TATE AND TATE-HOCHSCHILD COHOMOLOGY FOR FINITE DIMENSIONAL HOPF ALGEBRAS VAN C. NGUYEN 2 1 Abstract. LetAbeanyfinitedimensionalHopfalgebraoverafieldk. Wegeneralizethe 0 notionofTate cohomologyforA, whichis definedinbothpositive andnegativedegrees, 2 and compare it with the Tate-Hochschild cohomology of A that was presented in [3]. p We introduce cup products that make the Tate and Tate-Hochschild cohomology of A e become graded rings. We establish the relationship between these rings, which turns S out to be similar to that in the ordinary non-Tate cohomology case. As an example, 1 we explicitly compute the Tate-Hochschild cohomology for a finite dimensional (cyclic) 2 group algebra. In another example, we compute both the Tate and Tate-Hochschild ] cohomology for a Taft algebra, in particular, the Sweedler algebra H4. A R . h t a m 1. Introduction [ 1 Tate cohomology was first introduced by John Tate in 1952 [17] for group cohomology v arising from class field theory. Others then generalized it to the group ring RG where R is a 8 commutative ring and G is a finite group. Unlike the ordinary cohomology, this cohomology 8 8 theory is based on complete resolutions, which yield cohomology groups in both positive and 4 negative degrees. Duringthelast several decades, alot of efforts wereputinto studyingthis new . 9 cohomology whichwas brieflysummarizedin ([6], Ch.VI)or ([8], Ch.XII).In 1992, Benson and 0 Carlson provided some conditions on which the products in negative (Tate) cohomology vanish 2 [2]. Finally in a recent paper [3], Bergh and Jorgensen combined both the notions of Hochschild 1 : cohomology and Tate cohomology to present the Tate-Hochschild cohomology of an algebra A v such that its enveloping algebra Ae is two-sided Noetherian and Gorenstein. If the Gorenstein i X dimensionofAe is0,thiscohomologyagreeswiththeordinaryHochschildcohomologyinpositive r degrees. In this paper, we define the Tate and Tate-Hochschild cohomology specifically for a a finite dimensional Hopf algebra A over a field k. Since any finite dimensional Hopf algebra is a Frobenius algebra ([12], Theorem 2.1.3), we will be able to apply some of the results that appeared in [3]. We start out Section 2 by introducing some definitions and notations that will be used throughout the paper. In Section 3, applying a similar construction as in ([5], Section 5.15), we n generalize to the Tate cohomology H (A,k) of A, for all integers n. We explicitly show how to form an A-complete resolution P of k based on a projective resolution of k, by taking the k-dual of the projective resolution and spblicing the two complexes together. The Tate cohomology groups of A with coefficients in k can be defined as: Hn(A,k) := Extn(k,k) = Hn(Hom (P,k)). A A 1 b d 2 VANC.NGUYEN The Tate homology groups are defined in the same way by taking the Tor functor on an A- completeresolutionofk. WeobservethattheTatecohomologygroupssharesomeniceproperties such as the dimension-shifting in long exact sequences of cohomology groups, additivity of Ext, and moreover, they agree with the ordinary cohomology groups in positive degrees. d Let Ae = A ⊗k Aop be the enveloping algebra of A. We consider A as an Ae-module and obtain an Ae-complete resolution of A. Based on the definition in [3], we form the n-th Tate-Hochschild cohomology group of A: n n HH (A,A) := Ext (A,A), for all n ∈ Z. Ae It is known that thedordinary cohdomology of A is an algebra direct summand of its Hochschild cohomology ([11], Prop. 5.6 and Cor. 5.6). This motivates us to ask if the same assertion holds for the Tate cohomology version. Section 4 is where we start making the first comparison between the Tate and the Tate-Hochschild cohomology groups of A. In particular, n n by Proposition 4.2, H (A,k) is a vector space direct summand of HH (A,A), for all n > 0. By ([3], Cor. 3.8), if ν is the Nakayama automorphism of a Frobenius algebra A such that ν2 = 1, then there is an isomborphism: d HHn(A,A) ∼= HH−(n+1)(A,A), for all n ∈ Z. Following such symmetric result, we further study the Nakayama automorphism in Section 5. d d We compute the “formula” for the Nakayama automorphism of the Sweedler algebra H and 4 of the general Taft algebra, in an attempt to relate the Tate-Hochschild cohomology groups in positive degrees to those in negative degrees. This computation does not lead us to conclude the cohomology relation for a general finite dimensional Hopf algebra, However, we find that it is helpful in verifying our results in latter sections for the Tate-Hochschild cohomology of the Sweedler algebra H , as in this case we have ν2 = 1. 4 We approach a broader setting by establishing cup products for the two Tate cohomology ∗ ∗ types in Section 6. These multiplication structures turn H (A,k) and HH (A,A) into graded rings. Let Aad be A as an A-module via the left adjoint action. Using the ring structures, we prove the following isomorphism in Section 7: b d Theorem. Let A be a finite dimensional Hopf algebra over a field k. There exists an isomor- phism of algebras: HH∗(A,A) ∼=H∗(A,Aad). This result generates an impdortant relatiobn between the Tate and the Tate-Hochschild cohomology rings: ∗ Corollary. If A is a finite dimensional Hopf algebra over a field k, then H (A,k) is an algebra ∗ direct summand of HH (A,A). b Therefore, thedTate and the Tate-Hochschild cohomology share the same relation as that of the ordinary non-Tate cohomology. In the last two sections, we explicitly compute the Tate and the Tate-Hochschild cohomology for the (cyclic) group algebra kG and the Taft algebra, in particular, the Sweedler algebra H . These examples help us to understand the above relation 4 better, and once again, illustrate the symmetric result in [3]. TATE AND TATE-HOCHSCHILD COHOMOLOGY 3 2. Preliminaries and Notations Projective resolutions are commonly used to compute the cohomology of an algebra. We introduce a more general resolution, which involves both positive and negative degrees: Definition 2.1. Suppose R isatwo-sided Noetherian ring. A complete resolution of a finitely generated R-module M is an exact complex P = {{Pi}i∈Z,di : Pi → Pi−1} of finitely generated projective R-modules such that it satisfies the following properties: (1) The dual complex Hom (P,R) is also exact. R (2) There exists a projective resolution Q →ε M of M and a chain map P →ϕ Q where ϕ is n bijective for n ≥ 0. Some other authors also use the notation P →ϕ Q →ε M for a complete resolution of M. One should note that unlike projective resolutions, complete resolutions do not always exist. However, given the setting as in Definition 2.1, Theorems 3.1 and 3.2 in [1] guarantee the existence of such complete resolutions. Observe that when working in some appropriate context, if there exists a complete resolution, it can be constructed from a projective resolution as follows: the negative-degree component can be obtained by taking the dual of an ordinary projective resolution and splicing the two resolutions together. The detail of such construction will be described later. In his paper [17], Tate introduced a cohomology theory which used complete resolutions, and therefore, defined the cohomology in both positive and negative degrees. Let G be a finite group and R be a commutative ring. If ··· −d→3 P −d→2 P −d→1 P −→ε R → 0 2 1 0 is a RG-projective resolution of R, then apply Hom (−,R) to get a dual sequence: R 0→ R → Hom (P ,R) → Hom (P ,R) → Hom (P ,R) → ··· R 0 R 1 R 2 which is an exact sequence of RG-modules, since R ∼= Hom (R,R) and Hom (−,R) is an exact R R functor. Splicing these two sequences together, one forms