Gorenstein $\pi[T]$-projectivity with respect to a tilting module

TL;DR

This paper introduces Gorenstein $ ext{pi}[T]$-projective modules relative to a tilting module, explores their properties, and characterizes rings where all modules are Gorenstein $ ext{pi}[T]$-projective, especially on $T$-cocoherent rings.

Contribution

It defines Gorenstein $ ext{pi}[T]$-projectivity relative to a tilting module and provides characterizations of rings where all modules exhibit this property.

Findings

Gorenstein $ ext{pi}[T]$-projective modules are introduced and studied.

Characterizations of rings with all modules Gorenstein $ ext{pi}[T]$-projective are provided.

On $T$-cocoherent rings, Gorenstein $ ext{pi}[T]$-projectivity of all modules is equivalent to $ ext{pi}[T]$-projectivity of $ ext{sigma}[T]$-injective modules.

Abstract

Let $T$ be a tilting module. In this paper, Gorenstein $\pi[T]$-projective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein $\pi[T]$-projective are given. For instance, on the $T$-cocoherent rings, it is proved that the Gorenstein $\pi[T]$-projectivity of all $R$-modules is equivalent to the $\pi[T]$-projectivity of $\sigma[T]$-injective as a module.

Equations

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

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

$$0⟶M⟶C_0⟶C_1⟶\cdots ⟶C_{n-1}⟶C_n⟶0$$

$${\mathcal{E}}_T^n(A,M):=\frac{\ker {\delta }_{\ast }^n}{\mathrm{im}{\delta }_{\ast }^{n-1}},$$

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

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

$$\pi [T].pd(M)=\inf \{n:{\mathcal{E}}_T^{n+1}(M,N)=0\mathrm{for}\mathrm{every}N\in \pi [T]\}.$$

$$\mathbf{B}=\cdots ⟶B_1⟶B_0⟶B^0⟶B^1⟶\cdots$$

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

$$\begin{array}[]{ccccccccccccccccc}&0&0&0&\\ \hskip 28.45274pt\downarrow&\hskip 19.91684pt\downarrow&\downarrow&\\ o\longrightarrow A&\stackrel{{\scriptstyle\displaystyle f}}{{\longrightarrow}}B&\stackrel{{\scriptstyle\displaystyle g}}{{\longrightarrow}}C\longrightarrow 0\\ \hskip 42.67912pt\downarrow h_{0}^{{}^{\prime}}&\ \ \ \ \ \ \ \ \ \ \downarrow h_{0}&\ \ \downarrow h_{0}^{{}^{\prime\prime}}&\\ 0\longrightarrow T^{{}^{\prime}}_{0}&{\stackrel{{\scriptstyle\displaystyle\i_{0}}}{{\longrightarrow}}}\ \ \ \ \ T^{{}^{\prime}}_{0}\oplus T^{{}^{\prime\prime}}_{0}&{\stackrel{{\scriptstyle\displaystyle\pi_{0}}}{{\longrightarrow}}}\ \ T^{{}^{\prime\prime}}_{0}\longrightarrow 0\\ \hskip 28.45274pt\downarrow&\ \ \ \ \ \downarrow&{\downarrow}&\\ \end{array}$$

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

$$\mathbf{B}=\cdots ⟶B_1⟶B_0⟶B^0⟶B^1⟶\cdots$$

$$\cdots ⟶T_1⟶T_0⟶G⟶0$$

$$\cdots ⟶T_1⟶T_0⟶B^0⟶B^1⟶\cdots$$

$$0⟶G⟶M⟶I⟶0(1)$$

$$0\to I\to C^0\to C^1\to \cdots (2)$$

$$0\to G\to M\to C^0\to C^1\to \cdots ,$$

$$0\to G\to B^0\to \cdots \to B^{m-1}\to N\to 0.$$

$$0={\mathcal{E}}_T^r(G_j,U)\to {\mathcal{E}}_T^r(K_j,U)\to {\mathcal{E}}_T^{r+1}(K_{j-1},U)\to {\mathcal{E}}_T^{r+1}(G_j,U)=0$$

