Cograde conditions and cotorsion pairs

TL;DR

This paper investigates conditions under which certain functors preserve monomorphisms and epimorphisms in the context of rings and semidualizing bimodules, linking cograde conditions to cotorsion pairs and finitistic dimensions.

Contribution

It introduces new cograde conditions that characterize functor preservation and constructs complete cotorsion pairs related to these conditions.

Findings

Double functors preserve monomorphisms and epimorphisms under cograde conditions.

Constructs of two complete cotorsion pairs based on cograde assumptions.

Establishes relations between relative finitistic dimensions and projective dimensions of the bimodule.

Abstract

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We study when the double functor $\Tor^S_i(\omega, \Ext^i_{R}(\omega,-))$ preserves epimorphisms and the double functor $\Ext_{R}^i(\omega, \Tor_i^{S}(\omega,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $\omega$.

Equations

$$(-{)}_{\ast }:=\mathrm{Hom}(\omega ,-)\text{and}(-{)}^{\ast }:=\mathrm{Hom}(-,\omega ),$$

$${}_R{\omega }^{\perp }:=\{M\in \mathrm{Mod}R\mid {\mathrm{Ext}}_R^{⩾1}(\omega ,M)=0\},$$

$${{\omega }_S}^{\top }:=\{N\in \mathrm{Mod}S\mid {\mathrm{Tor}}_{⩾1}^S(\omega ,N)=0\}.$$

$${}_R{\omega }^{{\perp }_n}:=\{M\in \mathrm{Mod}R\mid {\mathrm{Ext}}_R^{1⩽i⩽n}(\omega ,M)=0\},$$

$${{\omega }_S}^{{\top }_n}:=\{N\in \mathrm{Mod}S\mid {\mathrm{Tor}}_{1⩽i⩽n}^S(\omega ,N)=0\};$$

$${\mathcal{F}}_{\omega }(R):=\{\omega {\otimes }_{S}F\mid F\mathrm{is}\mathrm{flat}\mathrm{in}\mathrm{Mod}S\},$$

$${\mathcal{P}}_{\omega }(R):=\{\omega {\otimes }_{S}P\mid P\mathrm{is}\mathrm{projective}\mathrm{in}\mathrm{Mod}S\},$$

$${\mathcal{I}}_{\omega }(S):=\{I_{\ast }\mid I\mathrm{is}\mathrm{injective}\mathrm{in}\mathrm{Mod}R\}.$$

$${\theta }_M:\omega {\otimes }_S{M}_{\ast }\to M$$

$${\mu }_N:N\to (\omega {\otimes }_{S}N{)}_{\ast }$$

$$\cdots \to X_n\to \cdots \to X_1\to X_0\to M\to 0$$

$$0\to X_n\to \cdots \to X_1\to X_0\to M\to 0$$

$$0\to K\to F\to M\to 0$$

$$0\to N\to X_0\to \cdots \to X_{n-1}\to M\to 0$$

$${\Omega }_{\mathcal{X}}^n(\mathcal{Y}):=\{N\in \mathcal{A}\mid N\text{is an}n\text{-}\mathcal{X}\text{-syzygy of some object in}\mathcal{Y}\},$$

$${\mathrm{co\Omega }}_{\mathcal{X}}^n(\mathcal{Y}):=\{M\in \mathcal{A}\mid M\text{is an}n\text{-}\mathcal{X}\text{-cosyzygy of some object in}\mathcal{Y}\}.$$

$${\Omega }_{\mathcal{A}}^n(S):={\Omega }_{{\mathcal{A}}_{\omega }(S)}^n(\mathrm{Mod}S),{\Omega }_{{\mathcal{I}}_{\omega }}^n(S):={\Omega }_{{\mathcal{I}}_{\omega }(S)}^n(\mathrm{Mod}S),$$

$${\Omega }_{{\mathcal{I}}_{\omega }}^n(R^{op}):={\Omega }_{{\mathcal{I}}_{\omega }(R^{op})}^n(\mathrm{Mod}{R}^{op}),$$

$${\mathrm{co\Omega }}_{\mathcal{B}}^n(R):={\mathrm{co\Omega }}_{{\mathcal{B}}_{\omega }(R)}^n(\mathrm{Mod}R),{\mathrm{co\Omega }}_{{\mathcal{F}}_{\omega }}^n(R):={\mathrm{co\Omega }}_{{\mathcal{F}}_{\omega }(R)}^n(\mathrm{Mod}R),$$

$${\mathrm{co\Omega }}_{{\mathcal{P}}_{\omega }}^n(R):={\mathrm{co\Omega }}_{{\mathcal{P}}_{\omega }(R)}^n(\mathrm{Mod}R),{\mathrm{co\Omega }}_{{\mathcal{P}}_{\omega }}^n(S^{op}):={\mathrm{co\Omega }}_{{\mathcal{P}}_{\omega }(S^{op})}^n(\mathrm{Mod}{S}^{op}).$$

$$N\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\omega {\otimes }_S{W_0}_{\ast }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\to {\mathrm{co\Omega }}^1(M{)}_{\ast }\stackrel{{\alpha }_{\ast }}{⟶}{I}^1(M{)}_{\ast }\stackrel{{g^1}_{\ast }}{⟶}{I}^2(M{)}_{\ast }$$

$$\omega {\otimes }_S{I}^0(M{)}_{\ast }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(1_{\omega }\otimes {g^1}_{\ast })(y)=(1_{\omega }\otimes \delta )\cdot (1_{\omega }\otimes \pi )\cdot (1_{\omega }\otimes \gamma )(y)=(1_{\omega }\otimes \delta )\cdot (1_{\omega }\otimes \pi )(x)=0.$$

Full text

Key words and phrases: Semidualizing bimodules; (Strong) $\operatorname{Ext}$-cograde, (Strong) $\operatorname{Tor}$-cograde; Double functors; $n$-$\mathcal{X}$-(co)syzygy; (Adjoint) $n$-cotorsionfree modules; Cotorsion pairs; Finitistic dimensions.

1. Introduction

Let $R$ be a left and right Noetherian ring and $n\geqslant 1$. It was proved by Auslander that the flat dimension of the $i$-th term in the minimal injective resolution of $R_{R}$ is at most $i$ for any $0\leqslant i<n$ if and only if the strong grade of $\operatorname{Ext}_{R}^{i}(M,R)$ is at least $i$ for any finitely generated left $R$-module $M$ and $1\leqslant i\leqslant n$; and this result is left-right symmetric ([16, Theorem 3.7]). In this case, $R$ is called Auslander $n$-Gorenstein. If $R$ is Auslander $n$-Gorenstein for all $n$, then it is said to satisfy the Auslander condition. This condition is a non-commutative version of commutative Gorenstein rings. It has been known that Auslander $n$-Gorenstein rings and the Auslander condition play a crucial role in homological algebra, representation theory of artin algebras and non-commutative algebraic geometry, see [5, 6, 10, 12, 15, 16, 20, 22, 24, 25, 26, 27, 28] and references therein. In particular, Auslander $n$-Gorenstein rings and some generalized versions were characterized in terms of the properties of the double functor $\operatorname{Ext}^{i}_{R^{op}}(\operatorname{Ext}^{i}_{R}(-,R),R)$ and certain (strong) grade conditions of $\operatorname{Ext}$-modules, and a series of cotorsion pairs were constructed under the Auslander condition ([24]).

It is well known that the (Auslander) transpose is one of the most powerful tools in representation theory of artin algebras and Gorenstein homological algebra, see [4, 7, 14]. To dualize this important and useful notion, we introduced in [32] the notion of the cotranspose of modules and then obtained many dual counterparts of interesting results ([32, 33, 34, 35]). As a dual of the notion of the (strong) grade of modules, we introduced in [32, 33] the notion of the (strong) cograde of modules, and obtained the dual versions of some results about the (strong) grade of modules. Let $R$ and $S$ be rings and ${}_{R}\omega_{S}$ a semidualizing bimodule. In this paper, we will study when the double functor $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,-))$ preserves epimorphisms and the double functor $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules and some related properties of the cotranspose of modules, and also investigate the relationship between certain cograde conditions of modules and complete cotorsion pairs. This paper is organized as follows.

In Section 2, we give some terminology and some preliminary results.

Let $R$ and $S$ be rings and ${}_{R}\omega_{S}$ a semidualizing bimodule. In Section 3, we study when $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,-))$ preserves epimorphisms and $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Let $n,k\geqslant 0$. We prove that the $\operatorname{Tor}$-cograde of $\operatorname{Ext}_{R}^{i+k}(\omega,M)$ with respect to $\omega$ is at least $i$ for any left $R$-module $M$ and $1\leqslant i\leqslant n$ if and only if $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is an epimorphism for any epimorphism of left $R$-modules $f:B\twoheadrightarrow C$ with $B,C$ being a $(k+1)$-cosyzygy and $0\leqslant i\leqslant n-1$ (Theorem  3.5); and that the $\operatorname{Ext}$-cograde of $\operatorname{Tor}^{S}_{i+k}(\omega,N)$ with respect to $\omega$ is at least $i$ for any left $S$-module $N$ and $1\leqslant i\leqslant n$ if and only if $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any monomorphism of left $S$-modules $g:B^{\prime}\rightarrowtail C^{\prime}$ with $B^{\prime},C^{\prime}$ being a $(k+1)$-yoke and $0\leqslant i\leqslant n-1$ (Theorem  3.7).

Moreover, we prove that the strong $\operatorname{Tor}$-cograde of $\operatorname{Ext}_{R}^{i+k}(\omega,M)$ with respect to $\omega$ is at least $i$ for any left $R$-module $M$ and $1\leqslant i\leqslant n$ if and only if for any exact sequence of left $R$-modules $0\to A\to B\stackrel{{\scriptstyle f}}{{\longrightarrow}}C\to 0$ with $A$ an $(i-1)$-$\mathcal{P}_{\omega}(R)$-syzygy of an $(i+k-1)$-cosyzygy, $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is an epimorphism for any $0\leqslant i\leqslant n-1$ (Theorem  3.8); and that the strong $\operatorname{Ext}$-cograde of $\operatorname{Tor}^{S}_{i+k}(\omega,N)$ with respect to $\omega$ is at least $i$ for any left $S$-module $N$ and $1\leqslant i\leqslant n$ if and only if for any exact sequence of left $S$-modules $0\to A\stackrel{{\scriptstyle g}}{{\rightarrow}}B\to C\to 0$ with $C$ an $(i-1)$-$\mathcal{I}_{\omega}(S)$-cosyzygy of an $(i+k-1)$-yoke, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$ (Theorem  3.9).

In Section 4, we introduce the notion of $\omega$ satisfying the (quasi) $n$-cograde condition in terms of the properties of the strong cograde of modules. By using the results obtained in Section 3, we give some equivalent characterizations for $\omega$ satisfying such conditions (Theorems 4.8 and 4.14). In particular, the $n$-cograde condition is left-right symmetric, but the quasi $n$-cograde condition is not. In addition, we prove that the $\operatorname{Tor}$-cograde of $\operatorname{Ext}^{i}_{R}(\omega,M)$ with respect to $\omega$ is at least $i-1$ for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$ if and only if the $\operatorname{Ext}$-cograde of $\operatorname{Tor}_{i}^{S}(\omega,N)$ with respect to $\omega$ is at least $i-1$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$ (Theorem  4.19).

In Section 5, we prove that if one of the equivalent conditions in Theorem  4.19 mentioned above is satisfied, then the right $S$-projective dimension $\mathop{\rm pd}\nolimits_{S^{op}}\omega$ of $\omega$ is at most $n-1$ if and only if $(\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R),{{}_{R}\omega^{\perp_{n}}})$ forms a complete cotorsion pair; and the left $R$-projective dimension $\mathop{\rm pd}\nolimits_{R}\omega$ of $\omega$ is at most $n-1$ if and only if $({{\omega_{S}}^{\top_{n}}},\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S))$ forms a complete cotorsion pair (Theorem  5.6); see Section 2 and 5 for the details of these notations. Then we apply this result to right quasi $(n-1)$-Gorenstein artin algebras (Corollary  5.8).

In Section 6, we introduce the finitistic $\mathcal{P}_{\omega}(R)$-injective dimension ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R$ of $R$ and the $\mathcal{I}_{\omega}(S)$-projective dimension ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S$ of $S$. We prove that if the $\operatorname{Tor}$-cograde of $\operatorname{Ext}^{i+k}_{R}(\omega,M)$ with respect to $\omega$ is at least $i$ for any $M\in\mathop{\rm Mod}\nolimits R$ and $i\geqslant 1$, then ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R\leqslant\mathop{\rm pd}\nolimits_{R}\omega\leqslant{\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R+k$; and if the $\operatorname{Ext}$-cograde of $\operatorname{Tor}_{i+k}^{S}(\omega,N)$ with respect to $\omega$ is at least $i$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $i\geqslant 1$, then ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S\leqslant\mathop{\rm pd}\nolimits_{S^{op}}\omega\leqslant{\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S+k$ (Theorem  6.3). As an application, we get that for an artin algebra $R$, if $R$ satisfies the Auslander condition, then $\mathop{\rm FPD}\nolimits R^{op}=\mathop{\rm FID}\nolimits R^{op}={\mathop{\rm id}\nolimits_{R^{op}}R}=\mathop{\rm id}\nolimits_{R}R=\mathop{\rm FPD}\nolimits R=\mathop{\rm FID}\nolimits R$; and if $R$ satisfies the right quasi Auslander condition, then $\mathop{\rm FPD}\nolimits R\leqslant\mathop{\rm FID}\nolimits R={\mathop{\rm id}\nolimits_{R^{op}}R}=\mathop{\rm id}\nolimits_{R}R\leqslant\mathop{\rm FPD}\nolimits R+1$, where $\mathop{\rm FID}\nolimits R$, $\mathop{\rm FPD}\nolimits R$, ${\mathop{\rm id}\nolimits_{R^{op}}R}$ and $\mathop{\rm id}\nolimits_{R}R$ are the finitistic injective dimension, the finitistic projective dimension, the right and left self-injective dimensions of $R$ respectively (Corollary  6.9).

2. Preliminaries

Throughout this paper, all rings are associative rings with units. For a ring $R$, $\mathop{\rm Mod}\nolimits R$ (resp. $\mathop{\rm mod}\nolimits R$) are the class of left (resp. finitely generated left) $R$-modules. Let $M\in\mathop{\rm Mod}\nolimits R$, we use $\mathop{\rm Add}\nolimits_{R}M$ to denote the subclass of $\mathop{\rm Mod}\nolimits R$ consisting of modules consisting of direct summands of direct sums of copies of $M$, and use $\mathop{\rm pd}\nolimits_{R}M$, $\mathop{\rm fd}\nolimits_{R}M$ and $\mathop{\rm id}\nolimits_{R}M$ to denote the projective, flat and injective dimensions of $M$ respectively.

Definition 2.1**.**

([2, 19]). Let $R$ and $S$ be rings. An ($R$-$S$)-bimodule ${}_{R}\omega_{S}$ is called semidualizing if the following conditions are satisfied.

1. (a1)

${}_{R}\omega$ admits a degreewise finite $R$-projective resolution. 2. (a2)

$\omega_{S}$ admits a degreewise finite $S$-projective resolution. 3. (b1)

The homothety map ${}_{R}R_{R}\stackrel{{{}_{R}\gamma}}{{\rightarrow}}\operatorname{Hom}_{S^{op}}(\omega,\omega)$ is an isomorphism. 4. (b2)

The homothety map ${}_{S}S_{S}\stackrel{{\scriptstyle\gamma_{S}}}{{\rightarrow}}\operatorname{Hom}_{R}(\omega,\omega)$ is an isomorphism. 5. (c1)

$\operatorname{Ext}_{R}^{\geqslant 1}(\omega,\omega)=0.$ 6. (c2)

$\operatorname{Ext}_{S^{op}}^{\geqslant 1}(\omega,\omega)=0.$

Wakamatsu in [37] introduced and studied the so-called generalized tilting modules, which are usually called Wakamatsu tilting modules, see [8, 29]. Note that a bimodule ${}_{R}\omega_{S}$ is semidualizing if and only if it is Wakamatsu tilting ([39, Corollary 3.2]). Examples of semidualizing bimodules are referred to [19, 31, 33, 35, 38].

From now on, $R$ and $S$ are arbitrary rings and we fix a semidualizing bimodule ${}_{R}\omega_{S}$. For convenience, We write

[TABLE]

[TABLE]

[TABLE]

For any $n\geqslant 1$, we write

[TABLE]

[TABLE]

in particular, ${}_{R}\omega^{\perp_{0}}=\mathop{\rm Mod}\nolimits R$ and ${\omega_{S}}^{\top_{0}}=\mathop{\rm Mod}\nolimits S$. Symmetrically, ${{\omega_{S}}^{\perp_{n}}}$ and ${{{}_{R}\omega}^{\top_{n}}}$ are defined. Following [19], set

[TABLE]

[TABLE]

[TABLE]

The modules in $\mathcal{F}_{\omega}(R)$, $\mathcal{P}_{\omega}(R)$ and $\mathcal{I}_{\omega}(S)$ are called $\omega$-flat, $\omega$-projective and $\omega$-injective respectively. Note that $\mathcal{P}_{\omega}(R)=\mathop{\rm Add}\nolimits_{R}\omega$ ([33, Proposition 3.4(2)]). Symmetrically, the classes of $\mathcal{F}_{\omega}(S^{op})$, $\mathcal{P}_{\omega}(S^{op})$ and $\mathcal{I}_{\omega}(R^{op})$ are defined.

Let $M\in\mathop{\rm Mod}\nolimits R$ and $N\in\mathop{\rm Mod}\nolimits S$. Then we have the following two canonical valuation homomorphisms

[TABLE]

defined by $\theta_{M}(x\otimes f)=f(x)$ for any $x\in\omega$ and $f\in M_{*}$; and

[TABLE]

defined by $\mu_{N}(y)(x)=x\otimes y$ for any $y\in N$ and $x\in\omega$. Recall that a module $M\in\mathop{\rm Mod}\nolimits R$ is called $\omega$-cotorsionless (resp. $\omega$-coreflexive) if $\theta_{M}$ is an epimorphism (resp. an isomorphism) ([32]); and a module $N\in\mathop{\rm Mod}\nolimits S$ is called adjoint $\omega$-cotorsionless (resp. adjoint $\omega$-coreflexive) if $\mu_{N}$ is a monomorphism (resp. an isomorphism) ([34]).

Definition 2.2**.**

([19]).

1. (1)

The Auslander class $\mathcal{A}_{\omega}(S)$ with respect to $\omega$ consists of all left $S$-modules $N$ satisfying the following conditions.

1. (A1)

$N\in{{\omega_{S}}^{\top}}$. 2. (A2)

$\omega\otimes_{S}N\in{{}_{R}\omega^{\perp}}$. 3. (A3)

$\mu_{N}$ is an isomorphism in $\mathop{\rm Mod}\nolimits S$.

1. (2)

The Bass class $\mathcal{B}_{\omega}(R)$ with respect to $\omega$ consists of all left $R$-modules $M$ satisfying the following conditions.

1. (B1)

$M\in{{}_{R}\omega^{\perp}}$. 2. (B2)

$M_{*}\in{{\omega_{S}}^{\top}}$. 3. (B3)

$\theta_{M}$ is an isomorphism in $\mathop{\rm Mod}\nolimits R$.

For a module $M\in\mathop{\rm Mod}\nolimits R$, we use

[TABLE]

to denote the minimal injective resolution of $M$. For any $n\geqslant 1$, $\mathop{{\rm co}\Omega}\nolimits^{n}(M):=\operatorname{Im}g^{n-1}$ is called the $n$-cosyzygy of $M$; in particular, $\mathop{{\rm co}\Omega}\nolimits^{0}(M)=M$. We use $\mathop{{\rm co}\Omega}\nolimits^{n}(R)$ to denote the subclass of $\mathop{\rm Mod}\nolimits R$ consisting of $n$-cosyzygy modules. Symmetrically, $\mathop{{\rm co}\Omega}\nolimits^{n}(S^{op})$ is defined.

