On the monoidal invariance of the cohomological dimension of Hopf algebras
aa r X i v : . [ m a t h . K T ] F e b ON THE MONOIDAL INVARIANCE OF THE COHOMOLOGICALDIMENSION OF HOPF ALGEBRAS
JULIEN BICHON
Abstract.
We discuss the question of whether the global dimension is a monoidal in-variant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidalcategories of comodules, then their global dimensions should be equal. We provide sev-eral positive new answers to this question, under various assumptions of smoothness,cosemisimplicity or finite dimension. We also discuss the comparison between the globaldimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimplecase, obtaining equality in the case the latter is finite. One of our main tools is the newconcept of twisted separable functor. introduction A classical invariant of an algebra A is its (right) global dimensionr . gldim( A ) = max { pd A ( M ) , M ∈ M A } ∈ N ∪ {∞} where for a (right) A -module M , pd A ( M ) stands for its projective dimension, i.e. thesmallest possible length for a resolution of M by projective A -modules.The global dimension is a key ingredient in the analysis of certain geometric propertiesof discrete groups [9, 14], and often serves as a good analogue of the dimension of a smoothaffine variety. However in some noncommutative situations, it is better to replace it bythe Hochschild cohomological dimension, which has similar geometric significance, and isdefined by :cd( A ) = max { n : H n ( A, M ) = 0 for some A − bimodule M } ∈ N ∪ {∞} = min (cid:8) n : H n +1 ( A, M ) = 0 for any A − bimodule M (cid:9) = pd A M A ( A )where H ∗ ( A, − ) denotes Hochschild cohomology and pd A M A ( A ) is the projective dimen-sion of A in the category of A -bimodules.Indeed, for example if A = A ( k ) is the first Weyl algebra ( k is, as in all the paper,an algebraically closed field), we have r . gldim( A ( k )) = 1 (in characteristic zero) andcd( A ( k )) = 2, while A ( k ) should definitively be considered as a 2-dimensional object.When A is a Hopf algebra, it is well-known that we haver . gldim( A ) = pd A ( k ε ) = cd( A ) = l . gldim( A ) = pd A ( ε k )where k ε and ε k denote the respective right and left trivial A -modules, and l . gldim( A ) isthe left global dimension. See [26] for the equalities at the extreme left and right, and,for example, [20] for the other equality.A general classical problem is whether the global dimension or the Hochschild coho-mological dimension remain preserved under various kind of “deformations” of A , andthe question we are particularly interested in, originally asked in [6] and suggested byexamples studied in [5], is the following one. Mathematics Subject Classification. uestion 1.1. If A and B are Hopf algebras having equivalent linear tensor categoriesof comodules, do we have cd( A ) = cd( B ) ? Notice that the word “tensor” is important in the above question, since this is whatcaptures information about the algebra structure inside the category of comodules. Drop-ping it would make the question meaningless, as shown by the example of group algebras:if two group algebras have equivalent categories of comodules, the only conclusion, in lackof additional information, is that the groups have the same cardinality.Partial positive answers to Question 1.1 were provided in [6, 7] when A , B are cosemisim-ple with antipode satisfying S = id, and by Wang, Yu and Zhang in [39], when A istwisted Calabi-Yau and B is homologically smooth.The aim of this paper is to provide several new positive answers to Question 1.1. Thefollowing theorem summarizes our contributions. Theorem 1.2.
Let A , B be Hopf algebras that have equivalent linear tensor categoriesof comodules: M A ≃ ⊗ M B . We have cd( A ) = cd( B ) if one of the following conditionsholds: (1) A , B are homologically smooth and have bijective antipode; (2) A , B are cosemisimple and cd( A ) , cd( B ) are finite; (3) A , B are finite-dimensional and char( k ) = 0 or char( k ) = p > d ϕ ( d )2 where d =dim( A ) , or A ∗ is unimodular. While the proof of the first statement (Theorem 8.1) is obtained by carefully inspectingarguments in [42, 39] and the third one (Theorem 8.3) is a rather direct consequence ofprevious results [24, 16, 1], the main effort in this paper is in proving the second statement(Theorem 4.6). Removing the assumption S = id from [7] (with instead the finitenessassumption on cohomological dimensions) enables us to compute cohomological dimensionin a number of new situations, see Section 7 for examples regarding universal cosovereignHopf algebras and free wreath products.Our method for proving (2) in Theorem 1.2 is based on the fact that if M A ≃ ⊗ M B as above, results by Schauenburg [35] ensures that there exists an A - B Galois object R , and then on proving that cd( A ) = cd( R ) = cd( B ). For this, one notices thatcd( A ) = pd R M BR ( R ), the projective dimension of R in the category of R -bimodules inside B -comodules, and then the main question is to compare pd R M BR ( R ) and pd R M R ( R ) =cd( R ). The main ingredient in this comparison is a twisted averaging trick, Lemma 4.2,that we believe to be quite non-straightforward. The averaging lemma leads to the con-cept of twisted separable functor we define in Section 3, a generalization of the notion ofseparable functor introduced in [30].Our initial idea to tackle Question 1.1 was, in [6], to use an auxiliary cohomologicaldimension for the Hopf algebra A , the Gestenhaber-Schack cohomological dimension,defined bycd GS ( A ) = max { n : Ext n YD AA ( k, V ) = 0 for some V ∈ Y D AA } ∈ N ∪ {∞} where Y D AA is the category of Yetter-Drinfeld modules over A and k is the trivial Yetter-Drinfeld module. It was shown in [6, Theorem 5.6, Corollary 5.7] that cd( A ) ≤ cd GS ( A )and that if A , B are Hopf algebras with M A ≃ ⊗ M B , thenmax(cd( A ) , cd( B )) ≤ cd GS ( A ) = cd GS ( B )Therefore, comparing cd( A ) and cd GS ( A ) can be a key step towards answers to Question1.1. In this direction, we obtain the following result. heorem 1.3. Let A be a cosemisimple Hopf algebra. If cd GS ( A ) is finite, we have cd( A ) = cd GS ( A ) . The above theorem has, as a corollary, a weak form of (2) in Theorem 1.2, which isprobably sufficient in dealing with numerous examples. Again the method of proof isbased on a twisted averaging trick and uses an appropriate twisted separable functor.We expect that the equality cd( A ) = cd GS ( A ) holds for any cosemisimple Hopf algebra,but as already pointed in [6], it cannot hold for any Hopf algebra over any field, aswe see by taking a semisimple non cosemisimple Hopf algebra over a field of positivecharacteristic, so we asked there whether the equality was true in characteristic zero.Etingof pointed out that it does not hold in characteristic zero even for the very simpleexample A = k [ x ] with x primitive. Hence we have now the following question. Question 1.4.
What are the Hopf algebras such that cd( A ) = cd GS ( A ) ? The paper is organized as follows. Section 2 collects preliminary notations and resultson cosemisimple Hopf algebras. Section 3 introduces the notion of twisted separablefunctor. After some preliminary material on categories of bimodules inside categoriesof comodules, Part (2) of Theorem 1.2 is proved in Section 4. Section 5 provides theproof of Theorem 1.3, together with the necessary material on Yetter-Drinfeld modules.Section 6 studies the behaviour of Gerstenhaber-Schack cohomological dimension underHopf subalgebras in the cosemisimple case. Section 7 is devoted to applications to someexamples. The final Section 8 reviews what is known about Question 1.1 outside thecosemisimple case, in the smooth case and in the finite-dimensional situation, providingthe proofs of (1) and (3) in Theorem 1.2. The reader only interested in those resultsmight go directly to this section.
Notations and conventions.
We work over an algebraically closed field k . We assumethat the reader is familiar with the theory of Hopf algebras and their tensor categoriesof comodules, as e.g. in [17, 22, 29], and with the basics of homological algebra [9, 40].If A is a Hopf algebra, as usual, ∆, ε and S stand respectively for the comultiplication,counit and antipode of A . We use Sweedler’s notations in the standard way. The categoryof right A -comodules is denoted M A , the category of right A -modules is denoted M A ,etc... The trivial (right) A -module is denoted k ε . The set of A -module morphisms (resp. A -comodule morphisms) between two A -modules (resp. two A -comodules) V and W isdenoted Hom A ( V, W ) (resp. Hom A ( V, W )).
