On the Gorensetein $(n,d)$-Flat and Gorensetein $(n,d)$-Injective Modules

TL;DR

This paper explores Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules over rings, providing their basic properties, characterizations of rings, and deriving known results as corollaries.

Contribution

It introduces and studies the properties of Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules, offering new characterizations of rings based on these modules.

Findings

Characterization of rings via Gorenstein (n,d)-flat modules

Properties of Gorenstein (n,d)-injective modules

Derivation of known results as corollaries

Abstract

Let R be a ring. In this paper, Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules and some of their basic properties are studied. Moreover, some characterizations of rings over Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules are given. Also, some known results can be obtained as corollaries

Equations

$$\begin{array}{ccccccc}R& \stackrel{F_R}{\to }& R& \stackrel{F_R}{\to }& R& \stackrel{F_R}{\to }& \cdots \end{array}$$

$$\mathbf{F}=\cdots ⟶F_1⟶F_0⟶F^0⟶F^1⟶\cdots$$

$$\mathbf{A}=\cdots ⟶A_1⟶A_0⟶A^0⟶A^1⟶\cdots$$

$$\mathbf{F}=\cdots ⟶F_1⟶F_0⟶F^0⟶F^1⟶\cdots$$

$$\mathbf{A}=\cdots ⟶A_1⟶A_0⟶A^0⟶A^1⟶\cdots$$

$$\mathbf{F}=\cdots ⟶F_1⟶F_0⟶F^0⟶F^1⟶\cdots$$

$$\mathbf{A}=\cdots ⟶A_1⟶A_0⟶A^0⟶A^1⟶\cdots$$

$$\cdots ⟶P_1⟶P_0⟶G⟶0,$$

$$\cdots ⟶P_1⟶P_0⟶B^0⟶B^1⟶\cdots$$

$$0⟶G⟶M⟶L⟶0(1)$$

$$0⟶L⟶(F^0{)}^{{}^{\prime }}⟶(F^1{)}^{{}^{\prime }}⟶(F^2{)}^{{}^{\prime }}⟶\cdots (2)$$

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

$$0⟶G⟶I^0⟶I^1⟶\cdots$$

$$\cdots ⟶A_1⟶A_0⟶I^0⟶I^1⟶\cdots$$

$$0⟶L⟶M⟶G⟶0(1)$$

$$\cdots ⟶A_2^{{}^{\prime }}⟶A_1^{{}^{\prime }}⟶A_0^{{}^{\prime }}⟶L⟶0(2)$$

$$\cdots \to A_2^{{}^{\prime }}\to A_1^{{}^{\prime }}\to A_0^{{}^{\prime }}\to M\to G\to 0,$$

$$0\to N\to A_{m-1}\to \cdots \to A_0\to G\to 0.$$

$$0\to B_m\to B_{m-1}\to \cdots \to B_0\to K\to 0.$$

$$0\to N\to A_0\to \cdots \to A_{m-1}\to A_m\to 0.$$

$$\cdots \to A_1\to A_0\to M\to 0.$$

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

$$\mathbf{I}=\cdots ⟶I_1⟶I_0⟶I^0⟶I^1⟶\cdots$$

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

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

$$0\to K_{n-1}\to F_{n-1}\to \cdots \to F_0\to F_1\to U\to 0,$$

$$0\to K_{n-1}{\otimes }_{R}S\to F_{n-1}{\otimes }_{R}S\to \cdots \to F_0{\otimes }_{R}S\to F_1{\otimes }_{R}S\to U{\otimes }_{R}S\to 0,$$

Full text

On the Gorensetein $(n,d)$-Flat and Gorensetein $(n,d)$-Injective Modules

M. Amini${}^{{}^{*}}$ ††∗Corresponding Author

Department of Mathematics, Payame Noor University, Tehran, Iran

E-mail: mostafa.amini@pnu.ac.ir

Abstract. Let $R$ be a ring. In this paper, Gorensetein $(n,d)$-flat and Gorenstein $(n,d)$-injective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over Gorensetein $(n,d)$-flat and Gorenstein $(n,d)$-injective modules are given. Also, some known results can be obtained as corollaries.

**2010 MSC :13D07; 16D40; 18G25;

Keywords :Gorenstein $(n,d)$-Flat module, Gorensetein $(n,d)$-Injective module, $n$-Coherent ring.

**

1. Basic Definitions and Notations

In this paper, we have assumed that $R$ is an associative ring with non-zero identity and modules are unitary. In this section, first some fundamental concepts and notations are stated. Let $n,d$ non-negative integers and $M$ a right $R$-module. Then

1. (1)

$M$ is said to be $n$-presented [14] if there is an exact sequence of right $R$-modules $F_{n}\rightarrow F_{n-1}\rightarrow\dots\rightarrow F_{1}\rightarrow F_{0}\rightarrow M\rightarrow$ , where each $F_{i}$ is a finitely generated free. 2. (2)

$R$ is called right $n$-coherent ring [14] if every $n$-presented right $R$-module is $(n+1)$-presented. 3. (3)

$M$ is called $(n,d)$-flat [13, 14] if ${Tor}_{d+1}^{R}(U,M)=0$ for every $n$-presented left $R$-module $U$. 4. (4)

$M$ is called $(n,d)$-injective [13, 14] if ${Ext}_{R}^{d+1}(U,M)=0$ for every $n$-presented right $R$-module $U$. 5. (5)

$M$ is said to be Gorenstein flat (resp.,Gorenstein injective) [2, 3, 6] if there is an exact sequence $\cdots\rightarrow I_{1}\rightarrow I_{0}\rightarrow I^{0}\rightarrow I^{1}\rightarrow\cdots$ of flat (resp., injective ) right modules with $M={\rm ker}(I^{0}\rightarrow I^{1})$ such that $U\otimes_{R}-$ (resp; ${Hom}(U,-)$) leaves the sequence exact whenever $U$ is an left injective (resp; right injective) module. 6. (6)

For any commutative ring $R$ of prime characteristic $p>0$, assume that $F_{R}:R\rightarrow R^{(e)}$ is the $e$-th iterated Frobenius map in which $R^{(e)}\cong R$. Then, the perfect closure [9] of $R$, denoted by $R_{\infty}$, is defined as the limit of the following direct system:

[TABLE] 7. (7)

Assume that $S\geq R$ is a unitary ring extension. Then, the ring $S$ is called $R$-projective [14, 15] in case, for any right $S$-module $M_{S}$ with an right $S$-module $N_{S}$, $N_{R}\mid M_{R}$ implies $N_{S}\mid M_{S}$, where $N\mid M$ means that $N$ is a direct summand of $M$. 8. (8)

A finite normalizing extension $S\geq R$ is called an almost excellent extension [14, 15] in case ${}_{R}S$ is flat, $S_{R}$ is projective, and the ring $S$ is $R$-projective.

