# Monic modules and semi-Gorenstein-projective modules

Pu Zhang
School of Mathematical Sciences, Shanghai Jiao Tong University
Shanghai 200240, P. R. China

###### Abstract.

The category ${\rm gp}(\Lambda)$ of Gorenstein-projective modules over tensor
algebra $\Lambda=A\otimes_{k}B$ can be described as the monomorphism category
${\rm mon}(B,{\rm gp}(A))$ of $B$ over ${\rm gp}(A)$. In particular,
Gorenstein-projective $\Lambda$-modules are monic. In this paper, we find the
similar relation between semi-Gorenstein-projective $\Lambda$-modules and
$A$-modules, via monic modules, namely, ${\rm mon}(B,\ ^{\perp}A)={\rm
mon}(B,A)\cap\ ^{\perp}\Lambda.$ Using this, it is proved that if $A$ is
weakly Gorenstein, then $\Lambda$ is weakly Gorenstein if and only each semi-
Gorenstein-projective $\Lambda$-modules are monic; and that if $B=kQ$ with $Q$
a finite acyclic quiver, then $\Lambda$ is weakly Gorenstein if and only if
$A$ is weakly Gorenstein. However, this relation itself does not answer the
question whether there exist double semi-Gorenstein-projective
$\Lambda$-modules which are not monic. Using the recent discovered examples of
double semi-Gorenstein-projective $A$-modules which are not torsionless, we
positively answer this question, by explicitly constructing a class of double
semi-Gorenstein-projective $T_{2}(A)$-modules with one parameter such that
they are not monic, and hence not torsionless. The corresponding results are
obtained also for the monic modules and semi-Gorenstein-projective modules
over the triangular matrix algebras given by bimodules.

MSC 2020:

Primary: 16G10; 16G50; secondary: 16E05; 16E65.

Key words and phrases: monic module, monomorphism category, (double) semi-
Gorenstein-projective module, Gorenstein-projective module, the canonical map,
torsionless module, reflexive module, (left) weakly Gorenstein algebra, semi-
Gorenstein-projective-free algebra

Supported by National Natural Science Foundation of China, Grant No. 12131015,
11971304.

pzhang$\char 64\relax$sjtu.edu.cn

##  1\. Introduction

Monic modules, defined on tensor products $\Lambda=A\otimes_{k}B$, or on
matrix algebras $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&B\end{smallmatrix}\right)$ of bimodule modules $M$, built a bridge between
Gorenstein-projective $\Lambda$-modules and Gorenstein-projective $A$-modules
(Theorems 2.6 and 2.13). In particular, Gorenstein-projective
$\Lambda$-modules are monic, in the both cases. This paper is to show that
they also play an important role in the study of semi-Gorenstein-projective
modules. In the both cases, we will give sufficient and necessary conditions
such that $\Lambda$ is weakly Gorenstein, and positively answer the question
whether there exist double semi-Gorenstein-projective $\Lambda$-modules which
are not monic, and hence not torsionless, and not Goresnstein-projective.

### 1.1.

Let $A$ be an Artin algebra. All the modules in this paper are finitely
generated, and we start from left modules. Let $A\mbox{-}{\rm mod}$ be the
category of left $A$-modules. For $M\in A\mbox{-}{\rm mod}$, denote by
$\operatorname{add}(M)$ the full subcategory of $A\mbox{-}{\rm mod}$ of direct
summands of a direct sum of copies of $M$; by ${}^{\perp}M$ the full
subcategory of $A\mbox{-}{\rm mod}$ of modules $X$ with
$\operatorname{Ext}_{A}^{i}(X,M)=0$ for $i\geq 1;$ and by $M^{\perp}$ the full
subcategory of $A\mbox{-}{\rm mod}$ of modules $X$ with
$\operatorname{Ext}_{A}^{i}(M,X)=0$ for $i\geq 1.$

Let $M^{*}$ denote the $A$-dual $\operatorname{Hom}_{A}(M,A)$ of $M$. Denote
by $\phi_{M}:M\longrightarrow M^{**}$ the canonical $A$-map, defined by
$\phi_{M}(m)(f)=f(m)$ for $m\in M$ and $f\in M^{*}$. A module $M$ is
torsionless if it is a submodule of a projective module, or, equivalently,
$\phi_{M}$ is a monomorphism; and $M$ is reflexive if $\phi_{M}$ is an
isomorphism.

A module $M$ is semi-Gorenstein-projective if $M\in{}^{\perp}A$; and $M$ will
be called double semi-Gorenstein-projective, if both $M$ and $M^{*}$ are semi-
Gorenstein-projective. By definition, a Gorenstein-projective module is double
semi-Gorenstein-projective and reflexive. This is introduced by Auslander and
Bridger [AB], under the name of modules of $G$-dimension zero, and it is
equivalent to the definition in terms of complete projective resolution given
by Enochs and Jenda ([EJ1], [EJ2]). For the equivalence we refer to [AM,
p.398] (where it is called a total reflexive module) and [Chr, Theorem 4.2.6].
Denote by ${\rm gp}(A)$ the full subcategory of $A\mbox{-}{\rm mod}$ of
Gorenstein-projective modules. Thus $\operatorname{add}(A)\subseteq{\rm
gp}(A)\subseteq{}^{\perp}A.$

### 1.2.

Avramov and Martsinkovsky [AM, p.398] has proposed the independence problem of
the total reflexivity. In fact, the known examples of semi-Gorenstein-
projective modules which are not Gorenstein-projective are few and
complicated. The first examples of reflexive semi-Gorenstein-projective
modules which are not Gorenstein-projective, and the first examples of
reflexive modules $M$ with $M^{*}$ semi-Gorenstein-projective such that $M$
are not semi-Gorenstein-projective, are discovered by Jorgensen and Şega [JS];
and the first examples of double semi-Gorenstein-projective modules which are
not torsionless, are recently founded in [RZ2, RZ3]. Putting together, this
solves the independence problem of the total reflexivity. Note that the first
examples of semi-Gorenstein-projective modules which are not Gorenstein-
projective over noncommutative algebras, are presented by Marczinzik [M2].

### 1.3.

Let $A$ and $B$ be finite-dimensional algebras over field $k$. Since Cartan-
Eilenberg [CE], modules over tensor algebra $\Lambda=A\otimes_{k}B$ have got
interest. They are complicated in the sense that $\Lambda$-mod can not be
controlled by $U\otimes_{k}V$ with $U\in A$-mod and $V\in B$-mod. However, if
$B$ is given by a bound quiver $(Q,I)$, one can study $\Lambda$-module by
taking the advantage of the representations of quivers over algebra $A$
([RS1-RS3], [S2-S5], [KLM1, KLM2], [LZ1, LZ2], [RZ1], [ZX]), i.e., any
$\Lambda$-module can be identified with a representation $(X_{i},\
X_{\alpha},\ i\in Q_{0},\ \alpha\in Q_{1})$ of $(Q,I)$ over $A$, where each
$X_{i}\in A$-mod, and each $X_{\alpha}:X_{s(\alpha)}\longrightarrow
X_{e(\alpha)}$ is an $A$-map, such that $X_{\alpha}$’s satisfy all the
relations which generate $I$.

When $Q$ is finite acyclic and $I$ is generated by monomial relations, this
identification permits us to define monic $\Lambda$-modules and monomorphism
category ${\rm mon}(B,\mathscr{C})={\rm mon}(Q,I,\mathscr{C})$ ([LZ1, LZ2],
[ZX]), for any additive full subcategory $\mathscr{C}$ of $A$-mod. This
definition is combinatorial and constructive, and it admits a homological
interpretation. In general, there is no longer the corresponding combinatorial
definition of a monic module, but this homological interpretation still makes
sense, and it is taken as the definition of the monomorphism category ${\rm
mon}(B,\mathscr{C})$ by Hu, Luo, Xiong and Zhou in [HLXZ]. See Subsection 2.1.

The study of the monomorphism categories can be traced to G. Birkhoff [Bir].
When $B$ is the path algebra of quiver $A_{n}$ with linear orientation, i.e.,
$B=T_{n}(k)=\left(\begin{smallmatrix}k&k&\cdots&k\\\ 0&k&\cdots&k\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&\cdots&k\end{smallmatrix}\right)$, then
$\Lambda=A\otimes_{k}B=T_{n}(A)$ and the monomorphism category ${\rm
mon}(B,A)={\rm mon}(B,A\mbox{-}{\rm mod})$ is exactly the submodule category
$\mathcal{S}_{n}(A)$ ([RS1-RS3]), or called the filtered chain category
([S1-S5]). They have Auslander-Reiten sequences ([RS2]) and the RSS
equivalence ([ZX]). Simson ([S2]-[S5]) has studied their representation type.
By Kussin, Lenzing and Meltzer [KLM1, KLM2] and Chen [Chen1], they are related
to the singularity theory.

### 1.4.

An important application of monomorphism categories is that they can describe
Gorenstein-projective modules as ${\rm gp}(\Lambda)={\rm mon}(B,\ {\rm
gp}(A))$ (cf. Theorem 2.6 below). Thus, Gorenstein-projective
$\Lambda$-modules are monic $\Lambda$-modules over Gorenstein-projective
$A$-modules. If $B$ is given by a finite acyclic quiver and monomial
relations, by the combinatorial definition of monic modules, this gives in
practice a reductive construction of Gorenstein-projective $\Lambda$-modules.

Question 1. Is there the similar relation between semi-Gorenstein-projective
$\Lambda$-modules and $A$-modules?

######  Theorem 1.1.

Let $A$ and $B$ be finite-dimensional $k$-algebras with ${\rm
gl.dim}B<\infty$, and $\Lambda=A\otimes_{k}B$. Then

${\rm mon}(B,\ ^{\bot}A)={\rm mon}(B,A)\cap\ ^{\bot}\Lambda.$

Moreover, if ${\rm inj.dim}\ A_{A}<\infty$, then ${}^{\bot}\Lambda={\rm
mon}(B,\ ^{\bot}A)$.

Theorem 1.1 will be proved in Section 3, as a special case of Theorem 3.1.

### 1.5.

An Artin algebra $A$ is Gorenstein, if ${\rm inj.dim}\ _{A}A<\infty$ and ${\rm
inj.dim}A_{A}<\infty.$ An Artin algebra $A$ is left weakly Gorenstein ([M1],
[RZ2]), if any left semi-Gorenstein-projective $A$-module is Gorenstein-
projective, i.e., ${}^{\bot}A={\rm gp}(A).$ It is open whether a left weakly
Gorenstein algebra is right weakly Gorenstein ([M1, $\S$ 5], [RZ2, 9.3]).
However, if no confusions caused, we will omit the word “left”.

By Enochs and Jenda [EJ2, 11.5.3], Gorenstein algebras are weakly Gorenstein.
By Yoshino [Y1, Theorem 5.5] and Beligiananis [Bel2, Corollary 5.11], if
${}^{\perp}A$ is of finite type, then $A$ is weakly Gorenstein. By Marczinzik
[M1, Theorem 3.5(3)], torsinless finite algebras are weakly Gorenstein. For
more information on weakly Gorenstein algebras we refer to [Bel1, Bel2], [M1],
and [RZ2, 1.2 - 1.4, 3.6].

Question 2. (i) Let $A$ and $B$ be Artin algebras, $M$ an $A$-$B$-bimodule
such that $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&B\end{smallmatrix}\right)$ is an Artin algebra. When $\Lambda$ is weakly
Gorenstein?

(ii) Let $A$ and $B$ be finite-dimensional $k$-algebras with ${\rm
gl.dim}B<\infty$. When the tensor product $A\otimes_{k}B$ is weakly
Gorenstein?

It turns out that, in the both cases, monic modules will play a crucial role.

######  Theorem 1.2.

Let $A$ and $B$ be Artin algebras, $M$ an $A$-$B$-bimodule with ${\rm
proj.dim}_{A}M<\infty$, and $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&B\end{smallmatrix}\right)$.

$(1)$ If ${\rm proj.dim}M_{B}<\infty$ and ${\rm D}(M_{B})\in\
(^{\perp}B)^{\perp}$, then $\Lambda$ is weakly Gorenstein if and only if each
semi-Gorenstein-projective $\Lambda$-module is monic respect to bimodule $M$,
and $A$ and $B$ are weakly Gorenstein.

$(2)$ If ${\rm proj.dim}M_{B}<\infty$ and $B$ is a Gorenstein algebra, then
$\Lambda$ is weakly Gorenstein if and only if each semi-Gorenstein-projective
$\Lambda$-module is monic respect to bimodule $M$ and $A$ is weakly
Gorenstein.

$(3)$ If ${}_{A}M$ is torsionless and $M_{B}$ is projective, then $\Lambda$ is
weakly Gorenstein if and only if $A$ and $B$ are weakly Gorenstein. In
particular, if ${}_{A}M$ and $M_{B}$ are projective, then $\Lambda$ is weakly
Gorenstein if and only if $A$ and $B$ are weakly Gorenstein.

Theorem 1.2 is the combination of Propositions 4.1, 4.2 and 4.4.

######  Theorem 1.3.

Let $A$ and $B$ be finite-dimensional $k$-algebras.

$(1)$ Assume that ${\rm gl.dim}B<\infty$ and $\Lambda=A\otimes_{k}B$. If
$\Lambda$ is weakly Gorenstein, then so is $A$. Conversely, if $A$ is weakly
Gorenstein, then a semi-Gorenstein-projective $\Lambda$-module $M$ is
Gorenstein-projective if and only if $M$ is monic.

Thus, if $A$ is weakly Gorenstein, then $\Lambda$ is weakly Gorenstein if and
only if each semi-Gorenstein-projective $\Lambda$-module is monic, or
equivalently, ${}^{\bot}\Lambda={\rm mon}(B,\ ^{\perp}A)$.

$(2)$ Let $Q$ be a finite acyclic quiver. Then $A\otimes_{k}kQ$ is weakly
Gorenstein if and only if $A$ is weakly Gorenstein.

The assumption ${\rm gl.dim}B<\infty$ holds automatically if $B$ is given by a
bound acyclic quiver. Theorems 1.3 is the combination of Propositions 4.7 and
4.8.

### 1.6.

An Artin algebra $A$ will be called left semi-Groenstein-projective-free, or
in short, lsgp-free, provided that each left semi-Gorenstein-projective
$A$-module is a projective module, i.e., ${}^{\perp}A=\operatorname{add}(A)$.
We do not know whether a lsgp-free algebra is right semi-Groenstein-
projective-free.

Recall that $A$ is left CM-free ([Chen 2]) if ${\rm
gp}(A)=\operatorname{add}(A)$. Thus, $A$ is lsgp-free if and only if $A$ is
left CM-free and left weakly Gorenstein. It is open whether a left CM-free
algebra is left weakly Gorenstein (or equivalently, lsgp-free). See [RZ2,
9.2]. Many algebras are lsgp-free. For example, this is the case if ${\rm
gl.dim.}A<\infty.$ There are also non Gorenstein algebras $A$ (thus, ${\rm
gl.dim.}A=\infty$) which are lsgp-free.

######  Theorem 1.4.

$(1)$ Assume that ${}^{\perp}B=\operatorname{add}(B)$ with ${\rm
proj.dim}M_{B}<\infty$ and that ${}_{A}M$ is torsionless with ${\rm proj.dim}\
_{A}M<\infty$. Then $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&B\end{smallmatrix}\right)$ is weakly Gorenstein if and only if $A$ is weakly
Gorenstein.

Moreover, if in addition ${}_{A}M$ is projective, then

${}^{\perp}\Lambda={\rm gp}(\Lambda)=\\{\binom{M\otimes_{B}P}{P}_{{\rm
Id}_{M\otimes_{B}P}}\oplus\binom{G}{0}\ |\ P\in\operatorname{add}(B),\ G\in\
^{\perp}A={\rm gp}(A)\\}$

and ${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$ if and only if
${}^{\perp}A=\operatorname{add}(A)$.

$(2)$ Let $I$ be an admissible ideal of $kQ$, and $\Lambda=A\otimes_{k}kQ/I$.
Then ${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$ if and only if
${}^{\perp}A=\operatorname{add}(A)$.

Theorems 1.4 is the combination of Propositions 4.9 and 4.11.

### 1.7.

If $\Lambda=A\otimes_{k}B$ with ${\rm gl.dim.}B<\infty$, or if
$\Lambda=\left(\begin{smallmatrix}A&M\\\ 0&B\end{smallmatrix}\right)$, then
Gorenstein-projective $\Lambda$-modules are always monic (cf. Theorems 2.6 and
1.1). Is this true for semi-Gorenstein-projective $\Lambda$-modules? One may
ask a stronger question:

Question 3. In the both cases, whether there exist double semi-Gorenstein-
projective $\Lambda$-modules which are not monic?

The positive answer will in particular gives double semi-Gorenstein-projective
modules which are not Gorenstein-projective. As mentioned in Subsection 1.2,
this is highly nontrivial.

To answer Question 3, we consider
$\Lambda=A\otimes_{k}k(\circ\longrightarrow\circ)=\left(\begin{smallmatrix}A&A\\\
0&A\end{smallmatrix}\right)=T_{2}(A)$. Any left $\Lambda$-module $M$ can be
identified with a triple $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$, where $\varphi:Y\longrightarrow X$ is a
left $A$-map. Thus, one has the exact sequence of left $A$-modules
$Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$, and there is a unique $A$-map $\beta:{\rm
Coker}\pi^{*}\longrightarrow Y^{*}$, such that the diagram with exact rows

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
Coker}\varphi)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{X^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{{\rm
Coker}\pi^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
Coker}\varphi)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{X^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$$\textstyle{Y^{*}.}$

commutes, where $p$ is the canonical $A$-epimorphism. So one has the left
$A$-map $\beta^{*}:Y^{**}\longrightarrow({\rm Coker}\pi^{*})^{*}$, and the
composition $\beta^{*}\phi_{Y}:Y\longrightarrow({\rm Coker}\pi^{*})^{*}$,
where $\phi_{Y}:Y\longrightarrow Y^{**}$ is the canonical map.

