Gorenstein homological invariant properties under Frobenius extensions
Zhao Zhibing
math.RT math.RA 13B02, 16E10, 16G10, 16G50

Gorenstein homological invariant properties under Frobenius extensions]
Gorenstein homological invariant properties under Frobenius extensions
Zhibing  Zhao]
Zhibing Zhao
Supported by Natural Science Foundation of China (No.11571329), the Natural Science Foundation of Anhui Province (No.1708085MA01) and Project of University Natural Science Research of Anhui Province (No.KJ2015A101).
Email: zbzhao64ahu.edu.cn[
    [
    December 30, 2023
=====================



School of Mathematical Sciences, Anhui University, Hefei, Anhui 230039


  We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying
 module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras, we obtain that
  the extension algebra is CM-finite (resp. CM-free) if and only if so is the base algebra. Furthermore, we prove that the reprensentation dimension of Artin algebras is invariant under separable Frobenius extensions.
 10pt

 2010 Mathematics Subject Classfication: 13B02, 16E10, 16G10, 16G50

Key words: Frobenius extensions, separable extensions, Gorenstein projective modules, reprensentation dimension



§ INTRODUCTION

 Frobenius extensions are firstly introduced by Kasch in <cit.> as a generalization of Frobenius algebra. The fundamental example is the group
 algebras induced by a finite index subgroup. After that, some natural generalizations of different kinds of Frobenius extensions are defined
by  Nakayama-Tzuzuku, Müller and Morita <cit.>. There are other examples of Frobenius extensions including Hopf subalgebras, finite extensions of
enveloping algebras of Lie super-algebra and finite extensions of enveloping algebras of Lie coloralgebras  etc <cit.>.

 Separable extensions are defined firstly by Hirata and Sugano in <cit.> as a generalization of separable algebras, and they made a thorough
 study of these connection with Galois theory for
 noncommutative rings and generalizations of Azumaya algebras.  Separable extensions are closely related to Frobenius extensions. In fact, Frobenius extensions
 is a generalization of Frobenius algebras, while a separable algebra is a Frobenius algebra if it is finite projective over the base ring. A ring
  extension that is both a separable extension and a Frobenius extension is called a separable Frobenius extension. Sugano proved that the central projective
  separable extensions are Frobenius
 extensions in <cit.>. More examples of separable Frobenius extensions can be found in Example <ref>.
  We refer to a lecture due to Kadison <cit.> for more details.

Recall from <cit.> that an A-module G is called Gorenstein projective if there is an exact sequence of projective A-modules

    𝐏:=⋯→ P_1→ P_0→ P^0→ P^1→⋯

in Mod-A which is still exact after applying the functor Hom_A(-,Q) for any projective A-module Q and G= Ker(P_0→ P^0). In this case, the sequence 𝐏 is called a complete projective resolution of G.
It is well-known that the notion of Gorenstein projective modules is a natural extension of  totally reflexive modules
to unnecessarily finitely generated modules (See <cit.>) .
The subcategory of all Gorenstein projective A-modules (resp. all finitely generated Gorenstein projective A-modules) is denoted by 𝒢𝒫(A) (resp. ℱ𝒢𝒫(A)).
The Gorenstein projective dimension of an A-module M, denoted by Gpd(M), is defined as
 Gpd(M)= Inf{n|∃ an exact sequence 0→ G_n→⋯ G_1→ G_0→ M→ 0 with G_i Gorenstein projective}. We set
 Gpd(M)=∞ if there is no such a Gorenstein projective resolution. As a hot topic of Gorenstein homological algebra, Gorenstein projectivity of modules and
 Gorenstein projective dimension of modules are studied widely (see <cit.>).

 The first motivation of this paper is inspired by the question in <cit.>. In <cit.>, Chen introduced a generalization of Frobenius extensions, called the totally reflexive extension,
 and he proved that totally reflexivity of modules is preserved under this kind of ring extension. At the end of <cit.>, Chen pointed out a question:
 is this true for Gorenstein projectivity in general?

 In <cit.>, Ren gave a partial answer for the above question. He proved that for a Frobenius extension, if a module over an extension ring is Gorenstein projective then
 it is also Gorenstein projective as a module over the base ring. Futhermore, the converse holds if the ring extension is a separable extension at the same time (see <cit.>).
 The first main result in this paper extends the result in <cit.>. See Theorem <ref>.

