HOCHSCHILD COHOMOLOGY AND SUPPORT VARIETIES FOR TAME HECKE ALGEBRAS

SIBYLLE SCHROLL AND NICOLE SNASHALL

Abstract. We give a basis for the Hochschild cohomology ring of tame Hecke algebras. We then show that the Hochschild cohomology ring modulo nilpotence is a finitely generated algebra of Krull dimension 2, and describe the support varieties of modules for these algebras.

Introduction

Hecke algebras play an important role in representation theory. They arise as deformations of the group algebras of finite Coxeter groups and appear as endomorphism algebras of induced representations of finite or p-adic Chevalley groups. They also give rise to the Kazhdan-Lusztig polynomials which appear in the expression of the canonical basis in terms of the natural basis of the Hecke algebra. In particular, the Hecke algebras of type A (which arise as a deformation of the group algebras of the symmetric group) have been well studied. A complete classification of the representation type of the blocks of the Hecke algebras of type A was obtained in [EN]. For the tame Hecke algebras of type A, it was shown in [J] that there are precisely two Morita equivalence classes of blocks and they are represented by H_{q}(S_{4}) and the principal block of H_{q}(S_{5}) with q = −1. Moreover, they are in the same derived equivalence class. This follows from [M] and see also [BHS], since they are generalized Brauer tree algebras of the same type.

The algebra H_{q}(S_{5}) with q = −1 is a Koszul symmetric special biserial algebra. Special biserial algebras occur in many aspects of representation theory and they are necessarily of finite or tame representation type. Symmetric special biserial algebras occur, for example, as Hopf algebras associated to infinitesimal groups whose principal block is tame [FS1, FS2], in the representation theory of U_{q}(sl_{2}) [P, Su, X], of Drinfeld doubles of generalized Taft algebras [EGST] (and see also [ST1]) and as socle deformations of the latter [ST2]. For the symmetric special biserial algebras in [ST1] and [ST2], the Hochschild cohomology ring modulo nilpotence has been shown to be a finitely generated algebra of Krull dimension 2. In this paper we show that the same phenomenon holds for the tame Hecke algebras, so the question naturally arises as to whether or not the Hochschild cohomology ring modulo nilpotence of any symmetric special biserial algebra is a finitely generated algebra of Krull dimension at most 2.

In this paper we let K be an algebraically closed field and study the Hochschild cohomology ring of the algebra A, which, in the case where the characteristic of K is not 2, is precisely the principal block of H_{q}(S_{5}) with q = −1. The algebra A is a symmetric special biserial algebra of tame representation type. Furthermore, it is Koszul with radical cube zero, and so is one of the algebras recently classified in [B]; it corresponds to the algebra associated to the simply laced extended Dynkin diagram of type T̃_{2} (we adopt the notation commonly used in articles in physics journals; note that in [B] and [ESo] this diagram is denoted by Z̃_{1}). It is well-known that the Hecke algebras of finite type arise from Brauer tree algebras, and thus [EH] shows that they are periodic algebras and describes their Hochschild cohomology ring. Hence the structure of the Hochschild cohomology ring for Hecke algebras of both tame and finite type is now known. Specifically, in Theorem 3.7, we prove that the Hochschild cohomology ring modulo nilpotence of a Hecke algebra has Krull dimension 1 if the algebra is of finite type and has Krull dimension 2 if the algebra is of tame type. For wild Hecke algebras, there has so far been very little progress in the study of Hochschild cohomology or support varieties.

Date: July 20, 2009.

2000 Mathematics Subject Classification. Primary: 16E40, 20C08; Secondary: 16S37.

The first author acknowledges support through a Leverhulme Early Career Fellowship.

The Hochschild cohomology ring modulo nilpotence was used in [SS] to define the concept of a support variety for any finitely generated module over a finite-dimensional algebra. It was shown in [EHSST] that, under certain reasonable finiteness conditions on the algebra, these support varieties have many of the analogous properties to those satisfied by modules over a group algebra for a finite group or over a cocommutative Hopf algebra. In addition, when these conditions hold, the Hochschild cohomology ring is itself a finitely generated algebra, and so the finiteness conjecture of [SS] holds concerning the Hochschild cohomology ring modulo nilpotence. We show that these finiteness conditions hold for the tame Hecke algebras, which has also been independently shown by Erdmann and Solberg in [ESo]. We then consider the consequences for support varieties of modules over A.

The paper is structured as follows. In section 1, we give a minimal projective bimodule resolution of A, and in section 2 a basis for each of the Hochschild cohomology groups HH^{n}(A) for n ≥ 0. This extends the results of [ESc], where the dimensions of the Hochschild cohomology groups were calculated for the Hecke algebra H_{q}(S_{4}) with q = −1. The advantage of considering A over the algebras in [ESc] is that A is a Koszul algebra, and so has a linear projective resolution as an A-A-bimodule. In addition, in section 1.1, we give the minimal projective bimodule resolution of a more general family of algebras which contains the algebra A. In section 3, we recall the finiteness conditions Fg1. and Fg2. of [EHSST] and use them to give new results on the support varieties of finitely generated A-modules. Finally, (in Theorem 3.7), we show that the Hochschild cohomology ring of A modulo nilpotence is a finitely generated commutative algebra of Krull dimension 2.