######  Theorem 1.5.

Let $A$ be an Artin algebra, $\Lambda=T_{2}(A)=\left(\begin{smallmatrix}A&A\\\
0&A\end{smallmatrix}\right),$ and $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ a left $\Lambda$-module. Then

$(1)$ There is a left $\Lambda$-module isomorphism
$\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}^{**}\cong\left(\begin{smallmatrix}X^{**}\\\
({\rm Coker}\pi^{*})^{*}\end{smallmatrix}\right)_{p^{*}}$, where $p^{*}:({\rm
Coker}\pi^{*})^{*}\longrightarrow X^{**}$ is the $A$-monomorphism induced by
$p:X^{*}\longrightarrow({\rm Coker}\varphi)^{*}$.

Taking this isomorphism as identity, then the canonical $\Lambda$-map
$\phi_{\binom{X}{Y}_{\varphi}}:\binom{X}{Y}_{\varphi}\longrightarrow\left(\begin{smallmatrix}X^{**}\\\
({\rm Coker}\pi^{*})^{*}\end{smallmatrix}\right)_{p^{*}}$ is given by
$\phi_{\binom{X}{Y}_{\varphi}}=\binom{\phi_{X}}{\beta^{*}\phi_{Y}}.$

$(2)$ $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is
torsionless and double semi-Gorenstein-projective if and only if
$\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is monic,
$X,\ Y$, and ${\rm Coker}\varphi$ are double semi-Gorenstein-projective, and
$X$ and $Y$ are torsionless.

$(3)$ $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is
double semi-Gorenstein-projective with epimorphism
$\phi_{\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}}$ if
and only if $\varphi^{*}:X^{*}\longrightarrow Y^{*}$ is an epimorphism, $X$
and $Y$ are double semi-Gorenstein-projective, $({\rm Coker}\varphi)^{*}$ is
semi-Gorenstein-projective, and $\phi_{X}$ and $\phi_{Y}$ are epimorphisms.

Theorem 1.5(1) is a summary of Lemma 5.3 and Proposition 5.4; and Theorem
1.5(2) and (3) will be clear after Proposition 5.7.

As remarked in [RZ4, 3.1], up to now, all the known examples have the
following property:

Double semi-Gorenstein-projective modules $M$ such that $\phi_{M}$ is a
monomorphism (an epimorphism, respectively) are Gorenstein-projective.

The following result shows that this property is preserved under the
$T_{2}$-extensions.

######  Theorem 1.6.

Let $A$ be an Artin algebra and
$\Lambda=T_{2}(A)=\left(\begin{smallmatrix}A&A\\\
0&A\end{smallmatrix}\right).$ Then

$(\rm 1)$ Any torsionless and double semi-Gorenstein-projective $A$-module is
Gorenstein-projective if and only if any torsionless and double semi-
Gorenstein-projective $\Lambda$-module is Gorenstein-projective.

$(\rm 2)$ Any double semi-Gorenstein-projective $A$-module $L$ with $\phi_{L}$
an epimorphism is Gorenstein-projective if and only if any double semi-
Gorenstein-projective $\Lambda$-module $M$ with $\phi_{M}$ an epimorphism is
Gorenstein-projective.

Theorem 1.6 will be proved in Subsection 5.9.

### 1.8.

The following result positively answers Question 3, and gives a construction
of double semi-Gorenstein-projective $T_{2}(A)$-modules which are not monic.

######  Theorem 1.7.

Suppose that $Y$ is a double semi-Gorenstein-projective $A$-module which is
not torsionless. Let $\varphi:Y\longrightarrow P$ be a left ${\rm
add}(A)$-approximation of $Y$. Then $\binom{P}{Y}_{\varphi}$ is a double semi-
Gorenstein-projective $T_{2}(A)$-module which is not monic. In particular,
$\binom{P}{Y}_{\varphi}$ is not torsionless.

Theorem 1.7 will be proved in Subsection 6.1. Using the algebra $A$ in [RZ2]
and the $A$-modules $M(1,-q,c)$ in [RZ3], by Theorem 1.7, we obtain a class of
double semi-Gorenstein-projective $T_{2}(A)$-modules with parameter $c$ as

$X(c):=\left(\begin{smallmatrix}{}_{A}A\\\
M(1,-q,c)\end{smallmatrix}\right)_{f_{1}}$

such that $X(c)$ not monic, and hence not torsionless; moreover, all the
canonical maps $\phi_{X(c)}:X(c)\longrightarrow X(c)^{**}$ are neither
monomorphisms nor epimorphisms, and $X(c)^{**}$ are not semi-Gorenstein-
projective. See Proposition 6.2.

##  2\. Preliminaries: Monic modules with relations to Gorenstein-projective
modules

###  2.1. Monic modules over tensor algebras

######  Definition 2.1.

([HLXZ, 3.1]) Let $A$ and $B$ be finite-dimensional $k$-algebras, and
$\Lambda=A\otimes_{k}B$.

(1) A left $\Lambda$-module $X$ is monic, if ${\rm
Tor}^{\Lambda}_{i}(A\otimes_{k}V,X)=0$ for all $i\geq 1$ and for all right
$B$-modules $V$.

Denote by ${\rm mon}(B,A)$ the full subcategory of $\Lambda$-mod consisting of
monic modules, which is called the monomorphism category of $B$ over $A$.

(2) Let $\mathscr{C}$ be an additive full subcategory of $A$-mod. An object
$X\in{\rm mon}(B,A)$ is a monic module over $\mathscr{C}$, if
$(A\otimes_{k}V)\otimes_{\Lambda}X\in\mathscr{C}$ for all right $B$-module
$V$.

Denote by ${\rm mon}(B,\mathscr{C})$ the full subcategory of ${\rm mon}(B,A)$
of monic modules over $\mathscr{C}$, which is called the monomorphism category
of $B$ over $\mathscr{C}$.

######  Lemma 2.2.

([HLXZ, Lemma 3.2(7)]; [ZX, Theorem 2.6(1)]) One has

$\displaystyle{\rm mon}(B,A)$ $\displaystyle=\\{X\in\Lambda\mbox{-}{\rm mod}\
|\ {\rm Tor}^{\Lambda}_{i}(A\otimes_{k}{\rm D}(S),X)=0,\ \forall\ i\geq 1,\
\forall\ \mbox{simple left}\ B\mbox{-module}\ S\\}$ $\displaystyle=\
^{\bot}({\rm D}(A_{A})\otimes_{k}B).$

######  Example 2.3.

$(1)$ If $B$ is the path algebra of the quiver $A_{n}\ \ (n\geq 2)$ with
linear orientation, then $B=T_{n}(k)=\left(\begin{smallmatrix}k&k&\cdots&k\\\
0&k&\cdots&k\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&k\end{smallmatrix}\right)$,
$\Lambda=A\otimes_{k}B=\left(\begin{smallmatrix}A&A&\cdots&A\\\
0&A&\cdots&A\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&A\end{smallmatrix}\right)=T_{n}(A)$, and ${\rm mon}(B,A)$ turns out
to be

$\mathcal{S}_{n}(A)=\\{\left(\begin{smallmatrix}X_{1}\\\ \vdots\\\ \\\
X_{n}\end{smallmatrix}\right)_{(\varphi_{i})}\in T_{n}(A)\mbox{-}{\rm mod}\ |\
\varphi_{i}:X_{i+1}\longrightarrow X_{i}\ \mbox{is a monomorphism},\ \forall\
1\leq i\leq n-1\\}.$

This submodule category has been studied in [A], [S1-S5], [RS1 - RS3], [Z1].

$(2)$ If $B$ is the path algebra $kQ$, where $Q=(Q_{0},Q_{1},s,e)$ is a finite
acyclic quiver, then a monical $\Lambda$-module has been defined in [LZ1] as a
representation $(X_{i},\ X_{\alpha},\ i\in Q_{0},\ \alpha\in Q_{1})$ of $Q$
over $A$, such that for each $i\in Q_{0}$ the $A$-map

$(X_{\alpha})_{\begin{smallmatrix}\alpha\in Q_{1}\\\
e(\alpha)=i\end{smallmatrix}}:\bigoplus_{\begin{smallmatrix}\alpha\in Q_{1}\\\
e(\alpha)=i\end{smallmatrix}}X_{s(\alpha)}\longrightarrow X_{i}$

is a monomorphism.

For any additive full subcategory $\mathscr{C}$ of $A$-mod, a monic
$\Lambda$-module over $\mathscr{C}$ has been defined in [ZX, 2.1], as a monic
$\Lambda$-module $(X_{i},\ X_{\alpha},\ i\in Q_{0},\ \alpha\in Q_{1})$
satisfying

$X_{i}/{\rm Im}(X_{\alpha})_{\begin{smallmatrix}\alpha\in Q_{1}\\\
e(\alpha)=i\end{smallmatrix}}\in\mathscr{C},\ \ \forall\ i\in Q_{0}.$

$(3)$ If $B=kQ/I$ with $I$ generated by monomial relations, then ${\rm
mon}(B,\mathscr{C})$ has also been defined combinatorially. For details see
[LZ2] and [ZX].

In all these monomorphism categories defined via quivers, “monomorphisms” are
visible, and they also admit the homological description in Definition 2.1
$($[Z1, Theorem 3.1], [LZ2, 2.1], [ZX, Theorem 2.6]$)$.

######  Lemma 2.4.

Let $\Lambda=A\otimes_{k}kQ$, where $Q$ is a finite acyclic quiver. Then
torsionless $\Lambda$-modules are monic.

Proof. Let $X=(X_{i},\ X_{\alpha})$ be a torsionless $\Lambda$-module. Then
$X$ is a submodule of a projective $\Lambda$-module, which is of the form
$P\otimes_{k}L$, where $P$ is a projective left $A$-module, and $L=(L_{i},\
L_{\alpha})$ is a projective left $kQ$-module. Thus there is a monomorphism
$(f_{i})_{i\in Q_{0}}:(X_{i},X_{\alpha})\longrightarrow(P\otimes_{k}L(i),\
{\rm Id}_{P}\otimes_{k}L_{\alpha})$ of $\Lambda$-modules. Hence, for each
$i\in Q_{0}$, the diagram of $A$-maps

$\textstyle{\bigoplus\limits_{\alpha\in
Q_{1},e(\alpha)=i}X_{s(\alpha)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigoplus\limits_{\alpha\in
Q_{1},e(\alpha)=i}f_{s(\alpha)}}$$\scriptstyle{(X_{\alpha})_{\alpha\in
Q_{1},e(\alpha)=i}}$$\textstyle{X_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{i}}$$\textstyle{\bigoplus\limits_{\alpha\in
Q_{1},e(\alpha)=i}P\otimes_{k}L_{s(\alpha)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\rm
Id}_{P}\otimes_{k}L_{\alpha})_{\alpha\in
Q_{1},e(\alpha)=i}}$$\textstyle{P\otimes_{k}L_{i}}$

commutes. Since both $\bigoplus\limits_{\alpha\in
Q_{1},e(\alpha)=i}f_{s(\alpha)}$ and $({\rm
Id}_{P}\otimes_{k}L_{\alpha})_{\alpha\in Q_{1},e(\alpha)=i}$ are
monomorphisms, it follows that $(X_{\alpha})_{\alpha\in Q_{1},e(\alpha)=i}$ is
a monomorphism, i.e., $X$ is a monic $\Lambda$-module. $\hfill\blacksquare$

######  Remark 2.5.

Lemma 2.4 is not true for $\Lambda=A\otimes_{k}(kQ/I)$, even if $I$ is
generated by monomial relations. For example, if $A=k$,
$Q=3\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}2\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}1$,
and $I=\langle\beta\alpha\rangle$, then the simple module $S(2)={\rm rad}P(3)$
is a torsionless $(kQ/I)$-module, but it is not monic.

###  2.2. Gorenstein-projective modules over tensor algebras

The relationship between Gorenstein-projective modules over
$\Lambda=A\otimes_{k}B$ and monomorphism categories of $B$ over $A$ is

######  Theorem 2.6.

([HLXZ, Theorem 4.5]) Let $A$ and $B$ be finite-dimensional $k$-algebras with
${\rm gl.dim}B<\infty$, and $\Lambda=A\otimes_{k}B$. Then ${\rm
gp}(\Lambda)={\rm mon}(B,\ {\rm gp}(A))$. In particular, a Gorenstein-
projective $\Lambda$-module is monic.

Theorem 2.6 is proved for $\Lambda=T_{n}(A)=A\otimes_{k}T_{n}(k)$ with $A$
Gorenstein in [Z1, Corollary 4.1(ii)]; it is proved for $B=kQ$ in [LZ1,
Theorem 5.1], and for $B=kQ/I$ in [LZ2, Theorem 4.1], where $Q$ is any finite
acyclic quiver, and $I$ is generated by monomial relations. In all these
cases, since ${\rm mon}(B,\ {\rm gp}(A))$ are defined via the combinatorics of
quivers, Theorem 2.6 provides in practice an inductive construction of
Gorenstein-projective modules.

###  2.3. Monic modules respect to bimodules

Let $A$ and $B$ be Artin algebras, and $M$ an $A$-$B$-bimodule such that
$\Lambda=\left(\begin{smallmatrix}A&M\\\ 0&B\end{smallmatrix}\right)$ is an
Artin algebra. This is equivalent to say that $A$ and $B$ are Artin
$R$-algebra, and $M$ is finitely generated over $R$ which acts centrally on
$M$, where $R$ is a commutative Artin ring ([ARS, Proposition 2.1, p.72]). Any
left $\Lambda$-module is identified with a triple $\binom{X}{Y}_{\varphi}$,
where $X$ is a left $A$-module, $Y$ is a left $B$-module, and
$\varphi:M\otimes_{B}Y\longrightarrow X$ is a left $A$-map.

######  Definition 2.7.

([XZZ, 2.1]) Let $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&B\end{smallmatrix}\right)$ be an Artin algebra. A $\Lambda$-module
$\binom{X}{Y}_{\varphi}$ is monic respect to bimodule ${}_{A}M_{B}$, provided
that $\varphi:M\otimes_{B}Y\longrightarrow X$ is a monomorphism.

Denote by $\mathcal{M}(A,M,B)$ the full subcategory of $\Lambda$-mod of monic
$\Lambda$-modules respect to bimodule ${}_{A}M_{B}$, which is called the
monomorphism category respect to bimodule ${}_{A}M_{B}$.

######  Example 2.8.

The monomorphism category ${\rm mon}(B,A)$ and the monomorphism category
$\mathcal{M}(A,M,B)$ are in different setting. Even if
$\Lambda=A\otimes_{k}B=\left(\begin{smallmatrix}A^{\prime}&M\\\
0&B^{\prime}\end{smallmatrix}\right)$, ${\rm
mon}(B,A)\neq\mathcal{M}(A^{\prime},M,B^{\prime})$ in general.

For example, consider $T_{n}(A)=A\otimes_{k}T_{n}(k)$. A $T_{n}(A)$-module
$X=\left(\begin{smallmatrix}X_{1}\\\ \vdots\\\ \\\
X_{n}\end{smallmatrix}\right)_{(\varphi_{i})}$ is a monic $T_{n}(A)$-module if
and only if $\varphi_{i}:X_{i+1}\longrightarrow X_{i}$ is a monomorphism for
all $1\leq i\leq n-1.$ Thus ${\rm mon}(T_{n}(k),A)=\mathcal{S}_{n}(A)$.

On the other hand, $T_{n}(A)=\left(\begin{smallmatrix}T_{n-1}(A)&M_{n-1}\\\
0&A\end{smallmatrix}\right)$ for $n\geq 2$, where
$M_{n-1}=\left(\begin{smallmatrix}A\\\ \vdots\\\ A\end{smallmatrix}\right)$
$($$n-1$ rows$)$ is a $T_{n-1}(A)$-$A$-bimodule, and $X$ is a monic
$T_{n}(A)$-module respect to bimodule $M_{n-1}$ if and only if

$\varphi_{i}\cdots\varphi_{n-1}:X_{n}\longrightarrow X_{i}$

is a monomorphism for all $1\leq i\leq n-1.$

Thus, a $T_{n}(A)$-module $X$ is a monic $T_{n}(A)$-module if and only if
$\left(\begin{smallmatrix}X_{1}\\\ \vdots\\\ \\\
X_{m}\end{smallmatrix}\right)_{(\varphi_{i})}$ is a monic $T_{m}(A)$-module
respect to bimodule $M_{m-1}$ for all $2\leq m\leq n$, where
$T_{m}(A)=\left(\begin{smallmatrix}T_{m-1}(A)&M_{m-1}\\\
0&A\end{smallmatrix}\right)$, and $M_{m-1}=\left(\begin{smallmatrix}A\\\
\vdots\\\ A\end{smallmatrix}\right)$ $($$m-1$ rows$)$; and a monic
$T_{n}(A)$-module $X$ respect to $M_{n-1}$ is not necessarily a monic
$T_{n}(A)$-module. In some sense, $\mathcal{M}(T_{n-1}(A),M_{n-1},A)$ can be
view as the local version of ${\rm mon}(T_{n}(k),A)$.

For example, let $n\geq 3$. Consider $T_{n}(A)$-module
$X=\left(\begin{smallmatrix}A\\\ \vdots\\\ A\\\ A\oplus A\\\
X_{n}=A\end{smallmatrix}\right)_{(\varphi_{i})}$, where

$\varphi_{n-1}=\binom{{\rm Id}_{A}}{0}:A\longrightarrow A\oplus A,\ \ \ \
\varphi_{n-2}=({\rm Id}_{A},0):A\oplus A\longrightarrow A$

and $\varphi_{i}={\rm Id}_{A}:X_{i+1}=A\longrightarrow A=X_{i}$ for all $1\leq
i\leq n-3.$ Then $X\notin{\rm mon}(T_{n}(k),A)$, but
$X\in\mathcal{M}(T_{n-1}(A),M_{n-1},A)$.

