The relative homological theory for graded rings was developed in [1, 2], where Gorenstein gr-injective, gr-projective, and gr-flat modules were introduced. Recent years, many authors have studied the existence of envelopes and covers in the category of graded modules and related them with the corresponding envelopes and covers in the category of modules (see [3, 10, 11, 20]).

The theory of cotorsion pairs is very important in relative homological theory. In this paper, we study the relation of cotorsion pairs in $R$-gr and $R$-mod when $R$ is a graded ring by a finite group $G$. Since cotorsion pairs have a close relationship with the theory of envelopes and covers, we get a relationship of the existence of envelopes and covers between the graded and ungraded cases.

In the Second section, we first consider the case of cotorsion pairs over strongly graded rings of finite group. Then we give the relationship of cotorsion pairs in $R$-mod and $S=R*G$-mod when $R$ is any ring and $G$ is a finite group.

In the Third section, we give the definition of rigid cotorsion pairs in $R$-gr and give a equivalent characterization of it. Then we study the relationship of cotorsion pairs between $R$-gr and $R$-mod when $R$ is a ring graded by a finite group $G$ with $|G|^{-1}\in R$.

In this paper, $G$ is always a finite multiplicative group with identity 1. A $G$-graded ring $R$ is a ring with identity 1, together with a direct sum $R =\oplus_{g\in G}R_{g}$ as additive subgroups, such that $R_{g}R_{h}\subseteq R_{gh}$ for all $g, h\in G$. It is well known that $R_{1}$ is a subring of $R$ and $1\in R_{1}$. If $R_{g}R_{h}=R_{gh}$ for all $g, h\in G$, then $R$ is called a strongly graded ring.

A graded left $R$-module $M$ is a left $R$-module endowed with an internal direct sum decomposition $M =\oplus_{\sigma\in G} M_{\sigma}$, where each $M_{\sigma}$ is a subgroup of the additive group of $M$ satisfying $R_{\sigma}M_{\tau}\subseteq M_{\sigma\tau}$ for all $\sigma, \tau\in G$. For graded left $R$-modules $M$ and $N$, ${\rm Hom}_{R-gr}(M, N)$ = $f : M \rightarrow N\ |\ f\ {\rm is} \ R{\rm -linear}\ {\rm and}\ f(M_{\sigma})\subseteq N_{\sigma}$ is the group of all morphisms from $M$ to $N$ in the category $R$-gr of all graded left $R$-modules. An $R$-linear map $f : M \rightarrow N$ is said to be a graded morphism of degree $\tau, \ \tau\in G$, if $f(M_{\sigma})\subseteq M_{\sigma\tau}$ for all $\sigma\in G$. Graded morphisms of degree $\sigma$ build an additive subgroup ${\rm Hom}_{R}(M, N)_{\sigma}$ of ${\rm Hom}_{R}(M, N)$. Then ${\rm Hom}_{R}(M, N) = \oplus_{\sigma\in G}{\rm Hom}_{R}(M, N)_{\sigma}$ is a graded abelian group of type $G$. We will denote ${\rm Ext}^{i}_{R-gr}$ and ${\rm Ext}^{i}_{R}$ as the right derived functors of ${\rm Hom}_{R-gr}$ and ${\rm Hom}_{R}$.

If $M =\oplus_{\sigma\in G}M_{\sigma}$ is a graded left $R$-module and $\sigma\in G$, then $M(\sigma)$ is the graded left $R$-module obtained by putting $M(\sigma)_{\tau}$ = $M_{\sigma\tau}$ for all $\tau\in G$, the graded module $M(\sigma)$ is called the $\sigma$-suspension of $M$. We can see the $\sigma$-suspension as an isomorphism of categories $T_{\sigma}: R-gr\rightarrow R-gr$, given on objects as $T_{\sigma}(M) = M(\sigma)$ for $M \in R-gr$.

There is a number of interesting functors relating the Grothendieck categories $R$-gr and $R$-mod (the category of all left $R$-modules). The forgetful functor $U:R - gr \rightarrow R-mod$ forgets the gradiation. This functor has a right adjoint $F$ which associated to $M\in R-mod$ the graded $R$-module $F(M)=\oplus_{\sigma\in G}(^{\sigma}M)$ (where each $^{\sigma}M$ is a copy of $M$, $^{\sigma}M=\{^{\sigma}m\ |\ m\in M\}$ with the structure of $R$-module given by $r*^{\sigma}m=^{\tau\sigma}m$ for $r\in R_{\tau}$). Following [13], when $G$ is finite, $(U, F)$ is an Frobenius pair (i.e., $(U, F)$ and $(F, U)$ are adjoint pairs). By [15], we know that the forgetful functor $U$ is always separable, and if $n=|G|$ is invertible in $R$, then $F$ is a separable functor.

We recall from [7] ([1]) that a (graded) left $R$-module $M$ is called Gorenstein (gr-) injective if there exists an exact sequence

$\mathscr{E} = \cdots \rightarrow E^{-2} \rightarrow E^{-1} \rightarrow E^{0} \rightarrow E^{1} \rightarrow \cdots$

of (gr-)injective modules such that $M={\rm ker}(E^{0}\rightarrow E^{1})$ and which remains exact whenever ${\rm Hom}_{R}(E, -)$ (${\rm Hom}_{R-gr}(E, -)$) is applied for any (gr-)injective module $E$. Dually we have the definition of Gorenstein (gr-)projective modules.