The principal block of H_{q}(S_{5}) for q = −1 was given by quiver and relations in [EN]. Let K be an algebraically closed field (with no restrictions placed on char K). Let Q be the quiver

[figure: 1; ε; α; 2; ε̄; ᾱ]

and let I be the ideal of KQ generated by {αε̄, εα, ᾱε, ε̄ᾱ, ε^{2} − αᾱ, ε̄^{2} − ᾱα}. Let A = KQ/I. Then A is the principal block of H_{q}(S_{5}) with q = −1 in the case where char K is not 2. The Hochschild cohomology ring of A is given by HH^{∗}(A) = Ext_{A^{e}}^{∗}(A, A) = ⊕_{n=0}^{∞} Ext_{A^{e}}^{n}(A, A) with the Yoneda product, where A^{e} = A^{op}⊗_{k}A denotes the enveloping algebra of A.

We denote the trivial path at the vertex i by e_{i}. We write paths from left to write. For any arrow a in the quiver Q, we write o(a) for the trivial path corresponding to the origin of a and t(a) for the trivial path corresponding to the terminus of a. Thus o(ε) = e_{1} = o(α), t(ε) = e_{1} and t(α) = e_{2} etc. There is an algebra isomorphism b : A → A induced by the involution given by e_{1} ↦ e_{2}, e_{2} ↦ e_{1}, α ↦ ᾱ, ᾱ ↦ α, ε ↦ ε̄ and ε̄ ↦ ε.

1. A minimal projective bimodule resolution

The terms of the minimal projective bimodule resolution (R_{•}, δ) of A can be calculated following [H]. We denote by P_{ij} the projective indecomposable A-A-bimodule Ae_{i} ⊗e_{j}A, for i, j = 1, 2. The projectives R_{n} are periodic of period 4. We have R_{0} = P_{11} ⊕ P_{22} and, for 4k + i ≥ 1,

(1)  R_{4k+i} ≅ \{ \begin{matrix}P_{11}^{2k+1} ⊕ P_{12}^{2k+i} ⊕ P_{21}^{2k+i} ⊕ P_{22}^{2k+1} & if i = 0, 1 \\ P_{11}^{2k+i−1} ⊕ P_{12}^{2k+2} ⊕ P_{21}^{2k+2} ⊕ P_{11}^{2k+i−1} & if i = 2, 3. \end{matrix}

We follow the approach of [GHMS] in defining the minimal projective A-A-bimodule resolution of the Koszul algebra A. As the starting point, we recall that [GSZ] explains how to recursively define sets - which we will denote by G^{n} rather than by f^{n} - for each n ≥ 0, in order to give an explicit construction of a minimal projective resolution (Q_{•}, d) of the right A-module A/J(A) where J(A) denotes the Jacobson radical of A. These sets have the following properties.

(i) For each n ≥ 0, Q_{n} = ⊕_{x∈G^{n}}t(x)A.

(ii) For each x ∈ G^{n}, there are unique elements r_{j} ∈ KQ such that x = ∑ h_{j}^{n−1}r_{j}

where the sum is over all elements h_{j}^{n−1} ∈ G^{n−1}. (iii) For each n ≥ 1, using the decomposition of (ii), for x ∈ G^{n}, the map d_{n} : Q_{n} →

Q_{n−1} is given by t(x)λ ↦ ∑_{j} r_{j}t(x)λ.

The minimal projective bimodule resolution of a Koszul algebra which was given in [GHMS] uses these same sets of [GSZ] in its construction.

Therefore, for our algebra A, we start by defining the set G^{0} to be the set of vertices of Q, labelled so that G^{0} = {g_{1}^{0} = e_{1}, ḡ_{1}^{0} = e_{2}}. Then, for n ≥ 1, we will recursively define sets G^{n} = {g_{i}^{n}, ḡ_{i}^{n}, f_{j}^{n}, f̄_{j}^{n}}_{i,j} in KQ for appropriate indices i and j and such that |G^{n}| = 2(n+1) for all n ≥ 0, so that these sets contain the requisite information to define the minimal projective A-A-bimodule resolution of A. The elements of the sets G^{n} will be uniform elements of KQ, that is, for each x ∈ G^{n} there are vertices v, w such that x = vxw. We set o(x) = v and t(x) = w.

Definition 1.1. For n ≥ 1, we define elements:

g_{1}^{n}  = \{ \begin{matrix}g_{1}^{n−1}ε − f_{1}^{n−1}ᾱ & if n = 4k or n = 4k + 2 \\ g_{1}^{0}ε & if n = 1 \\ g_{1}^{n−1}ε − f_{2}^{n−1}ᾱ & if n = 4k + 1 with k ≥ 1, or n = 4k + 3 \end{matrix}

g_{2l}^{n}  = \{ \begin{matrix}g_{2l}^{n−1}ε − f_{2l+1}^{n−1}ᾱ & 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 \\ g_{2l}^{n−1}ε + f_{2l−1}^{n−1}ᾱ & 1 ≤ l ≤ k if n = 4k + 1 or n = 4k + 3 \end{matrix}

g_{2l+1}^{n} = \{ \begin{matrix}g_{2l+1}^{n−1}ε + f_{2l}^{n−1}ᾱ & 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 \\ g_{2l+1}^{n−1}ε − f_{2l+2}^{n−1}ᾱ & 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3 \end{matrix}
g_{2k}^{4k}  = g_{2k}^{4k−1}ε
g_{2k+1}^{n} = \{ \begin{matrix}f_{2k}^{n−1}ᾱ & if n = 4k \\ g_{2k+1}^{n−1}ε & if n = 4k + 1 with k ≥ 1 \end{matrix}
g_{2k+2}^{4k+3} = f_{2k+1}^{4k+2}ᾱ

f_{1}^{n}  = \{ \begin{matrix}g_{1}^{0}α & if n = 1 \\ f_{1}^{n−1}ε̄+ g_{1}^{n−1}α & if n = 4k + 1 with k ≥ 1, or n = 4k + 3 \end{matrix}

