Tilting Modules Over Gorensetein $T_n^d$-Injective Gorensetein $T_n^d$-flat Modules

TL;DR

This paper investigates the properties of relative Gorenstein projective and injective modules in the context of tilting modules over certain Gorenstein rings, expanding understanding of their homological behavior.

Contribution

It introduces new results on Gorenstein modules relative to tilting modules over Gorenstein rings, highlighting their structure and properties.

Findings

Characterization of Gorenstein projective modules relative to tilting modules

Analysis of Gorenstein injective modules in the same context

New criteria for module classification based on Gorenstein properties

Abstract

Let T be a tilting module.In this paper, some relative Gorenstein projective and Gortenstein injective modules are studied.

Equations

$$\begin{array}[]{cccccccccccc}\vspace{-.10cm}0\longrightarrow M\longrightarrow T_{0}\longrightarrow T_{1}\longrightarrow\cdots\longrightarrow T_{n-1}\longrightarrow T_{n}\end{array}$$

$$\begin{array}[]{cccccccccccc}\vspace{-.10cm}0\longrightarrow M\longrightarrow T_{0}\longrightarrow T_{1}\longrightarrow\cdots\longrightarrow T_{n-1}\longrightarrow T_{n}\longrightarrow\cdots,\end{array}$$

$${\Gamma }_n^T(M,-):=\frac{\ker ({\delta }_n\otimes 1_B)}{\mathrm{im}({\delta }_{n+1}\otimes 1_B)};{\mathcal{E}}_T^n(-,N):=\frac{\ker {\delta }_{\ast }^n}{\mathrm{im}{\delta }_{\ast }^{n-1}},$$

$$\begin{array}[]{cccccccccc}\vspace{-.3cm}\hfil&&&\delta_{2}&&\delta_{1}&&\delta_{0}\\ \cdots&\longrightarrow&T_{2}&\longrightarrow&T_{1}&\longrightarrow&T_{0}&\longrightarrow&M&\longrightarrow 0\end{array}$$

$$\begin{array}[]{cccccccccc}\vspace{-.3cm}\hfil&\delta_{0}&&\delta_{1}&&\delta_{2}\\ 0\longrightarrow N&\longrightarrow&T^{0}&\longrightarrow&T^{1}&\longrightarrow&\cdots\end{array}$$

$$\mathbf{M}=\cdots ⟶M_1⟶M_0⟶M^0⟶M^1⟶\cdots$$

$$\mathbf{N}=\cdots ⟶N_1⟶N_0⟶N^0⟶N^1⟶\cdots$$

$$\begin{array}[]{ccccccccc}0\longrightarrow T^{{}^{\prime}}_{0}&\stackrel{{\scriptstyle\displaystyle i_{0}}}{{\longrightarrow}}T^{{}^{\prime}}_{0}\oplus T^{{}^{\prime\prime}}_{0}&\stackrel{{\scriptstyle\displaystyle\pi_{0}}}{{\longrightarrow}}T^{{}^{\prime\prime}}_{0}\longrightarrow 0\\ \hskip 39.83368pt\downarrow h_{0}^{{}^{\prime}}&\ \ \ \ \ \ \ \ \ \ \downarrow h_{0}&\ \ \downarrow h_{0}^{{}^{\prime\prime}}&\\ 0\longrightarrow A&{\stackrel{{\scriptstyle\displaystyle f}}{{\longrightarrow}}}\ \ \ \ \ B&{\stackrel{{\scriptstyle\displaystyle g}}{{\longrightarrow}}}\ \ C\longrightarrow 0\\ \hskip 28.45274pt\downarrow&\ \ \ \ \ \downarrow&{\downarrow}&\\ \hskip 28.45274pt0&\ \ \ \ \ 0&{0}&\end{array}$$

$$\mathrm{Hom}(T_0^{{}^{\prime }},T_0)\to \mathrm{Hom}(T_0^{{}^{\prime }},B)\to 0,$$

$$\mathbf{M}=\cdots ⟶M_1⟶M_0⟶M^0⟶M^1⟶\cdots$$

$$\mathbf{N}=\cdots ⟶N_1⟶N_0⟶N^0⟶N^1⟶\cdots$$

$$\mathbf{M}=\cdots ⟶M_1⟶M_0⟶M^0⟶M^1⟶\cdots$$

$$\mathbf{N}=\cdots ⟶N_1⟶N_0⟶N^0⟶N^1⟶\cdots$$

$$0⟶G⟶T^0⟶T^1⟶\cdots$$

$$\cdots ⟶M_1⟶M_0⟶T^0⟶T^1⟶\cdots$$

$$0⟶L⟶N⟶G⟶0(1)$$

$$\cdots ⟶M_2^{{}^{\prime }}⟶M_1^{{}^{\prime }}⟶M_0^{{}^{\prime }}⟶L⟶0(2)$$

$$\cdots \to M_2^{{}^{\prime }}\to M_1^{{}^{\prime }}\to M_0^{{}^{\prime }}\to N\to G\to 0,$$

