Representation dimension of artin algebras ^{1} ^{2}

Osamu Iyama

Abstract. We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will study a chain of rejective subcategories to construct modules with endomorphisms rings of finite global dimension. We apply these result to study a function r_{Λ} : mod Λ → ℕ_{≥0} which is a natural extension of Auslander’s representation dimension.

We study functorially finite subcategories [AS1] from the viewpoint of its resolution dimension (§1.1). Typical examples of functorially finite subcategories are given by morphisms of rings (§1.2) and by cotilting modules (§1.3). The subcategories of resolution dimension zero is called bireflective [St], and they correspond to ring epimorphisms (§1.4(1)). We introduce a special class of bireflective subcategories called rejective subcategories (§1.4(2)), and they correspond to factor algebras. Recently, rejective subcategories played a crucial role in the study of representation-finite orders [I1,2][Ru1,2]. In §2, we study certain chains of rejective subcategories called rejective chains (§2.1), which give a method to construct rings of finite global dimension (§2.1.1). Recently, in [I3,4], rejective chains were applied to give positive answer to Solomon’s conjecture on zeta functions of orders [S1,2] and the finiteness problem of the representation dimension of artin algebras [A][Xi1] (§2.2). Moreover, we show that rejective chains give quasi-hereditary algebras (§2.4) of Cline-Parshall-Scott [CPS1,2]. This provide us a categorical approach to quasi-hereditary algebras originally suggested by Dlab-Ringel [DR2,3].

The representation dimension of artin algebras Λ was introduced by M. Auslander [A] as a homological invariant to measure how far an artin algebra is from being representation-finite. We introduce a function r_{Λ}, which is given by the resolution dimension of certain subcategories (§3.1.2). The value r_{Λ}(Λ ⊕ DΛ) equals to Auslander’s representation dimension, and would be quite natural in Auslander’s philosophy, the homological approach to the representation theory. Our function r_{Λ} would give us much more information. For example, although rep.dim Λ does not distinguish tame hereditary algebras and wild hereditary algebras (§3.2.2(2)), the supremum |r_{Λ}| of r_{Λ} determines the representation type of hereditary algebras (§3.3.2). This is an application of Rouquier’s result [R] on exterior algebras (§3.6.2), which is proved in §4. We show that the value of r_{Λ}(Λ) is closely related to the reflexive-finiteness of Λ (§3.2.1).

All results in §1–3 were explained and proved in [I7].

1 2000 Mathematics Subject Classification. Primary 16E10; Secondary 16G10, 16G30

2 The detailed version of this paper will be submitted elsewhere.

0.1 Notations In this paper, any module is assumed to be a left module. For a ring Λ, we denote by J_{Λ} the Jacobson radical of Λ, and by mod Λ (resp. pr Λ) the category of finitely generated (resp. finitely generated projective) Λ-modules.

Let C be an additive category, C(X, Y) := Hom_{C}(X, Y), and fg the composition of f ∈ C(X, Y) and g ∈ C(Y, Z). We denote by J_{C} the Jacobson radical of C, and by ind C the set of isoclasses of indecomposable objects in C. In §1–§3, any subcategory is assumed to be full and closed under isomorphisms, direct sums and direct summands. For a collection S of objects in C, we denote by add S the smallest subcategory of C containing S, and by [S] the ideal of C consisting of morphisms which factor through some object in S. We call X ∈ C an additive generator of C if add X = C.

A C-module is a contravariant additive functors from C to the category of abelian groups. We denote by Mod C the category of C-modules, where (Mod C)(M, M′) consists of the natural transformations from M to M′. Then Mod C forms an abelian category. By Yoneda’s Lemma, C( , X) is a projective object in Mod C. We call M ∈ Mod C finitely presented if there exists an exact sequence C( , Y) → C( , X) → M → 0. We denote by mod C the category of finitely presented C-modules.

1 Functorially finite subcategory

1.1 Definition Let Λ be an artin algebra and C′ ⊆ C subcategories of mod Λ.

(1)[AS1] We call f ∈ C(Y, X) a right C′-approximation of X if Y ∈ C′ and C( , Y) →^{·f} C( , X) → 0 is exact on C′, or equivalently, C( , Y) →^{·f} [C′]( , X) → 0 is exact on C (§0.1). We call C′ a contravariantly finite subcategory of C if any X ∈ C has a right C′-approximation. Dually, a left C′-approximation and a covariantly finite subcategory are defined. We call C′ functorially finite if it is contravariantly and covariantly finite.

(2) A right C′-resolution of X ∈ C is a complex ⋯ → Y_{2} →^{f_{2}} Y_{1} →^{f_{1}} Y_{0} →^{f_{0}} X in C

such that Y_{i} ∈ C′ and ⋯ → C( , Y_{2}) →^{·f_{2}} C( , Y_{1}) →^{·f_{1}} C( , Y_{0}) →^{·f_{0}} C( , X) → 0 is exact

on C′. We write C′-resol.dim X ≤ n if X has a right C′-resolution with Y_{n+1} = 0, write C′-resol.dim C ≤ n if C′-resol.dim X ≤ n holds for any X ∈ C.^{3} We call them right resolution dimension. Dually, we define a left C′-resolution, left resolution dimension, C′ ^{op}-resol.dim X and C′ ^{op}-resol.dim C^{op}.

1.1.1 Proposition For a functorially finite subcategory C′ of C := mod Λ, 0 ≤ gl.dim(mod C′) − C′-resol.dim C ≤ 2, 0 ≤ gl.dim(mod C′ ^{op}) − C′ ^{op}-resol.dim C^{op} ≤ 2 and gl.dim(mod C′) = gl.dim(mod C′ ^{op}) hold.

1.2 Example Let ϕ : Λ → Γ be a morphism of artin algebras. We denote by ϕ^{∗} : mod Γ → mod Λ the natural induced functor. Then ϕ^{∗} has a left adjoint Γ⊗_{Λ} : mod Λ → mod Γ and a right adjoint Hom_{Λ}(Γ, ) : mod Λ → mod Γ.

(1) Define a (Λ, Λ)-module C_{ϕ} by an exact seqeuence Λ →^{ϕ} Γ →^{a} C_{ϕ} → 0. We have the associated complexes below given by d_{0}^{+} = d_{0}^{−} = ϕ, d_{i}^{+}(x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{i}) := 1 ⊗ a(x_{1}) ⊗ x_{2} ⊗ ⋯ ⊗ x_{i} and d_{i}^{−}(x_{1} ⊗ ⋯ ⊗ x_{i−1} ⊗ x_{i}) := x_{1} ⊗ ⋯ ⊗ x_{i−1} ⊗ a(x_{i}) ⊗ 1 for i > 0.

