Submitted exclusively to the London Mathematical Society DOI: 10.1112/S0000000000000000

SUPPORT VARIETIES AND HOCHSCHILD COHOMOLOGY RINGS

NICOLE SNASHALL and ØYVIND SOLBERG

Introduction

A powerful invariant in the modular representation theory of a finite group is the support variety of a module (see [6], [7], [26]). This is defined in terms of the maximal ideal spectrum of the group cohomology ring H^{∗}(G, k) = ⊕_{i≥0} Ext_{kG}^{i}(k, k) for a finite group G and a field k. The variety of a module M is given as the variety of the kernel of the homomorphism of graded rings − ⊗_{k} M: H^{∗}(G, k) → Ext_{kG}^{∗}(M, M), where Ext_{kG}^{∗}(M) = ⊕_{i≥0} Ext_{kG}^{i}(M, M). This morphism factors through the ring homomorphism HH^{∗}(kG) → Ext_{kG}^{∗}(M, M) given by − ⊗_{kG} M, where HH^{∗}(kG) is the Hochschild cohomology ring of kG. More generally for any k-algebra Λ, the map HH^{∗}(Λ) → Ext_{Λ}^{∗}(M, M) given by φ_{M} = − ⊗_{Λ} M is a homomorphism of graded rings. Our aim in this paper is to study and develop the theory of support varieties in this general setting using the ring homomorphisms φ_{M} for Λ-modules M. This is done by first considering varieties of pairs of modules.

Let Λ be an algebra over a commutative ring k, where we assume throughout that Λ is flat as a module over k. Denote by Λ^{e} the enveloping algebra Λ ⊗_{k} Λ^{op}. The heart of the matter is the graded ring HH^{∗}(Λ) = ⊕_{i≥0} Ext_{Λ^{e}}^{i}(Λ, Λ) and the homomorphism of graded rings φ_{M}: HH^{∗}(Λ) → Ext_{Λ}^{∗}(M, M) for any left Λ-module M. We begin by showing that the graded ring HH^{∗}(Λ) is graded commutative, that is: ηη′ = (−1)^{mn}η′η for two homogeneous elements η and η′ of degree m and n, respectively. If Λ is projective as a module over k, this is a well-known result, as then Ext_{Λ^{e}}^{∗}(Λ, Λ) is isomorphic to the Hochschild cohomology ring of Λ. Furthermore, we show that for each Λ-module M the morphism φ_{M}: HH^{∗}(Λ) → Ext_{Λ}^{∗}(M, M) induces a left and a right HH^{∗}(Λ)-module structure on Ext_{Λ}^{∗}(M, N) = ⊕_{i≥0} Ext_{Λ}^{i}(M, N) for each pair of Λ-modules M and N. In addition these module structures are graded commutative as φ_{N}(η)θ = (−1)^{mn}θφ_{M}(η) for an element η in HH^{n}(Λ) and θ in Ext_{Λ}^{m}(M, N). This enables us to show in section 2 that the annihilators of Ext_{Λ}^{∗}(M, N) for any graded subalgebra H of HH^{∗}(Λ) as a left and as a right H-module are equal and graded ideals. This is the basis for the theory of support varieties that we develop in this paper.

Let H = ⊕_{i≥0}H^{i} be a graded subring of HH^{∗}(Λ), where we assume that H^{0} is a local k-algebra. Since H is graded commutative, any maximal ideal in H contains all homogeneous nilpotent elements in H and rad(H^{0}). Hence, since we are interested in the maximal ideal spectrum in H, we observe that this spectrum is in 1-1 correspondence with the maximal ideal spectrum of \overline{H} = H/N_{H}, where N_{H} is the

2000 Mathematics Subject Classification 16E40, 16G10, 16P10, 16P20 (primary), 13D03, 16G70, 20J06 (secondary).

The first author was partially supported by a grant from the Research Council of Norway. The second author was supported by an LMS scheme 4 grant.

2 nicole snashall and Øyvind solberg

ideal in H generated by all the homogeneous nilpotent elements and rad(H^{0}). For an ideal I in \overline{H} denote by I′ the inverse image of I in H. Then for any pair M and N of Λ-modules, we define the support variety V_{H}(M, N) in \overline{H} by

V_{H}(M, N) = \{𝔪 ∈ MaxSpec \overline{H} | A_{H}(M, N) ⊆ 𝔪′\}.

We end section 2 by giving several basic properties of these varieties.

Very little is known about the graded ring HH^{∗}(Λ) in general for an arbitrary k-algebra Λ beyond being graded commutative. For artin algebras or even better for finite dimensional algebras over a field there are some results on the ring structure of HH^{∗}(Λ). Therefore in section 3 we restrict ourselves to considering artin algebras Λ over a commutative artinian ring k, where Λ is flat as a module over k. The Jacobson radical of Λ is denoted by 𝔯. Furthermore we consider only finitely generated modules, and we denote by mod Λ the category of finitely generated (left) Λ-modules. Since every finitely generated module has a finite (radical) filtration in (semi-) simple modules, the semisimple module Λ/𝔯 is important. After having shown that for any graded subalgebra H of HH^{∗}(Λ) we have V_{H}(M, Λ/𝔯) = V_{H}(M, M) = V_{H}(Λ/𝔯, M) for any module M in mod Λ, we define the support variety V_{H}(M) of a module M in mod Λ as V_{H}(M, Λ/𝔯). We develop the basic theory here generalising known results for group algebras. For a finite dimensional selfinjective algebra Λ over a field, we then show that the variety of a module M and its dual are equal, and so prove one of our main results showing that the variety is an invariant for a stable component of the Auslander-Reiten quiver.

All modules M with finite projective or finite injective dimension or with Ext_{Λ}^{i}(M, M) = (0) for i ≫ 0 have trivial variety. We do not know a characterisation of when a module has trivial variety for a general artin algebra. However, another natural question is when the support variety of every Λ-module is trivial, which we address in section 4. This is characterised by all homogeneous elements in H^{≥1} being nilpotent. Then, if all homogeneous elements in HH^{≥1}(Λ) are nilpotent and HH^{i}(Λ) ≠ (0) for infinitely many integers i > 0, it is shown that HH^{∗}(Λ) is infinitely generated as a graded algebra. Since the homogeneous nilpotent elements in H constitute uninteresting elements in terms of the support varieties which we would like to factor out, we investigate homogeneous nilpotent elements. In particular several classes of homogeneous nilpotent elements are given assuming certain conditions on Λ. For instance, if k is a field and Λ is a finite dimensional indecomposable k-algebra such that Λ/𝔯 is a separable algebra over k, we prove that the ideal generated by the homogeneous nilpotent elements is equal to \sqrt{Ker φ_{Λ/𝔯}}.

The results from section 4 naturally lead to the question of when the Hochschild cohomology ring is finitely generated as an algebra. It is unknown in general whether or not the ring HH^{∗}(Λ)/N is finitely generated as an algebra where N is the ideal of HH^{∗}(Λ) generated by all homogeneous nilpotent elements and rad(HH^{0}(Λ)). However we conjecture that HH^{∗}(Λ)/N is finitely generated as an algebra when Λ is an artin algebra over a commutative artinian ring k with Λ flat as a module over k. This has been answered affirmatively for finite dimensional selfinjective algebras of finite representation type over an algebraically closed field in [18]. Indeed in section 5 we show, for k a field and Λ = kQ/I a finite dimensional Nakayama algebra where I is an admissible ideal generated by a single relation, that HH^{∗}(Λ)/N is always finitely generated as a ring. Moreover for this class of rings the Hochschild cohomology ring exhibits all the possible behaviours, in that it can be finite dimensional, finitely generated as a ring, or infinitely generated as a ring. The above conjecture

support varieties and hochschild cohomology rings 3

is also known to be true for finite dimensional monomial algebras [17], any block of a group ring of a finite group [9, 27] and any block of a finite dimensional cocommutative Hopf algebra. In the last case, this can be seen using [15] and arguments from [8] which were pointed out to us by Ragnar-Olaf Buchweitz. An alternative reference would be [16], which was given to us by Eric M. Friedlander and Julia Pevtsova.

In section 6 we introduce the concept of complexity for a pair of modules and give some elementary properties of the complexity of modules.

The final section is devoted to comparing the support varieties for complete intersections defined by Avramov in [2], the support varieties for principal and non-principal blocks defined by Carlson and Linckelmann et al. (see [6], [22]) with the support varieties defined by a suitable graded subalgebra of the Hochschild cohomology ring of the ring as given in section 2. In these cases we recall or show that there exists a finite surjective map or an isomorphism from the support varieties defined by (a suitable graded subalgebra of) the Hochschild cohomology ring of the algebra and the support variety considered in these situations. Support varieties have also been studied for restricted Lie algebras; however we do not consider these, but refer the reader to [15].

1. Preliminary results

Let Λ and Γ be two algebras over a commutative ring k. Denote by Λ^{e} the enveloping algebra Λ ⊗_{k} Λ^{op}. Our main interest in this section is the graded ring Ext_{Λ^{e}}^{∗}(Λ, Λ), where Ext_{Λ^{e}}^{n}(Λ, Λ) is the set of homogeneous elements of degree n. If Λ is projective as a module over k, then the ring Ext_{Λ^{e}}^{∗}(Λ, Λ) is known to be isomorphic to the Hochschild cohomology ring HH^{∗}(Λ) of Λ. The Hochschild cohomology ring is known to be a graded commutative ring, that is: ηη′ = (−1)^{mn}η′η for two homogeneous elements η and η′ of degree m and n, respectively. Under the weaker assumption that Λ is flat as a module over k, we show that Ext_{Λ^{e}}^{∗}(Λ, Λ) is graded commutative. Furthermore, for each Λ-Γ-bimodule M there is a homomorphism of graded rings φ_{M}: Ext_{Λ^{e}}^{∗}(Λ, Λ) → Ext_{Λ⊗_{k}Γ^{op}}^{∗}(M, M) given by φ_{M}(η) = η ⊗_{Λ} M. This induces a left and a right Ext_{Λ^{e}}^{∗}(Λ, Λ)-module structure on Ext_{Λ⊗_{k}Γ^{op}}^{∗}(M, N) for each pair of Λ-Γ-bimodules M and N. For an element η in Ext_{Λ^{e}}^{n}(Λ, Λ) and θ in Ext_{Λ⊗_{k}Γ^{op}}^{m}(M, N) we prove that φ_{N}(η)θ = (−1)^{mn}θφ_{M}(η). This is the basis for the theory of support varieties developed later in the paper. By abuse of notation we denote the graded ring Ext_{Λ^{e}}^{∗}(Λ, Λ) by HH^{∗}(Λ) even when Λ is not projective over k.

We first prove the result from which all the results in this section are derived.

Theorem 1.1. Let Λ and Γ be two algebras over a commutative ring k. Assume that Λ is flat as a module over k. Let η be an element in HH^{n}(Λ), and let θ be an element in Ext_{Λ⊗_{k}Γ^{op}}^{m}(M, N) for two Λ-Γ-bimodules M and N. Then φ_{N}(η)θ = (−1)^{mn}θφ_{M}(η).

Proof. Let M and N be two Λ-Γ-bimodules. Let ⋯ → P^{n} → P^{n−1} → ⋯ → P^{1} → P^{0} → Λ → 0 be a projective Λ^{e}-resolution of Λ. Since Λ is flat over k, the enveloping algebra Λ^{e} is flat as a left and as a right Λ-module. Consequently any projective Λ^{e}-module is flat as a left and as a right Λ-module. Since the flat left and the flat right Λ-modules are closed under kernels of epimorphisms, all the syzygies 4 nicole snashall and Øyvind solberg

of Λ over Λ^{e} are also flat as a left and as a right Λ-module. Suppose η is given by 0 → Λ → E → P^{n−2} → ⋯ → P^{1} → P^{0} → Λ → 0.

First we consider the case when θ is in Hom_{Λ⊗_{k}Γ^{op}}(M, N). Since HH^{0}(Λ) = Hom_{Λ^{e}}(Λ, Λ) = Z(Λ) (the centre of Λ), the claim is clear for η in HH^{0}(Λ). Suppose η is in HH^{1}(Λ). Then there is a commutative diagram

[figure: 0; Λ ⊗_{Λ} M; id⊗θ; E ⊗_{Λ} M; id⊗θ; Λ ⊗_{Λ} M; 0; 0; Λ ⊗_{Λ} N; X; Λ ⊗_{Λ} M; id⊗θ; 0; 0; Λ ⊗_{Λ} N; E ⊗_{Λ} N; Λ ⊗_{Λ} N; 0]

which shows that (η ⊗_{Λ} N)θ = θ(η ⊗_{Λ} M). This argument is easily extended to any homogeneous element η in HH^{∗}(Λ).