Definition 2.3**.**

([32]). Let $M\in\mathop{\rm Mod}\nolimits R$ and $n\geqslant 1$.

1. (1)

$\mathop{\rm cTr}\nolimits_{\omega}M:=\mathop{\rm Coker}\nolimits({g^{0}}_{*})$ is called the cotranspose of $M$ with respect to ${}_{R}\omega_{S}$, where $g^{0}$ is as in (2.1). 2. (2)

$M$ is called $n$-$\omega$-cotorsionfree if $\mathop{\rm cTr}\nolimits_{\omega}M\in{{\omega_{S}}^{\top_{n}}}$; and is called $\infty$-$\omega$-cotorsionfree if it is $n$-$\omega$-cotorsionfree for all $n$.

We use $\mathop{{\rm c}\mathcal{T}}\nolimits^{n}_{\omega}(R)$ (resp. $\mathop{{\rm c}\mathcal{T}}\nolimits_{\omega}(R)$) to denote the subclass of $\mathop{\rm Mod}\nolimits R$ consisting of $n$-$\omega$-cotorsionfree modules (resp. $\infty$-$\omega$-cotorsionfree modules). Symmetrically, $\mathop{{\rm c}\mathcal{T}}\nolimits^{n}_{\omega}(S^{op})$ is defined. By [32, Proposition 3.2], we have that a module in $\mathop{\rm Mod}\nolimits R$ is $\omega$-cotorsionless (resp. $\omega$-coreflexive) if and only if it is in $\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$ (resp. $\mathop{{\rm c}\mathcal{T}}\nolimits^{2}_{\omega}(R)$).

Recall from [13] that a homomorphism $f:F\rightarrow N$ in $\mathop{\rm Mod}\nolimits S$ with $F$ flat is called a flat cover of $N$ if $\operatorname{Hom}_{S}(F^{\prime},f)$ is epic for any flat module $F^{\prime}$ in $\mathop{\rm Mod}\nolimits S$, and an endomorphism $h:F\rightarrow F$ is an automorphism whenever $f=fh$. Let $N\in\mathop{\rm Mod}\nolimits S$. Bican, El Bashir and Enochs proved in [9] that $N$ has a flat cover. We use

[TABLE]

to denote the minimal flat resolution of $N$ in $\mathop{\rm Mod}\nolimits S$, where each $F_{i}(N)\to\mathop{\rm Coker}\nolimits f_{i}$ is a flat cover of $\mathop{\rm Coker}\nolimits f_{i}$. For any $n\geqslant 1$, $\Omega^{n}_{\mathcal{F}}(N):=\operatorname{Im}f_{n-1}$ is called the $n$-yoke of $N$; in particular, $\Omega^{0}_{\mathcal{F}}(N)=N$. We use $\Omega^{n}_{\mathcal{F}}(S)$ to denote the subclass of $\mathop{\rm Mod}\nolimits S$ consisting of $n$-yoke modules. Symmetrically, $\Omega^{n}_{\mathcal{F}}(R^{op})$ is defined.

Definition 2.4**.**

([34]) Let $N\in\mathop{\rm Mod}\nolimits S$ and $n\geqslant 1$.

1. (1)

$\mathop{\rm acTr}\nolimits_{\omega}N:=\mathop{\rm Ker}\nolimits(1_{\omega}\otimes f_{0})$ is called the adjoint cotranspose of $N$ with respect to ${}_{R}\omega_{S}$, where $f_{0}$ is as in (2.2). 2. (2)

$N$ is called adjoint $n$-$\omega$-cotorsionfree if $\mathop{\rm acTr}\nolimits_{\omega}N\in{{}_{R}\omega^{\perp_{n}}}$; and is called adjoint $\infty$-$\omega$-cotorsionfree if it is adjoint $n$-$\omega$-cotorsionfree for all $n$.

We use $\mathop{{\rm ac}\mathcal{T}}\nolimits^{n}_{\omega}(S)$ (resp. $\mathop{{\rm ac}\mathcal{T}}\nolimits{\omega}(S)$) to denote the subclass of $\mathop{\rm Mod}\nolimits S$ consisting of adjoint $n$-$\omega$-cotorsionfree modules (resp. adjoint $\infty$-$\omega$-cotorsionfree modules). Symmetrically, $\mathop{{\rm ac}\mathcal{T}}\nolimits^{n}_{\omega}(R^{op})$ is defined. By [34, Proposition 3.2], we have that a module in $\mathop{\rm Mod}\nolimits S$ is adjoint $\omega$-cotorsionless (resp. adjoint $\omega$-coreflexive) if and only if it is in $\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$ (resp. $\mathop{{\rm ac}\mathcal{T}}\nolimits^{2}_{\omega}(S)$).

Definition 2.5**.**

([33])

1. (1)

Let $M\in\mathop{\rm Mod}\nolimits R$ and $n\geqslant 0$. The $\operatorname{Ext}$-cograde of $M$ with respect to $\omega$ is defined as $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}M:=\inf\{i\geqslant 0\mid\operatorname{Ext}_{R}^{i}(\omega,M)\neq 0\}$; and the strong $\operatorname{Ext}$-cograde of $M$ with respect to $\omega$, denoted by $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}M$, is said to be at least $n$ if $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}X\geqslant n$ for any quotient module $X$ of $M$. Symmetrically, the (strong) $\operatorname{Ext}$-cograde of a module in $\mathop{\rm Mod}\nolimits S^{op}$ is defined. 2. (2)

Let $N\in\mathop{\rm Mod}\nolimits S$ and $n\geqslant 0$. The $\operatorname{Tor}$-cograde of $N$ with respect to $\omega$ is defined as $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N:=\inf\{i\geqslant 0\mid\operatorname{Tor}^{S}_{i}(\omega,N)\neq 0\}$; and the strong $\operatorname{Tor}$-cograde of $N$ with respect to $\omega$, denoted by $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}N$, is said to be at least $n$ if $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}Y\geqslant n$ for any submodule $Y$ of $N$. Symmetrically, the (strong) $\operatorname{Tor}$-cograde of a module in $\mathop{\rm Mod}\nolimits R^{op}$ is defined.

Let $\mathcal{X}$ be a subclass of $\mathop{\rm Mod}\nolimits R$ and $M\in\mathop{\rm Mod}\nolimits R$. An exact sequence (of finite or infinite length):

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ is called an $\mathcal{X}$-resolution of $M$ if all $X_{i}$ are in $\mathcal{X}$. The $\mathcal{X}$-projective dimension $\mathcal{X}$-$\mathop{\rm pd}\nolimits_{R}M$ of $M$ is defined as $\inf\{n\mid$ there exists an $\mathcal{X}$-resolution

[TABLE]

of $M$ in $\mathop{\rm Mod}\nolimits R\}$. Dually, the notions of an $\mathcal{X}$-coresolution and the $\mathcal{X}$-injective dimension $\mathcal{X}$-$\mathop{\rm id}\nolimits_{R}M$ of $M$ are defined.

Let $\mathcal{F}$ be a subclass of $\mathop{\rm Mod}\nolimits R$. A module $M\in\mathop{\rm Mod}\nolimits R$ is said to have special $\mathcal{F}$-precover if there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $F\in\mathcal{F}$ and $\operatorname{Ext}^{1}_{R}(F^{\prime},K)=0$ for any $F^{\prime}\in\mathcal{F}$. The class $\mathcal{F}$ is called special precovering if any module in $\mathop{\rm Mod}\nolimits R$ has a special $\mathcal{F}$-precover. Dually, the notions of special $\mathcal{F}$-preenvelopes and special preenveloping classes are defined (see [14]).

Definition 2.6**.**

(cf. [17]) Let $\mathcal{U},\mathcal{V}$ be subclasses of $\mathop{\rm Mod}\nolimits R$. The pair $(\mathcal{U},\mathcal{V})$ is called a cotorsion pair if $\mathcal{U}={{}^{\bot_{1}}\mathcal{V}}:=\{U\in\mathop{\rm Mod}\nolimits R\mid\operatorname{Ext}_{R}^{1}(U,V)=0$ for any $V\in\mathcal{V}\}$ and $\mathcal{V}={\mathcal{U}^{\bot_{1}}}:=\{V\in\mathop{\rm Mod}\nolimits R\mid\operatorname{Ext}_{R}^{1}(U,V)=0$ for any $U\in\mathcal{U}\}$.

The following is the Salce’s lemma.

Lemma 2.7**.**

(cf. [17, Lemma 2.2.6])* Let $(\mathcal{U},\mathcal{V})$ be a cotorsion pair in $\mathop{\rm Mod}\nolimits R$. Then the following statements are equivalent.*

1. (1)

Any module in $\mathop{\rm Mod}\nolimits R$ has a special $\mathcal{U}$-precover. 2. (2)

Any module in $\mathop{\rm Mod}\nolimits R$ has a special $\mathcal{V}$-preenvelope.

In this case, the cotorsion pair $(\mathcal{U},\mathcal{V})$ is called complete.

Definition 2.8**.**

Let $\mathcal{X}$ be a subcategory of an abelian category $\mathcal{E}$ and $n\geqslant 1$. If there exists an exact sequence

[TABLE]

in $\mathcal{E}$ with all $X_{i}$ in $\mathcal{X}$, then $N$ is called an $n$-$\mathcal{X}$-syzygy of $M$ and $M$ is called an $n$-$\mathcal{X}$-cosyzygy of $N$.

For subcategories $\mathcal{X},\mathcal{Y}$ of an abelian category $\mathcal{E}$ and $n\geqslant 1$, we write

[TABLE]

[TABLE]

In particular, $\Omega^{0}_{\mathcal{X}}(\mathcal{Y})=\mathcal{Y}=\mathop{{\rm co}\Omega}\nolimits^{0}_{\mathcal{X}}(\mathcal{Y})$ and $\Omega^{-1}_{\mathcal{X}}(\mathcal{Y})=0=\mathop{{\rm co}\Omega}\nolimits^{-1}_{\mathcal{X}}(\mathcal{Y})$. For convenience, we write

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Lemma 2.9**.**

We have

1. (1)

$\Omega^{1}_{\mathcal{I}_{\omega}}(S)=\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$. 2. (2)

$\mathop{{\rm co}\Omega}\nolimits^{1}_{\mathcal{P}_{\omega}}(R)=\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$.

Proof.

(1) By [34, Proposition 3.8], we have $\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)\subseteq\Omega^{1}_{\mathcal{I}_{\omega}}(S)$. Now let $N\in\Omega^{1}_{\mathcal{I}_{\omega}}(S)$ and let $f^{0}:N\rightarrowtail I^{0}$ be a monomorphism in $\mathop{\rm Mod}\nolimits S$ with $I^{0}\in\mathcal{I}_{\omega}(S)$. Then from the following commutative diagram

[TABLE]

with $\mu_{I^{0}}$ an isomorphism, we get that $\mu_{N}$ is a monomorphism and $N\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$. It implies $\Omega^{1}_{\mathcal{I}_{\omega}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$.

(2) By [32, Proposition 3.7], we have $\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)\subseteq\mathop{{\rm co}\Omega}\nolimits^{1}_{\mathcal{P}_{\omega}}(R)$. Now let $M\in\mathop{{\rm co}\Omega}\nolimits^{1}_{\mathcal{P}_{\omega}}(R)$ and let $f_{0}:W_{0}\twoheadrightarrow M$ be an epimorphism in $\mathop{\rm Mod}\nolimits R$ with $W_{0}\in\mathcal{P}_{\omega}(R)$. Then from the following commutative diagram

[TABLE]

with $\theta_{W_{0}}$ an isomorphism, we get that $\theta_{M}$ is an epimorphism and $M\in\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$. It implies $\mathop{{\rm co}\Omega}\nolimits^{1}_{\mathcal{P}_{\omega}}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$. ∎

Let $\mathcal{C},\mathcal{E}$ be abelian categories and $\Delta:\mathcal{C}\to\mathcal{E}$ a functor. Recall that a sequence $\mathbb{T}$ in $\mathcal{C}$ is called $\Delta$-exact if $\Delta(\mathbb{T})$ is exact in $\mathcal{E}$.

3. (Strong) cograde conditions and double homological functors

In this section, we study when $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,-))$ preserves epimorphisms and $\operatorname{Ext}_{S^{op}}^{i}(\omega,\operatorname{Tor}_{i}^{R}(-,\omega))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules.

3.1. Cograde conditions

We begin with the following

Lemma 3.1**.**

2. (1)

Let $M\in\mathop{\rm Mod}\nolimits R$ with the minimal injective resolution as (2.1). Then there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ such that $1_{\omega}\otimes\pi$ is an isomorphism. 3. (2)

Let $N\in\mathop{\rm Mod}\nolimits S$ with the minimal flat resolution as (2.2). Then there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ such that $\sigma_{*}$ is an isomorphism.

Proof.

(1) Let $g^{0}=\alpha\cdot\beta$ (where $\beta:I^{0}(M)\twoheadrightarrow\mathop{{\rm co}\Omega}\nolimits^{1}(M)(=\operatorname{Im}g^{0})$ and $\alpha:\mathop{{\rm co}\Omega}\nolimits^{1}(M)\rightarrowtail I^{1}(M)$) be the natural epic-monic decomposition of $g^{0}$. Then we have the following commutative diagram with exact columns and rows

[TABLE]

in $\mathop{\rm Mod}\nolimits S$, where $C=I^{1}(M)_{*}/\mathop{{\rm co}\Omega}\nolimits^{1}(M)_{*}$, $\pi_{1}$ is the natural epimorphism, $\lambda$ and $\pi$ are induced homomorphisms. The rightmost column in the above diagram is exactly the exact sequence (3.1). Notice that

[TABLE]

is exact, so there exists a homomorphism $\delta:C\to I^{2}(M)_{*}$ in $\mathop{\rm Mod}\nolimits S$ such that ${g^{1}}_{*}=\delta\cdot\pi_{1}$, and hence ${g^{1}}_{*}=\delta\cdot\pi_{1}=\delta\cdot\pi\cdot\gamma$.

By [19, Lemma 4.1], for any injective module $I\in\mathop{\rm Mod}\nolimits R$, we have $\omega\otimes_{S}I_{*}\cong I$ canonically. So the upper row in the following commutative diagram

[TABLE]

is exact. Let $x\in\mathop{\rm Ker}\nolimits(1_{\omega}\otimes\pi)$. Then there exists $y\in\omega\otimes_{S}I^{1}(M)_{*}$ such that $x=(1_{\omega}\otimes\gamma)(y)$. It follows that

[TABLE]

So $y\in\mathop{\rm Ker}\nolimits(1_{\omega}\otimes{g^{1}}_{*})=\operatorname{Im}(1_{\omega}\otimes{g^{0}}_{*})$, and hence there exists $z\in\omega\otimes_{S}I^{0}(M)_{*}$ such that $y=(1_{\omega}\otimes{g^{0}}_{*})(z)$. Thus

[TABLE]

which implies that $1_{\omega}\otimes\pi$ is a monomorphism, and hence an isomorphism.

(2) Let $f_{0}=\alpha^{\prime}\cdot\beta^{\prime}$ (where $\beta^{\prime}:F_{1}(N)\twoheadrightarrow\Omega^{1}_{\mathcal{F}}(N)(=\operatorname{Im}f_{0})$ and $\alpha^{\prime}:\Omega^{1}_{\mathcal{F}}(N)\rightarrowtail F_{0}(N)$) be the natural epic-monic decomposition of $f_{0}$. Then we have the following commutative diagram with exact columns and rows

[TABLE]

in $\mathop{\rm Mod}\nolimits R$, where $\sigma$ and $\tau$ are induced homomorphisms. The leftmost column in the above diagram is exactly the exact sequence (3.2). Notice that

[TABLE]

is exact, so there exists a homomorphism $\phi:\omega\otimes_{S}F_{2}(N)\to\operatorname{Im}(1_{\omega}\otimes f_{1})$ in $\mathop{\rm Mod}\nolimits R$ such that $1_{\omega}\otimes f_{1}=\sigma_{1}\cdot\phi$, and hence $1_{\omega}\otimes f_{1}=\sigma_{1}\cdot\phi=\eta\cdot\sigma\cdot\phi$.

By [19, Lemma 4.1], for any flat module $F\in\mathop{\rm Mod}\nolimits S$, we have $F\cong(\omega\otimes_{S}F)_{*}$ canonically. So the upper row in the following commutative diagram is exact.

[TABLE]

Let $x\in(\mathop{\rm acTr}\nolimits_{\omega}N)_{*}$. Since $((1_{\omega}\otimes f_{0})_{*}\cdot\eta_{*})(x)=(((1_{\omega}\otimes f_{0})\cdot\eta)_{*})(x)=0$, we have that $\eta_{*}(x)\in\mathop{\rm Ker}\nolimits(1_{\omega}\otimes f_{0})_{*}=\operatorname{Im}(1_{\omega}\otimes f_{1})_{*}$ and there exists $y\in(\omega\otimes_{S}F_{2}(N))_{*}$ such that $\eta_{*}(x)=(1_{\omega}\otimes f_{1})_{*}(y)$. Thus

[TABLE]

As $\eta^{*}$ is monic, we have $x=\sigma_{*}(\phi_{*}(y))$. It means that $\sigma_{*}$ is an epimorphism, and hence an isomorphism. ∎

The following two lemmas are useful in this section.

Lemma 3.2**.**

Assume that $\mathop{{\rm co}\Omega}\nolimits^{n}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{m}_{\omega}(R)$ with $m,n\geqslant 0$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n+1}_{R}(\omega,M)\geqslant m$* for any $M\in\mathop{\rm Mod}\nolimits R$.* 2. (2)

$\mathop{{\rm co}\Omega}\nolimits^{n+1}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{m+1}_{\omega}(R)$.

Proof.

Because any injective module in $\mathop{\rm Mod}\nolimits R$ is in $\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$ by [32, Lemma 2.5(2)], we have that $\mathop{{\rm co}\Omega}\nolimits^{n+1}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$ for any $n\geqslant 0$, and the case for $m=0$ follows. Now suppose that $m\geqslant 1$ and $M\in\mathop{\rm Mod}\nolimits R$. By Lemma  3.1(1), there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ such that $1_{\omega}\otimes\pi$ is an isomorphism, where $C={I^{n+1}(M)}_{*}/{\mathop{{\rm co}\Omega}\nolimits^{n+1}(M)}_{*}$. Because $\mathop{{\rm co}\Omega}\nolimits^{n}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{m}_{\omega}(R)$ by assumption, we have that both $\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{n}(M)$ and $\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{n+1}(M)$ are in ${{\omega_{S}}^{\top_{m}}}$. It yields that

[TABLE]

for any $0\leqslant i\leqslant m-1$. In addition, we also have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$. By [19, Corollary 6.1], we have ${I^{n+2}(M)}_{*}\in{{\omega_{S}}^{\top}}$. So

[TABLE]

for any $0\leqslant i\leqslant m-1$. Thus we conclude that $\operatorname{Tor}_{0\leqslant i\leqslant m-1}^{S}(\omega,\operatorname{Ext}^{n+1}_{R}(\omega,M))=0$ if and only if $\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{n+1}(M)\in{{\omega_{S}}^{\top_{m+1}}}$, and if and only if $\mathop{{\rm co}\Omega}\nolimits^{n+1}(M)\in\mathop{{\rm c}\mathcal{T}}\nolimits^{m+1}_{\omega}(R)$. The proof is finished. ∎

Lemma 3.3**.**

Assume that $\Omega^{n}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{m}_{\omega}(S)$ with $m,n\geqslant 0$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n+1}^{S}(\omega,N)\geqslant m$* for any $N\in\mathop{\rm Mod}\nolimits S$.* 2. (2)

$\Omega^{n+1}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{m+1}_{\omega}(S)$.

Proof.