3 We borrowed the notation C′-resol.dim X in [AB]. Sikko [Si] denote C′-resol.dim (mod Λ) by gl.dim(C′, Λ).

X_{ϕ}^{+} : 0 → Λ ⟶^{d_{0}^{+}} Γ ⟶^{d_{1}^{+}} Γ ⊗_{Λ} C_{ϕ} ⟶^{d_{2}^{+}} Γ ⊗_{Λ} C_{ϕ} ⊗_{Λ} C_{ϕ} ⟶^{d_{3}^{+}} Γ ⊗_{Λ} C_{ϕ} ⊗_{Λ} C_{ϕ} ⊗_{Λ} C_{ϕ} → ⋯
X_{ϕ}^{−} : 0 → Λ ⟶^{d_{0}^{−}} Γ ⟶^{d_{1}^{−}} C_{ϕ} ⊗_{Λ} Γ ⟶^{d_{2}^{−}} C_{ϕ} ⊗_{Λ} C_{ϕ} ⊗_{Λ} Γ ⟶^{d_{3}^{−}} C_{ϕ} ⊗_{Λ} C_{ϕ} ⊗_{Λ} C_{ϕ} ⊗_{Λ} Γ → ⋯

(2) Define a (Γ, Γ)-module D_{ϕ} by an exact seqeuence 0 → D_{ϕ} →^{b} Γ ⊗_{Λ} Γ →^{c} Γ → 0, where c is the multiplication map. We have the associated complexes below given by e_{0}^{+} = e_{0}^{−} = c, e_{i}^{+}(x_{0} ⊗ x_{1} ⊗ ⋯ ⊗ x_{i}) := b(x_{0}x_{1}) ⊗ ⋯ ⊗ x_{i} ∈ Γ ⊗_{Λ} Γ ⊗_{Γ} D_{ϕ}^{⊗i−1} = Γ ⊗_{Λ} D_{ϕ}^{⊗i−1} and e_{i}^{−}(x_{0} ⊗ ⋯ ⊗ x_{i−1} ⊗ x_{i}) := x_{0} ⊗ ⋯ ⊗ b(x_{i−1}x_{i}) ∈ D_{ϕ}^{⊗i−1} ⊗_{Γ} Γ ⊗_{Λ} Γ = D_{ϕ}^{⊗i−1} ⊗_{Λ} Γ for i > 0.

Y_{ϕ}^{+} : ⋯ → Γ ⊗_{Λ} D_{ϕ} ⊗_{Γ} D_{ϕ} ⊗_{Γ} D_{ϕ} ⟶^{e_{3}^{+}} Γ ⊗_{Λ} D_{ϕ} ⊗_{Γ} D_{ϕ} ⟶^{e_{2}^{+}} Γ ⊗_{Λ} D_{ϕ} ⟶^{e_{1}^{+}} Γ ⊗_{Λ} Γ ⟶^{e_{0}^{+}} Γ → 0
Y_{ϕ}^{−} : ⋯ → D_{ϕ} ⊗_{Γ} D_{ϕ} ⊗_{Γ} D_{ϕ} ⊗_{Λ} Γ ⟶^{e_{3}^{−}} D_{ϕ} ⊗_{Γ} D_{ϕ} ⊗_{Λ} Γ ⟶^{e_{2}^{−}} D_{ϕ} ⊗_{Λ} Γ ⟶^{e_{1}^{−}} Γ ⊗_{Λ} Γ ⟶^{e_{0}^{−}} Γ → 0

1.2.1 Theorem (1) X_{ϕ} := add ϕ^{∗}(mod Γ) is a functorially finite subcategory of mod Λ. A left and right X_{ϕ}-resolutions of X ∈ mod Λ are given by X_{ϕ}^{+} ⊗_{Λ} X and Hom_{Λ}(X_{ϕ}^{−}, X) respectively.

(2) Y_{ϕ}^{+} := add Hom_{Λ}(Γ, mod Λ) is a covariantly finite subcategory of mod Γ. A left Y_{ϕ}^{+}-resolutions of Y ∈ mod Γ is given by an exact sequence Hom_{Γ}(Y_{ϕ}^{+}, Y).

(3) Y_{ϕ}^{−} := add Γ ⊗_{Λ} (mod Λ) is a contravariantly finite subcategory of mod Γ. A right Y_{ϕ}^{−}-resolutions of Y ∈ mod Γ is given by an exact sequence Y_{ϕ}^{−} ⊗_{Γ} Y.

1.3 Example Another well-known example of functorially finite subcategories is given by Auslander-Buchweitz theory [AB] and its application to the cotilting theory in [AR1]. Let Λ be an artin algebra and T a cotilting Λ-module with id_{Λ} T ≤ n [M].

1.3.1 Theorem X_{T} := {X ∈ mod Λ | Ext_{Λ}^{i}(X, T) = 0 for any i > 0} is a functorially finite subcategory of mod Λ with X_{T} -resol.dim (mod Λ) = n, X_{T}^{op}-resol.dim (mod Λ^{op}) = max{n − 2, 0} and n ≤ gl.dim(mod X_{T}) ≤ max{n, 2}.

1.4 Definition Let Λ be an artin algebra and C′ ⊆ C subcategories of mod Λ.

(1) We call C′ a coreflective (resp. reflective) subcategory of C if C′-resol.dim C = 0 (resp. C′ ^{op}-resol.dim C^{op} = 0) holds. This coincides with the usual definition [St], namely C′ is coreflective (resp. reflective) if and only if the inclusion functor C′ → C has a right (resp. left) adjoint [HS]. In this case, the counit ϵ^{−} (resp. unit ϵ^{+}) gives a right (resp. left) C′-approximation. We call C′ bireflective if it is reflective and coreflective.

(2) We call C′ a right (resp. left) rejective subcategory of C [I2,4] if the following equivalent conditions are satisfied.

(i) Any X ∈ C has a monic right (resp. epic left) C′-approximation.

(ii) The inclusion functor C′ → C has a right (resp. left) adjoint with a counit ϵ^{−} (resp. unit ϵ^{+}) such that ϵ_{X}^{−} is monic (resp. ϵ_{X}^{+} is epic) for any X ∈ C.

We call C′ rejective if it is left and right rejective. Any right (resp. left) rejective subcategories are coreflective (resp. reflective), but the converse does not hold in general.

1.4.1 A morphism ϕ : Λ → Γ of artin algebras is called a ring epimorphism if it is epic in the category of rings. This is equivalent to that the functor ϕ^{∗} : mod Γ → mod Λ is full. For example, the inclusion ( \begin{matrix}k & k \\ 0 & k \end{matrix} ) ⊂ M_{2}(k) is a ring epimorphism.

Theorem Let Λ be an artin algebra and C a subcategory of mod Λ.

