GORENSTEIN FLAT(COTORSION)DIMENSIONS AND HOPF ACTIONS

MENG Fan-yun

(School of Mathematical Sciences,Yangzhou University,Yangzhou 225002,China )

GORENSTEIN FLAT(COTORSION)DIMENSIONS AND HOPF ACTIONS

MENG Fan-yun

(School of Mathematical Sciences,Yangzhou University,Yangzhou 225002,China )

In this paper,we study the relationship of Gorenstein flat(cotorsion)dimensions between A-Mod and A#H-Mod.Using the properties of separable functors,we get that(1)Let A be a right coherent ring,assume that A#H/A is separable and φ:A→A#H is a splitting monomorphism of(A,A)-bimodules,then l.Gwd(A)=l.Gwd(A#H);(2)Assume that A#H/A is separable and φ:A→A#H is a splitting monomorphism of(A,A)-bimodules,then l.Gcd(A)= l.Gcd(A#H),which generalized the results in skew group rings.

coherent ring;Gorenstein flat module;Gorenstein cotorsion module

1 Introduction

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],Lpez-Ramos studied the relationship of Gorenstein injective(projective)dimensions between A-Mod and A#H-Mod. He showed that under some conditions,glGid(A)＜∞if and only if glGid(A#H)＜∞(resp. glGpd(A)＜∞if and only if glGpd(A#H)＜∞).

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 φ:A→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 φ:A→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 Δ:H?H→H,counit ε:H→k and antipode S:H→H.A k-algebra A is called a left H-module algebra if A is a left H-module such that h·(ab)=and h·1A=ε(h)1Afor all a,b∈A and h∈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?H,whose elements are denoted by a#h instead of a?h,with multiplication given by(a#h)(b#l)=Σa(h(1)·b)#h(2)l for a,b∈A and h,l∈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 C and D, a covariant functor F:C→D is said to be separable if for all M,N in C there are maps:HomD(F(M),F(N))→HomC(M,N))satisfying the following conditions.

1.For α∈HomC(M,N),we have

2.Given M′,N′∈C,α∈HomC(M,M′),β∈HomC(N,N′),f∈HomD(F(M),F(N)) and g∈HomD(F(M′),F(N′))such that the following diagram commutes

then the following diagram is also commutative

Let φ:A→A#H denote the inclusion map.We can associate to φ the restriction of scalars functorA(-):A#H-Mod→A-Mod,the induction functor A#H?A-=Ind(-): A-Mod→A#H-Mod and the coinduction functor HomA(A#H,-):A-Mod→A#H-Mod. It is well known that A#H?A-is left adjoint toA(-)and that HomA(A#H,-)is right adjoint toA(-).Since H is a finite-dimensional Hopf algebra,by[6,Theorem 5],the functor A#H?A-is isomorphic to HomA(A#H,-).So we have a double adjunctions (A#H?A-,A(-))and(A(-),A#H?A-).Now we consider the separability of functorsA(-)and A#H?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?A-=Ind(-)is separable if and only if φ splits as an A-bimodule map.

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

of flat left R-modules such that M=ker(F0→F1)and which remains exact whenever E?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

with every Fibeing Gorenstein flat.If no such finite sequence exsits,define GfdR(M)=∞; otherwise,if n is the least such integer,define GfdR(M)=n.In[3]left weak Gorenstein global dimension of R was define as l.Gwd(R)=sup{GfdR(M)|M is any left R-module}. A left R-module M is called Gorenstein cotorsion[8]if(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

with every Cibeing 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.

2 Gorenstein Flat Modules and Actions of Finite-Dimensional Hopf Algebras

In this paper,φ:A→A#H always denotes the inclusion map.If M∈A#H-Mod, thenAM will denote the image of M by the restriction of the scalars functorA(-):A#HMod→A-Mod.

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

Remark 2.2Let φ:A→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 φ)is also injective.

Proposition 2.3(1)If M∈A-Mod is Gorenstein flat,then A#H?AM is Gorenstein flat as a left A#H-module.

(2)If M∈A#H-Mod is Gorenstein flat,thenAM is Gorenstein flat as a left A-module.

Proof

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

of flat left A-modules such that M=ker(F0→F1)and which remains exact whenever E?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?A-preserves flat modules,we get that A#H?AF is an exact sequence of flat left A#H-modules and

Finally,let E′be any injective right A#H-module.Then E′?A#H(A#H?AF)(E′?A#HA#H)?AF is exact since E′?A#HA#HE′(as right A-modules)is injective by Remark 2.2.Thus A#H?AM is Gorenstein flat.

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

of flat left A#H-modules such that M=ker(F′0→F′1)and which remains exact whenever E?A#H-is applied for any injective right A#H-module E.ThenAF′is an exact sequence of flat left A-modules since the functorA(-)is exact and preserves flat modules.

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

Since H is a finite-dimensional Hopf algebra,by[6,Theorem 5],we can easily get that E′?AA#H is injective as a right A#H-module.By(?)we know that E′?A(AF′)is exact. ThereforeAM is Gorenstein flat.

Proposition 2.4Assume that A#H/A is separable and φ:A→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.

ProofLet{Fi}i∈Ibe a family of flat left A#H-modules,thenA(Fi)is flat as a left A-module for every i.If we consider the adjoint pair(A#H?A-,A(-)),we know thatpreserves inverse limits.ThusSince A is a right coherent ring,is flat as a left A-module.Then,we get thatFiis a flat left A#H-module.Thus A#H is a right coherent ring.