Acknowledgements.
I would like to thank Pavel Etingof for interesting discussions andpertinent remarks. 2. cosemisimple hopf algebras
In this section we collect some preliminary facts and notations on cosemisimple Hopfalgebras. Recall that a Hopf algebra is cosemisimple if and only if it admits a Haarintegral, i.e. a linear map h : A → k such that for any a ∈ A , we have h ( a (1) ) a (2) = h ( a ) = h ( a (2) ) a (1) and h (1) = 1The proof of the semisimplicity of the category of comodules from the existence of aHaar integral is a consequence of the following averaging construction, that we record forfuture use. roposition 2.1. Let V , W be right A -comodules over a cosemisimple Hopf algebra A ,and let f : V → W be a linear map. The map M V,W ( f ) : V −→ Wv h (cid:0) f ( v (0) ) (1) S ( v (1) ) (cid:1) f ( v (0) ) (0) is a morphism of comodules, with M ( f ) = f if and only if f is a morphism of comodulesand with, for any morphisms of comodules α : V ′ → V and β : W → W ′ , β ◦ M V,W ( f ) ◦ α = M V ′ ,W ′ ( β ◦ f ◦ α ) . The above construction therefore defines a projection M V,W :Hom(
V, W ) → Hom A ( V, W ) , that we call the averaging with respect to V and W . The Haar integral is not a trace in general, but satisfies a KMS type property, discoveredby Woronowicz [41] in the setting of compact quantum groups.
Theorem 2.2.
Let A be a cosemisimple Hopf algebra with Haar integral h . There exists aconvolution invertible linear map ψ : A → k , called a modular functional on A , satisfyingthe following conditions: • S = ψ ∗ id ∗ ψ − ; • σ := ψ ∗ id ∗ ψ is an algebra automorphism of A ; • we have h ( ab ) = h ( bσ ( a )) for any a, b ∈ A . The proof relies on the orthogonality relations, whose first occurence is due to Larson[23], and were completed by Woronowicz [41]. In all the treatment we are aware of[22, 31], the setting is over the field of complex numbers, but inspecting the proof showsthat it is valid for any cosemisimple Hopf algebra over any algebraically closed field.To conclude the section, we introduce a last piece of notation. If A is a cosemisimpleHopf algebra A with Haar integral h and modular functional ψ , we denote by θ thealgebra automorphism of A defined by θ = ψ ∗ id.3. Twisted separable functors
In this section we introduce the notion of twisted separable functor, as follows.
Definition 3.1.
Let C and D be some categories. We say that a functor F : C → D is twisted separable if there exist(1) an autoequivalence Θ of the category D ;(2) a generating subclass F of objects of C (i.e. for every object V of C , there existsan object P of F together with an epimorphism P → V ) together with, for anyobject P of F , an isomorphism θ P : F ( P ) → Θ F ( P );(3) a natural morphism M − , − : Hom D ( F ( − ) , Θ F ( − )) → Hom C ( − , − ) such that forany object P of F , we have M P,P ( θ P ) = id P .The naturality condition above means that for any morphisms α : V ′ → V , β : W → W ′ in C and any morphism f : F ( V ) → Θ F ( W ) in D , we have β ◦ M V,W ( f ) ◦ α = M V ′ ,W ′ (Θ F ( β ) ◦ f ◦ F ( α ))When F is the whole class of objects of C , the autoequivalence Θ is the identity andthe isomorphisms θ P all are the identity, we get the notion of separable functor from [30],which is known to be provide a convenient setting for various types of generalized Maschketheorems, see [11]. A basic example of a separable functor is, when A is a cosemisimpleHopf algebra, the forgetful functor M A → Vec k : this is the content of Proposition 2.1.Our motivation to introduce the present notion of twisted separable functor is thefollowing result. roposition 3.2. Let C and D be abelian categories having enough projective objects,and let F : C → D be a functor. Assume that the following conditions hold: (1) the functor F is exact and preserves projective objects; (2) the functor F is twisted separable and F , the corresponding class of objects of C ,contains a generating subclass F consisting of projective objects.Then, for any object V of C such that pd C ( V ) is finite, we have pd C ( V ) = pd D ( F ( V )) . As usual, the notation pd C ( V ) refers to the projective dimension of the object V , i.e.the smallest length of a resolution of V by projective objects in C , with, as wellpd C ( V ) = max { n : Ext n C ( V, W ) = 0 for some object W in C} We begin with some preliminaries.
Lemma 3.3.
Let C be an abelian category having enough projective objects, and let F be a generating subclass of C consisting of projective objects. If pd C ( V ) is finite, we have pd C ( V ) = max { n : Ext n C ( V, F ) = 0 for some object F in F } Proof.
Every object X fits into an exact sequence 0 → W → F → X → F anobject of F , hence projective. The result is thus obtained via a classical argument: if n = pd C ( V ), the long Ext exact sequence gives that the functor Ext n C ( V, − ) is right exact,and hence Ext n C ( V, F ) = 0 for some object F of F . (cid:3) Lemma 3.4.
Assume we are in the setting of Proposition 3.2. For any objects
X, W of C , we have a morphism Ext ∗D ( F ( X ) , Θ F ( W )) −→ Ext ∗C ( X, W ) which is surjective if W is an object of F .Proof. Start with a projective resolution · · · −→ P n d n −→ P n − d n − −→ · · · d −→ P d −→ P d −→ X → X by objects in C . Since the functor F is exact and preserves projectives, we get aprojective resolution · · · −→ F ( P n ) F ( d n ) −→ F ( P n − ) F ( d n − ) −→ · · · F ( d ) −→ F ( P ) F ( d ) −→ F ( P ) F ( d ) −→ F ( X ) → F ( X ) in D . For all i ≥
0, we have, by the naturality assumption, commutativediagrams Hom D ( F ( P i ) , Θ F ( W )) M Pi,W (cid:15) (cid:15) −◦ F ( d i +1 ) / / Hom D ( F ( P i +1 ) , Θ F ( W )) M Pi +1 ,W (cid:15) (cid:15) Hom C ( P i , W ) −◦ d i +1 / / Hom C ( P i +1 , W )that induce a morphism of complexes f M : Hom D ( F ( P ∗ ) , Θ F ( W )) → Hom C ( P ∗ , W )and hence a morphism between the corresponding cohomologies: H ∗ ( f M ) : Ext ∗D ( F ( X ) , Θ F ( W )) −→ Ext ∗C ( X, W )Assume now that W is an object of F , and let f ∈ Hom C ( P i , W ). We have M P i ,W ( θ W ◦ F ( f )) = M W,W ( θ W ) ◦ f = f nd if moreover f ◦ d i +1 = 0, we have also θ W ◦ F ( f ) ◦ F ( d i +1 ) = 0. This shows that H ∗ ( f M ) is surjective. (cid:3) Remark . Assume, as the setting of Proposition 3.2 allows us to, that in the proof ofthe previous lemma, we have started with a projective resolution · · · −→ P n d n −→ P n − d n − −→ · · · d −→ P d −→ P d −→ X → X by objects in F . Then, for f ∈ Hom C ( P i , W ), we have M P i ,W (Θ( F ( f )) ◦ θ P i ) = f ◦ M P i ,P i ( θ P i ) = f This shows that the morphism of complexes f M : Hom D ( F ( P ∗ ) , Θ F ( W )) → Hom C ( P ∗ , W )is surjective in general. However, since we see no reason that Θ F ( f ) ◦ θ P i ◦ F ( d i +1 ) = 0,we cannot conclude that the corresponding morphism in cohomology is surjective withoutour assumption on W . Proof of Proposition 3.2.