For more information about the relative cohence of rings and modules, see [4, 7, 9, 11]. In this paper, we introduce the Gorenstein $(n,d)$-flat modules and Gorenstein $(n,d)$-injective modules. A right module $G$ is said to be Gorenstein $(n,d)$-flat ($G_{n}^{d}$-flat for short), if there exists the following exact sequence of $(n,d)$-flat right $R$-modules

[TABLE]

with $G=\ker(F^{0}\rightarrow F^{1})$ such that ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})$ leaves this sequence exact whenever $U$ is an $n$-presented left $R$-module with ${fd}_{R}(U)<\infty$. A right module $G$ is said to be Gorenstein $(n,d)$-injective ($G_{n}^{d}$-injective for short), if there exists the following exact sequence of $(n,d)$-injective right $R$-modules

[TABLE]

with $G=\ker(A^{0}\rightarrow A^{1})$ such that ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})$ leaves this sequence exact whenever $U$ is an $n$-presented right $R$-module with ${pd}_{R}(U)<\infty$. The Gorenstein $(n,d)$-flat dimension (resp; Gorenstein $(n,d)$-injective dimension) of a module $M$ is denoted by $G_{n}^{d}.fd(M)$ (resp; $G_{n}^{d}.id(M)$). $G_{n}^{d}.fd(M)$ of an $R$-module $M$ is defined that $G_{n}^{d}.fd(M)\leq m$ (resp; $G_{n}^{d}.id(M)\leq m$) if and only if $M$ has a Gorenstein $(n,d)$-flat (resp; Gorenstein $(n,d)$-injective) resolution of length $m$.

In Section 2, we study some basic properties of the Gorenstein $(n,d)$-flat and Gorenstein $(n,d)$-injective modules. In section 3, we give sufficient conditions under which every module is Gorenstein $(n,d)$-injective. For instance, if $R$ is a $n$-coherent ring, then every $R$-module is Gorenstein $(n,d)$-injective if and only if $R$ is $(n,d)$-injective if and only if every Gorenstein $(n,0)$-injective is Gorenstein $(n,d)$-flat. Finally in section 4, some results of Gorenstein (n, d)-flat and Gorenstein (n, d)-injective modules on projective algebras are given. For instance, if $f:R\rightarrow S$ be a surjective ring homomorphism, $S$ a projective $R$-module and $G$ a right $S$-module, then $G_{R}$ is Gorenstein $(n,d)$-flat (resp; $G_{R}$ is Gorenstein (n, d)-injective ) if and only if $G_{S}$ is Gorenstein $(n,d)$-flat (resp; $G_{S}$ is Gorenstein (n, d)-injective ) if and only if $S\otimes_{R}G$ (resp; ${\rm Hom}_{R}(S,G)$) is Gorenstein $(n,d)$-flat (resp; Gorenstein $(n,d)$-injective ). Also, if $S\geq R$ is an almost excellent extension, then the perfect closure of $R$ is right Gorenstein $(n,d)$-injective if and only if the perfect closure of $S$ is right Gorenstein $(n,d)$-injective.

2. Gorensetein $(n,d)$-Flatnes and Gorensetein $(n,d)$-Injectivity

We start with the following definition.

Definition 2.1**.**

Let $G$ be a right $R$-module. Then

1. (1)

$G$ is called Gorenstein $(n,d)$-flat if there exists the following exact sequence of $(n,d)$-flat right $R$-modules:

[TABLE]

with $G=\ker(F^{0}\rightarrow F^{1})$ such that ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})$ leaves this sequence exact whenever $U$ is $n$-presented left $R$-module with ${\rm fd}_{R}(U)<\infty$. 2. (2)

$G$ is called Gorenstein $(n,d)$-injective if there exists the following exact sequence of $(n,d)$-injective right $R$-modules:

[TABLE]

with $G=\ker(A^{0}\rightarrow A^{1})$ such that ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})$ leaves this sequence exact whenever $U$ is $n$-presented right $R$-module with ${\rm pd}_{R}(U)<\infty.$

Lemma 2.2**.**

If $R$ is a left $n$-coherent ring and $U$ is an $n$-presented left $R$-module with ${\rm fd}_{R}(U)<\infty$, then ${\rm pd}_{R}(U)<\infty$.

Proof.

Let ${\rm fd}_{R}(U)=m<\infty$, then we must show that ${\rm pd}_{R}(U)\leq m$. Since $R$ is a left $n$-coherent ring and $U$ is an $n$-presented left $R$-module, the projective resolution $\cdots\rightarrow F_{n+1}\rightarrow F_{n}\rightarrow\cdots\rightarrow F_{0}\rightarrow U\rightarrow 0,$ where any $F_{i}$ is finitely generated free, exists. On the other hand, above exact squence is a flat resolution. So by [8, Proposition 8.17], $(m-1)$-syzygy is flat. Hence, the exact sequence $0\rightarrow K_{m-1}\rightarrow F_{m-1}\rightarrow\cdots\rightarrow F_{0}\rightarrow U\rightarrow 0$ is a flat resolution. If $n\geq m$ or $n\leq m$, then $K_{m-1}$ is finitely presented and consequencly by [8, Theorem 3.56], $K_{m-1}$ is projective and so, ${\rm pd}_{R}(U)\leq m$. ∎

In the following theorem, we show that in the case of $n$-coherent rings, the existence of complexs of a module is sufficient to be Gorenstein $(n,d)$-flat and Gorenstein $(n,d)$-injective.

Theorem 2.3**.**

Let $R$ be a ring and G a right $R$-module.

1. (1)

If $R$ is a left $n$-coherent ring, then $G$ is Gorenstein $(n,d)$-flat if and only if there is an exact sequence

[TABLE]

of $(n,d)$-flat right $R$-modules such that $G=\ker(F^{0}\rightarrow F^{1})$. 2. (2)

If $R$ is a right $n$-coherent ring, then $G$ is Gorenstein $(n,d)$-injective if and only if there is an exact sequence

[TABLE]

of $(n,d)$-injective right $R$-modules such that $G=\ker(A^{0}\rightarrow A^{1})$.

Proof.

(1) ($\Longrightarrow$) : This is a direct consequence of the definition.

($\Longleftarrow$) : By definition, it suffices to show that ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})$ is exact for every $n$-presented left $R$-module $U$ with ${\rm fd}_{R}(U)<\infty$. By Lemma 2.2, ${\rm pd}_{R}(U)<\infty$. We use the induction on $d$. Let $d=0$ and ${\rm pd}_{R}(U)=m<\infty$, then we show that $U\otimes_{R}{\mathbf{F}}$ is exact. To prove this, we use the induction on $m$. The case $m=0$ is clear. Assume that $m\geq 1$. There exists an exact sequence $0\rightarrow K\rightarrow P_{0}\rightarrow U\rightarrow 0$, where $P_{0}$ is projective. Now, from the $n$-coherence of $R$ and [15, Theorem 1], we deduce that $K$ is left $n$-presented. Also, ${\rm pd}_{R}(K)\leq m-1$. So, the following short exact sequence of complexes exists:

\begin{array}[]{ccccccccc}&\vdots&\vdots&\vdots&\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&K\otimes_{R}F_{1}&\longrightarrow P_{0}\otimes_{R}F_{1}&\longrightarrow U\otimes_{R}F_{1}\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&K\otimes_{R}F_{0}&\longrightarrow P_{0}\otimes_{R}F_{0}&\longrightarrow U\otimes_{R}F_{0}\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&K\otimes_{R}F^{0}&\longrightarrow P_{0}\otimes_{R}F^{0}&\longrightarrow U\otimes_{R}F^{0}\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ 0\longrightarrow&K\otimes_{R}F^{1}&\longrightarrow P_{0}\otimes_{R}F^{1}&\longrightarrow U\otimes_{R}F^{1}\longrightarrow 0\\ &\downarrow&\downarrow&\downarrow&\\ &\vdots&\vdots&\vdots&\\ &\parallel&\parallel&\parallel&\\ 0\longrightarrow&K\otimes_{R}{\mathbf{F}}&\longrightarrow P_{0}\otimes_{R}{\mathbf{F}}&\longrightarrow U\otimes_{R}{\mathbf{F}}\longrightarrow 0.\\ \end{array}