Because any flat module in $\mathop{\rm Mod}\nolimits S$ is in $\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$ by [34, Corollary 3.5(1)], we have that $\Omega^{n+1}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$ for any $n\geqslant 0$, and the case for $m=0$ follows. Now suppose that $m\geqslant 1$ and $N\in\mathop{\rm Mod}\nolimits S$. By Lemma  3.1(2), there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ such that $\sigma_{*}$ is an isomorphism. Because $\Omega^{n}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{m}_{\omega}(S)$ by assumption, we have that both $\mathop{\rm acTr}\nolimits_{\omega}\Omega^{n}_{\mathcal{F}}(N)$ and $\mathop{\rm acTr}\nolimits_{\omega}\Omega^{n+1}_{\mathcal{F}}(N)$ are in ${{}_{R}\omega^{\perp_{m}}}$. It yields that

[TABLE]

for any $0\leqslant i\leqslant m-1$. In addition, we also have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. By [19, Corollary 6.1], we have $\omega\otimes_{S}F_{n+2}(N)\in{{}_{R}\omega^{\perp}}$. So

[TABLE]

for any $0\leqslant i\leqslant m-1$. Thus we conclude that $\operatorname{Ext}^{0\leqslant i\leqslant m-1}_{R}(\omega,\operatorname{Tor}_{n+1}^{S}(\omega,N))=0$ if and only if $\mathop{\rm acTr}\nolimits_{\omega}\Omega^{n+1}_{\mathcal{F}}(N)\in{{}_{R}\omega^{\perp_{m+1}}}$, and if and only if $\Omega^{n+1}_{\mathcal{F}}(N)\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{m+1}_{\omega}(S)$. The proof is finished. ∎

Let $\mathcal{T}\subseteq\mathcal{W}$ be subcategories of an abelian category $\mathcal{E}$. Recall that $\mathcal{T}$ is called a generator (resp. cogenerator) for $\mathcal{W}$ if for any $W\in\mathcal{W}$, there exists an exact sequence

[TABLE]

in $\mathcal{E}$ with $T\in\mathcal{T}$ and $W^{\prime}\in\mathcal{W}$.

Lemma 3.4**.**

2. (1)

$\mathcal{P}_{\omega}(R)$* is a generator for $\mathcal{B}_{\omega}(R)$.* 3. (2)

$\mathop{{\rm co}\Omega}\nolimits^{n}(R)\subseteq\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{B}}(R)=\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{F}_{\omega}}(R)=\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{P}_{\omega}}(R)$* for any $n\geqslant 1$.*

Proof.

(1) Let $M\in\mathcal{B}_{\omega}(R)$. Then by [32, Theorem 3.9 and Proposition 3.7], there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with all $W_{i}\in\mathcal{P}_{\omega}(R)$ such that it remains exact after applying the functor $\operatorname{Hom}_{R}(\omega,-)$. Put $M_{1}:=\operatorname{Im}(W_{1}\to W_{0})$. Then $M_{1}\in\mathop{{\rm c}\mathcal{T}}\nolimits_{\omega}(R)$ by [32, Proposition 3.7]. Because both $M$ and $W_{0}$ are in ${{}_{R}\omega}^{\bot}$, we have $M_{1}\in{{}_{R}\omega}^{\bot}$. So $M_{1}\in\mathcal{B}_{\omega}(R)$ by [32, Theorem 3.9].

(2) Let $n\geqslant 1$. By [19, Lemma 4.1], we have that $\mathcal{B}_{\omega}(R)$ contains all injective left $R$-modules, which yields $\mathop{{\rm co}\Omega}\nolimits^{n}(R)\subseteq\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{B}}(R)$. Because ${\mathcal{B}_{\omega}}(R)\supseteq{\mathcal{F}_{\omega}}(R)\supseteq{\mathcal{P}_{\omega}}(R)$ by [19, Corollary 6.1], we have $\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{B}}(R)\supseteq\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{F}_{\omega}}(R)\supseteq\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{P}_{\omega}}(R)$. Because $\mathcal{B}_{\omega}(R)$ is closed under extensions by [19, Theorem 6.2], we have $\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{B}}(R)=\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{P}_{\omega}}(R)$ by (1) and [23, Corollary 5.4(2)]. ∎

In the following result, we characterize when the double functor $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,-))$ preserves epimorphisms in terms of the $\operatorname{Tor}$-cograde conditions of $\operatorname{Ext}$-modules.

Theorem 3.5**.**

The conditions (1)–(3) below are equivalent for any $n,k\geqslant 0$. If $k\geqslant 1$, then (1)–(4) are equivalent.

1. (1)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+k}_{R}(\omega,M)\geqslant i$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$.* 2. (2)

$\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$* is an epimorphism for any epimorphism $f:B\twoheadrightarrow C$ in $\mathop{\rm Mod}\nolimits R$ with $B,C\in\mathop{{\rm co}\Omega}\nolimits^{k+1}_{\mathcal{P}_{\omega}}(R)$ and $0\leqslant i\leqslant n-1$.* 3. (3)

$\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$* is an epimorphism for any epimorphism $f:B\twoheadrightarrow C$ in $\mathop{\rm Mod}\nolimits R$ with $B,C\in\mathop{{\rm co}\Omega}\nolimits^{k+1}(R)$ and $0\leqslant i\leqslant n-1$.* 4. (4)

$\mathop{{\rm co}\Omega}\nolimits^{i+k}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i+1}_{\omega}(R)$* for any $1\leqslant i\leqslant n$.*

Proof.

By using induction on $i$, $(1)\Leftrightarrow(4)$ follows from Lemma  3.2.

$(1)\Rightarrow(2)$ Let $f:B\twoheadrightarrow C$ be an epimorphism in $\mathop{\rm Mod}\nolimits R$ with $B,C\in\mathop{{\rm co}\Omega}\nolimits^{k+1}_{\mathcal{P}_{\omega}}(R)$. Then $C=\mathop{{\rm co}\Omega}\nolimits^{k+1}_{\mathcal{P}_{\omega}}(C^{\prime})$ for some $C^{\prime}\in\mathop{\rm Mod}\nolimits R$. By (1), we have

[TABLE]

for any $1\leqslant i\leqslant n-1$. Thus $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is epic. In the following, we will show that $1_{\omega}\otimes f_{*}$ is epic.

If $k\geqslant 1$, then $\mathop{{\rm co}\Omega}\nolimits^{k}_{\mathcal{P}_{\omega}}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$ by Lemma 2.9(2). So $\mathop{{\rm co}\Omega}\nolimits^{k+1}_{\mathcal{P}_{\omega}}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{2}_{\omega}(R)$ by Lemma  3.2, and hence $B,C\in\mathop{{\rm c}\mathcal{T}}\nolimits^{2}_{\omega}(R)$. It follows that $1_{\omega}\otimes f_{*}\cong f$ and $1_{\omega}\otimes f_{*}$ is epic.

Now suppose $k=0$. We have an epimorphism $p:W\twoheadrightarrow B$ in $\mathop{\rm Mod}\nolimits R$ with $W\in\mathop{\rm Add}\nolimits_{R}\omega$. From the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $M_{1}=\mathop{\rm Ker}\nolimits(f\cdot p)$, we get the following exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$. By (1). $\omega\otimes_{S}\operatorname{Ext}^{1}_{R}(\omega,M_{1})=0$. So $(1_{\omega}\otimes f_{*})\cdot(1_{\omega}\otimes p_{*})=1_{\omega}\otimes(f\cdot p)_{*}$ is epic, which implies that $1_{\omega}\otimes f_{*}$ is also epic.

By Lemma  3.4(2), we have $(2)\Rightarrow(3)$.

$(3)\Rightarrow(1)$ Let $M\in\mathop{\rm Mod}\nolimits R$. From the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$, we get the following exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$. Since $1_{\omega}\otimes f_{*}$ is an epimorphism by (2), we have that $\omega\otimes_{S}\operatorname{Ext}^{k+1}_{R}(\omega,M)=0$ and $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{k+1}_{R}(\omega,M)\geqslant 1$. In addition, for any $1\leqslant i\leqslant n-1$,

[TABLE]

is epic by (3), so we have

[TABLE]

Thus we conclude that $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+k+1}_{R}(\omega,M)\geqslant i+1$ for any $0\leqslant i\leqslant n-1$. ∎

Lemma 3.6**.**

2. (1)

$\mathcal{I}_{\omega}(S)$* is a cogenerator for $\mathcal{A}_{\omega}(S)$.* 3. (2)

$\Omega^{n}_{\mathcal{F}}(S)\subseteq\Omega^{n}_{\mathcal{A}}(S)=\Omega^{n}_{\mathcal{I}_{\omega}}(S)$* for any $n\geqslant 1$.*

Proof.

(1) Let $N\in\mathcal{A}_{\omega}(S)$. Then by [34, Theorem 3.11(1)], there exists an $(\omega\otimes_{S}-)$-exact exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with all $U^{i}\in\mathcal{I}_{\omega}(S)$. Put $N^{1}:=\operatorname{Im}(U^{0}\to U^{1})$. Then $N^{1}\in\mathop{{\rm ac}\mathcal{T}}\nolimits_{\omega}(S)$ by [34, Corollary 3.9]. Because both $N$ and $U^{0}$ are in ${\omega_{S}}^{\top}$, we have $N^{1}\in{\omega_{S}}^{\top}$. So $N^{1}\in\mathcal{A}_{\omega}(S)$ by [34, Theorem 3.11(1)] again.

(2) Let $n\geqslant 1$. By [19, Lemma 4.1], we have that $\mathcal{A}_{\omega}(S)$ contains all flat left $S$-modules, which yields $\Omega^{n}_{\mathcal{F}}(S)\subseteq\Omega^{n}_{\mathcal{A}}(S)$. Because $\mathcal{A}_{\omega}(S)$ is closed under extensions by [19, Theorem 6.2], we have $\Omega^{n}_{\mathcal{A}}(S)=\Omega^{n}_{\mathcal{I}_{\omega}}(S)$ by (1) and [23, Corollary 5.4(1)]. ∎

In the following result, we characterize when the double functor $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,-))$ preserves monomorphisms in terms of the $\operatorname{Ext}$-cograde conditions of $\operatorname{Tor}$-modules.

Theorem 3.7**.**

The conditions (1)–(3) below are equivalent for any $n,k\geqslant 0$. If $k\geqslant 1$, then (1)–(4) are equivalent.

1. (1)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+k}^{S}(\omega,N)\geqslant i$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$.* 2. (2)

$\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$* is a monomorphism for any monomorphism $g:B^{\prime}\rightarrowtail C^{\prime}$ in $\mathop{\rm Mod}\nolimits S$ with $B^{\prime},C^{\prime}\in\Omega^{k+1}_{\mathcal{I}_{\omega}}(S)$ and $0\leqslant i\leqslant n-1$.* 3. (3)

$\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$* is a monomorphism for any monomorphism $g:B^{\prime}\rightarrowtail C^{\prime}$ in $\mathop{\rm Mod}\nolimits S$ with $B^{\prime},C^{\prime}\in\Omega^{k+1}_{\mathcal{F}}(S)$ and $0\leqslant i\leqslant n-1$.* 4. (4)

$\Omega^{i+k}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i+1}_{\omega}(S)$* for any $1\leqslant i\leqslant n$.*

Proof.

By using induction on $i$, $(1)\Leftrightarrow(4)$ follows from Lemma  3.3.

$(1)\Rightarrow(2)$ Let $g:B^{\prime}\rightarrowtail C^{\prime}$ be a monomorphism in $\mathop{\rm Mod}\nolimits S$ with $B^{\prime},C^{\prime}\in\Omega^{k+1}_{\mathcal{I}_{\omega}}(S)$. Then $B^{\prime}=\Omega^{k+1}_{\mathcal{I}_{\omega}}(B^{\prime\prime})$ for some $B^{\prime\prime}\in\mathop{\rm Mod}\nolimits S$. By (1), we have

[TABLE]

for any $1\leqslant i\leqslant n-1$. Thus $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monic. In the following, we will show that $(1_{\omega}\otimes g)_{*}$ is monic.

If $k\geqslant 1$, then $\Omega^{k}_{\mathcal{I}_{\omega}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$ by Lemma 2.9(1). So $\Omega^{k+1}_{\mathcal{I}_{\omega}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{2}_{\omega}(S)$ by Lemma  3.3, and hence $B^{\prime},C^{\prime}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{2}_{\omega}(S)$. It follows that $(1_{\omega}\otimes g)_{*}\cong g$ and $(1_{\omega}\otimes g)_{*}$ is monic.

Now suppose $k=0$. We have a monomorphism $i:C^{\prime}\rightarrowtail U$ in $\mathop{\rm Mod}\nolimits S$ with $U\in\mathcal{I}_{\omega}(S)$. From the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $L_{1}=\mathop{\rm Coker}\nolimits(i\cdot g)$, we get the following exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. By (1), $(\operatorname{Tor}^{S}_{1}(\omega,L_{1}))_{*}=0$. So $(1_{\omega}\otimes i)_{*}\cdot(1_{\omega}\otimes g)_{*}=(1_{\omega}\otimes(i\cdot g))_{*}$ is monic, which implies that $(1_{\omega}\otimes g)_{*}$ is also monic.

By Lemma  3.6(2), we have $(2)\Rightarrow(3)$.

$(3)\Rightarrow(1)$ Let $N\in\mathop{\rm Mod}\nolimits S$. From the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$, we get the following exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. Since $(1_{\omega}\otimes g)_{*}$ is a monomorphism by (2), we have that $(\operatorname{Tor}^{S}_{k+1}(\omega,N))_{*}=0$ and $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{k+1}^{S}(\omega,N)\geqslant 1$. In addition, for any $1\leqslant i\leqslant n-1$,

[TABLE]

is monic by (3), so we have

[TABLE]

Thus we conclude that $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+k+1}^{S}(\omega,N)\geqslant i+1$ for any $0\leqslant i\leqslant n-1$. ∎

3.2. Strong cograde conditions

Compare the following result with Theorem  3.5.

Theorem 3.8**.**

For any $n\geqslant 1$ and $k\geqslant 0$, the following three statements are equivalent.

1. (1)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+k}_{R}(\omega,M)\geqslant i$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$.* 2. (2)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i+k-1}_{\mathcal{P}_{\omega}}(R))$, $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 3. (3)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i+k-1}(R))$, $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is an epimorphism for any $0\leqslant i\leqslant n-1$.

Moreover, if $k=0$, then any of the above statements is equivalent to the following

1. (4)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$, $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is an epimorphism for any $0\leqslant i\leqslant n-1$.

Proof.

$(1)\Rightarrow(2)$ Let $A=\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i+k-1}_{\mathcal{P}_{\omega}}(A^{\prime}))$ with $A^{\prime}\in\mathop{\rm Mod}\nolimits R$. For any $i\geqslant 0$, by dimension-shifting we have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$, which induces exact sequences

[TABLE]

and

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. Since $\mathop{\rm Coker}\nolimits(\operatorname{Ext}^{i}_{R}(\omega,f))\subseteq\operatorname{Ext}^{i+k+1}_{R}(\omega,A^{\prime})$, by (1) we have

[TABLE]

for any $0\leqslant i\leqslant n-1$. Moreover, it follows from (1) and the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ that $\operatorname{Tor}^{S}_{i-1}(\omega,\mathop{\rm Ker}\nolimits(\operatorname{Ext}^{i}_{R}(\omega,f)))=0$ for any $0\leqslant i\leqslant n-1$. Thus $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))=b\cdot a$ is an epimorphism for any $0\leqslant i\leqslant n-1$.

By Lemma  3.4(2), we have $(2)\Rightarrow(3)$.

$(3)\Rightarrow(1)$ Let $M\in\mathop{\rm Mod}\nolimits R$. Fix $i$ ($1\leqslant i\leqslant n$) and an $S$-submodule $L$ of $\operatorname{Ext}^{i+k}_{R}(\omega,M)$. Take an epimorphism $a:P\twoheadrightarrow L$ in $\mathop{\rm Mod}\nolimits S$ with $P$ projective and $a^{\prime}$ the composition

[TABLE]

Then $a^{\prime}$ can be lifted to $b:P\to\mathop{{\rm co}\Omega}\nolimits^{i+k}(M)_{*}$. Take the following pull-back diagram

[TABLE]

where $b^{\prime}$ is the composition

[TABLE]

It induces the following commutative diagram with exact rows

[TABLE]

In the following, we will proceed by induction on $i$. Let $i=1$. Since $1_{\omega}\otimes c_{*}$ is epic by (3), we have that $\omega\otimes_{S}L=0$ and $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{1+k}_{R}(\omega,M)\geqslant 1$.

Assume that the statement (1) holds for any $1\leqslant i\leqslant n-1$. Now consider the case for $i=n$. By the induction hypothesis, we have that $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+k}_{R}(\omega,M)\geqslant i$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n+k}_{R}(\omega,M)\geqslant n-1$. Then $\mathop{{\rm co}\Omega}\nolimits^{n+k-1}(M)\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n-1}_{\omega}(R)$ by Lemma  3.2. Because $\omega\otimes_{S}P\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n-1}_{\omega}(R)$ by [32, Proposition 3.7], it follows from [36, Lemma 4.3] that $X$ in the diagram (3.3) is in $\mathop{{\rm c}\mathcal{T}}\nolimits^{n-1}_{\omega}(R)$. By [32, Proposition 3.7] again, there exists $\operatorname{Hom}_{R}(\mathop{\rm Add}\nolimits_{R}\omega,-)$-exact exact sequences

[TABLE]

and

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with all $W^{\prime}_{j},W_{j}$ in $\mathop{\rm Add}\nolimits_{R}\omega$. Then both $Y$ and $Y^{\prime}$ are in ${{}_{R}\omega}^{\bot_{n-1}}$ and we get the following commutative diagram

[TABLE]

We can guarantee that $g$ is a monomorphism by adding a direct summand in $\mathop{\rm Add}\nolimits_{R}\omega$ (for example $W^{\prime}_{n-2}$) to $Y$ and $W_{n-2}$. Thus we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $Z=\mathop{\rm Coker}\nolimits g$. Since

[TABLE]

we obtain $L\cong\operatorname{Ext}^{n-1}_{R}(\omega,Z)$. Since $Y^{\prime}\in\Omega^{n-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{n+k-1}(R))$, by (3) we get that $\operatorname{Tor}^{S}_{n-1}(\omega,\operatorname{Ext}^{n-1}_{R}(\omega,h))$ is epic. So $\operatorname{Tor}_{n-1}^{S}(\omega,L)=0$ and $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n+k}_{R}(\omega,M)$ $\geqslant n$.

When $k=0$, the proof of $(3)\Rightarrow(1)\Rightarrow(2)$ is in fact that of $(4)\Leftrightarrow(1)$ by just removing the first sentence and putting $A^{\prime}=A$ in the beginning of the proof of $(1)\Rightarrow(2)$, ∎

Compare the following result with Theorem  3.7.

Theorem 3.9**.**

For any $n\geqslant 1$ and $k\geqslant 0$, the following three statements are equivalent.

1. (1)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+k}^{S}(\omega,N)\geqslant i$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$.* 2. (2)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i+k-1}_{\mathcal{I}_{\omega}}(S))$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$. 3. (3)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i+k-1}_{\mathcal{F}}(S))$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$.

Moreover, if $k=0$, then any of the above statements is equivalent to the following

1. (4)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$.

Proof.

$(1)\Rightarrow(2)$ Let $C=\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i+k-1}_{\mathcal{I}_{\omega}}(C^{\prime}))$ with $C^{\prime}\in\mathop{\rm Mod}\nolimits S$. For any $i\geqslant 0$, by dimension shifting we have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$, which induces exact sequences

[TABLE]

and

[TABLE]

in $\mathop{\rm Mod}\nolimits S$. Since $\mathop{\rm Ker}\nolimits(\operatorname{Tor}_{i}^{S}(\omega,g))$ is an $R$-quotient module of $\operatorname{Tor}_{i+k+1}^{S}(\omega,C^{\prime})$, by (1) we have

[TABLE]

Moreover, it follows from (1) and the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ that $\operatorname{Ext}_{R}^{i-1}(\omega,\mathop{\rm Coker}\nolimits(\operatorname{Tor}_{i}^{S}(\omega,g)))=0$ for any $0\leqslant i\leqslant n-1$. Thus $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))=b\cdot a$ is a monomorphism for any $0\leqslant i\leqslant n-1$.

By Lemma  3.6(2), we have $(2)\Rightarrow(3)$.