######  Lemma 2.9.

Let $\Lambda=\left(\begin{smallmatrix}A&M\\\ 0&B\end{smallmatrix}\right)$ be
an Artin algebra, where $M_{B}$ is projective. Then torsionless
$\Lambda$-modules are monic respect to bimodule ${}_{A}M_{B}$.

Proof. Let $L=\binom{X}{Y}_{\varphi}$ be a torsionless $\Lambda$-module. Then
$L$ is a submodule of a projective $\Lambda$-module, which is of the form
$\binom{P}{0}\oplus\binom{M\otimes_{B}Q}{Q}_{{\rm Id}_{M\otimes_{B}Q}}$, where
$P$ is a projective left $A$-module, and $Q$ is a projective left $B$-module.
Thus, there is a monomorphism
$\binom{\binom{f_{1}}{f_{2}}}{g}:\binom{X}{Y}_{\varphi}\longrightarrow\binom{P\oplus
M\otimes_{B}Q}{Q}_{\binom{0}{1}}.$ Since $M_{B}$ is projective, ${\rm
Id}_{M}\otimes_{B}g$ is a monomorphism. By the commutative diagram

$\textstyle{M\otimes_{B}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Id}_{M}\otimes_{B}g}$$\scriptstyle{\varphi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\binom{f_{1}}{f_{2}}}$$\textstyle{M\otimes_{B}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\binom{0}{1}}$$\textstyle{P\oplus
M\otimes_{B}Q}$

$\varphi$ is a monomorphism, i.e., $L$ is a monic $\Lambda$-module.
$\hfill\blacksquare$

######  Remark 2.10.

Lemma 2.9 is not true if $M_{B}$ is not projective. For example, let
$\Lambda=\left(\begin{smallmatrix}k&M\\\ 0&A\end{smallmatrix}\right)$, where
$A$ is the path algebra $k(2\longrightarrow 1)$, ${}_{k}M_{A}={\rm
D}(Ae_{1})=\operatorname{Hom}_{k}(Ae_{1},k)$ is a $k$-$A$-bimodule. Since
$Me_{2}=0$, $M\otimes_{A}Ae_{2}=Me_{2}\otimes_{A}e_{2}=0.$ Thus
$\binom{0}{Ae_{2}}$ is a left projective $\Lambda$-module. Let
$\sigma:Ae_{1}\longrightarrow Ae_{2}$ be the embedding. Then
$\binom{0}{\sigma}:\ \binom{0}{Ae_{1}}\longrightarrow\binom{0}{Ae_{2}}$ is a
$\Lambda$-monomorphism, and hence $\binom{0}{Ae_{1}}$ is a torsionless
$\Lambda$-module. But since $M\otimes_{A}Ae_{1}=M\otimes_{A}e_{1}Ae_{1}=k\neq
0$, $\binom{0}{Ae_{1}}$ is not monic respect to bimodule ${}_{k}M_{A}$.

###  2.4. (Semi-)Gorenstein-projective modules over triangular matrix algebras
of bimodules

For an Artin algebra $B$, let ${\rm D}$ denote the duality of $B$ ([ARS,
p.37]).

Let $A$ and $B$ be Artin algebras, $M$ an $A$-$B$-bimodule such that
$\Lambda=\left(\begin{smallmatrix}A&M\\\ 0&B\end{smallmatrix}\right)$ is an
Artin algebra. Under suitable conditions, semi-Gorenstein-projective
$\Lambda$-modules can be described as follows.

######  Theorem 2.11.

([XZ, Theorem 1.1]) Assume that ${\rm proj.dim}_{A}M<\infty$ and ${\rm
D}(M_{B})\in\ (^{\perp}(_{B}B))^{\perp}$. Then a $\Lambda$-module
$\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}\in\
^{\perp}\Lambda$ if and only if $Y\in\ ^{\perp}B$, the left $A$-map
$\varphi:M\otimes_{B}Y\longrightarrow X$ induces isomorphisms
$\operatorname{Ext}^{i}_{A}(X,A)\cong\operatorname{Ext}^{i}_{A}(M\otimes_{B}Y,A)$
for all $i\geq 1$, and $\varphi^{*}:X^{*}\longrightarrow(M\otimes_{B}Y)^{*}$
is a right $A$-epimorphism.

An $A$-$B$-bimodule $M$ is compatible ([Z2, Definition 1.1]), if the following
two conditions hold:

If $Q^{\bullet}$ is an exact sequence of projective $B$-modules, then
$M\otimes_{B}Q^{\bullet}$ is exact; and

If $P^{\bullet}$ is a complete $A$-projective resolution, then
$\operatorname{Hom}_{A}(P^{\bullet},M)$ is exact.

######  Lemma 2.12.

([Z2, Proposition 1.3(1)]) Let $M$ be an $A$-$B$-bimodule. If ${\rm
proj.dim}_{A}M<\infty$ and ${\rm proj.dim}M_{B}<\infty$, then $M$ is
compatible.

Under the condition of compatible bimodule, Gorenstein-projective
$\Lambda$-modules can be described as follows. In particular, again,
Gorenstein-projective $\Lambda$-modules are monic, but in the sense of respect
to bimodule ${}_{A}M_{B}$ (compare Theorem 2.6).

######  Theorem 2.13.

([Z2, Theorem 1.4]) Assume that $M$ is a compatible $A$-$B$-bimodule. Then
$\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\phi}\in\ {\rm
gp}(\Lambda)$ if and only if $\varphi:M\otimes_{B}Y\longrightarrow X$ is a
monomorphism, $\operatorname{Coker}\varphi\in{\rm gp}(A)$, and $Y\in\ {\rm
gp}(B)$.

If ths is the case, $X\in{\rm gp}(A)$ if and only if $M\otimes_{B}Y\in{\rm
gp}(A).$

######  Corollary 2.14.

Let $A$ be an Artin algebra, and
$\Lambda=T_{2}(A)=\left(\begin{smallmatrix}A&A\\\
0&A\end{smallmatrix}\right)$. Then

$(1)$ ${}^{\perp}\Lambda=\\{\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}\in\Lambda\mbox{-}{\rm mod}\ |\ X\in\
^{\perp}A,\ \ Y\in\ ^{\perp}A,\ \ \varphi^{*}:X^{*}\longrightarrow Y^{*}\
\mbox{is epic}\\}.$

$(2)$ ${\rm gp}(\Lambda)=\\{\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}\in\Lambda\mbox{-}{\rm mod}\ |\
\varphi:Y\longrightarrow X\ \mbox{is monic},\ \
\operatorname{Coker}\varphi\in{\rm gp}(A),\ \ Y\in{\rm gp}(A)\\}$

$=\\{\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}\in\Lambda\mbox{-}{\rm mod}\ |\
\varphi:Y\longrightarrow X\ \mbox{is monic},\ \
\operatorname{Coker}\varphi\in{\rm gp}(A),\ \ Y\in{\rm gp}(A),\ \ X\in{\rm
gp}(A)\\}.$

##  3\. Monomorphism categories over perpendicular categories

Let $A$ and $B$ be finite-dimensional $k$-algebras, and
$\Lambda=A\otimes_{k}B$. A relation between semi-Gorenstein-projective
$\Lambda$-modules and semi-Gorenstein-projective $A$-modules is contained in
the following general result.

######  Theorem 3.1.

Let $A$ and $B$ be finite-dimensional $k$-algebra with ${\rm gl.dim}B<\infty$,
$T$ an $A$-module, and $\Lambda=A\otimes_{k}B$. Then

${\rm mon}(B,\ ^{\bot}T)={\rm mon}(B,A)\cap\ ^{\bot}(T\otimes_{k}B).$

Moreover, if there is an exact sequence of left $A$-modules

$0\longrightarrow T_{m}\longrightarrow\cdots\longrightarrow
T_{0}\longrightarrow{\rm D}(A_{A})\longrightarrow 0$

with each $T_{j}\in{\rm add}(T)$, then ${\rm mon}(B,\ ^{\bot}T)=\
^{\bot}(T\otimes_{k}B)$.

In particular, there holds ${\rm mon}(B,\ ^{\bot}A)={\rm mon}(B,A)\cap\
^{\bot}\Lambda;$ and if ${\rm inj.dim}\ A_{A}<\infty$, then
${}^{\bot}\Lambda={\rm mon}(B,\ ^{\bot}A)$.

Proof. Let $X\in{\rm mon}(B,A)$. Since by definition ${\rm mon}(B,\
^{\bot}T)\subseteq{\rm mon}(B,A),$ it follows that, in order to prove ${\rm
mon}(B,\ ^{\bot}T)={\rm mon}(B,A)\cap\ ^{\bot}(T\otimes_{k}B),$ it suffices to
prove that $X\in\ ^{\bot}(T\otimes_{k}B)$ if and only if $X\in{\rm mon}(B,\
^{\bot}T)$, i.e., $(A\otimes_{k}V)\otimes_{\Lambda}X\in\ ^{\bot}T$ for all
right $B$-modules $V$.

Take a $\Lambda$-projective resolution

$P_{\bullet}:\ \ \cdots\longrightarrow P_{1}\longrightarrow
P_{0}\longrightarrow X\longrightarrow 0.$

Claim 1: $X\in\ ^{\bot}(T\otimes_{k}B)$ if and only if the complex

$\operatorname{Hom}_{\Lambda}(P_{\bullet},\operatorname{Hom}_{k}(S^{\prime},T))$

is exact, for each right simple $B$-module $S^{\prime}$.

Since ${\rm gl.dim}B<\infty$, it is clear that
${}^{\bot}(T\otimes_{k}B)=\bigcap\limits_{S}\ ^{\bot}(T\otimes_{k}S)$, where
$S$ ranges over all the left simple $B$-modules. To use the
Tensor-$\operatorname{Hom}$ adjoint pair later, we write a left simple
$B$-module $S$ as ${\rm D}(S^{\prime})$, where $S^{\prime}$ is a right simple
$B$-module. Thus, ${}^{\bot}(T\otimes_{k}B)=\bigcap\limits_{S^{\prime}}\
^{\bot}(T\otimes_{k}{\rm D}(S^{\prime}))$, where $S^{\prime}$ ranges over all
the right simple $B$-modules. Therefore, $X\in\ ^{\bot}(T\otimes_{k}B)$ if and
only if

$\operatorname{Hom}_{\Lambda}(P_{\bullet},T\otimes_{k}{\rm D}(S^{\prime}))$

is exact, for each right simple $B$-module $S^{\prime}$. Note that the
canonical $k$-linear isomorphism

$T\otimes_{k}{\rm
D}(S^{\prime})\longrightarrow\operatorname{Hom}_{k}(S^{\prime},T),\ t\otimes
f\mapsto``s^{\prime}\mapsto f(s^{\prime})t",\ \forall\ t\in T,\ f\in{\rm
D}(S^{\prime}),\ s^{\prime}\in S^{\prime}$

is a left $\Lambda$-isomorphism. Thus, $X\in\ ^{\bot}(T\otimes_{k}B)$ if and
only if
$\operatorname{Hom}_{\Lambda}(P_{\bullet},\operatorname{Hom}_{k}(S^{\prime},T))$
is exact, for each right simple $B$-module $S^{\prime}$.

Claim 2: $(A\otimes_{k}V)\otimes_{\Lambda}X\in\ ^{\bot}T$ for all right
$B$-modules $V$ if and only if the complex

$\operatorname{Hom}_{A}((A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{\bullet},T)$

is exact for each right simple $B$-module $S^{\prime}$.

By assumption $X\in{\rm mon}(B,A)$, i.e., ${\rm
Tor}^{\Lambda}_{i}(A\otimes_{k}V,X)=0$ for all $i\geq 1$ and for all right
$B$-modules $V$. It follows that the functor

$(A\otimes_{k}-)\otimes_{\Lambda}X:{\rm mod}B\longrightarrow A\mbox{-}{\rm
mod}$

is an exact functor. As a consequence, $(A\otimes_{k}V)\otimes_{\Lambda}X\in\
^{\bot}T$ for all right $B$-modules $V$ if and only if
$(A\otimes_{k}S^{\prime})\otimes_{\Lambda}X\in\ ^{\bot}T$ for each right
simple $B$-module $S^{\prime}$, since ${}^{\bot}T$ is extension closed. Since
${\rm Tor}^{\Lambda}_{i}(A\otimes_{k}S^{\prime},X)=0$ for all $i\geq 1$, it
follows that

$(A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{\bullet}:\ \
\cdots\longrightarrow(A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{1}\longrightarrow\cdots\longrightarrow(A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{0}\longrightarrow(A\otimes_{k}S^{\prime})\otimes_{\Lambda}X\longrightarrow
0$

is an exact sequence of left $A$-modules. Since each $P_{i}$ is a projective
left $\Lambda$-module, each
$(A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{i}\in\operatorname{add}(A\otimes_{k}S^{\prime}).$
Thus each $(A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{i}$ is projective as a
left $A$-module, and hence
$(A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{\bullet}$ is an $A$-projective
resolution of left $A$-module $(A\otimes_{k}S^{\prime})\otimes_{\Lambda}X$,
for each right simple $B$-module $S^{\prime}$. Therefore,
$(A\otimes_{k}S^{\prime})\otimes_{\Lambda}X\in\ ^{\bot}T$ for each right
simple $B$-module $S^{\prime}$ if and only if
$\operatorname{Hom}_{A}((A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{\bullet},T)$
is exact for each right simple $B$-module $S^{\prime}$.

Claim 3: There is an isomorphism of complexes

$\operatorname{Hom}_{A}((A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{\bullet},T)\cong\operatorname{Hom}_{\Lambda}(P_{\bullet},\operatorname{Hom}_{k}(S^{\prime},T))$

for each right simple $B$-module $S^{\prime}$.

Applying the Tensor-$\operatorname{Hom}$ adjoint pair
$((A\otimes_{k}S^{\prime})\otimes_{\Lambda}-,\
\operatorname{Hom}_{A}(A\otimes_{k}S^{\prime},-))$ between $\Lambda$-mod and
$A$-mod, one has the following isomorphism of complexes of $k$-spaces

$\operatorname{Hom}_{A}((A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{\bullet},T)\cong\operatorname{Hom}_{\Lambda}(P_{\bullet},\operatorname{Hom}_{A}(A\otimes_{k}S^{\prime},T)).$

Applying the adjoint pair $(A\otimes_{k}-,\ \operatorname{Hom}_{A}(A,-))$
between $k$-mod and $A$-mod, one has the isomorphisms of $k$-spaces

$\operatorname{Hom}_{A}(A\otimes_{k}S^{\prime},T)\cong\operatorname{Hom}_{k}(S^{\prime},\operatorname{Hom}_{A}(A,T))\cong\operatorname{Hom}_{k}(S^{\prime},T),$

which is clearly also an isomorphism of left $\Lambda$-modules. All together
we get an isomorphism of complexes

$\operatorname{Hom}_{A}((A\otimes_{k}S^{\prime})\otimes_{\Lambda}P_{\bullet},T)\cong\operatorname{Hom}_{\Lambda}(P_{\bullet},\operatorname{Hom}_{k}(S^{\prime},T))$

for each right simple $B$-module $S^{\prime}$.

It follows from Claim 1, Claim 2 and Claim 3 that $X\in\
^{\bot}(T\otimes_{k}B)$ if and only if $(A\otimes_{k}V)\otimes_{\Lambda}X\in\
^{\bot}T$ for all right $B$-module $V$. This proves ${\rm mon}(B,\
^{\bot}T)={\rm mon}(B,A)\cap\ ^{\bot}(T\otimes_{k}B)$.

Finally, assume that there is an exact sequence $0\longrightarrow
T_{m}\longrightarrow\cdots\longrightarrow T_{0}\longrightarrow{\rm
D}(A_{A})\longrightarrow 0$ with each $T_{j}\in{\rm add}(T)$. To show ${\rm
mon}(B,\ ^{\bot}T)=\ ^{\bot}(T\otimes_{k}B)$, it suffices to show
${}^{\bot}(T\otimes_{k}B)\subseteq\ {\rm mon}(B,A)$. By Lemma 2.2, ${\rm
mon}(B,A)=\ ^{\bot}({\rm D}(A_{A})\otimes_{k}B)$. Thus, it suffices to show
${}^{\bot}(T\otimes_{k}B)\subseteq\ ^{\bot}({\rm D}(A_{A})\otimes_{k}B)$. This
follows from the exact sequence $0\longrightarrow
T_{m}\otimes_{k}B\longrightarrow\cdots\longrightarrow
T_{0}\otimes_{k}B\longrightarrow{\rm D}(A_{A})\otimes_{k}B\longrightarrow 0$
with each $T_{j}\otimes_{k}B\in\mbox{add}(T\otimes_{k}B).$ This completes the
proof. $\hfill\blacksquare$

##  4\. Weakly Gorenstein algebras: Proof of Theorems 1.2, 1.3, and 1.4

###  4.1. When triangular matrix algebras of bimodules are weakly Gorenstein?

Let $A$ and $B$ be Artin algebras, $M$ an $A$-$B$-bimodule such that
$\Lambda=\left(\begin{smallmatrix}A&M\\\ 0&B\end{smallmatrix}\right)$ is an
Artin algebra. We will give various conditions for $\Lambda$ being a left
weakly Gorenstein algebra, i.e., $\ {}^{\perp}\Lambda={\rm gp}(\Lambda)$.

