Cotorsion Dimension of Weak Crossed Products

CHEN HuA-xIAND LIANG JIN-RONG

(1.Department of Mathematics and Physics,Bengbu College,Bengbu,Anhui,233000)

(2.Department of Basic Courses,Chuzhou Institute of Technology,Chuzhou,Anhui,239000)

Communicated by Du Xian-kun

1 Introduction

In 1996,Böhm and Szlachányi[1]introduced weak bialgebras(or weak Hopf algebras)as a generalization of ordinary bialgebras(or Hopf algebras).A general theory for these objects was subsequently developed in Böhmet al.[2].Brie fly,the axioms of a weak Hopf algebra are the same as the ones for a Hopf algebra,except that the coproduct of the unit,the product of the counit and the antipode conditions are replaced by weaker properties.The main motivation for studying weak Hopf algebras comes from quantum field theory,operator algebras and representation theory.It has turned out that many results of classical Hopf algebra theory can be generalized to weak Hopf algebras.Shen[3]extended the theory of crossed products were introduced independently by Blattner and Montgomery[4],Doi and Takeuchi[5]to more general Hopf structure:weak Hopf algebras.At the categorical level,Alonso Álvarez and González Rodríguez[6]introduced the notion of a weak crossed product and Alonso Álvarezet al.[7]investigated weak cleft theory and weak Galois extensions for weak Hopf algebras(see[8]and[9]).

In 2005,Mao and Ding[10]introduced the cotorsion dimension of modules and rings.Recently,Chenet al.[11]discussed the cotorsion dimension of the smash product A#H,which generalizes the result of group rings introduced by Bennis and Mahdou[12].It is now very natural to ask whether cotorsion dimension of the weak crossed products,the weak crossed products we consider here are generalizations of the crossed products and weak smash products.This question motivates the present research.

This paper is organized as follows:In Section 2,we recall some basic definitions and results such as cotorsion dimension,weak Hopf algebras,weak crossed products and so on.In Section 3,we mainly investigate the relationship between the global cotorsion dimension of the weak crossed product A#σH with the algebra A.

2 Preliminaries

Throughout this paper,we work over a commutative field k.All algebras,linear spaces etc.are over k;unadorned⊗ means⊗k.

2.1 Cotorsion Dimension

The cotorsion dimension of an A-module M denoted by cdA(M)is the least positive integer n satisfying(F,M)=0 for all flat A-modules F.In particular,if cdA(M)=0,then M is called cotorsion.The right global dimension of A is denoted by r.D(A).The left global cotorsion dimension of A,denoted by l.cot.D(A),is defined as the supremum of the cotorsion dimensions of A-modules(see[10]).

2.2 Weak Hopf Algebras

For the basic definitions and properties of weak Hopf algebras,see[2].Recall that a weak Hopf algebra H is an algebra(H,m,µ)and coalgebra(H,Δ,ε)such that for h,k,l∈ H,the following axioms hold:

(1)

(2)

(3)

(4)There exists a k-linear map S:H−→H,called the antipode,satisfying

We have idempotent mapsH−→H defined by εtand εsare called the target map and the source map,and their images Htand Hsare called the target and source space,which can be described as follows:

If H is a weak Hopf algebra,then for g,h∈H the following hold:

2.3 Weak Crossed Products A#σH

The notion of the crossed product with a Hopf algebroid was introduced by Böhm and Szlachányi[1].It is well-known that weak Hopf algebras provide examples of Hopf algebroids.So we deduce the notion of a weak crossed product over a weak Hopf algebra from it(see[3]).In the following,we assume that H has bijective antipode S.

Definition 2.1LetHbe a weak Hopf algebra andAan algebra.HmeasuresAif there exists ak-linear map,called a measuring,H⊗A → A,h⊗b→ h·bsuch that for allh∈H,l∈ Ht,b,b′∈ A,

Definition 2.2LetHbe a weak Hopf algebra measuringA.AnA-valued2-cocycleσonHis ak-linear mapH⊗HsH→Asatisfying

for allh,k,m∈H,l∈Ht.

An H-measured algebra A is called a σ-twisted left H-module if a 2-cocycle σ satis fies

Let H be a weak Hopf algebra measuring A.Let σ:H ⊗HsH → A be a map satisfying(2.5)and(2.6).Consider the k-space A⊗HtH where H is a left Ht-module via its multiplication and A is a right Ht-module via

Its multiplication is given by the following formula:

Then A⊗HtH is an associative algebra with unit 1A⊗Ht1Hif and only if σ is an A-valued 2-cocycle on H and A is a σ-twisted left H-module.The associative algebra is called a weak crossed product of A with H and is denoted by A#σH.

