ON THE HOCHSCHILD (CO)HOMOLOGY OF QUANTUM EXTERIOR ALGEBRAS

PETTER ANDREAS BERGH

Abstract. We compute the Hochschild cohomology and homology of the algebra Λ = k⟨x, y⟩/(x^{2}, xy + qyx, y^{2}) with coefficients in _{1}Λ_{ψ} for every degree preserving k-algebra automorphism ψ: Λ → Λ. As a result we obtain several interesting examples of the homological behavior of Λ as a bimodule.

Introduction

Throughout this paper, let k be a field and q ∈ k a nonzero element which is not a root of unity. Denote by Λ the k-algebra

Λ = k⟨x, y⟩/(x^{2}, xy + qyx, y^{2}),

and by Λ^{e} its enveloping algebra Λ^{op} ⊗_{k} Λ. All modules considered are assumed to be right modules.

During the last years, this 4-dimensional graded Koszul algebra, whose module category was classified in [7], has provided several examples (or rather counterexamples) giving negative answers to homological conjectures and questions. Among these are the conjecture of Auslander on local Ext-limitations (see [1, page 815], [6] and [8]) and the question of Happel on the relation between the global dimension and the vanishing of the Hochschild cohomology (see [5] and [2]).

We shall study the Hochschild cohomology and homology of Λ. More precisely, for every degree preserving k-algebra automorphism ψ: Λ → Λ we compute HH^{∗}(Λ,_{1}Λ_{ψ}) = Ext_{Λ^{e}}^{∗}(Λ,_{1}Λ_{ψ}) and HH_{∗}(Λ,_{1}Λ_{ψ}) = Tor_{∗}^{Λ^{e}}(Λ,_{1}Λ_{ψ}), that is, the Hochschild cohomology and homology of Λ with coefficients in the twisted bimodule _{1}Λ_{ψ} (the action of Λ^{e} on _{1}Λ_{ψ} is defined as λ(λ_{1} ⊗ λ_{2}) = λ_{1}λψ(λ_{2})). As a result we obtain several interesting examples, both in cohomology and homology, of the homological behavior of Λ as a bimodule.

1. The Hochschild Homology

Denote by D the usual k-dual Hom_{k}(−, k), and consider the map ϕ: _{Λ} Λ → D(Λ_{Λ}) of left Λ-modules defined by

ϕ(1)(α + βx + γy + δyx) =^{def} δ.

It is easy to show that this is an injective map and hence also an isomorphism since dim_{k} Λ = dim_{k} D(Λ), and therefore Λ is a Frobenius algebra by definition. Now take any element λ ∈ Λ, and consider the element ϕ(1) · λ ∈ D(Λ) (we consider D(Λ) as a Λ-Λ-bimodule). As ϕ is surjective, there is an element λ′ ∈ Λ such that λ′ · ϕ(1) = ϕ(λ′) = ϕ(1) · λ, and the map λ ↦ λ′ defines a k-algebra automorphism ν^{−1} : Λ → Λ whose inverse ν is called the Nakayama automorphism of Λ (with respect to the map ϕ). Straightforward calculations show that x · ϕ(1) = ϕ(1) · (−q^{−1}x) and y · ϕ(1) = ϕ(1) · (−qy), hence since x and y generate Λ over k we see that ν is the degree preserving map defined by

2000 Mathematics Subject Classification. 16E05, 16E40, 16P10.

x ↦ −q^{−1}x, y ↦ −qy.