$${\mathcal{E}}_T^{n+i}(G,U)\cong {\mathcal{E}}_T^{n+i-1}(K_0,U)\cong \cdots \cong {\mathcal{E}}_T^i(N,U),$$

$$0\to B_n\to B_{n-1}\to \cdots \to B_0\to N\to 0.$$

$$0\to M\to B^0\to B^1\to \cdots .$$

$$\mathbf{B}=\cdots \to T_1\to T_0\to T^0\to T^1\to \cdots ,$$

$$0⟶G⟶T^0⟶T^1⟶\cdots$$

$$gl.\pi [T].pd(R)=\sup \{\pi [T].pd(M)|M\mathrm{is}\mathrm{a}\mathrm{module}\}.$$

$$0⟶G⟶B^0⟶B^1⟶\cdots ⟶B^{m-2}⟶B^{m-1}⟶L⟶0$$

Full text

GORENSTEIN $\pi[T]$-PROJECTIVITY WITH RESPECT TO A TILTING MODULE

††thanks: Key Words: Coherent, Dimension, Gorenstein, Tilting ††thanks: 2010 Mathematics Subject Classification: 13D07; 16D40; 18G25;

M. Amini

**

***Department of Mathematics, Payame Noor University, Tehran, Iran.

  $\mathsf{mostafa.amini@pnu.ac.ir}$

Abstract. Let $T$ be a tilting module. In this paper, Gorenstein $\pi[T]$-projective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein $\pi[T]$-projective are given. For instance, on the $T$-cocoherent rings, it is proved that the Gorenstein $\pi[T]$-projectivity of all $R$-modules is equivalent to the $\pi[T]$-projectivity of $\sigma[T]$-injective as a module.

**1. Basic Definitions and Notations

**Throughout this paper, $R$ is an associative ring with non-zero identity, all modules are unitary left $R$-modules. First we recall some known notions and facts needed in the sequel. Let $R$ be a ring and $T$ an R-module. Then

1. (1)

We denote by ${Prod}T$ (resp. ${F.Prod}T$), the class of modules isomorphic to direct summands of direct product of copies (resp. finitely many copies) of $T$. 2. (2)

We denote by ${Add}T$ (resp. ${F.Add}T$), the class of modules isomorphic to direct summands of direct sum of copies (resp. finitely many copies) of $T$. 3. (3)

Following [3], a module $T$ is called tilting ($1$-tilting) if it satisfies the following conditions:

(a) pd$(T)\leq 1$, where $pd(T)$ denotes the projective dimension of $T$.

(b) Ext${}^{i}(T,T^{(\lambda)})=0$, for each $i>0$ and for every cardinal $\lambda$.

(c) There exists the exact sequence $0\rightarrow R\rightarrow T_{0}\rightarrow T_{1}\rightarrow 0$, where $T_{0},T_{1}\in{\rm Add}T$. 4. (4)

By $Copres^{n}T$ (resp. $F.Copres^{n}T$) and $Copres^{\infty}T$ (resp. $F.Copres^{\infty}T$), we denote the set of all modules $M$ such that there exists exact sequences

[TABLE]

and

[TABLE]

respectively, where $T_{i}\in{\rm Prod}T$ (resp. $T_{i}\in{\rm F.Prod}T$), for every $i\geq 0$. 5. (5)

A module $M$ is said to be cogenerated, by $T$, denoted by $M\in{Cogen}T$, (resp. generated, denoted $M\in{Gen}T$) by $T$ if there exists an exact sequence $0\rightarrow M\rightarrow T^{n}$ (resp. $T^{(n)}\rightarrow M\rightarrow 0$ ) , for some positive integer $n$. 6. (6)

Let $\mathcal{C}$ be a class of modules and $M$ be a module. A right (resp. left) $\mathcal{C}$-resolution of $M$ is a long exact sequence $0\rightarrow M\rightarrow C_{0}\rightarrow C_{1}\rightarrow\cdots$ (resp. $\cdots\rightarrow C_{1}\rightarrow C_{0}\rightarrow M\rightarrow 0$), where $C_{i}\in\mathcal{C}$, for all $i\geq 0$. It is said that a module $M$ has right $\mathcal{C}$-dimension $n$ (briefly, ${\mathcal{C}}.dim(M)=n$) if $n$ is the least non-negative integer such that there exists a long exact sequence