######  Proposition 4.1.

Assume that ${\rm proj.dim}_{A}M<\infty$, ${\rm proj.dim}M_{B}<\infty$, and
${\rm D}(M_{B})\in\ (^{\perp}(_{B}B))^{\perp}$. Then
$\Lambda=\left(\begin{smallmatrix}A&M\\\ 0&B\end{smallmatrix}\right)$ is
weakly Gorenstein if and only if each semi-Gorenstein-projective
$\Lambda$-module is monic respect to bimodule ${}_{A}M_{B}$, and $A$ and $B$
are weakly Gorenstein.

Proof. Since ${\rm proj.dim}_{A}M<\infty$ and ${\rm proj.dim}M_{B}<\infty$,
the $A$-$B$-bimodule $M$ is compatible (cf. Lemma 2.12). Thus, under the
assumptions, one can apply Theorems 2.11 and 2.13.

Assume that each semi-Gorenstein-projective $\Lambda$-module is monic respect
to bimodule ${}_{A}M_{B}$, and that $A$ and $B$ are weakly Gorenstein. Let
$\binom{X}{Y}_{\varphi}\in\ ^{\perp}\Lambda.$ We need to prove
$\binom{X}{Y}_{\varphi}\in\ {\rm gp}(\Lambda).$ By the assumption,
$\varphi:M\otimes_{B}Y\longrightarrow X$ is a monomorphism; together with
Theorem 2.11, one gets the conclusions:

$\bullet$ $\varphi:M\otimes_{B}Y\longrightarrow X$ is a monomorphism;

$\bullet$ $Y\in\ ^{\perp}B,$ and hence $Y\in{\rm gp}(B)$ (since by assumption
$B$ is weakly Gorenstein);

$\bullet$ $\varphi$ induces isomorphisms
$\operatorname{Ext}^{i}_{A}(X,A)\cong\operatorname{Ext}^{i}_{A}(M\otimes_{B}Y,A)$
for all $i\geq 1$;

$\bullet$ $\varphi^{*}:X^{*}\longrightarrow(M\otimes_{B}Y)^{*}$ is a right
$A$-epimorphism.

Applying $\operatorname{Hom}_{A}(-,A)$ to the exact sequence $0\longrightarrow
M\otimes_{B}Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\longrightarrow\operatorname{Coker}\varphi\longrightarrow
0$, since $\varphi^{*}:X^{*}\longrightarrow(M\otimes_{B}Y)^{*}$ is an
epimorphism and $\varphi$ induces isomorphisms
$\operatorname{Ext}^{i}_{A}(X,A)\cong\operatorname{Ext}^{i}_{A}(M\otimes_{B}Y,A)$
for all $i\geq 1$, it follows that $\operatorname{Coker}\varphi\in\ ^{\bot}A$.
Hence $\operatorname{Coker}\varphi\in{\rm gp}(A)$, since by assumption $A$ is
weakly Gorenstein. Thus, we get the following:

$\bullet$ $\varphi:M\otimes_{B}Y\longrightarrow X$ is a monomorphism;

$\bullet$ $\operatorname{Coker}\varphi\in\ {\rm gp}(A)$; and

$\bullet$ $Y\in{\rm gp}(B)$.

Applying Theorem 2.13, one gets $\binom{X}{Y}_{\varphi}\in{\rm gp}(\Lambda)$.
This proves the “if” part.

Conversely, assume that $\Lambda$ is weakly Gorenstein. Thus, any semi-
Gorenstein-projective $\Lambda$-module is Gorenstein-projective, and hence it
is monic respect to bimodule ${}_{A}M_{B}$, by Theorem 2.13. It remains to
prove that $A$ and $B$ are weakly Gorenstein. Let $X\in\ ^{\perp}A$. Applying
Theorem 2.11 one knows $\binom{X}{0}\in\ ^{\perp}\Lambda$, thus
$\binom{X}{0}\in{\rm gp}(\Lambda)$ by the assumption, and then by Theorem 2.13
one has $X\in{\rm gp}(A).$ This proves that $A$ is weakly Gorenstein.

Similarly, let $Y\in\ ^{\perp}B$. By Theorem 2.11 one knows
$\binom{M\otimes_{B}Y}{Y}_{{\rm Id}_{M\otimes_{B}Y}}\in\ ^{\perp}\Lambda$, and
hence $\binom{M\otimes_{B}Y}{Y}_{{\rm Id}_{M\otimes_{B}Y}}\in{\rm
gp}(\Lambda).$ Then by Theorem 2.13, $Y\in{\rm gp}(B).$ This proves that $B$
is weakly Gorenstein. $\hfill\blacksquare$

Taking $B$ to be a Gorenstein algebra in Proposition 4.1, we get

######  Proposition 4.2.

Assume that ${\rm proj.dim}_{A}M<\infty$, ${\rm proj.dim}M_{B}<\infty$, and
that $B$ is a Gorenstein algebra. Then
$\Lambda=\left(\begin{smallmatrix}A&M\\\ 0&B\end{smallmatrix}\right)$ is
weakly Gorenstein if and only if each semi-Gorenstein-projective
$\Lambda$-module is monic respect to bimodule ${}_{A}M_{B}$ and $A$ is weakly
Gorenstein.

Proof. Since by assumption $B$ is a Gorenstein algebra, it follows that $\
{}^{\perp}(_{B}B)={\rm gp}(B)$. Recall that for a Gorenstein algebra $B$,
$({\rm gp}(B),{\rm p}(B)^{<\infty})$ is a cotorsion pair (see e.g., [H],
[EJ2], [BR]), where ${\rm p}(B)^{<\infty}$ is the full subcategory of $B$-mod
consisting of modules of finite projective dimension. Thus

$\ (^{\perp}(_{B}B))^{\perp}={\rm gp}(B)^{\perp}={\rm p}(B)^{<\infty}.$

Since by assumption ${\rm proj.dim}M_{B}<\infty$, It follows that ${\rm
inj.dim}{\rm D}(M_{B})<\infty$. Since $B$ is Gorenstein, it follows that ${\rm
proj.dim}{\rm D}(M_{B})<\infty$, i.e., ${\rm D}(M_{B})\in{\rm
p}(B)^{<\infty}=\ (^{\perp}(_{B}B))^{\perp}$. Thus, the assertion follows from
Proposition 4.1. $\hfill\blacksquare$

Taking $B$ to be a field $k$ in Proposition 4.2, we get

######  Corollary 4.3.

Let $A$ be a finite-dimensional $k$-algebra.

$(1)$ Let $M$ be a finite-dimensional $A$-module. Assume that ${\rm
proj.dim}_{A}M<\infty$. Then $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&k\end{smallmatrix}\right)$ is weakly Gorenstein if and only if each semi-
Gorenstein-projective $\Lambda$-module is monic respect to bimodule
${}_{A}M_{k}$ and $A$ is weakly Gorenstein.

$(2)$ Let $P$ be a finite-dimensional projective left $A$-module, and
$\Lambda=\left(\begin{smallmatrix}A&P\\\ 0&k\end{smallmatrix}\right)$. Then

${}^{\perp}\Lambda=\\{\binom{G}{0}\oplus\binom{P\otimes_{k}V}{V}_{{\rm
Id}_{P\otimes_{k}V}}\ |\ G\in\ ^{\perp}A,\ V\in k\mbox{-}{\rm mod}\\}$ ${\rm
gp}(\Lambda)=\\{\binom{G}{0}\oplus\binom{P\otimes_{k}V}{V}_{{\rm
Id}_{P\otimes_{k}V}}\ |\ G\in\ {\rm gp}(A),\ V\in k\mbox{-}{\rm mod}\\}$

and $\Lambda$ is weakly Gorenstein if and only if $A$ is weakly Gorenstein.

Proof. $(2)$ Let $\binom{X}{V}_{\varphi}\in\ ^{\perp}\Lambda.$ By Theorem
2.11, one has

$\bullet$ $\varphi:P^{\oplus{\rm dim}V}\longrightarrow X$ induces isomorphisms
$\operatorname{Ext}^{i}_{A}(X,A)\cong\operatorname{Ext}^{i}_{A}(P^{\oplus{\rm
dim}V},A)=0,\ \forall\ i\geq 1$; and

$\bullet$ $\varphi^{*}:X^{*}\longrightarrow(P^{*})^{\oplus{\rm dim}V}$ is a
right $A$-epimorphism.

Thus $X\in\ ^{\perp}A$, and $\varphi^{*}$ is a splitting epimorphism. Hence
$\varphi^{**}$ is a splitting monomorphism. By the commutative diagram

$\textstyle{P^{\oplus{\rm
dim}V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\varphi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\textstyle{(P^{**})^{\oplus{\rm
dim}V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{**}}$$\textstyle{X^{**}}$

one sees that $\varphi$ is a splitting monomorphism. Thus $X=G\oplus
P^{\oplus{\rm dim}V}$ where $G$ is semi-Gorenstein-projective, and hence
$\binom{X}{V}_{\varphi}\cong\binom{G}{0}\oplus\binom{P\otimes_{k}V}{V}_{\rm
Id}$. This proves
${}^{\perp}\Lambda=\\{\binom{G}{0}\oplus\binom{P\otimes_{k}V}{V}_{{\rm
Id}_{P\otimes_{k}V}}\ |\ G\in\ ^{\perp}A,\ V\in k\mbox{-}{\rm mod}\\}$. Since
$\binom{P\otimes_{k}V}{V}_{\rm Id}$ is a projective $\Lambda$-module and
$\binom{G}{0}$ is a Gorenstein-projective $\Lambda$-module if and only if $G$
a Gorenstein-projective $A$-module, it follows that ${\rm
gp}(\Lambda)=\\{\binom{G}{0}\oplus\binom{P\otimes_{k}V}{V}_{{\rm
Id}_{P\otimes_{k}V}}\ |\ G\in\ {\rm gp}(A),\ V\in k\mbox{-}{\rm mod}\\}$.
Therefore $\Lambda$ is weakly Gorenstein if and only if $A$ is weakly
Gorenstein. $\hfill\blacksquare$

######  Proposition 4.4.

Assume that ${}_{A}M$ is torsionless with ${\rm proj.dim}\ _{A}M<\infty$ and
that $M_{B}$ is projective. Then $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&B\end{smallmatrix}\right)$ is weakly Gorenstein if and only if $A$ and $B$
are weakly Gorenstein.

In particular, if ${}_{A}M$ and $M_{B}$ are projective, then $\Lambda$ is
weakly Gorenstein if and only if $A$ and $B$ are weakly Gorenstein.

Proof. Since $M_{B}$ is projective, ${\rm D}(M_{B})$ is an injective
$B$-module, and hence ${\rm D}(M_{B})\in(^{\perp}B)^{\perp}$. Thus, the
assumption that ${\rm proj.dim.}_{A}M<\infty$ and $M_{B}$ is projective
guarantee that the conditions of Proposition 4.1 are satisfied. By Proposition
4.1, it suffices to prove that if $A$ and $B$ are weakly Gorenstein, then any
semi-Gorenstein-projective $\Lambda$-module $\binom{X}{Y}_{\varphi}$ is monic
respect to bimodule ${}_{A}M_{B}$.

Applying Theorem 2.11 to $\binom{X}{Y}_{\varphi}\in\ ^{\perp}\Lambda,$ one
gets the following conclusions:

$\bullet$ $Y\in\ ^{\perp}B,$ and hence $Y\in{\rm gp}(B)$ (since by assumption
$B$ is weakly Gorenstein);

$\bullet$ $\varphi:M\otimes_{B}Y\longrightarrow X$ induces isomorphisms
$\operatorname{Ext}^{i}_{A}(X,A)\cong\operatorname{Ext}^{i}_{A}(M\otimes_{B}Y,A),\
\forall\ i\geq 1$; and

$\bullet$ $\varphi^{*}:X^{*}\longrightarrow(M\otimes_{B}Y)^{*}$ is a right
$A$-epimorphism.

Since $Y\in{\rm gp}(B),$ ${}_{B}Y$ is a submodule of some projective
$B$-module ${}_{B}P$. Since $M_{B}$ is projective, it follows that
${}_{A}(M\otimes_{B}Y)$ is a submodule of ${}_{A}(M\otimes_{B}P)$. Since
${}_{B}P$ is projective, ${}_{A}(M\otimes_{B}P)\in\operatorname{add}(_{A}M)$.
Since by assumption ${}_{A}M$ is torsionless, it follows that $M\otimes_{B}P$
is a torsionless left $A$-module, and hence $M\otimes_{B}Y$ is a torsionless
left $A$-module. Thus, the canonical map
$\phi_{M\otimes_{B}Y}:M\otimes_{B}Y\longrightarrow(M\otimes_{B}Y)^{**}$ is a
monomorphism.

Since $\varphi^{*}:\ X^{*}\longrightarrow(M\otimes_{B}Y)^{*}$ is an
epimorphism, it follows that $\varphi^{**}:\
(M\otimes_{B}Y)^{**}\longrightarrow X^{**}$ is a monomorphism. From the
commutative diagram with monomorphism $\phi_{M\otimes_{B}Y}$

$\textstyle{M\otimes_{B}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{M\otimes_{B}Y}}$$\scriptstyle{\varphi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\textstyle{(M\otimes_{B}Y)^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{**}}$$\textstyle{X^{**}}$

one sees that $\varphi:M\otimes_{B}Y\longrightarrow X$ is a monomorphism,
i.e., $\binom{X}{Y}_{\varphi}$ is monic respect to bimodule ${}_{A}M_{B}$.
This completes the proof. $\hfill\blacksquare$

######  Remark 4.5.

The “only if” part in Proposition 4.4 does not need the condition that
${}_{A}M$ is torsionless.

###  4.2. When tensor algebras are weakly Gorenstein?

Let $A$ and $B$ be finite-dimensional $k$-algebras with ${\rm
gl.dim}B<\infty$, and $\Lambda=A\otimes_{k}B$. We first look at some
properties of a map ${\rm mon}(B,-)$.

######  Lemma 4.6.

Let $A$ and $B$ be finite-dimensional $k$-algebra, and
$\Lambda=A\otimes_{k}B$.

${\rm(i)}$ Let $\mathscr{C}$ be an additive full subcategory of $A$-mod closed
under direct summands, and $M\in A$-mod. Then $M\otimes_{k}B\in{\rm
mon}(B,\mathscr{C})$ if and only if $M\in\mathscr{C}.$

${\rm(ii)}$ If $M$ is a semi-Gorenstein-projective $A$-module which is not
Gorenstein-projective, then $M\otimes_{k}B$ is a semi-Gorenstein-projective
$\Lambda$-module which is not Gorenstein-projective.

${\rm(iii)}$ Let $\Omega$ $($respectively, $\Gamma$$)$ be the set of additive
full subcategories of $A$-mod $($respectively, $\Lambda$-mod$)$ closed under
direct summands. Then the map

${\rm mon}(B,-):\Omega\longrightarrow\Gamma,\ \ \ \mathscr{C}\mapsto{\rm
mon}(B,\mathscr{C})$

is an injective map.

Proof. ${\rm(i)}$ Assume that $M\in\mathscr{C}.$ For any right $B$-module $V$,
taking a $B$-projective resolution

$P_{\bullet}:\ \cdots\longrightarrow P_{1}\longrightarrow P_{0}\longrightarrow
V\longrightarrow 0$

of $V$, one has a projective resolution $A\otimes_{k}P_{\bullet}$ of right
$\Lambda$-module $A\otimes_{k}V$. By the isomorphisms

$(A\otimes_{k}V)\otimes_{\Lambda}(M\otimes_{k}B)\cong(A\otimes_{A}M)\otimes_{k}(V\otimes_{B}B)\cong
M\otimes_{k}V$

one sees that there is an isomorphism of complexes

$(A\otimes_{k}P_{\bullet})\otimes_{\Lambda}(M\otimes_{k}B)\cong
M\otimes_{k}P_{\bullet}$

and hence

${\rm Tor}^{\Lambda}_{i}(A\otimes_{k}V,M\otimes_{k}B)\cong{\rm
Tor}^{k}_{i}(M,V)=0.$

This shows $M\otimes_{k}B\in{\rm mon}(B,A).$ Further, by
$(A\otimes_{k}V)\otimes_{\Lambda}(M\otimes_{k}B)\cong
M\otimes_{k}V\in\mathscr{C},$ one gets $M\otimes_{k}B\in{\rm
mon}(B,\mathscr{C}).$

Conversely, if $M\otimes_{k}B\in{\rm mon}(B,\mathscr{C}),$ then by definition
$(A\otimes_{k}B)\otimes_{\Lambda}(M\otimes_{k}B)\cong
M\otimes_{k}B\in\mathscr{C}.$ Since $\mathscr{C}$ is closed under direct
summands, it follows that $M\in\mathscr{C}.$

${\rm(ii)}$ Assume that $M\in\ ^{\bot}A$ and $M\notin\ {\rm gp}(A)$. By (i),
$M\otimes_{k}B\in{\rm mon}(B,\ ^{\bot}A)\subseteq\ ^{\bot}\Lambda$, where the
inclusion follows from Theorem 3.1. Again by (i), $M\otimes_{k}B\notin{\rm
mon}(B,\ {\rm gp}(A))={\rm gp}(\Lambda)$, where the equality follows from
Theorem 2.6.

${\rm(iii)}$ Assume that $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are additive
full subcategories of $A$-mod closed under direct summands, such that ${\rm
mon}(B,\mathscr{C}_{1})={\rm mon}(B,\mathscr{C}_{2})$. We need to prove
$\mathscr{C}_{1}=\mathscr{C}_{2}$. Let $M\in\mathscr{C}_{1}$. By ${\rm(i)}$,
$M\otimes_{k}B\in{\rm mon}(B,\mathscr{C}_{1}).$ Thus $M\otimes_{k}B\in{\rm
mon}(B,\mathscr{C}_{2}).$ Again by ${\rm(i)}$, $M\in\mathscr{C}_{2}.$ This
completes the proof. $\hfill\blacksquare$

######  Proposition 4.7.

If $\Lambda$ is weakly Gorenstein, then so is $A$. Conversely, if $A$ is
weakly Gorenstein, then a semi-Gorenstein-projective $\Lambda$-module is
Gorenstein-projective if and only if it is monic.

