A class of special formal triangular matrix rings

Abstract

Let $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with C a semidualizing (S, R)-bimodule. It is proven that (1) A left S-module M in Bass class is C-torsionless (resp. C-reflexive) if and only if $\biggl (\begin{array}{c} \textrm{Hom}_{S}(C,M)\\ M \end{array}\biggr )$ is a torsionless (resp. reflexive) left T-module; (2) A left S-module M in Bass class is C-Gorenstein projective if and only if $\biggl (\begin{array}{c} \textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a Gorenstein projective left T-module; (3) If C is a faithfully semidualizing (S, R)-bimodule, then a left S-module M is C-n-tilting if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )$ is an n-tilting left T-module.

1 Introduction

Let R and S be rings and U an (S, R)-bimodule. $T=\biggl (\begin{matrix} R&{}0\\ U&{}S \end{matrix}\biggr )$ is known as a formal (or generalized) triangular matrix ring with usual matrix addition and multiplication. Formal triangular matrix rings play an important role in ring theory and the representation theory of algebra [3]. This kind of rings are noncommutative and are often used to construct examples and counterexamples. As a consequence of the classical results by Green [13], the module category over the formal triangular matrix ring T can be constructed from the categories of modules over R and S. So one can describe classes of modules over T from the corresponding classes of modules over R and S. Thus the properties of formal triangular matrix rings and modules over them make the theory of rings and modules more abundant and concrete, and have deserved more and more interest (see [1, 3, 7, 9, 13,14,15, 19, 30, 31]).

On the other hand, semidualizing modules were studied independently by Foxby [10], Golod [12] and Vasconcelos [28] over a commutative Noetherian ring. Later, Holm and White [18] extended the definition of semidualizing modules to general associative rings. Recall that an (S, R)-bimodule $_{S}C_{R}$ is semidualizing [18] if

• (1) $_{S}C$ admits a projective resolution $\cdots \rightarrow P_{1}\rightarrow P_{0}\rightarrow C\rightarrow 0$ such that each $P_{i}$ is a finitely generated projective left S-module;

• (2) $C_{R}$ admits a projective resolution $\cdots \rightarrow Q_{1}\rightarrow Q_{0}\rightarrow C\rightarrow 0$ such that each $Q_{i}$ is a finitely generated projective right R-module;

• (3) The homothety map $_{S}S_{S}\rightarrow \textrm{Hom}_{R}(C, C)$ is an isomorphism;

• (4) The homothety map $_{R}R_{R}\rightarrow \textrm{Hom}_{S}(C, C)$ is an isomorphism;

• (5) $\textrm{Ext}^{i}_{R}(C,C) = 0$ for all $i\ge 1$;

• (6) $\textrm{Ext}^{i}_{S}(C,C) = 0$ for all $i\ge 1$.

Examples of semidualizing modules can be found in [6, 18]. One basic subject on semidualizing modules is to extend the classical results in homological algebra to the relative setting with respect to a semidualizing module C. For example, C-torsionless (C-reflexive) modules, C-Gorenstein projective modules and C-tilting modules have been introduced and studied in [4, 17, 20, 25, 26, 29].

It is natural to ask what special properties the formal triangular matrix ring $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ admits when C is a semidualizing (S, R)-bimodule. In the present article, we will exhibit the connections between C-torsionless (resp. C-Gorenstein projective, C-tilting) left S-modules and torsionless (resp. Gorenstein projective, tilting) left T-modules. For example, we prove that (1) A left S-module M in the Bass class $\mathcal {B}_{C}(S)$ is C-torsionless (resp. C-reflexive) if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M) \\ M\end{array}\biggr )$ is a torsionless (resp. reflexive) left T-module (see Theorem 2.1); (2) A left S-module M in the Bass class $\mathcal {B}_{C}(S)$ is C-Gorenstein projective if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M) \\ M\end{array}\biggr )$ is a Gorenstein projective left T-module (see Theorem 2.3); (3) If C is a faithfully semidualizing (S, R)-bimodule, then a left S-module M is C-n-tilting if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M) \\ S\oplus M\end{array}\biggr )$ is an n-tilting left T-module (see Theorem 2.8).

Throughout this paper, all rings are nonzero associative rings with identity and all modules are unitary. For a ring R, we write R-Mod (resp. Mod-R) for the category of left (resp. right) R-modules. $_RM$ (resp. $M_{R}$) denotes a left (resp. right) R-module. The character module $\textrm{Hom}_{\mathbb {Z}}(M,\mathbb {Q}/\mathbb {Z})$ of a module M is denoted by $M^{+}$. pd(M) and id(M) denote the projective and injective dimensions of a module M respectively. Add(M) (resp. Prod(M)) denotes the class of modules isomorphic to direct summands of direct sums (resp. direct products) of copies of M.

Let $T=\biggl (\begin{matrix} R&{}0\\ U&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with R and S rings and U an (S, R)-bimodule. By [13, Theorem 1.5], the category T-Mod of left T-modules is equivalent to the category $\Omega$ whose objects are triples $M=\biggl (\begin{array}{c}M_{1} \\ M_{2}\end{array}\biggr )_{\varphi ^{M}}$, where $M_{1}\in R$-Mod, $M_{2}\in S$-Mod and $\varphi ^{M}: U\otimes _{R} M_{1} \rightarrow M_{2}$ is an S-morphism, and whose morphisms from $\biggl (\begin{array}{c} M_{1} \\ M_{2}\end{array}\biggr )_{\varphi ^{M}}$ to $\biggl (\begin{array}{c} N_{1}\\ N_{2}\end{array}\biggr )_{\varphi ^{N}}$ are pairs $\biggl (\begin{array}{c} f_{1}\\ f_{2}\end{array}\biggr )$ such that $f_{1}\in \textrm{Hom}_{R}(M_{1}, N_{1}), f_{2}\in \textrm{Hom}_{S}(M_{2}, N_{2})$ and $\varphi ^{N}(1\otimes f_{1})=f_{2}\varphi ^{M}$. The regular module $_{T}T$ corresponds to $\biggl (\begin{array}{c} R\\ U\oplus S\end{array}\biggr )_{\varphi ^{T}}$, where $\varphi ^{T}: U\otimes _{R} R\rightarrow U\oplus S$ is given by $\varphi ^{T}(u\otimes r)=(ur, 0)$ for $u\in U,r\in R$. Note that a sequence $0\rightarrow \biggl (\begin{array}{c} M'_{1}\\ M'_{2}\end{array}\biggr )_{\varphi ^{M'}}\rightarrow \biggl (\begin{array}{c} M_{1}\\ M_{2}\end{array}\biggr )_{\varphi ^{M}}\rightarrow \biggl (\begin{array}{c} M''_{1}\\ M''_{2}\end{array}\biggr )_{\varphi ^{M''}}\rightarrow 0$ of left T-modules is exact if and only if both sequences $0\rightarrow M'_{1}\rightarrow M_{1}\rightarrow M''_{1}\rightarrow 0$ and $0\rightarrow M'_{2}\rightarrow M_{2}\rightarrow M''_{2}\rightarrow 0$ are exact. Analogously, the category Mod-T of right T-modules is equivalent to the category $\Gamma$ whose objects are triples $W=(W_{1},W_{2})_{\varphi _{W}}$, where $W_{1}\in$ Mod-R, $W_{2}\in$ Mod-S and $\varphi _{W}: W_{2}\otimes _{S} U \rightarrow W_{1}$ is an R-morphism, and whose morphisms from $(W_{1},W_{2})_{\varphi _{W}}$ to $(X_{1},X_{2})_{\varphi _{X}}$ are pairs $(g_{1},g_{2})$ such that $g_{1}\in \textrm{Hom}_{R}(W_{1}, X_{1}), g_{2}\in \textrm{Hom}_{S}(W_{2}, X_{2})$ and $\varphi _{X}(g_{2}\otimes 1)=g_{1}\varphi _{W}$. In the rest of the paper, we shall identify T-Mod (resp. Mod-T) with this category $\Omega$ (resp. $\Gamma$) and, whenever there is no possible confusion, we shall omit $\varphi ^{M}$ (resp. $\varphi _{W}$). For example, for the left T-module $\biggl (\begin{array}{c} M_{1}\\ (U\otimes _{R} M_{1})\oplus M_{2}\end{array}\biggr )$, the S-morphism $U\otimes _{R} M_{1}\rightarrow (U\otimes _{R} M_{1})\oplus M_{2}$ is just the injection and for the left T-module $\biggl (\begin{array}{c} M_{1}\oplus \textrm{Hom}_{S}(U,M_{2})\\ M_{2}\end{array}\biggr )$, the R-morphism $M_{1}\oplus \textrm{Hom}_{S}(U,M_{2})\rightarrow \textrm{Hom}_{S}(U,M_{2})$ is just the projection.