f_{2l+1}^{n} = \{ \begin{matrix}f_{2l+1}^{n−1}ε̄− g_{2l+2}^{n−1}α & 0 ≤ l < k if n = 4k or n = 4k + 2 \\ f_{2l+1}^{n−1}ε̄+ g_{2l}^{n−1}α & 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3 \end{matrix}

f_{2l+2}^{n} = \{ \begin{matrix}f_{2l+2}^{n−1}ε̄+ g_{2l+1}^{n−1}α & 0 ≤ l < k if n = 4k or n = 4k + 2 \\ f_{2l+2}^{n−1}ε̄− g_{2l+3}^{n−1}α & 0 ≤ l < k if n = 4k + 1 or n = 4k + 3 \end{matrix}

f_{2k+1}^{n} = \{ \begin{matrix}g_{2k}^{n−1}α & if n = 4k + 1 with k ≥ 1 \\ f_{2k+1}^{n−1}ε̄ & if n = 4k + 2 \end{matrix}

f_{2k+2}^{n} = \{ \begin{matrix}g_{2k+1}^{n−1}α & if n = 4k + 2 \\ f_{2k+2}^{n−1}ε̄ & if n = 4k + 3. \end{matrix}

Applying the bar involution we define ḡ_{i}^{n} = b(g_{i}^{n}) and f̄_{j}^{n} = b(f_{j}^{n}) and we obtain in this way all the elements of G^{n}.

Remark. (1) Note that g_{1}^{1} = ε, f_{1}^{1} = α, ḡ_{1}^{1} = ε̄ and f̄_{1}^{1} = ᾱ so that G^{1} is the set of arrows of Q. Also, g_{1}^{2} = ε^{2} −αᾱ, f_{1}^{2} = αε̄ and f_{2}^{2} = εα, and so ḡ_{1}^{2} = ε̄^{2} −ᾱα, f̄_{1}^{2} = ᾱε and f̄_{2}^{2} = ε̄ᾱ. Thus G^{2} is a minimal set of uniform elements which generate the ideal I.

(2) It follows from above that

G^{n} = {g_{i}^{n}, ḡ_{i}^{n}, f_{j}^{n}, f̄_{j}^{n} | i = 1, … 2k + 1, j = 1, … n − 2k} for n = 4k or n = 4k + 2
G^{n} = \{g_{i}^{n}, ḡ_{i}^{n}, f_{j}^{n}, f̄_{j}^{n} | i = 1, … \frac{n+1}{2}, j = 1, … \frac{n+1}{2}\}  for n = 4k + 1 or n = 4k + 3

where g_{i}^{n} ∈ e_{1}KQe_{1}, f_{j}^{n} ∈ e_{1}KQe_{2}, f̄_{j}^{n} ∈ e_{2}KQe_{1}, and ḡ_{i}^{n} ∈ e_{2}KQe_{2}.

(3) For each n ≥ 1, the set G^{n} contains four monomials of length n. These are the four subpaths of length n of the path (εαε̄ᾱ)^{N} for N >> 0.

Thus

R_{n} = ⨁_{x∈G^{n}} Ao(x) ⊗ t(x)A
= (⨁_{i} Ao(g_{i}^{n}) ⊗ t(g_{i}^{n})A) ⊕ (⨁_{j} Ao(f_{j}^{n}) ⊗ t(f_{j}^{n})A)⊕
(⨁_{i} Ao(ḡ_{i}^{n}) ⊗ t(ḡ_{i}^{n})A) ⊕ (⨁_{j} Ao(f̄_{j}^{n}) ⊗ t(f̄_{j}^{n})A)

where, for each i, the summand Ao(g_{i}^{n}) ⊗ t(g_{i}^{n})A = P_{11} and the summand Ao(ḡ_{i}^{n}) ⊗ t(ḡ_{i}^{n})A = P_{22}, and, for each j, the summand Ao(f_{j}^{n}) ⊗ t(f_{j}^{n})A = P_{12} and the summand Ao(f̄_{j}^{n}) ⊗ t(f̄_{j}^{n})A = P_{21}.

Following [GHMS], in order to define the differential δ we need the following lemma so that we have two different ways of expressing the elements of the set G^{n} in terms of the elements of the set G^{n−1}. Namely, we write the elements of G^{n} in the form ∑ h_{i}^{n−1}p_{i} and in the form ∑ q_{i}h_{i}^{n−1} for appropriate p_{i}, q_{i} in the ideal of KQ which is generated by all the arrows, and where the sums are over all h_{i}^{n−1} ∈ G^{n−1}. The proof of Lemma 1.2 is straightforward and is omitted.

Lemma 1.2. For n ≥ 1,

g_{1}^{n}  = \{ \begin{matrix}εg_{1}^{0} & if n = 1 \\ εg_{1}^{n−1} − αf̄_{1}^{n−1} & if n ≥ 2 \end{matrix}

g_{2l}^{n}  = εg_{2l+1}^{n−1} + αf̄_{2l}^{n−1}  where 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 or 1 ≤ l ≤ k if n = 4k + 1 or n = 4k + 3

g_{2l+1}^{n} = εg_{2l}^{n−1} − αf̄_{2l+1}^{n−1}  where 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 or 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3
g_{2k}^{4k}  = αf̄_{2k}^{n−1}
g_{2k+1}^{n} = εg_{2k}^{n−1} if n = 4k, or if n = 4k + 1 with k ≥ 1
g_{2k+2}^{4k+3} = αf̄_{2k+2}^{n−1}