Thus, if $A$ is weakly Gorenstein, then $\Lambda$ is weakly Gorenstein if and
only if each semi-Gorenstein-projective $\Lambda$-module is monic, or
equivalently, ${}^{\bot}\Lambda={\rm mon}(B,\ ^{\perp}A)$.

Proof. If $\Lambda$ is weakly Gorenstein, then $A$ is weakly Gorenstein, by
Lemma 4.6(ii).

Assume that $A$ is weakly Gorenstein and $M$ is a semi-Gorenstein-projective
$\Lambda$-module. If $M$ is Gorenstein-projective, then $M$ is monic, by
Theorem 2.6. If $M$ is monic, then by Theorem 3.1 and Theorem 2.6 one has
$M\in{\rm mon}(B,A)\cap\ ^{\bot}\Lambda={\rm mon}(B,\ ^{\bot}A)={\rm mon}(B,\
{\rm gp}(A))={\rm gp}(\Lambda)$. $\hfill\blacksquare$

######  Proposition 4.8.

Let $Q$ be a finite acyclic quiver. Then $A\otimes_{k}kQ$ is weakly Gorenstein
if and only if $A$ is weakly Gorenstein.

In particular, $T_{n}(A)=\left(\begin{smallmatrix}A&A&\cdots&A&A\\\
0&A&\cdots&A&A\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&0&A\end{smallmatrix}\right)$ is weakly Gorenstein if and only if
$A$ is weakly Gorenstein.

Proof. By Proposition 4.7, it remains to prove the “if” part. Assume that $A$
is weakly Gorenstein. We will prove that $\Lambda=A\otimes_{k}kQ$ is weakly
Gorenstein, by using induction on $|Q_{0}|$. If $|Q_{0}|=1$, then $\Lambda=A$
is weakly Gorenstein, by the assumption.

Assume that $|Q_{0}|\geq 2$. We write the conjunction of paths of $Q$ from
left to right. Since $Q$ is an acyclic quiver, $Q$ has a source vertex, say,
$n$, and then

$kQ=\left(\begin{smallmatrix}kQ^{\prime}&{\rm rad}P(n)\\\
0&k\end{smallmatrix}\right)$

where $Q^{\prime}$ is the subquiver of $Q$ by deleting the source vertex $n$,
and $P(n)=kQe_{n}$. Then ${\rm rad}P(n)$ is a $kQ^{\prime}$-$k$-bimodule. Thus

$\Lambda=A\otimes_{k}kQ=\left(\begin{smallmatrix}A\otimes_{k}kQ^{\prime}&M\\\
0&A\end{smallmatrix}\right)$

where $M=A\otimes_{k}{\rm rad}P(n)$ is an
$(A\otimes_{k}kQ^{\prime})$-$A$-bimodule. Since $Q$ is acyclic, so is
$Q^{\prime}$. Hence ${\rm rad}P(n)$ is a projective left $kQ^{\prime}$-module.
Thus $M=A\otimes_{k}{\rm rad}P(n)$ is a projective left
$(A\otimes_{k}kQ^{\prime})$-module, and also $M=A\otimes_{k}{\rm rad}P(n)$ is
a projective right $A$-module.

Since $|Q^{\prime}_{0}|=|Q_{0}|-1$, by induction $A\otimes kQ^{\prime}$ is
weakly Gorenstein. Applying Proposition 4.4 to
$\left(\begin{smallmatrix}A\otimes kQ^{\prime}&M\\\
0&A\end{smallmatrix}\right)=\Lambda$, one sees that $\Lambda$ is weakly
Gorenstein. $\hfill\blacksquare$

###  4.3. Semi-Groenstein-projective-free algebras

Replacing the condition that $M_{B}$ is projective in Proposition 4.4 by
“${}^{\perp}B=\operatorname{add}(B)$ and ${\rm proj.dim}M_{B}<\infty$”, we
then get the following result on lsgp-free algebras.

######  Proposition 4.9.

Assume that ${}^{\perp}B=\operatorname{add}(B)$ with ${\rm
proj.dim}M_{B}<\infty$, and that ${}_{A}M$ is torsionless with ${\rm
proj.dim}\ _{A}M<\infty$. Then $\Lambda=\left(\begin{smallmatrix}A&M\\\
0&B\end{smallmatrix}\right)$ is left weakly Gorenstein if and only if $A$ is
left weakly Gorenstein.

Moreover, if in addition ${}_{A}M$ is projective, then

${}^{\perp}\Lambda={\rm gp}(\Lambda)=\\{\binom{M\otimes_{B}P}{P}_{{\rm
Id}_{M\otimes_{B}P}}\oplus\binom{G}{0}\ |\ P\in\operatorname{add}(B),\ G\in\
^{\perp}A={\rm gp}(A)\\}$

and ${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$ if and only if
${}^{\perp}A=\operatorname{add}(A)$.

Proof. Since ${}^{\perp}B=\operatorname{add}(B)$, it follows that
$(^{\perp}B)^{\perp}=B$-mod, and hence ${\rm D}(M_{B})\in(^{\perp}B)^{\perp}$.
So, the conditions of Proposition 4.1 are satisfied. To prove the first
assertion, by Proposition 4.1, it suffices to prove that if $A$ is left weakly
Gorenstein, then any semi-Gorenstein-projective $\Lambda$-module
$\binom{X}{Y}_{\varphi}$ is monic respect to bimodule ${}_{A}M_{B}$. In fact,
applying Theorem 2.11 to $\binom{X}{Y}_{\varphi}\in\ ^{\perp}\Lambda$ one gets
that $Y\in\ ^{\perp}B$, that $\varphi:M\otimes_{B}Y\longrightarrow X$ induces
isomorphisms
$\operatorname{Ext}^{i}_{A}(X,A)\cong\operatorname{Ext}^{i}_{A}(M\otimes_{B}Y,A)$
for all $i\geq 1$, and that
$\varphi^{*}:X^{*}\longrightarrow(M\otimes_{B}Y)^{*}$ is a right
$A$-epimorphism. Since $Y\in\ ^{\perp}B=\operatorname{add}(B)$, ${}_{B}Y$ is
projective. Thus ${}_{A}(M\otimes_{B}Y)\in\operatorname{add}(_{A}M)$. Since by
assumption ${}_{A}M$ is torsionless, it follows that ${}_{A}(M\otimes_{B}Y)$
is torsionless. Thus, the canonical map
$\phi_{M\otimes_{B}Y}:M\otimes_{B}Y\longrightarrow(M\otimes_{B}Y)^{**}$ is a
monomorphism. By the same argument as in the proof of Proposition 4.4 one
concludes that $\binom{X}{Y}_{\varphi}$ is monic respect to bimodule
${}_{A}M_{B}$.

Now, assume in addition that ${}_{A}M$ is projective. Continuing the argument
above, one knows that ${}_{A}(M\otimes_{B}Y)$ is projective, thus,
$\phi_{M\otimes_{B}Y}:M\otimes_{B}Y\longrightarrow(M\otimes_{B}Y)^{**}$ is an
isomorphism. By
$\operatorname{Ext}^{i}_{A}(X,A)\cong\operatorname{Ext}^{i}_{A}(M\otimes_{B}Y,A)=0$
for all $i\geq 1$, one has $X\in\ ^{\perp}A$. Since by assumption $A$ is
weakly Gorenstein, $X\in{\rm gp}(A)$, and hence $\phi_{X}:X\longrightarrow
X^{**}$ is an isomorphism. Since $\varphi^{*}:\
X^{*}\longrightarrow(M\otimes_{B}Y)^{*}$ is an epimorphism and
$(M\otimes_{B}Y)^{*}$ is a right projective $A$-module, it follows that
$\varphi^{*}$ is a splitting epimorphism, and hence $\varphi^{**}:\
(M\otimes_{B}Y)^{**}\longrightarrow X^{**}$ is a splitting monomorphism. From
the commutative diagram

$\textstyle{M\otimes_{B}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{M\otimes_{B}Y}}$$\scriptstyle{\cong}$$\scriptstyle{\varphi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\scriptstyle{\cong}$$\textstyle{(M\otimes_{B}Y)^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{**}}$$\textstyle{X^{**}}$

one sees that $\varphi:M\otimes_{B}Y\longrightarrow X$ is also a splitting
monomorphism. Thus $X\cong(M\otimes_{B}Y)\oplus X^{\prime}$ for some
$X^{\prime}\in{\rm gp}(A)$ and
$\binom{X}{Y}_{\varphi}=\binom{M\otimes_{B}Y}{Y}_{{\rm
Id}_{M\otimes_{B}Y}}\oplus\binom{X^{\prime}}{0}$.

Since ${}_{A}M$ is projective and ${\rm proj.dim}M_{B}<\infty$, the
$A$-$B$-bimodule $M$ is compatible (cf. Lemma 2.12). By Theorem 2.13,
$\binom{X^{\prime}}{0}_{\varphi}\in{\rm gp}(\Lambda)$, and hence
$\binom{X}{Y}_{\varphi}=\binom{M\otimes_{B}Y}{Y}_{{\rm
Id}_{M\otimes_{B}Y}}\oplus\binom{X^{\prime}}{0}\in{\rm gp}(\Lambda)$. This
proves

${}^{\perp}\Lambda={\rm gp}(\Lambda)=\\{\binom{M\otimes_{B}P}{P}_{{\rm
Id}_{M\otimes_{B}P}}\oplus\binom{G}{0}\ |\ P\in\operatorname{add}(B),\ G\in\
^{\perp}A={\rm gp}(A)\\}$

and from which one sees that ${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$
if and only if ${}^{\perp}A=\operatorname{add}(A)$. $\hfill\blacksquare$

######  Remark 4.10.

The “only if” part in Proposition 4.9 does not need the conditions that
${}^{\perp}B=\operatorname{add}(B)$ and ${}_{A}M$ is torsionless.

######  Proposition 4.11.

Let $Q$ be a finite acyclic quiver, $I$ an admissible ideal of $kQ$, and
$\Lambda=A\otimes_{k}kQ/I$. Then
${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$ if and only if
${}^{\perp}A=\operatorname{add}(A)$.

Proof. Assume that ${}^{\perp}A=\operatorname{add}(A)$. We will prove
${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$, again by using induction on
$|Q_{0}|$. If $|Q_{0}|=1$, then $\Lambda=A$, thus the assertion holds, by the
assumption ${}^{\perp}A=\operatorname{add}(A)$.

Assume that $|Q_{0}|\geq 2$. Similar as in the proof of Proposition 4.8, we
write $\Lambda$ as a triangular matrix algebra. However, in order to apply
Proposition 4.9, this time we need to use the subquiver $Q^{\prime}$ of $Q$ by
deleting a sink vertex, say, $1$, and the corresponding algebra
$kQ^{\prime}/I^{\prime}$. Then

$kQ/I=\left(\begin{smallmatrix}k&{\rm rad}(e_{1}kQ/I)\\\
0&kQ^{\prime}/I^{\prime}\end{smallmatrix}\right)$

where ${\rm rad}(e_{1}kQ/I)$ is a $k$-$(kQ^{\prime}/I^{\prime})$-bimodule.
Thus

$\Lambda=A\otimes_{k}(kQ/I)=\left(\begin{smallmatrix}A&M\\\
0&\Lambda^{\prime}\end{smallmatrix}\right)$

where $\Lambda^{\prime}=A\otimes_{k}(kQ^{\prime}/I^{\prime}),\
M=A\otimes_{k}{\rm rad}(e_{1}kQ/I)$ is a $A$-$\Lambda^{\prime}$-bimodule.
Since $|Q_{0}|=|Q_{0}|-1$, by induction one gets
${}^{\perp}\Lambda^{\prime}=\operatorname{add}(\Lambda^{\prime}).$

Since ${\rm proj.dim.}{\rm rad}(e_{1}kQ/I)_{kQ^{\prime}/I^{\prime}}<\infty$,
it follows that

${\rm proj.dim.}M_{\Lambda^{\prime}}<\infty.$

Also, ${}_{A}M$ is projective. Since we already known
${}^{\perp}\Lambda^{\prime}=\operatorname{add}(\Lambda^{\prime})$ by
induction, thus we can apply Proposition 4.9 to
$\Lambda=A\otimes_{k}(kQ/I)=\left(\begin{smallmatrix}A&M\\\
0&\Lambda^{\prime}\end{smallmatrix}\right)$ to get

${}^{\perp}\Lambda={\rm
gp}(\Lambda)=\\{\binom{M\otimes_{\Lambda^{\prime}}P}{P}_{{\rm
Id}_{M\otimes_{\Lambda^{\prime}}P}}\oplus\binom{G}{0}\ |\
P\in\operatorname{add}(\Lambda^{\prime}),\ G\in\ ^{\perp}A\\}.$

Since $G\in\ ^{\perp}A=\operatorname{add}(A)$, it follows that
$\binom{G}{0}\in\operatorname{add}(\Lambda)$, and hence
${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$.

Conversely, assume that ${}^{\perp}\Lambda=\operatorname{add}(\Lambda)$. Let
$X$ be an indecomposable $A$-module with $X\in\ ^{\perp}A$. For any
indecomposable projective $(kQ/I)$-module $P$, by the Cartan-Eilenberg
isomorphism ([CE, Thm. 3.1, p.209, p.205]) one has

${\rm
Ext}^{i}_{\Lambda}(X\otimes_{k}P,A\otimes_{k}kQ/I)=\bigoplus\limits_{p+q=i}({\rm
Ext}^{p}_{A}(X,A)\otimes_{k}{\rm Ext}^{q}_{kQ/I}(P,kQ/I))=0,\ \forall\ i\geq
1.$

So $X\otimes_{k}P\in\ ^{\perp}\Lambda=\operatorname{add}(\Lambda)$, and hence
$X\in\operatorname{add}(A).$ $\hfill\blacksquare$

###  4.4. Example and Problem

Let $A$ be the algebra given by quiver
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$$\textstyle{\begin{matrix}\
1\end{matrix}}$ and relations $\beta^{2},\ \alpha\beta$. The Auslander-Reiten
quiver of $A$ is

---
$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\begin{smallmatrix}2\\\
1\end{smallmatrix}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\begin{smallmatrix}&2\\\
2&&1\end{smallmatrix}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\begin{smallmatrix}2\\\
2\end{smallmatrix}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with indecomposable projective modules $P(1)=1$ and
$P(2)=\begin{smallmatrix}&2\\\ 2&&1\end{smallmatrix}$, and indecomposable
injective modules $I(1)=\begin{smallmatrix}2\\\ 1\end{smallmatrix}$ and
$I(2)=\begin{smallmatrix}2\\\ 2\end{smallmatrix}$. Since

$\operatorname{Ext}^{1}_{A}(2,\ \begin{smallmatrix}&2\\\
2&&1\end{smallmatrix})\neq 0,\ \ \
\operatorname{Ext}^{1}_{A}(\begin{smallmatrix}2\\\ 2\end{smallmatrix},\ 1)\neq
0,\ \ \ \operatorname{Ext}^{2}_{A}(\begin{smallmatrix}2\\\
1\end{smallmatrix},\ 1)=\operatorname{Ext}^{1}_{A}(\begin{smallmatrix}2\\\
1\end{smallmatrix},\ 2)\neq 0$

one sees that $A$ is lsgp-free, i.e., ${}^{\perp}A=\operatorname{add}(A)$.
Note that $A$ is not Gorenstein. By Proposition 4.11,
${}^{\perp}(A\otimes_{k}kQ/I)={\rm
gp}(A\otimes_{k}kQ/I)=\operatorname{add}(A\otimes_{k}kQ/I)$, for any finite
acyclic quiver $Q$ and any admissible ideal $I$.

Problem 1. Are there a left weakly Gorenstein algebra $A$, a finite acyclic
quiver $Q$, and an admissible ideal $I$ of $kQ$, such that $A\otimes_{k}kQ/I$
is not left weakly Gorenstein, or equivalently, such that there is a semi-
Gorenstein-projective $(A\otimes_{k}kQ/I)$-module which is not monic?

This problem is different from Question 4. Such an algebra $A$ (if there
exists) is not Gorenstein (otherwise, $A\otimes_{k}kQ/I$ is Gorenstein, by
[AR, Proposition 2.2]); such an $I\neq 0$, by Proposition 4.8; and also
${}^{\perp}A\neq\operatorname{add}(A)$, by Proposition 4.11.

##  5\. Canonical maps of modules over $T_{2}(A)$

Let $A$ be an Artin algebra, $\Lambda=T_{2}(A)=\left(\begin{smallmatrix}A&A\\\
0&A\end{smallmatrix}\right),$ and $M$ a $\Lambda$-module. We will give a
sufficient and necessary condition, such that the canonical $\Lambda$-map
$\phi_{M}:M\longrightarrow M^{**}$ is a monomorphism (an epimorphism, and
reflexive, respectively); and we will give a sufficient and necessary
condition such that $M$ is double semi-Gorenstein-projective with $\phi_{M}$ a
monomorphism (an epimorphism, respectively).

Recall that a left $\Lambda$-module $M$ is identified with the triple
$\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$, where
$\varphi:Y\longrightarrow X$ is a left $A$-map; and a right $\Lambda$-module
is identified with a triple $(U,V)_{\psi}$, where $\psi:U\longrightarrow V$ is
a right $A$-map. Using the identifications, we will determine the right
$\Lambda$-module $M^{*}=\operatorname{Hom}_{\Lambda}(M,\ _{\Lambda}\Lambda)$,
the left $\Lambda$-module $M^{**}=\operatorname{Hom}_{\Lambda}(M^{*},\
\Lambda_{\Lambda})$, and $\phi_{M}:M\longrightarrow M^{**}$.