Let V be an object of C , and let · · · −→ P n d n −→ P n − d n − −→ · · · d −→ P d −→ P d −→ V → V . Since the functor F is exact and preserves projectives,we get a projective resolution · · · −→ F ( P n ) F ( d n ) −→ F ( P n − ) F ( d n − ) −→ · · · F ( d ) −→ F ( P ) F ( d ) −→ F ( P ) F ( d ) −→ F ( V ) → F ( V ) in D . This shows that pd D ( F ( V )) ≤ pd C ( V ). To prove the converse in-equality, we can assume that n = pd D ( F ( V )) is finite. We then have in particu-lar Ext n +1 D ( F ( V ) , Θ F ( P )) = { } for any object P in F , and by Lemma 3.4, we haveExt n +1 C ( V, P ) = { } as well. Hence, assuming that pd C ( V ) is finite, Lemma 3.3 showsthat pd C ( V ) ≤ n , concluding the proof. (cid:3) In this paper we will not develop any more theory on twisted separable functors, andwill focus on applications of Proposition 3.2.4. comodule algebras and equivariant bimodule categories
Our aim in this section is to prove part (2) of Theorem 1.2. We begin with somepreliminaries on categories of bimodules in categories of comodules. Let A be a Hopfalgebra, let R be a right A -comodule algebra ( R is an algebra in the monoidal category M A ) and let R M AR be the category of A -bimodules in the category M A : the objects arethe A -comodules V with an R -bimodule structure having the Hopf bimodule compatibilityconditions( x · v ) (0) ⊗ ( x · v ) (1) = x (0) · v (0) ⊗ x (1) v (1) , ( v · x ) (0) ⊗ ( v · x ) (1) = v (0) · x (0) ⊗ v (1) x (1) for any x ∈ R and v ∈ V . The morphisms are the A -colinear and R -bilinear maps.The category R M AR is obviously abelian, and the tensor product of bimodules induces amonoidal strucure on it.The following basic property is certainly well-known, and a straightforward verification. Proposition 4.1.
Let A be a Hopf algebra and let R be an A -comodule algebra. (1) The forgetful functor Ω A : R M AR → M A has a left adjoint, which associates to acomodule V the object R ⊗ V ⊗ R whose bimodule structure is given by left andright multiplication of R and whose comodule structure is the tensor product ofthe underlying comodules. The forgetful functor Ω R : R M AR → R M R has a right adjoint, which associates toan R -bimodule V the object V ⊗ A whose R -bimodule structure is given by x · ( v ⊗ a ) = x (0) · v ⊗ x (1) a, ( v ⊗ a ) · x = v · x (0) ⊗ ax (1) and whose A -comodule structure is induced by the comultiplication of A . Objects in R M AR that are images of the above left adjoint functor are called free, theyare indeed free as bimodules. Any object in R M AR is a quotient of a free object. Thefollowing are direct consequences of the standard properties of pairs of adjoint functors.(1) The category R M AR has enough injective objects, since R M R has, and we have,for any object V in R M AR and any R -bimodule W :Ext ∗ R M R (Ω R ( V ) , W ) ≃ Ext ∗ R M AR ( V, W ⊗ A )(2) If M A has enough projective objects (in which case one says that A is co-Frobenius), so has R M AR . In that case, the previous isomorphism ensures thatfor any object V in R M AR , we havepd R M R (Ω R ( V )) ≤ pd R M AR ( V )(3) If A is cosemisimple, then R M AR has enough projective objects, and the projectiveobjects are the direct summands of the free ones.We now present our key averaging lemma for bimodules. If R is an A -comodule algebraover a cosemisimple Hopf algebra A , we denote by ρ the algebra automorphism of R defined by ρ = id ∗ ψ − , i.e. ρ ( x ) = ψ − ( x (1) ) x (0) , with ψ a modular functional as inSection 2. Lemma 4.2.
Let
V, W be objects of R M AR , where A is a cosemisimple Hopf algebra A and R is a right A -comodule algebra. If f : V → W is a linear map satisfying f ( v · x ) = f ( v ) · x and f ( x · v ) = ρ ( x ) · f ( v ) for any v ∈ V and x ∈ R , then M V,W ( f ) : V → W is a morphism in the category R M AR .Proof. We already know that M V,W ( f ) : V → W is colinear and there remains to provethat M V,W ( f ) is left and right R -linear as well. Let v ∈ V and x ∈ R . We have, using ourcondition on f and the compatibility between the comodule and right module structure: M V,W ( f )( v · x ) = h (cid:0) f (( v · x ) (0) ) (1) S (( v · x ) (1) ) (cid:1) f (( v · x ) (0) ) (0) = h (cid:0) f ( v (0) · x (0) ) (1) S ( v (1) x (1) ) (cid:1) f ( v (0) · x (0) ) (0) = h (cid:0) ( f ( v (0) ) · x (0) ) (1) S ( v (1) x (1) ) (cid:1) ( f ( v (0) ) · x (0) ) (0) = h (cid:0) ( f ( v (0) ) (1) x (1) S ( v (1) x (2) ) (cid:1) f ( v (0) ) (0) · x (0) = h (cid:0) ( f ( v (0) ) (1) x (1) S ( x (2) ) S ( v (1) ) (cid:1) f ( v (0) ) (0) · x (0) = M V,W ( f )( v ) · x Hence f is right R -linear. We also have, using our condition on f and the compatibilitybetween the comodule and left module structure: M V,W ( f )( x · v ) = h (cid:0) f (( x · v ) (0) ) (1) S (( x · v ) (1) ) (cid:1) f (( x · v ) (0) ) (0) = h (cid:0) ( f ( x (0) · v (0) )) (1) S ( x (1) v (1) ) (cid:1) f ( x (0) · v (0) ) (0) = ψ − ( x (1) ) h (cid:0) ( x (0) · f ( v (0) )) (1) S ( x (2) v (1) ) (cid:1) ( x (0) · f ( v (0) ) (0) = ψ − ( x (2) ) h (cid:0) x (1) f ( v (0) ) (1) S ( x (3) v (1) )) (cid:1) x (0) · f ( v (0) ) (0) sing the properties of the modular functional, this gives: M V,W ( f )( x · v ) = ψ − ( x (4) ) h (cid:0) f ( v (0) ) (1) S ( v (1) ) S ( x (5) ) ψ ( x (1) ) x (2) ψ ( x (3) ) (cid:1) x (0) · f ( v (0) ) (0) = h (cid:0) f ( v (0) ) (1) S ( v (1) ) S ( x (4) ) ψ ( x (1) ) x (2) ψ − ( x (3) ) (cid:1) x (0) · f ( v (0) ) (0) = h (cid:0) f ( v (0) ) (1) v (1) S ( x (2) ) S ( x (1) ) (cid:1) x (0) · f ( v (0) ) (0) = x · M V,W ( f )( v )and this shows that M V,W ( f ) is left R -linear as well. (cid:3) Lemma 4.3.
