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

TL;DR

This paper characterizes the structure of Cohen-Macaulay modules over certain hypersurfaces with countable representation type, showing they are generated by a small set of modules and their syzygies.

Contribution

It proves that all indecomposable maximal Cohen-Macaulay modules not locally free are generated by a single module and its syzygy, revealing a simplified module structure.

Findings

Indecomposable non-locally free modules are X and  X.

Any maximal Cohen-Macaulay module is an extension of X and  X.

Modules are dominated by a single module and its syzygy.

Abstract

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 representation type. In this paper, it is proved that the maximal Cohen-Macaulay R-modules which are locally free on the punctured spectrum are dominated by the maximal Cohen-Macaulay R-modules which are not locally free on the punctured spectrum. More precisely, there exists a single R-module X such that the indecomposable maximal Cohen-Macaulay R-modules not locally free on the punctured spectrum are X and its syzygy \Omega X and that any other maximal Cohen-Macaulay R-module is obtained from some extension of X and \Omega X.