Cluster tilting for one-dimensional hypersurface singularities

By I. Burban, Osamu Iyama, B. Keller, I. Reiten. 2007

141 citations recorded by the source corpus.

Abstract In this article we study Cohen–Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite-dimensional symmetric and satisfy τ 2 = id . In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.

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

View Paper: https://www.semanticscholar.org/paper/ecdb3f3143890cafb65d166d596072e4baba9694

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Citations in the corpus

Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday (2011, background): “er analogue of Auslander-Reiten theory, and the existence of (d−1)-cluster tilting objects is shown for quotient singularities in [Iya07a] and for some three dimensional hypersurface singularities in [BIKR08]. They are further investigated in [IY08, KR08, KMV08]. It is of interest to understand the ”
Representations of Algebras and Related Topics (2011, background): “s over a preprojective algebra of type Q associated with any element in the Coxeter group of Q (see [22], [23], [50], [51]); stable Cohen-Macaulay modules over isolated singularities (see [14], [60], [29]); and generalized cluster categories associated with ﬁnite dimensional algebras of global dimen”
Cluster categories (2010, background): “Actually, there may be maximal rigid objects, but no cluster tilting objects [30].”
Acyclic Calabi–Yau categories (2006, background): “The question arises as to whether, for a given quiver Q without oriented cycles, the cluster category is the ‘unique model’ of the associated cluster algebra.”
Maximal rigid subcategories in 2-Calabi-Yau triangulated categories (2010, unknown): “The converse is not true in general since the endomorphism algebras of maximal rigid objects in a 2−CY triangulated category may contain loops [BIKR][BMV].”
Flops and clusters in the homological minimal model programme (2014, background): “We also remark that the above greatly generalises and simplifies [7,14], which considered isolated cAn singularities with smooth minimal models and observed the connection to the Weyl group Sn , [39] which considered specific quotient singularities, again with smooth minimal models, and [24] which c”
Auslander–Reiten theory revisited (2008, methodology): “As a generalization of properties of n-Auslander algebras, we find a certain analogue of Auslander-Reiten theory for n-cluster tilting subcategories, namely the n-Auslander-Reiten translation, the n-Auslander-Reiten duality and n-almost split sequences.”
Non-Commutative Crepant Resolutions: Scenes from Categorical Geometry (2011, background): “ [Iya07b]. Here I will not say anything about cluster algebras or cluster categories; see [BM06] for an exposition, already somewhat out of date. Here is a direct deﬁnition of cluster tilting modules [BIKR08]. Deﬁnition N.3. Let R be a CM local ring and M a MCM R-module. Fix nÊ1. (i) Set M⊥n = n X ¯”
Cluster tilting for higher Auslander algebras (2008, background): “It also played an important role from the viewpoint of higher analogue of Auslander-Reiten theory [I3, I4, I5] in the study of rigid Cohen-Macaulay modules, Calabi-Yau algebras and categories, and non-commutative crepant resolutions [BIRSc, BIRSm, BIKR , GLS1, GLS2, GLS3, IR, IY, KR1, KR2, KV, KZ, ”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “ 2-cycles and then the quiver of the endomorphism algebra of µ1(T) is not determined by Q. Let us illustrate 34 BERNHARD KELLER this phenomenon on the following example (taken from Proposition 2.6 of [26]): Let Cbe the orbit category of the bounded derived category DD~ 6 under the action of the auto”
Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras (2011, unknown): “oncommutative crepant resolutions (=NCCRs) categorify 3-fold crepant resolutions. We then apply these results to the case of cAn singularities, generalizing some results of Burban–Iyama–Keller–Reiten [BIKR] and Dao–Huneke [DH10]. Contents 1. Introduction 1 1.1. Overview 1 1.2. Q-factorial results 3 ”