The (Gorenstein) homological properties and representation dimensions for skew group algebras, or more generally, for smash products and crossed products were discussed by several authors, for example in [4, 13, 14, 16, 17, 20]. In [13], López-Ramos studied the relationship of Gorenstein injective (projective) dimensions between $A$-Mod and $A\#H$-Mod. He showed that under some conditions, $glGid(A) < \infty$ if and only if $glGid(A\#H) < \infty$ (resp. $glGpd(A) < \infty$ if and only if $glGpd(A\#H) < \infty$).

The aim of this paper is to study the relationship of Gorenstein flat (cotorsion) dimensions between $A$-Mod and $A\#H$-Mod. First we prove that over a right coherent ring $A$, if $A\#H/A$ is separable and $\varphi:A\rightarrow A\#H$ is a splitting monomorphism of $(A, A)$-bimodules, $l.Gwd(A)=l.Gwd(A\#H)$. Then we study the relationship of Gorenstein cotorsion dimensions between $A$-Mod and $A\#H$-Mod. We prove that if $A\#H/A$ is separable and $\varphi:A\rightarrow A\#H$ is a splitting monomorphism of $(A, A)$-bimodules, $l.Gcd(A)=l.Gcd(A\#H)$.

Next we recall some notions and facts required in the following.

Throughout this paper, $H$ always denotes a finite-dimensional Hopf algebra over $k$ with comultiplication $\Delta: H\otimes H\rightarrow H$, counit $\varepsilon: H\rightarrow k$ and antipode $S : H\rightarrow H$. A $k$-algebra $A$ is called a left $H$-module algebra if $A$ is a left $H$-module such that $h \cdot(ab)=\sum(h_{(1)} \cdot a) (h_{(2)} \cdot b)$ and $h\cdot1_{A}=\varepsilon(h)1_{A}$ for all $a, b\in A$ and $h\in H$.

Let $A$ be a left $H$-module algebra, the smash product algebra (or semidirect product) of $A$ with $H$, denoted by $A\#H$, is the vector space $A\otimes H$, whose elements are denoted by $a\#h$ instead of $a\otimes h$, with multiplication given by $(a\#h)(b\#l)=\Sigma a(h_{(1)} \cdot b)\#h_{(2)}l$ for $a, b\in A$ and $h, l\in H$. The unit of $A\#H$ is $1\#1$ and we usually view $ah$ as $a\#h$ and $ha$ as $(1\#h)(a\#1)$. In this paper, $A$-Mod and $A\#H$-Mod denote the categories of left $A$-modules and left $A\#H$-modules, respectively.

The notion of separable functor was introduced in [15]. Consider categories $\mathcal {C}$ and $\mathcal {D}$, a covariant functor $F: \mathcal {C}\rightarrow\mathcal {D}$ is said to be separable if for all $M, N$ in $\mathcal {C}$ there are maps $\varphi^{F}_{M, N}:\mbox{Hom}_{\mathcal {D}}(F(M), F(N))\rightarrow \mbox{Hom}_{\mathcal {C}}(M, N))$ satisfying the following conditions.

1. For $\alpha\in {\rm{Hom}}_{\mathcal {C}}(M, N)$, we have $\varphi^{F}_{M, N}(F(\alpha))=\alpha$.

2. Given $M', N'\in\mathcal {C}$, $\alpha \in \mbox{Hom}_{\mathcal {C}}(M, M')$, $\beta \in \mbox{Hom}_{\mathcal {C}}(N, N')$, $f\in \mbox{Hom}_{\mathcal {D}}(F(M), F(N))$ and $g\in {\rm{Hom}}_{\mathcal {D}}(F(M'), F(N'))$ such that the following diagram commutes

then the following diagram is also commutative

Let $\varphi:\ A\rightarrow A\#H$ denote the inclusion map. We can associate to $\varphi$ the restriction of scalars functor $_{A}(-)$: $A\#H$-Mod $\rightarrow A$-Mod, the induction functor $A\#H\otimes_{A}-$ = $Ind(-)$: $A$-Mod $\rightarrow A\#H$-Mod and the coinduction functor $\mbox{Hom}_{A}(A\#H, - )$: $A$-Mod $\rightarrow A\#H$-Mod. It is well known that $A\#H\otimes_{A}- $ is left adjoint to $_{A}(-)$ and that $\mbox{Hom}_{A}(A\#H, - )$ is right adjoint to $_{A}(-)$. Since $H$ is a finite-dimensional Hopf algebra, by [6, Theorem 5], the functor $A\#H\otimes _{A}-$ is isomorphic to ${\rm{Hom}}_{A}(A\#H, \ -)$. So we have a double adjunctions $(A\#H\otimes_{A}-, \ _{A}(-) )$ and $(_{A}(-), \ A\#H\otimes_{A}-)$. Now we consider the separability of functors $_{A}(-)$ and $A\#H\otimes_{A}-$. From [15,Proposition 1.3], we have the following

1. $_{A}(-)$ is separable if and only if $A\#H/A$ is separable.

