Tate duality and transfer in Hochschild cohomology

Markus Linckelmann

Abstract 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.

1 Introduction

Let k be a field. For V a k-vector space, denote by V^{∨} its k-dual Hom_{k}(V, k). A finite-dimensional k-algebra A is called symmetric if A ≅ A^{∨} as A-A-bimodules. The image s ∈ A^{∨} of 1_{A} under such an isomorphism is called a symmetrising form for A. It is well-known that the Tate analogue ĤH^{∗}(A) of the Hochschild cohomology of a symmetric k-algebra A satisfies a duality (ĤH^{−n}(A))^{∨} ≅ ĤH^{n−1}(A), for every integer n. If A, B are two symmetric algebras and M is an A-B-bimodule which is finitely generated as a left A-module and as a right B-module, then M induces a transfer map tr_{M} : ĤH^{∗}(B) → ĤH^{∗}(A), and the dual M^{∨} induces a transfer map tr_{M^{∨}} : ĤH^{∗}(A) → ĤH^{∗}(B). These transfer maps depend on the choices of symmetrising forms for A and B. In positive degree they coincide with the transfer maps constructed in [11]. Similarly, for any two finitely generated B-modules V, W, there is a transfer map tr_{M^{∨}} = tr_{M^{∨}}(V, W) : Êxt_{A}^{n}(M ⊗_{B} V, M ⊗_{B} W) → Êxt_{B}^{n}(V, W). The underlying constructions are special cases of a general principle associating transfer maps with pairs of adjoint functors between triangulated categories; see [12, §7] or Section 5 below for a brief review.

Theorem 1.1. Let A, B be symmetric k-algebras, and let M be an A-B-bimodule which is finitely generated projective as a left A-module and as a right B-module. For any integer n we have a commutative diagram of k-vector spaces

\xymatrix{ĤH^{n−1}(A) \ar[r]^{^{tr^{M∨}}} \ar[d] & ĤH^{n−1}(B) \ar[d] \\ (ĤH^{−n}(A))^{∨} \ar[r]_{(tr_{M})^{∨}} & (ĤH^{−n}(B))^{∨}}

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_{X_{i}} 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 ĤH^{∗}(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 ĤH^{∗}(B). In particular, there is no loss of generality in stating Theorem 1.1 for bimodules rather than complexes.

Theorem 1.2. Let A, B be symmetric k-algebras, 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 V, W be finitely generated B-modules. For any integer n we have a commutative diagram of k-vector spaces

\xymatrix{Êxt_{B}^{n−1}(V, W) \ar[r]^{M⊗_{B}−} \ar[d] & Êxt_{A}^{n−1}(M ⊗_{B} V, M ⊗_{B} W) \ar[r]^{^{tr^{M∨}}} \ar[d] & Êxt_{B}^{n−1}(V, W) \ar[d] \\ Êxt_{B}^{−n}(W, V)^{∨} \ar[r]_{(tr_{M∨})^{∨}} & Êxt_{A}^{−n}(M ⊗_{B} W, M ⊗_{B} V)^{∨} \ar[r]_{(M⊗_{B}−)^{∨}} & Êxt_{B}^{−n}(W, V)^{∨}}

where tr_{M^{∨}} = tr_{M^{∨}}(V, W) and the vertical maps are the Tate duality isomorphisms.

Remark 1.3. Let G a finite group and H a subgroup of G. Tate duality for group cohomology is a canonical isomorphism Ĥ^{−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 ΔG the diagonal subgroup ΔG = {(x, x) | x ∈ G} of G × G. The induction functor Ind_{ΔG}^{G×G} sends the trivial kΔG-module to Ind_{ΔG}^{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 ≅ ΔG 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} : Ĥ^{∗}(G; k) → ĤH^{∗}(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_{H}^{G} : Ĥ^{−n}(H; k) → Ĥ^{−n}(G; k) with the restriction map res_{H}^{G} : 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_{A}(M, N) as a B-C-bimodule via (b · φ · c)(m) = φ(mb)c, where φ ∈ Hom_{A}(M, N), b ∈ B, c ∈ C, and m ∈ M. Similarly, we consider Hom_{B^{0}}(M, N′) as a C-A-bimodule via (c · ψ · a)(m) = cψ(am), where ψ ∈ Hom_{B^{0}}(M, N′), a ∈ A, c ∈ C, and m ∈ M.

2 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_{A}^{pr}(U, V) the space of A-homomorphisms from U to V which factor through a projective A-module, and we set \underline{Hom}_{A}(U, V) = Hom_{A}(U, V)/Hom_{A}^{pr}(U, V). The stable module category of A is the k-linear category \underline{mod}(A) having the same objects as mod(A), with morphism spaces \underline{Hom}_{A}(U, V) for any two finitely generated left A-modules, where the composition of morphisms in \underline{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 \underline{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 \underline{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 Êxt_{A}^{n}(U, V) = \underline{Hom}_{A}(U, Σ^{n}(V)). Tate duality for symmetric algebras states that for any integer n there is an isomorphism

2.1.

Êxt_{A}^{n−1}(V, U) ≅ Êxt_{A}^{−n}(U, V)^{∨} ,

which is natural in U and V. Equivalently, there is a natural nondegenerate bilinear form 2.2.

⟨−, −⟩ : Êxt_{A}^{n−1}(V, U) × Êxt_{A}^{−n}(U, V) → k .

The isomorphism 2.1 is equivalent to a natural isomorphism

2.3.