(1) The conditions below are equivalent, and there exists a bijection between rejective subcategories of mod Λ and factor algebras of Λ.

(i) C is a rejective subcategory of mod Λ.

(ii) C = X_{ϕ} := ϕ^{∗}(mod Γ) for a surjective ring morphism ϕ : Λ → Γ.

(iii) C is closed under submodules and factor modules.

(2) The conditions below are equivalent, and there exists a bijection between bireflective subcategories of mod Λ and ring epimorphisms from Λ to artin algebras (cf. [GD]).

(i) C is a bireflective subcategory of mod Λ.

(ii) C = X_{ϕ} := ϕ^{∗}(mod Γ) for a ring epimorphism ϕ : Λ → Γ between artin algebras.

(iii) C is functorially finite, and closed under kernels and cokernels.

1.5 If ϕ : Λ → Γ is not a ring epimorphism, it seems to be difficult to study the behavior of X_{ϕ} in general. But an interesting example is given by radical embeddings [EHIS], which also appeared in [N].

Theorem Let Λ ⊂^{ϕ} Γ be artin algebras with J_{Λ} = J_{Γ}.

(1) X_{ϕ} := add ϕ^{∗}(mod Γ) satisfies X_{ϕ}-resol.dim (mod Λ) ≤ 1 and X_{ϕ}^{op}-resol.dim (mod Λ^{op}) ≤ 1.

(2) Y_{ϕ}^{+} := add Hom_{Λ}(Γ, mod Λ) and Y_{ϕ}^{−} := add Γ ⊗_{Λ} (mod Λ) coincide with mod Γ.

(3) ϕ^{∗} induces a full faithful functor C′ := mod Γ/[mod Γ/J_{Γ}] → C := mod Λ/[mod Λ/J_{Λ}], and C′ forms a rejective subcategory of C.

2 Rejective chain

Let Λ be an artin algebra and C′ ⊆ C subcategories of mod Λ. We call C semisimple if J_{C} = 0 holds. We call C′ a cosemisimple subcategory of C if C /[C′] is semisimple, namely, any non-invertible morphism in C between indecomposable objects factor through an object in C′.

2.1 Definition Let Λ be an artin algebra and 0 = C_{m} ⊆ C_{m−1} ⊆ ⋯ ⊆ C_{0} = C a chain of subcategories of mod Λ. We call it a left (resp. right) rejective chain of length m if C_{n+1} is a cosemisimple left (resp. right) rejective subcategory of C_{n} for any n (0 ≤ n < m). In this case, let ϵ_{n}^{+} (resp. ϵ_{n}^{−}) be the unit (resp. counit) of the natural inclusion C_{n+1} → C_{n}. We call the chain above Λ-total if ϵ_{n,X}^{+} (resp. ϵ_{n,X}^{−}) is epic (resp. monic) in mod Λ for any X ∈ C_{n} and n (0 ≤ n < m). The following theorem plays an important role in §3.

2.1.1 Theorem Let Λ be an artin algebra and C a subcategory of mod Λ with # ind C < ∞. Assume that C has a Λ-total left (resp. right) rejective chain of length m > 0. Then C-resol.dim (mod Λ) < m (resp. C^{op}-resol.dim (mod Λ^{op}) < m) and gl.dim(mod C) ≤ m hold.

2.1.2 Example Let Λ be an artin algebra and m := LL(Λ) the Loewy length of Λ. Put C_{n} := add ⨁_{i=0}^{m−n} Λ/J_{Λ}^{i}. Then 0 = C_{m} ⊆ C_{m−1} ⊆ ⋯ ⊆ C_{0} gives a Λ-total left rejective chain. Put Γ := End_{Λ}(⨁_{i=0}^{m} Λ/J_{Λ}^{i}). By 2.1.1, we obtain Auslander’s result gl.dim Γ ≤ m [A]. We will show in 2.4.1 that Γ is quasi-hereditary, the result of Dlab-Ringel [DR2].

2.2 Generalizing the construction of the preprojective partition of Auslander-Smalo [AS2], we can prove that certain kind of subcategories has a left (resp. right) rejective chain. The author applied 2.2.1 to prove Solomon’s conjecture on zeta functions on orders [I3], and 2.2.2 to prove the finiteness of the representation dimension of artin algebras [I4] (see §3.1.1). See [I5] for their unified approach.

2.2.1 Theorem Let Λ be an artin algebra. For n ∈ ℕ, put C^{(n)} := add{X ∈ ind(mod Λ) | length_{Λ} X < n}. If # ind C^{(n)} < ∞, then C^{(n)} has a Λ-total left rejective chain and a Λ-total right rejective chain.

2.2.2 Theorem Let Λ be an artin algebra and M_{0} := M ∈ mod Λ. Put M_{n+1} := M_{n}J_{End_{Λ}(M_{n})} (resp. M_{n+1} := D((DM_{n})J_{End_{Λ}(DM_{n})})) inductively. Take m > 0 such that M_{m} = 0. Then C_{n} := add ⨁_{i=n}^{m} M_{i} gives a Λ-total right (resp. left) rejective chain 0 = C_{m} ⊆ C_{m−1} ⊆ ⋯ ⊆ C_{0}.

2.3 Let us recall quasi-hereditary algebras of Cline-Parshall-Scott [CPS1,2]. A twosided ideal I of an artin algebra Γ is called heredity if I^{2} = I, IJ_{Γ}I = 0 and I ∈ pr Γ. This condition is left-right symmetric since the last condition is equivalent to I ∈ pr Γ^{op}. An artin algebra Γ is called quasi-hereditary if it has a heredity chain, which is a chain 0 = I_{m} ⊆ I_{m−1} ⊆ ⋯ ⊆ I_{0} = Γ of ideals of Γ such that I_{n−1}/I_{n} is a heredity ideal of Γ/I_{n} for any n (0 < n ≤ m). Let us recall the following theorem in [CPS1,2].

2.3.1 Theorem Let Γ be an artin algebra. If I is a heredity ideal of Γ, then 0 ≤ gl.dim Γ − gl.dim Γ/I ≤ 2 holds. Consequently, if Γ is a quasi-hereditary algebra with a chain 0 = I_{m} ⊆ I_{m−1} ⊆ ⋯ ⊆ I_{0} = Γ, then gl.dim Γ ≤ 2m − 2 holds.

