Acyclic Calabi–Yau categories

By B. Keller, I. Reiten. Compositio Mathematica, 2006

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Abstract We prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. No direct combinatorial proof is known as yet. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c12bc6e493538abbb49e8d41e24506b2c785ee13

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Citations in the corpus

Almost complete cluster tilting objects in generalized higher cluster categories (2010, background): “he autoequivalence τ−1Σm. This higher cluster category is Homﬁnite, triangulated, and (m+1)-Calabi-Yau. It was ﬁrst mentioned in [17], and has been studied in more detail in several articles [2] [20] [21] [24] ... . Many results about cluster categories can be generalized to m-cluster categories. In”
Preprojective algebras and c‐sortable words (2010, background): “The cluster category is then equivalent to SubΛw with w = c(2), where c is a Coxeter element such that c(2) is a reduced expression [10, 18, 26].”
n-Abelian and n-exact categories (2014, background): “It is known that C is an algebraic triangulated category [37].”
On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon (2012, methodology): “If k is perfect, then by using the methods of [13], one can show that D is the orbit category D(mod Γ)/SΣ−2.”
Representations of Algebras and Related Topics (2011, background): “result due to Keller and Reiten ensures that the acyclic cluster categories are the only categories satisfying the properties needed to categorify acyclic cluster algebras. Theorem 2.9 (Keller-Reiten [71]). Let Cbe an algebraic triangulated 2-CY category with a cluster-tilting object T. If the quive”
Preprojective algebras and cluster algebras (2008, background): “n theorem of Fomin and Zelevinsky [11] there exists a complete rigid object of SubQJ whose endomorphism ring has a Gabriel quiver with stable part of Dynkin type. Using a theorem of Keller and Reiten [23], it follows that the stable category SubQJ is triangle equivalent to a cluster category of Dynk”
On algebras of finite Cohen-Macaulay type (2011, methodology): “Since T is rigid, by a result of Keller-Reiten [49], the functor H induces an equivalence between the stable category ` addT ‹addΩ´1T ˘L addT and mod-EndΛpT q op, hence we have an equivalence ` GprojΛqď1T L addT « mod-EndΛpT q op.”
Cluster algebra structures and semicanonical bases for unipotent groups (2007, methodology): “he assumptions above, the categories CM and CQop are triangle equivalent. Proof. We proved already that CM is an algebraic 2-Calabi-Yau category. According to an important result by Keller and Reiten [41], it is enough to construct a CM-maximal 1-orthogonal module T in CM such that the quiver of the”