Abstract. In this article we study Cohen-Macaulay modules over one-dimensional 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 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

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this paper we use the following definitions and notations.

this paper is to refine them in such a way that they could be applied to non--commutative Cohen--Macaulay 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 so--called "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 one-dimensional Cohen--Macaulay 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 Cohen-Macaulay 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 Cohen-Macaulay algebras. Likewise, the semicontinuity in other cases (finite-dimensional algebras or bim...

1. An easy example: vector bundles on P1 4 2. A simple example: projective configurations of type A 4 3. Elliptic curves are vector bundle tame 7

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