2.4 There exists a bijection between equivalence classes of Krull-Schmidt categories C with additive generators M and Morita-equivalence classes of semiperfect rings Γ, which is given by C ↦ C(M, M) and the converse is given by Γ ↦ pr Γ. In this case, the set of subcategories C′ of C and the set of idempotent ideals I of Γ correspond bijectively by C′ ↦ I := [C′](M, M). In particular, we have a bijection between semisimple rejective subcategories C′ of C and heredity ideals I of Γ. A heredity chain corresponds to a chain 0 = C_{m} ⊆ C_{m−1} ⊆ ⋯ ⊆ C_{0} = C of subcategories of C such that C_{n} /[C_{n+1}] is a semisimple rejective subcategory of C /[C_{n+1}] for any n (0 ≤ n < m). The following theorem gives a categorical approach to quasi-hereditary algebras.

2.4.1 Theorem Any Γ-total right (resp. left) rejective chain is a heredity chain.

3 Representation Dimension

3.1 Definition Let Λ be an artin algebra and M ∈ mod Λ. Put

g_{Λ}(M) := gl.dim End_{Λ}(M) and r_{Λ}(M) := inf{g_{Λ}(M ⊕ N) | N ∈ mod Λ}.

By 3.1.1 below, r_{Λ} gives a function mod Λ → ℕ_{≥0}. Put |r_{Λ}| := sup{r_{Λ}(M) | M ∈ mod Λ}. We call rep.dim Λ := r_{Λ}(Λ ⊕ DΛ) the representation dimension of Λ, introduced by Auslander [A].

3.1.1 Theorem r_{Λ}(M) < ∞ holds for any M ∈ mod Λ.

Proof Immediate from 2.1.1 and 2.2.2.

3.1.2 The following theorem gives a method to calculate r_{Λ} [A][EHIS] (cf. 1.1.1).

Theorem Let Λ be an artin algebra, M ∈ mod Λ and C := add M. If Λ ⊕ DΛ ∈ C, then C-resol.dim (mod Λ) = C^{op}-resol.dim (mod Λ^{op}) = max{g_{Λ}(M) − 2, 0}.

3.2 Immediately, we obtain Auslander’s theorem [A] below, which gives the reason why we call rep.dim Λ the representation dimension. Notice that it is easily checked that rep.dim Λ ≤ 1 and |r_{Λ}| ≤ 1 occurs only when Λ is semisimple.

Theorem Let Λ be an artin algebra. Then rep.dim Λ ≤ 2 if and only if |r_{Λ}| ≤ 2 if and only if Λ is representation-finite.

Proof Fix M ∈ mod Λ and assume that C := add M satisfies Λ ⊕ DΛ ∈ C. By 3.1.2, g_{Λ}(M) ≤ 2 is equivalent to C-resol.dim (mod Λ) = 0 which means that, for any

X ∈ mod Λ, there exists a morphism Y →^{f} X such that Y ∈ C and Hom_{Λ}( , Y) →^{·f} Hom_{Λ}( , X) is an isomorphism on C. This is equivalent to that f is an isomorphism by Λ ∈ C. Thus g_{Λ}(M) ≤ 2 is equivalent to C = mod Λ. Thus we obtain the assertion.

3.2.1 Theorem If Λ is 1-Gorenstein artin algebra [AR2][FGR], then r_{Λ}(Λ) ≤ 2 if and only if Λ is reflexive-finite, i.e. Λ has only finitely many isoclasses of indecomposable reflexive modules.

3.2.2 Example Let Λ be an artin algebra with the Loewy length LL(Λ).

(1) r_{Λ}(Λ) ≤ g_{Λ}(⨁_{i=0}^{LL(Λ)} Λ/J_{Λ}^{i}) ≤ LL(Λ) holds by 2.1.2. In particular, rep.dim Λ ≤ LL(Λ) holds if Λ is selfinjective [A].

(2) If Λ is hereditary, then one can check that rep.dim Λ ≤ g_{Λ}(Λ ⊕ DΛ) ≤ 3 holds [A]. On the other hand, if J_{Λ}^{2} = 0, then Λ is stably equivalent to a hereditary algebra [ARS], and rep.dim Λ ≤ 3 holds by 3.4(2) below [A][X]. This can be proved directly by rep.dim Λ ≤ g_{Λ}(Λ ⊕ Λ/J_{Λ} ⊕ DΛ) ≤ 3.

(3) As we shall see in 3.7, algebras with representation dimension at most 3 form an interesting class. Several classes of algebras are known to have the representation dimension at most 3 [Xi1][CP][H1,2][BHS]. The following theorem of Erdmann-Holm-Schröer and the author [EHIS] also gives such algebras.

3.2.3 Theorem Let Λ ⊂^{ϕ} Γ be artin algebras. Assume that Γ is representationfinite and J_{Λ} is an ideal (resp. left ideal, right ideal) of Γ. Then rep.dim Λ ≤ 3 (resp. r_{Λ}(Λ) ≤ 3, r_{Λ}(DΛ) ≤ 3) holds.

3.2.4 Example In [EHIS], the representation dimension of a special biserial algebras is shown to be at most 3 by applying 3.2.3. Now let us consider the representation dimension of a clannish algebra A over a field k [CB]. Then there exists a Bäckström k[[t]]- order Λ with a hereditary overorder Γ and an ideal I of Γ such that J_{Λ} = J_{Γ} and A = Λ/I by [I6;1.3(3)]. Since B := Γ/I is a cyclic Nakayama algebra with J_{A} = J_{B} ⊂ A ⊂ B, we obtain that B is representation-finite and rep.dim A ≤ 3 by 3.2.3.

3.3 Tame-Wild dichotomy and r_{Λ} By famous Drozd’s Theorem [D], finite dimensional algebras over an algebraically closed field is divided into two classes, tame and wild. Any representation-finite algebra is tame. A certain class of wild algebras called controlled wild algebras was introduced in [Ha]. Ringel conjectures that any wild algebra is controlled wild. We will prove the following theorem in 3.6.2.

Theorem If Λ is a controlled wild algebra, then |r_{Λ}| = ∞.

3.3.1 Question We conjecture that any wild algebra satifies |r_{Λ}| = ∞. This follows from Ringel’s conjecture. On the other hand, does any tame algebra satify |r_{Λ}| < ∞? As the theorem below shows, this is true for hereditary algebras. Also, it is an interesting question whether any tame algebra satifies rep.dim Λ ≤ 3 or not [BHS]. These questions can be regarded as a part of the study of tame algebras in terms of endomorphism rings.

3.3.2 Theorem Let Λ be a finite dimensional hereditary algebra. Then the value of |r_{Λ}| is given as follows.

associated valued quiver [DR1] Dynkin extended Dynkin

else

|r_{Λ}|  ≤ 2  3  ∞

3.4 Let Λ and Γ be artin algebras. We say that Λ is finitely equivalent to Γ if there exists subcategories X and X′ of mod Λ and mod Γ respectively such that # ind X < ∞, # ind X′ < ∞ and mod Λ/[X] is equivalent to mod Γ/[X′]. Especially, when X = pr Λ and X′ = pr Γ, we say that Λ is stably equivalent to Γ [ARS]. Xiangqian’s result (2) below is proved by an application of Auslander-Reiten theory. We need relative homological algebra of Auslander-Solberg to prove (1).