2 Modules over a special formal triangular matrix ring

A semidualizing (S, R)-bimodule C defines two important classes of modules. Following [18], the Auslander class with respect to C, denoted by $\mathcal {A}_{C}(R)$, consists of all left R-modules N satisfying

• (1) $\textrm{Tor}_{i}^{R}(C,N) = 0 = \textrm{Ext}^{i}_{S}(C,C\otimes _{R} N)$ for all $i\ge 1$;

• (2) The natural evaluation map $\mu _{N}: N \rightarrow \textrm{Hom}_{S}(C,C\otimes _{R} N)$ is an isomorphism.

Dually, the Bass class with respect to C, denoted by $\mathcal {B}_{C}(S)$, consists of all left S-modules M satisfying

• (1) $\textrm{Ext}^{i}_{S}(C,M)= 0 = \textrm{Tor}_{i}^{R}(C,\textrm{Hom}_{S}(C,M))$ for all $i\ge 1$;

• (2) The natural evaluation map $\nu _{M}: C\otimes _{R} \textrm{Hom}_{S}(C,M)\rightarrow M$ is an isomorphism.

Let C be a semidualizing (S, R)-bimodule. Recall that a left S-module M is C-torsionless (resp. C-reflexive) if the biduality map $\delta : M \rightarrow \textrm{Hom}_{R}(\textrm{Hom}_{S}(M, C), C)$, defined as $\delta (x)(f)=f(x)$ for each $x\in M$ and $f\in \textrm{Hom}_{S}(M,C)$, is a monomorphism (resp. isomorphism). If $C=R=S$, then a C-torsionless (resp. C-reflexive) module is exactly a torsionless (resp. reflexive) module.

We first exhibit the relation between C-torsionless (resp. C-reflexive) left S-modules and torsionless (resp. reflexive) left T-modules.

Theorem 2.1

Let $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with C a semidualizing (S, R)-bimodule and $M\in \mathcal {B}_{C}(S)$. Then

• (1) M is a C-torsionless left S-module if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a torsionless left T-module.

• (2) M is a C-reflexive left S-module if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a reflexive left T-module.

Proof

By [18, Theorem 6.4], we have

$$\begin{aligned} \textrm{Hom}_{R}(\textrm{Hom}_{S}(C,M),\textrm{Hom}_{S}(C,C))\cong \textrm{Hom}_{S}(M,C). \end{aligned}$$

So [22, Corollary 2.3] implies that

$$\begin{aligned}{} & {} \textrm{Hom}_{T}\left( \textrm{Hom}_{T}\left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr ),T\right) ,T\right) \\{} & {} \quad \cong \textrm{Hom}_{T}((\textrm{Hom}_{R}(\textrm{Hom}_{S}(C,M),R),0),T)\\{} & {} \quad \cong \textrm{Hom}_{T}((\textrm{Hom}_{R}(\textrm{Hom}_{S}(C,M),\textrm{Hom}_{S}(C,C)),0),T)\\{} & {} \quad \cong \textrm{Hom}_{T}((\textrm{Hom}_{S}(M,C),0),T) \\{} & {} \quad \cong \biggl (\begin{array}{c}\textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),R)\\ \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C)\end{array}\biggr ). \end{aligned}$$

(1) $``\Rightarrow "$ There is an exact sequence of left S-modules

$$\begin{aligned} 0\rightarrow M\rightarrow \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C), \end{aligned}$$

which induces the exact sequence

$$\begin{aligned} 0\rightarrow \textrm{Hom}_{S}(C,M)\rightarrow \textrm{Hom}_{S}(C,\textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C)). \end{aligned}$$

Note that $\textrm{Hom}_{S}(C,\textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C)) \cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),\textrm{Hom}_{S}(C,C))$

$$\begin{aligned} \cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),R). \end{aligned}$$

Thus $\textrm{Hom}_{S}(C,M)\rightarrow \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),R)$ is a monomorphism. Consequently, $\delta : \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )\rightarrow \textrm{Hom}_{T}\left( \textrm{Hom}_{T} \left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr ),T\right) ,T\right)$ is a monomorphism. So $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a torsionless left T-module.

“$\Leftarrow$” Since $\delta : \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )\rightarrow \textrm{Hom}_{T}\left( \textrm{Hom}_{T} \left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr ),T\right) ,T\right)$ is a monomorphism, $M\rightarrow \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C)$ is a monomorphism. So M is a C-torsionless left S-module.

(2) “$\Rightarrow$” Since $M\cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C)$, we get the isomorphisms

$$\begin{aligned} \textrm{Hom}_{S}(C,M)\cong \textrm{Hom}_{S}(C,\textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C))\cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),R). \end{aligned}$$

Hence $\delta : \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )\rightarrow \textrm{Hom}_{T}\left( \textrm{Hom}_{T} \left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr ),T\right) ,T\right)$ is an isomorphism. Thus $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a reflexive left T-module.

“$\Leftarrow$” Since $\delta : \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )\rightarrow \textrm{Hom}_{T} \left( \textrm{Hom}_{T}\left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr ),T\right) ,T\right)$ is an isomorphism, $M\rightarrow \textrm{Hom}_{R}(\textrm{Hom}_{S}(M,C),C)$ is an isomorphism. So M is a C-reflexive left S-module. $\square$

Let C be a semidualizing (S, R)-bimodule. According to [18], a left S-module is called C-projective (resp. C-flat) if it has the form $C\otimes _{R} M$ for some projective (resp. flat) left R-module M. A left R-module is called C-injective if it has the form $\textrm{Hom}_{S}(C,Y)$ for some injective left S-module Y.

Proposition 2.2

Let $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with C a semidualizing (S, R)-bimodule, $M\in \mathcal {B}_{C}(S)$ and $N\in \mathcal {A}_{C}(R)$. Then

• (1) M is a C-projective left S-module if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a projective left T-module.