[TABLE]

with $C_{i}\in\mathcal{C}$, for each $i\geq 0$. In particular, the ${\rm Prod}T$-dimension of $M$ is called $T$-injetive dimension of $M$ and is denoted by $T.i.dim(M)$. Note that for any tilting module $M$, if $M\in{\rm Cogen}T$, then [6, Proposition 2.1] implies that ${\rm Cogen}T={\rm Copres}^{\infty}T$. This shows that any module cogenerated by $T$ has an ${\rm Prod}T$-resolution. The ${\rm Prod}T$-resolutions and the relative homological dimension were studied by Nikmehr and Shaveisi in [6]. 7. (7)

For any homomorphism $f$, we denote by ${ker}f$ and ${im}f$, the kernel and image of $f$, respectively. Let $A$ and $M\in{\rm Cogen}T$ be two modules. We define the functor

[TABLE]

where

[TABLE]

${\rm Prod}T$-resolution of $M$ and $\delta_{*}^{n}={\rm Hom}(id_{B},\delta_{n})$, for every $i\geq 0$. see [6, 9] for more details. 8. (8)

Let $M\in{\rm Cogen}T$ and $N$ be two modules. A similar proof to that of [7, Lemma 2.11] shows that ${\mathcal{E}}^{0}_{T}(N,M)\cong{\rm Hom}(N,M)$. Moreover, ${\mathcal{E}}^{1}_{T}(-,M)=0$ implies that $M\in{\rm Prod}T$, and if $M\in{\rm Gen}T$, then ${\mathcal{E}}^{1}_{T}(M,-)=0$ implies that $M\in{\rm Add}T$. It is clear that ${\rm T.i.dim}(M)=n$ if and only if $n$ is the least non-negative integer such that ${\mathcal{E}}^{n+1}_{T}(A,M)=0$, for any module $A$, see [6, Remark 2.2] for more details. So, ${\rm T.i.dim}(M)=n$ if and only if ${\mathcal{E}}_{T}^{n+i}(A,M)=0$ for every module $A$ and every $i\geq 1$. A module with zero $T$-injective dimension (resp. $T$-projective dimension) is called $T$*-injective *(resp. $T$-projective). A similar proof to that of [7, Proposition 2.3] shows that the definition of ${\mathcal{E}}_{T}^{n}(C,M)$ is independent from the choice of ${\rm Prod}T$-resolutions. For unexplained concepts and notations, we refer the reader to [1, 6, 8]. 9. (9)

For module $T$, we denote by $\pi[T]$, the full subcategory of modules whose objects are of the form $\frac{B}{A}\leq\frac{T^{I}}{A}$, for some cardinal $I$ and some modules $A\leq B\leq T^{I}$. Also, the full subcategory $\sigma[T]$ of modules subgenerated by a given module $T$ (see [10]). 10. (10)

$G$ is called Gorenstein $\sigma[T]$-injective if there exists an exact sequence of $\sigma[T]$-injective modules

[TABLE]

with $G={\rm ker}(A^{0}\rightarrow A^{1})$ such that ${\rm Hom}(U,-)$ leaves this sequence exact whenever $U\in{\rm Pres}^{1}T$ with ${\rm T.p.dim}(U)<\infty$ (see [9]). 11. (11)

$M$ is said to be finitely cogenerated [1] if for every family $\{V_{k}\}_{J}$ of submodules of $M$ with ${\bigcap}_{J}V_{k}=0$, there is a finite subset $I\subset J$ with ${\bigcap}_{I}V_{k}=0$. 12. (12)

$M$ is said to be finitely copresented if there is an exact sequence of $R$-modules $0\rightarrow M\rightarrow E^{0}\rightarrow E^{1}$, where each $E^{i}$ is a finitely cogenerated injective module, see [2, 11, 12].