Theorem A. Let  A/S be a Frobenius extension of rings and M a right A-module. Then M_A is a Gorenstein projective module if and
only if M_S is a Gorenstein projective module.

Furthermore, we prove the following statements.

Theorem B. Let A be an algebra over a commutative Artin ring S. If A/S is a separable Frobenius extension, then we have

(1) A is CM-finite if and only if so is S;

(2) A is CM-free if and only if so is S.

It is well-known that determining the representation type of an algebra is fundamental and important in representation theory of Artin algebra. Auslander proved that
there exists a 1-1 correspondence between the Morita equivalent classes of Artin algebras of finite representation type and that of Artin algebras with global dimension
at most 2 and with dominant dimension at least 2 (see <cit.>). Motivated by this correspondence, Auslander introduced the notion of the representation dimension of Artin algebras. He
proved that an Artin algebra is of finite representation type if and only if its representation dimension is at most 2. In this sense, the representation dimension of an
Artin algebra is regarded as a trial to give a reasonable way of measuring homologically how far an Artin algebra is from being of finite representation type. Guo
 proved in <cit.> that the representation dimension of an Artin algebra is invariant under stable equivalences.

 The second motivation of this paper comes from the results in <cit.>. In <cit.>, Huang and Sun proved that the representation dimension of Artin algebras is preserved under excellent
 extensions. It follows from <cit.> that an excellent extension is a Frobenius extension. And an excellent extension is a separable Frobenius extension if the extension
 ring is commutative. But a separable Frobenius extension is not an excellent extension in general,
 see Example <ref> for more details.
 As a separable Frobenius extension version of Theorem 4.8 in <cit.>, we prove the following

Theorem C. Let S be a commutative Artin ring and A an S-algebra. If A is a separable Frobenius extension of S, then
rep.dim(A)= rep.dim(S).

The paper is organized as follows. In Section 2, we give some notations in our terminology and some preliminary results which are used in this paper. In Section 3, we
prove the main result: Theorem A and B, see Theorem <ref> and <ref> respectively.
In Section 4, the representation dimension under separable Frobenius extension of Artin algebras are studied, the main result Theorem C is proved.

Throughout this paper, all rings are associative rings with identity and all modules are unital right modules unless stated otherwise. Let A be a ring.
We denote the category of all right A-module (resp. finitely generated right A-module) by Mod-A (resp. mod-A).


§ PRELIMINARIES


A ring extension A/S is a ring homomorphism @C=0.5cm
   S [r]^l     A. A ring extension is an algebra if S is commutative and l factor as the composition S→ Z(A)↪ A where
   Z(A) is the center of A. The natural bimodule structure of _SA_S is given by s· a· s^':=l(s)· a· l(s^').
   Similarly, we can get some other module structures, for example A_S, _SA_A and _AA_S, etc.

For any ring extension A/S, there is a restriction functor R:  Mod-A→ Mod-S which sends M_A↦ M_S, given by m· s:=m· l(s).
In the opposite direction, there are two natural functors as follows:

(1) T=-_SA_A : Mod-S→ Mod-A is given by M_S↦ M_S A_A.

(2) H= Hom_S(_AA_S,-): Mod-S→ Mod-A is given by M_S↦ Hom_S(_AA_S,M_S).

It is easy to check that both (T,R) and (R,H) are adjoint pairs.
(<cit.>) A ring extension A/S is a Frobenius extension, provided that one of the following equivalent conditions holds:

(1) The functors T and H are naturally equivalent.

(2) _SA_A≅ Hom_S(_AA_S,_SS_S) and A_S is finitely generated projective.

(3) _AA_S≅ Hom_S^op(_SA_A,_SS_S) and _SA is finitely generated projective.

(4) There exists E∈ Hom_S-S(A,S), x_i, y_i∈ A such that ∀ a∈ A, one has ∑_ix_iE(y_ia)=a
and ∑_iE(ax_i)y_i=a.


 A ring extension A/S is a separable extension if and only if

    μ: A_SA→ A,    a⊗ b↦ ab,

is a split epimorphism of A-A-bimodules. If a ring extension A/S is both a Frobenius extension and a separable extension, then
it is called a separable Frobenius extension.


Let A/S be a ring extension and M∈Mod-A. Then M_S is a right S-module. There is a natural surjective map
 π: M_SA→ M given by m a↦ ma for any m∈ M and a∈ A. It is easy to check that π is split as a homomorphism
of S-modules, we denoted by M_S| M_SA_S that M is a direct summand of M_SA as right S-modules. However, π is not split as an A-homomorphism in general.
 The following lemma comes from <cit.>, which is analogous to the results in <cit.> for separable algebras over commutative rings.
(<cit.>) Let A/S be a ring extension. The following are equivalent:

(1) A/S is a separable extension.

