Gorenstein projective modules over Milnor squares of rings

1.   Introduction

Let $A_i$ and $A$ be four rings with units for $i = 0, 1, 2$. Let $\pi_i:A_i \rightarrow A_0$ and $p_i:A \rightarrow A_i$ be ring homomorphisms for $i = 1, 2$. We fix a commutative diagram of rings.

If the sequence ${ 0\longrightarrow A \xrightarrow{\left(p_1, p_2\right)} A_1\oplus A_2 \xrightarrow{\binom{\pi_1}{-\pi_2}} A_0 }$ is exact, it is called a pullback, and $A$ is called a pullback ring. For the given $\pi_n$ for $n = 1, 2$, we can always define a ring $A: = \{(x, y)\in A_1\times A_2\mid (x)\pi_1 = (y)\pi_2\}$, and define $p_n: A \rightarrow A_n$ as the canonical projections for $n = 1, 2$ such that it is a standard pullback. The ring $A$ is unique up to isomorphism, and we shall always consider the standard case throughout the paper. If, moreover, one of $\pi_1$ and $\pi_2$ is surjective, then it is called a Milnor square of rings (see ^[13]), which gives the short exact sequence

Note that if $\pi_1$ is surjective, then so is $p_2$.

Milnor squares of rings appear in many aspects in mathematics, such as algebraic $K$-theory, representation theory, and homological algebra (for instance, see ^[4,8,12,13]). In ^[13], they were studied by the author Milnor in his construction of projective $A$-modules from projective $A_1$- and $A_2$-modules, and then the establishment of the Mayer–Vietoris sequence of $K$-groups. As Milnor's philosophy of investigation of projective $A$-modules, Facchini and Vámos studied injective and flat $A$-modules from injective and flat $A_1$- and $A_2$-modules, respectively, in ^[4]. Herbara and Prihoda also investigated infinitely generated projective modules in ^[8] within the framework of Milnor squares of rings. In view of the development of Gorenstein homological algebra (see, for instance, ^[2,9]), it is essential to study Gorenstein-projective modules in the context of pullback rings, and the question of how to construct Gorenstein-projective $A$-modules from Gorenstein-projective $A_1$- and $A_2$-modules arises naturally.

In this paper, we give a method to answer this question under certain conditions. Given two Gorenstein-projective $A_1$- and $A_2$-modules $G_1$ and $G_2$, and an isomorphism $g: A_0\otimes_{A_1}G_1 \rightarrow A_0\otimes_{A_2}G_2$ of $A_0$-modules, we can define the standard pullback module $G$ of the triple $(G_1, G_2, g)$ (see Section 2.3). Our main result Theorem 1.1 shows that $G$ is a Gorenstein-projective $A$-module.

Theorem 1.1. Let $A_1\stackrel{{\pi_1}}{ \rightarrow} A_0 \stackrel{{\pi_2}}{ \leftarrow} A_2$ be homomorphisms of rings such that both $\pi_1$ and $\pi_2$ are surjective, and let $A$ be the pullback ring. Denote by $A_n-{\rm{GProj}}$ the category of Gorenstein-projective $A_n$-modules for $n = 1, 2.$ Assume that the following conditions hold.

(ⅰ) ${{\rm{Tor}}}^{A_2}_1(A_0, H) = 0$ for all $H \in A_2 -{\rm{GProj}}$.

(ⅱ) ${\rm{Add}}({\rm{Ker}}(\pi_n))\subseteq A_n-{\rm{GProj}}^{\bot}$ for $n = 1, 2.$

Let $(G_1, G_2, g)$ be a triple such that $G_n$ is a Gorenstein-projective $A_n$-module for $n = 1, 2$ and $g:A_0\otimes_{A_1} G_1\simeq A_0\otimes_{A_2}G_2$ is an $A_0$-isomorphism. Then the pullback module of the triple $(G_{1}, G_{2}, g)$ is a Gorenstein-projective $A$-module.

Moreover, when $A_2$ is strongly left CM-free (see Section 3), the conditions (ⅰ) and (ⅱ) in Theorem 1.1 can be weakened as follows:

Corollary 1.2. Assume that $A_2$ is strongly left CM-free. If ${\rm{Add}}({\rm{Ker}}(\pi_1))\subseteq {\rm{A_1}}-{\rm{GProj}}^{\bot},$ and $G_n$ is a Gorenstein-projective $A_n$-module for $n = 1, 2,$ then the pullback module of the triple $(G_{1}, G_{2}, g)$ is a Gorenstein-projective $A$-module.

An important class of examples of Milnor squares of rings with both $\pi_1$ and $\pi_2$ surjective can be provided by the Morita context rings with two bimodule homomorphisms zero. The current works (see ^[5,7]) about the construction of Gorenstein-projective modules over these kinds of rings are mainly concentrated in the settings of Artin algebras and Noetherian rings, while our results, Theorem 1.3, as an application of Theorem 1.1, extends these results to the general rings.

Theorem 1.3. Let $\Lambda_{(0, 0)} = \left(\begin{smallmatrix}\Lambda & N\\ M &B\end{smallmatrix}\right)_{(0, 0)}$ be a Morita context ring with two bimodule homomorphisms zero. Assume that the following conditions hold.

(1) ${{\rm{Tor}}}^{\Lambda}_1(M, H) = 0$ for all $H \in \Lambda -{\rm{GProj}}$, and ${\rm{ Add({}_BM)\subseteq B-{\rm{GProj}}^{\bot}}}$,

(2) ${{\rm{Tor}}}^{B}_1(N, G) = 0$ for all $G \in B -{\rm{GProj}}$, and ${\rm{Add({}_\Lambda N)\subseteq \Lambda -{\rm{GProj}}^{\bot}}}$.

Let $U\in \Lambda -{\rm{GProj}}$, $V \in B -{\rm{GProj}}$. If there exist a $\Lambda$-module $X$, a $B$-module $Y$, and two short exact sequences

and

in $\Lambda-{\rm{Mod}}$ and $B-{\rm{Mod}}$, respectively, then $(X, Y, (1_M\otimes s)f, (1_N\otimes t)g)$ is a Gorensteon-projective $\Lambda_{(0, 0)}$-module.

The rest of this paper is structured as follows: In Section 2, we fix some notations and recall basic facts for later use. In Section 3, we prove the main result Theorem 1.1 in Theorem 3.3, while Lemmas 3.1 and 3.2 are the preparations. In Section 4, we prove Theorem 1.3 in Corollary 4.1.

