Gorenstein Flat Modules of Hopf-Galois Extensions

Abstract

Let $A/B$ be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. Then the problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension will be studied. Finally, more relations for the crossed products and smash products will be obtained as applications.

1. Introduction

Hopf-Galois extensions were first studied in the 1960s in the articles of Chase et al. [1] and Chase and Sweedler [2]. In 1981, Kreimer and Takeuchi [3] developed their definition and since then it has undergone continuous development, as the definition is applied in different areas of mathematics. Hopf-Galois extensions contain some structures as examples, such as strongly graded algebras (here H is a group algebra), crossed products (just cleft extensions), smash products and so on.

Gorenstein homological algebra is a generalization of classical homological algebra. Gorenstein module and Gorenstein homological dimensions introduced by Enochs and Jenda [4] are the main research objects of Gorenstein homological algebra. This definition can also be traced back to the paper of Auslander and Bridger [5] for every finitely generated module. In the last 30 years, Gorenstein homological algebra has been developed in the singularity theory, tilting theory, the Tate cohomology, triangulated categories and so on. (see e.g., [6,7,8,9,10,11]).

The global and finitistic dimensions of Hopf-Galois extensions are considered in [12]. We want to use the homological properties of Hopf-Galois extensions used in [12] to study the relationship of Gorenstein flat dimensions and detail the preserving property of Gorenstein flat precovers in Hopf-Galois extensions.

We organized the paper as follows.

In Section 2, we recall some definitions and properties related to Hopf-Galois extension and Gorenstein flat (injective) modules.

In Section 3, we prove that if $A/B$ is a right H-Galois extension over a semisimple Hopf algebra H, then the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A are no more than that of B. We also study the preserving property of Gorenstein flat precovers and the complexity of Hopf-Galois extensions.

In Section 4, more relations for the crossed products and smash products will be obtained, building upon the work provided in Section 3.

2. Preliminaries

Throughout this paper, k is denoted as a field. All algebra structures, linear spaces, etc., will be over k, and I is denoted as the identity map on a linear space V; ⊗ means ${\otimes }_k$ and Hom is always supposed to be over k. For an algebra A, A-Mod and A-mod are denoted as the category of left A-modules and the category of of finitely generated left A-modules, respectively. For a left A-module M, add $\left(M\right)$ is denoted as the full subcategory of A-mod whose objects are direct summands of finite sums of copies of M. The reader is referred to [13,14] as general references for Hopf algebras.

2.1. Hopf Algebra

Following from [14], a Hopf algebra is a k-linear space H with an associative algebra structure $(H,m,1)$, that means $m(m\otimes I)=m(I\otimes m),a1=1a=a,$ for all $a\in A$ and a coassociative coalgebra structure $(H,\mathsf{\Delta },\epsilon )$, that means $(\mathsf{\Delta }\otimes I)\mathsf{\Delta }=(I\otimes \mathsf{\Delta })\mathsf{\Delta },(\epsilon \otimes I)\mathsf{\Delta }=(I\otimes \epsilon )\mathsf{\Delta }=I$, satisfying the following two conditions:

(i)

The comultiplications $\mathsf{\Delta }$ and $\epsilon$ are algebra morphisms:

$$\mathsf{\Delta }\left(gh\right)=\mathsf{\Delta }\left(g\right)\mathsf{\Delta }\left(h\right),\mathsf{\Delta }\left(1\right)=1\otimes 1,$$

$$\epsilon \left(gh\right)=\epsilon \left(g\right)\epsilon \left(h\right),\epsilon \left(1\right)=1.$$

(ii)

There exists a k-linear map $S:H\to H,$ called an antipode, satisfying

$$m(I\otimes S)\mathsf{\Delta }\left(h\right)=(\epsilon \otimes I)(h\otimes 1),$$

$$m(S\otimes I)\mathsf{\Delta }\left(h\right)=(I\otimes \epsilon )(1\otimes h),$$

for all $g,h\in H$.

We use the Sweedler-type notation for the comultiplication: $\mathsf{\Delta }\left(h\right)=h_1\otimes h_2$, for all $h\in H$.