A (graded) left $R$-module $M$ is called Gorenstein (gr-)flat [8] ([2]) if there exists an exact sequence

$\mathscr{F}=\cdots \rightarrow F^{-2} \rightarrow F^{-1} \rightarrow F^{0} \rightarrow F^{1} \rightarrow \cdots$

of (gr-)flat modules such that $M=ker(F^{0}\rightarrow F^{1})$ and which remains exact whenever $E\otimes_{R}-$ is applied for any (gr-)injective $R$-module $E$.

A cotorsion pair (or cotorsion theory) in an abelian category $\mathcal{A}$ (see [9]) is a pair of classes $(\mathcal{F}, \mathcal{C})$ of objects of $\mathcal{A}$ if the following properties are satisfied:

(1) ${\rm Ext}^{1}_{\mathcal{A}}(F, C) = 0$ for every $F \in\mathcal{F}, C\in \mathcal{C}$.

(2) ${\rm Ext}^{1}_{\mathcal{A}}(F, C) = 0$ for every $F \in \mathcal{F}$, implies $C \in \mathcal{C}$.

(3) ${\rm Ext}^{1}_{\mathcal{A}}(F, C) = 0$ for every $C \in \mathcal{C}$, implies $F \in \mathcal{F}$.

We use $\mathcal{C}^{\bot}$ (resp. $^{\bot}\mathcal{C}$) denote the the class of all objects such that ${\rm Ext}^{1}_{\mathcal{A}}(C, M)=0$ (resp. ${\rm Ext}^{1}_{\mathcal{A}}(M, C)=0$) for every $C\in\mathcal{C}$. A cotorsion pair $(\mathcal {F}, \mathcal {C})$ of objects of $\mathcal{A}$ is called complete (see [9, Corollary 1.2.7]) if every object has a special $\mathcal{C}$-preenvelope (and a special $\mathcal {F}$-precover). Throughout this paper, we denote ${\rm add}(\mathcal{F})$=$M\ |\ M$ is a direct summand of $\oplus_{i=1}^{i=n}F_{i}$, where $n$ is any nonnegative integer and each ${F_{i}} \in \mathcal{F}$, where $\mathcal{F}$ is any class of modules closed under isomorphisms.

We first consider the case of strongly graded ring. When $R$ is a strongly graded ring, $R-gr$ and $ R_{1}-mod$ are equivalent categories. By [6, Theorem 2.8], we know that the equivalence is given by the functors $(-)_{1}:R-gr\rightarrow R_{1}-{\rm mod}$, $(M)_{1}=M_{1}$ and $R\otimes_{R_{1}}-:R_{1}-mod\rightarrow R-gr$. Using these functors, we can easily get the following proposition, but for the completeness, we give a proof here.

Proposition 2.1 Assume that $R$ is a strongly graded ring.

(1) $(\mathcal {F}, \mathcal {C})$ is a cotorsion pair in $R_{1}$-mod if and only if $(R\otimes_{R_{1}}\mathcal {F}, R\otimes_{R_{1}}\mathcal {C})$ is a cotorsion pair in $R$-gr.

(2) $(\mathcal {F}, \mathcal {C})$ is a cotorsion pair in $R$-gr if and only if $((\mathcal {F})_{1}, (\mathcal {C})_{1})$ is a cotorsion pair in $R_{1}$-mod.

Proof We only prove (1), the proof of (2) is similar to that of (1).

For any $M\in \mathcal {F}$ and $N\in \mathcal{C}$, we have that

$Ext^{1}_{R-gr}(R\otimes_{R_{1}}M, R\otimes_{R_{1}}N)\cong Ext^{1}_{R_{1}}(M, (R\otimes_{R_{1}}N)_{1}) \cong Ext^{1}_{R_{1}}(M, N)=0$

since $(R\otimes_{R_{1}}, (-)_{1})$ is an Frobenius pair. Then $R\otimes_{R_{1}}\mathcal {F}\subseteq ^{\bot}{(R\otimes_{R_{1}}\mathcal {C})}$ and $R\otimes_{R_{1}}\mathcal {C}\subseteq (R\otimes_{R_{1}}\mathcal {F})^{\bot}$.

Next we prove that $^{\bot}(R\otimes_{R_{1}}\mathcal {C})\subseteq R\otimes_{R_{1}}\mathcal {F}$. Let $X\in{^{\bot}}(R\otimes_{R_{1}}\mathcal {C})$, we get that

$0=Ext^{1}_{R-gr}(X, R\otimes_{R_{1}}N)\cong Ext^{1}_{R_{1}}((X)_{1}, N)$

for any $N\in\mathcal {C}$. Hence $(X)_{1}\in{^{\bot}}\mathcal {C}=\mathcal {F}$. Then $R\otimes_{R_{1}}(X)_{1}\in R\otimes_{R_{1}}\mathcal {F}$. Since $X\cong R\otimes_{R_{1}}(X)_{1}$, $X\in R\otimes_{R_{1}}\mathcal {F}$.