The map ϕ induces a bimodule isomorphism _{1} Λ_{ν^{−1}} ≃ D(Λ), which in turn gives an isomorphism _{(νψ)^{−1}} Λ_{ν^{−1}} ≃_{(νψ)^{−1}} D(Λ)_{1} for any automorphism ψ of Λ. Furthermore, since _{(νψ)^{−1}} Λ_{ν^{−1}} is isomorphic to _{1} Λ_{ψ} and _{(νψ)^{−1}}D(Λ)_{1} = D(_{1}Λ_{(νψ)^{−1}}), we get an isomorphism _{1} Λ_{ψ} ≃ D(_{1}Λ_{(νψ)^{−1}}) of bimodules. Now from [3, Proposition VI.5.1] we get

HH^{n}(Λ,_{1}Λ_{ψ}) = Ext_{Λ^{e}}^{n}(Λ,_{1}Λ_{ψ})
≃ Ext_{Λ^{e}}^{n} (Λ, D(_{1}Λ_{(νψ)^{−1}}))
≃ D (Tor_{n}^{Λ^{e}}(Λ,_{1} Λ_{(νψ)^{−1}}))
= D (HH_{n}(Λ,_{1} Λ_{(νψ)^{−1}})) ,

thus when computing the (dimension of the) Hochschild cohomology group HH^{n}(Λ,_{1}Λ_{ψ}) we are also computing the Hochschild homology group HH_{n}(Λ,_{1} Λ_{(νψ)^{−1}}). Moreover, as ψ ranges over all degree preserving k-algebra automorphisms of Λ, so does (νψ)^{−1}.

2. The Cohomology Complex

We start by recalling the construction of the minimal bimodule projective resolution of Λ from [2]. Define the elements

f_{0}^{0} = 1, f_{0}^{1} = x, f_{1}^{1} = y,
f_{−1}^{n} = 0 = f_{n+1}^{n} for each n ≥ 0,

and for each n ≥ 2 define elements \{f_{i}^{n}\}_{i=0}^{n} ⊆ \underbrace{Λ ⊗_{k} ⋯ ⊗_{k} Λ}_{n copies} inductively by

f_{i}^{n} = f_{i−1}^{n−1} ⊗ y + q^{i}f_{i}^{n−1} ⊗ x.

Denote by P^{n} the Λ^{e}-projective module ⨁_{i=0}^{n} Λ ⊗_{k}f_{i}^{n} ⊗_{k} Λ, and by f̃_{i}^{n} the element 1 ⊗ f_{i}^{n} ⊗ 1 ∈ P^{n} (and f̃_{0}^{0} = 1 ⊗ 1). The set {f̃_{i}^{n}}_{i=0}^{n} generates P^{n} as a Λ^{e}-module. Now define a map δ_{n}: P^{n} → P^{n−1} by

f̃_{i}^{n} ↦ [xf̃_{i}^{n−1} + (−1)^{n}q^{i}f̃_{i}^{n−1}x] + [q^{n−i}yf̃_{i−1}^{n−1} + (−1)^{n}f̃_{i−1}^{n−1}y] .

It is shown in [2] that

(ℙ, δ): ⋯ → P^{n+1} ⟶^{δ_{n+1}} P^{n} ⟶^{δ_{n}} P^{n−1} → ⋯

is a minimal Λ^{e}-projective resolution of Λ. Denote the direct sum of n copies of _{1}Λ_{ψ} by _{1}Λ_{ψ}^{n}, and consider its standard k-basis {e_{i}^{n−1}, xe_{i}^{n−1}, ye_{i}^{n−1}, yxe_{i}^{n−1}}_{i=0}^{n−1}. Define a map d_{n}: _{1}Λ_{ψ}^{n} → _{1}Λ_{ψ}^{n+1} by

λe_{i}^{n−1} ↦ [xλ + (−1)^{n}q^{i}λψ(x)] e_{i}^{n} + [q^{n−i−1}yλ + (−1)^{n}λψ(y)] e_{i+1}^{n}.

Applying Hom_{Λ^{e}}(−,_{1}Λ_{ψ}) to the resolution (ℙ, δ), keeping in mind that Hom_{Λ^{e}}(P^{n},_{1}Λ_{ψ}) and _{1}Λ_{ψ}^{n+1} are isomorphic as k-vector spaces, we get the commutative diagram

\xymatrix{⋯ \ar[r] & Hom_{Λ^{e}}(P^{n−1},_{1}Λ_{ψ}) \ar[d]^{≀} \ar[r]^{δ_{n}^{∗}} & Hom_{Λ^{e}}(P^{n},_{1}Λ_{ψ}) \ar[d]^{≀} \ar[r]^{δ_{n+1}^{∗}} & ⋯ \\ ⋯ \ar[r] & _{1}Λ_{ψ}^{n} \ar[r]^{d_{n}} & _{1}Λ_{ψ}^{n+1} \ar[r]^{d_{n+1}} & ⋯}

of k-vector spaces.