Let $T$ be a tilting module. In this paper, we introduce the $\pi[T]$-projective modules, the $\pi[T]$-projective dimension and Gorenstein $\pi[T]$-projective modules.

Let $M\in{\rm Gen}T$. Then, $M$ is called $\pi[T]$-projective if the functor ${\mathcal{E}}_{T}^{1}(M,-)$ vanishes on $\pi[T]$. Also, the $\pi[T]$-projective dimension of $M$ is defined to be

[TABLE]

We define a module $G$ to be Gorenstein $\pi[T]$-projective ( $GT$-projective for short), if there exists an exact sequence of $\pi[T]$-projective modules

[TABLE]

with $G={\rm ker}(B^{0}\rightarrow B^{1})$ such that ${\rm Hom}(-,U)$ leaves this sequence exact whenever $U\in{\rm F.Copres}^{1}T$ with ${\rm T.i.dim}(U)<\infty$. In this paper, the $GT$-projective dimension of a module $G$ is denoted by $GT$-$pd(G)$.

In Section 2, we study some basic properties of the Gorenstein $\pi[T]$-projective modules. Recall that a ring $R$ is said to be cocoherent if every finitely cogenerated module is finitely copresented. So, $R$ is a cocoherent ring if and only if ${\rm Copres}^{0}R={\rm Copres}^{1}R$. For more information about the cocoherent rings, we refer the reader to [5]. As a cogeneralization of this concept, we call a ring $R$ to be $T$-cocoherent if ${\rm F.Copres}^{0}T={\rm F.Copres}^{1}T$.

Section 3 is devoted to some characterizations of $T$-cocoherent rings over which all modules are Gorenstein $\pi[T]$-projective. For instance, it is proved that every module is Gorenstein $\pi[T]$-projective if and if every $T$-injective module is $\pi[T]$-projective if and if every $\sigma[T]$-injective module is Gorenstein $\pi[T]$-projective. Finally, we give a sufficient condition under which every Gorenstein $\pi[T]$-projective module is $\pi[T]$-projective.

1 Gorenstein $\pi[T]$-Projectivity

We start with the following definition.

Definition 1

.* Let $T$ be a tilting module. Then*

1. (1)

$M$* is called $\pi[T]$-projective if ${\mathcal{E}}_{T}^{1}(M,N)=0$, for every $N\in\pi[T]$.* 2. (2)

Let $G\in{\rm Gen}T$. Then, $G$ is called Gorenstein $\pi[T]$-projective if there exists an exact sequence of $\pi[T]$-projective modules

[TABLE]

with $G={\rm ker}(B^{0}\rightarrow B^{1})$ such that ${\rm Hom}(-,U)$ leaves this sequence exact whenever $U\in{\rm F.Copres}^{1}T$ with ${\rm T.i.dim}(U)<\infty$

Remark 2

. Let $T$ be a tilting module. Then

1. (1)

${\mathcal{E}}_{T}^{1}(N,M)=0$ for any $\pi[T]$-projective module $N$ and any $M\in{\rm Copres}^{0}T$. 2. (2)

If $A\in{\rm Add}T$, then $A$ is $\pi[T]$-projective.

Lemma 3

.* Let $0\rightarrow A\stackrel{{\scriptstyle\displaystyle f}}{{\rightarrow}}B\stackrel{{\scriptstyle\displaystyle g}}{{\rightarrow}}C\rightarrow 0$ be an exact sequence. Then*

1. (1)

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

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

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

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

**Proof. **

(* $\Box$1) If $A$ is $T$-injective and $A,B,C\in{\rm Cogen}T$, then we deduce that the sequence*

[TABLE]

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

(2) We prove the assertion by induction on $n$. If $n=0$, then the commutative diagram with exact rows