Definition 2.3LetHbe a weak Hopf algebra measuring an algebraA.AnA-valued2-cocycleσonHis invertible if there exists ak-linear mapτ:H ⊗HtH → A(where the right and leftHt-module structures onHare given by right and left multiplication,respectively)satisfying

for allh,k ∈ Handl∈ Ht.Then the mapτis called an inverse ofσ.

Definition 2.4LetHbe a weak Hopf algebra andAan algebra.We define the following set:

Ifu∈WC(H,A),then we calluweak convolution invertible.

We obtain the following proposition from Theorem 4.11 and Theorem 4.12 in[7].

Proposition 2.1LetA#σHbe a weak crossed product.Thenγ:H → A#σHdefined byγ(h)=1#σhis weak convolution invertible if and only ifσis invertible.

3 Cotorsion Dimension of Weak Crossed Products

Lemma 3.1([7],Lemma 2.8)LetHbe a weak Hopf algebra andA#σHthe weak crossed product algebra,MandNbeA#σH-modules.The operationHomA(M,N)is a rightH-module defined by

Lemma 3.2([7],Theorem 2.11)LetHbe a weak Hopf algebra andA#σHa weak crossed product withσinvertible,MandNbeA#σH-modules.Then

whereHsbecomes a rightH-module viay·h=εs(yh)for anyy∈Hsandh∈H.

Lemma 3.3LetHbe a weak Hopf algebra andA#σHa weak crossed product withσinvertible,MandNbe twoA#σH-modules satisfying

Then

Proof.Let

be an exact sequence of A#σH-modules,where P is projective as an A#σH-module.Since(M,N)=0 for all p＞0,we have the exact sequence

so

Moreover,it is not hard to check that

by[13].By Lemma 3.2 and

we have

Moreover,

Finally,replacing M by C,we obtain the desired result by induction and the above diagrams.The proof is completed.

Theorem 3.1LetHbe a finite-dimensional weak Hopf algebra andA#σHa weak crossed product withσinvertible,Ma leftA#σH-module.Then we have

Moreover,ifHis semisimple,then

Proof.Suppose that cdA(M)=i＜∞and pd(HsH)=j＜∞,for any A#σH-module M,there is an exact sequence of A#σH-modules

where C0, ···,Ci−1are cotorsion A#σH-modules.Note that by Lemma 3.3,C is a cotorsion A-module.Thus,(F,C)=0 for all n＞0 and all flat A-modules F.Since pd(HsH)=j＜∞,by Lemma 3.3,

So we have

where Ci,···,Ci+jare cotorsion A#σH-modules.It follows that there is an exact sequence

with each Cicotorsion,and so

Theorem 3.2LetHbe a finite-dimensional semisimple weak Hopf algebra,A#σHbe a weak crossed product withσinvertible,andMbe a leftA#σH-module.Then

Proof.We only prove that

In fact,assume that

Let F be a flat A-module.Since A#σH is a free A-module,the product A#σH ⊗AF will be a flat A#σH-module.So

Note that

then we have

It follows that cdA(M)≤n.

Proposition 3.1LetHbe a finite dimensional weak Hopf algebra,A#σHa weak crossed product withσinvertible.Then

Moreover,ifHis semisimple,then

Proof.First,we have

Note that r.D(H)=pd(HsH)for any weak Hopf algebra H(see[13]and[14]),and so

The proof is completed.

Theorem 3.3LetHbe a finite dimensional semisimple weak Hopf algebra,A#σHa weak crossed product withσinvertible.Then

if one of the following conditions is satis fied:

(1)H∗is semisimple;

(2)Ais commutative.

Proof.By Proposition 3.1,we have

(1)Note that A#σH is a weak H∗-module algebra via

If H∗is semisimple,then by the duality theorem of[10]and Proposition 3.1 we have

(2)Suppose that l.cot.D(A#σH)=n is finite.By Theorem 19.2.5 of[10],we know that

Let F be a flat A-module.Then A#σH⊗AF is a flat A#σH-module.Since H is semisimple,we choosewith εt(l)=1.Choose c=1A.Then

By[15],we can obtain that the map δ:A#σH → A,δ(a#σh)= εt(h)·a is a split A#σH-epimorphism.Since A is commutative,A#σH is an A-bimodule defined by

It remains to show that δ is an A-bimodule homomorphism.In fact,

For this case we indeed have that A is a direct summand of A#σH as an A-bimodule.So there exists an A-bimodule B such that

Applying⊗AM to above the equation,we have

for any left A-module M.Then

Since(A#σH)⊗AM has a left A#σH-module structure and its any projective resolution is also a projective resolution over A,we have

and hence

The proof is completed.

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