REPRESENTATION DIMENSION OF ARTINALGEBRAS

STEFFEN OPPERMANN

In 1971, Auslander [1] has introduced the notion of representationdimension of an artin algebra. His definition is as follows (see Section 1for details on the notation):

Definition A. Let Λ be an artin algebra. We set

A(Λ) = Γ artin algebra | dom.dim Γ ≥ 2 and Λ ∼Morita

EndΓ(I0(Γ)).

Here I0(Γ) denotes the injective envelope of Γ as Γ-module.The representation dimension of Λ is defined as

repdim Λ =

1 if Λ is semi-simplemingl.dim Γ | Γ ∈ A(Λ) otherwise.

Auslander has shown that an algebra is of finite representation type,that is, it admits only finitely many indecomposable modules up toisomorphism, if and only if its representation dimension is at most 2.We will give a proof of this fact in Section 1 as Corollary 1.9. This ledAuslander to the expectation, “that this notion gives a reasonable wayof measuring how far an artin algebra is from being of finite represen-tation type.” [1, III.5, lines 2, 3]

Igusa and Todorov [6] have shown that there is a connection betweenAuslander’s representation dimension and the finitistic dimension con-jecture.

Finitistic dimension conjecture. Let Λ be an artin algebra, and

fin.dim Λ = suppdX | X ∈ mod Λ such that pdX <∞.

Then

fin.dim Λ <∞.

More precisely, they have shown that the finitistic dimension conjec-ture holds for any artin algebra of representation dimension at mostthree. Here we will prove this in Section 2.

Unfortunately, it turned out to be rather hard to compute the actualvalue of the representation dimension of a given algebra. However, in2003 Iyama [8] has shown that the representation dimension of a finitedimensional algebra is always finite. He did so by explicitly construct-ing an algebra Γ ∈ A(Λ) with gl.dim Γ < ∞, so the minimum in thedefinition above is always finite. We will explain his construction and

1

2 February 9, 2009

prove that it works in Section 3 here. When applied to a given al-gebra, this construction yields an upper bound for the representationdimension of this algebra.

By Auslander’s result mentioned above, it was known that any rep-resentation infinite algebra has representation dimension at least three.However, it was not known whether numbers greater than three canoccur as the representation dimension of a finite dimensional algebra,until Rouquier [12] has shown in 2005 that the representation dimen-sion of the exterior algebra of an n-dimensional vector space is alwaysn + 1. Here, in Section 4, we will take a slightly different approachto show that any number occurs as the representation dimension ofsome artin algebra. In order to do so we will give Rouquier’s defini-tion of dimension of a triangulated category [12, 13], and the authorsgeneralization to subcategories [10]. We will then explain what is theconnection of these dimensions to Auslander’s representation dimen-sion. Finally we will use this method to determine the representationdimension of the Beilinson algebras.

1. Different definitions and first properties

In this section we give three different definitions of representationdimension. We show that, provided the algebra is not semi-simple, allthree definitions are equivalent. We will give some indication as to whenwhich definition is most helpful. In particular we will prove Auslander’stheorem, saying that an algebra has representation dimension at mosttwo if and only if it is representation finite, using Definition C.

Notation. For an artin algebra Λ we denote by mod Λ the category offinitely generated left Λ-modules. We denote by D : mod Λ mod Λop

the standard duality.

Let us start by recalling Auslander’s original definition as given inthe introduction. First we define the notions involved.

Notation. Let Λ be an artin algebra. Let M ∈ mod Λ. Then we de-note by pdM and idM the projective dimension and injective dimen-sion of M , respectively. We denote by gl.dim Λ the global dimension ofΛ, that is the maximum over the projective dimensions of all modules.

1.1. Definition. Let Λ be an artin algebra, and let

Λ I0(Λ) I1(Λ) · · ·be a minimal injective resolution. Then the dominant dimension of Λis

dom.dim Λ = infn ∈ N | In(Λ) is not projective.

1.2. Definition. Let Λ and Γ be to artin algebras. We say Λ and Γare Morita equivalent (denoted by Λ ∼

MoritaΓ if there is a projective

February 9, 2009 3

generator P ∈ mod Λ (that is a projective module P ∈ proj Λ suchthat Λ ∈ addP ) such that Γ ∼= EndΛ(P ).