$$\cdots ⟶T_1⟶T_0⟶G⟶0,$$

$$\cdots ⟶T_1⟶T_0⟶N^0⟶N^1⟶\cdots$$

$$0⟶G⟶M⟶K⟶0(1)$$

$$0⟶K⟶(N^0{)}^{{}^{\prime }}⟶(N^1{)}^{{}^{\prime }}⟶(N^2{)}^{{}^{\prime }}⟶\cdots (2)$$

$$0\to G\to M\to (N^0{)}^{{}^{\prime }}\to (N^1{)}^{{}^{\prime }}\to (N^2{)}^{{}^{\prime }}\to \cdots ,$$

$$0\to L\to M_{m-1}\to \cdots \to M_0\to G\to 0.$$

$$0={\mathcal{E}}_R^i(U,G_{j-1})\to {\mathcal{E}}_R^i(U,K_j)\to {\mathcal{E}}_R^{i+1}(U,K_{j-1})\to {\mathcal{E}}_R^{i+1}(U,G_{j-1})=0$$

$${\mathcal{E}}_R^{m+i}(U,N)\cong {\mathcal{E}}_R^{m+i-1}(U,K_{m-1})\cong \cdots \cong {\mathcal{E}}_R^i(U,G),$$

$$0⟶\mathrm{Hom}(C,A)\stackrel{g^{\ast }}{⟶}\mathrm{Hom}(B,A)\stackrel{f^{\ast }}{⟶}\mathrm{Hom}(A,A)⟶{\mathcal{E}}_T^1(C,A)=0$$

$${\mathcal{E}}_T^1(N,T_0)={\mathcal{E}}_T^1(N,T_0^{{}^{\prime }}\oplus D)={\mathcal{E}}_T^1(N,T_0^{{}^{\prime }})\oplus {\mathcal{E}}_T^1(N,D)=0.$$

$$\cdots \to M_1\to M_0\to M\to 0.$$

$$\mathbf{A}=\cdots \to F_1\to F_0\to T^0\to T^1\to \cdots ,$$

Full text

1. (1)

If $A\in{\rm Pres}^{n}T$ and $C\in{\rm Pres}^{n}T$, then $B\in{\rm Pres}^{n}T.$ 2. (2)

If $A\in{\rm Pres}^{n}T$ and $B\in{\rm Pres}^{n+1}T$, then $C\in{\rm Pres}^{n+1}T.$ 3. (3)

If $B\in{\rm Pres}^{n}T$ and $C\in{\rm Pres}^{n+1}T$, then $A\in{\rm Pres}^{n}T.$

[TABLE]

exists, where $T^{\prime}_{0},T^{\prime\prime}_{0}\in{\rm Add}T$, $i_{0}$ is the inclusion map, $\pi_{0}$ is a canonical epimorphism and $h_{0}=fh_{0}^{\prime}$ is epimorphism, by Five Lemma. Let $K_{1}^{\prime}=\ker h_{0}^{\prime}$, $K_{1}=\ker h_{0}$ and $K_{1}^{\prime\prime}=\ker h_{0}^{\prime\prime}$. So, $B\in{\rm Pres}^{0}T$. It is clear that $K_{1}^{\prime},K_{1}^{\prime\prime}\in{\rm Pres}^{n}T$; so, the induction hypotises implies that $K_{1}\in{\rm Pres}^{n}T$. Hence $B\in{\rm Pres}^{n}T$.

(2) First assume that $n=0$. If $B\in{\rm Pres}^{1}T$ and $A\in{\rm Pres}^{0}T$, then the following commutative diagram with exact rows:

\begin{array}[]{ccccccccc}&\ \ \ \ \ \ T^{{}^{\prime}}_{0}&\longrightarrow A\longrightarrow 0\\ &\ \ \ \ \ \ \ \downarrow\gamma&\downarrow f&\\ {\hskip 113.81102ptT_{1}}&\stackrel{{\scriptstyle\displaystyle\alpha_{2}}}{{\longrightarrow}}T_{0}&\stackrel{{\scriptstyle\displaystyle\alpha_{1}}}{{\longrightarrow}}B\longrightarrow 0\\ &\ \ \ \ \ \|&\downarrow g&\\ {\hskip 113.81102ptT^{{}^{\prime}}_{0}\oplus T_{1}}&\stackrel{{\scriptstyle\displaystyle h}}{{\longrightarrow}}T_{0}&\stackrel{{\scriptstyle\displaystyle g\alpha_{1}}}{{\longrightarrow}}C\longrightarrow 0\\ &&\downarrow&\\ &&0&\end{array}