2.2. Hopf-Galois Extension

Following from [13], let A be a right H-comodule algebra over a Hopf algebra H, i.e., let A be an algebra equipped with an H-comodule structure ${\rho }_A:A\to A\otimes H$ (with notation $a\mapsto a_0\otimes a_1$) such that ${\rho }_A$ is an algebra map, that is, ${\rho }_A\left(ab\right)={\rho }_A\left(a\right){\rho }_A\left(b\right),{\rho }_A\left(1_A\right)=1_A\otimes 1_H,\forall a,b\in A$. Let B be the subalgebra of the H-coinvariant elements, i.e., $B:=A^{coH}:=\{a\in A|{\rho }_A\left(a\right)=a\otimes 1\}$. Then, the extension $A/B$ is the right H-Galois if the map

$$\beta :A{\otimes }_{B}A\to A\otimes H,\mathrm{defined}\mathrm{by}a{\otimes }_{B}b\mapsto (a\otimes 1)\rho \left(b\right)=a{b}_0\otimes b_1,$$

is bijective.

2.3. Gorenstein Injective Module and Gorenstein Flat Module

Following from [4], an A-module M is called a Gorenstein injective in A-Mod (resp. in A-mod), if there exists an exact sequence of injective modules in A-Mod (resp. in A-mod)

$${\mathcal{I}}^{•}=···\to E^{-1}\to E^0\to E^1\to E^2\to ···$$

with ${\mathrm{Hom}}_A(E,{\mathcal{I}}^{•})$ exact for any injective module E in A-Mod (resp. in A-mod), such that $M\cong \mathrm{Ker}(E^0\to E^1)$.

Following from [15], an A-module M is called a Gorenstein flat in A-Mod (resp. in A-mod), if there is an exact sequence of flat modules in A-Mod (resp. in A-mod)

$${\mathcal{F}}^{•}=\cdots ⟶F^{-1}⟶F^0⟶F^1⟶F^2⟶\cdots$$

with $I{\otimes }_A{\mathcal{F}}^{•}$ exact for any injective module I in Mod-A (resp. in mod-A), such that $M\cong \mathrm{Im}(F^{-1}\to F^0)$. Denote by A-GF (resp. A-Gf) the full subcategory of Gorenstein flat modules in $A\text{-}\mathrm{Mod}$ (resp. in $A\text{-}\mathrm{mod}$).

By [9], over a right coherent ring there is a connection between Gorenstein flat left modules and Gorenstein injective right modules.

Also following from [15], the Gorenstein flat dimension of a left A-module M, denoting GfdM, is defined as the smallest integer $n\ge 0$ such that M has a GF-resolution of length n. The global Gorenstein flat dimension of A denoting gl.Gfd $\left(A\right)=$ sup{Gfd$M|\forall M\in A$-Mod}.

Following from [9], the (left) finitistic Gorenstein flat dimension of an algebra A denoting FGFD (A) is defined as

$$\mathrm{FGFD}\left(A\right)=\sup \left\{{\mathrm{Gfd}}_{A}M\right|M\mathrm{is}\mathrm{a}\left(\mathrm{left}\right)A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\mathrm{G}\mathrm{f}\mathrm{d}}_{A}M<\infty \}.$$

3. Gorenstein Flat Dimensions for Hopf-Galois Extensions

Let $A/B$ be a right H-Galois extension over a semisimple Hopf algebra H. Then, in this section, we prove that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A are no more than that of B. Additionally, the problem of preserving the property of Gorenstein flat precovers for the Hopf-Galois extension will be studied.

First, consider the following two functors

$$\begin{array}{c}\hfill A{\otimes }_B-:B\text{-}\mathrm{Mod}\to A\text{-}\mathrm{Mod},M\mapsto A{\otimes }_{B}M,\\ \hfill {}_B(-):A\text{-}\mathrm{Mod}\to B\text{-}\mathrm{Mod},M\mapsto M,\end{array}$$

where ${}_B(-)$ is the restriction functor.

According to [12], we have the following lemma.

Lemma 1.

If H is finite-dimensional, then $(A{\otimes }_B-,{}_B(-))$ and $({}_B(-),A{\otimes }_B-)$ are both adjoint pairs.

Remark 1.