By induction, $P_{0}\otimes_{R}{\mathbf{F}}$ and $K\otimes_{R}{\mathbf{F}}$ are exact, hence $U\otimes_{R}{\mathbf{F}}$ is exact by [8, Theorem 6.10].

Let $d\geq 1$. Since, $U$ is $n$-presented, the exact sequence $0\rightarrow K^{{}^{\prime}}\rightarrow P_{0}^{{}^{\prime}}\rightarrow U\rightarrow 0$ with $P_{0}^{{}^{\prime}}$ is projective exists. Therefore, the following short exact sequence of complexes exists:

\begin{array}[]{ccccccccc}\vdots&\vdots&\\ \downarrow&\downarrow&\\ 0\longrightarrow{\rm Tor}_{d}^{R}(U,F_{1})&\longrightarrow{\rm Tor}_{d-1}^{R}(K^{{}^{\prime}},F_{1})\longrightarrow 0\\ \downarrow&\downarrow&\\ 0\longrightarrow{\rm Tor}_{d}^{R}(U,F_{0})&\longrightarrow{\rm Tor}_{d-1}^{R}(K^{{}^{\prime}},F_{0})\longrightarrow 0\\ \downarrow&\downarrow&\\ 0\longrightarrow{\rm Tor}_{d}^{R}(U,F^{0})&\longrightarrow{\rm Tor}_{d-1}^{R}(K^{{}^{\prime}},F^{0})\longrightarrow 0\\ \downarrow&\downarrow&\\ 0\longrightarrow{\rm Tor}_{d}^{R}(U,F^{1})&\longrightarrow{\rm Tor}_{d-1}^{R}(K^{{}^{\prime}},F^{1})\longrightarrow 0\\ \downarrow&\downarrow&\\ \vdots&\vdots&\\ \parallel&\parallel&\\ 0\longrightarrow{\rm Tor}_{d}^{R}(U,{\mathbf{F}})&\longrightarrow{\rm Tor}_{d-1}^{R}(K^{{}^{\prime}},{\mathbf{F}})\longrightarrow 0.\\ \end{array}

We know that $K^{{}^{\prime}}$ is $n$-presented with ${\rm pd}_{R}(K^{{}^{\prime}})<\infty$, and so by induction, ${\rm Tor}_{d-1}^{R}(K^{{}^{\prime}},{\mathbf{F}})$ is exact. Therefore, ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})$ is exact and hence, $G$ is Gorenstein $(n,d)$-flat.

(2) ($\Longrightarrow$) : This is a direct consequence of the definition.

($\Longleftarrow$) Let $U$ be a right $n$-presented $R$-module with ${\rm pd}_{R}(U)<\infty$. Then, a similar proof to that of (1), ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})$ is exact and hence $G$ is Gorenstein $(n,d)$-injective.

∎

Remark 2.4*.*

Let $R$ be a ring. Then:

1. (1)

Every flat right $R$-module is $(n,d)$-flat. 2. (2)

Every injective right $R$-module is $(n,d)$-injective. 3. (3)

Every right $R$-module is flat if and only if $(1,0)$-flat. 4. (4)

Every right $R$-module is injective if and only if $(0,0)$-injective. 5. (5)

Every $(m,d)$-injective right $R$-module is $(n,d)$-injective, for any $n\geq m.$ 6. (6)

Every $(m,d)$-flat right $R$-module is $(n,d)$-flat, for any $n\geq m.$ 7. (7)

Every $n$-presented left (resp; right) $R$-module is $m$-presented, for any $n\geq m.$ 8. (8)

Every $(n,d)$-flat right $R$-module is Gorenstein $(n,d)$-flat. 9. (9)

Every $(n,d)$-injective right $R$-module is Gorenstein $(n,d)$-injective.

Corollary 2.5**.**

Let $R$ be a left $n$-coherent ring and G a right $R$-module. Then the following assertions are equivalent:

1. (1)

$G$* is Gorenstein $(n,d)$-flat;* 2. (2)

There is an exact sequence $0\rightarrow G\rightarrow B^{0}\rightarrow B^{1}\rightarrow\cdots$ of right $R$-modules, where every $B^{i}$ is $(n,d)$-flat; 3. (3)

There is a short exact sequence $0\rightarrow G\rightarrow M\rightarrow L\rightarrow 0$ of right $R$-modules, where $M$ is $(n,d)$-flat and $L$ is Gorenstein $(n,d)$-flat.

Proof.

$(1)\Longrightarrow(2)$ and $(1)\Longrightarrow(3)$ follow from definition.

$(2)\Longrightarrow(1)$ For $R$-module $G$, there is an exact sequence

[TABLE]

where any $P_{i}$ is flat for any $i\geq 0$. By Remark 2.4, every $P_{i}$ is $(n,d)$-flat. Thus, the exact sequence

[TABLE]

of $(n,d)$-flat right modules exists, where $G={\rm ker}(B^{0}\rightarrow B^{1})$. Therefore by Theorem 2.3, $G$ is Gorenstein $(n,d)$-flat,

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

[TABLE]

exists, where $M$ is $(n,d)$-flat and $L$ is Gorenstein $(n,d)$-flat. Since $L$ is Gorenstein $(n,d)$-flat, there is an exact sequence

[TABLE]

where every $(F^{i})^{{}^{\prime}}$ is $(n,d)$-flat. Assembling the sequences $(1)$ and $(2)$, we get the exact sequence

[TABLE]

where $M$ and any $(F^{i})^{{}^{\prime}}$ are $(n,d)$-flat , as desired.

∎

Corollary 2.6**.**

Let $R$ be a right $n$-coherent ring and G a right $R$-module. Then the following assertions are equivalent:

1. (1)

$G$* is Gorenstein $(n,d)$-injective;* 2. (2)

There is an exact sequence $\cdots\rightarrow A_{1}\rightarrow A_{0}\rightarrow G\rightarrow 0$ of right $R$-modules, where every $A_{i}$ is $(n,d)$-injective; 3. (3)

There is a short exact sequence $0\rightarrow L\rightarrow M\rightarrow G\rightarrow 0$ of $R$-modules, where $M$ is $(n,d)$-injective and $L$ is Gorenstein $(n,d)$-injective.

Proof.