a doubly infinite sequence: ··· −d→3 P −d→2 P −d→1 P → Hom (P ,R) → Hom (P ,R) → ··· 2 1 0 R 0 R 1 Using the notation P := Hom (P ,R), the above sequence is a complete resolution of R: −(n+1) R n P: ··· −d→3 P −d→2 P −d→1 P → P → P → ··· 2 1 0 −1 −2 Let M be another (left) RG-module, apply Hom (−,M) to P and take the homology of RG this new complex. We thus obtain the Tate cohomology for RG: Hn(G,M) := Extn (R,M) = Hn(Hom (P,M)), for all n ∈ Z. RG RG We wouldblike to generadlize this definition to any finite dimensional Hopf algebra A over a field k. For the rest of this paper, we let k be our base field, tensor products will be over k unless stated otherwise. Let us recall some definitions and notations: Let A be a finite dimensional Hopf algebra over k with the antipode S. The k-dual Homk(−,k) is denoted by D(−) and the ring dual HomA(−,A) is denoted by (−)∗. This is an unfortunate notation, because to a Hopf algebraist, D(A) usually stands for the “Drinfeld 4 VANC.NGUYEN double” of A. However, to be consistent with the primary reference paper [3], we will adopt the above notation. We denote the counit ε :A → k and use Sweedler sigma notation for the comultiplication ∆ : A → A ⊗ A via ∆(a) = a ⊗ a , for a ∈ A. When the antipode S is (composition) 1 2 invertible, we denote its inversXe S. Observe that the dual D(A) of a finite dimensional Hopf algebra A is also a Hopf algebra, where the algebra structure of A becomes the coalgebra structure of D(A), and vice-versa, and the antipode of A translates into an antipode D(S) of D(A) in a canonical fashion. The opposite algebra of an algebra A, denoted by Aop, is the algebra A with the same set of elements and the same addition but with multiplication given by a∗b = ba, for all a,b ∈ A, where the right hand side is the usual multiplication of A. Let Ae = A⊗Aop be the enveloping algebra of A. Let σ : A → Ae be defined by σ(a) = a ⊗ S(a ). One can check that 1 2 σ is an injective algebra homomorphism. Consequently, we identify A with the subalgebra σ(A) of Ae. Moreover, we can also induce Ae-modules frPom A-modules as follows: let M be a left A-module. Consider Ae to be a right A-module via right multiplication by elements of σ(A), then using tensor product, Ae ⊗ M is a left Ae-module, with action of Ae given by A a·(b⊗ m)= ab⊗ m, for all a,b ∈ Ae,m ∈ M. A A Under this setting, k is the trivial (left) module of A via ε, that is, for a ∈ A,r ∈ k, then a·r = ε(a)r. We will use the following notation for the ordinary (non-Tate) cohomology and Hochschild cohomology of the Hopf algebra A, respectively: H∗(A,M) := Ext∗(k,M) = Extn(k,M) A A n≥0 M HH∗(A,M) := Ext∗ (A,M) = Extn (A,M) Ae Ae n≥0 M where M denotes a left A-module in the former case and an A-bimodule in the latter case. 3. Tate Cohomology for finite dimensional Hopf Algebras 3.1. Construction. We apply a similar construction to generalize the Tate cohomology to a finite dimensional Hopf algebra A. For any left A-module M, D(M) is a left A-module with action of A given by the antipode S: (a·f)(m)= f(S(a)m) for all a ∈ A, m ∈ M, and f ∈D(M). Let: ··· −d→3 P −d→2 P −d→1 P −d→0 k → 0 (1) 2 1 0 be an A-projective resolution of k, where Pi are finitely generated. Apply Homk(−,k) to get: D(d0) D(d1) D(d2) 0→ k −−−→ D(P )−−−→ D(P ) −−−→ D(P ) → ··· (2) 0 1 2 where D(Pi) = Homk(Pi,k), D(di) is the dual map of di, and k ∼= Homk(k,k). We claim that (2) is an exact sequence of projective A-modules. To see this, we note that exactness follows since Homk(−,k) is an exact functor. The dual modules D(Pi) are left A-modules as discussed above. D(P ) are injective since P are projective A-modules. However, D(P ) are also i i i projective. This is true because any finite dimensional Hopf algebra A is a Frobenius algebra TATE AND TATE-HOCHSCHILD COHOMOLOGY 5 ([12], Theorem 