Relative Gorenstein Dimensions over Triangular Matrix Rings

Abstract

Let A and B be rings, U a $(B,A)$-bimodule, and $T=\left(\begin{array}{cc}A& 0\\ U& B\end{array}\right)$ the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of $G_C$-projective modules over T. As an application, we study when a morphism in T-Mod is a special $G_{C}P(T)$-precover and when the class $G_{C}P(T)$ is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.

1. Introduction

Let A and B be rings and U be a $(B,A)$-bimodule. The ring $T=\left(\begin{array}{cc}A& 0\\ U& B\end{array}\right)$ is known as the formal triangular matrix ring with usual matrix addition and multiplication. Such rings play an important role in the representation theory of algebras. The modules over such rings can be described in a very concrete fashion. Therefore, formal triangular matrix rings and modules over them have proven to be a rich source of examples and counterexamples. Some important Gorenstein notions over formal triangular matrix rings have been studied by many authors (see [1,2,3]). For example, Zhang [1] introduced compatible bimodules and explicitly described the Gorenstein projective modules over triangular matrix Artin algebra. Enochs, Izurdiaga, and Torrecillas [2] characterized when a left module over a triangular matrix ring is Gorenstein projective or Gorenstein injective under the “Gorenstein regular” condition. Under the same condition, Zhu, Liu, and Wang [3] investigated Gorenstein homological dimensions of modules over triangular matrix rings. Mao [4] studied Gorenstein flat modules over T (without the “Gorenstein regular” condition) and gave an estimate of the weak global Gorenstein dimension of T.

Semidualizing modules were independently studied (under different names) by Foxby [5], Golod [6], and Vasconcelos [7] over a commutative Noetherian ring. Golod used these modules to study the $G_C$-dimension for finitely generated modules. Motivated (in part) by Enochs and Jenda’s extensions of the classical G-dimension given in [8], Holm and Jørgensen extended in [9] this notion to arbitrary modules. After that, several generalizations of semidualizing and the $G_C$-dimension have been made by several authors [10,11,12].

As the authors mentioned in [13], to study the Gorenstein projective modules and dimension relative to a semidualizing $(R,S)$-bimodule C, the condition ${\mathrm{End}}_S(C)\cong R$ seems to be too restrictive and in some cases unnecessary. Therefore, the authors introduced weakly Wakamatsu tilting as a weak notion of semidualizing, which made the theory of relative Gorenstein homological algebra wider and less restrictive, but still consistent. Weakly Wakamatsu tilting modules were the subject of many publications that showed how important these modules could become in developing the theory of relative (Gorenstein) homological algebra [13,14,15].

The main objective of the present paper is to study relative Gorenstein homological notions (w-tilting, relative Gorenstein projective modules, relative Gorenstein projective dimensions, and the relative global projective dimension) over triangular matrix rings.

This article is organized as follows:

In Section 2, we give some preliminary results.

In Section 3, we study how to construct w-tilting (tilting, semidualizing) over T using w-tilting (tilting, semidualizing) over A and B under the condition that U is relative (weakly) compatible. We introduce (weakly) C-compatible $(B,A)$-bimodules for a T-module C (Definition 4). Given two w-tilting modules ${}_A{C}_1$ and ${}_B{C}_2$, we prove in Proposition 2 that $C=\left(\begin{array}{c}{C}_1\\ (U{\otimes }_A{C}_1)\oplus C_2\end{array}\right)$ is a w-tilting T-module when U is weakly C-compatible.

In Section 4, we first describe relative Gorenstein projective modules over T. Let $C=\left(\begin{array}{c}{C}_1\\ (U{\otimes }_A{C}_1)\oplus C_2\end{array}\right)$ be a T-module. We prove in Theorem 1 that if U is C-compatible, then a T-module $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ is $G_C$-projective if and only if $M_1$ is a $G_{C_1}$-projective A-module, $\mathrm{Coker}{\phi }^M$ is a $G_{C_2}$-projective B-module, and ${\phi }^M:U{\otimes }_A{M}_1\to M_2$ is injective. As an application, we prove the converse of Proposition 2 and refine in the relative setting (Proposition 4), a result of when T is left (strongly) CM-free due to Enochs, Izurdiaga, and Torrecillas in [2]. Furthermore, when C is w-tilting, we characterize when a T-morphism is a special precover (see Proposition 5). Then, in Theorem 2, we prove that the class of $G_C$-projective T-modules is a special precovering if and only if so are the classes of $G_{C_1}$-projective A-modules and $G_{C_2}$-projective B-modules, respectively.

Finally, in Section 5, we give an estimate of the $G_C$-projective dimension of a left T-module and the left $G_C$-projective global dimension of T. First, it is proven that, given a T-module $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$, if $C=\mathbf{p}(C_1,C_2):=\left(\begin{array}{c}{C}_1\\ (U{\otimes }_A{C}_1)\oplus C_2\end{array}\right)$ is w-tilting, U is C-compatible, and:

$$S{G}_{C_2}-PD(B):=sup\{G_{C_2}-pd(U{\otimes }_{A}G)|G\in G_{C_1}P(A)\}<\infty ,$$

then:

$$max\{G_{C_1}-pd(M_1),(G_{C_2}-pd(M_2))-(S{G}_{C_2}-PD(B))\}$$

$$\le G_C-pd(M)\le$$

$$max\{(G_{C_1}-pd(M_1))+(S{G}_{C_2}-PD(B))+1,G_{C_2}-pd(M_2)\}.$$

As an application, we prove that, if $C=\mathbf{p}(C_1,C_2)$ is w-tilting and U is C-compatible, then:

$$max\{G_{C_1}-PD(A),G_{C_2}-PD(B)\}$$

$$\le G_C-PD(T)\le$$

$$max\{G_{C_1}-PD(A)+S{G}_{C_2}-PD(B)+1,G_{C_2}-PD(B)\}.$$

Some cases in which this estimation becomes an exact formula are also given.

The authors in [16] established a relationship between the projective dimension of modules over T and modules over A and B. Given an integer $n\ge 0$ and $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ a T-module, they proved that ${pd}_T(M)\le n$ if and only if ${pd}_A(M_1)\le n$, ${pd}_B({\overline{M}}_2)\le n$ and the map related to the n-th syzygy of M is injective. We end the paper by giving a counterexample to this result (Example 4).

2. Preliminaries

Throughout this paper, all rings are associative (not necessarily commutative) with identity, and all modules are, unless otherwise specified, unitary left modules. For a ring R, we use $\mathrm{Proj}(R)$ (resp., $\mathrm{Inj}(R)$) to denote the class of all projective (resp., injective) R-modules. The category of all left R-modules is denoted by R-Mod. For an R-module C, we use ${\mathrm{Add}}_R(C)$ to denote the class of all R-modules that are isomorphic to direct summands of direct sums of copies of C, and ${\mathrm{Prod}}_R(C)$ denotes the class of all R-modules that are isomorphic to direct summands of direct products of copies of C.

Given a class of modules $\mathcal{F}$ (which are always considered closed under isomorphisms), an $\mathcal{F}$-precover of $M\in R$-Mod is a morphism $\phi :F\to M$ ($F\in \mathcal{F}$) such that ${\mathrm{Hom}}_R(F^{\prime },\phi )$ is surjective for every $F^{\prime }\in \mathcal{F}$. If, in addition, any solution of the equation ${\mathrm{Hom}}_R(F,\phi )(g)=\phi$ is an automorphism of F, then $\phi$ is said to be an $\mathcal{F}$-cover. The $\mathcal{F}$-precover $\phi$ is said to be special if it is surjective and ${\mathrm{Ext}}^1(F,ker\phi )=0$ for every $F\in \mathcal{F}$. The class $\mathcal{F}$ is said to be a special (pre)covering if every module has a special $\mathcal{F}$-(pre)cover.

Given the class $\mathcal{F}$, the class of all modules N such that ${\mathrm{Ext}}_R^i(F,N)=0$ for every $F\in \mathcal{F}$ is denoted by ${\mathcal{F}}^{{\perp }_i}$ (similarly, ${}^{{\perp }_i}\mathcal{F}=\{N;{\mathrm{Ext}}_R^i(N,F)=0\forall F\in \mathcal{F}\}$). The right and left orthogonal classes ${\mathcal{F}}^{\perp }$ and ${}^{\perp }\mathcal{F}$ are defined as follows:

$${\mathcal{F}}^{\perp }={\cap }_{i\ge 1}{\mathcal{F}}^{{\perp }_i}\mathrm{and}{}^{\perp }\mathcal{F}={\cap }_{i\ge 1}{}^{{\perp }_i}\mathcal{F}$$

It is immediate to see that if C is any module, then ${\mathrm{Add}}_R{(C)}^{\perp }={\left\{C\right\}}^{\perp }$ and ${}^{\perp }{\mathrm{Prod}}_R(C)={}^{\perp }\left\{C\right\}$.

Given a class of R-modules $\mathcal{F}$, an exact sequence of R-modules:

$$\cdots \to X^1\to X^0\to X_0\to X_1\to \cdots$$

is called ${\mathrm{Hom}}_R(-,\mathcal{F})$-exact (resp., ${\mathrm{Hom}}_R(\mathcal{F},-)$-exact) if the functor ${\mathrm{Hom}}_R(-,F)$ (resp., ${\mathrm{Hom}}_R(F,-)$) leaves the sequence exact whenever $F\in \mathcal{F}$. If $\mathcal{F}=\left\{F\right\}$, we simply say ${\mathrm{Hom}}_R(-,F)$-exact. Similarly, we can define $\mathcal{F}{\otimes }_R$-exact sequences when $\mathcal{F}$ is a class of right R-modules.

We now recall some concepts needed throughout the paper.

Definition 1.

1. 

([17], Definition 2.1) A semidualizing bimodule is an $(R,S)$-bimodule C satisfying the following properties:

(a) 

${}_{R}C$ and $C_S$ both admit a degreewise finite projective resolution in the corresponding module categories (R-Mod and Mod-S);

(b) 

${\mathrm{Ext}}_R^{\ge 1}(C,C)={\mathrm{Ext}}_S^{\ge 1}(C,C)=0$;

(c) 

The natural homothety maps $R\stackrel{{}_R\gamma }{\to }{\mathrm{Hom}}_S(C,C)$ and $S\stackrel{{\gamma }_S}{\to }{\mathrm{Hom}}_R(C,C)$ both are ring isomorphisms.

2. 

([18], Section 3) A Wakamatsu tilting module, simply tilting, is an R-module C satisfying the following properties:

(a) 

${}_{R}C$ admits a degreewise finite projective resolution;

(b) 

${\mathrm{Ext}}_R^{\ge 1}(C,C)=0$;

(c) 

There exists a ${\mathrm{Hom}}_R(-,C)$-exact exact sequence of R-modules:

$$\mathbf{X}=0\to R\to C^0\to C^1\to \cdots$$

where $C^i\in {\mathrm{add}}_R(C)$ for every $i\in \mathbb{N}$.

It was proven in ([18], Corollary 3.2), that an $(R,S)$-bimodule C is semidualizing if and only if ${}_{R}C$ is tilting with $S={\mathrm{End}}_R(C)$. Therefore, the following notion, which is crucial in this paper, generalizes both concepts.

Definition 2

([13], Definition 2.1). An R-module C is weakly Wakamatsu tilting (w-tilting for short) if it has the following two properties:

1. 

${\mathrm{Ext}}_R^{i\ge 1}(C,C^{(I)})=0$ for every set $I;$

2. 

There exists a ${\mathrm{Hom}}_R(-,{\mathrm{Add}}_R(C))$-exact exact sequence of R-modules:

$$\mathbf{X}=0\to R\to A^0\to A^1\to \cdots$$

where $A^i\in {\mathrm{Add}}_R(C)$ for every $i\in \mathbb{N}$;

If C satisfies 1 but perhaps not 2, then C will be said to be Σ-self-orthogonal.

Definition 3

([13], Definition 2.2). Given any $C\in R$-Mod, an R-module M is said to be G${}_C$-projective if there exists a ${\mathrm{Hom}}_R(-,{\mathrm{Add}}_R(C))$-exact exact sequence of R-modules:

$$\mathbf{X}=\cdots \to P_1\to P_0\to A^0\to A^1\to \cdots$$

where the $P_i^{\prime }s$ are all projective, $A^i\in {\mathrm{Add}}_R(C)$ for every $i\in \mathbb{N}$, $M\cong \mathrm{Im}(P_0\to A^0)$.

We use $G_{C}P(R)$ to denote the class of all G${}_C$-projective R-modules.

It is immediate from the definitions that w-tilting modules can be characterized as follows.

Lemma 1.

An R-module C is w-tilting if and only if both C and R are $G_C$-projective modules.

Now, we recall some facts about triangular matrix rings. Let A and B be rings and U a $(B,A)$-bimodule. We shall denote by $T=\left(\begin{array}{cc}A& 0\\ U& B\end{array}\right)$ the generalized triangular matrix ring. By [19], Theorem 1.5, the category T-Mod of left T-modules is equivalent to the category ${}_T\Omega$ whose objects are triples $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$, where $M_1\in A$-Mod, $M_2\in B$-Mod, and ${\phi }^M:U{\otimes }_A{M}_1\to M_2$ is a B-morphism and whose morphisms from ${\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ to ${\left(\begin{array}{c}{N}_1\\ N_2\end{array}\right)}_{{\phi }^N}$ are pairs $\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)$ such that $f_1\in {\mathrm{Hom}}_A(M_1,N_1)$, $f_2\in {\mathrm{Hom}}_B(M_2,N_2)$ satisfying that the following diagram is commutative.