$(1)\Longrightarrow(2)$ and $(1)\Longrightarrow(3)$ follow from definition.

$(2)\Longrightarrow(1)$ For any module $G$, there is an exact sequence

[TABLE]

where every $I^{i}$ is injective for any $i\geq 0$. By Remark 2.4, each $I^{i}$ is $(n,d)$-injective. So, the exact sequence

[TABLE]

of $(n,d)$-injective right modules exists, where $G={\rm ker}(I^{0}\rightarrow I^{1})$. Therefore, $G$ is Gorenstein $(n,d)$-injective, by Theorem 2.3.

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

[TABLE]

exists, where $M$ is $(n,d)$-injective and $L$ is Gorenstein $(n,d)$-injective. Since $L$ is Gorenstein $(n,d)$-injective, there is an exact sequence

[TABLE]

where every $A_{i}^{{}^{\prime}}$ is $(n,d)$-injective. Assembling the sequences $(1)$ and $(2)$, we get the exact sequence

[TABLE]

where $M$ and $A_{i}^{{}^{\prime}}$ are $(n,d)$-injective, as desired. ∎

Proposition 2.7**.**

If $G$ is Gorenstein $(n,d)$-injective right $R$-module, then ${\rm Ext}_{R}^{i}(U,G)=0$ for any $i>d$ and every $n$-presented right $R$-module $U$ with ${\rm pd}_{R}(U)<\infty$.

Proof.

Let $G$ be a Gorenstein $(n,d)$-injective $R$-module and ${\rm pd}_{R}(U)=m<\infty$. Then by hypothesis, the following $(n,d)$-injective resolution of $G$ exists:

[TABLE]

So, ${\rm Ext}_{R}^{i}(U,A_{j})=0$ for every $0\leq j\leq m-1$ and any $i>d$, since $U$ is $n$-presented. Thus, we deduce that ${\rm Ext}_{R}^{i}(U,G)\cong{\rm Ext}_{R}^{m+i}(U,N)=0$.

∎

Next, we study the Gorenstein $(n,d)$-flatness and Gorenstein $(n,d)$-injectivity of modules, in short exact sequences.

Proposition 2.8**.**

Let $R$ be a left (resp; right) $n$-coherent ring.

1. (1)

Consider the exact sequence $0\rightarrow K\rightarrow B\rightarrow G\rightarrow 0$, where $B$ is $(n,d)$-flat. Then ${\rm G_{n}^{d}}$-${\rm fd}(G)\leq{\rm G_{n}^{d}}$-${\rm fd}(K)+1$. In particular, if $G$ is Gorenstein $(n,d)$-flat, so is $K$. 2. (2)

Consider the exact sequence $0\rightarrow G\rightarrow A\rightarrow N\rightarrow 0$, where $A$ is $(n,d)$-injective. Then ${\rm G_{n}^{d}}$-${\rm id}(G)\leq{\rm G_{n}^{d}}$-${\rm id}(N)+1$. In particular, if $G$ is Gorenstein $(n,d)$-injective, so is $N$.

Proof.

(1) We shall show that ${\rm G_{n}^{d}}$-${\rm fd}(G)\leq{\rm G_{n}^{d}}$-${\rm fd}(K)+1$. In fact, we may assume that ${\rm G_{n}^{d}}$-${\rm fd}(K)=m<\infty$. Then, by definition, $K$ admits a Gorenstein $(n,d)$-flat resolution:

[TABLE]

Assembling this sequence and the short exact sequence $0\rightarrow K\rightarrow B\rightarrow G\rightarrow 0$, the following commutative diagram is obtained:

\begin{array}[]{ccccccccccccccccc}0&\longrightarrow&B_{m}&\longrightarrow&\cdots&\longrightarrow&B_{1}&\longrightarrow&B_{0}&\longrightarrow&B&\longrightarrow&G&\longrightarrow&0\\ &&&&&&&&\downarrow&&\uparrow&&\\ &&&&&&&&K&={=}{={=}}&K&&\\ &&&&&&&&\downarrow&&\uparrow&&\\ &&&&&&&&0&&0&&\\ \end{array}

which shows that ${\rm G_{n}^{d}}$-${\rm fd}(G)\leq m+1$. The particular case follows from Corollary 2.5.

(2) We shall show that ${\rm G_{n}^{d}}$-${\rm id}(G)\leq{\rm G_{n}^{d}}$-${\rm id}(N)+1$. In fact, we may assume that ${\rm G_{n}^{d}}$-${\rm id}(N)=m<\infty$. Then, by definition, $N$ admits a Gorenstein $(n,d)$-injective resolution:

[TABLE]

Assembling this sequence and the short exact sequence $0\rightarrow G\rightarrow A\rightarrow N\rightarrow 0$, the following commutative diagram is obtained:

\begin{array}[]{ccccccccccccccccc}0&\longrightarrow&G&\longrightarrow&A&\longrightarrow&A_{0}&\longrightarrow&A_{1}&\longrightarrow&\cdots&\longrightarrow&A_{m}&\longrightarrow&0\\ &&&&\downarrow&&\uparrow&&\\ &&&&N&={=}{={=}}&N&&\\ &&&&\downarrow&&\uparrow&&\\ &&&&0&&0&&\\ \end{array}

which shows that ${\rm G_{n}^{d}}$-${\rm id}(G)\leq m+1$. The particular case follows from Corollary 2.6. ∎

Proposition 2.9**.**

Let $R$ be a left (resp; right) $n$-coherent ring.

1. (1)

Let $0\rightarrow K\rightarrow G\rightarrow B\rightarrow 0$ be an exact sequence of right $R$-modules. If $K$ is Gorenstein $(n,d)$-flat and $B$ is $(n,d)$-flat, then $G$ is Gorenstein $(n,d)$-flat. 2. (2)

Let $0\rightarrow A\rightarrow G\rightarrow N\rightarrow 0$ be an exact sequence of right $R$-modules. If $N$ is Gorenstein $(n,d)$-injective and $A$ is $(n,d)$-injective, then $G$ is Gorenstein $(n,d)$-injective.

Proof.

(1) $K$ is Gorenstein $(n,d)$-flat. So by Corollary 2.5, there exists an exact sequence of right $R$-modules $0\rightarrow K\rightarrow B^{{}^{\prime}}\rightarrow L\rightarrow 0$, where $B^{{}^{\prime}}$ is $(n,d)$-flat and $L$ is Gorenstein $(n,d)$-flat. Now, we consider the following diagram:

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

The exactness of the middle horizontal sequence with $B$ and $B^{{}^{\prime}}$ are $(n,d)$-flat, implies that $N$ is $(n,d)$-flat. Hence from the middle vertical sequence and Corollary 2.5, we deduce that $G$ is Gorenstein $(n,d)$-flat.

(2) Since $N$ is Gorenstein $(n,d)$-injective, by Corollary 2.6, there exists an exact sequence of of right $R$-modules $0\rightarrow K\rightarrow A^{{}^{\prime}}\rightarrow N\rightarrow 0$, where $A^{{}^{\prime}}$ is $(n,d)$-injective and $K$ is Gorenstein $(n,d)$-injective. Now, we consider the following diagram:

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

The exactness of the middle horizontal sequence with $A$ and $A^{{}^{\prime}}$ are $(n,d)$-injective, implies that $D$ is $(n,d)$-injective. Hence from the middle vertical sequence and Corollary 2.6, we deduce that $G$ is Gorenstein $(n,d)$-injective.