• (2) M is a C-flat left S-module if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a flat left T-module.

• (3) N is a C-injective left R-module if and only if $\biggl (\begin{array}{c}N\\ C\otimes _{R} N\end{array}\biggr )$ is an injective left T-module.

Proof

(1) follows from [15, Theorem 3.1]. (2) holds by [9, Proposition 1.14].

(3) follows from [14, Proposition 5.1]. $\square$

The origin of Gorenstein homological algebra may date back to 1960s when Auslander and Bridger introduced the concept of G-dimension for finitely generated modules over a two-sided Noetherian ring [2]. In 1990s, Enochs and Jenda extended the ideas of Auslander and Bridger and introduced the concepts of Gorenstein projective and injective modules over arbitrary rings [8]. The notions of Gorenstein projective and injective modules with respect to a semidualizing module were first introduced by Holm and Jørgensen in [17] and White in [29] for commutative rings. The noncommutative versions of them were given by Liu et al. [20].

Let C be a semidualizing (S, R)-bimodule. Recall that a left S-module M is C-Gorenstein projective [20] if there exists an exact sequence of left S-modules

$$\begin{aligned} \cdots \rightarrow P_{1}\rightarrow P_{0}\rightarrow C\otimes _{R} P^{0}\rightarrow C\otimes _{R} P^{1}\rightarrow \cdots \end{aligned}$$

such that all $P_{i}$ are projective left S-modules and $P^{i}$ are projective left R-modules, $M\cong \textrm{coker}(P_{1}\rightarrow P_{0})$ and $\textrm{Hom}_{S}(-, C\otimes _{R} X$) leaves the sequence exact whenever X is a projective left R-module.

A left R-module N is called C-Gorenstein injective [20] if there exists an exact sequence of left R-modules

$$\begin{aligned} \cdots \rightarrow \textrm{Hom}_{S}(C,E_{1})\rightarrow \textrm{Hom}_{S}(C,E_{0})\rightarrow E^{0}\rightarrow E^{1}\rightarrow \cdots \end{aligned}$$

such that all $E_{i}$ are injective left S-modules and $E^{i}$ are injective left R-modules, $N\cong \ker (E^{0}\rightarrow E^{1})$ and $\textrm{Hom}_{R}(\textrm{Hom}_{S}(C,Y),-)$ leaves the sequence exact whenever Y is an injective left S-module.

If $C=R=S$ in the above definitions, then we recover the categories of ordinary Gorenstein projective and injective modules in the sense of [8].

Next we give the Gorenstein version of Proposition 2.2.

Theorem 2.3

Let $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with C a semidualizing (S, R)-bimodule. If $M\in \mathcal {B}_{C}(S)$ and $N\in \mathcal {A}_{C}(R)$, then

• (1) M is a C-Gorenstein projective left S-module if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a Gorenstein projective left T-module.

• (2) N is a C-Gorenstein injective left R-module if and only if $\biggl (\begin{array}{c}N\\ C\otimes _{R} N\end{array}\biggr )$ is a Gorenstein injective left T-module.

Proof

(1) “$\Rightarrow$” There exists an exact sequence of left S-modules

$$\begin{aligned} \Theta : 0\rightarrow M\rightarrow C\otimes _{R} P^{0}\rightarrow C\otimes _{R} P^{1}\rightarrow \cdots \end{aligned}$$

such that all $P^{i}$ are projective left R-modules and the sequence $\textrm{Hom}_{S}(\Theta , C\otimes _{R} X$) is exact whenever X is a projective left R-module.

Since $M\in \mathcal {B}_{C}(S)$, $\textrm{Ext}^{i}_{S}(C,M)= 0$ for all $i\ge 1$. Since $P^{i}\in \mathcal {A}_{C}(R)$ by [18, Lemma 4.1], $\textrm{Hom}_{S}(C,C\otimes _{R} P^{i})\cong P^{i}$ and $\textrm{Ext}^{i}_{S}(C,C\otimes _{R} P^{i})=0$ for all $i\ge 1$. So we get the exact sequence of left T-modules

$$\begin{aligned} \Upsilon : 0\rightarrow \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )\rightarrow \biggl (\begin{array}{c} P^{0}\\ C\otimes _{R} P^{0}\end{array}\biggr )\rightarrow \biggl (\begin{array}{c} P^{1}\\ C\otimes _{R} P^{1}\end{array}\biggr )\rightarrow \cdots . \end{aligned}$$

On the one hand, for any projective left T-module $\biggl (\begin{array}{c} A_{1}\\ (C\otimes _{R} A_{1})\oplus A_{2}\end{array}\biggr )$, where $A_{1}$ is a projective left R-module and $A_{2}$ is a projective left S-module, by [18, Theorem 6.4], we have

$$\begin{aligned}{} & {} \textrm{Hom}_{T}\left( \biggl (\begin{array}{c} P^{i}\\ C\otimes _{R} P^{i}\end{array}\biggr ),\biggl (\begin{array}{c} A_{1}\\ (C\otimes _{R} A_{1})\oplus A_{2}\end{array}\biggr )\right) \cong \textrm{Hom}_{R}(P^{i}, A_{1})\\{} & {} \quad \cong \textrm{Hom}_{S}(C\otimes _{R} P^{i}, C\otimes _{R} A_{1}), \\{} & {} \textrm{Hom}_{T}( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr ),\biggl (\begin{array}{c} A_{1}\\ (C\otimes _{R} A_{1})\oplus A_{2}\end{array}\biggr ))\cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(C,M), A_{1})\\{} & {} \quad \cong \textrm{Hom}_{S}(M, C\otimes _{R} A_{1}). \end{aligned}$$

So the sequence $\textrm{Hom}_{T}(\Upsilon ,\biggl (\begin{array}{c} A_{1}\\ (C\otimes _{R} A_{1})\oplus A_{2}\end{array}\biggr ))\cong \textrm{Hom}_{S}(\Theta , C\otimes _{R} A_{1})$ is exact.

On the other hand, by [21, Lemma 3.2], for any $i\ge 1$, we have

$$\begin{aligned}{} & {} \textrm{Ext}^{i}_{T}\left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr ), \biggl (\begin{array}{c} A_{1}\\ (C\otimes _{R} A_{1})\oplus A_{2}\end{array}\biggr )\right) \cong \textrm{Ext}^{i}_{R}(\textrm{Hom}_{S}(C,M), A_{1})\\{} & {} \quad \cong \textrm{Ext}^{i}_{S}(M, C\otimes _{R} A_{1})=0. \end{aligned}$$

It follows that $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a Gorenstein projective left T-module by [16, Proposition 2.3].

“$\Leftarrow$” By [15, Theorem 3.1], there exists an exact sequence of left T-modules

$$\begin{aligned}{} & {} \Delta : \cdots \rightarrow \biggl (\begin{array}{c} P_{0}\\ (C\otimes _{R} P_{0})\oplus Q_{0}\end{array}\biggr )\rightarrow \biggl (\begin{array}{c} P^{0}\\ (C\otimes _{R} P^{0})\oplus Q^{0}\end{array}\biggr ){\mathop {\rightarrow }\limits ^{\biggl (\begin{array}{c} f^{0}\\ g^{0}\end{array}\biggr )}}\\{} & {} \quad \biggl (\begin{array}{c} P^{1}\\ (C\otimes _{R} P^{1})\oplus Q^{1}\end{array}\biggr )\rightarrow \cdots \end{aligned}$$

such that $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )\cong \ker \biggl (\begin{array}{c} f^{0}\\ g^{0}\end{array}\biggr )$, all $P_{i}$ and $P^{i}$ are projective left R-modules, $Q_{i}$ and $Q^{i}$ are projective left S-modules and the sequence $\textrm{Hom}_{T}(\Delta , W$) is exact whenever W is a projective left T-module.