2.   Preliminaries

In this section, we recall basic definitions and facts for later proofs.

Let $A$ be an associative ring with unit. We denote by $A {\mbox{-Mod}}$ the category of all left $A$-modules. As usual, $A-{\rm{Proj}}$ is the full subcategory of $A {\mbox{-Mod}}$ consisting of all projective modules. For a module $_AX$, we denote by ${{\rm{Add }}}(_AX)$ the full subcategory of $A {\mbox{-Mod}}$ consisting of modules isomorphic to direct summands of direct sums of $_AX$. Then we have $A-{\rm{Proj}}$ $= {{\rm{Add }}}(_AA)$.

Let $\cdots \rightarrow X_2 \rightarrow X_1 \rightarrow X_0 \rightarrow \cdots$ be a sequence in $A {\mbox{-Mod}}$ and $Y \in A {\mbox{-Mod}}$. It is called ${{\rm{Hom}}}_A(-, Y)$-exact if itself is exact and applying ${{\rm{Hom}}}_A(-, Y)$ preserves its exactness. Let $\cdots \rightarrow P_1 \stackrel{{d_1}}{ \rightarrow}P_0 \stackrel{{\delta}}{ \rightarrow}X \rightarrow0$ be a projective resolution of $_AX.$ We denote ${\rm{Ker}}(d_i)$ by $\Omega_i X$ for $i\geq 1$ and ${\rm{Ker}}(\delta)$ by $\Omega_0X.$

The composite of two homomorphisms $f: X\to Y$ and $g:Y\to Z$ will be denoted by $fg$ instead of $gf$. Thus, the image of $x\in X$ under $f$ is written as $(x)f$ or $xf$, and the image of $f$ is denoted by ${\rm{Im}}(f)$.

The composite of two functors $F: \mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{E}$ will be denoted by $G\circ F$, which is a functor from $\mathcal{C} \rightarrow \mathcal{E}.$ Let $(F, G)$ be an adjoint pair where $F: \mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$. We denote by $\eta:{\rm{Id}}_{\mathcal{C}} \rightarrow GF$ the unit, and by $\epsilon:FG \rightarrow {\rm{Id}}_{\mathcal{D}}$ the counit. We need the following fact.

Lemma 2.1. Let $(F, G)$ be an adjoint pair. Then

(1) The adjunction isomorphism $\eta_{X, Y}: {{\rm{Hom}}}_{\mathcal{D}}(FX, Y) \rightarrow {{\rm{Hom}}}_{\mathcal{C}}(X, GY)$ is defined by $f\mapsto \eta_X(Gf)$ with the inverse map $\eta_{X, Y}^{-1}$ defined by $g \mapsto (Fg)\epsilon_Y,$ where $\eta_X$ and $\epsilon_Y$ are the corresponding unit and counit maps.

(2) Let

be a commutative diagram in $\mathcal{C}$. Then the diagram

is commutative, and vice versa.

Next, we recall the definition of approximations. Let $\mathcal{D}$ be a full additive subcategory of an additive category $\mathcal{C}$ and $X$ an object in $\mathcal{C}$. A morphism $f: X\to D$ in $\mathcal{C}$ is called a left $\mathcal{D}$-approximation of $X$ if $D\in \mathcal{D}$ and ${{\rm{Hom}}}_{\mathcal{C}}(f, D'): {{\rm{Hom}}}_{\mathcal{C}}(D, D')\to {{\rm{Hom}}}_{\mathcal{C}}(X, D')$ is surjective for any object $D'\in\mathcal{D}$. Dually, a morphism $f: D\to X$ in $\mathcal{C}$ is called a right $\mathcal{D}$-approximation of $X$ if $D\in \mathcal{D}$ and ${{\rm{Hom}}}_{\mathcal{C}}(D', f): {{\rm{Hom}}}_{\mathcal{C}}(D', D)\to {{\rm{Hom}}}_{\mathcal{C}}(D', X)$ is surjective for any object $D'\in\mathcal{D}$. Note that left and right approximations are termed as preenvelopes and precovers in ring theory, respectively.

Let $0 \rightarrow X \stackrel{{\alpha}}{ \rightarrow} T \rightarrow Y \rightarrow 0$ be a short exact sequence with $_AT$ projective. Then $\alpha$ is a left ${{\rm{Add }}}(_AA)$-approximation of $X$ if and only if ${{\rm{Ext}}}^1_A(Y, P) = 0$ for all $P\in A-{\rm{Proj}}$ if and only if it is ${{\rm{Hom}}}_A(-, P)$-exact for every $P\in A-{\rm{Proj}}$.

The following homological facts are often used.

Lemma 2.2. Let $A$ and $B$ be unitary rings, and let $_AX$ be an $A$-module, $_BU$ a $B$-module, and $_BM_A$ a $B$-$A$-bimodule. If ${{\rm{Tor}}}^{A}_{1}(M, X) = {{\rm{Ext}}}^{1}_{B}(M, U) = 0$, then

Proof. Consider the short exact sequences

and

where $P\in A-{\rm{Proj}}$, $I\in B-{\rm{Inj}}$. Since ${{\rm{Tor}}}^{A}_{1}(M, X) = {{\rm{Ext}}}^{1}_{B}(M, U) = 0$ by assumption, applying $_BM\otimes_{A}-$ and ${{\rm{Hom}}}_{B}(M, -)$ yield two short exact sequences

and

respectively. Note that Hom is a functor of two variables. Then we have the following commutative diagram with exact rows and columns

In fact, the exactness of the left column follows from the commutative diagram with exact rows

Similarly, one can verify that the right column and the middle row are exact. Then by the Snake Lemma, we have ${{\rm{Ext}}}^{1}_{A}(X, {}_B(M, U))\simeq {{\rm{Ext}}}^{1}_{B}(M\otimes_{A}X, U)$. □

2.1.   Gorenstein-projective modules

Recall that a complete projective resolution is an exact sequence

such that ${{\rm{Hom}}}_A({P}^{\bullet}, P)$ is exact for every projective $A$-module $P$. An $A$-module $X$ is called Gorenstein-projective if there exits a complete projective resolution complex ${P}^{\bullet}$ such that $X\simeq {\rm{Im}}(d_0^P) = {\rm{Ker}}(d_P^0)$. One can observe that if ${P}^{\bullet}$ is a complete projective resolution, then all the images and kernels of ${P}^{\bullet}$ are Gorenstein-projective. We denote by $A-{\rm{GProj}}$ the category of all Gorenstein-projective $A$-modules, and let $A\mbox{-GProj}^{\bot}: = \{_AX\mid {{\rm{Ext}}}^1_A(M, X) = 0, \forall _AM \in A\mbox{-GProj} \}$. We list the following well-known properties for later use (see also ^[9]).