f_{1}^{n}  = \{ \begin{matrix}αḡ_{1}^{0} & if n = 1 \\ εf_{2}^{n−1} + αḡ_{1}^{n−1} & if n = 4k + 1 with k ≥ 1, or n = 4k + 3 \end{matrix}

f_{2l+1}^{n} = εf_{2l+2}^{n−1} + αḡ_{2l+1}^{n−1}  where 0 ≤ l < k if n = 4k or if n = 4k + 2 or 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3

f_{2l+2}^{n} = εf_{2l+1}^{n−1} − αḡ_{2l+2}^{n−1}  where 0 ≤ l < k if n = 4k or n = 4k + 2 or 0 ≤ l < k if n = 4k + 1 or n = 4k + 3
f_{2k+1}^{n} = αḡ_{2k+1}^{n−1} if n = 4k + 1 with k ≥ 1, or if n = 4k + 2
f_{2k+2}^{n} = εf_{2k+1}^{n−1} if n = 4k + 2 or n = 4k + 3.

and the analogous set of equations holds if we apply b to the equations above.

We now introduce some additional notation to enable us to distinguish the summands of R_{n}. First, let

\begin{aligned}𝔲 & := & e_{1} ⊗ e_{1} & ∈ & P_{11} = Ae_{1} ⊗ e_{1}A \\ 𝔲̄ & := & e_{2} ⊗ e_{2} & ∈ & P_{22} = Ae_{2} ⊗ e_{2}A \\ 𝔭 & := & e_{1} ⊗ e_{2} & ∈ & P_{12} = Ae_{1} ⊗ e_{2}A \\ 𝔭̄ & := & e_{2} ⊗ e_{1} & ∈ & P_{21} = Ae_{2} ⊗ e_{1}A. \end{aligned}

We add the superscript n to indicate that each of these elements lies in the projective module R_{n}, and then add subscripts in order to distinguish, within R_{n}, the different summands of the form Ae_{i} ⊗ e_{j}A. Specifically, 𝔲_{i}^{n} = e_{1} ⊗ e_{1} lies in the ith copy of P_{11} as a summand of R_{n}, that is, in the copy of P_{11} corresponding to the element g_{i}^{n} of G^{n}. Similarly, 𝔭_{j}^{n} = e_{1} ⊗ e_{2} lies in the jth copy of P_{12} as a summand of R_{n}, that is, in the copy of P_{12} corresponding to the element f_{j}^{n} of G^{n}.

The algebra isomorphism b : A → A extends to an algebra isomorphism b : A^{e} → A^{e} where b(a_{1} ⊗ a_{2}) = b(a_{1}) ⊗ b(a_{2}) for a_{1} ∈ A^{op}, a_{2} ∈ A. Thus b(𝔲) = 𝔲̄ and b(𝔭) = 𝔭̄.

For better readability we omit in the following definition the superscripts on all expressions except the left most one with the understanding that the terms on the right hand side are in R_{n−1}.

Definition 1.3. We define an A-A-bimodule homomorphism δ_{n} : R_{n} → R_{n−1}, for all n ≥ 1, by

𝔲_{1}^{n}  ↦ \{ \begin{matrix}𝔲_{1}ε − 𝔭_{1}ᾱ + ε𝔲_{1} − α𝔭̄_{1} & if n = 4k or n = 4k + 2 \\ 𝔲_{1}ε − ε𝔲_{1} & if n = 1 \\ 𝔲_{1}ε − 𝔭_{2}ᾱ − (ε𝔲_{1} − α𝔭̄_{1}) & if n = 4k + 1 with k ≥ 1, or n = 4k + 3 \end{matrix}

𝔲_{2l}^{n}  ↦ \{ \begin{matrix}𝔲_{2l}ε − 𝔭_{2l+1}ᾱ + ε𝔲_{2l+1} + α𝔭̄_{2l} & 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 \\ 𝔲_{2l}ε + 𝔭_{2l−1}ᾱ − (ε𝔲_{2l+1} + α𝔭̄_{2l}) & 1 ≤ l ≤ k if n = 4k + 1 or n = 4k + 3 \end{matrix}

𝔲_{2l+1}^{n} ↦ \{ \begin{matrix}𝔲_{2l+1}ε + 𝔭_{2l}ᾱ + ε𝔲_{2l} − α𝔭̄_{2l+1} & 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 \\ 𝔲_{2l+1}ε − 𝔭_{2l+2}ᾱ − (ε𝔲_{2l} − α𝔭̄_{2l+1}) & 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3 \end{matrix}
𝔲_{2k}^{4k}  ↦ 𝔲_{2k}ε + α𝔭̄_{2k}
𝔲_{2k+1}^{n} ↦ \{ \begin{matrix}𝔭_{2k}ᾱ + ε𝔲_{2k} & if n = 4k \\ 𝔲_{2k+1}ε − ε𝔲_{2k} & if n = 4k + 1 with k ≥ 1 \end{matrix}
𝔲_{2k+2}^{4k+3} ↦ 𝔭_{2k+1}ᾱ − α𝔭̄_{2k+2}

𝔭_{1}^{n}  ↦ \{ \begin{matrix}𝔲_{1}α − α𝔲̄_{1} & if n = 1 \\ 𝔭_{1}ε̄+ 𝔲_{1}α − ε𝔭_{1} − α𝔲̄_{1} & if n = 4k + 1 with k ≥ 1, or n = 4k + 3 \end{matrix}

