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76
Higher dimensional AuslanderReiten theory on maximal orthogonal subcategories
, 2005
"... We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them. ..."
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Cited by 37 (11 self)
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We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them.
Cluster structures for 2CalabiYau categories and unipotent groups
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This c ..."
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Cited by 32 (6 self)
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Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This class of 2CalabiYau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2CalabiYau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related
CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). I ..."
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Cited by 32 (5 self)
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Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents
The matrix factorisations of the Dmodel
, 2005
"... The fundamental matrix factorisations of the Dmodel superpotential are found and identified with the boundary states of the corresponding conformal field theory. The analysis is performed for both GSOprojections. We also comment on the relation of this analysis to the theory of surface singulariti ..."
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Cited by 16 (3 self)
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The fundamental matrix factorisations of the Dmodel superpotential are found and identified with the boundary states of the corresponding conformal field theory. The analysis is performed for both GSOprojections. We also comment on the relation of this analysis to the theory of surface singularities and their orbifold description.
Local Rings Of Finite CohenMacaulay Type
, 1997
"... this paper was partially supported by the National Science Foundation. ..."
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Cited by 14 (4 self)
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this paper was partially supported by the National Science Foundation.
Ascent of Finite CohenMacaulay Type
 J. Algebra
, 1999
"... this paper we prove the other direction. Specifically, we prove the following theorem. ..."
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Cited by 12 (7 self)
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this paper we prove the other direction. Specifically, we prove the following theorem.
Finite Gorenstein representation type implies simple singularity
 Adv. Math
"... Abstract. Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated sin ..."
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Cited by 9 (4 self)
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Abstract. Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive Rmodule is free.
Reconstruction algebras of type A
, 2007
"... Abstract. This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G = ..."
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Cited by 9 (4 self)
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Abstract. This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G =�n,q for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type A. As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (=GHilb) to the same level of difficulty as the toric case. Contents
Moduli of McKay quiver representations II: Gröbner basis techniques
, 2005
"... For a finite abelian group G ⊂ GL(n, k), let Yθ be the coherent component of the moduli space of θstable representations of the McKay quiver. We calculate the Gequivariant k[x1,...,xn]module parameterized by each point of Yθ via Gröbner bases. In the case Mθ ∼ = GHilb, we show that GHilb may b ..."
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Cited by 9 (3 self)
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For a finite abelian group G ⊂ GL(n, k), let Yθ be the coherent component of the moduli space of θstable representations of the McKay quiver. We calculate the Gequivariant k[x1,...,xn]module parameterized by each point of Yθ via Gröbner bases. In the case Mθ ∼ = GHilb, we show that GHilb may be reducible and its coherent component Yθ ∼ = Hilb G may be nonnormal, giving examples for G in GL(3, k) and GL(6, k) respectively. The latter answers a question of Nakamura.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 8 (7 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with 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 method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.