Lemma 2.3. Let $0 \rightarrow G' \rightarrow G \rightarrow G'' \rightarrow 0$ be a short exact sequence in $A-{\rm{Mod}}$.

(1) If $G$ and $G''$ are Gorenstein-projective, then $G'$ is Gorenstein-projective.

(2) If $G$, $G'$ are Gorenstein-projective, and ${{\rm{Ext}}}^1_A(G'', P) = 0$ for all projective $A$-modules $P$, then $G''$ is Gorenstein-projective.

Lemma 2.4. For any module $_AG$, the following two conditions are equivalent.

(1) $_AG$ is Gorenstein-projective.

(2) There exists a ${{\rm{Hom}}}_A(-, P)$-exact sequence $0 \rightarrow G \rightarrow Q^0 \rightarrow Q^1 \rightarrow\cdots$ for every projective $A$-module $P$, and ${{\rm{Ext}}}^i_A(G, P) = 0$ for all $i\geq1$ and all projective $A$-modules $P$.

Moreover, if $M \in A-{\rm{Mod}}$ such that ${{\rm{projdim}}} (_AM) < \infty,$ then $M \in A-{\rm{GProj}}^{\bot}$.

2.2.   Modules under change of rings

Let $f: A \rightarrow B$ be a homomorphism of rings. Then every $B$-module $U$ can be viewed as an $A$-module by defining $a\cdot u: = (a)f u$ for all $a\in A$ and $u\in U$. Thus we get the so-called restriction functor ${}_{A}(-): B {\mbox{-Mod}} \rightarrow A {\mbox{-Mod}}$. Moreover, there is an adjoint pair $(B\otimes_{A}-, {}_{A}(-))$ of functors whose unit is the canonical homomorphism of $A$-modules:

for $X\in A {\mbox{-Mod}}$, and the counit

for $U\in B {\mbox{-Mod}}$.

By Lemma 2.1, we have the following fact.

Lemma 2.5. Let $X \in A-{\rm{Mod}}$, $U\in B-{\rm{Mod}}$. Then

(1) There is an adjunction isomorphism

with the inverse map $\eta^{-1}_{X, U}$ defined by $(g)\eta^{-1}_{X, U} = (1_B\otimes g)\epsilon_U:b\otimes x\mapsto b((x)g), \ x\in X, \ b\in B.$

(2) The cannonical isomorphism $B\otimes_A(A)^I \rightarrow (B)^I$ maps $b\otimes (a_i)$ to $b(a_if)$ is the image of $f^I$ under $\eta^{-1}_{A^I, B^I},$ where $I$ denotes any cardinal number and $f^I$ denotes the diagonal map.

If $f$ is surjective, then $B\otimes_{A}X \simeq X/{\rm{Ker}}(f)X$ for $X \in A {\mbox{-Mod}}$, and we have the following lemma.

Lemma 2.6. Let $X \in A-{\rm{Mod}}$, $U\in B-{\rm{Mod}}$. Then

(1) The unit map $\eta_X: X \rightarrow B\otimes_AX$ is surjective.

(2) The counit map $\epsilon_U: B\otimes_{A}U \longrightarrow U$ is an isomorphism.

(2) Let $g \in {{\rm{Hom}}}_A(X, U).$ If $g$ is surjective, then so is $(g)\eta^{-1}_{X, U}$.

Remark that the unit map in Lemma 2.6(1) is actually the composite $X \rightarrow X/{\rm{Ker}}(f)X \simeq B\otimes_AX.$ Next we study when a ring extension preserves Gorenstein-projective modules.

Lemma 2.7. Let $f:A \rightarrow B$ be a surjective homomorphism of rings. Suppose that ${{\rm{Tor}}}^{A}_1(B, H) = 0$ for all $H \in A -{\rm{GProj}}$, and ${\rm{Add}}({\rm{Ker}}(f))\subseteq A-{\rm{GProj}}^{\bot}$. If $X\in A -{\rm{GProj}}$, then $B\otimes_A X \in B -{\rm{GProj}}$.

Proof. Since $X\in A -{\rm{Gproj}}$, there exists a complete projective resolution ${P}^{\bullet}$ such that ${\rm{Ker}}(d_P^0) = X.$ Since ${{\rm{Tor}}}^{A}_1(B, H) = 0$ for all Gorenstein-projective $A$-modules, we have that $B\otimes_A{P}^{\bullet}$ is exact with ${\rm{Ker}}(1_B \otimes d^0_P) = B\otimes_A X$.

Next we shall show that ${{\rm{Hom}}}_B({}B\otimes_A{P}^{\bullet}, Q)$ is exact for every $Q \in B -{\rm{Proj}}$, while by the isomorphism ${{\rm{Hom}}}_B({}B\otimes_A{P}^{\bullet}, Q)\cong {{\rm{Hom}}}_A({P}^{\bullet}, Q),$ we only need to show that ${{\rm{Hom}}}_A({P}^{\bullet}, Q)$ is exact. Then it is sufficient to show that ${}_AQ\subseteq A\mbox{-GProj}^{\bot}$. Since ${\rm{Add}}({\rm{Ker}}(f))\subseteq A\mbox{-GProj}^{\bot},$ by the short exact $0 \longrightarrow {\rm{Ker}}(f) \longrightarrow A \longrightarrow B \longrightarrow 0,$ we have ${{\rm{Add }}}({}_AB)\subseteq A\mbox{-GProj}^{\bot},$ and thus ${}_AQ\subseteq A\mbox{-GProj}^{\bot}.$ Then it follows that ${{\rm{Hom}}}_A({P}^{\bullet}, Q)$ is exact. □

2.3.   Modules over Milnor squares of rings

In this subsection, we mainly study the module category of a pullback ring.

Let $(*)$ be a Milnor square of rings. From whom we obtain four left adjoint functors listed in the following diagram

where $F$, $L$ denote the left adjoint functors $A_1\otimes_{A}-$ and $A_2\otimes_{A}-$, respectively, and $F_n$ denotes the left adjoint functors $A_0\otimes_{A_{n}}-$ for $n = 1, 2.$ We will fix the above notations in the rest of the paper. Note that there is a natural isomorphism of functors $\pi:F_1F \rightarrow A_0\otimes_A- \rightarrow F_2L.$ As for $F_n,$ we denote the unit map by $\eta_n:X_n \longrightarrow F_nX_n$ for $X_n \in A_n {\mbox{-Mod}}$, and the counit map $\epsilon_n:F_nY \longrightarrow Y$ for $Y\in A_0 {\mbox{-Mod}}$.