Conversely,let{Fi}i∈Ibe a family of flat left A-modules,since A#H?A-preserves flat modules,we know that A#H?AFiis flat as a left A#H-module for every i.If we consider the adjoint pair(A(-),A#H?A-),we know that A#H?A-preserves inverse limits.Thus

Since A#H is a right coherent ring,A# H?Amodule.Then,we get thatAis a flat left A-module.Since φ:A→A#H is a splitting monomorphism of(A,A)-bimodules,we get that the functor A#H?A-is separable by[15,Proposition 1.3].Consider the adjoint pair(A#H?A-,A(-)),by[9,Proposition 5]we know that the natural map ηM:M→A(A#H?AM)is a split monomorphism for every M∈A-Mod.Thenis a direct summand ofAHenceFiis 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 AMod and A#H-Mod when A is right coherent.

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

ProofFor every n,we need to show that GfdA(M)≤n for every left A-module M if and only if GfdA#H(N)≤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?A-andA(-)both preserve Gorenstein flat modules.Thus

Since A#H?A-is separable,M is a direct summand ofA(A#H?AM).Since A is a right coherent ring,by[2,Propositions 2.2 and 2.10]we know that GfdA(M)≤n.

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

Lemma 2.6(1)If N∈A-Mod is Gorenstein cotorsion,then A#H?AN is Gorenstein cotorsion as a left A#H-Mod.

(2)If N∈A#H-Mod is Gorenstein cotorsion,thenAN is Gorenstein cotorsion as a left A-module.

(3)Let M∈A#H-Mod and A#H/A be separable.Then M is Gorenstein cotorsion as a left A#H-module if and only ifAM 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(?)of left A#H-modules with P projective.SinceA(-)is exact and preserves Gorenstein flat and projective modules,we have an exact sequencewithAP projective andAF Gorenstein flat.Hence we have the following commutative diagram:

Note that σ1,σ2and σ3are isomorphisms by adjoint isomorphism.Hence

is an epimorphism.

Applying the functor HomA#H(-,A#H?AN)to(?),we get a long exact sequence

is an epimorphism,we know that(F,A#H?AN)=0 for any Gorenstein flat left A#H-module F.Hence A#H?AN is a Gorenstein cotorsion left A#H-module.

(2)Similarly,using the adjoint pair(A#H?A-,A(-))we can prove thatA(-)preserves Gorenstein cotorsion modules.

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

Conversely,ifAM is Gorenstein cotorsion as a left A-module,then by(1)we know that A#H?AAM is Gorenstein cotorsion.Since A#H/A is separable,the functorA(-)is separable by[15,Proposition 1.3].Consider the adjoint pair(A(-),A#H?A-),by[9,Proposition 5]we know that the natural map ηM:M→A#H?AAM is a split monomorphism for every left A#H-module M.Then M is a direct summand of A#H?AAM.Hence M is Gorenstein cotorsion as a left A#H-module since the class of Gorenstein cotorsion modules is closed under direct summands.

Proposition 2.7Let M∈A#H-Mod and N∈A-Mod.Then

(1)GcdA(AM)≤GcdA#H(M).

(2)GcdA#H(A#H?AN)≤GcdA(N).

Proof(1)Assume that GcdA#H(M)=n＜∞,then there exists an exact sequence of left A#H-modules

with every Cibeing Gorenstein cotorsion.By Lemma 2.6,A(-)preserves Gorenstein cotosion modules,we have an exact sequence of left A-modules

with every Cibeing Gorenstein cotorsion.Thus GcdA(AM)≤GcdA#H(M).

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

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

ProofLet M be any left A-module.Since φ:A→A#H is a splitting monomorphism of(A,A)-bimodules,M is a direct summand ofA(A#H?AM).Hence

By Proposition 2.7,

Thus l.Gcd(A)≤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?AAN.Hence

By Proposition 2.7,Thus l.Gcd(A#H)≤l.Gcd(A).

Corollary 2.9Let A be a k-algebra and G a finite group with|G|-1∈k.Then l.Gcd(A)=l.Gcd(A?G).

ProofBy 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∈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.

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Gorenstein平坦(余撓)維數(shù)和Hopf作用

孟凡云

(揚(yáng)州大學(xué)數(shù)學(xué)科學(xué)學(xué)院,江蘇揚(yáng)州225002)

設(shè)H是域k上的有限維Hopf代數(shù),A是左H-模代數(shù).本文研究了Gorenstein平坦(余撓)維數(shù)在A-模范疇和A#H-模范疇之間的關(guān)系.利用可分函子的性質(zhì),證明了(1)設(shè)A是右凝聚環(huán),若A#H/A可分且φ:A→A#H是可裂的(A,A)-雙模同態(tài),則l.Gwd(A)=l.Gwd(A#H);(2)若A#H/A可分且φ:A→A#H是可裂的(A,A)-雙模同態(tài),則l.Gcd(A)=l.Gcd(A#H),推廣了斜群環(huán)上的結(jié)果.

凝聚環(huán);Gorenstein平坦模;Gorenstein余撓模

O154.2

tion:16E10

A

0255-7797(2017)01-0083-08

?Received date:2015-09-04Accepted date:2016-02-18

Foundation item:Supported by the Natural Science Fund for Colleges and Universities in Jiangsu Province(15KJB110023)and the School Foundation of Yangzhou University(2015CJX002).

Biography:Meng Fanyun(1983–),female,born at Qufu,Shandong,lecture,major in Homological algebra.