Since we have the natural isomorphism:

$${\mathrm{Hom}}_B(U{\otimes }_A{M}_1,M_2)\cong {\mathrm{Hom}}_A(M_1,{\mathrm{Hom}}_B(U,M_2)),$$

there is an alternative way of defining T-modules and T-homomorphisms in terms of maps $\widetilde{{\phi }^M}:M_1\to {\mathrm{Hom}}_B(U,M_2)$ given by $\widetilde{{\phi }^M}(x)(u)={\phi }^M(u\otimes x)$ for each $u\in U$ and $x\in M_1.$

Analogously, the category Mod-T of right T-modules is equivalent to the category ${\Omega }_T$ whose objects are triples $M={\left(\begin{array}{c}{M}_1,M_2\end{array}\right)}_{{\psi }^M}$, where $M_1\in$ Mod-A, $M_2\in$ Mod-B, and ${\phi }^M:M_2{\otimes }_{B}U\to M_1$ is an A-morphism and whose morphisms from ${\left(\begin{array}{c}{M}_1,M_2\end{array}\right)}_{{\varphi }^M}$ to ${\left(\begin{array}{c}{N}_1,N_2\end{array}\right)}_{{\varphi }^N}$ are pairs $\left(\begin{array}{c}{f}_1,f_2\end{array}\right)$ such that $f_1\in {\mathrm{Hom}}_A(M_1,N_1)$, $f_2\in {\mathrm{Hom}}_B(M_2,N_2)$ satisfying that the following diagram:

is commutative.

In the rest of the paper, we shall identify T-Mod (resp. Mod-T) with ${}_T\Omega$ (resp. ${\Omega }_T$). Consequently, through the paper, a left (resp. right) T-module will be a triple $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ (resp. $M={\left(\begin{array}{c}{M}_1,M_2\end{array}\right)}_{{\varphi }^M}$), and whenever there is no possible confusion, we shall omit the morphisms ${\phi }^M$ and ${\varphi }^M$. For example, ${}_{T}T$ is identified with $\left(\begin{array}{c}A\\ U\oplus B\end{array}\right)$ and $T_T$ is identified with $\left(\begin{array}{c}A\oplus U,B\end{array}\right)$.

A sequence of left T-modules $0\to \left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)\to \left(\begin{array}{c}{M}_1^{\prime }\\ M_2^{\prime }\end{array}\right)\to \left(\begin{array}{c}{M}_1^{″}\\ M_2^{″}\end{array}\right)\to 0$ is exact if and only if both sequences $0\to M_1\to M_1^{\prime }\to M_1^{″}\to 0$ and $0\to M_2\to M_2^{\prime }\to M_2^{″}\to 0$ are exact.

Throughout this paper, $T=\left(\begin{array}{cc}A& 0\\ U& B\end{array}\right)$ is a generalized triangular matrix ring. Given a T-module $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$, the B-module $\mathrm{Coker}{\phi }^M$ is denoted as ${\overline{M}}_2$ and the A-module $\mathrm{Ker}\widetilde{{\phi }^M}$ as $\underline{M_1}$. A T-module $N={\left(\begin{array}{c}{N}_1\\ N_2\end{array}\right)}_{{\phi }^N}$ is a submodule of M if $N_1$ is a submodule of $M_1$, $N_2$ is a submodule of $M_2$, and ${\phi }^M{|}_{U{\otimes }_A{N}_1}={\phi }^N$.

As an interesting and special case of triangular matrix rings, we recall that the $T_2$-extension of a ring R is given by:

$$T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right)$$

and the modules over $T(R)$ are triples $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ where $M_1$ and $M_2$ are R-modules and ${\phi }^M:M_1\to M_2$ is an R-homomorphism.

There are some pairs of adjoint functors $(\mathbf{p},\mathbf{q})$, $(\mathbf{q},\mathbf{h})$ and $(\mathbf{s},\mathbf{r})$ between the category T-Mod and the product category A-Mod ×B-Mod, which are defined as follows:

• $\mathbf{p}$: A-Mod ×B-Mod$\to T$-Mod is defined as follows: for each 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 for any morphism $(f_1,f_2)$ in A-Mod$\times B$-Mod, let $\mathbf{p}(f_1,f_2)=\left(\begin{array}{c}{f}_1\\ (1_U{\otimes }_A{f}_1)\oplus f_2\end{array}\right)$;
• $\mathbf{q}$: T-Mod$\to A$-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 $\mathbf{q}(\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right))=(f_1,f_2)$;
• $\mathbf{h}$: A-Mod ×B-Mod$\to T$-Mod is defined as follows: for each object $(M_1,M_2)$ of A-Mod$\times B$-Mod, let $\mathbf{h}(M_1,M_2)=\left(\begin{array}{c}{M}_1\oplus {\mathrm{Hom}}_B(U,M_2)\\ M_2\end{array}\right)$ with the obvious map, and for any morphism $(f_1,f_2)$ in A-$Mod$B-$Mod$, let h$(f_1,f_2)=\left(\begin{array}{c}{f}_1\oplus {\mathrm{Hom}}_B(U,f_2)\\ f_2\end{array}\right)$;
• $\mathbf{r}$: A-Mod ×B-Mod$\to T$-Mod is defined as follows: for each object $(M_1,M_2)$ of A-Mod$\times B$-Mod, let $\mathbf{r}(M_1,M_2)=\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)$ with the zero map, and for any morphism $(f_1,f_2)$ in A-Mod$\times B$-Mod, let $\mathbf{r}(f_1,f_2)=\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)$;
• $\mathbf{s}$: T-Mod$\to A$-Mod $\times B$-Mod is defined, for each left T-module $\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)$ as $\mathbf{s}(\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right))=(M_1,{\overline{M}}_2)$ and for each morphism $\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)$ in T-Mod as $\mathbf{s}(\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right))=(f_1,{\overline{f}}_2)$, where ${\overline{f}}_2$ is the induced map.

It is easy to see that q is exact. In particular, p preserves projective objects and h preserves injective objects. Note that the pairs of adjoint functors $(\mathbf{p},\mathbf{q})$ and $(\mathbf{q},\mathbf{h})$ were defined in [2]. In general, the three pairs of adjoint functors defined above can be found in [20].

For a future reference, we list these adjointness isomorphisms:

$${\mathrm{Hom}}_T(\left(\begin{array}{c}{M}_1\\ (U{\otimes }_A{M}_1)\oplus M_2\end{array}\right),N)\cong {\mathrm{Hom}}_A(M_1,N_1)\oplus {\mathrm{Hom}}_B(M_2,N_2).$$

$${\mathrm{Hom}}_T(N,{\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_0)\cong {\mathrm{Hom}}_A(N_1,M_1)\oplus {\mathrm{Hom}}_B({\overline{N}}_2,M_2).$$

$${\mathrm{Hom}}_T(M,\left(\begin{array}{c}{N}_1\oplus {\mathrm{Hom}}_B(U,N_2)\\ N_2\end{array}\right))\cong {\mathrm{Hom}}_A(M_1,N_1)\oplus {\mathrm{Hom}}_B(M_2,N_2).$$

Now, we recall the characterizations of projective, injective, and finitely generated T-modules.

Lemma 2.

Let $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ be a T-module.

(1) 

([21], Theorem 3.1) M is projective if and only if $M_1$ is projective in A-Mod, ${\overline{M}}_2=Coker{\phi }^M$ is projective in B-Mod, and ${\phi }^M$ is injective.

(2) 

([22], Proposition 5.1) M is injective if and only if $\underline{M_1}=\mathrm{Ker}\widetilde{{\phi }^M}$ is injective in A-Mod, $M_2$ is injective in B-Mod, and $\widetilde{{\phi }^M}$ is surjective.

(3) 

([23]) M is finitely generated if and only if $M_1$ and ${\overline{M}}_2$ are finitely generated.

The following Lemma improves [24], Lemma 3.2.

Lemma 3.

Let $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ and $N={\left(\begin{array}{c}{N}_1\\ N_2\end{array}\right)}_{{\phi }^N}$ be two T-modules and $n\ge 1$ be an integer number. Then, we have the following natural isomorphisms:

1. 

If ${\mathrm{Tor}}_{1\le i\le n}^A(U,M_1)=0$, then ${\mathrm{Ext}}_T^n(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right),N)\cong {\mathrm{Ext}}_A^n(M_1,N_1);$

2. 

${\mathrm{Ext}}_T^n(\left(\begin{array}{c}0\\ M_2\end{array}\right),N)\cong {\mathrm{Ext}}_B^n(M_2,N_2);$

3. 

${\mathrm{Ext}}_T^n(M,\left(\begin{array}{c}{N}_1\\ 0\end{array}\right))\cong {\mathrm{Ext}}_A^n(M_1,N_1);$

4. 

If ${\mathrm{Ext}}_B^{1\le i\le n}(U,N_2)=0$, then ${\mathrm{Ext}}_T^n(M,\left(\begin{array}{c}{\mathrm{Hom}}_B(U,N_2)\\ N_2\end{array}\right))\cong {\mathrm{Ext}}_B^n(M_2,N_2).$

Proof. 

We prove only 1, since 2 is similar and 3 and 4 are dual. Assume that ${\mathrm{Tor}}_{1\le i\le n}^A(U,M_1)=0$, and consider an exact sequence of A-modules:

$$0\to K_1\to P_1\to M_1\to 0$$

where $P_1$ is projective. Therefore, there exists an exact sequence of T-modules:

$$0\to \left(\begin{array}{c}{K}_1\\ U{\otimes }_A{K}_1\end{array}\right)\to \left(\begin{array}{c}{P}_1\\ U{\otimes }_A{P}_1\end{array}\right)\to \left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)\to 0$$

where $\left(\begin{array}{c}{P}_1\\ U{\otimes }_A{P}_1\end{array}\right)$ is projective by Lemma 2.

Let $n=1$. By applying the functor ${\mathrm{Hom}}_T(-,N)$ to the above short exact sequence and since $\left(\begin{array}{c}{P}_1\\ U{\otimes }_A{P}_1\end{array}\right)$ and $P_1$ are projectives, we obtain a commutative diagram with exact rows:

where the first two columns are just the natural isomorphisms given by adjointness and the last two horizontal rows are epimorphisms. Thus, the induced map:

$${\mathrm{Ext}}_T^1(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right),N)\to {\mathrm{Ext}}_A^1(M_1,N_1)$$

is an isomorphism such that the above diagram is commutative.

Assume that $n>1$. Using the long exact sequence, we obtain a commutative diagram with exact rows:

where $\sigma$ is a natural isomorphism by induction, since ${\mathrm{Tor}}_k^A(U,K_1)=0$ for every $k\in \{1,\cdots ,n-1\}$ because of the exactness of the following sequence:

$$0={\mathrm{Tor}}_{k+1}^A(U,M_1)\to {\mathrm{Tor}}_k^A(U,K_1)\to {\mathrm{Tor}}_k^A(U,P_1)=0.$$

Thus, the composite map:

$$g\sigma f^{-1}:{\mathrm{Ext}}_T^n(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right),N)\to {\mathrm{Ext}}_A^n(M_1,N_1)$$

is a natural isomorphism, as desired. □

Since T can be viewed as a trivial extension (see [20,25] for more details), the following lemma can be easily deduced from [25], Theorems 3.1 and 3.4. For the convenience of the reader, we give a proof.

Lemma 4.

Let $X={\left(\begin{array}{c}{X}_1\\ X_2\end{array}\right)}_{{\phi }^X}$ be a T-module and $(C_1,C_2)\in A$-Mod $\times B$-Mod:

1. 

$X\in {\mathrm{Add}}_T(\mathit{p}(C_1,C_2))$ if and only if:

(i) 

$X\cong \mathit{p}(X_1,{\overline{X}}_2)$;

(ii) 

$X_1\in {\mathrm{Add}}_A(C_1)$ and ${\overline{X}}_2\in {\mathrm{Add}}_B(C_2).$

In this case, ${\phi }^X$ is injective;

2. 

$X\in {\mathrm{Prod}}_T(\mathit{h}(C_1,C_2))$ if and only if:

(i) 

$X\cong \mathit{h}(\underline{X_1},X_2)$;

(ii) 

$\underline{X_1}\in {\mathrm{Prod}}_A(C_1)$ and $X_2\in {\mathrm{Prod}}_B(C_2).$

In this case, $\widetilde{{\phi }^X}$ is surjective.

Proof. 

We only need to prove (1), since (2) is dual.