Next let θ: 0 → N → X → M → 0 be in Ext_{Λ⊗_{k}Γ^{op}}^{1}(M, N). Since all the syzygies of Λ over Λ^{e} are flat as a right Λ-module, we have an exact and commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Ω_{Λ^{e}}^{i}(Λ) ⊗_{Λ} N \ar[r] \ar[d] & P^{i−1} ⊗_{Λ} N \ar[r] \ar[d] & Ω_{Λ^{e}}^{i−1}(Λ) ⊗_{Λ} N \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ω_{Λ^{e}}^{i}(Λ) ⊗_{Λ} X \ar[r] \ar[d] & P^{i−1} ⊗_{Λ} X \ar[r] \ar[d] & Ω_{Λ^{e}}^{i−1}(Λ) ⊗_{Λ} X \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ω_{Λ^{e}}^{i}(Λ) ⊗_{Λ} M \ar[r] \ar[d] & P^{i−1} ⊗_{Λ} M \ar[r] \ar[d] & Ω_{Λ^{e}}^{i−1}(Λ) ⊗_{Λ} M \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

for all i, where Ω_{Λ^{e}}^{0}(Λ) = Λ. Denote the upper row by σ_{i}, the right most column by θ_{i}, the left most column by θ_{i+1} and the lower row by σ′_{i}. Then by [24, Lemma 3.1, Chapter VIII] the equality σ_{i}θ_{i} = −θ_{i+1}σ′_{i} holds for all i ≥ 1. Since η ⊗_{Λ} N = (0 → N → E ⊗_{Λ} N → Ω_{Λ^{e}}^{n−1}(Λ) ⊗_{Λ} N → 0)σ_{n−1} ⋯ σ_{2}σ_{1} and η ⊗_{Λ} M = (0 → M → E ⊗_{Λ} M → Ω_{Λ^{e}}^{n−1}(Λ) ⊗_{Λ} M → 0)σ′_{n−1} ⋯ σ′_{2}σ′_{1}, by iterated use of the above (η ⊗_{Λ} N)θ = (−1)^{n}θ(η ⊗_{Λ} M).

When θ is in Ext_{Λ⊗_{k}Γ^{op}}^{m}(M, N), the element θ can be viewed as a product of m short exact sequences. Then it follows easily that (η ⊗_{Λ} N)θ = (−1)^{mn}θ(η ⊗_{Λ} M) for η in HH^{n}(Λ) and θ in Ext_{Λ⊗_{k}Γ^{op}}^{m}(M, N).

As mentioned above, the ring Ext_{Λ^{e}}^{∗}(Λ, Λ) is isomorphic to the Hochschild cohomology ring of the algebra Λ when Λ is projective over k. Then the well-known fact that the Hochschild cohomology ring is graded commutative, is an immediate consequence of the above result by specialising to the case where Γ = Λ.

support varieties and hochschild cohomology rings 5

Corollary 1.2. Let Λ be an algebra over a commutative ring k, where Λ is flat as a module over k.

(a) The ring HH^{∗}(Λ) = Ext_{Λ^{e}}^{∗}(Λ, Λ) is graded commutative.

(b) If Λ is projective as a module over k, then the Hochschild cohomology ring of Λ is graded commutative.

Using that HH^{∗}(Λ) is graded commutative, it is straightforward to see that the square of any homogeneous element of odd degree is zero when the characteristic of k is different from 2.

The next result is the base on which the theory of support varieties developed later in the paper is built. It is another immediate consequence of Theorem 1.1.

Corollary 1.3. Let Λ be an algebra over a commutative ring k, where Λ is flat as a module over k. Then the left and the right HH^{∗}(Λ)-module structures induced by φ_{N} and φ_{M} on Ext_{Λ}^{∗}(M, N) are related as follows: if θ is an element in Ext_{Λ}^{m}(M, N) and η is in HH^{n}(Λ), then φ_{N}(η)θ = (−1)^{mn}θφ_{M}(η).

2. Support varieties

Throughout this section let Λ be an algebra over a commutative ring k, where Λ is flat as a module over k. Let H = ⊕_{i≥0}H^{i} be a graded subring of HH^{∗}(Λ). This section is devoted to defining the support variety of a pair of modules (M, N) using the ring homomorphism H ⟶^{inc} HH^{∗}(Λ) → Ext_{Λ}^{∗}(M, M), in a way similar to that for group rings of finite groups. Furthermore we develop some elementary properties of these varieties.

To each pair of Λ-modules (M, N) we associate two ideals A_{H}^{l}(M, N) and A_{H}^{r}(M, N) which are the annihilator of Ext_{Λ}^{∗}(M, N) as a left H-module and as a right H-module, respectively. First we show that these are graded ideals and also that they are equal.

Lemma 2.1. Let M and N be two Λ-modules. The ideals A_{H}^{l}(M, N) and A_{H}^{r}(M, N) are equal and graded ideals in H. Moreover, if Ext_{Λ}^{∗}(M, N) is nonzero, then A_{H}^{l}(M, N) = A_{H}^{r}(M, N) is a proper ideal.

Proof. Let x be in H. Then x = ∑_{i=0}^{N} x_{i} for some x_{i} in H^{i} and natural number N. Suppose xm = 0 for all m in Ext_{Λ}^{∗}(M, N). Then, in particular, xm = 0 for all homogeneous elements m, say m = m_{j} in Ext_{Λ}^{j}(M, N). Then xm = ∑_{i=0}^{N} x_{i}m_{j} = 0 if and only if x_{i}m_{j} = 0 for all i = 0, 1, …, N. Hence, if x Ext_{Λ}^{∗}(M, N) = (0), then all the homogeneous parts of x, the elements x_{i}, annihilate all homogeneous elements in Ext_{Λ}^{∗}(M, N) and therefore any element in Ext_{Λ}^{∗}(M, N). This shows that the left annihilator of Ext_{Λ}^{∗}(M, N) is a graded left ideal. Similarly the right annihilator of Ext_{Λ}^{∗}(M, N) is a graded right ideal. Using this and Corollary 1.3 we infer that the left and the right annihilator coincide, and therefore A_{H}^{l}(M, N) = A_{H}^{r}(M, N) is a graded ideal in H.

If Ext_{Λ}^{∗}(M, N) is nonzero, then Ext_{Λ}^{i_{0}}(M, N) ≠ (0) for some i_{0} ≥ 0. Therefore the identity in H is not in A_{H}^{l}(M, N), and hence A_{H}^{l}(M, N) = A_{H}^{r}(M, N) is a proper ideal.

We write A_{H}(M, N) for the graded ideal A_{H}^{l}(M, N) = A_{H}^{r}(M, N).

One of our standard assumptions throughout the rest of this section is that H^{0} is a local k-algebra. Under this assumption the maximal ideal 𝔞 = ⟨rad(H^{0}), H^{≥1}⟩ in H has a central role, as this ideal defines the modules with trivial variety. The modules with trivial variety over a group ring of a finite group correspond exactly to the projective modules. In our setting we see later in this section that the class of modules with trivial variety contains modules of finite injective dimension, of finite projective dimension and with no self extensions. Furthermore, note that H^{0} is a local k-algebra when Λ is an indecomposable artin k-algebra, k is a commutative artinian ring and H = HH^{∗}(Λ). (We keep the assumption that Λ is flat as a module over k.)

Next we show that the ideal 𝔞 contains the ideal A_{H}(M, N) for any pair of modules (M, N) whenever Ext_{Λ}^{∗}(M, N) ≠ (0).

Lemma 2.2. Let M and N be two Λ-modules.

(a) If Ext_{Λ}^{∗}(M, N) is nonzero, then the ideal A_{H}(M, N) is contained in the ideal 𝔞 = ⟨rad(H^{0}), H^{≥1}⟩ of H.

(b) If 𝔪 is a nonzero maximal ideal in H, then 𝔪 contains the ideal generated by rad(H^{0}).

Proof. (a) Let M and N be two nonzero left Λ-modules such that Ext_{Λ}^{∗}(M, N) is nonzero. Let y be an element in A_{H}(M, N), and write y = ∑_{i=0}^{n} y_{i}, where y_{i} is in H^{i} for all i. It is enough to prove that y_{0} is in rad(H^{0}). Since A_{H}(M, N) is a graded ideal in H, the element y_{0} is in A_{H}(M, N).

Assume that y_{0} is not in rad(H^{0}). Since H^{0} is a local ring, y_{0} is an invertible element. This contradicts the fact that y_{0} is in A_{H}(M, N), hence y_{0} is in rad(H^{0}). The claim in (a) follows from this.

(b) Let 𝔪 be a nonzero maximal ideal in H. Then the degree zero part 𝔪_{0} of 𝔪 is a proper ideal in H^{0}, and therefore 𝔪_{0} is contained in rad(H^{0}). Hence 𝔪 is contained in the proper ideal ⟨rad(H^{0}), ⊕_{i≥1}𝔪_{i}⟩. Since 𝔪 is a maximal ideal, 𝔪 = ⟨rad(H^{0}), ⊕_{i≥1}𝔪_{i}⟩ and ⟨rad(H^{0})⟩ is in particular contained in any nonzero maximal ideal 𝔪 in H.

Denote by N_{H} the ideal in H generated by the homogeneous nilpotent elements and rad(H^{0}), which is a graded ideal in H. Since H is graded commutative and from Lemma 2.2, any maximal ideal in H contains N_{H}. Therefore there is a 1-1 correspondence between the maximal ideals in H and the maximal ideals in \overline{H} = H/N_{H}. For an ideal I in \overline{H} denote by I′ the inverse image of I in H. Using these observations and Corollary 1.3 we are able to make the following definition.

Definition 2.3. Let M and N be any pair of Λ-modules. Define the support variety V_{H}(M, N) in \overline{H} associated with the pair (M, N) by

V_{H}(M, N) = \{𝔪 ∈ MaxSpec \overline{H} | A_{H}(M, N) ⊆ 𝔪′\}

whenever Ext_{Λ}^{∗}(M, N) is nonzero. If Ext_{Λ}^{∗}(M, N) is zero, then let V_{H}(M, N) = ∅.

In making this definition we remark that very little is known about the ring HH^{∗}(Λ) in general, and in particular it is unknown in general whether or not the

support varieties and hochschild cohomology rings 7

ring H/N_{H} is finitely generated as an algebra for H = HH^{∗}(Λ). We conjecture that H/N_{H} is finitely generated as an algebra for H = HH^{∗}(Λ), where Λ is an artin k-algebra and k is a commutative artinian ring with Λ flat as a k-module. This is answered affirmatively for finite dimensional selfinjective algebras of finite representation type over an algebraically closed field in [18] and for a finite dimensional monomial algebra in [17]. Furthermore, it is true for any block of a group ring of a finite group [9, 27]. In the same way as for group rings, referring to our earlier comments in the introduction, using [8, XI.8.2 and X.6.1], the Hochschild cohomology ring of a finite dimensional cocommutative Hopf algebra Λ over a field k is isomorphic to Ext_{Λ}^{∗}(k, Λ), where Λ is viewed as a left Λ-module via the map (id ⊗ S)Δ: Λ → Λ^{e} and the bimodule structure of Λ, that is, the adjoint representation of Λ. This isomorphism is also given in [16]. Thus by [15, Theorem 1.1], HH^{∗}(Λ) is a finitely generated module over Ext_{Λ}^{∗}(k, k). Consequently, as Ext_{Λ}^{∗}(k, k) is a finitely generated algebra, the Hochschild cohomology ring is a finitely generated algebra, and so our conjecture holds for any block of a finite dimensional cocommutative Hopf algebra. We return to this topic in section 5.

A consequence of the fact that the ring HH^{∗}(Λ) is graded commutative, is that the even part of HH^{∗}(Λ) is a commutative ring, and HH^{∗}(Λ) is commutative when the characteristic is 2. Furthermore, in characteristic different from 2, the algebra \overline{H} is isomorphic to the even part of H modulo the ideal generated by homogeneous nilpotent elements and rad(H^{0}). Hence the algebra \overline{H} is always commutative with no non-zero nilpotent elements and an algebra over the commutative ring H^{0}/ rad(H^{0}).

Motivated by Lemma 2.2 the variety of a pair of modules (M, N) is said to be trivial if V_{H}(M, N) = {𝔪_{gr}}, where 𝔪_{gr} is the maximal ideal ⟨N_{H}, H^{≥1}⟩/N_{H} in \overline{H}. We observe that 𝔪_{gr} = ⟨rad(H^{0}), H^{≥1}⟩/N_{H} and thus that 𝔪_{gr} = 𝔞/N_{H}. The class of pairs of modules with trivial variety is a natural class to try to characterise. Although we are at this point unable to do so, the following result describes some pairs of modules with trivial variety.

Proposition 2.4. Suppose one of the following conditions holds: (i) M is a Λmodule with Ext_{Λ}^{i}(M, M) = (0) for i ≫ 0, (ii) N is a Λ-module with Ext_{Λ}^{i}(N, N) = (0) for i ≫ 0, or (iii) Ext_{Λ}^{i}(M, N) = (0) for i ≫ 0 with Ext_{Λ}^{∗}(M, N) nonzero. Then V_{H}(M, N) = {𝔪_{gr}}.

Proof. Without loss of generality we can assume that one of the following conditions holds: (i) Ext_{Λ}^{i}(M, M) = (0) for i ≥ n, (ii) Ext_{Λ}^{i}(N, N) = (0) for i ≥ n, or (iii) Ext_{Λ}^{i}(M, N) = (0) for i ≥ n. In all these cases H^{≥n+1} is contained in A_{H}(M, N), since H^{≥n+1} · Ext_{Λ}^{∗}(M, N) = (0). In particular, we have that (H^{i})^{n+1} is contained in A_{H}(M, N) for all i = 1, 2, …, n. It follows that any maximal ideal 𝔪′ of H containing A_{H}(M, N) must contain H^{≥1}. But, from Lemma 2.2 (b), such an ideal 𝔪′ also contains rad(H^{0}). Hence, the ideal 𝔪_{gr} = ⟨rad(H^{0}), H^{≥1}⟩/N_{H} is contained in 𝔪 for any 𝔪 in V_{H}(M, N). Since 𝔪_{gr} is a maximal ideal, V_{H}(M, N) consists exactly of the maximal ideal 𝔪_{gr}. This completes the proof.