Definition A. Let Λ be an artin algebra. We set

A(Λ) = Γ artin algebra | dom.dim Γ ≥ 2 and Λ ∼Morita

EndΓ(I0(Γ)).

Here I0(Γ) denotes the injective envelope of Γ as Γ-module.The representation dimension of Λ is defined as

repdim Λ =

1 if Λ is semi-simplemingl.dim Γ | Γ ∈ A(Λ) otherwise.

1.3. Observations. • The condition dom.dim Γ ≥ 1 means that(up to multiplicity) I0(Γ) is the direct sum of all indecomposableprojective injective Λ-modules.• We set

S = X ∈ mod Γ |there is an exact sequence

X B0 B1 with

B0, B1 ∈ add I0(Γ).

Since I0(Γ) is injective the functor F : S mod Λ as depictedin the following diagram is an equivalence.

S mod EndΓ(I0(Γ)) mod Λ

add I0(Γ) inj EndΓ(I0(Γ)) inj Λ

DHomΓ(−,I0(Γ)) ≈

≈ ≈

F

• Since dom.dim Γ ≥ 2 we see that Γ ∈ S. In particular it makessense to consider F(Γ), and

EndΛ(F(Γ)) = EndΓ(Γ) = Γ.

• The Λ-module F(Γ) is a generator and cogenerator of mod Λ.That is, it contains all indecomposable projective and all inde-composable injective Λ modules as direct summands.

This motivates the following version of Auslander’s original defini-tion.

Definition B. Let Λ be an artin algebra. Then

repdim Λ = min gl.dim End(M) |M ∈ mod Λ

generator and cogenerator.

A generator cogenerator M realizing the minimum above is calledAuslander generator.

4 February 9, 2009

We have seen above that any algebra in A(Λ) is the endomorphismring of a generator cogenerator of mod Λ. Hence, in order to showthat the two definitions coincide for any non-semisimple algebra Λ, itsuffices to show than EndΛ(M) ∈ A(Λ) for any generator cogeneratorM of mod Λ.

Proof. Let M I0(M) I1(M) be the start of an injective resolutionof M . Applying the functor Hom(M,−) we obtain

EndΛ(M) HomΛ(M, I0(M)) HomΛ(M, I1(M)).

Now HomΛ(M,DΛ) = DHomΛ(Λ,M), and this is injective as EndΛ(M)-module since Λ ∈ addM . Hence the sequence above shows thatdom.dim EndΛ(M) ≥ 2, and moreover that

EndEndΛ(M)(I0(EndΛ(M))) = EndEndΛ(M)(HomΛ(M, I0(M)))

= EndΛ(I0(M)) ∼Morita

Λ.

Here the Morita equivalence follows from the fact that M is a cogen-erator.

1.4. Definition. ForM ∈ mod Λ andX ∈ mod Λ we say that f : M ′ Xis a right M-approximation, if any morphism ϕ : M X factors throughf .

It is called minimal right M-approximation, if it is moreover rightminimal in the sense of [3].

We denote by ΩMX the kernel of a minimal right M -approximationof X, and inductively we set Ωn+1

M X = ΩMΩnMX.

We set

M -resol.dimX = infn ∈ N | Ωn+1M X = 0, and

M -resol.dim(mod Λ) = supM -resol.dimX | X ∈ mod Λ.

1.5. Observations. • TheM -approximations and the derived con-structions only depend on addM .• For M = Λ the M -approximations are projective covers, and

ΩΛ = Ω is the usual syzygy.• If Λ ∈ addM then right M -approximations are always epimor-

phisms.• The functor

mod Λ mod EndΛ(M)

addM proj EndΛ(M)

G = HomΛ(M,−)

≈

maps– kernels to kernels, and– right M -approximations to projective covers.

February 9, 2009 5

In particular for any X ∈ mod Λ we have M -resol.dimX =pd G(X).• For any X ∈ mod Λ we there is a sequence

Ω2MX M1 M0 X

which is exact in M1. Therefore Ω2MX ∈ Im G.

• For X ∈ mod Λ we have G(X) ∈ Ω2(mod Γ) if and only if thereis an exact sequence X M0 M1 with M0,M1 ∈ addM .• If M is a cogenerator then we have G(X) ∈ Ω2(mod Γ) for anyX ∈ mod Λ.