(2) For any A-A-bimodule M, M_SA→ M is a split epimorphism of A-A-bimodules.

(3) There exists an element e∈ A_SA, such that μ(e)=1_A and ae=ea for any a∈ A.



(1) (<cit.>)  For any finite group G, ℤG/ℤ is a separable Frobenius extension. Indeed,
 let e=1/| G|∑_g∈ Gg_ℤg^-1∈ℤG_ℤℤG, where | G| is the order of G.
 Then e satiefies the condition (3) of the above lemma.

 (2) (<cit.>) Let K be a field and A=M_4(K). Let S be the subalgebra of A with K-basis consisting of the idempotents and matrix
 units,
 
    e_1=e_11+e_14, e_2=e_22+e_33, e_21, e_31, e_41, e_42, e_43.

 Then A/S is a separable Frobenius extension. But since A_S is not free as a right S-module, A/S is not an excellent extension.

 (3) Let S be a commutative algebra and A an Azumaya algebra over S. Then A/S is a separable Forbenius extension.
  See <cit.> for more details. We note that A is not free over S, so A/S is not an excellent extension in general.

 (4) Every strongly separable extension (see <cit.>) is a separable Frobenius extension. Specific examples of strongly separable extensions can be found in <cit.>.



 Let 𝒞 be a subcategory of mod-A and M∈mod-A. A homomorphism
 f:C→ M in mod-A is called a right 𝒞-approximation of M if C∈𝒞 and the sequence
 Hom_A(-,C)(-,f)⟶ Hom_A(-,M)⟶ 0
is exact in 𝒞. We say that an exact sequence
0⟶
C_nf_n⟶ C_n-1f_n-1⟶⋯f_1⟶ C_0f_0⟶ M⟶ 0
in mod-A is an n-𝒞-resolution of M if C_i∈𝒞 for any 0≤ i≤ n, and the sequence

    @C=0.5cm
      0 [r]     Hom_A(-,C_n) [rr]^(-,f_n)       Hom_A(-,C_n-1) [rr]^(-,f_n-1)      ⋯

  
    @C=0.5cm[rr]^(-,f_1)       Hom_A(-,C_0) [rr]^(-,f_0)       Hom_A(-,M) [r]     0

is exact in 𝒞 (See <cit.>).

We denoted by addM the full subcategory of mod-A consisting of all modules isomorphic to direct summands of finite direct sums of copies of M,
and denoted by GenM the full subcategory of mod-A consisting of all modules X such that there exists an epimorphism M_0↠ X with
M_0∈ addM. The following lemma comes from <cit.>.

(<cit.>) Let A be an Artin algebra and M, X∈ mod-A. If X=X_1⊕ X_2∈ GenM has an n-addM-resolution:

    @C=0.5cm
         0 [r]     M_n [rr]^f_n       M_n-1[rr]^f_n-1      ⋯[rr]^f_1       M_0 [rr]^f_0       X [r]     0 ,

then X_1 has an n-addM-resolution:

    @C=0.5cm
         0 [r]     M_n^'[rr]^f_n^'       M_n-1^'[rr]^f_n-1^'      ⋯[rr]^f_1^'       M_0^'[rr]^f_0^'       X_1 [r]     0 .



Let A be an Artin algebra and M∈mod-A.
Recall that M is called a generator-cogenerator for mod-A if every indecomposable projective module and also every indecomposable injective module in mod-A
are in addM. The following lemma comes from <cit.>.

(<cit.>) Let A be an Artin algebra and M a generator-cogenerator for mod-A.
Then the following statements are equivalent for any n≥ 3.

(1) Any indecomposable module in mod-A has an (n-2)-addM-resolution;

(2) gl.dim End(M_A)≤ n.





§ GORENSTEIN PROJECTIVE DIMENSIONS UNDER FROBENIUS EXTENSIONS

In this section, we will prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if
its underlying module over the base ring is Gorenstien projective. Moreover, we obtain some homological properties,
including the Gorenstein global dimension of rings and the  CM-finiteness and CM-freeness of Artin algebras,
 are invariant under separable Frobenius extensions.

