$\Omega$-Gorenstein Modules over Formal Triangular Matrix Rings

Abstract

Let A and B be rings and U a (B, A)-bimodule. Under some conditions, $\Omega$-Gorenstein module over the formal triangular matrix ring $T=\left( \begin{array}{cc} A \,\ &{}\quad 0 \\ U \ &{}\quad B \\ \end{array} \right)$ is explicitly described, where $\Omega$ is a class of left T-modules. As an application, it is shown that if $_BU$ has finite projective dimension and $U_A$ has finite flat dimension, then $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ is a Gorenstein projective left T-module if and only if $M_1$ is a Gorenstein projective left A-module,${{\text {Coker}}(}\varphi ^M)$ is a Gorenstein projective left B-module and $\varphi ^M:{U\otimes _{A}M_1}\rightarrow M_2$ is a monomorphism. This statement covers an earlier result of Enochs, Cortés-Izurdiaga and Torrecillas in this direction.

1 Introduction

It is well known that a commutative Noetherian local ring R is regular if and only if every R-module has finite projective dimension; see Auslander and Buchsbaum [3]. It is a crucial motivation for the study of homological dimensions of modules. In 1969, Auslander and Bridger [2] introduced an invariant, called Gorenstein dimension (G-dimension), for finitely generated modules over a commutative Noetherian ring and showed that R is Gorenstein if and only if every finitely generated R-module has finite G-dimension. It is an elegant characterization of Gorenstein rings. Over any associative rings, Enochs, Jenda, and Torrecillas [7] introduced the notion of Gorenstein flat modules in 1993 and Enochs and Jenda [5] introduced the concept of Gorenstein projective (injective) modules in 1995. The study of Gorenstein homological algebra often looks for clues from the classical homological algebra.

Formal triangular matrix rings play an important role in ring theory and the representation theory of algebra. Let A and B be any associative rings and U a (B, A)-bimodule. Enochs, Cortés-Izurdiaga and Torrecillas explicitly described the Gorenstein projective (injective) modules over the formal triangular matrix ring $T=\left( \begin{array}{cc} A \ \,&{}\quad 0 \\ U \ &{}\quad B \\ \end{array} \right)$. More precisely, if $_BU$ has finite projective dimension, $U_A$ has finite flat dimension and B is left Gorenstein regular, then $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ is a Gorenstein projective left T-module if and only if $M_1$ is a Gorenstein projective left A-module, ${{\text {Coker}}(}\varphi ^M)$ is a Gorenstein projective left B-module and $\varphi ^M:{U\otimes _{A}M_1}\rightarrow M_2$ is a monomorphism; see [4, Theorem 3.5]. If $_BU$ has finite projective dimension, $U_A$ has finite flat dimension and A is left Gorenstein regular, then $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ is a Gorenstein injective left T-module if and only if $M_2$ is a Gorenstein injective left B-module, ${{\text {Ker}}(}\widetilde{\varphi ^M})$ is a Gorenstein injective left A-module and $\widetilde{\varphi ^M}:M_1\rightarrow {\text {Hom}}_{B}(U,M_2)$ is an epimorphism; see [4, Theorem 3.8].

The main purpose of this paper is to extend the above results by removing the unnecessary condition “A and B are left Gorenstein regular”.

2 Notation and Terminology

Throughout this paper, all rings are associative with a unit. Let X be a complex of left R-modules (R-complex for short). With homological grading, X has the form

$$\begin{aligned} \cdots \longrightarrow X_{i+1} \xrightarrow []{\;{\partial }^{X}_{i+1}\;} X_i \xrightarrow []{\;{\partial }^{X}_{i}\;} X_{i-1} \longrightarrow \cdots . \end{aligned}$$

With cohomological grading by setting $X^i=X_{-i}$ and ${\partial }^{i}_{X}={\partial }^{X}_{-i}$, one gets

$$\begin{aligned} \cdots \longrightarrow X^{i-1} \xrightarrow []{\;{\partial }^{i-1}_{X}\;} X^i \xrightarrow []{\;{\partial }^{i}_{X}\;} X^{i+1} \longrightarrow \cdots . \end{aligned}$$

We use the notations ${\text {Z}}_{i}(X)$ for the kernel of differential ${\partial }^{X}_{i}$ and ${\text {C}}_{i}(X)$ for the cokernel of the differential ${\partial }^{X}_{i+1}$. An R-complex X is called acyclic if the homology complex H(X) is the zero complex. An acyclic complex P of projective R-modules is totally acyclic, if the complex ${\text {Hom}}_{R}(P,Q)$ is acyclic for every projective R-module Q. An acyclic complex U of injective R-modules is called a totally acyclic if the complex ${\text {Hom}}_{R}(J,U)$ is acyclic for every injective R-module J; see Enochs and Jenda [6].

