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25
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.
Moduli spaces for torsion free modules on curve singularities I
, 1993
"... this article is to construct coarse moduli spaces for torsion free Rmodules or, which amounts to the same, for CohenMacaulay representations of R. ..."
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Cited by 8 (4 self)
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this article is to construct coarse moduli spaces for torsion free Rmodules or, which amounts to the same, for CohenMacaulay representations of R.
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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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.
Matrix factorizations, minimal models and Massey products
, 2006
"... We present a method to compute the full non–linear deformations of matrix factorizations for ADE minimal models. This method is based on the calculation of higher products in the cohomology, called Massey products. The algorithm yields a polynomial ring whose vanishing relations encode the obstructi ..."
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Cited by 5 (0 self)
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We present a method to compute the full non–linear deformations of matrix factorizations for ADE minimal models. This method is based on the calculation of higher products in the cohomology, called Massey products. The algorithm yields a polynomial ring whose vanishing relations encode the obstructions of the deformations of the D–branes characterized by these matrix factorizations. This coincides with the critical locus of the effective superpotential which can be computed by integrating these relations. Our results for the effective superpotential are in agreement with those obtained from solving the A–infinity relations. We point out a relation to the superpotentials of Kazama–Suzuki models. We will illustrate
Local rings of countable CohenMacaulay type
 Proc. Amer. Math. Soc
"... Abstract. We prove (the excellent case of) Schreyer’s conjecture that a local ring with countable CM type has at most a onedimensional singular locus. Furthermore we prove that the localization of a CohenMacaulay local ring of countable CM type is again of countable CM type. Let (R,m) be a (commut ..."
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Abstract. We prove (the excellent case of) Schreyer’s conjecture that a local ring with countable CM type has at most a onedimensional singular locus. Furthermore we prove that the localization of a CohenMacaulay local ring of countable CM type is again of countable CM type. Let (R,m) be a (commutative Noetherian) local ring of dimension d. Recall that a nonzero Rmodule M is called maximal Cohen–Macaulay (MCM) provided it is finitely generated and there exists an Mregular sequence {x1,...,xd} in the maximal ideal m. We say that R itself is Cohen– Macaulay (CM) if it is MCM as a module over itself. The CM local rings of finite CMrepresentation type (meaning that they have only finitely many nonisomorphic indecomposable MCM modules) have been carefully studied over the last twenty years. The complete equicharacteristic hypersurfaces of finite CM type have been completely classified ([6], [4], [9]), as have the complete equicharacteristic 2dimensional normal domains ([2]). More generally, it is known that a CM local ring of finite CM type has at most an isolated singularity (proved by Auslander [1] in the complete case, LeuschkeWiegand [10] in the excellent case, and HunekeLeuschke [8] in general). Yoshino’s monograph [15] is a comprehensive source for information about rings of finite CM type. The related property of countable CM type has received much less attention. Buchweitz, Greuel, and Schreyer [4] classified the complete hypersurface singularities of countable CM type, but very little more has been learned since then. The open questions and conjectures in Schreyer’s 1987 survey article [12] have inspired work on both finite and countable CMrepresentation type. For example, Conjecture 7.3(a) states that a CM local ring R has finite CM type if and only if the madic completion has finite CM type; this was recently proved in case R is excellent in [10]. This paper is concerned with another of Schreyer’s conjectures:
A KrullSchmidt Theorem for Onedimensional Rings of Finite CohenMacaulay Type
, 2006
"... This paper determines when the KrullSchmidt property holds for all finitely generated modules and for maximal CohenMacaulay modules over onedimensional local rings with finite CohenMacaulay type. We classify all maximal CohenMacaulay modules over these rings, beginning with the complete rings w ..."
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This paper determines when the KrullSchmidt property holds for all finitely generated modules and for maximal CohenMacaulay modules over onedimensional local rings with finite CohenMacaulay type. We classify all maximal CohenMacaulay modules over these rings, beginning with the complete rings where the KrullSchmidt property is known to hold. We are then able to determine when the KrullSchmidt property holds over the noncomplete local rings and when we have the weaker property that any two representations of a maximal CohenMacaulay module as a direct sum of indecomposables have the same number of indecomposable summands. Keywords: KrullSchmidt, maximal CohenMacaulay, finite CohenMacaulay type 1
CONNECTIONS ON MODULES OVER SIMPLE CURVE SINGULARITIES
, 2006
"... Abstract. Let k be an algebraically closed field of characteristic 0, and let A be the complete local ring of a simple curve singularity defined over k. For any maximal CohenMacaulay Amodule M, we show that there exists an integrable connection on M, i.e. an Alinear homomorphism ∇ : Derk(A) → En ..."
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Abstract. Let k be an algebraically closed field of characteristic 0, and let A be the complete local ring of a simple curve singularity defined over k. For any maximal CohenMacaulay Amodule M, we show that there exists an integrable connection on M, i.e. an Alinear homomorphism ∇ : Derk(A) → Endk(M) that satisfy the Leibniz property and preserves the Lie product. 1.
MAXIMAL COHENMACAULAY MODULES OVER SURFACE SINGULARITIES
"... Abstract. This is a survey article about properties of CohenMacaulay modules over surface singularities. We discuss properties of the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Correspondence. Finally, we des ..."
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Abstract. This is a survey article about properties of CohenMacaulay modules over surface singularities. We discuss properties of the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Correspondence. Finally, we describe matrix factorizations corresponding to indecomposable CohenMacaulay modules over the nonisolated singularities A ∞ and D∞. 1. Introduction and