In order to compute HH^{n}(Λ,_{1}Λ_{ψ}) = Ext_{Λ^{e}}^{n}(Λ,_{1}Λ_{ψ}) for n > 0 we compute the cohomology Ker d_{n+1}/ Im d_{n} of the bottom complex in the above commutative diagram. We do this by finding dim_{k} Im d_{n}; once we know dim_{k} Im d_{n}, we obtain dim_{k} Ker d_{n} (and therefore also dim_{k} Ker d_{n+1}) from the equation

dim_{k} Ker d_{n} + dim_{k} Im d_{n} = dim_{k} _{1}Λ_{ψ}^{n} = 4n.

We then have

dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = dim_{k} Ker d_{n+1} − dim_{k} Im d_{n}.

Now let ψ be a degree preserving k-algebra automorphism of Λ. Then there are elements α_{1}, α_{2}, β_{1}, β_{2} ∈ k such that ψ(x) = α_{1}x+α_{2}y and ψ(y) = β_{1}x+β_{2}y. Since ψ(x^{2}) = ψ(y^{2}) = ψ(xy + qyx) = 0, we have the relations α_{1}α_{2} = β_{1}β_{2} = α_{2}β_{1} = 0. If α_{2} ≠ 0, then α_{1} = β_{1} = 0, implying x ∉ Im ψ. Similarly, if β_{1} ≠ 0, then α_{2} = β_{2} = 0, implying y ∉ Im ψ. Therefore α_{2} = β_{1} = 0, and this forces α_{1} and β_{2} to be nonzero. Thus the degree preserving k-algebra automorphisms of Λ are precisely those defined by

x ↦ αx, y ↦ βy

for two arbitrary nonzero elements α, β ∈ k. For such an automorphism, the result of applying d_{n} to the basis vectors {e_{i}^{n−1}, xe_{i}^{n−1}, ye_{i}^{n−1}, yxe_{i}^{n−1}}_{i=0}^{n−1} of _{1}Λ_{ψ}^{n} is

e_{i}^{n−1} ↦ [1 + (−1)^{n}q^{i}α] xe_{i}^{n} + [q^{n−i−1} + (−1)^{n}β] ye_{i+1}^{n}
xe_{i}^{n−1} ↦ [q^{n−i−1} + (−1)^{n+1}qβ] yxe_{i+1}^{n}
ye_{i}^{n−1} ↦ [−q + (−1)^{n}q^{i}α] yxe_{i}^{n}
yxe_{i}^{n−1} ↦ 0

for 0 ≤ i ≤ n − 1. Note that the inequality dim_{k} Im d_{n} ≤ 2n + 1 always holds.

3. The Hochschild Cohomology

We start by computing HH^{0}(Λ,_{1}Λ_{ψ}). Rather than computing this vector space directly using the identifications

HH^{0}(Λ,_{1}Λ_{ψ}) = {z ∈ _{1}Λ_{ψ} | λ · z = z · λ for all λ ∈ Λ}
= {z ∈ _{1}Λ_{ψ} | λz = zψ(λ) for all λ ∈ Λ}
= {z ∈ _{1}Λ_{ψ} | xz = zψ(x), yz = zψ(y), yxz = zψ(yx)}
= {z ∈ _{1}Λ_{ψ} | xz = αzx, yz = βzy, yxz = αβzyx},

we use our cohomology complex and the isomorphism HH^{0}(Λ,_{1}Λ_{ψ}) ≃ Ker d_{1}. From the above we see that the map d_{1} is defined by