Following ^[4], we study the category of $A {\mbox{-Mod}}$ by the so-called fiber product category $\mathcal{F}$ of $A_1 {\mbox{-Mod}}$ and $A_2 {\mbox{-Mod}}$. An object in $\mathcal{F}$ is a triple $(X_1, X_2, x)$, where $X_n\in A_n {\mbox{-Mod}}$, $n = 1, 2$, and $x:F_1X_1 \rightarrow F_2X_2$ is an $A_0$-isomorphism. A homomorphism from a triple $(X_1, X_2, x)$ to another triple $(Y_1, Y_2, y)$ in $\mathcal{F}$ is a pair $(\alpha_{1}, \alpha_2)$ of $A_n$-homomorphisms $\alpha_{n}:X_n \rightarrow Y_n$ such that the following square commutes.

Then for an $A$-module $X$, there is an associated triple $(FX, LX, \pi_X)$ where $\pi_X:F_1FX \rightarrow F_2LX$ is the composite of the cannonical $A_0$-isomorphisms $F_1FX\stackrel{{\simeq}}{ \rightarrow} A_0\otimes_AX\stackrel{{\simeq}}{ \rightarrow} F_2LX$.

Conversely, for a triple $(X_1, X_2, x)$ in $\mathcal{F}$, there is an associated pullback module ${{\rm{Pb}}}(X_1, X_2, x): = \{(x_1, x_2)\in X_1\times X_2\mid (1_{A_1}\otimes x_1)x = 1_{A_2}\otimes x_2\}$. Write $X$ for ${{\rm{Pb}}}(X_1, X_2, x)$ for brevity. Then $X$ gives the following standard pullback diagram of $A$-modules:

and the short exact sequence ${0\longrightarrow X \xrightarrow{(j_1, j_2)} X_1\oplus X_2\xrightarrow{\binom{\eta_1x}{-\eta_2}} F_2X_2\longrightarrow 0}$, where $j_n$ is the canonical projection for $n = 1, 2$. We denote by $\widetilde{j_1}$ and $\widetilde{j_2}$ the image of $j_1$ and $j_2$ under the isomorphism ${{\rm{Hom}}}_{A}(X, X_1) \simeq {{\rm{Hom}}}_{A_1}(FX, X_1)$ and ${{\rm{Hom}}}_{A}(X, X_2) \simeq {{\rm{Hom}}}_{A_2}(LX, X_2)$, respectively.

In this language, we summarize Milnor's classical description of projective modules over Milnor squares of rings as follows.

Lemma 2.8. ^[13] An $A$-module $P$ is projective if and only if there is a triple $(P_1, P_2, p)$ in $\mathcal{F}$ such that $P_n$ is projective as $A_n$-module for $n = 1, 2,$ and $P\simeq {{\rm{Pb}}}(P_1, P_2, p).$ Moreover, if $(P_1, P_2, p)$ is such a triple in $\mathcal{F}$ with $P$ its pullback module, then $FP\simeq P_1,$ and $LP\simeq P_2.$

In the following lemma, we discuss the case that for every triple $(X_1, X_2, x)$ with $X$ its pullback module, it holds that $FX\simeq X_1$ and $LX \simeq X_2$.

Lemma 2.9. If $\pi_n$ is surjective for $n = 1, 2,$ then both $\widetilde{j_1}:FX \rightarrow X_1$ and $\widetilde{j_2}:LX \rightarrow X_2$ are isomorphisms.

Proof. Since $\pi_n$ is surjective, $p_n$ is also surjective for $n = 1, 2$ by the pullback diagram $(*).$ Now consider the pullback diagram of $A$-modules with exact rows

where $K_2$ denotes ${\rm{Ker}}(\pi_2),$ and ${\rm{Ker}}(j_1) = (0, K_2X_2).$ By Lemma 2.6(1), the unit map $\eta_n$ is surjective for $n = 1, 2.$ Then $j_n$ is also surjective by the pullback diagram, and so is $\widetilde{j_n}$ by Lemma 2.6(3).

Next, we shall show that $\widetilde{j_n}$ is injective. We only prove the case for $n = 1,$ since the proof of $n = 2$ is similar. Applying $F$ to the upper row of the above diagram gives the exact sequence

We claim that ${\rm{Im}}(Fi_1) = 0.$ Actually, let $a_1\otimes (0, k_2x_2) \in F{\rm{Ker}}(j_1)$ with $a_1\in A_1,$ $k_2\in K_2,$ $x_2\in X_2.$ Since $\eta_1x$ is surjective, there exists a $x_{1}\in X_1$ such that $(x_{1}, x_{2})\in X$. Then $a_1\otimes (0, k_2x_2) = a_1\otimes (0, k_2)(x_1, x_2) = a_1(0, k_2)\otimes (x_1, x_2)$ in $FX,$ but $(0, k_2)$ obviously lies in ${\rm{Ker}}(p_1),$ thus $a_1(0, k_2) = 0,$ and our claim follows, which implies that $\widetilde{j_1}$ is an isomorphism. □

A module $_AX$ is called separated (see ^[12]), if there exists an $A$-monomorphism $X \rightarrow X_1\oplus X_2$ for some $A_n$-module $X_n$, $n = 1, 2$. By Lemma 2.8, a projective module is separated. Thus, a Gorenstein-projective module is also separated by the definition that it can be embedded into a projective module. The following lemma, which can be obtained from [12, Corollary 3.3], shows that a separated module must be a pullback module.

Lemma 2.10. Let $X$ be an $A$-module. Then the following are equivalent.

(1) $X$ is separated.

(2) $(p_1\otimes1_X, p_2\otimes1_X): X \longrightarrow FX \oplus LX$ is a monomorphism.

(3) There is a short exact sequence

(4) $X$ is isomorphic to a pullback module of a triple in $\mathcal{F}$.