Hence we have shown that Definition B is equivalent to the following:

Definition C. Let Λ be an artin algebra. Then

repdim Λ = minM -resol.dim(mod Λ) |M ∈ mod Λ

generator and cogenerator+ 2.

1.6. Remark. Motivated by this definition and the examples of alge-bras where the representation dimension is explicitly known (see The-orems 4.14 and 4.15) Ringel suggests to normalize the representationdimension, and set

n.repdim Λ = repdim Λ− 2.

Let us summarize what we have shown so far.

1.7. Theorem. Let Λ be an artin algebra, which is not semi-simple.Then Definitions A, B, and C are equivalent. (It is easy to see thatthey give 1, 0, and 2, respectively for semi-simple algebras.)

1.8. Remarks. • One advantage of Definition B is, that to useit we have to know only M , and not the rest of the modulecategory.• One advantage of Definition C is that, if we do not know M , we

can work in the module category of Λ and don’t have to con-struct projective resolutions in an unknown module category.

1.9. Corollary (Auslander). Let Λ be an artin algebra. Then repdim Λ ≤2 if and only if Λ has finite representation type.

Proof. Assume first that Λ has finite representation type. Then we canfind an additive generator M of mod Λ (that is a module M ∈ mod Λ,such that addM = mod Λ). One easily sees that M -resol.dimX = 0for any X ∈ mod Λ, and hence repdim Λ = 2.

Now assume that Λ is not representation finite, and let M be anAuslander generator. Then there is X ∈ mod Λ \ addM . For thisX we have M -resol.dimX ≥ 1, and hence M -resol.dim(mod Λ) ≥ 1.Therefore repdim Λ ≥ 3.

6 February 9, 2009

1.10. Example. Let Λ be the quiver algebra of linearly oriented A4,with Auslander-Reiten quiver as depicted below.

M1

M2

M3

projectives injectives

We obtain the following resolution dimensions:

M M -resol.dim(mod Λ)Λ⊕DΛ 1Λ⊕DΛ⊕M1 1Λ⊕DΛ⊕M2 2Λ⊕DΛ⊕M3 1Λ⊕DΛ⊕M1 ⊕M2 1Λ⊕DΛ⊕M1 ⊕M3 1Λ⊕DΛ⊕M2 ⊕M3 1Λ⊕DΛ⊕M1 ⊕M2 ⊕M3 0

This suggests that the M -resultion dimension tends to go down ifwe make M bigger. However this is not true in general, and in fact wehave

1.11. Theorem (Iyama [7, Theorem 4.6.2]). Let Λ be controlled wild,and n ∈ N. Then

∃B : ∀M generator and cognerator with B ∈ addM :

M -resol.dim(mod Λ) ≥ n

We now follow Ringel [11] to determine a large class of algebras whichhave representation dimension three.

1.12. Theorem. Assume the category

Sub Λ = X ∈ mod Λ | X Λn for some ncontains only finitely many indecomposables (such an algebra is calledtorsionless finite). Then repdim Λ ≤ 3. In particular, any represen-tation infinite torsionless finite algebra has representation dimensionthree.

Proof. See [11] for the proof that a torsionless finite algebra is alsocotorsionless finite, that is the category

FacDΛ = X ∈ ind Λ | DΛn X for some n

February 9, 2009 7

also contains only finitely many indecomposables.We choose M and M such that addM = Sub Λ and addM =

FacDΛ, and set M = M ⊕M .

Let X ∈ mod Λ. Let M ′f

X and M′ f

X be minimal rightM - and M -approximations of X. Since Λ ∈ addM the map f is onto.

Since addM is closed under factors f is into. We take the pullback asin the following diagram.

M X

K M

PB

Now, since addM is closed under subobjects we have K ∈ addM , andhence an M -resolution

K M ′ ⊕M ′X.

Hence M -resol.dim(mod Λ) ≤ 1, and the claim follows from Defini-tion C.

1.13. Examples. The following classes of algebras are torsionless finite,and hence have representation dimension at most three.

• hereditary algebras• concealed algebras• algebras with Rad2 = 0

2. The connection to the finitistic dimension conjecture

In this section we summarize Igusa and Tororov’s proof of the fol-lowing result.

2.1. Theorem (Igusa-Todorov [6]). Let Λ be an artin algebra of repre-sentation dimension at most three. Then

fin.dim Λ <∞.