Using the above result, observe that the variety of any pair of modules (M, N) with Ext_{Λ}^{∗}(M, N) nonzero where M or N has finite projective or finite injective dimension is trivial. In section 4 we characterise when the variety of all modules is trivial in terms of properties of H.

In computing the variety of a pair of modules the following three results give useful relationships between the varieties of modules occurring in an exact sequence.

Lemma 2.5. Let η: 0 → X_{1} → X_{2} → X_{3} → 0 be an exact sequence of Λmodules, and let M and N be two Λ-modules. Then the following statements hold.

(a) V_{H}(M, X_{i_{1}}) ⊆ V_{H}(M, X_{i_{2}}) ∪ V_{H}(M, X_{i_{3}}) whenever {i_{1}, i_{2}, i_{3}} = {1, 2, 3}.

(b) V_{H}(X_{i_{1}}, N) ⊆ V_{H}(X_{i_{2}}, N) ∪ V_{H}(X_{i_{3}}, N) whenever {i_{1}, i_{2}, i_{3}} = {1, 2, 3}.

Proof. (a) The short exact sequence η induces the exact sequence Ext_{Λ}^{∗}(M, X_{1}) → Ext_{Λ}^{∗}(M, X_{2}) → Ext_{Λ}^{∗}(M, X_{3}). It follows from this that A_{H}(M, X_{2}) ⊇ A_{H}(M, X_{1})A_{H}(M, X_{3}). Suppose that A_{H}(M, X_{2}) ⊆ 𝔪′ for some 𝔪 in MaxSpec \overline{H}. Since 𝔪 is a prime ideal, A_{H}(M, X_{1}) ⊆ 𝔪′ or A_{H}(M, X_{3}) ⊆ 𝔪′ and V_{H}(M, X_{2}) ⊆ V_{H}(M, X_{1}) ∪ V_{H}(M, X_{3}).

The sequence Ext_{Λ}^{∗}(M, X_{2}) → Ext_{Λ}^{∗}(M, X_{3}) → Ext_{Λ}^{∗+1}(M, X_{1}) is exact. Since A_{H}(M, X_{3}) ⊇ A_{H}(M, X_{1})A_{H}(M, X_{2}), it follows as above that V_{H}(M, X_{3}) ⊆ V_{H}(M, X_{1}) ∪ V_{H}(M, X_{2}).

The sequences Ext_{Λ}^{∗}(M, X_{3}) → Ext_{Λ}^{∗+1}(M, X_{1}) → Ext_{Λ}^{∗+1}(M, X_{2}) and 0 → Hom_{Λ}(M, X_{1}) → Hom_{Λ}(M, X_{2}) are exact. Similarly we have A_{H}(M, X_{1}) ⊇ A_{H}(M, X_{2})A_{H}(M, X_{3}), so that V_{H}(M, X_{1}) ⊆ V_{H}(M, X_{2}) ∪ V_{H}(M, X_{3}).

The proof of (b) is similar to that of (a) and is left to the reader.

Proposition 2.6. Let η: 0 → X′ → E → X → 0 be an exact sequence of Λ-modules where V_{H}(E, N) = V_{H}(M, E) = {𝔪_{gr}} for any two Λ-modules M and N. Then the following assertions hold.

(a) V_{H}(M, X) ∪ {𝔪_{gr}} = V_{H}(M, X′) ∪ {𝔪_{gr}} for any left Λ-module M, where 𝔪_{gr} is in one of the varieties V_{H}(M, X) or V_{H}(M, X′).

(b) V_{H}(X, N)∪{𝔪_{gr}} = V_{H}(X′, N)∪{𝔪_{gr}} for any left Λ-module N, where 𝔪_{gr} is in one of the varieties V_{H}(X, N) or V_{H}(X′, N).

Proof. (a) From Lemma 2.5 we infer that V_{H}(M, X) ⊆ V_{H}(M, X′) ∪ V_{H}(M, E), V_{H}(M, E) ⊆ V_{H}(M, X)∪V_{H}(M, X′) and V_{H}(M, X′) ⊆ V_{H}(M, X)∪V_{H}(M, E). The statement in (a) follows directly from this.

The claim in (b) is proved similarly to (a).

Proposition 2.7. Let M = ⊕_{i=1}^{r}M_{i} and N = ⊕_{i=1}^{s}N_{i}. Then

V_{H}(M, N) = ∪_{i,j=1}^{r,s}V_{H}(M_{i}, N_{j}).

Proof. Using the fact that Ext_{Λ}^{l}(M, N) = ⊕_{i,j=1}^{r,s} Ext_{Λ}^{l}(M_{i}, N_{j}) and Lemma 2.5 the claim follows.

3. Support varieties for artin algebras

Throughout this section let Λ be an artin algebra over a commutative artinian ring k, where Λ is flat as a k-module. Since k is noetherian and Λ is a finitely generated k-module, the last assumption is equivalent to requiring Λ to be a projective k-module. Denote by 𝔯 the radical of Λ. Let H be a graded subalgebra of HH^{∗}(Λ), where we assume throughout that H^{0} is a local k-algebra.

Here we introduce the support variety V_{H}(M) of a module M in mod Λ as

support varieties and hochschild cohomology rings 9

V_{H}(M, Λ/𝔯), after having shown that V_{H}(M, Λ/𝔯) = V_{H}(M, M) = V_{H}(Λ/𝔯, M). The additional assumption on Λ ensures that any finitely generated module M has a finite filtration in finitely generated semisimple modules, which we show implies that the variety of any module is contained in that of Λ/𝔯. Support varieties for left or right modules are defined in the same way, and they are both contained in MaxSpec \overline{H}. Furthermore, when k is a field we show that the varieties of M and D(M) are equal, where D denotes the usual duality Hom_{k}(−, k) from mod Λ to mod Λ^{op}. Since group rings of finite groups are selfinjective algebras, it is natural to believe that our more general theory should be particularly useful for selfinjective algebras. We show a first indication of this in proving that the variety is an invariant of a stable component in the Auslander-Reiten quiver of a finite dimensional selfinjective algebra over a field.

Our first aim is to prove that V_{H}(M, Λ/𝔯) = V_{H}(M, M) = V_{H}(Λ/𝔯, M) for M in mod Λ. The next result is the instrumental step in this direction.

Proposition 3.1. Let M and N be in mod Λ.

(a) V_{H}(M, N) ⊆ V_{H}(M, Λ/𝔯) ∩ V_{H}(Λ/𝔯, N) ⊆ V_{H}(Λ/𝔯, Λ/𝔯).

(b) If Ext_{Λ}^{∗}(M, N) is nonzero, then {𝔪_{gr}} ⊆ V_{H}(M, N) ⊆ V_{H}(Λ/𝔯, Λ/𝔯).

(c) V_{H}(M, N) ⊆ V_{H}(M, M) ∩ V_{H}(N, N) ⊇ V_{H}(N, M).

Proof. (a) For any simple Λ-module S we have that V_{H}(M, S) ⊆ V_{H}(M, Λ/𝔯) by Proposition 2.7. Using in addition Lemma 2.5 we infer that

V_{H}(M, N) ⊆ ∪_{i=0}^{l}V_{H}(M, 𝔯^{i}N/𝔯^{i+1}N) ⊆ V_{H}(M, Λ/𝔯),

where l is the radical length of Λ. Similarly we show that V_{H}(M, N) ⊆ V_{H}(Λ/𝔯, N). Applying what we have just shown, it follows that V_{H}(Λ/𝔯, N) ⊆ V_{H}(Λ/𝔯, Λ/𝔯). The claim is an immediate consequence of these observations. (b) Use (a) and Lemma 2.2 to obtain this.

(c) We clearly have that A_{H}(X, X) ⊆ A_{H}(M, N) and A_{H}(X, X) ⊆ A_{H}(N, M) for X = M and X = N. The claim follows directly from this.

Now the equalities V_{H}(M, Λ/𝔯) = V_{H}(M, M) = V_{H}(Λ/𝔯, M) for M in mod Λ are easy consequences of the above.

Theorem 3.2. V_{H}(M, Λ/𝔯) = V_{H}(Λ/𝔯, M) = V_{H}(M, M) for any module M in mod Λ.

Proof. By Proposition 3.1 (c) and (a)

V_{H}(M, Λ/𝔯) ⊆ V_{H}(M, M) ⊆ V_{H}(M, Λ/𝔯).

Hence V_{H}(M, Λ/𝔯) = V_{H}(M, M). Using the same arguments we similarly get V_{H}(Λ/𝔯, M) = V_{H}(M, M), and the claim follows.

Having the above result motivates the following definition of the support variety of a module in mod Λ.

Definition 3.3. The support variety V_{H}(M) of a module M in mod Λ with respect to H is given by V_{H}(M) = V_{H}(M, Λ/𝔯).

10 nicole snashall and Øyvind solberg

Following section 2, the variety of a module M is said to be trivial if V_{H}(M) = {𝔪_{gr}}. The ideal 𝔪_{gr} has the same central position here as it holds for a pair of modules, as the next result shows.

Proposition 3.4.

(a) {𝔪_{gr}} ⊆ V_{H}(M) ⊆ V_{H}(Λ/𝔯) for all non-zero modules M in mod Λ.

(b) If V_{H}(Λ/𝔯) = {𝔪_{gr}}, then V_{H}(M) = {𝔪_{gr}} for all modules M in mod Λ.

(c) If Ext_{Λ}^{i}(M, M) = (0) for i ≫ 0, or the projective or the injective dimension of M is finite, then V_{H}(M) = {𝔪_{gr}}.

(d) If 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence in mod Λ, then V_{H}(M_{i_{1}}) ⊆ V_{H}(M_{i_{2}}) ∪ V_{H}(M_{i_{3}}) whenever {i_{1}, i_{2}, i_{3}} = {1, 2, 3}.

(e) If 0 → M′ → E → M → 0 is an exact sequence with nonzero end terms in mod Λ where V_{H}(E) is trivial, then V_{H}(M) = V_{H}(M′). In particular, for any nonzero module M in mod Λ the equality V_{H}(M) = V_{H}(Ω_{Λ}^{n}(M)) holds for all integers n such that Ω_{Λ}^{n}(M) ≠ (0).

(f) If M = ⊕_{i=1}^{n}M_{i}, then V_{H}(M) = ∪_{i=1}^{n}V_{H}(M_{i}).

Proof. (a) Since Ext_{Λ}^{∗}(M, Λ/𝔯) is nonzero for a non-zero module M, the claim follows from Proposition 3.1.

(b) This is a direct consequence of (a).

The rest of the proposition is a direct consequence of Proposition 2.4, Lemma 2.5, Proposition 2.6 and Proposition 2.7.

In section 4 we characterise when all modules in mod Λ have a trivial variety in terms of properties of H.

From now on assume that k is a field, that is, Λ is a finite dimensional algebra over k. Moreover, we keep the following notation throughout the rest of this section. Let

P^{∗}: ⋯ → P^{n} ⟶^{d^{n}} P^{n−1} → ⋯ → P^{1} ⟶^{d^{1}} P^{0} ⟶^{d^{0}} Λ → 0 be a minimal Λ^{e}-projective

resolution of Λ. Under these assumptions P^{∗} ⊗_{Λ} M is a projective resolution of M for any Λ-module M. Denote by D the usual duality Hom_{k}(−, k) from mod Λ to mod Λ^{op}.

The varieties of the modules M and D(M) are defined by the kernel of the map H ⟶^{−⊗_{Λ}M} Ext_{Λ}^{∗}(M, M) and the kernel of the map H ⟶^{D(M)⊗_{Λ}−} Ext_{Λ^{op}}^{∗}(D(M), D(M)), respectively. Hence they are both contained in MaxSpec \overline{H}, so that it is possible to compare them and we show in the next result that in fact they are equal.

Proposition 3.5. Let Λ be a finite dimensional algebra over a field k. For any module M in mod Λ the varieties V_{H}(M) and V_{H}(D(M)) are equal.

Proof. Let P^{∗}: ⋯ → P^{n} ⟶^{d^{n}} P^{n−1} → ⋯ → P^{0} → Λ → 0 be a minimal

projective resolution of Λ over Λ^{e}. Let η be an element in H^{n} and identify this element as a map η: P^{n} → Λ. Assume that η is in A_{H}(M, M). In other words the map η ⊗_{Λ} M: P^{n} ⊗_{Λ} M → Λ ⊗_{Λ} M factors through the map P^{n} ⊗_{Λ} M ⟶^{d^{n}⊗_{Λ}M} P^{n−1} ⊗_{Λ} M. We want to show that η is in A_{H}(D(M), D(M)).