Finally, we can prove that $(R\otimes_{R_{1}}\mathcal {F})^{\bot}\subseteq R\otimes_{R_{1}}\mathcal {C}$. Hence $(R\otimes_{R_{1}}\mathcal {F}, R\otimes_{R_{1}}\mathcal {C})$ is a cotorsion pair in $R$-gr.

Using a proof similar to the above, we can easily prove that if $(R\otimes_{R_{1}}\mathcal {F}, R\otimes_{R_{1}}\mathcal {C})$ is a cotorsion pair in $R$-gr, then $(\mathcal {F}, \mathcal {C})$ is a cotorsion pair in $R_{1}$-mod.

Let $R$ be any ring and $S=R*G$ the skew group ring, as well known that $S$ is a strongly graded ring. By [16, 8] and [12, Example 2.2], if $G$ is a finite group and $|G|^{-1}\in R$, $S$ is an excellent extension of $R$. Now we recall the definition of excellent extension ([16, 8]). Let $R$ be a subring of the ring $S$ and they have the same identity. The ring $S$ is called an excellent extension of $R$ if the following two conditions are satisfied:

1. $S_{R}$ and $_{R}S$ are free modules with a basis $(1 = a_{1}, a_{2}, \cdots, a_{n})$ such that $a_{i}R= Ra_{i}$ for $i = 1 , \cdots, n$.

2. $S$ is left $R$-projective, that is, if $_{S}M$ is a submodule of $_{S}N$ and $_{R}M$ is a direct summand of $_{R }N$, then $_{S }M$ is a direct summand of $_{S }N$.

Now, we study the relationship of cotorsion pairs between $R$-mod and $S$-mod. Let $i: R\rightarrow S$ be the inclusion map, then we have the induction functor $S\otimes_{R}-: R-{\rm mod}\rightarrow S-{\rm mod}$ and restriction of the scalars functor $_{R}(-):S$-mod$\rightarrow R$-mod. If $M\in S$-mod, then $_{R}M$ will denote the image of $M$ by the restriction of the scalars functor.

Proposition 2.2 Let $R$ be any ring, $S=R*G$ be the skew group ring and $|G|^{-1}\in R$.

(1) If $(\mathcal {F}, \mathcal {C})$ is a cotorsion pair in $R$-mod and $_{R}(S\otimes_{R}\mathcal {F})\subseteq\mathcal {F}$, then $({\rm add}(S\otimes_{R}\mathcal {F}), {\rm add}(S\otimes_{R}\mathcal {C}))$ is a cotorsion pair in $S$-mod.

(2) If $(\mathcal {F'}, \mathcal {C'})$ is a cotorsion pair in $S$-mod and $S\otimes_{R} {_{R}}\mathcal {F'}\subseteq\mathcal {F'}$, then $( {\rm add}(_{R}\mathcal {F'}), {\rm add}(_{R}\mathcal {C'}))$ is a cotorsion pair in $R$-mod.

Proof (1) Assume that $(\mathcal {F}, \mathcal {C})$ is a cotorsion pair in $R$-mod and $_{R}(S\otimes_{R}\mathcal {F})\subseteq\mathcal {F}$, we will prove that $({\rm add}(S\otimes_{R}\mathcal {F}), {\rm add}(S\otimes_{R}\mathcal {C}))$ is a cotorsion pair in $S$-mod.

For any $M'\in {\rm add}(S\otimes_{R}\mathcal {F})$ and $N'\in {\rm add}(S\otimes_{R}\mathcal {C})$, if we want to prove that

${\rm Ext}^{1}_{S}(M', N')=0, $

we only need to prove that ${\rm Ext}^{1}_{S}(S\otimes_{R}M, S\otimes_{R}N)=0$ for any $M\in\mathcal {F}$ and $N\in\mathcal {C}$. By [4, Ⅲ, Proposition 4.14], we know $S\otimes_{R}-\cong {\rm Hom}_{R}(S, -)$ (In fact, the condition $R$ is artinian in [4, Ⅲ, Proposition 4.14] is not necessary). By the definition of excellent extension, $S$ is projective as right $R$-module. Then by [17, Excercise 9.21], we have that

$ {\rm Ext}^{1}_{S}(S\otimes_{R}M, S\otimes_{R}N)\cong {\rm Ext}^{1}_{S}(S\otimes_{R}M, {\rm Hom}_{R}(S, N))\\ \cong Ext^{1}_{R}({\rm Hom}_{S}(S, S\otimes_{R}M), N) \cong {\rm Ext}^{1}_{R}(S\otimes_{R}M, N)=0, $

since $_{R}(S\otimes_{R}\mathcal {F})\subseteq\mathcal {F}$. Thus

${\rm add}(S\otimes_{R}\mathcal {F})\subseteq^{\bot}({\rm add}(S\otimes_{R}\mathcal {C}))$

and

${\rm {\rm add}}(S\otimes_{R}\mathcal {C})\subseteq ({\rm add}(S\otimes_{R}\mathcal {F}))^{\bot}.$

Next we prove that $^{\bot}({\rm add}(S\otimes_{R}\mathcal {C}))\subseteq {\rm add}(S\otimes_{R}\mathcal {F})$. Let $X\in{^{\bot}}({\rm add}(S\otimes_{R}\mathcal {C}))$, then ${\rm Ext}^{1}_{S}(X, S\otimes_{R}N)=0$ for any $N\in\mathcal {C}$. We get that

$0={\rm Ext}^{1}_{S}(X, S\otimes_{R}N)\cong {\rm Ext}^{1}_{S}(X, {\rm Hom}_{R}(S, N))\cong {\rm Ext}^{1}_{R}(_{R}X, N)$

for any $N\in\mathcal {C}$. Hence $_{R}X\in{^{\bot}}\mathcal {C}=\mathcal {F}$. Then $S\otimes_{R} {_{R}}X\in S\otimes_{R}\mathcal {F}$. Since $S$ is left $R$-projective, by [19, Lemma 1.1], $X$ is a direct summand of $S\otimes_{R} {_{R}}X\in S\otimes_{R}\mathcal {F}$. Thus $X\in {\rm add}(S\otimes_{R}\mathcal {F})$.