in which the existence of $\gamma$ follows from the exactness of the sequence

[TABLE]

since $T^{{}^{\prime}}_{0}$ is $T$-projective. Also, $h$ is defined by $h(t^{{}^{\prime}}_{0},t_{1})=\gamma(t^{{}^{\prime}}_{0})+\alpha(t_{1})$. Therefore, we deduce that $C\in{\rm Pres}^{1}T$. For $n>0$, the assertion follows from induction.

(3) This is proved similarly.

• $\Box$*

1. (1)

If $G\in{\rm Cogen}T$, then $G$ is called Gorenstein $T_{n}^{d}$-injective if there exists the following exact sequence of ${\mathcal{ET}}$-modules:

[TABLE]

with $G=\ker(M^{0}\rightarrow M^{1})$ such that ${\mathcal{E}}_{T}^{d}(U,{\mathbf{M}})$ leaves this sequence exact whenever $U\in{\rm F.Pres}^{n}T$ with ${\rm T.pdim}(U)<\infty.$ 2. (2)

If $G\in{\rm Gen}T$, then $G$ is called Gorenstein $T_{n}^{d}$-flat if there exists the following exact sequence of ${\mathcal{FT}}$-modules:

[TABLE]

with $G=\ker(N^{0}\rightarrow N^{1})$ such that ${\Gamma}_{d}^{T}(U,{\mathbf{N}})$ leaves this sequence exact whenever $U\in{\rm F.Pres}^{n}T$ with ${\rm T.fdim}(U)<\infty$.

In the following theorem, we show that in the case of $(n,T)$-coherent rings, the existence of ${\mathcal{FT}}$-complex and ${\mathcal{ET}}$-complex of a module is sufficient to be Gorenstein $T_{n}^{d}$-flat and Gorenstein $T_{n}^{d}$-injective.

1. (1)

$G\in{\rm Cogen}T$* is Gorenstein $T_{n}^{d}$-injective if and only if there is an exact sequence*

[TABLE]

of ${\mathcal{ET}}$-modules such that $G=\ker(M^{0}\rightarrow M^{1})$. 2. (2)

$G\in{\rm Gen}T$* is Gorenstein $T_{n}^{d}$-flat if and only if there is an exact sequence*

[TABLE]

of ${\mathcal{FT}}$-modules such that $G=\ker(N^{0}\rightarrow N^{1})$.

($\Longleftarrow$) : By definition, it suffices to show that ${\mathcal{E}}_{T}^{d}(U,{\mathbf{M}})$ is exact for every module $U\in{\rm F.Pres}^{n}T$ with ${\rm T.pdim}(U)<\infty$. To prove this, we use the induction on $d$. Let $d=0$ and ${\rm T.pdim}(U)=m$, then we show that ${\rm Hom}(U,{\mathbf{M}})$ is exact. To prove this, we use the induction on $m$. The case $m=0$ is clear. Assume that $m\geq 1$. Since ${\rm T.pdim}(U)=m$, there exists an exact sequence $0\rightarrow L\rightarrow T_{0}\rightarrow U\rightarrow 0$ with $T_{0}\in{\rm F.Add}T\subseteq{\rm F.Pres}^{n-1}T$. Now, from the $(n,T)$-coherence of $R$ and Lemma 1, we deduce that $L,T_{0}\in{\rm F.Pres}^{n}T$. Also, ${\rm T.pdim}(L)\leq m-1$ and ${\rm T.pdim}(T_{0})=0$. So, the following short exact sequence of complexes exists:

\begin{array}[]{ccccccccc}&\vdots&\vdots&\vdots&\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&{\rm Hom}(U,M_{1})&\longrightarrow{\rm Hom}(T_{0},M_{1})&\longrightarrow{\rm Hom}(L,M_{1})\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&{\rm Hom}(U,M_{0})&\longrightarrow{\rm Hom}(T_{0},M_{0})&\longrightarrow{\rm Hom}(L,M_{0})\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&{\rm Hom}(U,M^{0})&\longrightarrow{\rm Hom}(T_{0},M^{0})&\longrightarrow{\rm Hom}(L,M^{0})\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&{\rm Hom}(U,M^{1})&\longrightarrow{\rm Hom}(T_{0},M^{1})&\longrightarrow{\rm Hom}(L,M^{1})\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ &\vdots&\vdots&\vdots&\\ &\parallel&\parallel&\parallel&\\ 0\longrightarrow&{\rm Hom}(U,{\mathbf{M}})&\longrightarrow{\rm Hom}(T_{0},{\mathbf{M}})&\longrightarrow{\rm Hom}(L,{\mathbf{M}})\longrightarrow 0.\\ \end{array}

By induction, ${\rm Hom}(L,{\mathbf{M}})$ and ${\rm Hom}(T_{0},{\mathbf{M}})$ are exact, hence ${\rm Hom}(U,{\mathbf{M}})$ is exact by [7, Theorem 6.10].