Let V be a comodule over the cosemisimple Hopf algebra A , let R be an A -comodule algebra and consider the map ρ V = ρ ⊗ id V ⊗ id R : R ⊗ V ⊗ R → R ⊗ V ⊗ R .We have M ( ρ V ) = id R ⊗ V ⊗ R , where M stands for averaging with respect to R ⊗ V ⊗ R .Proof. It is immediate that ρ V = ρ ⊗ id V ⊗ id R : R ⊗ V ⊗ R → R ⊗ V ⊗ R satisfies theassumption of Lemma 4.2, hence M ( ρ V ) is left and right R -linear. Since it is clear that M ( ρ V )(1 ⊗ v ⊗
1) = 1 ⊗ v ⊗ v ∈ V , we get the result by the R -bilinearity of M ( ρ V ). (cid:3) We now have all the ingredients to prove that for A cosemisimple and an A -comodulealgebra R , the forgetful functor Ω R : R M AR → R M R is twisted separable, and hence thefollowing result. Proposition 4.4. If A is a cosemisimple Hopf algebra and R is a right A -comodulealgebra, we have pd R M AR ( V ) = pd R M R ( V ) for any object V in R M AR such that pd R M AR ( V ) is finite. In particular, if pd R M AR ( R ) is finite, we have pd R M AR ( R ) = pd R M R ( R ) = cd( R ) .Proof. In order to show that the forgetful functor Ω R : R M AR → R M R is twisted separable,consider(1) the class F = F of free bimodules in R M AR ;(2) the autoequivalence Θ of the category R M R that associates to an R -bimodule W the R -bimodule ρ W having W as underlying vector space and R -bimodulestructure given by x · ′ w · ′ y = ρ ( x ) · w · y , and is trivial on morphisms;(3) for a free object R ⊗ V ⊗ R , the R -bimodule isomorphism ρ V : R ⊗ V ⊗ R → ρ ( R ⊗ V ⊗ R ) in Lemma 4.3;(4) for objects V, W in R M AR , the averaging map M V,W : Hom R M R ( V, ρ W ) → Hom R M AR ( V, W )from Lemma 4.2.It follows from Lemma 4.2, Lemma 4.3 and Proposition 2.1 that the functor Ω R : R M AR → R M R is indeed twisted separable. Moreover, as already said, the class F of free objectsconsists of projective objects, the projective objects in R M AR are direct summands offree objects and hence are preserved by Ω R , which is clearly exact. Hence we are in thesituation of Proposition 3.2, and we obtain the equality of projective dimensions. (cid:3) Recall that a left Galois object over a Hopf algebra A is a non-zero left A -comodulealgebra R such that the canonical map R ⊗ R −→ A ⊗ Rx ⊗ y x ( − ⊗ x (0) y is bijective. Similarly a right Galois object over A is a right A -comodule algebra suchthat the obvious analogue of the previous canonical map is bijective.Our main application of Proposition 4.4 is the following result. heorem 4.5. Let A be Hopf algebra and let R be a left or right A -Galois object. If A is cosemisimple and cd( A ) is finite, we have cd( A ) = cd( R ) Proof.
First recall [34] that it follows from the structure theorem for Hopf modules thatthe functor A M −→ A M AA V V ⊚ A is a monoidal equivalence of categories, where V ⊚ A is V ⊗ A as a vector space, has thetensor product left A -module structure and the right module and comodule structuresare induced by the multiplication and comultiplication of A respectively. This monoidalequivalence transforms the trivial module ε k into the A -bimodule A .Now let R be a left Hopf-Galois object over A . By the results in [35], there exists aHopf algebra B such that R is an A - B bi-Galois object and the cotensor product − (cid:3) A R induces a monoidal equivalence : M A ≃ ⊗ M B sending A to R , which in turn clearlyinduces an equivalence between the bimodule categories A M AA and R M BR . Composingwith the equivalence at the beginning of the proof, we get a monoidal equivalence A M ≃ ⊗ R M BR sending ε k to R , so that cd( A ) = pd A ( ε k ) = pd R M BR ( R ). Hence if cd( A ) is finite, weget that pd R M BR ( R ) is finite as well, and assuming moreover that A is cosemisimple, weconclude by Proposition 4.4 that cd( A ) = pd R M BR ( R ) = cd( R ), as claimed.If we start with a right Hopf-Galois object R over A , it is well-known that R op is a left A -Galois object in a natural way (if the antipode of A is bijective, which is the case heresince we assume cosemisimplicity), so that we can use the result for left A -Galois objectsto conclude that cd( A ) = cd( R ) as well. (cid:3) Theorem 4.6.
Let A , B be Hopf algebras that have equivalent linear tensor categoriesof comodules: M A ≃ ⊗ M B . If A and B are cosemisimple and cd( A ) , cd( B ) are finite,we have cd( A ) = cd( B ) .Proof. By the results in [35], such a monoidal equivalence arises from an A - B -bi-Galoisobject R , via the cotensor product. Hence Theorem 4.5 ensures, under the finitenessassumption on cd( A ) and cd( B ), that cd( A ) = cd( R ) = cd( B ). (cid:3) We finish the section by noticing that Proposition 4.4 can be strengthened in the case S = id. Proposition 4.7.
Let A be a cosemisimple Hopf algebra with S = id , and let R be aright A -comodule algebra. (1) The forgetful functor Ω R : R M AR → R M R is separable. We thus have pd R M AR ( V ) =pd R M R ( V ) for any object V in R M AR , and pd R M AR ( R ) = cd( R ) . (2) Let F : M A ≃ ⊗ M B be a monoidal equivalence with B satisfying S = id as well.We then have, for the B -comodule algebra S = F ( R ) , cd( R ) = cd( S ) . (3) Let F : M A −→ Vec k be a fibre functor. If cd( R ) is finite, we have, for the algebra S = F ( R ) , cd( R ) = cd( S ) .Proof. As in the proof of Lemma 4.2, using the properties of the modular functional, wesee that for any a, x ∈ Ah ( S ( a (1) ) xa (2) ) = ψ − ( a (2) ) h (cid:0) xa (3) S − ( a (1) ) (cid:1) t x = 1 this gives ε ( a ) = ψ − ( a (2) ) h (cid:0) a (3) S − ( a (1) ) (cid:1) . If S = id, then ψ − convolutioncommutes with the identity, hence we get ψ − = ε . Hence the automorphism ρ associatedto an A -comodule algebra R is the identity, the autoequivalence Θ in the proof of Propo-sition 4.4 is the identity, and the class F is the class of all objects, and it follows thatΩ R : R M AR → R M R is separable. The result about projective dimensions is then eitherwell-known or follows from the obvious improvement of Proposition 4.4 in the separablecase, having in mind that the conclusion of Lemma 3.4 now holds for any object.A monoidal equivalence F : M A ≃ ⊗ M B induces, as before, an equivalence betweenthe bimodule categories R M AR and S M BS for S = F ( R ), sending R to S , and then theassumption S = id on A and B ensures that cd( R ) = pd R M AR ( R ) = pd S M BS ( S ) = cd( S ).Start now with a fibre functor F : M A −→ Vec k , i.e. a k -linear monoidal exact faithfulfunctor that commutes with colimits. Such a functor induces, by Tannaka-Krein duality(see e.g. [21, 17]) or by the results in [35], a monoidal equivalence M A ≃ ⊗ M B for someHopf algebra B , with as well a monoidal equivalence R M AR ≃ ⊗ S M BS . The assumptionthat S = id for A then gives pd R M AR ( R ) = cd( R ). Since pd R M AR ( R ) = pd S M BS ( S ),Proposition 4.4 ensures, under the assumption that cd( R ) is finite, that pd S M BS ( S ) =cd( S ), and thus this gives the expected result. (cid:3) Example . Let σ : A ⊗ A → k be (Hopf, right) 2-cocycle on a Hopf algebra A (see [29]),i.e. σ is a convolution invertible linear map σ : A ⊗ A → k satisfying, for any a, b, c ∈ Aσ ( a,
1) = ε ( a )1 = σ (1 , a ) , σ ( a (2) , b (2) ) σ ( a (1) b (1) , c ) = σ ( a, b (1) c (1) ) σ ( b (2) , c (2) )If R is a right A -comodule algebra, we obtain a new (associative) algebra R σ by letting x.y = σ ( x (1) , y (1) ) x (0) y (0) We then have cd( R ) = cd( R σ ) if cd( R ) is finite. Proof.