Recall that fin.dim Λ = suppdX | X ∈ mod Λ∧pdX <∞. One ofthe most important conjectures about homological properties of artinalgebras is the following:

Finitistic dimension conjecture. Let Λ be an artin algebra, and

fin.dim Λ = suppdX | X ∈ mod Λ such that pdX <∞.Then

fin.dim Λ <∞.

Hence Igusa and Todorov’s theorem says that the finitistic dimensionconjecture holds for algebras of representation dimension at most three.

8 February 9, 2009

2.2. Remark. It can be very easy to show that repdim Λ ≤ 3 (usingDefinition B): If we have a generator cogenerator M it is easy to checkif gl.dim End(M) ≤ 3.

Idea of proof of 2.1. Let M be an Auslander generator. We have tofind an upper bound to pdX for any X ∈ mod Λ of finite projectivedimension.

By Definition C and the assumption of the theorem there is a shortexact sequence

M1 M0 X

with M1 and M0 ∈ addM . Hence there are also short exact sequences

ΩnM1 ΩnM0 ⊕ P ΩnX

for any n (with P projective). Since X has finite projective dimensioneventually we will have

ΩnM1∼= ΩnM0.

Now Igusa and Todorov apply the fitting lemma to show that there isφ(M) ∈ N such that for any M ′,M ′′ ∈ addM

ΩnM ′ ∼= ΩnM ′′ for some n ∈ N⇒Ωφ(M)M ′ ∼= Ωφ(M)M ′′

(see [6] for details).Now split Ωφ(M)M0 = Ωφ(M)M1 = A ⊕ B, with pdA < ∞ and B a

direct sum of indecomposable modules of infinite projective dimension.In the short exact sequence

A⊕B

(∗ ∗∗ fB∗ ∗

)A⊕B ⊕ P Ωφ(M)X

the map fB has to be an isomorphism (see [6]). Therefore there is ashort exact sequence

A A⊕ P Ωφ(M)X.

Hence

pdX ≤ φ(M) + pdA+ 1

≤ φ(M) + maxpdX | X ∈ add Ωφ(M)M and pdX <∞+ 1.

Since this upper bound for pdX does not depend on X it follows that

fin.dim Λ ≤ φ(M)+maxpdX | X ∈ add Ωφ(M)M and pdX <∞+1.

2.3. Example. Erdmann, Holm, Iyama, and Schroer [5] have shownthat any special biserial algebra has representation dimension at mostthree. With Theorem 2.1 they obtain as a corollary that any specialbiserial algebra has finite finitistic dimension.

February 9, 2009 9

2.4. Remarks. • The proof only requires the existence of a shortexact sequence M1 M0 X for any X, but does not needthis sequence to be an M -resolution, or M to be a cogenerator.• In order to show that the finitistic dimension is finite, it would

suffice to treat all X which are syzygies.

3. Iyama’s finiteness theorem

In this section we give a proof of the following result of Iyama.

3.1. Theorem (Iyama [7, 8]). Let Λ be an artin algebra. Then

repdim Λ <∞.

The strategy of the proof is to construct a generator cogeneratorM ∈ mod Λ such that M -resol.dim(mod Λ) <∞.

3.2. Remark. Iyama [8] has first shown that the generator cogeneratorM he constructed has quasi-hereditary endomorphism ring. Hence itsendomorphism ring has finite global dimension, and the representationdimension is finite by Definition B.

3.3. Construction. Let M0 be any generator cogenerator (typicallywe choose M0 = Λ⊕DΛ). Then set inductively

M i+1 = M i/ SocEndΛ(M i) Mi.

That means M i+1 is the image of the map

M i (M i)n

M i+1

(α1, . . . , αn)

where α1, . . . , αn generate Rad(M i,M i)End(M i), that is any radical en-domorphism of M i factors through (α1, . . . , αn). In particular the M i

have the properties

(1) Any radical map M i M i factors through M iproj

M i+1.(2) For any i we have SubM i+1 ⊆ SubM i.

Note that wheneverM i 6= 0 we have length(M i+1) length(M i). SinceM0 has finite length there is m such that Mm+1 = 0. We set

M = ⊕mi=0Mi.

Clearly this is a generator cogenerator.

3.4. Proposition. With M and m as above we have

M -resol.dim(mod Λ) ≤ m− 1.

In particularrepdim Λ ≤ m+ 1.

10 February 9, 2009

Clearly this implies Theorem 3.1.