e_{0}^{0} ↦ [1 − α]xe_{0}^{1} + [1 − β]ye_{1}^{1}
xe_{0}^{0} ↦ [1 + qβ]yxe_{1}^{1}
ye_{0}^{0} ↦ −[q + α]yxe_{0}^{1}
yxe_{0}^{0} ↦ 0,

and so calculation gives

dim_{k} HH^{0}(Λ,_{1}Λ_{ψ}) = \{ \begin{matrix}3 & when α = −q, β = −q^{−1} \\ 2 & when α = 1, β = 1 \\ 2 & when α = −q, β ≠ −q^{−1} \\ 2 & when α ≠ −q, β = −q^{−1} \\ 1 & otherwise \end{matrix}

when the characteristic of k is not 2. In the characteristic 2 case we replace −q, −q^{−1} and 1 in the above formula by ±q, ±q^{−1} and ±1, respectively.

Now we turn to the cohomology groups HH^{n}(Λ,_{1}Λ_{ψ}) for n > 0. To compute the dimension of Im d_{n}, we distinguish between four possible cases depending on whether or not α and β belong to the set

Σ = {±q^{i}}_{i∈ℤ}.

3.1. The case α, β ∉ Σ:

This is the easiest case; d_{n}(e_{i}^{n−1}), d_{n}(xe_{i}^{n−1}) and d_{n}(ye_{i}^{n−1}) are all nonzero for 0 ≤ i ≤ n − 1, hence dim_{k} Im d_{n} = 2n + 1 for all n. Then dim_{k} Ker d_{n} = 2n − 1, implying dim_{k} Ker d_{n+1} = 2n + 1 and therefore that HH^{n}(Λ,_{1}Λ_{ψ}) = 0 for n > 0.

Remark. We can relate the vanishing of cohomology to the conjecture of Tachikawa stating that over a selfinjective ring the only finitely generated modules having no self extensions are the projective ones. Namely, let M be a finitely generated Λmodule such that Λ has no bimodule extensions by Hom_{k}(M, M). Then from [3, Corollary IX.4.4] we get

Ext_{Λ}^{n}(M, M) ≃ HH^{n}(Λ, Hom_{k}(M, M)) = 0

for n > 0. Since Tachikawa’s conjecture holds for Λ (see [7, Proposition 4.2]), the module M must be projective and therefore (Λ is local) isomorphic to Λ^{t} for some t ∈ ℕ. This gives

Hom_{k}(M, M) ≃ (Λ ⊗_{k}D(Λ))^{t^{2}} ≃ (Λ^{e})^{t^{2}}.

In particular, if HH^{n}(Λ,_{1}Λ_{ψ}) = 0 for n > 0 then there cannot exist a Λ-module M such that Hom_{k}(M, M) is isomorphic to _{1}Λ_{ψ}, since this would imply the contradiction _{1}Λ_{ψ} ≃ (Λ^{e})^{t^{2}}.

3.2. The case α ∈ Σ, β ∉ Σ:

Since [q^{n−i−1} + (−1)^{n}β] and [q^{n−i−1} + (−1)^{n+1}qβ] are nonzero, we have d_{n}(e_{i}^{n−1}) ≠ 0 and d_{n}(xe_{i}^{n−1}) ≠ 0 for 0 ≤ i ≤ n−1. Therefore dim_{k} Im d_{n} ≥ 2n, and the problem is now whether or not the basis vector yxe_{0}^{n} belongs to Im d_{n}. This is the case if and only if d_{n}(ye_{0}^{n−1}) ≠ 0, that is, if and only if

(C1)  −q + (−1)^{n}α ≠ 0

holds. We now break down this case into three cases.

(i) The case α ≠ ±q:

Since (C1) holds we have dim_{k} Im d_{n} = 2n + 1, and so HH^{n}(Λ,_{1}Λ_{ψ}) = 0 for n > 0.

(ii) The case α = q:

When the characteristic of k is not 2, the condition (C1) holds if and only if n is odd. Therefore

dim_{k} Im d_{n} = \{ \begin{matrix}2n & for n even \\ 2n + 1 & for n odd, \end{matrix}

and so

dim_{k} Ker d_{n+1} = \{ \begin{matrix}2n + 1 & for n even \\ 2n + 2 & for n odd. \end{matrix}

This gives dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 1 for n > 0.

