On the structure of Cohen–Macaulay modules over hypersurfaces of countable Cohen–Macaulay representation type

Let R be a complete local hypersurface over an algebraically closed field of characteristic different from two, and suppose that R has countable Cohen–Macaulay (CM) representation type. In this paper, it is proved that the maximal Cohen–Macaulay (MCM) R-modules which are locally free on the punctured spectrum are dominated by the MCM R-modules which are not locally free on the punctured spectrum. More precisely, there exists a single R-module X such that the indecomposable MCM R-modules not locally free on the punctured spectrum are X and its syzygy ΩX and that any other MCM R-modules are obtained from extensions of X and ΩX.