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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.
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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Cited by 3 (0 self)
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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
AuslanderReiten theory revisited
 In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep
, 2008
"... Abstract. We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained. ..."
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Cited by 2 (2 self)
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Abstract. We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.
CohenMacaulay Modules over CohenMacaulay Algebras
"... this paper we use the following definitions and notations. ..."
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this paper we use the following definitions and notations.
Semicontinuity for representations of onedimensional CohenMacaulay Rings
, 1995
"... this paper is to refine them in such a way that they could be applied to noncommutative CohenMacaulay algebras, too. For this purpose we introduce the notion of "dense subrings" which seems rather technical but, nevertheless, useful. It enables the construction of "almost versal" families of mod ..."
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this paper is to refine them in such a way that they could be applied to noncommutative CohenMacaulay algebras, too. For this purpose we introduce the notion of "dense subrings" which seems rather technical but, nevertheless, useful. It enables the construction of "almost versal" families of modules for a given algebra (cf. Theorem 3.5) and the definition of the "number of parameters". Just as in the commutative case, it is important that the bases of these "almost versal" families are projective varieties. Once having this, we are able to prove an analogue of Knorrer's theorem (cf. Theorem 4.9) and a certain variant (cf. Theorem 4.11) which turns out to be useful, for instance, to extend the tameness criterion for commutative algebras [DG 2] to the case of characteristic 2. The semicontinuity implies, in particular, that the set of socalled "wild algebras" in any family is a countable union of closed subsets. A very exciting problem is whether it is actually closed, hence whether the set of tame algebras is open. However, Theorem 4.9, together with the results of [DG 2], imply that tame is indeed an open property for curve singularities (commutative onedimensional CohenMacaulay rings). An analogous procedure leads to the semicontinuity of the number of parameters in other cases, like representations of finite dimensional algebras or elements of finite dimensional bimodules. Though we do not consider here the problem of constructing moduli spaces for CohenMacaulay modules (cf. [GP]), we may rephrase the semicontinuity theorem by saying that the dimension of the moduli space for such modules of prescribed rank varies upper semicontinuosly in flat families of CohenMacaulay algebras. Likewise, the semicontinuity in other cases (finitedimensional algebras or bim...