When k is of characteristic 2 we see that (C1) never holds, hence dim_{k} Im d_{n} = 2n. Then dim_{k} Ker d_{n+1} = 2n + 2, giving dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 2 for n > 0.

(iii) The case α = −q:

As in the previous case, we get dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 1 for n > 0 when k is not of characteristic 2, and dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 2 for n > 0 in the characteristic 2 case.

Remark. From this case we obtain an example showing that symmetry in the vanishing of Ext over Λ^{e} does not hold. Namely, define ψ by

x ↦ q^{−1}x, y ↦ βy

for some element β not contained in Σ. For n > 0, case (i) above gives Ext_{Λ^{e}}^{n}(Λ,_{1}Λ_{ψ}) = 0, whereas from case (ii) we see that dim_{k} Ext_{Λ^{e}}^{n}(Λ,_{1}Λ_{ψ^{−1}}) is either 1 or 2, depending on the characteristic of k. Now it is easy to see that Ext_{Λ^{e}}^{n}(Λ,_{1}Λ_{ψ^{−1}}) is isomorphic to Ext_{Λ^{e}}^{n}(_{1}Λ_{ψ}, Λ).

3.3. The case α ∉ Σ, β ∈ Σ:

The algebra Λ is isomorphic to the algebra k⟨u, v⟩/(u^{2}, uv + q^{−1}vu, v^{2}) via the map

x ↦ v, y ↦ u,

hence this case is symmetric to the case α ∈ Σ, β ∉ Σ treated above. Namely, when β ≠ ±q^{−1} the result is as in (i), whereas when β = ±q^{−1} the result is as in (ii) and (iii).

3.4. The case α ∈ Σ, β ∈ Σ:

The basis vectors yxe_{0}^{n} and yxe_{n}^{n} can only be the image of ye_{0}^{n−1} and xe_{n−1}^{n−1}, respectively, whereas for 1 ≤ i ≤ n − 1 the basis vector yxe_{i}^{n} can be the image of both ye_{i}^{n−1} and xe_{i−1}^{n−1}. Therefore we break this case down into four cases, each depending on whether or not α = ±q and β = ±q^{−1}.

(i) The case α = ±q, β = ±q^{−1}:

We have

d_{n}(e_{i}^{n−1}) = [1 ± (−1)^{n}q^{i+1}] xe_{i}^{n} + [q^{n−i−1} ± (−1)^{n}q^{−1}] ye_{i+1}^{n},

and since i + 1 ≥ 1 the term [1 ± (−1)^{n}q^{i+1}] must be nonzero. Therefore d_{n}(e_{i}^{n−1}) ≠ 0. Applying d_{n} to xe_{i}^{n−1} and ye_{i}^{n−1} gives [q^{n−i−1} ± (−1)^{n+1}] yxe_{i+1}^{n} and [−q ± (−1)^{n}q^{i+1}] yxe_{i}^{n}, respectively, hence when the characteristic of k is not 2 we get

d_{n}(xe_{i}^{n−1}) = \{ \begin{matrix}0 & for i = n − 1, β = q^{−1}, n even \\ 0 & for i = n − 1, β = −q^{−1}, n odd \\ ≠ 0 & otherwise, \end{matrix}

d_{n}(ye_{i}^{n−1}) = \{ \begin{matrix}0 & for i = 0, α = q, n even \\ 0 & for i = 0, α = −q, n odd \\ ≠ 0 & otherwise. \end{matrix}

There are four possible pairs (α, β) to consider. If α = q and β = q^{−1}, then the above gives

dim_{k} Im d_{n} = \{ \begin{matrix}2n − 1 & for n even \\ 2n + 1 & for n odd, \end{matrix}

and therefore

dim_{k} Ker d_{n+1} = \{ \begin{matrix}2n + 1 & for n even \\ 2n + 3 & for n odd. \end{matrix}

This implies dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 2 for n > 0. Similar computation gives dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 2 for n > 0 also for the other three possible pairs.

