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ such that
$\Phi(X)\in\underline{\mathscr{S}}$ and $\Psi(X)\in\underline{M}^{\bot}$. Note
that $\underline{\mathscr{G}}$ contains $\underline{M}^{\bot}$ and is closed
under extensions of triangles in $A\mbox{{\rm-{\text@underline{Mod}}}}$, due
to Lemma 4.2. Since $\Psi(X)$ and $\Psi(X)[-1]$ lie in $\underline{M}^{\bot}$,
we see that $X\in\underline{\mathscr{G}}$ if and only if
$\Phi(X)\in\underline{\mathscr{G}}$. Thus
$\widetilde{\Phi}(\underline{\mathscr{G}})=\Phi(\underline{\mathscr{G}})\subseteq\underline{\mathscr{G}\cap\mathscr{S}}$.
Similarly, by the triangle $\Psi^{\prime}(X)\to
X\to\Phi^{\prime}(X)\to\Psi^{\prime}(X)[1]$ in Proposition 3.6,
$X\in\underline{\mathscr{G}}$ if and only if
$\Phi^{\prime}(X)\in\underline{\mathscr{G}}$. This implies
$\Phi^{\prime\prime}(\underline{\mathscr{G}\cap\mathscr{S}})=\Phi^{\prime}(\underline{\mathscr{G}\cap\mathscr{S}})\subseteq\underline{\mathscr{G}}$.

Next, we show that the functors in $(\sharp)$ induce triangle functors among
quotient categories.

By Corollary 3.13, $\mathscr{S}$ contains ${\rm Add}(M)$ and is closed under
taking $\Omega_{M}$ and $\Omega_{M}^{-}$ in $A\mbox{{\rm-Mod}}$. Since ${\rm
Add}(M)\subseteq\mathscr{G}$ and $\mathscr{C}$ is a triangulated category by
Lemma 3.17, $(\mathscr{G}\cap\mathscr{S})/[M]$ is a full triangulated
subcategory of $\mathscr{C}$. Note that $\Phi({\rm Add}(\underline{M}))={\rm
Add}(\underline{M})$ and $\Psi({\rm Add}(\underline{M}))=0$ due to
$\underline{M}\in\underline{\mathscr{S}}$. By Lemma 2.1(1), the adjoint pairs
$({\rm inc},\widetilde{\Phi})$ and $(\widetilde{\Psi},{\rm inc})$ in
$(\sharp)$ induce adjoint pairs $({\rm inc},\widetilde{\Phi}_{0})$ and
$(\widetilde{\Psi}_{0},\pi\circ{\rm inc})$ of additive functors among
triangulated categories:

$(\natural)\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.67778pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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13.53207pt\raise 5.33751pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.821pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.821pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.96667pt\hbox{$\scriptstyle{\;\widetilde{\Phi}_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 89.18645pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
7.67398pt\raise 8.3206pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
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89.18645pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
76.24678pt\raise 20.93755pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.33751pt\hbox{$\scriptstyle{{\rm inc}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
55.03984pt\raise 5.29541pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$

In this diagram, both ${\rm inc}$ and $\pi\circ{\rm inc}$ (see Lemma 4.2(2))
are fully faithful triangle functors. It is known that any left or right
adjoint of a triangle functor between triangulated categories is again a
triangle functor. Thus $\widetilde{\Phi}_{0}$ and $\widetilde{\Psi}_{0}$ are
triangle functors. In the following, we show that both $\pi\circ{\rm inc}$ and
$\widetilde{\Phi}_{0}$ have right adjoints.

According to Lemma 3.12(3), $\underline{\mathscr{T}}={\rm
Coloc}(\underline{\nu_{A}(M)})={\rm Im}(\Phi^{\prime})$. By the definition of
$\mathscr{T}$, if $X\in\underline{\mathscr{T}}$, then $\Psi^{\prime}(X)=0$ and
$\Phi^{\prime}(X)\simeq X$. This means that

$\Psi^{\prime}({\rm
Prod}(\underline{\nu_{A}(M)}))=0\quad\mbox{and}\quad\Phi^{\prime}({\rm
Prod}(\underline{\nu_{A}(M)}))={\rm Prod}(\underline{\nu_{A}(M)}).$

Since ${\rm add}(\nu_{A}(M))={\rm add}(M)$ and ${{}_{A}}M\in A\mbox{{\rm-
mod}}$, there holds ${\rm Prod}(\underline{\nu_{A}(M)})={\rm
Prod}(\underline{M})={\rm Add}(\underline{M})$. It follows that
$\Psi^{\prime}({\rm Add}(\underline{M}))=0$ and $\Phi^{\prime}({\rm
Add}(\underline{M}))={\rm Add}(\underline{M})$. Since ${\rm
Add}(\underline{M})\subseteq\underline{\mathscr{G}\cap\mathscr{S}}$ and
$\Phi^{\prime\prime}=\Phi^{\prime}\circ{\rm inc}$, we have
$\Phi^{\prime\prime}({\rm Add}(\underline{M}))={\rm Add}(\underline{M})$. Due
to Lemma 2.1(1),
$\Phi^{\prime\prime}:\underline{\mathscr{G}\cap\mathscr{S}}\to\underline{\mathscr{G}}$
induces a functor
$\Phi^{\prime\prime}_{0}:(\mathscr{G}\cap\mathscr{S})/[M]\to\mathscr{C}$ which
is a right adjoint of $\widetilde{\Phi}_{0}$, while
$\widetilde{\Psi^{\prime}}:\underline{\mathscr{G}}\to\underline{M}^{\bot}$
induces a functor
$\widetilde{\Psi^{\prime}}_{0}:\mathscr{C}\to\underline{M}^{\bot}$ which is a
right adjoint of $\pi\circ{\rm inc}$.

When acting on objects, $\widetilde{\Phi}_{0}$ and $\widetilde{\Phi}$ are the
same. So we denote $\widetilde{\Phi}_{0}$ by $\widetilde{\Phi}$ for
simplicity. Similarly, we denote $\widetilde{\Psi}_{0}$,
$\widetilde{\Psi^{\prime}}_{0}$ and $\Phi^{\prime\prime}_{0}$ by
$\widetilde{\Psi}$, $\widetilde{\Psi^{\prime}}$ and $\Phi^{\prime\prime}$,
respectively.

To show the existence of the recollement of $\mathscr{C}$ in Proposition 4.3,
it remains to show the existence of two canonical triangles in Definition
2.2(4). Let $X\in\mathscr{G}$. We have shown that there is a canonical
triangle ${\rm inc}\circ\widetilde{\Phi}(X)\to X\to{\rm
inc}\circ\widetilde{\Psi}(X)\to{\rm inc}\circ\widetilde{\Phi}(X)[1]$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ such that
$\widetilde{\Phi}(X)\in\underline{\mathscr{G}\cap\mathscr{S}}$ and
$\widetilde{\Psi}(X)\in\underline{M}^{\bot}$. Since
$\underline{M}^{\bot}={{}^{\bot}}\underline{M}\subseteq\underline{{{}^{\bot
1}}M}$ by Lemma 4.2(1), it follows from Lemma $\ref{M-stable category}(2)$
that this triangle induces a triangle ${\rm inc}\circ\widetilde{\Phi}(X)\to
X\to{\rm inc}\circ\widetilde{\Psi}(X)\to\Omega_{M}^{-}({\rm
inc}\circ\widetilde{\Phi}(X))$ in $\mathscr{C}$, which is the required second
triangle in Definition 2.2(4). Similarly, by the triangle $\Psi^{\prime}(X)\to
X\to\Phi^{\prime}(X)\to\Psi^{\prime}(X)[1]$ with
$\Psi^{\prime}(X)\in\underline{M}^{\bot}$ and
$\Phi^{\prime}(X)\in\underline{\mathscr{G}}$ and by the inclusion
$\underline{M}^{\bot}\subseteq\underline{M^{\bot 1}}$, we can obtain the first
triangle in Definition 2.2(4). Thus the proof of Proposition 4.3 is completed.
$\square$

Now, we define a full subcategory $\mathscr{E}$ of $\mathscr{G}$, which
contains both $\mathscr{G}_{0}$ and $A\mbox{{\rm-Proj}}$.

$\mathscr{E}:=\\{X\in\mathscr{G}\mid{\rm\underline{Hom}}_{A}(M,X),\,{\rm\underline{Hom}}_{A}(M[1],X)\in\Gamma\mbox{{\rm-
mod}}\\}.$

######  Lemma 4.4.

$(1)$
$\mathscr{E}=\\{X\in\mathscr{G}\mid{\rm\underline{Hom}}_{A}(M,X)\in\Gamma\mbox{{\rm-
mod}},\,{\rm\underline{Hom}}_{A}(X,M)\in\Gamma^{\rm op}\mbox{{\rm-mod}}\\}$.

$(2)$ $\mathscr{E}$ is a thick subcategory of $\mathscr{G}$ and
$\mathscr{E}/[M]$ is a full triangulated subcategory of $\mathscr{C}$.

Proof. $(1)$ As $A$ is an Artin algebra over an Artin commutative ring $R$ and
${{}_{A}}M$ is finitely generated, $\Gamma$ is an Artin algebra over $R$.
Moreover, a $\Gamma$-module $N$ is finitely generated if and only if ${}_{R}N$
is finitely generated as an $R$-module. Thus $(1)$ follows from
$D{\rm\underline{Hom}}_{A}(M[1],X)\simeq{\rm\underline{Hom}}_{A}(X,\nu_{A}(M))$
and ${\rm add}(\nu_{A}(M))={\rm add}(M)$.

$(2)$ Clearly, $\mathscr{E}$ is closed under direct summands in $\mathscr{G}$.
Now, let $0\to X_{1}\to X_{2}\to X_{3}\to 0$ be an exact sequence in
$A\mbox{{\rm-Mod}}$ with $X_{i}\in\mathscr{G}$ for $1\leq i\leq 3$. Since
$\underline{\mathscr{G}}=\underline{M}^{\bot\neq 0,-1}$ by Lemma 4.2, there is
a long exact sequence in $R\mbox{{\rm-mod}}$:

$0\to{\rm\underline{Hom}}_{A}(M[1],X_{1})\to{\rm\underline{Hom}}_{A}(M[1],X_{2})\to{\rm\underline{Hom}}_{A}(M[1],X_{3})\to{\rm\underline{Hom}}_{A}(M,X_{1})\to{\rm\underline{Hom}}_{A}(M,X_{2})\to{\rm\underline{Hom}}_{A}(M,X_{3})\to
0.$

This implies that $\mathscr{E}$ has the two out of three property in
$\mathscr{G}$. Thus $\mathscr{E}$ is a thick subcategory of $\mathscr{G}$.

By definition, $\mathscr{E}/[M]$ consists of all objects $X\in\mathscr{C}$
which is $M$-stably isomorphic to an object of $\mathscr{E}$. To show that
$\mathscr{E}/[M]$ is a full triangulated subcategory of $\mathscr{C}$, it
suffices to show that $\mathscr{E}/[M]$ is closed under taking $\Omega_{M}$
and $\Omega_{M}^{-}$ in $\mathscr{C}$.

Let $X\in\mathscr{E}$. Due to
${\rm\underline{Hom}}_{A}(M,X)\in\Gamma\mbox{{\rm-mod}}$, we see that $X$ has
a right ${\rm Add}(M)$-approximation $f:M_{0}\oplus P_{0}\to X$ such that
$M_{0}\in{\rm add}(M)$ and $P_{0}\in A\mbox{{\rm-Proj}}$. Note that
$\mathscr{E}$ contains ${\rm add}(M)$ and $A\mbox{{\rm-Proj}}$. This forces
$M_{0}\oplus P_{0}\in\mathscr{E}$, and further ${\rm Ker}(f)\in\mathscr{E}$.
It follows from $\Omega_{M}(X)\simeq{\rm Ker}(f)$ in $\mathscr{C}$ that
$\Omega_{M}(X)\in\mathscr{E}/[M]$. Since ${\rm\underline{Hom}}_{A}(X,M)$ is a
$\Gamma^{\rm op}$-module, we can show $\Omega_{M}^{-}(X)\in\mathscr{E}/[M]$.

The proof of $(2)$ also implies that if $X\in\mathscr{G}$, then
$X\in\mathscr{E}$ if and only if it has a complete ${\rm Add}(M)$-resolution
$M^{\bullet}_{X}:=(M_{X}^{i})_{i\in\mathbb{Z}}$ satisfying that
$M_{X}^{i}=N^{i}\oplus P^{i}$ with $N^{i}\in{\rm add}(M)$ and $P^{i}\in
A\mbox{{\rm-Proj}}$. $\square$

######  Corollary 4.5.

The recollement in Proposition 4.3 can be restricted a recollement of
$\mathscr{E}/[M]$:

$\textstyle{\underline{M}^{\bot}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\circ{\rm
inc}}$$\textstyle{\mathscr{E}/[M]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Psi^{\prime}}}$$\scriptstyle{\widetilde{\Phi}}$$\scriptstyle{\widetilde{\Psi}}$$\textstyle{(\mathscr{E}\cap\mathscr{S})/[M].\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{\prime\prime}}$$\scriptstyle{{\rm
inc}}$

Proof. Note that
$\big{(}(\mathscr{G}\cap\mathscr{S})/[M]\big{)}\cap(\mathscr{E}/[M])=(\mathscr{E}\cap\mathscr{S})/[M]$
and the image of the functor $\pi\circ{\rm
inc}:\underline{M}^{\bot}\to\mathscr{C}$ belongs to $\mathscr{E}/[M]$. Thus
Corollary 4.5 is a consequence of Proposition 4.3. $\square$

###  4.2 Compacts objects and representability of homological functors

In this subsection, we find out a special compact object in $\mathscr{C}$ (or
even in a bigger relative stable category) and establish a series of
homological functors from $\mathscr{C}$ to $\Gamma\mbox{{\rm-Mod}}$ (see
Theorem 4.14).

From now on, let ${\bf L}:A\mbox{{\rm-{\text@underline{Mod}}}}\to\mathscr{Y}$
be the left adjoint of the inclusion $\mathscr{Y}\subseteq
A\mbox{{\rm-{\text@underline{Mod}}}}$ and let ${\bf
R}:A\mbox{{\rm-{\text@underline{Mod}}}}\to\mathscr{X}$ be the right adjoint of
the inclusion $\mathscr{X}\subseteq A\mbox{{\rm-{\text@underline{Mod}}}}$.
Define

$H^{0}_{*}:=[1]\circ{\bf L}\circ[-1]\circ{\bf
R}:\;\;A\mbox{{\rm-{\text@underline{Mod}}}}\longrightarrow\mathscr{H},\;\;N:=H^{0}_{*}(M)\in\mathscr{H}\;\;\mbox{and}$
$\mathscr{H}^{\rm
fg}:=\\{X\in\mathscr{H}\mid{\rm\underline{Hom}}_{A}(M,X)\in\Gamma\mbox{{\rm-
mod}}\\}.$

Then $H^{0}_{*}$ is a homological functor, that is, for a triangle $X_{1}\to
X_{2}\to X_{3}\to X_{1}[1]$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$, the
sequence $H^{0}_{*}(X_{1})\to H^{0}_{*}(X_{2})\to H^{0}_{*}(X_{3})$ is exact
in $\mathscr{H}$. Moreover, $\mathscr{H}^{\rm
fg}=\mathscr{H}\cap\underline{\mathscr{E}}$ by Proposition 4.1 and Lemma
4.2(1). Note that
$D{\rm\underline{Hom}}_{A}(M,N[-1])\simeq{\rm\underline{Hom}}_{A}(N,\nu_{A}(M))$
by $(\lozenge)$. Since ${\rm add}(\nu_{A}(M))={\rm add}(M)$ and
$N\in\mathscr{H}$, we get ${\rm\underline{Hom}}_{A}(N,M)=0$. This implies
${\rm\underline{End}}_{A}(N)={\rm End}_{\mathscr{C}}(N)$.