𝔭_{2l+1}^{n} ↦ \{ \begin{matrix}𝔭_{2l+1}ε̄− 𝔲_{2l+2}α + ε𝔭_{2l+2} + α𝔲̄_{2l+1} & 0 ≤ l < k if n = 4k or if n = 4k + 2 \\ 𝔭_{2l+1}ε̄+ 𝔲_{2l}α − (ε𝔭_{2l+2} + α𝔲̄_{2l+1}) & 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3 \end{matrix}

𝔭_{2l+2}^{n} ↦ \{ \begin{matrix}𝔭_{2l+2}ε̄+ 𝔲_{2l+1}α + ε𝔭_{2l+1} − α𝔲̄_{2l+2} & 0 ≤ l < k if n = 4k or n = 4k + 2 \\ 𝔭_{2l+2}ε̄− 𝔲_{2l+3}α − (ε𝔭_{2l+1} + α𝔲̄_{2l+2}) & 0 ≤ l < k if n = 4k + 1 or n = 4k + 3 \end{matrix}

𝔭_{2k+1}^{n} ↦ \{ \begin{matrix}𝔲_{2k}α − α𝔲̄_{2k+1} & if n = 4k + 1 with k ≥ 1 \\ 𝔭_{2k+1}ε̄+ α𝔲̄_{2k+1} & if n = 4k + 2 \end{matrix}

𝔭_{2k+2}^{n} ↦ \{ \begin{matrix}𝔲_{2k+1}α + ε𝔭_{2k+1} & if n = 4k + 2 \\ 𝔭_{2k+2}ε̄− ε𝔭_{2k+1} & if n = 4k + 3 \end{matrix}

together with the correspondences given by applying b, that is, for any of the above correspondences 𝔵 ↦ δ(𝔵) we also have b(𝔵) ↦ b(δ(𝔵)).

The next result is now immediate from [GHMS, Theorem 2.1].

Proposition 1.4. The complex (R_{•}, δ) is a minimal projective A-A-bimodule resolution of A, where δ_{0} : R_{0} → A is the multiplication map.

1.1. A more general class of special biserial algebras. The algebra A belongs to the wider class of special biserial algebras whose quiver is Q but where we replace the relations by αε̄ = εα = ᾱε = ε̄ᾱ = 0, ε^{r} − (αᾱ)^{s} = 0 and ε̄^{r} − (ᾱα)^{s} = 0, where r, s are integers with r ≥ 2, s ≥ 1. Let I′ be the ideal of KQ generated by these relations and denote the resulting algebra by Λ = KQ/I′. We note that Λ is Koszul if and only if r = 2 and s = 1, whence Λ = A. Although Λ is, in general, not a Koszul algebra, we may still give an explicit minimal Λ-Λ-projective bimodule resolution of Λ.

The terms of the minimal projective bimodule resolution (R̃_{•}, ∂) of Λ follow the same pattern as those of the corresponding resolution of A given in (1), that is, for all r ≥ 2 and s ≥ 1, we have R̃_{0} = P̃_{11} ⊕ P̃_{22} and, for 4k + i ≥ 1,

R̃_{4k+i} ≅ \{ \begin{matrix}P̃_{11}^{2k+1} ⊕ P̃_{12}^{2k+i} ⊕ P̃_{21}^{2k+i} ⊕ P̃_{22}^{2k+1} & if i = 0, 1 \\ P̃_{11}^{2k+i−1} ⊕ P̃_{12}^{2k+2} ⊕ P̃_{21}^{2k+2} ⊕ P̃_{11}^{2k+i−1} & if i = 2, 3 \end{matrix}

where P̃_{ij} = Λe_{i} ⊗ e_{j}Λ for i, j = 1, 2. By abuse of notation, we write 𝔲 = e_{1} ⊗ e_{1} ∈ P̃_{11}, 𝔲̄ = e_{2} ⊗ e_{2} ∈ P̃_{22}, 𝔭 = e_{1} ⊗ e_{2} ∈ P̃_{12} and 𝔭̄ = e_{2} ⊗ e_{1} ∈ P̃_{21}, and denote by b the algebra homomorphism Λ^{e} → Λ^{e} induced by the involution e_{1} ↔ e_{2}, ε ↔ ε̄, α ↔ ᾱ on Λ. We keep the same conventions on the use of superscripts and subscripts as those preceding Definition 1.3, and make the following definition of ∂_{n}; again we omit the superscripts on the right hand side with the understanding that all the terms are in R̃_{n−1}.

Definition 1.5. We define a Λ-Λ-bimodule homomorphism ∂_{n} : R̃_{n} → R̃_{n−1}, for all n ≥ 1, by

𝔲_{1}^{n}  ↦ \{ \begin{matrix}𝔲_{1}ε^{r−1} − 𝔭_{1}ᾱ(αᾱ)^{s−1} + ε^{r−1}𝔲_{1} − (αᾱ)^{s−1}α𝔭̄_{1} & if n = 4k or n = 4k + 2 \\ 𝔲_{1}ε^{r−1} − ε^{r−1}𝔲_{1} & if n = 1 \\ 𝔲_{1}ε^{r−1} − 𝔭_{2}ᾱ(αᾱ)^{s−1} − (ε^{r−1}𝔲_{1} − (αᾱ)^{s−1}α𝔭̄_{1}) & if n = 4k + 1 with k ≥ 1, or n = 4k + 3 \end{matrix}

𝔲_{2l}^{n}  ↦ \{ \begin{matrix}𝔲_{2l}ε − 𝔭_{2l+1}ᾱ(αᾱ)^{s−1} + ε^{r−1}𝔲_{2l+1} + α𝔭̄_{2l} & 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 \\ 𝔲_{2l}ε^{r−1} + 𝔭_{2l−1}ᾱ − (ε^{r−1}𝔲_{2l+1} + α𝔭̄_{2l}) & 1 ≤ l ≤ k if n = 4k + 1 or if n = 4k + 3 \end{matrix}