When k is of characteristic 2 then

d_{n}(xe_{i}^{n−1}) = \{ \begin{matrix}0 & for i = n − 1 \\ ≠ 0 & otherwise, \end{matrix}
d_{n}(ye_{i}^{n−1}) = \{ \begin{matrix}0 & for i = 0 \\ ≠ 0 & otherwise, \end{matrix}

and so dim_{k} Im d_{n} = 2n − 1 for all n > 0. Consequently dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 4 for n > 0.

Remark. Note that, in this particular case, we have computed the (dimension of the) Hochschild homology HH_{∗}(Λ) = Tor_{∗}^{Λ^{e}}(Λ, Λ) of Λ. Namely, it follows from Section 1 that for each n > 0 the k-vector spaces HH^{n}(Λ,_{1} Λ_{ν^{−1}}) and D (HH_{n}(Λ)) are isomorphic, where ν is the Nakayama automorphism

x ↦ −q^{−1}x, y ↦ −qy

of Λ. Then ν^{−1} is defined by

x ↦ −qx, y ↦ −q^{−1}y,

hence _{1}Λ_{ν^{−1}} is precisely the sort of bimodule we have just considered in terms of Hochschild cohomology. Consequently dim_{k} HH_{n}(Λ) = 2 for n > 0 when the characteristic of k is not 2, whereas dim_{k} HH_{n}(Λ) = 4 for n > 0 in the characteristic 2 case.

(ii) The case α = ±q, β ≠ ±q^{−1}:

As in (i) the element d_{n}(e_{i}^{n−1}) is nonzero for 0 ≤ i ≤ n − 1. Moreover, since β ≠ ±q^{−1} the basis element yxe_{n}^{n} always lies in Im d_{n}, as does the basis elements yxe_{i}^{n} for 1 ≤ i ≤ n − 1. Therefore dim Im d_{n} ≥ 2n, and the question is whether or not yxe_{0}^{n} belongs to Im d_{n}, i.e. whether or not d_{n}(ye_{0}^{n−1}) is nonzero.

When the characteristic of k is not 2, then from (i) we see that

d_{n}(ye_{0}^{n−1}) = \{ \begin{matrix}0 & for α = q, n even \\ 0 & for α = −q, n odd \\ ≠ 0 & otherwise, \end{matrix}

and computation gives dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 1 for n > 0. However, when the characteristic of k is 2 then d_{n}(ye_{0}^{n−1}) = 0, giving dim_{k} Im d_{n} = 2n and consequently dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 2 for n > 0.

(iii) The case α ≠ ±q, β = ±q^{−1}:

Using the isomorphism Λ ≃ k⟨u, v⟩/(u^{2}, uv+q^{−1}vu, v^{2}) from the case α ∉ Σ, β ∈ Σ, we see that the present case is symmetric to the case (ii) above. Thus when the characteristic of k is not 2 then dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 1 for n > 0, whereas dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = 2 for n > 0 in the characteristic 2 case.

(iv) The case α ≠ ±q, β ≠ ±q^{−1}:

We now have α = ±q^{s} and β = ±q^{t} where s ∈ ℤ \ {1} and t ∈ ℤ \ {−1}, and therefore the basis elements yxe_{0}^{n} and yxe_{n}^{n} both lie in the image of d_{n}. To compute the dimension of Im d_{n}, we must find out when d_{n}(e_{i}^{n−1}) = 0 for some 0 ≤ i ≤ n−1 and when yxe_{i}^{n} ∉ Im d_{n} for some 1 ≤ i ≤ n − 1. We have

{d_{n}(e_{i}^{n−1}) = 0 for some 0 ≤ i ≤ n − 1}
(∗)  ⇕
{1 + (−1)^{n}q^{i}α = 0 and q^{n−i−1} + (−1)^{n}β = 0}

and

{yxe_{i}^{n} ∉ Im d_{n} for some 1 ≤ i ≤ n − 1}
(∗∗)  ⇕
{−q + (−1)^{n}q^{i}α = 0 and q^{n−i} + (−1)^{n+1}qβ = 0} ,

and when the characteristic of k is not 2 this happens precisely when we have the following:

(C2)  s ≤ 0, t ≥ 0, α = (−1)^{t−s}q^{s}, β = (−1)^{t−s}q^{t}.