[TABLE]

exists, where $T^{\prime}_{0},T^{\prime\prime}_{0}\in{\rm F.Prod}T$, $i_{0}$ is the inclusion map, $\pi_{0}$ is a canonical epimorphism and $h_{0}=i_{0}h_{0}^{\prime}$ is endomorphism, by Five Lemma. Let $K_{1}^{\prime}={\rm coker}(h_{0}^{\prime})$, $K_{1}={\rm coker}(h_{0})$ and $K_{1}^{\prime\prime}={\rm coker}(h_{0}^{\prime\prime})$. It is clear that $(T^{{}^{\prime}}_{0}\oplus T^{{}^{\prime\prime}})\in{\rm F.Prod}T$ and $K_{1}^{\prime},K_{1}^{\prime\prime}\in{\rm F.Copres}^{n-1}T$; so, the induction implies that $K_{1}\in{\rm F.Copres}^{n-1}T$. Hence $B\in{\rm F.Copres}^{n}T$.

(3) Let $B\in{\rm F.Pres}^{n+1}T$ and $C\in{\rm F.Pres}^{n}T$, 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}**

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

(4) Let $A\in{\rm F.Pres}^{n+1}T$ and $B\in{\rm F.Pres}^{n}T$, 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}**

where $T_{0},T_{0}^{{}^{\prime}}\in{\rm F.Prod}T$ and $L\in{\rm F.Copres}^{n-1}T$. 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 $N\in{\rm Cogen}T$, we have

[TABLE]

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

In the following theorem, we show that in the case of $T$-cocoherent rings, the existence of $\pi[T]$-projective complex of a module is sufficient to be Gorenstein $\pi[T]$-projective.

Theorem 4

.* Let $R$ be a $T$-cocoherent ring and $G\in{\rm Gen}T$ be a module. Then $G$ is Gorenstein $\pi[T]$-projective if and only if there is an exact sequence*

[TABLE]

of $\pi[T]$-projective modules such that $G=\ker(B^{0}\rightarrow B^{1})$.

**Proof. **

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

($\Longleftarrow$) : By definition, it suffices to show that ${\rm Hom}({\mathbf{B}},U)$ is exact for every module $U\in{\rm F.Copres}^{1}T$ with ${\rm T.i.dim}(U)=m<\infty$. To prove this, we use the induction on $m$. The case $m=0$ is clear. Assume that $m\geq 1$. Since $U\in{\rm F.Copres}^{1}T$, there exists an exact sequence $0\rightarrow U\rightarrow T_{0}\rightarrow I\rightarrow 0$ with $T_{0}\in{\rm F.Prod}T\subseteq{\rm F.Copres}^{0}T$. Now, from the $T$-cocoherence of $R$ and Lemma 3, we deduce that $I,T_{0}\in{\rm F.Copres}^{1}T$. Also, ${\rm T.i.dim}(I)\leq m-1$ and ${\rm T.i.dim}(T_{0})=0$. Thus by Remark 2, the following short exact sequence of complexes exists:

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

By induction, ${\rm Hom}({\mathbf{B}},T_{0})$ and ${\rm Hom}({\mathbf{B}},I)$ are exact, hence ${\rm Hom}({\mathbf{B}},U)$ is exact by [8, Theorem 6.10]. Therefore, $G$ is Gorenstein $\pi[T]$-projective.* $\Box$*

It is worthy to mention that the notion of $T$-injectivity ($T$-projectivity) is different from the notion of an $M$-injective ($M$-projective) module in [1].

Corollary 5

.* Let $R$ be a $T$-cocoherent ring and $G\in{\rm Gen}T$ be a module. Then the following assertions are equivalent:*

1. (1)

$G$* is Gorenstein $\pi[T]$-projective;* 2. (2)

There is an exact sequence $0\rightarrow G\rightarrow B^{0}\rightarrow B^{1}\rightarrow\cdots$ of modules, where every $B^{i}$ is $\pi[T]$-projective; 3. (3)

There is a short exact sequence $0\rightarrow G\rightarrow M\rightarrow I\rightarrow 0$ of modules, where $M$ is $\pi[T]$-projective and $I$ is Gorenstein $\pi[T]$-projective.

