Gorenstein Flat Modules of Hopf-Galois Extensions

: 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.


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.

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 ⊗ 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 (M) 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.

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 ρ A : A → A ⊗ H (with notation a → a 0 ⊗ a 1 ) such that ρ A is an algebra map, that is, Let B be the subalgebra of the H-coinvariant elements, i.e., B := A coH := {a ∈ A| ρ A (a) = a ⊗ 1}. Then, the extension A/B is the right H-Galois if the map is bijective.

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) 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) . Denote by A-GF (resp. A-Gf) the full subcategory of Gorenstein flat modules in A-Mod (resp. in A-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 ≥ 0 such that M has a GF-resolution of length n. The global Gorenstein flat dimension of A denoting gl.Gfd (A) = sup{GfdM| ∀ M ∈ A-Mod}.
Following from [9], the (left) finitistic Gorenstein flat dimension of an algebra A denoting FGFD (A) is defined as

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

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 ⊗ 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 − ⊗ 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
. We also find that A ⊗ 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 However, I B is a right injective B-module, and so I ⊗ B F • is exact. By the above, 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
The following proposition provides the relation between A-Gf and B-Gf.  It is shown in [9] that if A is right coherent, then the classical (left) finitistic flat dimension, FFD (A), is equal to the (left) finitistic Gorenstein flat dimension, FGFD (A) 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 (A) ≤ FFD (B).
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 F , is a morphism with F ∈ F then there is a morphism F f −→ F such that ϕ = ϕ f . Using F = A-GF, we obtain the notation of a Gorenstein flat precover. Finally, we study the complexity of Hopf-Galois extensions. Let A be a finite-dimensional algebra, and M ∈ A-mod with minimal projective resolution

Proposition 2. If H is semisimple and A/B is right H-Galois, then
Then, the complexity of M is defined as 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 ≤ cx A M. Conversely, let P : · · · −→ P j −→ · · · −→ P 0 −→ A M −→ 0 and P : · · · −→ P j −→ · · · −→ P 0 −→ B M −→ 0 be the minimal projective resolutions of M in A-mod and in B-mod, respectively. Then, by Remark, A ⊗ B P is a projective resolution of A ⊗ B M as an A-module. By Lemma 3, M is an A-direct summand of A ⊗ B M, so we obtain: dimP j ≤ dimA ⊗ B P j ≤ n · dimP j , for some n.
Thus cx A M ≤ 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 cx A ≤ cx B.

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 ⊗ A → A, h ⊗ a → h · a satisfying h · (ab) = (h 1 · a)(h 2 · b) and h · 1 = ε(h)1, for all h ∈ H and a, b ∈ A, then H is said to measure A.
Assume the Hopf algebra H measures the algebra A and the map σ ∈ Hom(H ⊗ H, A) is convolution invertible, we define a multiplications in the linear space A ⊗ H as (a ⊗ h)(b ⊗ l) = a(h 1 · b)σ(h 2 , l 1 ) ⊗ h 3 l 2 for all h, l ∈ H, a, b ∈ A. This structure is called the crossed product A# σ H of A with H and denoting a# σ h for the tensor product a ⊗ h in the following.
Proof. Based on the fact that A# σ H/A is a right H-Galois extension and Theorem 2, we immediately obtain this corollary.

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