2. $A\#H\otimes_{A}-$ = $\mbox{Ind}(-)$ is separable if and only if $\varphi$ splits as an $A$-bimodule map.

A left $R$-module $M$ is called Gorenstein flat [7] if there exists an exact sequence

$ \cdots \to {F^{-2}} \to {F^{-1}} \to {F^0} \to {F^1} \to \cdots $

of flat left $R$-modules such that $M=\mbox{ker}(F^{0}\rightarrow F^{1})$ and which remains exact whenever $E\otimes_{R}-$ is applied for any injective right $R$-module $E$. We will say that $M$ has Gorenstein flat dimension less than or equal to $n$ [10] if there exists an exact sequence

$ 0 \to {F_n} \to {F_{n-1}} \to \cdots \to {F_0} \to M \to 0 $

with every $F_{i}$ being Gorenstein flat. If no such finite sequence exsits, define $Gfd_{R}(M)=\infty$; otherwise, if $n$ is the least such integer, define $Gfd_{R}(M)=n$. In [3] left weak Gorenstein global dimension of $R$ was define as $l.Gwd(R)$=sup{$Gfd_{R}(M)\ |\ M$ is any left $R$-module}. A left $R$-module $M$ is called Gorenstein cotorsion [8] if $Ext^{1}_{R}(N, M)$ = 0 for all Gorenstein flat left $R$-modules $N$. We will say that $M$ has Gorenstein cotorsion dimension less than or equal to $n$ [12] if there exists an exact sequence

$ 0 \to M \to {C^0} \to {C^1} \to \cdots \to {C^n} \to 0 $

with every $C^{i}$ being Gorenstein cotorsion. The left global Gorenstein cotorsion dimension $l.Gcd(R)$ of $R$ is defined as the supremum of the Gorenstein cotorsion dimensions of left $R$-modules.

In this paper, $\varphi:\ A\rightarrow A\#H$ always denotes the inclusion map. If $M\in A\#H$-Mod, then $_{A}M$ will denote the image of $M$ by the restriction of the scalars functor $_{A}(-):A\#H$-Mod$\rightarrow A$-Mod.

Lemmma 2.1 (see[11, Corollary 3.6A]) Let $\eta:\ R\rightarrow S$ be a ring homomorphism such that $S$ becomes a flat left $R$-module under $\eta$. Then, for any injective module $M_{S}$, the right $R$-module $M$ (obtained by pullback along $\eta$) is also injective.

Remark 2.2 Let $\varphi:\ A\rightarrow A\#H$ be the inclusion map. Since $A\#H$ is free as a left $A$-module, then from Lemma 2.1 we know that for any injective right $A\#H$-module $M$, the right $A$-module $M$ (obtained by pullback along $\varphi$) is also injective.

Proposition 2.3 (1) If $M\in A$-Mod is Gorenstein flat, then $A\#H\otimes_{A}M$ is Gorenstein flat as a left $A\#H$-module.

(2) If $M\in A\#H$-Mod is Gorenstein flat, then $_{A}M$ is Gorenstein flat as a left $A$-module.

Proof

$(1)$ Since $M$ is a Gorenstein flat left $A$-module, we have an exact sequence

$ \mathfrak{F} \equiv \cdots \to {F^{-2}} \to {F^{-1}} \to {F^0} \to {F^1} \to \cdots $

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

Since $A\#H$ is free as a right $A$-module by [5, Proposition 6.1.7] and $A\#H\otimes_{A}-$ preserves flat modules, we get that $A\#H\otimes_{A}\mathfrak{F}$ is an exact sequence of flat left $A\#H$-modules and

$ A\# H{ \otimes _A}M = {\text{ker}}(A\# H{ \otimes _A}{F^0} \to A\# H{ \otimes _A}{F^1}). $

Finally, let $E'$ be any injective right $A\#H$-module. Then $E'\otimes_{A\#H}(A\#H\otimes_{A}\mathfrak{F})\cong (E'\otimes_{A\#H}A\#H)\otimes_{A}\mathfrak{F}$ is exact since $E'\otimes_{A\#H}A\#H\cong E'$ (as right $A$-modules) is injective by Remark 2.2. Thus $A\#H\otimes_{A}M$ is Gorenstein flat.

