From triangulated categories to abelian categories: cluster tilting in a general framework

By S. Koenig, Bin Zhu. 2006

219 citations recorded by the source corpus.

A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (i.e., a maximal 1-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.

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

View Paper: https://www.semanticscholar.org/paper/faddffb5bde837d41bddc6e5d6f01bcf4dfa9c90

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Citations in the corpus

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors (2019, result): “6 extends similar results from [19, 52, 55, 56] (see also Proposition 3.”
On algebras of finite Cohen-Macaulay type (2011, background): “Since addT is contravariantly finite, by [51] it follows that T is a cluster tilting object in GprojΛ.”
Representations of Algebras and Related Topics (2011, background): “s ﬁnite injective dimension as a module over itself) and are either of inﬁnite global dimension, or hereditary (see [70]). Other nice properties of the functor HomC(T,−) : C→modEndC(T) are studied in [73]. Example 2.5. Let Q be the quiver 1 →2 →3. The category CQ/add(T[1]) where T = π(1 )⊕π(2 1 )⊕π(”
On the cluster category of a marked surface without punctures (2010, background): “From [Koenig and Zhu 2008], we know that any sink (or source) map in C is again a sink (or source) map in mod J (Q,W ), therefore each AR-triangle in C”
Hearts of twin cotorsion pairs on extriangulated categories (2017, methodology): “as been introduced in the recent development of the theory of mutation, and of higher Auslander-Reiten theory. It gives a simultaneous generalization of t-structure [BBD], cluster tilting subcategory [KZ], co-t-structure, and is also related to notions such as n-cluster tilting subcategory. Quite of”
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 , ”
Maximal rigid subcategories in 2-Calabi-Yau triangulated categories (2010, unknown): “The first examples of 2−Calabi-Yau categories in which maximal rigid objects are not cluster tilting were given in [BIKR] (see also the example in Section 5 of [KZ] for the example of triangulated category in which maximal rigid objects are not cluster tilting).”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, result): “We refer to [19, 20, 23, 45, 47 , 13, 65, 59, 58] for more recent developments in the study of cluster categories and their generalization.”
Generalized cluster complexes via quiver representations (2006, background): “Remark 5.10 (1) From Corollary 5.8(2) for d = 1, combining with the result in [6] (see also [ 21 ]), in which the cluster tilting subcategories in D are proved to be in one-to-one correspondence with the cluster tilting modules in cluster categories by the projection π, we have an explanation on why”