弱Hopf-Galois扩张的Gorenstein维数

The study of Gorenstein projective modules can be traced back to Ref.[1], where the G-dimension of a finitely generated module over a two-sided Noetherian ring was introduced. Furthermore, the definition of Gorenstein projective modules was given in Ref.[2]. We knew already that a module of G-dimension zero is actually a Gorenstein projective module. Gorenstein projective modules play an important role in many areas. For example, they are widely used in the representation theory of Artin algebras, the theory of stable and singularity categories, and the cohomology theory of commutative rings, and so on.

Weak bialgebras and weak Hopf algebras introduced in Ref.[3] generalized the ordinary bialgebras and Hopf algebras by weakening the comultiplication of unit and the multiplication of counit. Comultiplication is allowed to be non-unital, but it is still coassociative. In exchange for coassociativity, the multiplicativity of the counit is replaced by a weaker condition, implying that the unit representation is not necessarily one-dimensional and irreducible. Weak Hopf algebras can provide us with a good framework for studying the symmetries of certain quantum field theories. Groupoid algebras, face algebras and generalized Kac algebras are examples of weak Hopf algebras.

The main purpose of this paper is to study the representation of weak Hopf algebras by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions. Let H be a weak Hopf algebra with a bijective antipode, A a weak right H-comodule algebra and B the H-coinvariant subalgebra of A. After recalling some basic definitions and giving a summary of the fundamental properties concerned with weak Hopf algebras, we study some properties of Gorenstein projective H-modules in representation category, and find that the Gorenstein global dimension of H is the same as the Gorenstein projective dimension of its left unital subalgebra. By applying the integral theory of weak Hopf algebras, on the one hand, we give a sufficient and necessary condition that a projective A-module is a projective B-module; on the other hand, we describe the separability of the functor A⊗B- and that of the restriction of scalar function B(-). Consequently, we investigate the Gorenstein global dimension of a weak Hopf-Galois extension.

1 Preliminaries

Throughout this article, k denotes a fixed field, and we will always work over k. The tensor product ⊗:=⊗k and Hom-functor are always assumed to be over k. We also use Sweedler’s notations for the terminologies on coalgebras and comodules. For a coalgebra C, we write the comultiplication Δ(c)=∑c1⊗c2 for any c∈C. For a right C-comodule M, we denote its coaction by ρ(m)=∑m(0)⊗m(1) for any m∈M. For any unexplained definitions and notations, one may refer to Ref.[4] or Ref.[5].

Definition 1[3] A weak Hopf algebra H is an algebra (H,μ,η) and a coalgebra (H,Δ,ε) such that

and there exists a linear map S: H→H, called an antipode, satisfying

for all x, y, z∈H, where Δ(1H)=∑11⊗12,

Let H be a weak Hopf algebra. The images Ht=εt(h) and Hs=εs(h) of the projections εt,εs: H→H defined by

are both separable subalgebras of H and commute with each other[3]. Ht and Hs are called left and right unital subalgebras, respectively.

Definition 2[3] A left integral in H is an element l∈H satisfying hl=εt(h)lfor all h∈H. Furthermore, if εt(l)=1H, then it is said to be normalized. A normalized right integral is defined similarly. A left or right integral in H is called non-degenerate if it defines a non-degenerate functional on H*, the dual space of H.

The spaces of left and right integrals in H are denoted as

and

respectively.

for all h∈H. Meanwhile, according to Ref.[6], the left integral λ∈

satisfies the following invariant properties for all h,g∈H,

If a weak Hopf algebra H is finite dimensional, then it has a bijective antipode S[6], in which inverse is denoted as S-1 in the following, and there exist non-degenerate left integrals

such that ∑l1λ(l 2)=1H[7]. Furthermore, the quasi-basis of λ is ∑l2⊗S-1(l1)[3]. Therefore, for all h∈H,

Let H be a weak Hopf algebra with an antipode S. According to Refs.[8-9], we have

Definition 3[10] Let H be a weak Hopf algebra, and A a right H-comodule, which is also an associative algebra. We call A a weak right H-comodule algebra if

Let H be a weak Hopf algebra, and A a weak right H-comodule algebra. Then, according to Ref.[8], we obtain that

Define the H-coinvariant subalgebra of A as

Then, we know from Ref.[11] that

Definition 4[12] Let A be a weak right H-comodule algebra and B the H-coinvariant subalgebra of A. The extension A/B is said to be weak Hopf-Galois, if the canonical map

is an isomorphism, where ⊗s:=⊗Hs is the relative tensor product over Hs

is a right Hs-module via y·a=∑ε(ya(1))a(0) for all y∈Hs, a∈A, and H is a left Hs-module via its multiplication).