For the “if” part: if $X_1\in {\mathrm{Add}}_A(C_1)$ and ${\overline{X}}_2\in {\mathrm{Add}}_B(C_2)$, then $X_1\oplus Y_1=C^{(I_1)}$ and ${\overline{X}}_2\oplus Y_2=C_2^{(I_2)}$ for some $(Y_1,Y_2)\in A$-Mod$\times B$-Mod and some sets $I_1$ and $I_2$. Without loss of generality, we may assume that $I=I_1=I_2.$ Then:

$$\begin{array}{ccc}\hfill X\oplus \mathbf{p}(Y_1,Y_2)& \cong & \mathbf{p}(X_1,{\overline{X}}_2)\oplus \mathbf{p}(Y_1,Y_2)\hfill \\ & =& \left(\begin{array}{c}{X}_1\\ (U{\otimes }_A{X}_1)\oplus {\overline{X}}_2\end{array}\right)\oplus \left(\begin{array}{c}{Y}_1\\ (U{\otimes }_A{Y}_1)\oplus Y_2\end{array}\right)\hfill \\ & \cong & \left(\begin{array}{c}{C}_1^{(I)}\\ (U{\otimes }_A{C}_1^{(I)})\oplus C_2^{(I)}\end{array}\right)\hfill \\ & \cong & {\left(\begin{array}{c}{C}_1\\ (U{\otimes }_A{C}_1)\oplus C_2\end{array}\right)}^{(I)}\hfill \\ & =& \mathbf{p}{(C_1,C_2)}^{(I)}.\hfill \end{array}$$

Hence, $X\in {\mathrm{Add}}_T(\mathbf{p}(C_1,C_2)).$

Conversely, let $X\in {\mathrm{Add}}_T(\mathbf{p}(C_1,C_2))$ and $Y={\left(\begin{array}{c}{Y}_1\\ Y_2\end{array}\right)}_{{\phi }^Y}$ be a T-module such that $X\oplus Y=\mathbf{p}{(C_1,C_2)}^{(I)}$ for some set I. Then, ${\phi }^X$ is injective, as X is a submodule of $C:=\mathbf{p}{(C_1,C_2)}^{(I)}$ and ${\phi }^C$ is injective. Consider now the split exact sequence:

$$0\to Y\stackrel{\left(\begin{array}{c}{\lambda }_1\\ {\lambda }_2\end{array}\right)}{⟶}C\stackrel{\left(\begin{array}{c}{p}_1\\ p_2\end{array}\right)}{⟶}X\to 0$$

which induces the following commutative diagram with exact rows and columns:

where $\overline{{\phi }^X}$, $\overline{{\phi }^C}$ and $\overline{{\phi }^X}$ are the canonical projections. Clearly, $p_1:C_1^{(I)}\to X_1$ and $\overline{p_2}:C_2^{(I)}\to {\overline{X}}_2$ are split epimorphisms. Then, $X_1\in {\mathrm{Add}}_A(C_1)$ and ${\overline{X}}_2\in {\mathrm{Add}}_B(C_2).$ It remains to prove that $X\cong \mathbf{p}(X_1,{\overline{X}}_2)$. For this, it suffices to prove that the short exact sequence:

$$0\to U{\otimes }_A{X}_1\stackrel{{\phi }^X}{\to }{X}_2\stackrel{\overline{{\phi }^X}}{\to }{\overline{X}}_2\to 0$$

splits. Let $r_2$ be the retraction of $\overline{p_2}$. If $i:C_2^{(I)}\to (U{\otimes }_A{C}_1^{(I)})\oplus C_2^{(I)}$ denotes the canonical injection, then $\overline{{\phi }^X}{p}_{2}i{r}_2=\overline{p_2}\overline{{\phi }^C}i{r}_2=\overline{p_2}{r}_2=1_{\overline{X_2}}$, and the proof is finished. □

Remark 1.

1. 

Since the class of projective modules over T is nothing but the class ${\mathrm{Add}}_T(T)$, when we take $C_1=A$ and $C_2=B$ in Lemma 4, we recover the characterization of projective T-modules. On the other hand, note that the class of injective T-modules coincides with the class ${\mathrm{Prod}}_T(T^{+})$. If we take $T_T=\left(\begin{array}{c}A\oplus U,B\end{array}\right)$, then the injective cogenerator T-module $T^{+}=\mathrm{Hom}(T,\mathbb{Q}/\mathbb{Z})$ can be identified with $\left(\begin{array}{c}{A}^{+}\oplus U^{+}\\ B^{+}\end{array}\right)\cong \mathit{q}(A^{+},B^{+})$. Therefore, by taking $C_1=A^{+}$ and $C_2=B^{+}$ in Lemma 4(A), we recover the characterization of injective T-modules;

2. 

Let $(C_1,C_2)$ be a module over A-Mod $\times B$-Mod. By Lemma 4(2), every module in ${\mathrm{Add}}_T(\mathit{p}(C_1,C_2))$ has the form $\mathit{p}(X_1,X_2)$ for some $X_1\in {\mathrm{Add}}_A(C_1)$ and $X_2\in {\mathrm{Add}}_B(C_2)$.

3. w-Tilting Modules

In this section, we study when the functor $\mathbf{p}$ preserves w-tilting modules.

It is well known that the functor $\mathbf{p}$ preserves projective modules. However, the functor $\mathbf{p}$ does not preserve w-tilting modules in general, as the following example shows.

Example 1.

Let Q be the quiver:

$$e_1\stackrel{\alpha }{⟶}{e}_2,$$

and let $R=kQ$ be the path algebra over an algebraic closed field k. Put $P_1=R{e}_1$, $P_2=R{e}_2$, $I_1={\mathrm{Hom}}_k(e_{1}R,k)$, and $I_2={\mathrm{Hom}}_k(e_{2}R,k).$ Note that $C_1$ and $C_2$ are projective and injective R-modules, respectively. By [12], Example 2.3,

$$C_1=P_1\oplus P_2(=R)\mathit{and}{C}_2=I_1\oplus I_2$$

are semidualizing $(R,R)$-bimodules and, then, w-tilting R-modules. Now, consider the triangular matrix ring:

$$T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right).$$

We claim that $\mathit{p}(C_1,C_2)$ is not a w-tilting $T(R)$-module. Note that $I_1$ is not projective. Since R is left hereditary by [26], Proposition 1.4, ${pd}_R(I_1)=1$. Hence, ${\mathrm{Ext}}_R^1(I_1,R)\ne 0.$ Using Lemma 3, we obtain that ${\mathrm{Ext}}_{T(R)}^1(\mathit{p}(C_1,C_2),\mathit{p}(C_1,C_2))\cong {\mathrm{Ext}}_R^1(C_1,C_1)\oplus {\mathrm{Ext}}_R^1(C_2,C_1)\oplus {\mathrm{Ext}}_R^1(C_2,C_2)\cong {\mathrm{Ext}}_R^1(I_1,R)\ne 0$. Therefore, $\mathit{p}(C_1,C_2)$ is not a w-tilting $T(R)$-module.

Motivated by the definition of compatible bimodules in [1], Definition 1.1, we introduce the following definition, which will be crucial throughout the rest of this paper.

Definition 4.

Let $(C_1,C_2)\in A$-Mod $\times B$-Mod and $C=\mathit{p}(C_1,C_2)$. The bimodule ${}_B{U}_A$ is said to be C-compatible if the following two conditions hold:

(a) 

The complex $U{\otimes }_A{\mathit{X}}_1$ is exact for every exact sequence in A-Mod:

$${\mathit{X}}_1:\cdots \to P_1^1\to P_1^0\to C_1^0\to C_1^1\to \cdots$$

where the $P_1^i$’s are all projective and $C_1^i\in {\mathrm{Add}}_A(C_1)$$\forall i$;

(b) 

The complex ${\mathrm{Hom}}_B({\mathit{X}}_2,U{\otimes }_A{\mathrm{Add}}_A(C_1))$ is exact for every

${\mathrm{Hom}}_B(-,{\mathrm{Add}}_B(C_2))$-exact exact sequence in B-Mod:

$${\mathit{X}}_2:\cdots \to P_2^1\to P_2^0\to C_2^0\to C_2^1\to \cdots$$

where the $P_2^i$’s are all projective and $C_2^i\in {\mathrm{Add}}_B(C_2)$$\forall i$.

Moreover, U is called weakly C-compatible if it satisfies $(b)$ and the following condition:

(a′) 

The complex $U{\otimes }_A{\mathit{X}}_1$ is exact for every ${\mathrm{Hom}}_A(-,{\mathrm{Add}}_A(C_1))$-exact exact sequence in A-Mod

$${\mathit{X}}_1:\cdots \to P_1^1\to P_1^0\to C_1^0\to C_1^1\to \cdots$$

where the $P_1^i$’s are all projective and $C_1^i\in {\mathrm{Add}}_A(C_1)$$\forall i$.

When $C={}_{T}T=\mathit{p}(A,B)$, the bimodule U will be called simply (weakly) compatible.

Remark 2.

1. 

It is clear by the definition that every C-compatible is weakly C-compatible;

2. 

The $(B,A)$-bimodule U is weakly compatible if and only if the functor $U{\otimes }_A-:A$-Mod $\to B$-Mod is weakly compatible (see [27]);

3. 

If A and B are Artin algebras, and since ${}_{T}T=\left(\begin{array}{c}A\\ U\oplus \end{array}\right)=\mathit{p}(A,B)$, it is easy to see that ${}_{T}T$-compatible bimodules are nothing but compatible $(B,A)$-bimodules as defined in [1].

The following can be applied to produce examples of (weakly) C-compatible bimodules later on.

Lemma 5.

Let $C=\mathit{p}(C_1,C_2)$ be a T-module:

• Assume that ${\mathrm{Tor}}_1^A(U,C_1)=0$. If ${\mathrm{fd}}_A(U)<\infty$, then U satisfies $(a)$;
• Assume that ${\mathrm{Ext}}_B^1(C_2,U{\otimes }_A{C}_1^{(I)})=0$ for every set I. If ${\mathrm{id}}_B(U{\otimes }_A{C}_1)<\infty$, then U satisfies $(b)$;
• If $U{\otimes }_A{C}_1\in {\mathrm{Add}}_B(C_2)$, then U satisfies $(b)$.

Proof. 

(3) is clear. We only prove (1), as (2) is similar. Consider an exact sequence of A-modules:

$${\mathbf{X}}_1:\cdots \to P_1^1\to P_1^0\to C_1^0\to C_1^1\to \cdots$$

where the $P_1^i$’s are all projective and $C_1^i\in {\mathrm{Add}}_A(C_1)$$\forall i$. We use induction on ${\mathrm{fd}}_{A}U$. If ${\mathrm{fd}}_{A}U=0,$ then the result is trivial. Now, suppose that ${\mathrm{fd}}_{A}U=n\ge 1$. Then, there exists an exact sequence of right A-modules:

$$0\to L\to F\to U\to 0$$

where ${\mathrm{fd}}_{A}L=n-1$ and F is flat. Applying the functor $-\otimes {\mathbf{X}}_1$ to the above short exact sequence, we obtain the commutative diagram with exact rows:

Since ${\mathrm{Tor}}_1^A(U,C_1)=0$, the above diagram induces an exact sequence of complexes:

$$0\to L{\otimes }_A{\mathbf{X}}_1\to F{\otimes }_A{\mathbf{X}}_1\to U{\otimes }_A{\mathbf{X}}_1\to 0.$$

By the induction hypothesis, the complexes $L{\otimes }_A{\mathbf{X}}_1$ and $F{\otimes }_A{\mathbf{X}}_1$ are exact. Thus, $U{\otimes }_A{\mathbf{X}}_1$ is exact, as well. □

Given a T-module $C=\mathbf{p}(C_1,C_2)$, we have simple characterizations of Conditions $(a^{\prime })$ and $(b)$ if $C_1$ and $C_2$ are w-tilting.

Proposition 1.

Let $C=\mathit{p}(C_1,C_2)$ be a T-module:

1. 

If $C_1$ is w-tilting, then the following assertions are equivalent:

(i) 

U satisfies $(a^{\prime })$;

(ii) 

${\mathrm{Tor}}_1^A(U,G_1)=0$, $\forall G_1\in G_{C_1}P(A)$;

(iii) 

${\mathrm{Tor}}_{i\ge 1}^A(U,G_1)=0$, $\forall G_1\in G_{C_1}P(A)$.

In this case, ${\mathrm{Tor}}_{i\ge 1}^A(U,C_1)=0;$

2. 

If $C_2$ is w-tilting, then the following assertions are equivalent:

(i) 

U satisfies $(b)$;

(ii) 

${\mathrm{Ext}}_B^1(G_2,U{\otimes }_A{X}_1)=0$, $\forall G_2\in G_{C_2}P(B)$, $\forall X_1\in {\mathrm{Add}}_A(C_1)$;

(iii) 

${\mathrm{Ext}}_B^{i\ge 1}(G_2,U{\otimes }_A{X}_1)=0$, $\forall G_2\in G_{C_2}P(B)$, $\forall X_1\in {\mathrm{Add}}_A(C_1)$;

In this case, ${\mathrm{Ext}}_B^{i\ge 1}(C_2,U{\otimes }_A{X}_1)=0$, $\forall X_1\in {\mathrm{Add}}_A(C_1).$

Proof. 

We only prove (1), since (2) is similar.

$(i)\Rightarrow (iii)$ Let $G_1\in G_{C_1}P(R)$. There exists a ${\mathrm{Hom}}_A(-,{\mathrm{Add}}_A(C_1))$-exact exact sequence in A-Mod:

$${\mathbf{X}}_1:\cdots \to P_1^1\to P_1^0\to C_1^0\to C_1^1\to \cdots$$

where the $P_1^i$’s are all projective, $G_1\cong \mathrm{Im}(P_1^0\to C_1^0)$ and $C_1^i\in {\mathrm{Add}}_A(C_1)$$\forall i$. By Condition $(a^{\prime })$, $U{\otimes }_A{\mathbf{X}}_1$ is exact, which means in particular that ${\mathrm{Tor}}_{i\ge 1}^A(U,G_1)=0.$

$(iii)\Rightarrow (ii)$ Clear.

$(ii)\Rightarrow (i)$ Follows by [13], Corollary 2.13.

Finally, to prove that ${\mathrm{Tor}}_{i\ge 1}^A(U,C_1)=0$, note that $C_1\in G_{C_1}P(A)$ by [13], Theorem 2.12.