######  Lemma 4.6.

Let $X\in\mathscr{X}$.

$(1)$ There is a canonical triangle $[1]\circ{\bf R}\circ[-1](X)\to
X\stackrel{{\scriptstyle\tau_{X}}}{{\longrightarrow}}H^{0}_{*}(X)\to[2]\circ{\bf
R}\circ[-1](X)$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$.

$(2)$ If $Y\in\mathscr{Y}[1]$, then
${\rm\underline{Hom}}_{A}(\tau_{X},Y):{\rm\underline{Hom}}_{A}(H^{0}_{*}(X),Y)\to{\rm\underline{Hom}}_{A}(X,Y)$
is an isomorphism. In particular, there is a natural isomorphism

${\rm\underline{Hom}}_{A}(\tau_{M},-):\;\;{\rm\underline{Hom}}_{A}(N,-)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,-):\;\;\mathscr{H}\longrightarrow\Gamma\mbox{{\rm-
Mod}}.$

Proof. Since $(\mathscr{X},\mathscr{Y})$ is a torsion pair in
$A\mbox{{\rm-{\text@underline{Mod}}}}$, each $A$-module $W$ is endowed with a
triangle ${\bf R}(W)\to W\to{\bf L}(W)\to{\bf R}(W)[1]$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. We take $W=X[-1]$ and get a triangle
${\bf R}(X[-1])\to X[-1]\to{\bf L}(X[-1])\to{\bf R}(X[-1])[1]$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. Shifting this triangle by $[1]$ yields
another triangle

$({\dagger})\quad{\bf R}(X[-1])[1]\to X\to{\bf L}(X[-1])[1]\to{\bf
R}(X[-1])[2].$

Due to $X\in\mathscr{X}$, we have ${\bf R}(X)\simeq X$. Thus
$H^{0}_{*}(X)=[1]\circ{\bf L}\circ[-1]\circ{\bf R}(X)\simeq{\bf L}(X[-1])[1].$
Then the triangle $({\dagger})$ can be rewritten as $[1]\circ{\bf
R}\circ[-1](X)\to
X\stackrel{{\scriptstyle\tau_{X}}}{{\longrightarrow}}H^{0}_{*}(X)\to[2]\circ{\bf
R}\circ[-1](X).$ This shows $(1)$.

Since $\mathscr{X}\subseteq A\mbox{{\rm-{\text@underline{Mod}}}}$ is closed
under $[1]$, it follows from ${\bf R}(X[-1])\in\mathscr{X}$ that both
$[1]\circ{\bf R}\circ[-1](X)$ and $[2]\circ{\bf R}\circ[-1](X)$ belong to
$\mathscr{X}$. As $\underline{{\rm Hom}}_{A}(X,Y)=0$ for $X\in\mathscr{X}$ and
$Y\in\mathscr{Y}$, the first part of $(2)$ holds by $(1)$, while the second
part of $(2)$ follows from $M\in\mathscr{X}$ and
$\mathscr{H}\subseteq\mathscr{Y}[1]$. $\square$

The next result follows from [13, Chap. III, Lemma 3.3 and Theorem 3.4]; see
also [24, Theorem 1.3(3)] for the assertion $(3)$.

######  Lemma 4.7.

$(1)$ For any $X\in\mathscr{X}$, the morphism $\tau_{X}$ in Lemma 4.6(1)
induces an isomorphism of $\Gamma$-modules:

${\rm\underline{Hom}}_{A}(M,\tau_{X}):{\rm\underline{Hom}}_{A}(M,X)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,H^{0}_{*}(X)).$

$(2)$ The functor $H^{0}_{*}$ induces an isomorphism
$\Gamma\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm\underline{End}}_{A}(N)$
of algebras such that $\gamma\,\tau_{M}=\tau_{M}H^{0}_{*}(\gamma)$ for any
$\gamma\in\Gamma$. In this sense, $\Gamma$ can be identified with
${\rm\underline{End}}_{A}(N)$.

$(3)$ $N$ is a small projective generator of $\mathscr{H}$ and the functor
${\rm\underline{Hom}}_{A}(N,-):\mathscr{H}\to\Gamma\mbox{{\rm-Mod}}$ is an
equivalence of abelian categories.

An easy observation is the following result, of which $(2)$ conveys that $N$
has only finitely many indecomposable direct summands in $\mathscr{C}$.

######  Corollary 4.8.

$(1)$ There is a fully faithful functor $\Theta:\Gamma\mbox{{\rm-
mod}}\to(\mathscr{E}\cap\mathscr{S})/[M]$ which sends $\Gamma$ to $N$.

$(2)$ Let $M=A\oplus\bigoplus_{i=1}^{m}M_{i}$, where $m\in\mathbb{N}$ and
$M_{i}$ are indecomposable and non-projective. Then
$N\simeq\bigoplus_{i=1}^{m}H_{*}^{0}(M_{i})$ in $\mathscr{C}$ and
$H_{*}^{0}(M_{i})$ are indecomposable in $\mathscr{C}$.

$(3)$ Let $X\in\mathscr{G}\cap\mathscr{S}$. Then there is a triangle $X_{0}\to
X\to H^{0}_{*}(X)\to\Omega_{M}^{-}(X_{0})$ in $\mathscr{C}$ such that
$X_{0}\in\mathscr{G}\cap\mathscr{S}$ and $\Omega_{M}(X_{0})\in\mathscr{H}$.
Further, if $X\in\mathscr{E}\cap\mathscr{S}$, then
$\Omega_{M}(X_{0}),H^{0}_{*}(X)\in\mathscr{H}^{\rm fg}$.

Proof. $(1)$ By Lemma 4.2(2)(b), the composition $U:\mathscr{H}^{\rm
fg}=\mathscr{H}\cap\underline{\mathscr{E}}\hookrightarrow\underline{\mathscr{E}\cap\mathscr{S}}\to(\mathscr{E}\cap\mathscr{S})/[M]$
is fully faithful. Moreover, from Lemmas 4.6(2) and 4.7(3) it follows that
${\rm\underline{Hom}}_{A}(M,-)\simeq{\rm\underline{Hom}}_{A}(N,-):\mathscr{H}^{\rm
fg}\to\Gamma\mbox{{\rm-mod}}$, which is an equivalence of abelian categories
sending $N$ to $\Gamma$. In particular, $N\in\mathscr{H}^{\rm fg}$. Now, let
$\Theta:\Gamma\mbox{{\rm-mod}}\to(\mathscr{E}\cap\mathscr{S})/[M]$ be the
composition of a quasi-inverse of ${\rm\underline{Hom}}_{A}(M,-)$ with $U$.
Then $\Theta$ is fully faithful and sends $\Gamma$ to $N$.

$(2)$ By Lemmas 4.6(2) and 4.7(1), $H_{*}^{0}$ induces an isomorphism
${\rm\underline{Hom}}_{A}(M,M_{i})\simeq{\rm\underline{Hom}}_{A}(N,H_{*}^{0}(M_{i}))$.
Thus
${\rm\underline{End}}_{A}(M_{i})\simeq{\rm\underline{End}}_{A}(H_{*}^{0}(M_{i}))$
as algebras. It then follows from $H_{*}^{0}(M_{i})\in\mathscr{H}$ that
${\rm\underline{End}}_{A}(H_{*}^{0}(M_{i}))\simeq{\rm
End}_{\mathscr{C}}(H_{*}^{0}(M_{i}))$ as algebras by Lemma 4.2(2). As $M_{i}$
is indecomposable and non-projective, it is also indecomposable in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. Thus $H_{*}^{0}(M_{i})$ is
indecomposable in $\mathscr{C}$.

$(3)$ Let $X_{0}:={\bf R}(X[-1])[1]$. Then $X_{0}[-1]\in\mathscr{X}$ and there
is a triangle $X_{0}\to
X\stackrel{{\scriptstyle\tau_{X}}}{{\longrightarrow}}H^{0}_{*}(X)\to X_{0}[1]$
in $A\mbox{{\rm-{\text@underline{Mod}}}}$ by Lemma 4.6(1). Thus
${\rm\underline{Hom}}_{A}(M,X_{0}[n])=0$ for all $n\geq 0$ by Proposition 4.1.
In particular, ${\rm Ext}_{A}^{1}(M,X_{0})=0$. Since both $X$ and
$H^{0}_{*}(X)$ lie in $\mathscr{G}\cap\mathscr{S}$, we see that
$X_{0}\in\mathscr{G}\cap\mathscr{S}$ and the above triangle induces a triangle
$X_{0}\to X\to H^{0}_{*}(X)\to\Omega_{M}^{-}(X_{0})$ in $\mathscr{C}$ by Lemma
3.16(2). Further, ${\rm\underline{Hom}}_{A}(M,X_{0}[-2])=0$ by Lemma 4.2(1),
and $X_{0}[-1]=\Omega_{A}(X_{0})\simeq\Omega_{M}(X_{0})\in\mathscr{G}$ by
${\rm\underline{Hom}}_{A}(M,X_{0})=0$. Thus $X_{0}[-1]\in\mathscr{H}$ by
Proposition 4.1. Suppose that $X$ is in $\mathscr{E}\cap\mathscr{S}$. Then
${\rm\underline{Hom}}_{A}(M[1],X_{0})\simeq{\rm\underline{Hom}}_{A}(M,X_{0}[-1])\simeq{\rm\underline{Hom}}_{A}(M,{\bf
R}(X[-1]))\simeq{\rm\underline{Hom}}_{A}(M,X[-1])\in\Gamma\mbox{{\rm-mod}}$,
where the last isomorphism follows from the fact that ${\bf
R}:A\mbox{{\rm-{\text@underline{Mod}}}}\to\mathscr{X}$ is the right adjoint of
the inclusion $\mathscr{X}\subseteq A\mbox{{\rm-{\text@underline{Mod}}}}$.
Now, it follows from $X_{0}\in\mathscr{G}$ and
${\rm\underline{Hom}}_{A}(M,X_{0})=0$ that $X_{0}\in\mathscr{E}$. Thus
$X_{0}\in\mathscr{E}\cap\mathscr{S}$. By Lemma 4.4(2), both
$\Omega_{M}(X_{0})$ and $H^{0}_{*}(X)$ belong to $\mathscr{H}^{\rm fg}$.
$\square$

######  Corollary 4.9.

The module ${{}_{A}}M$ is projective if and only if
$(\mathscr{E}\cap\mathscr{S})/[M]=0$ if and only if
$(\mathscr{G}\cap\mathscr{S})/[M]=0$.

Proof. If ${{}_{A}}M$ is projective, then $\mathscr{S}$ consists of projective
$A$-modules by Corollary 3.13, which implies
$(\mathscr{E}\cap\mathscr{S})/[M]=0=(\mathscr{G}\cap\mathscr{S})/[M]$. If
$(\mathscr{E}\cap\mathscr{S})/[M]=0$, then $\Gamma\mbox{{\rm-mod}}=0$ by
Corollary 4.8(1), that is, $\Gamma=0$, and therefore ${{}_{A}}M$ is
projective. $\square$

######  Lemma 4.10.

The object $N$ has a complete ${\rm Add}(M)$-resolution:

$\cdots\longrightarrow M^{-3}\longrightarrow M^{-2}\longrightarrow
P^{-1}\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}M\oplus
P^{0}\longrightarrow P^{1}\longrightarrow M^{2}\longrightarrow
M^{3}\longrightarrow\cdots$

satisfying

$(1)$ $N={\rm Coker}(\partial)$, $M^{i}\in{\rm Add}(M)$ for all $|i|\geq 2$
and $P^{i}\in{\rm Add}(A)$ for all $|i|\leq 1$; and

$(2)$ ${\rm Ker}(\partial)\in\mathscr{H}$ and ${\rm\underline{Hom}}_{A}(M,{\rm
Ker}(\partial))\simeq D\big{(}{\rm\underline{Hom}}_{A}(M,\nu_{A}(M))\big{)}$
as $\Gamma$-modules.

Proof. $(1)$ Let $\tau_{M}:M\to N$ be the morphism in $A$-Mod in Lemma 4.6(1).
Up to projective direct summand, we may assume that $\tau_{M}$ is a
homomorphism in $A\mbox{{\rm-Mod}}$ and a preimage of the $\tau_{M}$ in
$A$-Mod. Let $f:P^{0}\to{\rm Coker}(\tau_{M})$ be a projective cover of ${\rm
Coker}(\tau_{M})$. Then there exists a homomorphism $f_{0}:P^{0}\to N$ such
that $f$ is the composition of $f_{0}$ with the quotient map $N\to{\rm
Coker}(\tau_{M})$. Moreover, the map $\pi_{0}:=(\tau_{M},f_{0}):M\oplus
P^{0}\to N$ is a right ${\rm Add}(M)$-approximation of $N$. Let $K:={\rm
Ker}(\pi_{0})$ with a projective cover $\pi_{1}:P^{-1}\to K$. Then
$K\simeq\Omega_{M}(N)$ in $\mathscr{D}$. By the proof of Corollary 4.8(3), we
see that $K\in\mathscr{G}\cap\mathscr{S}$, ${\rm\underline{Hom}}_{A}(M,K)=0$
and $K[-1]=\Omega_{A}(K)\simeq\Omega_{M}(K)\in\mathscr{H}$. Consequently,
$\pi_{1}$ is a right ${\rm Add}(M)$-approximation of $K$. Now, let
$\partial:P^{-1}\to M\oplus P^{0}$ be the composition of $\pi_{1}$ with the
inclusion $\lambda_{1}:K\to M\oplus P^{0}$. Then ${\rm Coker}(\partial)=N$ and
${\rm Ker}(\partial)\simeq K[-1]$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$.
Since ${\rm\underline{Hom}}_{A}(N,M)=0$ and ${\rm Add}(M)={\rm Prod}(M)$, a
minimal injective envelope $N\to P^{1}$ is a left ${\rm Add}(M)$-approximation
of $N$. Thus $(1)$ holds.

$(2)$ Thanks to $N\in\mathscr{H}$, there holds
${\rm\underline{Hom}}_{A}(M,N[-1])=0={\rm\underline{Hom}}_{A}(M,N[-2])$ by
Proposition 4.1. This implies that
${\rm\underline{Hom}}_{A}(M,\lambda_{1}[-1]):{\rm\underline{Hom}}_{A}(M,K[-1])\to{\rm\underline{Hom}}_{A}(M,M[-1])$
is an isomorphism. By $(\lozenge)$,

${\rm\underline{Hom}}_{A}(M,M[-1])\simeq
D\big{(}{\rm\underline{Hom}}_{A}(M[-1],\nu_{A}(M)[-1])\big{)}\simeq
D\big{(}{\rm\underline{Hom}}_{A}(M,\nu_{A}(M))\big{)}$

as $\Gamma$-modules. Thus ${\rm\underline{Hom}}_{A}(M,{\rm
Ker}(\partial)\simeq{\rm\underline{Hom}}_{A}(M,K[-1])\simeq
D({\rm\underline{Hom}}_{A}(M,\nu_{A}(M)))$. $\square$

######  Corollary 4.11.

If $A$ is a symmetric algebra and $\Gamma$ is a Frobenius algebra, then
$\Omega_{M}^{2}(N)\simeq N$ in $\mathscr{C}$.