2 Gorenstein Global Dimensions for Weak Hopf Algebras

For a ring R, we denote the class of projective left R-modules by P, and for an object M in RM, the category of left R-modules, we denote the projective and injective dimension of M by p.dimRM and i.dimRM, respectively.

Definition 5[13] A left R-module M is Gorenstein projective if there exists an RHom(-,P)-exact sequence

such that M=Ker(P0→P1), where every Pi and Pi are projective.

Dually, Gorenstein injective left R-modules can be defined.

Definition 6 For a left R-module M, the Gorenstein projective dimension G.p.dimRM is at most n if there is an exact sequence

where every Gi is Gorenstein projective.

Dually, the Gorenstein injective dimension G.i.dimRM can be defined.

For any ring R, Ref.[14] shows us that

The common value is called the left Gorenstein global dimension of R and denoted as G.gl.dim(R).

We know that G.gl.dim(R)=0 if and only if the ring R is quasi-Frobenius[15]. In addition, R is left Gorenstein hereditary if every submodule of a projective left R-module is Gorenstein projective[16], i.e., G.gl.dim(R)≤1.

In what follows, we always assume that H is a weak Hopf algebra with a bijective antipode S. Then, the left H-module category HM, called the representation category, is an abelian monoidal category[17-18] such that

• Ht is the unit object with a left H-action via h·x=εt(hx) for all h∈H and x∈Ht;

• For M,N∈HM, the tensor product is M⊗tN:=M⊗HtN, where the right Ht-module structure on M is defined by m·x=S-1(x)·m for all m∈M and x∈Ht;

• The left H-module structure on M⊗tN is defined by the following diagnosing action h·(m⊗tn)=h1·m⊗th2·n, for all h∈H, m∈M, n∈N;

• For M,N∈HM, a homomorphism between M and N is left H-linear.

Proposition 1 If P is a Gorenstein projective left H-module, then so is P⊗tX for any left H-module X.

Proof If P is a Gorenstein projective left H-module, then there is an HHom(-,P)-exact sequence

such that M=Ker(P0→P1), where every Pi and Pi are projective. As Ht is separable, we can obtain an exact sequence

such that M⊗tX=Ker(P0⊗tX→P1⊗tX). Meanwhile, every Pi⊗tX and Pi⊗tX are projective left H-modules[19]. For any projective left H-module Q,

Hence, HHom(P⊗tX,Q) is exact, as desired.

Theorem 1 Let H be a weak Hopf algebra with a bijective antipode S. Then, G.gl.dim(H)=G.p.dimHHt.

Proof Obviously, G.gl.dim(H)≥G.p.dimHHt. Hence, we shall prove the reverse inequality. Assume that G.p.dimHHt=n<+∞. Then, there is an exact sequence

where every Gi is Gorenstein projective. For any left H-module X, we obtain an exact sequence

As Ht⊗tX≅X, and every Gi⊗tX is Gorenstein projective by Proposition 1, we obtain G.gl.dim(H)≤n. This shows that G.gl.dim(H)≤G.p.dimHHt.

3 Gorenstein Global Dimensions for Weak Hopf-Galois Extensions

Throughout this section,H is always assumed to be a weak Hopf algebra with a bijective antipode S, unless otherwise stated. Then, let

be two non-degenerate left integrals such that ∑l1λ(l2)=1H, and we fix a weak Hopf-Galois extension A/B and use the following formal notation for the inverse of β for all h∈H,

by definition. Such elements enjoy the following properties.