2.1.3). In fact, D(A) and A are isomorphic as left A-modules, implying that A is both projective and injective as a module over itself, so the same is true of dual modules. Splicing complexes (1) and (2) together, we obtain: ··· → P −d→2 P −d→1 P −→ξ D(P ) −D−(−d1→) D(P ) −D−(−d2→) D(P )→ ··· (3) 2 1 0 0 1 2 where the middle map ξ = D(d )◦d . Since (1) and (2) are exact, one can show that (3) is 0 0 an exact sequence of A-modules by proving the exactness at P and D(P ). That is, we want 0 0 to show Im(d ) = Ker(ξ) and Im(ξ) = Ker(D(d )). By exactness of (1), Im(d ) = Ker(d ), and 1 1 1 0 Im(d ) = k. By exactness of (2), Im(D(d )) = Ker(D(d )), and Ker(D(d )) = 0. Thus, we have 0 0 1 0 Im(ξ) = Im(D(d )) = Ker(D(d )), showing exactness at D(P ). 0 1 0 Let a ∈ Ker(ξ). Then ξ(a) = D(d )(d (a)) = 0, which implies d (a) ∈ Ker(D(d )) = {0}. 0 0 0 0 It follows that d (a) = 0, in other words, a ∈ Ker(d ). As a result, we see that Ker(ξ) ⊆ 0 0 Ker(d ) = Im(d ). Conversely, let b ∈ Ker(d ). Then d (b) = 0 implies D(d )(d (b)) = 0, since 0 1 0 0 0 0 D(d ) is a linear map, it takes 0 to 0. We have ξ(b) = 0 which shows b ∈ Ker(ξ). Hence, 0 Ker(ξ) = Ker(d )= Im(d ) showing exactness at P . Therefore, (3) is an exact sequence. 0 1 0 As before, we let P := D(P ) and obtain an A-complete resolution of k: −(n+1) n P : ··· → P −d→2 P −d→1 P −→ξ P −D−(−d1→) P −D−(−d2→) ··· 2 1 0 −1 −2 which we also call (3). For a left A-module M, apply Hom (−,M) to (3): A ··· −D\−(−d2→) P −D\−(−d1→) P −→ξb P −cd→1 P −cd→2 P −cd→3 ··· (4) −2 −1 0 1 2 where d(f) = f ◦ d and P = Hom (P ,M). We check that (4) is a complex of A-modules, i d A i d c c c that is, d◦d = 0. This follows from the exactness of (3) because the composition of any two consecubtive differential mabps is 0. We have (d◦d)(f)= f ◦d◦d= 0. b b The homology groups of this new complex (4) are the Tate cohomology groups: b b Hn(A,M) := Extn(k,M) = Hn(Hom (P,M)), for all n∈ Z. A A A A The Tate hobmology H (Ad,M) := Tor (k,M) is defined in the same manner by applying n n − ⊗ M to (3) for a left A-module M and taking the n-th homology of the new complex. A Here, we are only interestebd in the Tatedcohomology. Observe that in our context, naturally, the Tate (co)homology does not depend on the choice of the projective resolution of k (by the ordinary Comparison Theorem), and hence, is independent of the complete resolution of k ([1], Theorem 5.2 and Lemma 5.3). One can see this by applying the ordinary Comparison Theorem on the left side of two A-complete resolutions of k, which are just projective resolutions of k, to obtain a chain map between two resolutions in positive degrees. We take the linear dual Homk(−,k) = D(−) of the entire diagram, so the chain map now goes to the opposite direction. By definition of complete resolutions, the duals of complete resolutions are also exact sequences of projective A-modules. We apply the ordinary Comparison Theorem again on the left side of the diagram to obtain a chain map in negative degrees, using the fact that A is self-injective and for finitely generated free A-modules P , D(D(P )) ∼= P . Dualize the diagram back again i i i and we have a complete chain map going to both positive and negative degrees between two complete resolutions. This idea provides a version of the Comparison Theorem generalized to a complete resolution ([6], Prop. VI.3.3). 6 VANC.NGUYEN 3.2. Properties of Tate Cohomology. We make some comparison between the Tate coho- mology and the ordinary cohomology of A. Here are some important remarks: (a) For all n> 0, we have isomorphisms: Hn(A,M) ∼= Hn(A,M). This follows from the construction obf a complete resolution of k. The positive-degree components of a complete resolution of k arise from a projective resolution of k which is also used to form the ordinary cohomology groups Hn(A,M). 