∎

3. $(n,d)$-Injective Rings

A ring $R$ is right (resp; left) self-$(n,d)$-injective if $R$ is an $(n,d)$-injective right (resp; left) $R$-module. This section is devoted to $(n,d)$-injective rings over which every $R$-module is Gorenstein $(n,d)$-injective.

Proposition 3.1**.**

Let $R$ be a ring. Then, every right $R$-module is Gorenstein $(n,d)$-injective if and only if every projective right $R$-module is $(n,d)$-injective and ${\rm Ext}_{R}^{d+1}(U,N)=0$ for any right $R$-module $N$ and any $n$-presented right $R$-module $U$ with ${\rm pd}_{R}(U)<\infty$.

Proof.

$(\Longrightarrow)$ Let $M$ be a projective right $R$-module. Then by (1), $M$ is Gorenstein $(n,d)$-injective. So, the following $(n,d)$-injective resolution of $M$ exists:

[TABLE]

Since $M$ is projective, $M$ is $(n,d)$-injective as a direct summand of $A_{0}$. If $U$ is an $n$-presented right $R$-module with ${\rm pd}_{R}(U)<\infty$, then by Proposition 2.7 and (1), ${\rm Ext}_{R}^{d+1}(U,N)=0$ for any right $R$-module $N$.

$(\Longleftarrow)$ Choose an injective resolution $0\rightarrow G\rightarrow E^{0}\rightarrow E^{1}\rightarrow\cdots$ and a projective resolution $\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow G\rightarrow 0$ of right $R$-module $G$, where every $F_{i}$ is $(n,d)$-injective by (2). Assembling these resolutions, by Remark 2.4, we get the following $(n,d)$-injective resolotion:

[TABLE]

where $G={\rm ker}(E^{0}\rightarrow E^{1})$, $K^{i}={\rm ker}(E^{i}\rightarrow E^{i+1})$ and $K_{i}={\rm ker}(F_{i}\rightarrow F_{i-1})$ for any $i\geq 1$. Let $U$ be a $n$-presented right $R$-module with ${\rm pd}_{R}(U)<\infty$. Then by (2), ${\rm Ext}_{R}^{d+1}(U,G)={\rm Ext}_{R}^{d+1}(U,F_{i})={\rm Ext}_{R}^{d+1}(U,E^{i})=0$ for any $i\geq 0$. So, ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})$ is exact, and hence $G$ is Gorenstein $(n,d)$-injective. ∎

Theorem 3.2**.**

Let $R$ be a right $n$-coherent ring. Then the following are equivalent:

1. (1)

Every right $R$-module is Gorenstein $(n,d)$-injective; 2. (2)

Every projective right $R$-module is $(n,d)$-injective; 3. (3)

$R$* is right self-$(n,d)$-injective.*

Proof.

$(1)\Longrightarrow(2)$ and $(2)\Longrightarrow(3)$, is hold by Proposition 3.1.

$(3)\Longrightarrow(1)$ Let $G$ be a right $R$-module and $\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow G\rightarrow 0$ be any free resolution of $G$. Then by Proposition 3.1, each $F_{i}$ is $(n,d)$-injective. Hence Corollary 2.6 completes the proof. ∎

Example 3.3**.**

Let $R=k[x^{3},x^{2},x^{2}y,xy^{2},xy,y^{2},y^{3}]$ be a ring. We claim that $R$ is not $(n,0)$-injective. Suppose to the contrary, $R$ is $(n,0)$-injective. $\frac{R}{Rx^{2}}$ is $n$-presented, since $Rx^{2}\cong R$ is $n$-presented. Hence $\frac{R}{Rx^{2}}$ is $n$-presented and ${\rm pd}_{R}(\frac{R}{Rx^{2}})<\infty$. By Proposition 3.1 and Theorem 3.2, $\frac{R}{Rx^{2}}$ is propjective. Threrefore, the exact sequence $0\rightarrow Rx^{2}\rightarrow R\rightarrow\frac{R}{Rx^{2}}\rightarrow 0$ splits. Thus $Rx^{2}$ is a direct summand of $R$ and so, $x^{2}$ is an idempotent, a contradiction.

Proposition 3.4**.**

Let $R$ be a ring. Then:

1. (1)

Every Gorenstein $(n,d)$-injective right $R$-module is Gorenstein $(m,d)$-injective for any $m\geq n$. 2. (2)

Every Gorenstein $(n,d)$-flat right $R$-module is Gorenstein $(m,d)$-flat for any $m\geq n$. 3. (3)

If $R$ is a left $n$-coherent, then every Gorenstein $(m,d)$-flat right $R$-module is Gorenstein $(n,d)$-flat for any $m\geq n$. 4. (4)

If $R$ is a right $n$-coherent, then every Gorenstein $(m,d)$-injective right $R$-module is Gorenstein $(n,d)$-injective for any $m\geq n$.

Proof.

(1) Let $G$ be a Gorenstein $(n,d)$-injective right $R$-module. By Remark 2.4, every $(n,d)$-injective right $R$-module is $(m,d)$-injective for any $m\geq n$. Thus, there is an exact sequence

[TABLE]

of $(m,d)$-injective right $R$-modules, where $G={\rm ker}(I^{0}\rightarrow I^{1})$. By definition and Remark 2.4, ${\rm Ext}_{R}^{d}(U,{\mathbf{I}})$ leaves the sequence exact for any $m$-presented right $R$-module $U$ of finite projective dimension, since every $m$-presented right $R$-module is $n$-presented and $G$ is Gorenstein $(n,d)$-injective. Hence $G$ is Gorenstein $(m,d)$-injective.

(4) Let $G$ be a Gorenstein $(m,d)$-injective right $R$-module. Since $R$ is right $n$-coherent, every $n$-presented right $R$-module is $m$-presented for any $m\geq n$. Hence, any $(m,d)$-injective right $R$-module is $(n,d)$-injective. Thus, there is an exact sequence

[TABLE]

of $(n,d)$-injective right $R$-modules, where $G={\rm ker}(I^{0}\rightarrow I^{1})$. So by definition, ${\rm Ext}_{R}^{d}(U,{\mathbf{I}})$ leaves the sequence exact for any $n$-presented right $R$-module $U$ of finite projective dimension. Hence $G$ is Gorenstein $(n,d)$-injective.

(2), (3) Similar to proof (1), (4).

∎

A ring $R$ is called right $n$-regular, if every $n$-presented right $R$-module is projective. The following example show that Gorenstein $(m,d)$-injectivity does not imply Gorenstein $(n,d)$-injectivity for any $m\geq n$.