\underline{Hom}_{A}(V, Ω(U)) ≅ \underline{Hom}_{A}(U, V)^{∨} .

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 \underline{End}_{A}(U)-\underline{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. For τ ∈ Hom_{A}(U, A) and v ∈ V defined λ_{τ,v} ∈ Hom_{A}(U, V) by setting λ_{τ,v}(u) = τ(v)u for all u ∈ U. Note that λ_{τ,v} is the composition of the map τ : U → A followed by the map A → V sending 1_{A} to v; in particular, λ_{τ,v} factors through a projective A-module. This yields a map

Φ_{U,V} : Hom_{A}(U, A) ⊗_{A} V → Hom_{A}(U, V)

sending τ ⊗ v to the map λ_{τ,v}. The maps Φ_{U,V} are natural in U and V. By the above remarks, the image of Φ_{U,V} is equal to Hom_{A}^{pr}(U, V), and if U is finitely generated projective, then Φ_{U,V} is an isomorphism. The map sending τ ∈ Hom_{A}(U, A) to s ∘ τ is a natural isomorphism of right A-modules Hom_{A}(U, A) ≅ U^{∨}. Thus, for any finitely generated projective A-module P we have an isomorphism

P^{∨} ⊗_{A} V ≅ Hom_{A}(P, V)

sending s ∘ α ⊗ v to the map λ_{α,v} as defined above; that is, to the map x ↦ α(x)v, where α ∈ Hom_{A}(P, A), 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

2.4.

Hom_{A}(V, P) ≅ Hom_{A}(P, V)^{∨}

sending β ∈ Hom_{A}(V, P) to the unique map β̂ ∈ Hom_{A}(P, V)^{∨} satisfying β̂(λ_{α,v}) = s(α(β(v))). In the case A = k, viewed as symmetric algebra with Id_{k} as symmetrising form, the isomorphism 2.4 becomes the canonical isomorphism Hom_{k}(V, k) = V^{∨} ≅ Hom_{k}(k, V)^{∨}. Let

\xymatrix{P_{1} \ar[r]^{δ} & P_{0} \ar[r]^{π} & U \ar[r] & 0}

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_{A}(V, Ω(U)) → Hom_{A}(V, P_{0}). The map δ induces a map Hom_{A}(V, P_{1}) → Hom_{A}(V, P_{0}). 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_{A}^{pr}(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

\xymatrix{0 \ar[r] & Hom_{A}(U, V) \ar[r] & Hom_{A}(P_{0}, V) \ar[r] & Hom_{A}(P_{1}, V)}

which remains exact upon applying k-duality. Thus we obtain a commutative diagram 2.5.

\xymatrix{Hom_{A}(V, P_{1}) \ar[r]^{≅} \ar[d] & Hom_{A}(P_{1}, V)^{∨} \ar[d] \\ Hom_{A}(V, P_{0}) \ar[r]^{≅} & Hom_{A}(P_{0}, V)^{∨} \ar[d] \\ Hom_{A}(V, Ω(U)) \ar[u] \ar[r]_{T} & Hom_{A}(U, V)^{∨} \ar[d] \\  & 0}

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 \underline{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_{A}^{pr}(U, V). The elements in Hom_{A}^{pr}(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 all β̂, where β ∈ Hom_{A}(V, P_{0}) satisfies Im(β) ⊆ Ω(U), and hence T induces a surjective map Hom_{A}(V, Ω(U)) → \underline{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_{A}^{pr}(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:

2.6.

\xymatrix{\underline{Hom}_{A}(V, Ω(U)) \ar[r] \ar[d] & \underline{Hom}_{A}(U, V)^{∨} \ar[d] \\ \underline{Hom}_{A}(Σ(V), U) \ar[r] & \underline{Hom}_{A}(Σ(U), Σ(V))^{∨}}

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 ζ ∈ Êxt_{A}^{m+n−1}(W, U), η ∈ Êxt_{A}^{−m}(V, W), and τ ∈ Êxt_{A}^{−n}(U, V). Denote by ζη = Σ^{−m}(ζ) ∘ η and ητ = Σ^{−n}(η) ∘ τ the Yoneda products in Êxt_{A}^{n−1}(V, U) and Êxt_{A}^{−m−n}(U, W), respectively. Denote by T (ζ) and T (ζη) the images of ζ and of ζη in Ext_{A}^{−m−n}(U, W)^{∨} and Ext_{A}^{−n}(U, V)^{∨}, respectively, under the appropriate versions of the Tate duality isomorphism 2.1. We have

2.7.

T (ζη)(τ) = T (ζ)(ητ) ,

or equivalently,

2.8.

⟨ζη, τ⟩ = ⟨ζ, ητ⟩ .

To see this, consider the diagram

\xymatrix{Êxt_{A}^{n+m−1}(W, U) \ar[r]^{≅} \ar[d]_{≅} & Êxt_{A}^{−m−n}(U, W)^{∨} \ar[d]^{≅} \\ Êxt_{A}^{n−1}(Ω^{m}(W), U) \ar[r]^{≅} \ar[d]_{(η,U)} & Êxt_{A}^{−n}(U, Ω^{m}(W))^{∨} \ar[d]^{(U,η)^{∨}} \\ Êxt_{A}^{n−1}(V, U) \ar[r]_{≅} & Êxt_{A}^{−n}(U, V)^{∨}}

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

2.9.