Finally, let $Y\in ({\rm add}(S\otimes_{R}\mathcal {F}))^{\bot}$, ${\rm Ext}^{1}_{S}(S\otimes_{R}M, Y)=0$ for any $M\in\mathcal {F}$. Then we know that ${\rm Ext}^{1}_{S}(S\otimes_{R}M, Y)\cong {\rm Ext}^{1}_{R}(M, {\rm Hom}_{S}(S, Y))\cong {\rm Ext}^{1}_{R}(M, _{R}Y)$. Thus $_{R}Y\in\mathcal {F}^{\bot}=\mathcal {C}$, $S\otimes_{R}{_{R}}Y\in S\otimes_{R}\mathcal {C}$. Therefore $Y\in {\rm add}(S\otimes_{R}\mathcal {C}).$

Hence $({\rm add}(S\otimes_{R}\mathcal {F}), {\rm add}(S\otimes_{R}\mathcal {C}))$ is a cotorsion pair in $S$-mod.

(2) Assume that $(\mathcal {F'}, \mathcal {C'})$ is a cotorsion pair in $S$-mod and $S\otimes_{R} {_{R}}\mathcal {F'}\subseteq\mathcal {F'}$, we will prove that $({\rm add}(_{R}\mathcal {F'}), {\rm add}(_{R}\mathcal {C'}))$ is a cotorsion pair in $R$-mod.

If we want to prove that ${\rm Ext}^{1}_{R}(M, N)=0$ for any $M\in {\rm add}(_{R}\mathcal {F'})$ and $N\in {\rm add}(_{R}\mathcal {C'})$, we only need to prove that ${\rm Ext}^{1}_{R}(_{R}M, _{R}N)=0$ for any $M\in\mathcal {F'}$ and $N\in\mathcal {C'}$. By [17, Exercise 9.21] we have

${\rm Ext}^{1}_{R}(_{R}M, {_{R}}N)\cong {\rm Ext}^{1}_{R}(_{R}M, {\rm Hom}_{S}(S, N))\cong {\rm Ext}^{1}_{S}(S\otimes_{R}{_{R}}M, N).$

By hypothesis $S\otimes_{R} {_{R}}\mathcal {F'}\subseteq\mathcal {F'}$, we know that ${\rm Ext}^{1}_{R}({_{R}}M, {_{R}}N)\cong {\rm Ext}^{1}_{S}(S\otimes_{R}{{_{R}}}M, N)=0$. Thus ${\rm add}(_{R}\mathcal {F'})\subseteq {^{\bot}}({\rm add}(_{R}\mathcal {C'}))$, ${\rm add}(_{R}\mathcal {C'})\subseteq ({\rm add}(_{R}\mathcal {F'}))^{\bot}$.

Next we prove that

${^{\bot}}({\rm add}(_{R}\mathcal {C'}))\subseteq {\rm add}(_{R}\mathcal {F'}).$

Let $X\in{^{\bot}}({\rm add}(_{R}\mathcal {C'}))$, then

${\rm Ext}^{1}_{R}(X, {_{R}}N)=0$

for any $N\in\mathcal {C'}$. By [17, Excercise 9.21], we get that

${\rm Ext}^{1}_{R}(X, {_{R}}N)\cong {\rm Ext}^{1}_{R}(X, {\rm Hom}_{S}(S, N))\cong {\rm Ext}^{1}_{S}(S\otimes_{R}X, N)$

for any $N\in\mathcal {C'}$. Hence $S\otimes_{R}X\in{^{\bot}}\mathcal {C'}=\mathcal {F'}$. Then $_{R}(S\otimes_{R} X)\in {_{R}}\mathcal {F'}$. By the definition of excellent extension, $X$ is a direct summand of $_{R}(S\otimes_{R} X)$, $X\in {\rm add}(_{R}\mathcal {F'})$.

Finally, let $Y\in ({\rm add}(_{R}\mathcal {F'}))^{\bot}$, then ${\rm Ext}^{1}_{R}(_{R}M, Y)=0$ for any $M\in\mathcal {F'}$. By [17, Exercise 9.21] we know that

$0={\rm Ext}^{1}_{R}(_{R}M, Y)\cong {\rm Ext}^{1}_{R}(S\otimes_{S}M, Y)\cong {\rm Ext}^{1}_{S}(M, {\rm Hom}_{R}(S, Y))\cong {\rm Ext}^{1}_{S}(M, S\otimes_{R}Y)$

for any $M\in\mathcal {F'}$. Thus $S\otimes_{R}Y\in\mathcal {F'}^{\bot}=\mathcal {C'}$, ${_{R}}(S\otimes_{R}Y)\in {_{R}}\mathcal {C'}$ and $Y\in {\rm add}(_{R}\mathcal {C'})$.

