PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 5, May 1998, Pages 1345–1348 S 0002-9939(98)04210-5

A RELATION BETWEEN HOCHSCHILD HOMOLOGY AND COHOMOLOGY FOR GORENSTEIN RINGS

MICHEL VAN DEN BERGH

(Communicated by Lance W. Small)

Abstract. Let “HH” stand for Hochschild (co)homology. In this note we show that for many rings A there exists d ∈ ℕ such that for an arbitrary Abimodule N we have HH^{i}(N) = HH_{d−i}(N). Such a result may be viewed as an analog of Poincaré duality.

Combining this equality with a computation of Soergel allows one to compute the Hochschild homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo.

In the sequel the base field will be denoted by k. Let 𝔤 be a semisimple Lie algebra and let A be a regular minimal primitive quotient of U(𝔤). The Hochschild cohomology of A was computed by Soergel in [6] and shown to be equal to the cohomology of the corresponding flag variety. Soergel’s computation is rather ingenious and makes use of the Bernstein-Beilinson theorem together with the Riemann-Hilbert correspondence. The case of singular A is still open.

After Soergel’s result Patrick Polo asked whether perhaps the Hochschild homology of A also coincided with the homology of the underlying flag manifold. It is indeed rather likely that Soergel’s techniques can be adapted to this end.

In this note we answer Polo’s question with a different method. Indeed we show, using elementary homological algebra, that for many algebras A (in particular those considered above) we actually have

HH^{i}(N) = HH_{d−i}(N) (1)

for some d ∈ ℕ and for an arbitrary A-bimodule N (see Corollary 6). This can be considered a kind of Poincaré duality. In more general situations (1) may remain true provided we twist N on the right hand side by an automorphism or an invertible bimodule (see Theorem 1).

(1) may be used to complete the results in [7] where the Hochschild homology of a generic three dimensional type A regular algebra was computed (see [1] for terminology). Since we show below that (1) holds for such a algebras, we now also obtain the Hochschild cohomology without any extra work.

Our basic result is the following theorem, which is easily proved.

Theorem 1. Assume that A is an algebra such that there exists a d ∈ ℕ with the property that Ext_{A^{e}}^{i}(A, A^{e}) = 0 unless i = d. Put U = Ext_{A}(A, A^{e}), and assume

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors November 5, 1996.

1991 Mathematics Subject Classification. Primary 16E40.

Key words and phrases. Hochschild homology, Gorenstein rings.

The author is a senior researcher at the NFWO.

in addition that U is an invertible A-bimodule. Then for every A-bimodule N we have

HH^{i}(N) = HH_{d−i}(U ⊗_{A} N). (2)

Proof. To prove this it is convenient to make use of the derived category. We have

HH^{i}(N) = H^{i}(RHom_{A^{e}}(_{A^{e}}A,_{A^{e}}N)) = H^{i}(RHom_{A^{e}}(_{A^{e}}A, A^{e}) ⊗^{L}_{A^{e}} N)

= H^{i}(U[−d] ⊗^{L}_{A^{e}} N) = H^{i−d}(U ⊗^{L}_{A^{e}} N)

= H^{i−d}(A ⊗^{L}_{A^{e}} (U ⊗^{L}_{A} A)) = HH_{d−i}(U ⊗_{A} N).

We now give several situations in which the theorem can be applied. First from [8] we recall some results concerning Koszul algebras.

Proposition 2. Let A be a (graded) Koszul algebra of global dimension d such that A^{!} is Frobenius. Then the hypotheses of Theorem 1 are satisfied, and furthermore we have the following precise description of U.

Let ϕ^{!} be the graded automorphism of A^{!} such that (A^{!})^{∗} = A_{ϕ^{!}}^{!}. Furthermore let ϕ be the automorphism of A which is adjoint to ϕ^{!}. Finally let ϵ be the automorphism of A which is multiplication by (−)^{m} on A_{m}. Then

U = A_{ϕ^{−1}ϵ^{d+1}}.

In particular, (2) specializes to

HH^{i}(N) = HH_{d−i}(_{ϕϵ^{d+1}}N).

The simplest non-trivial Koszul algebras of finite global dimension are those of global dimension three. These have been studied extensively in [1, 2, 3]. Recall that they have a presentation k⟨x_{1}, x_{2}, x_{3}⟩/(f_{1}, f_{2}, f_{3}) in which x = (x_{1}, x_{2}, x_{3})^{t}, f = (f_{1}, f_{2}, f_{3})^{t} can be chosen in such a way that f = Mx, x^{t}M = (Qf)^{t} for 3 × 3 matrices M and Q, respectively with linear an scalar entries. It is shown in [8] that in this case the automorphism ϕ is given in terms of the generators by x ↦ Q^{t}x. In particular if A is a type A algebra (see [1]), then (1) holds with d = 3.