Êxt_{A}^{n−1}(V, U) ≅ Êxt_{A}^{−n}(U, V)^{∨} ≅ Êxt_{A}^{n−1}(V, U)^{∨∨}

yields the canonical double duality isomorphism, or equivalently, for ζ ∈ Êxt_{A}^{n−1}(V, U) and η ∈ Êxt_{A}^{−n}(U, V) we have

2.10.

⟨ζ, η⟩ = ⟨η, ζ⟩ .

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 ĤH^{n}(A) = \underline{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

2.11.

(ĤH^{−n}(A))^{∨} ≅ ĤH^{n−1}(A) ,

for any integer n.

3 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 verifications to the reader. The starting point is the standard tensor-Hom-adjunction; this is the adjoint pair of functors (M ⊗_{B} −, Hom_{A}(M, −)) between Mod(A) and Mod(B), with the natural isomorphism Hom_{A}(M ⊗_{B} V, U) ≅ Hom_{B}(V, Hom_{A}(M, U)) sending φ ∈ Hom_{A}(M ⊗_{B} V, U) to the map v ↦ (m ↦ φ(m ⊗ v)), where U is an A-module, V a B-module, v ∈ V and m ∈ M. The unit of this adjunction is represented by the B-B-bimodule homomorphism B → Hom_{A}(M, M) sending 1_{B} to Id_{M}; the counit of this adjunction is represented by the A-A-bimodule homomorphism M ⊗_{B} Hom_{A}(M, A) → A sending m ⊗ α to α(m), where m ∈ M and α ∈ Hom_{A}(M, A).

Since M is finitely generated projective as a left A-module, the canonical map Hom_{A}(M, A) ⊗_{A} U → Hom_{A}(M, U) sending α⊗u to the map m ↦ α(m)u is an isomorphism, where α ∈ Hom_{A}(M, A), u ∈ U, and m ∈ M. Under this isomorphism applied with U = M, the preimage of Id_{M} is an expression of the form ∑_{i∈I} α_{i} ⊗ m_{i}, where I is a finite indexing set, α_{i} ∈ Hom_{A}(M, A) and m_{i} ∈ M such that ∑_{i∈I} α_{i}(m′)m_{i} = m′ for all m′ ∈ M.

Since A is symmetric, the map sending α to s∘α is an isomorphism of B-A-bimodules Hom_{A}(M, A) ≅ M^{∨}. Similarly, the map sending β ∈ Hom_{B^{0}}(M, B) to t ∘ β is an isomorphism of B-A-bimodules Hom_{B^{0}}(M) ≅ M^{∨}. Combined with the standard adjunction these isomorphisms yield an adjunction

3.1.

Hom_{A}(M ⊗_{B} V, U) ≅ Hom_{B}(V, M^{∨} ⊗_{A} U)

sending λ_{γ,u} to the map v ↦ s ∘ γ_{v} ⊗ u, where γ ∈ Hom_{A}(M, A), u ∈ U, where λ_{γ,u} ∈ Hom_{A}(M ⊗_{B} V, U) is defined by λ_{γ,u}(m ⊗ v) = γ(m ⊗ v)u, and where γ_{v} ∈ Hom_{A}(M, A) is defined by γ_{v}(m) = γ(m⊗v), for all m ∈ M, v ∈ V. The unit and counit of this adjunction are represented by bimodule homomorphisms

3.2.

ϵ_{M} : B → M^{∨} ⊗_{A} M , 1_{B} ↦ ∑_{i∈I} (s ∘ α_{i}) ⊗ m_{i} ,

η_{M} : M ⊗_{B} M^{∨} → A , m ⊗ (s ∘ α) ↦ α(m) ,

where I, α_{i}, m_{i} are as before. Similarly, we have an adjunction isomorphism

3.3.

Hom_{B}(M^{∨} ⊗_{A} U, V) ≅ Hom_{A}(U, M ⊗_{B} V)

obtained from 3.1 by exchanging the roles of A and B and using M^{∨} instead of M together with the canonical double duality M^{∨∨} ≅ M. The adjunction unit and counit of this adjunction are represented by bimodule homomorphisms 3.4.

ϵ_{M^{∨}} : A → M ⊗_{B} M^{∨} , 1_{A} ↦ ∑_{j∈J} m_{j} ⊗ (t ∘ β_{j}) ,

η_{M^{∨}} : M^{∨} ⊗_{A} M → B , (t ∘ β) ⊗ m ↦ β(m) ,

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 ϵ_{M^{∨}} and η_{M^{∨}} in 3.4 to coincide with those obtained from 3.2 applied to M^{∨} instead of M we need to identify M and M^{∨∨}. One could avoid this by replacing the pair of bimodules (M, M^{∨}) by a pair of bimodules (M, N) which are dual to each other through a fixed choice of a nondegenerate bilinear map M × N → k; this is the point of view taken in [8].

The adjunction units and counits of the adjunctions 3.1 and 3.3 are also the units and counits of the corresponding adjunctions for right modules. More precisely, the maps ϵ_{M} and η_{M} represent the unit and counit of the adjoint pair (− ⊗_{B} M^{∨}, − ⊗_{A} M), and the maps ϵ_{M^{∨}} and η_{M^{∨}} represent the unit and counit of the adjoint pair (− ⊗_{A} M, − ⊗_{B} M^{∨}).

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

3.5.

