REPRESENTATION DIMENSION AND FINITELY GENERATED COHOMOLOGY

PETTER ANDREAS BERGH

Dedicated to Karin Erdmann on the occasion of her sixtieth birthday

Abstract. We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.

1. Introduction

In his 1971 notes [Au1], Auslander introduced the notion of the representation dimension of an Artin algebra. This invariant measures how far an algebra is from having finite representation type; it was introduced in order to study algebras of infinite representation type. A non-semisimple algebra is of finite type if and only if its representation dimension is exactly two, and of infinite type if and only if the representation dimension is at least three.

The representation dimension of an Artin algebra is always finite (cf. [Iya]). However, for a long time it was unclear whether there could exist algebras of representation dimension strictly greater than three. Moreover, Igusa and Todorov showed in [IgT] that if this was not the case, i.e. if the representation dimension could not exceed three, then the finitistic dimension conjecture would hold. However, in 2006 Rouquier showed in [Ro2] that the representation dimension of the exterior algebra on a d-dimensional vector space is d + 1, using the notion of the dimension of a triangulated category (cf. [Ro1]). Other examples illustrating this were subsequently given in [AvI], [BeO], [KrK], [Op1] and [Op2].

In this paper, we study selfinjective Artin algebras satisfying a certain finite generation hypothesis on its cohomology groups. Namely, we assume there exists a commutative graded Noetherian ring over which all the cohomology groups are finitely generated. Using a version of the Ghost Lemma, we obtain a lower bound for the dimension of the stable module category of an algebra satisfying this hypothesis. In particular, we show that the representation dimension of such an algebra is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing the known lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.

2000 Mathematics Subject Classification. 16G60, 16E30, 16E40.

Key words and phrases. Representation dimension, finitely generated cohomology, complexity.

The author was supported by NFR Storforsk grant no. 167130.

2. Representation dimension

Throughout this paper, we let k be a commutative Artin ring and Λ an Artin k-algebra with Jacobson radical 𝔯. We denote by mod Λ the category of finitely generated Λ-modules. The representation dimension of Λ, denoted repdim Λ, is defined as

repdim Λ =^{def} inf{gl. dim End_{Λ}(M) | M generates and cogenerates mod Λ},

where gl. dim denotes the global dimension of an algebra. Auslander showed that the representation dimension of a selfinjective algebra is at most its Loewy length, whereas Iyama showed in [Iya] that this invariant is finite for every Artin algebra.

In order to compute the representation dimension of exterior algebras, Rouquier used the notion of the dimension of a triangulated category, a concept he introduced in [Ro1]. We recall here the definitions. Let T be a triangulated category, and let C and D be subcategories of T . We denote by ⟨C⟩ the full subcategory of T consisting of all the direct summands of finite direct sums of shifts of objects in C. Furthermore, we denote by C ∗ D the full subcategory of T consisting of objects M such that there exists a distinguished triangle

C → M → D → C[1]

in T , with C ∈ C and D ∈ D. Finally, we denote the subcategory ⟨C ∗ D⟩ by C ⋄ D. Now define ⟨C⟩_{1} to be ⟨C⟩, and for each n ≥ 2 define inductively ⟨C⟩_{n} to be ⟨C⟩_{n−1} ⋄ ⟨C⟩. The dimension of T , denoted dim T , is defined as

dim T =^{def} inf{d ∈ ℤ | there exists an object M ∈ T such that T = ⟨M⟩_{d+1}}.

In other words, the dimension of T is the minimum number of layers needed to obtain T from one of its objects.

The key ingredient in the proof of our main result is the following lemma on compositions of natural transformations. The lemma is analogous to [Ro1, Lemma 4.11].

Lemma 2.1 (Ghost Lemma). Let T be a triangulated category, let H_{1}, …, H_{n+1} be cohomological functors on T , and for each 1 ≤ i ≤ n let H_{i} ⟶^{f_{i}} H_{i+1} be a natural transformation. Furthermore, let C_{1}, …, C_{n} be subcategories of T closed under shifts, and assume that for every object c ∈ C_{i} the map H_{i}(c[j]) ⟶^{f_{i}} H_{i+1}(c[j]) vanishes for j ≫ 0 (respectively, for j ≪ 0). Then for every object w ∈ C_{1} ⋄ ⋯ ⋄ C_{n} the map H_{1}(w[j]) ⟶^{f_{n}···f_{1}} H_{n+1}(w[j]) vanishes for j ≫ 0 (respectively, for j ≪ 0).

Proof. We may assume n ≥ 2. Let c_{1} → c → c_{2} → c_{1}[1] be a triangle in T with c_{1} ∈ C_{1} and c_{2} ∈ C_{2}. Then for every j ∈ ℤ, there is a commutative diagram

\xymatrix{H_{1}(c_{1}[j]) \ar[r] \ar[d]^{f_{1}} & H_{1}(c[j]) \ar[r] \ar[d]^{f_{1}} & H_{1}(c_{2}[j]) \ar[d]^{f_{1}} \\ H_{2}(c_{1}[j]) \ar[r] \ar[d]^{f_{2}} & H_{2}(c[j]) \ar[r] \ar[d]^{f_{2}} & H_{2}(c_{2}[j]) \ar[d]^{f_{2}} \\ H_{3}(c_{1}[j]) \ar[r] & H_{3}(c[j]) \ar[r] & H_{3}(c_{2}[j])}

with exact rows. By assumption, there is an integer j_{1} such that the vertical upper left map vanishes for j ≥ j_{1}, and an integer j_{2} such that the vertical lower right map vanishes for j ≥ j_{2}. An easy diagram chase shows that the vertical middle composition vanishes for j ≥ max{j_{1}, j_{2}}, hence for every object w ∈ C_{1} ⋄ C_{2} the map H_{1}(w[j]) ⟶^{f_{2}f_{1}} H_{3}(w[j]) vanishes for j ≫ 0. An induction argument now establishes the lemma. □

The triangulated category we shall use is the stable module category of Λ, in the case when Λ is selfinjective. This category, denoted \underline{mod}Λ, is defined as follows: the objects of \underline{mod}Λ are the same as in mod Λ, but two morphisms in mod Λ are equal in \underline{mod}Λ if their difference factors through a projective Λ-module. The cosyzygy functor Ω_{Λ}^{−1} : \underline{mod}Λ → \underline{mod}Λ is an equivalence of categories, and a triangulation of \underline{mod}Λ is given by using this functor as a shift and by letting short exact sequences in mod Λ correspond to triangles. Thus \underline{mod}Λ is a triangulated category, and its dimension is related to the representation dimension of Λ by the following result.

Proposition 2.2. [Ro2, Proposition 3.7] If Λ is selfinjective and not semisimple, then repdim Λ ≥ dim(\underline{mod}Λ) + 2.

This result was originally formulated for a finite dimensional algebra over a field, but it works just as well in our setting, i.e. for an Artin algebra which is not necessarily finite dimensional over a field.

The main result of this paper relates the representation dimension of Λ with the maximal complexity occurring among its finitely generated modules. Recall therefore that for a module M ∈ mod Λ with minimal projective resolution

⋯ → P_{2} → P_{1} → P_{0} → M → 0,

say, the complexity of M is defined as

cx M =^{def} inf{t ∈ ℕ ∪ {0} | ∃a ∈ ℝ such that ℓ_{k}(P_{n}) ≤ an^{t−1} for n ≫ 0}.

In general, the complexity of a module may be infinite, whereas it is zero if and only if the module is projective. The complexity of M can be computed as the rate of growth of the graded k-module Ext_{Λ}^{∗}(M, Λ/𝔯), and from the definition we also see that it equals the complexity of Ω_{Λ}^{i}(M) for any i ∈ ℕ. Moreover, given a short exact sequence

0 → X_{1} → X_{2} → X_{3} → 0

in mod Λ, it is well known that the inequality cx X_{u} ≤ sup{cx X_{v}, cx X_{w}} holds for {u, v, w} = {1, 2, 3}. In particular, induction on the length of a module shows that cx X ≤ cx Λ/𝔯 for every X ∈ mod Λ. We end this section with the following elementary lemma, which shows that a module generating \underline{mod}Λ must be of maximal complexity.

Lemma 2.3. Let Λ be selfinjective, let M ∈ mod Λ be a module, and suppose there exists a number n ∈ ℕ such that ⟨M⟩_{n} = \underline{mod}Λ. Then cx N ≤ cx M for every N ∈ mod Λ, in particular cx M = cx Λ/𝔯.

Proof. The result follows from the fact that triangles in \underline{mod}Λ correspond to short exact sequences in mod Λ. □

3. Finitely generated cohomology

We now introduce a certain “finite generation” assumption on the cohomology groups of Λ. Recall that for Λ-modules X and Y, the graded k-module Ext_{Λ}^{∗}(X, Y) is an Ext_{Λ}^{∗}(Y, Y) − Ext_{Λ}^{∗}(X, X)-bimodule via Yoneda products. Also recall that a graded k-module ⨁ V_{i} is of finite type if each V_{i} is a finitely generated k-module.

Assumption (Fg). There exists a commutative Noetherian graded k-algebra H = ⨁_{i=0}^{∞} H^{i} of finite type satisfying the following:

(i) For every M ∈ mod Λ there is a graded ring homomorphism

ϕ_{M}: H → Ext_{Λ}^{∗}(M, M).

(ii) For each pair (X, Y) of finitely generated Λ-modules, the scalar actions from H on Ext_{Λ}^{∗}(X, Y) via ϕ_{X} and ϕ_{Y} coincide, and Ext_{Λ}^{∗}(X, Y) is a finitely generated H-module.

In the assumption, why do we require that the left and right scalar multiplications on Ext_{Λ}^{∗}(X, Y) coincide? The reason is that this requirement is what makes the bifunctor Ext_{Λ}^{∗}(−, −) preserve maps. To see this, let f: M → M′ be a homomorphism in mod Λ. For every N ∈ mod Λ, this map induces a homomorphism f̂: Ext_{Λ}^{∗}(M′, N) → Ext_{Λ}^{∗}(M, N) of graded groups. The image of a homogeneous element

θ: 0 → N → X_{n} → ⋯ → X_{1} → M′ → 0

is the extension θf given by the commutative diagram

[figure: θf: 0; N; X_{n}; ⋯; X_{2}; Y; M; f; 0; θ: 0; N; X_{n}; ⋯; X_{2}; X_{1}; M′; 0]

in which the module Y is a pullback. For a homogeneous element η ∈ H we then get

f̂(θ · η) = f̂(η · θ)
= f̂(ϕ_{N}(η) ∘ θ)
= ϕ_{N}(η) ∘ (θf)
= η · f̂(θ)
= f̂(θ) · η,

showing f̂ is a homomorphism of H-modules. Similarly, Ext_{Λ}^{∗}(−, −) preserves maps in the second argument. The fact that Ext_{Λ}^{∗}(−, −) preserves maps is absolutely essential. Note that when using this property, induction on length of modules shows that the finite generation part of the assumption Fg is equivalent to Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯) being a finitely generated H-module.

It should also be noted that when Fg holds, then every finitely generated Λ-module has finite complexity, i.e. cx Λ/𝔯 < ∞. Namely, the H-module Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯) is finitely generated, and so its rate of growth is not more than that of H. The ring H is commutative Noetherian and of finite type, and hence its rate of growth is finite.

In the following examples we point out three important situations in which the assumption Fg holds.

Examples. (i) Suppose k is a field of positive characteristic p, and let G be a finite group whose order is divisible by p. Then by a theorem of Evens (cf. [Eve]), the graded commutative group cohomology ring H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) is Noetherian. Moreover, if X_{1} and X_{2} are finitely generated kG-modules, then Ext_{kG}^{∗}(X_{1}, X_{2}) is a finitely generated H^{∗}(G, k)-module via the maps

− ⊗_{k} X_{i}: H^{∗}(G, k) → Ext_{kG}^{∗}(X_{i}, X_{i}),

and the right and left scalar actions induced by these maps commute up to a graded sign. The even part ⨁ H^{2i}(G, k) of H^{∗}(G, k) is a commutative k-algebra, over which H^{∗}(G, k) is finitely generated as a module.

(ii) Let (A, 𝔪, k) be a commutative Noetherian local complete intersection. That is, the completion Â, with respect to the maximal ideal 𝔪, is of the form R/(x_{1}, …, x_{c}), where R is regular local and x_{1}, …, x_{c} is a regular sequence. We may without loss of generality assume that the length c of the defining regular sequence is the codimension of A, i.e. c = dim_{k} (𝔪 / 𝔪^{2}) − dim A. By [Avr, Section 1] there exists a polynomial ring Â[χ_{1}, …, χ_{c}] in commuting Eisenbud operators, such that for every finitely generated Â-module X there is a homomorphism

ϕ_{X}: Â[χ_{1}, …, χ_{c}] → Ext∗ ; Â(X, X)

of graded rings. Moreover, for every finitely generated Â-module Y, the left and right scalar actions on Ext∗ ; Â(X, Y) coincide, and the latter is a finitely generated Â[χ_{1}, …, χ_{c}]-module. Now if A is Artin, then it is a complete ring since 𝔪 is nilpotent. Thus Fg holds in this case.

(iii) Suppose Λ is projective as a k-module. Denote by Λ^{e} the enveloping algebra Λ ⊗_{k} Λ^{op} of Λ, and by HH^{∗}(Λ) its Hochschild cohomology ring. By [Yon, Proposition 3], this is a graded commutative ring, and equal to Ext_{Λ^{e}}^{∗}(Λ, Λ) since Λ is k-projective. If X_{1} and X_{2} are finitely generated Λ-modules, then the right and left scalar actions from HH^{∗}(Λ) on Ext_{Λ}^{∗}(X_{1}, X_{2}), via the maps

− ⊗_{Λ} X_{i}: HH^{∗}(Λ) → Ext_{Λ}^{∗}(X_{i}, X_{i}),

are graded commutative.

In [EHSST] the finite generation assumption imposed was the following: there exists a commutative Noetherian graded subalgebra S = ⨁_{i=0}^{∞} S^{i} of HH^{∗}(Λ), with S^{0} = HH^{0}(Λ), such that Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯) is a finitely generated S-module. However, having to deal with such an “unknown” subalgebra of the Hochschild cohomology ring is not satisfactory, and in fact it is not difficult to see (cf. [Sol, Proposition 5.7]) that the assumption is equivalent to the following one: the Hochschild cohomology ring HH^{∗}(Λ) is Noetherian and Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯) is a finitely generated HH^{∗}(Λ)- module. Therefore, as in the first example, we see that Fg holds by choosing H to be the even part ⨁ HH^{2i}(Λ) of HH^{∗}(Λ).

In particular, the assumption Fg holds for exterior algebras. Namely, suppose k is a field, let n be a number, and denote by Λ the algebra

k⟨x_{1}, …, x_{n}⟩/(x_{i}^{2}, x_{i}x_{j} + x_{j}x_{i}),

i.e. Λ is the exterior algebra on an n-dimensional vector space. Then by [Sol, Theorem 9.2 and Theorem 9.11], the Koszul dual Ext_{Λ}^{∗}(k, k) of Λ is the polynomial ring k[x_{1}, …, x_{n}], and via the map

− ⊗_{Λ} k: HH^{∗}(Λ) → Ext_{Λ}^{∗}(k, k)

this is a finitely generated HH^{∗}(Λ)-module. By choosing H to be the even part of the inverse image of k[x_{1}, …, x_{n}], we see that Fg holds.

Having pointed out these three examples where Fg hold, we now proceed with the following result. Together with the Ghost Lemma, it is the key ingredient in the proof of the main result.

Proposition 3.1. Suppose Λ is selfinjective and Fg holds. Then for every Λmodule M such that cx M ≥ 2, there exists a sequence

M = K_{0} ⟶^{f_{1}} K_{1} → ⋯ ⟶^{f_{c−1}} K_{c−1}

of modules and monomorphisms in mod Λ, where c = cx M, satisfying the following: (i) cx K_{j} = c − j for each 0 ≤ j ≤ c − 1,

(ii) for each 1 ≤ j ≤ c−1 the map Ext_{Λ}^{i}(K_{j}, M) ⟶^{(f_{j})^{∗}} Ext_{Λ}^{i}(K_{j−1}, M) vanishes

for i ≫ 0, (iii) the composition f_{c−1} ∘ ⋯ ∘ f_{1} is nonzero in \underline{mod}Λ.

Proof. First we recall some general facts. Let R be a positively graded commutative Noetherian k-algebra of finite type, and let G = ⊕_{n=0}^{∞}G^{n} be a finitely generated graded R-module. An application of the prime avoidance lemma gives the existence of a homogeneous element r ∈ R, of positive degree, such that the multiplication map

G^{n} ⟶^{·r} G^{n+|r|}

is injective for n ≫ 0 (cf. [Be1, Proposition 2.1]). Therefore, since Fg holds for Λ, given any Λ-modules X and Y there exists a homogeneous element η ∈ H^{+} such that the multiplication map

Ext_{Λ}^{i}(X, Y) ⟶^{·η} Ext_{Λ}^{i+|η|}(X, Y)

is injective for i ≫ 0.

Choose a homogeneous element η_{1} ∈ H^{+} such that the multiplication map

Ext_{Λ}^{i}(M, M ⊕ Λ/𝔯) ⟶^{·η_{1}} Ext_{Λ}^{i+|η_{1}|}(M, M ⊕ Λ/𝔯)

is injective for i ≫ 0. Applying the map ϕ_{M} to η_{1} gives a short exact sequence

ϕ_{M}(η_{1}): 0 → M ⟶^{f_{1}} K_{1} → Ω_{Λ}^{|η_{1}|−1}(M) → 0,

and the arguments used in the proof of [Be1, Theorem 3.2] shows that cx K_{1} = c−1. Next, if c ≥ 3, choose a homogeneous element η_{2} ∈ H^{+} such that the multiplication map

Ext_{Λ}^{i}(K_{1}, M ⊕ Λ/𝔯) ⊕ Ext_{Λ}^{i}(M, K_{1}) ⟶^{·η_{2}} Ext_{Λ}^{i+|η_{2}|}(K_{1}, M ⊕ Λ/𝔯) ⊕ Ext_{Λ}^{i+|η_{2}|}(M, K_{1})

is injective for i ≫ 0. Applying the map ϕ_{K_{1}} to η_{2} gives a short exact sequence

ϕ_{K_{1}}(η_{2}): 0 → K_{1} ⟶^{f_{2}} K_{2} → Ω_{Λ}^{|η_{2}|−1}(K_{1}) → 0,

in which cx K_{2} = c − 2. We continue this process until we end up with a module K_{c−1} of complexity 1. Thus we obtain homogeneous elements η_{1}, …, η_{c−1} ∈ H^{+}, and for each 1 ≤ j ≤ c − 1 a short exact sequence

ϕ_{K_{j−1}}(η_{j}): 0 → K_{j−1} ⟶^{f_{j}} K_{j} → Ω_{Λ}^{|η_{j}|−1}(K_{j−1}) → 0

with cx K_{j} = c − j (here K_{0} = M). For each j the element η_{j} is chosen in such a way that it is regular on Ext_{Λ}^{i}(K_{j−1}, M ⊕ Λ/𝔯) ⊕ Ext_{Λ}^{i}(M, K_{j−1}) for i ≫ 0.

We show that the sequence

M ⟶^{f_{1}} K_{1} → ⋯ ⟶^{f_{c−1}} K_{c−1}

of modules and maps satisfies (i), (ii) and (iii) in the statement of the proposition. It follows directly from the construction that cx K_{j} = c − j, hence (i) holds. Now for each 1 ≤ j ≤ c − 1 and i ≫ 0, the exact sequence ϕ_{K_{j−1}}(η_{j}) induces the two exact sequences

\xymatrix{Ext_{Λ}^{i}(K_{j}, M) \ar[r]^{(f_{j})^{∗}} & Ext_{Λ}^{i}(K_{j−1}, M) \ar[r]^{·η_{j}} & Ext_{Λ}^{i+|η_{j}|}(K_{j−1}, M)}

and

\xymatrix{Ext_{Λ}^{i}(M, K_{j−1}) \ar[r]^{(f_{j})_{∗}} & Ext_{Λ}^{i}(M, K_{j}) \ar[r] & Ext_{Λ}^{i−|η_{j}|+1}(M, K_{j−1}) \ar[r]^{·η_{j}} & Ext_{Λ}^{i+1}(M, K_{j−1}).}

From the upper exact sequence, we see that the map Ext_{Λ}^{i}(K_{j}, M) ⟶^{(f_{j})^{∗}} Ext_{Λ}^{i}(K_{j−1}, M) vanishes for i ≫ 0, since the multiplication map involving η_{j} is injective. Consequently (ii) holds. From the lower exact sequence we see that the map Ext_{Λ}^{i}(M, K_{j−1}) ⟶^{(f_{j})_{∗}} Ext_{Λ}^{i}(M, K_{j}) is surjective for i ≫ 0. This implies that when i is large, the maps f_{1}, …, f_{c−1} induce a chain

Ext_{Λ}^{i}(M, M) ⟶^{(f_{1})_{∗}} Ext_{Λ}^{i}(M, K_{1}) → ⋯ ⟶^{(f_{c−1})_{∗}} Ext_{Λ}^{i}(M, K_{c−1})

of epimorphisms. Now choose a homogeneous element η ∈ H^{+} which is regular on Ext_{Λ}^{i}(K_{c−1}, Λ/𝔯) for i ≫ 0. Applying ϕ_{K_{c−1}} to this element gives an element

ϕ_{K_{c−1}}(η): 0 → K_{c−1} → K → Ω_{Λ}^{|η|−1}(K_{c−1}) → 0

in Ext_{Λ}^{∗}(K_{c−1}, K_{c−1}), where the module K is projective. Using the arguments in the proof of [Be2, Corollary 3.2], we see that ϕ_{K_{c−1}}(η) cannot be nilpotent in Ext_{Λ}^{∗}(K_{c−1}, K_{c−1}). Consequently, given any w ∈ ℕ, there is an integer i ≥ w such that Ext_{Λ}^{i}(K_{c−1}, K_{c−1}) is nonzero. Using the exact sequences ϕ_{M}(η_{1}), ϕ_{K_{1}}(η_{2}), …, ϕ_{K_{c−2}}(η_{c−1}) we then see that given any w ∈ ℕ, there is an integer i ≥ w such that Ext_{Λ}^{i}(M, K_{c−1}) is nonzero. This shows that the composition

M ⟶^{f_{1}} K_{1} → ⋯ ⟶^{f_{c−1}} K_{c−1}

is nonzero in \underline{mod}Λ, and so (iii) holds. □

We now prove the main result: when Λ is selfinjective and Fg holds, then the dimension of the stable module category of Λ is at least cx Λ/𝔯 − 1.

Theorem 3.2. If Λ is selfinjective and Fg holds, then dim(\underline{mod}Λ) ≥ cx Λ/𝔯 − 1.

Proof. Denote the complexity of Λ/𝔯 by c. Let M ∈ mod Λ be a module generating \underline{mod}Λ, i.e. there exists a number n such that ⟨M⟩_{n} = \underline{mod}Λ. Our aim is to show that n ≥ c. If c ≤ 1, then there is nothing to prove, so we may assume c ≥ 2.

By Lemma 2.3 the module M must have maximal complexity, that is, the equality cx M = c holds. Choose a sequence

M = K_{0} ⟶^{f_{1}} K_{1} → ⋯ ⟶^{f_{c−1}} K_{c−1}

of modules and maps as in Proposition 3.1, and consider the functors \underline{Hom}_{Λ}(K_{j}, −) on \underline{mod}Λ, together with the natural transformations

\underline{Hom}_{Λ}(K_{c−1}, −) ⟶^{(f_{c−1})^{∗}} \underline{Hom}_{Λ}(K_{c−2}, −) → ⋯ ⟶^{(f_{1})^{∗}} \underline{Hom}_{Λ}(M, −).

Since the map Ext_{Λ}^{i}(K_{j}, M) ⟶^{(f_{j})^{∗}} Ext_{Λ}^{i}(K_{j−1}, M) vanishes for i ≫ 0, the map

\underline{Hom}_{Λ}(K_{j}, Ω_{Λ}^{i}(M)) ⟶^{(f_{j})^{∗}} \underline{Hom}_{Λ}(K_{j−1}, Ω_{Λ}^{i}(M)) vanishes for i ≪ 0. From Lemma 2.1 we conclude that for every module X ∈ ⟨M⟩_{c−1}, the map

\underline{Hom}_{Λ}(K_{c−1}, Ω_{Λ}^{i}(X)) ⟶^{(f_{c−1}∘···∘f_{1})^{∗}} \underline{Hom}_{Λ}(M, Ω_{Λ}^{i}(X))

vanishes for i ≪ 0. However, by [Be1, Theorem 2.3] the module K_{c−1} is periodic in \underline{mod}Λ, that is, there is an integer p ≥ 1 such that K_{c−1} ≃ Ω_{Λ}^{p}(K_{c−1}) in \underline{mod}Λ. Therefore, since the composition f_{c−1} ∘ ⋯ ∘ f_{1} is nonzero in \underline{mod}Λ, the map

\underline{Hom}_{Λ}(K_{c−1}, Ω_{Λ}^{ip}(K_{c−1})) ⟶^{(f_{c−1}∘···∘f_{1})^{∗}} \underline{Hom}_{Λ}(M, Ω_{Λ}^{ip}(K_{c−1}))

does not vanish for any i ∈ ℤ. This shows that the module K_{c−1} cannot be an element in X ∈ ⟨M⟩_{c−1}, and so n ≥ c. □

Using Proposition 2.2 and Auslander’s upper bound, we obtain the promised result on the representation dimension. We denote by ℓℓ(Λ) the Loewy length of our algebra Λ.

Theorem 3.3. If Λ is a non-semisimple selfinjective algebra and Fg holds, then

cx Λ/𝔯 + 1 ≤ repdim Λ ≤ ℓℓ(Λ).

Note that, in particular, this theorem shows that the Loewy length of a selfinjective Artin algebra satisfying Fg is strictly larger than the maximal complexity occurring among the modules. This is an analogue of Benson’s conjecture, which states that the Loewy length of a block of a group algebra in characteristic p is strictly larger than the p-rank of its defect group. The characteristic 2 case of Benson’s conjecture was proved by Rouquier in [Ro2], whereas the general case was proved by Oppermann in [Op1].

Rouquier showed that the representation dimension of the exterior algebra on an n-dimensional vector space is exactly n + 1. It therefore seems natural to ask the following:

Question. When Fg holds, what is the exact value of repdim Λ?

The following corollaries to Theorem 3.3 provide lower bounds for the representation dimension of the algebras given in the three examples prior to Theorem 3.2. In particular, we obtain [Op1, Corollary 19], half of [Ro2, Theorem 4.1] and the result of Avramov and Iyengar on the representation dimension of Artin complete intersections (cf. [AvI]).

Corollary 3.4. Suppose k is a field of positive characteristic p, and let G be a finite group whose order is divisible by p. Then repdim kG ≥ p − rank G + 1, that is, the representation dimension of kG is strictly greater than Krulldim H^{∗}(G, k).

Proof. By a result of Quillen (cf. [Qu1, Qu2]), the complexity of the trivial kGmodule, i.e. the Krull dimension of the cohomology ring H^{∗}(G, k), equals the p-rank of G. □

Corollary 3.5. Let A be a commutative Noetherian local complete intersection of codimension c. If A is Artin, then repdim A ≥ c + 1.

Proof. By a classical result of Tate (cf. [Tat, Theorem 6]), the complexity of the simple module over a complete intersection equals the codimension of the ring. □

Remark. Let A be a commutative Noetherian local complete intersection of codimension c. If A is complete, then the proof of Theorem 3.2 also applies to the stable category of finitely generated maximal Cohen-Macaulay A-modules. Namely, the dimension of this triangulated category is at least c − 1.

Corollary 3.6. Suppose k is a field and that Λ is a finite dimensional nonsemisimple selfinjective k-algebra, with the property that Λ/𝔯 ⊗_{k} Λ/𝔯 is semisimple (as happens for example when k is algebraically closed). Furthermore, suppose the Hochschild cohomology ring HH^{∗}(Λ) is Noetherian, and that Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯) is a finitely generated HH^{∗}(Λ)-module. Then repdim Λ ≥ Krulldim HH^{∗}(Λ) + 1.

Proof. The Krull dimension of HH^{∗}(Λ) is its rate of growth γ (HH^{∗}(Λ)) as a graded k-vector space. Therefore, since the HH^{∗}(Λ)-module Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯) is finitely generated, we see that

cx Λ/𝔯 = γ (Ext_{Λ}^{∗}(Λ/𝔯, Λ/𝔯)) ≤ γ (HH^{∗}(Λ)) = Krulldim HH^{∗}(Λ).

Denote the radical of Λ^{e} by 𝔯^{e}, and let B be a finitely generated bimodule (i.e. B ∈ mod Λ^{e}). If B is not simple, then choose an exact sequence

0 → S → B → B′ → 0

in which S is simple. This sequence induces an exact sequence

Ext_{Λ^{e}}^{∗}(Λ, S) → Ext_{Λ^{e}}^{∗}(Λ, B) → Ext_{Λ^{e}}^{∗}(Λ, B′)

of HH^{∗}(Λ)-modules, all of which are finitely generated by [EHSST, Proposition 2.4]. Consequently the inequality

γ (Ext_{Λ^{e}}^{∗}(Λ, B)) ≤ max{γ (Ext_{Λ^{e}}^{∗}(Λ, S)) , γ (Ext_{Λ^{e}}^{∗}(Λ, B′))}

holds, and so induction on length gives

γ (Ext_{Λ^{e}}^{∗}(Λ, B)) ≤ γ (Ext_{Λ^{e}}^{∗}(Λ, Λ^{e}/𝔯^{e})) = cx_{Λ^{e}} Λ.

In particular, the inequality γ (HH^{∗}(Λ)) ≤ cx_{Λ^{e}} Λ holds. But the complexity of Λ as a bimodule equals that of the Λ-module Λ/𝔯. Namely, applying − ⊗_{Λ} Λ/𝔯 to the minimal projective bimodule resolution of Λ gives the minimal Λ-projective resolution of Λ/𝔯. Therefore cx_{Λ^{e}} Λ = cx Λ/𝔯, and this shows that the Krull dimension of HH^{∗}(Λ) equals cx Λ/𝔯. □

We end this paper with two examples illustrating the main results. These provide new examples of classes of algebras with arbitrarily large representation dimension. In the first example we generalize Rouquier’s result on exterior algebras. Namely, we determine the exact value of the representation dimension of certain quantum exterior algebras.

Examples. (i) Let k be a field, let n ≥ 1 be an integer, and let Λ be the quantum exterior algebra

k⟨X_{1}, …, X_{n}⟩/(X_{i}^{2}, {X_{i}X_{j} − q_{ij}X_{j}X_{i}}_{i<j}),

where 0 ≠ q_{ij} ∈ k. This algebra is finite dimensional of dimension 2^{n}, and the complexity of k is n. Furthermore, this is a Frobenius algebra; the codimension two argument in the beginning of [BeE, Section 3] carries over. In particular, this algebra is selfinjective, and it was shown in [ErS] that Fg holds if and only if all the q_{ij} are roots of unity. Therefore, when this is the case, then the representation dimension of Λ is at least n + 1. However, the Loewy length of Λ is n + 1, and by Auslander’s result this is an upper bound of the representation dimension. Consequently, when all the q_{ij} are roots of unity, then repdim Λ = n + 1.

(ii) Let k be an algebraically closed field, and let R be a Noetherian Artin-Schelter regular Koszul k-algebra of dimension d. That is, R is graded connected of global dimension d, its Gelfand-Kirillov dimension is finite, and

Ext_{R}^{i}(k, R) ≃ \{ \begin{matrix}0 & i ≠ d \\ k & i = d (up to shift). \end{matrix}

If R is a finitely generated module over its center, then by [Sol, Proposition 9.15] the Koszul dual Λ of R is selfinjective, satisfies Fg, and cx Λ/𝔯 = d. Thus in this case the representation dimension of Λ is at least d + 1.

An example of such an algebra is obtained from the Sklyanin algebras (cf. [Smi, Section 8]: let E be an elliptic curve over k, and fix a point P ∈ E such that nP = 0 for some n ≥ 1. Denote by σ_{P} : E → E the corresponding translation automorphism. Furthermore, let d ≥ 1 be an integer, and let A_{d}(E, σ_{P}) be the d-dimensional Sklyanin algebra. This is an Artin-Schelter regular algebra of the above type.

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

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