(2) Let $M\in A\#H$-Mod be Gorenstein flat, then we have an exact sequence

$ \mathfrak{F}' \equiv \cdots \to F{'^{-2}} \to F{'^{-1}} \to F{'^0} \to F{'^1} \to \cdots $

of flat left $A\#H$-modules such that $M=\mbox{ker}(F'^{0}\rightarrow F'^{1})$ and which remains exact whenever $E\otimes_{A\#H}-$ is applied for any injective right $A\#H$-module $E$. Then $_{A}\mathfrak{F'}$ is an exact sequence of flat left $A$-modules since the functor $_{A}(-)$ is exact and preserves flat modules.

Finally, let $E'$ be any injective right $A$-module. Then

$ E'\otimes_{A}(_{A}\mathfrak{F'})\cong E'\otimes_{A}(A\#H\otimes_{A\#H}\mathfrak{F'})\cong (E'\otimes_{A}A\#H)\otimes_{A\#H}\mathfrak{F'}\ (*). $

Since $H$ is a finite-dimensional Hopf algebra, by [7, Theorem 5], we can easily get that $E'\otimes_{A}A\#H$ is injective as a right $A\#H$-module. By $(*)$ we know that $E'\otimes_{A}({_{A}}\mathfrak{F'})$ is exact. Therefore $_{A}M$ is Gorenstein flat.

Proposition 2.4 Assume that $A\#H/A$ is separable and $\varphi:A\rightarrow A\#H$ is a splitting monomorphism of $(A, A)$-bimodules. Then $A$ is a right coherent ring if and only if $A\#H$ is a right coherent ring.

Proof Let $\{F_{i}\}_{i\in I}$ be a family of flat left $A\#H$-modules, then $_{A}(F_{i})$ is flat as a left $A$-module for every $i$. If we consider the adjoint pair $(A\#H\otimes_{A}-, \ _{A}(-))$, we know that $_{A}(-)$ preserves inverse limits. Thus $_{A}(\prod F_{i})\cong \prod_{A}(F_{i})$. Since $A$ is a right coherent ring, $_{A}(\prod F_{i})\cong \prod_{A}(F_{i})$ is flat as a left $A$-module. Then, we get that $\prod F_{i}$ is a flat left $A\#H$-module. Thus $A\#H$ is a right coherent ring.

Conversely, let $\{F_{i}\}_{i\in I}$ be a family of flat left $A$-modules, since $A\#H\otimes_{A}-$ preserves flat modules, we know that $A\#H\otimes_{A}F_{i}$ is flat as a left $A\#H$-module for every $i$. If we consider the adjoint pair $(_{A}(-), \ A\#H\otimes_{A}-)$, we know that $A\#H\otimes_{A}-$ preserves inverse limits. Thus

$ A\#H\otimes_{A}(\prod F_{i})\cong \prod A\#H\otimes_{A}F_{i}. $

Since $A\#H$ is a right coherent ring, $A\#H\otimes_{A}(\prod F_{i})\cong \prod A\#H\otimes_{A}F_{i}$ is flat as a left $A\#H$-module. Then, we get that $_{A}(A\#H\otimes_{A}\prod F_{i})$ is a flat left $A$-module. Since $\varphi:A\rightarrow A\#H$ is a splitting monomorphism of $(A, A)$-bimodules, we get that the functor $A\#H\otimes_{A}-$ is separable by [15,Proposition 1.3]. Consider the adjoint pair $(\ A\#H\otimes_{A}-, \ _{A}(-))$, by [9,Proposition 5] we know that the natural map $\eta_{M}:M\rightarrow _{A}(A\#H\otimes_{A}M )$ is a split monomorphism for every $M\in A$-Mod. Then $\prod F_{i}$ is a direct summand of $_{A}(A\#H\otimes_{A}\prod F_{i})$. Hence $\prod F_{i}$ is flat as a left $A$-module since the class of flat modules is closed under direct summands. Thus $A$ is a right coherent ring.