Theorem Assume that Λ and Γ are not representation-finite.

(1) If Λ is finitely equivalent to Γ, then |r_{Λ}| = |r_{Γ}|.

(2)[X] If Λ is stably equivalent to Γ, then rep.dim Λ = rep.dim Γ.

3.5 Definition We shall introduce two homological invariant of Λ which is closely related to the function r_{Λ}. Let Λ and Γ be artin algebras.

(1) We write Λ ⪯ Γ if there exists P ∈ pr Γ such that End_{Γ}(P) is Morita-equivalent to Λ. Obviously, ⪯ gives a partial order on the set of Morita-equivalence classes of artin algebras. Define the expanded dimension of Λ by exp.dim Λ := inf{gl.dim Γ | Λ ⪯ Γ}. This concept first appeared in Auslander’s observation in [A] such that exp.dim Λ < ∞ by 2.1.2. For n = 0, 1, exp.dim Λ = n if and only if r_{Λ}(Λ) = n if and only if gl.dim Λ = n.

(2) Let C be a subcategory of mod Λ. We define the weak resolution dimension wresol.dim C as the minimal number n ≥ 0 which satisfies equivalent conditions below (cf. §1.1). (wresol.dim(mod Λ) + 2 coincides with rwrep.dim Λ in [R].)

(i) There exists M ∈ mod Λ such that, for any X ∈ C, there exists an exact sequence 0 → M_{n} → ⋯ → M_{0} → Y → 0 with M_{i} ∈ add M, Y ∈ C and X ∈ add Y.

(ii) There exists M ∈ mod Λ such that, for any X ∈ C, there exists an exact sequence 0 → Y → M_{0} → ⋯ → M_{n} → 0 with M_{i} ∈ add M, Y ∈ C and X ∈ add Y.

3.5.1 Let Λ be an artin algebra. One can easily check the following facts:

(1) exp.dim End_{Λ}(X) ≤ r_{Λ}(X) holds for any X ∈ mod Λ.

(2) wresol.dim(mod Λ) ≤ exp.dim Λ ≤ r_{Λ}(Λ) ≤ min{gl.dim Λ, rep.dim Λ, LL(Λ)}.

(3) wresol.dim add Ω^{2}(mod Λ) ≤ max{exp.dim Λ − 2, 0}.

(4) If a morphism ϕ : Λ → Γ of artin algebras satisfies add ϕ^{∗}(mod Γ) = mod Λ, then wresol.dim(mod Λ) ≤ wresol.dim(mod Γ).

3.6 Rouquier’s theorem below [R;6.10,6.9] gave the first example of algebras with representation dimension greater than 3. It follows from 4.2 and 4.3 below, which will be proved in §4.6. Consequently, exp.dim Λ and r_{Λ}(Λ) is also n + 1 by 3.5.1(2)(3).

Theorem Let k be a field and Λ = ∧(k^{n}) the exterior algebra with n > 0. Then wresol.dim(mod Λ) + 2 = rep.dim Λ = n + 1.

3.6.1 Example We can obtain many artin algebras with large representation dimension by using 3.6. Again let Λ = ∧(k^{n}) be the exterior algebra with n > 0.

(1) An artin algebra Γ with Λ ⪯ Γ satisfies n + 1 = exp.dim Λ ≤ exp.dim Γ ≤ rep.dim Γ by 3.5.1(2).

(2) If ϕ : Λ → Γ is a split monomorphism of artin algebras, then n−1 = wresol.dim(mod Λ) ≤ wresol.dim(mod Γ) ≤ rep.dim Γ by 3.5.1(4).

3.6.2 Proof of 3.3 For any artin algebra Γ, there exists M ∈ mod Λ and an ideal I of End_{Λ}(M) such that End_{Λ}(M) = Γ ⊕ I [Ha;2.3]. Considering the case Γ is the exterior algebra over n-dimensional vector space, we obtain n−1 ≤ wresol.dim(mod End_{Λ}(M)) ≤ exp.dim End_{Λ}(M) ≤ r_{Λ}(M) ≤ |r_{Λ}| by 3.6.1(2) and 3.5.1(1)(2). Thus |r_{Λ}| = ∞ holds.

3.7 Finitistic dimension conjecture Let Λ be an artin algebra and fin.dim Λ := sup{pd X | X ∈ mod Λ, pd X < ∞} the finitistic dimension of Λ [B]. The finitistic dimension conjecture (FDC) asserts that fin.dim Λ < ∞ holds for any artin algebra Λ.

We refer to [Z] for known results and the relationship to other homological conjecture. Recently, Igusa-Todorov [IT] introduced a function ψ_{Λ} and applied it to prove (FDC) for artin algebras with rep.dim Λ ≤ 3. We refer [EHIS] and [Xi2,3] for approach to (FDC) using Igusa-Todorov’s theorem.

3.7.1 Lemma Let Λ be an artin algebra. Then there exists a function ψ_{Λ} : mod Λ → ℕ_{≥0} with the following properties.

(i) If pd X < ∞, then ψ_{Λ}(X) = pd X.

(ii) add X ⊆ add Y implies ψ_{Λ}(X) ≤ ψ_{Λ}(Y).

(iii) If 0 → X → Y → Z → 0 is exact with pd Z < ∞, then pd Z ≤ ψ_{Λ}(X ⊕ Y) + 1.

3.7.2 Theorem Let Λ be an artin algebra. If wresol.dim add Ω^{n}(mod Λ) ≤ 1 holds for some n, then fin.dim Λ < ∞. Thus rep.dim Λ ≤ 3 (resp. r_{Λ}(Λ) ≤ 3, exp.dim Λ ≤ 3) implies fin.dim Λ < ∞.

Proof By 3.5.1(2)(3), we only have to show the former assertion. Let M ∈ mod Λ be in 3.5(2)(i). For any X ∈ mod Λ with pd X < ∞, take an exact sequence 0 → M_{1} → M_{0} → Y → 0 with M_{i} ∈ add M and Ω^{n}X ∈ add Y. Then pd X ≤ pd Y + n ≤ ψ_{Λ}(M_{1} ⊕ M_{0}) + n + 1 = ψ_{Λ}(M) + n + 1 holds by 3.7.1(ii)(iii).

4 Dimensions of triangulated categories

In this section, we shall give a proof of Rouquier’s theorem 3.6.