Let A and B be any associative rings and U a (B, A)-bimodule. Then, $T=\left( \begin{array}{cc} A \ \,&{}\quad 0 \\ U \ &{}\quad B \\ \end{array} \right)$ always denotes a formal triangular matrix ring. By Green [10, Theorem 1.5], the category T-Mod of left T-modules is equivalent to the category ${\mathfrak {C}}$ whose objects are triples $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$, where $M_1\in A$-Mod (category of left A-modules), $M_2\in B$-Mod (category of left B-modules) and $\varphi ^M:{U\otimes _{A}M_1}\rightarrow M_2$ is a B-morphism, and whose morphism from $\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ to $\left( \begin{array}{c} N_1 \\ N_2 \\ \end{array} \right) _{\varphi ^N}$ is pairs $\left( \begin{array}{c} f_1 \\ f_2 \\ \end{array} \right) _{\varphi ^M}$ such that $f_1\in {\text {Hom}}_{A}(M_1,N_1), f_2\in {\text {Hom}}_{A}(M_2,N_2)$ satisfying that the following diagram

is commutative. Given such a triple $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ in ${\mathfrak {C}}$, we denote by $\widetilde{\varphi ^M}:M_1\rightarrow {\text {Hom}}_{B}(U,M_2)$ given by $\widetilde{\varphi ^M}(x)(u)=\varphi ^M({u\otimes _{A}x})$ for any $u\in U$ and $x\in M_1$. In the rest of the paper, we shall identify the category of left T-modules with this category ${\mathfrak {C}}$ and, whenever there is no possible confusion, we will omit the morphism $\varphi ^M$. Consequently, a left T-module will be a pair $\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right)$. Notice that a sequence of left T-modules

$$\begin{aligned} 0\rightarrow \left( \begin{array}{c} L_1 \\ L_2 \\ \end{array} \right) _{\varphi ^L}\rightarrow \left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}\rightarrow \left( \begin{array}{c} N_1 \\ N_2 \\ \end{array} \right) _{\varphi ^N}\rightarrow 0 \end{aligned}$$

is exact if and only if both sequences $0\rightarrow L_1\rightarrow M_1\rightarrow N_1\rightarrow 0$ and $0\rightarrow L_2\rightarrow M_2\rightarrow N_2\rightarrow 0$ are exact.

Recall from [4, Page 1547] that there are some functors between the category T-Mod and the product category A-Mod$\times B$-Mod:

$\bullet \ \mathbf{p }: A$-Mod$\times B$-Mod$\rightarrow T$-Mod is defined as follows: for an object $(M_1,M_2)$ of A-Mod$\times B$-Mod, let $\mathbf{p }(M_1,M_2)=\left( \begin{array}{c} M_1 \\ ({U\otimes _{A}M_1})\oplus M_2 \\ \end{array} \right)$ with the obvious map and let $\mathbf{p} (f_1,f_2)=\left( \begin{array}{c} f_1 \\ ({1_U\otimes _{A}f_1})\oplus f_2 \\ \end{array} \right)$ for any morphism $(f_1,f_2)$ in A-Mod$\times B$-Mod.

$\bullet \ \mathbf{h} : A$-Mod$\times B$-Mod$\rightarrow T$-Mod is defined as follows: for each object $(M_1,M_2)$ of A-Mod$\times B$-Mod, let $\mathbf{h} (M_1,M_2)=\left( \begin{array}{c} M_1\oplus {\text {Hom}}_{B}(U,M_2) \\ M_2 \\ \end{array} \right)$ with the obvious map and let $\mathbf{h} (f_1,f_2)=\left( \begin{array}{c} f_1\oplus {\text {Hom}}_{B}(U,f_2) \\ f_2 \\ \end{array} \right)$ for any morphism $(f_1,f_2)$ in A-Mod$\times B$-Mod.