Let $d\geq 1$ and $U\in{\rm F.Pres}^{n}T$. Consider the exact sequence $0\rightarrow K\rightarrow T_{0}\rightarrow U\rightarrow 0$, where $T_{0}\in{\rm F.Add}T$. So the following short exact sequence of complexes exists:

\begin{array}[]{ccccccccc}\vdots&\vdots&\\ \downarrow&\downarrow&\\ 0\longrightarrow{\mathcal{E}}_{T}^{d-1}(K,M_{1})&\longrightarrow{\mathcal{E}}_{T}^{d}(U,M_{1})\longrightarrow 0\\ \downarrow&\downarrow&\\ 0\longrightarrow{\mathcal{E}}_{T}^{d-1}(K,M_{0})&\longrightarrow{\mathcal{E}}_{T}^{d}(U,M_{0})\longrightarrow 0\\ \downarrow&\downarrow&\\ 0\longrightarrow{\mathcal{E}}_{T}^{d-1}(K,M^{0})&\longrightarrow{\mathcal{E}}_{T}^{d}(U,M^{0})\longrightarrow 0\\ \downarrow&\downarrow&\\ 0\longrightarrow{\mathcal{E}}_{T}^{d-1}(K,M^{1})&\longrightarrow{\mathcal{E}}_{T}^{d}(U,M^{1})\longrightarrow 0\\ \downarrow&\downarrow&\\ \vdots&\vdots&\\ \parallel&\parallel&\\ 0\longrightarrow{\mathcal{E}}_{T}^{d-1}(K,{\mathbf{M}})&\longrightarrow{\mathcal{E}}_{T}^{d}(U,{\mathbf{M}})\longrightarrow 0.\\ \end{array}

By induction, ${\mathcal{E}}_{T}^{d-1}(K,{\mathbf{M}})$ is exact. So, ${\mathcal{E}}_{T}^{d}(U,{\mathbf{M}})$ is exact and hence, $G$ is Gorenstein $T_{n}^{d}$-flat.

(2) A similar proof to that of (1).

• $\Box$*

1. (1)

If $U\in{\rm F.Pres}^{n}T$, then $U\in{\rm F.Pres}^{m}T$ for any $n\geq m.$ 2. (2)

Every $T_{m}^{d}$-injective $R$-module is $T_{n}^{d}$-injective, for any $n\geq m.$ 3. (3)

Direct sum of $T_{n}^{d}$-injective $R$-modules is $T_{n}^{d}$-injective. 4. (4)

Every $T_{m}^{d}$-flat $R$-module is $T_{n}^{d}$-flat, for any $n\geq m.$

1. (1)

$G$* is Gorenstein $T_{n}^{d}$-injective;* 2. (2)

There is an exact sequence $\cdots\rightarrow M_{1}\rightarrow M_{0}\rightarrow G\rightarrow 0$ of modules, where every $M_{i}\in{\mathcal{ET}}$; 3. (3)

There is a short exact sequence $0\rightarrow L\rightarrow N\rightarrow G\rightarrow 0$ of modules, where $N\in{\mathcal{ET}}$ and $L$ is Gorenstein $T_{n}^{d}$-injective.

$(2)\Longrightarrow(1)$ For any module $G\in{\rm Cogen}T$, there is an exact sequence

[TABLE]

where any $T^{i}\in{\rm Prod}T\subseteq{\mathcal{ET}}$. So, the exact sequence

[TABLE]

of ${\mathcal{ET}}$- modules exists, where $G={\rm ker}(T^{0}\rightarrow T^{1})$. Therefore, $G$ is Gorenstein $T_{n}^{d}$-injective, by Theorem 3.