In the characteristic 2 case we may relax this condition; in this case (∗) and (∗∗) occur precisely when we have the following:

(C3)  s ≤ 0, t ≥ 0.

However, both when the characteristic of k is not 2 and (C2) holds, and in the characteristic 2 case when (C3) holds, we see that (∗) occurs for n = t − s + 1, whereas (∗∗) occurs for n = t − s + 2.

Therefore, when the characteristic of k is not 2 the dimension of Im d_{n} is given by

dim_{k} Im d_{n} = \{ \begin{matrix}2n & when (C2) holds and n = t − s + 1 \\ 2n & when (C2) holds and n = t − s + 2 \\ 2n + 1 & otherwise, \end{matrix}

implying the dimension of Ker d_{n+1} is given by

dim_{k} Ker d_{n+1} = \{ \begin{matrix}2n + 2 & when (C2) holds and n = t − s \\ 2n + 2 & when (C2) holds and n = t − s + 1 \\ 2n + 1 & otherwise. \end{matrix}

Consequently , the dimension of HH^{n}(Λ,_{1}Λ_{ψ}) for n > 0 is given by

dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = \{ \begin{matrix}1 & when (C2) holds and n = t − s \\ 2 & when (C2) holds and n = t − s + 1 \\ 1 & when (C2) holds and n = t − s + 2 \\ 0 & otherwise. \end{matrix}

When the characteristic of k is 2, we obtain the exact same formulas, but with (C2) replaced by (C3).

Remark. (i) In the case considered above, we see that the cohomology is zero except possibly in three degrees, depending on what conditions s, t, α and β satisfy. As a consequence, we construct a counterexample to the following conjecture by Auslander (see [1, page 815]): if M is a finitely generated module over an Artin algebra Γ, then there exists a number n_{M} such that for any finitely generated module N we have

Ext_{Γ}^{i}(M, N) = 0 for i ≫ 0 ⇒ Ext_{Γ}^{i}(M, N) = 0 for i ≥ n_{M}.

The first counterexample to this conjecture appeared in [6], where the algebra considered was a finite dimensional commutative Noetherian local Gorenstein algebra. A counterexample over our algebra Λ = k⟨x, y⟩/(x^{2}, xy + qyx, y^{2}) was given in [8].

As for a counterexample using Hochschild cohomology, define for each natural number t an automorphism ψ: Λ → Λ by x ↦ q^{−t}x and y ↦ q^{t}y, and denote the bimodule _{1}Λ_{ψ} by M_{t}. Then condition (C2)/(C3) is satisfied (with s = −t, α = q^{−t} and β = q^{t}), and so

Ext_{Λ^{e}}^{n}(Λ, M_{t}) = \{ \begin{matrix}≠ 0 & for n = 2t + 2 \\ 0 & for n > 2t + 2. \end{matrix}

(ii) Even though the above conjecture of Auslander fails to hold in general, Auslander himself proved (unpublished, see [1, page 815]) that if the conjecture holds for the enveloping algebra Γ^{e} of a finite dimensional algebra Γ over a field, then the finitistic dimension

sup{pd_{Γ} X | X finitely generated Γ-module with pd_{Γ} X < ∞}

of Γ is finite. In view of the above remark, we see that the converse to this result does not hold; our algebra Λ, being selfinjective, trivially has finite finitistic dimension, whereas the conjecture of Auslander does not hold for Λ^{e}.

