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Tate duality and transfer in Hochschild cohomology

We show that dualising transfer maps in Hochschild cohomology of symmetric algebras commutes with Tate duality. This extends a well-known result in group cohomology.

where the vertical maps are the Tate duality isomorphisms.
Theorem 1.1 holds with M replaced by a bounded complex of A-B-bimodules X whose components X i are finitely generated projective as left and right modules. This follow, for instance, from the formula tr X = i (−1) i tr Xi in [11, 2.11. (ii)], with i running over the integers for which X i is nonzero. Alternatively, if U is a finitely generated projective A-B-bimodule, then tr U is zero on HH * (B). By a standard argument due to Rickard (appearing at the end of the proof of [14, 2.1]), X is quasi-isomorphic to a complex with at most one nonprojective component. Thus there is a bimodule M such that tr X = tr M on HH * (B). In particular, there is no loss of generality in stating Theorem 1.1 for bimodules rather than complexes. Remark 1.3. Let G a finite group and H a subgroup of G. Tate duality for group cohomology is a canonical isomorphismĤ −n (G; k) ∨ ∼ =Ĥ n−1 (G; k), for any integer n. Any k(G × G)-module can be viewed as a kG-kG-bimodule through the isomorphism k(G × G) ∼ = kG ⊗ k (kG) 0 sending (x, y) ∈ G × G to x ⊗ y −1 . Denote by ∆G the diagonal subgroup ∆G = {(x, x) | x ∈ G} of G × G. The induction functor Ind G×G ∆G sends the trivial k∆G-module to Ind G×G ∆G (k) ∼ = kG, the latter viewed as a k(G × G)-module with (x, y) ∈ G × G acting by left multiplication with x and right multiplication with y −1 . Combined with the canonical isomorphism G ∼ = ∆G and the interpretation of k(G × G)-modules as kG-kG-bimodules, this functor sends the trivial kG-module to the kG-kG-bimodule kG, and induces a split injective graded algebra homomorphism δ G : H * (G; k) → HH * (kG); similarly for H instead of G. Following [11, 4.6, 4.7], the restriction and transfer maps between H * (G; k) and H * (H; k) extend to transfer maps between HH * (kG) and HH * (kH) induced by the kG-kH-bimodule kG H and its dual H kG; a similar statement holds for their Tate analogues. The above theorems can be used to show the well-known fact that Tate duality identifies the k-dual of the usual transfer map tr G H :Ĥ −n (H; k) →Ĥ −n (G; k) with the restriction map res G H :Ĥ n−1 (G; k) →Ĥ n−1 (H; k). This fact is applied, for instance, in Benson's approach [2] to Greenlees' local cohomology spectral sequence [10]. Theorem 1.1 might provide one of the technical ingredients towards constructing similar local cohomology spectral sequences in Hochschild cohomology of symmetric algebras. The background motivation is the question whether the Castelnuovo-Mumford regularity of the Hochschild cohomology of symmetric algebras is invariant under separable equivalences (cf. [13, 3.1, 3.2]).
The proofs of the above theorems are formal verifications, based on explicit descriptions of Tate duality for symmetric algebras (reviewed in §2) and of well-known adjunction maps (reviewed in §3). These are used (in §4) to show that Tate duality and adjunction are compatible. After a brief review of transfer maps in Tate-Hochschild cohomology (in §5) the proofs of Theorem 1.1 and Theorem 1.2 are completed in §6 and §7, respectively. We conclude with some remarks on extending results of Benson and Carlson [3] on products in negative group cohomology to the Hochschild cohomology of symmetric algebras in §8. The results in this last section have independently been obtained in work of Bergh, Jorgensen, and Oppermann [6, §3].
If not stated otherwise, modules are unitary left modules. For any k-algebra A we denote by Mod(A) the category of left A-modules, and by mod(A) the subcategory of finitely generated A-modules. Right A-modules are identified with A 0 -modules, where A 0 is the opposite algebra of A. Given two k-algebras A, B, we adopt the convention that for any A-B-bimodule M , the left and right k-vector space structure on M induced by the unit maps k → A and k → B of A and B coincide. Thus we may consider the A-B-bimodule M as an A ⊗ k B 0 -module or as a right A 0 ⊗ k B-modules, whichever is more convenient. We denote by perf(A, B) the category of A-B-bimodules which are finitely generated projective as left A-modules and as right B-modules. Given three k-algebras A, B, C, an A-B-bimodule M , an A-C-bimodule N , and a C-B-bimodule N ′ , we consider as usual Hom