Proof. Since $A$ is symmetric, $\nu_{A}(M)\simeq M$ as
$A$-$\Lambda$-bimodules. By Lemma 4.10(2), ${\rm\underline{Hom}}_{A}(M,{\rm
Ker}(\partial))\simeq D\big{(}{\rm\underline{Hom}}_{A}(M,M)\big{)}=D(\Gamma)$
as $\Gamma$-modules. Note that ${\rm\underline{Hom}}_{A}(M,N)\simeq\Gamma$ by
Lemmas 4.6(2) and 4.7(2). Since $\Gamma$ is a Frobenius algebra, $\Gamma\simeq
D(\Gamma)$ as $\Gamma$-modules. Moreover, ${\rm Ker}(\partial)\in\mathscr{H}$
and ${\rm Ker}(\partial)\simeq\Omega_{M}^{2}(N)$ in $\mathscr{C}$ by Lemma
4.10. It follows from Lemmas 4.6(2) and 4.7 that ${\rm Ker}(\partial)\simeq N$
in $\mathscr{H}$. Thus $\Omega_{M}^{2}(N)\simeq N$ in $\mathscr{C}$. $\square$

######  Lemma 4.12.

Let $X\in{{}^{\bot>0}}M$ and $l_{X}:X\to M^{X}$ be a left ${\rm
Add}(M)$-approximation of $X$. Then there is an exact sequence of
$\Gamma$-modules:

$0\longrightarrow{\rm\underline{Hom}}_{A}(N,X)\stackrel{{\scriptstyle(\tau_{M})_{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,X)\stackrel{{\scriptstyle(l_{X})^{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,M^{X}).$

Proof. Keep all notations introduced in the proof of Lemma 4.10. Recall that
${\rm Prod}(M)={\rm Add}(M)$ and ${\rm\underline{Hom}}_{A}(N,M)=0$. If
$M_{0}\in{\rm Add}(M)$, then ${\rm\underline{Hom}}_{A}(N,M_{0})=0$. It follows
from $M^{X}\in{\rm Add}(M)$ that ${\rm\underline{Hom}}_{A}(N,M^{X})=0$. This
implies that the composition of $(\tau_{M})_{*}$ with $(l_{X})^{*}$ is $0$.
Thus ${\rm Im}((\tau_{M})_{*})\subseteq{\rm Ker}((l_{X})^{*})$.

Applying ${\rm\underline{Hom}}_{A}(-,X)$ to the triangle $K\to
M\stackrel{{\scriptstyle\tau_{M}}}{{\longrightarrow}}N\to K[1]$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ yields an exact sequence
${\rm\underline{Hom}}_{A}(K[1],X)\to{\rm\underline{Hom}}_{A}(N,X)\stackrel{{\scriptstyle(\tau_{M})_{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,X)$.
By $(\lozenge)$, we have
$\underline{{{}^{\bot>0}}M}=\underline{M}^{\bot\leq-2}=\mathscr{Y}[2]$.
Moreover, $\underline{K}[-1]\in\mathscr{H}\subseteq\mathscr{X}$ by the proof
of Lemma 4.10. Thus $\underline{K}[1]\in\mathscr{X}[2]$. Since
$X\in\underline{{{}^{\bot>0}}M}$ and $(\mathscr{X},\mathscr{Y})$ is a torsion
pair in $A\mbox{{\rm-{\text@underline{Mod}}}}$, we have
${\rm\underline{Hom}}_{A}(K[1],X)=0$. Thus $(\tau_{M})_{*}$ is injective. It
remains to show ${\rm Ker}((l_{X})^{*})\subseteq{\rm Im}((\tau_{M})_{*})$.

Suppose that
$g\in{\rm\underline{Hom}}_{A}(M,X)={\rm\underline{Hom}}_{A}(M\oplus P_{0},X)$
with $gl_{X}=0$. Let $0\to
X\stackrel{{\scriptstyle\lambda_{0}}}{{\longrightarrow}}Q^{0}\stackrel{{\scriptstyle\mu_{0}}}{{\longrightarrow}}\Omega_{A}^{-}(X)\to
0$ be a short exact sequence of $A$-modules in which $\lambda_{0}$ is an
injective envelope. We claim that the pair $(g\lambda_{0},0)$ with
$g\lambda_{0}:M\oplus P^{0}\to Q^{0}$ and $0:N\to\Omega_{A}^{-}(X)$ can be
completed into the commutative diagram

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{-3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{-3}}$$\textstyle{M^{-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{-2}}$$\textstyle{P^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{f^{-1}}$$\textstyle{M\oplus
P^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\scriptstyle{g\lambda_{0}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{0}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q^{-3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q^{-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{0}}$$\textstyle{\Omega_{A}^{-}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

in which the first arrow is an ${\rm Add}(M)$-resolution of $N$ (see Lemma
4.10) and the second arrow is a minimal projective resolution of
$\Omega_{A}^{-}(X)$. In other words, there is a chain map
$f^{\bullet}:=(\cdots,f^{-3},f^{-2},f^{-1},g\lambda_{0},0).$ To show the
existence of the chain map, one splits the long exact sequences into a series
of short exact sequences, and then construct relevant homomorphisms together
with commutative diagrams between these sequences. We carry out the details as
follows.

For $K:={\rm Ker}(\pi_{0})$, let $\lambda_{1}:K\to M\oplus P^{0}$ be the
canonical inclusion, and let $\alpha:K\to X$ be the composition of
$\lambda_{1}$ with $g:M\oplus P^{0}\to X$. Since $P^{-1}$ is projective, there
exists a map $f^{-1}:P^{-1}\to Q^{-1}$ making the diagram commute:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{A}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega_{A}(\alpha)}$$\textstyle{P^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{-1}}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{A}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

To construct $f^{-2}$, we will show that
${\rm\underline{Hom}}_{A}(M,\alpha[-1]):{\rm\underline{Hom}}_{A}(M,K[-1])\to{\rm\underline{Hom}}_{A}(M,X[-1])$
is zero, where $[-1]$ denotes $\Omega_{A}$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$.

By the proof of Lemma 4.10(2), ${\rm\underline{Hom}}_{A}(M,\lambda_{1}[-1])$
is an isomorphism. It is enough to show that the map
${\rm\underline{Hom}}_{A}(M,g[-1]):{\rm\underline{Hom}}_{A}(M,M[-1])\to{\rm\underline{Hom}}_{A}(M,X[-1])$
is zero. Applying $D:\Gamma\mbox{{\rm-Mod}}\to\Gamma^{\rm op}\mbox{{\rm-Mod}}$
to this map, we see that $(\lozenge)$ yields
$D{\rm\underline{Hom}}_{A}(M,g[-1])\simeq{\rm\underline{Hom}}_{A}(g,\nu_{A}(M)):{\rm\underline{Hom}}_{A}(X,\nu_{A}(M))\to{\rm\underline{Hom}}_{A}(M,\nu_{A}(M))$.
Although $D$ may not be an equivalence in general, it is always exact and
reflects zero objects. Thus ${\rm\underline{Hom}}_{A}(M,g[-1])=0$ if and only
if ${\rm\underline{Hom}}_{A}(g,\nu_{A}(M))=0$. Since ${\rm
add}(\nu_{A}(M))={\rm add}(M)={\rm Prod}(M)$,
${\rm\underline{Hom}}_{A}(g,\nu_{A}(M))=0$ is equivalent to saying that
${\rm\underline{Hom}}_{A}(g,M^{\prime}):{\rm\underline{Hom}}_{A}(X,M^{\prime})\to{\rm\underline{Hom}}_{A}(M,M^{\prime})$
is $0$ for any $M^{\prime}\in{\rm Prod}(M)$. By assumption, $l_{X}$ is a left
${\rm Add}(M)$-approximation of $X$ and $gl_{X}=0$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. This leads to
${\rm\underline{Hom}}_{A}(g,M^{\prime})=0$. Thus
${\rm\underline{Hom}}_{A}(M,\alpha[-1])=0$, and therefore
${\rm\underline{Hom}}_{A}(M^{-2},\alpha[-1])=0$, due to $M^{-2}\in{\rm
Add}(M)$. Hence there are two homomorphisms $f^{-2}$ and $\beta$ such that the
diagram is commutative:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{M}(\Omega_{A}(K))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{M^{-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{-2}}$$\textstyle{\Omega_{A}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega_{A}(\alpha)}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{A}^{2}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q^{-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{A}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since $X\in\underline{{{}^{\bot>0}}M}=\underline{M}^{\bot\leq-2}$, we have
${\rm\underline{Hom}}_{A}(M,\Omega_{A}^{n}(X))={\rm\underline{Hom}}_{A}(M,X[-n])=0$
for all $n\geq 2$. Then
${\rm\underline{Hom}}_{A}(M^{-n-1},\Omega_{A}^{n}(X))=0$, due to
$M^{-n-1}\in{\rm Add}(M)$. Consequently, the components $f^{-n-1}:M^{-n-1}\to
Q^{-n-1}$ for $n\geq 2$ in $f^{\bullet}$ can be constructed.

Since
$N\in\mathscr{H}\subseteq\underline{M}^{\bot>0}\cap\underline{\mathscr{S}}$ by
Proposition 4.1, it follows from Proposition 3.8(1) that $f^{\bullet}=0$ in
${\mathscr{K}}(A)$. Therefore there are homomorphisms $h:N\to Q^{0}$ and
$s^{0}:M\oplus P^{0}\to Q^{-1}$ such that
$g\lambda_{0}=s^{0}\mu_{1}\lambda_{0}+\pi_{0}h$ and $h\mu_{0}=0$. Since
$\lambda_{0}$ is the kernel of $\mu_{0}$, there is a unique map $h_{0}:N\to X$
satisfying $h=h_{0}\lambda_{0}$. Since $\lambda_{0}$ is injective,
$g=s^{0}\mu_{1}+\pi_{0}h_{0}$. This forces $g=\tau_{M}h_{0}$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ and shows $g\in{\rm
Im}((\tau_{M})_{*})$. Thus ${\rm Ker}((l_{X})^{*})\subseteq{\rm
Im}((\tau_{M})_{*})$. Finally, via the isomorphism
$\Gamma\simeq{\rm\underline{End}}_{A}(N)$ in Lemma 4.7(2),
${\rm\underline{Hom}}_{A}(N,X)$ becomes a $\Gamma$-module and $(\tau_{M})_{*}$
is a homomorphism of $\Gamma$-modules. $\square$

######  Corollary 4.13.

The object $N$ is compact in $\underline{{{}^{\bot>0}}M}$.

Proof. It suffices to show that
${\rm\underline{Hom}}_{A}(N,-):\underline{{{}^{\bot>0}}M}\to\Gamma\mbox{{\rm-
Mod}}$ commutes with direct sums. Let $\\{X_{i}\\}_{i\in I}$ be a set of
$A$-modules in ${{}^{\bot>0}}M$ with $I$ an index set. For each $i$, let
$g_{i}:X_{i}\to M_{i}$ be a left ${\rm Add}(M)$-approximation of $X_{i}$.
According to ${\rm Add}(M)={\rm Prod}(M)$, the direct sum $g=(g_{i})_{i\in
I}:\bigoplus_{i\in I}X_{i}\to\bigoplus_{i\in I}M_{i}$ of all these $g_{i}$ is
a left ${\rm Add}(M)$-approximation of $\bigoplus_{i\in I}X_{i}$. Note that
${\rm\underline{Hom}}_{A}(M,-):A\mbox{{\rm-{\text@underline{Mod}}}}\to\Gamma\mbox{{\rm-
Mod}}$ commutes with direct sums since $M$ is compact in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. By Lemma 4.12, we can construct the
following commutative diagram with exact arrows and canonical vertical maps:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{i\in
I}{\rm\underline{Hom}}_{A}(N,X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{i\in
I}{\rm\underline{Hom}}_{A}(M,X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{i\in
I}{\rm\underline{Hom}}_{A}(M,M_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm\underline{Hom}}_{A}(N,\bigoplus_{i\in
I}X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm\underline{Hom}}_{A}(M,\bigoplus_{i\in
I}X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm\underline{Hom}}_{A}(M,\bigoplus_{i\in
I}M_{i}).}$

Thus the first vertical map is an isomorphism. $\square$

######  Theorem 4.14.

$(1)$ The object $N$ belongs to $(\mathscr{E}\cap\mathscr{S})/[M]$ and is
compact in $({{}^{\bot>0}}M)/[M]$. In particular, $N$ is compact in
$\mathscr{C}$.

$(2)$ For each $n\in\mathbb{Z}$, there exists a natural isomorphism of
homological functors:

$H^{n}\big{(}{\rm\underline{Hom}}_{A}(M,M^{\bullet}_{-})\big{)}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}{\rm\underline{Hom}}_{M}\big{(}\Omega_{M}^{n}(N),-\big{)}:\;\;\mathscr{C}\longrightarrow\Gamma\mbox{{\rm-
Mod}}.$

Proof $(1)$ By Lemma 4.2, $N\in(\mathscr{G}\cap\mathscr{S})/[M]$. Note that
${\rm\underline{Hom}}_{A}(M[1],N)=0$ by Proposition 4.1 and
${\rm\underline{Hom}}_{A}(M,N)\simeq{\rm\underline{Hom}}_{A}(M,M)$ by Lemma
4.7(1). This shows $N\in\mathscr{E}$. Since ${{}^{\bot>0}}M$ contains ${\rm
Add}(M)$ and is closed under direct sums in $A\mbox{{\rm-Mod}}$, we see that
${{}^{\bot>0}}M$, as an additive category, has coproducts. Moreover, by Lemma
2.1(2), the quotient functor
$\underline{{{}^{\bot>0}}M}\to({{}^{\bot>0}}M)/[M]$ preserves coproducts and
compact objects. Now $(1)$ follows from Corollary 4.13.

