Finitistic Dimension of Faithfully Flat Weak Hopf-Galois Extension
aa r X i v : . [ m a t h . R T ] M a r Finitistic Dimension of Faithfully Flat WeakHopf-Galois Extension ⋆ Aiping ZhangSchool of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, ChinaE-mail: [email protected]
Abstract.
Let H be a finite-dimensional weak Hopf algebra over a field k and A/B be a right faithfully flat weak H -Galois extension. We provethat if the finitistic dimension of B is finite, then it is less than or equalto that of A . Moreover, suppose that H is semisimple. If the finitisticdimension conjecture holds, then the finitistic dimension of B is equalto that of A . Keywords: finitistic dimension, weak Hopf-Galois extension,
Throughout the paper, we work over a field k , vector spaces, algebras, coalgebrasand unadorned ⊗ are over k . Given an algebra A , we denote by A -Mod and A -mod the categories of left A -modules and of finitely generated left A -modules,respectively. For a left A -module M , we denote by pd M the projective dimensionof M .Weak Hopf algebras are generalizations of ordinary Hopf algebras. Examples ofweak Hopf algebras are groupoid algebras, face algebras [7], quantum groupoids [9], ⋆ Supported by the NSF of China (Grant No. 11601274). k -bialgebra H is both a k -algebra ( m, µ )and a k -coalgebra (∆ , ε ) such that ∆( hk ) = ∆( h )∆( k ), and∆ (1) = 1 ⊗ ′ ⊗ ′ = 1 ⊗ ′ ⊗ ′ ,ε ( hkl ) = ε ( hk ) ε ( k l ) = ε ( hk ) ε ( k l ) , for all h, k, l ∈ H , where 1 ′ stands for another copy of 1. Here we use Sweedler ′ snotation for the comultiplication. Namely, for h ∈ H, ∆( h ) = h ⊗ h , where weomit the summation symbol and indices. The maps ε t , ε s : H → H defined by ε t ( h ) = ε (1 h )1 ; ε s ( h ) = 1 ε ( h )are called the target map and source map, and their images H t and H s are calledthe target and source space.A weak Hopf algebra H is a weak bialgebra together with a k -linear map S : H → H (called the antipode) satisfying S ∗ id H = ε s , id H ∗ S = ε t , S ∗ id H ∗ S = S, where ∗ is the convolution product.We remark that a weak Hopf algebra is a Hopf algebra if and only if ∆(1) = 1 ⊗ ε is a homomorphism of algebras.Let H be a weak Hopf algebra and A be an algebra. By [6], A is a right weak H -comodule algebra if there is a right H -comodule structure ρ A : A → A ⊗ H, a a ⊗ a such that ρ ( ab ) = ρ ( a ) ρ ( b ) for each a, b ∈ A and 1 ⊗ ∈ A ⊗ H t . The coinvariants defined by B := A coH := { a ∈ A | ρ A ( a ) = 1 a ⊗ } is a subalgebra of A . We say the extension A/B is right weak H -Galois if the map β : A ⊗ B A → A ⊗ t H, given by a ⊗ B b ab ⊗ t b is bijective, where A ⊗ t H := ( A ⊗ H ) ρ (1) = { a ⊗ h △ = a ⊗ t h | a ∈ A, h ∈ H } . A , is defined asfin.dim A = sup { pd A M | M ∈ A -mod, pd A M < ∞} . The famous finitistic dimension conjecture says that fin.dim
A < ∞ for anyfinite dimensional algebra A and it has been proved that several classes of algebrashave finite finitistic dimension. The finitistic dimension conjecture is still opennow, for details, we refer to [11,12,14,16] and the references therein.The purpose of this paper is to study the relationship of finitistic dimensionsunder right faithfully flat weak Hopf-Galois extension. The main motivation forthe study is that zhou and Li proved in [15]: let H be a finite dimensional weakHopf algebra and A/B be a right faithfully flat weak H -Galois extension, if H issemisimple, then the finitistic dimension of A is less than or equal to that of B .Thus the question arises: whether the finitistic dimension of B is less or equal tothat of A when A/B is a right faithfully flat weak H -Galois extension? This isof particular interest because of the close relationship to the finitistic dimensionconjecture.Our main result is as follows. Theorem.