(iii) The computation of the Hochschild cohomology HH^{∗}(Λ) = Ext_{Λ^{e}}^{∗}(Λ, Λ) of Λ is covered by the last of the above cases. When ψ is the identity automorphism we have s = t = 0, and the condition (C2)/(C3) is satisfied. This gives

dim_{k} HH^{n}(Λ) = \{ \begin{matrix}2 & for n = 1 \\ 1 & for n = 2 \\ 0 & for n ≥ 3, \end{matrix}

and so our algebra Λ is a counterexample to the following question raised by Happel in [5]: if the Hochschild cohomology groups of a finite dimensional algebra vanish in high degrees, does the algebra have finite global dimension? The counterexample above first appeared in [2], where it was shown that the generating function ∑_{n=0}^{∞} HH^{n}(Λ)t^{n} of HH^{n}(Λ) is 2 + 2t + t^{2}.

The converse to the question of Happel is always true when the algebra modulo its Jacobson radical is separable over the ground field. More specifically, if Γ is a finite dimensional algebra over a field K with Jacobson radical 𝔯, and the semisimple algebra Γ/ 𝔯 is separable over K, then by [4, §3] the implication

gl. dim Γ < ∞ ⇒ pd_{Γ^{e}} Γ < ∞

holds. In particular the Hochschild cohomology groups HH^{n}(Γ) = Ext_{Γ^{e}}^{n}(Γ, Γ) and the homology groups HH_{n}(Γ) = Tor_{n}^{Γ^{e}}(Γ, Γ) of Γ vanish for n ≫ 0 when the global dimension of Γ is finite. It is not known whether the vanishing of the Hochschild homology groups in high degrees for a finite dimensional algebra implies the global dimension of the algebra is finite.

(iv) Because of the equality dim_{k} HH^{n}(Λ,_{1}Λ_{ψ}) = dim_{k} HH_{n}(Λ,_{1} Λ_{(νψ)^{−1}}) which follows from Section 1, the somewhat strange behavior in Hochschild cohomology revealed in the last case considered above can also be transferred to Hochschild homology. When the automorphism ψ is given by ψ(x) = ±q^{s}x and ψ(y) = ±q^{t}y, then the automorphism θ =^{def} (νψ)^{−1}, where ν is the Nakayama automorphism, is given by

x ↦ ∓q^{1−s}x, y ↦ ∓q^{−(t+1)}y.

Thus for such an automorphism θ, when s ∈ ℤ \ {1} and t ∈ ℤ \ {−1} we get

dim_{k} HH_{n}(Λ,_{1}Λ_{θ}) = \{ \begin{matrix}1 & when (C2)/(C3) holds and n = t − s \\ 2 & when (C2)/(C3) holds and n = t − s + 1 \\ 1 & when (C2)/(C3) holds and n = t − s + 2 \\ 0 & otherwise. \end{matrix}

when n > 0. In these formulas condition (C2) applies when the characteristic of k is not 2, and condition (C3) applies in the characteristic 2 case.

Acknowledgement

I would like to thank my supervisor Øyvind Solberg for valuable suggestions and comments.

References

[1] Auslander, M. (1999). Selected works, part 1. Reiten, Smalø, Solberg (editors). American Mathematical Society.

[2] Buchweitz, R.-O., Green, E., Madsen, D., Solberg, Ø. (2005). Finite Hochschild cohomology without finite global dimension. Math. Res. Lett. 12(5-6):805-816.

[3] Cartan, H., Eilenberg, S. (1956). Homological algebra. Princeton University Press.

[4] Eilenberg, S., Ikeda, M., Nakayama, T. (1955). On the dimension of modules and algebras, I. Nagoya Math. J. 8:49-57.

[5] Happel, D. (1989). Hochschild cohomology of finite-dimensional algebras. In: Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988). Lecture Notes in Mathematics 1404:108-126.

[6] Jorgensen, D., Sega, L. (2004). Nonvanishing cohomology and classes of gorenstein rings. Adv. Math. 188(2):470-490.

[7] Schulz, R. (1986). Boundedness and periodicity of modules over QF rings. J. Algebra 101:450-469.

[8] Smalø, S. (2006). Local limitations of the Ext functor do not exist. Bull. London Math. Soc. 38(1):97-98.

Petter Andreas Bergh, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

   E-mail address: bergh@math.ntnu.no