0 (b) The group H (A,M) is a quotient of H0(A,M). 0 By definibtion, H0(A,M) = Ker(d ) while H (A,M) = Ker(d )/Im(ξ). 1 1 (c) For all n < −1, since HomA(Pn,Mb) ∼= P−(nb+1) ⊗A M by thebdualitybisomorphism ([6], Prop. I.8.3c), we have isomorphisms: Hn(A,M) ∼= H (A,M). −(n+1) With n < −1,Pn = D(P−(n+1)), sbo for m ∈ M,p ∈ P−(n+1),f ∈ Pn, the duality isomorphism τ : P ⊗ M → Hom (P ,M) is given by τ(p ⊗ m)(f) = mf(p). −(n+1) A A n A This isomorphism induces the isomorphism on cohomology. Tate cohomology also enjoys standard properties of ordinary cohomology such as the dimension-shifting: (d) If 0 → M → M′ → M′′ → 0 is a short exact sequence of (left) A-modules, then there is a doubly infinite long exact sequence of Tate cohomology groups: ··· → Hn(A,M) → Hn(A,M′)→ Hn(A,M′′) → Hn+1(A,M) → ··· whosecohomoblogy longexabctsequenceisbthedesiredlonbgexact sequence([1],Prop.5.4). (e) If (N ) is a finite family of (left) A-modules and (M ) is any family of A-modules, j j∈J i i∈I then there are natural isomorphisms, for all n ∈ Z: Extn( N ,M) ∼= Extn(N ,M) A j A j j∈J j∈J M Y d d Extn(N, M )∼= Extn(N,M ). A i A i i∈I i∈I Y Y d d TheideaofthisproofissimilartothatoftheordinaryExtn,usingtheanalogousrelation A for Hom ([1], Prop. 5.7). In a current paper [3], the notion of Tate cohomology was combined with the notion of Hochschild cohomology and extended once again to the so-called Tate-Hochschild coho- mology. This new object gives rise to a question: Is there a relationship between the Tate cohomology of a finite dimensional Hopf algebra A and its Tate-Hochschild cohomology? In the following sections, we will examine this new construction and try to make some connection between these two Tate cohomology structures. TATE AND TATE-HOCHSCHILD COHOMOLOGY 7 4. Tate-Hochschild Cohomology for finite dimensional Hopf Algebras Definition 4.1. Let k,A, and Ae be defined as before (such that Ae is two-sided Noetherian and Gorenstein). Let M be an A-bimodule, for any integer n ∈ Z, the n-th Tate-Hochschild cohomology group is defined as: n n HH (A,M) := Ext (A,M), for all n ∈ Z. Ae When Ae is a two-sdided Noetheriadn and Gorenstein ring, by ([1], Theorems 3.1, 3.2), every finitely generated Ae-module admits a complete resolution. Hence, we obtain a complete resolution X for A as an Ae-module. Moreover, any bimodule M of A can be viewed as a left Ae-module by setting (a⊗b)·m = amb, for a⊗b ∈ Ae and m ∈M. The n-th Tate-Hochschild cohomology group is the n-th homology group of the complex HomAe(X,M). As noted in [3], the assumption that Ae is two-sided Noetherian is not necessary in the finite dimensional case. This also intrigues us to focus on finite dimensional Hopf algebras A. We recall the fact that a finite dimensional Hopf algebra A is Frobenius which is self- injective. ([3], Lemmas 3.1 and 3.2) show that Aop,Ae are also Frobenius, hence self-injective. By definition of Gorenstein ring, we see that finite dimensional self-injective algebras A,Aop,Ae are Gorenstein of Gorenstein dimension 0. Therefore, the Tate-Hochschild cohomology groups of A agree with the ordinary Hochschild cohomology groups in all positive degrees: HHn(A,M) ∼= HHn(A,M), for all n> 0. We make our first attempt to compare the Tate cohomology and Tate-Hochschild cohomology d in positive degrees: n Proposition 4.2. Let A be a finite dimensional Hopf algebra over a field k, then H (A,k) is a n n vector space direct summand of HH (A,A), for all n > 0. That means, we may view H (A,k) n b as (isomorphic to) a quotient of HH (A,A). d b Proof. For n > 0,Hn(A,k) ∼= Hdn(A,k) by the property in Section 3.2, and HHn(A,A) ∼= HHn(A,A) by the above discussion. Moreover, for a finite dimensional Hopf algebra A, its antipode S is bijectbive ([12], Theorem 2.1.3). Apply ([14], Lemma 7.2), we