□

In the following proposition, we study when $\mathbf{p}$ preserves w-tilting (tilting) modules.

Proposition 2.

Let $C=\mathit{p}(C_1,C_2)$ be a T-module and assume that U is weakly C-compatible. If $C_1$ and $C_2$ are w-tilting (tilting), then $\mathit{p}(C_1,C_2)$ is w-tilting (tilting).

Proof. 

By Lemma 2, the functor $\mathbf{p}$ preserves finitely generated modules, so we only need to prove the statement for w-tilting. Assume that $C_1$ and $C_2$ are w-tilting, and let I be a set. Then, ${\mathrm{Ext}}_A^{i\ge 1}(C_1,C_1^{(I)})=0$ and ${\mathrm{Ext}}_B^{i\ge 1}(C_2,C_2^{(I)})=0$. By Proposition above, we have ${\mathrm{Ext}}_B^{i\ge 1}(C_2,U{\otimes }_A{C}_1^{(I)})=0$ and ${\mathrm{Tor}}_{i\ge 1}^A(U,C_1)=0$. Using Lemma 3, for every $n\ge 1$, we obtain that:

$$\begin{array}{ccc}\hfill {\mathrm{Ext}}_T^n(C,C^{(I)})& =& {\mathrm{Ext}}_T^n(\mathbf{p}(C_1,C_2),\mathbf{p}{(C_1,C_2)}^{(I)})\hfill \\ & \cong & {\mathrm{Ext}}_A^n(C_1,C_1^{(I)})\oplus {\mathrm{Ext}}_B^n(C_2,U{\otimes }_A{C}_1^{(I)})\oplus {\mathrm{Ext}}_B^n(C_2,C_2^{(I)})\hfill \\ & =& 0.\hfill \end{array}$$

Moreover, there exist exact sequences:

$${\mathbf{X}}_1:0\to A\to C_1^0\to C_1^1\to \cdots$$

and:

$${\mathbf{X}}_2:0\to B\to C_2^0\to C_2^1\to \cdots$$

which are ${\mathrm{Hom}}_A(-,{\mathrm{Add}}_A(C_1))$-exact and ${\mathrm{Hom}}_B(-,{\mathrm{Add}}_B(C_2))$-exact, respectively, and such that $C_1^i\in {\mathrm{Add}}_A(C_1)$ and $C_2^i\in {\mathrm{Add}}_B(C_2)$ for every $i\in \mathbb{N}.$ Since U is weakly C-compatible, the complex $U{\otimes }_A{\mathbf{X}}_1$ is exact. Therefore, we construct in T-Mod the exact sequence:

$$\mathbf{p}({\mathbf{X}}_1,{\mathbf{X}}_2):0\to T\to \mathbf{p}(C_1^0,C_2^0)\to \mathbf{p}(C_1^1,C_2^1)\to \cdots$$

where $\mathbf{p}(C_1^i,C_2^i)=\left(\begin{array}{c}{C}_1^i\\ (U{\otimes }_A{C}_1^i)\oplus C_2^i\end{array}\right)\in {\mathrm{Add}}_T(\mathbf{p}(C_1,C_2))$, $\forall i\in \mathbb{N},$ by Lemma 4(1).

Let $X\in {\mathrm{Add}}_T(\mathbf{p}(C_1,C_2))$. As a consequence of Lemma 4(1), $X=\mathbf{p}(X_1,X_2)$ where $X_1\in {\mathrm{Add}}_A(C_1)$ and $X_2\in {\mathrm{Add}}_B(C_2).$ Using the adjointness $(\mathbf{p},\mathbf{q})$, we obtain an isomorphism of complexes:

$${\mathrm{Hom}}_T(\mathbf{p}({\mathbf{X}}_1,{\mathbf{X}}_2),X)\cong {\mathrm{Hom}}_A({\mathbf{X}}_1,X_1)\oplus {\mathrm{Hom}}_B({\mathbf{X}}_2,U\otimes X_1)\oplus {\mathrm{Hom}}_B({\mathbf{X}}_2,X_2).$$

However, the complexes ${\mathrm{Hom}}_A({\mathbf{X}}_1,X_1)$ and ${\mathrm{Hom}}_B({\mathbf{X}}_2,X_2)$ are exact, and the complex ${\mathrm{Hom}}_B({\mathbf{X}}_2,U\otimes X_1)$ is also exact since U is weakly C-compatible. Then, ${\mathrm{Hom}}_T(\mathbf{p}({\mathbf{X}}_1,{\mathbf{X}}_2),X)$ is exact, as well, and the proof is finished. □

Now, we illustrate Proposition 2 with two applications.

Corollary 1.

Let $C=\mathit{p}(C_1,C_2)$ be a T-module and $A^{\prime }$ and $B^{\prime }$ be two rings such that ${}_A{C}_{A^{\prime }}$ and ${}_A{C}_{B^{\prime }}$ are bimodules, and assume that U is weakly C-compatible. If ${}_A{C}_{A^{\prime }}$ and ${}_A{C}_{B^{\prime }}$ are semidualizing bimodules, then $\mathit{p}(C_1,C_2)$ is a semidualizing $(T,{\mathrm{End}}_T(C))$-bimodule.

Proof. 

This follows by Proposition 2 and [18], Corollary 3.2. □

Corollary 2.

Let R and S be rings, $\theta :R\to S$ be a homomorphism with $S_R$ flat, and $T=T(\theta )=:\left(\begin{array}{cc}R& 0\\ S& S\end{array}\right)$. Let $C_1$ be an R-module such that $S{\otimes }_R{C}_1\in {\mathrm{Add}}_R(C_1)$ (for instance, if R is commutative or $R=S$). If ${}_R{C}_1$ is w-tilting, then:

1. 

$S{\otimes }_R{C}_1$ is a w-tilting S-module;

2. 

$C=\left(\begin{array}{c}{C}_1\\ (S{\otimes }_R{C}_1)\oplus (S{\otimes }_R{C}_1)\end{array}\right)$ is a w-tilting $T(\theta )$-module.

Proof. 

1. Let $C_2=S{\otimes }_R{C}_1$, and note that $C=\mathbf{p}(C_1,C_2)$ and that ${}_S{S}_R$ is C-compatible. Therefore, by Proposition 2, we only need to prove that $C_2$ is a w-tilting S-module. Since ${}_R{C}_1$ is w-tilting, there exist ${\mathrm{Hom}}_R(-,{\mathrm{Add}}_R(C_1))$-exact exact sequences:

$$\mathbf{P}:\cdots \to P_1\to P_0\to C\to 0$$

and:

$$\mathbf{X}:0\to R\to C_0\to C_1\to \cdots$$

with each ${}_R{P}_i$ projective and ${}_R{C}_i\in {\mathrm{Add}}_R(C_1)$. Since $S_R$ is flat, we obtain an exact sequence:

$$S{\otimes }_R\mathbf{P}:\cdots \to S{\otimes }_R{P}_1\to S{\otimes }_R{P}_0\to S{\otimes }_{R}C\to 0$$

and:

$$S{\otimes }_R\mathbf{X}:0\to S\to S{\otimes }_R{C}_0\to S{\otimes }_R{C}_1\to \cdots$$

with each $S{\otimes }_R{P}_i$ a projective S-module and $S{\otimes }_R{C}_i\in {\mathrm{Add}}_R(C_2)$.

We prove now that $S{\otimes }_R\mathbf{P}$ and $S{\otimes }_R\mathbf{X}$ are ${\mathrm{Hom}}_S-(,{\mathrm{Add}}_S(C_2))$-exact. Let I be a set. Then, ${\mathrm{Hom}}_S(S{\otimes }_R\mathbf{P},S{\otimes }_R{C}_1^{(I)})\cong {\mathrm{Hom}}_R(\mathbf{P},{\mathrm{Hom}}_S(S,S{\otimes }_R{C}_1^{(I)}))\cong {\mathrm{Hom}}_R(\mathbf{P},S{\otimes }_R{C}_1^{(I)})$ is exact since $S{\otimes }_R{C}_1^{(I)}\in {\mathrm{Add}}_R(C_1)$. Similarly, $S{\otimes }_R\mathbf{X}$ is ${\mathrm{Hom}}_S(-,{\mathrm{Add}}_S(C_2))$-exact;

2. This assertion follows from Proposition 2. We only need to note that S is weakly C-compatible since $S_R$ is flat and $S{\otimes }_R{C}_1\in {\mathrm{Add}}_R(C_2)$. □

We end this section with an example of a w-tilting module that is neither projective nor injective.

Example 2.

Take R and $C_2$ as in Example 1. Therefore, by Corollary 2, $C=\left(\begin{array}{c}{C}_2\\ C_2\oplus C_2\end{array}\right)$ is a w-tilting $T(R)$-module. By Lemma 2, C is not projective since $C_2$ is not, and it is not injective since the map $\widetilde{{\phi }^C}:C_2\to C_2\oplus C_2$ is not surjective.

Moreover, by [26], Proposition 2.6, $gl.dim(T(R))=gl.dim(R)+1\le 2$. Therefore, if $0\to T(R)\to E^0\to E^1\to E^2\to 0$ is an injective resolution of $T(R)$, then $C^1=E^0\oplus E^1\oplus E^2$ is a w-tilting $T(R)$-module. Note that $T(R)$ has at least three w-tilting modules, $C^1$, $C^2=T(R)$ and $C^3=C$.

4. Relative Gorenstein Projective Modules

In this section, we describe $G_C$-projective modules over T. Then, we use this description to study when the class of $G_C$-projective T-modules is a special precovering class.

Clearly, the functor $\mathbf{p}$ preserves the projective module. Therefore, we start by studying when the functor $\mathbf{p}$ also preserves relative Gorenstein projective modules. However, first, we need the following:

Lemma 6.

Let $C=\mathit{p}(C_1,C_2)$ be a T-module and U be weakly C-compatible:

• If $M_1\in G_{C_1}P(A)$, then $\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1;\end{array}\right)\in G_{C}P(T).$
• If $M_2\in G_{C_2}P(B)$, then $\left(\begin{array}{c}0\\ M_2\end{array}\right)\in G_{C}P(T).$

Proof. 

$1.$ Suppose that $M_1\in G_{C_1}P(A)$. There exists a ${\mathrm{Hom}}_A(-,{\mathrm{Add}}_A(C_1))$-exact exact sequence:

$${\mathbf{X}}_1:\cdots \to P_1^1\to P_1^0\to C_1^0\to C_1^1\to \cdots$$

where the $P_1^i$’s are all projective, $C_1^i\in {\mathrm{Add}}_A(C_1)$$\forall i$ and $M_1\cong \mathrm{Im}(P_1^0\to C_1^0)$. Using the fact that U is weakly C-compatible, we obtain that the complex $U{\otimes }_A{\mathbf{X}}_1$ is exact in B-Mod, which implies that the complex $\mathbf{p}({\mathbf{X}}_1,0):$

$$\cdots \to \left(\begin{array}{c}{P}_1^1\\ U{\otimes }_A{P}_1^1\end{array}\right)\to \left(\begin{array}{c}{P}_1^0\\ U{\otimes }_A{P}_1^0\end{array}\right)\to \left(\begin{array}{c}{C}_1^0\\ U{\otimes }_A{C}_1^0\end{array}\right)\to \left(\begin{array}{c}{C}_1^1\\ U{\otimes }_A{C}_1^1\end{array}\right)\to \cdots$$

is exact with

$$\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)\cong \mathrm{Im}(\left(\begin{array}{c}{P}_1^0\\ U{\otimes }_A{P}_1^0\end{array}\right)\to \left(\begin{array}{c}{C}_1^0\\ U{\otimes }_A{C}_1^0\end{array}\right)).$$

Clearly, $\mathbf{p}(P_1^i,0)=\left(\begin{array}{c}{P}_1^i\\ U{\otimes }_A{P}_1^i\end{array}\right)\in \mathrm{Proj}(T)$ and $\mathbf{p}(C_1^i,0)=\left(\begin{array}{c}{C}_1^i\\ U{\otimes }_A{C}_1^i\end{array}\right)\in {\mathrm{Add}}_T(C)$$\forall i\in \mathbb{N}$ by Lemmas 2(1) and 4(1). If $X\in {\mathrm{Add}}_T(C)$, then $X_1\in {\mathrm{Add}}_A(C_1)$ by Lemma 4(1), and using the adjointness, we obtain that the complex

${\mathrm{Hom}}_T(\mathbf{p}({\mathbf{X}}_1,0),X)\cong {\mathrm{Hom}}_A({\mathbf{X}}_1,X_1)$ is exact. Hence, $\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)$ is $G_C$-projective;

$2.$ Suppose that $M_2$ is $G_{C_2}$-projective. There exists a ${\mathrm{Hom}}_B(-,{\mathrm{Add}}_B(C_2))$-exact exact sequence:

$${\mathbf{X}}_2:\cdots \to P_2^1\to P_2^0\to C_2^0\to C_2^1\to \cdots$$

where the $P_2^i$’s are all projective, $C_2^i\in {\mathrm{Add}}_B(C_2)$$\forall i$ and $M_2\cong \mathrm{Im}(P_2^0\to C_2^0).$ Clearly, the complex:

$$\mathbf{p}(0,{\mathbf{X}}_2):\cdots \to \left(\begin{array}{c}0\\ P_2^1\end{array}\right)\to \left(\begin{array}{c}0\\ P_2^0\end{array}\right)\to \left(\begin{array}{c}0\\ C_2^0\end{array}\right)\to \left(\begin{array}{c}0\\ C_2^1\end{array}\right)\to \cdots$$

is exact with $\left(\begin{array}{c}0\\ M_2\end{array}\right)\cong \mathrm{Im}(\left(\begin{array}{c}0\\ P_2^1\end{array}\right)\to \left(\begin{array}{c}0\\ C_2^0\end{array}\right))$, $\mathbf{p}(0,P_2^i)=\left(\begin{array}{c}0\\ P_2^i\end{array}\right)\in \mathrm{Proj}(T)$ and $\mathbf{p}(0,C_2^i)=\left(\begin{array}{c}0\\ C_2^i\end{array}\right)\in {\mathrm{Add}}_T(C)$$\forall i,$ by Lemmas 2(1) and 4(1). Let $X\in {\mathrm{Add}}_T(C)$. Then, by Lemma 4(1), $X=\mathbf{p}(X_1,X_2)$ where $X_1\in {\mathrm{Add}}_A(C_1)$ and $X_2\in {\mathrm{Add}}_B(C_2)$. Using adjointness, we obtain that:

$${\mathrm{Hom}}_T(\mathbf{p}(0,{\mathbf{X}}_2),X)\cong {\mathrm{Hom}}_B({\mathbf{X}}_2,U{\otimes }_A{X}_1)\oplus {\mathrm{Hom}}_B({\mathbf{X}}_2,X_2)$$

The complex ${\mathrm{Hom}}_B({\mathbf{X}}_2,X_2)$ is exact, and since U is weakly C-compatible, the complex ${\mathrm{Hom}}_B({\mathbf{X}}_2,U{\otimes }_A{X}_1)$ is also exact. This means that ${\mathrm{Hom}}_T(\mathbf{p}(0,{\mathbf{X}}_2),X)$ is exact as well and $\left(\begin{array}{c}0\\ M_2\end{array}\right)$ is $G_C$-projective. □

Proposition 3.