###  5.1. The $\Lambda$-dual of a left $\Lambda$-module

For a left $\Lambda$-module $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$, we will determine the right
$\Lambda$-module $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)^{*}_{\varphi}=\operatorname{Hom}_{\Lambda}(\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi},\ _{\Lambda}\Lambda)$. As a left
$\Lambda$-module, ${}_{\Lambda}\Lambda=\left(\begin{smallmatrix}A\\\
0\end{smallmatrix}\right)\oplus\left(\begin{smallmatrix}A\\\
A\end{smallmatrix}\right)_{{\rm Id}_{A}}=\left(\begin{smallmatrix}A\oplus A\\\
A\end{smallmatrix}\right)_{\binom{0}{{\rm Id}_{A}}}$. Thus, any $\Lambda$-map

$f\in\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)^{*}_{\varphi}=\operatorname{Hom}_{\Lambda}(\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi},\left(\begin{smallmatrix}A\oplus A\\\
A\end{smallmatrix}\right)_{\binom{0}{{\rm Id}_{A}}})$

is of the form
$\left(\begin{smallmatrix}\left(\begin{smallmatrix}\alpha_{1}\\\
\alpha_{2}\end{smallmatrix}\right)\\\ \beta\end{smallmatrix}\right)$, where
$\alpha_{1},\alpha_{2}\in X^{*}=\operatorname{Hom}_{A}(X,A),\ \beta\in Y^{*}$,
such that the square

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\varphi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\binom{\alpha_{1}}{\alpha_{2}}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\binom{0}{1}}$$\textstyle{A\oplus
A}$

commutes. So $\alpha_{1}\varphi=0$, $\beta=\alpha_{2}\varphi.$ Thus, there is
a unique $g\in({\rm Coker}\varphi)^{*}=\operatorname{Hom}_{A}({\rm
Coker}\varphi,A)$ such that $\alpha_{1}=g\pi,$ where $\pi:X\longrightarrow{\rm
Coker}\varphi$ is the canonical $A$-epimorphism.

######  Lemma 5.1.

Let $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ be a
left $\Lambda$-module with $\varphi:Y\longrightarrow X$ a left $A$-map. Then

$(\rm i)$ Any $f\in\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)^{*}_{\varphi}$ is of the form
$\left(\begin{smallmatrix}\left(\begin{smallmatrix}g\pi\\\
\alpha_{2}\end{smallmatrix}\right)\\\
\alpha_{2}\varphi\end{smallmatrix}\right)$, where $g\in({\rm
Coker}\varphi)^{*}$, $\pi:X\longrightarrow{\rm Coker}\varphi$ is the canonical
$A$-epimorphism, and $\alpha_{2}\in X^{*}$.

$(\rm ii)$ There is a unique right $\Lambda$-module isomorphism
$h:\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}^{*}\cong(({\rm Coker}\varphi)^{*},\
X^{*})_{\pi^{*}}$, given by

$f=\left(\begin{smallmatrix}\left(\begin{smallmatrix}g\pi\\\
\alpha_{2}\end{smallmatrix}\right)\\\
\alpha_{2}\varphi\end{smallmatrix}\right)\mapsto(g,\alpha_{2})$

where $\pi^{*}:({\rm Coker}\varphi)^{*}\longrightarrow X^{*}$ is the right
$A$-monomorphism induced by $\pi$.

Proof. $(\rm ii)$ We claim that $h$ is a right $\Lambda$-map, i.e.,

$h(f\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right))=h(f)\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right),\ \ \forall\
f=\left(\begin{smallmatrix}\left(\begin{smallmatrix}g\pi\\\
\alpha_{2}\end{smallmatrix}\right)\\\
\alpha_{2}\varphi\end{smallmatrix}\right)\in\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}^{*},\ \ \forall\
\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right)\in\Lambda.$

In fact, for any $\binom{x}{y}\in\binom{X}{Y}_{\varphi}$, since
$\pi\varphi=0$, one has

$\displaystyle(f\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right))\binom{x}{y}$
$\displaystyle=(f\binom{x}{y})\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right)=\left(\begin{smallmatrix}(g\pi)(x)&\alpha_{2}(x)\\\
0&\alpha_{2}(\varphi(y))\end{smallmatrix}\right)\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right)$
$\displaystyle=\left(\begin{smallmatrix}(g\pi)(x)a_{1}&(g\pi)(x)a_{2}+\alpha_{2}(x)a_{3}\\\
0&\alpha_{2}(\varphi(y))a_{3}\end{smallmatrix}\right)=\left(\begin{smallmatrix}((ga_{1})\pi)(x)&((ga_{2})\pi+\alpha_{2}a_{3})(x)\\\
0&((ga_{2})\pi+\alpha_{2}a_{3})(\varphi(y))\end{smallmatrix}\right)$
$\displaystyle=\left(\begin{smallmatrix}\left(\begin{smallmatrix}(ga_{1})\pi\\\
(ga_{2})\pi+\alpha_{2}a_{3}\end{smallmatrix}\right)\\\
((ga_{2})\pi+\alpha_{2}a_{3})\varphi\end{smallmatrix}\right)\binom{x}{y}$

Thus $f\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right)=\left(\begin{smallmatrix}\left(\begin{smallmatrix}(ga_{1})\pi\\\
(ga_{2})\pi+\alpha_{2}a_{3}\end{smallmatrix}\right)\\\
((ga_{2})\pi+\alpha_{2}a_{3})\varphi\end{smallmatrix}\right),$ and hence

$h(f\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right))=(ga_{1},(ga_{2})\pi+\alpha_{2}a_{3}).$

One the other hand, by the right $\Lambda$-module structure of $(({\rm
Coker}\varphi)^{*},X^{*})_{\pi^{*}}$, one has

$\displaystyle h(f)\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right)$
$\displaystyle=(g,\alpha_{2})\left(\begin{smallmatrix}a_{1}&a_{2}\\\
0&a_{3}\end{smallmatrix}\right)=(ga_{1},\pi^{*}(ga_{2})+\alpha_{2}a_{3})$
$\displaystyle=(ga_{1},(ga_{2})\pi+\alpha_{2}a_{3}).$

This proves the claim.

Since the map

$(({\rm
Coker}\varphi)^{*},X^{*})_{\pi^{*}}\longrightarrow\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}^{*},\ \ \ (g,\alpha_{2})\mapsto
f=\left(\begin{smallmatrix}\left(\begin{smallmatrix}g\pi\\\
\alpha_{2}\end{smallmatrix}\right)\\\
\alpha_{2}\varphi\end{smallmatrix}\right)$

is the inverse of $h$, $h$ is a right $\Lambda$-isomorphism.
$\hfill\blacksquare$

###  5.2. The $\Lambda$-dual of a right $\Lambda$-module

Similarly, one can determine the $\Lambda$-dual of a right $\Lambda$-module
$(U,V)_{\psi}$, where $\psi:U\longrightarrow V$ is a right $A$-map. As a right
$\Lambda$-module, $\Lambda_{\Lambda}=(A,A)_{{\rm
Id}_{A}}\oplus(0,A)=(A,A\oplus A)_{\binom{{\rm Id}_{A}}{0}}$. So any
$\Lambda$-map

$f\in(U,V)_{\psi}^{*}=\operatorname{Hom}_{\Lambda}((U,V)_{\psi},(A,A\oplus
A)_{\binom{{\rm Id}_{A}}{0}})$

is of the form $(\alpha,\left(\begin{smallmatrix}\beta_{1}\\\
\beta_{2}\end{smallmatrix}\right))$, where $\alpha\in
U^{*}=\operatorname{Hom}_{A}(U,A_{A}),\ \beta_{1},\ \beta_{2}\in V^{*}$, such
that

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\psi}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\binom{\beta_{1}}{\beta_{2}}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\binom{1}{0}}$$\textstyle{A\oplus
A}$

commutes. Thus $\alpha=\beta_{1}\psi,\ \beta_{2}\psi=0.$ Hence, there is a
unique $g\in({\rm Coker}\psi)^{*}=\operatorname{Hom}_{A}({\rm Coker}\psi,\
A_{A})$ such that $\beta_{2}=g\pi,$ where $\pi:V\longrightarrow{\rm
Coker}\psi$ is the canonical $A$-map. By the similar argument one has

######  Lemma 5.2.

Let $(U,V)_{\psi}$ be a right $\Lambda$-module with $\psi:U\longrightarrow V$
a right $A$-map. Then

$(\rm i)$ Any $f\in(U,V)_{\psi}^{*}$ is of the form
$(\beta_{1}\psi,\left(\begin{smallmatrix}\beta_{1}\\\
g\pi\end{smallmatrix}\right))$, where $\beta_{1}\in V^{*}$, $g\in({\rm
Coker}\psi)^{*}$, and $\pi:V\longrightarrow{\rm Coker}\psi$ is the canonical
$A$-epimorphism.

$(\rm ii)$ There is a unique left $\Lambda$-module isomorphism
$h^{\prime}:(U,V)_{\psi}^{*}\cong\left(\begin{smallmatrix}V^{*}\\\ ({\rm
Coker}\psi)^{*}\end{smallmatrix}\right)_{\pi^{*}}$, given by

$f=(\beta_{1}\psi,\left(\begin{smallmatrix}\beta_{1}\\\
g\pi\end{smallmatrix}\right))\mapsto\binom{\beta_{1}}{g}$

where $\pi^{*}:({\rm Coker}\psi)^{*}\longrightarrow V^{*}$ is the left
$A$-monomorphism induced by $\pi$.

###  5.3. The left $\Lambda$-module $\binom{X}{Y}^{**}_{\varphi}$

For any left $\Lambda$-module $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ with left $A$-map
$\varphi:Y\longrightarrow X$, by Lemma 5.1, one has the right module
isomorphism

$None$ $h:\binom{X}{Y}^{*}_{\varphi}\cong(({\rm Coker}\varphi)^{*},\
X^{*})_{\pi^{*}},\ \ \
f=\left(\begin{smallmatrix}\left(\begin{smallmatrix}g\pi\\\
\alpha_{2}\end{smallmatrix}\right)\\\
\alpha_{2}\varphi\end{smallmatrix}\right)\mapsto(g,\alpha_{2})$

where $\pi^{*}:({\rm Coker}\varphi)^{*}\longrightarrow X^{*}$ is the right
$A$-monomorphism induced by $\pi:X\longrightarrow{\rm Coker}\varphi$. Applying
Lemma 5.2 to $(({\rm Coker}\varphi)^{*},\ X^{*})_{\pi^{*}}$, we then get

######  Lemma 5.3.

$(\rm i)$ Any $f\in(({\rm Coker}\varphi)^{*},\ X^{*})^{*}_{\pi^{*}}$ is of the
form $(\beta_{1}\pi^{*},\left(\begin{smallmatrix}\beta_{1}\\\
gp\end{smallmatrix}\right))$, where $\beta_{1}\in X^{**}$, $g\in({\rm
Coker}\pi^{*})^{*}$, and $p:X^{*}\longrightarrow{\rm Coker}\pi^{*}$ is the
canonical $A$-epimorphism.

$(\rm ii)$ There is a unique left $\Lambda$-module isomorphism
$\widetilde{h}:\ \left(\begin{smallmatrix}X^{**}\\\ ({\rm
Coker}\pi^{*})^{*}\end{smallmatrix}\right)_{p^{*}}\cong\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}^{**}$, given by

$\binom{\beta_{1}}{g}\mapsto
h^{*}((\beta_{1}\pi^{*},\left(\begin{smallmatrix}\beta_{1}\\\
gp\end{smallmatrix}\right)))$

where $p^{*}:({\rm Coker}\pi^{*})^{*}\longrightarrow X^{**}$ is the
$A$-monomorphism induced by $p$, $h:\binom{X}{Y}^{*}_{\varphi}\cong(({\rm
Coker}\varphi)^{*},\ X^{*})_{\pi^{*}}$ is given in $(5.1)$, and $h^{*}:(({\rm
Coker}\varphi)^{*},\
X^{*})_{\pi^{*}}^{*}\longrightarrow\binom{X}{Y}^{**}_{\varphi}$ is induced by
$h$.

###  5.4. The canonical $\Lambda$-map
$\phi_{\binom{X}{Y}_{\varphi}}:\binom{X}{Y}_{\varphi}\longrightarrow\binom{X}{Y}_{\varphi}^{**}$

For a left $\Lambda$-module $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$, one has an exact sequence
$Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$ of left $A$-modules. Applying
$\operatorname{Hom}_{A}(-,\ _{A}A)$, one gets an exact sequence of right
$A$-modules

$0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}X^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}Y^{*}$

and the exact sequence

$0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}X^{*}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}{\rm Coker}\pi^{*}\longrightarrow 0.$

Thus, there is a unique $A$-map $\beta:{\rm Coker}\pi^{*}\longrightarrow
Y^{*}$ such that the diagram

$None$
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
Coker}\varphi)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{X^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{{\rm
Coker}\pi^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
Coker}\varphi)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{X^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$$\textstyle{Y^{*}.}$

commutes, i.e., $\varphi^{*}=\beta p.$ Thus, $\varphi^{*}$ is an epimorphism
if and only if so is $\beta$, and if and only if $\beta$ is an isomorphism. So
one has the $A$-map $\beta^{*}:Y^{**}\longrightarrow({\rm Coker}\pi^{*})^{*}.$
Consider the composition

$\beta^{*}\phi_{Y}:Y\longrightarrow({\rm Coker}\pi^{*})^{*}$

where $\phi_{Y}:Y\longrightarrow Y^{**}$ is the canonical map. By the
definition of $\phi_{Y}$ and $\beta^{*}$, one knows that
$\beta^{*}\phi_{Y}:Y\longrightarrow({\rm Coker}\pi^{*})^{*}$ is given by

$y\mapsto``g\mapsto(\beta(g))(y)",\ \ \forall\ g\in{\rm Coker}\pi^{*}$

i.e., $((\beta^{*}\phi_{Y})(y))(g)=(\beta(g))(y),\ \forall\ y\in Y.$

######  Proposition 5.4.

For any left $\Lambda$-module $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ with left $A$-map
$\varphi:Y\longrightarrow X$, with the notations above one has

$(\rm i)$
$\binom{\phi_{X}}{\beta^{*}\phi_{Y}}:\binom{X}{Y}_{\varphi}\longrightarrow\binom{X^{**}}{({\rm
Coker}\pi^{*})^{*}}_{p^{*}}$ is left $\Lambda$-map, where
$\phi_{X}:X\longrightarrow X^{**}$ and $\phi_{Y}:Y\longrightarrow Y^{**}$ are
the canonical $A$-maps, $\beta:{\rm Coker}\pi^{*}\longrightarrow Y^{*}$ is the
canonical $A$-map such that $\varphi^{*}=\beta p$, and
$\beta^{*}:Y^{**}\longrightarrow({\rm Coker}\pi^{*})^{*}$ is induced by
$\beta$.

$(\rm ii)$ The canonical $\Lambda$-map
$\phi_{\binom{X}{Y}_{\varphi}}:\binom{X}{Y}_{\varphi}\longrightarrow\binom{X}{Y}_{\varphi}^{**}$
is given by

$\phi_{\binom{X}{Y}_{\varphi}}=\widetilde{h}\circ\binom{\phi_{X}}{\beta^{*}\phi_{Y}}.$

where $\widetilde{h}:\ \left(\begin{smallmatrix}X^{**}\\\ ({\rm
Coker}\pi^{*})^{*}\end{smallmatrix}\right)_{p^{*}}\longrightarrow\binom{X}{Y}_{\varphi}^{**}$
is the isomorphism given in Lemma 5.3.