N^{∨} ⊗_{B} M^{∨} ≅ (M ⊗_{B} N)^{∨}

sending (t∘β)⊗μ to the map m⊗n ↦ μ(mβ(n)), where μ ∈ M^{∨}, β ∈ Hom_{B}(N, B) (hence t∘β ∈ N^{∨}), and where m ∈ M, n ∈ N. This is obtained as the composition of the natural isomorphisms

N^{∨} ⊗_{B} M^{∨} ≅ Hom_{B}(N, M^{∨}) ≅ (M ⊗_{B} N)^{∨}

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.

\xymatrix{A \ar[d] \ar[r]^{ϵ_{M^{∨}}} & M ⊗_{B} M^{∨} \ar[d] \\ A^{∨} \ar[r]_{(η_{M})^{∨}} & (M ⊗_{B} M^{∨})^{∨}}

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.

\xymatrix{M^{∨} ⊗_{A} M \ar[d] \ar[r]^{η_{M^{∨}}} & B \ar[d] \\ (M^{∨} ⊗_{A} M)^{∨} \ar[r]_{(ϵ_{M})^{∨}} & B^{∨}}

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

3.8.

τ : Hom_{k}(U, k) ≅ Hom_{A}(U, A^{∨})

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

3.9.

β : Hom_{A}(A^{∨}, U) ≅ Hom_{k}(k, U)

sending φ ∈ Hom_{A}(A^{∨}, U) to the unique linear map sending 1 ∈ k to φ(s).

4 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

\xymatrix{Hom_{A}(U, A) \ar[r]^{σ} \ar[d] & Hom_{k}(U, k) \ar[d] \ar[r]^{τ} & Hom_{A}(U, A^{∨}) \ar[d] \\ Hom_{A}(A, U)^{∨} \ar[r]_{α^{∨}} & Hom_{k}(k, U)^{∨} \ar[r]_{β^{∨}} & Hom_{A}(A^{∨}, U)^{∨}}

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

\xymatrix{Hom_{A}(M ⊗_{B} V, P) \ar[r] \ar[d] & Hom_{B}(V, M^{∨} ⊗_{A} P) \ar[d] \\ Hom_{A}(P, M ⊗_{B} V)^{∨} \ar[r] & Hom_{B}(M^{∨} ⊗_{A} P, V)^{∨}}

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:

\xymatrix{Hom_{A}(M ⊗_{B} V, A) \ar[r] \ar[d] & Hom_{k}(M ⊗_{B} V, k) \ar[r] \ar[d] & Hom_{B}(V, M^{∨}) \ar[d] \\ Hom_{A}(A, M ⊗_{B} V)^{∨} \ar[r] & Hom_{k}(k, M ⊗_{B} V)^{∨} \ar[r] & Hom_{B}(M^{∨}, V)^{∨}}

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

\xymatrix{\underline{Hom}_{A}(M ⊗_{B} V, Ω(U)) \ar[r] \ar[d] & \underline{Hom}_{B}(V, M^{∨} ⊗_{A} Ω(U)) \ar[d] \\ \underline{Hom}_{A}(U, M ⊗_{B} V)^{∨} \ar[r] & \underline{Hom}_{B}(M^{∨} ⊗_{A} U, V)^{∨}}

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 Ω(M^{∨} ⊗_{A} V) = M^{∨} ⊗_{A} Ω(U), with Ω denoting either Ω_{A} or Ω_{B}.

Proof. Let

\xymatrix{P_{1} \ar[r]^{δ} & P_{0} \ar[r]^{π} & U \ar[r] & 0}

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

\xymatrix{M^{∨} ⊗_{A} P_{1} \ar[r]^{Id⊗δ} & M^{∨} ⊗_{A} P_{0} \ar[r]^{Id⊗π} & M^{∨} ⊗_{A} U \ar[r] & 0}

is exact. Since M^{∨} is also finitely generated as a left B-module, it follows that the B-modules M^{∨}⊗_{A} P_{1} and M^{∨} ⊗_{A} P_{0} are finitely generated projective. In particular, we may identify Ω(M^{∨} ⊗_{A} U) = ker(Id ⊗ π) = M^{∨} ⊗_{A} Ω(U). Combining the commutative diagram 2.5, used twice (with A, U, M ⊗_{B} V and with B, M^{∨} ⊗_{A} U, V, respectively), with the commutative square from 4.2, also used twice (with P_{1} and P_{0} instead of P), yields the result.

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

\xymatrix{\underline{Hom}_{A⊗_{k}C^{0}}(M ⊗_{B} V, Ω(U)) \ar[r] \ar[d] & \underline{Hom}_{B⊗_{k}C^{0}}(V, M^{∨} ⊗_{A} Ω(U)) \ar[d] \\ \underline{Hom}_{A⊗_{k}C^{0}}(U, M ⊗_{B} V)^{∨} \ar[r] & \underline{Hom}_{B⊗_{k}C^{0}}(M^{∨} ⊗_{A} U, V)^{∨}}

where the horizontal isomorphisms are the canonical adjunction isomorphisms, the vertical isomorphisms are the Tate duality isomorphisms from 2.3, and where we identify Ω(M^{∨} ⊗_{A} U) = M^{∨} ⊗_{A} Ω(U), with Ω denoting either Ω_{B⊗_{k}C^{0}} or Ω_{A⊗_{k}C^{0}}.