For a ring (or an algebra) A, we denoted by  𝒫(A) the full subcategory of Mod-A consisting of all
projective right A-modules. For a right A-module M, we denote
the projective dimension of M by  pd(M_A). In order to prove that the Gorenstein projectivity of modules is
 preserved under Frobenius extensions, we need the following lemma.




(<cit.>) Let A/S be a Frobenius extension of rings and M a right A-module.
If the underlying module M_S is Gorenstein projective, then the following statements hold.

(1) For any projective right A-module P and any i≥ 1, Ext^i_A(M,P)=0;

(2) M_SA_A≅ Hom_S(_AA,M) is a Gorenstein projective right A-module.

Let  A/S be a Frobenius extension of rings and M a right A-module. Then M_A is a Gorenstein projective module if and
only if M_S is a Gorenstein projective module.


Proof. The necessity holds by Lemma 2.2 in <cit.>. For the sake of completeness, we give the proof as follows.
  Assume that M_A is Gorenstein projective as a right A-module. There exists a complete projective resolution
𝐏:=⋯→ P_1→ P_0→ P^0→ P^1→⋯ in Mod-A such that Hom_A(𝐏,Q) is also exact
for any projective A-module Q and M= Im(P_0→ P^0). By the assumption, A_S is finitely generated projective as a right S-module. Hence
(P_i)_S≅ P_i_AA_S is
projective as an S-module for any i. For any projective S-module L, we have Hom_S(_AA,L)≅ L_SA_A is also projective as a right A-module.
Hence Hom_S(𝐏,L)≅ Hom_S(𝐏_AA_S,L)≅ Hom_A(𝐏,  Hom_S(_AA,L)) is exact. So
𝐏≅𝐏_AA_S in Mod-S is a complete projective resolution of M_S and therefore M_S is a Gorenstein projective module.

Conversely, if M_S is Gorenstein projective as a right S-module. Then we have Ext^i_A(M,P)
=0 for any projective A-module P and any i≥ 1 and M_SA_A≅ Hom_S(_AA,M) are Gorenstein projective as right A-modules by Lemma <ref>. We need to construct a complete projective
resolution of M_A in Mod-A.

Since M_SA_A is Gorenstein projective, there exists a short exact sequence
0⟶
 M_SA_A f⟶  P_0
⟶ G⟶ 0
 in Mod-A such  that P_0∈𝒫(A) and G∈𝒢𝒫(A).
  Applying the restriction functor, we get the exact sequence @C=0.5cm
  0 [r]     M_SA_S @C=0.5cm[rr]^f       P_0 [r]     G [r]     0 in Mod-S such  that P_0∈𝒫(S)
  and G,  M_SA∈𝒢𝒫(S) by the necessity. Let Q be any projective right S-module, and g: M_SA_S→ Q any S-homomorphism.
   
    Q     
    
      0[r]    M_SA [u]_g[r]^f     P_0@..>[ul]_∃ h[r]     G [r]     0

There exists an S-homomorphism h:P_0→ Q such that hf=g since G is Gorenstein projective as an S-module.

  Note that there are two maps i:M→ Hom_S(A,M) given by i(m)(a)=ma and π:M_SA→ M given by π(m a)=ma for any m∈ M and a∈ A.
Since M_SA_A ≅ Hom_S(_AA,M), there is an A-monomorphism, which still denoted it by i, i:M↪ M_SA and it is split
as a homomorphism of S-modules.
It follows that we get an exact sequence @C=0.5cm
  0 [r]     M_A [rr]^fi       P_0 [r]     G_0 [r]     0 in Mod-A where G_0= Coker(fi). Applying the restriction functor, we have an exact sequence
 @C=0.5cm
  0 [r]     M_S [rr]^fi       P_0 [r]     G_0 [r]     0 in Mod-S such that M_S∈𝒢𝒫(S) (since M_S| M_SA_S and M_SA_S is Gorenstein projective) and P_0∈𝒫(S).
 We claim that G_0∈𝒢𝒫(S).

 Let Q be any projective S-module and α:M→ Q any S-homomorphism. Then απ is an S-homomorphism from M_SA to Q. Hence,
 there exists an S-homomorphism β:P_0→ Q such that β f=απ. And so β (fi)=α (π i)=α.
 