$(2)$ Let $M^{\bullet}_{X}:=(M_{X}^{n},d_{X}^{n})_{n\in\mathbb{Z}}$ be a
complete ${\rm Add}(M)$-resolution of $X$. Then $X\simeq{\rm
Coker}(d_{X}^{-1})$ and ${\rm Hom}_{A}(M,M^{\bullet}_{X})$ is acyclic.
Clearly, there is a canonical ring homomorphism
$\Lambda\twoheadrightarrow\Gamma$ and the functor
${\rm\underline{Hom}}_{A}(M,-):A\mbox{{\rm-Mod}}\to\Gamma\mbox{{\rm-Mod}}$ is
naturally isomorphic to the composition of $G$ with
$\Gamma\otimes_{\Lambda}-$. Then
${\rm\underline{Hom}}_{A}(M,M^{\bullet}_{X})\simeq\Gamma\otimes_{\Lambda}{\rm
Hom}_{A}(M,M^{\bullet}_{X})$ as complexes. Since $\Gamma\otimes_{\Lambda}-$ is
right exact, the sequence

${\rm\underline{Hom}}_{A}(M,M^{n-1}_{X})\stackrel{{\scriptstyle(d_{X}^{n-1})^{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,M^{n}_{X})\longrightarrow{\rm\underline{Hom}}_{A}(M,\Omega_{M}^{-n}(X))\longrightarrow
0$

is exact for all $n$. As the inclusion $\lambda_{n}:\Omega_{M}^{-n}(X)\to
M^{n+1}_{X}$ is a left ${\rm Add}(M)$-approximation of $\Omega_{M}^{-n}(X)$,
it follows from Lemma 4.12 that the sequence

$0\longrightarrow{\rm\underline{Hom}}_{A}(N,\Omega_{M}^{-n}(X))\stackrel{{\scriptstyle(\tau_{M})_{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,\Omega_{M}^{-n}(X))\stackrel{{\scriptstyle(\lambda_{n})^{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M,M_{X}^{n+1})$

is exact. Consequently,
$H^{n}({\rm\underline{Hom}}_{A}(M,M^{\bullet}_{X}))\simeq
H^{n}(\Gamma\otimes_{\Lambda}{\rm
Hom}_{A}(M,M^{\bullet}_{X}))\simeq{\rm\underline{Hom}}_{A}(N,\Omega_{M}^{-n}(X)).$
Since ${\rm\underline{Hom}}_{A}(N,M^{\prime})=0$ for any $M^{\prime}\in{\rm
Add}(M)$, we have

${\rm\underline{Hom}}_{A}(N,\Omega_{M}^{-n}(X))={\rm\underline{Hom}}_{M}(N,\Omega_{M}^{-n}(X))\simeq{\rm\underline{Hom}}_{M}(\Omega_{M}^{n}(N),X).$

This shows $(2)$.

###  4.3 Compact objects from left approximations

In this subsection, we characterize compact objects in $\mathscr{C}$ in terms
of $M$-filtered modules (see Proposition 4.17) and show that all objects of
$(\mathscr{E}\cap\mathscr{S})/[M]$ are compact in $\mathscr{C}$. We then
establish a connection between $(\mathscr{G}\cap\mathscr{S})/[M]$ and
$(\mathscr{E}\cap\mathscr{S})/[M]$ by employing strong generators (see
Corollary 4.18).

As a preparation, we recall a construction of ${\bf L}(X)$ and ${\bf R}(X)$
from the proof of [13, Theorem III.2.3].

Let $\mathfrak{A}:=\\{\underline{M}[n]\mid n\geq 0\\}\subseteq\mathscr{X}$.
Denote by ${\rm Add}(\mathfrak{A})$ the full subcategory of
$A\mbox{{\rm-{\text@underline{Mod}}}}$ consisting of direct summands of
arbitrary direct sums of objects of $\mathfrak{A}$. For a full subcategory
$\mathscr{U}$ of $A\mbox{{\rm-{\text@underline{Mod}}}}$, we denote by
$\mathscr{U}^{\star n}=\mathscr{U}\star\mathscr{U}\star\cdots\star\mathscr{U}$
($n$-factors) the category of $n$-extensions of $\mathscr{U}$ by $\mathscr{U}$
in $A\mbox{{\rm-{\text@underline{Mod}}}}$.

Let $X\in A\mbox{{\rm-Mod}}$. We construct a right ${\rm
Add}(\mathfrak{A})$-approximation $f_{1}:Q_{1}\to X$ of $X$ as follows:
Consider the set $I_{X}$ of the union of $\underline{{\rm Hom}}_{A}(P,X)$ with
$P$ running over $\mathfrak{A}$, define $Q_{1}=\bigoplus_{\lambda\in
I_{X}}P_{\lambda}$ and take $f_{1}$ to be the morphism induced by $I_{X}$,
where $\lambda$ is morphism from $P_{\lambda}$ to $X$. Then we extend $f_{1}$
to a triangle $Q_{1}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}X\stackrel{{\scriptstyle
g_{1}}}{{\longrightarrow}}X_{1}\longrightarrow Q_{1}[1]$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. Now, we can repeat this construction
by replacing $X$ by $X_{1}\in A\mbox{{\rm-Mod}}$. In general, for each $n\geq
0$, we can inductively construct a triangle

$Q_{n+1}\stackrel{{\scriptstyle
f_{n+1}}}{{\longrightarrow}}X_{n}\stackrel{{\scriptstyle
g_{n+1}}}{{\longrightarrow}}X_{n+1}\longrightarrow Q_{n+1}[1]$

in $A\mbox{{\rm-{\text@underline{Mod}}}}$ such that $f_{n+1}$ is a right ${\rm
Add}(\mathfrak{A})$-approximation of $X_{n}$ with $X_{0}:=X$. Setting
$T_{1}:=Q_{1}$ and $h_{1}:=f_{1}$, we then construct inductively a tower of
objects
$T_{1}\stackrel{{\scriptstyle\tau_{1}}}{{\longrightarrow}}T_{2}\stackrel{{\scriptstyle\tau_{2}}}{{\longrightarrow}}T_{3}\longrightarrow\cdots$
which is embedded into the following tower of triangles in
$A\mbox{{\rm-{\text@underline{Mod}}}}$:

$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{\tau_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{T_{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{1}[1]}$$\textstyle{(\star\star)}$$\textstyle{T_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\scriptstyle{\tau_{2}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}g_{2}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{3}}$$\textstyle{T_{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}[1]}$$\textstyle{T_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{3}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}g_{2}g_{3}}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{3}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$

Applying the Octahedral Axiom of triangulated categories for each $n$ yields a
series of triangles

$(\diamond\diamond)\quad
T_{n}\stackrel{{\scriptstyle\tau_{n}}}{{\longrightarrow}}T_{n+1}\stackrel{{\scriptstyle\sigma_{n}}}{{\longrightarrow}}Q_{n+1}\longrightarrow
T_{n}[1].$

This implies $T_{n}\in{\rm Add}(\mathfrak{A})^{\star n}\subseteq\mathscr{X}$
for $n\geq 1$.

Let $\underrightarrow{{\rm Hocolim}}(T_{n})$ be the _homotopy colimit_ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ of the tower of objects

$T_{1}\stackrel{{\scriptstyle\tau_{1}}}{{\longrightarrow}}T_{2}\stackrel{{\scriptstyle\tau_{2}}}{{\longrightarrow}}T_{3}\stackrel{{\scriptstyle\tau_{3}}}{{\longrightarrow}}\cdots\longrightarrow
T_{n}\stackrel{{\scriptstyle\tau_{n}}}{{\longrightarrow}}T_{n+1}\longrightarrow\cdots$

defined by the triangle

$(\sharp)\quad\quad\bigoplus_{n\geq
1}T_{n}\stackrel{{\scriptstyle(1-\tau_{*})}}{{\longrightarrow}}\bigoplus_{n\geq
1}T_{n}\longrightarrow\underrightarrow{{\rm
Hocolim}}(T_{n})\longrightarrow\bigoplus_{n\geq 1}T_{n}[1]$

where the morphism $(1-\tau_{*})$ is induced by $({\rm
Id}_{T_{n}},-\tau_{n}):T_{n}\to T_{n}\oplus
T_{n+1\hookrightarrow}\bigoplus_{n\geq 1}T_{n}$. Now, we choose $\tau_{n}$ as
a representative in $A\mbox{{\rm-Mod}}$ and denote by
$\lim\limits_{\longrightarrow}T_{n}$ the colimit of the direct system
$\\{(T_{n},\tau_{n})\mid n\geq 1\\}$ of $A$-modules. Then there is a short
exact sequence

$0\to\bigoplus_{n\geq
1}T_{n}\stackrel{{\scriptstyle(1-\tau_{*})}}{{\longrightarrow}}\bigoplus_{n\geq
1}T_{n}\longrightarrow\lim\limits_{\longrightarrow}T_{n}\longrightarrow 0$

which induces a canonical triangle in $A\mbox{{\rm-{\text@underline{Mod}}}}$:

$\bigoplus_{n\geq
1}T_{n}\stackrel{{\scriptstyle(1-\tau_{*})}}{{\longrightarrow}}\bigoplus_{n\geq
1}T_{n}\longrightarrow\lim\limits_{\longrightarrow}T_{n}\longrightarrow\bigoplus_{n\geq
1}T_{n}[1].$

This implies that $\underrightarrow{{\rm
Hocolim}}(T_{n})\simeq\lim\limits_{\longrightarrow}T_{n}$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. Since the homotopy colimit of a tower
of triangles in $A\mbox{{\rm-{\text@underline{Mod}}}}$ is a triangle (for
example, see [35, Lemma 4.2]), there is a triangle in
$A\mbox{{\rm-{\text@underline{Mod}}}}$:

$({\ddagger})\quad\quad\underrightarrow{{\rm Hocolim}}(T_{n})\longrightarrow
X\longrightarrow\underrightarrow{{\rm
Hocolim}}(X_{n})\longrightarrow\underrightarrow{{\rm Hocolim}}(T_{n})[1].$

By the proof of [13, Theorem III.2.3 and Remark III.2.7], we can show
$\underrightarrow{{\rm Hocolim}}(T_{n})\in\mathscr{X}$ and
$\underrightarrow{{\rm Hocolim}}(X_{n})\in\mathscr{Y}$. Since
$(\mathscr{X},\mathscr{Y})$ is a torsion pair in
$A\mbox{{\rm-{\text@underline{Mod}}}}$, there are isomorphisms

${\bf R}(X)\simeq\underrightarrow{{\rm Hocolim}}(T_{n})\;\;\mbox{and}\;\;{\bf
L}(X)\simeq\underrightarrow{{\rm Hocolim}}(X_{n}).$

Recall that $T:\mathscr{D}\to\mathscr{C}$ stands for the left adjoint of the
inclusion $\mathscr{C}\to\mathscr{D}$ (see Lemma 3.20(3)) and $N$ is defined
to be $H_{*}^{0}(M)$.

######  Lemma 4.15.

$(1)$ If $X\in{{}^{\bot>0}M}$, then ${\bf R}(X)$ lies in
$\underline{\mathscr{G}\cap\mathscr{S}}$ and is isomorphic in $\mathscr{C}$ to
$T(T_{3})$.

$(2)$ If $X\in\mathscr{G}$, then ${\bf L}(X)\in\underline{M}^{\bot}$. If
further $X\in\mathscr{E}$, then $T_{3}\in
A\mbox{{\rm-{\text@underline{mod}}}}$.

$(3)$ $\Omega_{M}^{-}(N)\simeq\Omega_{A}^{-}(N)\simeq T(\Omega_{A}^{-}(M))$ in
$\mathscr{C}$.

Proof. $(1)$ Recall from Section 4.2 that
$\mathscr{X}=\underline{M}^{\bot>0}\cap\underline{\mathscr{S}}$ and
$\mathscr{Y}={{}^{\bot\geq-1}}\underline{M}\subseteq{{}^{\bot>0}}\underline{M}=\underline{{}^{\bot>0}M}$.
Let $X\in{{}^{\bot>0}M}$. Since there is a triangle ${\bf R}(X)\to X\to{\bf
L}(X)\to{\bf R}(X)[1]$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$ with ${\bf
R}(X)\in\mathscr{X}$ and ${\bf L}(X)\in\mathscr{Y}$, we see that ${\bf R}(X)$
lies in $\underline{{}^{\bot>0}M}$, and therefore in
$\underline{\mathscr{G}\cap\mathscr{S}}$.

Set
$\Lambda_{n}:=\\{i\in\mathbb{N}\mid{\rm\underline{Hom}}_{A}(M[i],X_{n})\neq
0\\}$ for $n\geq 0$. Since ${\rm add}({{}_{A}}M)={\rm add}(\nu_{A}(M))$, it is
clear that $\underline{{{}^{\bot>0}}M}=\underline{M}^{\bot\leq-2}$ by
$(\lozenge)$. This implies $\Lambda_{0}\subseteq\\{0,1\\}$. So we can choose
$Q_{1}=M^{(I_{1,0})}\oplus M[1]^{(I_{1,1})}$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ for some index sets $I_{1,0}$ and
$I_{1,1}$. For a natural number $n$, we apply
${\rm\underline{Hom}}_{A}(M[i],-)$ for $i\geq 0$ to the triangle

$Q_{n+1}\stackrel{{\scriptstyle
f_{n+1}}}{{\longrightarrow}}X_{n}\stackrel{{\scriptstyle
g_{n+1}}}{{\longrightarrow}}X_{n+1}\longrightarrow Q_{n+1}[1]$

in<EMAIL_ADDRESS>This yields an exact sequence of
abelian groups:

${\rm\underline{Hom}}_{A}(M[i],Q_{n+1})\stackrel{{\scriptstyle(f_{n+1})^{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M[i],X_{n})\stackrel{{\scriptstyle(g_{n+1})^{*}}}{{\longrightarrow}}{\rm\underline{Hom}}_{A}(M[i],X_{n+1})\longrightarrow{\rm\underline{Hom}}_{A}(M[i],Q_{n+1}[1]).$

Since $f_{n+1}$ is a right ${\rm Add}(\mathfrak{A})$-approximation of $X_{n}$,
the map $(f_{n+1})^{*}$ is always surjective, and therefore there is an
injection
${\rm\underline{Hom}}_{A}(M[i],X_{n+1})\hookrightarrow{\rm\underline{Hom}}_{A}(M[i],Q_{n+1}[1])$
for $i\geq 0$. Using the fact ${\rm
Add}(\underline{M})\subseteq\underline{\mathscr{G}}=\underline{M}^{\bot\neq
0,-1}$, we then can show $\Lambda_{n}\subseteq\\{i\in\mathbb{N}\mid n\leq
i\leq 2n+1\\}$ by induction on $n$ and choose
$Q_{n+1}=\bigoplus_{j\in\Lambda_{n}}M[j]^{(I_{n+1,j})}$ for some index sets
$I_{n+1,j}$.

Consider $n\geq 3$. Then ${\rm\underline{Hom}}_{A}(Q_{n+1},M[1])=0$. It
follows from ($\diamond\diamond$) and Lemma 3.16(2) that there is a triangle
$T_{n}\to T_{n+1}\to Q_{n+1}\to\Omega_{M}^{-}(T_{n})$ in $\mathscr{D}$. We
apply the functor $T$ to this triangle and produce another triangle
$T(T_{n})\to T(T_{n+1})\to T(Q_{n+1})\to\Omega_{M}^{-}(T(T_{n}))$ in
$\mathscr{C}$. Since ${\rm\underline{Hom}}_{A}(M[m],\mathscr{G})=0$ for any
$m\geq 2$, the minimal injective envelope $M[m]\to I^{m}$ of $M[i]$ is a left
$\mathscr{G}$-approximation of $M[m]$. This means $T(M[m])\simeq I^{m}=0$ in
$\mathscr{C}$ for any $m\geq 2$. Since $T$ commutes with direct sums, we have
$T(Q_{n+1})\simeq\bigoplus_{j\in\Lambda_{n}}T(M[j])^{(I_{n+1,j})}=0$. Thus
$T(T_{n})\simeq T(T_{n+1})$. Consequently, the homotopy colimit
$\underrightarrow{{\rm Hocolim}}(T(T_{n}))$ in $\mathscr{C}$ of
$\\{(T(T_{n}),T(\tau_{n})\mid n\geq 1\\}$ is isomorphic to $T(T_{3})$.
Moreover, due to ${\bf R}(X)\in{{}^{\bot 1}M}$, we see from Lemma 3.16(2) that
${\bf R}(X)$ is also the homotopy colimit in $\mathscr{D}$ of $\\{T_{n}\\}$.
Since ${\bf R}(X)\in\mathscr{G}$ and $T$ commutes with homotopy colimits by
Lemma 3.20(3), there are isomorphisms ${\bf R}(X)\simeq T({\bf
R}(X))\simeq\underrightarrow{{\rm Hocolim}}(T(T_{n}))\simeq T(T_{3})$ in
$\mathscr{C}$.