**Proof. **

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

$(2)\Longrightarrow(1)$ For module $G\in{\rm Gen}T$, [6, Proposition 2.1] implies that ${\rm Gen}T={\rm Pres}^{\infty}T$. So, there is an exact sequence

[TABLE]

where any $T_{i}$, $\pi[T]$-projective by Remark 2. Thus, the exact sequence

[TABLE]

of $\pi[T]$-projective modules exists, where $G={\rm ker}(B^{0}\rightarrow B^{1})$. Therefore, $G$ is Gorenstein $\pi[T]$-projective, by Theorem 4.

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

[TABLE]

exists, where $M$ is $\pi[T]$-projective and $I$ is Gorenstein $\pi[T]$-projective. Since $I$ is Gorenstein $\pi[T]$-projective, there is an exact sequence

[TABLE]

where every $C^{i}$ is $\pi[T]$-projective. Assembling the sequences $(1)$ and $(2)$, we get the exact sequence

[TABLE]

where $M$ and every $C^{i}$ is $\pi[T]$-projective, as desired.* $\Box$*

Proposition 6

.* For any module $G\in{\rm Gen}T$, the following statements hold.*

1. (1)

If $G$ is Gorenstein $\pi[T]$-projective, then ${\mathcal{E}}_{T}^{i}(G,U)=0$ for all $i>0$ and every module $U\in{\rm F.Copres}^{1}T$ with ${\rm T.i.dim}(U)<\infty$. 2. (2)

If $0\rightarrow N\rightarrow G_{n-1}\rightarrow\cdots\rightarrow G_{0}\rightarrow G\rightarrow 0$ is an exact sequence of modules where every $G_{i}$ is a Gorenstein $\pi[T]$-projective and $G_{i}\in{\rm Gen}T$, then ${\mathcal{E}}_{T}^{i}(N,U)={\mathcal{E}}_{T}^{n+i}(G,U)$ for any $i>0$ and any module $U\in{\rm F.Copres}^{1}T$ with ${\rm T.i.dim}(U)<\infty$.

**Proof. **

(1) Let $G$ be a Gorenstein $\pi[T]$-projective module, and ${\rm T.i.dim}(U)=m<\infty$. Then by hypothesis, the following $\pi[T]$-projective resolution of $G$ exists:

[TABLE]

By Remark 2, ${\mathcal{E}}_{T}^{i}(B_{j},U)=0$ for every $i>0$ and every $0\leq j\leq m-1$. Since ${\rm T.i.dim}(U)=m$, we deduce that ${\mathcal{E}}_{T}^{i}(G,U)\cong{\mathcal{E}}_{T}^{m+i}(N,U)=0$.

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

[TABLE]

exists and so ${\mathcal{E}}_{T}^{r}(K_{j},U)\cong{\mathcal{E}}_{T}^{r+1}(K_{j-1},U)$, for every $r\geq 0$. Since $K_{n-1}=N$, we have

[TABLE]

as desired.* $\Box$*

Next, we study the Gorenstein $\pi[T]$-projectivity of modules on $T$-cocoherent rings, in short exact sequences.

Proposition 7

.* Let $R$ be $T$-cocoherent and consider the exact sequence $0\rightarrow N\rightarrow B\rightarrow G\rightarrow 0$, where $B$ is $\pi[T]$-projective. Then ${\rm GT}$-${\rm pd}(G)\leq{\rm GT}$-${\rm pd}(N)+1$. In particular, if $G$ is Gorenstein $\pi[T]$-projective, so is $N$.*

**Proof. **

We shall show that ${\rm GT}$-${\rm pd}(G)\leq{\rm GT}$-${\rm pd}(N)+1$. In fact, we may assume that ${\rm GT}$-${\rm pd}(N)=n<\infty$. Then, by definition, $N$ admits a Gorenstein $\pi[T]$-projective resolution:

[TABLE]

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

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

which shows that ${\rm GT}$-${\rm pd}(G)\leq n+1$. The particular case follows from Corollary 5.* $\Box$*