$(3)\Rightarrow(1)$ Let $N\in\mathop{\rm Mod}\nolimits S$. Fix $i$ ($1\leqslant i\leqslant n$) and an $R$-quotient module $H$ of $\operatorname{Tor}_{i+k}^{S}(\omega,N)$. Take a monomorphism $a:H\rightarrowtail I$ in $\mathop{\rm Mod}\nolimits R$ with $I$ injective and $a^{\prime}$ the composition

[TABLE]

Then $a^{\prime}$ can be extended to $b:\omega\otimes_{S}\Omega^{i+k}_{\mathcal{F}}(N)\to I$. Take the following push-out diagram:

[TABLE]

where $b^{\prime}$ is the composition

[TABLE]

It induces the following commutative diagram with exact rows

[TABLE]

In the following, we will proceed by induction on $i$. Let $i=1$. Since $(1_{\omega}\otimes c)_{*}$ is monic by (2), we have that $H_{*}=0$ and $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{1+k}^{S}(\omega,N)\geqslant 1$.

Assume that the statement (1) holds for any $1\leqslant i\leqslant n-1$. Now consider the case for $i=n$. By the induction hypothesis, we have that $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+k}^{S}(\omega,N)\geqslant i$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n+k}^{S}(\omega,N)\geqslant n-1$. Then $\Omega^{n+k-1}_{\mathcal{F}}(N)\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n-1}_{\omega}(S)$ by Lemma  3.3. Because $I_{*}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n-1}_{\omega}(S)$ by [34, Propposition], it follows from the dual result of [36, Lemma 4.3] that $Y$ in the diagram (3.5) is in $\mathop{{\rm ac}\mathcal{T}}\nolimits^{n-1}_{\omega}(S)$. By [34, Propposition] again, there exist $(\omega\otimes_{S}-)$-exact exact sequences

[TABLE]

and

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with all $U^{i},V^{i}$ in $\mathcal{I}_{\omega}(S)$. Then both $X$ and $X^{\prime}$ are in ${\omega_{S}}^{\top_{n-1}}$ and we get the following commutative diagram

[TABLE]

We can guarantee that $f$ is an epimorphism by adding a direct summand in $\mathcal{I}_{\omega}(S)$ (for example $V^{n-2}$) to $X$ and $U^{n-2}$. Thus we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $Z=\mathop{\rm Ker}\nolimits f$. Since

[TABLE]

we obtain $H\cong\operatorname{Tor}_{n-1}^{S}(\omega,Z)$. Since $X^{\prime}\in\mathop{{\rm co}\Omega}\nolimits^{n-1}_{\mathcal{I}_{\omega}}(\Omega^{n+k-1}_{\mathcal{F}}(S))$, by (3) we get that $\operatorname{Ext}_{R}^{n-1}(\omega,\operatorname{Tor}_{n-1}^{S}(\omega,h))$ is a monomorphism. So $\operatorname{Ext}_{R}^{n-1}(\omega,H)=0$ and $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n+k}^{S}(\omega,N)\geqslant n$.

When $k=0$, the proof of $(3)\Rightarrow(1)\Rightarrow(2)$ is in fact that of $(4)\Leftrightarrow(1)$ by just removing the first sentence and putting $C^{\prime}=C$ in the beginning of the proof of $(1)\Rightarrow(2)$, ∎

4. (Quasi) $n$-cograde condition

In this section, we introduce and study the (quasi) $n$-cograde condition of semidualizing bimodules.

4.1. The $n$-cograde condition

Definition 4.1**.**

For any $n\geqslant 1$, $\omega$ is said to satisfy the right $n$-cograde condition if $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$; and $\omega$ is said to satisfy the left $n$-cograde condition if $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{R}(M^{\prime},\omega)\geqslant i$ for any $M^{\prime}\in\mathop{\rm Mod}\nolimits R^{op}$ and $1\leqslant i\leqslant n$.

As a consequence of Theorems 3.8 and 3.9, we get the following equivalent characterizations for $\omega$ satisfying the right $n$-cograde condition.

Corollary 4.2**.**

For any $n\geqslant 1$, the following statements are equivalent.

1. (1)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$.* 2. (2)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$.* 3. (3)

$\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}({\omega},-))$* preserves epimorphisms in $\mathop{\rm Mod}\nolimits R$ for $0\leqslant i\leqslant n-1$.* 4. (4)

$\operatorname{Ext}^{i}_{R}(\omega,\operatorname{Tor}^{S}_{i}({\omega},-))$* preserves monomorphisms in $\mathop{\rm Mod}\nolimits S$ for $0\leqslant i\leqslant n-1$.* 5. (5)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{P}_{\omega}}(R))$, $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 6. (6)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i-1}(R))$, $\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 7. (7)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i-1}_{\mathcal{I}_{\omega}}(S))$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$. 8. (8)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i-1}_{\mathcal{F}}(S))$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$.

Proof.

By [33, Theorem 6.9], we have $(1)\Leftrightarrow(2)$. By Theorems 3.8 and 3.9, we have $(1)\Leftrightarrow(3)\Leftrightarrow(5)\Leftrightarrow(6)$ and $(2)\Leftrightarrow(4)\Leftrightarrow(7)\Leftrightarrow(8)$ respectively. ∎

Symmetrically, we have the following equivalent characterizations for $\omega$ satisfying the left $n$-cograde condition.

Corollary 4.3**.**

For any $n\geqslant 1$, the following statements are equivalent.

1. (1)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{S^{op}}(\omega,N^{\prime})\geqslant i$* for any $N^{\prime}\in\mathop{\rm Mod}\nolimits S^{op}$ and $1\leqslant i\leqslant n$.* 2. (2)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{R}(M^{\prime},\omega)\geqslant i$* for any $M^{\prime}\in\mathop{\rm Mod}\nolimits R^{op}$ and $1\leqslant i\leqslant n$.* 3. (3)

$\operatorname{Tor}^{R}_{i}(\operatorname{Ext}^{i}_{S^{op}}({\omega},-),\omega)$* preserves epimorphisms in $\mathop{\rm Mod}\nolimits S^{op}$ for $0\leqslant i\leqslant n-1$.* 4. (4)

$\operatorname{Ext}^{i}_{S^{op}}(\omega,\operatorname{Tor}^{R}_{i}(-,{\omega}))$* preserves monomorphisms in $\mathop{\rm Mod}\nolimits R^{op}$ for $0\leqslant i\leqslant n-1$.* 5. (5)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S^{op}$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{P}_{\omega}}(S^{op}))$, $\operatorname{Tor}^{R}_{i}(\operatorname{Ext}^{i}_{S^{op}}(\omega,f),\omega)$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 6. (6)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S^{op}$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i-1}(S^{op}))$, $\operatorname{Tor}^{R}_{i}(\operatorname{Ext}^{i}_{S^{op}}(\omega,f),\omega)$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 7. (7)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R^{op}$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i-1}_{\mathcal{I}_{\omega}}(R^{op}))$, $\operatorname{Ext}^{i}_{S^{op}}(\omega,\operatorname{Tor}^{R}_{i}(g,{\omega}))$ is a monomorphism for any $0\leqslant i\leqslant n-1$. 8. (8)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R^{op}$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i-1}_{\mathcal{F}}(R^{op}))$, $\operatorname{Ext}^{i}_{S^{op}}(\omega,\operatorname{Tor}^{R}_{i}(g,{\omega}))$ is a monomorphism for any $0\leqslant i\leqslant n-1$.

In the following, we will establish the left-right symmetry of the $n$-cograde condition.

Lemma 4.4**.**

Let

[TABLE]

be an exact sequence in $\mathop{\rm Mod}\nolimits R$ such that $A$ is superfluous in $B$. Then the following assertions hold.

1. (1)

Let $L\in\mathop{\rm Mod}\nolimits R^{op}$. If $L^{\prime}\otimes_{R}C=0$ for any submodule $L^{\prime}$ of $L$, then $L\otimes_{R}B=0$. 2. (2)

Let $M\in\mathop{\rm Mod}\nolimits R$. If $\operatorname{Hom}_{R}(C,M^{\prime})=0$ for any quotient module $M^{\prime}$ of $M$, then $\operatorname{Hom}_{R}(B,M)=0$.

Proof.

(1) If $L\otimes_{R}B\neq 0$, then there exists $x\in L$ such that $xR\otimes_{R}B\neq 0$. Since $xR\cong R/I$ for some right ideal $I$ of $R$, we have that

[TABLE]

and $IB\lneqq B$. In view of the assumption that $A$ is superfluous in $B$, it follows that $IB+A\lneqq B$ and

[TABLE]

It contradicts the assumption.

(2) If $\operatorname{Hom}_{R}(B,M)\neq 0$, then there exists a non-zero homomorphism $f\in\operatorname{Hom}_{R}(B,M)$. Pick the kernel $L$ of $f$ such that $\operatorname{Im}f$ $\cong$ $B/L$. Because $A$ is superfluous in $B$ and $f\neq 0$, we have $A+L\lneqq B$. Then there exists a non-zero natural epimorphism $\pi:B/A(\cong C)\twoheadrightarrow B/(A+L)$. Note that the inclusions $(A+L)/L\subseteq B/L\subseteq M$ induce an embedding homomorphism

[TABLE]

Then $0\neq i\cdot\pi\in\operatorname{Hom}_{R}(C,\frac{M}{(A+L)/L})$, which contradicts the assumption. ∎

It is straightforward to verify the following observation.

Lemma 4.5**.**

2. (1)

If $P\in\mathop{\rm Mod}\nolimits R$ is finitely generated projective, then $\mathop{\rm pd}\nolimits_{S^{op}}P^{*}=\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P$. 3. (2)

If $Q\in\mathop{\rm Mod}\nolimits S^{op}$ is finitely generated projective, then $\mathop{\rm pd}\nolimits_{R}Q^{*}=\mathcal{P}_{\omega}(S^{op})$-$\mathop{\rm id}\nolimits_{S^{op}}Q$.

Lemma 4.6**.**

Let $P\in\mathop{\rm Mod}\nolimits R$ be finitely generated projective and $t\geqslant 0$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm pd}\nolimits_{S^{op}}P^{*}\leqslant t$. 2. (2)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P\leqslant t$. 3. (3)

$\operatorname{Ext}^{t+1}_{S^{op}}(\omega,H)\otimes_{R}P=0$* for any $H\in\mathop{\rm Mod}\nolimits S^{op}$.* 4. (4)

$\operatorname{Hom}_{R}(P,\operatorname{Tor}^{S}_{t+1}(\omega,N))=0$* for any $N\in\mathop{\rm Mod}\nolimits S$.*

Proof.

By Lemma  4.5(1), we have $(1)\Leftrightarrow(2)$.

$(1)\Leftrightarrow(3)$ Let $H\in\mathop{\rm Mod}\nolimits S^{op}$ and

[TABLE]

be an injective resolution of $H$ in $\mathop{\rm Mod}\nolimits S^{op}$. Because $P\in\mathop{\rm Mod}\nolimits R$ is finitely generated projective by assumption, the functor $-\otimes_{R}P$ is exact. Then we have

[TABLE]

Now the assertion follows easily.

$(1)\Leftrightarrow(4)$ Since $\mathop{\rm pd}\nolimits_{S^{op}}P^{*}=\mathop{\rm fd}\nolimits_{S^{op}}P^{*}$, the assertion follows from [35, Lemma 7.6]. ∎

Recall from [30] that a ring $R$ is called semiregular if $R/J(R)$ is von Neumann regular and idempotents can be lifted modulo $J(R)$, where $J(R)$ is the Jacobson radical of $R$. The class of semiregular rings includes: (1) von Neumann regular rings; (2) semiperfect rings; (3) left cotorsion rings; and (4) right cotorsion rings. See [18] for the definitions of left cotorsion rings and right cotorsion rings.

If $R$ is a semiregular ring, then any finitely presented left or right $R$-module has a projective cover by [30, Theorem 2.9]. In this case, since ${}_{R}\omega$ admits a degreewise finite $R$-projective resolution by Definition  2.1, we may assume that

[TABLE]

is the minimal projective resolution of ${}_{R}\omega$ in $\mathop{\rm mod}\nolimits R$. Put $\omega_{i}:=\operatorname{Im}(P_{i}(\omega)\to P_{i-1}(\omega))$ for any $i\geqslant 1$ and $\omega_{0}:=\omega$. Analogously, if $S$ is a semiregular ring, then we assume that

[TABLE]

is the minimal projective resolution of $\omega_{S}$ in $\mathop{\rm mod}\nolimits S^{op}$. By Lemma  4.6, we have the following

Proposition 4.7**.**

Let $R$ be a semiregular ring and $m,n\geqslant 1$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm pd}\nolimits_{S^{op}}P_{i}(\omega)^{*}\leqslant m-1$* for any $0\leqslant i\leqslant n-1$.* 2. (2)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P_{i}(\omega)\leqslant m-1$* for any $0\leqslant i\leqslant n-1$.* 3. (3)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\geqslant n$* for any $N^{\prime}\in\mathop{\rm Mod}\nolimits S^{op}$.* 4. (4)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{m}^{S}(\omega,N)\geqslant n$* for any $N\in\mathop{\rm Mod}\nolimits S$.*

Proof.

By [35, Proposition 7.7] and Lemma  4.6, we have $(4)\Leftrightarrow(1)\Leftrightarrow(2)$.

$(3)\Rightarrow(1)$ We proceed by induction on $n$. Let $N^{\prime}\in\mathop{\rm Mod}\nolimits S^{op}$. Suppose $n=1$. Because $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\geqslant 1$ by (3), we have $L^{\prime}\otimes_{R}\omega=0$ for any submodule $L^{\prime}$ of $\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})$ in $\mathop{\rm Mod}\nolimits R^{op}$. It follows from Lemma  4.4(1) that $\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\otimes_{R}P_{0}(\omega)=0$. Therefore by Lemma  4.6 we get $\mathop{\rm pd}\nolimits_{S^{op}}P_{0}(\omega)^{*}\leqslant m-1$ and the case for $n=1$ is proved.

Now suppose $n\geqslant 2$. Let $X$ be a submodule of $\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})$ in $\mathop{\rm Mod}\nolimits R^{op}$. By (3), we have $\operatorname{Tor}_{0\leqslant i\leqslant n-1\emph{}}^{R}(X,\omega)=0$. Then for any $0\leqslant i\leqslant n-2$, we have

[TABLE]

For any $i\geqslant 0$, from the exact sequence

[TABLE]

we get the following exact sequence

[TABLE]

By the induction hypothesis, we have $\mathop{\rm pd}\nolimits_{S^{op}}P_{i}(\omega)^{*}\leqslant m-1$ for any $0\leqslant i\leqslant n-2$. Then it follows from Lemma  4.6 that $\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\otimes_{R}P_{n-2}(\omega)=0$ and hence $X\otimes_{R}P_{n-2}(\omega)=0$. So it is derived from (4.1) that $X\otimes_{R}\omega_{n-1}=0$. Notice that $P_{n-1}(\omega)$ is the projective cover of $\omega_{n-1}$, so $\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\otimes_{R}P_{n-1}(\omega)=0$ by Lemma  4.4(1). It follows from Lemma  4.6 that $\mathop{\rm pd}\nolimits_{S^{op}}P_{n-1}(\omega)^{*}\leqslant m-1$.

$(1)\Rightarrow(3)$ Let $X$ be a submodule of $\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})$ in $\mathop{\rm Mod}\nolimits R^{op}$. Then by (1) and Lemma  4.6, we have $\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\otimes_{R}(\oplus_{i=0}^{n-1}P_{i}(\omega))=0$, and hence $X\otimes_{R}(\oplus_{i=0}^{n-1}P_{i}(\omega))=0$. Since $\omega_{i}$ is a quotient module of $P_{i}(\omega)$ for any $i\geqslant 0$, we then have $X\otimes_{R}(\oplus_{i=0}^{n-1}\omega_{i})=0$.

If $n=1$, then $X\otimes_{R}\omega=0$ and $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\geqslant 1$. If $n\geqslant 2$, then from (4.1) we get $\operatorname{Tor}^{R}_{1}(X,\oplus_{i=0}^{n-2}\omega_{i})=0$. Since $\operatorname{Tor}_{i+1}^{R}(X,\omega)\cong\operatorname{Tor}_{1}^{R}(X,\omega_{i})$ for any $i\geqslant 0$, we have that $\operatorname{Tor}^{R}_{0\leqslant i\leqslant n-1}(X,\omega)=0$ and $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{m}_{S^{op}}(\omega,N^{\prime})\geqslant n$. ∎

The following result means that the $n$-cograde condition is left-right symmetric.

Theorem 4.8**.**

Let $R$ be semiregular and $n\geqslant 1$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm pd}\nolimits_{S^{op}}P_{i}(\omega)^{*}\leqslant i$* for any $0\leqslant i\leqslant n-1$.* 2. (2)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P_{i}(\omega)\leqslant i$* for any $0\leqslant i\leqslant n-1$.* 3. (3)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$.* 4. (4)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$.* 5. (5)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{S^{op}}(\omega,N^{\prime})\geqslant i$* for any $N^{\prime}\in\mathop{\rm Mod}\nolimits S^{op}$ and $1\leqslant i\leqslant n$.* 6. (6)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{R}(M^{\prime},\omega)\geqslant i$* for any $M^{\prime}\in\mathop{\rm Mod}\nolimits R^{op}$ and $1\leqslant i\leqslant n$.* 7. (7)

$\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}({\omega},-))$* preserves epimorphisms in $\mathop{\rm Mod}\nolimits R$ for $0\leqslant i\leqslant n-1$.* 8. (8)

$\operatorname{Ext}^{i}_{R}(\omega,\operatorname{Tor}^{S}_{i}({\omega},-))$* preserves monomorphisms in $\mathop{\rm Mod}\nolimits S$ for $0\leqslant i\leqslant n-1$.* 9. (9)

$\operatorname{Tor}^{R}_{i}(\operatorname{Ext}^{i}_{S^{op}}({\omega},-),\omega)$* preserves epimorphisms in $\mathop{\rm Mod}\nolimits S^{op}$ for $0\leqslant i\leqslant n-1$.* 10. (10)

$\operatorname{Ext}^{i}_{S^{op}}(\omega,\operatorname{Tor}^{R}_{i}(-,{\omega}))$* preserves monomorphisms in $\mathop{\rm Mod}\nolimits R^{op}$ for $0\leqslant i\leqslant n-1$.*

Proof.

By Proposition  4.7, we have $(1)\Leftrightarrow(2)\Leftrightarrow(4)\Leftrightarrow(5)$. By Corollaries  4.2 and  4.3, we have $(3)\Leftrightarrow(4)\Leftrightarrow(7)\Leftrightarrow(8)$ and $(5)\Leftrightarrow(6)\Leftrightarrow(9)\Leftrightarrow(10)$. ∎

As a consequence, we get the following

Corollary 4.9**.**

Let $R$ and $S$ be semiregular and $n\geqslant 1$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm pd}\nolimits_{S^{op}}P_{i}(\omega)^{*}\leqslant i$* for any $0\leqslant i\leqslant n-1$.* 2. (2)

$\mathop{\rm pd}\nolimits_{R}Q_{i}(\omega)^{*}\leqslant i$* for any $0\leqslant i\leqslant n-1$.* 3. (3)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P_{i}(\omega)\leqslant i$* for any $0\leqslant i\leqslant n-1$.* 4. (4)

$\mathcal{P}_{\omega}(S^{op})$-$\mathop{\rm id}\nolimits_{S^{op}}Q_{i}(\omega)\leqslant i$* for any $0\leqslant i\leqslant n-1$.*

Proof.

By the symmetric version of Proposition  4.7, we have

[TABLE]

Now the assertion follows from Theorem 4.8. ∎

4.2. The quasi $n$-cograde condition

Definition 4.10**.**

For any $n\geqslant 1$, $\omega$ is said to satisfy the right quasi $n$-cograde condition if $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+1}^{S}(\omega,N)\geqslant i$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$; and $\omega$ is said to satisfy the left quasi $n$-cograde condition if $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+1}^{R}(M^{\prime},\omega)\geqslant i$ for any $M^{\prime}\in\mathop{\rm Mod}\nolimits R^{op}$ and $1\leqslant i\leqslant n$.