$\bullet \ \mathbf{q} : T$-Mod$\rightarrow A$-Mod$\times B$-Mod is defined, for each left T-module $\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right)$ as $\mathbf{q} \left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) =(M_1,M_2)$, and for each morphism $\left( \begin{array}{c} f_1 \\ f_2 \\ \end{array} \right)$ in T-Mod as $\mathbf{q} \left( \begin{array}{c} f_1 \\ f_2 \\ \end{array} \right) =(f_1,f_2)$.

Notice that $(\mathbf{p} ,\mathbf{q} )$ and $(\mathbf{q} ,\mathbf{h} )$ are adjoint pairs. We list a few lemmas for later use.

Lemma 1

[12, Theorem 3.1] A left T-module $P=\left( \begin{array}{c} P_1 \\ P_2 \\ \end{array} \right) _{\varphi ^P}$ is projective if and only if $P_1$ is a projective left A-module, ${{\text {Coker}}(}\varphi ^P)$ is a projective left B-module and $\varphi ^P:{U\otimes _{A}P_1}\rightarrow P_2$ is a monomorphism.

Lemma 2

[11, Proposition 5.1] and [1, Page 956] A left T-module $I=\left( \begin{array}{c} I_1 \\ I_2 \\ \end{array} \right) _{\widetilde{\varphi ^I}}$ is injective if and only if $I_2$ is an injective left B-module, ${{\text {Ker}}(}\widetilde{\varphi ^I})$ is an injective left A-module and $\widetilde{\varphi ^I}:I_1\rightarrow {\text {Hom}}_{B}(U,I_2)$ is an epimorphism.

Lemma 3

[8, Proposition 1.14] A left T-module $F=\left( \begin{array}{c} F_1 \\ F_2 \\ \end{array} \right) _{\varphi ^F}$ is flat if and only if $F_1$ is a flat left A-module, ${{\text {Coker}}(}\varphi ^F)$ is a flat left B-module and $\varphi ^F$ is a monomorphism.

Throughout, let ${\mathcal {X}}$ and ${\mathcal {Y}}$ denote a class of left A-modules and $0\in {\mathcal {X}}$, and left B-modules and $0\in {\mathcal {Y}}$, respectively. Recall the following definition from Geng and Ding [9, Definition 2.2].

Definition 1

Let ${\mathcal {X}}$ be a class of left A-modules. A left A-module M is called ${\mathcal {X}}$-Gorenstein if there exists an acyclic complex

$$\begin{aligned} X= \cdots \rightarrow X_1\rightarrow X_0\rightarrow X_{-1}\rightarrow X_{-2}\rightarrow \cdots \end{aligned}$$

of modules in ${\mathcal {X}}$ such that $M={{\text {Ker}}(}X_0\rightarrow X_{-1})$, and ${\text {Hom}}_{R}({\mathcal {X}},X)$ and ${\text {Hom}}_{R}(X,{\mathcal {X}})$ are acyclic.

We denote by ${\text {G}}(\Omega )$, ${\text {G}}({\mathcal {X}})$, and ${\text {G}}({\mathcal {Y}})$ the class of $\Omega$-Gorenstein left T-modules, ${\mathcal {X}}$-Gorenstein left A-modules, and ${\mathcal {Y}}$-Gorenstein left B-modules, respectively.

Definition 2

Let M be a left A-module. ${\mathcal {X}}{\text {-dim}}_{A}{M}$ is defined as $\inf \{n\ge 0|\,{\text {there exists an exact sequence}}\ 0\rightarrow X_n\rightarrow \cdots \rightarrow X_1\rightarrow X_0\rightarrow M\rightarrow 0\ {\text {with}}\ X_i\in {\mathcal {X}}\}$. We set ${\mathcal {X}}{\text {-dim}}_{A}{M}$ infinity if no such integer exists. Similarly, one can define ${\mathcal {Y}}{\text {-dim}}_{B}{N}$ and $\Omega {\text {-dim}}_{T}{L}$, where N is a left B-module and L is a left T-module.

3 $\Omega$-Gorenstein Modules

In this section, we characterize $\Omega$-Gorenstein modules over the formal triangular matrix ring $T=\left( \begin{array}{cc} A \ \,&{}\quad 0 \\ U \ &{}\quad B \\ \end{array} \right)$. Recall that $\Omega$ is self-orthogonal if

$$\begin{aligned} {\text {Ext}}_{T}^{i\ge 1}(W,W')=0 \end{aligned}$$

for any $W,W'\in \Omega$.