Proof. $(\rm i)$ One needs to prove the diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\beta^{*}\phi_{Y}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\textstyle{({\rm
Coker}\pi^{*})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}}$$\textstyle{X^{**}}$

commutes, i.e., $p^{*}\beta^{*}\phi_{Y}=\phi_{X}\varphi$. In fact, since
$\varphi^{*}=\beta p$, one has $\varphi^{**}=p^{*}\beta^{*}.$ By the
functorial property of the canonical map $\phi_{X}:X\longrightarrow X^{**}$
one has the commutative diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\phi_{Y}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\textstyle{Y^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{**}}$$\textstyle{X^{**}.}$

It follows that $p^{*}\beta^{*}\phi_{Y}=\varphi^{**}\phi_{Y}=\phi_{X}\varphi.$

$(\rm ii)$ We need to prove
$\phi_{\binom{X}{Y}_{\varphi}}\binom{x}{y}=\widetilde{h}(\binom{\phi_{X}}{\beta^{*}\phi_{Y}}\binom{x}{y}),\
\ \forall\ \binom{x}{y}\in\binom{X}{Y}_{\varphi}.$ For this, let
$f\in\binom{X}{Y}_{\varphi}^{*}$. By Lemma 5.1(i),
$f=\left(\begin{smallmatrix}\left(\begin{smallmatrix}g\pi\\\
\alpha_{2}\end{smallmatrix}\right)\\\
\alpha_{2}\varphi\end{smallmatrix}\right)$, where $g\in({\rm
Coker}\varphi)^{*}$, $\pi:X\longrightarrow{\rm Coker}\varphi$ is the canonical
$A$-epimorphism, and $\alpha_{2}\in X^{*}$. By the definition of
$\phi_{\binom{X}{Y}_{\varphi}}$ one has

$\phi_{\binom{X}{Y}_{\varphi}}\binom{x}{y}(f)=f\binom{x}{y}=\left(\begin{smallmatrix}\left(\begin{smallmatrix}g\pi\\\
\alpha_{2}\end{smallmatrix}\right)\\\
\alpha_{2}\varphi\end{smallmatrix}\right)\binom{x}{y}=\left(\begin{smallmatrix}(g\pi)(x)&\alpha_{2}(x)\\\
0&\alpha_{2}(\varphi(y))\end{smallmatrix}\right)\in\Lambda.$

On the other hand, by the definitions of $\beta^{*}\phi_{Y}$ and
$\widetilde{h}$ one has

$\displaystyle\widetilde{h}(\binom{\phi_{X}}{\beta^{*}\phi_{Y}}\binom{x}{y})(f)=\widetilde{h}(\binom{\phi_{X}(x)}{\beta^{*}\phi_{Y}(y)})(f)$
$\displaystyle=h^{*}((\phi_{X}(x)\pi^{*},\left(\begin{smallmatrix}\phi_{X}(x)\\\
(\beta^{*}\phi_{Y}(y))p\end{smallmatrix}\right)))(f)=(\phi_{X}(x)\pi^{*},\left(\begin{smallmatrix}\phi_{X}(x)\\\
(\beta^{*}\phi_{Y}(y))p\end{smallmatrix}\right))(h(f))$
$\displaystyle=(\phi_{X}(x)\pi^{*},\left(\begin{smallmatrix}\phi_{X}(x)\\\
(\beta^{*}\phi_{Y}(y))p\end{smallmatrix}\right))(g,\alpha_{2})=\left(\begin{smallmatrix}\phi_{X}(x)(\pi^{*}(g))&\phi_{X}(x)(\alpha_{2})\\\
0&\beta^{*}\phi_{Y}(y)(p(\alpha_{2}))\end{smallmatrix}\right)$
$\displaystyle=\left(\begin{smallmatrix}\pi^{*}(g)(x)&\alpha_{2}(x)\\\
0&\phi_{Y}(y)(\beta(p(\alpha_{2})))\end{smallmatrix}\right)=\left(\begin{smallmatrix}\pi^{*}(g)(x)&\alpha_{2}(x)\\\
0&\beta(p(\alpha_{2}))(y)\end{smallmatrix}\right)$
$\displaystyle=\left(\begin{smallmatrix}g(\pi(x))&\alpha_{2}(x)\\\
0&\varphi^{*}(\alpha_{2})(y)\end{smallmatrix}\right)=\left(\begin{smallmatrix}(g\pi)(x)&\alpha_{2}(x)\\\
0&\alpha_{2}(\varphi(y))\end{smallmatrix}\right)\in\Lambda.$

This completes the proof. $\hfill\blacksquare$

###  5.5. Torsionless $\Lambda$-modules and reflexive $\Lambda$-modules

######  Corollary 5.5.

Let $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ be a
left $\Lambda$-module, where $\varphi:Y\longrightarrow X$ is a left $A$-map.
Then

$(\rm i)$ $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$
is a torsionless $\Lambda$-module if and only if it is monic $($i.e.,
$\varphi$ is a monomorphism$)$, $X$ and $Y$ are torsionless $A$-modules.

$(\rm ii)$ $\phi_{\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}}$ is a $\Lambda$-epimorphism if and only
if $\phi_{X}$ and $\beta^{*}\phi_{Y}:Y\longrightarrow({\rm Coker}\pi^{*})^{*}$
are $A$-epimorphisms.

$(\rm iii)$ $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is a reflexive $\Lambda$-module if and
only if $\varphi$ is a monomorphism, $X$ is reflexive $A$-module, and
$\beta^{*}\phi_{Y}$ is an isomorphism.

Proof. By Proposition 5.4,
$\phi_{\binom{X}{Y}_{\varphi}}:\binom{X}{Y}_{\varphi}\longrightarrow\binom{X}{Y}_{\varphi}^{**}$
is given by

$\phi_{\binom{X}{Y}_{\varphi}}=\widetilde{h}\circ\binom{\phi_{X}}{\beta^{*}\phi_{Y}}$

where $\widetilde{h}:\ \left(\begin{smallmatrix}X^{**}\\\ ({\rm
Coker}\pi^{*})^{*}\end{smallmatrix}\right)_{p^{*}}\longrightarrow\binom{X}{Y}_{\varphi}^{**}$
is an isomorphism. Since
$\binom{\phi_{X}}{\beta^{*}\phi_{Y}}:\binom{X}{Y}_{\varphi}\longrightarrow\binom{X^{**}}{({\rm
Coker}\pi^{*})^{*}}_{p^{*}}$ is a left $\Lambda$-map, the diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\beta^{*}\phi_{Y}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\textstyle{({\rm
Coker}\pi^{*})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}}$$\textstyle{X^{**}}$

commutes, i.e., $p^{*}\beta^{*}\phi_{Y}=\phi_{X}\varphi$.

$(\rm i)$ Assume that $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is a torsionless $\Lambda$-module, i.e.,
$\phi_{\binom{X}{Y}_{\varphi}}:\binom{X}{Y}_{\varphi}\longrightarrow\binom{X}{Y}_{\varphi}^{**}$
is a $\Lambda$-monomorphism. Thus $\phi_{X}$ and $\beta^{*}\phi_{Y}$ are
monomorphisms, in particular $\phi_{Y}$ is a monomorphism, so $X$ and $Y$ are
torsionless.

Since $p^{*}\beta^{*}\phi_{Y}=\phi_{X}\varphi$ and $p^{*}:({\rm
Coker}\pi^{*})^{*}\longrightarrow X^{**}$ is a monomorphism, it follows that
$\phi_{X}\varphi$ is a monomorphism, and hence $\varphi$ is a monomorphism,
i.e., $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is
monic.

Conversely, assume that $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is monic, $X$ and $Y$ are torsionless
$A$-modules, i.e., $\varphi$ is a monomorphism, $\phi_{X}$ and $\phi_{Y}$ are
monomorphisms. By $p^{*}\beta^{*}\phi_{Y}=\phi_{X}\varphi$,
$p^{*}\beta^{*}\phi_{Y}$ is a monomorphism, and hence $\beta^{*}\phi_{Y}$ is a
monomorphism. Thus
$\phi_{\binom{X}{Y}_{\varphi}}=\widetilde{h}\circ\binom{\phi_{X}}{\beta^{*}\phi_{Y}}$
is a monomorphism, i.e., $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is a torsionless $\Lambda$-module.

The assertion $(\rm ii)$ a direct consequence of the formula
$\phi_{\binom{X}{Y}_{\varphi}}=\widetilde{h}\circ\binom{\phi_{X}}{\beta^{*}\phi_{Y}}$.

The assertion $(\rm iii)$ a direct consequence of $(\rm i)$ and
$\phi_{\binom{X}{Y}_{\varphi}}=\widetilde{h}\circ\binom{\phi_{X}}{\beta^{*}\phi_{Y}}$.
$\hfill\blacksquare$

###  5.6. Double semi-Gorenstein-projective $\Lambda$-modules

For a left $\Lambda$-module $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ with $\varphi:Y\longrightarrow X$ a left
$A$-map, by Corollary 2.14(i), $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}\in\ ^{\perp}A$ if and only if the
following conditions (1)-(3) hold:

(1) $X\in\ ^{\perp}A;$

(2) $Y\in\ ^{\perp}A;$

(3) $\varphi^{*}:X^{*}\longrightarrow Y^{*}$ is an epimorphism.

By Lemma 5.1, $\binom{X}{Y}^{*}_{\varphi}\cong(({\rm Coker}\varphi)^{*},\
X^{*})_{\pi^{*}}$ as right $\Lambda$-modules, where $\pi^{*}:({\rm
Coker}\varphi)^{*}\longrightarrow X^{*}$ is the right $A$-monomorphism induced
by $\pi:X\longrightarrow{\rm Coker}\varphi$. Thus, by the right module version
of Corollary 2.14(i), $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}^{*}\in\ ^{\perp}A$ if and only if the
following conditions (4)-(6) hold:

(4) $({\rm Coker}\varphi)^{*}\in\ ^{\perp}A;$

(5) $X^{*}\in\ ^{\perp}A;$

(6) $\pi^{**}:X^{**}\longrightarrow({\rm Coker}\varphi)^{**}$ is an
epimorphism.

######  Lemma 5.6.

Let $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ be a
left $\Lambda$-module, where $\varphi:Y\longrightarrow X$ is a left $A$-map.
Then $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is
double semi-Gorenstein-projective if and only if the conditions $(1)-(6)$
above hold, and if and only if the conditions $(1)-(8)$ hold, where

$(7)$ $Y^{*}\in\ ^{\perp}A;$

$(8)$ The canonical $A$-map $\beta:{\rm Coker}\pi^{*}\longrightarrow Y^{*}$ is
an isomorphism.

Proof. It remains to show that the conditions (1) - (6) imply the conditions
(7) and (8).

Assume that the conditions (1) - (6) hold. Since
$\varphi^{*}:X^{*}\longrightarrow Y^{*}$ is an epimorphism, $\beta:{\rm
Coker}\pi^{*}\longrightarrow Y^{*}$ is an isomorphism (cf. the diagram (5.2)).
Applying $\operatorname{Hom}_{A}(-,\ _{A}A)$ to the exact sequence
$0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}X^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}Y^{*}\longrightarrow
0,$ by the assumption that $\pi^{**}:X^{**}\longrightarrow({\rm
Coker}\varphi)^{**}$ is an epimorphism and by the assumptions $X^{*}\in\
^{\perp}A$ and $({\rm Coker}\varphi)^{*}\in\ ^{\perp}A$, one sees that
$Y^{*}\in\ ^{\perp}A.$ $\hfill\blacksquare$

###  5.7. A double semi-Gorenstein-projective $\Lambda$-module $M$ with
$\phi_{M}$ monomorphism or epimorphism

######  Proposition 5.7.

Let $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ be a
left $\Lambda$-module with left $A$-map $\varphi:Y\longrightarrow X$. Then

$(\rm i)$ $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$
is torsionless and double semi-Gorenstein-projective if and only if
$\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is monic
$($i.e. $\varphi$ is a monomorphism$)$, $X,\ Y$, and ${\rm Coker}\varphi$ are
double semi-Gorenstein-projective, and $X$ and $Y$ are torsionless.

$(\rm ii)$ $\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$
is double semi-Gorenstein-projective with epimorphism
$\phi_{\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}}$ if
and only if the following conditions are satisfied$:$

$\bullet$ $\varphi^{*}:X^{*}\longrightarrow Y^{*}$ is an epimorphism;

$\bullet$ All the five modules $X,\ Y,\ X^{*},\ Y^{*},\ ({\rm
Coker}\varphi)^{*}$ are semi-Gorenstein-projective;

$\bullet$ $\phi_{X}$ and $\phi_{Y}$ are epimorphisms.

$(\rm iii)$ (Corollary 2.14) $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is Gorenstein-projective if and only if
$\varphi$ is a monomorphism, $Y$ and ${\rm Coker}\varphi$ are Gorenstein-
projective. If this is the case, then $X$ is Gorenstein-projective.

Proof. $(\rm i)$ Assume that $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is torsionless and double semi-
Gorenstein-projective. By Corollary 5.5$(\rm i)$, $\varphi$ is a monomorphism,
and $X$ and $Y$ are torsionless. By Lemma 5.6, all the conditions (1)-(8)
hold. Applying $\operatorname{Hom}_{A}(-,\ _{A}A)$ to the exact sequence
$0\longrightarrow
Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$, since $\varphi^{*}:X^{*}\longrightarrow
Y^{*}$ is an epimorphism, and since $X$ and $Y$ are semi-Gorenstein-
projective, it follows that ${\rm Coker}\varphi$ is semi-Gorenstein-
projective.

Conversely, assume that $\varphi$ is a monomorphism, $X$, $Y$ and ${\rm
Coker}\varphi$ are double semi-Gorenstein-projective, and that $X$ and $Y$ are
torsionless. By Corollary 5.5$(\rm i)$, $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is torsionless. Again applying
$\operatorname{Hom}_{A}(-,\ _{A}A)$ to the exact sequence $0\longrightarrow
Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$, since ${\rm Coker}\varphi$ is semi-
Gorenstein-projective, $\varphi^{*}:X^{*}\longrightarrow Y^{*}$ is an
epimorphism and $0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}X^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}Y^{*}\longrightarrow
0$ is an exact sequence. Since $Y^{*}$ is semi-Gorenstein-projective,
$\pi^{**}:X^{**}\longrightarrow({\rm Coker}\varphi)^{**}$ is an epimorphism.
Thus, all the conditions (1)-(6) hold. By Lemma 5.6,
$\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is double
semi-Gorenstein-projective.

$(\rm ii)$ Assume that $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is double semi-Gorenstein-projective and
$\phi_{\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}}$ is
an epimorphism. By Lemma 5.6, all the conditions (1)-(8) are satisfied. By
Corollary 5.5$(\rm ii)$, $\phi_{X}$ and $\beta^{*}\phi_{Y}$ are epimorphisms,
where $\beta:{\rm Coker}\pi^{*}\longrightarrow Y^{*}$ is the canonical $A$-map
such that $\varphi^{*}=\beta p$, $p:X^{*}\longrightarrow{\rm Coker}\pi^{*}$ is
the canonical $A$-epimorphism, and $\beta^{*}:Y^{**}\longrightarrow({\rm
Coker}\pi^{*})^{*}$ is induced by $\beta$. It remains to show that $\phi_{Y}$
is an epimorphism. In fact, by Condition (8), $\beta^{*}$ is an isomorphism,
hence $\phi_{Y}$ is an epimorphism.

Conversely, assume that $\varphi^{*}:X^{*}\longrightarrow Y^{*}$ is an
epimorphism, all the five modules $X$, $Y$, $X^{*}$, $Y^{*}$, $({\rm
Coker}\varphi)^{*}$ are semi-Gorenstein-projective, and that $\phi_{X}$ and
$\phi_{Y}$ are epimorphisms. Since $\varphi^{*}$ is an epimorphism,
$\beta:{\rm Coker}\pi^{*}\longrightarrow Y^{*}$ is an isomorphism (cf.
Subsection 5.4), and hence $\beta^{*}$ is an isomorphism. Thus
$\beta^{*}\phi_{Y}$ is an epimorphism. By Corollary 5.5$(\rm ii)$,
$\phi_{\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}}$ is
an epimorphism.

Applying $\operatorname{Hom}_{A}(-,\ _{A}A)$ to the exact sequence
$Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$, since $\varphi^{*}:X^{*}\longrightarrow
Y^{*}$ is an epimorphism, it follows that $0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}X^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}Y^{*}\longrightarrow
0$ is an exact sequence. Since $Y^{*}$ is semi-Gorenstein-projective,
$\pi^{**}:X^{**}\longrightarrow({\rm Coker}\varphi)^{**}$ is an epimorphism.
Thus, all the conditions (1)-(6) hold. By Lemma 5.6,
$\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is double
semi-Gorenstein-projective.

$(\rm iii)$ This is just Corollary 2.14. We rewrite here, because in the
setting of (i) and (ii), it admits a simple proof. The “if” part follows from
$(\rm i)$ and $(\rm ii)$ and the fact that Gorenstein-projective modules are
closed under extensions.

Assume that $\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ is Gorenstein-projective. Then by $(\rm
i)$ and $(\rm ii)$, $\varphi$ is a monomorphism, $X$ and $Y$ are Gorenstein-
projective, and ${\rm Coker}\varphi$ is double semi-Gorenstein-projective.
Moreover, the diagram

$None$
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\phi_{Y}}$$\scriptstyle{\cong}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\phi_{X}}$$\scriptstyle{\cong}$$\textstyle{{\rm
Coker}\varphi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{{\rm
Coker}\varphi}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{**}}$$\textstyle{X^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{**}}$$\textstyle{({\rm
Coker}\varphi)^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

commutes with exact rows. So $\phi_{{\rm Coker}\varphi}$ is an isomorphism,
and thus ${\rm Coker}\varphi$ is Gorenstein-projective. $\hfill\blacksquare$

###  5.8. Problems

As remarked in [RZ4, 3.1], all known examples of double semi-Gorenstein-
projective modules $M$ such that $\phi_{M}$ is a monomorphism (an epimorphism,
respectively) are Gorenstein-projective.

Problem 2. Is there a torsionless and double semi-Gorenstein-projective module
$M$ such that $M$ is not Gorenstein-projective?

Problem 3. Is there a double semi-Gorenstein-projective module $M$ with
$\phi_{M}$ an epimorphism such that $M$ is not semi-Gorenstein-projective?

Theorem 1.6 is a result in this direction.

###  5.9. Proof of Theorem 1.6

$(\rm 1)$ Assume that any torsionless and double semi-Gorenstein-projective
$A$-module is Gorenstein-projective. Let $M=\left(\begin{smallmatrix}X\\\
Y\end{smallmatrix}\right)_{\varphi}$ be a torsionless and double semi-
Gorenstein-projective $\Lambda$-module. We need to show that $M$ is
Gorenstein-projective.

By Proposition 5.7$(\rm i)$, $\varphi$ is a monomorphism, $X$, $Y$, and ${\rm
Coker}\varphi$ are double semi-Gorenstein-projective, and $X$ and $Y$ are
torsionless. By the assumption, $X$ and $Y$ are Gorenstein-projective.

By Lemma 5.6, $\varphi^{*}$ and $\pi^{**}$ are epimorphisms. Thus, one again
has the commutative diagram (5.3) with exact rows, from which one knows that
$\phi_{{\rm Coker}\varphi}$ is also an isomorphism, and hence ${\rm
Coker}\varphi$ is Gorenstein-projective. Thus, $M$ is Gorenstein-projective,
by Proposition 5.7$(\rm iii)$.