    Q     
    
      0[r]    M [u]_α[r]^fi     P_0@..>[ul]_∃β[r]     G_0 [r]     0 
       M_SA[u]_π[ur]_f

 It follows from the exact sequence 
    @C=0.5cm
       0 [r]     Hom_S(G_0,Q) [r]     Hom_S(P_0,Q) [r]     Hom_S(M,Q)
       [r]     Ext_S^1(G_0,Q) [r]     0
 that
   Ext_S^1(G_0,Q)=0. Hence G_0 is Gorenstein projective as an S-module by <cit.>.

 Note that G_0 is  Gorenstein projective as a right S-module, then Ext^i_A(G_0,P)=0 for any projective A-module P and any i≥ 1 and G_0_SA_A≅ Hom_S(_AA,G_0) is  Gorenstein projective as a right A-module by Lemma <ref>. Repeating this process, we get an exact sequence
 𝐐:0→ M→ P_0→ P_1→ P_2→⋯ in Mod-A with P_i projective and  Hom_A(𝐐,P) is also exact for any
 projective right A-module P. Linking up the projective resolution of M_A and 𝐐, we get a complete projective resolution of M_A.
 The proof is completed.□

A functor F𝒞→𝒟 is called a Frobenius functor if there exists a functor
G𝒟→𝒞 such that both (F,G)
and (G,F) are adjoint pairs (see <cit.>). By Definition <ref>, we know that the functors -_SA_A≅ Hom_S(_AA_S,-) induced by
 the Frobenius bimodule (see <cit.>) _AA_S are  Frobenius functors.

Let  A/S be a Frobenius extension of rings. Then the Frobenius bimodule _AA_S induces a Frobenius functor from
𝒢𝒫(A) to 𝒢𝒫(S).

Proof. Put that T=-_SA_A: Mod-S→ Mod-A given by M_S↦ M_S A_A. Then T≅ H= Hom_S(_AA_S,-)
is a Frobenius functor with the restriction functor R as a left and right adjoint  at the same time.

By Lemma <ref> and Theorem <ref>, we get T|_𝒢𝒫(S)⊆𝒢𝒫(A)
and R|_𝒢𝒫(A)⊆𝒢𝒫(S), respectively. Hence T is a Frobenius functor from
𝒢𝒫(A) to 𝒢𝒫(S).□

 Let A/S be a Frobenius extension of rings and M a right A-module. Then Gpd(M_S)≤ Gpd(M_A). Moreover, if  A/S is separable, then
 Gpd(M_S)=  Gpd(M_A).
 
 Proof. Without loss of the generality, we assume that Gpd(M_A)=n<∞. There is a Gorenstein projective resolution of M_A as follows
 
    0→ G_n→ G_n-1→⋯→ G_1→ G_0→ M→ 0

 in Mod-A with G_i Gorenstein projective for any 0≤ i≤ n. Applying the restriction functor, we get the following exact sequence
 
    0→ (G_n)_S→ (G_n-1)_S→⋯→ (G_1)_S→ (G_0)_S→ M_S→ 0

in Mod-S with (G_i)_S≅ G_i_AA_S Gorenstein projective as a right S-module for 0≤ i≤ n
 by Theorem <ref>. Hence Gpd(M_S)≤ n=  Gpd(M_A).

Furthermore, if A/S is separable, then M_A| M_SA_A by Lemma <ref>. Without loss of the generality, we assume that Gpd(M_S)=m≤∞. There exists
a Gorenstein projective resolution of M_S as follows

    0→ L_m→ L_m-1→⋯→ L_1→ L_0→ M→ 0

in Mod-S such that L_i is Gorenstein projective as a right S-module for any 0≤ i≤ m. Applying the functor -_SA_A, we get the following exact sequence

    0→ L_m_SA_A→ L_m-1_SA_A→⋯→ L_1_SA_A→ L_0_SA_A→ M_SA_A→ 0

in Mod-A and L_i_SA_A is Gorenstein projective as a right A-module for 0≤ i≤ m by Lemma <ref>. And so
 Gpd(M_SA_A)≤ m. Hence, by the  <cit.>, Gpd(M_A)≤ m=  Gpd(M_S).□

 Recall that the right Gorenstein global dimension of a ring A, denoted by r. Ggl.dim(A), defined as r.  Ggl.dim(A)=sup{ Gpd(M_A)| M is a right A-module} (see <cit.>). It follows from Proposition <ref> that the right Gorenstein global dimension of rings is preserved under  separable Frobenius extensions.

Let A/S be a separable Frobenius extension of rings. Then r. Ggldim(A)
= r. Ggldim(S).