Assume that $(F,G)$ is an adjoint pair of functors of Abelian categories. If G is exact, then F preserves projective objects; if F is exact, then G preserves injective objects. In line with [3], if $A/B$ is right H-Galois over a finite-dimensional Hopf algebra H, then A is projective as a left and right B-module. This means that the above functors $A{\otimes }_B-$ and ${}_B(-)$ are both exact, and so they preserve projective objects and injective objects.

We remark here that Gorenstein flat modules are defined by $-{\otimes }_A-$ and Gorenstein projective modules by Hom ${}_A(-,-)$, so they need to be handled in different ways. However, over a right coherent ring there is a good connection between Gorenstein flat left modules and Gorenstein injective right modules (see [9]). So, in the following, we always assume that A is right coherent (and so is B).

Lemma 2.

If $A/B$ is right H-Galois over a finite-dimensional Hopf algebra H, then

(i) 

Let M be a (left) A-module. If M is Gorenstein flat (resp. Gorenstein injective) as an A-module, then it is also Gorenstein flat (resp. Gorenstein injective) as a B-module.

(ii) 

Let M be a (left) B-module. If M is Gorenstein flat (resp. Gorenstein injective) as a B-module, then $A{\otimes }_{B}M$ is also Gorenstein flat (resp. Gorenstein injective) as an A-module.

Proof. 

For a Gorenstein injective case, the proof is similar to the proof of [16] (Lemma 3.1) using Lemma 1 and Remark 1. For a Gorenstein flat case,

(i)

M is a Gorenstein flat (left) A-module if and only if the Pontryagin dual Hom${}_Z(M,Q/Z)$ is a Gorenstein injective (right) A-module, if A is right coherent.

(ii)

If $M\in B$-Mod is Gorenstein flat, then there is a complete flat resolution ${\mathcal{F}}^{•}$, as follows:

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

such that $M\cong$ Im $(F^{-1}\to F^0)$. Since A is projective as a right B-module, we find that $A{\otimes }_B{\mathcal{F}}^{•}$ is exact in A-Mod and $A{\otimes }_{B}M\cong$ Im $(A{\otimes }_B{F}^{-1}\to A{\otimes }_B{F}^0)$. We also find that $A{\otimes }_B{F}^i$ is flat as an A-module for each i according to the Remark. Then, for any injective A-module I, we have

$$I{\otimes }_A\left(A{\otimes }_B{\mathcal{F}}^{•}\right)\cong I{\otimes }_B{\mathcal{F}}^{•}.$$

However, $I_B$ is a right injective B-module, and so $I{\otimes }_B{\mathcal{F}}^{•}$ is exact. By the above, $I{\otimes }_A\left(A{\otimes }_B{\mathcal{F}}^{•}\right)$ is exact. Therefore, $A{\otimes }_{B}M$ is a Gorenstein flat left A-module.

□

The following lemma is provided in [12].

Lemma 3.

If H is semisimple and $A/B$ is right H-Galois, then for any A-module M, M is an A-direct summand of $A{\otimes }_{B}M$.

The following proposition provides the relation between A-Gf and B-Gf.

Proposition 1.

If H is semisimple and $A/B$ is right H-Galois, then A-Gf = add $(A{\otimes }_B(B$-Gf)).

Proof. 

Let F be a Gorenstein flat left B-module. Then, by Lemma 2, $A{\otimes }_{B}F$ is a Gorenstein flat A-module.

Let $F^{\prime }$ be a Gorenstein flat left A-module. Then, by Lemma 3.2, it is also a Gorenstein flat B-module. By Lemma 3, $F^{\prime }$ is a direct summand of $A{\otimes }_B{F}^{\prime }$ as a left A-module.

Thus, A-Gf = add $(A{\otimes }_B(B$-Gf)). □

Lemma 4.

If H is semisimple and $A/B$ is right H-Galois, then for each A-module M, Gfd ${}_{A}M=$ Gfd ${}_{B}M$.

Proof. 