$(3)\Longrightarrow(2)$ Assume that the exact sequence

[TABLE]

exists, where $N\in{\mathcal{ET}}$ and $L$ is Gorenstein $T_{n}^{d}$-injective. Since $L$ is Gorenstein $T_{n}^{d}$-injective, there is an exact sequence

[TABLE]

where every $M_{i}^{{}^{\prime}}\in{\mathcal{ET}}$. Assembling the sequences $(1)$ and $(2)$, we get the exact sequence

[TABLE]

where $N,M_{i}^{{}^{\prime}}\in{\mathcal{ET}}$, as desired.* $\Box$*

1. (1)

$G$* is Gorenstein $T_{n}^{d}$-flat;* 2. (2)

There is an exact sequence $0\rightarrow G\rightarrow N^{0}\rightarrow N^{1}\rightarrow\cdots$ of $R$-modules, where every $N^{i}\in{\mathcal{FT}}$; 3. (3)

There is a short exact sequence $0\rightarrow G\rightarrow M\rightarrow K\rightarrow 0$ of $R$-modules, where $M\in{\mathcal{FT}}$ and $K$ is Gorenstein $T_{n}^{d}$-flat.

$(2)\Longrightarrow(1)$ For any $R$-module $G\in{\rm Gen}T$, there is an exact sequence

[TABLE]

where any $T_{i}\in{\rm Add}T\subseteq{\mathcal{FT}}$. Thus, the exact sequence

[TABLE]

of ${\mathcal{FT}}$-modules exists, where $G={\rm ker}(N^{0}\rightarrow N^{1})$. Therefore by Theorem 3, $G$ is Gorenstein $T_{n}^{d}$-flat,

$(3)\Longrightarrow(2)$ Assume that the exact sequence

[TABLE]

exists, where $M\in{\mathcal{FT}}$ and $K$ is Gorenstein $T_{n}^{d}$-flat. Since $K$ is Gorenstein $T_{n}^{d}$-flat, there is an exact sequence

[TABLE]

where every $(N^{i})^{{}^{\prime}}\in{\mathcal{FT}}$. Assembling the sequences $(1)$ and $(2)$, we get the exact sequence

[TABLE]

where $M,(N^{i})^{{}^{\prime}}\in{\mathcal{FT}}$, as desired.

• $\Box$*

If $G\in{\rm Cogen}T$ is Gorenstein $T_{n}^{d}$-injective, then ${\mathcal{E}}_{T}^{i}(U,G)=0$ for any $i>d$ and every $U\in{\rm F.Pres}^{n}T$ with ${\rm T.pdim}(U)<\infty$.

If $0\rightarrow G\rightarrow G_{0}\rightarrow G_{1}\rightarrow\cdots\rightarrow G_{m-1}\rightarrow N\rightarrow 0$ is an exact sequence of modules where every $G_{j}$ is a Gorenstein $T_{n}^{d}$-injective and $G_{j}\in{\rm Cogen}T$, then ${\mathcal{E}}_{T}^{i}(U,N)={\mathcal{E}}_{T}^{m+i}(U,G)$ for any $i>d$ with ${\rm T.pdim}(U)<\infty$.

If $G\in{\rm Gen}T$ is Gorenstein $T_{n}^{d}$-flat, then ${\Gamma}_{i}^{T}(U,G)=0$ for any $i>d$ and every $U\in{\rm F.Pres}^{n}T$ with ${\rm T.fdim}(U)<\infty$.

If $0\rightarrow N\rightarrow G_{m-1}\rightarrow G_{m-2}\rightarrow\cdots\rightarrow G_{0}\rightarrow G\rightarrow 0$ is an exact sequence of modules where every $G_{i}$ is a Gorenstein $T_{n}^{d}$-flat and $G_{i}\in{\rm Gen}T$, then ${\Gamma}_{i}^{T}(U,N)={\Gamma}_{m+i}^{T}(U,G)$ with ${\rm T.fdim}(U)<\infty$.

[TABLE]

So, ${\rm{\mathcal{E}}}_{T}^{i}(U,M_{j})=0$ for every $0\leq j\leq m-1$ and any $i>d$, since $U\in{\rm F.Pres}^{n}T$ and any $M_{j}\in{\mathcal{ET}}.$ Thus by [5, Proposition 2.2], we deduce that ${\rm{\mathcal{E}}}_{T}^{i}(U,G)\cong{\rm{\mathcal{E}}}_{T}^{m+i}(U,L)$. Therefore ${\mathcal{E}}_{T}^{i}(U,G)=0$, since ${\rm T.pdim}(U)=m<\infty$.

(2) Setting $G_{m}=N$ and $K_{j-1}=\ker(G_{j-1}\rightarrow G_{j})$, for every $0\leq j\leq m$, the short exact sequence $0\rightarrow K_{j-1}\rightarrow G_{j-1}\rightarrow K_{j}\rightarrow 0$ exists. Thus by (1), the induced exact sequences

[TABLE]

exists and so ${\rm{\mathcal{E}}}_{T}^{i}(U,K_{j})\cong{\rm{\mathcal{E}}}_{R}^{i+1}(U,K_{j-1})$. Since $K_{0}=G$, we have

[TABLE]

as desired.