Proof. $(1) \Rightarrow (2)$ Since $X$ is separated, there exists a monomorphism $(c_1, c_2):X \rightarrow X_1 \oplus X_2$ with $A_n$-module $X_n,$ Note that $p_1\otimes1_X$ and $p_2\otimes1_X$ are actually the unit maps corresponding to the left adjoint functions $F$ and $L$, respectively. Then $(c_1, c_2)$ factors through $FX\oplus LX$; that is, there is a commutative diagram.

where $c = \big(\begin{smallmatrix}(c_1)\eta_{X, X_1}^{-1} & 0\\ 0 & (c_2)\eta_{X, X_2}^{-1} \end{smallmatrix}\big)$. This shows that $(p_1\otimes1_X, p_2\otimes1_X)$ is indeed a monomorphism.

$(2) \Rightarrow (3)$ This follows from the commutative diagram

where the upper row is obtained from applying $-\otimes_{A}X$ to ($\dagger$), and $X'$ denotes ${\rm{Ker}} \binom{\eta_1\pi_X}{-\eta_2}.$ Since $(p_1\otimes1_X, p_2\otimes1_X)$ is injective, it follows that $X\simeq X'.$

$(3) \Rightarrow (4)$ $(3)$ implies that $X$ is isomprphic to ${\rm{Ker}}\binom{\eta_1\pi_X}{-\eta_2}$, which is the pullback module of the triple $(FX, LX, \pi_X).$

$(4) \Rightarrow (1)$ is trivial. □

Finally, we have an analogue of Lemma 2.8.

Proposition 2.11. Suppose that $\pi_n$ is surjective for $n = 1, 2$. Then an $A$-module $X$ is separated if and only if there is a triple $(X_1, X_2, x)$ in $\mathcal{F}$ such that $X\simeq {{\rm{Pb}}}(X_1, X_2, x).$ Moreover, if $(X_1, X_2, x)$ is a triple in $\mathcal{F}$ with $X$ its pullback module, then $FX \simeq X_1$ and $LX\simeq X_2.$

Proof. This follows from Lemmas 2.10 and 2.9.□

3.   Proof of the main results

In this section, we always assume that $(*)$ is a Milnor square of rings such that $\pi_n$ is surjective for $n = 1, 2.$

We call a triple $(G_1, G_2, g)$ in $\mathcal{F}$ a Gorenstein triple if $G_n \in A_n-{\rm{GProj}}$ for $n = 1, 2.$ Our purpose is to construct a Gorenstein-projective $A$-module from a given Gorenstein triple.

We know that each Gorenstein-projective module can be embedded into a projective module such that the cokernel is also Gorenstein-projective by its definition, while the following lemma will lift this property to a whole Gorenstein triple, which will be crucial to our later proof.

Lemma 3.1. Assume that ${\rm{Add(_{A_n}{\rm{Ker}}(\pi_n))\subseteq A_n-{\rm{GProj}}^{\bot}}}$ for $n = 1, 2.$ Then for a Gorenstein triple $(G_1, G_2, g),$ there exists a $(T_1, T_2, t)$ and a homomorphism $(\alpha_1, \alpha_2):(G_1, G_2, g) \rightarrow (T_1, T_2, t)$ in $\mathcal{F}$ such that $_{A_n}T_n\in {{\rm{Add }}}(_{A_n}A_n)$ and $\alpha_n$ is a left ${{\rm{Add }}}(_{A_n}A_n)$-approximation of $G_n,$ for $n = 1, 2$.

Proof. Our proof proceeds in two steps:

Step 1. Construction of $(T_1, T_2, t).$ Since $G_1 \in A_1 -{\rm{GProj}}$, there is a monomorphism $a:G_1 \rightarrow P_1$ which is a left ${{\rm{Add }}}(_{A_1}A_1)$-approximation of $G_1.$ By choosing a projective module $Q_1,$ we can modify $a$ as $a_1:G_1 \stackrel{{(a, 0)}}{ \rightarrow} P_1 \oplus Q_1 \simeq (A_1)^I,$ where $I$ denotes any cardinal number. Similarly, there is also a monomorphism $a_2:G_2 \rightarrow (A_2)^J$, which is a left ${{\rm{Add }}}(_{A_2}A_2)$-approximation of $G_2.$ Now write $T_1$ for $(A_1)^I\oplus (A_1)^J$ and $T_2$ for $(A_2)^I \oplus (A_2)^J.$ Considering the canonical isomorphism

where $t_n^I$ denotes the image of $(\pi_n)^I$ under the isomorphism ${{\rm{Hom}}}_{A_n}((A_n)^I, (A_0)^I) \rightarrow {{\rm{Hom}}}_{A_0}(F_n(A_n)^I, (A_0)^I)$, which is an isomorphism by Lemma 2.5(2), and $t_n^J$ is defined in the same way for $n = 1, 2.$ Then we naturally get a triple $(T_1, T_2, t)$ in $\mathcal{F}.$

Step 2. Construction of $(\alpha_1, \alpha_2).$ Our aim is to construct two homomorphisms $b_1: G_1 \rightarrow(A_1)^J$ and $b_2:G_2 \rightarrow (A_2)^I$ such that $(\alpha_1, \alpha_2): = ((a_1, b_1), (b_2, a_2)):(G_1, G_2, g) \rightarrow (T_1, T_2, t)$ is a homomorphism in $\mathcal{F}$. If so, it will follow from $a_n$ being an ${{\rm{Add }}}(_{A_n}A_n)$-approximation that $\alpha_n$ is also an ${{\rm{Add }}}(_{A_n}A_n)$-approximation for $n = 1, 2,$ and then we will complete the proof.

Actually, by the definition of $t$ and by the diagram (2.1), our desired $b_n$ should make the following diagram commute.

that is, $b_1: G_1 \rightarrow(A_1)^J$ and $b_2: G_2 \rightarrow(A_2)^I$ should satisfy $(F_1b_1)t_1^J(t_2^J)^{-1} = gF_2a_2,$ and $(F_1a_1)t_1^I(t_2^I)^{-1} = gF_2b_2$ respectively.