Background material on Tate duality
Tate duality for symmetric algebras is a special case of Auslander-Reiten duality. We need an explicit description of Tate duality in order to relate it to the adjunction units and counits arising in the definition of transfer maps on Hochschild cohomology. This is a specialisation to symmetric algebras of arguments and results due to Auslander and Reiten in [1]. (By taking into account the Nakayama functor, this description yields the corresponding duality for selfinjective algebras, but we will not need this degree of generality in this paper; see e.g. [5] for more details). Let A be a symmetric k-algebra; that is, A ∼ = A ∨ as A-A-bimodules. Choose a symmetrising form s ∈ A ∨ ; that is, s is the image of 1 A under a chosen bimodule isomorphism Φ : A ∼ = A ∨ . Since A is generated by 1 A as a left or right A-module and since a · 1 A = a = a · 1 A , it follows that the map Φ sends a ∈ A to the linear form a · s defined by (a · s)(a ′ ) = s(aa ′ ) = s(a ′ a) = (s · a)(a ′ ) for all a ′ ∈ A. For any two A-modules U , V , we denote by Hom pr A (U, V ) the space of A-homomorphisms from U to V which factor through a projective A-module, and we set Hom A (U, V ) = Hom A (U, V )/Hom pr A (U, V ). The stable module category of A is the k-linear category mod(A) having the same objects as mod(A), with morphism spaces Hom A (U, V ) for any two finitely generated left A-modules, where the composition of morphisms in mod(A) is induced by the composition of A-homomorphisms. For any finitely generated left A-module U choose a projective A-module P U , a surjective Ahomomrphism π U : P U → U , an injective A-module I U and an injective A-homomorphism ι U : U → I U . Set Ω A (U ) = ker(π U ), and Σ A = coker(ι U ). If no confusion arises, we simply write Ω and Σ instead of Ω A and Σ A . The operators Σ and Ω induce inverse self-equivalences, still denoted Ω and Σ, on mod(A); these functors do not depend on the choice of the (P U , π U ) and (I U , ι U ) in the sense that any other choice yields functors which are isomorphic to Ω and Σ through uniquely determined isomorphisms of functors. The category mod(A), together with the self-equivalence Σ and triangles induced by short exact sequences in mod(A) is triangulated. Let U , V be finitely generated left A-modules. For any integer n set Ext n A (U, V ) = Hom A (U, Σ n (V )). Tate duality for symmetric algebras states that for any integer n there is an isomorphism Indeed, the isomorphism 2.3 is the special case n = 0 of the isomorphism 2.1, and conversely, 2.1 follows from 2.3 applied with Ω n (V ) instead of V . The naturality implies in particular that 2.1 and 2.3 are isomorphisms of End A (U )-End A (V )-bimodules. As mentioned before, we will need an explicit description of the isomorphism 2.3 in order to compare it to transfer maps and their duals.
Thus, for any finitely generated projective A-module P we have an isomorphism , v ∈ V , and x ∈ P . Dualising the left term and applying the standard adjunction and double duality P ∨∨ ∼ = P yields an isomorphism (P ∨ ⊗ A V ) ∨ ∼ = Hom A (V, P ). Together with the previous isomorphism, we obtain an isomorphism ). In the case A = k, viewed as symmetric algebra with Id k as symmetrising form, the isomorphism 2.4 becomes the canonical isomorphism Hom be an exact sequence of A-modules, with P 0 = P U and P 1 = P Ω(U) projective. Then ker(π) = Im(δ) = Ω(U ). The inclusion Ω(U ) ֒→ P 0 induces an injective map Hom ). An A-homomorphism from U to V factors through a projective module if and only if it factors through the map δ, viewed as a map from P 1 to the submodule Ω(U ) of P 0 . It follows that the image of the map Hom A (V, P 1 ) → Hom A (V, P 0 ) can be identified with the subspace Hom pr A (V, Ω(U )) of Hom A (V, Ω(U )), where Hom A (V, Ω(U )) is viewed as a subspace of Hom A (V, P 0 ). Applying the contravariant functor Hom A (−, V ) to the previous exact sequence yields an exact sequence which remains exact upon applying k-duality. Thus we obtain a commutative diagram 2.5.
in which the right column is exact, and where the two horizontal isomorphisms are from 2.4. The map T induces the desired Tate duality isomorphism. To see this, note first that since Hom A (U, V ) is a quotient of Hom A (U, V ), its dual can be identified to the subspace of Hom A (U, V ) ∨ which annihilates Hom pr A (U, V ). The elements in Hom pr A (U, V ) are finite sums of maps of the form λ κ,v , where κ ∈ Hom A (U, A) and v ∈ V . The image of this space in Hom A (P 0 , V ) obtained from precomposing with π consists of finite sums of maps λ τ,v , where τ ∈ Hom A (P 0 , V ) and v ∈ V such that τ factors through π, or equivalently, such that τ annihilates the submodule Ω(U ) of P 0 . But this is exactly the subspace of allβ, where β ∈ Hom A (V, P 0 ) satisfies Im(β) ⊆ Ω(U ), and hence T induces a surjective map Hom A (V, Ω(U )) → Hom A (U, V ) ∨ . The kernel of this map consists of all homomorphisms in the image of the map Hom A (V, P 1 ) → Hom A (V, P 0 ), which by the above is Hom pr A (V, Ω(U )). This yields an isomorphism as stated in 2.3. A diagram chase shows that this isomorphism 'commutes' with isomorphisms obtained from applying the equivalence Σ; that is, the following diagram is commutative: where the horizontal isomorphisms are the Tate duality isomorphisms from 2.3, where the vertical isomorphisms are induced by Σ, and where we have identified Σ(Ω(U )) = U = Ω(Σ(U )) in the lower left corner of this diagram.
Using the naturality of Tate duality, we obtain a compatibility of Tate duality and Yoneda products as follows. Let U , V , W be finitely generated A-modules, let m, n be integers, and let T (ζη)(τ ) = T (ζ)(ητ ) , or equivalently, To see this, consider the diagram where the horizontal maps are the Tate duality isomorphisms, where the vertical isomorphisms are induced by Ω m , and where the two remaining vertical maps are induced by (pre-) composing with η. The upper square is commutative by 2.6. The lower square is commutative by the naturality of Tate duality. The image of ζ under the two left vertical maps is ζη, and the image of T (ζ) under the right two vertical maps is the map τ → T (ζ)(ητ ). By the commutativity of this diagram this map is equal to T (ζη), whence 2.7 and 2.8. Tate duality is dual to its own inverse: combining two Tate duality isomorphisms This can be seen by observing that if P , Q are two finitely generated projective A-modules, then the composition of the two consecutive isomorphisms Hom A (P, Q) ∼ = Hom A (Q, P ) ∨ ∼ = Hom A (P, Q) ∨∨ obtained from 2.4 is equal to the canonical double duality isomorphism.
The opposite algebra A 0 of A is again symmetric, with the same symmetrising form s. Thus the algebra A ⊗ k A 0 is symmetric as well. The Tate-Hochschild cohomology of A is defined by HH n (A) = Hom A⊗ k A 0 (A, Σ n (A)), for any integer n, where here Σ = Σ A⊗ k A 0 . Tate duality for Tate-Hochschild cohomology is thus a canonical isomorphism