4.1 Definition [R;3.1] Let T be a triangulated category, and C and C′ subcategories of T . We denote by C ∗ C′ the subcategory of T consisting of X such that there exists a triangle Y → X → Y′ → ⋯ such that Y ∈ C and Y′ ∈ C′. We denote by ⟨C⟩ the smallest full subcategory of T containing C and closed under finite direct sums, direct summands and shifts. Put C ⋄ C′ := ⟨C ∗ C′⟩. Put ⟨C⟩_{0} := 0, ⟨C⟩_{i} := ⟨C⟩_{i−1} ⋄ ⟨C⟩ for i > 0 and ⟨C⟩_{∞} := ⋃_{i≥0}⟨C⟩_{i}. We sometimes denote ⟨C⟩_{i} by ⟨C⟩_{T ,i}. The dimension dim T of T is the minimal integer d ≥ 0 such that there exists M ∈ T with T = ⟨M⟩_{d+1}.

4.1.1 Example This concept is motivated by examples in algebraic geometry. Let k be a field and X a scheme over k.

(1) If X is smooth quasi-projective, then dim D^{b}(coh X) ≤ 2 dim X [R;5.8].

(2) If X is smooth affine and of finite type, then dim D^{b}(coh X) = dim X [R;5.37].

(3) If X is separated and of finite type, then dim D^{b}(coh X) < ∞ [R;5.38]. This is due to Kontsevich, Bondal and Van den Bergh for non-singular case [BV].

4.2 Let Λ be a selfinjective artin algebra. Then the stable category \underline{mod}Λ [ARS] forms a triangulated category. The following proposition gives the relationship between the dimension of stable categories and the representation dimension.

Proposition [R;6.9] dim \underline{mod}Λ ≤ wresol.dim(mod Λ) ≤ rep.dim Λ − 2 ≤ LL(Λ) − 2.

Proof Since Λ is selfinjective, Ω^{2}(mod Λ) = mod Λ holds. By 3.5.1(2)(3), it suffices to show the left inequality. Put n := wresol.dim(mod Λ) and take M ∈ mod Λ in 3.5(2)(i). For any X ∈ mod Λ, there exists an exact sequence 0 → M_{n} → ⋯ → M_{0} → Y → 0 with M_{i} ∈ add M and X ∈ add Y. Put Y_{0} := Y and Y_{n} := M_{n}. Then we have an exact sequence 0 → Y_{i} → M_{i−1} → Y_{i−1} → 0 with M_{i−1} ∈ ⟨M⟩ (0 < i ≤ n). Inductively, Y_{i} ∈ ⟨M⟩_{n−i+1} holds for any i. Thus Y ∈ ⟨M⟩_{n+1} and ⟨M⟩_{n+1} = \underline{mod}Λ hold.

4.3 Theorem [R;6.10] Let Λ := ∧(k^{n}) be the exterior algebra with n > 0. Then dim \underline{mod}Λ = n − 1.

4.4 Definition Let A be an abelian category with an autofunctor 𝔽 : A → A. Define a new category (A, 𝔽) as follows: An object is a apir (M, d) consisting of an object M of A and d ∈ A(M, 𝔽M) such that d𝔽(d) = 0. Put (A, 𝔽)((M, d), (M′, d′)) = {f ∈ A(M, M′) | d𝔽(f) = fd′}. We regard A as a full subcategory of (A, 𝔽) by M ↦ (M, 0). We call a morphism f in (A, 𝔽) quasi-isomorphism if the induced map Ker d/ Im d → Ker d′/ Im d′ is an isomorphism in A. Now (A, 𝔽) has the structure of exact category, where an exact sequence is a sequence which is a split exact as a sequence in A. Then (A, 𝔽) forms a Frobenius category, and its homotopy category H(A, 𝔽) is defined as usual. We define the derived category D(A, 𝔽) as the localization of H(A, 𝔽) by quasiisomorphisms. Later we shall use the three categories below.

(1)[R;5.1.3] Let Γ be a k-algebra. Then the category of differential Γ-modules is diff Γ := (mod Γ, 1). Put D diff Γ := D(mod Γ, 1). Then D diff Γ(M, M′) = ∏_{n≥0} Ext_{Γ}^{n}(M, M′) holds for any M = (M, 0), M′ = (M′, 0) ∈ D diff Γ.

(2)[K] Let Γ be a graded k-algebra. Then the category of differential graded Γmodules is diffgr Γ := (grmod Γ, 𝔽), where 𝔽 is the shift. Put D diffgr Γ := D(grmod Γ, 𝔽).

(3) Let Γ be a commutative graded k-algebra, X := Proj Γ and O_{x} the local ring at the point x ∈ X. For the shift 𝔽 : coh X → coh X, we put diffcoh X := (coh X, 𝔽) and D diffcoh X := D(coh X, 𝔽). We have the following commutative diagram:

diffgr Γ \underrightarrow{_{( )}} diffcoh X ⟶^{( )_{x}} diff O_{x}
↓  ↓  ↓
D diffgr Γ\underrightarrow{_{( )}}D diffcoh X⟶^{( )_{x}}D diff O_{x}

4.5 Proposition [R;5.11] Let Γ be a k-algebra and L a Γ-module with pd_{Γ} L ≥ n. Then L ∉ ⟨Γ⟩_{D diff Γ,n}.

Proof (i)[R;5.10] Put Γ^{en} := Γ ⊗_{k} Γ^{op}. We will show that there exist Γ^{en}-modules M_{0} = Γ, M_{1}, ⋯, M_{n} which are projective as left and as right Γ-modules, and ζ_{i} ∈ Ext_{Γ^{en}}^{1}(M_{i}, M_{i+1}) (0 ≤ i < n) such that (ζ_{0} ⋯ ζ_{n−1}) ⊗_{Γ} 1_{L} is a non-zero element of Ext_{Γ}^{n}(L, M_{n} ⊗_{Γ} L).

Let ⋯ → P_{1} →^{f_{1}} P_{0} →^{f_{0}} Γ → 0 be a projective resolution of the Γ^{en}-module Γ. Put M_{i+1} := Ker f_{i}. Then we have an exact sequence 0 → M_{i+1} → P_{i} → M_{i} → 0, which gives an element ζ_{i} ∈ Ext_{Γ^{en}}^{1}(M_{i}, M_{i+1}). We have a projective resolution 0 → M_{n} ⊗_{Γ} L → P_{n−1} ⊗_{Γ} L → ⋯ → P_{0} ⊗_{Γ} L → L → 0 of the Γ-module L, and the assertion follows.