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^{0}- bimodule 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 of B ⊗_{k} C^{0}-modules (M ⊗_{k} C) ⊗_{B⊗_{k}C^{0}} V ≅ M ⊗_{B} V sending (m ⊗ c) ⊗ v to m ⊗ vc, where m ∈ M, v ∈ V, and c ∈ C. Moreover, since C is symmetric, the choice of a symmetrising form on C yields an isomorphism of (B ⊗_{k} C^{0})-(A ⊗_{k} C^{0})-bimodules (M ⊗_{k} C)^{∨} ≅ M^{∨} ⊗_{k} C^{∨} ≅ M^{∨} ⊗_{k} C. This, in turn, yields an isomorphism of B ⊗_{k} C^{0}-modules (M ⊗_{k} C)^{∨} ⊗_{A⊗_{k}C^{0}} Ω(U) ≅ M^{∨} ⊗_{A} Ω(U). With these identifications, the commutative diagram under consideration takes the form as stated.

5 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 ∈ ĤH^{n}(B) is represented by a B-B-bimodule homomorphism, abusively denoted by the same letter, ζ : B → Σ^{n}(B). We denote by tr_{M}(ζ) the element in ĤH^{n}(A) represented by the A-A-bimodule homomorphism

\xymatrix{M ⊗_{B} M^{∨} = M ⊗_{B} B ⊗_{B} M^{∨} \ar[r]^{Id_{M} ⊗ζ⊗Id_{M^{∨}}} & M ⊗_{B} Σ^{n}(B) ⊗_{B} M^{∨} = Σ^{n}(M ⊗_{B} M^{∨})}

precomposed with the adjunction unit ϵ_{M^{∨}} : A → M ⊗_{B} M^{∨} and composed with the ‘shifted’ adjunction counit Σ^{n}(η_{M}) : Σ^{n}(M ⊗_{B} M^{∨}) → Σ^{n}(A). The identification M ⊗_{B} Σ^{n}(B) ⊗_{B} M^{∨} = Σ^{n}(M ⊗_{B} M^{∨}) is to be understood as the canonical isomorphism in \underline{mod}(A ⊗_{k} A^{0}), using the fact that the functor M ⊗_{B} − ⊗_{B} M^{∨} sends a projective resolution of the B-B-bimodule B to a projective resolution of the A-A-bimodule M ⊗_{B} M^{∨}. Modulo this identification, we thus have

tr_{M}(ζ) = Σ^{n}(η_{M}) ∘ (Id_{M} ⊗ ζ ⊗ Id_{M^{∨}}) ∘ ϵ_{M^{∨}} .

In this way, tr_{M} becomes a graded k-linear but not necessarily multiplicative map from ĤH^{∗}(B) to ĤH^{∗}(A). We will need the following alternative description of transfer maps.

Lemma 5.1. For any integer n, the transfer map tr_{M} makes the following diagram commutative:

\xymatrix{\underline{Hom}_{B⊗_{k}B^{0}}(B, Σ^{n}(B)) \ar[d] \ar[rr]^{^{tr^{M}}} &  & \underline{Hom}_{A⊗_{k}A^{0}}(A, Σ^{n}(A)) \\ \underline{Hom}_{B⊗_{k}B^{0}}(M^{∨} ⊗_{A} M, Σ^{n}(B)) \ar[r]_{≅} & \underline{Hom}_{A⊗_{k}B^{0}}(M, Σ^{n}(M)) \ar[r]_{≅} & \underline{Hom}_{A⊗_{k}A^{0}}(A, Σ^{n}(M) ⊗_{B} M^{∨}) \ar[u]}

where the lower horizontal isomorphisms are adjunction isomorphisms, the left vertical map us induced by precomposing with the adjunction counit M^{∨} ⊗_{A} M → B, and the right vertical map is induced by composing with the map obtained from applying Σ^{n} to the adjunction counit M ⊗_{B} M^{∨} → A.

Proof. The main theorem on adjoint functors describes adjunction isomorphisms in terms of adjunction units and counits. Applied to the diagram in the statement it implies that the composition of the two maps

\xymatrix{\underline{Hom}_{B⊗_{k}B^{0}}(B, Σ^{n}(B)) \ar[r] & \underline{Hom}_{B⊗_{k}B^{0}}(M^{∨} ⊗_{A} M, Σ^{n}(B)) \ar[r]_{≅} & \underline{Hom}_{A⊗_{k}B^{0}}(M, Σ^{n}(M))}

is equal to the map sending ζ ∈ \underline{Hom}_{B⊗_{k}B^{0}}(B, Σ^{n}(B)) to Id_{M} ⊗ ζ, where we identify M ⊗_{B} B = M and Σ^{n}(M) = M⊗_{B}Σ^{n}(B), and where we use abusively the same letters for module homomorphisms and their classes in the stable category. Similarly, the next adjunction isomorphism

\xymatrix{\underline{Hom}_{A⊗_{k}B^{0}}(M, Σ^{n}(M)) \ar[r]_{≅} & \underline{Hom}_{A⊗_{k}A^{0}}(A, Σ^{n}(M) ⊗_{B} M^{∨})}

sends Id_{M} ⊗ ζ to (Id_{M} ⊗ ζ ⊗ Id_{M^{∨}}) ∘ ϵ_{M^{∨}}, where ϵ_{M^{∨}} : A → M ⊗_{B} M^{∨} is the adjunction counit. The right vertical map is induced by composition with Σ^{n}(η_{M}), and hence the image of (Id_{M} ⊗ ζ ⊗ Id_{M^{∨}}) ∘ ϵ_{M^{∨}} is equal to Σ^{n}(η_{M}) ∘ (Id_{M} ⊗ ζ ⊗ Id_{M^{∨}}) ∘ ϵ_{M^{∨}}. By the remarks preceding this Lemma, this is equal to tr_{M}(ζ).