On adjunction for symmetric algebras
Let A, B be symmetric algebras with symmetrising forms s ∈ A ∨ and t ∈ B ∨ . The field k is trivially a symmetric k-algebra, and it is always understood being endowed with Id k as symmetrising form.
Let M be an A-B-bimodule which is finitely generated projective as a left A-module and as a right B-module. It is well-known that the functors M ⊗ B − and M ∨ ⊗ A − are left and right adjoint to each other; see e.g. Broué [7] or [8, §6]. We will need the explicit description from [8] of this adjunction in order to identify certain isomorphisms as special cases of this adjunction. We briefly sketch this, but leave detailed

3.1.
Hom The unit and counit of this adjunction are represented by bimodule homomorphisms where I, α i , m i are as before. Similarly, we have an adjunction isomorphism where J is a finite indexing set, β j ∈ Hom B 0 (M, B), m j ∈ M , such that j∈J m j β j (m ′ ) = m ′ for all m ′ ∈ M , where m ∈ M and β ∈ Hom B 0 (M, B). Note the slight abuse of notation: for the maps ǫ M ∨ and η M ∨ in 3.4 to coincide with those obtained from 3. Duality is compatible with tensor products: if N is a B-C-bimodule, where C is another symmetric k-algebra, such that N is finitely generated projective as a left B-module and as a right C-module, then we have a natural isomorphism of C-A-bimodules , and where m ∈ M , n ∈ N . This is obtained as the composition of the natural isomorphisms where the second isomorphism is the standard adjunction with k instead of A. Using this isomorphism (applied to C = A and N = M ∨ ) we obtain that the adjunction units and counits from the left and right adjunction of the functors M ∨ ⊗ A − and M ⊗ B − are dual to each other. More precisely, we have a commutative diagram