Proof. It is easy to see r. Ggl.dim(A)≤ r. Ggl.dim(S) by Proposition <ref>.
Let M_S be any S-module. Also by Proposition <ref>,
Gpd(M_SA_S)≤ Gpd(M_SA_A).  Since M_S| M_SA_S, Gpd(M_S)≤ Gpd(M_SA_A) by <cit.>.
It follows that r. Ggl.dim(S)≤ r. Ggl.dim(A).□

5pt




 Recall from <cit.> that an Artin algebra A is Cohen-Macaulay finite, or simply, CM-finite, if A has
only finitely many isomorphism classes of indecomposable finitely generated Gorenstein projective modules in mod-A. It is easy to see that A is CM-finite if and only if
there exists a module G∈mod-A such that ℱ𝒢𝒫(A)= addG. Clearly, A is CM-finite if A is of finite representation type. Some other examples of CM-finite algebra can
be found in <cit.>.
 An Artin algebra A is called Cohen-Macaulay free, or simply, CM-free, if any Gorenstein projective module in mod-A is projective. It is well-known that
 𝒢𝒫(A)=𝒫(A) if gl.dim(A)<∞, and A is CM-free if gl.dim(A)<∞.

 In order to  prove that the  CM-finiteness and CM-freeness of Artin algebras are preserved under separable
 Frobenius extensions, we need the following easy observation, which is maybe known.

  Let A be a separable Frobenius extension over a commutative Artin ring S and M a right A-module. Then we have

(1) pd(M_A)= pd(M_S);

(2) r. gl.dim(A)=r. gl.dim(S).

Proof. (1) Without loss of generality, we assume that pd(M_A)=n<∞ and N_S is any right S-module.
Then Ext^n+i_S(M,N)≅ Ext^n+i_S(M_AA_S,N)≅ Ext^n+i_A(M, Hom_S(_AA_S,N))
=0 for any i≥ 1.
Hence pd(M_S)≤ n= pd(M_A).
 Conversely, we can assume that pd(M_S)=m<∞ and W_A is any right A-module. Since A/S is a separable extension ,
M_A| (M_SA)_A by Lemma <ref>. It follows from
Ext^m+i_A(M_SA_A,W)≅ Ext^m+i_S(M_S, Hom_A(_SA_A,W))=0 that Ext^m+i_A(M,W)=0 for any i≥ 1. Thus
pd(M_A)≤ m= pd(M_S).

(2) It follows from (1) that r. gl.dim(A)≤ r. gl.dim(S). Conversely, let X_S be any right S-module. Then X_S| X_SA_S.
Since pd(X_SA)_S= pd(X_SA)_A by (1), pd(X_S)≤ pd(X_SA)_A. It follws that r. gl.dim(S)≤ r. gl.dim(A).□


Let A be an algebra over a commutative Artin ring S. If A/S is a separable Frobenius extension, then we have

(1) A is CM-finite if and only if so is S;

(2) A is CM-free if and only if so is S.

Proof. (1) Assume that S is CM-finite, there exists a module G∈mod-S such that ℱ𝒢𝒫(S)= addG.
We claim that ℱ𝒢𝒫(A)= add(G_SA)_A. By Lemma <ref>, (G_SA)_A is Gorenstein projective as a right A-module,
and so add(G_SA)_A⊆ℱ𝒢𝒫(A). Let M_A be any indecomposable Gorenstein projective A-module in mod-A. By Proposition <ref>,
M_S is also Gorenstein projective as a right S-module. There exists a positive integer n such that M_S| G_S^n. Hence M_SA_A| (G_SA_A)^n. Since
M_A| M_SA_A by Lemma <ref>, M_A| (G_SA_A)^n. Thus ℱ𝒢𝒫(A)⊆ add(G_SA)_A.

Conversely, if A is CM-finite, then there exists a module W∈mod-A such that ℱ𝒢𝒫(A)= addW. It suffices to prove that
ℱ𝒢𝒫(S)= addW_S. By Proposition <ref>, W_S is Gorenstien projective as a right S-module. Hence addW_S⊆ℱ𝒢𝒫(S).
Let X_S be any indecomposable Gorenstein projective S-module in mod-S. By Lemma <ref>, X_SA_A is also Gorenstein projective as a right
A-module, and so X_SA_A| W_A^n for some positive integer n. Applying the restriction functor, we get X_SA_S| W_S^n. It follows from X_S| X_SA_S that
X_S| W_S^n. Hence ℱ𝒢𝒫(S)⊆ addT_S.