Next we consider the relationship of the left weak Gorenstein global dimensions in $A$-Mod and $A\#H$-Mod when $A$ is right coherent.

Theorem 2.5 Let $A$ be a right coherent ring. Assume that $A\#H/A$ is separable and $\varphi:A\rightarrow A\#H$ is a splitting monomorphism of $(A, A)$-bimodules. Then $l.Gwd(A)=l.Gwd(A\#H)$.

Proof For every $n$, we need to show that $Gfd_{A}(M)\leq n$ for every left $A$-module $M$ if and only if $Gfd_{A\#H}(N)\leq n$ for every left $A\#H$-module $N$.

Suppose that $l.Gwd(A\#H)=n$ and let $M$ be any $A$-module. From Proposition 2.3 we know that $A\#H\otimes_{A}-$ and $_{A}(-)$ both preserve Gorenstein flat modules. Thus

$ Gfd_{A\#H}(A\#H\otimes_{A}M)\leq n, Gfd_{A}(_{A}(A\#H\otimes_{A}M))\leq n. $

Since $A\#H\otimes_{A}-$ is separable, $M$ is a direct summand of $_{A}(A\#H\otimes_{A}M)$. Since $A$ is a right coherent ring, by [2, Propositions 2.2 and 2.10] we know that $Gfd_{A}(M)\leq n$.

Since $A\#H/A$ is separable, $_{A}(-)$ is separable by [15, Proposition 1.3]. Similarly, we can prove that if $l.Gwd(A)\leq n$ then $l.Gwd(A\#H)\leq n$.

Lemma 2.6 (1) If $N\in A$-Mod is Gorenstein cotorsion, then $A\#H\otimes_{A}N$ is Gorenstein cotorsion as a left $A\#H$-Mod.

(2) If $N\in A\#H$-Mod is Gorenstein cotorsion, then $_{A}N$ is Gorenstein cotorsion as a left $A$-module.

(3) Let $M\in A\#H$-Mod and $A\#H/A$ be separable. Then $M$ is Gorenstein cotorsion as a left $A\#H$-module if and only if $_{A}M$ is Gorenstein cotorsion as a left $A$-module.

Proof (1) Let $N$ be any Gorenstein cotorsion left $A$-module and $F$ any Gorenstein flat left $A\#H$-module. For $F$ we have an exact sequence $0 \to K \to P \to F \to 0\left( * \right)$ of left $A\#H$-modules with $P$ projective. Since $_{A}(-)$ is exact and preserves Gorenstein flat and projective modules, we have an exact sequence $0{ \to _A}K{ \to _A}P{ \to _A}F \to 0$ with $_{A}P$ projective and $_{A}F$ Gorenstein flat. Hence we have the following commutative diagram:

Note that $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$ are isomorphisms by adjoint isomorphism. Hence

$ {\text{Ho}}{{\text{m}}_{A\# H}}(P,A\# H{ \otimes _A}N) \to {\text{ Ho}}{{\text{m}}_{A\# H}}(K,A\# H{ \otimes _A}N) $

is an epimorphism.

Applying the functor ${\text{Hom}}_{A\#H}(-, A\#H\otimes_{A} N)$ to $(*)$, we get a long exact sequence $ 0 \to {\text{Ho}}{{\text{m}}_{A\# H}}(F, A\# H{ \otimes _A}N) \to {\text{Ho}}{{\text{m}}_{A\# H}}(P, A\# H{ \otimes _A}N) \to {\text{Ho}}{{\text{m}}_{A\# H}}(K, A\# H{ \otimes _A}N) \to Ext_{A\# H}^1(F, A\# H{ \otimes _A}N){\text{ }} \to Ext_{A\# H}^1(P, A\# H{ \otimes _A}N) = 0.$

Since

$ {\text{Ho}}{{\text{m}}_{A\# H}}(P,A\# H{ \otimes _A}N) \to {\text{Ho}}{{\text{m}}_{A\# H}}(K,A\# H{ \otimes _A}N) $

is an epimorphism, we know that $Ext^{1}_{A\#H}(F, A\#H\otimes_{A} N)=0$ for any Gorenstein flat left $A\#H$-module $F$. Hence $A\#H\otimes_{A} N$ is a Gorenstein cotorsion left $A\#H$-module.