Let V, W be finitely generated B-modules. An element in Êxt_{A}^{n}(M ⊗_{B}V, M ⊗_{B}W) is represented by an A-homomorphism η : M ⊗_{B} V → M ⊗_{B} Σ^{n}(W), where we identify Σ^{n}(M ⊗_{B} W) = M ⊗_{B} Σ^{n}(W) and where we use the same letter Σ for either Σ_{A} or Σ_{B}. The transfer map tr_{M^{∨}} = tr_{M^{∨}}(V, W) sends η to the element tr_{M^{∨}}(η) in Ext_{B}^{n}(V, W) represented by the B-homomorphism

\xymatrix{V \ar[r]^{ϵ_{M}} & M^{∨} ⊗_{A} M ⊗_{B} V \ar[r]^{Id_{M^{∨}} ⊗η} & M^{∨} ⊗_{A} M ⊗_{B} Σ^{n}(W) \ar[r]^{η_{M^{∨}}} & Σ^{n}(W)}

The transfer map tr_{M^{∨}} admits the two following descriptions.

Lemma 5.2. For any integer n, the transfer map tr_{M^{∨}} = tr_{M^{∨}}(V, W) makes the following diagram commutative:

\xymatrix{ & Ext_{B}^{n}(V, M^{∨} ⊗_{A} M ⊗_{B} W) \ar[dr]^{(V,η_{M^{∨}})} \\ Ext_{A}^{n}(M ⊗_{B} V, M ⊗_{B} W) \ar[ur]^{≅} \ar[dr]_{≅} \ar[rr]^{_{tr_{M^{∨}}}} &  & Êxt_{B}^{n}(V, W) \\  & Êxt_{B}^{n}(M^{∨} ⊗_{A} M ⊗_{B} V, W) \ar@{-->}[ur]_{(ϵ_{M} ,W)}}

Here the left two isomorphisms are the adjunction isomorphisms, and the maps labelled (V, η_{M^{∨}}) and (ϵ_{M}, W) are induced by composition and precomposition with η_{M^{∨}} and ϵ_{M}, respectively.

Proof. This follows using the same arguments as in the proof of Lemma 5.1.

Remark 5.3. Let n be an integer. The elements of ĤH^{n}(A) are morphisms from A to Σ^{n}(A) in the stable category \underline{perf}(A, A) of A-A-bimodules which are finitely generated projective as left and right A-modules. The category \underline{perf}(A, A) is a thick subcategory of \underline{mod}(A ⊗_{k} A^{0}). Following [11, 3.1.(iii)] or [12, 5.1], an element ζ ∈ HH^{n}(A) is called M-stable if there is η ∈ HH^{n}(B) such that Id_{M} ⊗ζ = η⊗Id_{M} : M → Σ^{n}(M) in \underline{perf}(A, B), where we identify as usual Σ^{n}(M) = M ⊗_{B}Σ^{n}(B) = Σ^{n}(A) ⊗_{A} M. Suppose that M and M^{∨} induce a stable equivalence of Morita type between A and B. The functor M ⊗_{B} − ⊗_{B} M^{∨} induces an equivalence of triangulated categories \underline{perf}(B, B) ≅ \underline{perf}(A, A) sending B to the bimodule M ⊗_{B} M^{∨} which is isomorphic to A in \underline{perf}(A, A). It follows

that this functor induces a graded algebra isomorphism Φ_{M} : ĤH^{∗}(B) ≅ ĤH^{∗}(A). The adjunction maps ϵ_{M}, η_{M}, ϵ_{M^{∨}}, η_{M^{∨}} are isomorphisms in the appropriate stable categories of bimodules. Thus every element in ĤH^{∗}(A) is M-stable. It follows from [11, 3.6] that the isomorphism Φ_{M} is equal to the analogue for stable categories of the normalised transfer map Tr_{M} as defined in [11, 3.1.(ii)].

6 Proof of Theorem 1.1

We use the notation from the statement of Theorem 1.1. We identify ΩΣ^{n}(A) = Σ^{n−1}(A) and ΩΣ^{n}(B) = Σ^{n−1}(B). The left vertical isomorphism in the diagram in Theorem 1.1 is the composition

\underline{Hom}_{A⊗_{k}A^{0}}(A, Σ^{n−1}(A)) ≅ \underline{Hom}_{A⊗_{k}A^{0}}(Σ^{n}(A), A))^{∨} ≅ \underline{Hom}_{A⊗_{k}A^{0}}(A, Σ^{−n}(A)))^{∨} ,

where the first isomorphism is the Tate duality isomorphism 2.3 applied to U = Σ^{n}(A) and V = A, and where the second isomorphism is induced by the equivalence Σ^{n} on \underline{mod}(A ⊗_{k} A^{0}). Thus, the commutativity of the diagram in Theorem 1.1 is equivalent to the commutativity of the diagram

6.1.