It is trivial that $\omega$ satisfies the right (resp. left) quasi $n$-cograde conditions if it satisfies the right (resp. left) $n$-cograde condition. But the converse does not hold true in general, see Subsection 4.4 below.

The following lemma is useful in the sequel.

Lemma 4.11**.**

For any $n\geqslant 0$, the following assertions hold.

1. (1)

Let $M\in\mathop{\rm Mod}\nolimits R$. If $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}M\geqslant n$ and $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n+1$, then $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}M\geqslant n+1$. 2. (2)

Let $N\in\mathop{\rm Mod}\nolimits S$. If $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N\geqslant n$ and $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,N)\geqslant n+1$, then $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N\geqslant n+1$.

Proof.

We proceed by induction on $n$.

(1) If $n=0$, then $\omega\otimes_{S}M_{*}=0$ by assumption. It follows from [33, Lemma 6.1(1)] that $M_{*}=0$ and $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}M\geqslant 1$.

Let $n\geqslant 1$. Consider an injective resolution

[TABLE]

of $M$ in $\mathop{\rm Mod}\nolimits R$. Put $M^{\prime}=\operatorname{Im}(I^{n-1}\to I^{n})$. Since $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}M\geqslant n$ by the induction hypothesis, applying the functor $(-)_{*}$ to the above exact sequence yields the following exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$. Because $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n+1$ by assumption, we have $\operatorname{Tor}^{S}_{0\leqslant i\leqslant n}(\omega,\operatorname{Ext}^{n}_{R}(\omega,M))=0$. Then by [11, Proposition VI.5.1], we have

[TABLE]

for any $0\leqslant i\leqslant n$ and $j\geqslant 0$, and hence

[TABLE]

It implies that the exact sequence

[TABLE]

splits and hence $\operatorname{Ext}_{R}^{n}(\omega,M)$ is a direct summand of ${M^{\prime}}_{*}$. Since ${M^{\prime}}_{*}$ is adjoint 1-$\omega$-cotorsionfree, so is $\operatorname{Ext}_{R}^{n}(\omega,M)$. Thus, applying [34, Proposition 3.2], the T-cograde condition on $\operatorname{Ext}_{R}^{n}(\omega,M)$ proves $\operatorname{Ext}_{R}^{n}(\omega,M)=0$. Consequently we have $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}M\geqslant n+1$ and the assertion follows.

(2) If $n=0$, then $(\omega\otimes_{S}N)_{*}=0$ by assumption. It follows from [33, Lemma 6.1(2)] that $\omega\otimes_{S}N=0$ and $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N\geqslant 1$.

Let $n\geqslant 1$. Consider a projective resolution

[TABLE]

of $N$ in $\mathop{\rm Mod}\nolimits S$. Put $N^{\prime}=\operatorname{Im}(P_{n}\to P_{n-1})$. Since $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N\geqslant n$ by the induction hypothesis, applying the functor $\omega\otimes_{S}-$ to the above exact sequence yields the following exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. Because $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,N)\geqslant n+1$ by assumption, we have $\operatorname{Ext}^{0\leqslant i\leqslant n}_{R}(\omega,\operatorname{Tor}_{n}^{S}(\omega,N))=0$. Notice that $\omega\otimes_{S}P\in\mathop{\rm Add}\nolimits_{R}\omega$ for any projective module $P$ in $\mathop{\rm Mod}\nolimits S$, so $\operatorname{Ext}^{0\leqslant i\leqslant n}_{R}(\omega\otimes_{S}P_{j},\operatorname{Tor}_{n}^{S}(\omega,N))=0$ for any $j\geqslant 0$, and hence

[TABLE]

It induces an exact sequence

[TABLE]

Because $\omega\otimes_{S}N^{\prime}\in\mathop{{\rm c}\mathcal{T}}\nolimits_{\omega}^{1}(R)$ by [33, Lemma 6.1(2)], there exists an epimorphism $U\twoheadrightarrow\omega\otimes_{S}N^{\prime}$ in $\mathop{\rm Mod}\nolimits R$ with $U\in\mathop{\rm Add}\nolimits_{R}\omega$ by [32, Lemma 3.6(1)]. Because $(\operatorname{Tor}_{n}^{S}(\omega,N))_{*}=0$, we have $\operatorname{Hom}_{R}(U,\operatorname{Tor}_{n}^{S}(\omega,N))=0$. It follows that $\operatorname{Hom}_{R}(\omega\otimes_{S}N^{\prime},\operatorname{Tor}_{n}^{S}(\omega,N))=0$ and $\operatorname{Hom}_{R}(\operatorname{Tor}_{n}^{S}(\omega,N),\operatorname{Tor}_{n}^{S}(\omega,N))=0$, which implies $\operatorname{Tor}_{n}^{S}(\omega,N)=0$. So $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N\geqslant n+1$ and the assertion follows. ∎

We have the following equivalent characterizations for $\omega$ satisfying the right quasi $n$-cograde condition.

Proposition 4.12**.**

For any $n\geqslant 1$, the following statements are equivalent.

1. (1)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+1}^{S}(\omega,N)\geqslant i$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$.* 2. (2)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$.* 3. (3)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i}_{\mathcal{I}_{\omega}}(S))$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$. 4. (4)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i}_{\mathcal{F}}(S))$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$. 5. (5)

$\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$* is an epimorphism for any epimorphism $f:B\twoheadrightarrow C$ in $\mathop{\rm Mod}\nolimits R$ with $B,C\in\mathop{{\rm co}\Omega}\nolimits^{1}_{\mathcal{P}_{\omega}}(R)$ and $0\leqslant i\leqslant n-1$.* 6. (6)

$\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$* is an epimorphism for any epimorphism $f:B\twoheadrightarrow C$ in $\mathop{\rm Mod}\nolimits R$ with $B,C\in\mathop{{\rm co}\Omega}\nolimits^{1}(R)$ and $0\leqslant i\leqslant n-1$.* 7. (7)

$\mathop{{\rm co}\Omega}\nolimits^{i}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i+1}_{\omega}(R)$* for any $1\leqslant i\leqslant n$.*

Proof.

By Theorems 3.9 and 3.5, we have $(1)\Leftrightarrow(3)\Leftrightarrow(4)$ and $(2)\Leftrightarrow(5)\Leftrightarrow(6)\Leftrightarrow(7)$ respectively. In the following, we will prove $(1)\Leftrightarrow(2)$ by induction on $n$.

$(1)\Rightarrow(2)$ Let $M\in\mathop{\rm Mod}\nolimits R$. By Lemma  3.1(1), for any $n\geqslant 1$, there exist exact sequences

[TABLE]

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ such that $1_{\omega}\otimes\pi$ is an isomorphism, where $C=I^{n}(M)_{*}/\mathop{{\rm co}\Omega}\nolimits^{n}(M)_{*}$. Because $I^{n+1}(M)_{*}\in{\omega_{S}}^{\top}$ by [19, Corollary 6.1], it follows from the exact sequence (4.3) that $\operatorname{Tor}_{i}^{S}(\omega,C)\cong\operatorname{Tor}_{i+1}^{S}(\omega,\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{n}(M))$ for any $i\geqslant 1$.

If $n=1$, then from the exact sequence (4.2) we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. Because $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{2}^{S}(\omega,\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{1}(M))\geqslant 1$ by assumption, we have $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\omega\otimes_{S}\operatorname{Ext}^{1}_{R}(\omega,M))\geqslant 1$. It is derived from Lemma  4.11(2) that $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{1}_{R}(\omega,M)\geqslant 1$.

Now suppose $n\geqslant 2$. Then $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n-1$ by the induction hypothesis. It follows from Theorem  3.5 that $\mathop{{\rm co}\Omega}\nolimits^{i}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)$ for any $1\leqslant i\leqslant n$. So $\mathop{{\rm co}\Omega}\nolimits^{n-1}(M)\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n-1}_{\omega}(R)$, and hence $\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{n-1}(M)$ $\in{\omega_{S}}^{\top_{n-1}}$. Thus from the exact sequences (4.2) and (4.3) we get the following exact sequence

[TABLE]

By (1), we have $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n-1}^{S}(\omega,\operatorname{Ext}^{n}_{R}(\omega,M))\geqslant n$. It follows from Lemma  4.11(2) that $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n$.

$(2)\Rightarrow(1)$ Let $N\in\mathop{\rm Mod}\nolimits S$ and $X$ a quotient module of $\operatorname{Tor}_{n+1}^{S}(\omega,N)$ in $\mathop{\rm Mod}\nolimits R$, and let $\beta:\operatorname{Tor}_{1}^{S}(\omega,\Omega_{\mathcal{F}}^{n}(N))(\cong\operatorname{Tor}_{n+1}^{S}(\omega,N))\twoheadrightarrow X$ be an epimorphism in $\mathop{\rm Mod}\nolimits R$. By Lemma  3.1(2), we have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ such that $\sigma_{*}$ is an isomorphism. Then we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$, where $f=\beta\cdot\tau$. It is easy to see that $\eta_{*}$ is an isomorphism.

Let $n=1$. Because $\Omega_{\mathcal{F}}^{1}(N)\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(S)$ by [34, Corollary 3.5(1)], we have $\mathop{\rm acTr}\nolimits_{\omega}\Omega_{\mathcal{F}}^{1}(N)\in{{}_{R}\omega}^{\bot_{1}}$. Then the exact sequence (4.4) gives that $X_{*}\cong\operatorname{Ext}_{R}^{1}(\omega,\mathop{\rm Ker}\nolimits f)$. So $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}X_{*}\geqslant 1$ by (2), and hence $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}X\geqslant 1$ by Lemma  4.11(1). The case for $n=1$ is proved.

Now suppose $n\geqslant 2$. Then $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+1}^{S}(\omega,N)\geqslant i$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n+1}^{S}(\omega,N)\geqslant n-1$ by the induction hypothesis. So $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}X\geqslant n-1$.

By Theorem  3.7, we have $\Omega^{i}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(S)$ for any $1\leqslant i\leqslant n$. So $\Omega^{n}_{\mathcal{F}}(N)\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n}_{\omega}(S)$ and $\mathop{\rm acTr}\nolimits_{\omega}\Omega^{n}_{\mathcal{F}}(N)\in{{}_{R}\omega}^{\bot_{n}}$. It follows from the exact sequence (4.4) that $\operatorname{Ext}_{R}^{n-1}(\omega,X)\cong\operatorname{Ext}_{R}^{n}(\omega,\mathop{\rm Ker}\nolimits f)$. Then by (2), we have $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n-1}_{R}(\omega,X)=\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,\mathop{\rm Ker}\nolimits f)\geqslant n$. Thus $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}X\geqslant n$ by Lemma  4.11(1). ∎

We also have the following

Proposition 4.13**.**

For any $n\geqslant 1$, the following statements are equivalent.

1. (1)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+1}_{S^{op}}(\omega,N^{\prime})\geqslant i$* for any $N^{\prime}\in\mathop{\rm Mod}\nolimits S^{op}$ and $1\leqslant i\leqslant n$.* 2. (2)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{R}(M^{\prime},\omega)\geqslant i$* for any $M^{\prime}\in\mathop{\rm Mod}\nolimits R^{op}$ and $1\leqslant i\leqslant n$.* 3. (3)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S^{op}$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{P}_{\omega}}(S^{op}))$, $\operatorname{Tor}^{R}_{i}(\operatorname{Ext}^{i}_{S^{op}}(\omega,f^{\prime}),\omega)$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 4. (4)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S^{op}$ with $A\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i}(S^{op}))$, $\operatorname{Tor}^{R}_{i}(\operatorname{Ext}^{i}_{S^{op}}(\omega,f^{\prime}),\omega)$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 5. (5)

$\operatorname{Ext}_{S^{op}}^{i}(\omega,\operatorname{Tor}_{i}^{R}(g^{\prime},\omega))$* is a monomorphism for any monomorphism $g^{\prime}:B^{\prime}\rightarrowtail C^{\prime}$ in $\mathop{\rm Mod}\nolimits R^{op}$ with $B^{\prime},C^{\prime}\in\Omega^{1}_{\mathcal{I}_{\omega}}(R^{op})$ and $0\leqslant i\leqslant n-1$.* 6. (6)

$\operatorname{Ext}_{S^{op}}^{i}(\omega,\operatorname{Tor}_{i}^{R}(g^{\prime},\omega))$* is a monomorphism for any monomorphism $g^{\prime}:B^{\prime}\rightarrowtail C^{\prime}$ in $\mathop{\rm Mod}\nolimits R^{op}$ with $B^{\prime},C^{\prime}\in\Omega^{1}_{\mathcal{F}}(R^{op})$ and $0\leqslant i\leqslant n-1$.* 7. (7)

$\Omega^{i}_{\mathcal{F}}(R^{op})\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i+1}_{\omega}(R^{op})$* for any $1\leqslant i\leqslant n$.*

Proof.

By the symmetric versions of Theorems 3.8 and 3.7, we have $(1)\Leftrightarrow(3)\Leftrightarrow(4)$ and $(2)\Leftrightarrow(5)\Leftrightarrow(6)\Leftrightarrow(7)$ respectively. In the following, we will prove $(1)\Leftrightarrow(2)$ by induction on $n$.

$(1)\Rightarrow(2)$ Let $M^{\prime}\in\mathop{\rm Mod}\nolimits R^{op}$ and let

[TABLE]

be the minimal flat resolution of $M^{\prime}$ in $\mathop{\rm Mod}\nolimits R^{op}$. By Lemma  3.1(2), for any $n\geqslant 1$, there exist exact sequences

[TABLE]

[TABLE]

in $\mathop{\rm Mod}\nolimits S^{op}$ such that $\sigma_{*}$ is an isomorphism. Because $F_{n+1}(M^{\prime})\otimes_{R}\omega\in{\omega_{S}}^{\bot}$ by [19, Corollary 6.1], it follows from the exact sequence (4.6) that $\operatorname{Ext}^{i}_{S^{op}}(\omega,\operatorname{Im}(1_{\omega}\otimes f_{n}))\cong\operatorname{Ext}^{i+1}_{S^{op}}(\omega,\mathop{\rm acTr}\nolimits_{\omega}\Omega_{\mathcal{F}}^{n}(M^{\prime}))$ for any $i\geqslant 1$.

If $n=1$, then from the exact sequence (4.5) we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R^{op}$. Because $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{2}_{S^{op}}(\omega,\mathop{\rm acTr}\nolimits_{\omega}\Omega_{\mathcal{F}}^{1}(M^{\prime}))\geqslant 1$ by assumption, we have $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}(\operatorname{Tor}_{1}^{R}(M^{\prime},\omega))_{*}\geqslant 1$. It is derived from Lemma  4.11(1) that $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{1}^{R}(M^{\prime},\omega)\geqslant 1$.

Now suppose $n\geqslant 2$. Then $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{R}(M^{\prime},\omega)\geqslant i$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{R}(M^{\prime},\omega)\geqslant n-1$ by the induction hypothesis. It follows from Theorem  3.7 that $\Omega^{i}_{\mathcal{F}}(R^{op})\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(R^{op})$ for any $1\leqslant i\leqslant n$. So $\Omega^{n-1}_{\mathcal{F}}(M^{\prime})\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n-1}_{\omega}(R^{op})$ and $\mathop{\rm acTr}\nolimits_{\omega}\Omega^{n-1}_{\mathcal{F}}(M^{\prime})\in{\omega_{S}}^{\bot_{n-1}}$. Thus from the exact sequences (4.5) and (4.6) we get the following exact sequence

[TABLE]

By (1), we have $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n-1}_{S^{op}}(\omega,\operatorname{Tor}^{R}_{n}(M^{\prime},\omega))\geqslant n$. It follows from Lemma  4.11(1) that $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{R}(M^{\prime},\omega)\geqslant n$.

$(2)\Rightarrow(1)$ Let $N^{\prime}\in\mathop{\rm Mod}\nolimits S^{op}$ and $Y$ a submodule of $\operatorname{Ext}^{n+1}_{S^{op}}(\omega,N^{\prime})$ in $\mathop{\rm Mod}\nolimits R^{op}$, and let $\alpha:Y\rightarrowtail\operatorname{Ext}^{1}_{S^{op}}(\omega,\mathop{{\rm co}\Omega}\nolimits^{n}(N^{\prime}))(\cong\operatorname{Ext}^{n+1}_{S^{op}}(\omega,N^{\prime}))$ be a monomorphism in $\mathop{\rm Mod}\nolimits R^{op}$. By Lemma  3.1(1), we have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R^{op}$ such that $\pi\otimes 1_{\omega}$ is an isomorphism. Then we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R^{op}$, where $g=\lambda\cdot\alpha$. It is easy to see that $\rho\otimes 1_{\omega}$ is an isomorphism.

Let $n=1$. Because $\mathop{{\rm co}\Omega}\nolimits^{1}(N^{\prime})\in\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(S^{op})$ by [32, Lemma 2.5(2)], we have $\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{1}(N^{\prime})\in{\omega_{S}}^{\top_{1}}$. Then the exact sequence (4.7) gives that $Y\otimes_{R}\omega\cong\operatorname{Tor}^{R}_{1}(\mathop{\rm Coker}\nolimits g,\omega)$. So $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}Y\otimes_{R}\omega\geqslant 1$ by (2), and hence $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}Y\geqslant 1$ by Lemma  4.11(2). The case for $n=1$ is proved.

Now suppose $n\geqslant 2$. Then $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+1}_{S^{op}}(\omega,N^{\prime})\geqslant i$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n+1}_{S^{op}}(\omega,N^{\prime})\geqslant n-1$ by the induction hypothesis. So $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}Y$ $\geqslant n-1$.

By Theorem  3.5, we have $\mathop{{\rm co}\Omega}\nolimits^{i}(R^{op})\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R^{op})$ for any $1\leqslant i\leqslant n$. So $\mathop{{\rm co}\Omega}\nolimits^{n}(N^{\prime})\in\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R^{op})$ and $\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{n}(N^{\prime})\in{{}_{R}\omega}^{\top_{n}}$. It follows from the exact sequence (4.7) that $\operatorname{Tor}^{R}_{n-1}(Y,\omega)\cong\operatorname{Tor}^{R}_{n}(\mathop{\rm Coker}\nolimits g,\omega)$. Then by (2), we have $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}^{R}_{n-1}(Y,\omega)=\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}^{R}_{n}(\mathop{\rm Coker}\nolimits g,\omega)\geqslant n$. Thus $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}Y$ $\geqslant n$ by Lemma  4.11(2). ∎

Now we are in a position to state the following

Theorem 4.14**.**

Let $R$ be semiregular and $n\geqslant 1$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm pd}\nolimits_{S^{op}}P_{i}(\omega)^{*}\leqslant i+1$* for any $0\leqslant i\leqslant n-1$.* 2. (2)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P_{i}(\omega)\leqslant i+1$* for any $0\leqslant i\leqslant n-1$.* 3. (3)

$\mathop{\rm{s.T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+1}_{S^{op}}(\omega,N^{\prime})\geqslant i$* for any $N^{\prime}\in\mathop{\rm Mod}\nolimits S^{op}$ and $1\leqslant i\leqslant n$.* 4. (4)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+1}^{S}(\omega,N)\geqslant i$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$.* 5. (5)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$.* 6. (6)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{R}(M^{\prime},\omega)\geqslant i$* for any $M^{\prime}\in\mathop{\rm Mod}\nolimits R^{op}$ and $1\leqslant i\leqslant n$. * 7. (7)

$\operatorname{Tor}^{S}_{i}(\omega,\operatorname{Ext}^{i}_{R}(\omega,f))$* is an epimorphism for any epimorphism $f:B\twoheadrightarrow C$ in $\mathop{\rm Mod}\nolimits R$ with $B,C\in\mathop{{\rm co}\Omega}\nolimits^{1}(R)$ and $0\leqslant i\leqslant n-1$. * 8. (8)

$\operatorname{Ext}_{S^{op}}^{i}(\omega,\operatorname{Tor}_{i}^{R}(f^{\prime},\omega))$* is a monomorphism for any monomorphism $f^{\prime}:B^{\prime}\rightarrowtail C^{\prime}$ in $\mathop{\rm Mod}\nolimits R^{op}$ with $B^{\prime},C^{\prime}\in\Omega^{1}_{\mathcal{F}}(R^{op})$ and $0\leqslant i\leqslant n-1$.* 9. (9)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S^{op}$ with $A^{\prime}\in\Omega^{i-1}_{\mathcal{P}_{\omega}}(\mathop{{\rm co}\Omega}\nolimits^{i}(S^{op}))$, $\operatorname{Tor}^{R}_{i}(\operatorname{Ext}^{i}_{S^{op}}(\omega,g^{\prime}),\omega)$ is an epimorphism for any $0\leqslant i\leqslant n-1$. 10. (10)

For any exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in\mathop{{\rm co}\Omega}\nolimits^{i-1}_{\mathcal{I}_{\omega}}(\Omega^{i}_{\mathcal{F}}(S))$, $\operatorname{Ext}_{R}^{i}(\omega,\operatorname{Tor}_{i}^{S}(\omega,g))$ is a monomorphism for any $0\leqslant i\leqslant n-1$. 11. (11)

$\mathop{{\rm co}\Omega}\nolimits^{i}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i+1}_{\omega}(R)$* for any $1\leqslant i\leqslant n$.* 12. (12)

$\Omega^{i}_{\mathcal{F}}(R^{op})\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i+1}_{\omega}(R^{op})$* for any $1\leqslant i\leqslant n$.*

Proof.