Conversely, assume that any torsionless and double semi-Gorenstein-projective
$\Lambda$-module is Gorenstein-projective. Let $L$ be a torsionless and double
semi-Gorenstein-projective $A$-module. We need to prove that $L$ is
Gorenstein-projective.

Since $L$ is torsionless, a left ${\rm add}(A)$-approximation
$\varphi:L\longrightarrow P$ of $L$ is a monomorphism, where $P$ is a
projective $A$-module. Since both $P$ and $L$ are semi-Gorenstein-projective
and $\varphi$ is a left ${\rm add}(A)$-approximation, it follows that ${\rm
Coker}\varphi$ is also semi-Gorenstein-projective. Consider the
$\Lambda$-module $\left(\begin{smallmatrix}P\\\
L\end{smallmatrix}\right)_{\varphi}$. Since $P^{*}$ and $L^{*}$ are semi-
Gorenstein-projective and

$0\longrightarrow({\rm Coker}\varphi)^{*}\longrightarrow
P^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}L^{*}\longrightarrow
0$

is an exact sequence, it follows that $({\rm Coker}\varphi)^{*}$ is also semi-
Gorenstein-projective. Thus, by Proposition 5.7$(\rm i)$,
$\left(\begin{smallmatrix}P\\\ L\end{smallmatrix}\right)_{\varphi}$ is a
torsionless and double semi-Gorenstein-projective $\Lambda$-module. By the
assumption, $\left(\begin{smallmatrix}P\\\
L\end{smallmatrix}\right)_{\varphi}$ is Gorenstein-projective. Hence $L$ is
Gorenstein-projective, by Proposition 5.7$(\rm iii)$.

$(\rm 2)$ Assume that any double semi-Gorenstein-projective $A$-module $L$
with $\phi_{L}$ an epimorphism is Gorenstein-projective. Let
$M=\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ be a
double semi-Gorenstein-projective $\Lambda$-module such that $\phi_{M}$ is an
epimorphism. We need to prove that $M$ is Gorenstein-projective.

By Proposition 5.7$(\rm ii)$, $\varphi^{*}:X^{*}\longrightarrow Y^{*}$ is an
epimorphism, all the five modules $X$, $Y$, $X^{*}$, $Y^{*}$, $({\rm
Coker}\varphi)^{*}$ are semi-Gorenstein-projective, and $\phi_{X}$ and
$\phi_{Y}$ are epimorphisms. By the assumption, $X$ and $Y$ are Gorenstein-
projective, in particular, $\phi_{X}$ and $\phi_{Y}$ are isomorphisms.

We claim that $\varphi:Y\longrightarrow X$ is a monomorphism and ${\rm
Coker}\varphi$ is reflexive. In fact, applying $\operatorname{Hom}_{A}(-,A)$
to
$Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$, since $\varphi^{*}:X^{*}\longrightarrow
Y^{*}$ is an epimorphism, it follows that

$0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}X^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}Y^{*}\longrightarrow
0$

is an exact sequence. Since $Y^{*}$ is semi-Gorenstein-projective,

$0\longrightarrow
Y^{**}\stackrel{{\scriptstyle\varphi^{**}}}{{\longrightarrow}}X^{**}\stackrel{{\scriptstyle\pi^{**}}}{{\longrightarrow}}({\rm
Coker}\varphi)^{**}\longrightarrow 0$

is an exact sequence. Thus, by the functorial property of $\phi$ one has the
commutative diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\phi_{Y}}$$\scriptstyle{\cong}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\phi_{X}}$$\scriptstyle{\cong}$$\textstyle{{\rm
Coker}\varphi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{{\rm
Coker}\varphi}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{**}}$$\textstyle{X^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{**}}$$\textstyle{({\rm
Coker}\varphi)^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with exact rows. Since both $\phi_{Y}$ and $\varphi^{**}$ are monomorphisms,
$\varphi$ is a monomorphism. Also, this commutative diagram shows that
$\phi_{{\rm Coker}\varphi}$ is an isomorphism, i.e., ${\rm Coker}\varphi$ is
reflexive. This proves the claim.

Applying $\operatorname{Hom}_{A}(-,\ _{A}A)$ to the exact sequence
$0\longrightarrow
Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$, since $\varphi^{*}:X^{*}\longrightarrow
Y^{*}$ is an epimorphism and $X$ and $Y$ are semi-Gorenstein-projective, it
follows that ${\rm Coker}\varphi$ is also semi-Gorenstein-projective. So,
${\rm Coker}\varphi$ is double semi-Gorenstein-projective and reflexive, i.e.,
${\rm Coker}\varphi$ is Gorenstein-projective. By Proposition 5.7$(\rm iii)$,
$M=\left(\begin{smallmatrix}X\\\ Y\end{smallmatrix}\right)_{\varphi}$ is
Gorenstein-projective.

Conversely, assume that any double semi-Gorenstein-projective $\Lambda$-module
$M$ with $\phi_{M}$ an epimorphism is Gorenstein-projective. Let $L$ be a
double semi-Gorenstein-projective $A$-module such that $\phi_{L}$ is an
epimorphism. We need to show that $L$ is Gorenstein-projective.

Take a left ${\rm add}(A)$-approximation $\varphi:L\longrightarrow P$ of $L$.
Applying $\operatorname{Hom}_{A}(-,\ _{A}A)$ to the exact sequence
$L\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}P\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$, since $\varphi$ is left ${\rm
add}(A)$-approximation, $\varphi^{*}:P^{*}\longrightarrow L^{*}$ is an
epimorphism and

$0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}P^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}L^{*}\longrightarrow
0$

is an exact sequence. Since $L^{*}$ and $P^{*}$ are semi-Gorenstein-
projective, so is $({\rm Coker}\varphi)^{*}$. Thus, by Proposition 5.7$(\rm
ii)$, $\left(\begin{smallmatrix}P\\\ L\end{smallmatrix}\right)_{\varphi}$ is a
double semi-Gorenstein-projective $\Lambda$-module such that
$\phi_{\left(\begin{smallmatrix}P\\\ L\end{smallmatrix}\right)_{\varphi}}$ is
an epimorphism. By the assumption, $\left(\begin{smallmatrix}P\\\
L\end{smallmatrix}\right)_{\varphi}$ is Gorenstein-projective. Hence by
Proposition 5.7$(\rm iii)$, $L$ is Gorenstein-projective. $\hfill\blacksquare$

##  6\. Double semi-Gorenstein-projective modules which are not monic

###  6.1. Proof of Theorem 1.7

Since by assumption $Y$ is not torsionless and $\varphi:Y\longrightarrow P$ is
a left ${\rm add}(A)$-approximation of $Y$, it follows that $\varphi$ is not a
monomorphism. Thus $\binom{P}{Y}_{\varphi}$ is not a monic $T_{2}(A)$-module.
By Corollary 5.5(i), $\binom{P}{Y}_{\varphi}$ is not torsionless.

Apply $\operatorname{Hom}_{A}(-,\ _{A}A)$ to the exact sequence
$Y\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}P\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}{\rm
Coker}\varphi\longrightarrow 0$. Since $\varphi$ is a left ${\rm
add}(A)$-approximation of $Y$, $\varphi^{*}:P^{*}\longrightarrow Y^{*}$ is an
epimorphism and

$0\longrightarrow({\rm
Coker}\varphi)^{*}\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}P^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}Y^{*}\longrightarrow
0$

is an exact sequence. Since $Y^{*}$ and $P^{*}$ are semi-Gorenstein-
projective, it follows that $({\rm Coker}\varphi)^{*}$ is semi-Gorenstein-
projective and $\pi^{**}:P^{**}\longrightarrow({\rm Coker}\varphi)^{**}$ is an
epimorphism. Thus, all the conditions (1) - (6) in Subsection 5.6 are
satisfied. By Lemma 5.6, $\left(\begin{smallmatrix}P\\\
Y\end{smallmatrix}\right)_{\varphi}$ is a double semi-Gorenstein-projective
$T_{2}(A)$-module. $\hfill\blacksquare$

###  6.2. A class of double semi-Gorenstein-projective modules which are not
torsionless

From now on, $A$ is the algebra $\Lambda(q)$, which has been studied in [RZ2,
RZ3], i.e.,

$A=k\langle x,y,z\rangle/\langle x^{2},\ y^{2},\ z^{2},\ yz,\ xy+qyx,\ xz-zx,\
zy-zx\rangle$

where $q$ is an non-zero element in field $k$, and $q$ is of multiplicative
order $\infty$. Then $A$ is a short local algebra of wild representation type,
with a basis $1,\ x,\ y,\ z,\ yx,\ zx$, and with Hilbert type
$(|J/J^{2}|,|J^{2}|)=(3,2)$, where $J$ is the Jacobson radical of $A$. For the
studies on short local algebras, we refer to e.g. [L], [Y2], [CV], [AIS],
[RZ5, RZ6].

When $(a,b,c)$ ranges over $k^{3}\setminus\\{0\\}$, left $A$-modules

$M(a,b,c):={}_{A}A/[A(ax+by+cz)+{\rm soc}A]=A\bar{1}$

give all the $3$-dimensional local $A$-modules. They are $(A/J^{2})$-modules.
Since $A/J^{2}$ is commutative, ${\rm
D}(M(a,b,c))=\operatorname{Hom}_{k}(M(a,b,c),k)$ are also left
$(A/J^{2})$-modules, and hence left $A$-modules. [RZ3, Proposition A.1]
asserts that $M(a,b,c)$ and ${\rm D}(M(a,b,c))$ give all the indecomposable
left $A$-modules of dimension $3$.

For $(a,b,c)\in k^{3}\setminus\\{0\\}$, we also consider right $A$-modules

$M^{\prime}(a,b,c):=A/[(ax+by+cz)A+{\rm soc}A]=\bar{1}A.$

######  Lemma 6.1.

([RZ3, 1.7]) An indecomposable $A$-module $M$ of dimension at most $3$ is
double semi-Gorenstein-projective which are not torsionless if and only if
$M\cong M(1,-q,c)$ for some $c\in k.$ Moreover,

$M(1,-q,c)^{*}\cong(x-y)A\cong A/(x-q^{-1}y)A=M^{\prime}(1,-q^{-1},0)$

where the first isomorphism is given by $f\mapsto f(\bar{1})$, and the second
isomorphism is given by $x-y\mapsto\bar{1}$.

###  6.3. A class of double semi-Gorenstein-projective $T_{2}(A)$-modules
which are not monic

In order to apply Theorem 1.7 to get a family of double semi-Gorenstein-
projective $T_{2}(A)$-modules which are not monic, we look for a left ${\rm
add}(A)$-approximation of $M(1,-q,c)=A/A(x-qy+cz)=A\bar{1}$. Any $f\in
M(1,-q,c)^{*}=\operatorname{Hom}_{A}(A\bar{1},\ A)$ is the right
multiplication by $f(\bar{1})$. Since ${\rm Im}f\in J$,
$f(\bar{1})=c_{1}x+c_{2}y+c_{3}z+c_{4}yx+c_{5}zx$ with $c_{i}\in k$, such that
$(x-qy+cz)f(\bar{1})=0$. Thus $c_{1}+c_{2}=0,\ c_{3}=0.$ Hence
$f(\bar{1})\in(x-y)A$ and $M(1,-q,c)^{*}$ has a $k$-basis $f_{1},\ f_{2},\
f_{3}$, where $f_{i}:M(1,-q,c)\longrightarrow A$ is the left $A$-map given by

$f_{1}(\bar{1})=x-y,\ \ \ f_{2}(\bar{1})=yx,\ \ \ f_{3}(\bar{1})=zx.$

Therefore $M(1,-q,c)^{*}=f_{1}A$ and

$f_{1}:M(1,-q,c)\longrightarrow A$

is a left ${\rm add}(A)$-approximation of $M(1,-q,c)$. Applying Theorem 1.7
one gets

######  Proposition 6.2.

For all $c\in k$, the $T_{2}(A)$-modules

$X(c):=\left(\begin{smallmatrix}{}_{A}A\\\
M(1,-q,c)\end{smallmatrix}\right)_{f_{1}}$

where $f_{1}:M(1,-q,c)\longrightarrow A$ is the left $A$-map given by
$f_{1}(\bar{1})=x-y,$ are double semi-Gorenstein-projective, but not monic,
and hence not torsionless.

Moreover, one has

$(\rm i)$ $X(c)^{*}\cong((x-q^{-1}y)A,\ A_{A})_{\sigma}$, where
$\sigma:(x-q^{-1}y)A\longrightarrow A$ is the embedding;
$X(c)^{**}\cong\left(\begin{smallmatrix}A\\\
A(x-y)A\end{smallmatrix}\right)_{\iota}$ is not semi-Gorenstein-projective,
where $\iota:A(x-y)\longrightarrow A$ is the embedding. In particular,
$X(c)^{*}$ is not Gorenstein-projective.

$(\rm ii)$ If one identifies $X(c)^{**}$ with $\left(\begin{smallmatrix}A\\\
A(x-y)A\end{smallmatrix}\right)_{\iota}$, then the canonical $\Lambda$-map
$\phi_{X(c)}:X(c)\longrightarrow X(c)^{**}$ reads as

$\binom{{\rm Id}_{A}}{r_{x-y}}:\left(\begin{smallmatrix}{}_{A}A\\\
M(1,-q,c)\end{smallmatrix}\right)_{f_{1}}\longrightarrow\left(\begin{smallmatrix}{}_{A}A\\\
A(x-y)A\end{smallmatrix}\right)_{\iota}$

where $r_{x-y}$ is the right multiplication by $x-y$. Thus, $\phi_{X(c)}$ is
neither a monomorphism nor an epimorphism.

Proof. It remains to prove $(\rm i)$ and $(\rm ii)$.

$(\rm i)$ Note that there are isomorphisms

$({\rm Coker}f_{1})^{*}=(A/A(x-y))^{*}\cong(x-q^{-1}y)A\cong A/(x-q^{-2}y)A$

as right $A$-modules, where the first isomorphism is given by $g\mapsto
g(\bar{1})$, and the second isomorphism is given by $x-q^{-1}y\mapsto\bar{1}$.
We stress that

$A/A(x-y)\ncong M(1,-1,0)=A/[A(x-y)+{\rm soc}A]$

and that $(A/A(x-y))^{*}\cong(x-q^{-1}y)A\cong M(1,-1,0)^{*}$. By Lemma
5.1$(\rm ii)$, there is a right $\Lambda$-module isomorphism

$h:\ X(c)^{*}\cong((x-q^{-1}y)A,\ A_{A})_{\sigma}$

where $\sigma:(x-q^{-1}y)A\longrightarrow A$ is the embedding.

Note that $({\rm Coker}\sigma)^{*}=(A/(x-q^{-1}y)A)^{*}\cong A(x-y)A$ as left
$A$-modules, with the isomorphism given by $g\mapsto g(\bar{1})$. By Lemma
5.3$(\rm ii)$, there is a left $\Lambda$-module isomorphism

$\left(\begin{smallmatrix}A\\\ A(x-y)A\end{smallmatrix}\right)_{\iota}\cong
X(c)^{**}$

where $\iota:A(x-y)A\longrightarrow A$ is the embedding.

Note that $A(x-y)A=A(x-y)\oplus kzx$ is decomposable left $A$-module of
dimension $3$. By [RZ3, Theorem 1.5], $A(x-y)A$ not semi-Gorenstein-
projective. It follows from Corollary 2.14(1) that
$\left(\begin{smallmatrix}A\\\ A(x-y)A\end{smallmatrix}\right)_{\iota}$ is not
semi-Gorenstein-projective, and hence $X(c)^{**}$ is not semi-Gorenstein-
projective. In particular, $X(c)^{*}$ is not Gorenstein-projective.

$(\rm ii)$ To get $\phi_{X(c)}$, we apply Proposition 5.4 to
$X(c)=\left(\begin{smallmatrix}{}_{A}A\\\
M(1,-q,c)\end{smallmatrix}\right)_{f_{1}}$. It is clear that $\phi_{A}={\rm
Id}_{A}$. Since $X(c)$ is double semi-Gorenstein-projective, it follows from
Lemma 5.6 that the canonical $A$-map $\beta:{\rm Coker}\sigma\longrightarrow
M(1,-q,c)^{*}$ appeared in Proposition 5.4, is an isomorphism. Without loss of
generality, one may regarded $\beta$ as the identity. Note that

$M(1,-q,c)^{**}\cong M^{\prime}(1,-q^{-1},0)^{*}\cong A(x-y)A$

where the first isomorphism is given in Lemma 6.1, and the second isomorphism
is given by $g\mapsto g(\bar{1})$. By Proposition 5.4(ii), if we identify
$X(c)^{**}$ with $\left(\begin{smallmatrix}A\\\
A(x-y)A\end{smallmatrix}\right)_{\iota}$, then $\phi_{X(c)}$ reads as

$\binom{{\rm Id}_{A}}{\phi_{M(1,-q,c)}}:\left(\begin{smallmatrix}{}_{A}A\\\
M(1,-q,c)\end{smallmatrix}\right)_{f_{1}}\longrightarrow\left(\begin{smallmatrix}{}_{A}A\\\
A(x-y)A\end{smallmatrix}\right)_{\iota}.$

Since the diagram

$\textstyle{M(1,-q,c)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{M(1,-q,c)}}$$\scriptstyle{f_{1}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A(x-y)A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{A}$

commutes, it follows that $\phi_{M(1,-q,c)}:M(1,-q,c)\longrightarrow A(x-y)A$
is just the right multiplication $r_{x-y}$. Thus, $\phi_{M(1,-q,c)}$ is
neither a monomorphism nor an epimorphism, and hence $\phi_{X(c)}$ is neither
a monomorphism nor an epimorphism. $\hfill\blacksquare$

Acknowledgement: The author sincerely thanks Claus Michael Ringel for his
helpful discussions and comments, and the anonymous referee for suggestions on
the presentation of the paper.

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