Acyclic Calabi-Yau categories are cluster categories Bernhard Keller (joint work with Idun Reiten)

Let k be a field and Q a finite quiver without oriented cycles. Let kQ be the path algebra of Q and mod kQ the category of k-finite-dimensional right kQ-modules. The cluster category C_{Q} was introduced in [1] (for general Q) and, independently, in [4] (for Q of type A_{n}). It is defined as the orbit category of the bounded derived category D^{b}(mod kQ) under the action of the automorphism Σ^{−1} ∘ S^{2}, where S is the suspension (=shift) functor of the derived category and Σ its Serre functor, characterized by the Serre duality formula

D Hom(X, Y) = Hom(Y, ΣX) ,

where D is the duality functor Hom_{k}(?, k). The motivation behind this definition was to find a ‘categorification’ of the cluster algebras introduced by Fomin-Zelevinsky in [6]. This program has been quite successful, cf. e.g. [3] [5] and the references given there. The cluster category has the following properties (explained below in more detail):

a) C_{Q} is a triangulated category. In fact, it is even an algebraic triangulated category, i.e. there is a triangle equivalence between C_{Q} and the stable category \underline{E} of a Frobenius category E.

b) C_{Q} is Hom-finite (i.e. all its morphism spaces are finite-dimensional) and Calabi-Yau of CY-dimension 2. By this, one means that it admits a Serre functor Σ (which is induced by that of the derived category) and that there is an isomorphism of triangle functors between Σ and S^{2}. Note that this last property holds almost by definition of C_{Q}.

c) If T_{Q} denotes the image of the free module kQ under the projection from the derived category to the cluster category, then T_{Q} is a cluster-tilting object in C_{Q}, i.e. we have c1) Hom(T_{Q}, ST_{Q}) = 0 and c2) for each object X, if we have Hom(T_{Q}, SX) = 0, then X vanishes.

d) The endomorphism algebra of T_{Q} is isomorphic to kQ. In particular, its

ordinary quiver does not admit oriented cycles. These properties were proved in [1] except for a), which was proved in [9]. We say that a k-linear category is a 2-Calabi-Yau category if it satisfies a) and b). Our main result is that properties a) to d) characterize the cluster category if k is algebraically closed:

Theorem. Suppose that k is algebraically closed. If C is an algebraic 2-Calabi-Yau category and admits a cluster-tilting object T such that the ordinary quiver Q of the endomorphism algebra of T does not contain oriented cycles, then there is a triangle equivalence from C_{Q} to C which takes the object T_{Q} to T.

The theorem allows one to show that cluster categories, whose definition may seem artifical at first glance, do occur in nature: Let k be an algebraically closed field of characteristic 0, S the completed power series algebra k[[X, Y, Z]] and G the cyclic group of order three acting linearly on S such that a generator of G multiplies the three variables by the same primitive third root of unity. It is not hard to show that the fixed point algebra R = S^{G} is a Gorenstein complete local normal domain that has an isolated singularity. We consider the Frobenius category E = CM(R) of its maximal Cohen-Macaulay modules. By a theorem of Auslander’s, the stable category C = \underline{E} is 2-Calabi-Yau. Work of Iyama [8] shows that T = S considered as an R-module is a cluster-tilting object in C. Its ring of R-linear endomorphisms is isomorphic to the skew group algebra S ∗ G and its endomorphism ring in \underline{E} is the path algebra of the generalized Kronecker quiver Q with three arrows. Thus the hypotheses of the theorem are satisfied and we obtain a triangle equivalence between C_{Q} and \underline{CM}(R). In particular, this allows us to compute the Auslander-Reiten quiver of \underline{CM}(R). It also shows that Yoshino’s classification of the rigid Cohen-Macaulay modules [12] in \underline{CM}(R) is equivalent to the classification of the cluster-tilting objects in the cluster category of the generalized Kronecker quiver with three arrows and thus [5] to that of the cluster variables in the corresponding cluster algebra [11].

Assume that k is algebraically closed. Let C be a 2-Calabi-Yau category admitting a cluster-tilting object T. One can show that the number of pairwise indecomposable non isomorphic direct factors of T does not depend on the choice of T, cf. [10]. We call this number the rank of C. We say that C is acyclic if it admits a cluster tilting object the quiver of whose endomorphism algebra does not have oriented cycles (or equivalently, if it is triangle equivalent to a cluster category).

Conjecture. If C is of rank at most three and the quivers of the endomorphism algebras of its tilting objects admit neither loops nor 2-cycles, then it is acyclic.

It was shown in [7] that if C is the stable module category \underline{mod} Λ(Δ) of a preprojective algebra associated with a simply laced Dynkin diagram Δ, then the quivers of the endomorphism algebras of its cluster-tilting objects admit neither loops nor 2-cycles. It follows that this property also holds if C is constructed as a ‘CY-subquotient’ of \underline{mod} Λ(Δ) (for the ‘CY-subquotient’ construction, cf. section 2 of [2] and section 5.4 of [5]). Thus the conjecture implies that any CY-subquotient of rank ≤ 3 of \underline{mod} Λ(Δ) is acyclic. This holds indeed in all the examples we have checked.

References

[1] Aslak Bakke Buan, Robert J. Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Advances in Mathematics, to appear.

[2] Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster mutation via quiver representations, arXiv:math.RT/0412077.

[3] Aslak Bakke Buan, Robert J. Marsh, Idun Reiten, and Gordana Todorov, Clusters and seeds in acyclic cluster algebras, arXiv:math.RT/0510359.

[4] Philippe Caldero, Frédéric Chapoton, and Ralf Schiffler, Quivers with relations arising from clusters (A_{n} case), arXiv:math.RT/0401316.

[5] Philippe Caldero and Bernhard Keller, From triangulated categories to cluster algebras II, arXiv:math.RT/0510251.

[6] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529 (electronic).

[7] Christof Geiß, Bernard Leclerc, and Jan Schröer, Rigid modules over preprojective algebras, arXiv:math.RT/0503324.

[8] Osamu Iyama, Higher dimensional Auslander correspondence, arXiv:math.RT/0411631.

[9] Bernhard Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581.

[10] Yann Palu, On algebraic Calabi-Yau categories, Ph. D. thesis in preparation.

[11] Paul Sherman and Andrei Zelevinsky, Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. Math. J. 4 (2004), no. 4, 947–974, 982.

[12] Yuji Yoshino, Rigid Cohen-Macaulay modules over a three dimensional Gorenstein ring, Oberwolfach Report 6, 2005, pp. 345–347.