The algebra R σ is the image of R under the fibre functor M A → Vec k which hasthe forgetful functor as underlying functor and monoidal constraint V ⊗ W → V ⊗ W , v ⊗ w σ ( v (1) , w (1) ) v (0) ⊗ w (0) . The result is thus a consequence of Proposition 4.7. (cid:3) Remark . If the Hopf algebra A σ (see [15, 35]) satisfies as well S = id, we can concludethat cd( R ) = cd( R σ ) without the finiteness assumption. This applies in particular, when A is a group algebra, to the 2-cocycle twisting of a group-graded algebra, which thereforehas the same Hochschild cohomological dimension as the original algebra. This wasprobably well-known, but we are not aware of an explicit reference for this fact.5. yetter-drinfeld modules over cosemimple hopf algebras Our aim in this section is to prove Theorem 1.3. Recall that a (right-right) Yetter-Drinfeld module over a Hopf algebra A is a right A -comodule and right A -module V satisfying the condition, ∀ v ∈ V , ∀ a ∈ A ,( v · a ) (0) ⊗ ( v · a ) (1) = v (0) · a (2) ⊗ S ( a (1) ) v (1) a (3) The category of Yetter-Drinfeld modules over A is denoted Y D AA : the morphisms are the A -linear and A -colinear maps. The category Y D AA is obviously abelian, and, endowedwith the usual tensor product of modules and comodules, is a tensor category, with unitthe trivial Yetter-Drinfeld module, denoted k .The forgetful functor Ω A : Y D AA → M A has a left adjoint [10], the free Yetter-Drinfeldmodule functor, which sends a comodule V to the Yetter-Drinfeld module V ⊠ A , which s a vector space is V ⊗ A , has the right module structure given by multiplication on theright, and right coaction given by( v ⊗ a ) (0) ⊗ ( v ⊗ a ) (1) = v (0) ⊗ a (2) ⊗ S ( a (1) ) v (1) a (3) A Yetter-Drinfeld module isomorphic to some V ⊠ A as above is said to be free. Let usrecord the following facts, that are straightforward consequences of standard propertiesof pairs of adjoint functors.(1) Every Yetter-Drinfeld module is a quotient of a free Yetter-Drinfeld module. In-deed, for a Yetter-Drinfeld V , the A -module structure of V induces a surjectivemorphism Ω A ( V ) ⊠ A → V .(2) If the category M A has enough projective objects, then so has Y D AA .(3) If A is cosemisimple, then Y D AA has enough projective objects, and the projectiveobjects are precisely the direct summands of the free Yetter-Drinfeld modules.Similarly, the forgetful functor Ω A : Y D AA → M A has a right adjoint [10], the cofreeYetter-Drinfeld module functor, which sends a module V to the Yetter-Drinfeld module V A , which as a vector space is V ⊗ A , has the right comodule structure given by thecomultiplication of A on the right, and right A -module structure given by( v ⊗ a ) · b = v · b (2) ⊗ S ( b (1) ) ab (3) Again, as a consequence of general properties of adjoint functors, it follows that thecategory
Y D AA has enough injective objects, since M A has.Recall that we have defined the Gerstenhaber-Schack cohomological dimension of aHopf algebra A bycd GS ( A ) = max { n : Ext n YD AA ( k, V ) = 0 for some V ∈ Y D AA } ∈ N ∪ {∞} The name comes from the fact, proved in [37], that if V is a Yetter-Drinfeld over A , thenExt ∗YD AA ( k, V ) is isomorphic with H ∗ GS ( A, V ), the Gerstenhaber-Schack cohomology of A with coefficients in V [19, 36].Notice that since Y D AA has enough injective objects, the above Ext can computed usinginjective resolutions of V , and if Y D AA has enough projective objects, using projectiveresolutions of k in Y D AA . Another consequence of general properties of pairs of adjointfunctors is that we have, for any Yetter-Drinfeld module V and any A -module W, naturalisomorphisms Ext ∗ A (Ω A ( V ) , W ) ≃ Ext ∗YD AA ( V, W A )This is what proves that cd( A ) ≤ cd GS ( A ) [6].We now present an averaging lemma for Yetter-Drinfeld modules over cosemisimpleHopf algebras, in the same spirit as Lemma 4.2, which will be the key tool towards theproof of Theorem 1.3. Lemma 5.1.
Let
V, W be Yetter-Drinfeld modules over a cosemisimple Hopf algebra A .If f : V → W is a linear map satisfying f ( v · a ) = f ( v ) · θ ( a ) for any v ∈ V and a ∈ A ,then M V,W ( f ) : V → W is a morphism of Yetter-Drinfeld modules.Proof. We already know that M V,W ( f ) : V → W is colinear and there remains to provethat M V,W ( f ) is A -linear as well. Let v ∈ V and a ∈ A . We have, using our condition n f and the Yetter-Drinfeld property: M V,W ( f )( v · a ) = h (cid:0) f (( v · a ) (0) ) (1) S (( v · a ) (1) ) (cid:1) f (( v · a ) (0) ) (0) = h (cid:0) f ( v (0) · a (2) ) (1) S ( S ( a (1) ) v (1) a (3) ) (cid:1) f ( v (0) · a (2) ) (0) = h (cid:0) ( f ( v (0) ) · θ ( a (2) )) (1) S ( S ( a (1) ) v (1) a (3) ) (cid:1) ( f ( v (0) ) · θ ( a (2) )) (0) = ψ ( a (2) ) h (cid:0) ( f ( v (0) ) · a (3) ) (1) S ( S ( a (1) ) v (1) a (4) ) (cid:1) ( f ( v (0) ) · a (3) ) (0) = ψ ( a (2) ) h (cid:0) S ( a (3) ) f ( v (0) ) (1) a (5) S ( S ( a (1) ) v (1) a (6) ) (cid:1) f ( v (0) ) (0) · a (4) = ψ ( a (2) ) h (cid:0) S ( a (3) ) f ( v (0) ) (1) a (5) S ( a (6) ) S ( v (1) ) S ( a (1) ) (cid:1) f ( v (0) ) (0) · a (4) = ψ ( a (2) ) h (cid:0) S ( a (3) ) f ( v (0) ) (1) S ( v (1) ) S ( a (1) ) (cid:1) f ( v (0) ) (0) · a (4) Using the properties of the modular functional, and since σ ◦ S = σ − = ψ − ∗ S ∗ ψ − because σ is an algebra map, this gives: M V,W ( f )( v · a ) = ψ ( a (2) ) h (cid:0) f ( v (0) ) (1) S ( v (1) ) S ( a (1) ) σ ( S ( a (3) ) (cid:1) f ( v (0) ) (0) · a (4) = ψ ( a (2) ) h (cid:0) f ( v (0) ) (1) S ( v (1) ) S ( a (1) ) ψ − ( a (3) ) S ( a (4) ) ψ − ( a ) (cid:1) f ( v (0) ) (0) · a (6) = h (cid:0) f ( v (0) ) (1) S ( v (1) ) S ( a (1) ) ψ ( a (2) ) S ( a (3) ) ψ − ( a )) (cid:1) f ( v (0) ) (0) · a (5) = h (cid:0) f ( v (0) ) (1) S ( v (1) ) S ( a (1) ) S ( a ) (cid:1) f ( v (0) ) (0) · a (3) = h (cid:0) f ( v (0) ) (1) S ( v (1) ) (cid:1) f ( v (0) ) (0) · a = M V,W ( f )( v ) · a and this shows that M V,W ( f ) is A -linear. (cid:3) Lemma 5.2.