\xymatrix{\underline{Hom}_{A⊗_{k}A^{0}}(A, Σ^{n−1}(A)) \ar[r]^{^{tr^{M∨}}} \ar[d] & \underline{Hom}_{B⊗_{k}B^{0}}(B, Σ^{n−1}(B)) \ar[d] \\ (\underline{Hom}_{A⊗_{k}A^{0}}(Σ^{n}(A), A))^{∨} \ar[r]_{(Σ^{n}∘tr_{M} ∘Σ^{−n})^{∨}} & (\underline{Hom}_{B⊗_{k}B^{0}}(Σ^{n}(B), B))^{∨}}

where the vertical maps are appropriate versions of the Tate duality isomorphism 2.3. Lemma 5.1 describes the horizontal maps in this diagram as a composition of four maps. The commutativity of this diagram will therefore be established by combining four diagrams as follows. In all four of those diagrams, the vertical maps are the relevant Tate duality isomorphisms from 2.3. Consider the diagram

6.2.

\xymatrix{\underline{Hom}_{A⊗_{k}A^{0}}(A, Σ^{n−1}(A)) \ar[r] \ar[d] & \underline{Hom}_{A⊗_{k}A^{0}}(M ⊗_{B} M^{∨}, Σ^{n−1}(A)) \ar[d] \\ Hom_{A⊗_{k}A^{0}}(Σ^{n}(A), A)^{∨} \ar[r] & Hom_{A⊗_{k}A^{0}}(Σ^{n}(A), M ⊗_{B} M^{∨})^{∨}}

where the horizontal maps are induced by (pre-)composing with the adjunction counit M ⊗_{B} M^{∨} → A. The commutativity of the diagram 6.2 follows from the naturality of Tate duality. Consider next the diagram

6.3.

\xymatrix{\underline{Hom}_{A⊗_{k}A^{0}}(M ⊗_{B} M^{∨}, Σ^{n−1}(A)) \ar[r] \ar[d] & \underline{Hom}_{B⊗_{k}A^{0}}(M^{∨}, Σ^{n−1}(M^{∨})) \ar[d] \\ Hom_{A⊗_{k}A^{0}}(Σ^{n}(A), M ⊗_{B} M^{∨})^{∨} \ar[r] & Hom_{B⊗_{k}A^{0}}(Σ^{n}(M^{∨}), M^{∨})^{∨}}

where the horizontal maps are adjunction isomorphisms, modulo identifying M^{∨} ⊗_{A} Σ^{n−1}(A) ≅ Σ^{n−1}(M^{∨}) and M^{∨} ⊗_{A} Σ^{n}(A) ≅ Σ^{n}(M^{∨}). The commutativity of 6.3 is a special case of the compatibility 4.4 of Tate duality with adjunction. Similarly, the analogous version of 4.4 for the adjoint pair (− ⊗_{A} M, − ⊗_{B} M^{∨}) yields the commutativity of the diagram 6.4.

\xymatrix{\underline{Hom}_{B⊗_{k}A^{0}}(M^{∨}, Σ^{n−1}(M^{∨})) \ar[r] \ar[d] & \underline{Hom}_{B⊗_{k}B^{0}}(B, Σ^{n−1}(M^{∨}) ⊗_{A} M) \ar[d] \\ Hom_{B⊗_{k}A^{0}}(Σ^{n}(M^{∨}), M^{∨})^{∨} \ar[r] & Hom_{B⊗_{k}B^{0}}(Σ^{n}(M^{∨}) ⊗_{A} M, B)^{∨}}

where the the horizontal maps are adjunction isomorphisms. Finally, consider the diagram

6.5.

\xymatrix{\underline{Hom}_{B⊗_{k}B^{0}}(B, Σ^{n−1}(M^{∨}) ⊗_{A} M) \ar[r] \ar[d] & \underline{Hom}_{B⊗_{k}B^{0}}(B, Σ^{n−1}(B)) \ar[d] \\ Hom_{B⊗_{k}B^{0}}(Σ^{n}(M^{∨}) ⊗_{A} M, B)^{∨} \ar[r] & \underline{Hom}_{B⊗_{k}B^{0}}(Σ^{n}(B), B)}

where the horizontal maps are induced by composition with the adjunction counit M^{∨} ⊗_{A} M → B, shifted by Σ^{n−1} or Σ^{n} as appropriate. The commutativity of 6.5 follows again from the naturality of Tate duality 2.3. Concatenating the four diagrams 6.2, 6.3, 6.4, and 6.5 horizontally yields the commutativity of the diagram 6.1, which completes the proof of Theorem 1.1.

7 Proof of Theorem 1.2

We use the notation from the statement of Theorem 1.2. It suffices to show the commutativity of the second of the two squares in the diagram in the statement of 1.2, since the first is obtained by duality, thanks to the fact that applying Tate duality twice yields the canonical double duality. After replacing W by Σ^{n}(W), it suffices to show the commutativity of the diagram

7.1.

\xymatrix{\underline{Hom}_{A}(M ⊗_{B} V, M ⊗_{B} Ω(W)) \ar[r]^{^{tr^{M^{∨}}}} \ar[d] & \underline{Hom}_{B}(V, Ω(W)) \ar[d] \\ \underline{Hom}_{A}(M ⊗_{B} W, M ⊗_{B} V)^{∨} \ar[r]_{(M⊗_{B}−)^{∨}} & \underline{Hom}_{B}(W, V)^{∨}}

where the vertical maps are versions of the Tate duality isomorphism 2.3. Lemma 5.2 describes the map tr_{M^{∨}} as a composition of two maps, and hence the commutativity of this diagram will be established by combining the following two diagrams. By 4.3, we have a commutative diagram

7.2.