$(2)$ If $X\in\mathscr{G}$, then ${\bf L}(X)\in\underline{M}^{\bot>0}$. This
implies ${\bf L}(X)\in\underline{M}^{\bot}$ because ${\bf
L}(X)\in\mathscr{Y}=\underline{M}^{\bot\leq 0}$. Let

$\mathscr{A}:=\\{U\in
A\mbox{{\rm-{\text@underline{Mod}}}}\mid\bigoplus_{n=0}^{\infty}{\rm\underline{Hom}}_{A}(M[n],U)\in\Gamma\mbox{{\rm-
mod}}\\}.$

Since $\Gamma$ is an Artin algebra over a commutative Artin ring $R$, a
$\Gamma$-module $N$ is finitely generated if and only if $N$ is a finitely
generated $R$-module. This implies that an $A$-module $U$ lies in
$\mathscr{A}$ if and only if
$\bigoplus_{n=0}^{\infty}{\rm\underline{Hom}}_{A}(M[n],U)\in R\mbox{{\rm-
mod}}$. The latter is equivalent to saying that there is a non-negative
integer $\delta_{U}$ such that ${\rm\underline{Hom}}_{A}(M[n],U)=0$ for
$n>\delta_{U}$ and ${\rm\underline{Hom}}_{A}(M[n],U)\in R\mbox{{\rm-mod}}$ for
$0\leq n\leq\delta_{U}$. Moreover, $\mathscr{A}$ is closed under direct
summands, finite direct sums, the shift $[-1]$ and extensions of triangles in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. Now, it follows from
$M\in\underline{M}^{\bot\neq 0,-1}$ and ${\rm Hom}_{A}(M[1],M)\simeq
D{\rm\underline{Hom}}_{A}(M,\nu_{A}(M))$ that $M[j]\in\mathscr{A}$ for any
$j\in\mathbb{Z}$. Consequently, for $U\in\mathscr{A}$, there is a right ${\rm
Add}(\mathfrak{A})$-approximation $f_{U}:Q_{U}\to U$ of $U$ such that
$Q_{U}\in{\rm add}(\bigoplus_{n=0}^{m}M[n])$ for some $m\geq 0$ and the third
term $C_{U}$ of the triangle $Q_{U}\stackrel{{\scriptstyle
f_{U}}}{{\longrightarrow}}U\to C_{U}\to Q_{U}[1]$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ still lies in $\mathscr{A}$.

By Lemma 4.2, there holds $\underline{\mathscr{E}}\subseteq\mathscr{A}$. If
$X\in\mathscr{E}$, then $I_{n+1,j}$ can be chosen to be finite sets and
$Q_{n+1}\in{\rm add}(\bigoplus_{j\in\Lambda_{n}}M[j])\subseteq
A\mbox{{\rm-{\text@underline{mod}}}}$.

$(3)$ In the proof of $(1)$, we take $X=N$. Then $X\in\mathscr{E}$ by Theorem
4.14(1). Recall that $X\in\mathscr{H}\subseteq\underline{M}^{\bot\neq 0}$ by
Proposition 4.1, and
${\rm\underline{Hom}}_{A}(M,M)\simeq{\rm\underline{Hom}}_{A}(M,X)$ by Lemma
4.7(1). Further, we even have

$(a)$ $\Lambda_{0}\subseteq\\{0\\}$, $Q_{1}=M$ and $X_{1}[-2]\in\mathscr{H}$
(see the proof of Lemma 4.10); and

$(b)$ $\Lambda_{n}\subseteq\\{i\in\mathbb{N}\mid n+1\leq i\leq 2n\\}$ and
$Q_{n+1}\in{\rm add}(\bigoplus_{i=n+1}^{2n}M[i])\in
A\mbox{{\rm-{\text@underline{mod}}}}$ for $n\geq 1$.

Since $M$ lies in $\underline{\mathscr{G}}=\underline{M}^{\bot\neq 0,-1}$,
there holds ${\rm\underline{Hom}}_{A}(Q_{3},M[1])=0$. By Lemma 3.16(2), there
exists a right triangle
$T_{2}\stackrel{{\scriptstyle\tau_{2}}}{{\longrightarrow}}T_{3}\stackrel{{\scriptstyle\sigma_{2}}}{{\longrightarrow}}Q_{3}\rightarrow\Omega_{M}^{-}(T_{2})$
in $\mathscr{D}$. This gives rise to another triangle $T(T_{2})\to T(T_{3})\to
T(Q_{3})\to\Omega_{M}^{-}(T(T_{2}))$ in $\mathscr{C}$. Clearly, $T(Q_{3})=0$
by $Q_{3}\in{\rm add}(M[3]\oplus M[4])$. Thus $T(T_{2})\simeq T(T_{3})$. It
then follows from (1) and $X\in\mathscr{H}\subseteq\mathscr{X}$ that
$X\simeq{\bf R}(X)\simeq T(T_{2})$ in $\mathscr{C}$.

By Lemma 4.10, $X[1]=\Omega_{A}^{-}(X)\simeq\Omega_{M}^{-}(X)$ in
$\mathscr{C}$. Further, we will show
$T_{2}[1]=\Omega_{A}^{-}(T_{2})\simeq\Omega_{M}^{-}(T_{2})$ in $\mathscr{C}$.
Actually, it suffices to prove that any homomorphism from $T_{2}$ to a module
in ${\rm Add}(M)$ factorizes through the injective envelope of $T_{2}$, or
equivalently, ${\rm\underline{Hom}}_{A}(T_{2},M)=0$. Thanks to ${\rm
add}(M)={\rm add}(\nu_{A}(M))$, we will show
${\rm\underline{Hom}}_{A}(T_{2},\nu_{A}(M))=0$.

Indeed, ${\rm\underline{Hom}}_{A}(T_{2},\nu_{A}(M))\simeq
D{\rm\underline{Hom}}_{A}(M[1],T_{2})$ by $(\lozenge)$. We apply
${\rm\underline{Hom}}_{A}(M[1],-)$ to the triangle $X_{2}[-1]\to T_{2}\to X\to
X_{2}$ (see ($\star\star$)) and obtain an isomorphism
${\rm\underline{Hom}}_{A}(M[1],X_{2}[-1])\simeq{\rm\underline{Hom}}_{A}(M[1],T_{2})$.
Here we use the fact $X\in\mathscr{H}\subseteq\underline{M}^{\bot\neq 0}$. So,
it is enough to show ${\rm\underline{Hom}}_{A}(M[1],X_{2}[-1])=0$. Recall that
the triangle $Q_{2}\stackrel{{\scriptstyle f_{2}}}{{\longrightarrow}}X_{1}\to
X_{2}\to Q_{2}[1]$ has the following properties: $Q_{2}\in{\rm add}(M[2])$ by
$(b)$, $X_{1}\in\mathscr{H}[2]$ by $(a)$, and there exists an injection
${\rm\underline{Hom}}_{A}(M[2],X_{2})\hookrightarrow{\rm\underline{Hom}}_{A}(M[2],Q_{2}[1])$
by the proof of $(1)$. It follows from ${\rm\underline{Hom}}_{A}(M,M[1])=0$
that
${\rm\underline{Hom}}_{A}(M[1],X_{2}[-1])\simeq{\rm\underline{Hom}}_{A}(M[2],X_{2})=0$.
Thus ${\rm\underline{Hom}}_{A}(M[1],T_{2})=0$, and therefore
${\rm\underline{Hom}}_{A}(T_{2},\nu_{A}(M))=0$.

By the facts that $Q_{2}\in{\rm add}(M[2])$ and
${\rm\underline{Hom}}_{A}(M[2],M)=0$, we have

${\rm
Ext}_{A}^{1}(Q_{2}[1],M)\simeq{\rm\underline{Hom}}_{A}(Q_{2}[1],M[1])\simeq{\rm\underline{Hom}}_{A}(Q_{2},M)=0.$

By Lemma 3.16(2), the triangle
$M[1]\stackrel{{\scriptstyle\tau_{1}[1]}}{{\longrightarrow}}T_{2}[1]\stackrel{{\scriptstyle\sigma_{1}[1]}}{{\longrightarrow}}Q_{2}[1]\longrightarrow
M[2]$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$ can be extended to a right
triangle
$M[1]\stackrel{{\scriptstyle\tau_{1}[1]}}{{\longrightarrow}}T_{2}[1]\stackrel{{\scriptstyle\sigma_{1}[1]}}{{\longrightarrow}}Q_{2}[1]\longrightarrow\Omega_{M}^{-}(M[1])$
in $\mathscr{D}$. This yields the triangle in $\mathscr{C}$

$T(M[1])\stackrel{{\scriptstyle
T(\tau_{1}[1])}}{{\longrightarrow}}T(T_{2}[1])\stackrel{{\scriptstyle
T(\sigma_{1}[1])}}{{\longrightarrow}}T(Q_{2}[1])\longrightarrow\Omega_{M}^{-}(T(M[1]))$

by Lemma 3.20(3). It then follows from $T(M[3])=0$ that $T(Q_{2}[1])=0$ and
$T(M[1])\simeq T(T_{2}[1])$ in $\mathscr{C}$. Moreover, $T(T_{2}[1])\simeq
T(\Omega_{M}^{-}(T_{2}))\simeq\Omega_{M}^{-}(T(T_{2}))\simeq\Omega_{M}^{-}(X)$
in $\mathscr{C}$, where the second isomorphism is due to the fact that $T$
commutes with $\Omega_{M}^{-}$. Thus
$T(\Omega_{A}^{-}(M))=T(M[1])\simeq\Omega_{M}^{-}(X)$ in $\mathscr{C}$.
$\square$

Now, we state a property of finitely $M$-filtered $A$-modules introduced in
Definition 1.1.

######  Lemma 4.16.

If $X$ is a finitely $M$-filtered $A$-module, then $X\in\mathscr{G}^{\bot>0}$
and has an ${\rm add}(M)$-resolution of finite length. Moreover, if the
$A$-module $X$ lies in $\mathscr{G}$, then $X\in{\rm add}(M)$.

Proof. Since $\mathscr{G}^{\bot>0}$ contains $M$ and is closed under
cosyzygies and extensions in $A\mbox{{\rm-Mod}}$, we have
$X\in\mathscr{G}^{\bot>0}$. Now, let $\mathscr{M}$ be the full subcategory of
$A\mbox{{\rm-mod}}$ consisting of all those modules having an ${\rm
add}(M)$-resolution of finite length. Then $Y\in\mathscr{M}$ if and only if
${\rm Hom}_{A}(M,Y)\in\mathscr{P}^{<\infty}(\Lambda)$. Since ${{}_{A}}M$ is a
self-orthogonal generator, $Y\in\mathscr{M}$ if and only if there is an exact
sequence $0\to M_{n}\to M_{n-1}\to\cdots\to M_{0}\to Y\to 0$ in $A\mbox{{\rm-
mod}}$ for some $n\in\mathbb{N}$ such that $M_{j}\in{\rm add}(M)$ for all
$0\leq j\leq n$. Clearly, $\mathscr{M}\subseteq M^{\bot>0}\subseteq M^{\bot
1}$. As $\mathscr{P}^{<\infty}(\Lambda)$ is always closed under extensions in
$\Lambda\mbox{{\rm-mod}}$, $\mathscr{M}$ is closed under extensions in
$A\mbox{{\rm-mod}}$. So, to show that the finitely $M$-filtered module $X$
belongs to $\mathscr{M}$, it suffices to show
$\Omega_{A}^{-i}(M)\in\mathscr{M}$ for each $i\geq 0$. However, this follows
from the exact sequence $0\to M\to I^{0}\to\cdots\to
I^{i-1}\to\Omega_{A}^{-i}(M)\to 0$, where $I^{j}$ is injective and therefore
in ${\rm add}(M)$ for $0\leq j\leq i-1$. As to the last statement in the
lemma, we notice $\mathscr{M}\cap{{}^{\bot>0}}M={\rm add}(M)$,
$\mathscr{G}\subseteq{{}^{\bot>0}}M$ and $\mathscr{G}\cap\mathscr{M}={\rm
add}(M)$. $\square$

In the following, we describe zero objects and compact objects in
$\mathscr{C}$. This is related to pure-projective modules. It is known that a
module over an Artin algebra is _pure-projective_ if and only if it is a
direct sum of finitely generated, indecomposable modules.

######  Proposition 4.17.

The following hold for $X\in\mathscr{E}\cap\mathscr{S}$.

$(1)$ $X$ is compact in $\mathscr{C}$ and isomorphic in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ to an $M$-filtered module.

$(2)$ If $X^{\bot 1}$ is closed under direct sums of countably many finitely
$M$-filtered $A$-modules in $A\mbox{{\rm-Mod}}$, then $X\in{\rm Add}(M)$. In
particular, if $X$ is pure-projective, then $X\in{\rm Add}(M)$.

$(3)$ If $X\in\mathscr{H}\cap A\mbox{{\rm-{\text@underline{mod}}}}$, then $X$
is projective.

Proof. Let $X\in\mathscr{E}\cap\mathscr{S}$. We keep all notations in the
proof of Lemmas 4.15(1)-(2).

$(1)$ By Lemma 4.2, $X\in\mathscr{X}$ and $X\simeq{\bf R}(X)$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. Moreover, by Lemmas 4.15(1)-(2), ${\bf
R}(X)\simeq T(T_{3})$ in $\mathscr{C}$ with $T_{3}\in
A\mbox{{\rm-{\text@underline{mod}}}}$. Each object of
$A\mbox{{\rm-{\text@underline{mod}}}}$ is compact in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ and the quotient functor
$A\mbox{{\rm-{\text@underline{Mod}}}}\to\mathscr{D}$ preserves compact objects
by Lemma 2.1(2). Hence $T_{3}$ is compact in $\mathscr{D}$. As $T$ preserves
compact objects by Lemma 3.20(3), both $T(T_{3})$ and $X$ are compact in
$\mathscr{C}$.

It follows from $X\in\mathscr{E}$ that the proofs of Lemmas 4.15(1)-(2) yield
$Q_{n+1}=\bigoplus_{j\in\Lambda_{n}}M[j]^{(I_{n+1,j})}$, where
$\Lambda_{n}\subseteq\\{i\in\mathbb{N}\mid n\leq i\leq 2n+1\\}$,
$M[j]=\Omega_{A}^{-j}(M)$ and $I_{n+1,j}$ are finite sets. This implies
$T_{n}\in A\mbox{{\rm-{\text@underline{mod}}}}$ for $n\geq 1$. Since triangles
in $A\mbox{{\rm-{\text@underline{mod}}}}$ (up to isomorphism) are induced from
short exact sequences, we can add finitely generated projective modules to
$T_{n}$ and assume that the associated map $\tau_{n}:T_{n}\to T_{n+1}$ is
injective and ${\rm Coker}(\tau_{n})$ is isomorphic to a finite direct sum of
modules in $\\{A\\}\cup\\{\Omega_{A}^{-i}(M)\mid i\in\mathbb{N}\\}$. Now, let
$X^{\prime}$ be the colimit of the direct system $\\{(T_{n},\tau_{n})\mid
n\geq 1\\}$ in $A\mbox{{\rm-Mod}}$. Then the canonical map $T_{n}\to
X^{\prime}$ is injective. Thus $T_{n}$ can be regarded as a submodule of
$X^{\prime}$ and $T_{n}\subseteq T_{n+1}$. Moreover,
$X^{\prime}=\bigcup_{n=0}^{\infty}T_{n}$ and is $M$-filtered. Now (1) follows
from $X\simeq{\bf R}(X)\simeq X^{\prime}$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$.