Let H be a finite-dimensional weak Hopf algebra over a field k and A/B be a right faithfully flat weak H -Galois extension. If the finitistic dimensionof B is finite, then it is less or equal to that of A . In this paper, we always assume H is finite dimensional and we follow thestandard terminology and notation used in the representation theory of algebrasand quantum groups, see [1,2,10]. The following well known Lemma shows the relations between the projectivedimensions of the three modules in a short exact sequence, which will be used later.
Lemma 2.1. If → A → A → A → is exact and two of the modules havefinite projective dimension, then so does the third. Moreover, if n < ∞ , pd A = n ,and pd A ≤ n , then pd A = n . 3et A/B be a right weak H -Galois extension. Consider the following two func-tors: A ⊗ B − : B − Mod → A − Mod, M A ⊗ B M, B ( − ) : A − Mod → B − Mod, M M, where B ( − ) is the restriction functor. It is known that ( A ⊗ B − , B ( − )) is anadjoint pair. Let ( F, G ) be an adjoint pair of functors of abelian categories. If F isexact, then G preserves injective objects, if G is exact, then F preserves projectiveobjects. Lemma 2.2.
Let
A/B be a right weak H -Galois extension for a finite-dimensionalweak Hopf algebra H . Then for each (finitely generated) B-module M, pd B ( A ⊗ B M ) ≤ pd A ( A ⊗ B M ) ≤ pd B M . Proof.
According to [4, Corallary 4.3], A B and B A are both finitely generatedprojective. It follows that pd B ( A ⊗ B M ) ≤ pd A ( A ⊗ B M ) by the change of ringtheorem. Assume that pd B M = n < ∞ , and let P be a projective resolution of M as a B -module of length n. Now consider the adjoint pair ( A ⊗ B − , B ( − )). Since B ( − ) is exact, the functor A ⊗ B − preserves projective objects. It follows that A ⊗ B P is a projective resolution of A ⊗ B M as an A -module. This implies thatpd A ( A ⊗ B M ) ≤ pd B M . The proof is completed. ✷ Now we can prove the main result of the paper.
Theorem 2.3.
Let
A/B be a right faithfully flat weak H -Galois extension fora finite-dimensional weak Hopf algebra H . If fin.dim B < ∞ , then fin.dim B ≤ fin.dim A . Proof.
For any left B -module M , there is a map ϕ : M → A ⊗ B M givenby m ⊗ m. Then it is a preliminary fact that ϕ is monic since A is a rightfaithfully flat B -module.Suppose fin.dim B = n < ∞ , we choose B M with pd B M = n . There is an B -exact sequence 0 → M → A ⊗ B M → C → . By Lemma 2.2., pd B ( A ⊗ B M ) ≤ pd B M , so pd B C is also finite and pd B C ≤ n sincefin.dim B = n . Now Lemma 2.1. gives pd B ( A ⊗ B M ) = n . By Lemma 2.2 again,4ince pd B ( A ⊗ B M ) ≤ pd A ( A ⊗ B M ) ≤ pd B M , we conclude that pd A ( A ⊗ B M ) = n. Therefore, it follows directly that fin.dim B ≤ fin.dim A . The proof is com-pleted. ✷ Corollary 2.4.
Let
A/B be a right faithfully flat weak H -Galois extension for afinite dimensional weak Hopf algebra H . Suppose H is semisimple. If the finitisticdimension conjecture holds, then fin.dim B = fin.dim A . Proof.
According to [15], Let
A/B be a right faithfully flat weak H -Galoisextension for a finite dimensional weak Hopf algebra H . Suppose H is semisimple,then fin.dim A ≤ fin.dim B . By Theorem 2.3., the proof is completed. ✷ Note that if H is an ordinary Hopf algebra, then the weak H -Galois extensionis just an H -Galois extension. So we may ask the following question: Let A/B be a right faithfully flat right H -Galois extension for a finite dimensional Hopfalgebra H . Is the finitistic dimension of B less or equal to that of A ? Of course,An affirmative answer to the finitistic dimension conjecture would also give anaffirmative answer to this question. References [1] I.Assem, D.Simson, A.Skowronski,
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