ON THE HOCHSCHILD COHOMOLOGY OF TAME HECKE ALGEBRAS

KARIN ERDMANN AND SIBYLLE SCHROLL

Abstract. We explicitly calculate a projective bimodule resolution for a special biserial algebra giving rise to the Hecke algebra H_{q}(S_{4}) when q = −1. We then determine the dimensions of the Hochschild cohomology groups.

1. Introduction

In this paper we are interested in Hochschild cohomology of finite-dimensional algebras; the main motivation is to generalize group cohomology to larger classes of algebras. If suitable finite generation holds, one can define support varieties of modules as introduced by [SS]. Furthermore, when the algebra is self-injective, many of the properties of group representations generalize to this setting as was shown in [EHSST]. Although Hecke algebras do not have a Hopf algebra structure, one may expect that being deformations of group algebras, they should have good homological properties. Furthermore, the study of Hochschild cohomology for blocks of group algebras with cyclic or dihedral defect groups has made use of the fact that the basic algebras in this case are special biserial [Ho1, Ho2]. These results suggest that Hochschild cohomology of special biserial algebras might be accessible more generally. This paper is concerned with the self-injective special biserial algebra A which occurs as the basic algebra of the Hecke algebra H_{q}(S_{4}) when q = −1. For this algebra we will explicitly give a minimal projective bimodule resolution and use it to calculate the dimensions of the Hochschild cohomology. By [EN] any tame block of some Hecke algebra H_{q}(S_{n}) is derived equivalent to A. Since Hochschild cohomology is invariant under derived equivalence, our result gives information for arbitrary tame blocks Hecke algebras of type A. In order for the Hochschild cohomology ring to be finitely generated, the dimensions of HH^{n}(A) must have at most polynomial growth. Therefore our results give evidence towards the finite generation of the Hochschild cohomology. Further evidence towards this comes from [ES], which shows by a different approach, via derived equivalence, that the finite generation hypothesis in [EHSST] holds for the special biserial algebra under consideration in this paper.

The second author acknowledges support through an EPSRC Postdoctoral Fellowship EP/D077656/1 as well as through a Leverhulme Early Career Fellowship.

2000 Mathematics Subject Classification: 16E40, 20C08, 16E10 (primary).

The algebra A can be described by a quiver and relations (cf. [EN]). Let k be a field and let Q be the quiver described in figure 1 with relations

Figure 1. The quiver of A

(1)  εα = βε = 0, (αβ)^{2} = ε^{2}.

Then A = kQ/I where I is generated by the relations in (1). We denote the trivial path at the vertex i, i = 1, 2, by e_{i} and the corresponding simple A-module by S_{i}. We read paths from left to right and, unless stated differently, all the modules considered will be (finitely generated) right A-modules. Let radA be the Jacobson radical of A and denote the enveloping algebra of A by A^{e} = A^{op} ⊗_{k} A. For a path a in A we denote by o(a) the idempotent at the vertex from which the path leaves and by t(a) the idempotent at the vertex at which the path arrives. We say a path p in kQ is right (resp. left) uniform if it is a linear combination of paths ending in (resp. beginning at) a single vertex.

2. Construction of a minimal projective bimodule resolution

In this section we construct a minimal projective A-A-bimodule resolution of A, that is, we construct an exact complex

R_{•} : … ⟶ R_{n} ⟶^{δ_{n}} R_{n−1} ⟶^{δ_{n−1}} … ⟶ R_{1} ⟶^{δ_{1}} R_{0} ⟶^{δ_{0}} A ⟶ 0

with R_{n} = ⨁ P_{ij} and where P_{ij} is the projective A-A-bimodule Ae_{i} ⊗ e_{j}A. By [H] the projective P_{ij} occurs in R_{n} exactly as many times as dimExt^{n}(S_{i}, S_{j}). Minimal projective resolutions of simple modules in this case can be calculated, for example similar to the Appendix in [R]. Thus the terms of the projective bimodule resolution (R_{•}, δ) are periodic of period 4 and, for k ≥ 0,

(2)  \begin{matrix}R_{4k} ≅ & P_{11}^{2k+1} ⊕ P_{22} \\ R_{4k+1} ≅ & P_{11}^{2k+1} ⊕ P_{12} ⊕ P_{21} \\ R_{4k+2} ≅ & P_{11}^{2k+1} ⊕ P_{12} ⊕ P_{21} \\ R_{4k+3} ≅ & P_{11}^{2k+2} ⊕ P_{22}. \end{matrix}