Lemma 1[20] For all a∈A and h∈H, we have

Consider the two functors:

where B(-) is the restriction of the scalars functor.

Lemma 2 (A⊗B-,B(-)) and (B(-),A⊗B-) are double adjunctions.

Proof Note that the element ∑λ(a(1))a(0)∈B[21] for a∈A. Then, for all N∈BM, we have a well-defined map φ:A⊗BN→BHom(A,N) given by

Then, it is easy to check that φ is a morphism of left A-modules, where BHom(A,N)∈AM via (a·f)(b)=f(ba) for all a,b∈A and f∈BHom(A,N). Meanwhile, we claim that φ is a bijection with the inverse given by

Indeed, on the one hand, for all a,b∈A and n∈N, by Lemma 1, we have

∑∑ili(S-1(l ))⊗Bλ(ri(S-1a(1))ri(S-1a(0)·n=

∑∑ili(S-1(l ))ri(S-1a(0)λ(ri(S-1a(1))⊗Bn=

∑∑ili(S-1(l 2))ri(S-1(12))a(0)λ(S-1(l 1)a(1))⊗Bn=

∑1(0)a(0)λ(S-1(l )1(1)a(1))⊗Bn

On the other hand, for all b∈A and f∈BHom(A,N), by Lemma 1, we have

∑∑iλ(b(1)li(S-1(l ))(1))b(0)li(S-1(l ))(0)·

∑∑iλ(b(1)l 2)f(b(0)li(S-1(l 1))ri(S-1(l 1)))=

∑λ(l 2)ε(S-1(l 1)b(1)1(1))f(b(0)1(0))=

∑λ(l 2)ε(S-1(l 1)b(1))f(b(0))n=f(b)

Hence, the adjoint isomorphism theorem, (A⊗B-,B(-)) and (B(-),A⊗B-) are double adjunctions[12].

By Lemma 2, we have the following assertion.

Corollary 1 If P∈BM is projective, then A⊗BP∈AM is projective.

Lemma 3 If H is semisimple and P∈AM, then P is projective as a left A-module if and only if P is projective as a left B-module.

Proof As H is semisimple, we know that H is finite dimensional and there is

which is normalized[3], i.e., εs(J)=1H. Then, for all a∈A,

In fact, by Lemma 1, we have

Consider that P is a projective left B-module and let M,N∈AM. Suppose that δ:M→N and ξ:P→N are left A-module morphisms such that δ is surjective. Since P is projective as a left B-module, there is a left B-module morphism f:P→M such that

Define

Obviously,

is a morphism of left A-modules by (16). Moreover,

∑ili(J)ri(J)·ξ(p)=∑ε(J1(1))1(0)·ξ(p)=

∑ε(εs(J)1(1))1(0)·ξ(p)=∑ε(1(1))1(0)·ξ(p)=

where the third equality follows[8]. Hence,

Thus, P is projective as a left A-module.

Conversely, suppose that P is projective as a left A-module. It follows from Lemma 2 that (A⊗B-,B(-)) is an adjoint pair. Since A is projective as a right B-module[22], we obtain that the functor A⊗B- is exact. Hence, B(-) preserves projective objects. Therefore, P is projective as B-module.

Definition 7[23] Let C and D be categories. A covariant functor F:C→D is separable if for all objects M,N∈C, there are maps

satisfying the following conditions:

1) For α∈HomC(M,N), we have

2) Given M′,N′∈C, f∈HomD(F(M),F(N)),g∈HomD(F(M′),F(N′)), α∈HomC(M,M′), β∈HomC(N,N′), we have

Lemma 4 1) If H is semisimple, then B(-) is separable.

2) If H* is semisimple, then A⊗B- is separable.

Proof 1) Let J∈

such that εs(J)=1H. By the proof of Lemma 3, we can know that the element ∑ili(J)⊗Bri(J) is exactly the separability idempotent, i.e., the extension A/B is separable. Hence, the restriction of scalars functor B(-) is separable[23].