$(2)$ Since all $T_{n}$ are finitely $M$-filtered and $X\in\mathscr{G}$, we
have $T_{n}\in X^{\bot 1}$ by Lemma 4.16. Let $Z:=\bigoplus_{n\geq 1}T_{n}$.
The assumption of $(2)$ implies ${\rm Ext}_{A}^{1}(X,Z)=0$. So the exact
sequence $0\to Z\stackrel{{\scriptstyle(1-\tau_{*})}}{{\longrightarrow}}Z\to
X^{\prime}\to 0$ induced by $\\{\tau_{n}\mid n\geq 1\\}$ splits, and therefore
$X^{\prime}$ is isomorphic to a direct summand of $Z$. Since $T_{n}$ is a
finitely generated $A$-module, it is a direct sum of finitely many
indecomposable submodules with the local endomorphism rings. It follows from
[1, Corollary 26.6] that $X^{\prime}\simeq\bigoplus_{n\geq 1}T_{n}^{\prime}$,
where $T_{n}^{\prime}$ is a direct summand of $T_{n}$. Moreover, by Lemma
4.16, $T_{n}$ has an ${\rm add}(M)$-resolution of finite length, and so does
$T_{n}^{\prime}$. It follows from $X\simeq X^{\prime}$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ and $X\in\mathscr{G}$ that
$X^{\prime}\in\mathscr{G}$, and therefore $T_{n}^{\prime}\in\mathscr{G}$.
Consequently, $T_{n}^{\prime}\in{\rm add}(M)$ by the proof of Lemma 4.16.
Hence $X^{\prime}\in{\rm Add}(M)$ and $X\in{\rm Add}(M)$.

For any $Z\in A\mbox{{\rm-mod}}$, $Z$ is clearly finite presented. So $Z^{\bot
1}$ is always closed under arbitrary direct sums in $A\mbox{{\rm-Mod}}$. This
implies that if $X$ is pure-projective, then $X^{\bot 1}$ is closed under
arbitrary direct sums in $A\mbox{{\rm-Mod}}$, and therefore $X\in{\rm
Add}(M)$.

$(3)$ By Lemmas 4.2(1) and 4.4(1), $\mathscr{H}\cap
A\mbox{{\rm-{\text@underline{mod}}}}\subseteq\underline{\mathscr{E}\cap\mathscr{S}}$.
It follows from $X\in A\mbox{{\rm-{\text@underline{mod}}}}$ that $X^{\bot 1}$
is closed under arbitrary direct sums in $A\mbox{{\rm-Mod}}$. Further,
$X\in{\rm Add}(M)$ by $(2)$, and ${\rm\underline{Hom}}_{A}(N,M)=0$ by Lemma
4.10. Due to ${\rm Add}(M)={\rm Prod}(M)$, we have
${\rm\underline{Hom}}_{A}(N,X)=0$. It follows from Lemma 4.7(3) and
$X\in\mathscr{H}$ that $X\simeq 0$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$.
Thus $X$ is projective. $\square$

Now, we describe compact generators of the right-hand side in the recollement
of Theorem 1.5.

Let $\mathcal{T}$ be a triangulated category. For a full subcategory
$\mathcal{U}$ of $\mathcal{T}$, we denote by $\langle\mathcal{U}\rangle$ the
smallest full subcategory of $\mathcal{T}$ containing $\mathcal{U}$ and being
closed under finite coproducts, direct summands and shifts. Let $\mathcal{V}$
be another full subcategory of $\mathcal{T}$. Define
$\mathcal{U}\star\mathcal{V}$ to be the full subcategory of $\mathcal{T}$
consisting of objects $X$ such that there is a triangle $U\to X\to V\to U[1]$
with $U\in\mathcal{U}$ and $V\in\mathcal{V}$. Following [37, 3.1], we set
$\mathcal{U}\diamond\mathcal{V}:=\langle\mathcal{U}\star\mathcal{V}\rangle$,
and then define $\langle\mathcal{U}\rangle_{0}:=0$ and
$\langle\mathcal{U}\rangle_{n+1}:=\langle\mathcal{U}\rangle_{n}\diamond\langle\mathcal{U}\rangle$
for $n\geq 0$ inductively. Clearly, the objects of
$\langle\mathcal{U}\rangle_{n+1}$ are the _direct summands_ of the objects
obtained by taking $(n+1)$-fold extensions of finite direct sums of shifts of
objects of $\mathcal{U}$. Following [37, Definition 3.1], the _dimension_ of
$\mathcal{T}$, denoted by $\dim(\mathcal{T})$, is defined to be the minimal
natural number $n$ such that there is an object $X\in\mathcal{T}$ such that
$\mathcal{T}=\langle X\rangle_{n+1}$. If no such $X$ exists, one defines
$\dim(\mathcal{T})=\infty$. If $\dim(\mathcal{T})$ is finite, then such an
object $X$ is called a _strong generator_ of $\mathcal{T}$.

For a pair of integers $i\leq j$, we denote by
$\langle\mathcal{U}\rangle_{n+1}^{[i,j]}$ the full subcategory of
$\langle\mathcal{U}\rangle_{n+1}$ consisting of all objects which are obtained
by taking $(n+1)$-fold extensions of finite direct sums of objects in the
class $\\{U[-s]\mid U\in\mathcal{U},s\in\mathbb{Z},i\leq s\leq j\\}$. Here, we
do not require taking both direct summands and arbitrary shifts in
$\langle\mathcal{U}\rangle_{n+1}^{[i,j]}$.

######  Corollary 4.18.

$(1)$ Let $\mathcal{S}$ be the finite set of isomorphism classes of simple
objects of $\mathscr{H}$, and let $n$ be the Loewy length of the algebra
$\Gamma$. Then $(\mathscr{G}\cap\mathscr{S})/[M]$ is a compactly generated
triangulated category:

$\big{(}\mathscr{G}\cap\mathscr{S}\big{)}/[M]=\langle{\rm
Add}(\mathcal{S})\rangle_{2n}^{[-1,0]}\;\;\mbox{and}\;\;\big{(}(\mathscr{G}\cap\mathscr{S}\big{)}/[M])^{\rm
c}=(\mathscr{E}\cap\mathscr{S})/[M]=\langle\mathcal{S}\rangle_{2n}^{[-1,0]}.$

$(2)$ If $A$ is symmetric and $\Gamma$ is semisimple, then there are triangle
equivalences

$(\mathscr{G}\cap\mathscr{S})/[M]={\rm
Add}\big{(}N\oplus\Omega_{A}^{-}(N)\big{)}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\Gamma\times\Gamma)\mbox{{\rm-
Mod}},$ $(\mathscr{E}\cap\mathscr{S})/[M]={\rm
add}\big{(}N\oplus\Omega_{A}^{-}(N)\big{)}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\Gamma\times\Gamma)\mbox{{\rm-
mod}},$

where $(\Gamma\times\Gamma)\mbox{{\rm-Mod}}$, as a triangulated category, has
the shift functor induced from the automorphism of the algebra
$\Gamma\times\Gamma$ by permutating the first and second coordinate.

$(3)$ If $T(X)\in\mathscr{E}/[M]$ for any $X\in A\mbox{{\rm-mod}}$, then the
recollement in Proposition 4.3 restricts to a half recollement of triangulated
categories:

$\textstyle{(\underline{M}^{\bot})^{\rm
c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi\circ\rm
inc}}$$\textstyle{\mathscr{C}^{\rm
c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Phi}}$$\scriptstyle{\widetilde{\Psi}}$$\textstyle{(\mathscr{E}\cap\mathscr{S})/[M].\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
inc}}$

Proof. $(1)$ Let $\mathscr{R}:=(\mathscr{G}\cap\mathscr{S})/[M]$. By Lemma
4.2(2)(b), we can identify $\mathscr{H}$ with its (essential) image in
$\mathscr{R}$ under the quotient functor
$\underline{\mathscr{G}\cap\mathscr{S}}\to\mathscr{R}$. It follows from the
first part of Corollary 4.8(3) that
$\mathscr{R}=\mathscr{H}[1]\star\mathscr{H}$, where $[1]$ denotes the functor
$\Omega_{M}^{-}$. To characterize $\mathscr{R}$ in terms of $\mathcal{S}$, we
use the functor ${\rm\underline{Hom}}_{A}(N,-):\mathscr{H}\to\Gamma\mbox{{\rm-
Mod}}$ which is an equivalence of abelian categories sending $N$ to $\Gamma$,
due to Lemma 4.7(2)(3). Recall that $\mathscr{H}^{\rm
fg}:=\\{X\in\mathscr{H}\mid{\rm\underline{Hom}}_{A}(M,X)\in\Gamma\mbox{{\rm-
mod}}\\}=\mathscr{H}\cap\underline{\mathscr{E}}$. By the proof of Corollary
4.8(1), the functor ${\rm\underline{Hom}}_{A}(N,-)$ restricts to an
equivalence: $\mathscr{H}^{\rm
fg}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Gamma\mbox{{\rm-mod}}$.
This equivalence clearly sends simple objects of $\mathscr{H}$ to simple
$\Gamma$-modules. Since $\Gamma$ is an Artin algebra, it has only finitely
many isomorphism classes of simple modules and each (respectively, finitely
generated) $\Gamma$-module is generated by simple modules under arbitrary
(respectively, finite) direct sums and taking $n$-fold extensions.
Consequently, $\mathcal{S}$ is a finite set and each object of $\mathscr{H}$
(respectively, $\mathscr{H}^{\rm fg}$) is generated by $\mathcal{S}$ under
arbitrary (respectively, finite) direct sums and taking $n$-fold extensions.
Note that each short exact sequence in $\mathscr{H}$ induces a triangle in
$\mathscr{R}$ by Lemma 3.16(2). Now, by
$\mathscr{R}=\mathscr{H}[1]\star\mathscr{H}$, the first equality in $(1)$
holds, and therefore $\mathscr{R}={\rm Loc}(\mathcal{S})$. Similarly, from the
equality $(\mathscr{E}\cap\mathscr{S})/[M]=\mathscr{H}^{\rm
fg}[1]\star\mathscr{H}^{\rm fg}$ by the second part of Corollary 4.8(3), we
see that $(\mathscr{E}\cap\mathscr{S})/[M]$ is generated by $\mathcal{S}$
under taking the shifts $[i]$ for $i=0,1$ and $(2n)$-fold extensions. This
implies the third equality in $(1)$. By Proposition 4.17(1), each object of
$\mathscr{H}^{\rm fg}$ is compact in $\mathscr{C}$ and thus also in
$\mathscr{R}$. Hence $\mathscr{R}$ is compactly generated. By [33, Theorem
4.4.9], we have $\mathscr{R}^{\rm c}={\rm thick}(\mathcal{S})$. Clearly, ${\rm
thick}(\mathcal{S})\subseteq(\mathscr{E}\cap\mathscr{S})/[M]=\langle\mathcal{S}\rangle_{2n}^{[-1,0]}\subseteq{\rm
thick}(\mathcal{S})$. Thus $\mathscr{R}^{\rm
c}=\langle\mathcal{S}\rangle_{2n}^{[-1,0]}$.

$(2)$ Suppose that $A$ is symmetric and $\Gamma$ is semisimple. Then $\Gamma$
is symmetric. By Corollary 4.11, $\Omega_{M}^{2}(N)\simeq N$ in $\mathscr{C}$.
Moreover, by Lemma 4.10, $\Omega_{M}^{-}(N)\simeq\Omega_{A}^{-}(N)$ in
$\mathscr{C}$. It follows that $\Omega_{M}^{2i}(N)\simeq N$ and
$\Omega_{M}^{2i+1}(N)\simeq\Omega_{A}^{-}(N)$ for any $i\in\mathbb{Z}$. Let
$W:=N\oplus\Omega_{A}^{-}(N)$ and $\mathscr{W}:={\rm Add}(W)$. Then
$\mathscr{W}=\Omega_{M}(\mathscr{W})$. Clearly, $\mathscr{W}$ contains $N$ and
is closed under direct sums in $\mathscr{C}$. As $\Gamma$ is semisimple, there
holds ${\rm add}(N)={\rm add}(\mathcal{S})\subseteq\mathscr{R}$. To show
$\mathscr{R}=\mathscr{W}$, it suffices to show that $\mathscr{W}$ is a
triangulated subcategory of $\mathscr{C}$. Since $\mathscr{W}$ is closed under
$\Omega_{M}$, we only need to show that, for any morphism $f:X_{1}\to X_{2}$
in $\mathscr{W}$ and triangle $X_{1}\stackrel{{\scriptstyle
f}}{{\rightarrow}}X_{2}\rightarrow X_{3}\rightarrow\Omega_{M}^{-}(X_{1})$ in
$\mathscr{C}$, the term $X_{3}$ belongs to $\mathscr{W}$.

Since $\mathscr{H}$ is the heart of the torsion pair
$(\mathscr{X},\mathscr{Y})$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$, ${\rm
Ext}_{\mathscr{H}}^{j}(U,V)\simeq{\rm\underline{Hom}}_{A}(U,V[j])$ for any
$U,V\in\mathscr{H}$ and $j=0,1$. Since $N$ is a projective object in
$\mathscr{H}$ by Lemma 4.7(3), we have
${\rm\underline{Hom}}_{A}(N,N[1])\simeq{\rm Ext}_{\mathscr{H}}^{1}(N,N)=0$.
Further, by Lemma 3.17 and [16, Lemma 3.5], if $V_{1}$ and $V_{2}$ lie in
$\mathscr{G}$, then ${\rm\underline{Hom}}_{A}(V_{1},V_{2}[n])$
$\simeq{\rm\underline{Hom}}_{M}(V_{1},\Omega_{M}^{-n}(V_{2}))$ for all $n\geq
1$. This implies
${\rm\underline{Hom}}_{A}(N,N[1])\simeq{\rm\underline{Hom}}_{M}(N,\Omega_{M}^{-}(N))$,
and therefore
${\rm\underline{Hom}}_{M}(N,\Omega_{M}^{-}(N))=0={\rm\underline{Hom}}_{M}(\Omega_{M}^{-}(N),N)$.
Let $B:={\rm\underline{End}}_{A}(W)$. Then $B\simeq\Gamma\oplus\Gamma$ as
algebras. Since $B$ is semisimple and $W$ is compact in $\mathscr{C}$ by Lemma
4.14(1), the functor ${\rm\underline{Hom}}_{M}(W,-):\mathscr{W}\to
B\mbox{{\rm-Mod}}$ is an additive equivalence. By this equivalence and the
fact that $B\mbox{{\rm-Mod}}$ is a semisimple abelian category, it can be
proved that $f$ as a morphism in $\mathscr{W}$ is isomorphic to a direct sum
of the identity map of $Z_{1}$ with the zero map $Z_{2}\to Z_{3}$, where
$Z_{i}\in\mathscr{W}$ for $1\leq i\leq 3$. Consequently,
$X_{3}\simeq\Omega_{M}^{-}(Z_{2})\oplus Z_{3}$. It then follows from
$\mathscr{W}=\Omega_{M}(\mathscr{W})$ that $X_{3}\in\mathscr{W}$. Thus
$\mathscr{W}$ is a triangulated subcategory of $\mathscr{C}$, and therefore
$\mathscr{R}=\mathscr{W}$. By $(1)$, $\mathscr{R}^{\rm c}={\rm add}(W)$.