Hence $({\rm add}(_{R}\mathcal {F'}), {\rm add}(_{R}\mathcal {C'}))$ is a cotorsion pair in $R$-mod

Lemma 3.1 Let $(\mathcal{F}, \mathcal{C})$ be a cotorsion pair in $R$-gr. The following statements are equivalent:

(1) If $M\in\mathcal{F}$, then $M(\sigma)\in\mathcal{F}$ for every $\sigma\in G$.

(2) If $N\in\mathcal{C}$, then $N(\tau)\in\mathcal{C}$ for every $\tau\in G$.

Proof $(1)\Rightarrow(2)$ For any $M\in\mathcal{F}$, $N\in\mathcal{C}$ and any $\tau\in G$, by hypothesis we have

${\rm Ext}^{1}_{R-gr}(M, N(\tau))\cong {\rm Ext}^{1}_{R-gr}(M(\tau^{-1}), N)=0.$

Then $N(\tau)\in \mathcal{F}^{\bot}=\mathcal{C}$ for any $\tau\in G.$

$(2)\Rightarrow(1)$ is similar to that of $(1)\Rightarrow(2)$.

Remark 3.2 (1) A cotorsion pair satisfying the equivalent conditions in Lemma 3.1 is said to be a rigid cotorsion pair.

(2) $(gr-\mathcal{P}(R), R-gr)$ and $(R-gr, gr-\mathcal{I}(R))$ are rigid cotorsion pairs in $R$-gr, where $gr-\mathcal{P}(R)$ (resp. $gr-\mathcal{I}(R)$) denotes the class of gr-projective (resp. gr-injective) left $R$-modules.

Proposition 3.3 Let $(\mathcal{F}, \mathcal{C})$ be a cotorsion pair in $R$-gr, then $(\mathcal{F}, \mathcal{C})$ is a rigid cotorsion pair if and only if ${\rm EXT}^{1}_{R}(M, N)=0$ for all $M\in\mathcal{F}$ and $N\in\mathcal{C}$.

Proof Since ${\rm Hom}_{R}(-, N)_{\sigma}={\rm Hom}_{R-gr}(-, N(\sigma))$ for every $\sigma\in G$, we get that

${\rm Ext}^{1}_{R}(M, N)_{\sigma}\cong {\rm Ext}^{1}_{R-gr}(M, N(\sigma))$

for any $\sigma\in G$.

Let $M\in \mathcal {F}$ and $N\in \mathcal{C}$, if $(\mathcal{F}, \mathcal{C})$ is a rigid cotorsion pair, then $N(\sigma)\in \mathcal {C}$ for any $\sigma\in G$. Thus ${\rm Ext}^{1}_{R}(M, N)_{\sigma}=0$ for any $\sigma\in G$, and hence ${\rm Ext}^{1}_{R}(M, N)=0$.

Conversely, for any $M\in \mathcal {F}$ and $N\in \mathcal{C}$, we have ${\rm Ext}^{1}_{R-gr}(M, N(\sigma))\cong {\rm Ext}^{1}_{R}(M, N)_{\sigma}=0$ for any $\sigma\in G$. Then for any $N\in \mathcal{C}$ and any $\sigma\in G$, we know $N(\sigma)\in \mathcal {F}^{\bot}=\mathcal{C}$. Hence $(\mathcal{F}, \mathcal{C})$ is a rigid cotorsion pair.

By [5], we know when $R$ is a graded ring by a finite group $G$, $R\#k[G]^{*}$ is the free right and left $R$-module with basis $p_{g}| g \in G$, and there is a category isomorphism between $R$-gr and $R\#k [G]^{*}$-mod given by the functors $(-)_{gr}: R\#k[G]^{*}-{\rm mod}\rightarrow R-gr$, $(-)^{\#}: R-gr \rightarrow R\#k[G]^{*}-{\rm mod}$. In [5], Cohen and Montgomery also proved the duality theorem for Coactions. The Duality Theorem for Coactions says that if $R$ is graded by $G$, then there is an action of $G$ on $R\#k[G]^{*}$, and $(R\#k[G]^{*}) * G \cong M_{n}(R)$ the $n \times n$ matrix ring over $R$, where $n =|G|$.

Theorem 3.4  Let $R$ be a ring graded by a finite group $G$ with $|G|^{-1}\in R$.

(1) If $(\mathcal {F}, \mathcal {C})$ is a cotorsion pair in $R$-mod and $U(F(\mathcal {F}))\subseteq\mathcal {F}$, then $({\rm add}(F(\mathcal {F})), {\rm add}(F(\mathcal {C})))$ is a cotorsion pair in $R$-gr.

(2) If $(\mathcal {F'}, \mathcal {C'})$ is a cotorsion pair in $R$-gr and $F(U( \mathcal {F'}))\subseteq\mathcal {F'}$, then $({\rm add}(U(\mathcal {F'})), {\rm add}(U(\mathcal {C'})))$ is a cotorsion pair in $R$-mod.