idendtify H∗(A,k) with a subalgebra of HH∗(A,A), and use appropriate isomorphism maps to obtain the desired result. (cid:3) By ([3], Cor. 3.8), if ν is the Nakayama automorphism of a Frobenius algebra A such that ν2 = 1, then there is an isomorphism: HHn(A,A) ∼= HH−(n+1)(A,A), for all n ∈ Z. Thus,givingthissymmetricstructure,wecansimplifytherelationbetweentheTatecohomology d d in degrees n > 0 and the Tate-Hochschild cohomology in degrees n ∈ Z\{0,−1}. Namely, for n > 0,Hn(A,k) is a vector space direct summand of HHn(A,A) ∼= HH−(n+1)(A,A). This leads us to examine when the Nakayama automorphism ν of a finite dimensional Hopf algebra is havingbthe property ν2 = 1. This is trivial if A is sydmmetric, as νdis the identity in this case. We then assume A is a finite dimensional Hopf algebra that is not symmetric. 8 VANC.NGUYEN 5. Nakayama Automorphism Recall that for any Hopf algebra A, its dual D(A) obtains the structure of A-module via a left action a ⇁ f and a right action f ↽ a of A on D(A) as follows: for a,b ∈ A,f ∈ D(A), the actions are given by: (a ⇁ f)(b) := f(S(a)b) and (f ↽ a)(b) := f(bS(a)) = (S(a) ⇀ f)(b). Therefore, D(A) is a left and right A-module. There are also a natural left action f ⇀ a := f(a )a and a right action a ↼ f := f(a )a of D(A) on A. 2 1 1 2 P The counit ε : A→ k gives rise toPdefining the space of left integrals of A: l := {x ∈ A|hx =ε(h)x, for all h ∈ A} ZA and the space of right integrals: r := {x ∈ A|xh = ε(h)x, for all h ∈ A}. ZA l r When A is a finite dimensional Hopf algebra (hence Frobenius), we have dimk = dimk = A A l r dimk D(A) = dimk D(A) = 1, ([12], Theorem 2.1.3). This shows the existeRnce of nonR-zero integrals in any finite dimensional Hopf algebra A and also in its dual D(A). R R l Let 0 6= f ∈ be a non-zero left integral of D(A), the non-degenerate associative D(A) bilinear Frobenius form B : A × A → k of A is given by: B(x,b) = f(xb), for all x,b ∈ A. R The Nakayama automorphism ν : A → A satisfies B(x,b) = B(b,ν(x)), for all x,b ∈ A. Replacing B with a new Frobenius form B′ defined by a unit element u ∈ A gives us a new automorphism ν′ = I ◦ν, where I is the inner automorphism r 7→ uru−1. The Nakayama u u automorphism ν is unique up to composition with an inner automorphism. Equivalently, it is a well-defined element of the group of outer automorphisms of A. r r Let t ∈ , we have at ∈ , for all a ∈ A. The (right) modular function for A is an A A algebra homomorphism α ∈ D(A) such that at = α(a)t, for all a ∈ A. By ([10], Lemma 1.5), R R the Nakayama automorphism ν :A → A of a finite dimensional Hopf algebra has the form: 2 2 ν(a) = S (a ↼ α) = (S a) ↼ α and its inverse is defined by ν−1(a) = S2(a ↼ α−1) = (S2a)↼ α−1. This formula for ν also shows that ν has a finite order dividing 2·dimk(A). Consider for every a ∈ A: ν2(a) = S2((S2a) ↼ α) ↼ α= (S4a) ↼ α2. Therefore, to get ν2 = 1, we need the antipode S to be of order 4 (since S is bijective, S is also of order 4), and the right modular function α to be of order 2. Keeping in mind that these are sufficient but not necessary conditions for ν2 = 1, we examine several known cases for the former condition: • If A is commutative or cocommutative then S2 = 1, ([12], Cor. 1.5.12). • If A is a finite-dimensional semisimple and cosemisimple Hopf algebra over a field k of characteristic 0 or of characteristic p > (dimkA)2, then S2 = 1, ([12], Theorem 2.5.3). TATE AND TATE-HOCHSCHILD COHOMOLOGY 9 Example 5.1 (Sweedler algebra). The smallest non-commutative, non-cocommutative Hopf algebra with dimension 4 over a given field k of characteristic 6= 2 was described by Sweedler as follows: H4 = Spank{1,g,x,gx|g2 = 1,x2 = 0,xg = −gx} with the Hopf algebra structure: ∆(g) = g⊗g, ∆(x)= 1⊗x+x⊗g, ε(g) = 1, ε(x) = 0, S(g) = g = g−1, S(x) = −xg. The antipode S has order 4. The integral spaces of H are: 4 l r = khx+gxi, = khx−gxi. ZH4 ZH4 To find the right modular function for H , we take the right integral t = x−gx and multiply 4 this element by all the basis elements of H on the left to see what the right modular function 4 looks like. The desired modular function α ∈ D(H ) should satisfy: at = α(a)t, for all a ∈ H . 