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 Böhm et 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 Álvarez et 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,Chen et 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

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

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.

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

Definition 2.2 Let H be a weak Hopf algebra measuring A.An A-valued 2-cocycle σ on H is a k-linear map H⊗HsH→A satisfying

for all h,k,m∈H,l∈Ht.

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Definition 2.1 Let H be a weak Hopf algebra and A an algebra.H measures A if there exists a k-linear map,called a measuring,H⊗A → A,h⊗b→ h·b such that for all h∈H,l∈ Ht,b,b′∈ A,

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

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 2.3 Let H be a weak Hopf algebra measuring an algebra A.An A-valued 2-cocycle σ on H is invertible if there exists a k-linear map τ:H ⊗HtH → A(where the right and left Ht-module structures on H are given by right and left multiplication,respectively)satisfying

for all h,k ∈ H and l∈ Ht.Then the map τ is called an inverse of σ.

Definition 2.4 Let H be a weak Hopf algebra and A an algebra.We define the following set:

If u∈WC(H,A),then we call u weak convolution invertible.

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

Proposition 2.1 Let A#σH be a weak crossed product.Then γ:H → A#σH defined by γ(h)=1#σh is weak convolution invertible if and only if σ is invertible.

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3 Cotorsion Dimension of Weak Crossed Products

Lemma 3.1([7],Lemma 2.8) Let H be a weak Hopf algebra and A#σH the weak crossed product algebra,M and N be A#σH-modules.The operation HomA(M,N)is a right H-module defined by

Lemma 3.2([7],Theorem 2.11) Let H be a weak Hopf algebra and A#σH a weak crossed product with σ invertible,M and N be A#σH-modules.Then

where Hsbecomes a right H-module via y·h=εs(yh)for any y∈Hsand h∈H.

Lemma 3.3 Let H be a weak Hopf algebra and A#σH a weak crossed product with σ invertible,M and N be two A#σH-modules satisfying

Then

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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,if H is semisimple,then

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 3.1 Let H be a finite-dimensional weak Hopf algebra and A#σH a weak crossed product with σ invertible,M a left A#σH-module.Then we have

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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

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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,

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So we have

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

with each Cicotorsion,and so

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Proof.We only prove that

In fact,assume that

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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

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then we have

It follows that cdA(M)≤n.

Proposition 3.1 Let H be a finite dimensional weak Hopf algebra,A#σH a weak crossed product with σ invertible.Then

Theorem 3.2 Let H be a finite-dimensional semisimple weak Hopf algebra,A#σH be a weak crossed product with σ invertible,and M be a left A#σH-module.Then

Moreover,if H is 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.

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Theorem 3.3 Let H be a finite dimensional semisimple weak Hopf algebra,A#σH a weak crossed product with σ invertible.Then

if one of the following conditions is satis fied:

(1)H∗is semisimple;

(2)A is commutative.

Proof.By Proposition 3.1,we have

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

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:

(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

翻转机构的运动分为顺时针翻转180°过程和逆时针翻转180°过程，顺时针翻转完成待打磨铸锭上面和下面的变换，逆时针翻转则回到翻转机构的初始位置。翻转机构的运动形式为旋转运动，采用SCCA曲线形式定义伺服电机的加速度轮廓曲线，如图4所示，其中0～2 s为顺时针翻转过程，2～4 s为逆时针翻转过程。

and hence

The proof is completed.

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