By Proposition  4.7, we have $(1)\Leftrightarrow(2)\Leftrightarrow(3)\Leftrightarrow(4)$. By Propositions  4.12 and  4.13, we have $(4)\Leftrightarrow(5)\Leftrightarrow(7)\Leftrightarrow(10)\Leftrightarrow(11)$ and $(3)\Leftrightarrow(6)\Leftrightarrow(8)\Leftrightarrow(9)\Leftrightarrow(12)$ respectively. ∎

For the right quasi 1-cograde condition, we have some additional interesting equivalent characterizations.

Proposition 4.15**.**

Let $R$ be a semiregular ring. Then the following statements are equivalent.

1. (1)

$\mathop{\rm pd}\nolimits_{S^{op}}P_{0}(\omega)^{*}\leqslant 1$. 2. (2)

$\mathop{\rm{s.E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{2}^{S}(\omega,N)\geqslant 1$* for any $N\in\mathop{\rm Mod}\nolimits S$.* 3. (3)

$\theta_{M}$* is a superfluous epimorphism for any $M\in\mathop{{\rm co}\Omega}\nolimits^{1}(R)$.* 4. (4)

$\mu_{M^{\prime}}$* is an essential monomorphism for any $M^{\prime}\in\Omega^{1}_{\mathcal{F}}(R^{op})$.*

Proof.

By Theorem 4.14, we have $(1)\Leftrightarrow(2)$.

$(1)\Rightarrow(3)$ Let $M\in\mathop{{\rm co}\Omega}\nolimits^{1}(R)$. By [32, Lemma 2.5(2)], we have $\mathop{{\rm co}\Omega}\nolimits^{1}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$. So $M\in\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$ and $\theta_{M}$ is an epimorphism. Because $\mathop{\rm Ker}\nolimits\theta_{M}\cong$ $\operatorname{Tor}_{2}^{S}(\omega,\mathop{\rm cTr}\nolimits_{\omega}M)$ by [32, Proposition 3.2], we have

[TABLE]

by (1) and Lemma  4.6. It follows easily that $X_{*}=0$ for any quotient module $X$ of $\mathop{\rm Ker}\nolimits\theta_{M}$. Let $A$ be a submodule of $\omega\otimes_{S}M_{*}$ in $\mathop{\rm Mod}\nolimits R$ such that $\mathop{\rm Ker}\nolimits\theta_{M}+A=\omega\otimes_{S}M_{*}$. Then $(\mathop{\rm Ker}\nolimits\theta_{M}+A)/A(\cong\mathop{\rm Ker}\nolimits\theta_{M}/(A\cap\mathop{\rm Ker}\nolimits\theta_{M}))$ is isomorphic to a quotient module of $\mathop{\rm Ker}\nolimits\theta_{M}$, and so $((\mathop{\rm Ker}\nolimits\theta_{M}+A)/A)_{*}=0$. Since $\omega\otimes_{S}M_{*}\in\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$ by [33, Lemma 6.1(2)], $(\mathop{\rm Ker}\nolimits\theta_{M}+A)/A\in\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R)$ by [32, Corollary 3.8]. It follows that $\theta_{(\mathop{\rm Ker}\nolimits\theta_{M}+A)/A}:\omega\otimes_{S}((\mathop{\rm Ker}\nolimits\theta_{M}+A)/A)_{*}\to(\mathop{\rm Ker}\nolimits\theta_{M}+A)/A$ is epic and $(\mathop{\rm Ker}\nolimits\theta_{M}+A)/A=0$. It induces that $A=\mathop{\rm Ker}\nolimits\theta_{M}+A=\omega\otimes_{S}M_{*}$ and $\theta_{M}$ is a superfluous epimorphism.

$(3)\Rightarrow(2)$ Let $f:B\twoheadrightarrow C$ be an epimorphism in $\mathop{\rm Mod}\nolimits R$ with $B,C\in\mathop{{\rm co}\Omega}\nolimits^{1}(R)(\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{1}_{\omega}(R))$. Then $\theta_{C}\cdot(1_{\omega}\otimes f_{*})=f\cdot\theta_{B}$ is epic. Because $\theta_{C}$ is a superfluous epimorphism by (3), it follows from [1, Corollary 5.15] that $1_{\omega}\otimes f_{*}$ is epic. Now the assertion follows from Theorem 4.14.

$(1)\Rightarrow(4)$ Let $M^{\prime}\in\Omega^{1}_{\mathcal{F}}(R^{op})$. By [34, Corollary 3.5(1)], we have $\Omega^{1}_{\mathcal{F}}(R^{op})\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(R^{op})$. So $M^{\prime}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(R^{op})$ and $\mu_{M^{\prime}}$ is a monomorphism. Because $\mathop{\rm Coker}\nolimits\mu_{M^{\prime}}\cong\operatorname{Ext}^{2}_{S^{op}}(\omega,\mathop{\rm acTr}\nolimits_{\omega}M^{\prime})$ by [34, Proposition 3.2], we have

[TABLE]

by (1) and Lemma  4.6. It follows easily that $Y\otimes_{R}\omega=0$ for any submodule $Y$ of $\mathop{\rm Coker}\nolimits\mu_{M^{\prime}}$. Let $A^{\prime}$ be a submodule of $(M^{\prime}\otimes_{R}\omega)_{*}$ in $\mathop{\rm Mod}\nolimits R^{op}$ with $A^{\prime}\cap M^{\prime}=0$. Then $A^{\prime}\cong A^{\prime}/A^{\prime}\cap M^{\prime}\cong(A^{\prime}+M^{\prime})/M^{\prime}$ is isomorphic to a submodule of $\mathop{\rm Coker}\nolimits\mu_{M^{\prime}}$, and so $A^{\prime}\otimes_{R}\omega=0$. Since $(M^{\prime}\otimes_{R}\omega)_{*}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(R^{op})$ by [33, Lemma 6.1(1)], $A^{\prime}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(R^{op})$ by [34, Corollary 3.3(1)]. It follows that $\mu_{A^{\prime}}:A^{\prime}\to(A^{\prime}\otimes_{R}\omega)_{*}$ is monic, It induces that $A^{\prime}=0$ and $\mu_{M^{\prime}}$ is an essential monomorphism.

$(4)\Rightarrow(2)$ Let $g:B^{\prime}\rightarrow C^{\prime}$ be a monomorphism in $\mathop{\rm Mod}\nolimits R^{op}$ with $B^{\prime},C^{\prime}\in\Omega^{1}_{\mathcal{F}}(R^{op})(\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{1}_{\omega}(R^{op}))$. Then $(g\otimes 1_{\omega})_{*}\cdot\mu_{B^{\prime}}=\mu_{C^{\prime}}\cdot g$ is monic. Because $\mu_{B^{\prime}}$ is an essential monomorphism by (4), it follows from [1, Corollary 5.13] that $(g\otimes 1_{\omega})_{*}$ is monic. Now the assertion follows from Theorem 4.14. ∎

4.3. The equivalence of certain cograde condition of modules

We have the following facts: for the strong $\operatorname{Tor}$-cograde condition of modules in Theorem  3.8(1) and the strong $\operatorname{Ext}$-cograde condition of modules in Theorem  3.9(1), they are equivalent when $k=0$ by Theorems  4.8; but they are not equivalent when $k=1$ by Theorem  4.14 and Subsection 4.4 below. Also from Theorem  4.14 and Subsection 4.4 below we know that the $\operatorname{Tor}$-cograde condition of modules in Theorem  3.5(1) and the $\operatorname{Ext}$-cograde condition of modules in Theorem  3.7(1) are not equivalent when $k=0$. In this subsection, we will show that these two cograde conditions of modules are equivalent when $k=1$.

For any $i\geqslant 1$, by [34, Proposition 3.8] we have $\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(S)\subseteq\Omega^{i}_{\mathcal{I}_{\omega}}(S)$. The following result characterizes when they are identical.

Proposition 4.16**.**

For any $n\geqslant 1$, the following statements are equivalent.

1. (1)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i-1$* for any $N\in\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{A}}(S)$ and $1\leqslant i\leqslant n$.* 2. (2)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i-1$* for any $N\in\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{I}_{\omega}}(S)$ and $1\leqslant i\leqslant n$.* 3. (3)

$\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(S)=\Omega^{i}_{\mathcal{A}}(S)$* for any $1\leqslant i\leqslant n$.* 4. (4)

$\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(S)=\Omega^{i}_{\mathcal{I}_{\omega}}(S)$* for any $1\leqslant i\leqslant n$.*

Proof.

Because $\mathcal{I}_{\omega}(S)\subseteq\mathcal{A}_{\omega}(S)$, we have $(1)\Rightarrow(2)$. By Lemma  3.6(2), we have $(3)\Leftrightarrow(4)$.

$(2)\Rightarrow(4)$ By [34, Proposition 3.8], it suffices to prove $\Omega^{i}_{\mathcal{I}_{\omega}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(S)$ for any $1\leqslant i\leqslant n$. We proceed by induction on $n$. The case for $n=1$ follows from Lemma 2.9(1).

Now let $N\in\Omega^{n}_{\mathcal{I}_{\omega}}(S)$ with $n\geqslant 2$ and let

[TABLE]

be an exact sequence in $\mathop{\rm Mod}\nolimits S$ with all $I^{i}$ in $\mathcal{I}_{\omega}(S)$. By the induction hypothesis, we have $\operatorname{Im}f^{1}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n-1}_{\omega}(S)$. Applying the functor $\omega\otimes_{S}-$ to (4.8) gives an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. Set $M:=\operatorname{Im}({1_{\omega}\otimes f^{0})}$ and let $1_{\omega}\otimes f^{0}:=\alpha\cdot\pi$ (where $\pi:\omega\otimes_{S}N\twoheadrightarrow M$ and $\alpha:M\hookrightarrow\omega\otimes_{S}I^{0}$) be the natural epic-monic decomposition of $1_{\omega}\otimes f^{0}$. Then we have the following commutative diagram with exact rows

[TABLE]

Since $\mu_{\operatorname{Im}f^{1}}$ is a monomorphism by the above argument, it follows from the snake lemma that $g$ is an epimorphism. On the other hand, we have

[TABLE]

As $\alpha_{*}$ is monic, we get that $\pi_{*}\cdot\mu_{N}=g$ and $\pi_{*}$ is epic. Consider the following commutative diagram with exact rows

[TABLE]

Because $(\operatorname{Tor}_{n}^{S}(\omega,\mathop{\rm Coker}\nolimits f^{n-1}))_{*}=0$ by assumption, we have that $\pi_{*}$ is an isomorphism. So $\mu_{N}$ is epic by the diagram (4.11), and hence an isomorphism. Thus $N\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{2}_{\omega}(S)$ and the case for $n=2$ follows.

Now suppose $n\geqslant 3$. By the induction hypothesis, we have that $\operatorname{Im}f^{1}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n-1}_{\omega}(S)$ and $\mu_{\operatorname{Im}f^{1}}$ is an isomorphism. So $\operatorname{Ext}^{1}_{R}(\omega,M)=0$ by the diagram (4.10). In addition, we have $\omega\otimes_{S}\operatorname{Im}f^{1}\in{{}_{R}\omega}^{\bot_{n-3}}$ by [34, Corollary 3.3(3)]. Because $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,\mathop{\rm Coker}\nolimits f^{n-1})\geqslant n-1$ (by assumption) and $\omega\otimes_{S}I^{0}\in{{}_{R}\omega}^{\bot}$, applying the dimension shifting to (4.9) we obtain $\omega\otimes_{S}N\in{{}_{R}\omega}^{\bot_{n-2}}$. Therefore we conclude that $N\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n}_{\omega}(S)$ by [34, Corollary 3.3(3)] again.

$(3)\Rightarrow(1)$ We proceed by induction on $n$. The case for $n=1$ is trivial. Let $N\in\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{A}}(S)$ with $n\geqslant 2$. Then there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with all $A_{i}$ in $\mathcal{A}_{\omega}(S)$. By (3), we have $H\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n}_{\omega}(S)$. By the induction hypothesis, we have that $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i-1$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,N)\geqslant n-2$.

Put $M:=\mathop{\rm Ker}\nolimits(1_{\omega}\otimes f)$. Because $A_{i}\in\mathop{{\rm ac}\mathcal{T}}\nolimits_{\omega}(S)$ by [34, Theorem 3.11(1)], we obtain that $M_{*}\cong H(\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{n}_{\omega}(S))$ and $M\in{{}_{R}\omega}^{\bot_{n-2}}$. By [35, Proposition 5.1], we have the following exact sequences

[TABLE]

[TABLE]

such that $\theta_{M}=\lambda\cdot\pi$. Since $\mu_{M_{*}}$ is an isomorphism, it follows from [33, Lemma 6.1(1)] that $(\theta_{M})_{*}$ is also an isomorphism. Then both $\lambda_{*}$ and $\pi_{*}$ are isomorphisms.

From the exact sequence (4.13), we get $\operatorname{Im}\theta_{M}\in{{}_{R}\omega}^{\bot_{n-2}}$. Because $\omega\otimes_{S}M_{*}\in{{}_{R}\omega}^{\bot_{n-2}}$ by [34, Corollary 3.3], from the exact sequence (4.12) it yields that $\operatorname{Ext}_{R}^{n-2}(\omega,\operatorname{Tor}_{n}^{S}(\omega,N))=0$. Thus we have $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,N)\geqslant n-1$. ∎

For any $i\geqslant 1$, by [32, Proposition 3.7] we have $\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)\subseteq\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{P}_{\omega}}(R)$. The following result characterizes when they are identical.

Proposition 4.17**.**

For any $n\geqslant 1$, the following statements are equivalent.

1. (1)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i-1$* for any $M\in\Omega^{i}_{\mathcal{B}}(R)$ and $1\leqslant i\leqslant n$.* 2. (2)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i-1$* for any $M\in\Omega^{i}_{\mathcal{F}_{\omega}}(R)$ and $1\leqslant i\leqslant n$.* 3. (3)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i-1$* for any $M\in\Omega^{i}_{\mathcal{P}_{\omega}}(R)$ and $1\leqslant i\leqslant n$.* 4. (4)

$\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)=\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{B}}(R)$* for any $1\leqslant i\leqslant n$.* 5. (5)

$\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)=\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{F}_{\omega}}(R)$* for any $1\leqslant i\leqslant n$.* 6. (6)

$\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)=\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{P}_{\omega}}(R)$* for any $1\leqslant i\leqslant n$.*

Proof.

Because ${\mathcal{B}_{\omega}}(R)\supseteq{\mathcal{F}_{\omega}}(R)\supseteq{\mathcal{P}_{\omega}}(R)$, we have $(1)\Rightarrow(2)\Rightarrow(3)$. By Lemma  3.4(2), we have $(4)\Leftrightarrow(5)\Leftrightarrow(6)$.

$(3)\Rightarrow(6)$ By [32, Proposition 3.7], it suffices to prove $\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{P}_{\omega}}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)$ for any $1\leqslant i\leqslant n$. We proceed by induction on $n$. The case for $n=1$ follows from Lemma 2.9(2).

Now let $M\in\mathop{{\rm co}\Omega}\nolimits^{n}_{\mathcal{P}_{\omega}}(R)$ with $n\geqslant 2$ and let

[TABLE]

be an exact sequence in $\mathop{\rm Mod}\nolimits R$ with all $W_{i}$ in $\mathcal{P}_{\omega}(R)$. By the induction hypothesis, we have $\operatorname{Im}f_{1}\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n-1}_{\omega}(R)$. Applying the functor $(-)_{*}$ to (4.14) gives an exact sequence

[TABLE]

Set $N:=\operatorname{Im}({f_{0}}_{*})$ and let ${f_{0}}_{*}:=\alpha\cdot\pi$ (where $\pi:{W_{0}}_{*}\twoheadrightarrow N$ and $\alpha:N\hookrightarrow M_{*}$) be the natural epic-monic decompositions of ${f_{0}}_{*}$. Then we have the following commutative diagram with exact rows

[TABLE]

So we have

[TABLE]

Because $1_{\omega}\otimes\pi$ is epic, we have $\theta_{M}\cdot(1_{\omega}\otimes\alpha)=g$ and the following commutative diagram with exact rows

[TABLE]

Since $\theta_{\operatorname{Im}f_{1}}$ is an epimorphism by the above argument, it follows from the snake lemma that $g$ is an isomorphism. Thus $1_{\omega}\otimes\alpha$ is a monomorphism. Because $\omega\otimes_{S}\operatorname{Ext}^{n}_{R}(\omega,\mathop{\rm Ker}\nolimits f_{n-1})=0$ by assumption, we have that $\theta_{M}$ is an isomorphism and $M\in\mathop{{\rm c}\mathcal{T}}\nolimits^{2}_{\omega}(R)$ by the diagram (4.17). It means that the assertion holds true for $n=2$. If $n\geqslant 3$, then the fact that $\operatorname{Im}f_{1}\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n-1}_{\omega}(R)$ implies $\theta_{\operatorname{Im}f_{1}}$ is an isomorphism. So $\operatorname{Tor}^{S}_{1}(\omega,N)=0$ by the diagram (4.16). In addition, we have $(\operatorname{Im}f_{1})_{*}\in{\omega_{S}}^{\top_{n-3}}$ by [32, Corollary 3.4(3)]. Because $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,\mathop{\rm Ker}\nolimits f_{n-1})\geqslant n-1$ by assumption, applying the dimension shifting to (4.15) we obtain $M_{*}\in{\omega_{S}}^{\top_{n-2}}$. Therefore we conclude that $M\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n}_{\omega}(R)$ by [32, Corollary 3.4(3)] again.

$(4)\Rightarrow(1)$ We proceed by induction on $n$. The case for $n=1$ is trivial. Let $M\in\Omega^{n}_{\mathcal{B}}(R)$ with $n\geqslant 2$ and let

[TABLE]

be an exact sequence with all $B_{i}$ in $\mathcal{B}_{\omega}(R)$. By (4), we have $L\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n}_{\omega}(R)$. By the induction hypothesis, we have $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i-1$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n-2$.

Put $N:=\mathop{\rm cTr}\nolimits_{\omega}\mathop{\rm Ker}\nolimits f$. Because $B_{i}\in\mathop{{\rm c}\mathcal{T}}\nolimits_{\omega}(R)$ by [32, Theorem 3.9], we obtain that $\omega\otimes_{S}N\cong L(\in\mathop{{\rm c}\mathcal{T}}\nolimits^{n}_{\omega}(R))$ and $N\in{\omega_{S}}^{\top_{n-2}}$. By [33, Proposition 6.7], we have the following exact sequences

[TABLE]

[TABLE]

such that $\mu_{N}=\lambda\cdot\pi$. Since $\theta_{\omega\otimes_{S}N}$ is an isomorphism, it follows from [33, Lemma 6.1(2)] that $1_{\omega}\otimes\mu_{N}$ is also an isomorphism. Then both $1_{\omega}\otimes\lambda$ and $1_{\omega}\otimes\pi$ are isomorphisms.