Let V be a right comodule over the cosemisimple Hopf algebra A , andconsider the linear map θ V = id V ⊗ θ : V ⊠ A → V ⊠ A . We have M ( θ V ) = id V ⊠ A , where M ( θ V ) stands for M V ⊠ A,V ⊠ A ( θ V ) .Proof. It is immediate that id V ⊗ θ : V ⊠ A → V ⊠ A satisfies the assumption of Lemma5.1, hence M (id V ⊗ θ ) is A -linear. Since it is clear that M (id V ⊗ θ )( v ⊗
1) = v ⊗ v ∈ V , we get the result by the A -linearity of M (id V ⊗ θ ). (cid:3) We now have all the ingredients to prove that for A cosemisimple, the forgetful functorΩ A : Y D AA → M A is twisted separable, and hence the following result, which will haveTheorem 1.3 as an immediate corollary. Proposition 5.3. If A is a cosemisimple Hopf algebra, we have pd YD AA ( V ) = pd A ( V ) forany Yetter-Drinfeld module V such that pd YD AA ( V ) is finite.Proof. In order to show that the forgetful functor Ω A : Y D AA → M A is twisted separable,consider(1) the class F = F of free Yetter-Drinfeld modules;(2) the autoequivalence Θ of the category M A that associates to a right A -module W the A -module W θ having W as underlying vector space and A -module structuregiven by w · ′ a = w · θ ( a ), and is trivial on morphisms;(3) for a free Yetter-Drinfeld module V ⊠ A , the A -module isomorphism θ V : V ⊗ A → ( V ⊗ A ) θ in Lemma 5.2.(4) for Yetter-Drinfeld modules V, W , the averaging map M V,W : Hom A ( V, W θ ) → Hom YD AA ( V, W )from Lemma 5.1. t follows from Lemma 5.1, Lemma 5.2 and Proposition 2.1 that the functor Ω A : Y D AA →M A is indeed twisted separable. Moreover, as already said, the class F of free Yetter-Drinfeld modules consists of projective objects, the projective objects in Y D AA are directsummands of free objects and hence are preserved by Ω A , which is exact. Hence we are inthe situation of Proposition 3.2, and we obtain the equality of projective dimensions. (cid:3) Proof of Theorem 1.3.
Let A be a cosemimple Hopf algebra. Since cd GS ( A ) = pd YD AA ( k )and cd( A ) = pd A ( k ε ), we have cd( A ) = cd GS ( A ) if cd GS ( A ) is finite, by Proposition5.3. (cid:3) We get the following weak form of Theorem 4.6, whose formulation is probably useful.
Corollary 5.4.
Let A , B be Hopf algebras such that M A ≃ ⊗ M B . If A and B arecosemisimple and cd GS ( A ) is finite, we have cd( A ) = cd( B ) . As in Section 4, Proposition 5.3 can be strengthened when S = id. Theorem 5.5.
Let A be Hopf algebra. The forgetful functor Ω A : Y D AA → M A isseparable if and only if A is cosemisimple and S = id , and in that case we have pd YD AA ( V ) = pd A ( V ) for any Yetter-Drinfeld module V .Proof. If A is cosemisimple and S = id, we see, as in the proof of Proposition 4.7, thatthe automorphism θ of A is the identity, and that Ω A : Y D AA → M A is indeed separable,and the assertion on projective dimensions, which was already proved in [7, Section 6],follows similarly.Assume now that Ω A : Y D AA → M A is separable. Since Ω A admits the right adjoint − A , the characterization of separability for functors that admit a right adjoint in [33]gives in particular an A -colinear and A -linear map η : k A → k with η (1) = 1By the A -colinearity and η (1) = 1, we have that η = h is a Haar integral on A , which isthus cosemisimple. The A -linearity of h gives, for any a, x ∈ A , h ( S ( a (1) ) xa (2) ) = ε ( a ) h ( x )We have seen in the proof of Proposition 4.7 that for any a, x ∈ A , h ( S ( a (1) ) xa (2) ) = ψ − ( a (2) ) h (cid:0) xa (3) S − ( a (1) ) (cid:1) Hence we have for any a, x ∈ Ah (cid:0) x ( ε ( a ) − ψ − ( a (2) ) a (3) S − ( a (1) )) (cid:1) = 0The non-degeneracy of the Haar integral (which follows from the orthogonality relations)then gives, for any a ∈ A ε ( a )1 = ψ − ( a (2) ) a (3) S − ( a (1) )Hence applying ε gives ε = ψ − , and we thus have S = id. (cid:3) hopf subalgebras and cohomological dimension Let B ⊂ A be a Hopf subalgebra. Under the assumption of faithful flatness of A as a B -module, which holds in many situations and in particular if A is cosemisimple [12], we havecd( B ) ≤ cd( A ) [6, Proposition 3.1]. In this section we prove, in view of an example in thenext section, an analogue inequality for Gerstenhaber-Schack cohomological dimension,in the cosemisimple case. Of course, if the conclusion of Theorem 1.3 was known to holdfor any cosemisimple Hopf algebra, this would become trivial. e begin with some results of independent interest. Recall [6] that a Yetter-Drinfeldmodule is said to be relative projective if it is a direct summand in a free one. Proposition 6.1.
Let A be a Hopf algebra, let V be a Yetter-Drinfeld over A and let W be a right A -comodule. Then we have an isomorphism of Yetter-Drinfeld modules (Ω A ( V ) ⊗ W ) ⊠ A ≃ V ⊗ ( W ⊠ A ) In particular, if P is a relative projective Yetter-Drinfeld module, so is the Yetter-Drinfeldmodule V ⊗ P .Proof. The map (Ω A ( V ) ⊗ W ) ⊠ A V ⊗ ( W ⊠ A ) v ⊗ w ⊗ a v · a (1) ⊗ w ⊗ a (2) is easily seen to be a morphism of Yetter-Drinfeld modules, and its inverse is given by v ⊗ w ⊗ a v · S ( a (1) ) ⊗ w ⊗ a (2) . If P is relative projective, let W be a right A -comodule and Q be a Yetter-Drinfeld module such that W ⊠ A ≃ P ⊕ Q . We then have( V ⊗ P ) ⊕ ( V ⊗ Q ) ≃ V ⊗ ( W ⊠ A ) ≃ (Ω A ( V ) ⊗ W ) ⊠ A , which proves that V ⊗ P isrelative projective. (cid:3) Corollary 6.2. If A is a cosemisimple Hopf algebra, we have cd GS ( A ) = pd YD AA ( k ) = max n n : Ext n YD AA ( k, V ) = 0 for some V ∈ Y D AA o = max n pd YD AA ( V ) , V ∈ Y D AA o = max n n : Ext n YD AA ( V, W ) = 0 for some V, W ∈ Y D AA o = min n n : Ext n +1 YD AA ( V, W ) = 0 for any
V, W ∈ Y D AA o = max n injd YD AA ( V ) , V ∈ Y D AA o where injd YD AA is the injective dimension in the category Y D AA .Proof. The first two equalities have already been discussed. Let P ∗ → k be resolution of k by projective objects, of length n = pd YD AA ( k ). Since A is cosemisimple, the projectiveobjects are the relative projectives, so if V is a Yetter-Drinfeld module, tensoring theabove resolution with V yields, by Proposition 6.1, a length n resolution of V by pro-jective objects. This gives the third equality, and the other ones then follow by classicalarguments. (cid:3) Let B ⊂ A be a Hopf subalgebra. Recall [7] that there is a pair of adjoint functors Y D AA −→ Y D BB Y D BB −→ Y D AA X X ( B ) V V ⊠ B A where(1) for a Yetter-Drinfeld module X over A , X ( B ) = { x ∈ X | x (0) ⊗ x (1) ∈ X ⊗ B } has the restricted B -module structure;(2) for a Yetter-Drinfeld module V over B , V ⊠ B A is the induced module V ⊗ B A ,with A -comodule structure given by( v ⊗ B a ) (0) ⊗ ( v ⊗ B a ) (1) = v (0) ⊗ B a (2) ⊗ S ( a (1) ) v (1) a (3) Lemma 6.3.
Let B ⊂ A be a Hopf subalgebra, and assume that A is cosemisimple. Let V be a Yetter-Drinfeld module over B . Then V is isomorphic to a direct summand of ( V ⊠ B A ) ( B ) . roof. It is immediate to check that we have a morphism of Yetter-Drinfeld modules i : V → ( V ⊠ B A ) ( B ) , v v ⊗ B A is cosemisimple. Then, by the proof of Theorem 2.1 in [12], thereexists a sub- B -bimodule T ⊂ A , which is as well a subcoalgebra, such that A = B ⊕ T .Let E : A → B be the corresponding projection: E ( b ) = b for b ∈ B and E ( a ) = 0 for a ∈ T . By construction E is a B -bimodule map and a coalgebra map, and it is immediateto check that we have for any a ∈ AS ( E ( a ) (1) ) ⊗ E ( a ) (2) ⊗ E ( a ) (3) = S ( a (1) ) ⊗ E ( a (2) ) ⊗ a (3) From this, we see that the map( V ⊠ B A ) ( B ) → V, v ⊗ B a → v · E ( a )is a morphism of Yetter-Drinfeld modules. Since this map is clearly a retraction to i , thisproves the lemma. (cid:3) We now have all the ingredients to prove the expected result.