(3) If $(\mathcal {F}, \mathcal {C})$ is a rigid cotorsion pair in $R$-gr, then $({\rm add}(U(\mathcal {F})), {\rm add}(U(\mathcal {C})))$ is a cotorsion pair in $R$-mod.

Proof (1) By [15], the functor ${\rm Col}_{G}(-):R-{\rm mod}\rightarrow M_{n}(R)-{\rm mod}$ is an category equivalent functor, then $({\rm Col}_{G}(\mathcal {F}), {\rm Col}_{G}(\mathcal {C}))$ is a cotorsion pair in $M_{n}(R)-{\rm mod}$. Let $i: R\#k[G]^{*}\rightarrow M_{n}(R)\cong(R\#k[G]^{*}) * G $ be inclusion map, we have the restriction functor $i_{*}:M_{n}(R)-{\rm mod}\rightarrow R\#k[G]^{*}-{\rm mod}$ and the induction functor

$i^{*}:R\#k[G]^{*}-{\rm mod}\rightarrow M_{n}(R)-{\rm mod} .$

Since $(R\#k[G]^{*}) * G \cong M_{n}(R)$ is a skew group ring of $R\#k[G]^{*}$, by Proposition 2.2, we know if $i^{*}(i_{*}({\rm Col}_{G}(\mathcal {F})))\subseteq {\rm Col}_{G}(\mathcal {F})$, then $({\rm add}(i_{*}({\rm Col}_{G}(\mathcal {F})), {\rm add}(i_{*}({\rm Col}_{G}(\mathcal {C})))$ is a cotorsion pair in $R\#k[G]^{*}-{\rm mod}$. Since $(-)_{gr}: R\#k[G]^{*}-{\rm mod}\rightarrow R-gr$ is an category equivalent functor, we get that if $i^{*}(i_{*}({\rm Col}_{G}(\mathcal {F})))\subseteq {\rm Col}_{G}(\mathcal {F})$, then $({\rm add}((i_{*}({\rm Col}_{G}(\mathcal {F}))_{gr}), {\rm add}((i_{*}({\rm Col}_{G}(\mathcal {C}))_{gr}))$ is a cotorsion pair in $R$-gr.

Let $H=(-)_{gr}\circ i_{*}\circ Col_{G}(-)$, then $({\rm add}(H(\mathcal {F})), {\rm add}(H(\mathcal {C})))$ is a cotorsion pair in $R$-gr. By [15, Lemma 3.2], we know that the functors $F$ and $H$ are isomorphic. Since $H=(-)_{gr}\circ i_{*}\circ Col_{G}(-)$, we know that $H'={{\rm Col}_{G}}^{-1}(-)\circ i^{*} \circ(-)^{\#}$ is the left adjoint functor of $H$, where ${{\rm Col}_{G}}^{-1}(-): M_{n}(R)-{\rm mod}\rightarrow R-{\rm mod}$ is the inverse of the functor ${\rm Col}_{G}(-)$. Hence the functor $H'$ is isomorphic to $U$. Since ${\rm Col}_{G}(-):R-{\rm mod}\rightarrow M_{n}(R)-{\rm mod}$ is an equivalent functor, we can prove that $U(F(\mathcal {F}))\subseteq\mathcal {F}$ if and only if $i^{*}(i_{*}({\rm Col}_{G}(\mathcal {F})))\subseteq {\rm Col}_{G}(\mathcal {F})$. Hence $({\rm add}(F(\mathcal {F})), {\rm add}(F(\mathcal {C})))$ is a cotorsion pair in $R$-gr.

(2) By a proof similar to that of (1), we get the desired results.

(3) Since $G$ is finite, by the remark following [14, I.2.12], we have

${\rm EXT}^{1}_{R}(M, N)\cong {\rm Ext}^{1}_{R}(M, N)$

for any $M\in\mathcal {F}$ and $N\in \mathcal {C}$. By Proposition 3.3, we know that $Ext^{1}_{R}(M, N)=0$. Hence

${\rm add}(U(\mathcal {F}))\subseteq {^{\bot}}{\rm add}(U(\mathcal {C}))$

and

${\rm add}(U(\mathcal {C}))\subseteq({\rm add}(U(\mathcal {F})))^{\bot}.$

The rest of the proof is similar to that of Proposition 2.2, but for the completeness, we give a proof here. Next we prove that ${^{\bot}}({\rm add}(U(\mathcal {C})))\subseteq {\rm add}(U(\mathcal {F}))$. Let $X\in{^{\bot}}({\rm add}(U(\mathcal {C})))$, then ${\rm Ext}^{1}_{R}(X, U(N))=0$ for any $N\in\mathcal {C}$. Since $(U, F)$ is an Frobenius pair, we get that

${\rm Ext}^{1}_{R}(X, U(N))\cong {\rm Ext}^{1}_{R-gr}(F(X), N)$

for any $N\in\mathcal {C}$. Hence $F(X)\in{^{\bot}}\mathcal {C}=\mathcal {F}$, $U(F(X))\in U(\mathcal {F})$. Since $F$ is separable, by [11, Proposition 5], $X$ is a direct summand of $U(F(X))$, $X\in {\rm add}(U(\mathcal {F}))$.