ON THE HOCHSCHILD COHOMOLOGY OF TAME HECKE ALGEBRAS 3

2.1. From one-sided to two-sided resolutions. The construction of the differential δ is based on the construction of a minimal projective resolution of A/radA as a left and as a right A-module as developed in [GSZ]. We then use the differentials of the left and right module resolutions of A/radA to explicitly construct the differential of a minimal A-A-bimodule resolution of A. Our approach follows that of [GS] (see also [ST]) and, for Koszul algebras, that of [GHMS]. We begin by briefly recalling the construction of projective resolutions as developed in [GSZ]. Let Λ be a finite dimensional algebra of the form Λ = kQ/I. In [GSZ] the authors then define sets of paths g^{n} in kQ for n ≥ 0 where g^{0} is given by the trivial paths in kQ, g^{1} corresponds to the arrows of Q and g^{2} is a minimal set of uniform generators of I. For n ≥ 3, g^{n} is constructed recursively such that the elements in g^{n} are uniform and satisfy the property: for x ∈ g^{n}, x = ∑_{p∈g^{n−1}} pλ_{p} = ∑_{q∈g^{n−2}} qμ_{q} for unique elements λ_{p}, μ_{q} ∈ kQ and μ_{q} ∈ I such that there is a minimal projective Λ-module resolution

P_{•} : … ⟶ P_{n} ⟶^{d_{n}} P_{n−1} ⟶^{d_{n−1}} … ⟶ P_{1} ⟶^{d_{1}} P_{0} ⟶ Λ/radΛ ⟶ 0

such that

(1) P_{n} = ⨁_{x∈g^{n}} t(x)Λ for all n ≥ 0

(2) the map d_{n} : P_{n} → P_{n−1} is given by

t(x)a ↦ ∑_{p∈g^{n−1}} λ_{p}t(x)a

where as above x = ∑_{p∈g^{n−1}} pλ_{p} is determined by unique ele-
ments λ_{p} ∈ kQ.

The collection of elements of g^{n} is written as g^{n} = {g_{i}^{n}}.

In [GS] the sets g^{n} are then used to give an explicit construction of the first three differentials of a minimal projective Λ-Λ-bimodule resolution of Λ. Based on this approach we first construct the sets g^{n} for the minimal right A-module resolution of A/radA and then proceed to construct a A-A-bimodule resolution of A.

2.2. A minimal projective right A-module resolution of A/radA. In this paragraph we construct a minimal projective right A-module resolution of A/radA. To facilitate the notations we slightly change the notation of [GSZ] and define the sets defining the differentials as follows:

Definition 2.1. For the algebra A we set g_{1}^{0} = e_{1} and f_{22}^{0} = e_{2} and we define recursively, for all i = 0, 1, 2, 3 and all k ≥ 0 and for all n ≥ 4,

g_{r}^{n}  = \{ \begin{matrix}g_{1}^{n−1}ε − g_{2}^{n−1}αβ & if r = 1 \\ g_{r−1}^{n−1}ε − g_{r+1}^{n−1}αβ & if r odd, 1 < r ≤ 2k − 1 \\ g_{r+1}^{n−1}ε + g_{r−1}^{n−1}αβ & if r even, 1 < r ≤ 2k \end{matrix}

g_{2k+1}^{4k+i} = \{ \begin{matrix}g_{2k+1}^{4k+i−1}ε & if i = 1 and k = 0 \\ g_{2k}^{4k+i−1}ε & if i = 0, 1 and k > 0 \\ g_{2k+1}^{4k+i−1}ε − f_{12}^{4k+i−1}βαβ & if i = 2, 3 and k = 0 \\ g_{2k}^{4k+i−1}ε − f_{12}^{4k+i−1}βαβ & if i = 2, 3 and k > 0 \end{matrix}

g_{2k}^{4k} = g_{2k−1}^{4k−1}αβ if k > 0

g_{2k+2}^{4k+3} = g_{2k+1}^{4k+2}αβ if k > 0

f_{12}^{4k+i} = g_{2k+1}^{4k+i−1}α if i = 1, 2

f_{21}^{4k+i} = \{ \begin{matrix}f_{22}^{4k}β & if i = 1 \\ f_{21}^{4k+1}ε & if i = 2 \end{matrix}

f_{22}^{4k+i} = \{ \begin{matrix}f_{22}^{4k−1}(βα)^{2} & if i = 0 \\ f_{21}^{4k+2}α & if i = 3 \end{matrix}

We define g^{n} to be the set containing all elements of the form g_{∗}^{n} and f_{∗}^{n}.