For any A-module M, it is clear that Gfd ${}_{B}M\le$ Gfd ${}_{A}M$, since any GF-resolution of ${}_{A}M$ is a GF-resolution of ${}_{B}M$ by Lemma 2. Conversely, we may assume Gfd ${}_{B}M=n<\infty$; so, there exists a GF-resolution ${\mathcal{G}}^{•}$ of ${}_{B}M$ of length n. Then, $A{\otimes }_B{\mathcal{G}}^{•}$ is a GF-resolution of $A{\otimes }_{B}M$ as an A-module by Lemma 2 and Remark. This follows Gfd $A{\otimes }_{B}M\le n$. By Lemma 3, M is an A-direct summand of $A{\otimes }_{B}M$, so Gfd ${}_{A}M\le$ Gfd ${}_A\left(A{\otimes }_{B}M\right)\le n$. Thus, Gfd ${}_{A}M\le$ Gfd ${}_{B}M$. □

Now, we obtain the main result of this section using Lemma 4.

Theorem 1.

If H is semisimple and $A/B$ is right H-Galois, then gl.Gfd $\left(A\right)$≤ gl.Gfd $\left(B\right)$, FGFD $\left(A\right)\le$ FGFD $\left(B\right)$.

It is shown in [9] that if A is right coherent, then the classical (left) finitistic flat dimension, FFD $\left(A\right)$, is equal to the (left) finitistic Gorenstein flat dimension, FGFD $\left(A\right)$ of an algebra A. So, we obtain the following corollary.

Corollary 1.

If H is semisimple and $A/B$ is right H-Galois, then FFD $\left(A\right)\le$ FFD $\left(B\right)$.

Next, we study the preserving property of Gorenstein flat precovers of Hopf-Galois extensions. We refer to [15] for the notions of precover. Given a class of A-modules $\mathcal{F}$, an F-precover of an A-module M is a morphism $F\stackrel{\phi }{⟶}M$ with $F\in \mathcal{F}$, such that if $F^{\prime }\stackrel{{\phi }^{\prime }}{⟶}M$ is a morphism with $F^{\prime }\in \mathcal{F}$ then there is a morphism $F^{\prime }\stackrel{f}{⟶}F$ such that ${\phi }^{\prime }=\phi f$. Using $\mathcal{F}=A$-GF, we obtain the notation of a Gorenstein flat precover.

Proposition 2.

If H is semisimple and $A/B$ is right H-Galois, then

(i) 

For any left B-module M, M has a Gorenstein flat precover if and only if it is $A{\otimes }_{B}M$.

(ii) 

For any left A-module M, ${}_{A}M$ has a Gorenstein flat precover if and only if it is ${}_{B}M$.

Proof. 

(i)

First, if M has a Gorenstein flat precover, then by the dual case of [17] (Proposition 2.5) and Lemma 1 $A{\otimes }_{B}M$ has a Gorenstein flat precover. Conversely, if $A{\otimes }_{B}M$ has a Gorenstein flat precover, then by [18] (Proposition 6) and Lemma 1, M has a Gorenstein flat precover.

(ii)

Similar to (i).

□

Now, by using Proposition 2, we obtain the preserving property of Gorenstein flat precovers for the Hopf-Galois extension $A/B$.

Theorem 2.

If H is semisimple and $A/B$ is right H-Galois, then every $M\in A$-Mod has a Gorenstein flat precover if and only if every $M\in B$-Mod has a Gorenstein flat precover.

Finally, we study the complexity of Hopf-Galois extensions.

Let A be a finite-dimensional algebra, and M$\in A$-mod with minimal projective resolution

$${\mathcal{P}}^{•}:···\stackrel{f_{n+1}}{⟶}{P}_n\stackrel{f_n}{⟶}···\stackrel{f_2}{⟶}{P}_1\stackrel{f_1}{⟶}{P}_0\stackrel{f_0}{⟶}M⟶0.$$

Then, the complexity of M is defined as

$$\mathrm{c}\mathrm{x}M=\mathrm{i}\mathrm{n}\mathrm{f}\{n\in \mathbb{N}|\dim P_i\le c{i}^{n-1}\mathrm{for}\mathrm{some}\mathrm{positive}c\in \mathbb{Q}\mathrm{and}\mathrm{almost}\mathrm{all}i\ge 0\}$$

We refer to the complexity of M as infinite, if no such n exists. In particular, M is projective if and only if cx M = 0, so it is a measurement of how far M is from being projective. Notice that over the group algebra $A=kG$ of a finite group, every finite dimensional A-module has finite complexity by the Alperin–Evens theorem. The complexity of A is defined as

$$\mathrm{c}\mathrm{x}A=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\mathrm{c}\mathrm{x}M\right|M\in A\text{-}\mathrm{m}\mathrm{o}\mathrm{d}\}$$

Lemma 5.