Example 3.5**.**

Let $K$ be a field and $E$ be a $k$-vector space with infinite rank. Let $R=K\propto E$. The trivial extension of $K$ by $E$. Then, every right $R$-module is Gorenstein $(2,0)$-injective. We claim that there is right $R$-module $N$ so that $N$ is not Gorenstein $(1,0)$-injective. Suppose to the contrary, every right $R$-module is Gorenstein $(1,0)$-injective. By Proposition 3.1, we see that ${\rm Ext}_{R}^{1}(U,M)=0$ for any right $R$-module $M$ and any $R$-module $1$-presented $U$. So, every $1$-presented right $R$-module is projective and it follows that $R$ is $1$-regular or regular, a cotradiction. Since by [13, Example 3.8], $R$ is not regular.

Theorem 3.6**.**

Let $R$ be right and left $n$-coherent ring. Then the following statements are equivalent:

1. (1)

$R$* is two-sided self-$(n,d)$-injective;* 2. (2)

Every Gorenstein $(n,d)$-flat $R$-module ( right and left) is Gorenstein $(n,d)$-injective; 3. (3)

Every Gorenstein flat $R$-module ( right and left) is Gorenstein $(n,d)$-injective; 4. (4)

Every flat $R$-module ( right and left) is Gorenstein $(n,d)$-injective; 5. (5)

Every Gorenstein projective $R$-module ( right and left) is Gorenstein $(n,d)$-injective; 6. (6)

Every projective $R$-module ( right and left) is Gorenstein $(n,d)$-injective; 7. (7)

Every Gorenstein injective $R$-module ( right and left) is Gorenstein $(n,d)$-flat; 8. (8)

Every injective $R$-module (right and left) is Gorenstein $(n,d)$-flat; 9. (9)

Every Gorenstein $(1,0)$-injective $R$-module ( right and left) is Gorenstein $(n,d)$-flat; 10. (10)

Every Gorenstein $(n,0)$-injective $R$-module ( right and left) is Gorenstein $(n,d)$-flat.

Proof.

$(1)\Longrightarrow(2)$, $(1)\Longrightarrow(3)$, $(1)\Longrightarrow(4)$, $(1)\Longrightarrow(5)$ and $(1)\Longrightarrow(6)$ follow immediately from Theorem 3.2.

$(3)\Longrightarrow(4)$, $(4)\Longrightarrow(6)$ and $(5)\Longrightarrow(6)$ are trivial.

$(3)\Longrightarrow(1)$ Assume that $G$ is a projective right (resp; left) $R$-module. Then $G$ is flat and so, $G$ is Gorenstein $(n,d)$-injective by (3). So, similar to proof $(\Longrightarrow)$ of Proposition 3.1, $G$ is $(n,d)$-injective. Thus, the assertion follows from Theorem 3.2.

$(6)\Longrightarrow(1)$ This is similar to proof $(3)\Longrightarrow(1)$.

$(1)\Longrightarrow(9)$ By [12, Proposition 2.21 and Remark 2.22], every $(1,0)$-injective right (resp; left) $R$-module is $(n,d)$-flat. Suppose that $G$ is Gorenstein $(1,0)$-injective. So, the exact sequence

[TABLE]

of $(n,d)$-flat right (resp; left) $R$-modules exists, where $G={\rm ker}(M^{0}\rightarrow M^{1})$. Let $U$ be a $n$-presented left (resp; right) $R$-module with ${\rm f.d(U)}<\infty$. Then similar to proof Theorem 2.3(1), ${\rm Tor}_{d}^{R}(U,{\mathbf{M}})$ is exact.

$(9)\Longrightarrow(7)$ Let $M$ be a Gorenstein injective right (resp; left) $R$-module. Then, the exact sequence

[TABLE]

of injective right (resp; left) $R$-modules exists, where $M={\rm ker}(M^{0}\rightarrow M^{1})$. By Remark 2.4, every injective $R$-module is $(1,0)$-injective. Also, ${\rm Hom}_{R}(U,{\mathbf{M}})$ leaves the sequence exact for any $n$-presented right (resp; left) $R$-module $U$ with ${\rm p.d(U)}<\infty$, and so $M$ is Gorenstein $(1,0)$-injective.

$(7)\Longrightarrow(8)$ is trivial, since every injective $R$-module is Gorenstein injective.

$(8)\Longrightarrow(1)$ Let $M$ be an injective right $R$-module. Since $M$ is Gorenstein $(n,d)$-flat, we have a long exact sequence:

[TABLE]

where any $M_{i}$ is a $(n,d)$-flat right $R$-module and $M={\rm ker}(M^{0}\rightarrow M^{1})$. Then, the split exact sequence $0\rightarrow M\rightarrow M^{0}\rightarrow L\rightarrow 0$ implies that $M$ is $(n,d)$-flat, and hence by [12, Proposition 2.21 and Remark 2.22], we deduce that $R$ is left self-$(n,d)$-injective. Similarly, $R$ is right self-$(n,d)$-injective.

$(1)\Longrightarrow(10)$ Suppose that $G$ is a Gorenstein $(n,0)$-injective right (resp; left) $R$-module. Then by Propsition 3.4(4), $G$ is Gorenstein $(1,0)$-injective. Also, by [12, Proposition 2.21 and Remark 2.22], every $(1,0)$-injective right (resp; left) $R$-module is $(n,d)$-flat. Thus, the exact sequence

[TABLE]

of $(n,d)$-flat right (resp; left) $R$-modules exists, where $G={\rm ker}(N^{0}\rightarrow N^{1})$. Then similar to proof Theorem 2.3(1), (10) follows.

$(10)\Longrightarrow(9)$ By Proposition 3.4(1), every Gorenstein $(1,0)$-injective right (resp; left) $R$-module is Gorenstein $(n,0)$-injective. So by (10), (9) is hold. ∎

Example 3.7**.**

Let $R$ be a commutative $n$-regular ring. Then every $R$-module is Gorenstein $(n,d)$-injective and Gorenstein $(n,d)$-flat, since by [13, Theorem 3.9], $R$ is self-$(n,0)$-injective and $n$-coherent. But, there is an $R_{\infty}$-module $N$ such that $N$ is not Gorenstein $(n,0)$-injective and Gorenstein $(n,0)$-flat. Since if every $R_{\infty}$-module is Gorenstein $(n,0)$-injective and Gorenstein $(n,0)$-flat, then by Proposition 3.1, every $n$-presented right $R_{\infty}$-module with finite dimention projective is projective and consequenclay $R_{\infty}$ is $n$-regular and $n$-coherent, a contradiction. Because $R=K[x^{3},x^{3}y,xy^{3},y^{3}]$ is $n$-regular and if $K$ is a field of characteristic $p>5$, then by [10, Example 6], $R_{\infty}$ is not $n$-coherent.

4. Gorenstein $(n,d)$-flat and

Gorenstein $(n,d)$-injective modules on projective algebras

Lemma 4.1**.**

Let $f:R\rightarrow S$ be a surjective ring homomorphism, $S_{R}$ a projective right $R$-module and ${}_{R}S$ a projective left $R$-module. Then:

1. (1)

If $U$ is an $n$-presented right (resp; left) $R$-module, then $U\otimes_{R}S$ is an $n$-presented right $S$-module. 2. (2)

If $N$ is an $(n,d)$-flat right $R$-module, then $N\otimes_{R}S$ is an $(n,d)$-flat right $S$-module . 3. (3)

If $N$ is an $(n,d)$-injective right $R$-module, then ${\rm Hom}_{R}(S,N)$ is an $(n,d)$-injective right $S$-module .