Finally, let $Y\in ({\rm add}(U(\mathcal {F})))^{\bot}$, then ${\rm Ext}^{1}_{R}(U(M), Y)=0$ for any $M\in\mathcal {F}$. Since

${\rm Ext}^{1}_{R}(U(M), Y)\cong {\rm Ext}^{1}_{R-gr}(M, F(Y))$

for any $M\in\mathcal {F}$, $F(Y)\in\mathcal {F}^{\bot}=\mathcal {C}$, $U(F(Y))\in U(\mathcal {C})$. Since $Y$ is a direct summand of $U(F(Y))$, $Y\in {\rm add}(U(\mathcal {C}))$.

Corollary 3.5 Let $R$ be a ring graded by a finite group $G$ with $|G|^{-1}\in R$.

(1) If $(\mathcal {F}, \mathcal {C})$ is a complete cotorsion pair in $R$-mod and $U(F(\mathcal {F}))\subseteq\mathcal {F}$, then $({\rm add}(F(\mathcal {F})), {\rm add}(F(\mathcal {C})))$ is a complete cotorsion pair in $R$-gr.

(2) If $(\mathcal {F'}, \mathcal {C'})$ is a complete cotorsion pair in $R$-gr and $F(U( \mathcal {F'}))\subseteq\mathcal {F'}$, then $({\rm add}(U(\mathcal {F'})), {\rm add}(U(\mathcal {C'})))$ is a complete cotorsion pair in $R$-mod.

Proof By a proof similar to that of [21, Theorem 3.7], we get the desired results.

From Corollary 3.5, we get a relationship of the existence of envelopes and covers between the graded and ungraded cases.

Lemma 3.6 Let $R$ be a ring graded by a finite group $G$, $M$ a graded left $R$-module and $N$ a left $R$-module.

(1) If $N$ is a Goreinstein injective left $R$-module, then $F(N)$ is Gorenstein gr-injective. If $M$ is Gorenstein gr-injective, then $U(M)$ is Gorenstein injective.

(2) If $N$ is a Goreinstein projective left $R$-module, then $F(N)$ is Gorenstein gr-projective. If $M$ is Goreinstein gr-projective, then $U(M)$ is Gorenstein projective.

(3) Suppose that $R$ is an $n$-FC ring. If $M$ is Gorenstein gr-flat, then $U(M)$ is Gorenstein flat. If $N$ is a Gorenstein flat left $R$-module, then $F(N)$ is Gorenstein gr-flat.

Proof (1) By [1, Proposition 3.6], we know that if $N$ is a Gorenstein injective left $R$-module, then $F(N)$ is Gorenstein gr-injective. Since $G$ is finite, $(U, F)$ is an Frobenius pair. Using a proof similar to that of [1, Proposition 3.6], we can prove that if $M$ is Gorenstein gr-injective, then $U(M)$ is Gorenstein injective.

(2) By [1, Proposition 4.5], we know that if $N$ is a Gorenstein projective left $R$-module, then $F(N)$ is Gorenstein gr-projective. Since $(U, F)$ is an Frobenius pair, we can similarly prove that if $M$ is Gorenstein gr-projective, then $U(M)$ is Gorenstein projective.

(3) By [2, Proposition 3.1 and Theorem 3.3], we get the results.

Let $gr-\mathcal {G}\mathcal {P}(R)$, $gr-\mathcal {G}\mathcal {I}(R)$ and $gr-\mathcal {G}\mathcal {F}(R)$ denote the class of Gorenstein $gr$-projective, Gorenstein $ gr$-injective and Gorenstein $gr$-flat graded left $R$-modules respectively. As a application, we have the following proposition:

Proposition 3.7 If $R$ is a ring graded by a finite group $G$ with $|G|^{-1}\in R$, then we have

(1) $(\mathcal {G}\mathcal {P}(R), \mathcal {G}\mathcal {P}(R)^{\bot})$ is a (complete) cotorsion pair in $R$-mod if and only if $(gr-\mathcal {G}\mathcal {P}(R), gr-\mathcal {G}\mathcal {P}(R)^{\bot})$ is a (complete) cotorsion pair in $R$-gr.

(2) $(^{\bot}\mathcal {G}\mathcal {I}(R), \mathcal {G}\mathcal {I}(R))$ is a (complete) cotorsion pair in $R$-mod if and only if $(^{\bot}gr-\mathcal {G}\mathcal {I}(R), gr-\mathcal {G}\mathcal {I}(R))$ is a (complete) cotorsion pair in $R$-gr.

(3) Suppose that $R$ is an $n$-FC ring, then $(\mathcal {G}\mathcal {F}(R), \mathcal {G}\mathcal {F}(R)^{\bot})$ is a (complete) cotorsion pair in $R$-mod if and only if $(gr-\mathcal {G}\mathcal {F}(R), gr-\mathcal {G}\mathcal {F}(R)^{\bot})$ is a (complete) cotorsion pair in $R$-gr.

Proof (1) Let $(\mathcal{G}\mathcal {P}(R), \mathcal{G}\mathcal{P}(R)^{\bot})$ be a cotorsion pair in $R$-mod, by Theorem 3.4 and Lemma 3.6, $({\rm add}(F(\mathcal{G}\mathcal {P}(R))), {\rm add}(F(\mathcal{G}\mathcal{P}(R)^{\bot}))$ is a cotorsion pair. So we only need to prove that ${\rm add}(F(\mathcal{G}\mathcal {P}(R)))=gr-\mathcal{G}\mathcal {P}(R)$ and ${\rm add}(F(\mathcal{G}\mathcal {P}(R)^{\bot}))=gr-\mathcal{G}\mathcal {P}(R)^{\bot}$.