Remark.We note that g_{1}^{1} = ε, f_{12}^{1} = α and f_{21}^{1} = β, hence the set corresponding to g^{1} is the set of arrows of Q. Also g_{1}^{2} = ε^{2} − αβαβ, f_{12}^{2} = εα and f_{21}^{2} = βε so that the set corresponding to g^{2} is a minimal set of uniform elements generating I.

Set P_{n} = ⨁_{x∈g^{n}} t(x)A for all n ≥ 0, that is, P_{4k+i} = (e_{1}A)^{2k+2} ⊕ e_{2}A for i = 1, 2, 3, 4. Let d_{n} : P_{n} → P_{n−1} be the map defined by t(x)a ↦ ∑_{p∈g^{n−1}} λ_{p}t(x)a. It then follows directly from [GSZ] that

Proposition 2.2. The complex (P_{•}, d) is a minimal projective right A-module resolution of A/radA.

2.3. A Bimodule resolution. In order to define a projective A-A-bimodule resolution of A we use the sets g^{n} defined in the previous paragraph. We note that we can write g_{1}^{3} = g_{1}^{2}ε − f_{12}^{2}βαβ = εg_{1}^{2} − βαβf_{12}^{2} and g_{1}^{4k+i} = g_{1}ε − g_{2}αβ = εg_{1} − αβg_{3}. Continuing in this way we will be able to define the differential of a projective A-A-bimodule

ON THE HOCHSCHILD COHOMOLOGY OF TAME HECKE ALGEBRAS 5

resolution of A. But first we need to introduce some notation: we write the elements e_{i} ⊗ e_{j} of Ae_{i} ⊗ e_{j}A as follows

\begin{aligned}𝔞 & := & e_{1} ⊗ e_{1} & ∈ & P_{11} = Ae_{1} ⊗ e_{1}A \\ 𝔟 & := & e_{1} ⊗ e_{2} & ∈ & P_{12} = Ae_{1} ⊗ e_{2}A \\ 𝔠 & := & e_{2} ⊗ e_{1} & ∈ & P_{21} = Ae_{2} ⊗ e_{1}A \\ 𝔡 & := & e_{2} ⊗ e_{2} & ∈ & P_{22} = Ae_{2} ⊗ e_{2}A. \end{aligned}

The terms of R_{•} in each degree contain multiple copies of P_{11} and always at most one copy of P_{12}, P_{21} and P_{22} (see (2)). We need to distinguish the different copies of P_{11} in R_{n}, for n ≥ 0. We do this by adding superscripts. As usual, the subscripts determine in which degree of the complex an element lies. For example, 𝔞_{i}^{r} = e_{1} ⊗ e_{1} lies in the rth copy of P_{11} as a summand of R_{i}, 𝔞_{i+1}^{r+1} = e_{1} ⊗ e_{1} lies in the r + 1th copy of P_{11} as a summand of R_{i+1} and 𝔟_{4k+1} = e_{1} ⊗ e_{2} (resp. 𝔠_{4k+1} = e_{2} ⊗ e_{1}) lies in the copy of P_{12} (resp. P_{21}) as a summand of R_{4k+1}.

Definition 2.3. The map δ_{0} : R_{0} → A is given by multiplication. With the above notation, for n ≥ 1, we define the differential δ_{n} : R_{n} → R_{n−1} by the following recursively defined maps