Proof.

(1) Suppose $U$ is $n$-presented right $R$-module. Then, the exact sequence

[TABLE]

of right $R$-modules exists, where $K_{n-1}$ is finitely generated and each $F_{i}$ is finitely generated projective for $0\leq i\leq n-1$. ${}_{R}S$ is projective, hence the exact sequence

[TABLE]

of right $S$-modules exists, where $K_{n-1}\otimes_{R}S$ is finitely generated and each $F_{i}\otimes_{R}S$ is finitely generated projective for $0\leq i\leq n-1$, since by [8, Theorem 2.75], we have ${\rm Hom}_{R}(F_{i},-)\cong{\rm Hom}_{S}(F_{i}\otimes_{R}S,-)$. So, $U\otimes_{R}S$ is an $n$-presented right $S$-module.

(2) Let $U$ be an $n$-presented left $S$-module. Then by [5, Lemma 3.10 and Lemma 3.11], $U$ is $n$-presented left $R$-module. By [8, Corollary 10.61], ${\rm Tor}_{d+1}^{R}(N,U)\cong{\rm Tor}_{d+1}^{S}(N\otimes_{R}S,U)$, and so $N\otimes_{R}S$ is an $(n,d)$-flat right $S$-module, since $N$ is $(n,d)$-flat right $R$-module.

(3) Let $U$ be an $n$-presented right $S$-module. Then $U$ is an $n$-presented right $R$-module. We claim that ${\rm Ext}_{S}^{k}(U,{\rm Hom}_{R}(S,N))\cong{\rm Ext}_{R}^{k}(U,N)$. We use the induction on $k$. Let $k=0$. Then by [8, Theorem 2.75],${\rm Hom}_{S}(U,{\rm Hom}_{R}(S,N))\cong{\rm Hom}_{R}(U,N)$. Let $k>0$, then by induction hypothesis and the exact sequence $0\rightarrow K\rightarrow F_{0}\rightarrow U\rightarrow 0$, where $F_{0}$ is a finitely generated free, we deduce that ${\rm Ext}_{S}^{k}(U,{\rm Hom}_{R}(S,N))\cong{\rm Ext}_{R}^{k}(U,N)$, and so, (3) is follows.

∎

Proposition 4.2**.**

Let $f:R\rightarrow S$ be a surjective ring homomorphism, $S_{R}$ a projective right $R$-module and ${}_{R}S$ a projective left $R$-module. Then:

1. (1)

If $U$ is a left $S$-module with ${\rm fd}_{S}(U)<\infty$, then ${\rm fd}_{R}(U)<\infty$. 2. (2)

If $U$ is a left $R$-module with ${\rm fd}_{R}(U)<\infty$, then ${\rm fd}_{S}(U\otimes_{R}S)<\infty$. 3. (3)

If $U$ is a right $S$-module with ${\rm pd}_{S}(U)<\infty$, then ${\rm pd}_{R}(U)<\infty$. 4. (4)

If $U$ is a right $R$-module with ${\rm pd}_{R}(U)<\infty$, then ${\rm pd}_{S}(U\otimes_{R}S)<\infty$.

Proof.

(1) By [5, Lemma 3.10] and [8, Corollary 10.61],

[TABLE]

. So, ${\rm fd}_{R}(U)<\infty$.

(2) By [8, Corollary 10.61], ${\rm Tor}_{n}^{R}(U,-)\cong{\rm Tor}_{n}^{S}(U\otimes_{R}S,-)$. So, (2) holds.

(3) Similar to proof (3) of lemma 4.1, ${\rm Ext}_{S}^{n}(U,{\rm Hom}_{R}(S,-))\cong{\rm Ext}_{R}^{n}(U,-).$ So, (2) holds.

(4) By [8, Theorem 2.76], ${\rm Ext}_{R}^{n}(U.-)\cong{\rm Ext}_{S}^{n}(U\otimes_{R}S.-)$. So, ${\rm pd}_{S}(U\otimes_{R}S)<\infty$.

∎

Theorem 4.3**.**

Let $f:R\rightarrow S$ be a surjective ring homomorphism, $S_{R}$ a projective right $R$-module, ${}_{R}S$ a projective left $R$-module and $G$ a left $S$-module. Then, the following statements are equivalent:

1. (1)

$G_{R}$* is Gorenstein $(n,d)$-flat;* 2. (2)

$G\otimes_{R}S$* is Gorenstein $(n,d)$-flat;* 3. (3)

$G_{S}$* is Gorenstein $(n,d)$-flat.*

Proof.

$(1)\Longrightarrow(2)$ Suppose that $N$ is an $(n,d)$-flat right $R$-module. Then by Lemma 4.1, $N\otimes_{R}S$ is an $(n,d)$-flat right $S$-module. But, ${}_{R}S_{S}$ is a flat. So if $G$ is an Gorenstein $(n,d)$-flat right $R$-module, then the exact sequence

[TABLE]

of $(n,d)$-flat right $R$-modules, where $G={\rm ker}(F^{0}\rightarrow F^{1})$, induces the following exact sequence of $(n,d)$-flat right $S$-modules :

[TABLE]

where $G\otimes_{R}S={\rm ker}(F^{0}\otimes_{R}S\rightarrow F^{1}\otimes_{R}S)$. Let $U$ be an $n$-presented left $S$-module with ${\rm fd}_{S}(U)<\infty$. We show that ${\rm Tor}_{d}^{S}(U,{\mathbf{F}}\otimes_{R}S)$ is exact. By Proposition 4.2, ${\rm fd}_{R}(U)<\infty$. Also, $U$ is an $n$-presented left $R$-module. Thus by hypothesis, ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})$ is exact. Therefore ${\rm Tor}_{d}^{S}(U,{\mathbf{F}}\otimes_{R}S)$ is exact, since by [8, Corollary 10.61], ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})\cong{\rm Tor}_{d}^{R}(U\otimes_{R}S,{\mathbf{F}})\cong{\rm Tor}_{d}^{S}(U,{\mathbf{F}}\otimes_{R}S)$. Hence, $G\otimes_{R}S$ is Gorenstein $(n,d)$-flat.

$(2)\Longrightarrow(3)$ By [5, Lemma 3.10], $G\otimes_{R}S\cong G_{S}$, and hence $G_{S}$ is Gorenstein $(n,d)$-flat.