$(3)$ By Lemma 3.20(4) and [33, Theorem 4.4.9], the assumption of Corollary
4.18(3) implies $\mathscr{C}^{\rm c}\subseteq\mathscr{E}/[M]$. Combining
Corollary 4.5 with Proposition 4.17(1), we see that $\widetilde{\Phi}$ sends
compact objects of $\mathscr{C}$ to objects of
$(\mathscr{E}\cap\mathscr{S})/[M]$ which are also compact in $\mathscr{C}$.
Note that $\widetilde{\Psi}:\mathscr{C}\to\underline{M}^{\bot}$ always
preserves compact generating sets. Thus the first two lines of functors in the
recollement of Corollary 4.5 restrict to the half recollement in Corollary
4.18(3). $\square$

######  Corollary 4.19.

$(1)$ $\dim\big{(}(\mathscr{E}\cap\mathscr{S})/[M])\big{)}\leq
2\;LL(\Gamma)-1$, where $LL(\Gamma)$ is the Loewy length of $\Gamma$.

$(2)$ $\dim\big{(}(\mathscr{E}\cap\mathscr{S})/[M])\big{)}\leq 2\,{\rm
gl.dim}(\Gamma)+1$, where ${\rm gl.dim}(\Gamma)$ is the global dimension of
$\Gamma$ .

Proof. $(1)$ follows from Corollary 4.18(1).

$(2)$ If ${\rm gl.dim}(\Gamma)$ is infinite, then the inequality in $(2)$
holds trivially. Now, let $m:={\rm gl.dim}(\Gamma)<\infty$. In the following,
we follow the notation in the proof of Corollary 4.18(1) and show that
$(\mathscr{E}\cap\mathscr{S})/[M]=\langle{\rm
add}(N)\rangle_{2m+2}^{[-m-1,0]}$. This implies
$\dim\big{(}(\mathscr{E}\cap\mathscr{S})/[M])\big{)}\leq 2m+1$.

By the second part of Corollary 4.8(3),
$(\mathscr{E}\cap\mathscr{S})/[M]=\mathscr{H}^{\rm fg}[1]\star\mathscr{H}^{\rm
fg}$. So, it suffices to control the objects of $\mathscr{H}^{\rm fg}$ by $N$.
We take an object $X\in\mathscr{H}^{\rm fg}$. Since the functor
${\rm\underline{Hom}}_{A}(N,-):\mathscr{H}^{\rm fg}\to\Gamma\mbox{{\rm-mod}}$
is an equivalence of abelian categories sending $N$ to $\Gamma$ and since each
finitely generated $\Gamma$-module has a projective resolution of length $m$
by finitely generated projective $\Gamma$-modules, there is a long exact
sequence $(\ast):\;0\to N_{m}\to\cdots\to N_{1}\to N_{0}\to X\to 0$ in
$\mathscr{H}^{\rm fg}$ with $N_{i}\in{\rm add}(N)$ for $0\leq i\leq m$. Note
that each short exact sequence $0\to X\to Y\to Z\to 0$ in $\mathscr{H}$ gives
rise to a triangle $X\to Y\to Z\to\Omega_{A}^{-1}(X)$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ with the terms $X,Y,Z\in\mathscr{H}$.
Since
$\mathscr{H}\subseteq\underline{\mathscr{G}\cap\mathscr{S}}\subseteq\underline{{{}^{\bot
1}}M}$ by Lemma 4.2(1), this triangle induces a triangle $X\to Y\to Z\to X[1]$
in $\mathscr{C}$ by Lemma 3.16(2). So, we divide the sequence $(\ast)$ into a
series of short exact sequences in $\mathscr{H}^{\rm fg}$ and then obtain
$X\in\langle{\rm add}(N)\rangle_{m+1}^{[-m,0]}$. Thus
$(\mathscr{E}\cap\mathscr{S})/[M]=\langle{\rm
add}(N)\rangle_{2m+2}^{[-m-1,0]}$. $\square$

Proof of Theorem 1.5. The existence of the recollement in $(1)$ follows from
Proposition 4.3, while $(2)$ and (3) are exactly Corollaries 4.5 and 4.19,
respectively. $\square$

Proof of Proposition 1.6. $(2)$ and $(3)$ follow from Corollary 4.18(1), while
the first part of $(1)$ is Proposition 4.17(1). Let
$\mathscr{E}_{0}:=\mathscr{E}\cap\mathscr{S}\cap A\mbox{{\rm-mod}}$. Clearly,
${\rm add}(M)\subseteq\mathscr{E}_{0}$. If $X\in\mathscr{E}_{0}$, then
$X\in{\rm Add}(M)$ by Proposition 4.17(2) because finitely generated
$A$-modules are pure-projective. As $X$ is finitely generated, it lies in
${\rm add}(M)$. Thus $\mathscr{E}_{0}={\rm add}(M)$. This implies the second
part of $(1)$. $\square$

##  5 Tachikawa’s second conjecture

In this section we prove Theorem 1.2 and Corollary 1.4. As a consequence, we
show that the Nakayama conjecture is true for Gorenstein-Morita algebras (see
Definition 1.3(ii)). Moreover, we introduce two homological conditions
${\rm(CI)}$ and ${\rm(CII)}$). They are connected with finitistic dimension
(see Lemma 5.1). Also, the invariance of ${\rm(CII)}$ under different types of
equivalences between algebras is discussed in Corollary 5.3.

Let $B$ be an Artin algebra. The _dominant dimension_ of $B$, denoted by ${\rm
dom.dim}(B)$, is by definition the largest natural number $n$ or $\infty$ such
that, in a minimal injective coresolution $0\to{{}_{B}}B\to I^{0}\to
I^{1}\to\cdots\to I^{n}\to\cdots$, all these module $I^{i}$ are projective for
$0\leq i<n$. The unsolved Nakayama Conjecture says that an Artin algebra is
self-injective whenever its dominant dimension is infinite. Related to this
conjecture, Tachikawa proposed two conjectures in [40, p. 115-116].

(TC1): If an Artin algebra $B$ satisfies ${\rm Ext}^{n}_{B}(D(B),B)=0$ for all
$n\geq 1$, then $B$ is self-injective.

(TC2): Let $B$ be a self-injective algebra and $Y$ a finitely generated
$B$-module. If ${\rm Ext}^{n}_{B}(Y,Y)=0$ for all $n\geq 1$, then $Y$ is
projective.

As pointed in the introduction, the two conjectures (TC1) and (TC2) hold true
for all algebras if and only if so does the Nakayam conjecture for all
algebras. Moreover, it was shown in [31] that, given a pair $(B,Y)$ with $B$ a
self-injective algebra and $Y$ a finitely generated, self-orthogonal
$B$-module, the algebra ${\rm End}_{B}(B\oplus Y)$ satisfies the Nakayama
conjecture if and only if $Y$ is projective.

By [31], algebras of dominant dimension at least $2$ are exactly endomorphism
algebras of generator-cogenerators over algebras. Recall that a finitely
generated module $X$ over an algebra $C$ is called a _generator-cogenerator_
if $C\oplus D(C)\in{\rm add}({{}_{C}}X)$. If $B$ is the endomorphism algebra
of a generator-cogenerator $X$ over an algebra $C$, then, by Müller’s theorem
(see [31, Lemma 3]), ${\rm dom.dim}(B)=n>1$ if and only if ${\rm
Ext}_{C}^{i}(X,X)=0$ for $1\leq i<n-1$. In particular, ${\rm
dom.dim}(B)=\infty$ if and only if $X$ is self-orthogonal, that is, ${\rm
Ext}_{C}^{i}(X,X)=0$ for all $i\geq 1$.

Proof of Theorem 1.2. $(1)\Rightarrow(2)\mbox{-}(5)$. Suppose ${{}_{A}}M$ is
projective. Then $\mathscr{G}=A\mbox{{\rm-Mod}}$ and $W=\Omega_{A}^{-}(M)=0$.
This implies $(4)$ and $(5)$. Since each $A$-module is always a filtered
colimit of finitely generated $A$-modules, $(2)$ holds. Note that $A$-filtered
modules are projective. Thus $(3)$ holds.

$(2)\Rightarrow(4)$. This is clear since $W\in\mathscr{G}$.

$(4)\Rightarrow(5)$. Let $f:\Omega_{A}^{-}(M)\to W$ be the minimal left
$\mathscr{G}$-approximation of $\Omega_{A}^{-}(M)$ with $W\in\mathscr{G}$.
Assume that $W$ is a filtered colimit of $\\{W_{i}\mid i\in I\\}$ with $I$ an
essentially small, filtered category and with all $W_{i}\in\mathscr{G}_{0}$.
We show that $W^{\bot 1}$ is closed under countable direct sums in
$A\mbox{{\rm-Mod}}$. This implies $(4)$.

Let $\lambda_{i}:W_{i}\to W$ be the canonical homomorphism of the colimit $W$.
Since $\Omega_{A}^{-}(M)$ is finitely generated and even finitely presented,
the canonical map $\lim\limits_{\longrightarrow}{\rm
Hom}_{A}(\Omega_{A}^{-}(M),W_{i})\to{\rm Hom}_{A}(\Omega_{A}^{-}(M),W)$
induced by $\\{{\rm Hom}_{A}(\Omega_{A}^{-}(M),\lambda_{i})\mid i\in I\\}$ is
an isomorphism. As $I$ is a filtered category, the colimt
$\lim\limits_{\longrightarrow}{\rm Hom}_{A}(\Omega_{A}^{-}(M),W_{i})$ of
abelian groups is a quotient group of the direct sum $\bigoplus_{i\in I}{\rm
Hom}_{A}(\Omega_{A}^{-}(M),W_{i})$ and each of its elements can be represented
by a homomorphism of ${\rm Hom}_{A}(\Omega_{A}^{-}(M),W_{i})$ for some index
$i\in I$. Consequently, there is an index $i\in I$ and a homomorphism
$f_{i}:\Omega_{A}^{-}(M)\to W_{i}$ such that $f=f_{i}\lambda_{i}$. Further,
due to $W_{i}\in\mathscr{G}$, the approximation implies that there is a
homomorphism $g_{i}:W\to W_{i}$ such that $f_{i}=fg_{i}$. It follows that
$f=fg_{i}f_{i}$. Since $f$ is left minimal, $g_{i}f_{i}$ is an isomorphism.
Thus $g_{i}$ is split-injective. Since $W_{i}$ is finitely generated, $W$ is
also finitely generated (presented). Hence, $W^{\bot 1}$ is closed under
arbitrary direct sums in $A\mbox{{\rm-Mod}}$.

$(5)\Rightarrow(1)$. By Lemma 4.15(3),
$T(\Omega_{A}^{-}(M))\simeq\Omega_{M}^{-}(N)$ in $\mathscr{C}$. Thus
$W\simeq\Omega_{M}^{-}(N)$ in $\mathscr{C}$. By Lemma 3.16(1), there are
$M_{1},M_{2}\in{\rm Add}(M)$ such that $W\oplus
M_{1}\simeq\Omega_{M}^{-}(N)\oplus M_{2}$. Let $\mathscr{M}$ be the full
subcategory of $A\mbox{{\rm-mod}}$ consisting of modules with finite ${\rm
add}(M)$-resolutions. By $M\in\mathscr{G}^{\bot>0}$ and $W\in\mathscr{G}$, we
have $\mathscr{M}\subseteq\mathscr{G}^{\bot>0}\subseteq W^{\bot 1}$. For
$X\in{\rm Add}(M)$, since ${{}_{A}}M$ is self-orthogonal and finitely
presented, $X^{\bot 1}$ contains $\mathscr{M}$ and is closed under arbitrary
direct sums in $A\mbox{{\rm-Mod}}$. Thus $(3)$ implies that
$\Omega_{M}^{-}(N)^{\bot 1}$ is closed under countable direct sums in
$A\mbox{{\rm-Mod}}$ of modules in $\mathscr{M}$. By Lemma 4.16,
$\Omega_{M}^{-}(N)^{\bot 1}$ is closed under countable direct sums in
$A\mbox{{\rm-Mod}}$ of finitely $M$-filtered $A$-modules. Further, it follows
from $N\in\mathscr{E}\cap\mathscr{S}$, Corollary 3.13 and Lemma 4.4(2) that
$\Omega_{M}^{-}(N)\in\mathscr{E}\cap\mathscr{S}$. By Proposition 4.17(2),
$\Omega_{M}^{-}(N)\in{\rm Add}(M)$. This shows $\Omega_{M}^{-}(N)=0$ (and thus
also $N=0$) in $\mathscr{C}$. By
${\rm\underline{End}}_{A}(N)={\rm\underline{End}}_{M}(N)$, we have $N=0$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$. Since
${\rm\underline{End}}_{A}(M)\simeq{\rm\underline{End}}_{A}(N)$ by Lemma
4.7(2), $M=0$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$. In other words,
${{}_{A}}M$ is projective.

$(3)\Rightarrow(1)$. By Proposition 1.6(1), the module $N$ is $M$-compact
(that is, compact in $\mathscr{C}$) and isomorphic in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ to an $M$-filtered module $X$. Then
there are projective $A$-modules $P$ and $Q$ with $N\oplus P\simeq X\oplus Q$.
Since projective $A$-modules are zero in $\mathscr{C}$, $X$ is $M$-compact. By
$(3)$, $X$ lies in ${\rm Add}(M)$, and therefore $N=0$ in $\mathscr{C}$. By
Lemma 4.7(2), we have $M=0$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$. Thus
${{}_{A}}M$ is projective. $\square$

Let $\mathscr{P}^{<\infty}(B)$ be the category of finitely generated
$B$-modules with finite projective dimension, and let $B\mbox{{\rm-
GProj}}_{\omega}$ be the category of _countably generated, compactly
Gorenstein-projective_ $B$-modules. Clearly, finitely generated, Gorenstein-
projective $B$-modules are in $B\mbox{{\rm-GProj}}_{\omega}$. Note that the
category of countably generated $B$-modules is a Serre subcategory of
$B\mbox{{\rm-Mod}}$. This is due to the fact: A ring $R$ has the property that
each submodule of each countably generated left $R$-module is countably
generated if and only if each left ideal of $R$ is countably generated.

We consider the following two homological conditions:

${\rm(CI)}$ The direct sum of countably many $B$-modules from
$\mathscr{P}^{<\infty}(B)$ belongs to $B\mbox{{\rm-
GProj}}_{\omega}^{\,\,\bot>0}$.

${\rm(CII)}$ Any compactly Gorenstein-projective, compactly filtered
$B$-module is projective.

The _finitistic dimension_ of an Artin algebra $B$ is the supremum of
projective dimensions of all $B$-modules in $\mathscr{P}^{<\infty}(B)$. The
well-known finitistic dimension conjecture says that an Artin algebra $B$
should always have finite finitistic dimension. The validity of this
conjecture for $B$ implies the one of the Nakayama conjecture for $B$.
However, the finitistic dimension conjecture is still open.