Let $C=\mathit{p}(C_1,C_2)$ be a T-module. If ${}_B{U}_A$ is weakly C-compatible, then the functor $\mathit{p}$ sends $G_{(C_1,C_2)}$-projectives to $G_C$-projectives. The converse holds provided that $C_1$ and $C_2$ are w-tilting.

In particular, $\mathit{p}$ preserves Gorenstein projective modules if and only if U is weakly compatible.

Proof. 

Note that:

$$\mathbf{p}(M_1,M_2)=\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)\oplus \left(\begin{array}{c}0\\ M_2\end{array}\right).$$

Therefore, this direction follows from Lemma 6 and [13], Proposition 2.5.

Conversely, assume that $C_1$ and $C_2$ are w-tilting. By Proposition 1, it suffices to prove that ${\mathrm{Tor}}_1^A(U,G_{C_1}P(A))=0={\mathrm{Ext}}_B^1(G_{C_2}P(B),U{\otimes }_A{\mathrm{Add}}_A(C_1))$.

Let $G_1\in G_{C_1}P(A)$. By [13], Corollary 2.13, there exits an exact and a ${\mathrm{Hom}}_A(-,{\mathrm{Add}}_A(C_1))$-exact sequence $0\to L_1\stackrel{ı}{\to }{P}_1\to G_1\to 0$, where ${}_A{P}_1$ is projective and $L_1$ is $G_{C_1}$-projective. Note that $A,C_1\in G_{C_1}P(A)$ and $B,C_2\in G_{C_2}P(B)$ by Lemma 1. Then, ${}_{T}T=\mathbf{p}(A,B)$ and $C=\mathbf{p}(C_1,C_2)$ are $G_C$-projective, which imply by Lemma 1 that C is w-tilting. Moreover $\left(\begin{array}{c}{L}_1\\ U{\otimes }_A{L}_1\end{array}\right)=\mathbf{p}(L_1,0)$ is also $G_C$-projective, and by [13], Corollary 2.13, there exists a short exact sequence:

$$0\to \left(\begin{array}{c}{L}_1\\ U{\otimes }_A{L}_1\end{array}\right)\to {\left(\begin{array}{c}{X}_1\\ X_2\end{array}\right)}_{{\phi }^X}\to {\left(\begin{array}{c}{H}_1\\ H_2\end{array}\right)}_{{\phi }^H}\to 0$$

where $X={\left(\begin{array}{c}{X}_1\\ X_2\end{array}\right)}_{{\phi }^X}\in {\mathrm{Add}}_T(C)$ and $H={\left(\begin{array}{c}{H}_1\\ H_2\end{array}\right)}_{{\phi }^H}$ is $G_C$-projective.

Since $X_1\in {\mathrm{Add}}_A(C_1)$, we have the following commutative diagram with exact rows:

Therefore, if we apply the functor $U{\otimes }_A-$ to the above diagram, we obtain the following commutative diagram with exact rows:

The commutativity of this diagram implies that the map $1_U\otimes ı$ is injective, and since $P_1$ is projective, ${\mathrm{Tor}}_1^A(U,G_1)=0$.

Now, let $G_2\in G_{C_2}P(B)$ and $Y_2\in {\mathrm{Add}}_A(C_1)$. By hypothesis, $\left(\begin{array}{c}0\\ G_2\end{array}\right)=\mathbf{p}(0,G_2)$ is $G_C$-projective, and by Lemma 4, $\left(\begin{array}{c}{Y}_1\\ U\otimes Y_1\end{array}\right)=\mathbf{p}(Y_1,0)\in {\mathrm{Add}}_T(C)$. Hence, ${\mathrm{Ext}}_B^1(G_2,U{\otimes }_A{Y}_1)={\mathrm{Ext}}_T^1(\left(\begin{array}{c}0\\ G_2\end{array}\right),\left(\begin{array}{c}{Y}_1\\ U\otimes Y_1\end{array}\right))=0$ by Lemma 3 and [13], Proposition 2.4. □

Theorem 1.

Let $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ and $C=\mathit{p}(C_1,C_2)$ be two T-modules. If U is C-compatible, then the following assertions are equivalent:

1. 

M is $G_C$-projective;

2. 

(i) 

${\phi }^M$ is injective;

(ii) 

$M_1$ is $G_{C_1}$-projective and ${\overline{M}}_2:=\mathrm{Coker}{\phi }^M$ is $G_{C_2}$-projective.

In this case, if $C_2$ is Σ-self-orthogonal, then $U{\otimes }_A{M}_1$ is $G_{C_2}$-projective if and only if $M_2$ is $G_{C_2}$-projective.

Proof. 

$2.\Rightarrow 1.$ Since ${\phi }^M$ is injective, there exists an exact sequence in T-Mod:

$$0\to \left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)\to M\to \left(\begin{array}{c}0\\ {\overline{M}}_2\end{array}\right)\to 0$$

Note that $\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)and\left(\begin{array}{c}0\\ {\overline{M}}_2\end{array}\right)$ are $G_C$-projective T-modules by Lemma 6. Therefore, M is $G_C$-projective by [13], Proposition 2.5.

$1.\Rightarrow 2.$ There exists a ${\mathrm{Hom}}_T(-,{\mathrm{Add}}_T(C))$-exact sequence in T-Mod:

$$\mathbf{X}=\cdots \to {\left(\begin{array}{c}{P}_1^1\\ P_2^1\end{array}\right)}_{{\phi }^{P^1}}\to {\left(\begin{array}{c}{P}_1^0\\ P_2^0\end{array}\right)}_{{\phi }^{P^0}}\to {\left(\begin{array}{c}{C}_1^0\\ C_2^0\end{array}\right)}_{{\phi }^{C^0}}\to {\left(\begin{array}{c}{C}_1^1\\ C_2^1\end{array}\right)}_{{\phi }^{C^1}}\to \cdots$$

where $C^i={\left(\begin{array}{c}{C}_1^i\\ C_2^i\end{array}\right)}_{{\phi }^{C^i}}\in {\mathrm{Add}}_T(C)$, $P^i={\left(\begin{array}{c}{P}_1^i\\ P_2^i\end{array}\right)}_{{\phi }^{P^i}}\in \mathrm{Proj}(T)$$\forall i\in \mathbb{N}$, and such that $M\cong \mathrm{Im}(P^0\to C^0).$ Then, we obtain the exact sequence:

$${\mathbf{X}}_1=\cdots \to P_1^1\to P_1^0\to C_1^0\to C_1^1\to \cdots$$

where $C_1^i\in {\mathrm{Add}}_A(C_1)$, $P_1^i\in \mathrm{Proj}(A)$$\forall i\in \mathbb{N}$ by Lemmas 2(1) and 4(1) and such that $M_1\cong \mathrm{Im}(P_1^0\to C_1^0).$ Since U is C-compatible, the complex $U{\otimes }_A{\mathbf{X}}_1$ is exact with $U{\otimes }_A{M}_1\cong Im(U{\otimes }_A{P}_1^0\to U{\otimes }_A{C}_1^0).$ If ${\iota }_1:M_1\to C_1^0$ and ${\iota }_2:M_2\to C_2^0$ are the inclusions, then $1_U\otimes {\iota }_1$ is injective, and the following diagram commutes:

By Lemma 4(1), ${\phi }^{C^0}$ is injective, then ${\phi }^M$ is also injective. Moreover, for every $i\in \mathbb{N}$, ${\phi }^{P^i}$ and ${\phi }^{C^i}$ are injective by Lemmas 2 and 4(1). Then, the following diagram with exact columns:

is commutative. Since the first row and the second row are exact, we obtain the exact sequence of B-modules:

$${\overline{\mathbf{X}}}_2:\cdots \to \overline{P_2^1}\to \overline{P_2^0}\to \overline{C_2^0}\to \overline{C_2^1}\to \cdots$$

where $\overline{P_2^i}\in \mathrm{Proj}(B)$, $\overline{C_2^i}\in {\mathrm{Add}}_B(C_2)$ by Lemmas 2 and 4(1) and such that ${\overline{M}}_2=$$\mathrm{Im}(\overline{P_2^0}\to \overline{C_2^0})$. It remains to see that ${\mathbf{X}}_1$ and ${\overline{\mathbf{X}}}_2$ are ${\mathrm{Hom}}_A(-,\mathrm{Add}(C_1))$-exact and ${\mathrm{Hom}}_B(-,{\mathrm{Add}}_B(C_2))$-exact, respectively. Let $X_1\in {\mathrm{Add}}_A(C_1)$ and $X_2\in {\mathrm{Add}}_B(C_2)$. Then, $\mathbf{p}(X_1,0)=\left(\begin{array}{c}{X}_1\\ U{\otimes }_A{X}_1\end{array}\right)\in {\mathrm{Add}}_T(C)$ and $\mathbf{p}(0,X_2)=\left(\begin{array}{c}0\\ X_2\end{array}\right)\in {\mathrm{Add}}_T(C)$ by Lemma 4(1). Therefore, by using adjointness, we obtain that ${\mathrm{Hom}}_B({\overline{\mathbf{X}}}_2,X_2)\cong {\mathrm{Hom}}_T(\mathbf{X},\left(\begin{array}{c}0\\ X_2\end{array}\right))$ is exact. Using adjointness again, we obtain that:

$${\mathrm{Hom}}_T(\mathbf{X},\left(\begin{array}{c}0\\ U{\otimes }_A{X}_1\end{array}\right))\cong {\mathrm{Hom}}_B({\overline{\mathbf{X}}}_2,U{\otimes }_A{X}_1)$$

and:

$${\mathrm{Hom}}_T(\mathbf{X},\left(\begin{array}{c}{X}_1\\ 0\end{array}\right))\cong {\mathrm{Hom}}_A({\mathbf{X}}_1,X_1).$$

Note that $C^i\cong \mathbf{p}(C_1^i,\overline{C_2^i})$ by Lemma 4(1). Hence, ${\mathrm{Ext}}_T^1(C^i,\left(\begin{array}{c}0\\ U{\otimes }_A{X}_1\end{array}\right))\cong {\mathrm{Ext}}_B^1(\overline{C_2^i},U{\otimes }_A{X}_1)=0$ by Lemma 3. Therefore, if we apply the functor ${\mathrm{Hom}}_T(\mathbf{X},-)$ to the sequence:

$$0\to \left(\begin{array}{c}0\\ U{\otimes }_A{X}_1\end{array}\right)\to \left(\begin{array}{c}{X}_1\\ U{\otimes }_A{X}_1\end{array}\right)\to \left(\begin{array}{c}{X}_1\\ 0\end{array}\right)\to 0,$$

we obtain the following exact sequence of complexes:

$$0\to {\mathrm{Hom}}_B({\overline{\mathbf{X}}}_2,U{\otimes }_A{X}_1)\to {\mathrm{Hom}}_T(\mathbf{X},\left(\begin{array}{c}{X}_1\\ U{\otimes }_A{X}_1\end{array}\right))\to {\mathrm{Hom}}_A({\mathbf{X}}_1,X_1)\to 0.$$

Since U is C-compatible, it follows that ${\mathrm{Hom}}_B({\overline{\mathbf{X}}}_2,U{\otimes }_A{X}_1)$ is exact, and since C is w-tilting, ${\mathrm{Hom}}_T(\mathbf{X},\left(\begin{array}{c}{X}_1\\ U{\otimes }_A{X}_1\end{array}\right))$ is also exact. Thus, ${\mathrm{Hom}}_A({\mathbf{X}}_1,X_1)$ is exact, and the proof is finished. □

The following consequence of the above theorem gives the converse of Proposition 2.

Corollary 3.

