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Cluster tilting for one-dimensional hypersurface singularities. (English) Zbl 1143.13014
In this important paper the authors apply methods of associative cluster tilting theory in order to investigate connections between Cohen-Macauley modules over 1-dimensional hypersurface singularities and the representation theory of associative algebras. One of the main result of the paper is Theorem 1.3. In this theorem the authors describe the number of rigid objects, basic cluster tilting objects, basic maximal rigid objects and indecomposable summands of basic maximal rigid objects in the stable category \(\underline{\mathrm{CM}}(R)\) of maximal Cohen-Macauley module over a simple one-dimensional singularity of dimension \(\geq 1\) over an algebraic closed field of characteristic 0.
Two proofs for this theorem are presented in Section 2 (using additive functions on the AR quiver) and in Section 3 (using the algorithm Singular). In the next section a large class of 1-dimensional hypersurface singularities having a cluster tilting object is constructed. In Section 5 and Section 6 connections between cluster tilting theory and birational geometry are exhibited. The main result concerning this subject are presented in Theorem 1.5 and Theorem 1.6. In the end of the paper the authors present an application to finite-dimensional algebras (Theorem 1.7).

MSC:
13C14 Cohen-Macaulay modules
16G10 Representations of associative Artinian rings
18E30 Derived categories, triangulated categories (MSC2010)
14E05 Rational and birational maps
Software:
SINGULAR
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