3.6.
A where the left vertical isomorphism is induced by s (sending a ∈ A to the linear map a · s defined by (a · s)(a ′ ) = s(aa ′ ) for all a ∈ A) and where the right vertical isomorphism combines the isomorphism (M ⊗ B M ∨ ) ∨ ∼ = M ∨∨ ⊗ B M ∨ from (3.5) and the canonical isomorphism M ∨∨ ∼ = M . The commutativity is verified by chasing 1 A through this diagram. Similarly, we have a commutative diagram 3.7.
where the right vertical isomorphism is induced by t and the left vertical isomorphism is (3.5) combined with M ∨∨ ∼ = M as before.
For later reference, we mention the special case of the adjunction isomorphism 3.1 with the algebras k, A instead of A, B, respectively, the k-A-bimodule A instead of M , and k and U instead of U and V , respectively. This yields a natural isomorphism sending γ ∈ Hom k (U, k) to the map u → (a → γ(au)), where u ∈ U and a ∈ A. Similarly, the special case of the adjunction 3.3 with the algebras k, A instead of A, B, respectively, the A-k-bimodule A instead of M , and the modules U , k instead of V , U , respectively, yields a natural isomorphism sending ϕ ∈ Hom A (A ∨ , U ) to the unique linear map sending 1 ∈ k to ϕ(s).

Tate duality and adjunction
As in the preceding section, let A, B be symmetric k-algebras, with a fixed choice of symmetrising forms s ∈ A ∨ and t ∈ B ∨ . Let M be an A-B-bimodule which is finitely generated projective as a left A-module and as a right B-module. Tate duality is induced by the isomorphisms in 2.4, and so we need to show that these are compatible with the adjunctions from §3.
Lemma 4.1. Let U be a finitely generated A-module. We have a commutative diagram of k-linear isomorphisms where the map σ is induced by composing with the symmetrising form s, the map τ is the adjunction isomorphism, the map α ∨ is the dual of the canonical isomorphism α : Hom k (k, U ) ∼ = U ∼ = Hom A (A, U ), the map β ∨ is the dual of the adjunction isomorphism, and where the vertical isomorphisms are given by 2.4, with k considered as symmetric algebra having Id k as symmetrising form.
Proof. Let ϕ ∈ Hom A (U, A). The left vertical isomorphism sends ϕ to the map λ α,u → s(α(ϕ(u))), where u ∈ U , α ∈ Hom A (A, A) and λ α,u (a) = α(a)u = aα(1)u. Thus λ α,u = λ Id,u ′ , where Id is the identity map on A and u ′ = α(1)u. It follows that the left vertical isomorphism sends ϕ to the unique map sending λ Id,u to s(ϕ(u)). The upper horizontal isomorphism sends ϕ to s • ϕ. Similarly, the middle vertical isomorphism sends s • ϕ to the map sending λ Id k ,u to s(ϕ(u)). This shows the commutativity of the left square in the diagram. The commutativity of the right square can be verified directly using the explicit descriptions of τ and β from 3.8 and 3.9. Alternatively, it is easy to see that τ • σ and α • β are both induced by the isomorphism A ∼ = A ∨ sending 1 A to s. Thus the outer rectangle (that is, with the vertical arrow in the middle removed) is commutative by the naturality of the isomorphism 2.4. Since all involved maps are isomorphism, the commutativity of the right square follows.
Lemma 4.2. Let P be a finitely generated projective A-module, and let V be a finitely generated B-module. Then M ∨ ⊗ A P is a finitely generated projective B-module, and we have a commutative diagram of k-linear isomorphisms where the horizontal isomorphisms are given by the adjunction isomorphisms, and where the vertical isomorphisms are from 2.4.
Proof. The maps in this diagram are natural in P . Thus it suffices to show the commutativity for P = A. More explicitly, we will show the commutativity of the following diagram of linear isomorphisms: The vertical arrows are isomorphisms from 2.4. The upper two horizontal maps are adjunction isomorphisms, and their composition is the upper isomorphism of the diagram in the statement (with P = A). Similarly, the lower two horizontal maps are dual to adjunction isomorphisms, and their composition is the lower horizontal isomorphism of the diagram in the statement (with P = A). The commutativity of the left square in this diagram follows from that of the left square in Lemma 4.1. For the commutativity it suffices, by naturality, to show this for M = B, which is a special case of the right square in Lemma 4.1.
Proposition 4.3. Let U be a finitely generated A-module, and let V be a finitely generated Bmodule. We have a commutative diagram of k-linear isomorphisms where the horizontal isomorphisms are given by the adjunction isomorphisms, the vertical isomorphisms are the Tate duality isomorphisms from 2.3, and where we identify Proof. Let be an exact sequence of A-modules, with P 0 = P U and P 1 = P Ω(U) projective, so that ker(π) = Im(δ) = Ω(U ). Since M ∨ is finitely generated as a right A-module, the sequence of B-modules For the proof of Theorem 1.1 we will need the following bimodule version of 4.3.
Corollary 4.4. Let C be a symmetric k-algebra with a fixed choice of a symmetrising form, U a finitely generated A ⊗ k C 0 -module, and let V be a finitely generated B ⊗ k C 0 -module. We have a commutative diagram of k-linear isomorphisms where the horizontal isomorphisms are the canonical adjunction isomorphisms, the vertical isomorphisms are the Tate duality isomorphisms from 2.3, and where we identify Proof. We will show that this diagram is isomorphic to the commutative diagram from 4.3 applied to the algebras A ⊗ k C 0 , B ⊗ k C 0 instead of A, B, respectively, and to the A ⊗ k C 0 -B ⊗ k C 0bimodule M ⊗ k C, respectively. In this commutative diagram, we identify the terms through the following isomorphisms. We consider M ⊗ k C as an (A ⊗ k C 0 )-(B ⊗ k C 0 )-bimodule as follows. The left A ⊗ k C 0 -module structure on M ⊗ k C is given by left multiplication with A on M and by right multiplication with C on C. Similarly, the right B ⊗ k C 0 -module structure on M ⊗ k C is given by right multiplication with B on M and by left multiplication with C on C. We have an isomorphism and c ∈ C. Moreover, since C is symmetric, the choice of a symmetrising form on C yields an isomorphism of (B With these identifications, the commutative diagram under consideration takes the form as stated.