Proposition 8

.* Let $R$ be a $T$-cocoherent ring and $0\rightarrow N\rightarrow G\rightarrow B\rightarrow 0$ be an exact sequence, where $N,B\in{\rm Gen}T$. If $N$ is Gorenstein $\pi[T]$-projective and $B$ is $\pi[T]$-projective, then $G$ is Gorenstein $\pi[T]$-projective.*

**Proof. **

Since $N$ is Gorenstein $\pi[T]$-projective, by Corollary 5, there exists an exact sequence of $0\rightarrow N\rightarrow B^{{}^{\prime}}\rightarrow K\rightarrow 0$, where $B^{{}^{\prime}}$ is $\pi[T]$-projective and $K$ is Gorenstein $\pi[T]$-projective. Now, we consider the following diagram:

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

The exactness of the middle horizontal sequence with $B$ and $B^{{}^{\prime}}$, $\pi[T]$-projective, implies that $D$ is $\pi[T]$-projective. Hence from the middle vertical sequence and Corollary 5, we deduce that $G$ is Gorenstein $\pi[T]$-projective.

• $\Box$*

2 Gorensetein $\pi[T]$-projective Modules on $T$-Cocoherent Rings

This section is devoted to $T$-cocoherent rings over which every module is Gorenstein $\pi[T]$-projective.

Lemma 9

.* Let $T$ be a tilting module and $G\in{Gen}T$. Then, $G\in{Cogen}T$.*

**Proof. **

Let $G\in{Gen}T$. Then, the short exact sequence $0\rightarrow K\rightarrow T^{(I)}\rightarrow G\rightarrow 0$ exists. We have $K\subseteq T^{(I)}\subseteq T^{I}$. So, $K\in{\rm Cogen}T$. By [6, Proposition 2.1], ${\rm Cogen}T={\rm Copres}^{\infty}T$, since $T$ is tilting. Thus by Lemma 3, $G\in{\rm Copres}^{m}T$, and hence $G\in{Cogen}T$.

$\Box$**

Proposition 10

.* Let $R$ be a ring. The following assertions are equivalent:*

1. (1)

Every module belong ${\rm Gen}T$, is Gorenstein $\pi[T]$-projective; 2. (2)

The ring satisfies the following two conditions:

(i)* Every $T$-injective module is $\pi[T]$-projective.*

(ii)* ${\mathcal{E}}_{T}^{1}(N,U)=0$ for any $N\in{\rm Gen}T$ and any $U\in{\rm F.Copres}^{n}T$ with ${\rm T.i.dim}(U)<\infty$.*

**Proof. **

$(1)\Longrightarrow(2)$ The condition $(i)$ follows from this fact that every $T$-injective module $M$ is Gorenstein $\pi[T]$-projective. So, the following $\pi[T]$-projective resolution of $M$ exists:

[TABLE]

Since $M$ is $T$-injective, $M$ is $\pi[T]$-projective as a direct summand of $B^{0}$. Also, Proposition 6(1) and (1) imply that ${\mathcal{E}}_{T}^{1}(N,U)=0$ for any module $N\in{\rm Gen}T$ and any module $U\in{\rm F.Copres}^{1}T$ with finite $T$-injective dimension. So the condition $(ii)$ follows.

$(2)\Longrightarrow(1)$ Let $G\in{\rm Gen}T$ be.Then by Lemma 9, $G\in{Cogen}T$. So, a ${\rm Add}T$-resolution $\cdots\rightarrow T_{1}\rightarrow T_{0}\rightarrow G\rightarrow 0$ and a ${\rm Prod}T$- resolution $0\rightarrow G\rightarrow T^{0}\rightarrow T^{1}\rightarrow\cdots$ of $G$ exists. By Remark 2, any $T_{i}$ is $\pi[T]$-projective and any $T^{i}$ is $T$-injective. Hence by (2), every $T^{i}$ is $\pi[T]$-projective. Assembling these resolutions, we get the following exact sequence of $\pi[T]$-projective modules:

[TABLE]

where $G={\rm ker}(T^{0}\rightarrow T^{1})$. So by (2)(ii), ${\rm Hom}({\mathbf{B}},U)$ is exact for any module $U\in{\rm F.Copres}^{1}T$ with finite $T$-injective dimension. Hence $G$ is Gorenstein $\pi[T]$-projective.* $\Box$*

The next theorem shows that if $R$ is a $T$-cocoherent ring and every $\sigma[T]$-injective module is Gorenstein $\pi[T]$-projective, then every module is Gorenstein $\pi[T]$-projective.