𝔞_{n}^{r} ↦ \{ \begin{matrix}𝔞_{n−1}^{1}ε − ε𝔞_{n−1}^{1} & n = 1, r = 1 \\ 𝔞_{n−1}^{1}ε − 𝔟_{n−1}βαβ + ε𝔞_{n−1}^{1} − αβα𝔠_{n−1} − αβ𝔟_{n−1}β − α𝔠_{n−1}αβ & n = 2, r = 1 \\ 𝔞_{n−1}^{1}ε − 𝔟_{n−1}βαβ − ε𝔞_{n−1}^{1} + αβα𝔠_{n−1} & n = 3, r = 1 \\ 𝔞_{n−1}^{1}αβ − αβ𝔞_{n−1}^{1} − ε𝔟_{n−1}β + α𝔠_{n−1}ε & n = 3, r = 2 \end{matrix}

and, for all n = 4k + i with i = 0, 1, 2, 3 and k ≥ 1,

𝔞_{4k+i}^{r} ↦ \{ \begin{matrix}𝔞^{1}ϵ − 𝔞^{2}αβ + ϵ𝔞^{1} − αβ𝔞^{2} & if r = 1 and i = 0, 2 \\ 𝔞^{r+1}ε + 𝔞^{r−1}αβ + ε𝔞^{r} − αβ𝔞^{r+2} & if 1 < r < 2k, r even and i = 0, 2 \\ 𝔞^{r−1}ε − 𝔞^{r+1}αβ + ε𝔞^{r} + αβ𝔞^{r−2} & if 1 < r < 2k, r odd and i = 0, 2 \\ 𝔞^{r−1}αβ + ε𝔞^{r} + (−1)^{k}αβα𝔡β & if r = 2k and i = 0 \\ 𝔞^{r−1}ε + αβ𝔞^{r−2} + (−1)^{k}α𝔡βαβ & if r = 2k + 1 and i = 0 \\ 𝔞^{r+1}ε + 𝔞^{r−1}αβ + ε𝔞^{r} + (−1)^{k+1}αβα𝔠 − αβ𝔟β & if r = 2k and i = 2 \\ 𝔞^{r−1}ε + ε𝔞^{r} + αβ𝔞^{r−2} − 𝔟βαβ + (−1)^{k+1}α𝔠αβ & if r = 2k + 1 and i = 2 \\ 𝔞^{1}ε − 𝔞^{2}αβ − ε𝔞^{1} + αβ𝔞^{3} & if r = 1 and i = 1, 3 \\ 𝔞^{3}ε + 𝔞^{1}αβ − ε𝔞^{2} − αβ𝔞^{1} & if r = 2 and i = 1, 3 \\ 𝔞^{r−1}ε − 𝔞^{r+1}αβ − ε𝔞^{r} + αβ𝔞^{r+2} & if 1 < r ≤ 2k and r is odd \\ 𝔞^{r+1}ε + 𝔞^{r−1}αβ − ε𝔞^{r} − αβ𝔞^{r−2} & if 1 < r ≤ 2k and r is even \\ 𝔞^{r−1}ε − ε𝔞^{r} & if r = 2k + 1 and i = 1 \\ 𝔞^{r−1}ε − 𝔟βαβ − ε𝔞^{r} + (−1)^{k}αβα𝔠 & if r = 2k + 1 and i = 3 \\ 𝔞^{r−1}αβ − αβ𝔞^{r−2} + (−1)^{k}α𝔠ε − ε𝔟β & if r = 2k + 2 and i = 3 \end{matrix}

where on the right hand side we omit the subscript with the understanding that these are all elements in R_{4k+i−1}. Finally, for n = 4k + i with i = 0, 1, 2, 3 and k ≥ 0, we set

\begin{matrix}𝔡_{4k} & ↦ & 𝔡_{4k−1}(βα)^{2} + (βα)^{2}𝔡_{4k−1} + (−1)^{k}β𝔞_{4k−1}^{2k−1}α \\ 𝔟_{4k+1} & ↦ & 𝔞_{4k}^{2k+1}α + (−1)^{k+1}α𝔡_{4k} \\ 𝔠_{4k+1} & ↦ & 𝔡_{4k}β + (−1)^{k+1}β𝔞_{4k}^{2k} \\ 𝔟_{4k+2} & ↦ & 𝔞_{4k+1}^{2k+1}α + ε𝔟_{4k+1} \\ 𝔠_{4k+2} & ↦ & 𝔠_{4k+1}ε + (−1)^{k}β𝔞_{4k+1}^{2k+1} \\ 𝔡_{4k+3} & ↦ & 𝔠_{4k+2}α + (−1)^{k+1}β𝔟_{4k+2}. \end{matrix}

It is easily checked that if we set R_{n} = ⨁_{x∈g^{n}} Ao(x) ⊗ t(x)A we obtain exactly the expressions as stated at the beginning of section 2. Furthermore, we have that,

Theorem 2.4. The complex (R_{•}, δ) is a minimal projective resolution of A as an A-A-bimodule.