If H is semisimple and $A/B$ is right H-Galois, then for each finite dimensional left A-module M, cx ${}_{A}M$= cx ${}_{B}M$.

Proof. 

First, since ${}_B(-)$ is exact (see Remark), we know that any projective resolution of M in A-mod is also a projective resolution of M in B-mod. It follows that cx ${}_{B}M\le$ cx ${}_{A}M$.

Conversely, let

$$\mathcal{P}:\cdots ⟶P_j⟶\cdots ⟶P_0⟶{}_{A}M⟶0$$

and

$${\mathcal{P}}^{\prime }:\cdots ⟶P_j^{\prime }⟶\cdots ⟶P_0^{\prime }⟶{}_{B}M⟶0$$

be the minimal projective resolutions of M in A-mod and in B-mod, respectively. Then, by Remark, $A{\otimes }_B{\mathcal{P}}^{\prime }$ is a projective resolution of $A{\otimes }_{B}M$ as an A-module. By Lemma 3, M is an A-direct summand of $A{\otimes }_{B}M$, so we obtain:

$$\dim P_j\le \dim A{\otimes }_B{P}_j^{\prime }\le n·\dim P_j^{\prime },\mathrm{for}\mathrm{some}n.$$

Thus cx ${}_{A}M\le$ cx ${}_{B}M$. □

Following from Lemma 5, we immediately obtain the following result.

Theorem 3.

If H is semisimple and $A/B$ is right H-Galois, then $cxA\le cxB$.

4. Applications

In this section, as the applications of the work explored in Section 3, we obtain more relations for the crossed products and smash products.

The following definitions about crossed products can be found in [13,19]. Let H be a Hopf algebra and A an algebra, if there exists a k-linear map $H\otimes A\to A,h\otimes a\mapsto h·a$ satisfying $h·\left(ab\right)=(h_1·a)(h_2·b)$ and $h·1=\epsilon \left(h\right)1$, for all $h\in H$ and $a,b\in A$, then H is said to measure A.

If the maps $\sigma ,\tau \in$ Hom$(H\otimes H,A)$ satisfy $(\sigma \ast \tau )(h\otimes l)=\sigma (h_1,l_1)\tau (h_2,l_2)=\epsilon \left(h\right)\epsilon \left(l\right)1_A$ and $(\tau \ast \sigma )(h\otimes l)=\tau (h_1,l_1)\sigma (h_2,l_2)=\epsilon \left(h\right)\epsilon \left(l\right)1_A$, for all $h,l\in H$, then we call the map $\sigma$ a convolution invertible with the inverse $\tau$.

Assume the Hopf algebra H measures the algebra A and the map $\sigma \in$ Hom$(H\otimes H,A)$ is convolution invertible, we define a multiplications in the linear space $A\otimes H$ as

$$(a\otimes h)(b\otimes l)=a(h_1·b)\sigma (h_2,l_1)\otimes h_3{l}_2$$

for all $h,l\in H,a,b\in A$. This structure is called the crossed product $A{\#}_{\sigma }H$ of A with H and denoting $a{\#}_{\sigma }h$ for the tensor product $a\otimes h$ in the following.

Remark 2.

 

1. 

If A is a twisted H-module, i.e., $1·a=a,h·(l·a)=\sigma (h_1,l_1)(h_2{l}_2·a){\sigma }^{-1}(h_3,l_3),$ and σ is a cocycle, i.e., $\sigma (h,1)=\sigma (1,h)=\epsilon \left(h\right)1,(h_1·\sigma (l_1,t_1))\sigma (h_2,l_2{t}_2)=\sigma (h_1,l_1)\sigma (h_2{l}_2,t),$ for all $h,l,t\in H,a\in A$, then the crossed product $A{\#}_{\sigma }H$ forms an associative algebra with the unit element $1{\#}_{\sigma }1$.

2. 