Let $C=\mathit{p}(C_1,C_2)$ and assume that U is C-compatible. Then, C is w-tilting if and only if $C_1$ and $C_2$ are w-tilting.

Proof. 

An easy application of Proposition 1 and Theorem 1 on the T-modules $C=\left(\begin{array}{c}{C}_1\\ (U{\otimes }_A{C}_1)\oplus C_2\end{array}\right)$ and ${}_{T}T=\left(\begin{array}{c}A\\ U\oplus B\end{array}\right)$. □

One would like to know if every w-tilting T-module has the form $\mathbf{p}(C_1,C_2)$ where $C_1$ and $C_2$ are w-tilting. The following example gives a negative answer to this question.

Example 3.

Let R be a quasi-Frobenius ring and $T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right)$. Consider the exact sequence of T-modules:

$$0\to T\to \left(\begin{array}{c}R\oplus R\\ R\oplus R\end{array}\right)\to \left(\begin{array}{c}R\\ 0\end{array}\right)\to 0.$$

By Lemma 2, $I^0=\left(\begin{array}{c}R\oplus R\\ R\oplus R\end{array}\right)$ and $I^1=\left(\begin{array}{c}R\\ 0\end{array}\right)$ are both injective $T(R)$-modules. Note that $T(R)$ is Noetherian ([23], Proposition 1.7), and then, we can see that $C:=I^0\oplus I^1$ is a w-tilting $T(R)$-module, but does not have the form $\mathit{p}(C_1,C_2)$ where $C_1$ and $C_2$ are w-tilting by Lemma 4 since $I^1\in {\mathrm{Add}}_{T(R)}(C)$ and ${\phi }^{I^1}$ is not injective.

As an immediate consequence of Theorem 1, we have the following.

Corollary 4.

Let R be a ring and $T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right)$. If $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ and $C=\mathit{p}(C_1,C_1)$ are two $T(R)$-modules with $C_1$Σ-self-orthogonal, then the following assertions are equivalent:

1. 

M is $G_C$-projective $T(R)$-module;

2. 

$M_1$ and ${\overline{M}}_2$ are $G_{C_1}$-projective R-modules, and ${\phi }^M$ is injective;

3. 

$M_1$ and $M_2$ are $G_{C_1}$-projective R-modules, and ${\phi }^M$ is injective.

An Artin algebra $\Lambda$ is called Cohen–Macaulay-free (CM-free) if any finitely generated Gorenstein projective module is projective. The authors in [2] extended this definition to arbitrary rings and defined strongly CM-free as rings over which every Gorenstein projective module is projective. Now, we introduce a relative notion of these rings and give a characterization of when T is such rings.

Definition 5.

Let R be a ring. Given an R-module C, R is called CM-free (relative to C) if $G_{C}P(R)\cap R\text{-}\mathrm{mod}={\mathrm{add}}_R(C)$, and it is called strongly CM-free (relative to C) if $G_{C}P(R)={\mathrm{Add}}_R(C)$.

Remark 3.

Let R be a ring and C a Σ-self-orthogonal R-module. Then, ${\mathrm{Add}}_R(C)\subseteq G_{C}P(R)$ and ${\mathrm{add}}_R(C)\subseteq G_{C}P(R)\cap R\text{-}\mathrm{mod}$ by [13], Propositions 2.5 and 2.6 and Corollary 2.10, then R is CM-free (relative to C) if and only if every finitely generated $G_C$-projective is in ${\mathrm{add}}_R(C)$, and it is strongly CM-free (relative to C) if every $G_C$-projective is in ${\mathrm{Add}}_R(C).$

Using the above results, we refine and extend [2], Theorem 4.1, to our setting. Note that the condition B is left Gorenstein regular is not needed.

Proposition 4.

Let ${}_A{C}_1$ and ${}_B{C}_2$ be Σ-self-orthogonal, and $C=\mathit{p}(C_1,C_2)$. Assume that U is weakly C-compatible, and consider the following assertions:

1. 

T is (strongly) CM-free relative to C;

2. 

A and B are (strongly) CM-free relative to $C_1$ and $C_2$, respectively.

Then, $1.\Rightarrow 2.$ If U is C-compatible, then $1.\iff 2.$

Proof. 

We only prove the result for relative strongly CM-free, since the case of relative CM-free is similar.

$1.\Rightarrow 2.$ By the remark above, we only need to prove that $G_{C_1}P(A)\subseteq {\mathrm{Add}}_A(C_1)$ and $G_{C_2}P(B)\subseteq {\mathrm{Add}}_B(C_2)$. Let $M_1$ be a $G_{C_1}$-projective A-module and ${}_B{M}_2$ a $G_{C_2}$-projective B-module. By the assumption and Proposition 3, $\mathbf{p}(M_1,M_2)\in G_{C}P(T)={\mathrm{Add}}_T(C)$. Hence, $M_1\in {\mathrm{Add}}_A(C_1)$ and $M_2\in {\mathrm{Add}}_B(C_2)$ by Lemma 4.

$2.\Rightarrow 1.$ Assume U is C-compatible. Clearly, C is $\mathsf{\Sigma }$-self-orthogonal, then by Remark above, we only need to prove that $G_{C}P(T)\subseteq {\mathrm{Add}}_T(C)$. Let $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ be a $G_C$-projective T-module. By the assumption and Theorem 1, $M_1\in G_{C_1}P(A)={\mathrm{Add}}_A(C_1)$ and ${\overline{M}}_2\in G_{C_2}P(B)={\mathrm{Add}}_B(C_2)$, and the map ${\phi }^M$ is injective. By the assumption, we can easily see that ${\mathrm{Ext}}_B^{i\ge 1}(U{\otimes }_A{M}_1,{\overline{M}}_2)=0$. Therefore, the map $0\to U{\otimes }_A{M}_1\stackrel{{\phi }^M}{\to }{M}_2\to {\overline{M}}_2\to 0$ splits. Hence, $M\cong \mathbf{p}(M_1,{\overline{M}}_2)\in {\mathrm{Add}}_T(C)$ by Lemma 4. □

Our aim now is to study special $G_{C}P(T)$-precovers in T-Mod. We start with the following result.

Proposition 5.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting, U be C-compatible, and $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ and $G={\left(\begin{array}{c}{G}_1\\ G_2\end{array}\right)}_{{\phi }^G}$ two T-modules with G$G_C$-projective. Then:

$$f=\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right):G⟶M$$

is a special $G_{C}P(T)$-precover if and only if:

(i) 

$G_1\stackrel{f_1}{\to }{M}_1$ is a special $G_{C_1}P(A)$-precover;

(ii) 

$G_2\stackrel{f_2}{\to }{M}_2$ is surjective with its kernel lying in $G_{C_2}P{(B)}^{{\perp }_1}$.

In this case, if $G_2\in G_{C_2}P(B)$, then $G_2\stackrel{f_2}{\to }{M}_2$ is a special $G_{C_2}P(B)$-precover.

Proof. 

First of all, let $K=\mathrm{Ker}f={\left(\begin{array}{c}{K}_1\\ K_2\end{array}\right)}_{{\phi }^K}$, and note that, since $C_1$ is w-tilting, ${\mathrm{Tor}}_1^A(U,H_1)=0$ for every $H_1\in G_{C_1}P(A)$ by Proposition 1(1).

⇒ Since the map f is surjective, so are $f_1$ and $f_2$. Let $H_1\in G_{C_1}P(A)$ and $H_2\in G_{C_2}P(B)$. Then, $\left(\begin{array}{c}{H}_1\\ U{\otimes }_A{H}_1\end{array}\right),\left(\begin{array}{c}0\\ H_2\end{array}\right)\in G_{C}P(T)$ by Theorem 1. Using Lemma 3 and the fact that K lies in $G_{C}P{(R)}^{{\perp }_1}$, we obtain that:

$${\mathrm{Ext}}_A^1(H_1,K_1)\cong {\mathrm{Ext}}_T^1(\left(\begin{array}{c}{H}_1\\ U{\otimes }_A{H}_1\end{array}\right),K)=0$$

and:

$${\mathrm{Ext}}_B^1(H_2,K_2)\cong {\mathrm{Ext}}_T^1(\left(\begin{array}{c}0\\ H_2\end{array}\right),K)=0.$$

It remains to see that $G_1\in G_{C_1}P(A)$, which is true by Theorem 1, since G is $G_C$-projective.

⇐ The morphism f is surjective since $f_1$ and $f_2$ are. Therefore, we only need to prove that K lies in $G_{C}P{(R)}^{{\perp }_1}$. Let $H\in G_{C}P(R)$. By Theorem 1, we have the short exact sequence of T-modules:

$$0\to \left(\begin{array}{c}{H}_1\\ U{\otimes }_A{H}_1\end{array}\right)\to H\to \left(\begin{array}{c}0\\ {\overline{H}}_2\end{array}\right)\to 0$$

where $H_1$ is $G_{C_1}$-projective and ${\overline{H}}_2$ is $G_{C_2}$-projective. Therefore, by hypothesis and Lemma 3, we obtain that ${\mathrm{Ext}}_T^1(\left(\begin{array}{c}{H}_1\\ U{\otimes }_A{H}_1\end{array}\right),K)\cong {\mathrm{Ext}}_A^1(H_1,K_1)=0$ and ${\mathrm{Ext}}_T^1(\left(\begin{array}{c}0\\ {\overline{H}}_2\end{array}\right),K)\cong {\mathrm{Ext}}_B^1({\overline{H}}_2,K_2)$ = 0. Then, the exactness of this sequence:

$${\mathrm{Ext}}_T^1(\left(\begin{array}{c}{H}_1\\ U{\otimes }_A{H}_1\end{array}\right),K)\to {\mathrm{Ext}}_T^1(H,K)\to {\mathrm{Ext}}_T^1(\left(\begin{array}{c}0\\ {\overline{H}}_2\end{array}\right),K)$$

implies that ${\mathrm{Ext}}_T^1(H,K)=0.$ □

Theorem 2.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting and U C-compatible. Then, the class $G_{C}P(T)$ is special precovering in T-Mod if and only if the classes $G_{C_1}P(A)$ and $G_{C_2}P(B)$ are special precovering in A-Mod and B-Mod, respectively.

Proof. 

⇒ Let $M_1$ be an A-module and ${\left(\begin{array}{c}{G}_1\\ G_2\end{array}\right)}_{{\phi }^G}\to \left(\begin{array}{c}{M}_1\\ 0\end{array}\right)$ be a special $G_{C}P(T)$-precover in T-Mod. Then, by Proposition 5, $G_1\to M_1$ is a special $G_{C_1}P(A)$-precover in A-Mod.

Let $M_2$ be a B-module and $\left(\begin{array}{c}0\\ f_2\end{array}\right):{\left(\begin{array}{c}{G}_1\\ G_2\end{array}\right)}_{{\phi }^G}\to \left(\begin{array}{c}0\\ M_2\end{array}\right)$ be a special $G_{C}P(T)$-precover in T-Mod. By Proposition 5, $G_1\to 0$ is a special $G_{C_1}P(A)$-precover. Then, ${\mathrm{Ext}}_A^1(G_{C_1}P(A),G_1)=0$. On the other hand, by [13], Proposition 2.8, there exists an exact sequence of A-modules:

$$0\to G_1\to X_1\to H_1\to 0$$

where $X_1\in {\mathrm{Add}}_A(C_1)$ and $H_1$ is $G_{C_1}$-projective. However, this sequence splits, since ${\mathrm{Ext}}_A^1(H_1,G_1)=0$, which implies that $G_1\in {\mathrm{Add}}_A(C_1)$. Let $K={\left(\begin{array}{c}{K}_1\\ K_2\end{array}\right)}_{{\phi }^K}$ be the kernel of $\left(\begin{array}{c}0\\ f_2\end{array}\right)$. Note that $K_1=G_1$. Therefore, there exists a commutative diagram:

Using the snake lemma, there exists an exact sequence of B-modules:

$$0\to {\overline{K}}_2\to {\overline{G}}_2\to M_2\to 0$$

where ${\overline{G}}_2$ is $G_{C_2}$-projective by Theorem 1. It remains to see that ${\overline{K}}_2$ lies in $G_{C_2}P{(B)}^{{\perp }_1}$. Let $H_2\in G_{C_2}P(B)$. Then, ${\mathrm{Ext}}_B^1(H_2,K_2)=0$ by Proposition 5 and ${\mathrm{Ext}}_B^{i\ge 1}(H_2,U{\otimes }_A{G}_1)=0$ by Proposition 1(2). From the above diagram, ${\phi }^K$ is injective. Therefore, if we apply the functor ${\mathrm{Hom}}_B(H_2,-)$ to the short exact sequence:

$$0\to U{\otimes }_A{G}_1\to K_2\to {\overline{K}}_2\to 0,$$

we obtain an exact sequence:

$${\mathrm{Ext}}_B^1(H_2,K_2)\to {\mathrm{Ext}}_B^1(H_2,{\overline{K}}_2)\to {\mathrm{Ext}}_B^2(H_2,U{\otimes }_A{G}_1)$$

which implies that ${\mathrm{Ext}}_B^1(H_2,{\overline{K}}_2)=0$.

⇐ Note that the functor $U{\otimes }_A-:A$-Mod$\to B$-Mod is $G_{C_1}P(A)$-exact since ${\mathrm{Tor}}_1^A(U,G_{C_1}P(A))=0$ by Proposition 1. Therefore, this direction follows by [27], Theorem 1.1, since $G_{C}P(T)=\mathsf{\{}M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}\in T\text{-}\mathrm{Mod}\mathsf{|}{M}_1\in G_{C_1}P(A),\overline{M_2}\in G_{C_2}P(B)\mathrm{and}{\phi }^M\mathrm{is}\mathrm{injective}\mathsf{\}}$ by Theorem 1. □

Corollary 5.