(3) and (4) are similar to the proof of (1) and (2).

• $\Box$*

1. (1)

If $A$ is $T$-injective and $A,B,C\in{\rm Cogen}T$, then $B=A\oplus C$. 2. (2)

If $A\in{\rm Copres}^{n}T$ and $C\in{\rm Copres}^{n}T$, then $B\in{\rm Copres}^{n}T.$ 3. (3)

If $C\in{\rm Copres}^{n}T$ and $B\in{\rm Copres}^{n+1}T$, then $A\in{\rm Copres}^{n+1}T.$ 4. (4)

If $B\in{\rm Copres}^{n}T$ and $A\in{\rm Copres}^{n+1}T$, then $C\in{\rm Copres}^{n}T.$

[TABLE]

is exact. So, there exists $h:B\rightarrow A$ such that $hf=1_{A}.$

(2) It is similar to the proof of Lemma 1, Part (1).

(3) Let $B\in{\rm Copres}^{n+1}T$ and $C\in{\rm Copres}^{n}T$, then the following commutative diagram with exact rows:

\begin{array}[]{ccccccccc}&{0}&{0}&\\ &{\downarrow}&{\downarrow}&\\ 0\longrightarrow A&={=}A&&&\\ \downarrow&\downarrow&\\ 0\longrightarrow B\longrightarrow T_{0}\longrightarrow L\longrightarrow 0\\ \downarrow&&\downarrow&\parallel&\\ 0\longrightarrow C\longrightarrow D\longrightarrow L\longrightarrow 0&\\ &&\downarrow&\downarrow&\\ &&0&0\end{array}

exists, where $T_{0}\in{\rm Prod}T$ and $L\in{\rm Copres}^{n}T$. By (2), $D\in{\rm Copres}^{n}T$. So, we deduce that $A\in{\rm Copres}^{n+1}T$.

(4) Let $A\in{\rm Copres}^{n+1}T$ and $B\in{\rm Copres}^{n}T$, then the following commutative diagram with exact rows:

\begin{array}[]{ccccccccc}&{0}&{0}&\\ &{\downarrow}&{\downarrow}&\\ 0\longrightarrow A\longrightarrow T_{0}^{{}^{\prime}}\longrightarrow L^{{}^{\prime}}\longrightarrow 0\\ \downarrow&\downarrow&\\ 0\longrightarrow B\longrightarrow T_{0}\longrightarrow L\longrightarrow 0\\ \downarrow&&\downarrow&\parallel&\\ 0\longrightarrow C\longrightarrow D\longrightarrow L\longrightarrow 0&\\ &&\downarrow&\downarrow&\\ &&0&0\end{array}

exists, where $T_{0},T_{0}^{{}^{\prime}}\in{\rm Prod}T$ and $L\in{\rm Copres}^{n-1}T$. Since $T_{0}^{{}^{\prime}}$ is $T$-injective, we have that $T_{0}=T_{0}^{{}^{\prime}}\oplus D$ by (1), and $D\in{\rm Cogen}T$. Thus for any module $N$, we have

[TABLE]

Hence $D\in{\rm Prod}T$. On the other hand, $L\in{\rm Copres}^{n-1}T$. Therefore, we conclude that $C\in{\rm Copres}^{n}T.$

• $\Box$*

1. (1)

Let $0\rightarrow M\rightarrow G\rightarrow N\rightarrow 0$ be an exact sequence. If $N\in{\rm Cogen}T$ is Gorenstein $T_{n}^{d}$-injective and $M\in{\mathcal{ET}}$, then $G$ is Gorenstein $T_{n}^{d}$-injective. 2. (2)

Let $0\rightarrow K\rightarrow G\rightarrow N\rightarrow 0$ be an exact sequence. If $K\in{\rm Gen}T$ is Gorenstein $T_{n}^{d}$-flat and $N\in{\mathcal{FT}}$, then $G$ is Gorenstein $T_{n}^{d}$-flat.

\begin{array}[]{ccccccccc}&&&&0&&0&&\\ &&&&\downarrow&&\downarrow&&\\ &&&&K&={=}&K&&\\ &&&&\downarrow&&\downarrow&&\\ 0&\longrightarrow&M&\longrightarrow&D&\longrightarrow&M^{\prime}&\longrightarrow&0\\ &&\parallel&&\downarrow&&\downarrow&&\\ 0&\longrightarrow&M&\longrightarrow&G&\longrightarrow&N&\longrightarrow&0\\ &&&&\downarrow&&\downarrow&&\\ &&&&0&&0&&\\ \end{array}

The exactness of the middle horizontal sequence with $M,M^{{}^{\prime}}\in{\mathcal{ET}}$, implies that $D\in{\mathcal{ET}}$. Hence from the middle vertical sequence and Corollary 5, we deduce that $G$ is Gorenstein $T_{n}^{d}$-injective.