Let us now exhibit another situation in which Theorem 1 applies and which occurs frequently in practice.

Proposition 3. Let A be a k-algebra equipped with a k-linear ascending filtration (F_{n}A)_{n≥0} such that gr_{F} A is commutative of finite type over k. Assume that gr A is Gorenstein of dimension d. Then the hypotheses of Theorem 1 hold, and furthermore U carries an A-bimodule filtration such that gr U is equal to ω_{gr A/k}^{−1}, the inverse of the dualizing module of gr A.

Proof. We use the standard spectral sequence

(3) Ext_{(gr A)^{e}}^{i}(gr A, (gr A)^{e}) ⇒ Ext_{A^{e}}^{i}(A, A^{e}),

which converges because we have positive filtrations.

Let C is a commutative Cohen-Macaulay ring of dimension d. From the adjunction formula one has

RHom_{C^{e}}(C, ω_{C^{e}}) = ω_{C}[−d].

Assume now that C is Gorenstein. Then ω_{C^{e}} = ω_{C} ⊗_{k} ω_{C} and ω_{C} are invertible, and hence

RHom_{C^{e}}(C, ω_{C^{e}}) = RHom_{C^{e}}(C, C^{e}) ⊗_{C^{e}} ω_{C^{e}}.

Combining this yields

RHom_{C^{e}}(C, C^{e}) = ω_{C}[−d] ⊗_{C^{e}} ω_{C^{e}}^{−1} = ω_{C}^{−1}[−d].

Applying this with C = gr A shows that (3) degenerates, and we obtain

RHom_{A^{e}}(A, A^{e}) = U[−d]

where U lives in one degree and where, from a functoriality argument using (3), one obtains that the filtration on U is an A^{e} filtration, hence an A bimodule filtration. Furthermore for this filtration, gr U = ω_{C}^{−1}. It now suffices to apply lemma 4 below to deduce that U is invertible.

Lemma 4. Assume that A is a ring equipped with a positive ascending filtration, and let U be an A-bimodule, equipped with a left limited A-bimodule filtration. If gr U is an invertible gr A-bimodule, then U is an invertible A-bimodule.

Proof. We have to show [4, §1] that U is left and right projective and that the canonical maps

(4)  A → End_{A}(_{A}U),
(5)  A → End_{A}(A_{U})

are surjective. The fact that U is projective is standard (see [5, Cor. D.VII.6]). Therefore let us concentrate on (4). (5) is similar. We use the standard spectral sequence

Ext^{i}(gr(_{A}U), gr(_{A}U)) ⇒ Ext_{A}^{i}(U, U)

which is compatible with (4) and which obviously degenerates. From this we deduce that

gr End_{A}(_{A}U) = End_{A}(gr(_{A}U)).

Combining this with the hypotheses that gr U is invertible yields that

gr A → gr End_{A}(_{A}U)

is surjective. Hence (4) is also surjective.

We use the following lemma below.

Lemma 5. Let the notation be as in the previous lemma. Assume that gr U is left free of rank one and let θ be the automorphism of gr A such that gr U = (gr A)_{θ}. Then there exists a filtered automorphism ψ of A such that U = A_{ψ} and gr ψ = θ.

Proof. Let ū be a generator of gr U such that

ūy = θ(y)ū (6)

and let u be a lifting to U. It is easy to see that u is a basis for U. Thus there is an automorphism ψ of A such that ux = ψ(x)u. If we take the image of this equality in gr A, we find that ūx̄ = \overline{ψ(x)}ū. Comparing with (6) yields θ(x̄) = \overline{ψ(x)}. From this we easily deduce that ψ is filtered, and gr ψ = θ.

Corollary 6. Assume that A is equipped with a filtration (F_{n}A)_{n∈ℕ} such that F_{0}A = k and such that gr A is commutative and generated by gr_{1} A. Put 𝔤 = gr_{1} A with Lie algebra structure deduced from the commutator on A. Assume furthermore that gr A is Gorenstein and [𝔤, 𝔤] = 𝔤. Then

HH^{i}(N) = HH_{d−i}(N).

Proof. Since gr A is graded local, we have ω_{A} = A, as A-bimodules, and hence we know from lemma 5 that H^{i}(N) = H_{d−i}(_{ψ^{−1}}N) for some filtered automorphism of A such that gr ψ = id_{gr A}. Let us consider 𝔤 as a subspace of F_{1}A. For x, y ∈ 𝔤 we have

yx − xy = [y, x] + α (7)

for some α ∈ k. We also have ψ(x) = x + β(x), where β ∈ 𝔤^{∗}. Applying ψ to (7) yields

yx − xy = [y, x] + β([y, x]) + α

and thus β | [𝔤, 𝔤] = 0. Hence β = 0, and we are done.

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Departement WNI, Limburgs Universitair Centrum, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium

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