a characterization of cluster categories

Quivers and derived categoriesThe cluster category, and the main theorem

ExamplesOn the proofs

Summary

A characterization of cluster categories

Bernhard Keller1 Idun Reiten2

1Department of Mathematics and Institute of MathematicsUniversity Paris 7

2Norwegian University of Science and TechnologyTrondheim

Bernhard Keller, Idun Reiten A characterization of cluster categories



Summary




Summary

Outline

1 Quivers and derived categories

2 The cluster category, and the main theorem

3 Examples

4 On the proofs




Summary

A quiver is an oriented graph

DefinitionA quiver Q is an oriented graph: It is given by

a set Q0 (the set of vertices)a set Q1 (the set of arrows)two maps

s : Q1 → Q0 (taking an arrow to its source)t : Q1 → Q0 (taking an arrow to its target).

RemarkA quiver is a ‘category without composition’.




Summary

A quiver can have loops, cycles, several components.

Example

The quiver ~A3 : 1 2αoo 3

βoo is an orientation of the Dynkin

diagram A3 : 1 2 3 .

Example

Q : 3λ

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1 ν// 2

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We have Q0 = {1, 2, 3, 4, 5, 6}, Q1 = {α, β, . . .}.α is a loop, (β, γ) is a 2-cycle, (λ, µ, ν) is a 3-cycle.




Summary

representation of a quiver = diagram of vector spaces

Let k be an algebraically closed field.Let Q be a finite quiver (the sets Q0 and Q1 are finite).

DefinitionA representation of Q is a diagram of finite-dimensional vectorspaces of the shape given by Q.

Example

A representation of ~A2 : 1α // 2 is a diagram of two

finite-dimensional vector spaces linked by one linear map

V : V1Vα // V2 .




Summary

The category of representations of Q is abelian.

Definitionrep(Q) = category of representations of Q.

RemarksA morphism of representations of Q is a morphism ofdiagrams.The category of representations is a k -linear abeliancategory with enough projectives (it is even a modulecategory).




Summary

Definition of the derived category DQ

DefinitionDQ = bounded derived category of rep(Q)

objects: bounded complexes V : . . . → V p → V p+1 → . . .of representationsmorphisms: obtained from morphisms of complexes byformally inverting all quasi-isomorphismssuspension functor: S : DQ → DQ, V 7→ SV = V [1]

triangles: U ′ → V ′ → W ′ → SU ′ obtained from short exactsequences of complexes 0 → U → V → W → 0.

RemarkDQ is k -linear. It is abelian iff Q has no arrows.




Summary

Objects of DQ decompose into indecomposables.

DefinitionAn object V of DQ is indecomposable if V 6= 0 and in eachdecomposition V ∼= V ′ ⊕ V ′′, we have V ′ = 0 or V ′′ = 0.

Decomposition theorem(Azumaya-Fitting-Krull-Remak-Schmidt- . . . )

a) An object of DQ is indecomposable iff its endomorphismring is local.

b) Each object of DQ decomposes into a finite sum ofindecomposables, unique up to isomorphism andpermutation.




Summary

Indecomposable objects and irreducible morphisms

Let A be any k -linear category where the decompositiontheorem holds. We will assign a quiver ΓA to A.

The vertices of ΓA will be the isomorphism classes of theindecomposables of A.To define the arrows, let

R(X , Y ) = {non invertible morphisms f : X → Y} ,

where X , Y are indecomposable in A. Then R is an ideal(namely, the radical) of the category indA ofindecomposables of A.




Summary

The quiver of a category with decomposition

DefinitionThe quiver of A is the quiver ΓA with

vertices: representatives X of the isoclasses ofindecomposables of Aarrows: the number of arrows from X to Y equals thedimension of the space

irr(X , Y ) = R(X , Y )/R2(X , Y ) = {irreducible morphisms}

of ‘morphisms without non trivial factorization’.




Summary

The quiver of the derived category

TheoremSuppose that Q does not have oriented cycles.

a) The quiver of the category PQ of projectives of rep(Q) isthe opposite quiver Qop.

b) (Happel, 1986) If the underlying graph of Q is a Dynkindiagram of type An, Dn or En, the quiver of DQ is therepetition ZQop of the opposite quiver: It has

vertices: (p, x), for p ∈ Z, x ∈ Q0,arrows: for each arrow α : x → y of Qop, we have arrows

(p, α) : (p, x)→ (p, y), p ∈ Z, andσ(p, α) : (p − 1, y)→ (p, x), p ∈ Z.




Summary

The example ~A3

Example

Q = ~A3 : 1 2oo 3oo

ΓPQ = Qop :

ΓDQ = ZQop :

. . .

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Summary

The Serre functor

Blanket assumptionsQ is a finite quiver without oriented cycles.All categories and functors are k -linear.

Theorem (Happel, 1986)DQ admits a Serre functor (=Nakayama functor), i.e. anautoequivalence ν : DQ

∼→ DQ such that

D Hom(X , ?) ∼→ Hom(?, νX )

for all X ∈ DQ, where D = Homk (?, k).




Summary

Calabi-Yau categories

Let d be an integer and T a triangulated category withfinite-dimensional Hom-spaces.

Definition (Kontsevich)

T is d-Calabi-Yau if it has a Serre functor ν and ν ∼→ Sd astriangle functors.