(2) Assume that S is CM-free and G_A is any Gorenstein projective right A-module. By Proposition <ref>, G_S is
Gorenstein projective as a right S-module. Hence G_S is projective by assumption. It follows from Lemma <ref>(1) that G_A is also projective as
a right A-module. Therefore A is CM-free.

Conversely, if A is CM-free and Q_S is any Gorenstein projective right S-module. By Lemma <ref>, Q_SA_A is Gorenstein projective as
a right A-module. Hence Q_SA_A is projective by assumption.  It follows from Lemma <ref>(1) that Q_SA_S is  projective as a right S-module.
Hence Q_S is a projective right S-module as a direct summand of Q_SA_S. So S is CM-free. □







§ REPRESENTATION DIMENSIONS UNDER SEPARABLE FROBENIUS EXTENSIONS

In this section, we will show that the representation dimension of Artin algebras is invariant under separable Frobenius extensions.

 Recall that a module M∈mod-A is called an additive generator for mod-A if any indecomposable module in mod-A is in addM. Obviously, an Artin algebra
A is of finite representation type if and only if A has an additive generator.

Let A be an algebra over a commutative Artin ring S. If A/S is a separable Frobenius extension, 
then A is of finite representation type if and only if so is S.

Proof. We assume that A is of finite representation type. Then there is an additive generator M_A for mod-A. We claim that M_S≅ M_AA_S
is an additive generator for mod-S. Let X_S∈mod-S be any indecomposable right S-module. Then X_SA_A| M_A^n for some positive integer n. Applying the
restriction functor, we get X_SA_S| M_S^n. It follows from X_S| X_SA_S that X_S| M_S^n. Thus, M_S is an additive generator for mod-S.

Conversely, if S is of finite representation type and G_S is an additive generator for mod-S. We claim that G_SA_A is an additive generator for mod-A.
Let N_A be any indecomposable right A-module in mod-A. Then N_S≅ N_AA_S| G_S^m for some positive integer m. And we get N_SA_A| (G_SA)^m_A.
It follows from N_A| N_SA_A that N_A| (G_SA)^m_A. Therefore G_SA_A is an additive generator for mod-A. □

As a homological dimension of measuring homologically how far an Artin algebra is from being of finite representation type, the representation dimension of an Artin algebra A,
denoted by rep.dim(A), was defined as follows in <cit.>.

  Let A be an Artin algebra. The representation dimension rep.dim(A) of A is defined as inf{ gl.dimEnd(M_A)| M is
 a generator-cogenerator for mod-A} if A is non-semisimple; and rep.dim(A)=1 if A is semisimple.
 

 Let A, S be Artin algebras and A/S a separable Frobenius extention. By Lemma <ref>(2), rep.dim(A)=1 if and only if rep.dim(S)=1.
And it follows from Proposition <ref> that rep.dim(A)≤2 if and only if rep.dim(S)≤ 2. In general, we have the following main result of this section.
 Let S be a commutative Artin ring and A an S-algebra. If A is a separable Frobenius extension of S, then
rep.dim(A)= rep.dim(S).


Proof. By the definition of Frobenius extensions, A is an Artin algebra. Then the assertion holds true provided either rep.dim(A)
or rep.dim(S) is at most 2 by Lemma <ref>(2) and Proposition <ref>.

Now assume that rep.dim(A)=n(≥ 3) and M_A is a generator-cogenerator for mod-A such that gl.dimEnd(M_A)=n.
Let X∈mod-S be indecomposable. It follows from X_S| X_SA_S that it is easy to see M_S is a generator-cogenerator for mod-S.
Since X⊗_SA_A∈mod-A, by Lemma <ref>,
we have the following exact sequence

    0→ M_n-2→ M_n-3→⋯→ M_0→ X⊗_SA → 0

in mod-A (and hence in mod-S) with M_i∈ addM_A such that

    0→ Hom_A(M,M_n-2)→ Hom_A(M,M_n-2)→⋯


    → Hom_A(M,M_0)→ Hom_A(M, X⊗_SA)→ 0

is also exact. By assumption, A/S is a Frobenius extension, _SA is finitely generated projective. Then
we get the following exact sequence

    0→ Hom_A(M,M_n-2)⊗_SA → Hom_A(M,M_n-2)⊗_SA →⋯


    → Hom_A(M,M_0)⊗_SA → Hom_A(M, X⊗_SA)⊗_SA → 0.(*)

Since ( Hom_A(_SA,-)(≅-_AA_S), -⊗_SA_A) is an adjoint pair, for any N∈mod-A (also in mod-S) we get
 
    Hom_A(M,N)_SA
       ≅ Hom_A(M, N_SA)  
       ≅_S(_A(A,M),N) 
       ≅_S(M,N),

where the first isomorphism comes from <cit.> and the second isomorphism holds by the adjoint isomorphism.
Hence from the exact sequence (*), we get the following exact sequence

    0→ Hom_S(M,M_n-2)→ Hom_S(M,M_n-2)→⋯


    → Hom_S(M,M_0)→ Hom_S(M, X⊗_SA)→ 0.