Proof: It is straightforward to verify that Imδ_{n} ⊆ Kerδ_{n−1} and therefore (R_{•}, δ) is a complex. In order to show that the complex is exact we first note that there is an isomorphism of complexes of right A-modules (A/radA ⊗_{A} R_{•}, id ⊗ δ) ≃ (P_{•}, d). This holds since it is easily verified that for all n, we have A/radA ⊗_{A} R_{n} ≃ P_{n} and furthermore, the diagram

(3)  \xymatrix{A/radA ⊗_{A} R_{n} \ar[d]^{≃} \ar[r]^{id⊗δ_{n}} & A/radA ⊗_{A} R_{n−1} \ar[d]^{≃} \\ P_{n} \ar[r]^{d_{n}} & P_{n−1}}

commutes for all n ≥ 1. Now we apply the same arguments as in [GS, Proposition 2.8] and see also [ST, Theorem 1.6]. Suppose that Kerδ_{n−1} ⊈ Imδ_{n} for some n ≥ 1. Then there exists a simple A-A-bimodule S ⊗ T composed of a simple left A-module S and a simple right A module T and there exists a non-zero map f : Kerδ_{n−1} → S ⊗ T. We have seen (Proposition 2.2 and (3)) that the complex (A/radA ⊗_{A} R_{•}, id ⊗ δ) is a minimal projective resolution of A/radA as a right A-module. Therefore the following isomorphisms of right A-modules hold A/radA ⊗_{A} Imδ_{n} ≃ Im(id ⊗ δ_{n}) ≃ Ker(id⊗δ_{n−1}) ≃ A/radA⊗_{A}Kerδ_{n−1} and we obtain a non-zero map as the composition

A/radA⊗_{A}R_{n} ⟶^{id⊗δ_{n}} A/radA⊗_{A}Imδ_{n} ⟶^{≃} A/radA⊗_{A}Kerδ_{n−1} ⟶^{id⊗f} S⊗T.

ON THE HOCHSCHILD COHOMOLOGY OF TAME HECKE ALGEBRAS 7

But this is the same as the functor A/radA ⊗_{A} − applied to the sequence of maps R_{n} →^{δ_{n}} Kerδ_{n−1} →^{f} S ⊗ T. However, the map f ∘ δ_{n} is zero. This gives a contradiction and therefore the complex (R_{•}, δ_{n}) is exact. Since we calculated the terms of R_{•} following [H], the minimality follows. □

3. Dimensions of the Hochschild cohomology groups

The Hochschild cohomology of A is defined by HH^{∗}(A) = Ext_{A^{e}}^{∗}(A, A). In each degree n, n ≥ 0, there is an exact sequence

(4) 0 → Hom_{A^{e}}(Ω^{n}(A), A) → Hom_{A^{e}}(R_{n}, A) → Hom_{A^{e}}(Ω^{n+1}(A), A) → Ext_{A^{e}}^{n+1}(A, A) → 0.

and since we have an explicit description of the terms R_{n} of the A-A-bimodule resolution of A, it is straightforward to calculate the dimensions of the complex Hom_{A^{e}}(R^{•}, A). These dimensions are for, k ≥ 0,

dim Hom_{A^{e}}(R_{4k}, A) = 8k + 7
dim Hom_{A^{e}}(R_{4k+1}, A) = 8k + 8
dim Hom_{A^{e}}(R_{4k+2}, A) = 8k + 8
dim Hom_{A^{e}}(R_{4k+3}, A) = 8k + 11.

Furthermore, HH^{0}(A) is given by the center Z(A) of A. As an algebra, Z(A) is generated by

{e_{1} + e_{2}, ε, αβ + βα, ε^{2}, βαβα}.