Theorem 1

Let be a left T-module. Assume that the following conditions hold:

• (1) ${U\otimes _{A}X}$ is exact for any acyclic complex X of modules in ${\mathcal {X}}$.

• (2) ${U\otimes _{A}{\mathcal {X}}}\subseteq {\mathcal {Y}}$.

• (3) $\Omega$ is self-orthogonal and Hom$_T(\Omega ,-)$ is exact.

Then, the following conditions are equivalent.

• (1) M is an $\Omega$-Gorenstein left T-module.

• (2) $M_1$ is an ${\mathcal {X}}$-Gorenstein left A-module, ${{\text {Coker}}(}\varphi ^M)$ is a ${\mathcal {Y}}$-Gorenstein left B-module, and $\varphi ^M:{U\otimes _{A}M_1} \rightarrow M_2$ is a monomorphism.

Proof

$(1)\Rightarrow (2)$: Let M be an $\Omega$-Gorenstein left T-module. There exists an exact sequence of left T-modules

such that $\left( \begin{array}{c} W_1^i \\ W_2^i \\ \end{array} \right) _{\varphi ^i}\in \Omega$ for $i\in {\mathbb {Z}}$, ${\text {Hom}}_{T}\Big {(}W,\Omega \Big {)}$ and ${\text {Hom}}_{T}\Big {(}\Omega ,W\Big {)}$ are acyclic and $M={{\text {Ker}}\left( \begin{array}{c} d_1^0 \\ d_2^0 \\ \end{array} \right) }$.

Next we show that $M_1$ is an ${\mathcal {X}}$-Gorenstein left A-module. There is an exact sequence of left A-modules

such that each $W_1^i\in {\mathcal {X}}$ for $i\in {\mathbb {Z}}$, $M_1={{\text {Ker}}d_1^0}$ and ${U\otimes _{A}W_1}$ is exact by assumption. Let $\tau _1: M_1\rightarrow W_1^0$ and $\tau _2: M_2\rightarrow W_2^0$ be the inclusions. Consider the following commutative diagram of left B-modules

where $\varphi ^0$ is a monomorphism and so is $\varphi ^M$. Let $X\in {\mathcal {X}}$ be a left A-module. There is an exact sequence of left T-modules

$$\begin{aligned} 0\rightarrow \left( \begin{array}{c} 0 \\ {U\otimes _{A}X} \\ \end{array} \right) \rightarrow \left( \begin{array}{c} X \\ {U\otimes _{A}X} \\ \end{array} \right) \rightarrow \left( \begin{array}{c} X \\ 0 \\ \end{array} \right) \rightarrow 0, \end{aligned}$$

where $\left( \begin{array}{c} X \\ {U\otimes _{A}X} \\ \end{array} \right) \in \Omega$, and $\left( \begin{array}{c} 0 \\ {U\otimes _{A}X} \\ \end{array} \right) \in \Omega$ as ${U\otimes _{A}{\mathcal {X}}}\subseteq {\mathcal {Y}}$. Therefore, $\Omega {\text {-dim}}_{T}{\left( \begin{array}{c} X \\ 0 \\ \end{array} \right) }<\infty$. As $(\mathbf{p} ,\mathbf{q} )$ and $(\mathbf{q} ,\mathbf{h} )$ are adjoint pairs, there are isomorphisms

$$\begin{aligned} {\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} X \\ {U\otimes _{A}X} \\ \end{array} \right) ,\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) \Big {)}\cong {\text {Hom}}_{A}(X,W_1) \end{aligned}$$

and ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) ,\left( \begin{array}{c} X \\ 0 \\ \end{array} \right) \Big {)}\cong {\text {Hom}}_{A}(W_1,X)$. Now ${\text {Hom}}_{A}(X,W_1)$ is exact, and so is ${\text {Hom}}_{A}(W_1,X)$ by [13, Theorem 5.8] as $\Omega$ is self-orthogonal. It follows that $M_1$ is an ${\mathcal {X}}$-Gorenstein left A-module.