######  Lemma 5.1.

$(1)$ If ${\rm(CI)}$ holds, then so does ${\rm(CII)}$.

$(2)$ If an Artin algebra $B$ has finite finitistic dimension or is a
virtually Gorenstein algebra, then ${\rm(CI)}$ holds.

Proof. $(1)$ Let $Y$ be a compactly Gorenstein-projective $B$-module which is
compactly filtered by a sequence $\\{Y_{i}\mid i\in\mathbb{N}\\}$ of
submodules of $Y$. Set $X:=\bigoplus_{i\in\mathbb{N}}Y_{i}$. Then there is a
canonical exact sequence $0\to X\to X\to Y\to 0$ in $B\mbox{{\rm-Mod}}$. Since
$Y_{i}$ are finitely generated for all $i$, the module $Y$ is countably
generated. This implies $Y\in B\mbox{{\rm-GProj}}_{\omega}$. Moreover, since
$Y_{i}\in\mathscr{P}^{<\infty}(B)$ for all $i\in\mathbb{N}$, the condition
(CI) implies that the sequence splits and $Y$ is isomorphic to a direct
summand of $X$. Note that $Y_{i}$ is a direct sum of finitely many
indecomposable submodules with the local endomorphism rings. By [1, Corollary
26.6], $Y\simeq\bigoplus_{i\in\mathbb{N}}Y_{i}^{\prime}$, where
$Y_{i}^{\prime}$ is a direct summand of $Y_{i}$ for each $i$. Since
$\mathscr{P}^{<\infty}(B)\cap B\mbox{{\rm-GProj}}={\rm add}(B)$,
$Y_{i}^{\prime}$ is projective for all $i$. Thus $Y$ is projective. This shows
that (CII) holds.

$(2)$ Clearly, $B\mbox{{\rm-GProj}}^{\bot>0}$ contains all $B$-modules of
finite projective dimension. For a virtually Gorenstein algebra $B$,
$B\mbox{{\rm-GProj}}^{\bot>0}={{}^{\bot>0}}B\mbox{{\rm-GInj}}$, and therefore
$B\mbox{{\rm-GProj}}^{\bot>0}$ is closed under arbitrary direct sums in
$B\mbox{{\rm-Mod}}$. Thus (2) holds. $\square$

######  Lemma 5.2.

Let $B$ and $C$ be Artin algebras and $X$ a finitely generated
$C$-$B$-bimodule. Suppose that ${{}_{C}}X$ and $X_{B}$ are projective and the
tensor functor $X\otimes_{B}-:B\mbox{{\rm-Mod}}\to C\mbox{{\rm-Mod}}$ induces
a triangle equivalence $B\mbox{-}\underline{\rm GProj}\to
C\mbox{-}\underline{\rm GProj}$. If $C$ satisfies ${\rm(CII)}$, then so does
$B$.

Proof. Let $F:={{}_{C}}X\otimes_{B}-$. Since both ${{}_{C}}X$ and $X_{B}$ are
projective, the functor $F:B\mbox{{\rm-Mod}}\to C\mbox{{\rm-Mod}}$ is exact
and preserves projective modules. This yields
$F(\mathscr{P}^{<\infty}(B))\subseteq\mathscr{P}^{<\infty}(C)$. Since $F$
commutes with filtered colimits, it sends compactly filtered $B$-modules to
compactly filtered $C$-modules. Further, since $F$ induces a triangle
equivalence $B\mbox{-}\underline{\rm GProj}\to C\mbox{-}\underline{\rm
GProj}$, it reflects projective modules and sends compactly Gorenstein-
projective $B$-modules to compactly Gorenstein-projective $C$-modules. This
implies Lemma 5.2. $\square$

Next, we point out that the condition ${\rm(CII)}$ is preserved by several
classes of equivalences between algebras. For the unexplained notions below of
stable equivalences of adjoint type and singular equivalences of Morita type
with level, we refer to [43] and [41], respectively. Given a finitely
generated $B$-module $N$, we denote by ${\rm thick}({{}_{B}}N)$ the smallest
thick subcategory of $B\mbox{{\rm-mod}}$ which contains $N$.

######  Corollary 5.3.

Let $B$ and $C$ be finite-dimensional algebras over a field. Suppose that

$(a)$ $B$ and $C$ are derived equivalent, or

$(b)$ $B$ and $C$ are stably equivalent of adjoint type, or

$(c)$ $B$ and $C$ are singularly equivalent of Morita type with level defined
by a pair of bimodules $({{}_{C}}X{{}_{B}},{{}_{B}}Y_{C})$ such that ${\rm
Hom}_{C}(X,C)\in{\rm thick}({{}_{B}}B\oplus D(B))$ and ${\rm
Hom}_{B}(Y,B)\in{\rm thick}({{}_{C}}C\oplus D(C))$.

Then $B$ satisfies ${\rm(CII)}$ if and only if so does $C$.

Proof. $(a)$ follows from Lemma 5.2 and [25, Example 4.7 and Corollary 5.4].
The case $(b)$ is a special case of $(c)$. In the case $(c)$, the modules
${{}_{C}}X$ and $X_{B}$ are finitely generated and projective by the
definition of singular equivalences of Morita type with level. By Lemma 5.2,
it suffices to show that the functor $F:={{}_{C}}X\otimes_{B}-:B\mbox{{\rm-
Mod}}\to C\mbox{{\rm-Mod}}$ induces a triangle equivalence
$B\mbox{-}\underline{\rm GProj}\to C\mbox{-}\underline{\rm GProj}$. Note that
for finitely generated Gorenstein-projective modules, the triangle equivalence
$B\mbox{-}\underline{\rm Gproj}\to C\mbox{-}\underline{\rm Gproj}$ is known in
[41, Proposition 4.5] under a stronger assumption that ${\rm
Hom}_{C}(X,C)\in\mathscr{P}^{<\infty}(B)$ and ${\rm
Hom}_{B}(Y,B)\in\mathscr{P}^{<\infty}(C)$. Since our discussions involve
infinitely generated Gorenstin-projective modules, we have to check whether
$F$ and $G$ can be regraded as functors between $B\mbox{{\rm-GProj}}$ and
$C\mbox{{\rm-GProj}}$.

Let $U\in B\mbox{{\rm-GProj}}$ with a complete projective resolution
$P^{\bullet}$. Then $F(P^{\bullet}):=(F(P^{i})_{i\in\mathbb{Z}})$ is an exact
complex of projective $C$-modules. Moreover, ${\rm
Hom}^{\bullet}_{C}(F(P^{\bullet}),C)\simeq{\rm
Hom}^{\bullet}_{B}(P^{\bullet},{\rm Hom}_{C}(X,C))$ as complexes. By [14,
Theorem 2], ${\rm thick}({{}_{B}}B\oplus D(B))=B\mbox{{\rm-
GProj}}^{\bot>0}\cap B\mbox{{\rm-mod}}$. Since each cocycle of $P^{\bullet}$
lies in $B\mbox{{\rm-GProj}}$ and ${\rm Hom}_{C}(X,C)\in{\rm
thick}({{}_{B}}B\oplus D(B))$, the complex ${\rm
Hom}^{\bullet}_{B}(P^{\bullet},{\rm Hom}_{C}(X,C))$ is exact. Consequently,
$F(P^{\bullet})$ is a compete projective resolution of $F(U)$, and therefore
$F(U)\in C\mbox{{\rm-GProj}}$. In other words, $F$ restricts to a functor
$B\mbox{{\rm-GProj}}\to C\mbox{{\rm-GProj}}$. Similarly, the functor
$G:={{}_{B}}Y\otimes_{C}-:C\mbox{{\rm-Mod}}\to B\mbox{{\rm-Mod}}$ also
restricts to a functor $C\mbox{{\rm-GProj}}\to B\mbox{{\rm-GProj}}$.

Recall that $B\mbox{-}\underline{\rm GProj}$ is a triangulated category with
the shift functor $\Sigma_{B}$ which is a quasi-inverse of the syzygy functor
$\Omega_{B}$. This implies
$Y\otimes_{C}X\otimes_{B}-\simeq\Omega^{n}_{B}(-)\simeq\Sigma_{B}^{-n}:B\mbox{-}\underline{\rm
GProj}\longrightarrow B\mbox{-}\underline{\rm GProj},$ where $n$ is the level
of the singular equivalence in $(c)$ and $\Sigma_{B}^{n}$ denotes the $n$-th
shift functor of $B\mbox{-}\underline{\rm GProj}$. Similarly, there are
equivalences
$X\otimes_{B}Y\otimes_{C}-\simeq\Omega^{n}_{C}(-)\simeq\Sigma_{C}^{-n}:C\mbox{-}\underline{\rm
GProj}\rightarrow C\mbox{-}\underline{\rm GProj}.$ Thus $F$ induces a triangle
equivalences $B\mbox{-}\underline{\rm GProj}\longrightarrow
C\mbox{-}\underline{\rm GProj}$ with a quasi-inverse $\Sigma_{B}^{n}\circ G$:
$C\mbox{-}\underline{\rm GProj}\to B\mbox{-}\underline{\rm GProj}$. $\square$

Recall from Definition 1.3(ii) that an algebra $B$ is said to be _compactly
Gorenstein_ if ${\rm(CII)}$ holds; and _Gorenstein-Morita_ if $B$ is both
strongly Morita and compactly Gorenstein. By Lemma 5.1, a strongly Morita
algebra is Gorenstein-Morita if it is virtually Gorenstein or has finite
finitistic dimension.

Proof of Corollary 1.4. Suppose that $\Lambda$ is a Gorenstin-Morita algebra.
Then there is a self-injective algebra $A$ and a finitely generated $A$-module
$M$ which is a generator such that $\Lambda={\rm End}_{A}(M)$ and ${\rm
add}(M)={\rm add}(\nu_{A}(M))$. Further, suppose ${\rm
dom.dim}(\Lambda)=\infty$. Then ${{}_{A}}M$ is self-orthogonal by [31, Lemma
3].

In the following, we show that $N:=H_{*}^{0}(M)=0$ in $\mathscr{C}$ (see
Section 4.2). If it is the case, then
${\rm\underline{End}}_{A}(M)\simeq{\rm\underline{End}}_{A}(N)$ by Lemma
4.7(2), and therefore $M=0$ in $A\mbox{{\rm-{\text@underline{Mod}}}}$, that
is, ${{}_{A}}M$ is projective. Thus $\Lambda$ is Morita equivalent to $A$ and
must be self-injective, and therefore the first part of Corollary 1.4 is
proved.

Now, let $M=A\oplus\bigoplus_{i=1}^{m}M_{i}$ with $m\in\mathbb{N}$ and $M_{i}$
indecomposable and non-projective for $1\leq i\leq m$. By Lemma 4.8(2),
$N\simeq\bigoplus_{i=1}^{m}H_{*}^{0}(M_{i})$ in $\mathscr{C}$ and
$H_{*}^{0}(M_{i})$ are indecomposable in $\mathscr{C}$ for $1\leq i\leq m$.
Clearly, $H_{*}^{0}(M_{i})$ as an $A$-module does not contain non-projective
direct summands in ${\rm Add}(M)$, due to ${\rm\underline{Hom}}_{A}(N,M)=0$.
So we can assume that $H_{*}^{0}(M_{i})=P_{i}\oplus N_{i}$ as $A$-modules,
where $P_{i}$ is a projective $A$-module, and $N_{i}$ is either zero or an
indecomposable $A$-module that does not lie in ${\rm Add}(M)$. Now, we turn to
show $N_{i}=0$ for all $i$. This implies that $N$ is projective, and thus
$N=0$ in $\mathscr{C}$.

By Theorem 4.14(1) and Proposition 4.17(1), $N_{i}$ is isomorphic in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ to an $M$-filtered $A$-module $X$. Now,
suppose that $X$ is filtered by a sequence $\\{X_{n}\mid n\in\mathbb{N}\\}$ of
submodules of $X$ (see Definition 1.1). Then
$X=\lim\limits_{n\rightarrow\infty}X_{n}=\bigcup_{n=0}^{\infty}X_{n}$. By
Lemma 4.16, $X_{n}\in M^{\bot>0}$ and has an ${\rm add}(M)$-resolution of
finite length. Applying $G:={\rm Hom}_{A}(M,-)$ to the inclusions $X_{n}\to
X_{n+1}$, we obtain $G(X_{n})\in\mathscr{P}^{<\infty}(\Lambda)$ and injective
homomorphisms $G(X_{n})\to G(X_{n+1})$. Since ${{}_{A}}M$ is finitely
presented, $G$ commutes with filtered colimits. It follows that
$G(X)=\lim\limits_{n\rightarrow\infty}G(X_{n})=\bigcup_{n=0}^{\infty}G(X_{n})$.
In other words, $G(X)$ is compactly filtered.

Assume $N_{i}\neq 0$. Then $N_{i}$ is indecomposable and does not belong to
${\rm Add}(M)$. It follows from $X\simeq N_{i}$ in
$A\mbox{{\rm-{\text@underline{Mod}}}}$ and $N_{i}\in\mathscr{G}$ that
$X\in\mathscr{G}$ and
$\Omega_{M}^{-}\Omega_{M}(X)\simeq\Omega_{M}^{-}\Omega_{M}(N_{i})$. By Remark
3.19, $\Omega_{M}^{-}\Omega_{M}(N_{i})\simeq N_{i}$, and therefore $X\simeq
N_{i}\oplus M_{0}$ for some $M_{0}\in{\rm Add}(M)$. As $N$ is compact in
$\mathscr{C}$ by Theorem 4.14(1), $X$ is also compact in $\mathscr{C}$. It
follows from Lemma 3.17(2) that $G(X)$ is compactly Gorenstein-projective, but
not projective. Since $\Lambda$ is compactly Gorenstein, $G(X)$ must be
projective. This leads to a contradiction. Thus $N_{i}=0$.

If $\Lambda$ is gendo-symmetric, then $A$ can be assumed to be symmetric.
Since gendo-symmetric, virtually Gorenstein algebras are Gorenstein-Morita,
the second part of Corollary 1.4 follows from the first part of Corollary 1.4.
$\square$

To end this section, we mention the following questions related to the results
in this paper.

Question 1\. Find necessary and sufficient conditions for Artin algebras to be
compactly Gorenstein.

Finitely generated Gorenstein-projective modules over an Artin algebra are
compactly Gorenstein-projective. The next question is about countably
generated Gorenstein-projective modules.

Question 2\. Describe the class of countably generated, compactly Gorenstein-
projective modules over an Artin algebra.

Question 3\. Find more new examples of Gorenstein-Morita algebras.

Acknowledgements. Both authors would like to thank Ming Fang for discussions
on gendo-symmetric algebras and stable categories. The research work was
supported partially by the National Natural Science Foundation of China (Grant
12031014, 12122112 ).

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Hongxing Chen,

School of Mathematical Sciences & Academy for Multidisciplinary Studies,
Capital Normal University, 100048 Beijing,

P. R. China

Email<EMAIL_ADDRESS>(H.X.Chen)

Changchang Xi,

School of Mathematical Sciences, Capital Normal University, 100048 Beijing, P.
R. China; and School of Mathematics and Statistics, Shaanxi Normal University,
710119 Xi’an, P. R. China

Email<EMAIL_ADDRESS>(C.C.Xi)