By (4), in order to calculate the dimensions of HH^{n}(A) for all n ≥ 0 we need to calculate the dimensions of Hom_{A^{e}}(Ω^{n}(A), A) for all n ≥ 1. Using the description

Hom_{A^{e}}(Ω^{n}(A), A) = {γ_{n} ∈ Hom_{A^{e}}(R_{n}, A)| γ_{n}(Ω(A)^{n+1}) = 0}

we will calculate a basis for Hom_{A^{e}}(Ω^{n}(A), A). Since R_{•} is exact the images of the differential yield the generators of Ω^{n}(A) in each degree n. Thus, for k ≥ 0, we have

\begin{aligned}Ω^{4k}(A) & = & < δ_{4k}(𝔞_{4k}^{1}), …, δ_{4k}(𝔞_{4k}^{2k+1}), δ_{4k}(𝔡_{4k}) > \\ Ω^{4k+1}(A) & = & < δ_{4k+1}(𝔞_{4k+1}^{1}), …, δ_{4k+1}(𝔞_{4k+1}^{2k+1}), δ_{4k+1}(𝔟_{4k+1}), δ_{4k+1}(𝔠_{4k+1}) > \\ Ω^{4k+2}(A) & = & < δ_{4k+2}(𝔞_{4k+2}^{1}), …, δ_{4k+2}(𝔞_{4k+2}^{2k+1}), δ_{4k+2}(𝔟_{4k+2}), δ_{4k+2}(𝔠_{4k+2}) > \\ Ω^{4k+3}(A) & = & < δ_{4k+3}(𝔞_{4k+3}^{1}), …, δ_{4k+3}(𝔞_{4k+3}^{2k+2}), δ_{4k+3}(𝔡_{4k+3}) > . \end{aligned}

For each of the four cases above, we will obtain a system of equations whose solution space gives a basis of Hom_{A^{e}}(Ω^{n}(A), A).

3.1. Characteristic of k ≠ 2. Suppose that the characteristic of k ≠ 2. Then

Lemma 3.1. We have the following dimensions for k ≤ 0:

dim Hom_{A^{e}}(Ω^{4k+i}(A), A) = \{ \begin{matrix}5k + 5 & if i = 0 \\ 5k + 5 & if i = 1 \\ 5k + 6 & if i = 2 \\ 5k + 6 & if i = 3 \end{matrix}

Proof: Let k ≥ 1, we start by calculating the dimension of Hom_{A^{e}}(Ω^{4k}(A), A). Let γ_{4k} : R_{4k} → A be the map defined by γ_{4k}(𝔞_{4k}^{j}) = a_{1}^{j}e_{1} + b_{1}^{j}ε + c_{1}^{j}αβ + d_{1}^{j}ε^{2} for 1 ≥ j ≥ 2k + 1 and γ_{4k}(𝔡_{4k}) = a_{2}e_{2} + b_{2}βα + c_{2}βαβα with a_{1}^{j}, b_{1}^{j}, …, a_{2}, b_{2} ∈ k. With this notation we have that γ_{4k}(x) = 0 for all generators x of Ω^{4k+1}(A) if and only if for all even n, such that 2 ≤ n ≤ 2k,

a_{1}^{n} = a_{1}^{n+1}  b_{1}^{n} = b_{1}^{n+1}  c_{1}^{n} = c_{1}^{n+1}

and

(−1)^{k+1}a_{2} = a_{1}^{2k+1} (−1)^{k+1}b_{2} = c_{1}^{2k+1}.

Thus the dimension of Hom_{A^{e}}(Ω^{4k}(A), A) is 5k + 5.

Similarly for the dimension of Hom_{A^{e}}(Ω^{4k+1}(A), A), let γ_{4k+1} : R_{4k+1} → A be the map defined by γ_{4k+1}(𝔞_{4k+1}^{i}) = a_{1}^{i}e_{1} +b_{1}^{i}ε+c_{1}^{i}αβ +d_{1}^{i}ε^{2} for 1 ≥ i ≥ 2k+1 and γ_{4k+1}(𝔟_{4k+1}) = a_{12}α + b_{12}αβα and γ_{4k+1}(𝔠_{4k+1}) = a_{21}β + b_{21}βαβ with a_{1}^{i}, b_{1}^{i}, …, a_{12}, b_{21}, … ∈ k. Then for all generators x of Ω^{4k+2}(A), we have that γ_{4k+1}(x) = 0 if and only if the following equations hold

a_{1}^{1} = a_{1}^{2} = … = a_{1}^{2k+1} = a_{1}^{2k+1} = 0
b_{1}^{1} = c_{1}^{2}
b_{1}^{2k} + b_{1}^{2k+1} + c_{1}^{2k−1} − a_{12} + (−1)^{k+1}a_{21} = 0
b_{1}^{n} + b_{1}^{n+1} + c_{1}^{n−1} − c_{1}^{n+2} = 0