\xymatrix{\underline{Hom}_{A}(M ⊗_{B} V, M ⊗_{B} Ω(W)) \ar[r]^{≅} \ar[d] & \underline{Hom}_{B}(V, M^{∨} ⊗_{A} M ⊗_{B} Ω(W)) \ar[d] \\ \underline{Hom}_{A}(M ⊗_{B} W, M ⊗_{B} V)^{∨} \ar[r]_{≅} & \underline{Hom}_{B}(M^{∨} ⊗_{A} M ⊗_{B} W, V)^{∨}}

where the horizontal isomorphisms are adjunction isomorphisms, and the vertical isomorphisms are Tate duality isomorphisms. Using the naturality of Tate duality applied with the couint η_{M^{∨}} tensored by either Id_{W} or Id_{Ω(W)} yields a commutative diagram 7.3.

\xymatrix{\underline{Hom}_{B}(V, M^{∨} ⊗_{A} M ⊗_{B} Ω(W) \ar[r] \ar[d] & \underline{Hom}_{B}(V, Ω(W)) \ar[d] \\ \underline{Hom}_{B}(M^{∨} ⊗_{A} M ⊗_{B} W, V)^{∨} \ar[r] & \underline{Hom}_{B}(W, V)^{∨}}

Concatenating the two diagrams 7.2 and 7.3 yields the diagram 7.1, where we use the description of tr_{M^{∨}} from Lemma 5.2. This proves Theorem 1.2.

8 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 Êxt_{A}^{n−1}(V, U), then there is a nonzero element η in Êxt_{A}^{−n}(U, V) such that the Yoneda product ζη is nonzero in Êxt_{A}^{−1}(U, U).

Proof. By Tate duality, if ζ is nonzero in Êxt_{A}^{n−1}(V, U), then there is τ ∈ Ext_{A}^{−n}(U, V) such that

⟨ζ, τ⟩ ≠ 0. Denote by ι_{U} the image of Id_{U} in \underline{End}_{A}(U) = Êxt_{A}^{0}(U, U). Applying the appropriate version of 2.7 shows that ⟨ζη, ι_{U}⟩ = ⟨ζ, η⟩ ≠ 0, hence in particular, ζη ≠ 0.

Lemma 8.2. Let U, V, W be finitely generated A-modules, and let m, n be integers. Let ζ ∈ Êxt_{A}^{n−1}(V, U) and η ∈ Êxt_{A}^{m−1}(W, V) such that the Yoneda product ζη is nonzero in Êxt_{A}^{m+n−2}(W, U). Then there is τ ∈ Êxt_{A}^{−m−n+1}(U, W) such that the Yoneda product ητ is nonzero in Êxt_{A}^{−n}(U, V).

Proof. By Lemma 8.1 there is τ such that ζητ is nonzero in Êxt_{A}^{−1}(U, U). Then necessarily ητ is nonzero, whence the result.

For U, V two A-modules, we denote by \overline{Ext}_{A}^{∗}(U, V) the nonnegative part of Êxt_{A}^{∗}(U, V). That is, for n > 0 we have \overline{Ext}_{A}^{n}(U, V) = Êxt_{A}^{n}(U, V) = Ext_{A}^{n}(U, V), for n < 0 we have \overline{Ext}_{A}^{n}(U, V) = Ext_{A}^{n}(U, V) = \{0\}, and \overline{Ext}_{A}^{0}(U, V) = Êxt_{A}^{0}(U, V) = \underline{Hom}_{A}(U, V), while Ext_{A}^{0}(U, V) = Hom_{A}(U, V).

Proposition 8.3. Let U be a finitely generated A-module such that \overline{Ext}_{A}^{∗}(U, U) is graded-commutative. Suppose that there are negative integers m, n such that Êxt_{A}^{m}(U, U) · Êxt_{A}^{n}(U, U) ≠ 0. Then \overline{Ext}_{A}^{∗}(U, U) has depth at most one.

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 ∈ Êxt_{A}^{m}(U, U) and η ∈ Êxt_{A}^{n}(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 ∈ Êxt_{A}^{−m−n−1}(U, U) such that ητ ≠ is nonzero in Êxt_{A}^{−m−1}(U, U). Note that since m is negative, we have −m − 1 ≥ 0. Since ζ_{1} is not a zero divisor, we have ζ_{1}^{a}ζη ≠ 0, hence ζ_{1}^{a}ζ ≠ 0 for all nonnegative integers a. Choose a maximal such that deg(ζ_{1}^{a}ζ) < 0. Then 0 ≤ deg(ζ_{1}^{a+1}ζ) < d_{1}. In particular, ζ^{a+1}ζ is contained in \overline{Ext}_{A}^{∗}(U, U) but not in the ideal ζ_{1}\overline{Ext}_{A}^{∗}(U, U). Thus the image of ζ_{1}^{a+1}ζ in the quotient \overline{Ext}_{A}^{∗}(U, U)/ζ_{1}\overline{Ext}_{A}^{∗}(U, U) is non zero. However, for some sufficiently large integer b we have ζ_{2}^{b}ζ_{1}^{a+1}ζ = ζ_{1}(ζ_{1}^{a}ζζ_{2}^{b}), hence ζ_{2} is not regular on this quotient. This contradiction shows that \overline{Ext}_{A}^{∗}(U, U) has no regular sequence of length two.

Since the Hochschild cohomology of an algebra is graded-commutative we obtain the following consequence:

Corollary 8.4. Suppose that there are negative integers m, n such that ĤH^{m}(A) · ĤH^{n}(A) ≠ 0. Then \overline{HH}^{∗}(A) has depth at most one.

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