To obtain $b_1: G_1 \rightarrow(A_1)^J$, we need to consider the composite map $g(F_2a_2)t_2^J:F_1G_1 \rightarrow (A_0)^J$ in $A_0 {\mbox{-Mod}}$, and denote it by $g_0.$ Let $\widetilde{g_0}$ be the image of $g_0$ under the isomorphism $\eta_{G_1, (A_0)^J}: {{\rm{Hom}}}_{A_0}(F_1G_1, (A_0)^J)\simeq {{\rm{Hom}}}_{A_1}(G_1, (A_0)^J).$ By the assumption that ${\rm{Add(_{A_1}{\rm{Ker}}}}(\pi_1))\subseteq {\rm{A_1{-GProj}^{\bot}}}$, we have ${{\rm{Ext}}}_{A_1}^1(G_1, {\rm{Ker}}(\pi_1)^J) = 0.$ Then applying ${{\rm{Hom}}}_{A_1}(G_1, -)$ to the short exact sequence

preserves its exactness. Thus there exists a $b_1: G_1 \rightarrow(A_1)^J$ such that $b_1(\pi_1)^J = \widetilde{g_0}.$ Note that by Lemma 2.5(1), we have $g_0 = (F_1\widetilde{g_0})\epsilon_1 = (F_1b_1)(F_1(\pi_1)^J)\epsilon_1$ with $\epsilon_1$ the counit map $F_1(A_0)^J \rightarrow (A_0)^J,$ but again by Lemma 2.5(1) and the definition of $t_1^J$, we have $(F_1(\pi_1)^J)\epsilon_1 = t_1^J,$ and then it follows that $(F_1b_1)t_1^J = g_0: = g(F_2a_2)t^J_2,$ as desired. Similarly, by considering the composite map $g^{-1}(F_1a_1)t_1^I,$ the existence of the desired $b_2$ can also be proved. This completes the proof. □

The following lemma exhibits a situation that makes an $A$-module $X$ lie in the left Ext-orthogonal group of ${{\rm{Add }}}(_AA),$ which is a necessary condition for $_AX$ to be Gorenstein-projective.

Lemma 3.2. Suppose that $F_2$ sends Gorenstein-projective $A_2$-modules to Gorenstein-projective $A_0$-modules. Let $0 \rightarrow X \rightarrow T \rightarrow Y \rightarrow 0$ be a short exact sequence in $A-{\rm{Mod}}$ where $T$ is projective and both $X$ and $Y$ are pullback modules of two Gorenstein triples. If ${{\rm{Tor}}}_{1}^{A}(A_n, X) = 0,$ and ${{\rm{Tor}}}_{1}^{A}(A_n, Y) = 0$ for $n = 1, 2$, then we have ${{\rm{Ext}}}^{1}_{A}(X, P) = 0$ for all $P\in A-{\rm{Proj}}$.

Proof. Take $P\in A-{\rm{Proj}}$ and write $P_n$ for $A_n\otimes_AP\in A_n -{\rm{Proj}}$ for $n = 0, 1, 2.$ By applying $-\otimes_AP$ to $(\dagger),$ we have the short exact sequence

Applying ${{\rm{Hom}}}_A(Y, -)$ to the above sequence yields the long exact sequence

Then we only need to show that ${{\rm{Ext}}}^{1}_{A}(Y, P_0) = 0,$ and ${{\rm{Ext}}}^{2}_{A}(Y, P_1\oplus P_2)\simeq {{\rm{Ext}}}^{1}_{A}(X, P_1\oplus P_2) = 0.$ If so, it will follow that ${{\rm{Ext}}}_A^2(Y, P) = 0,$ and thus ${{\rm{Ext}}}_A^1(X, P) = 0$ by the dimension shifting.

We first show that ${{\rm{Ext}}}^{1}_{A}(X, P_n) = 0$ for $n = 1, 2.$ Assume that $X$ is a pullback module of a Gorenstein triple $(X_1, X_2, x).$ By Lemma 2.9, we have $FX \simeq X_1,$ and $LX \simeq X_2,$ and thus $FX\in A_1 -{\rm{GProj}}$ and $LX\in A_2 -{\rm{GProj}}$. Then, by Lemma 2.2 and the assumption that ${{\rm{Tor}}}_{1}^{A}(A_n, X) = 0$ for $n = 1, 2$, we have ${{\rm{Ext}}}_A^1(X, P_1)\simeq {{\rm{Ext}}}_{A_1}^1(FX, P_1) = 0$, and ${{\rm{Ext}}}_A^1(X, P_2)\simeq {{\rm{Ext}}}_{A_2}^1(LX, P_2) = 0.$

Next, we show that ${{\rm{Ext}}}^{1}_{A}(Y, P_0) = 0.$ Since $Y$ is also a pullback module of a Gorenstein triple, we similarly have $FY\in A_1 -{\rm{GProj}}$ and $LY\in A_2 -{\rm{GProj}}$. Then, by the assumption on $F_2,$ we have $A_0\otimes_AY \simeq F_2LY \in A_0 -{\rm{GProj}}$. Applying $-\otimes_{A}Y$ to $(\dagger)$ yields a long exact sequence

Since $Y$ is separated, it follows from Lemma 2.10(2) that ${{\rm{Tor}}}^A_1(A_0, Y) \simeq {{\rm{Tor}}}^A_1(A_1, Y) \oplus {{\rm{Tor}}}^A_1(A_2, Y) = 0.$ Then by Lemma 2.2, we have ${{\rm{Ext}}}_A^1(Y, P_0)\simeq {{\rm{Ext}}}_{A_0}^1(A_0\otimes_AY, P_0) = 0.$ □

Now it is time to prove our main result.

Theorem 3.3. Let $(G_1, G_2, g)$ be a Gorenstein triple in $\mathcal{F}$, and $G: = {{\rm{Pb}}}(G_{1}, G_{2}, g).$ Assume that the following conditions hold.

(ⅰ) ${{\rm{Tor}}}^{A_2}_1(A_0, H) = 0$ for all $H \in A_2 -{\rm{GProj}}$.

(ⅱ) ${\rm{Add}}({\rm{Ker}}(\pi_n))\subseteq A_n-{\rm{GProj}}^{\bot}$ for $n = 1, 2.$

Then we have that

${\rm{(a)}}$ There exists a short exact sequence $0 \rightarrow G \rightarrow T \rightarrow G^1 \rightarrow 0$ in $A-{\rm{Mod}}$ where $T$ is projective, and $G^1$ is a pullback module of a Gorenstein triple in $\mathcal{F}$, and ${{\rm{Tor}}}^A_1(A_n, G^1) = 0$ for $n = 1, 2.$

${\rm{(b)}}$ $\Omega_0 G$ is a pullback module of a Gorenstein triple in $\mathcal{F}$, and ${{\rm{Tor}}}^A_1(A_n, G) = 0$ for $n = 1, 2.$

${\rm{(c)}}$ $G\in A-{\rm{GProj}}$.

Proof. By the conditions (ⅰ), and (ⅱ) for $n = 2,$ it follows from Lemma 2.7 that $F_2$ sends Gorenstein-projective $A_2$-modules to $A_0$-modules, which will be used as a fact in the following proof.

(a) By Lemma 3.1, we have a triple $(T_1, T_2, t)$ and a homomorphism $(\alpha_1, \alpha_2)$ from $(G_1, G_2, g)$ to $(T_1, T_2, t)$ in $\mathcal{F}$ with $\alpha_n$ an ${{\rm{Add }}}(A_n)$-approximation of $G_n.$ Since $G_n$ can be embedded into a projective module, the embedding map must factor through $\alpha_n$, and it follows that $\alpha_n$ is injective, and we have the short exact sequence in $A_{n} {\mbox{-mod}}$

where $L_{n}\in A_{n} -{\rm{GProj}}$ by Lemma 2.3(2) for $n = 1, 2$. Then, by applying $F_n$ to (3.2) and by the condition (ⅰ) and diagram (2.2), we obtain the commutative diagram with an exact lower row

where the induced map $l$ is an isomorphism by the Five Lemma. Denote by $T$ the pullback module of the triple $(T_1, T_2, t).$ By the definition of pullback modules, we obtain the left two columns of the following commutative diagram with exact columns and the lower two rows

where $G^1$ denotes ${\rm{Ker}} \binom{\eta_1l}{-\eta_2}.$ In fact, the commutativity of the lower two squares follows from (3.3) and the functoriality of $\eta_n$ for $n = 1, 2$. Then, by the Snake Lemma, we obtain our desired short exact sequence, the induced first row

In fact, from the exact third column in (3.4), we see that $G^1$ is the pullback module of the Gorenstein triple $(L_1, L_2, l)$ in $\mathcal{F},$ and from the upper two rows in diagram (3.4), we deduce the commutative two diagrams by Lemma 2.1(2)

where all the vertical isomorphisms follow from Lemma 2.9, which implies that both $F\alpha$ and $L\alpha$ are injective, and thus ${{\rm{Tor}}}^A_1(A_n, G^1) = 0$ for $n = 1, 2$.

(b) Choose a short exact sequence $0 \rightarrow \Omega_0 G \rightarrow P \stackrel{{\beta}}{ \rightarrow} G \rightarrow 0$ with $_AP$ projective. Since both $G$ and $P$ are separated, we have, by Lemma 2.10(3), the two exact columns of the following commutative diagram with exact rows

where $C_1$ and $C_2$ are kernels of $F\beta$ and $L\beta$, respectively, both of which are Gorenstein-projective modules by Lemma 2.3(1). $c$ is induced from the diagram

where the commutativity of the right square follows from the functoriality of $\pi:F_1F \rightarrow F_2L$, and the exactness of the lower row follows from the condition (ⅰ). Then, by the Snake Lemma, the first column is exact. This shows that $\Omega_0 G$ is isomorphic to the pullback module of the Gorenstein triple $(C_1, C_2, c)$ in $\mathcal{F}$. Now, by a similar proof in (a), we have ${{\rm{Tor}}}^A_1(A_n, G) = 0$ for $n = 1, 2.$

(c) Step 1. We show that ${{\rm{Ext}}}^1_A(G, P) = 0$ for every $P\in A-{\rm{Proj}}$.

Actually, this follows from Lemma 3.2 and the short exact sequence

in ${\rm{(a)}},$ since $G$ is a pullback module of a Gorenstein triple by the assumption, and ${{\rm{Tor}}}^A_1(A_n, G) = 0$ for $n = 1, 2$ by $\rm{(b)},$ and $G^1$ is a pullback module of a Gorenstein triple such that ${{\rm{Tor}}}^A_1(A_n, G^1) = 0$ for $n = 1, 2$ by ${\rm{(a)}}$.

Step 2. We show that there is a ${{\rm{Hom}}}_A(-, P)$-exact sequence

for every $P\in A-{\rm{Proj}}$.

Since $G^1$ is a pullback module of a Gorenstein triple, again by $(a)$ and by induction on $i\geq 1$, we obtain a series of short exact sequences of $A$-modules

with $T^i$ projective and $G^i$ the pullback module of a Gorenstein triple such that ${{\rm{Tor}}}^A_1(A_n, G^i) = 0$ for $n = 1, 2.$ Then, by Lemma 3.2, we have ${{\rm{Ext}}}^1_A(G^i, P) = 0$ for all $i\geq 1$ and $P\in A-{\rm{Proj}}$. Thus, by splicing these short exact sequences for all $i\geq 1$, together with (3.6), we obtain the desired $P^+.$

Step 3. We show that ${{\rm{Ext}}}^{i}_A(G, P) = 0$ for all $i\geq 2$ and $P\in A-{\rm{Proj}}$.

This is equivalent to showing that ${{\rm{Ext}}}^1_A(\Omega_iG, P) = 0$ for all $i\geq 0$ and $P\in A-{\rm{Proj}}$. But this similarly follows from the induction on $\Omega_i G$ by $\rm{(b)}$ for $i\geq 0$, and applying Lemma 3.2 to the short exact sequences $0 \rightarrow \Omega_{i+1} G \rightarrow T_i \rightarrow \Omega_i G \rightarrow 0$ and $0 \rightarrow \Omega_{0} G \rightarrow T_0 \rightarrow G \rightarrow 0$ with $T_0, T_i$ projective.

Finally, by Lemma 2.4, we get that $G$ is a Gorenstein-projective $A$-module. □

A ring is said to be strongly left CM-free (see ^[1]) if each Gorenstein-projective left module over it is projective. We have the following corollary.

Corollary 3.4. Let $(G_1, G_2, g)$ be a Gorenstein triple in $\mathcal{F}$, and $G: = {{\rm{Pb}}}(G_{1}, G_{2}, g).$ Assume that $A_2$ is strongly left CM-free, and ${\rm{Add}}({\rm{Ker}}(\pi_1))\subseteq A_1-{\rm{GProj}}^{\bot}.$ Then $G\in A-{\rm{GProj}}$.

4.   Applications to Morita context rings

Morita context rings play an important role in ring theory. They provide many important examples and frameworks of kinds of problems (for instance, see ^[6,11]). In this section we construct Gorenstein-projective modules over Morita context rings with two bimodule homomorphisms zero $\Lambda_{(0, 0)} = \left(\begin{smallmatrix}\Lambda & N\\ M &B\end{smallmatrix}\right)_{(0, 0)},$ of which the multiplication is defined as

A left $\Lambda_{(0, 0)}$-module is identified with a quadruple $({}_\Lambda X, {}_BY, f, g)$, where $f\in {{\rm{Hom}}}_B(M\otimes_\Lambda X, Y)$ and $g\in {{\rm{Hom}}}_\Lambda(N\otimes_BY, X),$ and a homomorphism from a triple $(X, Y, f, g)$ to another triple $(X', Y', f', g')$ is a pair $(\alpha, \beta)$ with $\alpha\in {{\rm{Hom}}}_\Lambda(X, X')$ and $\beta\in {{\rm{Hom}}}_B(Y, Y')$ such that the two diagrams are commutative

For more details about Morita context rings, we refer the reader to ^[6], for instance.