for n ∈ {2, 4, …, 2k − 2}. Thus the dimension of Hom_{A^{e}}(Ω^{4k+1}(A), A) is 5k + 5. In the same way we obtain the following sets of equations for Hom_{A^{e}}(Ω^{4k+2}(A), A) for all even n, such that 2 ≤ n ≤ 2k

a_{1}^{n} = a_{1}^{n+1}  b_{1}^{n} = b_{1}^{n+1}  c_{1}^{n} = c_{1}^{n+1}

and

a_{12} = (−1)^{k}a_{21} and b_{12} = (−1)^{k}b_{21}

giving dim Hom_{A^{e}}(Ω^{4k+2}(A), A) = 5k + 6. Finally for Hom_{A^{e}}(Ω^{4k+3}(A), A) we obtain

a_{1}^{1} = a_{1}^{2} = … = a_{1}^{2k+2} = 0
b_{1}^{2} = c_{1}^{2}
b_{1}^{n} + b_{1}^{n+1} + c_{1}^{n−1} − c_{1}^{n+2} = 0

for n ∈ {2, 4, …, 2k} and

b_{1}^{2k+2} = (−1)^{k}a_{2} and c_{1}^{2k+2} = (−1)^{k}2a_{2}

such that dim Hom_{A^{e}}(Ω^{4k+3}(A), A) = 5k + 6.

From Definition 2.3 we see that for n ≤ 3 pattern for the expressions of the differential is not regular. Therefore in these small cases we have to calculate the dimension of Hom_{A^{e}}(Ω^{n}(A), A) by hand. These calculations show that dim Hom_{A^{e}}(Ω^{1}(A), A) = 5, dim Hom_{A^{e}}(Ω^{2}(A), A) = 6 and dim Hom_{A^{e}}(Ω^{3}(A), A) = 6.

□

The dimensions of the Hochschild cohomology groups of A then follow directly from above.

Theorem 3.2. The dimensions of the Hochschild cohomology groups of A are as follows

dim HH^{4k+i}(A) = \{ \begin{matrix}2k + 3 & i = 0, k > 0 \\ 2k + 3 & i = 1, k ≥ 0 \\ 2k + 3 & i = 2, k ≥ 0 \\ 2k + 4 & i = 3, k ≥ 0 \end{matrix}

and the set {e_{1} + e_{2}, ε, αβ + βα, ε^{2}, βαβα} is a k-basis of HH^{0}(A).

3.2. Characteristic of k = 2. Suppose the characteristic of k = 2. Then

Lemma 3.3. We have the following dimensions for k > 0:

dim Hom_{A^{e}}(Ω^{4k+i}(A), A) = \{ \begin{matrix}5k + 5 & if i = 0 \\ 5k + 8 & if i = 1 \\ 5k + 6 & if i = 2 \\ 5k + 9 & if i = 3 \end{matrix}

and dim Hom_{A^{e}}(Ω^{1}(A), A) = 6, dim Hom_{A^{e}}(Ω^{2}(A), A) = 6, and dim Hom_{A^{e}}(Ω^{3}(A), A) = 8.

Proof: The proof is analogous to the proof of Lemma 3.1. □

Again the dimensions of the Hochschild cohomology groups of A then directly follow.

Theorem 3.4. The dimensions of the Hochschild cohomology groups of A are as follows

dim HH^{4k+i}(A) = \{ \begin{matrix}2k + 6 & i = 0, k > 1 \\ 2k + 6 & i = 1, k > 0 \\ 2k + 6 & i = 2, k > 0 \\ 2k + 7 & i = 3, k > 0 \end{matrix}

and dim HH^{1}(A) = 4, dim HH^{2}(A) = 4, dim HH^{3}(A) = 6 and dim HH^{4}(A) = 7. The set {e_{1} + e_{2}, ε, αβ + βα, ε^{2}, βαβα} is a k-basis of HH^{0}(A).

Acknowledgements. The authors would like to thank Nicole Snashall for her very helpful suggestions and comments and for the careful reading of this manuscript.

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Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford OX1 3LB, UK Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

E-mail address: erdmann@maths.ox.ac.uk, schroll@mcs.le.ac.uk