Therefore we get the exact sequence of left S-modules

$$\begin{aligned} \Lambda : \cdots \rightarrow (C\otimes _{R} P_{0})\oplus Q_{0}\rightarrow (C\otimes _{R} P^{0})\oplus Q^{0}{\mathop {\rightarrow }\limits ^{g^{0}}}(C\otimes _{R} P^{1})\oplus Q^{1}\rightarrow \cdots \end{aligned}$$

such that $M\cong \ker (g^{0})$. For any projective left R-module X, $\biggl (\begin{array}{c} X\\ C\otimes _{R} X\end{array}\biggr )$ is a projective left T-module. By [18, Theorem 6.4], the sequence $\textrm{Hom}_{S}(\Lambda , C\otimes _{R} X)\cong \textrm{Hom}_{T}(\Delta ,\biggl (\begin{array}{c} X\\ C\otimes _{R} X\end{array}\biggr ))$ is exact. Thus M is a C-Gorenstein projective left S-module by [20, Theorem 2.9].

(2) $``\Rightarrow "$ There exists an exact sequence of left R-modules

$$\begin{aligned} \Psi : \cdots \rightarrow \textrm{Hom}_{S}(C,E_{2})\rightarrow \textrm{Hom}_{S}(C,E_{1})\rightarrow \textrm{Hom}_{S}(C,E_{0})\rightarrow N\rightarrow 0 \end{aligned}$$

such that all $E_{i}$ are injective left S-modules and the sequence $\textrm{Hom}_{R}(\textrm{Hom}_{S}(C,Y),\Psi )$ is exact whenever Y is an injective left S-module.

Since $N\in \mathcal {A}_{C}(R)$, $\textrm{Tor}_{i}^{R}(C,N) = 0$ for all $i\ge 1$. Since $E_{i}\in \mathcal {B}_{C}(S)$ by [18, Lemma 4.1], $C\otimes _{R}\textrm{Hom}_{S}(C,E_{i})\cong E_{i}$ and $\textrm{Tor}_{i}^{R}(C,\textrm{Hom}_{S}(C,E_{i}))=0$ for all $i\ge 1$. So we get the exact sequence of left T-modules

$$\begin{aligned}{} & {} \Sigma : \cdots \rightarrow \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,E_{2})\\ E_{2}\end{array}\biggr )\rightarrow \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,E_{1})\\ E_{1}\end{array}\biggr )\rightarrow \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,E_{0})\\ E_{0}\end{array}\biggr )\\{} & {} \quad \rightarrow \biggl (\begin{array}{c}N\\ C\otimes _{R} N\end{array}\biggr )\rightarrow 0. \end{aligned}$$

On the one hand, for any injective left T-module $\biggl (\begin{array}{c} B_{1}\oplus \textrm{Hom}_{S}(C,B_{2})\\ B_{2}\end{array}\biggr )$, where $B_{1}$ is an injective left R-module and $B_{2}$ is an injective left S-module, by [18, Theorem 6.4], we have

$$\begin{aligned}{} & {} \textrm{Hom}_{T}\Bigg (\biggl (\begin{array}{c} B_{1}\oplus \textrm{Hom}_{S}(C,B_{2})\\ B_{2}\end{array}\biggr ),\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,E_{i})\\ E_{i}\end{array}\biggr )\cong \textrm{Hom}_{S}(B_{2},E_{i})\\{} & {} \quad \cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(C,B_{2}), \textrm{Hom}_{S}(C,E_{i})),\\{} & {} \textrm{Hom}_{T}\Bigg (\biggl (\begin{array}{c} B_{1}\oplus \textrm{Hom}_{S}(C,B_{2})\\ B_{2}\end{array}\biggr ),\biggl (\begin{array}{c}N\\ C\otimes _{R} N\end{array}\biggr ))\cong \textrm{Hom}_{S}(B_{2},C\otimes _{R} N)\\{} & {} \quad \cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(C,B_{2}), N). \end{aligned}$$

Therefore the sequence $\textrm{Hom}_{T}(\biggl (\begin{array}{c} B_{1}\oplus \textrm{Hom}_{S}(C,B_{2})\\ B_{2}\end{array}\biggr ),\Sigma )\cong \textrm{Hom}_{R}(\textrm{Hom}_{S}(C,B_{2}), \Psi )$ is exact.

On the other hand, by [21, Lemma 3.2], for any $i\ge 1$, we have

$$\begin{aligned}{} & {} \textrm{Ext}^{i}_{T}\left( \biggl (\begin{array}{c} B_{1}\oplus \textrm{Hom}_{S}(C,B_{2})\\ B_{2}\end{array}\biggr ),\biggl (\begin{array}{c}N\\ C\otimes _{R} N\end{array}\biggr )\right) \cong \textrm{Ext}^{i}_{S}(B_{2}, C\otimes _{R} N)\\{} & {} \quad \cong \textrm{Ext}^{i}_{R}(\textrm{Hom}_{S}(C,B_{2}), N)=0. \end{aligned}$$

Therefore $\biggl (\begin{array}{c}N\\ C\otimes _{R} N\end{array}\biggr )$ is a Gorenstein injective left T-module.

“$\Leftarrow$” By [14, Proposition 5.1], there exists an exact sequence of left T-modules

$$\begin{aligned}{} & {} \Xi : \cdots \rightarrow \biggl (\begin{array}{c} G_{0}\oplus \textrm{Hom}_{S}(C,H_{0})\\ H_{0}\end{array}\biggr )\rightarrow \biggl (\begin{array}{c} G^{0}\oplus \textrm{Hom}_{S}(C,H^{0})\\ H^{0}\end{array}\biggr )\biggr ){\mathop {\rightarrow }\limits ^{\biggl (\begin{array}{c} f^{0}\\ g^{0}\end{array}\biggr )}}\\{} & {} \quad \biggl (\begin{array}{c} G^{1}\oplus \textrm{Hom}_{S}(C,H^{1})\\ H^{1}\end{array}\biggr )\rightarrow \cdots \end{aligned}$$

such that $\biggl (\begin{array}{c}N\\ C\otimes _{R} N\end{array}\biggr )\cong \ker \biggl (\begin{array}{c} f^{0}\\ g^{0}\end{array}\biggr )$, all $G_{i}$ and $G^{i}$ are injective left R-modules, $H_{i}$ and $H^{i}$ are injective left S-modules and the sequence $\textrm{Hom}_{T}(K,\Xi )$ is exact whenever K is an injective left T-module. So we get the exact sequence of left R-modules

$$\begin{aligned}{} & {} \Phi : \cdots \rightarrow G_{0}\oplus \textrm{Hom}_{S}(C,H_{0})\rightarrow G^{0}\oplus \textrm{Hom}_{S}(C,H^{0}){\mathop {\rightarrow }\limits ^{f^{0}}}\\{} & {} \quad G^{1}\oplus \textrm{Hom}_{S}(C,H^{1})\rightarrow \cdots \end{aligned}$$

such that $N\cong \ker (f^{0})$. For any injective left S-module Y, $\biggl (\begin{array}{c} \textrm{Hom}_{S}(C,Y)\\ Y\end{array}\biggr )$ is an injective left T-module. By [18, Theorem 6.4], the sequence

$$\begin{aligned} \textrm{Hom}_{R}(\textrm{Hom}_{S}(C,Y), \Phi )\cong \textrm{Hom}_{T}\left( \biggl (\begin{array}{c} \textrm{Hom}_{S}(C,Y)\\ Y\end{array}\biggr ),\Xi \right) \end{aligned}$$

is exact. Thus N is a C-Gorenstein injective left R-module by the dual of [20, Theorem 2.9]. $\square$