Proof. Let X ∈ mod Λ not contain any injective direct summands.Then we show the following by induction:

(Ak) There is a radical monomorphism ΩkMX (Mk)n for some n.

(Bk) There is anM -approximationM ′ ΩkMX withM ′ ∈ add⊕mi=k+1M

i.

(A0) is true (take an injective envelope). We will show (Ak) ⇒(2)

(Bk)

⇒(1)

(Ak+1).

For implication (1) note that, in the setup of (Bk), we have

Ωk+1M X

∈ RadM ′ (Mk+1)n

for some n by Property (2) in Construction 3.3 above.For implication (2) let s be the biggest number such that there is

an M -approximation M ′ ΩkM with M ′ ∈ add⊕mi=s+1M

i. We mayassume M ′ to be of the form ⊕mi=s+1(M i)ni for some ni. If we assumes < k we have the following diagram,

⊕mi=s+1(M i)ni (M s+2)ns+1 ⊕⊕mi=s+2(M i)ni

ΩkMX (M s+1)n

M -approx

∈ Rad

where the upper map is projection in the components M s+1 M s+2

and identity elsewhere. By Property (1) of Construction 3.3 we cancomplete this to a commutative square. Since the image of the rightvertical map is the same as the image of the composition from leftupper to right lower corner, which is Ωk

MX, we obtain the followingcommutative diagram.

⊕mi=s+1(M i)ni (M s+2)ns+1 ⊕⊕mi=s+2(M i)ni

ΩkMX (M s+1)n

M -approx

∈ Rad

Now the diagonal map is also an M -approximation, contradicting theminimality of s. Hence s ≥ k, and the inductive statements are shown.

Finally we see that by (Bm) we have ΩmMX = 0, and henceM -resol.dimX ≤

m− 1.

February 9, 2009 11

3.5. Example (Result by Auslander). Let Λ with LL Λ = 2 (LL denotesthe Loewy length).

M0 = P ⊕DΛ (P projective non-injective)

M1 = P ⊕DΛ/ SocDΛ︸ ︷︷ ︸semisimple

M2 = semisimple

M3 = 0 ⇒ repdim Λ ≤ 3.

3.6. Example (Result by Auslander). Let Λ be self-injective. M0 =Λ. Inductively one sees that the indecomposable direct summands ofM i have pairwise non-isomorphic simple tops, and hence LLM i+1 <LLM i. Therefore MLL Λ = 0, and repdim Λ ≤ LL Λ.

3.7. Example (Iyama). Let Λ be the Beilinson-algebra kQ/I with

Q =

x0

xn

x0

xn

x0

xn

x0

xn

x0

xn

and

I = (xixj − xjxi).

Starting with M0 = Λ ⊕ DΛ one obtains Mn+2 = 0, and hencerepdim Λ ≤ n+ 2.

4. Lower bounds

In this section we will restrict ourselves to finite dimensional k-algebras.

4.1. Idea (Rouquier). M -resol.dimX big ≈ it takes many short exactsequences to build X from M .

Translation to the triangulated world gives the dimension of a (sub-category of a) triangulated category.

4.2. Construction. Let T be a triangulated category, and M ∈ T .

〈M〉 = 〈M〉1 = addM [i] | i ∈ Z〈M〉n+1 = addX | ∃M ′ X M ′′ M ′[1] with

M ′ ∈ 〈M〉 ,M ′′ ∈ 〈M〉n

The subcategory 〈M〉n is also called “nth thickening of M”. It containsall objects that can be constructed from M using triangles at mostn− 1 times.

4.3. Definition. The dimension of a triangulated category T is

dim T = infn | ∃M ∈ T : T = 〈M〉n+1.

12 February 9, 2009

For a subcategory C ⊆ T the dimension is defined to be

dimT C = infn | ∃M ∈ T : C ⊆ 〈M〉n+1.We will omit the index T when there is no danger of confusion.

Of particular interest to us are dimDb(mod Λ) and dim mod Λ =dimDb(mod Λ) mod Λ.

4.4. Lemma. Let M ∈ mod Λ be a generator, and X ∈ mod Λ. Thenfor any n we have

M -resol.dimX ≤ n⇒ X ∈ 〈M〉n+1 .

In particular

M -resol.dim(mod Λ) ≥ dim mod Λ, and

repdim Λ ≥ dim mod Λ + 2.