Proposition 6.4.
Let B ⊂ A be a Hopf subalgebra. If A is cosemisimple, we have cd GS ( B ) ≤ cd GS ( A ) .Proof. We can assume that cd GS ( A ) = n is finite. Since A is cosemisimple, [12, Theorem2.1] ensures that A is flat as a left B -module, and B ⊂ A is coflat. Hence, by [7,Proposition 3.3] we haveExt ∗YD AA ( V ⊠ B A, X ) ≃ Ext ∗YD BB ( V, X ( B ) )for any Yetter-Drinfeld module X over A , and any Yetter-Drinfeld module V over B .Hence, for V = k , Corollary 6.2 yieldsExt n +1 YD BB ( k, X ( B ) ) ≃ Ext n +1 YD AA ( k ⊠ B A, X ) = { } for any Yetter-Drinfeld module X over A . Lemma 6.3 ensures that any Yetter-Drinfeldmodule over B is a direct summand in one of type X ( B ) , so we get cd GS ( B ) ≤ n , asrequired. (cid:3) examples We now use the previous results to examine some examples that were not covered bythe literature.7.1.
Universal cosovereign Hopf algebras.
In this subsection we complete some ofthe results of [7] on the cohomological dimension of the universal cosovereign Hopf alge-bras. Recall that for n ≥ F ∈ GL n ( k ), the algebra H ( F ) is the algebra generatedby ( u ij ) ≤ i,j ≤ n and ( v ij ) ≤ i,j ≤ n , with relations: uv t = v t u = I n ; vF u t F − = F u t F − v = I n , where u = ( u ij ), v = ( v ij ) and I n is the identity n × n matrix. The algebra H ( F ) has aHopf algebra structure defined by∆( u ij ) = X k u ik ⊗ u kj , ∆( v ij ) = X k v ik ⊗ v kj ,ε ( u ij ) = ε ( v ij ) = δ ij , S ( u ) = v t , S ( v ) = F u t F − . We refer the reader to [4, 7] for more information and background on the universalcosovereign Hopf algebras H ( F ). ecall [7] that we say that a matrix F ∈ GL n ( k ) is • normalizable if tr( F ) = 0 and tr( F − ) = 0 or tr( F ) = 0 = tr( F − ); • generic if it is normalizable and the solutions of the equation q − p tr( F )tr( F − ) q +1 = 0 are generic, i.e. are not roots of unity of order ≥ • an asymmetry if there exists E ∈ GL n ( k ) such that F = E t E − . Theorem 7.1.
Let F ∈ GL n ( k ) , n ≥ . If F is an asymmetry or F is generic, we have cd( H ( F )) = 3 .Proof. We know from [7, Theorem 2.1], that cd( H ( F )) = 3 if F is an asymmery and thatcd GS ( H ( F ) = 3 if F is generic, in which case H ( F ) is cosemisimple [4], so Theorem 1.3gives the result in that case. (cid:3) As an illustration of Theorem 4.5, consider, for E ∈ GL n ( k ) and F ∈ GL m ( k ), n, m ≥ H ( E, F ) presented by generators u ij , v ij , 1 ≤ i ≤ m, ≤ j ≤ n , and relations uv t = I m = vF u t E − ; v t u = I n = F u t E − v. Theorem 7.2. If E , F are generic, tr( E ) = tr( F ) and tr( E − ) = tr( F − ) , then we have cd( H ( E, F )) = 3 .Proof.
The assumption tr( E ) = tr( F ) and tr( E − ) = tr( F − ) ensures that H ( E, F ) isan H ( E )- H ( F )-bi-Galois object [4]. Hence, since the genericity assumption ensures that H ( E ) cosemisimple and we know from the previous result that cd( H ( E )) and cd( H ( F ))are finite, the result follows from Theorem 4.5. (cid:3) Free wreath products.
In this subsection we assume that the base field is k = C ,since the monoidal equivalences on which we rely [18, 25] were obtained in this framework.Before going to the general setting of Theorem 7.4, we feel it is probably worth to presenta particular example. So for n, p ≥
1, consider, following the notation of [2], the algebra A ph ( n ) presented by generators u ij , 1 ≤ i, j ≤ n , and relations n X j =1 u pij = 1 = n X j =1 u pji , u ij u ik = 0 = u ji u ki , for k = j, At p = 1, A h ( n ) = A s ( n ), the coordinate algebra of Wang’s quantum permutation group[38]. In general A ph ( n ) is a Hopf algebra with [3]∆( u ij ) = X k u ik ⊗ u kj , ε ( u ij ) = δ ij , S ( u ij ) = u p − ji The following result, for which the p = 1 case was obtained in [6] (see [8] as well, whereit is shown that A s ( n ) is Calabi-Yau of dimension 3), will be a particular instance of theforthcoming Theorem 7.4. Theorem 7.3.
We have, for p ≥ and n ≥ , cd( A ph ( n )) = 3 . Let A be a Hopf algebra, and consider A ∗ n , the free product algebra of n copies of A , which inherits a natural Hopf algebra structure such that the canonical morphisms ν i : A −→ A ∗ n are Hopf algebras morphisms. The free wreath product A ∗ w A s ( n ) [3] isthe quotient of the algebra A ∗ n ∗ A s ( n ) by the two-sided ideal generated by the elements: ν k ( a ) u ki − u ki ν k ( a ) , ≤ i, k ≤ n , a ∈ A. he free wreath product A ∗ w A s ( n ) admits a Hopf algebra structure given by∆( u ij ) = n X k =1 u ik ⊗ u kj , ∆( ν i ( a )) = n X k =1 ν i ( a (1) ) u ik ⊗ ν k ( a (2) ) ,ε ( u ij ) = δ ij , ε ( ν i ( a )) = ε ( a ) , S ( u ij ) = u ji , S ( ν i ( a )) = n X k =1 ν k ( S ( a )) u ki . When A is a compact Hopf algebra (i.e. arises from a compact quantum, we do not needthe precise definition here), the free wreath product is as well a compact Hopf algebra.In that case the monoidal categories of comodules have been described for n ≥ S = id and Fima-Pittau [18] in general.Taking A to be the group algebra C [ Z /p Z ], we have A ph ( n ) ≃ C [ Z /p Z ] ∗ w A s ( n ) by [3,Example 2.5], hence Theorem 7.3 is a particular instance of the following result. Theorem 7.4.