(2) By Lemma 1, $G\in{\rm Gen}T$, since $K,N\in{\rm Gen}T$. $K$ is Gorenstein $T_{n}^{d}$-flat. So by Corollary 6, there exists an exact sequence of $0\rightarrow K\rightarrow N^{{}^{\prime}}\rightarrow L\rightarrow 0$, where $N^{{}^{\prime}}\in{\mathcal{FT}}$ and $L$ is Gorenstein $T_{n}^{d}$-flat. Now, we consider the following diagram:

\begin{array}[]{ccccccccc}&&0&&0&&\\ &&\downarrow&&\downarrow&&\\ &0\longrightarrow&K&\longrightarrow&G&\longrightarrow&N&\longrightarrow&0\\ &&\downarrow&&\downarrow&&\parallel&&\\ &0\longrightarrow&N^{\prime}&\longrightarrow&E&\longrightarrow&N&\longrightarrow&0\\ &&\downarrow&&\downarrow&&\\ &&L&={=}&L&&\\ &&\downarrow&&\downarrow&&\\ &&0&&0&&\\ \end{array}

The exactness of the middle horizontal sequence with $N,N^{{}^{\prime}}\in{\mathcal{FT}}$, implies that $E\in{\mathcal{FT}}$. Hence from the middle vertical sequence and Corollary 6, we deduce that $G$ is Gorenstein $T_{n}^{d}$-flat.

$\Box$**

In this part, we show that which conditions under every module in ${\rm Cogen}T$ is Gorenstein $T_{n}^{d}$-injective.

1. (1)

Every module in ${\rm Cogen}T$, is Gorenstein $T_{n}^{d}$-injective; 2. (2)

The ring satisfies the following two conditions:

(i)* Every $T$-projective module is $T_{n}^{d}$-injective.*

(ii)* ${\mathcal{E}}_{T}^{d+1}(U,N)=0$ for any $N\in{\rm Cogen}T$ and any $U\in{\rm F.Pres}^{n}T$ with ${\rm T.pdim}(U)<\infty$.*

[TABLE]

Since $M$ is $T$-projective, $M$ is $T_{n}^{d}$-injective as a direct summand of $M_{0}$. Also, by Proposition 7 and (1), the condition $(ii)$ follows.

$(2)\Longrightarrow(1)$ Since $T$ is tilting, the exact sequence $0\rightarrow R\rightarrow T_{0}\rightarrow T_{1}\rightarrow 0$ exists, where $T_{0},T_{1}\in{\rm Add}T$. So $T_{0},T_{1}\in{\rm Gen}T$. Hence, the exact sequence $0\rightarrow K\rightarrow T^{(m)}\rightarrow T_{i}\rightarrow 0$ exists for $i=0,1$. On the other hand, $K\subseteq T^{(m)}\subseteq T^{m}$. So, $K,T^{(m)}\in{\rm Cogen}T$. Thus by [5, Proposition 2.1], $K,T^{(m)}\in{\rm Copres}^{\infty}T$. Therefore by Lemma 8, $T_{i},R\in{\rm Copres}^{k}T$ and hence $R\in{\rm Cogen}T$. Let $G\in{\rm Cogen}T$. Choose a ${\rm Prod}T$-resolution $0\rightarrow G\rightarrow T^{0}\rightarrow T^{1}\rightarrow\cdots$ of $G$ and a free resolution $\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow G\rightarrow 0$, where every $F_{i}\in{\rm Cogen}T$. Also by Lemma 1 and [5, Proposition 2.1], we get that $F_{i}\in{\rm Gen}T={\rm Pres}^{\infty}T$, since $T_{0},T_{1}\in{\rm Gen}T$. Every projective in ${\rm Gen}T$ is $T$-projective. So by (2), every $F_{i}$ is $T_{n}^{d}$-injective. Assembling these resolutions, by Remark 4 and (2)(i), we get the following ${\mathcal{ET}}$-resolotion:

[TABLE]

where $G={\rm ker}(T^{0}\rightarrow T^{1})$, $K^{i}={\rm ker}(T^{i}\rightarrow T^{i+1})$ and $K_{i}={\rm ker}(F_{i}\rightarrow F_{i-1})$ for any $i\geq 1$. By Lemma 8, $K_{i},K^{i}\in{\rm Cogen}T$, since $G,T^{i},F_{i}\in{\rm Cogen}T$. Let $U\in{\rm F.Pres}^{n}T$ with ${\rm T.p.dim}(U)<\infty$. Then by (2), ${\mathcal{E}}_{T}^{d+1}(U,G)={\mathcal{E}}_{T}^{d+1}(U,F_{i})={\mathcal{E}}_{T}^{d+1}(U,T^{i})=0$ for any $i\geq 0$. So, ${\mathcal{E}}_{T}^{d}(U,{\mathbf{A}})$ is exact, and hence $G$ is Gorenstein $T_{n}^{d}$-injective.* $\Box$*

1. (1)

Every module in ${\rm Cogen}T$, is Gorenstein $T_{n}^{d}$-injective; 2. (2)

Every $T$-projective module is $T_{n}^{d}$-injective; 3. (3)

$R$* is $T_{n}^{d}$-injective;* 4. (4)

Every Gorenstein $T_{n}^{d}$-flat is Gorenstein $T_{n}^{d}$-injective; 5. (5)

Every $T$-flat module is Gorenstein $T_{n}^{d}$-injective; 6. (6)

Every $T$-projective module is Gorenstein $T_{n}^{d}$-injective.

$(3)\Longrightarrow(1)$ Let $G\in{\rm Cogen}T$ be a module and $\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow G\rightarrow 0$ be any free resolution of $G$. Then, similar to proof ($(2)\Longrightarrow(1)$) of Proposition 10, each $F_{i}\in{\mathcal{ET}}$. Hence Corollary 5 completes the proof.

$(3)\Longrightarrow(4)$ Let $N\in{\rm Gen}T$ is Gorenstein $T_{n}^{d}$-flat. Similar to proof ($(2)\Longrightarrow(1)$) from Proposition 10, $N\in{\rm Cogen}T$. So, (2) follows immediately from (1).

$(4)\Longrightarrow(5)$ every $T$-flat is $T_{n}^{d}$-flat and every $T_{n}^{d}$-flat is Gorenstein $T_{n}^{d}$-flat. So by (4), (5) is hold.

$(5)\Longrightarrow(6)$ Is clear, since every $T$-projective is $T$-flat.

$(6)\Longrightarrow(3)$ Similar to proof ($(1)\Longrightarrow(2)$) from Proposition 10, every $T$-projective module is $T_{n}^{d}$-injective. Also, the exact sequence $0\rightarrow R\rightarrow T_{0}\rightarrow T_{1}\rightarrow 0$ exists, where $T_{0},T_{1}\in{\rm Add}T$. So $T_{0},T_{1}\in{\rm Gen}T$. Thus by [5, Proposition 2.1], $T_{i}\in{\rm Gen}T={\rm Pres}^{\infty}T$ for $i=0,1$. Hence by Lemma 1, $R\in{\rm Gen}T$. Therefore $R$ is $T$-projective and hence, it is $T_{n}^{d}$-injective.

$\Box$**

Let $R$ be a ring. Then $R$ is $n$-regular if every $n$-presented $R$-module is projective. $M$ is called $(n,d)$-flat if ${\rm Tor}_{d+1}^{R}(U,M)=0$ for every $n$-presented $R$-module $U$. $M$ is called $(n,d)$-injective if ${\rm Ext}_{R}^{d+1}(U,M)=0$ for every $n$-presented $R$-module $U$ (see, [11, 12]). In particular, if $T=R$, then every $T_{n}^{d}$-flat module is $(n,d)$-flat, every $T_{n}^{d}$-injective module is $(n,d)$-injective, every Gorenstein $T_{n}^{d}$-flat module is Gorenstein $(n,d)$-flat and every Gorenstein $T_{n}^{d}$-injective module Gorenstein $(n,d)$-injective.

1. (1)

Let $R$ be a $1$-Gorenstein ring and $0\rightarrow R\rightarrow E^{0}\rightarrow E^{1}\rightarrow 0$ be the minimal injective resolution of $R$. Then, $T=E_{0}\oplus E_{1}$ is Gorenstein $T_{n}^{d}$-injective and Gorenstein $T_{n}^{d}$-flat, since by [3], $T$ is a tilting module. 2. (2)

Let $R$ be an $n$-regular ring. Then replacing $T$ by $R$ as an $R$-module of Theorem 11, every $R$-module is Gorenstein $(n,0)$-injective and Gorenstein $(n,0)$-flat, since by [11, Theorem 3.9], $R$ is $(n,0)$-injective.