Proof. This follows immediately from the fact that short exact se-quences in mod Λ turn into triangles in Db(mod Λ).

4.5. Lemma.repdim Λ ≥ dimDb(mod Λ)

Proof. See [9, 12].

If we want to use these inequalities to establish lower bounds forthe representation dimension, we need to find a method to prove lowerbounds for dimensions of triangulated categories or their subcategories.One key ingredient is the ghost lemma.

4.6. Definition. Let C ′ ⊂ C be categories, f : X Y in C is called

C ′-ghost if any composition C ′ XfY with C ′ ∈ C ′ vanishes.

Here we look at 〈M〉 ⊂ Db(mod Λ).

4.7. Lemma (Ghost lemma). Let T be a triangulated category, M ∈ T ,and

X0

f1X1

f2X2

f3· · ·

fnXn

a sequence of 〈M〉-ghosts, such that the composition f1 · · · fn 6= 0.Then X0 6∈ 〈M〉n.

Proof. See, for instance, [9].

4.8. Example. Let T = Db(mod Λ), M = Λ, and X ∈ mod Λ withpdX = n. Then X 6∈ 〈M〉n.

Proof. The projective resolution

ΩnX Pn−1 Pn−2 P1 P0 X

Ωn−1X ΩX

February 9, 2009 13

gives rise to a sequence of maps

X ΩX[1] · · · Ωn−1X[n− 1] ΩnX[n]

in Db(mod Λ). They are all 〈Λ〉-ghosts, and their composition in non-zero. Hence the claim follows from the ghost lemma.

4.9. Remark. For this example it suffices for Λ to be a left noetherianring.

4.10. Problem. In order to show that dimDb(mod Λ) or dim mod Λ isbig we need to find such a sequence of 〈M〉-ghosts for an arbitrary M .

This problem can be solved “in a different world”:Let X be a reduced scheme of finite type over k. We look at cohX

instead of mod Λ.

4.11. Remark. The easiest (and most important) examples are:

An : cohAn = mod k[x1, . . . , xn]

Pn : cohPn = modgr k[x0, . . . , xn]/(modules of finite length)

The important difference between mod Λ and cohX is that in thegeometric setup we have an extra tool: localization. For any closedpoint p we have the local ring Op, and for any M ∈ cohX, the corre-sponding module Mp ∈ modOp.

We denote by kp = Op/p the simple sheaf concentrated in point p.(This is called a “skyscraper sheaf”.)

4.12. Lemma. Let X be a reduced scheme of finite type over k, andM ∈ cohX. Then Mp is projective over Op for all p in a dense openset of closed points in X.

4.13. Corollary.

dimDb(coh X)kp | p a closed point in X ≥ dimX

Proof. Choose a closed point p in an irreducible component of X whichhas the same dimension as X, such that Mp is projective over Op. Then

kp ∈ 〈M〉n ⇒ kp ∈ 〈Mp〉n ⊆ 〈Op〉n⇒

Example4.8

n pdOpkp ≥ dimX.

Back to finite dimensional algebras. The first examples of algebrasof representation dimension strictly bigger then three have been theexterior algebras.

4.14. Theorem (Rouquier). Let Λ be the exterior algebra of an n-dimensional vector space. Then

repdim Λ = n+ 1.

14 February 9, 2009

Vague idea of proof. (see [12] for details) Use Koszul duality

Db(mod Λ)RHom(k,−)

D(dgmod k[x1, . . . , xn])

and relate the latter category to the derived category of coherent sheaveson Pn−1.

Here we want to look at the Beilinson algebras in more detail.

4.15. Theorem. Let Λ be the Beilinson algebra (see Example 3.7) kQ/Iwith

Q =

x0

xn

x0

xn

x0

xn

x0

xn

x0

xn

and

I = (xixj − xjxi).Then

repdim Λ = n+ 2.

Proof. The sheaf T = O⊕O(1) · · · ⊕O(n) is a tilting bundle in cohPnwith End(T ) = Λ. Hence it induces a derived equivalence

Db(cohPn)RHom(T,−)

Db(mod Λ).