4 4 After checking for each basis element, we have: α(x) = α(gx) = α(xg) = 0, α(g) = −1, α(1) = 1, and ν(g) = −g, ν(x)= −x. By direct computation or by using the fact that S has order 4, we obtain: ν2(a) = (S4a)↼ α2 = a, for all a ∈ H . 4 Therefore, ν2 = 1 on H . It follows from the above discussion that the Tate-Hochschild coho- 4 mology of H is symmetric in the sense that: 4 HHn(H ,H )∼= HH−(n+1)(H ,H ), for all n ∈Z. 4 4 4 4 An explicit computation of the Tate-Hochschild cohomology of H in Section 9 will illustrate d d 4 this result again. We examine a more general case, known as the Taft algebra, as described in [16] and ([13], Example 2.9): Example 5.2. Let N ≥ 2 be a positive integer. Assume the field k contains a primitive N-th root of unity ω. Let A be the Taft algebra generated over k by two elements g and x, subject to the following relations: A = Spank{1,g,x,gx|gN = 1,xN = 0,xg = ωgx}. A is a Hopf algebra with comultiplication ∆, counit ε, and antipode S given by: ∆(g) = g⊗g, ∆(x) = 1⊗x+x⊗g, ε(g) = 1, ε(x) = 0, S(g) = g−1, S(x) = −xg−1. 10 VANC.NGUYEN N−1 Note that A is not semisimple and is of dimension N2. Using t = xN−1gj as a right integral j=0 X of A, we can check that the modular function α ∈ D(A) is defined as: α(x) = α(gx) = α(xg) = 0, α(g) = ω, α(1) = 1. The Nakayama map ν : A→ A is given by: ν(g) = ωg, ν(x)= ωx, which yields: 1 a = 1 ν2(a) = ω2a a = g,x  ω4a a = xg So ν2 is not the identity map unless N =2, which we have seen in Example 5.1. We remark  that under some other settings (for example, using S instead of S, or using the left integral and left modular function), one might obtain different “formulas” for the Nakayama automorphism, such as ν′(g) = ωg and ν′(x) = ω−1x as described in ([13], Example 2.9). It is not easy to classify all non-symmetric finite dimensional Hopf algebras A such that ν2 = 1. What we have done here does not effectively yield a general connection between the Tate cohomology and the Tate-Hochschild cohomology of a finite dimensional Hopf algebra. In the next section, we observe that these two Tate-cohomology versions obtain ring structures which can help us to develop a deeper understanding of their relation as algebras. 6. Cup Products 6.1. Cup Product on Tate Cohomology. SupposePisanA-completeresolutionofk. Based on the discussion in ([6], Section VI.5), we also note the following difficulties in constructing the cup product on Tate cohomology: First of all, P⊗P is not a complete resolution of k⊗k ∼= k, as (P⊗P) is not the same as + the tensor product of resolutions P ⊗P , where P = {P } . Consequently, using the map + + + n n≥0 Hom (P,M)⊗Hom (P,N) → Hom (P⊗P,M ⊗N) would not obviously induce a cohomology A A A product in Tate cohomology as it does in the ordinary non-Tate cohomology. Secondly, when applying the diagonal approximation (a chain map that preserves augmentation) Γ :P → P⊗P, for any n ∈ Z, there are infinitely many (i,j) such that i+j = n, and the dimension-shifting property in Section 3.2 suggests that the corresponding cup products should all be non-trivial. So Γ should have a non-trivial component Γ , for all (i,j). Hence, the range of Γ should be ij the graded module in which in the dimension n is P ⊗P , rather than P ⊗P . This i j i j i+j=n i+j=n Y M discussion motivates us to the following definitions: Definition 6.1. If B and B′ are graded modules, their complete tensor product B⊗B′ is defined by: (B⊗B′) = B ⊗B′. b n i j i+j=n Y b