Thus X⊗_SA as an S-module has an (n-2)-addM_S-resolution. Since X_S| (X⊗_SA)_S, X_S has an
(n-2)-addM_S-resolution by Lemma <ref>. So gl.dimEnd(M_S)≤ n by Lemma <ref>
and therefore rep.dim(S)≤ n.

Conversely, assume that rep.dim(S)=m(≥ 3) and Q_S is a generator-cogenerator for mod-S such that gl.dimEnd(Q_S)=m.
Since S∈ addQ_S, A_A≅ S_SA_A∈ add(Q_SA)_A. It follows that (Q_SA)_A is a generator for mod-A.
On the other hand, if Y∈mod-A, then Y is also a right S-module. Hence there exists a postive integer t such that 0→ Y_S→ Q_S^(t) is exact
in mod-S, and so 0→ Y_SA→ (Q_SA)^(t) is exact in mod-A.
By the assumption, A/S is a separable Frobenius extension. So Y_A| (Y_SA)_A, and hence (Q_SA)_A is a cogenerator for mod-A. Thus
(Q_SA)_A is a generator-cogenerator for mod-A.

Let V∈mod-A be indecomposable. By Lemma <ref>, V_S has an (m-2)-addQ_S-resolution as an S-module

    @C=0.5cm
      0 [r]     Q_m-2[rr]^f_m-2       Q_m-3[rr]^f_m-3      ⋯[rr]^f_1       Q_0 [rr]^f_0       V_S [r]     0 .

We claim that the following sequence

    0→ Q_m-2_SA→ Q_m-3_SA→⋯→ Q_0_SA→ V_SA→ 0.(**)

is an (m-2)-add(Q_SA)_A-resolution of (V_SA)_A.

It is easy to see that Q_i_SA_A∈addQ_SA_A. Let K_i= Kerf_i for any 0≤ i≤ m-2 and K_-1=V. Because _SA is a finitely generated projective S-module,
 we have the following exact sequence

    0→ K_i_SA → Q_i_SA→  K_i-1_SA→ 0,

which is exact both as right A-modules and as right S-modules for any 0≤ i≤ m-2. So the sequence (**) is exact in mod-A. On the other hand, we have the following sequence

    0→ Hom_S(Q,K_i)→ Hom_S(Q,Q_i)→ Hom_S(Q,K_i-1)→ 0

in mod-S, which induces the following exact sequence

    0→ Hom_S(Q,K_i)_SA→ Hom_S(Q,Q_i)_SA→ Hom_S(Q,K_i-1)_SA→ 0

for any 0≤ i≤ m-2. By Lemma 3.2.4 in <cit.>, Hom_S(Q,L)_SA≅ Hom_A(Q_SA,L_SA) for any L∈mod-S. Hence the sequence

    0→ Hom_A(Q_SA,K_i_SA)→ Hom_A(Q_SA,Q_i_SA)


    → Hom_A(Q_SA,K_i-1_SA)→ 0

is also exact for any 0≤ i≤ m-2, which implies that the following sequence

    0→ Hom_A(Q_SA,Q_m-2_SA)→ Hom_A(Q_SA,Q_m-3_SA)→⋯


    Hom_A(Q_SA,Q_0_SA)→ Hom_A(Q_SA,V_SA)→ 0

is exact. The claim is proved.

Notice that V_A| (V_SA)_A, so V_A has an (m-2)-add(Q_SA)_A-resolution by Lemma <ref>. Thus
gl.dimEnd((Q_SA)_A)≤ m by Lemma <ref> and therefore rep.dim(A)≤ m. The proof is finished.
□



Acknowledgements

The research was completed during the author's visit at the University of Washington. He would like to thank Professor James Zhang for his hospitality.
The author thanks the referees for the helpful comments and valuable suggestions.



50
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