From the exact sequence (4.18), we get $\operatorname{Im}\mu_{N}\in{\omega_{S}}^{\top_{n-2}}$. Because $(\omega\otimes_{S}N)_{*}\in{\omega_{S}}^{\top_{n-2}}$ by [32, Corollary 3.4], from the exact sequence (4.19) it yields that $\operatorname{Tor}_{n-2}^{S}(\omega,\operatorname{Ext}^{n}_{R}(\omega,M))=0$. Thus we have $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n-1$. ∎

Lemma 4.18**.**

For any $n\geqslant 0$, the following statements are equivalent.

1. (1)

$\omega\otimes\operatorname{Ext}^{2}_{R}(\omega,-)$* vanishes on $\mathop{\rm Mod}\nolimits R$.* 2. (2)

$(\operatorname{Tor}_{2}^{S}(\omega,-))_{*}$* vanishes on $\mathop{\rm Mod}\nolimits S$.* 3. (3)

$M_{*}\in\mathop{{\rm ac}\mathcal{T}}\nolimits^{2}_{\omega}(S)$* for any $M\in\mathop{\rm Mod}\nolimits R$.* 4. (4)

$\omega\otimes_{S}N\in\mathop{{\rm c}\mathcal{T}}\nolimits^{2}_{\omega}(R)$* for any $N\in\mathop{\rm Mod}\nolimits S$.*

Proof.

By [35, Corollary 6.6], we have $(3)\Leftrightarrow(4)$.

$(1)\Leftrightarrow(4)$ Assume that (1) holds true. Let $N\in\mathop{\rm Mod}\nolimits S$. By [33, Lemma 6.1(2)], we have

[TABLE]

It follows that $\theta_{\omega\otimes_{S}N}$ is a split epimorphism and

[TABLE]

So $\theta_{\omega\otimes_{S}N}$ is a monomorphism, and hence an isomorphism.

Conversely, assume that (4) holds true. Let $M\in\mathop{\rm Mod}\nolimits R$. By [33, Lemma 6.1(2)] again, we have

[TABLE]

It follows that

[TABLE]

$(2)\Leftrightarrow(3)$ Assume that (2) holds true. Let $M\in\mathop{\rm Mod}\nolimits R$. By [33, Lemma 6.1(1)], we have

[TABLE]

It follows that $\mu_{M_{*}}$ is a split monomorphism and

[TABLE]

So $\mu_{M_{*}}$ is an epimorphism, and hence an isomorphism.

Conversely, assume that (3) holds true. Let $N\in\mathop{\rm Mod}\nolimits S$. By [33, Lemma 6.1(1)] again, we have

[TABLE]

It follows that

[TABLE]

∎

The following result establishes the left-right symmetry of certain cograde condition of modules.

Theorem 4.19**.**

For any $n\geqslant 1$, the following statements are equivalent.

1. (1)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i-1$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n$.* 2. (2)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i-1$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n$.* 3. (3)

$\mathop{{\rm co}\Omega}\nolimits^{i}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)=\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{B}}(R)$* for any $1\leqslant i\leqslant n$.* 4. (4)

$\mathop{{\rm co}\Omega}\nolimits^{i}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)=\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{F}_{\omega}}(R)$* for any $1\leqslant i\leqslant n$.* 5. (5)

$\mathop{{\rm co}\Omega}\nolimits^{i}(R)\subseteq\mathop{{\rm c}\mathcal{T}}\nolimits^{i}_{\omega}(R)=\mathop{{\rm co}\Omega}\nolimits^{i}_{\mathcal{P}_{\omega}}(R)$* for any $1\leqslant i\leqslant n$.* 6. (6)

$\Omega^{i}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(S)=\Omega^{i}_{\mathcal{A}}(S)$* for any $1\leqslant i\leqslant n$.* 7. (7)

$\Omega^{i}_{\mathcal{F}}(S)\subseteq\mathop{{\rm ac}\mathcal{T}}\nolimits^{i}_{\omega}(S)=\Omega^{i}_{\mathcal{I}_{\omega}}(S)$* for any $1\leqslant i\leqslant n$.*

Proof.

By Theorem  3.5 and Proposition  4.17, we have $(1)\Leftrightarrow(3)\Leftrightarrow(4)\Leftrightarrow(5)$. By Theorem  3.7 and Proposition  4.16, $(2)\Leftrightarrow(6)\Leftrightarrow(7)$.

In the following, we will prove $(1)\Leftrightarrow(2)$ by induction on $n$. The case for $n=1$ is trivial and the case for $n=2$ follows from Lemma  4.18. Now suppose $n\geqslant 3$.

$(1)\Rightarrow(2)$ Let $N\in\mathop{\rm Mod}\nolimits S$. By the induction hypothesis, we have that $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i-1$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,N)\geqslant n-2$. By Lemma 3.1(2), there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ such that $\sigma_{*}$ is an isomorphism. By Theorem 3.7, we have that $\Omega_{\mathcal{F}}^{n-1}(N)\in\mathop{{\rm ac}\mathcal{T}}\nolimits_{\omega}^{n-1}(S)$ and $\mathop{\rm acTr}\nolimits_{\omega}\Omega_{\mathcal{F}}^{n-1}(N)\in{{}_{R}\omega}^{\bot_{n-1}}$. So

[TABLE]

[TABLE]

Then $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n-2}_{R}(\omega,\operatorname{Tor}^{S}_{n}(\omega,N))\geqslant n-1$ by (1). It follows from Lemma  4.11(1) that $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,N)\geqslant n-1$.

$(2)\Rightarrow(1)$ Let $M\in\mathop{\rm Mod}\nolimits R$. By the induction hypothesis, we have that $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i-1$ for any $1\leqslant i\leqslant n-1$ and $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n-2$. By Lemma 3.1(1), there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ such that $1_{\omega}\otimes\pi$ is an isomorphism. By Theorem 3.5, we have that $\mathop{{\rm co}\Omega}\nolimits^{n-1}(M)\in\mathop{{\rm c}\mathcal{T}}\nolimits_{\omega}^{n-1}(R)$ and $\mathop{\rm cTr}\nolimits_{\omega}\mathop{{\rm co}\Omega}\nolimits^{n-1}(M)\in{\omega_{S}}^{\top_{n-1}}$. So

[TABLE]

[TABLE]

Then $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n-2}^{S}(\omega,\operatorname{Ext}^{n}_{R}(\omega,M))\geqslant n-1$ by (2). It follows from Lemma  4.11(2) that $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,M)\geqslant n-1$. ∎

4.4. Examples

In this subsection, we give some examples for $\omega$ satisfying the (quasi) $n$-cograde condition.

Let $R$ be an artin algebra. Recall that $R$ is called Auslander $n$-Gorenstein if $\mathop{\rm pd}\nolimits_{R^{op}}I^{i}(R_{R})\leqslant i$ for any $0\leqslant i\leqslant n-1$; equivalently $\mathop{\rm pd}\nolimits_{R}I^{i}(_{R}R)\leqslant i$ for any $0\leqslant i\leqslant n-1$ ([16, 26]); and $R$ is called left (resp. right) quasi $n$-Gorenstein if $\mathop{\rm pd}\nolimits_{R}I^{i}(_{R}R)$ (resp. $\mathop{\rm pd}\nolimits_{R^{op}}I^{i}(R_{R})\leqslant i+1$ for any $0\leqslant i\leqslant n-1$ ([22]).

Let $D$ be the ordinary duality between $\mathop{\rm mod}\nolimits R$ and $\mathop{\rm mod}\nolimits R^{op}$. Then $D(R)$ is a semidualizing $(R,R)$-bimodule. Because

[TABLE]

[TABLE]

we have

Example 4.20**.**

2. (1)

$R$ is Auslander $n$-Gorenstein if and only if $D(R)$ satisfies the $n$-cograde condition. 3. (2)

$R$ is left (resp. right) quasi $n$-Gorenstein if and only if $D(R)$ satisfies the left (resp. right) quasi $n$-cograde condition.

So, if putting ${{}_{R}\omega_{S}}={{}_{R}D(R)_{R}}$ in Theorem  4.8 (resp. Theorem  4.14), then all the conditions there are equivalent to that $R$ is Auslander $n$-Gorenstein (resp. right quasi $n$-Gorenstein). Note that the notion of quasi $n$-Gorenstein algebras is not left-right symmetric ([5, p.11]). So, contrary to the $n$-cograde condition, the quasi $n$-cograde condition is not left-right symmetric.

Example 4.21**.**

Let $Q$ be the quiver

[TABLE]

and $R=KQ/<\beta\alpha-\delta\gamma,\varepsilon\gamma>$ with $K$ a field. Take

[TABLE]

By [3, Example VI.2.8(a)], we have that $\omega_{R}$ is a non-injective tilting module with $\mathop{\rm pd}\nolimits_{R}\omega=1$. Thus it is a semidualizing $(R,\mathop{\rm End}\nolimits_{R}(\omega))$-bimodule. It is straightforward to verify that the projective cover $P_{0}(\omega)$ of $\omega$ is $P(1)\oplus P(4)^{2}\oplus P(5)^{2}$. So $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P_{0}(\omega)=0$, and hence $\omega$ satisfies the left and right 1-cograde conditions by Theorem  4.8. Since $\mathop{\rm pd}\nolimits_{R}\omega=1$, we have $\operatorname{Ext}_{R}^{\geqslant 2}(\omega,M)=0$ for any $M\in\mathop{\rm Mod}\nolimits R$. By Theorem  4.8 again, we have that $\omega$ satisfies the left and right $n$-cograde conditions for any $n\geqslant 1$.

5. Two cotorsion pairs

In this section, we will construct two complete cotorsion pairs under any of the equivalent conditions in Theorem  4.19.

For any $n\geqslant 0$, set $\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n}(R):=\{M\in\mathop{\rm Mod}\nolimits R\mid\mathcal{P}_{\omega}(R){\text{-}}\mathop{\rm id}\nolimits_{R}M\leqslant n\}$.

Lemma 5.1**.**

Let $M\in{{}_{R}\omega^{\perp_{n-1}}}$ with $n\geqslant 1$. If $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}_{R}^{n}(\omega,M)\geqslant n-1$, then there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $X\in{{}_{R}\omega^{\perp_{n}}}$ and $Y\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$.

Proof.

Let $M\in{{}_{R}\omega^{\perp_{n-1}}}$. From the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$, we get the following commutative diagram with exact rows

[TABLE]

where the upper sequence is a projective resolution of $\operatorname{Ext}_{R}^{n}(\omega,M)$ in $\mathop{\rm Mod}\nolimits S$. Taking the mapping cone of the diagram (5.1), we get an exact sequence

[TABLE]

Since $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}_{R}^{n}(\omega,M)\geqslant n-1$, we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$. Then we get the following commutative diagram with exact columns and rows

[TABLE]

where

[TABLE]

[TABLE]

Then $Y\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. From the exactness of (5.2) and the middle column in the diagram (5.3), we know that $X\in{{}_{R}\omega^{\perp_{n}}}$. So the top row in the diagram (5.3) is the desired exact sequence. ∎

For any $n\geqslant 0$, set $\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n}(S):=\{N\in\mathop{\rm Mod}\nolimits S\mid\mathcal{I}_{\omega}(S){\text{-}}\mathop{\rm pd}\nolimits_{S}N\leqslant n\}$.

Lemma 5.2**.**

Let $N\in{{\omega_{S}}^{\top_{n-1}}}$ with $n\geqslant 1$. If $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}^{S}_{n}(\omega,N)\geqslant n-1$, then there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $X^{\prime}\in{{\omega_{S}}^{\top_{n}}}$ and $Y^{\prime}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$.

Proof.

Let $N\in{{\omega_{S}}^{\top_{n-1}}}$. From the exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$, we get the following commutative diagram with exact rows

[TABLE]

where the lower sequence is an injective resolution of $\operatorname{Tor}_{n}^{S}(\omega,N)$ in $\mathop{\rm Mod}\nolimits R$. Taking the mapping cone of diagram (5.4), we get an exact sequence

[TABLE]

Since $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}^{S}_{n}(\omega,N)\geqslant n-1$, we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$. Then we get the following commutative diagram with exact columns and rows

[TABLE]

where

[TABLE]

[TABLE]

Then $Y^{\prime}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$. From the exactness of (5.5) and the middle column in the diagram (5.6), we know that $X^{\prime}\in{{\omega_{S}}^{\top_{n}}}$. So the bottom row in the diagram (5.6) is the desired exact sequence. ∎

Lemma 5.3**.**

For any $n\geqslant 0$, we have

1. (1)

$\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n}(R)$* is closed under direct summands and closed under extensions.* 2. (2)

$\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n}(S)$* is closed under direct summands and closed under extensions.*

Proof.

(1) By [33, Lemma 4.6], $\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n}(R)$ is closed under direct summands.

Let

[TABLE]

be an exact sequence in $\mathop{\rm Mod}\nolimits R$ with $A,C\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n}(R)$. It is easy to see that it is $\operatorname{Hom}_{R}(-,\mathcal{P}_{\omega}(R))$-exact. Then $B\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n}(R)$ by the generalized horseshoe lemma (c.f. [23, Lemma 3.1(2)]).

(2) By [34, Lemma 4.7], $\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n}(S)$ is closed under direct summands.

Let

[TABLE]

be an exact sequence in $\mathop{\rm Mod}\nolimits S$ with $A,C\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n}(S)$. It is easy to see that it is $(\omega\otimes_{S}-)$-exact; equivalently it is $\operatorname{Hom}_{R}(-,\mathcal{I}_{\omega}(S))$-exact by [34, p.298, Observation]. Then $B\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n}(S)$ by the generalized horseshoe lemma (c.f. [23, Lemma 3.1(1)]). ∎

Proposition 5.4**.**

Let $n,k\geqslant 1$ and $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+k}_{R}(\omega,M)\geqslant i$ for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n-1$. Then for any $M\in\mathop{\rm Mod}\nolimits R$ and $0\leqslant i\leqslant n-1$, there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $X\in{{}_{R}\omega^{\perp_{i+1}}}$ and $Y\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant i}(R)$.

Proof.

We proceed by induction on $n$. The case for $n=1$ follows from Lemma  5.1. Now suppose $n\geqslant 2$. By the induction hypothesis, for any $0\leqslant i\leqslant n-2$ there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $X_{i}\in{{}_{R}\omega^{\perp_{i+1}}}$ and $Y_{i}\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant i}(R)$. Then

[TABLE]

So $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n}_{R}(\omega,X_{n-2})=\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n+k-1}_{R}(\omega,M)\geqslant n-1$ by assumption. Applying Lemma  5.1, we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $X_{n-1}\in{{}_{R}\omega^{\perp_{n}}}$ and $Y_{n-1}\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. Consider the following push-out diagram

[TABLE]

By Lemma  5.3(1), we have $Y\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. So the middle row in this diagram is the desired sequence. ∎

Proposition 5.5**.**

Let $n,k\geqslant 1$ and $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+k}^{S}(\omega,N)\geqslant i$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n-1$. Then for any $N\in\mathop{\rm Mod}\nolimits S$ and $0\leqslant i\leqslant n-1$, there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $X^{\prime}\in{{\omega_{S}}^{\top_{i+1}}}$ and $Y^{\prime}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant i}(S)$.

Proof.

We proceed by induction on $n$. The case for $n=1$ follows from Lemma  5.2. Now suppose $n\geqslant 2$. By the induction hypothesis, for any $0\leqslant i\leqslant n-2$ there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $X_{i}^{\prime}\in{{\omega_{S}}^{\top_{i+1}}}$ and $Y_{i}^{\prime}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant i}(S)$. Then

[TABLE]

So $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n}^{S}(\omega,X^{\prime}_{n-2})=\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n+k-1}^{S}(\omega,N)\geqslant n-1$ by assumption. Applying Lemma  5.2, we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $X_{n-1}^{\prime}\in{{\omega_{S}}^{\top_{n}}}$ and $Y^{\prime}_{n-1}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{n-1}(S)$. Consider the following pull-back diagram

[TABLE]

By Lemma  5.3(2), we have $Y^{\prime}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$. So the middle column in this diagram is the desired sequence. ∎

Based on the equivalence of (1) and (2) in Theorem  4.19, we have the following

Theorem 5.6**.**

For any $n\geqslant 1$, we have

1. (1)

If one of the equivalent conditions in Theorem  4.19 is satisfied, then the following statements are equivalent.

1. (1.1)

$\mathop{\rm pd}\nolimits_{S^{op}}\omega\leqslant n-1$. 2. (1.2)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}R\leqslant n-1$. 3. (1.3)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P\leqslant n-1$* for any projective $P$ in $\mathop{\rm Mod}\nolimits R$.* 4. (1.4)

$(\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R),{{}_{R}\omega^{\perp_{n}}})$* forms a complete cotorsion pair.*

1. (2)

If one of the equivalent conditions in Theorem  4.19 is satisfied, then the following statements are equivalent.

1. (2.1)

$\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}Q\leqslant n-1$* for some injective cogenerator $Q$ in $\mathop{\rm Mod}\nolimits S$.* 2. (2.2)

$\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}I\leqslant n-1$* for any injective module $I$ in $\mathop{\rm Mod}\nolimits S$.* 3. (2.3)

$({{\omega_{S}}^{\top_{n}}},\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S))$* forms a complete cotorsion pair.*

If $R$ and $S$ are artin algebras, then the statements (2.1)–(2.3) are equivalent to the following

1. (2.4)

$\mathop{\rm pd}\nolimits_{R}\omega\leqslant n-1$.

Proof.

By Lemma  4.5(1), we have $(1.1)\Leftrightarrow(1.2)$.

If $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}R\leqslant n-1$, then $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}F\leqslant n-1$ for any free module $F$ in $\mathop{\rm Mod}\nolimits R$ by [19, Proposition 5.1(b)]. It follows from Lemma  5.3(1) that $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}P\leqslant n-1$ for any projective $P$ in $\mathop{\rm Mod}\nolimits R$. This proves $(1.2)\Leftrightarrow(1.3)$.

$(1.3)\Rightarrow(1.4)$ It is easy to verify that $\operatorname{Ext}_{R}^{1}(A,B)=0$ for any $A\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$ and $B\in{{}_{R}\omega^{\perp_{n}}}$.

Let $M\in\mathop{\rm Mod}\nolimits R$. By Lemma 5.1 when $n=1$ or taking $k=1$ in Proposition  5.4 when $n\geqslant 2$, we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $B\in{{}_{R}\omega^{\perp_{n}}}$ and $A\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. It implies that $M$ has a special ${{}_{R}\omega^{\perp_{n}}}$-preenvelope and ${{}_{R}\omega^{\perp_{n}}}$ is special preenveloping in $\mathop{\rm Mod}\nolimits R$. If $M\in({\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R))^{\bot_{1}}}$, then the exact sequence (5.7) splits. It follows that $M$ is a direct summand of $B$ and $M\in{{}_{R}\omega^{\perp_{n}}}$.

Let

[TABLE]

be an exact sequence in $\mathop{\rm Mod}\nolimits R$ with $P$ projective. By (1.3), we have $P\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. By Lemma 5.1 when $n=1$ or by Proposition  5.4 when $n\geqslant 2$, we have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $B^{\prime}\in{{}_{R}\omega^{\perp_{n}}}$ and $A^{\prime}\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. Consider the following push-out diagram

[TABLE]

Since $\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$ is closed under extensions by Lemma  5.3(1), it follows from the middle column in the above diagram that $A^{\prime\prime}\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. If $M\in{{}^{\bot_{1}}{(_{R}\omega^{\perp_{n}}})}$, then the middle row in the above diagram splits and $M$ is a direct summand of $A^{\prime\prime}$. By Lemma  5.3(1), we have $M\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$. It follows from Lemma  2.7 that $(\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R),{{}_{R}\omega^{\perp_{n}}})$ forms a complete cotorsion pair.

$(1.4)\Rightarrow(1.2)$ By (1.4), we immediately have that ${{}_{R}R}\in\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n-1}(R)$ and $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}R\leqslant n-1$.