(ii) Assume L ∈ ⟨Γ⟩_{n}. Then there exist triangles L_{j−1} →^{a_{j}} L_{j} →^{b_{j}} N_{j} → ⋯ (0 < j < n) such that L_{0}, N_{j} ∈ ⟨Γ⟩ (0 < j < n) and L is a direct summand of L_{n−1}. By 4.4(1), we regard ζ_{i} ⊗ 1_{N} ∈ Ext_{Γ}^{1}(M_{i} ⊗_{Γ} N, M_{i+1} ⊗_{Γ} N) ⊆ D diff Γ(M_{i} ⊗_{Γ} N, M_{i+1} ⊗_{Γ} N) for any Γ-module N. Thus we have the following morphism of triangles in D diff Γ:

M_{i} ⊗_{Γ} L_{j−1} ⟶^{1_{M_{i}}⊗a_{j}} M_{i} ⊗_{Γ} L_{j} ⟶^{1_{M_{i}}⊗b_{j}} M_{i} ⊗_{Γ} N_{j} ⟶ ⋯
↓^{ζ_{i}⊗1_{L_{j−1}}}  ↓^{ζ_{i}⊗1_{L_{j}}}  ↓^{ζ_{i}⊗1_{N_{j}}}
M_{i+1} ⊗_{Γ} L_{j−1}⟶^{1_{M_{i+1}}⊗a_{j}}M_{i+1} ⊗_{Γ} L_{j}⟶^{1_{M_{i+1}}⊗b_{j}}M_{i+1} ⊗_{Γ} N_{j}⟶ ⋯

Since the right-hand side map is zero by N_{j} ∈ ⟨Γ⟩, the morphism ζ_{i} ⊗ 1_{L_{j}} factors through 1_{M_{i+1}} ⊗ a_{j}. Thus we have the following commutative diagram:

M_{0} ⊗_{Γ} L_{n−1}
[figure: ζ_{0}⊗1]
M_{1} ⊗_{Γ} L_{n−2} ⟶^{1⊗a_{n−1}} M_{1} ⊗_{Γ} L_{n−1}
[figure: ζ_{1}⊗1; ζ_{1}⊗1]
…  …  …
[figure: ζ_{n−3}⊗1; ζ_{n−3}⊗1]
M_{n−2} ⊗_{Γ} L_{1}⟶^{1⊗a_{2}}⋯⟶^{1⊗a_{n−2}}M_{n−2} ⊗_{Γ} L_{n−2}⟶^{1⊗a_{n−1}}M_{n−2} ⊗_{Γ} L_{n−1}
[figure: ζ_{n−2}⊗1; ζ_{n−2}⊗1; ζ_{n−2}⊗1]
M_{n−1} ⊗_{Γ} L_{0}⟶^{1⊗a_{1}}M_{n−1} ⊗_{Γ} L_{1}⟶^{1⊗a_{2}}⋯⟶^{1⊗a_{n−2}}M_{n−1} ⊗_{Γ} L_{n−2}⟶^{1⊗a_{n−1}}M_{n−1} ⊗_{Γ} L_{n−1}
[figure: ζ_{n−1}⊗1=0; ζ_{n−1}⊗1; ζ_{n−1}⊗1; ζ_{n−1}⊗1]
M_{n} ⊗_{Γ} L_{0} ⟶^{1⊗a_{1}} M_{n} ⊗_{Γ} L_{1} ⟶^{1⊗a_{2}}⋯⟶^{1⊗a_{n−2}} M_{n} ⊗_{Γ} L_{n−2} ⟶^{1⊗a_{n−1}} M_{n} ⊗_{Γ} L_{n−1}

Since ζ_{n−1}⊗1_{L_{0}} = 0 holds by L_{0} ∈ ⟨Γ⟩, the right-hand side composition (ζ_{0} ⋯ ζ_{n−1})⊗ 1_{L_{n−1}} in the above diagram is zero, a contradiction to (i) and 4.4(1).

4.6 Proof of 4.3 In the rest of this paper, let Λ := ∧(k^{n}) be the exterior algebra and Γ := k[x_{1}, ⋯, x_{n}] the polynomial ring. By 4.2 and LL(Λ) = n + 1, we only have to show n − 1 ≤ dim \underline{mod}Λ. The usual Koszul duality gives an equivalence D^{b}(grmod Λ) → D^{b}(grmod Γ) of triangulated categories, which induces an equivalence \underline{grmod}Λ → D^{b}(coh P^{n−1}) of triangulated categories [BBG]. But we need the following version of Koszul duality due to Keller.

4.6.1 Theorem [K;10.5] There exists an equivalence D^{b}(mod Λ) → ⟨Γ⟩_{D diffgr Γ,∞} of triangulated categories, which induces an equivalence \underline{mod}Λ → ⟨Γ⟩_{D diffgr Γ,∞}/⟨k⟩_{D diffgr Γ,∞} of triangulated categories.

4.6.2 Proposition For any (F, d) ∈ D diffcoh P^{n−1}, there exists a point x ∈ P^{n−1} such that (F_{x}, d_{x}) ∈ ⟨O_{x}⟩_{D diff O_{x}}.

Proof Put G := Ker d(1)/ Im d and take a point x ∈ P^{n−1} such that G_{x} is a projective O_{x}-module. We have an exact sequence (F_{x}, d_{x}) → Ker d_{x} →^{f} G_{x} → 0 in diff O_{x}, and we can take g ∈ Hom_{O_{x}}(G_{x}, Ker d_{x}) such that gf = 1_{G_{x}}. Since the composition G_{x} →^{g} Ker d_{x} ⊂ (F_{x}, d_{x}) is a quasi-isomorphism, we obtain the assertion.

4.6.3 Since Γ-modules of finite length vanishes by the functor diffgr Γ → diffcoh P^{n−1}, we have the following functors of triangulated categories for any x ∈ P^{n−1} by 4.4 and 4.6.1:

\underline{mod}Λ ⟶^{∼} T := ⟨Γ⟩_{D diffgr Γ,∞}/⟨k⟩_{D diffgr Γ,∞} ⟶ D diffcoh P^{n−1} ⟶ D diff O_{x}

Assume that M ∈ T satisfies T = ⟨M⟩_{T ,n−1}. Let (F, d) ∈ D diffcoh P^{n−1} be the image of M, x ∈ P^{n−1} the point obtained by 4.6.2, and ℘ the homogeneous prime ideal of Γ which defines x. Then Γ/℘ is contained in T by the regularity of Γ. Since the image of Γ/℘ in D diff O_{x} is the simple O_{x}-module k_{x}, we obtain k_{x} ∈ ⟨(F_{x}, d_{x})⟩_{D diff O_{x},n−1} ⊆ ⟨O_{x}⟩_{D diff O_{x},n−1} by the choice of x. This contradicts to 4.5 by pd_{O_{x}} k_{x} = n − 1.

References

[A] M. Auslander: Representation dimension of Artin algebras. Lecture notes, Queen Mary College, London, 1971.

