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Triangulated categories of singularities and Dbranes in LandauGinzburg models
 Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262
, 2005
"... Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend ..."
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Cited by 206 (7 self)
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Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend
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 38 (15 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.
NONCOMMUTATIVE RESOLUTION, FBLOWUPS AND Dmodules
, 2009
"... We explain the isomorphism between the GHilbert scheme and the Fblowup from the noncommutative viewpoint after Van den Bergh. In doing this, we immediately and naturally arrive at the notion of Dmodules. We also find, as a byproduct, a canonical way to construct a noncommutative resolution at le ..."
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Cited by 7 (3 self)
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We explain the isomorphism between the GHilbert scheme and the Fblowup from the noncommutative viewpoint after Van den Bergh. In doing this, we immediately and naturally arrive at the notion of Dmodules. We also find, as a byproduct, a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic.
Representation dimension and Solomon zeta function
, 2003
"... ClineParshallScott introduced the concept of quasihereditary algebras (§2.5) to study highest weight categories in the representation theory of Lie algebras and algebraic groups [CPS1,2]. Quasihereditary algebras were effectively applied in the representation theory of artin algebras as well by ..."
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Cited by 2 (2 self)
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ClineParshallScott introduced the concept of quasihereditary algebras (§2.5) to study highest weight categories in the representation theory of Lie algebras and algebraic groups [CPS1,2]. Quasihereditary algebras were effectively applied in the representation theory of artin algebras as well by DlabRingel [DR1,2,3] and many other authors. On the other hand, in the representation theory of orders, the concept of overorders and overrings (§1.1), a noncommutative analogy of the normalization in the commutative ring theory, plays a crucial role. From an overring Γ of an order Λ, we naturally obtain a full subcategory latΓ of lat Λ. Formulating this correspondence Γ ↦ → latΓ categorically, we obtain the concept of the rejection (§1,§2). Recently it was effectively applied to study orders of finite representation type by the author [I1,2,3] and Rump [Ru1,2,3]. Originally DrozdKirichenkoRoiter found the onepoint rejection (§1.3) in their theory of Bass orders [DKR], and later HijikataNishida applied the fourpoints rejection (§1.5) to local orders of finite representation type and suggested a possibility of generalization [HN1,2,3]. In this paper, we will show that there exists a close relationship between quasihereditary algebras and the rejection from the viewpoint of the approximation theory of AuslanderSmalo [AS2]. As an application, we will solve two open problems [I4,5]. One concerns the representation dimension of artin algebras introduced by M. Auslander about 30 years ago [A1], and another concerns the Solomon zeta functions of orders introduced by L. Solomon about 25 years ago [S1,2]. It will turn out that the rejection relates these two quite different problems with each other closely. orders (Krull dimension one) artin algebras (Krull dimension zero) overrings of an order Λ factor algebras of an artin algebra Λ
CLASSIFICATION OF SIMPLE PLANE CURVE SINGULARITIES AND THEIR AUSLANDERREITEN QUIVER
, 2011
"... First and foremost, I thank my supervisor, Dr. Daniel Chan. After much patience on his part, I eventually gained perspective and direction for my approach to this thesis. Often, my progress in any endeavour comes in quantum leaps after periods of stagnation, like the flick of a switch which first ne ..."
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First and foremost, I thank my supervisor, Dr. Daniel Chan. After much patience on his part, I eventually gained perspective and direction for my approach to this thesis. Often, my progress in any endeavour comes in quantum leaps after periods of stagnation, like the flick of a switch which first needs to be located in a dark room. Daniel was patient in this respect while I found my way in the dark, and his trusty flashlight which he used from time to time was instrumental to my efforts and much appreciated. The category theory contained in this thesis was inspired by Assoc. Prof. Jie Du. I took a course at UNSW taught by him entitled Representations of Quivers in which he had an excellent treatment of the language of categories and functors. Anthony Christie, 17 September 2011. This thesis is largely a consolidation of parts of lectures given by Yoshino, Y. at Tokyo Metropolitan University in 1987 (cf. [17]). We aim to investigate the CohenMacaulay modules of simple plane curve singularities. We
The Maranda Theorem and Liftings of Modules.
, 2005
"... Throughout this paper let be a noetherian Ralgebra where R is a complete local ring with maximal ideal m and let x in m be a regular element. This setting will be referred to as being general or the general case. We denote by mod the category of all nitely generated modules. ..."
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Throughout this paper let be a noetherian Ralgebra where R is a complete local ring with maximal ideal m and let x in m be a regular element. This setting will be referred to as being general or the general case. We denote by mod the category of all nitely generated modules.
ON THE ZD ∞ CATEGORY
, 2004
"... Abstract. In this paper we give a direct proof of the properties of the ZD∞ category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author. 1. ..."
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Abstract. In this paper we give a direct proof of the properties of the ZD∞ category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author. 1.