Theorem 11

.* Let $R$ be a $T$-cocoherent ring. Then the following are equivalent:*

1. (1)

Every module is Gorenstein $\pi[T]$-projective; 2. (2)

Every Gorenstein $\sigma[T]$-injective module is Gorenstein $\pi[T]$-projective; 3. (4)

Every $\sigma[T]$-injective module is Gorenstein $\pi[T]$-projective; 4. (5)

Every $T$-injective module is $\pi[T]$-projective.

**Proof. **

$(1)\Longrightarrow(2)$ This is a clear.

$(2)\Longrightarrow(4)$ Let $G$ be a $\sigma[T]$-injective module. Every $\sigma[T]$-injective module is Gorenstein $\sigma[T]$-injective (see,[9]). Since $G$ is Gorenstein $\sigma[T]$-injective, impelis that $G$ is Gorenstein $\pi[T]$-projective by hypothesis.

$(4)\Longrightarrow(5)$ Let $G$ be a $T$-injective module. Since $G$ is $\sigma[T]$-injective, impelis that $G$ is Gorenstein $\pi[T]$-projective by hypothesis. By Corollary 5, there exists an exact sequence $0\rightarrow G\rightarrow B\rightarrow N\rightarrow 0,$ where $B$ is $\pi[T]$-projective. Thus the sequence splits. Hence $G$ is $\pi[T]$-projective as a direct summand of $B$.

$(5)\Longrightarrow(1)$ Let $G\in{\rm Gen}T$. Then by Lemma 9, there is an exact sequence

[TABLE]

where any $T^{i}$ is $T$-injective. Then by (5), every $T^{i}$ is $\pi[T]$-projective. Hence Corollary 5 completes the proof.

$\Box$**

We denote the right $\pi[T]$-projective dimension of any module $M$ by $\pi[T].pd(M)$, and $\pi[T].pd(M)=\inf\{n:\ \mathcal{E}^{n+1}_{T}(M,N)=0\ {\rm for\ every}\ N\in\pi[T]\}.$

Example 12

. 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, $\pi[T].pd(E^{0})=\pi[T].pd(E^{1})=0$. Since by [4], $T=E_{0}\oplus E_{1}$ is a tilting module. So, any $E^{i}$ is $\pi[T]$-projective and hence, any $E^{i}$ is Gorenstein $\pi[T]$-projective for $i=0,1$.

Definition 13

.* We define the global $\pi[T]$-projective dimension of any ring $R$ to be:*

[TABLE]

Clearly, every $\pi[T]$-projective module is Gorenstein $\pi[T]$-projective. But the converse is not true in general. We finish this paper with the following theorem which determines a sufficient condition under which the converse holds.

Theorem 14

.* If $gl.\pi[T].pd(R)<\infty$, then every Gorenstein $\pi[T]$-projective module is $\pi[T]$-projective.*

**Proof. **

Suppose that $gl.\pi[T]$.${\rm pd}(R)=m<\infty$, and $G$ is a Gorenstein $\pi[T]$-projective module. If $m=0$, then $\mathcal{E}^{1}_{T}(M,N)=0$ for any $N\in\pi[T]$ , and hence $G$ is $\pi[T]$-projective. For $m\geq 1$, since $G$ is Gorenstein $\pi[T]$-projective, there exists an exact sequence $0\rightarrow G\rightarrow B^{0}\rightarrow B^{1}\rightarrow\cdots$ with each $B^{i}$ is $\pi[T]$-projective. Let $L={\rm coker}(B^{m-2}\rightarrow B^{m-1})$ . Then

[TABLE]

is exact, and hence $G$ is $\pi[T]$-projective since $\pi[T]$.${\rm pd}(L)\leq m$.* $\Box$*