2) Note that A is a weak left H*-module algebra under the action φ⇀a=∑φ(a(1))a(0) for a∈A, φ∈H*[24]. As H* is semisimple, there exists a normalized left integral

Then, λ⇀1A=1A, and the map F:A→B defined by F(a)=λ⇀a is a B-bimodule projection[21]. This assumes that B is a direct summand of A as a B-bimodule. Hence, the unit

of the adjunction (A⊗B-,B(-)) is a split monomorphism[25]. So, the induction function A⊗B- is separable[11].

Corollary 2 1) If H is semisimple, then M is a left A-module direct summand of A⊗BM for all M∈AM.

2) If H* is semisimple, then N is a left B-module direct summand of A⊗BN for all N∈BM.

Proof 1) As H is semisimple, we know from Lemma 4 that B(-) is separable. Hence, the counit width=5,height=8,dpi=110

of the adjunction (A⊗B-,B(-)) is a cosplit epimorphism[11], that is, there is a natural transformation

for all M∈AM. Thus, M is a left A-module direct summand of A⊗BM.

2) As H* is semisimple, we have that A⊗B- is separable by Lemma 4. Hence, we also know that the counit

of the adjunction (B(-),A⊗B-) is a cosplit epimorphism[11]. Thus, N is a left B-module direct summand of A⊗BN for all N∈BM.

Lemma 5 If N∈BM is Gorenstein projective, then A⊗BN∈AM is Gorenstein projective.

Proof Suppose that N is a Gorenstein projective left B-module. Then, there is a BHom(-,P)-exact sequence

such that N=Ker(P0→P1), where every Pi and Pi are projective. Since A is projective as a right B-module, we obtain that A⊗BP is exact and A⊗BN=Ker(A⊗BP0→A⊗BP1). We also obtain that A⊗BPi is projective for every i because of the assertion that B(-) is exact and A⊗B- is a left adjoint[12].

Let us suppose finally that Q∈AM is Gorenstein projective. Then,

However, BQ is projective since B(-) is a left adjoint of A⊗B- which is exact. Thus, BHom(P,Q) is exact since N is Gorenstein projective, and so, AHom(A⊗BP,Q) is also exact, which implies that A⊗BN is Gorenstein projective.

Theorem 2 If H is a semisimple and cosemisimple weak Hopf algebra, then G.gl.dim(A)=G.gl.dimB.

Proof Assume that G.gl.dimB=n<+∞. For any M∈AM, as a left B-module, there is a Gorenstein projective resolution:

where every Gi is Gorenstein projective. Since the functor A⊗B- is exact, it induces a left A-module exact sequence

From Lemma 5, we know that every A⊗BGi is Gorenstein projective, and thus, G.gl.dimA(A⊗BM)≤n. Since M is a direct summand of A⊗BM as left A-modules by Corollary 2, G.gl.dimAM≤n[26]. Hence, G.gl.dim(A)≤n.

Suppose that G.gl.dim(A)=q≤n=G.gl.dimB. Then, G.gl.dimB(A⊗BN)≤q for any N∈BM. Therefore, BExti(A⊗BN,U)=0 for all i>q and all projective left B-module U[26]. Since N is a left B-module direct summand of A⊗BN and the functor BExt(-,U) preserves finite direct sums, BExt(N,U)=0 for all i>q and all projective left B-module U, and, hence, G.gl.dimBN≤q. This implies that G.gl.dim(A)≤G.gl.dimB. Therefore, in view of the above discussion, we have G.gl.dim(A)=G.gl.dimB.

Corollary 3 Let H be a semisimple and cosemisimple weak Hopf algebra. Then, A is quasi-Frobenius (rep. left Gorenstein hereditary) if and only if so is B.

In particular, if R#H is a weak smash product of a weak left H-module algebra R (see Ref.[24] for the definitions), then R#H is quasi-Frobenius (rep. left Gorenstein hereditary) if and only if so is R.

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Gorenstein dimensions for weak Hopf-Galois extensions

1 Preliminaries

2 Gorenstein Global Dimensions for Weak Hopf Algebras

3 Gorenstein Global Dimensions for Weak Hopf-Galois Extensions

弱Hopf-Galois扩张的Gorenstein维数