Next we prove that ${{\text {Coker}}(}\varphi ^M)$ is a ${\mathcal {Y}}$-Gorenstein left B-module. Consider the following commutative diagram with exact rows

Since the first column and the second column are exact, the third column is exact with ${{\text {Coker}}(\varphi ^i)}\in {\mathcal {Y}}$ for $i\in {\mathbb {Z}}$. Let $Y\in {\mathcal {Y}}$ be a left B-module. Then, ${\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) }\in \Omega$. Applying the functor ${\text {Hom}}_{B}(-,Y)$ to the exact sequence , one has the exact sequence

$$\begin{aligned} 0\rightarrow {\text {Hom}}_{B}({{\text {Coker}}(\varphi ^M)},Y)\rightarrow {\text {Hom}}_{B}(M_2,Y)\rightarrow {\text {Hom}}_{B}({U\otimes _{A}M_1},Y) \end{aligned}$$

and so ${\text {Hom}}_{B}({{\text {Coker}}(\varphi ^M)},Y)\cong {{\text {Ker}}(}{\text {Hom}}_{B}(M_2,Y)\rightarrow {\text {Hom}}_{B}({U\otimes _{A}M_1},Y))\cong {\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M},\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) _{0}\Big {)}$, where the last isomorphism follows by a direct calculation. Similarly, ${\text {Hom}}_{B}({{\text {Coker}}(\varphi ^i)},Y)\cong {\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} W_1^i \\ W_2^i \\ \end{array} \right) _{\varphi ^i},\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) _{0}\Big {)}$ for $i\in {\mathbb {Z}}$. Hence, the sequence

$$\begin{aligned} {{\text {Coker}}(\varphi ^{W})}: \cdots \rightarrow {{\text {Coker}}(\varphi ^{-1})}\rightarrow {{\text {Coker}}(\varphi ^0)}\rightarrow {{\text {Coker}}(\varphi ^1)}\rightarrow {{\text {Coker}}(\varphi ^2)}\rightarrow \cdots \end{aligned}$$

is ${\text {Hom}}_{B}(-,Y)$-exact. Since $(\mathbf{p} ,\mathbf{q} )$ is an adjoint pair, there is an isomorphism ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) ,\left( \begin{array}{c} 0 \\ {{\text {Coker}}(\varphi ^{W})} \\ \end{array} \right) \Big {)}\cong {\text {Hom}}_{B}(Y,{{\text {Coker}}(\varphi ^{W})})$. Therefore, ${\text {Hom}}_{B}(Y,{{\text {Coker}}(\varphi ^{W})})$ is acyclic by assumption. Therefore, ${{\text {Coker}}(}\varphi ^M)$ is a ${\mathcal {Y}}$-Gorenstein left B-module.

$(2)\Rightarrow (1)$: Let $M_1$ be an ${\mathcal {X}}$-Gorenstein left A-module, ${{\text {Coker}}(}\varphi ^M)$ a ${\mathcal {Y}}$-Gorenstein left B-module, and $\varphi ^M:{U\otimes _{A}M_1} \rightarrow M_2$ be a monomorphism. There is an exact sequence of left T-modules

$$\begin{aligned} 0\rightarrow \left( \begin{array}{c} M_1 \\ {U\otimes _{A}M_1} \\ \end{array} \right) \rightarrow \left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) \rightarrow \left( \begin{array}{c} 0 \\ {{\text {Coker}}(}\varphi ^M) \\ \end{array} \right) \rightarrow 0\,. \end{aligned}$$

We will show that $\left( \begin{array}{c} M_1 \\ {U\otimes _{A}M_1} \\ \end{array} \right)$ and $\left( \begin{array}{c} 0 \\ {{\text {Coker}}(}\varphi ^M) \\ \end{array} \right)$ are $\Omega$-Gorenstein left T-modules. Then, it follows from [15, Corollary 4.5] that M is an $\Omega$-Gorenstein left T-module as $\Omega$ is self-orthogonal.

Since $M_1$ is an ${\mathcal {X}}$-Gorenstein left A-module, there exists an exact sequence

such that $X^i\in {\mathcal {X}}$ for $i\in {\mathbb {Z}}$, $M_1={{\text {Ker}}d_0}$, and the complexes ${\text {Hom}}_{A}(X,{\mathcal {X}})$ and ${\text {Hom}}_{A}({\mathcal {X}},X)$ are acyclic. By assumption, the complex ${U\otimes _{A}X}$ is acyclic. Now there is an exact sequence of left T-modules