Let R be a ring, $T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right)$, and $C=\mathit{p}(C_1,C_1)$ a w-tilting $T(R)$-module. Then, $G_{C}P(T(R))$ is a special precovering class if and only if $G_{C_1}P(R)$ is a special precovering class.

5. Relative Global Gorenstein Dimension

In this section, we investigate the $G_C$-projective dimension of T-modules and the left $G_C$-projective global dimension of T.

Let R be a ring. Recall [13] that a module M is said to have a $G_C$-projective dimension less than or equal to n, $G_C-pd(M)\le n$, if there is an exact sequence:

$$0\to G_n\to \cdots \to G_0\to M\to 0$$

with $G_i\in G_{C}P(R)$ for every $i\in \{0,\cdots ,n\}$. If n is the least nonnegative integer for which such a sequence exists, then $G_C-pd(M)=n$, and if there is no such n, then $G_C-pd(M)=\infty$.

The left $G_C$-projective global dimension of R is defined as:

$$G_C-PD(R)=sup\left\{G_C-pd(M)\right|M\mathrm{is}\mathrm{an}R\text{-}\mathrm{module}\}$$

Lemma 7.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting and U C-compatible.

1. 

$G_{C_2}-pd(M_2)=G_C-pd(\left(\begin{array}{c}0\\ M_2\end{array}\right)).$

2. 

$G_{C_1}-pd(M_1)\le G_C-pd(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)),$ and the equality holds if

${\mathrm{Tor}}_{i\ge 1}^A(U,M_1)=0.$

Proof. 

1. Let $n\in \mathbb{N}$, and consider an exact sequence of B-modules:

$$0\to K_2^n\to G_2^{n-1}\to \cdots \to G_2^0\to M_2\to 0$$

where each $G_2^i$ is $G_{C_2}$-projective. Thus, there exists an exact sequence of T-modules:

$$0\to \left(\begin{array}{c}0\\ K_2^n\end{array}\right)\to \left(\begin{array}{c}0\\ G_2^{n-1}\end{array}\right)\to \cdots \to \left(\begin{array}{c}0\\ G_2^0\end{array}\right)\to \left(\begin{array}{c}0\\ M_2\end{array}\right)\to 0$$

where each $\left(\begin{array}{c}0\\ G_2^i\end{array}\right)$ is $G_C$-projective by Theorem 1. Again, by Theorem 1, $\left(\begin{array}{c}0\\ K_2^n\end{array}\right)$ is $G_C$-projective if and only if $K_2^n$ is $G_{C_1}$-projective, which means that $G_C-pd(\left(\begin{array}{c}0\\ M_2\end{array}\right))\le n$ if and only if $G_{C_2}-pd(M_2)\le n$ by [13], Theorem 3.8. Hence $G_C-pd(\left(\begin{array}{c}0\\ M_2\end{array}\right))=G_{C_2}-pd(M_2);$

2. We may assume that $n=G_C-pd(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right))<\infty$. By Definition, there exists an exact sequence of T-modules:

$$0\to G^n\to G^{n-1}\to \cdots \to G^0\to \left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)\to 0$$

where each $G^i={\left(\begin{array}{c}{G}_1^i\\ G_2^i\end{array}\right)}_{{\phi }^{G^i}}$ is $G_C$-projective. Thus, there exists an exact sequence of A-modules:

$$0\to G_1^n\to G_1^{n-1}\to \cdots \to G_1^0\to M_1\to 0$$

where each $G_1^i$ is $G_{C_1}$-projective by Theorem 1. Therefore, $G_{C_1}-pd(M_1)\le n$. Conversely, we prove that $G_C-pd(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right))\le G_{C_1}-pd(M_1)$. We may assume that $m:=G_{C_1}-pd(M_1)<\infty .$ The hypothesis means that if:

$${\mathbf{X}}_1:0\to K_1^m\to P_1^{m-1}\to \cdots \to P_1^0\to M_1\to 0$$

is an exact sequence of A-modules where each $P_1^i$ is projective, then the complex $U{\otimes }_A{\mathbf{X}}_1$ is exact. Since $C_1$ is w-tilting, each $P_i$ is $G_{C_1}$-projective by [13], Proposition 2.11, and then, $K^m$ is $G_{C_1}$-projective by [13], Theorem 3.8. Thus, there exists an exact sequence of T-modules

$0\to \left(\begin{array}{c}{K}_1^m\\ U{\otimes }_A{K}_1^m\end{array}\right)\to \left(\begin{array}{c}{P}_1^{m-1}\\ U{\otimes }_A{P}_1^{m-1}\end{array}\right)\to \cdots \to \left(\begin{array}{c}{P}_1^0\\ U{\otimes }_A{P}_1^0\end{array}\right)\to \left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right)\to 0$, where $\left(\begin{array}{c}{K}_1^m\\ U{\otimes }_A{K}_1^m\end{array}\right)$ and all $\left(\begin{array}{c}{P}_1^i\\ U{\otimes }_A{P}_1^i\end{array}\right)$ are $G_C$-projectives by Theorem 1. Therefore, $G_C-pd(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_1\end{array}\right))\le m=G_{C_1}-pd(M_1)$. □

Given a T-module $C=\mathbf{p}(C_1,C_2)$, we introduce a strong notion of the $G_{C_2}$-projective global dimension of B, which will be crucial when we estimate the $G_C$-projective dimension of a T-module and the left global $G_C$-projective dimension of T. Set:

$$S{G}_{C_2}-PD(B)=sup\{G_{C_2}-p{d}_B(U{\otimes }_{A}G)|G\in G_{C_1}P(A)\}.$$

Remark 4.

1. 

Clearly, $S{G}_{C_2}-PD(B)\le G_{C_2}-PD(B)$;

2. 

Note that ${pd}_B(U)=sup\left\{{pd}_B(U{\otimes }_{A}P)\right|{}_{A}P\mathrm{is}\mathrm{projective}\}$. Therefore, in the classical case, the strong left global dimension of B is nothing but the projective dimension of ${}_{B}U$.

Theorem 3.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting, U C-compatible, $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ a T-module, and $S{G}_{C_2}-PD(B)<\infty$. Then:

$$max\{G_{C_1}-p{d}_A(M_1),(G_{C_2}-p{d}_B(M_2))-(S{G}_{C_2}-PD(B))\}$$

$$\le G_C-pd(M)\le$$

$$max\{(G_{C_1}-p{d}_A(M_1))+(S{G}_{C_2}-PD(B))+1,G_{C_2}-p{d}_B(M_2)\}$$

Proof. 

First of all, note that $C_1$ and $C_2$ are w-tilting by Proposition 3, and let $k:=S{G}_{C_2}-PD(B).$

Let us first prove that:

$$max\{G_{C_1}-pd(M_1),G_{C_2}-pd(M_2)-k\}\le G_C-pd(M).$$

We may assume that $n:=G_C-pd(M)<\infty .$ Then, there exists an exact sequence of T-modules:

$$0\to G^n\to G^{n-1}\to \cdots \to G^0\to M\to 0$$

where each $G^i={\left(\begin{array}{c}{G}_1^i\\ G_2^i\end{array}\right)}_{{\phi }^{G^i}}$ is $G_C$-projective. Thus, there exists an exact sequence of A-modules:

$$0\to G_1^n\to G_1^{n-1}\to \cdots \to G_1^0\to M_1\to 0$$

where each $G_1^i$ is $G_{C_1}$-projective by Theorem 1. Therefore, $G_{C_1}-pd(M_1)\le n$. By Theorem 1, for each i, there exists an exact sequence of B-modules:

$$0\to U{\otimes }_A{G}_1^i\to G_2^i\to \overline{G_2^i}\to 0$$

where $\overline{G_2^i}$ is $G_{C_2}$-projective. Then, $G_{C_2}-pd(G_2^i)=G_{C_2}-pd(U{\otimes }_A{G}_1^i)\le k$ by [13], Proposition 3.11. Therefore, using the exact sequence of B-modules:

$$0\to G_2^n\to G_2^{n-1}\to \cdots \to G_2^0\to M_2\to 0$$

and [13], Proposition 3.11(4), we obtain that $G_{C_2}-pd(M_2)\le n+k.$

Next we prove that:

$$G_C-pd(M)\le max\{G_{C_1}-pd(M_1)+k+1,G_{C_2}-pd(M_2)\}.$$

We may assume that:

$$m:=max\{G_{C_1}-pd(M_1)+k+1,G_{C_2}-pd(M_2)\}<\infty .$$

Then, $n_1:=G_{C_1}-pd(M_1)<\infty$ and $n_2:=G_{C_2}-pd(M_2)<\infty$. Since $G_{C_1}-pd(M_1)$

$=n_1\le m-k-1$, there exists an exact sequence of A-modules:

$$0\to G_1^{m-k-1}\to \cdots \to G_1^{n_2-k}\to \cdots \stackrel{f_1^1}{\to }{G}_1^0\stackrel{f_1^0}{\to }{M}_1\to 0$$

where each $G_1^i$ is $G_{C_1}$-projective. Since $C_2$ is w-tilting, there exists an exact sequence of B-modules $G_2^0\stackrel{g_2^0}{\to }{M}_2\to 0$ where $G_2^0$ is $G_{C_2}$-projective by [13], Corollary 2.14. Let $K_1^i=\mathrm{Ker}{f}_1^i$, and define the map $f_2^0:U{\otimes }_A{G}_1^0\oplus G_2^0\to M_2$ to be $({\phi }^M(1_U\otimes f_1^0))\oplus g_2^0$. Then, we obtain an exact sequence of T-modules:

$$0\to {\left(\begin{array}{c}{K}_1^1\\ K_2^1\end{array}\right)}_{{\phi }^{K^1}}\to \left(\begin{array}{c}{G}_1^0\\ (U{\otimes }_A{G}_1^0)\oplus G_2^0\end{array}\right)\stackrel{\left(\begin{array}{c}{f}_1^0\\ f_2^0\end{array}\right)}{\to }M\to 0.$$

Similarly, there exists an exact sequence of B-modules $G_2^1\stackrel{g_2^1}{\to }{K}_2^1\to 0$ where $G_2^1$ is $G_{C_2}$-projective, and then, we obtain an exact sequence of T-modules:

$$0\to {\left(\begin{array}{c}{K}_1^2\\ K_2^2\end{array}\right)}_{{\phi }^{K^2}}\to \left(\begin{array}{c}{G}_1^1\\ (U{\otimes }_A{G}_1^1)\oplus G_2^1\end{array}\right)\to {\left(\begin{array}{c}{K}_1^1\\ K_2^1\end{array}\right)}_{{\phi }^{K^1}}\to 0.$$

Repeating this process, we obtain the exact sequence of T-modules:

$$0\to \left(\begin{array}{c}0\\ K_2^{m-k}\end{array}\right)\to \left(\begin{array}{c}{G}_1^{m-k-1}\\ (U{\otimes }_A{G}_1^{m-k-1})\oplus G_2^{m-k-1}\end{array}\right)\stackrel{\left(\begin{array}{c}{f}_1^{m-k-1}\\ f_2^{m-k-1}\end{array}\right)}{⟶}$$

$$\cdots \to \left(\begin{array}{c}{G}_1^1\\ (U{\otimes }_A{G}_1^1)\oplus G_2^1\end{array}\right)\stackrel{\left(\begin{array}{c}{f}_1^1\\ f_2^1\end{array}\right)}{⟶}\left(\begin{array}{c}{G}_1^0\\ (U{\otimes }_A{G}_1^0)\oplus G_2^0\end{array}\right)\stackrel{\left(\begin{array}{c}{f}_1^0\\ f_2^0\end{array}\right)}{⟶}M\to 0$$

Note that $G_{C_2}-pd((U{\otimes }_A{G}_1^i)\oplus G_2^i)=G_{C_2}-pd(U{\otimes }_A{G}_1^i)\le k$, for every $i\in \{0,\cdots ,m-k-1\}$. Therefore, by [13], Proposition 3.11(2), and the exact sequence $0\to K_2^{m-k}\to (U{\otimes }_A{G}_1^{m-k-1})\oplus G_2^{m-k-1}\stackrel{f_2^{m-k-1}}{⟶}\cdots \to (U{\otimes }_A{G}_1^0)\oplus G_2^0\stackrel{f_2^0}{\to }{M}_2\to 0$, we obtain that $G_{C_2}-pd(K_2^{m-k})\le k$. This means that there exists an exact sequence of B-modules:

$$0\to G_2^m\to \cdots \to G_2^{m-k+1}\to G_2^{m-k}\to K_2^{m-k}\to 0.$$

Thus, there exists an exact sequence of T-modules:

$$0\to \left(\begin{array}{c}0\\ G_2^m\end{array}\right)\to \cdots \to \left(\begin{array}{c}0\\ G_2^{m-k+1}\end{array}\right)\to$$

$$\left(\begin{array}{c}0\\ G_2^{m-k}\end{array}\right)\to \left(\begin{array}{c}{G}_1^{m-k-1}\\ (U{\otimes }_A{G}_1^{m-k-1})\oplus G_2^{m-k-1}\end{array}\right)\stackrel{\left(\begin{array}{c}{f}_1^{m-k-1}\\ f_2^{m-k-1}\end{array}\right)}{⟶}$$

$$\cdots \to \left(\begin{array}{c}{G}_1^1\\ (U{\otimes }_A{G}_1^1)\oplus G_2^1\end{array}\right)\stackrel{\left(\begin{array}{c}{f}_1^1\\ f_2^1\end{array}\right)}{⟶}\left(\begin{array}{c}{G}_1^0\\ (U{\otimes }_A{G}_1^0)\oplus G_2^0\end{array}\right)\stackrel{\left(\begin{array}{c}{f}_1^0\\ f_2^0\end{array}\right)}{⟶}M\to 0.$$

By Theorem 1, all $\left(\begin{array}{c}{G}_1^i\\ (U{\otimes }_A{G}_1^i)\oplus G_2^i\end{array}\right)$ and all $\left(\begin{array}{c}0\\ G_2^j\end{array}\right)$ are $G_C$-projectives. Thus, $G_C-pd(M)\le m$. □

The following consequence of Theorem 3 extends [2], Proposition 2.8(1), and [3], Theorem 2.7(1), to the relative setting.