ExampleX a smooth projective variety of dimension d .T the bounded derived category of coherent sheaves on X .Then ν =?⊗ ω[d ] and

X is Calabi-Yau ⇔ ω ∼→ O ⇔ T is d-Calabi-Yau




Summary

The cluster category

DefinitionThe cluster category CQ is the universal 2-Calabi-Yau categoryunder the derived category DQ:

DQ

(P,π)��

(F ,φ)

&&MMMMMMMMMMMMM

CQ // T

CQ, T 2-Calabi-YauP, F triangle functorsπ : P ◦ ν ∼→ ν ◦ Pφ : F ◦ ν ∼→ ν ◦ F

RemarkCQ is due to Buan-Marsh-Reineke-Reiten-Todorov for Q, toCaldero-Chapoton-Schiffler for ~An.




Summary

Explicit construction of the cluster category

Remarks1) Strictly speaking the definition should be formulated in the

homotopy category of enhanced triangulated categories.2) Explicitly, CQ is the orbit category of DQ under the action of

the automorphism ν−1 ◦ S2. It hasobjects: same as those of DQ

morphisms:

CQ(X , Y ) =⊕p∈Z

DQ(X , (ν−1 ◦ S2)pY ).




Summary

The quiver of the cluster category

Theorem (BMRRT)The decomposition theorem holds in CQ and its quiver isisomorphic to the quotient of the quiver of DQ under the actionof the automorphism induced by ν−1 ◦ S2.

Example

For Q = ~An = ( 1 2oo 3oo . . .oo noo ) the quiver ofCQ is a Moebius strip of width n with n(n + 3)/2 vertices.




Summary

F = ν−1 ◦ S2

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Summary

The canonical cluster-tilting subcategory

Recall that we have functors rep(Q) // DQ // CQ . Let TQ

be the image of the category PQ of projectives of rep(Q).

Theorem (BMRRT)

a) The quiver of TQ is isomorphic to Qop.b) TQ ⊂ CQ is a cluster-tilting subcategory, i.e.

1) We have Ext1(T , T ′) = 0 for all T , T ′ in TQ ,2) if X ∈ CQ satisfies Ext1(T , X ) = 0 for all T in TQ , then X

belongs to TQ .

DefinitionTQ is the canonical cluster-tilting subcategory.




Summary




Summary

The main theorem

RemarkThe objects of TQ generate CQ as a triangulated category but forgeneral T , T ′ in T , we have Exti(T , T ′) 6= 0 for infinitely many i .

Main theoremLet

C be an algebraic 2-CY category,T ⊂ C a cluster-tilting subcategory,Q the opposite quiver of T .

If Q does not have oriented cycles, then CQ∼→ C.




Summary

Example: Cohen-Macaulay modules

Let k = C, ζ a primitive third root of 1. Let G = Z/3Z act onS = k [[X , Y , Z ]] by multiplying the generators by ζ. Then

R = SG is an isolated singularity of dimension 3 andGorenstein.The category CM(R) of maximal Cohen-Macaulaymodules is Frobenius.The stable category CM(R) is 2-Calabi-Yau (Auslander).Decompose S = SG ⊕ T1 ⊕ T2 over R. Then the directsums of copies of the Ti form a cluster-tilting subcategoryT of C (Iyama).The quiver of T is the generalized Kronecker quiver

Q : 1 ////// 2 .




Summary

Cluster categories occur in nature

Conclusion

The stable category of Cohen-Macaulay modules over SG istriangle equivalent to the cluster category CQ.

MoralCluster categories occur in nature.




Summary

Example: 2-Calabi-Yau categories with finitely manyindecomposables

Theorem (Amiot)Let C be an algebraic 2-Calabi-Yau category with finitely manyindecomposables. Suppose that C maximal 2-Calabi-Yau. ThenC is triangle equivalent to C∆ for a unique Dynkin diagram ∆ oftype An, Dn or En.

RemarkTo prove the theorem, one first shows that C has the expectedquiver. This was done independently by Xiao-Zhu and Amiot.




Summary

Example: Modules over the preprojective algebra oftype A4

Let C be the stable category of finite-dimensional modules overthe preprojective algebra of type A4.Then C is 2-Calabi-Yau (Crawley-Boevey).By work of Geiss-Leclerc-Schröer, C contains a cluster tiltingsubcategory T ′ with quiver

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It has lots of oriented cycles!Bernhard Keller, Idun Reiten A characterization of cluster categories



Summary

Quiver mutations help

By quiver mutation theory (google ‘quiver mutation’ !), C alsocontains a cluster-tilting subcategory T with quiver

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(Known) ConclusionC is triangle equivalent to the cluster category CD6 .




Summary

The (best) proof of the main theorem uses theuniversal property

Michel Van den Bergh: We use the universal property!

DQ

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(F ,φ)

&&CQF // C

where φ : F ◦ ν ∼→ ν ◦ F .

1) Construct F via (F , φ). Subtle: Construct φ !2) F is an equivalence by the




Summary

Beilinson’s lemma has a cluster analogue

C′ G // C

T ′

OO

G|T ′// T

OO

Lemma (cluster-Beilinson)

G : C′ → C a triangle functor between 2-CY categoriesT ′ ⊂ C′ a cluster tilting subcategory.

Then G is an equivalence iff T = G(T ′) is a cluster-tiltingsubcategory and the restriction of G to T ′ is fully faithful.




Summary

Summary

Cluster categories occur in nature.Google ‘quiver mutation’!


a characterization of cluster categories

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