We have that D(M) ⊗_{Λ} X and D(Hom_{Λ}(X, M)) are isomorphic as right Λ-

support varieties and hochschild cohomology rings 11

modules for a Λ-Λ-bimodule X. Then the map

Hom_{Λ^{op}}(D(M) ⊗_{Λ} P^{n−1}, D(M) ⊗_{Λ} Λ) → Hom_{Λ^{op}}(D(M) ⊗_{Λ} P^{n}, D(M) ⊗_{Λ} Λ)

is isomorphic to

Hom_{Λ^{op}}(D(Hom_{Λ}(P^{n−1}, M)), D(Hom_{Λ}(Λ, M))) → Hom_{Λ^{op}}(D(Hom_{Λ}(P^{n}, M)), D(Hom_{Λ}(Λ, M))).

This is in turn isomorphic to the map

Hom_{Λ}(Hom_{Λ}(Λ, M), Hom_{Λ}(P^{n−1}, M)) → Hom_{Λ}(Hom_{Λ}(Λ, M), Hom_{Λ}(P^{n}, M))

and to

Hom_{Λ}(P^{n−1} ⊗_{Λ} Hom_{Λ}(Λ, M), M) → Hom_{Λ}(P^{n} ⊗_{Λ} Hom_{Λ}(Λ, M), M).

Under these isomorphisms the homomorphism id_{D(M)} ⊗_{Λ} η: D(M) ⊗_{Λ} P^{n} → D(M) ⊗_{Λ} Λ corresponds to η ⊗_{Λ} M: P^{n} ⊗_{Λ} M → Λ ⊗_{Λ} M in Hom_{Λ}(P^{n} ⊗_{Λ} M, M). By assumption it follows that there exists a homomorphism g in Hom_{Λ^{op}}(D(M)⊗_{Λ} P^{n−1}, D(M)⊗_{Λ} Λ) mapping onto the homomorphism D(M)⊗_{Λ} η: D(M)⊗_{Λ} P^{n} → D(M) ⊗_{Λ} Λ. This shows that D(M) ⊗_{Λ} η is in A_{H}(D(M), D(M)). Hence we have that A_{H}(M, M) is contained in A_{H}(D(M), D(M)). Applying this again we infer that A_{H}(M, M) = A_{H}(D(M), D(M)) and therefore V_{H}(M) = V_{H}(D(M)).

Group rings of finite groups and finite dimensional cocommutative Hopf algebras are all selfinjective finite dimensional algebras. A crucial property in both these situations is the finite generation of Ext^{∗}(M, N) as a module over the ring defining the varieties, in our case the graded subalgebra H of HH^{∗}(Λ). In the forthcoming work [12] we show that if Ext_{Λ}^{∗}(M, N) is a finitely generated module over a Noetherian commutative subalgebra H of HH^{∗}(Λ), then Λ is a Gorenstein algebra, that is, id_{Λ}Λ and idΛ_{Λ} are both finite. Here id denotes the injective dimension of a module. A selfinjective algebra is in particular Gorenstein, and, in this case, under the assumption on finite generation above, the variety of a module is trivial if and only if the module is projective. This indicates that one should expect a rich theory in the case of selfinjective algebras. In the next two results we specialise to this case.

For a selfinjective artin algebra Λ the functor ( )^{∗} given as Hom_{Λ}( , Λ): mod Λ → mod Λ^{op} is a duality. Furthermore, it shares the same property as the general duality D: mod Λ → mod Λ^{op} by preserving the variety of the module.

Proposition 3.6. Let Λ be a finite dimensional selfinjective algebra over a field k. Then V_{H}(M) = V_{H}(M^{∗}) for any M in mod Λ.

Proof. Let P^{∗}: ⋯ → P^{n} ⟶^{d^{n}} P^{n−1} → ⋯ → P^{0} → Λ → 0 be a minimal

projective resolution of Λ over Λ^{e}. Let η be an element in H^{n} and identify this element as a homomorphism η: P^{n} → Λ. Assume that η is in A_{H}(M, M). In other words the homomorphism η ⊗_{Λ} M: P^{n} ⊗_{Λ} M → Λ ⊗_{Λ} M factors through the map P^{n} ⊗_{Λ} M ⟶^{d^{n}⊗_{Λ}M} P^{n−1} ⊗_{Λ} M. We want to show that η is in A_{H}(M^{∗}, M^{∗}).

We want to investigate the element M^{∗} ⊗_{Λ} η: M^{∗} ⊗_{Λ} P^{n} → M^{∗} ⊗_{Λ} Λ. The homomorphism

Hom_{Λ^{op}}(M^{∗} ⊗_{Λ} P^{n−1}, M^{∗} ⊗_{Λ} Λ) → Hom_{Λ^{op}}(M^{∗} ⊗_{Λ} P^{n}, M^{∗} ⊗_{Λ} Λ)

is isomorphic to the homomorphism

Hom_{Λ}(M, Hom_{Λ^{op}}(M^{∗} ⊗_{Λ} P^{n−1}, Λ)) → Hom_{Λ}(M, Hom_{Λ^{op}}(M^{∗} ⊗_{Λ} P^{n}, Λ))

using that ( )^{∗} is a duality since Λ is selfinjective. Using the adjointness isomorphism and Λ being selfinjective the above homomorphism is isomorphic to the homomorphism

Hom_{Λ}(M, Hom_{Λ}(P^{n−1}, M)) → Hom_{Λ}(M, Hom_{Λ}(P^{n}, M))

which in turn is isomorphic to

Hom_{Λ}(P^{n−1} ⊗_{Λ} M, M) → Hom_{Λ}(P^{n} ⊗_{Λ} M, M).

Under these isomorphisms the homomorphism M^{∗} ⊗_{Λ} η in Hom_{Λ^{op}}(M^{∗} ⊗_{Λ} P^{n}, M^{∗} ⊗_{Λ} Λ) corresponds to η ⊗_{Λ} M in Hom_{Λ}(P^{n} ⊗_{Λ} M, M). By assumption there is a homomorphism g in Hom_{Λ}(P^{n−1} ⊗_{Λ} M, M) mapping to η ⊗_{Λ} M. Hence it follows that there is a g′ in Hom_{Λ^{op}}(M^{∗} ⊗_{Λ} P^{n−1}, M^{∗} ⊗_{Λ} Λ) mapping to M^{∗} ⊗_{Λ} η. Therefore M^{∗} ⊗_{Λ} η is zero and η is in A_{H}(M^{∗}, M^{∗}). Applying this again and using that M^{∗∗} ≃ M we infer that V_{H}(M^{∗}) = V_{H}(M).

For a definition and a detailed treatment of almost split sequences we refer the reader to [1]. Let τ denote the Auslander-Reiten translate in mod Λ. Recall that τ = D Tr, where Tr is the transpose of a module. The Auslander-Reiten quiver is an invariant of the algebra, and the shapes or possible shapes of the components of this quiver have been studied extensively for different classes of selfinjective artin algebras. We end this section by showing that all the modules in a stable component of the Auslander-Reiten quiver have the same variety, that is, the variety is an invariant of the stable component. This fact was pointed out to us by Karin Erdmann, as a consequence of the first assertion of the next result.

Theorem 3.7. Let Λ be a finite dimensional selfinjective algebra over a field k.

(a) V_{H}(M) = V_{H}(τM) for any module M in mod Λ.

(b) All indecomposable modules in a component of the stable AR-quiver have the same variety.

Proof. (a) Let M be a non-projective Λ-module. Let P_{1} → P_{0} → M → 0 be a minimal projective presentation of M. Then Tr M is given by the exact sequence 0 → M^{∗} → P_{0}^{∗} → P_{1}^{∗} → Tr M → 0. Since P_{i}^{∗} is projective for i = 0, 1, we have by Proposition 3.4 (e) that V_{H}(M^{∗}) = V_{H}(Tr M). By Proposition 3.6 and Proposition 3.5 we conclude that V_{H}(M) = V_{H}(D Tr M).

(b) Let C be an indecomposable non-projective Λ-module. Let 0 → τC → ⊕_{i=1}^{n}B_{i} → C → 0 be the almost split sequence with B_{i} indecomposable for i = 1, 2, …, n. By Proposition 3.4 the variety V_{H}(⊕_{i=1}^{n}B_{i}) = ∪_{i=1}^{n}V_{H}(B_{i}) ⊆ V_{H}(C) ∪ V_{H}(τC) = V_{H}(C). In particular V_{H}(B_{i}) ⊆ V_{H}(C) for all i = 1, 2, …, n.

Let 0 → τB_{i} → ⊕_{j=1}^{m}X_{j} → B_{i} → 0 be the almost split sequence for a nonprojective B_{i}. Similarly as above V_{H}(X_{j}) ⊆ V_{H}(B_{i}) for all j = 1, 2, …, m. Recall that for any pair of non-projective indecomposable modules M and N, there is an almost split sequence 0 → τN → M ⊕ M′ → N → 0 if and only if there is an almost split sequence 0 → τM → τN ⊕ N′ → M → 0 (see chapters V and VII in [1]). It follows from this that one of the X_{j}’s is isomorphic to τC. Consequently we infer that V_{H}(C) = V_{H}(B_{i}) for all non-projective modules among the B_{i}’s. The claim follows from this.

4. The Hochschild cohomology ring

Throughout this section let Λ be an artin algebra over a commutative artinian ring k, where Λ is flat as a module over k. Let H be a graded subalgebra of HH^{∗}(Λ), where H^{0} is a local k-algebra. In this case H^{0} is an artin algebra. Note that rad(H^{0}) is nilpotent in this case, so that N_{H} is the ideal generated by the homogeneous nilpotent elements.

A natural question is, when is the support variety of any Λ-module trivial? This is characterised by all homogeneous elements in H^{≥1} being nilpotent. For this reason and that we consider the factor ring H/N_{H} in defining the support varieties, further information on nilpotent elements is useful. We show that if all homogeneous elements in HH^{≥1}(Λ) are nilpotent and HH^{i}(Λ) ≠ (0) for infinitely many integers i > 0, then HH^{∗}(Λ) is infinitely generated as a graded algebra. Under extra assumptions on Λ, for instance, when k is a field and Λ is an indecomposable artin k-algebra such that Λ/𝔯 is a separable algebra over k, we show that any homogeneous element in Ker φ_{Λ/𝔯} has nilpotency index at most the radical length of Λ. Then we use this information and the relationship between Ext_{Λ}^{n}(S, S′) (for simple modules S and S′) and which projective Λ^{e}-modules occur in the projective resolution of Λ (see [19]), to give classes of homogeneous elements in HH^{∗}(Λ) which are nilpotent. Moreover we show that \sqrt{Ker φ_{Λ/𝔯}} = N_{H} when H = HH^{∗}(Λ).

We first address the question of when the support variety of every Λ-module is trivial. In Proposition 2.4 we saw a sufficient condition for this to happen. Now we give a characterisation. Recall that 𝔪_{gr} = ⟨rad(H^{0}), H^{≥1}⟩/N_{H} and 𝔞 = ⟨rad(H^{0}), H^{≥1}⟩.

Proposition 4.1. Let Λ be an artin k-algebra which is flat as a module over k where k is a commutative artinian ring. The following are equivalent.

(a) Any homogeneous element in H^{≥1} is nilpotent.

(b) V_{H}(M) = {𝔪_{gr}} for all Λ-modules M in mod Λ, and H is a local ring.

(c) V_{H}(Λ/𝔯) = {𝔪_{gr}}, and H is a local ring.

Proof. (a) ⇒ (b). Assume that any homogeneous element in H^{≥1} is nilpotent. Then H^{≥1} is contained in any maximal ideal in H. Since Λ is an artin algebra, rad H^{0} is nilpotent. Thus 𝔪′_{gr} = 𝔞 is contained in any maximal ideal in H. Therefore H is a local ring and V_{H}(M) = {𝔪_{gr}} for all Λ-modules M.

(b) ⇒ (c). This implication is trivial.

(c) ⇒ (a). Assume that H^{≥1} is generated by the elements G = {g_{i}}_{i∈Σ} for some index set Σ over H^{0}, where ⟨g_{i}⟩_{i∈Σ\{i_{0}}} ∩ ⟨g_{i_{0}}⟩ ⊆ ⟨rad H^{0}⟩ for any i_{0} in Σ. Let g = g_{i_{0}} be in G, and assume that g^{i} is not in A_{H}(Λ/𝔯, Λ/𝔯) for any i ≥ 1. Denote by H′ the ring H^{0}[g] ⊆ H. Since A_{H′}(Λ/𝔯, Λ/𝔯) is a graded ideal in H′ and g^{i} is not in A_{H}(Λ/𝔯, Λ/𝔯) for any i ≥ 1, the ideal A_{H′}(Λ/𝔯, Λ/𝔯) ⊆ ⟨rad H^{0}⟩. Therefore V_{H′}(Λ/𝔯) = MaxSpec H′.

Let 𝔠 = ⟨g_{i}⟩_{i∈Σ\{i_{0}}} ⊆ H. Then H/𝔠 ≃ H^{0}[g]/𝔠∩H^{0}[g] where 𝔠∩H^{0}[g] ⊆ ⟨rad H^{0}⟩. Hence there is an epimorphism H/𝔠 ⟶^{π} (H^{0}/ rad H^{0})[g]. The ring (H^{0}/ rad H^{0}) is a division ring and the ideal generated by (g−1) is a maximal ideal in (H^{0}/ rad H^{0})[g], 14 nicole snashall and Øyvind solberg

since (H^{0}/ rad H^{0})[g] is isomorphic to a polynomial ring in one indeterminate over a division ring. Let p: H → (H^{0}/ rad H^{0})[g] be the composition of the natural projections H → H/𝔠 and π. Then 𝔪 = p^{−1}(g − 1) is a maximal ideal in H with A_{H}(Λ/𝔯, Λ/𝔯) ⊆ 𝔪, which is a contradiction. Since any element in H^{≥1} is a polynomial expression involving only a finite number of the generators in G, all elements in H^{≥1} to some power are in A_{H}(Λ/𝔯, Λ/𝔯). Or in other words, \sqrt{A_{H}(Λ/𝔯, Λ/𝔯)} contains 𝔪′_{gr}. Since 𝔪′_{gr} is a maximal ideal, \sqrt{A_{H}(Λ/𝔯, Λ/𝔯)} = 𝔪′_{gr}. This shows that any homogeneous element in H^{≥1} is nilpotent.

Results on nilpotent elements give information on the finite generation or otherwise of the Hochschild cohomology ring as the following proposition shows. This question will be further considered in section 5.

Proposition 4.2. Let Λ be an artin k-algebra which is flat as a module over k where k is a commutative artinian ring. Suppose that all elements in HH^{≥1}(Λ) are nilpotent and that HH^{∗}(Λ) is finitely generated as a graded algebra. Then HH^{i}(Λ) = (0) for i ≫ 0.

Proof. Suppose that HH^{≥1}(Λ) is generated by the elements {g_{1}, …, g_{t}} over HH^{0}(Λ) where g_{1}, …, g_{t} are homogeneous elements in HH^{∗}(Λ). Let N_{0} be the maximal degree of the elements {g_{1}, …, g_{t}} and let N_{1} be the maximal nilpotency index of the elements {g_{1}, …, g_{t}}. Then any monomial m on {g_{1}, …, g_{t}} of degree at least tN_{0}N_{1} must contain g_{i}^{N_{1}} for some i, and hence m = 0. Thus HH^{i}(Λ) = (0) for i ≥ tN_{0}N_{1}.

This has the following immediate consequence which we apply in the next section.

Corollary 4.3. Let Λ be an artin k-algebra which is flat as a module over k where k is a commutative artinian ring. Suppose that all elements in HH^{≥1}(Λ) are nilpotent and that HH^{i}(Λ) ≠ (0) for infinitely many integers i ≥ 0. Then HH^{∗}(Λ) is infinitely generated as a graded algebra.

For the rest of this section we assume that k is a field and restrict to finite dimensional k-algebras Λ where Λ/𝔯 ⊗_{k} Λ/𝔯 is semisimple. We note that Λ/𝔯 ⊗_{k} Λ/𝔯 is semisimple when Λ is an indecomposable finite dimensional algebra over k such that Λ/𝔯 is a separable algebra over k, and that, whenever Λ/𝔯⊗_{k} Λ/𝔯 is semisimple, then the Jacobson radical of Λ^{e} is equal to Λ ⊗_{k} 𝔯 + 𝔯 ⊗_{k} Λ.

We have seen that the support variety of any module in mod Λ is contained in the support variety of Λ/𝔯. The graded ideal defining the support variety of Λ/𝔯 is generated by homogeneous nilpotent elements, which is a consequence of the next result. We denote the ideal N_{H} by N when H = HH^{∗}(Λ).

Proposition 4.4. Let k be a field and let Λ be a finite dimensional algebra over k where Λ/𝔯 ⊗_{k} Λ/𝔯 is semisimple.

(a) Let η be in HH^{n}(Λ) and assume that η ⊗_{Λ} Λ/𝔯 is zero in Ext_{Λ}^{n}(Λ/𝔯, Λ/𝔯). Then η is nilpotent in HH^{∗}(Λ) with nilpotency index at most the radical length of Λ.

(b) Ker φ_{Λ/𝔯} ⊆ N.

support varieties and hochschild cohomology rings 15

Proof. (a) Let η be in HH^{n}(Λ), and assume that η ⊗_{Λ} Λ/𝔯 is zero in Ext_{Λ}^{n}(Λ/𝔯, Λ/𝔯). Represent η as a map η: P^{n} → Λ, where P^{n} is the n-th projective module in a minimal projective resolution (P^{∗}, d^{∗}) of Λ over Λ^{e}. Then P^{∗} ⊗_{Λ} Λ/𝔯 is a minimal projective resolution of Λ/𝔯 over Λ, hence \overline{d^{n}}: P^{n} ⊗_{Λ} Λ/𝔯 → P^{n−1} ⊗_{Λ} Λ/𝔯 has its image in 𝔯(P^{n−1} ⊗_{Λ} Λ/𝔯). Since η ⊗_{Λ} Λ/𝔯 is assumed to be zero, there exists a homomorphism g: P^{n−1} ⊗_{Λ} Λ/𝔯 → Λ/𝔯 such that η ⊗_{Λ} Λ/𝔯 = g\overline{d^{n}}. Clearly Im g\overline{d^{n}} ⊆ 𝔯Λ/𝔯 = (0), thus η ⊗_{Λ} Λ/𝔯 = 0 and therefore Im η ⊆ 𝔯.

Now consider η as a homomorphism η: Ω_{Λ^{e}}^{n}(Λ) → Λ. Construct a lifting of η to the n-th syzygy Ω^{n}(η): Ω_{Λ^{e}}^{2n}(Λ) → Ω_{Λ^{e}}^{n}(Λ). Tensoring this lifting with − ⊗_{Λ} Λ/𝔯 and using that η ⊗_{Λ} Λ/𝔯 = 0, there exists a homomorphism g: P^{2n−1} ⊗_{Λ} Λ/𝔯 → Ω_{Λ^{e}}^{n}(Λ) ⊗_{Λ} Λ/𝔯 such that Ω^{n}(η) ⊗_{Λ} Λ/𝔯 = g(δ ⊗_{Λ} Λ/𝔯), where δ: Ω_{Λ^{e}}^{2n}(Λ) → P^{2n−1} is the natural inclusion. Since Im δ ⊆ 𝔯P^{2n−1} + P^{2n−1}𝔯, we infer that Im Ω^{n}(η) ⊗_{Λ} Λ/𝔯 ⊆ 𝔯(Ω_{Λ^{e}}^{n}(Λ)/Ω_{Λ^{e}}^{n}(Λ)𝔯). Consequently Im Ω^{n}(η) ⊆ 𝔯Ω_{Λ^{e}}^{n}(Λ)+Ω_{Λ^{e}}^{n}(Λ)𝔯. Similarly, Im Ω^{in}(η) ⊆ 𝔯Ω_{Λ^{e}}^{in}(Λ) + Ω_{Λ^{e}}^{in}(Λ)𝔯 for i ≥ 1, so that Im ηΩ^{n}(η) ⋯ Ω^{in}(η) ⊆ 𝔯^{i} for all i ≥ 1. If l is the radical length of Λ, then η^{l} = 0.

(b) Since Ker φ_{Λ/𝔯} = ⟨η ∈ HH^{∗}(Λ) | η ∈ HH^{n}(Λ) for some n and η ⊗_{Λ} Λ/𝔯 = 0⟩, the claim is immediate from (a).

For H = HH^{∗}(Λ) and with the above hypotheses, Proposition 4.1 now has the following consequence.

Corollary 4.5. Let k be a field and let Λ be a finite dimensional algebra over k where Λ/𝔯 ⊗_{k} Λ/𝔯 is semisimple. Then any homogeneous element in HH^{≥1}(Λ) is nilpotent if and only if V_{HH^{∗}(Λ)}(Λ/𝔯) = {𝔪_{gr}}.

Proof. The proof is immediate from Proposition 4.1 once we have shown that if V_{HH^{∗}(Λ)}(Λ/𝔯) = {𝔪_{gr}} then HH^{∗}(Λ) is local. So suppose V_{HH^{∗}(Λ)}(Λ/𝔯) = {𝔪_{gr}}, and suppose for contradiction that HH^{∗}(Λ) is not a local ring. Let 𝔪 ≠ 𝔪_{gr} be a maximal ideal of HH^{∗}(Λ)/N. Let H = HH^{∗}(Λ). Then A_{H}(Λ/𝔯, Λ/𝔯) ⊈ 𝔪′ so there is some η ∈ A_{H}(Λ/𝔯, Λ/𝔯) with η ∉ 𝔪′. But from Proposition 4.4(a), η ∈ A_{H}(Λ/𝔯, Λ/𝔯) implies that η is nilpotent and thus η lies in every maximal ideal of HH^{∗}(Λ). Hence η ∈ 𝔪′ which gives a contradiction. Thus HH^{∗}(Λ) is local as required.

A minimal projective resolution (P^{∗}, d^{∗}) of Λ over Λ^{e} is usually lengthy and hard to calculate. However the projective modules in the resolution are easy to describe using [19]. There it is shown that an indecomposable projective Λ^{e}-module Λe ⊗_{k} fΛ occurs with multiplicity dim_{k} Ext_{Λ^{e}}^{n}(Λe/𝔯e, Λf/𝔯f) in the n-th projective in a minimal projective resolution of Λ as an Λ^{e}-module.

Remark. The converse of (a) in Proposition 4.4 is not true. Consider the follow-

ing example. Let Q be the quiver \xymatrix{1 \ar[r]^{a} & 2 \ar[l]^{b}} and consider the relation ρ = \{ab\}. Let

Λ = kQ/⟨ρ⟩ for a field k. We denote the trivial paths by e_{1} and e_{2}. The projective resolution of Λ over Λ^{e} is

0 → Λe_{1} ⊗_{k} e_{1}Λ ⟶ ⊕_{i=1}^{2}Λe_{i} ⊗_{k} e_{i+1}Λ ⟶ ⊕_{i=1}^{2}Λe_{i} ⊗_{k} e_{i}Λ ⟶ Λ → 0

where e_{3} = e_{1}. The element η: Λe_{1} ⊗_{k} e_{1}Λ → Λ given by η(e_{1} ⊗ e_{1}) = e_{1} represents a nonzero nilpotent element in HH^{∗}(Λ), since the projective dimension of Λ as a Λ^{e}-module is 2. Moreover η is not in the kernel of φ_{Λ/𝔯}.

These results allow us to give the following characterisation of homogeneous nilpotent elements. Furthermore we show that \sqrt{Ker φ_{Λ/𝔯}} = N.

Proposition 4.6. Let k be a field and let Λ be a finite dimensional algebra over k where Λ/𝔯 ⊗_{k} Λ/𝔯 is semisimple.

(a) A homogeneous element η in HH^{∗}(Λ) is nilpotent if and only if η is in \sqrt{Ker φ_{Λ/𝔯}}.

(b) \sqrt{Ker φ_{Λ/𝔯}} = N, where N is the ideal in HH^{∗}(Λ) generated by all homogeneous nilpotent elements.

Proof. (a) Let η be in HH^{n}(Λ). Assume that η is nilpotent. Then η^{m} = 0 in HH^{∗}(Λ) for some m. In particular, η^{m} ⊗_{Λ} Λ/𝔯 = (η ⊗_{Λ} Λ/𝔯)^{m} = 0. Hence η^{m} is in Ker φ_{Λ/𝔯} and therefore η is in \sqrt{Ker φ_{Λ/𝔯}}.

Assume that η is in \sqrt{Ker φ_{Λ/𝔯}}. For some m the element η^{m} is in Ker φ_{Λ/𝔯}. By Proposition 4.4 the element η^{m} is nilpotent. Thus η is a nilpotent element.

(b) The radical of a graded ideal is again a graded ideal, hence \sqrt{Ker φ_{Λ/𝔯}} is a graded ideal. Therefore it follows from (a) that \sqrt{Ker φ_{Λ/𝔯}} is contained in N.

By definition N is a graded ideal. Let x be a homogeneous element in N. Then x can be written as ∑_{i=1}^{t} r_{i}m_{i}, where r_{i} in HH^{∗}(Λ) is homogeneous and m_{i} is nilpotent for all i. Let N be the maximal nilpotency degree of m_{i}. Then in the expression for x^{tN} any m_{i} occurs with exponent at least N, and hence x^{tN} = 0 and x is in \sqrt{Ker φ_{Λ/𝔯}}. Since N is a graded ideal, we infer that N is contained in \sqrt{Ker φ_{Λ/𝔯}}. From this we conclude that \sqrt{Ker φ_{Λ/𝔯}} = N.

For a Λ^{e}-homomorphism ϕ: Λe ⊗_{k} fΛ → Λ where e and f are primitive idempotents such that Λe ≄ Λf, then the image of ϕ must be in the radical of Λ. Utilising this we give classes of homogeneous elements in HH^{∗}(Λ) which are nilpotent. Note that part (b) (i) of the next result is independent of the characteristic of k. Here rl(Λ) denotes the radical length of the ring Λ.

Proposition 4.7. Let k be a field and let Λ be a finite dimensional algebra over k where Λ/𝔯 ⊗_{k} Λ/𝔯 is semisimple.

(a) Assume that Ext_{Λ}^{m}(S, S) = (0) for all simple Λ-modules S and for some m > 0. Then any element in HH^{r}(Λ) is nilpotent when m = rq for some q in ℕ with nilpotency index at most rl(Λ)q.

(b) Let Λ = kΓ/⟨ρ⟩ for a quiver Γ with admissible relations ρ over a field k.

(i) If Γ does not have any loops, then any element in HH^{1}(Λ) is nilpotent with nilpotency index at most rl(Λ).

(ii) If ρ does not have any minimal relations starting and ending in the same vertex, then any element in HH^{i}(Λ) for i = 1, 2 is nilpotent with nilpotency index at most rl(Λ).

Proof. (a) Assume that Ext_{Λ}^{m}(S, S) = (0) for all simple Λ-modules S. Then, from [19], in the minimal projective resolution of Λ over Λ^{e}, there are no copies of Λe ⊗_{k} eΛ in P^{m} for any primitive idempotent e in Λ. If η is in HH^{m}(Λ) and represented by a map η: P^{m} → Λ, then Im η ⊆ 𝔯. Consequently η is nilpotent in HH^{∗}(Λ) with nilpotency index at most rl(Λ) by Proposition 4.4. The claim in (a) follows from this.

support varieties and hochschild cohomology rings 17

(b) The second projective P^{1} in a minimal projective resolution of Λ over Λ^{e} is a direct sum of projective Λ^{e}-modules Λe_{i} ⊗_{k} e_{j}Λ for each arrow a: i → j in Γ. The statement in (i) is then a consequence of (a).

The third projective P^{2} in a minimal projective resolution of Λ over Λ^{e} is a direct sum of projective Λ^{e}-modules Λe_{i} ⊗_{k} e_{j}Λ for each minimal relation σ in ρ, where σ starts in i and ends in j. The minimal relations ρ can be chosen uniform, that is, a sum of paths that all start in one vertex and all end in one vertex. Since ρ does not contain any minimal relations starting and ending in the same vertex, there cannot be any loops in Γ. Using this the claim in (ii) follows directly from (a).

The following is a direct consequence of Proposition 4.7 (a). Recall that pd and id denote the projective and the injective dimension of a module, respectively.

Proposition 4.8. Let k be a field and let Λ be a finite dimensional algebra over k where Λ/𝔯⊗_{k}Λ/𝔯 is semisimple. Let {S_{1}, …, S_{n}} be a complete set of isomorphism classes of simple Λ-modules. Let S_{1} = {i | pd_{Λ}S_{i} < ∞} and S_{2} = {i | id_{Λ}S_{i} < ∞}. If S_{1} ∪ S_{2} = {1, 2, …, n}, then any element in HH^{≥1}(Λ) is nilpotent.

5. Nakayama algebras with a single relation

In this section we discuss a special class of Nakayama algebras to illustrate all possible different behaviours concerning finite generation of the Hochschild cohomology ring of a finite dimensional algebra. Recall that an artin algebra is a Nakayama algebra if all the indecomposable projective and all the indecomposable injective modules are uniserial. First we make a general observation concerning a sufficient condition for the Hochschild cohomology ring to be finite dimensional.

Let Λ be a finite dimensional k-algebra of finite global dimension over a field k. Then Ext_{Λ^{e}}^{i}(Λ, Λ) = (0) for i > gldimΛ and hence HH^{∗}(Λ) is finite dimensional. It is an open problem whether HH^{i}(Λ) = (0) for all i ≫ 0, or equivalently HH^{∗}(Λ) is finite dimensional, implies that gldimΛ is finite. Let Σ be a quiver, and let J denote the ideal generated by the arrows in kΣ. One example of a partial answer to the above question can be obtained from [23], where it is shown for a truncated path algebra kΣ/J^{t} that the Hochschild cohomology ring HH^{∗}(kΣ/J^{t}) is finite dimensional if and only if Σ has no oriented cycles. It follows from [5] for t ≥ 2 that gldimkΣ/J^{t} is finite if and only if Σ has no oriented cycles.

Now let k be a field and Λ = kQ/I be a finite dimensional indecomposable path algebra where I is an admissible ideal. Then Λ is a Nakayama algebra if and only if the quiver Q is the Dynkin diagram 𝔸_{n} for some n with linear orientation, or an extended Dynkin (Euclidean) diagram 𝔸̃_{n} for some n with cyclic orientation. In the first case Λ always has finite global dimension, so that this case is covered by the above remark. The 𝔸_{n} case is characterised by Λ having a projective simple module. So now we restrict to the case when Q is the quiver 𝔸̃_{n} with cyclic orientation for some n and I is generated by a single relation.

Let N denote the ideal of HH^{∗}(Λ) which is generated by all the homogeneous nilpotent elements. We determine the structure of the Hochschild cohomology ring HH^{∗}(Λ). In fact we show that this class of algebras contains all the different behaviours for the Hochschild cohomology ring; it can be finite dimensional, finitely generated as a ring, or infinitely generated as a ring. However the factor ring 18 nicole snashall and Øyvind solberg

HH^{∗}(Λ)/N is always finitely generated as a ring, even though HH^{∗}(Λ) is not. We write ⊗ for ⊗_{k} throughout this section.

In discussing the 𝔸̃_{n} case, recall that Λ is given by the following quiver Q

[figure: 1; 2; a_{1}; 3; a_{2}; 4; a_{3}; m − 1; m; a_{m−1}; 1; a_{m}]

and I is an admissible ideal generated by a single relation. We write paths from left to right, and in this section we consider right modules.

In the case m = 1, the quiver Q has a single loop α and thus Λ ≅ k[α]/⟨α^{r}⟩ for some r ≥ 2. Moreover the algebra Λ is selfinjective. The structure of HH^{∗}(Λ) was given in [20] from whence it follows that HH^{∗}(Λ) is finitely generated as a ring.

Thus we assume now that m ≥ 2. Let ρ be the single generator of the ideal I so ρ is a path of length l at least 2. By 𝔬(ρ) we denote the vertex where the path ρ starts, and by 𝔱(ρ) the vertex where ρ ends. The trivial path at the vertex i is denoted by e_{i}. We have the following three cases to consider.

Case I: l ≤ m

For this case

pdS_{i} = \begin{cases}2 & if i = 𝔬(ρ) \\ 1 & otherwise, \end{cases}

and hence gldimΛ = 2. Thus HH^{i}(Λ) = (0) for i ≥ 3 and so HH^{∗}(Λ) is a finite dimensional algebra.

Case II: l = rm + r̄ with r ≥ 1 and 1 ≤ r̄ ≤ m − 1

Let σ = a_{1}a_{2} … a_{m} and t = r̄ so that, without loss of generality, we may take ρ = σ^{r}a_{1} … a_{t} for some r ≥ 1 and 1 ≤ t ≤ m − 1. Let σ̄ = a_{t+1} … a_{m}a_{1} … a_{t}.

Let

⋯ → P^{2n+1} ⟶^{δ^{2n+1}} P^{2n} ⟶^{δ^{2n}} ⋯ ⟶^{δ^{4}} P^{3} ⟶^{δ^{3}} P^{2} ⟶^{δ^{2}} P^{1} ⟶^{δ^{1}} P^{0} → Λ → 0

be a minimal projective Λ^{e} resolution of Λ, where the projectives are given by

P^{0} = ⨁_{i=1}^{m}Λe_{i} ⊗ e_{i}Λ,

P^{1} = ⨁_{i=1}^{m}Λe_{i} ⊗ e_{i+1}Λ where we define e_{m+1} = e_{1},

and

P^{i} = Λe_{𝔬(ρ)} ⊗ e_{𝔱(ρ)}Λ for i ≥ 2.

support varieties and hochschild cohomology rings 19

By hypothesis 𝔬(ρ) ≠ 𝔱(ρ) and thus if η in HH^{i}(Λ) for i ≥ 2 is represented by a map η: P^{i} → Λ, then Im η ⊆ 𝔯. Hence η is nilpotent by Proposition 4.4. Moreover every element in HH^{1}(Λ) is nilpotent since the quiver contains no loops (Proposition 4.7) and thus every element in ⊕_{i≥1} HH^{i}(Λ) is nilpotent. Let

z = a_{1} … a_{m} + a_{2} … a_{m}a_{1} + … + a_{m}a_{1} … a_{m−1} ∈ 𝔯.

Then HH^{0}(Λ) has basis {1, z, …, z^{r}} if t = 1 and basis {1, z, …, z^{r+1}} if t ≠ 1. Since z is nilpotent, we have HH^{∗}(Λ)/N ≅ k and so is clearly finitely generated as a ring.

To determine the rest of the structure of HH^{∗}(Λ), we note that the maps in the projective resolution are given by

\begin{matrix}δ^{1} : & e_{i} ⊗ e_{i+1} & ↦ & e_{i} ⊗ a_{i} − a_{i} ⊗ e_{i+1} \\ δ^{2} : & e_{1} ⊗ e_{t+1} & ↦ & \begin{matrix}∑_{i=0}^{r−1} σ^{i}( ∑_{j=0}^{m−1} a_{1} … a_{j} ⊗ a_{j+2} … a_{m})σ^{r−i−1}a_{1} … a_{t} \\ +σ^{r}(∑_{j=0}^{t−1} a_{1} … a_{j} ⊗ a_{j+2} … a_{t}) \end{matrix} \\ δ^{2n+1} : & e_{1} ⊗ e_{t+1} & ↦ & e_{1} ⊗ σ̄ − σ ⊗ e_{t+1} for n ≥ 1 \\ δ^{2n}: & e_{1} ⊗ e_{t+1} & ↦ & ∑_{i=0}^{r}σ^{i} ⊗ σ̄^{r−i} for n ≥ 2. \end{matrix}

For n ≥ 2, it is easy to verify using length arguments that the map θ_{n}: P^{2n} → Λ given by e_{1} ⊗ e_{t+1} ↦ a_{1} … a_{t} represents a non-zero element of HH^{2n}(Λ). Thus HH^{i}(Λ) ≠ (0) for infinitely many values of i. However each of these elements are nilpotent and thus HH^{∗}(Λ) is infinitely generated as a graded algebra (Corollary 4.3).

Case III: l = rm with r ≥ 2

Let σ = a_{1}a_{2} … a_{m} so that, without loss of generality, we may take ρ = σ^{r} for some r ≥ 2.

We consider the following minimal projective Λ^{e} resolution of Λ

⋯ → P^{2n+1} ⟶^{δ^{2n+1}} P^{2n} ⟶^{δ^{2n}} ⋯ ⟶^{δ^{4}} P^{3} ⟶^{δ^{3}} P^{2} ⟶^{δ^{2}} P^{1} ⟶^{δ^{1}} P^{0} → Λ → 0

where the projectives are given by

P^{0} = ⨁_{i=1}^{m}Λe_{i} ⊗ e_{i}Λ,

P^{1} = ⨁_{i=1}^{m}Λe_{i} ⊗ e_{i+1}Λ where we define e_{m+1} = e_{1},

and

P^{i} = Λe_{1} ⊗ e_{1}Λ for i ≥ 2,

and the maps are given by

\begin{matrix}δ^{1} : & e_{i} ⊗ e_{i+1} & ↦ & e_{i} ⊗ a_{i} − a_{i} ⊗ e_{i+1}, \\ δ^{2} : & e_{1} ⊗ e_{1} & ↦ & ∑_{i=0}^{r−1} σ^{i}( ∑_{j=0}^{m−1} a_{1} … a_{j} ⊗ a_{j+2} … a_{m})σ^{r−i−1}, \\ δ^{2n+1} : & e_{1} ⊗ e_{1} & ↦ & e_{1} ⊗ σ − σ ⊗ e_{1} for n ≥ 1, \\ δ^{2n}: & e_{1} ⊗ e_{1} & ↦ & ∑_{i=0}^{r−1} σ^{i} ⊗ σ^{r−i−1} for n ≥ 2. \end{matrix}

First note that the centre of Λ, HH^{0}(Λ) = sp{1, z, …, z^{r}} where

z = a_{1} … a_{m} + a_{2} … a_{m}a_{1} + … + a_{m}a_{1} … a_{m−1} ∈ 𝔯.

Since e_{1}Λe_{1} is spanned by the set {e_{1}, σ, …, σ^{r−1}}, it is clear, for n ≥ 2, that Hom(P^{2n}, Λ) = Ker δ^{2n+1∗} and thus HH^{2n}(Λ) = Hom(P^{2n}, Λ)/ Im δ^{2n∗}. Now HH^{2n}(Λ) is generated over Z(Λ) by the element η_{n}: P^{2n} → Λ given by e_{1}⊗e_{1} ↦ e_{1}. Indeed, noting that σ^{r−1}η_{n} represents the zero element in HH^{2n}(Λ), we have that HH^{2n}(Λ) = sp{η_{n}, ση_{n}, …, σ^{r−2}η_{n}} for n ≥ 2. The elements σ^{i}η_{n} are necessarily nilpotent, for i ≥ 1, since σ^{i} ∈ 𝔯 (Proposition 4.4). For HH^{2}(Λ), we let η_{1} : P^{2} → Λ be given by e_{1} ⊗ e_{1} ↦ e_{1} and then HH^{2}(Λ) = sp{η_{1}, ση_{1}, …, σ^{r−1}η_{1}}. We also have that η_{1}^{2} = η_{2}, and, more generally, that η_{1}^{n} = η_{n}, so it is clear that η_{1} is not nilpotent. Thus ⊕_{n≥1} HH^{2n}(Λ) is generated by HH^{2}(Λ).

Let N^{even} denote the ideal of HH^{even}(Λ) = ⊕_{n≥0} HH^{2n}(Λ) which is generated by all the homogeneous nilpotent elements. Hence

HH^{even}(Λ)/N^{even} ≅ k[η_{1}]

where η_{1} is in degree 2.

To determine the remaining structure of HH^{∗}(Λ) we have to take into consideration the characteristic of the field k.

Since the quiver has no loops, every element of HH^{1}(Λ) is nilpotent (Proposition 4.7). Indeed HH^{1}(Λ) = sp{θ, σθ, …, σ^{r−1}θ} where θ: P^{1} → Λ is given by e_{1} ⊗e_{2} ↦ a_{1} and e_{i} ⊗ e_{i+1} ↦ 0 for i ≠ 1.

For n ≥ 1, let σψ_{n}: P^{2n+1} → Λ be given by e_{1} ⊗ e_{1} ↦ σ. Then it may be shown that sp{σψ_{n}, …, σ^{r−1}ψ_{n}} ⊆ HH^{2n+1}(Λ). Moreover all these elements of HH^{2n+1}(Λ) are represented by a map P^{2n+1} → Λ where the image of e_{1} ⊗ e_{1} is in e_{1}𝔯e_{1}, and hence are nilpotent.

It may also be shown that σ^{i}ψ_{1} ∘ η_{1} = σ^{i}ψ_{2}, and, more generally, that σ^{i}ψ_{1} ∘ η_{1}^{n−1} = σ^{i}ψ_{n} for i ≥ 1. Moreover θ ∘ η_{1} = σψ_{1}.

First suppose that char k ∤ r.

The element P^{2n+1} → Λ given by e_{1} ⊗ e_{1} ↦ e_{1} is not in Ker δ^{2n+2∗} and thus HH^{2n+1}(Λ) = sp{σψ_{n}, …, σ^{r−1}ψ_{n}} for n ≥ 1. Since it has already been shown that HH^{1}(Λ) is nilpotent, it follows that every element of HH^{2n+1}(Λ) is nilpotent for n ≥ 0. Thus for char k ∤ r,

HH^{∗}(Λ)/N ≅ HH^{even}(Λ)/N^{even} ≅ k[η_{1}]

where η_{1} is in degree 2, and

HH^{∗}(Λ) ≅ k[z, θ, η_{1}]/⟨z^{r+1}, θ^{2}, z^{r}θ, z^{r}η_{1}, z^{r−1}θη_{1}, z^{r−1}η_{1}^{2}⟩

where z is in degree 0, θ is in degree 1 and η_{1} is in degree 2.

Now consider the case where char k | r.

In this case, for n ≥ 1, the element P^{2n+1} → Λ defined by e_{1} ⊗ e_{1} ↦ e_{1} is in HH^{2n+1}(Λ) since the composition with δ^{2n+2} has image rσ^{r−1} = 0. For n ≥ 1, let ψ_{n}: P^{2n+1} → Λ be defined by e_{1} ⊗ e_{1} ↦ e_{1}. Then HH^{2n+1}(Λ) = sp{ψ_{n}, σψ_{n}, …, σ^{r−1}ψ_{n}} and also ψ_{1} ∘ η_{1}^{n−1} = ψ_{n}. It now remains to determine whether or not ψ_{1} is nilpotent.

If char k ≠ 2, then every element of HH^{2n+1}(Λ) is nilpotent as we note after Corollary 1.2 and thus ψ_{1} is nilpotent with ψ_{1}^{2} = 0.

Hence for char k | r and char k ≠ 2 we have

HH^{∗}(Λ) ≅ k[z, θ, η_{1}, ψ_{1}]/⟨z^{r+1}, θ^{2}, z^{r}θ, z^{r}η_{1}, z^{r−1}θη_{1}, z^{r−1}η_{1}^{2}, θη_{1} − σψ_{1}, ψ_{1}^{2}⟩

where z is in degree 0, θ is in degree 1, η_{1} is in degree 2 and ψ_{1} is in degree 3, and also

HH^{∗}(Λ)/N ≅ k[η_{1}]

where η_{1} is in degree 2.

If char k = 2, then we have 2 | r so r = 2q for some q. Then there are liftings so that ψ_{1}^{2} is represented by the map P^{6} → Λ given by e_{1} ⊗ e_{1} ↦ qσ^{r−2} and so ψ_{1}^{2} = qσ^{r−2}η_{1}^{3}. Thus ψ_{1}^{2}, and hence ψ_{1} itself, is nilpotent if r > 2 since then qσ^{r−2} ∈ 𝔯.

Hence for char k = 2 and r = 2q with q > 1 we have

HH^{∗}(Λ) ≅ k[z, θ, η_{1}, ψ_{1}]/⟨z^{r+1}, θ^{2}, z^{r}θ, z^{r}η_{1}, z^{r−1}θη_{1}, z^{r−1}η_{1}^{2}, θη_{1} − σψ_{1}, ψ_{1}^{2} − qσ^{r−2}η_{1}^{3}⟩

where z is in degree 0, θ is in degree 1, η_{1} is in degree 2 and ψ_{1} is in degree 3, and also

HH^{∗}(Λ)/N ≅ k[η_{1}]

where η_{1} is in degree 2.

Finally if char k = 2 and r = 2, then ψ_{1}^{2} is represented by the map P^{6} → Λ given by e_{1} ⊗ e_{1} ↦ e_{1}. Thus ψ_{1}^{2} = η_{1}^{3} and hence ψ_{1} is not nilpotent.

Thus for char k = 2 and r = 2 we have

HH^{∗}(Λ) ≅ k[z, θ, η_{1}, ψ_{1}]/⟨z^{r+1}, θ^{2}, z^{r}θ, z^{r}η_{1}, z^{r−1}θη_{1}, z^{r−1}η_{1}^{2}, θη_{1} − σψ_{1}, ψ_{1}^{2} − η_{1}^{3}⟩

where z is in degree 0, the element θ is in degree 1, the element η_{1} is in degree 2 and ψ_{1} is in degree 3, but in this case

HH^{∗}(Λ)/N ≅ k[η_{1}, ψ_{1}]/⟨ψ_{1}^{2} − η_{1}^{3}⟩

where η_{1} is in degree 2 and ψ_{1} is in degree 3.

It should be noted that, for all the algebras in Case III, irrespective of the characteristic of the field, we have shown HH^{∗}(Λ) is finitely generated as a ring.

We summarise the above in the following theorem.

Theorem 5.1. Let Λ = kQ/I be a finite dimensional Nakayama algebra where 22 nicole snashall and Øyvind solberg

k is a field and I is an admissible ideal generated by a single relation ρ of length l. Let N denote the ideal of HH^{∗}(Λ) which is generated by all the homogeneous nilpotent elements. Then HH^{∗}(Λ)/N is finitely generated as a ring.

If Λ has a projective simple module, then HH^{∗}(Λ) is a finite dimensional kalgebra. Furthermore, if Λ has no projective simple module and if Q has m vertices and m arrows, then HH^{∗}(Λ) is finitely generated as a ring if and only if l ≤ m − 1 or m | l.

6. Complexity

Throughout this section let Λ be an artin algebra over a commutative artinian ring k with Λ flat as a module over k. For group rings of finite groups over a field, the dimension of the variety of a module M is equal to the so-called complexity of the module. In this section we investigate elementary properties of a natural generalisation of this notion to our more general setting. Through examples we show that our notion of complexity seems to be as closely related to the same properties of the modules as it is in the group ring case.

Motivated by the definition for group rings and our definition of support varieties using a graded subalgebra of HH^{∗}(Λ), we make the following definition of complexity.

Definition 6.1.

(a) For a pair of Λ-modules M and N, the complexity c(M, N) is inf{b ∈ ℕ∪{0} | ℓ(Ext_{Λ}^{n}(M, N)) ≤ an^{b−1} for some a ∈ ℝ and for all n ≫ 0}, where ℓ(X) denotes the length of a k-module X.

(b) The complexity of a module M is defined as c(M) = c(M, Λ/𝔯).

We first give some elementary properties of the complexity of modules.

Proposition 6.2. For Λ-modules M, N

(a) c(M ⊕ N) = max{c(M), c(N)}.

(b) c(M) ≤ c(Λ/𝔯).

(c) c(M) = c(Ω_{Λ}(M)).

(d) Let 0 → M_{1} → M_{2} → M_{3} → 0 be an exact sequence of Λ-modules. Then c(M_{r}) ≤ max{c(M_{s}), c(M_{t})} whenever {r, s, t} = {1, 2, 3}.

(e) c(M) = 0 if and only if M has finite projective dimension.

Proof. (a) Since the minimal projective resolution of M ⊕ N is the direct sum of the minimal projective resolution of M and N, the claim follows easily.

(b) Using the radical filtration of M we conclude that ℓ(Ext_{Λ}^{n}(M, Λ/𝔯)) ≤ ℓ(M)ℓ(Ext_{Λ}^{n}(Λ/𝔯, Λ/𝔯)). This implies that c(M) ≤ c(Λ/𝔯).

(c) It is easy to see that ℓ(Ext_{Λ}^{i}(Ω_{Λ}(M), Λ/𝔯)) ≤ ai^{b−1} if and only if ℓ(Ext_{Λ}^{i}(M, Λ/𝔯)) ≤ a′i^{b−1} for real numbers a and a′. Hence c(M) = c(Ω_{Λ}(M)).

(d) Using the Horseshoe Lemma it is immediate that c(M_{2}) ≤ max{c(M_{1}), c(M_{3})}. Moreover there exists a projective module Q and an exact sequence 0 → Ω_{Λ}(M_{2}) → Ω_{Λ}(M_{3})⊕Q → M_{1} → 0. By our previous observation and (c), c(M_{3}) ≤ max{c(M_{1}), c(M_{2})}. Similarly we have c(M_{1}) ≤ max{c(M_{2}), c(M_{3})}.

(e) This is an immediate consequence of the definition.

The next examples show that, at least in some situations, the support variety of a module M and the complexity of M are related.

Example 6.3. Let Λ be a Nakayama algebra over an algebraically closed field k. If M is an indecomposable module, then Ω_{Λ}^{i}(M) is indecomposable with simple top for all i in ℕ whenever it is nonzero. Hence, if M is indecomposable, dim_{k} Ext_{Λ}^{i}(M, Λ/𝔯) ≤ 1 for all i in ℕ. Consequently, c(M) ≤ 1.

If Λ = kQ/I is a Nakayama algebra where k is a field and I is an admissible ideal generated by a single relation, we showed in section 5, for H = HH^{∗}(Λ), that \overline{H} = HH^{∗}(Λ)/N is either isomorphic to k or of Krull dimension 1. Only in Case III is \overline{H} non-trivial and of Krull dimension 1.

Using the above notation we have that the following conditions are equivalent for an indecomposable module in Case III: (i) c(M) = 0 and (ii) the variety of M is trivial. Moreover the complexity c(M) = 1 if and only if M is a periodic module. Here an indecomposable module M is said to be periodic if Ω_{Λ}^{m}(M) ≃ Ω_{Λ}^{n}(M) for some positive integers m and n with m ≠ n.

Example 6.4. Let Λ be a finite dimensional indecomposable algebra over a field k. Suppose Λ is a direct summand of Ω_{Λ^{e}}^{n}(Λ) for some n ≥ 1. Then M. C. R. Butler has observed that Λ is selfinjective and Ω_{Λ^{e}}^{n}(Λ) ≃ Λ; a proof of this can be found in [18]. If k is an algebraically closed field, then \overline{H} = HH^{∗}(Λ)/N is isomorphic to k[x] by [18, Proposition 1.1]. The condition Λ ≃ Ω_{Λ^{e}}^{n}(Λ) for some n ≥ 1 is known to be satisfied for selfinjective Nakayama algebras, Möbius algebras and preprojective algebras whenever Λ is a finite dimensional indecomposable algebra over an algebraically closed field (see [10, 11, 13]).

Let Λ be as above and suppose that Ω_{Λ^{e}}^{n}(Λ) ≃ Λ for some n ≥ 1. Then an indecomposable module is either projective or periodic. Thus an indecomposable module M is projective if and only if c(M) = 0, whilst M is periodic if and only if c(M) = 1.

7. Support varieties for group rings and complete intersections

This section is devoted to comparing the support varieties for complete intersections defined and studied by Avramov et al. in [2], [3], the support varieties for principal and non-principal blocks defined by Carlson and Linckelmann et al. (see [6], [22]) with the support varieties defined by a suitable graded subalgebra of the Hochschild cohomology ring of the ring as given in section 2. In particular there is a finite surjective map or an isomorphism from the support varieties defined by (a suitable graded subalgebra of) the Hochschild cohomology ring of the algebra to the support varieties considered in these situations.

Support varieties of modules have also been studied for finite dimensional cocommutative Hopf algebras (see [15]), and, as in the group case, there is a finite surjective map from the support varieties defined by the Hochschild cohomology ring to these varieties.

We start by considering varieties for modules over the principal block of a finite group. Let G be a finite group, and let k be a field with char k dividing the order of G. The variety of a module M over the principal block b_{0} is defined using the map γ_{M} = − ⊗_{k} M: H^{∗}(G, k) → Ext_{kG}^{∗}(M, M), where H^{∗}(G, k) is the group 24 nicole snashall and Øyvind solberg

cohomology ring Ext_{kG}^{∗}(k, k). The variety V_{G}(M) for the module M is defined to be the subvariety of MaxSpec H^{∗}(G, k) defined by the ideal Ker(−⊗_{k} M) in H^{∗}(G, k). For a module M over the principal block we have the commutative diagram

\xymatrix{H^{∗}(G, k) \ar[r]^{γ_{M}} \ar[d]^{ψ} & Ext_{kG}^{∗}(M, M) \ar[d]^{≀} \\ HH^{∗}(kGb_{0}) \ar[r]^{φ_{M}} & Ext_{kGb_{0}}^{∗}(M, M)}

where ψ is the composition H^{∗}(G, k) → HH^{∗}(kG) → HH^{∗}(kGb_{0}) with H^{∗}(G, k) → HH^{∗}(kG) induced by the induction functor Ind_{ΔG}^{G×G} (identifying k(G×G) with kG^{e} via (g, h) ↦ g ⊗ h^{−1}) and the map HH^{∗}(kG) → HH^{∗}(kGb_{0}) being the multiplication by b_{0}. Since HH^{∗}(kG) is a Noetherian module over H^{∗}(G, k), the module HH^{∗}(kGb_{0}) is also a Noetherian module over H^{∗}(G, k). Hence the map ψ induces a finite surjective map V_{HH^{∗}(kGb_{0})}(M) → V_{G}(M). For more details on this we refer the reader to [6].

For a non-principal block b of kG, let P_{γ} be a defect pointed group of b. Then Linckelmann considers the cohomology ring H^{∗}(G, b, P_{γ}) of the block b with respect to the defect pointed group P_{γ}. Moreover, there is a map of graded algebras from H^{∗}(G, b, P_{γ}) to HH^{∗}(kGb), where HH^{∗}(kGb) is a Noetherian module over H^{∗}(G, b, P_{γ}). The variety of a module M over the block b is defined using the compositions of the maps H^{∗}(G, b, P_{γ}) → HH^{∗}(kGb) → Ext_{kGb}^{∗}(M, M). The variety V_{G,b}(M) for a module M over the block b is defined to be the subvariety of MaxSpec H^{∗}(G, b, P_{γ}) defined by the kernel of this map. Since HH^{∗}(kGb) is a Noetherian module over H^{∗}(G, b, P_{γ}), we have as for a principal block a finite surjective map V_{HH^{∗}(kGb)}(M) → V_{G,b}(M). For a more detailed presentation of the above we refer the reader to [22].

Now we turn our attention to support varieties for complete intersections. Let k be an algebraically closed field, and let Q = k⟦x_{1}, …, x_{n}⟧ be the formal power series ring in the indeterminants {x_{1}, …, x_{n}}. Let {a_{1}, …, a_{t}} be a regular sequence in the maximal ideal of Q, and denote by 𝔞 the ideal in Q generated by this regular sequence. Let R be the complete intersection Q/𝔞. Then t cohomological operators {χ_{j}}_{j=1}^{t} are defined in [25] with an action on the left and the right of Ext_{R}^{∗}(M, N) for two finitely generated R-modules M and N via a graded ring homomorphism φ_{X}: R[χ_{1}, …, χ_{t}] → Ext_{R}^{∗}(X, X) for an R-module X. These ring homomorphisms satisfy the following properties: (i) Im φ_{X} is contained in the centre of the Extalgebra Ext_{R}^{∗}(X, X) and (ii) φ_{N}(χ_{j})η = ηφ_{M}(χ_{j}) for any element η in Ext_{R}^{n}(M, N) and j in {1, …, t}. Denote by A′_{H}(M, N) the annihilator Ann_{H}(Ext_{R}^{∗}(M, N) ⊗_{R} Q/(x_{1}, …, x_{n})), where H = R[χ_{1}, …, χ_{t}] ⊗_{R} Q/(x_{1}, …, x_{n}). Define

V′_{R}(M, N) = {(b_{1}, …, b_{t}) ∈ k^{t} | φ(b_{1}, …, b_{t}) = 0, ∀ φ ∈ A′_{H}(M, N)}.

The operators χ_{j} for j = 1, …, t are defined as follows (see [25]). For further details on different ways of describing these operators see [4]. Let M be a finitely generated R-module. Let 0 → B → G → M → 0 be an exact sequence of Q-modules with G a projective Q-module. Applying the functor R ⊗_{Q} − to this sequence gives rise to the exact sequence

0 → 𝔞/𝔞^{2} ⊗_{R} M → R ⊗_{Q} B → R ⊗_{Q} G → M → 0.

Since \{a_{1}, …, a_{t}\} is a regular sequence in Q, the factor 𝔞/𝔞^{2} ≃ ⊕_{i=1}^{t}R\overline{a_{i}}, where \overline{a_{i}} = a_{i} + 𝔞^{2}. Let a_{j}^{∗}: 𝔞/𝔞^{2} → R be the dual basis of \overline{a_{j}} for j = 1, …, t. Then consider the commutative diagram

[figure: 0; 𝔞/𝔞^{2} ⊗_{R} M; a_{j}^{∗}⊗M; R ⊗_{Q} B; R ⊗_{Q} G; M; 0; 0; R ⊗_{R} M; E_{j}^{M}; R ⊗_{Q} G; M; 0.]

Denote the lower row by E_{j}^{M}. Let η: 0 → N → B_{1} → ⋯ → B_{n} → M → 0 be an extension in Ext_{R}^{n}(M, N) for two R-modules M and N. Then ηχ_{j} is by definition the Yoneda product of η and E_{j}^{M}, and χ_{j}η is by definition the Yoneda product of E_{j}^{N} and η. It is known that ηχ_{j} = χ_{j}η, and the homomorphism φ_{M}: R[χ_{1}, …, χ_{t}] → Ext_{R}^{∗}(M, M) given by φ_{M}(χ_{j}) = E_{j}^{M} has its image in the centre of Ext_{R}^{∗}(M, M).

Next we interpret the operators χ_{j} for j = 1, …, t as an action of a subalgebra of the Hochschild cohomology ring on Ext_{R}^{∗}(M, M).

Let 0 → K → Q ⊗_{k} Q → Q → 0 be exact, where the rightmost morphism is multiplication. For any R-module M this gives rise to the short exact sequence 0 → K ⊗_{Q} M → Q ⊗_{k} M → M → 0 where G = Q ⊗_{k} M is a projective Q-module. Applying the functor R ⊗_{Q} − to this exact sequence with M = R induces the exact sequence of R-R-bimodules

0 → 𝔞/𝔞^{2} → R ⊗_{Q} K ⊗_{Q} R → R ⊗_{k} R → R → 0

We also have the commutative diagram

[figure: 0; Ω_{R^{e}}^{2}(R); γ; P_{1}; R ⊗_{k} R; R; 0; 0; 𝔞/𝔞^{2}; R ⊗_{Q} K ⊗_{Q} R; R ⊗_{k} R; R; 0.]

Denote the upper row in this diagram by E. On tensoring the exact sequence E with − ⊗_{R} M and taking the pushout by the map (a_{j}^{∗} ⊗ M)(γ ⊗ M) we obtain the exact sequence E_{j}^{M}. Taking the pushout by the map a_{j}^{∗}γ gives rise to an exact sequence E_{j}^{R}, and then applying −⊗_{R} M we also obtain the exact sequence E_{j}^{M}; hence E_{j}^{M} = E_{j}^{R} ⊗_{R} M. Let ψ: R[χ_{1}, …, χ_{t}] → HH^{even}(R) be given by ψ(χ_{j}) = E_{j}^{R}. Then ψ is a ring homomorphism, and the action of χ_{j} on θ in Ext_{R}^{∗}(M, N) is given by χ_{j}θ = (ψ(χ_{j}) ⊗_{R} N)θ = θ(ψ(χ_{j}) ⊗_{R} M). This shows that the support varieties defined by Avramov are given by choosing the graded subalgebra R[E_{1}^{R}, …, E_{t}^{R}] of HH^{even}(R) and using the definition of support varieties given in section 2. Furthermore that the operators satisfy the properties they have, is a straightforward consequence of Corollary 1.3.

Acknowledgement

The first author thanks the Department of Mathematical Sciences at NTNU, Trondheim for their generous hospitality and the Research Council of Norway for their financial support. The second author thanks the London Mathematical Society for the support during the preparation of this work and also the Department of Mathematics and Computer Science at the University of Leicester for their very kind hospitality and support for his visit there. Both the authors thank Luchezar 26 support varieties and hochschild cohomology rings

L. Avramov, Ragnar-Olaf Buchweitz and Karin Erdmann for their interest and comments on earlier versions of this paper.

References

1. Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995.

2. Avramov, L. L., Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71–101.

3. Avramov, L. L. and Buchweitz, R.-O., Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), 285–318.

4. Avramov, L. L. and Sun, L.-C., Cohomology operators defined by a deformation, J. Algebra 204 (1998), 684–710.

5. Bardzell, M. J., The alternating syzygy behavior of monomial algebras, J. Algebra 188 (1997), no. 1, 69–89.

6. Carlson, J. F., The complexity and varieties of modules, in Lecture Notes in Mathematics 882, 415–422, Springer-Verlag, 1981.

7. Carlson, J. F., Varieties and the cohomology ring of a module, J. Algebra 85 (1983), 104–143.

8. Cartan, H. and Eilenberg, S., Homological algebra, Princeton University Press, 1956.

9. Evens, L., The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961) 224– 239.

10. Erdmann, K. and Holm, T., Twisted bimodules and Hochschild cohomology for self-injective algebras of class A_{n}, Forum Math. 11 (1999), 177–201.

11. Erdmann, K., Holm, T. and Snashall, N., Twisted bimodules and Hochschild cohomology for self-injective algebras of type A_{n} II, Algebras and Representation Theory 5 (2002), 457–482.

12. Erdmann, K., Holm, T., Snashall, N., Solberg, Ø. and Taillefer, R., Support varieties for selfinjective algebras, preprint in preparation.

13. Erdmann, K. and Snashall, N., Preprojective algebras of Dynkin type: periodicity and the second Hochschild cohomology, Canad. Math. Soc. Conference Proceedings 24 (1998), 183–193.

14. Friedlander, E. M. and Parshall, B. J., Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), 553–562.

15. Friedlander, E. M. and Suslin, A., Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270.

16. Ginzburg, V. and Kumar, S., Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), no. 1, 179–198.

17. Green, E. L., Snashall, N. and Solberg, Ø., The Hochschild cohomology ring modulo nilpotence of a monomial algebra, preprint in preparation.

18. Green, E. L., Snashall, N. and Solberg, Ø., The Hochschild cohomology ring of a selfinjective algebra of finite representation type, to appear in Proc. Amer. Math. Soc.

19. Happel, D., Hochschild cohomology of finite-dimensional algebras, Springer Lecture Notes in Mathematics 1404 (1989), 108–126.

20. Holm, T., Hochschild cohomology rings of algebras k[X]/(f), Beiträge Algebra Geom. 41 (2000), no. 1, 291–301.

21. Hovey, M. and Palmieri, H. H., Stably thick subcategories of modules over Hopf algebras, Math. Proc. Camb. Phil. Soc. 130 (2001), 441–474.

22. Linckelmann, M., Varieties in block theory, J. Algebra 215 (1999), 460–480.

23. Locateli, A. C., Hochschild cohomology of truncated quiver algebras, Comm. Alg. 27 (1999), no. 2, 645–664.

24. MacLane, S., Homology, Classics in Mathematics, Springer-Verlag, 1995.

25. Metha, V., Endomorphisms of complexes and modules over Golod rings, PhD-thesis, University of California, Berkeley, CA, 1976.

26. Quillen, D., The spectrum of an equivariant cohomology ring I, Ann. Math. 94 (1971), 549– 572.

27. Venkov, B. B., Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk SSSR 127 (1959) 943–944.

Nicole Snashall Department of Mathematics and

Computer Science University of Leicester University Road Leicester, LE1 7RH

njs5@mcs.le.ac.uk Øyvind Solberg Institutt for matematiske fag NTNU N–7491 Trondheim Norway

oyvinso@math.ntnu.no

SUPPORT VARIETIES AND HOCHSCHILD COHOMOLOGY RINGS

NICOLE SNASHALL and ØYVIND SOLBERG

Abstract for On-Line Publication

We define a support variety for a finitely generated module over an artin algebra Λ over a commutative artinian ring k, with Λ flat as a module over k, in terms of the maximal ideal spectrum of the algebra HH^{∗}(Λ) of Λ. This is modelled on what is done in modular representation theory, and the varieties defined in this way are shown to have many of the same properties as for group rings. In fact the notion of a variety in our sense and for principal and non-principal blocks are related by a finite surjective map of varieties. For a finite dimensional selfinjective algebra over a field the variety is shown to be an invariant of the stable component of the Auslander-Reiten quiver. Moreover we give information on nilpotent elements in HH^{∗}(Λ), give a thorough discussion of the ring HH^{∗}(Λ) on a class of Nakayama algebras, a brief discussion on a possible notion of complexity and make a comparison with support varieties for complete intersections.