Let C be a semidualizing (S, R)-bimodule. A left S-module M is said to be C-Gorenstein flat if there exists an exact sequence of left S-modules

$$\begin{aligned} \cdots \rightarrow F_{1}\rightarrow F_{0}\rightarrow C\otimes _{R} F^{0}\rightarrow C\otimes _{R} F^{1}\rightarrow \cdots \end{aligned}$$

such that all $F_{i}$ are flat left S-modules and $F^{i}$ are flat left R-modules, $M\cong \textrm{coker}(F_{1}\rightarrow F_{0})$ and $\textrm{Hom}_{R}(C, Y)\otimes _{S}-$ leaves the sequence exact whenever Y is an injective right R-module.

Recall that a semidualizing (S, R)-bimodule C is faithfully semidualizing [18] if it satisfies the following conditions for all modules $_{S}M$ and $N_{R}$: (1) If $\textrm{Hom}_{S}(C,M) = 0$, then $M = 0$; (2) If $\textrm{Hom}_{R}(C,N)=0$, then $N = 0$.

Lemma 2.4

Let R be a right coherent ring and C a faithfully semidualizing (S, R)-bimodule. Then a left S-module M is C-Gorenstein flat if and only if $M^{+}$ is C-Gorenstein injective.

Proof

“$\Rightarrow$” There exists an exact sequence of left S-modules

$$\begin{aligned} \cdots \rightarrow F_{1}\rightarrow F_{0}\rightarrow C\otimes _{R} F^{0}\rightarrow C\otimes _{R} F^{1}\rightarrow \cdots \end{aligned}$$

such that all $F_{i}$ are flat left S-modules and $F^{i}$ are flat left R-modules, $M\cong \textrm{coker}(F_{1}\rightarrow F_{0})$ and $\textrm{Hom}_{R}(C, Y)\otimes _{S}-$ leaves the sequence exact whenever Y is an injective right R-module. Then we get the exact sequence

$$\begin{aligned} \cdots \rightarrow (C\otimes _{R} F^{1})^{+}\rightarrow ( C\otimes _{R} F^{0})^{+}\rightarrow F_{0}^{+}\rightarrow F_{1}^{+}\rightarrow \cdots \ \ \ \ (*) \end{aligned}$$

such that $M^{+}\cong \ker (F_{0}^{+}\rightarrow F_{1}^{+})$ and $\textrm{Hom}_{S}(\textrm{Hom}_{R}(C, Y),-)$ leaves the sequence exact. Since $(*)$ is equivalent to the exact sequence

$$\begin{aligned} \cdots \rightarrow \textrm{Hom}_{R}(C,(F^{1})^{+})\rightarrow \textrm{Hom}_{R}(C,(F^{0})^{+})\rightarrow F_{0}^{+}\rightarrow F_{1}^{+}\rightarrow \cdots \end{aligned}$$

and all $F_{i}^{+}$ and $(F^{i})^{+}$ are injective, $M^{+}$ is C-Gorenstein injective.

“$\Leftarrow$” For any injective right R-module Y and any $i\ge 1$,

$$\begin{aligned} \textrm{Tor}_{i}^{S}(\textrm{Hom}_{R}(C,Y),M)^{+}\cong \textrm{Ext}^{i}_{S}(\textrm{Hom}_{R}(C,Y),M^{+})=0. \end{aligned}$$

Thus $\textrm{Tor}_{i}^{S}(\textrm{Hom}_{R}(C,Y),M)=0$.

There exists an epimorphism $\textrm{Hom}_{R}(C,E_{0})\rightarrow M^{+}$ with $E_{0}$ an injective right R-module, which yields the monomorphism $M^{++}\rightarrow \textrm{Hom}_{R}(C,E_{0})^{+}\cong C\otimes _{R}E_{0}^{+}$ by [24, Lemma 3.60]. Since there is a monomorphism $M\rightarrow M^{++}$, one gets a monomorphism $M\rightarrow C\otimes _{R}E_{0}^{+}$. Since $E_{0}^{+}$ is flat by [5, Theorem 1], M has a monic C-flat preenvelope $M\rightarrow C\otimes _{R}F^{0}$ with $F^{0}$ a flat left R-module by [18, Proposition 5.3(d)]. Thus we get the exact sequence

$$\begin{aligned} 0\rightarrow M\rightarrow C\otimes _{R}F^{0}\rightarrow L^{1}\rightarrow 0. \end{aligned}$$

For any injective right R-module Q, $Q^{+}$ is flat by [5, Theorem 1]. So we get the exact sequence

$$\begin{aligned}{} & {} 0\rightarrow \textrm{Hom}_{S}(L^{1},C\otimes _{R}Q^{+})\rightarrow \textrm{Hom}_{S}( C\otimes _{R}F^{0},C\otimes _{R}Q^{+})\rightarrow \\{} & {} \quad \textrm{Hom}_{S}(M,C\otimes _{R}Q^{+})\rightarrow 0. \end{aligned}$$

Since $\textrm{Hom}_{R}(C,Q)^{+}\cong C\otimes _{R}Q^{+}$, one gets the exact sequence

$$\begin{aligned}{} & {} 0\rightarrow \textrm{Hom}_{R}(C,Q)\otimes _{S}M\rightarrow \textrm{Hom}_{R}(C,Q)\otimes _{S}(C\otimes _{R} F^{0})\rightarrow \\{} & {} \quad \textrm{Hom}_{R}(C,Q)\otimes _{S}L^{1}\rightarrow 0. \end{aligned}$$

[18, Theorem 6.4(c)] implies that

$$\begin{aligned} \textrm{Tor}_{1}^{S}(\textrm{Hom}_{R}(C,Q),C\otimes _{R}F^{0})\cong \textrm{Tor}_{1}^{R}(Q, F^{0})=0. \end{aligned}$$

Thus we have $\textrm{Tor}_{1}^{S}(\textrm{Hom}_{R}(C,Q),L^{1})=0$. Note that

$$\begin{aligned} \textrm{Ext}^{1}_{S}(\textrm{Hom}_{R}(C,Q),(L^{1})^{+})\cong \textrm{Tor}_{1}^{S}(\textrm{Hom}_{R}(C,Q),L^{1})^{+}=0. \end{aligned}$$

Since $(C\otimes _{R} F^{0})^{+}$ and $M^{+}$ are C-Gorenstein injective, the induced exact sequence

$$\begin{aligned} 0\rightarrow (L^{1})^{+}\rightarrow (C\otimes _{R} F^{0})^{+}\rightarrow M^{+}\rightarrow 0 \end{aligned}$$

implies that $(L^{1})^{+}$ is C-Gorenstein injective from the dual version of [29, Corollary 3.8]. Thus the above proof gives rise to the exact sequence

$$\begin{aligned} 0\rightarrow L^{1}\rightarrow C\otimes _{R} F^{1}\rightarrow L^{2}\rightarrow 0 \end{aligned}$$

such that $\textrm{Hom}_{R}(C,Q)\otimes _{S}-$ leaves it exact, $F^{1}$ is flat and $(L^{2})^{+}$ is C-Gorenstein injective. Continuing this process yields the exact sequence

$$\begin{aligned} 0\rightarrow M\rightarrow C\otimes _{R} F^{0}\rightarrow C\otimes _{R} F^{1}\rightarrow C\otimes _{R} F^{2} \rightarrow \cdots \end{aligned}$$

such that $\textrm{Hom}_{R}(C,Q)\otimes _{S}-$ leaves it exact and each $F^{i}$ is flat.