Corollary 6.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting, U C-compatible and $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ a T-module. If $S{G}_{C_2}-PD(B)<\infty ,$ then $G_C-pd(M)<\infty$ if and only if $G_{C_1}-pd(M_1)<\infty$ and $G_{C_2}-pd(M_2)<\infty .$

The following theorem gives an estimate of the left $G_C$-projective global dimension of T.

Theorem 4.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting and U C-compatible. Then:

$$max\{G_{C_1}-PD(A),G_{C_2}-PD(B)\}$$

$$\le G_C-PD(T)\le$$

$$max\{G_{C_1}-PD(A)+S{G}_{C_2}-PD(B)+1,G_{C_2}-PD(B)\}.$$

Proof. 

We prove first that $max\{G_{C_1}-PD(A),G_{C_2}-PD(B)\}\le G_C-PD(T)$. We may assume that $n:=G_C-PD(T)<\infty .$ Let $M_1$ be an A-module and $M_2$ be a B-module. Since $G_C-pd(\left(\begin{array}{c}{M}_1\\ U{\otimes }_A{M}_2\end{array}\right)\le n$ and $G_C-pd(\left(\begin{array}{c}0\\ M_2\end{array}\right)\le n$, $G_{C_1}-pd(M_1)\le n$ and $G_{C_2}-pd(M_2)\le n$ by Lemma 7. Thus, $G_{C_1}-PD(A)\le n$ and $G_{C_2}-PD(B)\le n$.

Next, we prove that:

$$G_C-PD(T)\le max\{G_{C_1}-PD(A)+1+S{G}_{C_2}-PD(B),G_{C_2}-PD(B)\}.$$

We may assume that:

$$m:=max\{G_{C_1}-PD(A)+1+S{G}_{C_2}-PD(B),G_{C_2}-PD(B)\}<\infty .$$

Then, $n_1:=G_{C_1}-PD(A)<\infty$ and $k:=S{G}_{C_2}-PD(B)\le n_2:=G_{C_2}-PD(B)<\infty$

Let $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ be a T-module. By Theorem 3,

$$G_C-pd(M)\le max\{n_1+k+1,n_2\}\le m.$$

□

Corollary 7.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting and U C-compatible. Then,

$G_C-PD(T)<\infty$ if and only if $G_{C_1}-PD(A)<\infty$ and $G_{C_2}-PD(B)<\infty$

Recall that a ring R is called left Gorenstein regular if the category R-Mod is Gorenstein ([2], Definition 2.1, and [28], Definition 2.18).

We know by [29], Theorem 1.1, that the following equality holds:

$$sup\left\{{Gpd}_R(M)\right|M\in R\text{-}\mathrm{Mod}\}=sup\left\{{\mathrm{Gid}}_R(M)\right|M\in R\text{-}\mathrm{Mod}\}.$$

and this common value is call the left global Gorenstein dimension of R, denoted by $l.\mathrm{Ggldim}(R)$. As a consequence of [28], Theorem 2.28, a ring R is left Gorenstein regular if and only if the global Gorenstein dimension of R is finite.

We shall say that a ring R is left n-Gorenstein regular if $n=l.\mathrm{Ggldim}(R)<\infty .$

Enochs, Izurdiaga, and Torrecillas characterized in [2], Theorem 3.1, when T is left Gorenstein regular under the conditions that ${}_{B}U$ has finite projective dimension and $U_A$ has finite flat dimension. As a direct consequence of Corollary 7, we refine this result.

Corollary 8.

Assume that U is compatible. Then, T is left Gorenstein regular if and only if so are A and B.

There are some cases when the estimate in Theorem 4 becomes an exact formula, which computes left the $G_C$-projective global dimension of T.

Recall that an injective cogenerator E in R-Mod is said to be strong if any R-module embeds in a direct sum of copies of E.

Corollary 9.

Let $C=\mathit{p}(C_1,C_2)$ be w-tilting and U C-compatible.

1. 

If $U=0$ then:

$$G_C-PD(T)=max\{G_{C_1}-PD(A),G_{C_2}-PD(B)\};$$

2. 

If A is left Noetherian and ${}_A{C}_1$ is a strong injective cogenerator, then:

$$G_C-PD(T)=\left\{\begin{array}{cc}{G}_{C_2}-PD(B)\hfill & ifU=0\hfill \\ max\{S{G}_{C_2}-PD(B)+1,G_{C_2}-PD(B)\}\hfill & ifU\ne 0.\hfill \end{array}\right.$$

Proof. 

1. Using a similar way as we do in the proof of Theorems 3 and 4, we can prove this statement. We only need to notice that if $U=0$, then a T-module $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ is $G_C$-projective if and only if $M_1$ is $G_{C_1}$-projective and $M_2$ is $G_{C_2}$-projective (since ${\phi }^M$ is always injective and $M_2={\overline{M}}_2$) by Theorem 1;

2. Note first that $G_{C_1}-PD(A)=0$ by [14], Corollary 2.3. Then, the case $U=0$ follows by 1. Assume that $U\ne 0$. Note that by Theorem 1, $\left(\begin{array}{c}A\\ 0\end{array}\right)$ is not $G_C$-projective since $U\ne 0$. Hence, $G_{C_2}-PD(B)\ge {G_C-pd}_T(\left(\begin{array}{c}A\\ 0\end{array}\right))\ge 1$.

By Theorem 4, we have the inequality:

$$G_{C_2}-PD(B)\le G_C-PD(T)\le max\{S{G}_{C_2}-PD(B)+1,G_{C_2}-PD(B)\}.$$

Therefore, the case $S{G}_{C_2}-PD(B)+1\le G_{C_2}-PD(B)$ is clear, and we only need to prove the result when $S{G}_{C_2}-PD(B)+1>n:=G_{C_2}-PD(B)$. Since $G_{C_2}-pd(U{\otimes }_{A}G)\le G_{C_2}-PD(B)=n$ for every $G\in G_{C_1}P(A)$, $S{G}_{C_2}-PD(B)=n$. Let $G_1$ be a $G_{C_1}$-projective A-module with $G_{C_2}-pd(U{\otimes }_A{G}_1)=n$, and consider the following short exact sequence:

$$0\to \left(\begin{array}{c}0\\ U{\otimes }_A{G}_1\end{array}\right)\to \left(\begin{array}{c}{G}_1\\ U{\otimes }_A{G}_1\end{array}\right)\to \left(\begin{array}{c}{G}_1\\ 0\end{array}\right)\to 0.$$

By Theorem 1, $\left(\begin{array}{c}{G}_1\\ U{\otimes }_A{G}_1\end{array}\right)$ is $G_C$-projective and by Lemma 7:

$$G_C-pd(\left(\begin{array}{c}0\\ U{\otimes }_A{G}_1\end{array}\right))=G_{C_2}-pd(U{\otimes }_{A}G)=n.$$

Thus, by [13], Proposition 3.11(4):

$$G_C-pd(\left(\begin{array}{c}{G}_1\\ 0\end{array}\right))=G_C-pd(\left(\begin{array}{c}0\\ U{\otimes }_A{G}_1\end{array}\right))+1=n+1=S{G}_{C_2}-PD(B)+1.$$

This shows that $G_C-PD(T)=S{G}_{C_2}-PD(B)+1$, and the proof is finished. □

Corollary 10.

Let R be a ring, $T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right)$ and $C=\mathit{p}(C_1,C_1)$ where $C_1$ is w-tilting. Then:

$$G_C-PD(T(R))=G_{C_1}-PD(R)+1.$$

Proof. 

Note first that C is a w-tilting $T(R)$-module, R is C-compatible, and $S{G}_{C_1}-PD(R)=0$. Therefore, by Theorem 4,

$$G_{C_1}-PD(R)\le G_C-PD(T(R))\le G_{C_1}-PD(R)+1.$$

The case $G_{C_1}-PD(R)=\infty$ is clear. Assume that $n:=G_{C_1}-PD(R)<\infty .$

There exists an R-module M with $G_{C_1}-pd(M)=n$ and ${\mathrm{Ext}}_R^n(M,X)\ne 0$ for some $X\in {\mathrm{Add}}_R(C_1)$ by [13], Theorem 3.8. If we apply the functor ${\mathrm{Hom}}_{T(R)}(-,\left(\begin{array}{c}0\\ X\end{array}\right))$ to the exact sequence of $T(R)$-modules:

$$0\to \left(\begin{array}{c}0\\ M\end{array}\right)\to {\left(\begin{array}{c}M\\ M\end{array}\right)}_{1_M}\to \left(\begin{array}{c}M\\ 0\end{array}\right)\to 0$$

we obtain an exact sequence

$$\cdots \to {\mathrm{Ext}}_{T(R)}^n(\left(\begin{array}{c}M\\ M\end{array}\right),\left(\begin{array}{c}0\\ X\end{array}\right))\to {\mathrm{Ext}}_{T(R)}^n(\left(\begin{array}{c}0\\ M\end{array}\right),\left(\begin{array}{c}0\\ X\end{array}\right))\to$$

$${\mathrm{Ext}}_{T(R)}^{n+1}(\left(\begin{array}{c}M\\ 0\end{array}\right),\left(\begin{array}{c}0\\ X\end{array}\right))\to {\mathrm{Ext}}_{T(R)}^{n+1}(\left(\begin{array}{c}M\\ M\end{array}\right),\left(\begin{array}{c}0\\ X\end{array}\right))\to \cdots$$

By Lemma 3, ${\mathrm{Ext}}_{T(R)}^{i\ge 1}(\left(\begin{array}{c}M\\ M\end{array}\right),\left(\begin{array}{c}0\\ X\end{array}\right))\cong {\mathrm{Ext}}_R^{i\ge 1}(M,0)=0$. Again, by Lemma 3 and the above exact sequence,

$${\mathrm{Ext}}_{T(R)}^{n+1}(\left(\begin{array}{c}M\\ 0\end{array}\right),\left(\begin{array}{c}0\\ X\end{array}\right))\cong {\mathrm{Ext}}_{T(R)}^n(\left(\begin{array}{c}0\\ M\end{array}\right),\left(\begin{array}{c}0\\ X\end{array}\right))\cong {\mathrm{Ext}}_R^n(M,X)\ne 0.$$

since $\left(\begin{array}{c}0\\ X\end{array}\right)\in {\mathrm{Add}}_{T(R)}(C)$ by Lemma 4(1), it follows that $n<G_C-pd(\left(\begin{array}{c}M\\ 0\end{array}\right))$ by [13], Theorem 3.8. However, $G_C-pd(\left(\begin{array}{c}M\\ 0\end{array}\right))\le G_C-PD(T(R))\le n+1$. Thus, $G_C-pd(\left(\begin{array}{c}M\\ 0\end{array}\right))=n+1$, which means that $G_C-PD(T(R))=n+1.$ □

Corollary 11.

Let R be a ring, $T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right)$, and $n\ge 0$ an integer. Then, $T(R)$ is left $(n+1)$-Gorenstein regular if and only if R is left n-Gorenstein regular .

The authors in [16] established a relationship between the projective dimension of modules over T and modules over A and B. Given an integer $n\ge 0$ and $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ a T-module, they proved that ${pd}_T(M)\le n$ if and only if ${pd}_A(M_1)\le n$, ${pd}_B({\overline{M}}_2)\le n$, and the map related to the n-th syzygy of M is injective. The following example shows that this is not true in general.

Example 4.

Let R be a left hereditary ring that is not semisimple, and let $T(R)=\left(\begin{array}{cc}R& 0\\ R& R\end{array}\right)$. Then, $lD(T(R))=lD(R)+1=2$ by [24], Corollary 3.4(3). This means that there exists a $T(R)$-module $M={\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)}_{{\phi }^M}$ with ${pd}_{T(R)}(M)=2$. If $K^1={\left(\begin{array}{c}{K}_1^1\\ K_2^1\end{array}\right)}_{{\phi }^{K^1}}$ is the first syzygy of M, then there exists an exact sequence of $T(R)$-modules:

$$0\to K^1\to P\to M\to 0$$

where $P={\left(\begin{array}{c}{P}_1\\ P_2\end{array}\right)}_{{\phi }^P}$ is projective. Then, we obtain the following commutative diagram:

By the snake lemma, ${\phi }^{K^1}$ is injective. On the other hand, since $lD(R)=1$, ${pd}_R(M_1)\le 1$ and ${pd}_R({\overline{M}}_2)\le 1$. However, ${pd}_{T(R)}(M)=2>1$.