(2) Similarly, using the adjoint pair $(A\#H\otimes_{A}-, \ _{A}(-))$ we can prove that $_{A}(-)$ preserves Gorenstein cotorsion modules.

(3) The "only if" part can be gotten directly by (2).

Conversely, if$_{A}M$ is Gorenstein cotorsion as a left $A$-module, then by (1) we know that $A\#H\otimes_{A}{_{A}M}$ is Gorenstein cotorsion. Since $A\#H/A$ is separable, the functor $_{A}(-)$ is separable by [15, Proposition 1.3]. Consider the adjoint pair $(_{A}(-), \ A\#H\otimes_{A}-)$, by [9, Proposition 5] we know that the natural map $\eta_{M}:M\rightarrow A\#H\otimes_{A}{_{A}}M $ is a split monomorphism for every left $A\#H$-module $M$. Then $M$ is a direct summand of $A\#H\otimes_{A}$ $_{A}M$. Hence $M$ is Gorenstein cotorsion as a left $A\#H$-module since the class of Gorenstein cotorsion modules is closed under direct summands.

Proposition 2.7 Let $M\in A\#H$-Mod and $N\in A$-Mod. Then

(1) $Gcd_{A}({_{A}}M)\leq Gcd_{A\#H}(M)$.

(2) $Gcd_{A\#H}(A\#H\otimes_{A}N)\leq Gcd_{A}(N)$.

Proof (1) Assume that $Gcd_{A\#H}(M)=n < \infty$, then there exists an exact sequence of left $A\#H$-modules

$ 0 \to M \to {C^0} \to {C^1} \to \cdots \to {C^n} \to 0 $

with every $C^{i}$ being Gorenstein cotorsion. By Lemma 2.6, ${_{A}}(-)$ preserves Gorenstein cotosion modules, we have an exact sequence of left $A$-modules

$ 0{ \to _A}M{ \to _A}{C^0}{ \to _A}{C^1} \to \cdots { \to _A}{C^n} \to 0 $

with every $C^{i}$ being Gorenstein cotorsion. Thus $Gcd_{A}({_{A}}M)\leq Gcd_{A\#H}(M)$.

(2) Similarly, using Lemma 2.6, we can get that

$ Gcd_{A\#H}(A\#H\otimes_{A}N)\leq Gcd_{A}(N). $

Theorem 2.8 Assume that $A\#H/A$ is separable and $\varphi:A\rightarrow A\#H$ is a splitting monomorphism of $(A, A)$-bimodules, then $l.Gcd(A)=l.Gcd(A\#H)$.

Proof Let $M$ be any left $A$-module. Since $\varphi:A\rightarrow A\#H$ is a splitting monomorphism of $(A, A)$-bimodules, $M$ is a direct summand of $_{A}(A\#H\otimes_{A}M)$. Hence

$ Gcd_{A}(M)\leq Gcd_{A}({_{A}}(A\#H\otimes_{A}M)). $

By Proposition 2.7,

$ Gcd_{A}({_{A}}(A\#H\otimes_{A}M))\leq Gcd_{A\#H}(A\#H\otimes_{A}M)\leq l.Gcd(A\#H). $

Thus $l.Gcd(A)\leq l.Gcd(A\#H).$

Let $N$ be any left $A\#H$-module. Since $A\#H/A$ is separable, $N$ is a direct summand of $A\#H\otimes_{A}{_{A}}N$. Hence

$ Gcd_{A\#H}(N)\leq Gcd_{A\#H}(A\#H\otimes_{A}{_{A}}N). $

By Proposition 2.7,

$ Gcd_{A\#H}(A\#H\otimes_{A}{_{A}}N)\leq Gcd_{A}(_{A}N)\leq l.Gcd(A). $

Thus $l.Gcd(A\#H)\leq l.Gcd(A).$

Corollary 2.9 Let $A$ be a $k$-algebra and $G$ a finite group with $|G|^{-1}\in k$. Then $l.Gcd(A)=l.Gcd(A*G)$.

Proof By the definition of the skew group ring, we know that $A$ is a left $H$-module algebra and $A*G=A\#H$, where $H=kG$. Since $G$ a finite group with $|G|^{-1}\in k$, $H$ is semisimple. Then from [19], we know that $A\#H/A$ is separable. By [1, Lemma 4.5], we know that $A$ is a direct summand of $A\#H$ as $(A, A)$-bimodule. By Theorem 2.8 we immediately get the desired result.