It follows that M is C-Gorenstein flat. $\square$

Theorem 2.5

Let $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with R and S right coherent rings and C a faithfully semidualizing (S, R)-bimodule. If $M\in \mathcal {B}_{C}(S)$, then M is a C-Gorenstein flat left S-module if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a Gorenstein flat left T-module.

Proof

Since R and S are right coherent rings and $C_{R}$ is finitely presented, T is a right coherent ring by [9, Corollary 2.3]. So M is a C-Gorenstein flat left S-module if and only if $M^{+}$ is a C-Gorenstein injective right S-module by Lemma 2.4 if and only if $(M^{+}\otimes _{S}C,M^{+})$ is a Gorenstein injective right T-module by Theorem 2.3 if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ M\end{array}\biggr )$ is a Gorenstein flat left T-module by [16, Theorem 3.6]. $\square$

The following lemma is a noncommutative version of [27, Theorem 2.8].

Lemma 2.6

Let C be a faithfully semidualizing (S, R)-bimodule. Then

• (1) $M\in \mathcal {B}_{C}(S)$ if and only if $\textrm{Hom}_{S}(C, M)\in \mathcal {A}_{C}(R)$.

• (2) $N\in \mathcal {A}_{C}(R)$ if and only if $C\otimes _{R}N\in \mathcal {B}_{C}(S)$.

Proof

(1) “$\Rightarrow$” It follows from [18, Proposition 4.1].

“$\Leftarrow$” Since $\textrm{Hom}_{S}(C, M)\in \mathcal {A}_{C}(R)$, $\mu _{\textrm{Hom}_{S}(C,M)}: \textrm{Hom}_{S}(C, M)\rightarrow \textrm{Hom}_{S}(C,C\otimes _{R} \textrm{Hom}_{S}(C, M))$ is an isomorphism. Note that

$$\begin{aligned} \textrm{Hom}_{S}(C,\nu _{M})\mu _{\textrm{Hom}_{S}(C,M)}=id_{\textrm{Hom}_{S}(C,M)}. \end{aligned}$$

So $\textrm{Hom}_{S}(C,\nu _{M})$ is an isomorphism. The exact sequence

$$\begin{aligned} 0\rightarrow \ker (\nu _{M})\rightarrow C\otimes _{R} \textrm{Hom}_{S}(C, M){\mathop {\rightarrow }\limits ^{\nu _{M}}}M \end{aligned}$$

induces the exact sequence

$$\begin{aligned}{} & {} 0\rightarrow \textrm{Hom}_{S}(C,\ker (\nu _{M}))\rightarrow \textrm{Hom}_{S}(C,C\otimes _{R} \textrm{Hom}_{S}(C, M)){\mathop {\rightarrow }\limits ^{\textrm{Hom}_{S}(C, \nu _{M})}}\\{} & {} \quad \textrm{Hom}_{S}(C, M). \end{aligned}$$

Thus $\textrm{Hom}_{S}(C,\ker (\nu _{M}))=0$. Since C a faithfully semidualizing (S, R)-bimodule, $\ker (\nu _{M})=0$. So we get the exact sequence

$$\begin{aligned} 0\rightarrow C\otimes _{R} \textrm{Hom}_{S}(C, M){\mathop {\rightarrow }\limits ^{\nu _{M}}}M\rightarrow \textrm{coker}(\nu _{M})\rightarrow 0, \end{aligned}$$

which induces the exact sequence

$$\begin{aligned}{} & {} 0\rightarrow \textrm{Hom}_{S}(C,C\otimes _{R} \textrm{Hom}_{S}(C, M)){\mathop {\rightarrow }\limits ^{\textrm{Hom}_{S}(C, \nu _{M})}}\textrm{Hom}_{S}(C, M)\rightarrow \\{} & {} \quad \textrm{Hom}_{S}(C,\textrm{coker}(\nu _{M}))\rightarrow 0. \end{aligned}$$

Thus $\textrm{Hom}_{S}(C,\textrm{coker}(\nu _{M}))=0$. So $\textrm{coker}(\nu _{M})=0$. Hence $\nu _{M}$ is an isomorphism. Therefore $M\in \mathcal {B}_{C}(S)$.

The proof of (2) is similar. $\square$

Recall that a left R-module M is n-tilting [11] if $pd(M)\le n$, $\textrm{Ext}_{R}^{i}(M, M^{(\kappa )}) =0$ for any $i\ge 1$ and any cardinal $\kappa$, there exist an integer $r \ge 0$ and an exact sequence of left R-modules $0\rightarrow$ $_{R}R\rightarrow X^{0}\rightarrow X^{1}\rightarrow \cdots \rightarrow X^{r}\rightarrow 0$ with each $X^{i}\in$ Add(M). Dually we have the concept of an n-cotilting module.

Recently, the notion of tilting modules with respect to a semidualizing module over a commutative ring was introduced by Salimi in [25]. Here we extend it to the noncommutative setting.

Definition 2.7

Let C be a semidualizing (S, R)-bimodule.

A left S-module M is called C-n-tilting if it satisfies the following conditions:

• (1) $pd_{R}(\textrm{Hom}_{S}(C, M))\le n$;

• (2) $\textrm{Ext}_{R}^{i}(\textrm{Hom}_{S}(C,M), \textrm{Hom}_{S}(C,M)^{(\kappa )}) =0$ for any $i\ge 1$ and any cardinal $\kappa$;

• (3) There exist an integer $r \ge 0$ and an exact sequence of left S-modules $0\rightarrow$ $_{S}C\rightarrow X^{0}\rightarrow X^{1}\rightarrow \cdots \rightarrow X^{r}\rightarrow 0$ with each $X^{i}\in$ Add(M).

Dually, a left R-module N is called C-n-cotilting if it satisfies the following conditions:

• (1) $id_{S}(C\otimes _{R}N)\le n$;

• (2) $\textrm{Ext}_{S}^{i}((C\otimes _{R}N)^{\kappa }, C\otimes _{R}N) =0$ for any $i\ge 1$ and any cardinal $\kappa$;

• (3) There exist an integer $r \ge 0$ and an exact sequence of left R-modules $0\rightarrow X_{r}\rightarrow \cdots \rightarrow X_{1}\rightarrow X_{0}\rightarrow C^{+}\rightarrow 0$ with each $X_{i}\in$ Prod(N).