We have cd( A ∗ w A s ( n )) = max(cd( A ) , for any compact Hopf algebra A such that cd( A ) = cd GS ( A ) and any n ≥ .Proof. First notice that there is a Hopf algebra map π : A ∗ w A s ( n ) → A s ( n ) such that π ( u ij ) = u ij and π ( a ) = ε ( a ), hence A s ( n ) stands as Hopf subalgebra of A ∗ w A s ( n ). Wethus have, by [6, Proposition 3.1], 3 = cd( A s ( n )) ≤ cd( A ∗ w A s ( n )). Similarly the naturalmap A ∗ n → A ∗ w A s ( n ) has a retraction, and hence A ∗ n stands as left coideal ∗ -subalgebraof A ∗ w A s ( n ). By the results in [13], A ∗ w A s ( n ) is thus faithfully flat as A ∗ n -module, henceprojective [28]. We then have, using [7, Corollary 5.3], cd( A ) = cd( A ∗ n ) ≤ cd( A ∗ w A s ( n )),since restricting a resolution by projective A ∗ w A s ( n )-modules to A ∗ n -modules remainsa projective resolution. Hence we havemax(cd( A ) , ≤ cd( A ∗ w A s ( n ))The converse inequality obviously holds is cd( A ) is infinite, hence we can assume thatcd( A ) is finite, and hence, in view of our assumption, that cd GS ( A ) is finite.The results in [18, 25] ensure the existence, for q satisfying q + q − = √ n , of a monoidalequivalence between the category of comodules over A ∗ w A s ( n ) and the category ofcomodules over a certain Hopf subalgebra H of the free product A ∗ O ( SU q (2)). We have,combining Proposition 6.4 and [7, Corollary 5.10]cd GS ( H ) ≤ cd GS ( A ∗ O ( SU q (2))) = max(cd GS ( A ) , cd GS ( O ( SU q (2)))Since cd GS ( O ( SU q (2)) = 3 by [5, 6], we get cd GS ( H ) ≤ max(cd GS ( A ) , GS ( A ) is finite, we get that cd GS ( H ) is finite. Hence by Corollary 5.4 andTheorem 1.3, we getcd( A ∗ w A s ( n )) = cd( H ) = cd GS ( H ) ≤ max(cd GS ( A ) ,
3) = max(cd( A ) , (cid:3) Remark . At n = 2, using the simple description of the free wreath product as a crossedcoproduct in [3], it is not difficult to show directly that cd( A ∗ w A s (2)) = max(cd( A ) , A is non trivial. Remark . Fima-Pittau [18] define more generally a free wreath product A ∗ w A aut ( R, ϕ ),for suitable pairs (
R, ϕ ) consisting of a finite-dimensional C ∗ -algebra and a faithful state,and prove a similar monoidal equivalence result, so that Theorem 7.4 should generalizeto this setting. . question 1.1 outside the cosemisimple case In this last section we prove the remaining assertions of Theorem 1.2.8.1.
The smooth case.
In this subsection we remark that the results in [42, 39] enableus to have a positive answer to Question 1.1 only assuming that the Hopf algebras arehomologically smooth, while [39, Theorem 2.4.5] assumed furthermore that one of theHopf algebras is twisted Calabi-Yau, and then proved that the other one is twisted Calabi-Yau as well. Recall that an algebra is said to be homologically smooth if the trivialbimodule has a finite resolution by finitely generated projective bimodules. For Hopfalgebras, this is equivalent to say that the trivial left or right A -module has a finiteresolution by finitely generated projective modules. Theorem 8.1.
Let A , B be Hopf algebras that have equivalent linear tensor categoriesof comodules: M A ≃ ⊗ M B . If A and B are homologically smooth and have bijectiveantipode, we have cd( A ) = cd( B ) .Proof. The assumption on the bijectivity of the antipodes enables us to use results in[42, 39]. By the results in [35] such a monoidal equivalence arises from an A - B -bi-Galoisobject R ( R is an A - B -bicomodule algebra, is left Hopf-Galois over A and right Hopf-Galois over B ). Assuming that A is homologically smooth ensures that cd( A ) is finite, soas in Lemma 3.3 we have cd( A ) = max { n : Ext n A M ( ε k, F ) = 0 for some free module F } ,and that the functor Ext ∗ A M ( ε k, − ) commutes with direct colimits (see e.g. [9, ChapterVIII]), hence cd( A ) = pd A M ( ε k ) = max { n : Ext n A M ( ε k, A ) = 0 } The algebra R is homologically smooth since A is, by [42, Lemma 2.4], hence we havesimilarly cd( R ) = pd R M R ( R ) = max { n : Ext n R M R ( R, R ⊗ R ) = 0 } Using again smoothness, we have by [42, Lemma 2.1, Lemma 2.2]Ext ∗ R M R ( R, R ⊗ R ) ≃ Ext n A M ( ε k, A A ⊗ R ) ≃ Ext ∗ A M ( ε k, A A ) ⊗ R where the left A -module structure on A A ⊗ R is simply multiplying in A on the left.Hence cd( A ) = cd( R ). That we have as well cd( B ) = cd( R ) follows in a symmetricmanner from [39, Lemma 2.3.2.ii]. (cid:3) Remark . The argument in the proof shows in fact that if A is a smooth Hopf algebrawith bijective antipode, then for any left or right A -Galois object R , we have cd( A ) =cd( R ). Notice that here, as the Weyl algebra example shows (which is a Galois object over k [ x, y ]), the good dimension to consider is indeed the Hochschild cohomological dimension,and not the global dimension. Note also that a positive answer to the question of whethera Galois object has the same cohomological dimension as the Hopf algebra definitivelyneeds some assumption on the Hopf algebra, as the example of the Taft algebra H n shows,which admits the matrix algebra M n ( k ) as a Galois object [27].8.2. The finite-dimensional situation.
Etingof pointed out that, as a step towards ananswer to Question 1.1, one should understand the finite-dimensional case first. However,the situation is not so clear to us, and we only have the following partial result. Recallthat a Hopf algebra A is said to be unimodular if there is a non-zero two-sided integralin A , i.e. there exists a non-zero t ∈ A such that ta = at = ε ( a ) t for any a . If A iscosemisimple and finite-dimensional, then A ∗ is unimodular. Theorem 8.3.
Let A , B be finite-dimensional Hopf algebras such that M A ≃ ⊗ M B .Then we have cd( A ) = cd( B ) if one of the following condition holds. The characteristic of k is zero, or satisfies p > d ϕ ( d )2 , where d = dim( A ) . (2) A ∗ is unimodular.Proof. First notice that since a finite-dimensional Hopf algebra is self-injective (projectivemodules are injective), we have cd( A ) , cd( B ) ∈ { , ∞} and hence there are only few casesto consider. Moreover, for the Drinfeld double D ( A ), we have cd( D ( A )) = 0 if and onlyif D ( A ) is semisimple, if and only if A is semisimple and cosemisimple [32, Proposition7], and cd( D ( A )) = ∞ otherwise. Moreover, we have cd( D ( A )) = cd( D ( B )) since ourmonoidal equivalence M A ≃ ⊗ M B induces a monoidal equivalence between the monoidalcenters of these categories (notice that cd( D ( A )) = cd GS ( A )).If k has characteristic zero or satisfies p > d ϕ ( d )2 , then by [24, Theorem 3.3] and [16,Theorem 4.2] respectively, we have that A is semisimple if and only if A is semisimpleand cosemisimple, if and only if cd( D ( A )) = 0. Hence under one of these assumptionswe have cd( A ) = cd( B ) because cd( D ( A )) = cd( D ( B )).Since M A ≃ ⊗ M B and A , B are finite-dimensional, We have, by [35, Corollary 5.9], B ≃ A σ for some Hopf 2-cocycle σ . At the dual level this means that B ∗ ≃ ( A ∗ ) J forsome Drinfeld twist J . Hence if cd( A ) = 0, i.e. A is semisimple, we have that A ∗ iscosemisimple, and assuming that A ∗ is unimodular, we have that B ∗ is cosemisimple aswell by [1, Corollary 3.6], and hence B is semisimple, so that cd( B ) = 0, as needed. Theassumption that A ∗ is unimodular is stable under Drinfeld twist since the multiplicationdoes not change, thus B ∗ is unimodular as well, and hence we also have cd( B ) = 0 ⇒ cd( A ) = 0, concluding the proof. (cid:3) As we see in the proof of the previous theorem, a complete answer to Question 1.1 inthe finite-dimensional case reduces to the question whether the class of finite-dimensionalcosemisimple Hopf algebras is stable under Drinfeld twists. Remark 3.9 in [1] claimedthat this is expected to be true, and would follow from a weak form of an importantconjecture of Kaplansky saying that a finite-dimensional cosemisimple Hopf algebra isunimodular (the strong form says that a cosemisimple Hopf algebra satisfies S = id),but we are not aware of a proof since then. References [1] E. Aljadeff, P. Etingof, S. Gelaki, D. Nikshych, On twisting of finite-dimensional Hopf algebras,
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