𝔲_{2l+1}^{n} ↦ \{ \begin{matrix}𝔲_{2l+1}ε^{r−1} + 𝔭_{2l}ᾱ(αᾱ)^{s−1} + ε𝔲_{2l} − (αᾱ)^{s−1}α𝔭̄_{2l+1} & 1 ≤ l < k if n = 4k or 1 ≤ l ≤ k if n = 4k + 2 \\ 𝔲_{2l+1}ε − 𝔭_{2l+2}ᾱ(αᾱ)^{s−1} − (ε𝔲_{2l} − (αᾱ)^{s−1}α𝔭̄_{2l+1}) & 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3 \end{matrix}
𝔲_{2k}^{4k}  ↦ 𝔲_{2k}ε + α𝔭̄_{2k}
𝔲_{2k+1}^{n} ↦ \{ \begin{matrix}𝔭_{2k}ᾱ + ε𝔲_{2k} & if n = 4k \\ 𝔲_{2k+1}ε − ε𝔲_{2k} & if n = 4k + 1 with k ≥ 1 \end{matrix}
𝔲_{2k+2}^{4k+3} ↦ 𝔭_{2k+1}ᾱ − α𝔭̄_{2k+2}

𝔭_{1}^{n}  ↦ \{ \begin{matrix}𝔲_{1}α − α𝔲̄_{1} & if n = 1 \\ 𝔭_{1}ε̄^{r−1} + 𝔲_{1}α − ε^{r−1}𝔭_{2} − α𝔲̄_{1} & if n = 4k + 1 with k ≥ 1 or n = 4k + 3 \end{matrix}

𝔭_{2l+1}^{n} ↦ \{ \begin{matrix}𝔭_{2l+1}ε̄− 𝔲_{2l+2}α(αᾱ)^{s−1} + ε^{r−1}𝔭_{2l+2} + α𝔲̄_{2l+1} & 0 ≤ l < k if n = 4k or if n = 4k + 2 \\ 𝔭_{2l+1}ε̄^{r−1} + 𝔲_{2l}α − (ε^{r−1}𝔭_{2l+2} + α𝔲̄_{2l+1}) & 1 ≤ l < k if n = 4k + 1 or 1 ≤ l ≤ k if n = 4k + 3 \end{matrix}

𝔭_{2l+2}^{n} ↦ \{ \begin{matrix}𝔭_{2l+2}ε̄^{r−1} + 𝔲_{2l+1}α + ε𝔭_{2l+1} − (αᾱ)^{s−1}α𝔲̄_{2l+2} & 0 ≤ l < k if n = 4k or if n = 4k + 2 \\ 𝔭_{2l+2}ε̄− 𝔲_{2l+3}α(αᾱ)^{s−1} − (ε𝔭_{2l+1} − (αᾱ)^{s−1}α𝔲̄_{2l+2}) & 0 ≤ l < k if n = 4k + 1 or n = 4k + 3 \end{matrix}

𝔭_{2k+1}^{n} ↦ \{ \begin{matrix}𝔲_{2k}α − α𝔲̄_{2k+1} & if n = 4k + 1 with k ≥ 1 \\ 𝔭_{2k+1}ε̄+ α𝔲̄_{2k+1} & if n = 4k + 2 \end{matrix}

𝔭_{2k+2}^{n} ↦ \{ \begin{matrix}𝔲_{2k+1}α + ε𝔭_{2k+1} & if n = 4k + 2 \\ 𝔭_{2k+2}ε̄^{r−1} − ε𝔭_{2k+1} & if n = 4k + 3 \end{matrix}

together with the correspondences given by applying b, that is, for any 𝔵 ↦ ∂(𝔵) given above, we also have b(𝔵) ↦ b(∂(𝔵)).

Proposition 1.6. The complex (R̃_{•}, ∂) is a minimal projective Λ-Λ-bimodule resolution of Λ, where ∂_{0} : R̃_{0} → Λ is the multiplication map.

Proof: A straightforward verification shows that (R̃_{•}, ∂) is a complex. The fact that this complex is exact follows from an analogous argument to the one in [GS, Proposition 2.8]. □

2. A basis of the Hochschild cohomology ring

In this section we return to the algebra A and give a basis of the Hochschild cohomology ring HH^{∗}(A) in the case where char K ≠ 2. By [ESc] the dimensions of the Hochschild cohomology groups HH^{n}(A) are as follows

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

and the set {e_{1} + e_{2}, ε, ε̄, ε^{2}, ε̄^{2}} is a K-basis of HH^{0}(A).

We describe the elements of a basis of each Hochschild cohomology group HH^{n}(A), for n ≥ 1, in terms of cocycles in Hom(R_{n}, A). We write δ_{n} for both the map δ_{n}: R_{n} → R_{n−1} and for the induced map Hom(R_{n−1}, A) → Hom(R_{n}, A). Thus our basis is given in terms of a set of elements in Ker δ_{n+1} as a subset of Hom(R^{n}, A), such that the corresponding cosets in Ker δ_{n+1}/ Im δ_{n} form a basis of HH^{n}(A) = Ker δ_{n+1}/ Im δ_{n}. We keep the notation of section 1. When describing a cocycle in Hom(R_{n}, A), we simply write the images of the generators 𝔲_{i}^{n}, 𝔲̄_{i}^{n}, 𝔭_{j}^{n} or 𝔭̄_{j}^{n} in R_{n} where those images are non-zero.