such that $\left( \begin{array}{c} M_{1} \\ {U\otimes _{A}M_1}\\ \end{array} \right) \cong {{\text {Ker}}\left( \begin{array}{c} d^{0} \\ {1\otimes _{A}d^0}\\ \end{array} \right) }$ and each $\left( \begin{array}{c} X^i \\ {U\otimes _{A}X^i} \\ \end{array} \right) \in \Omega$ for $i\in {\mathbb {Z}}$. Let $\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) _{\varphi ^W}\in \Omega$ be a left T-module. Since $(\mathbf{p},\,q )$ is an adjoint pair, there is an isomorphism ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} X^i \\ {U\otimes _{A}X^i} \\ \end{array} \right) ,\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) \Big {)}\cong {\text {Hom}}_{A}(X^i,W_1)$ for $i\in {\mathbb {Z}}$. It follows that ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} X \\ {U\otimes _{A}X} \\ \end{array} \right) ,\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) \Big {)}\cong {\text {Hom}}_{A}(X,W_1)$ is acyclic. By assumption, ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) ,\left( \begin{array}{c} X \\ {U\otimes _{A}X} \\ \end{array} \right) \Big {)}$ is acyclic. Therefore, $\left( \begin{array}{c} M_1 \\ {U\otimes _{A}M_1} \\ \end{array} \right)$ is an $\Omega$-Gorenstein left T-module.

It remains to show that $\left( \begin{array}{c} 0 \\ {{\text {Coker}}(}\varphi ^M) \\ \end{array} \right)$ is an $\Omega$-Gorenstein left T-module. Since ${{\text {Coker}}(}\varphi ^M)$ is a ${\mathcal {Y}}$-Gorenstein left B-module, there is an exact sequence

such that $Y^i\in {\mathcal {Y}}$ for $i\in {\mathbb {Z}}$, ${{\text {Coker}}(}\varphi ^M)={{\text {Ker}}d^0}$, and ${\text {Hom}}_{B}({\mathcal {Y}},Y)$ and ${\text {Hom}}_{B}(Y,{\mathcal {Y}})$ are acyclic. Now there is an exact sequence of left T-modules

with each $\left( \begin{array}{c} 0 \\ Y^i \\ \end{array} \right) \in \Omega$ for $i\in {\mathbb {Z}}$ and $\left( \begin{array}{c} 0 \\ {{\text {Coker}}(\varphi ^M)}\\ \end{array} \right) ={{\text {Ker}}\left( \begin{array}{c} 0 \\ d^0\\ \end{array} \right) }$. Let $\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) _{\varphi ^W}\in \Omega$ be a left T-module. Since $(\mathbf{p} ,\mathbf{q} )$ is an adjoint pair, there are isomorphisms

$$\begin{aligned} {\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) ,\left( \begin{array}{c} 0 \\ {{\text {Coker}}(}\varphi ^W) \\ \end{array} \right) \Big {)}\cong {\text {Hom}}_{B}(Y,{{\text {Coker}}(}\varphi ^W)) \end{aligned}$$

and ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) ,\left( \begin{array}{c} W_1 \\ {U\otimes _{A}W_1} \\ \end{array} \right) \Big {)}\cong {\text {Hom}}_{B}(Y,{U\otimes _{A}W_1})$. Since ${U\otimes _{A}{\mathcal {X}}}\subseteq {\mathcal {Y}}$, the complex ${\text {Hom}}_{B}(Y,{U\otimes _{A}W_1})$ is acyclic. Now there is an exact sequence of left T-modules

$$\begin{aligned} 0\rightarrow \left( \begin{array}{c} W_1 \\ {U\otimes _{A}W_1} \\ \end{array} \right) \rightarrow \left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) \rightarrow \left( \begin{array}{c} 0 \\ {{\text {Coker}}(}\varphi ^W) \\ \end{array} \right) \rightarrow 0, \end{aligned}$$

which yields an exact sequence of complexes

as Hom$_T(\Omega ,-)$ is exact, where $(-,-)$ denotes Hom$_T(\Omega ,-)$. Notice that the complexes ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) ,\left( \begin{array}{c} W_1 \\ {U\otimes _{A}W_1} \\ \end{array} \right) \Big {)}$ and are acyclic, and so is ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) ,\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) \Big {)}$. Now ${\text {Hom}}_{T}\Big {(}\left( \begin{array}{c} W_1 \\ W_2 \\ \end{array} \right) ,\left( \begin{array}{c} 0 \\ Y \\ \end{array} \right) \Big {)}$ is acyclic by assumption. Therefore, $\left( \begin{array}{c} 0 \\ {{\text {Coker}}(}\varphi ^M) \\ \end{array} \right)$ is an $\Omega$-Gorenstein left T-module. $\square$

Now in Theorem 1, if one takes ${\mathcal {X}}$ and ${\mathcal {Y}}$ to be the class of projective left A-modules and projective left B-modules, respectively, then $\Omega$ is the class of projective left T-modules by Lemma 1. Hence, one has the next result which covers [4, Theorem 3.5].