Since ExtiPn(O(j), kp) = 0 for i ≥ 1, for any j, and any closed point p,the functor RHom(T,−) maps kp | p closed point to mod Λ. Hence

dimDb(mod Λ) mod Λ ≥ dimDb(mod Λ) RHom(T, kp | p closed point)= dimDb(coh X)kp | p closed point≥

Corollary 4.13n

Now repdim Λ ≥ n+ 2 by Lemma 4.4. We have seen repdim Λ ≤ n+ 2in Example 3.7.

4.16. Remark. The proof above (showing that repdim Λ ≥ n + 2)works for any algebra Λ that comes up as the endomorphism ring of atilting object in cohX, where X is a reduced scheme of dimension n.

4.17. Example. For X = P1 × · · · × P1︸ ︷︷ ︸n copies

there is a tilting bundle

T = ⊕i∈0,1nO(i1)⊗ · · · ⊗ O(in).

By Remark 4.16 the endomorphism ring of T has representation di-mension at least n + 2. Using Iyama’s upper bound (Proposition 3.4)one easily sees that repdim Λ = n+ 2.

The algebras arising here have the following shape:

n = 1: End(T ) = k[ ]

February 9, 2009 15

n = 2: End(T ) = kQ/I with

Q =

x1

y1

x1

y1

x2 y2 x2 y2

and

I = (x1x2 − x2x1, x1y2 − y2x1, y1x2 − x2y1, y1y2 − y2y1).

n = 3: End(T ) = kQ/I with

Q = and

I = (similar commutation relations).

and similar hypercubes with commutation relations for larger n.

We conclude by giving two more general results giving lower boundsfor the representation dimension by relating the module category tosome commutative setup:

Setup for the lattice theorem: Let R be a finitely generated commuta-tive algebra over k without zero divisors, and let Λ be a finite dimen-sional k-algebra. Let L be a Λ⊗k R-lattice (that is a bimodule, whichis projective over R). Then the functor

L⊗R − : finite length modules over R mod Λ

is exact, and hence induces maps

(L⊗R −)Extd : ExtdR(X, Y ) ExtdΛ(L⊗R X,L⊗R Y )

for R-modules X and Y of finite length.

4.18. Theorem (Lattice theorem – see [10]). Let Λ, R and L be asabove, and d ∈ N. Assume

(L⊗R −)Extd(ExtdR(R/p, R/p)) 6= 0

for all p in a dense subset of maximal ideals of R. Then

repdim Λ ≥ d+ 2.

16 February 9, 2009

4.19. Theorem (Bergh [4]). Assume Λ is self-injective, and the evenHochschild cohomology ring satisfies a finite generation hypothesis (see[4]). Then

repdim Λ ≥ Krull.dim HH2∗(Λ) + 1.

(Here Krull.dim HH2∗(Λ) denotes the Krull dimension of the (commu-tative) even Hochschild cohomology ring.)

References

1. Maurice Auslander, Representation dimension of Artin algebras, Queen MaryCollege Mathematics Notes, 1971, republished in [2].

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

3. Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation the-ory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36,Cambridge University Press, Cambridge, 1997, Corrected reprint of the 1995original.

4. Petter Andreas Bergh, Representation dimension and finitely generated coho-mology, Adv. Math. 219 (2008), no. 1, 389–400.

5. Karin Erdmann, Thorsten Holm, Osamu Iyama, and Jan Schroer, Radical em-beddings and representation dimension, Adv. Math. 185 (2004), no. 1, 159–177.

6. Kiyoshi Igusa and Gordana Todorov, On the finitistic global dimension conjec-ture for Artin algebras, Representations of algebras and related topics, FieldsInst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 201–204.

7. Osamu Iyama, Rejective subcategories of artin algebras and orders, preprint,arXiv:math.RT/0311281.

8. , Finiteness of representation dimension, Proc. Amer. Math. Soc. 131(2003), no. 4, 1011–1014.

9. Henning Krause and Dirk Kussin, Rouquier’s theorem on representation dimen-sion, Trends in representation theory of algebras and related topics, Contemp.Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 95–103.

10. Steffen Oppermann, Lower bounds for Auslander’s representation dimension,preprint, 2007.

11. Claus Michael Ringel, The torsionless modules of an artin algebra, 2008.12. Raphael Rouquier, Representation dimension of exterior algebras, Invent.

Math. 165 (2006), no. 2, 357–367.13. , Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2,

193–256.

Institutt for matematiske fag, NTNU, 7491 Trondheim, NorwayE-mail address: Steffen.Oppermann@math.ntnu.no