If $\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}Q\leqslant n-1$ for some injective cogenerator $Q$ in $\mathop{\rm Mod}\nolimits S$, then any direct product of $Q$ is in $\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$ by [19, Proposition 5.1(c)]. It follows from Lemma  5.3(2) that $\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}I\leqslant n-1$ for any injective module $I$ in $\mathop{\rm Mod}\nolimits S$. This proves $(2.1)\Leftrightarrow(2.2)$.

$(2.2)\Rightarrow(2.3)$ It is easy to verify that $\operatorname{Ext}_{S}^{1}(C,D)=0$ for any $C\in{{\omega_{S}}^{\top_{n}}}$ and $D\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$.

Let $N\in\mathop{\rm Mod}\nolimits S$. By Lemma 5.2 when $n=1$ or taking $k=1$ in Proposition  5.5 when $n\geqslant 2$, we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C\in{{\omega_{S}}^{\top_{n}}}$ and $D\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$. It implies that $N$ has a special ${{\omega_{S}}^{\top_{n}}}$-precover and ${{\omega_{S}}^{\top_{n}}}$ is precovering in $\mathop{\rm Mod}\nolimits S$. If $N\in{{}^{\bot_{1}}(\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)})$, then the exact sequence (5.8) splits. It follows that $N$ is a direct summand of $C$ and $N\in{{\omega_{S}}^{\top_{n}}}$.

Let

[TABLE]

be an exact sequence in $\mathop{\rm Mod}\nolimits S$ with $I$ injective. By (2.2), we have $I\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$. By Lemma 5.2 when $n=1$ or by Proposition  5.5 when $n\geqslant 2$, we have an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $C^{\prime}\in{{\omega_{S}}^{\top_{n}}}$ and $D^{\prime}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$. Consider the following pull-back diagram

[TABLE]

Since $\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$ is closed under extensions by Lemma  5.3(2), it follows from the middle row in the above diagram that $D^{\prime\prime}\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$. If $N\in({{\omega_{S}}^{\top_{n}}})^{\bot_{1}}$, then the middle column in the above diagram splits and $N$ is a direct summand of $D^{\prime\prime}$. By Lemma  5.3(2), we have $N\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$. It follows from Lemma  2.7 that $({{\omega_{S}}^{\top_{n}}},\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S))$ forms a complete cotorsion pair.

$(2.3)\Rightarrow(2.2)$ For any injective module $I$ in $\mathop{\rm Mod}\nolimits S$, by (2.3) we have that $I\in\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n-1}(S)$ and $\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}I\leqslant n-1$.

If $R$ and $S$ are artin algebras, then $\mathop{\rm pd}\nolimits_{R}\omega=\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}D(S_{S})$ by [34, Lemma 4.9]. Because $D(S_{S})$ is an injective cogenerator in $\mathop{\rm Mod}\nolimits S$, $(2.1)\Leftrightarrow(2.4)$ follows. ∎

Observation 5.7**.**

Let $R$ be an artin algebra and ${{}_{R}\omega_{S}}={{}_{R}D(R)_{R}}$. Then we have

1. (1)

$\mathop{\rm pd}\nolimits_{R}\omega=\mathop{\rm id}\nolimits_{R^{op}}R$ and $\mathop{\rm pd}\nolimits_{R^{op}}\omega=\mathop{\rm id}\nolimits_{R}R$. 2. (2)

$\mathcal{P}_{\omega}(R)$ is exactly the subclass of $\mathop{\rm Mod}\nolimits R$ consisting of injective modules. It implies that

1. (2.1)

$\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}M=\mathop{\rm id}\nolimits_{R}M$ for any $M\in\mathop{\rm Mod}\nolimits R$. 2. (2.2)

$\mathcal{P}_{\omega}\text{-}\mathop{\rm id}\nolimits^{\leqslant n}(R)=\mathcal{I}^{\leqslant n}(R):=\{M\in\mathop{\rm Mod}\nolimits R\mid\mathop{\rm id}\nolimits_{R}M\leqslant n\}$. 3. (3)

$\mathcal{I}_{\omega}(R)$ is exactly the subclass of $\mathop{\rm Mod}\nolimits R$ consisting of projective modules. It implies that

1. (3.1)

$\mathcal{I}_{\omega}(R)$-$\mathop{\rm pd}\nolimits_{R}N=\mathop{\rm pd}\nolimits_{R}N$ for any $N\in\mathop{\rm Mod}\nolimits R$. 2. (3.2)

$\mathcal{I}_{\omega}\text{-}\mathop{\rm pd}\nolimits^{\leqslant n}(R)=\mathcal{P}^{\leqslant n}(R):=\{N\in\mathop{\rm Mod}\nolimits R\mid\mathop{\rm pd}\nolimits_{R}N\leqslant n\}$. 4. (4)

By [11, Proposition VI.5.3], it is easy to see that ${{\omega_{R}}^{\top_{n+1}}}={{}^{\bot_{n+1}}{{}_{R}R}}$. 5. (5)

If $R$ is right quasi $(n-1)$-Gorenstein, then all conditions in Theorem  4.19 are satisfied; see Theorem  4.14 and Example  4.20(2).

As an application of Theorem  5.6, we have the following

Corollary 5.8**.**

Let $R$ be a right quasi $(n-1)$-Gorenstein artin algebra with $n\geqslant 1$. Then the following statements are equivalent.

1. (1)

$\mathop{\rm id}\nolimits_{R}R\leqslant n-1$. 2. (2)

$\mathop{\rm id}\nolimits_{R^{op}}R\leqslant n-1$. 3. (3)

$(\mathcal{I}^{\leqslant n-1}(R),{{}_{R}D(R)^{\perp_{n}}})$* forms a complete cotorsion pair.* 4. (4)

$({{}^{\bot_{n}}{{}_{R}R}},\mathcal{P}^{\leqslant n-1}(R))$* forms a complete cotorsion pair.*

Proof.

By Theorem  5.6 and Observation  5.7, we have $(1)\Leftrightarrow(3)$ and $(2)\Leftrightarrow(4)$.

$(1)\Leftrightarrow(2)$ Let $\mathop{\rm id}\nolimits_{R}R\leqslant n-1$. By [6, Theorem 4.7] and the symmetric version of [21, Theorem], we have $\mathop{\rm id}\nolimits_{R^{op}}R\leqslant(n-1)+(n-2)=2n-3$. Conversely, let $\mathop{\rm id}\nolimits_{R^{op}}R\leqslant n-1$. By [35, Theorem 7.5], we have $\mathop{\rm id}\nolimits_{R}R\leqslant n-1$. Now the assertion follows from [40, Lemma A]. ∎

As a consequence of Corollary  5.8, we have the following

Corollary 5.9**.**

For any artin algebra $R$, the following conditions are equivalent.

1. (1)

$\mathop{\rm id}\nolimits_{R}R\leqslant 1$. 2. (2)

$\mathop{\rm id}\nolimits_{R^{op}}R\leqslant 1$.

Furthermore, if $R$ is right quasi 1-Gorenstein, then they are equivalent to each of the following two statements.

1. (3)

$(\mathcal{I}^{\leqslant 1}(R),{{}_{R}D(R)^{\perp_{2}}})$* forms a complete cotorsion pair.* 2. (4)

$({{}^{\bot_{2}}{{}_{R}R}},\mathcal{P}^{\leqslant 1}(R))$* forms a complete cotorsion pair.*

Proof.

The first assertion follows from [21, Corollary 2]. If $R$ is right quasi 1-Gorenstein, then we get the second assertion by putting $n=2$ in Corollary  5.8. ∎

We use $\mathcal{I}(R)$ and $\mathcal{P}(R)$ to denote the subclasses of $\mathop{\rm Mod}\nolimits R$ consisting of injective and projective modules respectively. Putting $n=1$ in Corollary  5.8, we have the following

Corollary 5.10**.**

For any artin algebra $R$, the following statements are equivalent.

1. (1)

$R$* is self-injective.* 2. (2)

$(\mathcal{I}(R),{{}_{R}D(R)^{\perp_{1}}})$* forms a complete cotorsion pair *$($in this case, ${{}_{R}D(R)^{\perp_{1}}}=\mathcal{I}(R)^{\perp_{1}})$. 3. (3)

$({{}^{\bot_{1}}{{}_{R}R}},\mathcal{P}(R))$* forms a complete cotorsion pair *$($in this case, ${{}^{\bot_{1}}{{}_{R}R}}={{}^{\perp_{1}}}{\mathcal{P}(R)})$.

6. Relative finitistic dimensions

In this section, we introduce and study the finitistic $\mathcal{P}_{\omega}(R)$-injective dimension and the $\mathcal{I}_{\omega}(S)$-projective dimension of rings.

The finitistic $\mathcal{P}_{\omega}(R)$-injective dimension ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R$ of $R$ is defined as

[TABLE]

and the finitistic $\mathcal{I}_{\omega}(S)$-projective dimension ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S$ of $S$ is defined as

[TABLE]

Lemma 6.1**.**

For any $n\geqslant 0$ and $k\geqslant 1$, we have

1. (1)

Let $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+k}_{R}(\omega,M)\geqslant i$ for any $M\in\mathop{\rm Mod}\nolimits R$ and $1\leqslant i\leqslant n+1$. If ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R=n$, then $\mathop{\rm pd}\nolimits_{R}\omega\leqslant n+k$. 2. (2)

Let $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+k}^{S}(\omega,N)\geqslant i$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $1\leqslant i\leqslant n+1$. If ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S=n$, then $\mathop{\rm pd}\nolimits_{S^{op}}\omega\leqslant n+k$.

Proof.

(1) Let $M\in\mathop{\rm Mod}\nolimits R$. By Proposition  5.4, there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with $X\in{{}_{R}{\omega}}^{\bot_{n+2}}$ and $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}Y\leqslant n+1$. If ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R=n$, then $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}Y\leqslant n$. Thus we have that

[TABLE]

and $\mathop{\rm pd}\nolimits_{R}\omega\leqslant n+k$.

(2) Let $N\in\mathop{\rm Mod}\nolimits S$. By Proposition  5.5, there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with $X^{\prime}\in{\omega_{S}}^{\top_{n+2}}$ and $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{S}Y^{\prime}\leqslant n+1$. If ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S=n$, then $\mathcal{I}_{\omega}(R)$-$\mathop{\rm pd}\nolimits_{S}Y^{\prime}\leqslant n$. Thus we have that

[TABLE]

and $\mathop{\rm pd}\nolimits_{S^{op}}\omega=\mathop{\rm fd}\nolimits_{S^{op}}\omega\leqslant n+k$. ∎

Lemma 6.2**.**

For any $n\geqslant 0$, we have

1. (1)

Let ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R\leqslant n$ and $N\in\mathop{\rm Mod}\nolimits S$. If $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N\geqslant n+1$, then $N=0$. 2. (2)

Let ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S\leqslant n$ and $H\in\mathop{\rm Mod}\nolimits R$. If $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}H\geqslant n+1$, then $H=0$.

Proof.

(1) Consider a projective resolution

[TABLE]

of $N$ in $\mathop{\rm Mod}\nolimits S$. If $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}N\geqslant n+1$, then we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$, where $M=\mathop{\rm Ker}\nolimits(\omega\otimes_{S}Q_{n+1}\to\omega\otimes_{S}Q_{n})$. By [34, Corollary 3.5], $Q\cong(\omega\otimes_{S}Q)_{*}$ canonically for any projective $Q$ in $\mathop{\rm Mod}\nolimits S$, so $N\cong\operatorname{Ext}^{n+1}_{R}(\omega,M)$. Because ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R\leqslant n$ by assumption, we have that $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}M\leqslant n$ and $N\cong\operatorname{Ext}^{n+1}_{R}(\omega,M)=0$.

(2) Consider an injective resolution

[TABLE]

of $H$ in $\mathop{\rm Mod}\nolimits R$. If $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}H\geqslant n+1$, then we get an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$, where $N=\mathop{\rm Coker}\nolimits({I^{n}}_{*}\to{I^{n+1}}_{*})$. By [32, Lemma 2.5(2)], $\omega\otimes_{S}I_{*}\cong I$ canonically for any injective $I$ in $\mathop{\rm Mod}\nolimits R$, so $H\cong\operatorname{Tor}_{n+1}^{S}(\omega,N)$. Because ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S\leqslant n$ by assumption, we have that $\mathcal{I}_{\omega}(R^{op})$-$\mathop{\rm pd}\nolimits_{S}N\leqslant n$ and $H\cong\operatorname{Tor}_{n+1}^{S}(\omega,N)=0$. ∎

The following is the main result in this section.

Theorem 6.3**.**

For any $k\geqslant 0$, we have

1. (1)

If $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+k}_{R}(\omega,M)\geqslant i$ for any $M\in\mathop{\rm Mod}\nolimits R$ and $i\geqslant 1$, then ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R\leqslant\mathop{\rm pd}\nolimits_{R}\omega\leqslant{\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R+k$. 2. (2)

If $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+k}^{S}(\omega,N)\geqslant i$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $i\geqslant 1$, then ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S\leqslant\mathop{\rm pd}\nolimits_{S^{op}}\omega\leqslant{\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S+k$.

Proof.

(1) Let $\mathop{\rm pd}\nolimits_{R}\omega=n(<\infty)$ and $M\in\mathop{\rm Mod}\nolimits R$ with $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}M=m(<\infty)$. Then there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with all $\omega^{i}$ in $\mathcal{P}_{\omega}(R)$. Since $\mathcal{P}_{\omega}(R)\subseteq\mathcal{B}_{\omega}(R)$ by [19, Corollary 6.1], we have $\mathcal{B}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}M\leqslant\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}M<\infty$. If $m>n$, then it follows from [33, Theorem 4.2] that $\mathcal{B}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}M\leqslant n$ and $\operatorname{Im}f^{n}\in\mathcal{B}_{\omega}(R)$. On the other hand, we have the following exact and split sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with all ${\omega^{i}}_{*}$ projective. So $(\operatorname{Im}f^{n})_{*}$ is projective, and hence $\operatorname{Im}f^{n}\in\mathcal{P}_{\omega}(R)$ by [19, Lemma 5.1(2)]. It yields that $\mathcal{P}_{\omega}(R)$-$\mathop{\rm id}\nolimits_{R}M\leqslant n$, a contradiction. This proves ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R\leqslant\mathop{\rm pd}\nolimits_{R}\omega$.

In the following, we will prove $\mathop{\rm pd}\nolimits_{R}\omega\leqslant{\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R+k$. The case for $k\geqslant 1$ follows from Lemma  6.1(1). Now suppose that $k=0$ and ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R=n(<\infty)$. Let $M\in\mathop{\rm Mod}\nolimits R$. Then $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{n+1}_{R}(\omega,M)\geqslant n+1$ by assumption. It follows from Lemma  6.2(1) that $\operatorname{Ext}^{n+1}_{R}(\omega,M)=0$ and $\mathop{\rm pd}\nolimits_{R}\omega\leqslant n$.

(2) Let $\mathop{\rm pd}\nolimits_{S^{op}}\omega=n(<\infty)$ and $N\in\mathop{\rm Mod}\nolimits S$ with $\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}N=m(<\infty)$. Then there exists an exact sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits S$ with all $U_{i}$ in $\mathcal{I}_{\omega}(S)$. Since $\mathcal{I}_{\omega}(S)\subseteq\mathcal{A}_{\omega}(S)$ by [19, Corollary 6.1], we have $\mathcal{A}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}N<\infty$. If $m>n$, then it follows from the dual result of [33, Theorem 4.2] that $\mathcal{A}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}N\leqslant n$ and $\operatorname{Im}g_{n}\in\mathcal{A}_{\omega}(S)$. On the other hand, we have the following exact and split sequence

[TABLE]

in $\mathop{\rm Mod}\nolimits R$ with all $\omega\otimes_{S}U_{i}$ injective. So $\omega\otimes_{S}\operatorname{Im}g_{n}$ is injective, and hence $\operatorname{Im}g_{n}\in\mathcal{I}_{\omega}(S)$ by [19, Lemma 5.1(3)]. It yields that $\mathcal{I}_{\omega}(S)$-$\mathop{\rm pd}\nolimits_{S}N\leqslant n$, a contradiction. This proves ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S\leqslant\mathop{\rm pd}\nolimits_{S^{op}}\omega$.

In the following, we will prove $\mathop{\rm pd}\nolimits_{S^{op}}\omega\leqslant{\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S+k$. The case for $k\geqslant 1$ follows from Lemma  6.1(2). Now suppose that $k=0$ and ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S=n$. Let $N\in\mathop{\rm Mod}\nolimits S$. Then $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{n+1}^{S}(\omega,N)\geqslant n+1$ by assumption. It follows from Lemma  6.2(2) that $\operatorname{Tor}_{n+1}^{S}(\omega,N)=0$ and $\mathop{\rm pd}\nolimits_{S^{op}}\omega=\mathop{\rm fd}\nolimits_{S^{op}}\omega\leqslant n$. ∎

Putting $k=0$ in Theorem  6.3, we immediately get the following

Corollary 6.4**.**

2. (1)

If $\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i}_{R}(\omega,M)\geqslant i$ for any $M\in\mathop{\rm Mod}\nolimits R$ and $i\geqslant 1$, then ${\rm F}\mathcal{P}_{\omega}{\text{-}}\mathop{\rm id}\nolimits R=\mathop{\rm pd}\nolimits_{R}\omega$. 3. (2)

If $\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i}^{S}(\omega,N)\geqslant i$ for any $N\in\mathop{\rm Mod}\nolimits S$ and $i\geqslant 1$, then ${\rm F}\mathcal{I}_{\omega}{\text{-}}\mathop{\rm pd}\nolimits S$ $=\mathop{\rm pd}\nolimits_{S^{op}}\omega$.

The following is an immediate consequence of Corollaries  4.2 and 6.4.

Corollary 6.5**.**

If $\omega$ satisfies the $n$-cograde condition for all $n$, then

[TABLE]

Combining Theorem  4.19 with the case for $k=1$ in Theorem  6.3, we get the following

Corollary 6.6**.**

We have

[TABLE]

[TABLE]

if either of the following conditions is satisfied.

1. (1)

$\mathop{\rm{T{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Ext}^{i+1}_{R}(\omega,M)\geqslant i$* for any $M\in\mathop{\rm Mod}\nolimits R$ and $i\geqslant 1$.* 2. (2)

$\mathop{\rm{E{\text{-}}cograde}}\nolimits_{\omega}\operatorname{Tor}_{i+1}^{S}(\omega,N)\geqslant i$* for any $N\in\mathop{\rm Mod}\nolimits S$ and $i\geqslant 1$.*

Corollary 6.7**.**

If $\omega$ satisfies the right quasi $n$-cograde condition for all $n$, then

[TABLE]

Proof.

The former equality follows from Proposition  4.12 and Corollary  6.4(1), and the later inequalities follow from the definition of the right quasi $n$-cograde condition and Corollary  6.6. ∎

Observation 6.8**.**

Let $R$ be an artin algebra and ${{}_{R}\omega_{S}}={{}_{R}D(R)_{R}}$. Then we have

1. (1)

By Observation 5.7, we have

[TABLE]

[TABLE] 2. (2)

If $R$ is right (or left) quasi $n$-Gorenstein for all $n$, then $\mathop{\rm id}\nolimits_{R^{op}}R=\mathop{\rm id}\nolimits_{R}R$ ([21, Corollary 4]).

As a consequence of the above results, we have the following

Corollary 6.9**.**

Let $R$ be an artin algebra. Then we have

1. (1)

If $R$ satisfies the Auslander condition (that is, $R$ is Auslander $n$-Gorenstein for all $n$), then

[TABLE] 2. (2)

If $R$ satisfies the right quasi Auslander condition (that is, $R$ is right quasi $n$-Gorenstein for all $n$), then

[TABLE]

Proof.

In view of Example  4.20, Observations 5.7 and 6.8, the assertions follow from Corollaries  6.5 and 6.7 respectively. ∎

Acknowledgements. This research was partially supported by NSFC (Grant Nos. 11571164, 11501144), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and NSF of Guangxi Province of China (Grant No. 2016GXNSFAA380151). The authors thank the referee for the useful suggestions.