Corollary 1

Let $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ be a left T-module. If $_BU$ has finite projective dimension and $U_A$ has finite flat dimension, then the following conditions are equivalent.

• (1) M is a Gorenstein projective left T-module.

• (2) $M_1$ is a Gorenstein projective left A-module, ${{\text {Coker}}(}\varphi ^M)$ is a Gorenstein projective left B-module and $\varphi ^M:{U\otimes _{A}M_1}\rightarrow M_2$ is a monomorphism.

In this case, ${U\otimes _{A}M_1}$ is a Gorenstein projective left B-module if and only if $M_2$ is a Gorenstein projective left B-module.

Remark 1

Note that if one takes ${\mathcal {X}}$ and ${\mathcal {Y}}$ to be the class of flat left A-modules and flat left B-modules, respectively, then $\Omega$ in Theorem 1 is the class of flat left T-modules by Lemma 3. It is not hard to see that there is a Gorenstein flat version of Corollary 1.

Corollary 2

Let be a class of left T-modules and $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ be a left T-module. Assume that the following conditions hold:

• (1) ${U\otimes _{A}X}$ is exact for any acyclic complex X of modules in ${\mathcal {X}}$.

• (2) ${U\otimes _{A}{\mathcal {X}}}\subseteq {\mathcal {Y}}$.

• (3) $\Omega$ is self-orthogonal and Hom$_T(\Omega ,-)$ is exact.

Then, each $\Omega$-Gorenstein left T-module is in $\Omega$ if and only if each ${\mathcal {X}}$-Gorenstein left A-module is in ${\mathcal {X}}$ and every ${\mathcal {Y}}$-Gorenstein left B-module is in ${\mathcal {Y}}$.

Proof

Let $M_1$ be an ${\mathcal {X}}$-Gorenstein left A-module and $M_2$ a ${\mathcal {Y}}$-Gorenstein left B-module. By Theorem 1, $\mathbf{p} (M_1,M_2)$ is an $\Omega$-Gorenstein left T-module and so it is in $\Omega$. It follows that $M_1\in {\mathcal {X}}$ and $M_2\in {\mathcal {Y}}$.

Conversely, let $M=\left( \begin{array}{c} M_1 \\ M_2 \\ \end{array} \right) _{\varphi ^M}$ be an $\Omega$-Gorenstein left T-module. By Theorem 1, $M_1$ is an ${\mathcal {X}}$-Gorenstein left A-module, ${{\text {Coker}}(}\varphi ^M)$ is a ${\mathcal {Y}}$-Gorenstein left B-module and $\varphi ^M$ is a monomorphism. Hence, $M_1\in {\mathcal {X}}$ and ${{\text {Coker}}(}\varphi ^M)\in {\mathcal {Y}}$. It follows that $M\in \Omega$. $\square$

Remark 2

Notice that there are dual versions of Theorem 1, Corollary 1 and Corollary 2, where , and assume that the following conditions hold:

• (1) ${\text {Hom}}_{B}(U,Y)$ is exact for any acyclic complex Y of modules in ${\mathcal {Y}}$.

• (2) ${\text {Hom}}_{B}(U,{\mathcal {Y}})\subseteq {\mathcal {X}}$.

• (3) $\Omega$ is self-orthogonal and Hom$_T(-,\Omega )$ is exact.

Remark 3

Let $_BU$ be a finitely presented module. Note that if one takes ${\mathcal {X}}$ and ${\mathcal {Y}}$ to be the class of FP-injective left A-modules and FP-injective left B-modules, respectively, then $\Omega$ in Remark 2 is the class of FP-injective left T-modules by [14, Theorem 3.3].

Remark 4

As an application, one can discuss $\Omega$-Gorenstein dimensions of a left T-module over formal triangular matrix rings by using Definition 2. It is standard.