Transfer in Tate-Hochschild cohomology
Following [12], a pair of adjoint functors between triangulated categories induces transfer maps between the graded centers of these categories as well as Ext-groups. We briefly review this, specialised to Tate-Hochschild cohomology (cf. [12, §7.1]). Let A, B be symmetric k-algebras, with a fixed choice of symmetrising forms s ∈ A ∨ and t ∈ B ∨ , and let M be an A-B-bimodule which is finitely generated projective as a left A-module and as a right B-module. Let n be an integer. We will write Σ instead of Σ A⊗ k A 0 or Σ B⊗ k B 0 . An element ζ ∈ HH The right vertical map is induced by composition with Σ n (η M ), and hence the image of ( By the remarks preceding this Lemma, this is equal to tr M (ζ).

Products in negative Tate cohomology
Let A be a symmetric k-algebra. The results of this section have been obtained independently by Bergh, Jorgensen, and Oppermann [6, §3]. They are generalisations to symmetric algebras of results due to Benson and Carlson in [3], and the proofs we present here are straightforward adaptations of those given in [3]. See also [4] for connections with Steenrod operations.
Lemma 8.1. Let U , V be finitely generated A-modules, and let n be an integer. If ζ is a nonzero element in Ext Proof. We follow the proof of [3, Theorem 3.1]. Suppose that Ext * A (U, U ) has a regular sequence of length 2, consisting of homogeneous elements ζ 1 , ζ 2 of positive degrees d 1 , d 2 , respectively. Let ζ ∈ Ext m A (U, U ) and η ∈ Ext n A (U, U ) such that ζη = 0. By Lemma 8.2, applied with m + 1, n + 1 instead of m, n, respectively, there is an element τ ∈ Ext −m−n−1 A (U, U ) such that ητ = is nonzero in Ext −m−1 A (U, U ). Note that since m is negative, we have −m − 1 ≥ 0. Since ζ 1 is not a zero divisor, we have ζ a 1 ζη = 0, hence ζ a 1 ζ = 0 for all nonnegative integers a. Choose a maximal such that deg(ζ a 1 ζ) < 0. Then 0 ≤ deg(ζ a+1 1 ζ) < d 1 . In particular, ζ a+1 ζ is contained in Ext * A (U, U ) but not in the ideal ζ 1 Ext * A (U, U ). Thus the image of ζ a+1 1 ζ in the quotient Ext * A (U, U )/ζ 1 Ext * A (U, U ) is non zero. However, for some sufficiently large integer b we have ζ b 2 ζ a+1 1 ζ = ζ 1 (ζ a 1 ζζ b 2 ), hence ζ 2 is not regular on this quotient. This contradiction shows that Ext * A (U, U ) has no regular sequence of length two.

Proof. By Tate duality, if ζ is nonzero in Ext
Since the Hochschild cohomology of an algebra is graded-commutative we obtain the following consequence: