A lower bound for the representation dimension of kC_{p}^{n}

Steffen Oppermann

The representation dimension of a finite dimensional algebra has been introduced by Auslander in his Queen Mary College notes [1]. Rouquier has shown ([8]) that the representation dimension of the exterior algebra of an n-dimensional vector space is n + 1, thus giving the first example of an algebra known to have representation dimension strictly larger than 3. To do so, he proved that the representation dimension is bounded below by the dimension of the stable module category plus two, and analyzed the latter with the help of Koszul duality. In characteristic 2, the exterior algebra is just kC_{2}^{n}, so Rouquier in particular has determined the representation dimension of these algebras. He pointed out that his result implies the case p = 2 of the following conjecture of Benson: The Loewy length of a block of a group algebra in characteristic p is strictly larger than the p-rank of its defect group.

Here we generalize Rouquier’s result to group algebras of elementary abelian groups. More precisely we will prove the following:

Theorem. Let k be a field of characteristic p, C_{p}^{n} the elementary abelian group of order p^{n}. Then

dim kC_{p}^{n} -\underline{mod} ≥ n − 1

One crucial idea of the proof of the p = 2 case in [8] is to transfer the problem to commutative algebra with the help of Koszul duality. In contrast, we will work directly in the module category and show that certain morphisms vanish in the stable category.

By Rouquier’s result mentioned above the following lower bound for the representation dimension follows:

Corollary 1. Let k be a field of characteristic p. Then repdim kC_{p}^{n} > n.

Following arguments due to Rouquier, we can deduce a lower bound for the representation dimension of any block of a group algebra. The p-rank of a group G is the maximal n, such that G has an elementary abelian subgroup of order p^{n}.

Corollary 2. Let k be a field of characteristic p. Let G be a finite group, B a block of kG and D a defect group of B. Then

repdim B > p -rank(D).

Finally, by an argument of Auslander, the representation dimension of a self-injective algebra is bounded above by its Loewy length, that is the minimal n such that Rad^{n} vanishes. Using this, Bensons’s conjecture follows.

Corollary 3 (Benson’s conjecture). Let k be a field of characteristic p. Let G be a finite group, B a block of kG, ll B its Loewy length, and D a defect group of B. Then

ll B > p -rank(D).

In the first section we will recall the definitions and some basic properties of the representation dimension (due to Auslander) and the dimension of a triangulated category (due to Rouquier).

In the second section we explain the idea of the proof of the Theorem. We will assume to have a module M generating (as will be defined in Section 1) the stable module category in a finite number of steps. We will explain a method to show that another object N is not in the category generated by M in a fixed number of steps. To apply this method, we need to find a module N with a certain chain of endomorphisms having a non-zero composition in the stable module category, but such that every single endomorphism in the chain annihilates all morphisms from M.

For an infinite field, we will in Section 3 explicitly find a module N and endomorphisms meeting our requirements. More precisely, we will construct a family of modules in such a way that we can, depending on M, choose an adequate N (Proposition 11). We will show that this module N cannot be in the category generated by M in to few steps. This provides the lower bound for the dimension of the stable module category.

In the fourth section we will show that our result also holds for a finite field. We look at the algebraic closure, where we may use the result for infinite fields, and then see that everything actually happens in a finite extension of the given field.

Finally, in Section 5, we will see that our result also induces a lower bound for the representation dimension of any block of a group algebra. This implies Benson’s conjecture.

Acknowledgements. This paper is part of the auther’s PhD project. I would like to thank Professor Koenig for suggesting this subject and for many helpful questions, comments, and suggestions.

1. Definitions of the dimensions

Representation dimension:

Definition. Let Λ be a finite dimensional algebra. Then the representation dimension of Λ is defined to be

repdim Λ = min{gld End_{Λ}(M) | M generates and cogenerates Λ -mod}.

The representation dimension of a finite dimensional algebra has been introduced by Auslander [1] in order to measure how far an algebra is from being representation finite. He showed that an algebra is representation infinite if and only if its representation dimension is at least three. To date, no general method is known for calculating the representation dimension of a given algebra, and in fact the only examples known by now to have representation dimension greater than three are the exterior algebras ([8]) and the algebras in the family considered by Krause and Kussin ([6]). However, it has been shown by Iyama [5], that the representation dimension is always finite.

Since we know the module category of Λ better than the one of End_{Λ}(M), the following lemma may help us to understand the representation dimension.

Lemma 4 ([4, Lemma 2.1]). Let Λ not semisimple, M ∈ Λ -mod a generator and cogenerator, n ∈ ℕ. Then the following are equivalent:

1. gld End_{Λ}(M) = n

2. For every N ∈ Λ -mod there is an exact sequence

0 ^{}M_{n−2}  ^{}⋯ ^{}M_{0}  ^{}N ^{}0,

such that the induced sequence

0 ^{}Hom_{Λ}(M, M_{n−2}) ^{}⋯ ^{}Hom_{Λ}(M, N) ^{}0

is also exact.

Such a sequence will be called a universal M-resolution of N.

This means, that the representation dimension of a non-semisimple algebra can also be defined to be the minimal n, such that there is a generator cogenerator M having the property that every module has a universal Mresolution of length at most n − 2.

In order to get upper bounds for the representation dimension of some algebra, one can choose a generator cogenerator M and calculate the maximal length of a universal M-resolution of a module. The following lemma is an example of this technique.

Lemma 5 ([1, III.5, p.55], improved in [8, Proposition 3.9]). Let Λ be self-injective. Then

repdim Λ ≤ ll Λ,

where ll Λ is the Loewy length of Λ.

Proof. Take M = ⨁_{i} Λ/ Rad^{i} Λ. Let us denote for a moment the kernel of the universal M-cover of a module N by Ω_{M}N. Now note that in every step of the resolution, the Loewy length of the module decreases by at least one, that means ll Ω_{M}^{i+1}N ≤ ll Ω_{M}^{i}N − 1. If N is projective then it is in add M, therefore we may assume that ll N < ll Λ. Putting this together we find ll Ω_{M}^{ll Λ−1}N = 0, so Ω_{M}^{ll Λ−1}N = 0. Therefore repdim Λ ≤ ll Λ as claimed.

Alternatively, see [1, III.5, p.55], but note that if the Loewy length of the module is ll Λ, then the module is projective, so this case can be excluded.

Dimension of a triangulated category: The notion of dimension of a triangulated category has been introduced by Rouquier in [7].

Let T be a triangulated category, I ⊂ ObT . Then let ⟨I⟩ be the full subcategory of T of all direct summands of finite direct sums of shifts of objects in I. For two subclasses I_{1}, I_{2} ⊂ ObT let I_{1} ∗ I_{2} be the class of all extensions between them, that is the elements of I_{1} ∗ I_{2} are exactly the M, such that there is a distinguished triangle M_{1} ^{}M ^{}M_{2} ^{}M_{1}[1] in T with M_{i} ∈ I_{i}. Now set

I_{1} ⋄ I_{2} = ⟨I_{1} ∗ I_{2}⟩

and

⟨I⟩_{0} = 0,
⟨I⟩_{1} = ⟨I⟩
⟨I⟩_{i+1} = ⟨I⟩_{i} ⋄ ⟨I⟩ .

Definition. The dimension of a triangulated category T is the minimal d such that there is an object M ∈ T with T = ⟨M⟩_{d+1}.

The following lemma is an immediate consequence of this definition:

Lemma 6 ([7, Lemma 3.4]). Let S ^{F}^{}T be a triangle functor between two triangulated categories. Assume any object in T is a direct summand of an object in the image of F. Then

dim T ≤ dim S.

Here we will only be looking at the case T = Λ -\underline{mod} of the stable module category of a self-injective algebra Λ. Since short exact sequences in Λ -mod become triangles in Λ -\underline{mod}, any N having an M resolution of length n is contained in ⟨M⟩_{n+1}. Therefore we have the following lemma:

Lemma 7 ([8, part of Proposition 3.6]). Let Λ be a self-injective algebra. Then

repdim Λ ≥ dim Λ -\underline{mod} +2.

In particular this shows that the Theorem implies Corollary 1.

2. Outline of the proof the Theorem:

Let k be a field of characteristic p, V an n-dimensional k-vectorspace. Let

Λ = S(V)/(v^{p} | v ∈ V)

be the symmetric algebra modulo all p-th powers. Then Λ ≅ kC_{p}^{n}.

We will use the following lemma to get a lower bound for the dimension of Λ -\underline{mod}:

Lemma 8 ([6, Lemma 2.3] and [7, Lemma 4.11]). Let T be triangulated category and let

H_{1} ^{f_{1}}^{}H_{2} ^{f_{2}}^{}⋯ ^{f_{n−1}}^{}H_{n−1} ^{f_{n−1}}^{}H_{n}

be a sequence of morphisms between cohomological functors T^{op} ^{}Ab. For every i, let I_{i} be a subcategory of T closed under shifts and on which f_{i} vanishes. Then f_{1} ⋯ f_{n−1} vanishes on I_{1} ⋄ ⋯ ⋄ I_{n−1}.

Proof. This can easily be shown by induction on n. See [6] or [7].

To make use of it, we let M ∈ Ob(Λ -\underline{mod}) such that M realizes the minimal d in the definition of dim Λ -\underline{mod}, so ⟨M⟩_{dim(Λ -\underline{mod})+1} = Λ -\underline{mod}. We will find a module N, depending on M, and morphisms f_{i} : N ^{}N such that Hom(−, N) ^{f_{i∗}}^{}Hom(−, N) is 0 on ⟨M⟩ but f_{1} ⋯ f_{n−1} ≠ 0. Thus, by the above lemma, ⟨M⟩_{n−1} cannot be the entire stable category. Therefore dim(Λ -\underline{mod}) + 1 > n − 1.

3. Proof of the Theorem in the case k is infinite

In this section we assume the field k to be infinite. Whenever we are talking about open or closed sets we are referring to the classical Zariski topology.

Lemma 9. Let Y ⊂ mod- Λ be a finite set of right Λ-modules. Then there is an open, nonempty subset U ⊂ V, such that for any u ∈ U, any y ∈ Y ∈ Y, and any 1 ≤ s < p we have

yu = 0 ⇒ y · Rad^{s} Λ ⊂ Y u^{s}.

Proof. Fix 1 ≤ s < p and Y ∈ Y. Any v ∈ V induces a linear map

ρ_{v}^{s} : Y ^{}Y : y ^{}yv^{s}.

Now we can find a set with the desired property for our fixed s and Y:

U_{s,Y} = {v ∈ V | rk ρ_{v}^{s} maximal}.

By choosing a basis for V and Y, the maps ρ_{v}^{s} induce a polynomial map

k^{dim V} ≅ V ^{}End_{k} Y ≅ k^{dim Y ×dim Y} .

We then compose this map with taking subdeterminants of size r, where r is the maximal rank in the definition of U_{s,Y} above. This results in polynomial maps k^{dim V} ^{}k, such that U_{s,Y} is just the set where not all of these polynomials are zero. Thus U_{s,Y} is open and obviously it is non-empty.

Now fix u ∈ U_{s,Y}. Let {a_{i} | 1 ≤ i ≤ A} be a basis of Ker ρ_{u}^{1}, complement it to a basis {a_{i}, b_{j} | 1 ≤ i ≤ A, 1 ≤ j ≤ B} of Ker ρ_{u}^{s}, and finally to a basis {a_{i}, b_{j}, c_{l} | 1 ≤ i ≤ A, 1 ≤ j ≤ B, 1 ≤ l ≤ C} of Y.

Let v ∈ V. The rank of ρ_{u}^{s} is maximal, so in particular rk ρ_{u+εv}^{s} ≤ rk ρ_{u}^{s} for any ε. Fix 1 ≤ i ≤ A. Therefore, for all ε, the tuple

(a_{i}ρ_{u+εv}^{s}, c_{1}ρ_{u+εv}^{s}, …, c_{C}ρ_{u+εv}^{s})

is linearly dependent. Since a_{i}u = 0, for ε ≠ 0 the tuple

(a_{i}ρ_{v}^{s}, c_{1}ρ_{u+εv}^{s}, …, c_{C}ρ_{u+εv}^{s})

also is linearly dependent. But the set of all ε such that it is linearly dependent is closed (the subdeterminants of (a_{i}ρ_{v}^{s}, c_{l}ρ_{u+εv}^{s}) are polynomials in ε), hence it has to be all of k. Therefore, especially (a_{i}ρ_{v}^{s}, c_{l}ρ_{u}^{s} | 1 ≤ l ≤ C) is linearly dependent, so a_{i}ρ_{v}^{s} ∈ ⟨c_{l}ρ_{u}^{s} | 1 ≤ l ≤ C⟩. But the v^{s} generate Rad^{s} Λ as a k-vector space, since s is strictly smaller than p.

Finally set U = ⋂_{Y ∈Y} ⋂_{s=1}^{p−1} U_{s,Y}.

Lemma 10. Let X ⊂ Λ -mod be a finite set of Λ-modules. Then there is an open, nonempty subset U ⊂ V, such that for any u ∈ U, any φ ∈ Hom_{Λ}(X, Λ) with X ∈ X, and any 1 ≤ s < p we have

φu = 0 ⇒ φ · Rad^{s} Λ ⊂ Hom_{Λ}(X, Λ)u^{s}.

Proof. Set Y = {Hom_{Λ}(X, Λ) | X ∈ X } in Lemma 9.

Proposition 11. Let X ⊂ Λ -mod be a finite set of Λ-modules. Let u ∈ U of Lemma 10, N = Λu^{p−1}. Then, for any X ∈ X and any f ∈ Rad^{p−1} Λ, any composition X ^{}N ^{·f}^{}N factors through Λ -proj.

Proof. Fix X ∈ X. Since N is a submodule of Λ, we may identify

Hom_{Λ}(X, N) = {φ ∈ Hom_{Λ}(X, Λ) | φ(X) ⊂ N}
= {φ ∈ Hom_{Λ}(X, Λ) | φ(X) ⊂ Λu^{p−1}}
= {φ ∈ Hom_{Λ}(X, Λ) | φ(X)u = 0}

Let φ ∈ Hom_{Λ}(X, N), that is φ ∈ Hom_{Λ}(X, Λ) with φu = 0. Then, by Lemma 10, φf ∈ Hom_{Λ}(X, Λ)u^{p−1}.

The projective cover of N is induced by the endomorphism u^{p−1} of Λ. Therefore, the maps X ^{}N factoring through a projective module are exactly the elements of Hom_{Λ}(X, Λ)u^{p−1}.

Proposition 12. Let v ∈ V, N = Λv^{p−1}. Then the composition

N [figure] hd N ≅ k ≅ Soc N^{}^{}N

does not factor through Λ -proj.

Proof. Let {v_{1}, v_{2}, …, v_{n}} be a basis of V, v_{1} = v. Then the composition above is (up to a scalar) multiplication with ∏_{i≠1} v_{i}^{p−1}. Let f be any map Λ ^{}N. It is defined by the image of 1_{Λ} in N ⊂ Λ, which we will also call f. Now assume that the following diagram commutes.

N \overline{∏_{i≠1} v_{i}^{p−1}_{}}N
[figure: f]
Λ^{∩}

The image of v^{p−1} has to be the same on both ways, that is v^{p−1}f = ∏_{i} v_{i}^{p−1}. Therefore f − ∏_{i≠1} v_{i}^{p−1} is a multiple of v. Since f ∈ N, f is also a multiple of v. So ∏_{i≠1} v_{i}^{p−1} would be a multiple of v, but that is not true in Λ.

Therefore the morphism cannot factor through Λ -proj.

Proof of the Theorem for infinite fields. Let M be a Λ-module realizing the minimal d in the definition of the dimension of the stable module category. Let X = {M, ℧M}, where ℧ is the cosyzygy functor, which is the shift in the stable category. Then choose u ∈ U as in Lemma 10, and complement it to a basis {u_{1}, … u_{n}} of V with u_{1} = u. Let N = Λu^{p−1}. Then we have the following sequence of cohomological functors Λ -\underline{mod} ^{}k -mod.

\underline{Hom}_{Λ}(−, N) ^{·u_{2}^{p−1}}^{}\underline{Hom}_{Λ}(−, N) ⋯ ^{·u_{n}^{p−1}}^{}\underline{Hom}_{Λ}(−, N)

Its composition is nonzero by Proposition 12. So by Lemma 8 we only need

to show that Hom(℧^{i}M, N) ^{·u_{j}^{p−1}}^{}Hom(℧^{i}M, N) is zero for any i ∈ ℕ and any 2 ≤ j ≤ n.

The following diagrams have short exact rows and commute for any f ∈ Λ.

[figure: N; Λ; Λ/(u^{p−1}); Λ/(u^{p−1}); ·u; Λ; ·u^{p−1}; N]

[figure: N; ·f; Λ; ·f; Λ/(u^{p−1}); ·f; Λ/(u^{p−1}); ·f; ·u; Λ; ·f; ·u^{p−1}; N; ·f]

Therefore ℧N = Λ/(u^{p−1}), ℧(Λ/(u^{p−1})) = N and ℧( ^{·f}^{}) = ^{·f}^{}. So the following diagram commutes.

\underline{Hom}_{Λ}(℧^{i}M, N)  \overline{·u_{j}^{p−1}  ^{}}\underline{Hom}_{Λ}(℧^{i}M, N)

[figure: \underline{Hom}_{Λ}(℧^{i+2n}M, ℧^{2n}N); ≅; ℧^{2n}(·u_{j}^{p−1}); \underline{Hom}_{Λ}(℧^{i+2n}M, ℧^{2n}N); ≅]

[figure: \underline{Hom}_{Λ}(℧^{i+2n}M, N); ≅; ·u_{j}^{p−1}; \underline{Hom}_{Λ}(℧^{i+2n}M, N); ≅]

By choosing n appropriately we can get i + 2n ∈ {0, 1}, so ℧^{i+2n}M ∈ X and the claim follows from Proposition 11.

4. The case of a finite field

Now let k be finite, \overline{k} its algebraic closure. Denote by \overline{Λ} = \overline{k}⊗_{k} Λ the induced algebra. For any Λ-module X let \overline{X} = \overline{k}⊗_{k} X ∈ \overline{Λ} -mod, and for X ⊂ Λ -mod let \overline{X} = \{\overline{X} | X ∈ X \}. Whenever we are talking about an extension field k̂ of k let Λ̂, X̂ and X̂ be the obvious variations of the above.

Lemma 13. Let X ⊂ Λ -mod finite. Then there is a finite extension k̂ of k and u ∈ V̂ such that any φ ∈ HomΛ̂(X̂, Λ̂) with X ∈ X, and any 1 ≤ s < p we have

φu = 0 ⇒ φ · Rad^{s} Λ̂ ⊂ HomΛ̂(X̂, Λ̂)u^{p−1}.

Proof. By Lemma 10 there is u ∈ \overline{V} such that for any φ ∈ Hom\overline{Λ}(\overline{X}, \overline{Λ}) and any 1 ≤ s < p we have φu = 0 ⇒ φ·Rad^{s} \overline{Λ} ⊂ Hom\overline{Λ}(\overline{X}, \overline{Λ})u^{s}. Choose k̂ finite over k such that u ∈ V̂. Since \overline{k} ⊗k̂ HomΛ̂(X̂, Λ̂) = Hom\overline{Λ}(\overline{X}, \overline{Λ}) ([3, 29.5]), we may identify HomΛ̂(X̂, Λ̂) with the subset of morphisms in Hom\overline{Λ}(\overline{X}, \overline{Λ}) mapping X̂ to Λ̂. Therefore, for any φ ∈ HomΛ̂(X̂, Λ̂) with φu = 0, we have φ·Rad^{s} Λ̂ ⊂ Hom\overline{Λ}(\overline{X}, \overline{Λ})u^{s} ∩HomΛ̂(X̂, Λ̂) = HomΛ̂(X̂, Λ̂)u^{s}. The right equality holds because Λ̂ is a direct summand of \overline{Λ} as Λ̂-module.

Proposition 14. Let X ⊂ Λ -mod finite, u and k̂ as in Lemma 13, and set N = Λ̂u^{p−1}. Then, for any X ∈ X and any f ∈ Rad^{p−1} Λ̂, any composition X̂ ^{}N ^{·f}^{}N of Λ̂-morphisms factors through Λ̂ -proj.

Proof. This is just the proof of Proposition 11, replacing the reference to Lemma 10 by a reference to Lemma 13.

Proposition 15. Let X ⊂ Λ -mod finite, u and k̂ as in Lemma 13, and set N = Λ̂u^{p−1}. Then, for any X ∈ X and any f ∈ Rad^{p−1} Λ̂, any composition X ^{}N ^{·f}^{}N of Λ-morphisms factors through Λ -proj.

Proof. Any Λ-morphism φ : X ^{}N lifts to a Λ̂-morphism φ̂ : X̂ ^{}N as indicated in the following diagram.

N \overline{u_{i}^{p−1} ^{}}N
[figure: φ]
[figure: X̂; φ̂; Λ̂]

[figure]

X

The dashed arrow now exists by Proposition 14 making the square commutative, so the composition factors through Λ̂. Clearly this is a projective Λ-module.

We note that Proposition 12 does not depend on the field at all. But we need the morphism to be non-zero in Λ -\underline{mod}, not just in Λ̂ -\underline{mod}.

Proposition 16. Let k̂ be a finite extension field of k, v ∈ V̂. Then the composition ψ : Λ̂v^{p−1} ^{}k̂ ^{}Λ̂v^{p−1} does not factor through a projective Λ-module.

Proof. Clearly Λ̂ is a projective Λ-module. So we only have to check that there is no Λ-morphism φ : Λ̂ ^{}Λ̂v^{p−1} which makes the following triangle commutative.

Λ̂v^{p−1}  \overline{ψ^{}}Λ̂v^{p−1}
[figure: φ]
Λ̂^{∩}

Assume such a φ exists. By what has been said above this φ cannot be Λ̂-linear. But we can define

φ̂ = ∑_{i=1}^{|k̂|−1} α^{i}φα^{−i}

where α is a generator of the cyclic group k̂^{×} (so one can see the sum as one over the multiplicative group of k̂). Then −φ̂ also makes the triangle commutative, because

v^{p−1}φ̂ = ∑ v^{p−1}α^{i}φα^{−i} = ∑ v^{p−1}α^{i}ψα^{−i} = ∑ v^{p−1}ψ
= (|k̂| − 1)v^{p−1}ψ = −v^{p−1}ψ

We claim that φ̂ is a Λ̂-morphism. It is obviously Λ-linear, so we only need to check that it is also k̂-linear. But

αφ̂ = ∑_{i=1}^{|k̂|−1} α^{i+1}φα^{−i} = ∑_{i=2}^{|k̂|} α^{i}φα^{−i+1} = φ̂α,

where the last equality holds because α^{|k̂|} = α. But we already know that no Λ̂-linear map makes the triangle commutative, so such a φ cannot exist.

Proof of the Theorem for finite fields. The argument is the same as the one in the proof of the Theorem for infinite fields on page 7, with references to Propositions 11 and 12 replaced by references to Propositions 15 and 16 respectively. However, when we choose M as before and set X = {M, ℧M}, we need to check that X̂ is indeed {M̂, ℧Λ̂M̂}, or, in other words, that taking cosyzygies and tensoring with k̂ commutes. This is the case because tensoring with k̂ is exact and k̂ ⊗ Λ is projective over itself.

5. Applications

The applications here are obtained by applying the ideas of [8] to the more general result.

Proposition 17 (implicit in [8, Theorem 4.9]). Let H ≤ G be finite groups. Then

dim kG -\underline{mod} ≥ dim kH -\underline{mod} .

Proof. We have the exact functors

res : kG -mod ^{}kH -mod and
ind : kH -mod ^{}kG -mod .

Since both of them map projective modules to projective ones there are induced triangle functors kG -\underline{mod} [figure] kH -\underline{mod}. By Lemma 6, it suffices to show that every kH-module is a direct summand of a module in the image of res. But kH is a direct summand of kG as kH-kH-bimodule, so 1_{kH -mod} is a direct summand of res ∘ ind.

Corollary 18. Let G be a finite group, k a field of characteristic p, p | |G|. Then

ll kG ≥ repdim kG ≥ dim kG -\underline{mod} +2 > p -rank(G).

Proof. The first inequality is Lemma 5, the second one is Lemma 7. The third inequality follows from the Theorem and Proposition 17.

Proposition 19 ([8, Proposition 4.7]). Let G be a finite group and B a block of kG. Let D be a defect group of B. Then dim B -\underline{mod} = dim kD -\underline{mod}.

The proof is similar to the one of Proposition 17. See [8].

Corollary 20. Let G be a finite group, B a block of kG, char k = p. Let D be a defect group of B. Then

ll(B) ≥ repdim B ≥ dim B -\underline{mod} +2 = dim D -\underline{mod} +2 > p -rank(D).

References

[1] Maurice Auslander, Representation dimension of Artin algebras, Queen Mary College Mathematics Notes (1971), republished in [2].

[2] , Selected works of Maurice Auslander. Part 1, American Mathematical Society, Providence, RI, 1999, Edited and with a foreword by Idun Reiten, Sverre O. Smalø, and Øyvind Solberg.

[3] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988, Reprint of the 1962 original, A Wiley-Interscience Publication.

[4] Karin Erdmann, Thorsten Holm, Osamu Iyama, and Jan Schröer, Radical embeddings and representation dimension, Adv. Math. 185 (2004), no. 1, 159–177.

[5] Osamu Iyama, Finiteness of representation dimension, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1011–1014 (electronic).

[6] Henning Krause and Dirk Kussin, Rouquier’s theorem on representation dimension, preprint.

[7] Raphaël Rouquier, Dimensions of triangulated categories, preprint.

[8] , Representation dimension of exterior algebras, preprint, to appear in Inventiones Math.