$(3)\Longrightarrow(1)$ Suppose that $N$ is an $(n,d)$-flat right $S$-module. Then by [5, Lemma 3.12], $N$ is an $(n,d)$-flat right $R$-module. So, if $G$ be a Gorenstein $(n,d)$-flat right $S$-module, then the exact sequence

[TABLE]

of $(n,d)$-flat right $R$-modules exists, where $G={\rm ker}(F^{0}\rightarrow F^{1})$. Let $U$ be an $n$-presented left $R$-module with ${\rm fd}_{R}(U)<\infty$. It is sufficient to prove that ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})$ is exact. $U\otimes_{R}S$ is an $n$-presented left $S$-module by Lemma 4.1, and by Proposition 4.2, ${\rm fd}_{S}(U\otimes_{R}S)<\infty$. So by hypothesis, ${\rm Tor}_{d}^{S}(U\otimes_{R}S,{\mathbf{F}})$ is exact. On the other hand, by [8, Corollary 10.61], we have ${\rm Tor}_{d}^{S}(U\otimes_{R}S,{\mathbf{F}})\cong{\rm Tor}_{d}^{R}(U,{\mathbf{F}}\otimes_{S}S)\cong{\rm Tor}_{d}^{R}(U,{\mathbf{F}})$. Therefore, ${\rm Tor}_{d}^{R}(U,{\mathbf{F}})$ is exact, and hence $G_{R}$ is Gorenstein $(n,d)$-flat.

∎

Theorem 4.4**.**

Let $f:R\rightarrow S$ be a surjective ring homomorphism, $S_{R}$ a projective right $R$-module, ${}_{R}S$ a projective left $R$-module and $G$ a left $S$-module. Then, the following statements are equivalent:

1. (1)

$G_{R}$* is Gorenstein $(n,d)$-injective;* 2. (2)

${\rm Hom}_{R}(S,G)$* is Gorenstein $(n,d)$-injective;* 3. (3)

$G_{S}$* is Gorenstein $(n,d)$-injective.*

Proof.

$(1)\Longrightarrow(2)$ Suppose that $N$ is an $(n,d)$-injective right $R$-module. Then by Lemma 4.1, ${\rm Hom}_{S}(S,N)$ is an $(n,d)$-injective right $S$-module. But, $S$ is a projective $R$-module. So if $G$ is Gorenstein $(n,d)$-injective right $R$-module , then the exact sequence

[TABLE]

of $(n,d)$-injective right $R$-modules, where $G={\rm ker}(A^{0}\rightarrow A^{1})$, induces the following exact sequence of $(n,d)$-injective right $S$-modules :

[TABLE]

where ${\rm Hom}_{R}(S,G)={\rm ker}({\rm Hom}_{R}(S,A^{0})\rightarrow{\rm Hom}_{R}(S,A^{1}))$. Let $U$ be an $n$-presented right $S$-module with ${\rm pd}_{S}(U)<\infty$. Then by [5, Lemmas 3.10 and 3.11], $U\cong U\otimes_{R}S$ is an $n$-presented right $R$-module, and by Proposition 4.2, ${\rm pd}_{R}(U)<\infty$. Also, similar to proof (3) of Lemma 4.1, we have that

[TABLE]

By hypothesis, ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})$ is exact. Therefore ${\rm Ext}_{S}^{d}(U,{\rm Hom}_{R}(S,{\mathbf{A}}))$ is exact, and hence ${\rm Hom}_{R}(S,G)$ is Gorenstein $(n,d)$-injective.

$(2)\Longrightarrow(3)$ By [8, Proposition 8.33], ${\rm Hom}_{R}(S,G)={\rm Hom}_{S}(S,G)\cong G$, since $f$ is surjective. So, we deduce that $G_{S}$ is Gorenstein $(n,d)$-injective.

$(3)\Longrightarrow(1)$ Suppose that $N$ is an $(n,d)$-injective right $S$-module. Then by [5, Lemma 3.12], $N$ is an $(n,d)$-injective right $R$-module. So, if $G$ be a Gorenstein $(n,d)$-injective right $S$-module, then the exact sequence

[TABLE]

of $(n,d)$-injective right $R$-modules exists, where $G={\rm ker}(A^{0}\rightarrow A^{1})$. Let $U$ be an $n$-presented right $R$-module with ${\rm pd}_{R}(U)<\infty$. It is sufficient to prove that ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})$ is exact. $U\otimes_{R}S$ is an $n$-presented right $S$-module by Lemma 4.1. Also by Proposition 4.2, ${\rm pd}_{S}(U\otimes_{R}S)<\infty$. So by hypothesis, ${\rm Ext}_{S}^{d}(U\otimes_{R}S,{\mathbf{A}})$ is exact. On the other hand, similar to proof (4) of Proposition 4.2, we have ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})\cong{\rm Ext}_{S}^{d}(U\otimes_{R}S,{\mathbf{A}})$. Therefore, ${\rm Ext}_{R}^{d}(U,{\mathbf{A}})$ is exact, and hence $G_{R}$ is a Gorenstein $(n,d)$-injective.

∎

we have the following corollary.

Corollary 4.5**.**

Let $f:R\rightarrow S$ be a surjective ring homomorphism, $S$ a projective $R$-module and $M$ a $S$-module. Then :

1. (1)

$G_{d}^{n}.fd(M_{R})=G_{d}^{n}.fd(M_{S})=G_{d}^{n}.fd(S\otimes_{R}M).$** 2. (2)

$G_{d}^{n}.id(M_{R})=G_{d}^{n}.id(M_{S})=G_{d}^{n}.id({\rm Hom}_{R}(S,M)).$**

In the following propositon, Gorenstein $(n,d)$-injectivity of modules over almost excellent extensions is studied.

Proposition 4.6**.**

Let $S\geq R$ be an almost excellent extension. Then

1. (1)

$R$* is right Gorenstein $(n,d)$-injective if and only if right $S$ is Gorenstein $(n,d)$-injective.* 2. (2)

$R_{\infty}$* is right Gorenstein $(n,d)$-injective if and only if $S_{\infty}$ is right Gorenstein $(n,d)$-injective.*

Proof.

(1) ($\Longrightarrow$): Let $R$ be a right Gorenstein $(n,d)$-injective, then the exact sequence

[TABLE]

of $(n,d)$-injective right $R$-modules exists, where $R={\rm ker}(A^{0}\rightarrow A^{1})$. Since the exact sequence $0\rightarrow K\rightarrow A_{0}\rightarrow R\rightarrow 0$ is split, $R$ is right $(n,d)$-injective. Thus by [14, Proposition 5.1], $S$ is right $(n,d)$-injective and hence by Remark 2.4, $S$ is right Gorenstein $(n,d)$-injective.

($\Longleftarrow$): Similar to proof ($\Longrightarrow$).

(2) If $S\geq R$ is an almost excellent extension, then $S_{\infty}\geq R_{\infty}$ is an almost excellent extension. Hence by (1), (2) is hold.

∎

From Theorem 3.6, Proposition 4.6 and [14, Theorem 5.3] we immediately have the following corollary.

Corollary 4.7**.**

Let $S\geq R$ be an almost excellent extension and $R$ be $n$-coherent. Then the following statements are equivalent:

1. (1)

$R_{R}$* is Gorenstein $(n,d)$-injective;* 2. (2)

Every Gorenstein $(n,d)$-flat right $R$-module is Gorenstein $(n,d)$-injective right $R$-module; 3. (3)

Every Gorenstein $(n,d)$-flat right $S$-module is Gorenstein $(n,d)$-injective right $S$-module.