If σ is trivial in the crossed product $A{\#}_{\sigma }H$, that is for $h,k\in H$, $\sigma (h,k)=\epsilon \left(h\right)\epsilon \left(k\right)1$, then (1) holds only A is an H-module, and (2) holds obviously. It follows that A is a left H-module algebra. In the case in which we obtain the multiplication in a smash product (see Definition 4.1.3 of [13]),

3. 

Let $A{\#}_{\sigma }H$ be a crossed product. We consider the extension $A{\#}_{\sigma }H/A$ and find that $A{\#}_{\sigma }H/A$ is exactly a right H-Galois extension and has the normal basis property; thus, $A{\#}_{\sigma }H/A$ is a right H-Galois extension (see [20] (Theorem 1.18)). As a special case, $A\#H/A$ is also a right H-Galois extension.

Proposition 3.

If H and its dual $H^{*}$ are semisimple and $A{\#}_{\sigma }H$ is a crossed product, then gl.Gfd $\left(A{\#}_{\sigma }H\right)$= gl.Gfd $\left(A\right)$, FGFD $\left(A{\#}_{\sigma }H\right)=$ FGFD $\left(A\right)$.

Proof. 

Following from Theorem 3 and the fact $A{\#}_{\sigma }H/A$ is a right H-Galois extension, we obtain gl.Gfd $\left(A{\#}_{\sigma }H\right)$≤ gl.Gfd $\left(A\right)$. Conversely, $A{\#}_{\sigma }H$ is a left $H^{*}$-module algebra via

$$g·\left(a{\#}_{\sigma }h\right)=a{\#}_{\sigma }(g\rightharpoonup h)=\langle g,h_2\rangle a{\#}_{\sigma }{h}_1,$$

for $a{\#}_{\sigma }h\in A{\#}_{\sigma }H,g\in H^{*}$ (see [19]). Thus, we may form the smash product algebra $\left(A{\#}_{\sigma }H\right)\#H^{*}$. So, $\left(A{\#}_{\sigma }H\right)\#H^{*}/A{\#}_{\sigma }H$ is also right $H^{*}$-Galois. By the above conclusion and semisimplicity of $H^{*}$, we have gl.Gfd $(\left(A{\#}_{\sigma }H\right)\#H^{*})\le$ gl.Gfd $\left(A{\#}_{\sigma }H\right)$. By [19] (Theorem 2.2), $\left(A{\#}_{\sigma }H\right)\#H^{*}\cong M_n\left(A\right)$, where n = dim H, so it is Morita equivalent to A. This means gl.Gfd $\left(A\right)=$ gl.Gfd $(\left(A{\#}_{\sigma }H\right)\#H^{*})$. Thus,

$$\mathrm{g}\mathrm{l}.\mathrm{G}\mathrm{f}\mathrm{d}\left(A\right)=\mathrm{g}\mathrm{l}.\mathrm{G}\mathrm{f}\mathrm{d}(\left(A{\#}_{\sigma }H\right)\#H^{*})\le \mathrm{g}\mathrm{l}.\mathrm{G}\mathrm{f}\mathrm{d}\left(A{\#}_{\sigma }H\right)\le \mathrm{g}\mathrm{l}.\mathrm{G}\mathrm{f}\mathrm{d}\left(A\right).$$

Therefore, gl.Gfd $\left(A{\#}_{\sigma }H\right)$ = gl.Gfd $\left(A\right)$. The other can be proved in a similar way. □

Corollary 2.

Let $A{\#}_{\sigma }H$ be a crossed product over a semisimple Hopf algebra H. Then, every $M\in A{\#}_{\sigma }H$-Mod has a Gorenstein flat precover if and only if every $M\in A$-Mod has a Gorenstein flat precover.

Proof. 

Based on the fact that $A{\#}_{\sigma }H/A$ is a right H-Galois extension and Theorem 2, we immediately obtain this corollary. □

5. Conclusions

The main objective of this paper is to study Gorenstein flat modules of Hopf-Galois extensions, and we draw the following conclusions.

(1)

We prove that if $A/B$ is a right H-Galois extension over a semisimple Hopf algebra H, then the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B.

(2)

We show that the Hopf-Galois extensions preserve Gorenstein flat precovers.