[AB] M. Auslander, R. O. Buchweitz: The homological theory of maximal Cohen-Macaulay approximations, Soc. Math. de France, Mem. no. 38 (1989) 5–37.

[AR1] M. Auslander, I. Reiten: Applications of contravariantly finite subcategories. Adv. Math. 86 (1991), no. 1, 111–152.

[AR2] M. Auslander, I. Reiten: k-Gorenstein algebras and syzygy modules. J. Pure Appl. Algebra 92 (1994), no. 1, 1–27.

[ARS] M. Auslander, I. Reiten, S. O. Smalo: Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995.

[AS1] M. Auslander, S. O. Smalo: Almost split sequences in subcategories. J. Algebra 69 (1981), no. 2, 426–454.

[AS2] M. Auslander, S. O. Smalo: Preprojective modules over Artin algebras. J. Algebra 66 (1980), no. 1, 61–122.

[ASo] M. Auslander, O. Solberg: Relative homology and representation theory. I. Relative homology and homologically finite subcategories. Comm. Algebra 21 (1993), no. 9, 2995–3031.

[B] H. Bass: Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960) 466-488.

[BBG] J. N. Bernstein, I. M. Gel’fand, S. I. Gel’fand: Algebraic bundles over bfP^{n−1} and problems of linear algebra. Funct. Anal. and its Appl. 12 (1978) 212-214.

[BHS] R. Bocian, T. Holm, A. Skowronski: The representation dimension of domestic weakly symmetric algebras, to appear in Central European Journal of Mathematics, to appear.

[BV] A. Bondal, M. Van den Bergh: Generators and representabilityof functors in commutative and noncommutative geometry, preprint.

[CB] W. W. Crawley-Boevey: Functorial filtrations. II. Clans and the Gelfand problem. J. London Math. Soc. (2) 40 (1989), no. 1, 9–30.

[CP] F. U. Coelho, M. I. Platzeck : On the representation dimension of some classes of algebras, preprint.

[CPS1] E. Cline, B. Parshall, L. Scott: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391 (1988), 85–99.

[CPS2] E. Cline, B. Parshall, L. Scott: Algebraic stratification in representation categories. J. Algebra 117 (1988), no. 2, 504–521.

[DR1] V. Dlab, C. M. Ringel: Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc. 6 (1976), no. 173.

[DR2] V. Dlab, C. M. Ringel: Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring. Proc. Amer. Math. Soc. 107 (1989), no. 1, 1–5.

[DR3] V. Dlab, C. M. Ringel: Auslander algebras as quasi-hereditary algebras. J. London Math. Soc. (2) 39 (1989), no. 3, 457–466.

[D] Y. A. Drozd: Tame and wild matrix problems. (Russian) Representations and quadratic forms (Russian), pp. 39–74, 154, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979.

[EHIS] K. Erdmann, T. Holm, O. Iyama, J. Schröer: Radical embeddings and representation dimension, to appear in Advances in Mathematics.

[FGR] R. M. Fossum, P. Griffith, I. Reiten: Trivial extensions of abelian categories. Homological algebra of trivial extensions of abelian categories with applications to ring theory. Lecture Notes in Mathematics, Vol. 456. Springer-Verlag, Berlin-New York, 1975.

[GD] P. Gabriel, J. A. de la Pena: Quotients of representation-finite algebras. Comm. Algebra 15 (1987), no. 1-2, 279–307.

[Ha] Y. Han: Controlled wild algebras. Proc. London Math. Soc. (3) 83 (2001), no. 2, 279–298.

[HS] P. J. Hilton, U. Stammbach: A course in homological algebra. Graduate Texts in Mathematics,

4. Springer-Verlag, New York, 1997.

[H1] T. Holm: Representation dimension of some tame blocks of finite groups, to appear in Algebra Colloquium.

[H2] T. Holm: The representation dimension of Schur algebras: the tame case, preprint.

[I1] O. Iyama: A generalization of rejection lemma of Drozd-Kirichenko. J. Math. Soc. Japan 50 (1998), no. 3, 697–718.

[I2] O. Iyama: τ-categories I,II,III, to appear in Algebras and Representation Theory.

[I3] O. Iyama: A proof of Solomon’s second conjecture on local zeta functions of orders, J. Algebra 259 (2003), no. 1, 119-126.

[I4] O. Iyama: Finiteness of Representation dimension, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1011-1014.

[I5] O. Iyama: Representation dimension and Solomon zeta function, to appear in Fields Institute Communications.

[I6] O. Iyama: Quadratic bimodules and Quadratic orders, preprint.

[I7] O. Iyama: Rejective subcategories of artin algebras and orders, preprint.

[IT] K. Igusa, G. Todorov: On the finitistic global dimension conjecture, preprint.

[K] B. Keller: Deriving DG categories, Ann. Sci. Ec. Norm. Sup. 27 (1994), 63-102.

[M] Y. Miyashita: Tilting modules of finite projective dimension. Math. Z. 193 (1986), no. 1, 113–146.

[N] H. Nagase: Non-strictly wild algebras. J. London Math. Soc. (2) 67 (2003), no. 1, 57–72.

[R] R. Rouquier, Dimensions of triangulated categories, preprint.

[Ru1] W. Rump: Lattice-finite rings, preprint.

[Ru2] W. Rump: The category of lattices over a lattice-finite ring, preprint.

[Si] S. Sikko: Resolutions with nth syzygies. Comm. Algebra 23 (1995), no. 10, 3729–3739.

[S1] L. Solomon: Zeta functions and integral representation theory. Advances in Math. 26 (1977), no. 3, 306–326.

[S2] L. Solomon: Partially ordered sets with colors. Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978), pp. 309–329, Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979.

[St] B. Stenström: Rings of quotients Die Grundlehren der Mathematischen Wissenschaften, Band 217. An introduction to methods of ring theory. Springer-Verlag, New York-Heidelberg, 1975.

[Xi1] C. C. Xi: Representation dimension and quasi-hereditary algebras. Adv. Math. 168 (2002), no. 2, 193–212.

[Xi2] C. C. Xi: On the finitistic dimension conjecture I: related to representation-finite algebras, preprint.

[Xi3] C. C. Xi: On the finitistic dimension conjecture II: related to finite global dimension, preprint.

[X] G. Xiangqian: Representation dimension: an invariant under stable equivalence, preprint.

[Z] H. B. Zimmermann: The finitistic dimension conjectures—a tale of 3.5 decades. Abelian groups and modules (Padova, 1994), 501–517, Math. Appl., 343, Kluwer Acad. Publ., Dordrecht, 1995.

Department of Mathematics, Himeji Institute of Technology, Himeji, 671-2201, Japan iyama@sci.himeji-tech.ac.jp