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In the recent paper "Mutation in triangulated categories and rigid Cohen-Macaulay modules" Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data.
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O. Iyama and Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules
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