Since $F(\mathcal{G}\mathcal {P}(R))\subseteq gr-\mathcal{G}\mathcal {P}(R)$ and $gr-\mathcal{G}\mathcal {P}(R)$ is closed under finite direct sums and direct summands, ${\rm add}(F(\mathcal{G}\mathcal {P}(R)))\subseteq gr-\mathcal{G}\mathcal{P}(R)$. Let $M\in gr-\mathcal{G}\mathcal {P}(R)$, since $M$ is a direct summand of $FU(M)$, $M\in {\rm add}(F(\mathcal{G}\mathcal {P}(R)))$ (since $U(M)\in\mathcal{G}\mathcal {P}(R)$ by Lemma 3.6). Hence ${\rm add}(F(\mathcal{G}\mathcal {P}(R)))=gr-\mathcal{G}\mathcal {P}(R)$.

Then we prove that ${\rm add}(F(\mathcal{G}\mathcal {P}(R)^{\bot}))=gr-\mathcal{G}\mathcal {P}(R)^{\bot}$.

Let $N \in \mathcal{G}\mathcal {P}(R)^{\bot}$ and $M \in gr-\mathcal{G}\mathcal {P}(R)$, since $U(M)\in \mathcal{G}\mathcal {P}(R)$, we have

${\rm Ext}^{1}_{R-gr}(M, F(N))\cong {\rm Ext}^{1}_{R}(U(M), N)=0.$

Then $F(N)\in (gr-\mathcal{G}\mathcal {P}(R))^{\bot}$, and ${\rm add}(F(\mathcal{G}\mathcal {P}(R)^{\bot}))\subseteq gr-\mathcal{G}\mathcal {P}(R)^{\bot}$ since $gr-\mathcal{G}\mathcal {P}(R)^{\bot}$ is closed under finite direct sums and direct summands.

Let $N' \in gr-\mathcal{G}\mathcal {P}(R)^{\bot}$ and $M' \in \mathcal{G}\mathcal {P}(R)$, we have

$ {\rm Ext}^{1}_{R}(M', U(N'))\cong {\rm Ext}^{1}_{R-gr}(F(M'), N')=0$

and then $FU(N')\in F(\mathcal{G}\mathcal {P}(R)^{\bot})$. Since $N'$ is a direct summand of $FU(N')$, we know that $N' \in {\rm add}(F(\mathcal{G}\mathcal {P}(R)^{\bot}))$. Thus ${\rm add}(F(\mathcal{G}\mathcal {P}(R)^{\bot}))=gr-\mathcal{G}\mathcal {P}(R)^{\bot}$, we get the desired results.

Conversely, let $(gr-\mathcal{G}\mathcal {P}(R), gr-\mathcal{G}\mathcal{P}(R)^{\bot})$ be a cotorsion pair in $R-gr$. Since $FU(gr-\mathcal{G}\mathcal {P}(R))\subseteq gr-\mathcal{G}\mathcal {P}(R)$, $({\rm add}(U(gr-\mathcal{G}\mathcal {P}(R)), {\rm add}(U(gr-\mathcal{G}\mathcal{P}(R)^{\bot}))$ is a cotorsion pair in $R$-mod. Using Lemma 3.6 and a proof similar to the above, we can prove that

${\rm add}(U(gr-\mathcal{G}\mathcal {P}(R))=\mathcal{G}\mathcal {P}(R).$

Next we prove that

${\rm add}(U(gr-\mathcal{G}\mathcal{P}(R)^{\bot}))=\mathcal{G}\mathcal {P}(R)^{\bot}.$

For any $M\in\mathcal{G}\mathcal {P}(R)$ and $N\in gr-\mathcal{G}\mathcal {P}(R)^{\bot}$, we have the following isomorphisms

${\rm Ext}^{1}_{R}(M, U(N))\cong {\rm Ext}^{1}_{R-gr}(F(M), N)=0, $

since ${\rm add}(F(\mathcal{G}\mathcal {P}(R)))=gr-\mathcal{G}\mathcal {P}(R)$. Hence ${\rm add}(U(gr-\mathcal{G}\mathcal {P}(R)^{\bot}))\subseteq\mathcal{G}\mathcal {P}(R)^{\bot}$ since $\mathcal{G}\mathcal {P}(R)^{\bot}$ is closed under finite direct sums and direct summands. Similarly, we can prove that $\mathcal{G}\mathcal {P}(R)^{\bot}\subseteq {\rm add}(U(gr-\mathcal{G}\mathcal {P}(R)^{\bot}))$. Thus ${\rm add}(U(gr-\mathcal{G}\mathcal {P}(R)^{\bot}))=\mathcal{G}\mathcal {P}(R)^{\bot}$. Hence $(\mathcal{G}\mathcal {P}(R), \mathcal{G}\mathcal {P}(R)^{\bot})$ is a cotorsion pair in $R$-mod.

By Corollary 3.5, we get the desired results.

Using Theorem 3.4, Corollary 3.5, Lemma 3.6 and a proof similar to that of (1), we get (2) and (3).