It is easy to verify that the following elements are indeed cocycles in Hom(R_{n}, A). They are used to define a basis of the Hochschild cohomology groups in Theorem 2.1.

(i) Suppose i = 0 and k ≥ 1 or i = 2 and k ≥ 0.

(a) Let ϕ^{4k+i} : \begin{cases}𝔲_{1}^{4k+i} ↦ e_{1}, \\ 𝔲̄_{1}^{4k+i} ↦ e_{2}. \end{cases}

(b) For 1 ≤ l ≤ k, let θ_{l}^{4k+i} : \{ \begin{matrix}𝔲_{2l}^{4k+i} ↦ e_{1}, & 𝔲_{2l+1}^{4k+i} ↦ e_{1}, \\ 𝔲̄_{2l}^{4k+i} ↦ e_{2}, & 𝔲̄_{2l+1}^{4k+i} ↦ e_{2}. \end{matrix}

(c) For 1 ≤ l ≤ k, let ψ_{l}^{4k+i} : 𝔲_{2l}^{4k+i} ↦ ε^{2}.

(ii) Suppose i = 1 or i = 3 and that k ≥ 0. For 1 ≤ l ≤ 2k + 1 if i = 1 and 1 ≤ l ≤ 2k + 2 if i = 3, let

χ_{l}^{4k+i} : \{ \begin{matrix}𝔲_{l}^{4k+i} ↦ ε, & 𝔲̄_{l}^{4k+i} ↦ ε̄, \\ 𝔭_{l}^{4k+i} ↦ (−1)^{l+1}α, & 𝔭̄_{l}^{4k+i} ↦ (−1)^{l+1}ᾱ. \end{matrix}

Theorem 2.1. Suppose char K ≠ 2, and let k ≥ 0 and 0 ≤ i ≤ 3 with 4k + i > 0.

(1) For i = 0 or 2, HH^{4k+i}(A) has basis {ϕ^{4k+i}, εϕ^{4k+i}, ε̄ϕ^{4k+i}, ψ_{l}^{4k+i}, θ_{l}^{4k+i} | 1 ≤

l ≤ k}.

(2) For i = 1 or 3, HH^{4k+i}(A) has basis {χ_{l}^{4k+i}, εχ_{1}^{4k+i}, ε̄χ_{1}^{4k+i} | 1 ≤ l ≤ m_{i}}, where

m_{i} = 2k + 1 if i = 1 and m_{i} = 2k + 2 if i = 3.

We omit the proof since it is straightforward to check these maps represent linearly independent non-zero elements in HH^{4k+i}(A).

3. Fg1. and Fg2.

Throughout this section, suppose again that char K is arbitrary. In [SS], the Hochschild cohomology ring was used to define the support variety of a finitely generated module over any finite-dimensional algebra. It was shown in [EHSST] that, when certain (reasonable) finiteness conditions hold, then the support varieties of [SS] share many of the analogous properties of support varieties for finite group rings or co-commutative Hopf algebras. We start by stating these finiteness conditions and showing that they hold in our setting.

Let J(A) denote the Jacobson radical of A. Then the Yoneda algebra or Ext algebra of A is given by E(A) = Ext^{∗}(A/J(A), A/J(A)) with the Yoneda product. We use the notation E(A)^{n} = Ext^{n}(A/J(A), A/J(A)) for the nth-graded component of E(A). The finiteness conditions of [EHSST] are:

Fg1. There is a graded subalgebra H of HH^{∗}(A) such that H is a commutative Noe-

therian ring and H^{0} = HH^{0}(A). Fg2. E(A) is a finitely generated H-module.

The algebra A is Koszul, so we know from [BGSS] that the image of the natural ring homomorphism HH^{∗}(A) → E(A) is precisely the graded centre Z_{gr}(E(A)) of E(A), where Z_{gr}(E(A)) is the subring of E(A) generated by all homogeneous elements z in E(A)^{n} (n ≥ 0) such that xy = (−1)^{nm}yx for all y ∈ E(A)^{m}. Moreover, E(A) is given by quiver and relations as E(A) = KQ/I^{⊥} where I^{⊥} = ⟨ε^{2} + αᾱ, ε̄^{2} + ᾱα⟩ (from [GM, Theorem 2.2]). Thus E(A) is the preprojective algebra associated to the simply laced extended Dynkin graph of type T̃_{2}.

Now, it is easy to check that the elements

x = ε^{2} + ε̄^{2}

and

z = εαε̄ᾱ + αε̄ᾱε + ε̄ᾱεα + ᾱεαε̄

are both in Z_{gr}(E(A)).

Theorem 3.1. As a left K[x, z]-module, E(A) is finitely generated with generating set

S = {e_{1}, e_{2}, ε, α, ε̄, ᾱ, εα, ε̄ᾱ, αε̄, ᾱε, εαε̄, ε̄ᾱε, ᾱεα, αεᾱ, εαε̄ᾱ, ε̄ᾱεα}.

Before proving Theorem 3.1, we consider the conditions Fg1. and Fg2.. The element ϕ^{2} ∈ HH^{2}(A) is a pre-image of x and the element θ_{1}^{4} ∈ HH^{4}(A) is a pre-image of z. Let H be the graded subalgebra of HH^{∗}(A) generated by HH^{0}(A), ϕ^{2} and θ_{1}^{4}, so that H is a pre-image of K[x, z] in HH^{∗}(A). Then we have the following immediate consequence of Theorem 3.1.

Corollary 3.2. The conditions Fg1. and Fg2. hold for the algebra A with respect to the subring H of HH^{∗}(A).