The following theorem establishes the connection between C-tilting left S-modules (resp. C-cotilting left R-modules) and tilting (resp. cotilting) left T-modules.

Theorem 2.8

Let $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with C a faithfully semidualizing (S, R)-bimodule. Then

• (1) M is a C-n-tilting left S-module if and only if $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )$ is an n-tilting left T-module.

• (2) N is a C-n-cotilting left R-module if and only if $\biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )$ is an n-cotilting left T-module.

Proof

(1) “$\Rightarrow$” Since $pd_{R}(\textrm{Hom}_{S}(C, M))\le n$, $\textrm{Hom}_{S}(C, M)\in \mathcal {A}_{C}(R)$ by [18, Corollary 6.2]. So $M\in \mathcal {B}_{C}(S)$ by Lemma 2.6.

Note that $pd_{T}\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )=pd_{R}(\textrm{Hom}_{S}(C, M))\le n$ by [23, Theorem 2.4].

By [21, Lemma 3.2], for any $i\ge 1$ and any cardinal $\kappa$, we have

$$\begin{aligned}{} & {} \textrm{Ext}_{T}^{i}\left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr ), \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )^{(\kappa )}\right) \\{} & {} \quad \cong \textrm{Ext}_{R}^{i}(\textrm{Hom}_{S}(C,M), \textrm{Hom}_{S}(C,M)^{(\kappa )}) =0. \end{aligned}$$

There exist an integer $r\ge 0$ and an exact sequence of left S-modules

$$\begin{aligned} 0\rightarrow C\rightarrow X^{0}\rightarrow X^{1} \rightarrow \cdots \rightarrow X^{r}\rightarrow 0 \end{aligned}$$

with each $X^{i}\in$ Add(M). Since $\textrm{Ext}^{i}_{S}(C,M)= 0=\textrm{Ext}^{i}_{S}(C,C)$, we get the exact sequence of left R-modules

$$\begin{aligned} 0\rightarrow \textrm{Hom}_{S}(C,C)\rightarrow \textrm{Hom}_{S}(C,X^{0})\rightarrow \textrm{Hom}_{S}(C,X^{1}) \rightarrow \cdots \rightarrow \textrm{Hom}_{S}(C,X^{r})\rightarrow 0. \end{aligned}$$

Thus there is an exact sequence of left T-modules

$$\begin{aligned}{} & {} 0\rightarrow \biggl (\begin{array}{c}R\\ S\oplus C\end{array}\biggr )\rightarrow \biggl (\begin{array}{c} \textrm{Hom}_{S}(C,X^{0})\\ S\oplus X^{0}\end{array}\biggr )\rightarrow \biggl (\begin{array}{c} \textrm{Hom}_{S}(C,X^{1})\\ X^{1}\end{array}\biggr )\rightarrow \cdots \rightarrow \\{} & {} \quad \biggl (\begin{array}{c} \textrm{Hom}_{S}(C,X^{r})\\ X^{r}\end{array}\biggr )\rightarrow 0 \end{aligned}$$

with $\biggl (\begin{array}{c} \textrm{Hom}_{S}(C,X^{0})\\ S\oplus X^{0}\end{array}\biggr )$,$\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,X^{i})\\ X^{i}\end{array}\biggr )\in$ Add$\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr ), 1\le i\le r$.

It follows that $\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )$ is an n-tilting left T-module.

“$\Leftarrow$” Note that $pd_{R}(\textrm{Hom}_{S}(C, M))\le pd_{T}\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )\le n$. Thus $\textrm{Hom}_{S}(C, M)\in \mathcal {A}_{C}(R)$ by [18, Corollary 6.2]. So $M\in \mathcal {B}_{C}(S)$ by Lemma 2.6.

By [21, Lemma 3.2], for any $i\ge 1$ and any cardinal $\kappa$, we have

$$\begin{aligned}{} & {} \textrm{Ext}_{R}^{i}(\textrm{Hom}_{S}(C,M), \textrm{Hom}_{S}(C,M)^{(\kappa )})\\{} & {} \quad \cong \textrm{Ext}_{T}^{i}\left( \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr ), \biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )^{(\kappa )}\right) =0. \end{aligned}$$

There exist an integer $r\ge 0$ and an exact sequence of left T-modules

$$\begin{aligned} 0\rightarrow \biggl (\begin{array}{c}R\\ S\oplus C\end{array}\biggr )\rightarrow \biggl (\begin{array}{c} X^{0}_{1}\\ X^{0}_{2}\end{array}\biggr )_{\varphi ^{0}}\rightarrow \biggl (\begin{array}{c} X^{1}_{1}\\ X^{1}_{2}\end{array}\biggr )_{\varphi ^{1}}\rightarrow \cdots \rightarrow \biggl (\begin{array}{c} X^{r}_{1}\\ X^{r}_{2}\end{array}\biggr )_{\varphi ^{r}}\rightarrow 0 \end{aligned}$$

with each $\biggl (\begin{array}{c}X^{i}_{1}\\ X^{i}_{2}\end{array}\biggr )_{\varphi ^{i}}\in$ Add$\biggl (\begin{array}{c}\textrm{Hom}_{S}(C,M)\\ S\oplus M\end{array}\biggr )$. So we get the exact sequence of left R-modules

$$\begin{aligned} 0\rightarrow R\rightarrow X^{0}_{1}\rightarrow X^{1}_{1} \rightarrow \cdots \rightarrow X^{r}_{1}\rightarrow 0 \end{aligned}$$

with each $X^{i}_{1}\in$ Add$(\textrm{Hom}_{S}(C,M))$. Since $\textrm{Tor}_{i}^{R}(C,\textrm{Hom}_{S}(C,M))=0$ for all $i\ge 1$, we obtain the exact sequence of left S-modules

$$\begin{aligned} 0\rightarrow C\rightarrow C\otimes _{R}X^{0}_{1}\rightarrow C\otimes _{R}X^{1}_{1} \rightarrow \cdots \rightarrow C\otimes _{R}X^{r}_{1}\rightarrow 0 \end{aligned}$$

with each $C\otimes _{R}X^{i}_{1}\in$ Add(M).

Thus M is a C-n-tilting left S-module.

(2) “$\Rightarrow$” Since $id_{S}(C\otimes _{R}N)\le n$, $C\otimes _{R}N\in \mathcal {B}_{C}(S)$ by [18, Corollary 6.2]. So $N\in \mathcal {A}_{C}(R)$ by Lemma 2.6.

Note that $id_{T}\biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )=id_{S}(C\otimes _{R}N)\le n$ by [23, Theorem 2.4].