Let $A_1$ be the upper triangular matrix ring $\left(\begin{smallmatrix}\Lambda & N\\ 0 &B\end{smallmatrix}\right)$, $A_2$ be the lower triangular matrix ring $\left(\begin{smallmatrix}\Lambda & 0\\ M &B\end{smallmatrix}\right)$, and $A_0$ be the diagonal matrix ring $\left(\begin{smallmatrix}\Lambda & 0\\ 0 &B\end{smallmatrix}\right)$. Then $\Lambda_{(0, 0)}$ can be interpreted as a pullback ring through the following commutative diagram.

where $p_n$ and $\pi_n$ are canonical projections for $n = 1, 2$. By the characterization of Gorenstein-projective modules over triangular matrix rings in ^[3], we obtain the following corollary, which allows us to obtain Gorenstein-projective $\Lambda_{(0, 0)}$-modules from Gorenstein-projective $\Lambda$-modules and $B$-modules.

Corollary 4.1. Assume that ${\rm{(1)}}$ ${{\rm{Tor}}}^{\Lambda}_1(M, H) = 0$ for all $H \in \Lambda -{\rm{GProj}}$, and ${\rm{Add({}_BM)\subseteq B-{\rm{GProj}}^{\bot}}}$,

${\rm{(2)}}$ ${{\rm{Tor}}}^{B}_1(N, G) = 0$ for all $G \in B -{\rm{GProj}}$, and ${\rm{Add({}_\Lambda N)\subseteq \Lambda -{\rm{GProj}}^{\bot}}}$.

Let $U\in \Lambda -{\rm{GProj}}$, $V \in B -{\rm{GProj}}$. If there exist a $\Lambda$-module $X$, a $B$-module $Y$, and two short exact sequences

and

in $\Lambda-{\rm{Mod}}$ and $B-{\rm{Mod}}$, respectively, then $(X, Y, (1_M\otimes s)f, (1_N\otimes t)g) \in \Lambda_{(0, 0)} -{\rm{Gproj}}$.

Proof. Write $W$ for $(X, Y, (1_N\otimes t)g, (1_M\otimes s)f)$, $W_1$ for $(X, V, 0, g)$, $W_2$ for $(U, Y, f, 0)$, and $W_0$ for $(U, V, 0, 0).$ Then $W_0\simeq A_0\otimes_{A_1} W_1 \simeq A_0\otimes_{A_2}W_2$, and one can verify that the following commutative diagram is a pullback diagram,

By the condition (2) and the short exact sequence

it follows from Theorem [3, Theorem 2.5] that $W_1: = (X, V, g) \in A_1 -{\rm{GProj}}$. Dually, by the condition $(1)$ and the short exact sequence

we have $W_2: = (U, Y, f) \in A_2 -{\rm{GProj}}$. Now we only need to show that the conditions (ⅰ) and (ⅱ) in Theorem 3.3 are satisfied here, and then it will follow that $W\in \Lambda_{(0, 0)} -{\rm{GProj}}$.

Actually, by [5, Lemma 3.9], it follows from ${\rm{Add({}_\Lambda N)\subseteq \Lambda{-GProj}^{\bot}}}$ that ${\rm{Add({}_\Lambda N, 0, 0)\subseteq A_1{-GProj}^{\bot}}}$, and similarly we have ${\rm{Add(0, {}_BM, 0)\subseteq A_2{-GProj}^{\bot}}}$. By [11, Proposition 6.1] or [7, Lemma 3.7], it follows from ${{\rm{Tor}}}^{\Lambda}_1(M, H) = 0$ for all $H \in \Lambda -{\rm{GProj}}$ that ${{\rm{Tor}}}^{A_2}_1((M_\Lambda, 0, 0), H') = 0$ for all $H' \in A_2 -{\rm{GProj}}$. Then, by the short exact sequence

we have ${{\rm{Tor}}}^{A_2}_1(A_0, H') = 0$ for all $H' \in A_2 -{\rm{GProj}}$. Actually, for every $H' \in A_2 -{\rm{GProj}}$, we have ${{\rm{Tor}}}^{A_2}_1(A_0, H') = {{\rm{Tor}}}^{A_2}_2(A_0, \mho H') = {{\rm{Tor}}}^{A_2}_1((M_\Lambda, 0, 0), \mho H') = 0,$ where $\mho H'$ lies in the short exact sequence $0 \rightarrow H' \rightarrow P \rightarrow \mho H' \rightarrow 0$ taken from the complete projective resolution of $H'$. □

Corollary 4.1 extends both the construction of Gorenstein-projective modules over Morita context algebras in [5, Theorem 3.10($\alpha$)] and the sufficient conditions for a module to be Gorenstein-projective over Noetherian rings in [7, Proposition 3.14(2)] to the general rings. The conditions ${\rm{(1)}}$ and ${\rm{(2)}}$ in Corollary 4.1 can be realized by ${{\rm{flatdim}}} M_\Lambda < \infty,$ ${{\rm{flatdim}}} N_B < \infty,$ ${{\rm{projdim}}}_\Lambda N < \infty$, ${{\rm{projdim}}} _BM < \infty.$

Let $\Delta_{(0, 0)}$ be the special Morita context rings where $\Lambda = N = M = B$ in $\Lambda_{(0, 0)}.$ By Corollary 4.1, we have the following, which extends the results in [5, Corollary 3.12], also in [10, Example 4.11].

Corollary 4.2. Let $U, V\in \Lambda-{\rm{GProj}}$. If there exist two $\Lambda$-modules $X$ and $Y$ and two short exact sequences

and

in $\Lambda-{\rm{Mod}}$, then $(X, Y, tg, sf), (Y, X, sf, tg) \in \Delta_{(0, 0)} -{\rm{Gproj}}$.

Acknowledgments

The author would like to thank Professors Changchang Xi and Wei Hu for their guidance. Also, the author is grateful to the referees for their helpful suggestions.

Conflict of interest

The author declares no conflicts of interest in this paper.