This result was independently shown by Erdmann and Solberg in [ESo], where they consider all radical cube zero weakly symmetric algebras. We observe that Corollary 3.2 does not require the explicit structure of either Z_{gr}(E(A)) or HH^{∗}(A), and that it now follows from [So] that Fg1. and Fg2. also hold with respect to HH^{even}(A) ⊃ H.

In order to prove Theorem 3.1, we need the following lemma.

Lemma 3.3. For any h ∈ S and any arrow x in Q we have hx = ∑_{s∈S} z_{s}s where z_{s} ∈ Z_{gr}(E(A)).

Proof: We will show the result for x = α and x = ε; the cases x = ᾱ and x = ε̄ are analogous.

Clearly the result holds for α, εα, ε̄ᾱα, ᾱεα and ε̄ᾱεα . We note that ᾱα = −xe_{2}, ε^{2}α = xα and αε̄ᾱα = −ε^{2}αε̄ = −xαε̄. Finally, εαε̄ᾱα = −ε^{3}αε̄ = −xεαε̄. Again the result immediately holds for ε, ᾱε and ε̄ᾱε. Furthermore, ε^{2} = xe_{1}, ᾱε^{2} = xε̄ᾱ, αε̄ᾱε = ze_{1} − εαε̄ᾱ and εαε̄ᾱε = zε − xαε̄ᾱ. □

Proof of Theorem 3.1: A proof by induction will show that S is a generating set of E(A). More precisely, we show by induction on n that each element of E(A)^{n} can be expressed as a linear combination of elements in S with coefficients in Z_{gr}(E(A)).

The base case holds since e_{1} and e_{2} are in S. Suppose that the result holds for E(A)^{n}, that is, every x ∈ E(A)^{n} is of the form x = ∑_{s∈S} z_{s}s with z_{s} ∈ Z_{gr}(E(A)). Since E(A) is Koszul, an element x ∈ E(A)^{n+1} is a linear combination of elements of the form yγ where y ∈ E(A)^{n} and γ is an arrow of Q. By induction y = ∑_{s∈S} z_{s}s with z_{s} ∈ Z_{gr}(E(A)) and thus yγ = ∑_{s∈S} z_{s}sγ. By Lemma 3.3 we have y = ∑_{s∈S} z_{s}(∑_{t∈S} z′_{t}t) with z′_{t} ∈ Z_{gr}(E(A)) and therefore y = ∑_{s∈S} ∑_{t∈S}(z_{s}z′_{t})t. Thus x has the required form. □

Following [SS], the variety of a finitely generated A-module M with respect to H is given by

V_{H}(M) = MaxSpec(H/A_{H}(M, M))

where A_{H}(M, M) is the annihilator of Ext_{A}^{∗}(M, M) as an H-module. Moreover, V_{H}(A/J(A)) = MaxSpec H. Corollary 3.2 and [EHSST, Theorems 2.5(b), 5.3] have the following direct consequences for our algebra A.

Proposition 3.4. (1) HH^{∗}(A) is a finitely generated K-algebra.

(2) For M ∈ mod A, the variety of M is trivial if and only if M is projective.

(3) For M ∈ mod A and M indecomposable, the variety of M is a line if and only if M is periodic.

We now consider the simple A-modules S_{1} and S_{2} (corresponding to the vertices 1 and 2 respectively of the quiver Q). From [SS, Proposition 3.4], we have that V_{H}(A/J(A)) = V_{H}(S_{1}) ∪ V_{H}(S_{2}) and V_{H}(S_{1}) = V_{H}(rad(e_{1}A)). Using the exact sequence 0 → S_{1} → rad(e_{1}A) → A/J(A) → 0, we have V_{H}(A/J(A)) ⊆ V_{H}(S_{1}) ∪ V_{H}(rad(e_{1}A)) = V_{H}(S_{1}). Thus V_{H}(A/J(A)) = V_{H}(S_{1}). A similar argument for S_{2} proves the following proposition.

Proposition 3.5. V_{H}(S_{1}) = V_{H}(S_{2}) = MaxSpec H.

This is, in fact, a special case of the following result.

Proposition 3.6. Suppose that Ã is a selfinjective algebra and that H̃ is a graded subalgebra of HH^{∗}(Ã) containing HH^{0}(Ã). Let S be the simple module corresponding to the indecomposable projective P, so that P/ rad P ≅ S. If soc P ≅ S and if Ã/J(Ã) is a summand of rad P/ soc P then VH̃(S) = VH̃(Ã/J(Ã)) = MaxSpec H̃.

Finally, we remark that it can be verified that Z_{gr}(E(A)) and K[x, z] are isomorphic as algebras. Since A is Koszul, we know from [BGSS] that the Hochschild cohomology ring of A modulo nilpotence is isomorphic to Z_{gr}(E(A)) modulo nilpotence. Hence HH^{∗}(A)/N ≅ K[x, z], where N denotes the ideal of HH^{∗}(A) generated by all nilpotent elements. In contrast, for a Hecke algebra B of finite type, the results of [EH] show that the Hochschild cohomology ring modulo nilpotence of B is isomorphic to K[y], where y is in degree m ≥ 1 and m is minimal such that Ω_{B^{e}}^{m}(B) ≅ B as B-B-bimodules. Thus we have the following description of the Hochschild cohomology ring modulo nilpotence for all Hecke algebras of tame and of finite type.

Theorem 3.7. The Hochschild cohomology ring modulo nilpotence of a Hecke algebra has Krull dimension 1 if the algebra is of finite type and has Krull dimension 2 if the algebra is of tame type.

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Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

E-mail address: schroll@mcs.le.ac.uk

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

E-mail address: N.Snashall@mcs.le.ac.uk