By [21, Lemma 3.2], for any $i\ge 1$ and any cardinal $\kappa$,

$$\begin{aligned} \textrm{Ext}_{T}^{i}\left( \biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )^{\kappa }, \biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )\right) \cong \textrm{Ext}_{S}^{i}((C\otimes _{R}N)^{\kappa }, C\otimes _{R}N) =0. \end{aligned}$$

There exist an integer $r\ge 0$ and an exact sequence of left R-modules

$$\begin{aligned} 0\rightarrow X_{r}\rightarrow \cdots \rightarrow X_{1} \rightarrow X_{0}\rightarrow C^{+}\rightarrow 0 \end{aligned}$$

with each $X_{i}\in$ Prod(N). Since $\textrm{Tor}_{i}^{R}(C,N)= 0=\textrm{Tor}_{i}^{R}(C,C^{+})$ for any $i\ge 1$ by [18, Theorem 6.4], we get the exact sequence of left S-modules

$$\begin{aligned} 0\rightarrow C\otimes _{R} X_{r}\rightarrow \cdots \rightarrow C\otimes _{R} X_{1} \rightarrow C\otimes _{R} X_{0}\rightarrow C\otimes _{R} C^{+}\rightarrow 0. \end{aligned}$$

Since $C\otimes _{R} C^{+}\cong \textrm{Hom}_{R}(C,C)^{+}\cong S^{+}$, we get the exact sequence of left T-modules

$$\begin{aligned} 0\rightarrow \biggl (\begin{array}{c}X_{r}\\ C\otimes _{R} X_{r}\end{array}\biggr )\rightarrow \cdots \rightarrow \biggl (\begin{array}{c} X_{1}\\ C\otimes _{R} X_{1} \end{array}\biggr )\rightarrow \biggl (\begin{array}{c} R^{+}\oplus X_{0} \\ C\otimes _{R} X_{0} \end{array}\biggr )\rightarrow \biggl (\begin{array}{c} R^{+}\oplus C^{+}\\ S^{+}\end{array}\biggr )\rightarrow 0 \end{aligned}$$

with $\biggl (\begin{array}{c} R^{+}\oplus X_{0}\\ C\otimes _{R} X_{0}\end{array}\biggr ),\biggl (\begin{array}{c}X_{i}\\ C\otimes _{R} X_{i}\end{array}\biggr )\in$ Prod$\biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr ), 1\le i\le r$.

Since $\biggl (\begin{array}{c} R^{+}\oplus C^{+}\\ S^{+}\end{array}\biggr )\cong (R\oplus C, S)^{+}=(T_{T})^{+}$, $\biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )$ is an n-cotilting left T-module.

“$\Leftarrow$” Note that $id_{S}(C\otimes _{R} N)\le id_{T}\biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )\le n$. Thus $C\otimes _{R} N\in \mathcal {B}_{C}(S)$ by [18, Corollary 6.2], which implies that $N\in \mathcal {A}_{C}(R)$ by Lemma 2.6.

By [21, Lemma 3.2], for any $i\ge 1$ and any cardinal $\kappa$, we have

$$\begin{aligned} \textrm{Ext}_{S}^{i}((C\otimes _{R}N)^{\kappa }, C\otimes _{R}N)\cong \textrm{Ext}_{T}^{i}\left( \biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )^{\kappa }, \biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )\right) =0. \end{aligned}$$

There exist an integer $r\ge 0$ and an exact sequence of left T-modules

$$\begin{aligned} 0\rightarrow \biggl (\begin{array}{c}X^{r}_{1}\\ X^{r}_{2}\end{array}\biggr )_{\varphi ^{r}}\rightarrow \cdots \rightarrow \biggl (\begin{array}{c} X^{1}_{1}\\ X^{1}_{2}\end{array}\biggr )_{\varphi ^{1}}\rightarrow \biggl (\begin{array}{c} X^{0}_{1}\\ X^{0}_{2}\end{array}\biggr )_{\varphi ^{0}}\rightarrow \biggl (\begin{array}{c} R^{+}\oplus C^{+}\\ S^{+}\end{array}\biggr )\rightarrow 0 \end{aligned}$$

with each $\biggl (\begin{array}{c}X^{i}_{1}\\ X^{i}_{2}\end{array}\biggr )_{\varphi ^{i}}\in$ Prod$\biggl (\begin{array}{c}R^{+}\oplus N\\ C\otimes _{R} N\end{array}\biggr )$. So we get the exact sequence of left S-modules

$$\begin{aligned} 0\rightarrow X^{r}_{2} \rightarrow \cdots \rightarrow X^{1}_{2}\rightarrow X^{0}_{2}\rightarrow S^{+}\rightarrow 0 \end{aligned}$$

with each $X^{i}_{2}\in$ Prod$(C\otimes _{R} N)$. Since $\textrm{Ext}_{S}^{i}(C,C\otimes _{R} N)=0$ for all $i\ge 1$, we get the exact sequence of left R-modules

$$\begin{aligned}{} & {} 0\rightarrow \textrm{Hom}_{S}(C, X^{r}_{2})\rightarrow \cdots \rightarrow \textrm{Hom}_{S}(C, X^{1}_{2}) \rightarrow \textrm{Hom}_{S}(C, X^{0}_{2})\rightarrow \\{} & {} \quad \textrm{Hom}_{S}(C,S^{+})\rightarrow 0 \end{aligned}$$

with each $\textrm{Hom}_{S}(C, X^{i}_{2})\in$ Prod(N).

Since $\textrm{Hom}_{S}(C,S^{+})\cong (S\otimes _{S}C)^{+}\cong C^{+}$, N is a C-n-cotilting left R-module. $\square$

Finally, we give a computing formula of the self-injective dimension of a formal triangular matrix ring $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ with C a faithfully semidualizing (S, R)-bimodule.

Proposition 2.9

Let $T=\biggl (\begin{matrix} R&{}0\\ C&{}S \end{matrix}\biggr )$ be a formal triangular matrix ring with C a faithfully semidualizing (S, R)-bimodule. Then

• (1) $id_{T}\biggl (\begin{array}{c}0\\ C\end{array}\biggr )=max\{id(_{S}C), id(_{R}R)+1\}$.

• (2) $id(_{T}T)=max\{id(_{S}C), id(_{S}S), id_{R}(\textrm{Hom}_{S}(C,S))+1\}.$

Proof

(1) follows from [23, Theorem 2.4].

(2) By [23, Theorem 2.4], $id_{T}\biggl (\begin{array}{c}R\\ C\end{array}\biggr )=id(_{S}C)$.

Next we prove that $id_{T}\biggl (\begin{array}{c} 0\\ S\end{array}\biggr )$ = max$\{id(_{S}S), id_{R}(\textrm{Hom}_{S}(C,S))+1\}$.

(i) If $id(_{S}S)<\infty$, then $S\in \mathcal {B}_{C}(S)$ by [18, Corollary 6.2]. Therefore [23, Theorem 2.4] implies that $id_{T}\biggl (\begin{array}{c} 0\\ S\end{array}\biggr )$ = max$\{id(_{S}S), id_{R}(\textrm{Hom}_{S}(C,S))+1\}.$

(ii) If $id(_{S}S)=\infty$, then it is clear that $id_{T}\biggl (\begin{array}{c} 0\\ S\end{array}\biggr )= \infty$.

It follows that $id(_{T}T)=id_{T}\biggl (\begin{array}{c}R\\ C\oplus S\end{array}\biggr )$ = max$\{id_{T}\biggl (\begin{array}{c}R\\ C\end{array}\biggr ),id_{T}\biggl (\begin{array}{c} 0\\ S\end{array}\biggr )\}$

= max$\{id(_{S}C), id(_{S}S), id_{R}(\textrm{Hom}_{S}(C,S))+1\}.$ $\square$

Remark 2.10

Recall that a semidualizing (S, R)-bimodule C is dualizing if $id(_{S}C)<\infty$ and $id(C_{R})<\infty$. By Proposition 2.9, if $id(_{T}T)<\infty$ and $id(T_{T})<\infty$, then C is dualizing.